Model based control of a flight simulator motion system - DCSC

Model based control of a flight simulator motion
system

Model based control of a flight simulator motion
system
PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir K.F. Wakker
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op maandag 10 december 2001 om 16.00 uur
door
Sjirk Holger KOEKEBAKKER
werktuigkundig ingenieur
geboren te Ermelo

Dit proefschrift is goedgekeurd door de promotor:
Prof. ir O.H. Bosgra
Samenstelling promotiecommissie:
Rector Magnificus voorzitter
Prof. ir O.H. Bosgra Technische Universiteit Delft, promotor
Dr ir A.J.J. Van der Weiden Technische Universiteit Delft, toegevoegd promotor
Prof. dr ir M. Steinbuch Technische Universiteit Eindhoven
Prof. dr ir J.A. Mulder Technische Universiteit Delft
Prof. dr ir P.M.J. van den Hof Technische Universiteit Delft
Prof. dr ir J.H. de Leeuw University of Toronto
Ir P.C. Teerhuis Technische Universiteit Delft
Ir P.C. Teerhuis heeft als begeleider in belangrijke mate aan de
totstandkoming van het proefschrift bijgedragen.
Keywords: model based control, flight simulation, parallel motion systems
ISBN 90-370-0194-7
Copyright c 2001 by S.H. Koekebakker
All rights reserved. No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical, including photocopying, recording or
by any information storage and retrieval system, without written permission from the copyright owner.
Printed in the Netherlands by Ponsen & Looijen b.v.

To faith
We’re on a road to nowhere
Come on inside
Takin’ that ride to nowhere
We’ll take that ride
Talking Heads, 1985
Running is no sport
but a way of travelling
displacing body and mind
Vrij naar Jan Knippenberg

Voorwoord
In de ruwweg zeven jaren, maanden, dagen en uren tussen de start van dit onderzoek en het
moment waarop het verdedigd zal worden is veel gebeurd. Beginnend vanuit een kamertje
in een studentenhuis in Delft kon ik me weinig voorstellen bij de grote, relatief eenzaam
verrichte inspanningen waar kond van werd gedaan in het voorwoord van de verschillende
proefschriften. Onderzoek doen is leuk en zeker in het kader van een multidisciplinair
project werk je daarbij met veel mensen samen. Pas nadat het onderzoek na vier jaar
afgerond maar nog niet helemaal opgeschreven was en naast een aantrekkelijke voltijds
baan en een fantastisch gezin met inmiddels twee dochters vanuit Venlo het werk in groter
geheel vastgelegd moest worden, kon ik mij vinden in vele van die eerder gelezen frasen.
Alhoewel een groot aantal mensen je op een of andere manier bijstaan tijdens de gehele periode,
moet het schrijfwerk toch grotendeels alleen werkend achter de computer gebeuren.
Hier een punt achter zettend is eindelijk de tijd ook gekomen om al deze mensen uitdrukkelijk
te bedanken voor de hulp die ze me op verschillende wijze hebben geboden. Een
aantal mensen zal ik alleen als groep noemen gezien de vele veranderingen die zich over
deze lange periode hebben voorgedaan en het risico daarbij individuele namen over te slaan.
Ten eerste wil ik mijn promotor Okko Bosgra bedanken die me in staat heeft gesteld
mijn promotieonderzoek te verrichten binnen zijn vakgroep. Vervolgens bedank ik mijn
directe begeleiders, Ton van der Weiden met zijn directheid en politiek gevoel voor tact
waar nodig in dit project en Piet Teerhuis met zijn aanstekelijke liefde voor de techniek en
in het bijzonder de hydraulische systemen.
Zoals in de referenties terug te vinden is, heeft een grote groep studenten het laatste (anderhalf
of meer) jaar van de ingenieursopleiding een afstudeerproject in het kader van mijn
onderzoek kunnen verrichten. Dit heb ik als zeer prettig ervaren. Zeker op het moment dat
er gelijktijdig vijf, zes enthousiaste mensen de ’Simona motion control room’ bij werktuigbouwkunde
bevolkten en we door slim tactisch spel een (audiovisio)hardloopwedstrijd konden
winnen zonder als eerste aan te komen. Boris Rijnten, Vasken der Kevorkian, Sander
Bettendorf, Philippe Piatkiewitz, Jan van Hulzen, Maris Franken, Riad Al-Saidi, Etienne
van Zuijlen en Arne Scheffer bedankt voor jullie mede- en vooral samenwerking.
Alle groeps- en ex-groepsleden van de vakgroep Systeem- en regeltechniek wil ik bedanken
voor de collegialiteit, ook in de periode dat ik al uit Delft weg was. Met name door
de combinatie van een grote groep promovendi, de ervaren staf en de rechtdoorzee support
groep was er sprake van een levendige en productieve sectie. Marco Dettori bracht daar
als langstzittende kamergenoot nog wat Italiaanse cultuur in die gecombineerd met Carsten
Scherers internationale en vooral ook wiskundige achtergrond tot prettig pittig smakende
lunchdiscussies leidden. Thomas de Hoog wil ik ook speciaal noemen vanwege het feit dat
vii

viii Voorwoord
ik de template van zijn proefschrift en allerhande LateX-advies goed heb kunnen gebruiken.
Door het interfacultaire karakter van het simulatorproject is er interactie met veel mensen
die ik als zeer leerzaam heb ervaren. Ik bedank alle mensen betrokken bij Simona, vooral
ook door de eigenlijk nooit ontbrekende en ook zeker noodzakelijke passie voor het maken
van een technisch complex apparaat waarbij altijd vele hobbels genomen moeten worden.
Ik wil een aantal mensen hiervan met name noemen waarmee ik door hun directe betrokkenheid
bij mijn onderzoek nauw samengewerkt heb. Sunjoo Advani, die met zijn enthousiasme
het Simonaproject als projectleider lange tijd over pieken en dalen getrokken
heeft en ook tot nu nog grote betrokkenheid bij dit onderzoek heeft getoond. Gert van
Schothorst, die, twee jaar eerder als werktuigkundig promovendus bij het simulatorproject
begonnen, veel werk op het gebied van de modelvorming rond hydraulische systemen heeft
verricht waarop ik met mechanica en regeling verder kon bouwen. Met Peter Valk kon ik,
gegeven ook de gelijkgestemdheid op sportief gebied, eindeloze discussies voeren. Rens
de Keijzer, Kees Slinkman, Rolf van Overbeek, Fred den Hoedt, John Dukker en Ad van
der Geest brachten allen vanuit onze faculteit een praktisch technische insteek waar ik nog
steeds niet aan kan tippen. Ook de wizardous meet- en communicatiekastjes van de prettig
nuchtere Henk Huisman moet ik niet vergeten.
Buiten het werk brachten in de eerste jaren vooral de gezamenlijk gelopen ettelijk
honderden trainingskilometers, tientallen etentjes en (estafette)wedstrijden onder andere in
Londen, Parijs, Barcelona en een tweede keer ook weer tactisch winnend rond het IJsselmeer
met het losse en toch ook weer vaste groepje jonge oude Delvers de nodige afleiding
en zeker ook vriendschap. Daarnaast leefden ook veel andere vrienden, kennissen en
familie mee. Allen bedankt!
Alhoewel de frequentie verlaagd is houdt nu een groep enthousiaste collega’s de sportiviteit
en de felle en gelukkig lekker oeverloze en daarmee gevarieerde discussies tijdens de
gezamenlijke etentjes erin. Ik wil graag hierbij ook mijn hoofd bij de researchafdeling van
Océ, Jo Geraedts, bedanken voor zijn steun ten aanzien van mijn werk voor dit onderzoek
en dan met name zijn motivatie om er vooral op een goede manier een punt achter te zetten.
Tenslotte kan ik van hieruit nooit mijn lieve Els, Hanneke en Ingrid voldoende bedanken
voor het geduld dat ze met me en het vertrouwen dat ze in me hebben gehad. Ik zal van
’pappa z’n werk’ met vier PC’s en meer dan twintig meter stapels papieren archief op twaalf
vierkante meter maar (g?)een museum maken.
Sjirk Koekebakker
Venlo, oktober 2001.

Summary
The extremely high safety requirements in air transportation require a proper understanding
of human pilot behaviour under extreme weather conditions. Advanced research in this
area must utilize flight simulator equipment that is able to reproduce critical flight conditions
with great fidelity. The turbulence and wind shear phenomena in the lower part of
the atmosphere contain a velocity spectrum that induces large forces on the aircraft over a
wide frequency range. Thus, motion reproduction and generation of desired accelerations
in a high quality flight simulator constitutes a complicated control task that must exploit
the capabilities of an available actuation system to its limits. Predominately, these systems
are comprised of six hydraulically actuated linear servo motors, similar to many robotic
manipulators. The control of modern robotic devices employs advanced model based control
strategies that allow higher performance to be achieved than less structured approaches,
and can provide more insight into the system limitations. In flight simulator motion control,
application of such methods has been almost absent. One of the reasons might be that
the motion systems have a degree of complexity, which does not allow exact modelling or
detailed models as part of the controller. Application has required an intermediate step of
extracting a reasonably detailed model, which described the most relevant dynamic characteristics.
The central problem of this research was to investigate which relevant system knowledge
should be used in a model based control strategy, and to quantify the extent to which this improves
the controlled dynamics of the simulator. First, the full design process was defined,
structured and evaluated. This began with modelling and analysis, followed by control synthesis
and, finally, testing the system. Many solutions and procedures of the steps taken
were given in literature, but an integral approach, having the specific properties and requirements
of flight simulation motion generation in mind, was still lacking. In this research,
integratibility and applicability were the main arguments in choosing the most suitable alternative
or, if necessary, newly proposed variant, of each step in the system/control design
procedure.
First, a relevant model structure for analysis and control through physical modelling was
derived. Next, the structure was evaluated by experiments. Model parameters were identified
and boundaries were provided to the extent for which the model validity holds. Then,
the most appropriate model based robot control strategy was chosen and, given the particular
system properties and performance requirements, modified for this application. The next
step was actual application within the simulator environment, after which the testing procedure
was defined in order to quantify the performance with respect to the requirements, and
to conventional systems.
ix

x Summary
Structural insight into the system kinematics and dynamics was required in order to
maintain at least the level of robustness of conventional control strategies, and simultaneously
increase motion performance and predictability. The physical modelling began with
an analysis of the system kinematics and, as a contribution of this thesis, a method was proposed
to exclude singular points from the operational workspace and, subsequently, safely
and efficiently determine platform pose from the actuator positions in real time. Calibration
of the physical system improved further its positioning accuracy by an order of magnitude.
The dynamics of the Stewart Platform of this simulator were expressed by a set of explicit
differential equations using platform coordinates. It was shown in theory and practice
that the rigid body modes resulting from interaction of hydraulics and mechanics form the
most relevant motion system dynamics in simulator applications. The platform to actuator
coordinate jacobian is a central operator here. The research showed also how the parallel
actuation can be viewed from a new set of coordinates as six nearly independent hydraulic
systems driving a single mass.
The applied model based controller attained several goals, which could not be implemented
conventionally. First, it directs the required forces for the desired accelerations
along the appropriate vectors experiencing largely different masses. Then, the feedback
paths of the positional errors also use these decoupled directions. Finally, the interaction
between hydraulics and mechanics is minimised over the feed forward path primarily by
applying compensation of the required oil flow given the desired velocity. Implementation
in practice showed an even higher bandwidth, less peaking, and a more equalised response
of the model based controller over each degree of freedom compared with a conventional
strategy. Even more can be gained since model based control can more effectively use
predictive information from the vehicle simulation model.
Performance limitations to this technique can be found at frequencies where parasitic
effects such as structural flexibility, and hydraulic valve and transmission line dynamics
become significant, or when the system no longer behaves as being rigidly attached to the
floor. However, through the application of a light weight and rigid design of the moving
platform, and with the control strategy proposed in this thesis, the system bandwidth can
be raised to a sufficiently high level, thereby enabling a wider range of motion based flight
simulation research to be executed than heretofore.

Contents
Voorwoord vii
Summary ix
1 Introduction 1
1.1 Flight simulation . .............................. 1
1.1.1 The art of simulation . . ....................... 2
1.1.2 Mechanism of motion perception . . . ............... 4
1.1.3 Historical perspective . . ....................... 8
1.1.4 Motion simulation . . . ....................... 9
1.1.5 SIMONA project . . . . ....................... 12
1.2 Motion control . . .............................. 17
1.2.1 Motion control structure ....................... 17
1.2.2 Motion control specifications . . ................... 19
1.2.3 Conventional simulator motion control ............... 21
1.2.4 Robot control . . . . . . ....................... 22
1.3 Problem statement .............................. 25
1.3.1 General problem statement . . . ................... 25
1.3.2 Structuring the problem statement;
approach in research . . ....................... 25
1.4 Outline of the thesis .............................. 28
2 Mechanics of parallel driven motion systems 31
2.1 Parallel motion systems . . . . . ....................... 32
2.2 Kinematics .................................. 33
2.2.1 Notation and definitions ....................... 33
2.2.2 Calculating velocity and acceleration by differentiation . . . .... 34
2.2.3 Parametrising orientation by euler angles . . . . . . ........ 36
2.2.4 Parametrising orientation by euler parameters . . . . ........ 38
2.2.5 Jacobian matrices . . . . ....................... 40
2.2.6 Parallel, serial and singular configurations . . . . . . ........ 41
2.2.7 Stewart platform definitions and assumptions . . . . ........ 43
2.2.8 Stewart platform kinematics . . ................... 44
xi

xii Contents
2.2.9 Interpretation and use of the jacobian matrix, J l�x .......... 45
2.2.10 Velocity and acceleration of the actuator joints . . . ........ 47
2.2.11 The Simona flight simulator motion system kinematics . . . .... 49
2.3 Analysis of the Stewart platform kinematics . . ............... 50
2.3.1 Convergence NR-iteration . . . ................... 53
2.3.2 NR-convergence Stewart platform . . . ............... 54
2.3.3 Lipschitz condition on jacobian ................... 56
2.3.4 Exclusion of singular points . . ................... 58
2.3.5 Sufficient update frequency . . ................... 59
2.4 Modelling the mechanical part of the system dynamics . . . ........ 60
2.4.1 General theory in modelling the dynamics of mechanical systems . 60
2.4.2 Example in modelling using Kane’s method . . . . . ........ 64
2.4.3 Stewart platform dynamics . . . ................... 66
2.4.4 Basic Stewart platform mechanical model dynamics ........ 66
2.4.5 Influence of the actuator inertial forces ............... 67
2.4.6 The Stewart platform model including actuator inertia . . . .... 69
2.4.7 Parasitic mechanical aspects: modelled as linear flexibility . .... 70
2.5 Chapter Resume . . .............................. 73
3 Hydraulically driven motion systems 75
3.1 The basic structure of hydraulic actuators . . . ............... 76
3.1.1 Leakage . . .............................. 78
3.1.2 Basic hydraulically driven mechanical system model ........ 79
3.2 Extensions . .................................. 80
3.2.1 Servo valve .............................. 80
3.2.2 Transmission lines . . . ....................... 85
3.3 Integrated system . .............................. 89
3.3.1 Passive input/output pairs in flight simulator motion systems .... 90
3.3.2 The 1-d.o.f. hydraulically driven mechanical system ........ 94
3.3.3 Additional mechanical modes . ................... 97
3.4 Hydraulically driven Stewart platform . ................... 101
3.4.1 Basic model structure hydraulically driven systems . ........ 102
3.4.2 Dynamical and kinematical properties of the SRS-motion system;
Design aspects . . . . . ....................... 104
3.4.3 Actuator inertial effects . ....................... 110
3.4.4 Analyzing the SRS hydraulically driven motion system model . . . 112
3.4.5 Connection of a flexible foundation to the SRS system model . . . 116
3.5 Chapter Resume . . .............................. 118
4 Parameter identification and model validation 119
4.1 Calibration .................................. 120
4.1.1 Calibrating the Stewart platform ................... 121

Contents xiii
4.1.2 Redundant measurement of the platform pose . . . . ........ 123
4.1.3 Stewart platform pose measurement in practice . . . ........ 126
4.1.4 Identification of the kinematical parameters . . . . . ........ 128
4.1.5 Results in calibrating the SRS motion system . . . . ........ 131
4.2 Stewart platform model parameter identification ............... 133
4.2.1 Gravitational force determination . . . ............... 133
4.2.2 Identification of the inertial properties . ............... 136
4.3 Frequency response model validation . . ................... 141
4.3.1 Frequency response dummy platform . ............... 142
4.3.2 Additional dynamics into the higher frequency area . ........ 147
4.4 Chapter Resume . . .............................. 161
5 Model based control of the flight simulator motion system 163
5.1 Introduction .................................. 163
5.2 Another look at the control problem . . ................... 164
5.3 Control Strategy . . .............................. 166
5.4 Inner loop pressure control . . . ....................... 168
5.5 Multivariable feedback linearisation . . ................... 173
5.5.1 Feedback linearising robotic manipulators . . . . . . ........ 173
5.5.2 Feedback linearisation of a Stewart platform . . . . ........ 174
5.5.3 Implicit state measurement requirements . . . . . . ........ 176
5.5.4 Outer loop control . . . ....................... 177
5.6 Reference model based feed forward . . ................... 178
5.6.1 Reference model based predictors . . . ............... 179
5.6.2 Construction of the reference model based controller ........ 181
5.7 Implementational issues . . . . . ....................... 185
5.8 Performance quantification . . . ....................... 185
5.8.1 Motion system evaluation methods and requirements ........ 187
5.8.2 New test procedure . . . ....................... 189
5.9 Experimentally evaluating performance ................... 191
5.9.1 Experimental set up . . . ....................... 191
5.9.2 Characteristics of the Simona motion system . . . . ........ 193
5.10 Chapter Resume . . .............................. 200
6 Review and discussion on the results 203
6.1 Flight simulation . .............................. 203
6.2 Motion system specifications . . ....................... 204
6.3 Theoretical modelling . . . . . . ....................... 205
6.4 Experimental modelling and validation . ................... 206
6.5 Control strategy and evaluation . ....................... 208
7 Conclusion and recommendations 211

xiv Contents
7.1 Conclusion .................................. 211
7.2 Recommendations . .............................. 213
A Frequency domain measurements 215
B Derivation of actuator inertial properties 227
Bibliography 231
Glossary of symbols 241
Samenvatting en CV 247

Chapter 1
Introduction
”As man travels, he vibrates. For as he travels, he generally experiences accelerations
not perfectly uniform in magnitude nor direction, and these he feels as varying forces or
vibrations. Travelling faster, these vibrations become more severe, until he learns new
ways to reduce them. But the pioneers will vibrate considerably.” [23] These phrases by
Clark, originally used to motivate research into the tolerable limits of motion cues for human
beings in the early development of the U.S. space program, exactly point at the aspect of
piloted simulation, which is the subject of this research. The motion cues, which are a
consequence of travelling, can be stimulated without actually travelling, thereby enhancing
perceived realism in a simulator. Further, since proper control of a vehicle requires sufficient
reduction of vibration, the actual sensation of these resonances forms an essential part of
basic skill training.
The tool to achieve this, is a motion system. By automatic control, the dynamics of
the motion system will have to be transformed into the dynamics of the vehicle with its
environment. This requires knowledge of both systems. With the developments in computer
technology, more and more information can be directly incorporated in the controller. Given
feasible trajectories generated by a reference generating system of e.g. an aircraft model in
our case, this research considers the use of motion system knowledge in a model based
controller.
In this chapter, first an overview of the most important aspects of flight simulation will be
given in Section 1.1. Then, the specific requirements and approaches towards motion control
will be discussed in Section 1.2. This will lead to the problem statement and approach of
this research in Section 1.3. Finally, the outline of this thesis will be given in Section 1.4.
1.1 Flight simulation
First, the different aspects of the complex system of flight simulation will be discussed.
With an emphasis on the motion aspect, the evolution of these systems over the years will
be sketched. Then, the Simona project centred around the development of a modern research
simulator, in which this investigation was embedded, will be introduced. Finally, the specific
features of the six degrees-of-freedom Stewart-Gough Platform motion system, nowadays
generally used in flight simulation, will be stated.
1

2 1 Introduction
1.1.1 The art of simulation
The fact that simulation is often still considered an art instead of a science [25],[131], reflects
the fact that this complex mechanism of creating a virtual environment is not fully
understood yet. The flight simulator or more specifically piloted simulation is, however,
widely used nowadays, especially in the area of aerospace and astronautics. Following Advani
[7], several areas of application can be recognised.
Area of application
Engineering simulators are used to evaluate the characteristics of a vehicle. In the development
of a new aeroplane, the simulator is currently intensively used during the whole
process of design, due to their ability to predict problems and support flight clearance. With
simulation, large reduction in flight testing could be achieved, which saved costs and allowed
earlier certification [131]. Most engineering simulators are fixed base i.e. without
inertial motion since motion cues are not always important and necessary.
However, the motion system is essential in predicting handling qualities, e.g. a pilot
induced oscillation (PIO) due the dynamics of the vehicle, in a proper way [121]. Low
frequent modes of the aircraft can be damped more appropriately with motion, while vibrations
at higher frequencies amplified by the pilot in the loop, e.g. often modes due to
structural aeroelasticity, can usually only be detected if the controlled motion system leaves
these modes unaltered.
Training simulators are used to train pilots both in the procedural (e.g. flight management)
as well as the basic skill tasks (e.g. manual control). In military applications the procedural
training is often done in fixed base simulators. The basic skill training is mostly performed
in-flight since the discrepancy between the simulator and aircraft is considered too large in
extreme manoeuvres and of course actual flight training is much more exciting. In commercial
airlines, the economic attractiveness of motion base simulators is preferred over
actual flight time and many hours are spent in training simulators. Proficiency checks and
re-currency training i.e. skill examinations and transfer to another type of plane, are allowed
by the regulating authorities without actually flying for pilots with over 500 hours experience
on an aircraft of the same group or 1500 (Europe) to 2500 hours (USA) grand total
[25]. There is a tendency to further reduce this prescribed number of hours [143], which
allow this so-called Zero-Flight-Time (ZFT) training, but exact knowledge of the required
simulation fidelity is still lacking.
Research simulators are often applied for fundamental investigation into pilot/vehicle interaction
and human perception research. These kind of simulators most often require the
highest level of performance w.r.t. the motion system as the resulting sensory input is essential
in the manual control task [47].
Apart from this fundamental scientific work, the simulator has also proved valuable
for accident investigations [131]. Research simulators have been used to study the circum-

1.1 Flight simulation 3
stances of incidents due to clear air turbulence and other types of severe turbulence like gust,
and come up with more beneficial strategies in control and better operational procedures.
In these experiments, the motion system had to be capable of producing vibrations which
made the pilot unable to read the instruments.
Advantages of simulators
Reviewing the area of application reveals the various specific qualities of the flight simulator.
Over the years it has proved to have several advantages as compared to flying an actual
aircraft.
-Cost effectiveness. Both with respect to initial investment as to operational costs, the flight
simulator, though still requiring a considerable amount of money (often over 10M$
and about 1k$/h), is still an order of a magnitude less expensive than most public
transportation aircrafts. Further, the effectiveness of training and testing is in many
cases even better.
-Safety requirements. The simulator is a safe tool to test and train in conditions, which
are dangerous in actual flight. Training simulators can safely be used to show pilots
the best operating procedures in unusual circumstances. Inexperienced pilots can
be taught the bulk of basic skills and procedures avoiding the risk of putting them
into the air. Many potential flight design errors can be detected in advance. Further,
the fact that accidents can be analysed in detail with a simulator, enhances flight
safety. An indirect safety aspect is the cost effectiveness enabling a thorough and still
economically feasible training program.
-Environmental considerations. Every hour spent in a simulator instead of an aircraft,
directly saves a considerable amount of kerosine burnt in the air. Indirectly, environmentally
less harmful flight operation can be tested and trained in a simulator.
Since the objective is not to duplicate, but to simulate an environment which creates the
best and most effective training and testing results [25],[117], the flight simulator should
also be considered as a training or testing tool in its own right, rather than as a substitute for
the aircraft [143]. In this respect the simulator is often more effective.
Training can be split up in parts and these modules can be taught separately, in a step by
step approach. The simulator provides an objective tool to evaluate different subjects under
equivalent conditions as the repeatability of the system is much higher. A simulator is very
flexible in performing specific tests e.g. under desired weather conditions or at a certain
airport.
Flight simulator components
In piloted simulation, a complex environment is created, which has to reflect the impression
of travelling in the actual vehicle with an emphasis on the perceived realism of the task to
be investigated or learned. This environment can be divided into several subsystems:
-Interior. First of all the static environment of the pilot has preferably the look of the interior
of the vehicle cockpit.

4 1 Introduction
-Visual system. The exterior out-of-the-window sight is simulated with an image generating
system in case of civil aircraft simulation often projected via a wide angle mirror
attached in front of the simulator cockpit.
-Instruments. The instruments in the cockpit give information about the state of the simulated
vehicle and should actively be changed accordingly.
-Motion system. The required inertial motion, the actual accelerations of the system, can
be generated to some extent by actively moving the system.
-Control loading. The pilot controls to the vehicle often responds to the given input by
feedback of forces which reflect some of the external forces on the vehicle e.g. the
aerodynamic force on the rudder, etc. These forces have to be generated artificially
and in this case called control loading in a simulator and vehicles, where this has been
decoupled mechanically (fly-by-wire).
-Audio system. The generation of the sound a vehicle together with its environment produces
adds to the perceived realism in the simulator and, above this, is used in the
control task (gas throttle). In some cases, the audio system can also mask the parasitic
noise produced by other sub systems [10].
Apart from the static interior, all these subsystems add to the motion awareness and all cues
(noticeable stimuli), which can be generated by these subsystems, have to be considered in
evaluating a manual control task.
The main unknown remains to what extent the simulator environment reflects the inthe-air
conditions and to what extent this is required to achieve a certain training or test
objective. First, the simulator is only as good as the data it contains i.e. the quality of
the model representing the aircraft and the flight environment to be simulated. Secondly,
inherent limitations of the system used to simulate can be the cause of mismatches. The
motion system of a simulator does posses one of the most obvious constraints with its very
limited amount of travel. Fortunately, the pilot can be made to perceive the motion desired
if the main cues are correctly simulated, allowing for some deviation in the others.
1.1.2 Mechanism of motion perception
Perception of motion has been studied for more than hundred years [88]. At that time,
mainly experimental data was obtained. In the late 60’s of the past century models were
proposed, which described the human behaviour in a motion control loop [69],[94]. With
the crossover model of McRuer et al. [94], the pilot acts according to a describing function,
H(j!), over the frequency, !,
H(j!)=Kh 1j! +1
2j! +1 e; dj! : (1.1)
This model assumes only one main cueing input and considered additional dynamics of the
sensory and response system as a simple time delay, e ; dj! . The parameters, gain, Kh, lead
time constant, 1, and lag time constant, 2, are adapted such that a slope of -20 dB/decade

1.1 Flight simulation 5
in the describing function frequency plot around the crossover frequency, ! c is attained of
the open loop system of human and the system to be controlled, which usually results in
a sufficient stability margin in the closed loop. This frequency is approximately 2 rad/s to
5 rad/s in case of a flight control task (involving tracking and disturbance rejection) [53, 94,
149]. With aircraft vehicle control, on which this model was experimentally validated, the
open loop transfer function with a double integrator, 1=(j!) 2 , can be controlled with PD,
proportional/differential action in which 2 < 1=(!c) < 1.
The optimal control model of Kleinman et al.[69] assumes that the pilot operates in an
optimal way, given all the perceived cues with their respective limitations. This motivated
the research into the mechanism of motion perception ([159] and the references therein) with
a model based approach in which the interaction between the different stimuli, e.g. inertial
motion and visual inputs, is taken into account. These models are still being refined [53]
as the system is incompletely understood. Though the quantitative values are seen to be
dependent on the subject, task and environment involved, the qualitative control behaviour
of a well trained subject is reasonable consistent and reproducible.
Motion results in stimuli inputs to the sensory organs. Above some sensory threshold,
the sensory organs start to fire pulses to the central nervous system. The subject perceives
motion from the set of cues generated in this way. These cues have to be reasonably consistent
relative to each other. Otherwise disorientation and motion sickness can occur. Some
deviations can be tolerated as long as the coherence functions of the subject from each cue to
the corresponding movement have sufficient overlap [148]. Advantage from these tolerable
differences is taken in simulation, which allows perceived motion to be simulated without
the need for very large simulator excursions.
The main sensory mechanisms to detect motion as given in the AGARD Advisory report
159, [2] are:
- Semicircular canals and Otoliths are called the vestibular organs. Both reside in the
inner ear. The function of the semicircular canals is roughly that of rate gyros, from
which angular velocity can be measured. The otoliths sense the specific force which
is the combination of inertial acceleration and gravity.
- Tactile or somatosensory receptors permit sensing a change of force on the body, e.g. due
to a change in orientation. These sensors have a high-pass characteristic, which causes
sustained uniform pressure to be faded to a certain reference level. Especially the fingertips
are very capable detecting high frequency vibrations.
- Proprioceptive and kinesthetic senses lead to information about the relative positions of
the parts of the body as well as their movements i.e. the body acceleration. This is
attained through the double sided muscular coupling of force and velocity/position
and an internal model of the body
- The eyes primarily detect motion through the peripheral retina. With uniform motion of
a wide visual field, it is possible to create a so-called self motion sensation, which
is called vection. The foveal area of the retina or central part is primarily associated
with scanning and recognition i.e. the cognitive sense of self-motion.

6 1 Introduction
Advani [7] also mentions the auditory system in the ears, from which through sound motion
can be detected.
With the visual and auditory system of the flight simulator, the pilot can be isolated from
detecting inertial motion through sound or sight. With the other organs, inertial motion cues
will be perceived as motion. In case of required high forces, the proprioceptive and tactile
senses can also be stimulated with a so-called g-suit, mainly used in military applications.
Control forces and vibrations provided by the control loading system will also influence the
tactile senses.
The vestibular system is considered to generate the main cues in the frequency area
around the crossover frequency (:3 to 1Hz), essential to learn skill based control behaviour.
This is the central issue in motion-based flight simulation. In order to recreate the skill
based behaviour, the pilot in the simulator must be provided with well coordinated visual
and vestibular cues. Motion cues around the crossover frequency frequency area can only
be stimulated with actual inertial motion and this is the main reason a motion system is
essential in case of piloted control tasks in flight simulation.
In the low frequency area ( :1 Hz), motion is much easier to detect through the visual
information of position and orientation at the instrument displays and exterior screen. Some
mismatch with inertial motion, with the already low accelerations related to these frequencies,
is allowed, even slightly superthreshold, without the pilot noticing discrepancies.
Schroeder [126] notes that the review of different studies resulted in a wide range of
threshold values. This is confirmed by the study reported in [54] in which it is shown that a
threshold depends on several factors such as the frequency contents of the stimulus and the
task the subject has to fulfil i.e. mental load. Values range for the translational accelerations
from :08 m=s 2 in heave, no task involved to :7 m=s 2 in a very busy situation experiencing
surge (longitudinal direction). Rotational thresholds are found to be in a closer range of
:8 deg =s 2 to 1:5 deg =s 2 for pitch and roll, though Schroeder [126] reports a range of
:0035 deg =s 2 to 2 deg =s 2 for yaw motion (vertical axis rotation).
Hosman [53] provides some transfer functions for the vestibular system. Both the semicircular
channels as the otholits are modelled according to a function specified in the frequency
domain:
xout
=
xin
K( Lj! +1)
� (1.2)
( 1j! + 1)( 2j! +1)
where the ratio between (actual) input acceleration, x in, and noticed acceleration (xout in
impulses per second, ips), is given by a second order transfer function with gain, K, two
lag time constants, 1 and 2, and a lead time constant, L.
Although the models of the two vestibular organs, which give the sensitivity over the frequency,
have the same structure, the characteristic is different. This is due to the parameters,
which are given by Hosman [53] as in Table 1.1.
This means the sensitivity of the semicircular channels drops over the range from 0.025 Hz
to 1.4 Hz after which it starts to drop again at 30 Hz, while the otholits become more sensitive
from .16 Hz to .3 Hz, after which it starts to drop again at 10 Hz. This opposite effect
in the relative important frequency range could maybe explain the fact that different conclusions
can be drawn towards the question whether translational accelerations or angular

1.1 Flight simulation 7
Semicircular Channels Otholits
K 2 ips=( =s 2 ) 3:4 ips=(m=s 2 )
L 0:11 s 1 s
1 5:9 s 0:5 s
2 0:005 s 0:016 s
Table 1.1: The parameters of the models of the vestibular organs
accelerations are more important. In case of roll and lateral motion in large aircrafts and yaw
and lateral motion in helicopters, lateral motion cues are shown to be more important [126].
In other cases reported in [54], [126] rotational cues are considered more relevant. Chung
et al. [22] conclude that the translational and rotational motion relative to each other should
not differ in phase nor amplitude since these types of motion are coupled as a function of
place.
The motion cues provided through the vestibular system with inertial motion, change
the describing function of the pilot [53], [149]. With the introduction of motion, the time
delay in the describing function is seen to drop from 0.23 s to 0.12 s in case of a disturbance
rejection task. The crossover frequency a pilot can achieve is lower in the absence of inertial
motion and, as a result, also the performance is. Hall [47] argues and confirms by an
example that, even with equivalent performance, the pilot can develop a different and false
strategy if the cues in a simulator deviate from those in the aircraft. Especially in case of
relatively inexperienced pilots, all cues are important in the learning process of picking the
most relevant [143].
There is no doubt inertial motion is necessary in the appropriate perception of motion.
Not much is known to what extent the inertial motion of the simulator is allowed to differ
from the aircraft. In the low frequency area it is concluded that high pass filtering at .08 Hz is
acceptable [149]. The crossover region between .1 Hz and 1 Hz is likely to be most sensitive
to deviations. At higher frequencies, the situation is less clear. Pilots are seen to provide
lead well over 3 Hz and consequently still respond to motion with increasing gain in this
frequency area. Further, pilots respond much earlier when inertial motion is present. In the
more general requirements of the international standard ISO 2631 [138], which evaluates
human exposure to whole-body vibration, it is noted that humans are most sensitive to heave
accelerations in the frequency area between 4 Hz and 8 Hz. The bandwidth of many motion
systems in this frequency area makes these systems prone to errors exactly in the area where
the level of comfort is most sensitive.
Over the years, many discrepancies in the motion systems of simulators have been considered
acceptable. There was no economical or technical feasible alternative. Other subsystems
were also unable to represent reality in a satisfactory manner. A pilot was required
to fly an aircraft on the basis of instrument displays and external sight without motion cues.
Being able to fly a simulator without appropriate inertial motion qualified since it is more
difficult than with an aircraft. In-flight training still formed a considerable part of the learning
process. This point of view is changing as can be seen by putting the current state of the
art in a historical perspective.

8 1 Introduction
1.1.3 Historical perspective
In [4], Adorian et al. provide a nice overview of the development of the flight simulators over
the years. Almost directly with the introduction of the first aeroplanes, the first simulators
were built. Already in those days, a motion system was part of the simulator. Around 1910,
these motion systems had to be driven by hand by the instructor, or wind, and mainly had
to give the experience of a changing orientation around one to three axes. Soon the inertial
motion was provided automatically. The Ruggles Orientator, developed in 1917, had full rotation
around three axes and additionally vertical movement, all driven by electrical motors.
The most successful device, the Link trainer, developed in 1927-29, was used well into the
50’s. It was driven pneumatically and operated by the stick and rudder in the cockpit. The
simulated effects were adjusted by trial and error independently for ailerons, elevators and
rudder. In this way, the aircraft’s coordinated behaviour could not be represented properly.
After 1930, for thirty years, the main developments concerned other subsystems than the
motion system of the simulator. Early in the Second World Ward, hundreds of the so-called
Celestrial Navigation Trainers, massive bomber crew trainers, were built. A moving dome
provided for the ’correct’ out-of-the-window sight in case of a nightly atlantic ocean crossing.
During the war, the visual system was developed further into actual image projection
systems.
With the instruments in the cockpit, all basic flying behaviour, all engine, electrical and
hydraulic systems could be simulated. The computation, at first pneumatically, could solve
the equations of motion of an aircraft with the introduction of the analog computer.
The first full aircraft simulator operated by an airline was installed in 1948 (a Boeing
377 by PanAm). Most full aircraft simulators did not have a motion system at that time,
which was justified by the statement that ”modern pilots should not fly by the seat of their
pants.” Partly, the control loading system was thought to compensate for the lack of motion
and provide for a realistic feel.
With the improvement of flight test data and the increasing complexity of the aircrafts,
the analog computer became the bottle-neck in the simulator. The demand for increased
fidelity and reliability motivated the introduction of the digital computer in the flight simulator
system.
However, correlation was poor with flight tests concerning low damping, stability or
characteristics of pilot induced oscillations. Further, strong visual cues supplied by the
improved wide field of view visual system were disorienting in the absence of motion cues.
In 1959, the first motion research simulator was built at NASA, which also indicated the
importance of motion cues [10]. Analysis of loss of control in extreme operations motivated
the research with centrifuge simulators [23]. The many motion deficiencies, due to the
very limited vertical stroke, the vibration noise and incorrect centrifugal forces, were so
distracting that most of the research was still performed with fixed base simulators.
Motion systems with alternative construction were designed. The introduction of the
wide body transport aircraft, such as the B747, required lateral acceleration and led to the
four and six degrees-of-freedom (d.o.f.) motion systems. The first six-degrees-of-freedom
research simulator at NASA Ames was put into operation in 1964. These systems were often
given too much stroke (up to 30 m), which was not necessary given the limited velocity and
resulted in inferior dynamics of lengthy cables (low eigenfrequency). Also the turn-around-

1.1 Flight simulation 9
bump due to coulomb friction prohibited unnoticeable fade to a central position.
In the late sixties, the synergetic, fully parallel, hydraulically driven motion system came
into operation. The most commonly used is the Stewart-Gough platform [140]. Through
the hydrostatic bearings introduced by Viersma [153] in 1969 in the first (3-d.o.f.) flight
simulator motion system at Delft University of Technology [12], these systems had virtually
no coulomb friction. The first commercially available 6-d.o.f. motion system in 1977 with
hydrostatic bearings was reported on by Baret in [13]. Nowadays, almost every simulator
motion system operates in this manner.
In the late 70’s, an attempt was made to classify the characteristics of motion systems in
flight simulators [1], [67]. Travelling limits, response time (latency), roughness and noise
were considered most important. The regulating authorities such as the FAA (USA) and
JAR (Europe), given confidence in the use of simulation, came up with required motion
system specifications, which reflect the abilities of motion systems at that time [3], [25],
[118], [139]. It is argued that the fact that these standards have been basically unchanged
for the past 15 years, has stopped virtually all fundamental development in flight simulators
[143]. Further, since there are still no objective tests, no prescribed manoeuvres and no
criteria for accurate motion cueing, makes it difficult to establish new standards. There is,
however, a market demand from the airlines to come up with additional knowledge to enable
a decrease in required flight experience for zero flight time training.
In the 80’s and 90’s, the main developments concerned the rapid growing abilities in
the visual and computational systems. As the main selling argument in the 60’s came from
the all but perfect though impressive motion systems, now most interest stems from the fast
improving, marketable, visual system. As in the past, higher standards for other subsystems,
such as the quick response of the image generating systems now available [100], the
increased computational power enabling to include aeroflexible modes in the aircraft model
and more advance control strategies for motion control, will be a drive for higher fidelity
with respect to motion.
In [10] it is not expected that any major changes in motion travel will be made in flight
simulation motion systems. However, within the current structure improvements can be attained
[7] and more often multi-stage, cascaded, motion systems can be observed, in which
the need for motion cues (with the current 6-d.o.f. structure) and travel (additional for lateral
and longitudinal travel over rails as in the National Advanced Driving Simulator [20], [92])
have been decoupled but still have considerable interaction in the dynamics. Reduction of
time delays in motion system response and interaction is needed for solving more complex
problems with visual/motion/model mismatches [10]. At this moment, high frequency motion
cues ( 4 Hz) require special effects included in the motion system controller and
phase lag in the crossover region is compensated for by additional lead (differential action)
on the reference signals [10], [24], [143]. A more structural model based based approach
will be strived for in this research.
1.1.4 Motion simulation
As already mentioned, in flight simulation, the pilot relies on the perception of self-motion
through several stimuli, and uses this perception to exercise control over the aircraft. From

10 1 Introduction
Experimentator
Situation
Aircraft
Model
Wash-Out
Filter
Pilot
Motion
Controller
Simulation computer Motion Computer
Motion
System
Continuous World
Fig. 1.1: A block diagram representation of motion simulation.
the pilots point of view, the simulator should be able to replace the aircraft.
In the control task, the pilot uses his visual perception to make a good estimation of
the aircraft’s long term attitude and velocity. The approximate frequency response of visual
perception can be modelled as a first order low-pass filter with a break frequency of 0.1 Hz
[92]. Fortunately, the inertial motion of the simulator has a minor role below this frequency.
There is a potential payoff in tuning the motion cues to match the human physiological
sensors’ responses, rather than attempting to match the aircraft motion themselves. If done
properly, the effective provision of motion can be obtained in an envelope substantially
reduced from that of the aircraft motion.
The stimulation through actual inertial motion with a simulator can usually be represented
by the block scheme given in Fig. 1.1. The pilot responds to simulator cues and
the task at hand provided through the experiment leader. With a model of the aircraft, it is
calculated what the actual aircraft motion would have done due to the action of the pilot,
the situation (on the ground the aircraft has different dynamics in interacting with the track
than in the air) and the situation dependent disturbances such as turbulence.
With washout, the calculated aircraft motion has to be translated into feasible motion
system trajectories regarding the limited stroke, velocity and accelerations. Washout must
provide some form of high pass filtering to limit the simulator excursions given the unlimited
aircraft travel. Washout does usually separate the motion cues into high (”onset”)
and low (”sustained”) frequency components. Further, by also introducing a scaling factor
of simulator motion, all onsets can be performed by the simulator without attaining the
inherent limitations of the motion system.
Most sustained cues can not be represented through inertial motion of the simulator and
have to be ”washed out”. Sustained forward or lateral acceleration is sometimes represented
through a trick already mentioned by Johnson in 1931 [4]. Through the help of the gravity
force, a backward tilt gives the same sensation as a forward linear acceleration. Problem

1.1 Flight simulation 11
Fig. 1.2: A typical modern flight simulator equipped with a synergistic (hexapod) motion
system. 1
with this so-called tilt coordination is to attain the tilted pose without the pilot noticing
i.e. pitching or rolling with subthreshold angular motion. In a critical survey, Advani [7]
notices that the mostly used classical linear washout leads to considerable false cues in the
important frequency area between 0.1 Hz and 1 Hz. In an overview of wash-out techniques
in [92] by Martin, also non-linear wash-out is mentioned, which however has not led to
completely acceptable solutions so far.
The feasible washed-out motion trajectories are the reference input to the motion controller.
The task of the motion controller here is to attain a one-to-one correspondence
between the actual motion system trajectories and the desired feasible washed-out motion
trajectories. In the usual configuration, the references contain positional information, which
is also measured and can be used in feedback. Further, the desired acceleration profiles are
provided for, which can be applied in generating an appropriate feed forward.
As it is physically impossible to generate immediate acceleration with the motion system,
imperfect cues are generated, which lag the desired cues. Enlarging the bandwidth
decreases the lag but has to be compromised with sufficient attenuation of false cues resulting
from noise and excitation of parasitic dynamics, which mainly reside in the high
frequency area. Considering a more general setting than the one given in Fig. 1.1, can possibly
lead to a less severe compromise by e.g. allowing available predictive information in
the simulation model to be sent to the motion controller.
The flight simulator motion systems, nowadays, have almost invariably a so-called 6d.o.f.
synergetic construction. A motion system of this type is shown in Fig. 1.2. 1 Six
1 Picture courtesy of Rexroth-Hydraudyne.

12 1 Introduction
actuators are connected in parallel to a base frame at the foundation with one gimbal, a
freely rotating joint, and at the other end to the simulator with another gimbal. Kinematically,
the six lengths of the actuators can be driven independently. However, dynamically,
this system is highly interactive i.e. an elongation force of one actuator generally induces reaction
forces in all others. Further, prescribed platform motion requires coordinated motion
of all actuators, which varies with the platform position.
The main advantages of this construction are the parallel connections resulting in a system
with higher stiffness given a specific actuator and its simplicity. Six identical system
are easy to build and maintain. The stiffness is due to the fact that springs in parallel result
in higher stiffness while springs in series will result in a decrease. This is important in flight
simulator motion systems since the payload to be accelerated is usually relatively high. With
the synergetic construction the static load per actuator will be lower and the eigenfrequencies
will shift to higher values. The forces to be applied with the actuators are usually still
considerably high. Most flight simulator motion systems are therefore driven by hydraulic
servo actuators, since these systems provide superior performance in generating high power
long stroke linear motions [124]. Further, the use of hydrostatic bearings results in very low
coulomb friction forces, which minimises the velocity reversal bump. As simulator wash
out motion reversal and aircraft motion do not coincide, an unnoticeable change of velocity
sign is very important in flight simulation.
With the introduction of improved linear electrical drives, there is a trend to use electrically
driven motion system if a simulator has relatively low payload up to 2500 kg
[24].
1.1.5 SIMONA project
Through a preliminary investigation for an upgrade programme of the older DUT 3-d.o.f.
simulator, a new international centre for research into flight simulation techniques, Simona,
was initiated in 1992. Simona stands for Simulation, Motion and Navigation, which reflects
the areas of interest in the research programme of the three groups, who initiated this centre.
Instead of an upgrade of the older simulator, which has been dismantled, the core of Simona
will be a full scale 6-d.o.f. flight simulator, the Simona Research Simulator (SRS) [6]. The
development of the SRS is also performed within Simona. The actual fabrication of the
simulator was started in 1994, after a large donation of the Dutch Government to enable a
multi-discipline international centre with a high fidelity simulator.
The three groups involved, each have their specific research interest and contribution to
the project.
The Simulation and Control Group of the Faculty of Aerospace Engineering performs
studies into flight control, the man-machine interaction of the pilot in the aircraft
and modelling and real-time simulation of the aircraft and its systems. In the, now
ongoing, realisation phase of the simulator the group takes part in the development
of simulation software, the interior and the visual system. The Group of Production
Technology within Aerospace Engineering did the design of the integrated platformcockpit
shuttle and planned connections of projection platform and image projection
screens, which will be connected in future.

1.1 Flight simulation 13
Fig. 1.3: Artist’s impression of what will be the full operational Simona flight simulator

14 1 Introduction
Fig. 1.4: Simona motion system with empty shuttle on top, configuration B.

1.1 Flight simulation 15
The Mechanical Engineering Systems and Control Group of the Faculty of Design,
Engineering and Production performs research into the design, development, modelling
and control of hydraulic servo and motion systems. The 6-d.o.f. hydraulically
driven motion system has been realised and tested and so have the first model based
controllers as part of this research. In future, alternative control concepts will have to
be developed. For one, to interconnect simulation models and motion control in an
optimal way. Further, to adapt to the changing characteristics of the motion system
with e.g. a, to some extent flexibly attached, visual system.
The Telecommunications and Traffic-Control Systems and Services Group of the Faculty
of Information Technology and Systems does research into navigation technology
and advanced cockpit display systems. The operational SRS can serve as an experimental
set-up to test new navigation techniques and approaches to simulate these
techniques appropriately. The group has been involved in the development of a realtime
simulation platform.
An impression of what will be the full operational flight simulator is given in Fig. 1.3.
The motion system consists of six hydraulic servo actuators which are connected in parallel
at the base frame and simulator. The actuators together can drive the simulator in all three
translational and rotational degrees of freedom. At the simulator side, the actuators are connected
to the bottom of the shuttle, an integrated platform/cockpit made of the light weight
material TWARON/carbon. In front of the shuttle, a wide angle visual mirror projection
screen will be attached. With projectors on top of the shuttle and a back projection screen
in between, the pilot in the cockpit can be provided with a out-of-the-window view via the
mirror in front of the shuttle.
The new design of the integrated platform/cockpit and the use of light weight materials
will lead to a relative low centre of gravity and considerably less mass to be accelerated
as compared to conventional simulators. These properties strived for in the design of the
construction, will have to result in favourable motion characteristics. In what sense the
expectedly 4 tons of the SRS instead of the 12 to 15 tons often reported for other systems
[24], [92] can be taken advantage of, is one of the issues to be discussed in this research.
After separate testing of all six hydraulic actuators in a special experimental set-up in
1994 [124], the integration of the motion system was complete early in 1997, and the first
free-run tests of the integrated Simona motion platform were performed. The motion system
was tested in three configurations in the Central Workshop at the faculty of Mechanical
Engineering.
A. The first tests were performed with a steel platform, the dummy platform of 2200 kg.
B. In the autumn of 1997 the empty (only chairs) shuttle of 1700 kg was put on top of the
motion system, as shown in Fig. 1.4.
C. Finally, tests were performed with the estimated final load of 4000 kg with the dummy
platform and additional masses (6 times 250 kg) on top, shown in Fig. 1.5, in the
months of May and June, 1998.
In the early spring of 1999, the motion system was to be operated at the building especially
made to house the Simona centre, eventually with the full operational simulator. Main

16 1 Introduction
Fig. 1.5: Simona motion system with dummy platform and additional masses, configuration
C.

1.2 Motion control 17
w z
- -
- Psp u
K
Fig. 1.6: A standard control design structure
efforts are now in progress to complete the cockpit interior, the real-time simulation environment
and the visual system.
1.2 Motion control
Having sketched the flight simulation process and the place motion generation has in this
environment, the motion control problem can be defined. First the process of motion generation
is put into a generalised framework. From this perspective the motion control specifications
are stated and the conventional approach towards the design of motion control in
flight simulators is discussed. Then realising that the motion system of a flight simulator can
be considered a robotic system, of course with its specific demands and properties, some of
the modern control strategies developed in robot control are put forward, since these could,
with some modification, possibly be of benefitinflight simulation.
1.2.1 Motion control structure
A standard control framework, adopted in many textbooks on modern control [17], [89],
[160], is given in Fig. 1.6. A controller, K, is provided with measurement signals, y, and
has to stabilise a plant, Psp, with input signals, u, such that the cost variables, z, are minimal
in some sense, despite the disturbance signals, w.
The plant, Psp, is often called the generalised plant or standard plant since it usually
does not only consists of the plant to be controlled, the motion system in flight simulation
motion control, but can also contain weightings e.g. sensory thresholds weighting on the
cost variable specific force or a reference model e.g. the aircraft to be simulated. Also the
other entities can be viewed in a generalised way, e.g. reference signals can be incorporated
as disturbances w and additional feedback paths can be taken in case of a robust control
problem to describe a set of systems i.e. uncertainty.
In a general simulation motion control problem, as shown in Fig. 1.7, the performance
cost variables, zp, will be the difference of motion in the aircraft and in the simulator with
the pilot in-the-loop weighted with the ability of the pilot to perceive these differences,
Wp. Further, motion system limitations should be penalised by additional cost variables,
zc. So the standard plant will consist of the motion system, the simulation model (Sim),
the aircraft, the pilot and weighting functions. The generalised disturbances are both the
y

18 1 Introduction
wda
wr
wdm
u
-
-
-
-
-
Aircraft
Pilot
Pilot
Motion System
Sim - Ww
Motion Control
e? - Wp -
6
Fig. 1.7: The flight simulation motion control problem put in the generalised plant structure
tasks to be tracked, wr and the external disturbances to be rejected. There will be both
external disturbances on the aircraft, wda, such as forces due to turbulence, which should
be simulated, and external disturbances on the motion system, w dm, such as friction forces,
which are to be rejected.
The controller receives information from both the motion system, y m, which has to
be stabilised and the other subsystems of the generalised plant to construct an appropriate
reference to be tracked, yr. In case of the motion controller in the conventional setting as
given in Fig. 1.1 this will be the desired motion reconstructed by the wash-out filters, W w.
If the reconstruction of a feasible trajectory is taken as a part of the control problem, as is
done by Idan et al. in [58], the aircraft motion can be taken as the reference and the wash
out model becomes part of the motion controller to be designed. When the whole inertial
motion simulation is considered, the ’simulation controller’ is fed by actions of the pilot and
the experiment leader. The controller can act through the actuators of the motion system, u.
To calculate an appropriate controller, a mathematical model of the subsystems in the
standard plant can be derived. If these models are sufficiently simple, e.g. linear time-
+
-
yr
ym
zp
zc
-

1.2 Motion control 19
invariant, an optimal controller can be synthesised, which for instance minimises some
maximal amplification on some norm on the cost, z, given a bound on the norm of the
disturbances, w [160]. The flight simulation motion control problem is yet too complex to
be solved in this way, but some observations can be made from the structure of the standard
plant given in Fig. 1.7.
The task of tracking a reference model is a particular element in flight simulation,
which requires special consideration. Several techniques, like model following [145], model
matching [156], the servo mechanism [33], etc., exist to achieve such tasks. Tracking results
through the servo-mechanism with linear time-invariant systems if the internal model
principle [40] is satisfied, which states that the (unstable) parts of the system to be tracked
have to be part of the controller. Also with the other techniques, knowledge of the system
to be matched or followed is incorporated in the controller (design). Although the reference
aircraft model is available in flight simulation, the specific properties of these models are
not yet taken into account in the motion controller design.
The reference tracking problem typically leads to a two degree of freedom controller
design problem [152], as given in Fig. 1.8, in which a feedforward, C 1, and a feedback part,
C2, can be distinguished. With feedforward, ideally, the input, u, required to achieve the
desired reference, r, should be generated, i.e. an inverse model of the system, G. Usually,
however, the system does not have a causal and/or stable inverse and therefore an inverse
model cannot be used. Further, the feedforward should not unnecessarily excite unmodelled
dynamics in the system. Typically, this dynamics resides in the high frequency area, where
an inverse model would, moreover, usually result in high amplification.
The feedback part should take care of uncertainties such as unmodelled dynamics and
other external disturbances and, if necessary, stabilisation of the plant using the error, e, the
difference between the measured output, y, and the reference, r. Where feedforward can
be seen as specific shaping of the plant w.r.t. reference signals only, the feedback shapes
the sensitivity of the plant towards all disturbances. Feedback usually results in a frequency
area (often the lower frequencies) where the controlled plant becomes less sensitive, an area
where sensitivity is higher (around the bandwidth) and an area with unchanged characteristics
(high frequency area) [89].
De Roover argues in [30] that a third degree of freedom in the design of the controller
should be considered. This should take care of the generation of an appropriate reference
signal. In case of the flight simulation motion control, the causality problem in the mechanics
is often overcome by generating an acceleration reference signal instead of position
only. Calculating the forces required given a position reference would lead to a non-causal
transfer function. This is not the case with acceleration. It is questionable, whether an acceleration
reference signal is sufficient to fully overcome the causality problem since this
would imply that immediate force generation is possible.
1.2.2 Motion control specifications
The main task to be accomplished after wash-out, tracking of the reference acceleration
signals should not only be accomplished w.r.t. the asymptotic behaviour at infinitely long
time spans but especially the transient behaviour, the generation of appropriate onset mo-

20 1 Introduction
-
C1
r + e +
- e - C2 - e?
u
y
- G
-
+
6
Fig. 1.8: Two degree of freedom control structure
tion, is most relevant. By using (model) knowledge of the reference generating system, the
disturbed aircraft, the pilot and the plant, the motion system, use can be made of the two
additional degrees of freedom in the controller to enhance performance [32].
As acceleration is the perceived unit, the performance of such motion control is much
more sensitive as (disturbance) signals become faster, as compared to systems with position
as cost variable. Attaining higher bandwidths than required can make the system
unnecessarily sensitive. Though it is known that below approximately 0.1 Hz motion is not
perceived through inertial movement, the required bandwidth is yet unknown. Of course,
the crossover frequency area of the pilot/aircraft loop around 1 Hz should be simulated with
close correspondence but the question of whether the vestibular organs, which are sensitive
up to 30 Hz, should be stimulated over this whole frequency area is not seen to be
answered in literature. It can be expected that striving for a bandwidth of 30 Hz will lead
to compromising false cues due to parasitic dynamics of the motion system with required
cues.
In summary, inertial motion control of the cockpit in flight simulators is directed at
providing appropriate cues in the following areas:
- The onset of the aircraft response to the actions of the pilot. Both shape (amplitude) and
timing (phase) of the response is important here. As the bandwidth of the pilot usually
does not exceed 1 Hz, in a first attempt, a bandwidth of an order of a magnitude higher
can be strived for in this respect.
- A realistic generation of disturbances resulting from e.g. turbulence. At higher frequencies,
above 2-3 Hz, the shape of these disturbances becomes most important while
(linear) phase lags, small time delays, are not so relevant.
- The dynamics related to the aircraft resulting in vibrations. This can be both due to the
pilot in the loop as the disturbances. This dynamics can change drastically with the
situation e.g. on the ground or airborne. This also means sufficient suppression or at
least minimal excitation of simulator related dynamics i.e. parasitic dynamics, which
do not correspond with the pilot in-the-loop with the actual aircraft.
- Given feasible reference trajectories, the position control should be such that errors in

1.2 Motion control 21
the actuator position do not result in running the actuators out-of-stroke. The corrective
action should not lead to false cues and should therefore typically be of a low
frequency nature and thus have very limited bandwidth.
Not only the quality of the motion control is important, but also the predictability. In doing
pilot/aircraft interaction experiments or in design of a training programm, it is relevant
to know to what degree the simulated environment corresponds with the actual situation.
Finally, the main requirement in doing piloted simulation experiments is safety, which demands
stability of the motion control loop at all time.
1.2.3 Conventional simulator motion control
Most motion control schemes, reported on in flight simulation, focus on the kinematics of
the mechanical system and the dynamics of independent hydraulically driven masses [12,
24, 46, 51, 82]. Given a desired simulator trajectory, the required actuator trajectories can be
calculated with the inverse kinematics (end-effector to actuator coordinate transformation)
and with a fully parallel driven robot these relations can be given explicitly. By neglecting
the interaction between the actuators, each feedback loop can be designed given the transfer
function of a hydraulic actuator driving a mass. The linearised version from valve input,
i, to position, q, as in [99, 153] consists of a lightly damped second order system in series
with an integrator. This is described in the frequency domain as
q
i =
K
s(s2 =! 2 o +2 s=!o
� (1.3)
+1)
considering s = j!.
So at low frequencies the system acts as a velocity generator and at higher values the
resonant p characteristics of second order system plays a role with the eigenfrequency, ! o =
c=m, which depends on the driven mass, m, and the limited oil stiffness, c, often with
poor damping,

22 1 Introduction
the characteristics as described for the one degree-of-freedom system. Pressure feedback,
therefore, still results in damping only. As the eigenfrequencies can vary considerably for
each direction and simulator position, the relative damping provided by pressure feedback
will differ. The positional feedback will have to be tuned according to the lowest occurring
system eigenfrequency. The resulting controlled system has a bandwidth corresponding to
this frequency and is often somewhat lower since sufficient damping of the faster modes can
require overdamped characteristics of the slowest.
By using the acceleration reference signals as lead feedforward tuned according to a
neutral simulator position 2 [24, 46], the bandwidth up to which the system reacts properly
to these references can be extended. In principle, the second degree of freedom in the controller
is used to create a local inverse of the controlled system up to a frequency somewhat
higher than the bandwidth attained with feedback.
Although the parallel structure used in flight simulator motion systems requires some special
consideration, the system can be viewed as a robotic manipulator for which a wide field
of literature exists on control strategies, which make more extensive use of system/model
knowledge. Not until very recently, this is seen to be applied in simulator motion systems
[20, 58, 82]. It seems worthwhile to review the robot control strategies, which can be of use.
1.2.4 Robot control
Although the dynamics of robot manipulators is highly nonlinear, a very specific structure
can be recognised. Most robot control strategies take advantage of this structure. In [136], a
selection of articles provides a state of the art and an overview is given of the developments
in robot control over the years. There are also numerous textbooks on the subject. In recent
works like [11, 31, 81, 102, 113, 115], modern issues such as passivity, robustness, tracking,
elasticity and flexibility, are treated.
Computed torque
One of the earliest applications of nonlinear control of robot manipulators was the method of
computed torque, which started in the early seventies. Using the Euler-Lagrange equations
of a rigid robot manipulator with n input torques, and output positions, q given by
M (q)q + C(q� _q)_q +g(q) = � (1.5)
the required input torque can be calculated, compensating the coriolis/centripetal terms, C,
and gravity, g, measuring the positions, q, and velocities, _q, and filling in a new input vector,
a, instead of q, to end up with
M (q)q = M (q)a: (1.6)
2This position usually corresponds with all the actuators half stroke and is the position to which the wash-out
fades

1.2 Motion control 23
Since the inertia matrix, M, can be inverted, linear, parallel (decoupled) double integrator
systems result from a to q,
q = 1
s 2 In n a (1.7)
The double integrator systems can easily be controlled with any desired bandwidth by a
simple proportional/derivative (PD) controller.
a = Kpe + Kd _e + q d
(1.8)
with the positional error, e =(qd ; q), the desired positions, qd, can be tracked with step
and ramp inputs.
In [77], several control strategies like the computed torque method, feedback linearisation,
resolved acceleration method and inverse dynamics are shown to be very similar and
unifiable. In all cases, exact linearisation and decoupling is strived for. This can be done
both in end effector or joint coordinates.
The main disadvantages of the computed torque method are its robustness properties.
Both with inexact cancellation due to parameter uncertainties (e.g. an inexactly known inertia
matrix) [9] and unmodelled dynamics, such as joint elasticity [85], this controller can
go unstable.
Robust robot control
Robustness can be regained by adding an outer loop such as in the sliding mode control
of Slotine and Li [133] or the saturating controller applied by Spong [135]. Both methods
employ local high gain, which is sufficiently high to track in the presence of uncertainties
but on the other hand is bounded. With the discontinuous sliding mode feedback method,
high frequency unmodelled dynamics is easily excited and chattering is often reported in
practice. Transient response in saturating control is worse. Both properties do not seem
most suitable for flight simulation motion control.
Yet another robustifying approach is to compensate for the main nonlinearities and consider
the system linear with some, possibly varying, uncertainty as was done by Van de Linden
[146]. Robust control techniques developed for linear systems as described in textbooks
as [132, 160] can then be used. Main problem is to arrive at a nonconservative description
of an uncertainty in the nonlinear plant at the level of the outer loop.
Passive robot control
Alternatively, use can be made of the passivity (’no energy generating’) property of a robotic
system [11, 14]. Feedback with a passive controller results in a passive closed loop system,
which under some assumptions has specific stability properties. The feedback controller can
e.g. have a PD structure with possibly time varying gains (dependent on estimated mass matrix).
W.r.t. feedforward, the computed torque structure can still be employed. This strategy
even generalises to a much larger class of systems [147] among which are the mechanical
systems with possibly flexible structures [63, 66].
Most textbooks on robotic systems deal with rigid body structures only. The complexity
of the system increases drastically in including deformations. The flexible structure has

24 1 Introduction
more degrees of freedom than control inputs. Trajectory tracking of a robot end-effector becomes
very difficult and requires accurate models [31]. The eigenfrequency of the flexible
mode is usually a hard constraint for the attainable performance such as bandwidth. With
passive control, though stable, performance is often hard to specify.
Actuator dynamics
Also passivity (control) of robot manipulators with dynamics in the actuators is addressed
as by Arimoto [11]. The DC-actuator employed there is equivalent with a linearised version
of a hydraulic actuator [137]. Although the structure of the (voltage driven) DC-motor and
the hydraulic actuator may be equivalent, the parameters usually vary largely. Therefore,
the DC-motor can often be considered relatively fast in comparison with the robot mechanics.
On the contrary, the (finite oil stiffness of the) hydraulic actuator forms a relevant part
of the dynamics in e.g. a flight simulator motion system around the achievable bandwidth.
The influence of the mechanics can nominally be decoupled in the hydraulics by local
velocity compensation [127]. By an inner feedback loop the hydraulic force generation
becomes relatively fast compared to the mechanics. This cascaded approach was successfully
implemented by Heintze and Van Schothorst [48, 124]. In another cascaded approach
suggested by Qu and Dawson [113], actuator-level robust control is achieved through a
recursive or backstepping design [78]. Though this method is going through rapid development
[128], not much practical applications have seen to be reported yet.
Parallel systems
Parallel driven systems, such as the flight simulator Stewart platform motion system in
Fig. 1.2, are usually not discussed in considering robotic systems. Only recently, in the late
eighties, the advantages of parallel driven robotic manipulators were recognised and now
become more widely applied in industry [5, 50, 96]. Such systems are inherently stiffer
and due to the parallel actuation, less actuator weight has to be moved, which makes them
suitable for fast assembly lines.
The computed torque control laws for Stewart platforms presented by Liu et al. in [83],
reveal some of the additional problems with application of model based controller for the
parallel systems. The linearising control in end-effector coordinates requires matrix inversion
(of the jacobian) and the forward kinematical problem has to be solved on-line.
This means computing the platform position based on joint measurements, which is easy
in serial robots. No closed form explicit solution, however, has seen to be found for this
problem for Stewart platforms. Choice of linearising in joint coordinates does not solve
any of these problems but actually adds the requirement of calculating derivative jacobian
matrices (solved in [38] by Dutre et al.).
Considering general forms of parallel or closed chain mechanisms, the situation becomes
even more difficult as discussed by Ghorbel in [44], where several parallel mechanisms
and control strategies are classified. In some cases, it is not possible to describe such
a system with a minimal number of coordinates globally and due to this fact and the specific
singularities, which only appear in parallel systems, global stability results are generally
hard to derive, especially for tracking problems.
Many control strategies presented for parallel systems are designed for appropriate in-

1.3 Problem statement 25
teraction with an environment. Force or hybrid position/force control and creating a passive
compliance are often strived for [96, 103, 110, 116, 142]. Another field of application is the
area of active vibration control, mostly in space structure applications [42, 108, 93]. This
requires high bandwidth controllers, which, however, have very limited stroke and thus do
not have to include typical position dependent nonlinearities.
Summarising, in principle, most of the robot control strategies take advantage of the
specific structure in the dynamics of mechanical systems. Though not often seen, the model
based schemes are applicable to the parallel driven flight simulation motion systems, but
all require some modifications in analysis and design. The main mechanical nonlinearities
can be compensated for by a computed torque strategy, possible feedforwarded. Further,
several alternatives exist to robustify the feedback loop. An important source of additional
relevant dynamics can be expected from the hydraulic servo actuators, which are of main
consideration in the conventional flight simulator motion control and usually not taken into
account in robot control.
1.3 Problem statement
Due to the rapid progress in all kinds of flight simulator subsystems such as simulation
computing power, image generating systems, etc., also enhanced quality of flight simulator
inertial motion generation is desired. At least, high fidelity motion generation would enable
research into the specific requirements of motion simulation, which are still not exactly
known. In similar systems, such as robotic manipulators, advanced model based control
strategies exist, which can attain higher performance than less structured methods, provide
more insight into the systems limitations and possibly point at constructional properties,
which can be taken advantage of. In flight simulator motion control, application of such
methods has been almost absent. One of the reasons might be that the motion systems have
a degree of complexity, which does not allow exact modelling or detailed models as part of
the controller. Application requires an intermediate step of extracting a reasonably detailed
model, which describes the most relevant dynamic characteristics.
1.3.1 General problem statement
The foregoing leads to the following problem statement for this research:
Investigate what relevant system knowledge should be used in a model based control
strategy and to what extent does this improve the controlled dynamics of a flight
simulator motion system.
1.3.2 Structuring the problem statement;
approach in research
To solve this problem, the full design process has to be defined, structured and evaluated.
This process starts with modelling and analysis, proceeds with control synthesis and finally
testing of the system. Many solutions and procedures of the steps to be taken are given

26 1 Introduction
in literature, but an integral approach, having the specific properties and requirements of
flight simulation motion generation in mind, still lacks. In this research, integratibility and
applicability will be the main arguments in choosing the most suitable alternative or, if
necessary, newly proposed variant, of each step in the system/control design procedure.
Since the overall investment is high and the fidelity of the motion system is considered
important, the full motion control design process of a flight simulator is allowed a fairly
elaborate procedure optimised towards the specific properties of the system at hand, as
compared to the industrial robot control design. Only, of course, if this leads to enhanced
quality of the resulting system.
In the control design procedure employed in this research, the following fairly standard
subproblems in application directed systems and control, will be distinguished:
- Obtain a relevant model structure for analysis and control through physical modelling.
- Evaluate the structure by experiments, identify model parameters and provide boundaries
for the extent to which the model validity holds.
- Choose and, if necessary, modify the most appropriate model based robot control strategy,
given the systems properties and requirements. Design and implement this controller
on the motion system of the Simona flight simulator.
- Define a test to quantify performance of the controlled system and use this test to compare
with the usual control design approach.
Related to the flight simulation application and the properties of the motion systems used,
specific elements have to be emphasised in each of these steps.
Physical modelling
Modelling according to physical relations such as balance relations i.e. force balance equations
in mechanics and flow balance equations in hydraulics, has two important advantages
as compared to the so-called black box experimental identification. First, as the structure
of the model points at the underlying physical properties of the system, analysis and (constructional)
design considerations are performed much easier. Secondly, modelling of a
nonlinear system such as the flight simulator motion platform, does not impose additional
difficulties.
As already mentioned, the mechanics and hydraulics are considered the main subsystems
to be taken into account. These subsystems specifically, and the integration of the
two, will form the basis of the models to be derived. The mechanical system is, even if
the parts are considered rigid, relatively complex. This complexity mainly stems from the
parallel kinematical structure, which, however, has its influence on the systems dynamics.
The kinematics of parallel systems, and of the Stewart platform specifically, will have to be
analysed.
Nowadays, the equations of motion of complex rigid body systems can often be generated
automatically in symbolic form. This way of working will be evaluated as opposed to
the derivation ’by hand’ for this system.
The models describing the dynamics of the mechanical system will have to be evaluated
for use in a real-time model based control structure. Further, it will have to be analysed, how

1.3 Problem statement 27
choices in the design of the construction resulting in the specific kinematical and inertial
(mass) properties, influence the dynamics.
The hydraulic servo actuators of the Simona motion system have been modelled and
evaluated in detail in [124] by Van Schothorst. In this research, only models will be discussed,
which describe the most relevant dynamics for control. Finally, the properties of the
integrated hydraulically driven mechanical system will have to be analysed.
Experimental validation
Since the kinematical structure of the system forms the basis of the dynamical properties of
this motion system, some attention will have to be paid to the validity of the structure and the
identification of the parameters to be used. An, ’after construction’, calibration procedure
will be evaluated as opposed to setting tight (and often expensive) tolerance specifications
in fabrication.
Experimental identification procedures will have to be developed and applied to reconstruct
the parameters such as dissipative terms, which are difficult to predict in advance,
and evaluate others such as the inertial properties. The models will be evaluated through
comparison with the experimental data.
The area of validity of the models is also an important aspect, which has to be checked.
The cause and severity of possible parasitic effects will be investigated. Especially flexibility
is known to constrain the achievable control performance and difficult to predict.
Control strategy
To maintain structure in a model based controller for a complex system such as the motion
system under consideration and to let a controller result, which can be implemented on a
real-time application, modularization of the tasks is desired. The physical structure of the
system provides a reasonable way to achieve this with a multi level controller in which for
each level specific tasks are defined in close relation with the other levels.
It should be possible to turn the hydraulic actuators into fast and smooth force generators
by local inner loop pressure control as done by Heintze in [48]. The applicability of this
design step, which e.g. assumes accurate velocity measurement, should be evaluated.
Any standard robot control strategy, which assumes direct torque, can be put on top of
this lower level. As first basic step, the computed torque schemes will have to be made fit for
use in parallel systems. This mainly concerns the choice of an appropriate set of coordinates
and evaluation of safe (convergent and stable) and sufficiently fast reconstruction of these
coordinates.
In the outer level, the two degrees of freedom, smooth reference feedforward, possibly
taking reference model knowledge into account, and position feedback for stabilisation and
preventing the system from running out of stroke, should not interfere with each other in
performing their basic task.
Performance evaluation
In evaluation of the model based control strategy the feedback oriented conventional decentralised
controller will be taken as a reference. It is understood that, in practise, this
conventional controller can be improved by design of an additional feed forward path, but

28 1 Introduction
this usually leads to local improvement only and is either an ad hoc approach or takes a
similar strategy as in the more structured model based control design.
Evaluation test procedures exist [1], but are outdated and should be modified to current
standards of control performance quantification. In this research, the following approach
will be taken. By assuming the high fidelity controller being capable of achieving a reasonably
linear system response, the frequency response measurements are considered most
important. A set of other tests is mainly developed to evaluate the assumption of closed
loop linear system response.
Further, it will be observed that the existing evaluation procedures have two main limitations.
First, they have a local character and consequently do not show whether the controller
does adequately respond to the nonlinear characteristics of the system over the workspace.
Secondly, the system is not evaluated performing its actual task, generating inertial motion
cues in flight simulation manoeuvres. A modified test will, as well as possible, have to
overcome these limitations.
1.4 Outline of the thesis
The thesis roughly follows the line of the reasoning for the approach sketched in the preceding
section, to go through the full controller design process. Subsequently the subproblems
in modelling, identification, control and evaluation of the motion system/control design with
the flight simulation application in mind, will be treated.
Physical modelling
By considering the laws of physics, dynamical models of the flight simulator motion system
will be obtained. In Chapter 2, the mechanical part will be discussed. Specifically, attention
will be paid to the kinematics and dynamics of a parallel rigid body system. In Chapter 3,
the basic structure of the hydraulic servo actuators and extensions to this model will first be
discussed. The integration of the hydraulic mechanical system, the full model of the motion
system and its properties are the subject of Section 3.3 and its specifics in a Stewart platform
construction are pointed at in Section 3.4.
’Open loop’ experimental verification
Chapter 4 deals with the experimental validation of the physical models, the identification
of model parameters and the parasitic ’additional’ dynamics. The identification of the
static parameters in the kinematical structure, calibration through redundant measurements
is discussed in Section 4.1. Dynamic (inertial) parameter identification by modal analysis
techniques is the subject of Section 4.2. Frequency response measurements are the main
ingredients in the model validation considered in Section 4.3. This includes getting track
of the parasitic dynamics caused by flexibility in the system and its environment in Section
4.3.2, which concludes the discussion on the ’open loop’ experiments taken in this
chapter.
Control strategy
In Chapter 5, the control strategy is built up by defining the general approach for multi level

1.4 Outline of the thesis 29
control in Section 5.3 after a short introduction to the main control aspects in Section 5.1
and Section 5.2. Then, each part of this controller will be treated. In Section 5.4, the local
pressure control of the hydraulic servo systems is outlined. Section 5.5 discusses the computed
torque control control level to compensate for the multivariable, nonlinear mechanics.
This requires an iterative coordinate reconstruction and is followed by the feedback path defined
by the problem of position stabilisation. Interconnection with the trajectory generating
system with reference model based control, treated in Section 5.6, defines the forward path
at the outer level.
Closed loop experimental evaluation
The evaluation of the controller on the actual experimental set-up is the subject of the second
part of Chapter 5 starting with discussing the implementational issues in Section 5.7. First,
the procedure to test the performance of the controlled system is defined in Section 5.8.
Then, this procedure is applied to the Simona motion system with the newly proposed controller,
which is discussed in Section 5.9. As a reference, comparing with a conventional
controller strategy.
Final considerations
The main aspects in the design of flight simulator motion system controllers evaluating the
experience of going through the full system/control design process defined are treated in
Chapter 6 giving a review and discussion on the results. Directed at the problem statement,
defined in Section 1.3, the final Chapter 7 states conclusions towards the opportunities given
in this research by the application of a model based controller in flight simulation motion
control. Also recommendations are provided.

Chapter 2
Mechanics of parallel driven
motion systems
The, fully parallel, hydraulically driven construction of a Stewart platform is often applied
for flight simulator motion systems. In this research, the use of model based control for such
systems is investigated. In this chapter, the first step in modelling and analysis is performed
by considering the mechanics.
In general, models of parallel robots result in combined algebraic and differential equations,
which causes difficulties with simulation, analysis and model based control. This
problem will be stated in discussing some literature on parallel manipulators and modelling
of mechanical systems in Section 2.1.
In Section 2.2, first a short introduction into basic kinematics will be given. Then, the
structure of kinematic parallel and serial singularities will be treated. The, less standard,
parallel singularity causes local loss of controllability and observability of the mechanical
system and should be avoided. In Section 2.3, a new method for the Stewart platform is
introduced to ensure exclusion of these points from the working area of the manipulator.
Moreover, this is also a necessary condition for guaranteed sufficiently fast convergence of
an iterative scheme in reconstructing the platform coordinates. For the motion system at
hand, sufficient accurate real time convergence is proven, which enables safe use of this
scheme in model based control later on in Chapter 5.
The platform coordinates are a proper choice in parametrization of the Stewart platform,
thereby circumventing difficulties with combined algebraic and differential equations. In
Section 2.4, which deals with the dynamics of the mechanical part of the motion system,
it is shown that an explicit differential model results from this choice. Thereby, the use of
a suitable projection method, following Kane [64], is exploited to arrive at the equations
of motion. In this way, a proper starting point for simulation, analysis, model reduction
and model based control is provided for. Some extensions, e.g. parasitic modes of the
foundation, are also included.
31

32 2 Mechanics of parallel driven motion systems
2.1 Parallel motion systems
The use of robot manipulators is widely spread in industry nowadays. Most of these manipulators
are constructed as a series connection of joints and links. The dual form of these
robots, the parallel manipulator, is less often seen to be applied. In (flight)simulation motion
systems however, the parallel construction is almost invariably in use. The Stewart platform
is a 6 degrees-of-freedom (d.o.f.) parallel manipulator, which is applied in most of the current
high fidelity flight simulators. It was named after Stewart [140] who illustrated the use
of such a parallel structure for flight simulation in 1965. Nowadays, it is also referred to
as Gough-platform, since it was Gough who presented the practical use of such a system
somewhat earlier in 1949 rediscovering the structure already described around 1800 by the
mathematician Cauchy [95]. As a manufacturing and fabrication robot it is also named a
hexapod.
There are several advantages in applying a parallel construction. This kind of manipulators
have higher rigidity and accuracy due to the parallel force path and averaged link to
end-effector error. The inverse kinematics (from end-effector to link coordinates) which is
a problem in path generation of serial manipulators is easily solved in parallel robots. There
are also disadvantages. The dual forward kinematics is a complex algebraic problem and
has in general more than one solution [114]. Modelling the dynamics is also less straight
forward.
For several reasons, feedback control of these motion systems is still decentralized
i.e. per actuator without taking mechanical coupling into account. Setting higher standards
of motion realism in simulation will involve modern control strategies in order to fully benefit
from recent constructional and computational improvements in flight simulators e.g. light
weight, low centre-of-gravity (c.o.g.) platforms, high frequency airplane dynamics simulation
[6]. Also use of a Stewart platform as a more general robot, as presented by Nguyen
et al. [104], will be enhanced with high performance control of motion, as these systems
usually require minimal time tasks.
By incorporating more structural system information i.e. a model into the controller,
it is possible to achieve higher performance. Most modern control strategies are therefore
model based in some way (directly, in design or evaluation). In this case the quality of
motion depends on the fidelity of the model. Deriving a model of the mechanics of the
Stewart platform manipulator for analysis, design and control will be the subject of this
chapter. To arrive at a model with structure from which insight can be gained, modelling
will be done based on physical laws.
Modelling the dynamics of a Stewart platform as a multibody system has been done
by Lee et al. [80] who claim to be the first to present a complete model and with more
simplifications by Do et al. [36] and Liu et al. [83]. Modelling the mechanics of this platform
can be done in several ways and with various objectives in mind. The equations of motion
can be derived by using the classical approach of Lagrange [80] or Newton-euler [36].
In general, deriving the equations of motion of a parallel manipulator results in combined
differential and algebraic (constraint) equations (see e.g. Roberson and Schwertassek
[120]). In simulation and control this formulation can cause difficulties, usually referred to
as index problems [19]. Here it is shown that an explicit differential model for the Stewart

2.2 Kinematics 33
platform results if one makes the right choices in parametrization. Dependent variables are
explicit functions of the integrable differential equations. In this way index problems, etc.
are circumvented.
By using a modern projection or dual basis method [79] like Kane’s [64], which have
the advantages of both the Newton-euler and Lagrange formulation but without the corresponding
disadvantages [56], it will be shown that applying this approach can result in a
model from which more insight can be gained.
Together with the alternative parametrization, this is advantageous over the models earlier
presented in literature, if one wants to apply a model for both analysis, simulation and
control. Model based feedback, however, is still more complex for parallel manipulators,
since the dynamics are only described in end-effector coordinates and the measured signals
are link related. Next to the fairly low performance requirements of flight simulator motion
systems in the past, this will probably be the reason that in most of these motion systems,
feedback controllers are decentralized (one SISO loop per parallel link i.e. actuator).
Through analysis of the derived model, some of the disadvantages of decentralized control
can be revealed. To apply a simple, but accurate model based controller in practise, one
would like to quantify the errors made by undermodelling, to be able to do robust analysis
of the control scheme. The modelling approach taken here aims at a model from which
the influences of different system parameters, like masses, inertias, velocity, gravity can be
clearly separated.
2.2 Kinematics
After stating some notational issues, the fundamental formulas of mechanics to describe the
kinematics will be introduced. Then the Stewart platform will be defined in order to derive
a model of its specific kinematics. After some model analysis with the control objective in
mind, this part will be concluded by a resume.
2.2.1 Notation and definitions
Capital symbols, X are used for matrices, x for vectors, x for scalars. With some scalar
(energy) functions X is used. x y denotes the vector product which can also be written
as ~ Xy = ( ~ Y ) T x where the vector product matrix, ~ X, is a skew symmetric (X = ;X T )
matrix parametrized by the vector, x T = x1 x2 x3 , such that the result is the vector
product.
~X =
2
4 0 ;x3 x2
x3 0 ;x1
;x2 x1 0
3
5 (2.1)
X Y denotes vector wise product of the columns stacked in the matrices.
p
The index xn is used for the normalising operation xn = x= j x j with j x j= xT x.
Pxn denotes the projector to the (hyper)plane with normal vector x n and can be constructed

34 2 Mechanics of parallel driven motion systems
Fig. 2.1: Projection of arbitrary vector, s onto the plane with normalised normal vector, x n.
The projection matrix, Pxn, can be composed from the vector product (matrices,
~Xn).
from the vector product matrix
Pxn =(I ; xnx T n )=(~ Xn) 4 = ~ Xn ~ X T n = ;( ~ Xn) 2 : (2.2)
Projection matrices have some properties, which can be taken advantage of, like
Pxn = P T xn = P k xn� (2.3)
with k>0 and integer. In some cases it will be appropriate to use such a projection plane
for construction of a frame as done in Fig. 2.1.
Motion can be described w.r.t. various frames. A matrix or vector described in some
frame can have a superscript referring to this frame. For the inertial frame, G, or ground
coordinates, the index x g will be used. As a function of the moving end-effector or platform
frame, M, vectors will be denoted x m . If a (rotation) matrix maps a vector into another
frame it will be denoted as g R m if R maps from M to G.
The subscript index like ai will be used to refer to the i th -actuator if also non actuator
dependent variables appear in the equation.
2.2.2 Calculating velocity and acceleration by differentiation
An important part of kinematics is the ability to derive velocity and acceleration vectors of
any part of a (partly) moving construction. The basic formulas to do this will be derived in
this section.
The motion of a point (mass particle, joint, etc.) is usually most conveniently and invariantly
defined w.r.t. the body frame to whom it’s connected. The motion of a frame put
in another frame generally consists of a translation, which can be described by a vector, t,
and a rotation for which a matrix, R, can be used. The orientation of a whole frame can
be described by a rotation matrix. A rotation matrix consists of perpendicular unit vectors
which describe the basis of the frame into the other frame. As a result a rotation matrix, R

2.2 Kinematics 35
has the following property:
R T R = I (2.4)
Any real 3 3-matrix with this property and the property det(R) = 1 is a rotation matrix,
which set is often referred as the SO(3)-group [77]. SO(3) stands for the Special
(det(R) =1) Orthonormal (RT R = I) group of matrices of size 3x3. With det(R) =;1
also the mirror operation is included (transformation of right hand frames to left hand frames
and vice versa). The position of a point pm in frame M can now be described in frame G
by:
p g = t g + g R m p m
(2.5)
To describe the velocity of this point in the other frame one can simply differentiate this
equation. Some properties of the time derivative of the rotation matrix can be derived by
differentiating (2.4). This results in skew symmetric matrices which can be parametrized by
the vector product matrix of the (thereby defined) angular velocity ! m .
with
~ m =
R T _ R = ; _ R T R = ~ m
2
4 0 ;!3 !2
!3 0 ;!1
;!2 !1 0
(2.6)
3
5 (2.7)
Now, for the vector, which is rigidly attached to the frame M, _p m = 0. Hence, by differentiating
(2.5) and using (2.6), one obtains
_p g = _ t g + g R m ~ m p m = _ t g + ~ g p g = _ t g +! g
p g
(2.8)
where the change of frame for the matrix ~ ,isgivenby ~ g = g R m ~ mm R g and using
the inverse rotation given by m R g =( g R m ) T with (2.4).
By differentiating (2.8) the acceleration of the point, still rigidly attached to the frame
M(_p m = 0), can be calculated, using ~ P m ! m = ~ m p m :
p g = t g + g R m ( ~ P m ) T _! m + g R m ( ~ m ) 2 p m
= t g + ( ~ P g ) T _! g +( ~ g ) 2 p g
= t g ; p g _! g +! g
(! g
p g ) (2.9)
In other cases, the point considered can already be moving in the frame M (p m ,v m m =
_p m ,a m m = pm ). Here, by differentiation of
_p g = _ t g + g R m ( ~ ) m p m + g R m v m p
the coriolis acceleration appears as the third term in
p g = a g p + g R m a m m +2g R m ~ m v m m
= a g p +ag m +2~ g v g m =a g p +ag m +2(!g
(2.10)
v g m ): (2.11)
Where p is a point connected to M momentarily at the same position as p. Its acceleration,
a g p , is given by (2.9).

36 2 Mechanics of parallel driven motion systems
Fig. 2.2: Physical structure of the euler angle representation.
2.2.3 Parametrising orientation by euler angles
In stating the equations of motion, the state which describes the orientation, usually only
appears in the rotation matrix. It is possible to parametrise the rotation matrix in several
ways.
Although the rotation matrix consists of nine entries, its properties, given by (2.4), put
constraints on these entries. The orientation, i.e. the SO(3)-group, can locally be represented
by 3 parameters. It can, however, not be covered by a single coordinate chart of this
size [77]. Several parametrizations are used in literature. The euler angle description consists
of three subsequent simple planar rotations around momentary axes. The euler angles
are not convenient for use in modelling the Stewart platform but are merely discussed for
reference since it is a popular description in the aerospace community.
An often used example of euler angle rotation is physically represented in Fig. 2.2. Both
three unit vectors nG�M of the ground frame, G, and moving frame, M, are drawn. A simple
x�y�z
planar rotation around the x-axis has the following structure
Tx =
2
4
1 0 0
0 cos( ) ; sin( )
0 sin( ) cos( )
A vector, x m , quantified in the axes of M can be represented in G by
x g = Rx m = Tz�Ty�Tx�xm 3
5 (2.12)
(2.13)
Since the subsequent rotations are not commutative, it is important to define the order of
rotation. The order defined in (2.13) is the usual order applied in the aerospace community.

2.2 Kinematics 37
This means that the airplane, at first positioned so that the body axis is parallel to the fixed
inertial frame, G, has to be rotated as follows:
- First rotate over an angle , the yaw angle, around the inertial n g z-axis. In taxiing, this
typically will be the main angle of rotation.
- Then pitch over an angle around the intermediate y � -axis. This will usually be the main
angle to be changed in take-off. Note, e.g. from Fig. 2.2 that this axis is in general not
an axis of the inertial or the moving frame.
- Finally roll around the airplanes’ longitudinal axis, n m x , over an angle to obtain its actual
orientation.
The fully parametrized rotation matrix is then as follows:
R =
2
4
c c c s s ; s c s s + c s c
s c c c + s s s s s c ; c s
;s c s c c
3
5 (2.14)
In this equation s = sin( ), c = cos( ) and so on. The three euler angles have the
disadvantage of a highly non-linear appearance in both the rotation matrix and the euler
angle velocity to angular velocity transformation. The latter can even become singular.
Since, although any rotation matrix can be described by this parametrization, in some states
(consider = =2 in Fig. 2.2) this chart does not allow three independent changes of
orientation, i.e. the kinematical relationship between =[ ] T and the angular velocity
in the body frame, ! m , given by
! m =
2
4 p
q
r
2
4 1
0
0
3
5 = ( ) _ =
3 2
5 _ T
+ Tx0 2
4
4 0
1
0
1 0 ;s
0 c c s
0 ;s c c
3
5 _ T
+ Tx0T T y0 2
4 0
0
1
32
5
4 _
_
_
3
5 =
3
5 _ (2.15)
becomes singular. p, q and r are the components of the angular velocity, ! m , measured in
moving body frame. In an airplane this will usually be the output of the gyros.
In some experiments, a rotation matrix can be identified. With ; =2 < � � < =2,
the euler angles can be calculated from such rotation matrix as follows:
sin( ) = ;R(3� 1) (2.16)
sin( ) = R(2� 1)= cos( )
sin( ) = R(3� 2)= cos( )
As already said, the euler angle description of rotation is not convenient in describing the
model of the Stewart platform. This is mainly because evaluation of the high number of sine
and cosine functions is both analytically and numerically relatively troublesome and this can
fully be avoided in modelling the motion system mechanics by using euler parameters.

38 2 Mechanics of parallel driven motion systems
G
nµ
n
g
z
n
n
g
x
g
y
s m
µ
s g
nµ x sm
µ
s
s
g
m
2sin(1/2 µ )
( nµ x sm)xnµ
Fig. 2.3: Visualising the euler parameter representation by a rotation around the unit vector,
n , over an angle, . The lower right schematic drawing is the plane orthogonal
to n through the points sg and sm.
2.2.4 Parametrising orientation by euler parameters
An attractive alternative of euler angles are the unit quaternions [130] called euler parameters.
No singularities and, even more important, numerically convenient relationships with
both the rotation matrix and the angular velocity and as with the euler angles, an interpretation
which can be visualised. This at the cost of an additional fourth parameter and
consequently an extra constraint equation.
The orientation of one frame w.r.t. another frame can always be described by a single
rotation over an angle , about a unique axis with direction n . The four euler parameters,
T T
, parametrise and n as follows:
e = 0
The constraint equation,
= 1 2 3
0 = cos(1=2 ) (2.17)
T = sin(1=2 )n (2.18)
T
e e =1� (2.19)
is a direct result from this definition. By not using the angle of rotation itself as a coordinate
but the sine and cosine instead, these geometric functions do not have to be evaluated anymore.
And as a result we will see in the subsequent sections that in a full mechanical model
of the Stewart platform not one geometric function will appear.

2.2 Kinematics 39
With Fig. 2.3 the rotation of a vector, sm, in the body frame, M, to its direction in the
inertial frame, G, given as sg, can be described using the unit vector, n , along the axis of
rotation and the angle of rotation, .
The following derivation starts with constructing s g from sm adding the two component
vectors in the plane given in Fig. 2.3 and is directed towards the specification of the rotation
matrix.
sg = sm ; (1 ; cos( ))(n sm) n + sin( )(n sm)
= sm + 2 sin 2 (1=2 )n (n sm) +2cos(1=2 )sin(1=2 )(n sm)
= I +2~~ +2 0~ sm
= 2
=
2
2
4 2
0
2
+ 1 ; 1=2 1 2 ; 0 3 1 3 + 0 2
2
1 2 + 0 3 0 + 2 2 ; 1=2 2 3 ; 0 1
1 3 ; 0 2 2 3 + 0 1 ; 1=2
4 ; 1 0 ; 3 2
; 2 3 0 ; 1
; 3 ; 2 1 0
3 2
5
2
0
+ 2
3
3
5 sm
4 ; 1 0 3 ; 2
; 2 ; 3 0 1
; 3 2 ; 1 0
3T
5 sm
= G( e)L T ( e)sm = R( e)sm (2.20)
So, rotation appears to be equal to two simple subsequent transformations, G and L,
which are linear in the euler parameters. By Nikravesh et al. [107] it is shown that this
parametrization and using _ RR T = ~! g leads to the following relation to calculate _ e or _
from ! and e,
_e = 1
2 GT ( e)! g
In only constructing the variable, _, a reduced version is given by
(2.21)
_ = 1
2 GT s ( e)! g � (2.22)
where the matrix, Gs, is given by the last three columns of G. This equation can be used to
get from the angular velocity to the rotation matrix with help of an integration routine and
initial conditions on . Further, the fairly simple relations like
! g =2G( e)_ e� (2.23)
! m =2L_ e and e =1=2G T _! g can be derived.
With angles ; < < , can be used as the (orientation) state from which 0 =
p 1 ; T is solved. In that case, a three parameter setting is used at the cost of possible
singularities. The nonsingular envelope is, however, twice as large as in case of using euler
angles.

40 2 Mechanics of parallel driven motion systems
Fig. 2.4: A 2-d.o.f. serial system in a singular configuration
It is possible to calculate euler parameters from any rotation matrix where j j< .
0 = 1p
(1 + tr(R))
2
(2.24)
1 = (R(3� 2) ; R(2� 3))=(4 0) (2.25)
2 = (R(1� 3) ; R(3� 1))=(4 0) (2.26)
3 = (R(2� 1) ; R(1� 2))=(4 0) (2.27)
where tr(R) = P i=3
i=1 R(i� i) is the trace of the rotation matrix. Only in case of 0 = 0
another solution strategy has to be applied. With the earlier given relationships, (2.14) and
(2.16) between R and and vice versa, euler parameters can be calculated from euler angles
and vice versa.
2.2.5 Jacobian matrices
Connected parts of a mechanical system will result in less freedom of motion reflected by
constraint equations which can be explicit as in y = f(x) or implicit as in g(x� y) = 0.
Such equations also constrain the time derivatives of these coordinates, i.e. the velocities
and accelerations. In describing a mechanical system, e.g. stating its equations of motion,
it is often convenient to state the motion of the system as a function of a limited number of
(generalised) variables (coordinates and/or velocities).
If some variations or velocities can be described as a product of a (position-dependent)
matrix and a vector of other variations this matrix will be called a (semi-)Jacobian matrix.
The jacobian matrix between two sets of variables usually comes out naturally by a time
differentiated version of an equation in which one of the sets is explicitly stated as in y(t) =

2.2 Kinematics 41
f(x(t)),
_y(t) = @f
@x (x(t)) _x(t) =Jy�x(x(t)) _x(t) (2.28)
In mechanical systems such mappings often result in an indirect way from differentiation
of the rotation matrix and using ! as velocity term. The angular velocity, !, however, is no
time derivative of any representation of attitude. E.g. given (2.8),
_p g = I g R m ( ~ P m ) T
_t g
! m = Jp g �x _x (2.29)
Also matrices like Jp g �x will be termed jacobian, although this is not precise i.e. not a result
of (2.28), but just states one set of velocities as a function of another set.
2.2.6 Parallel, serial and singular configurations
In kinematics, jacobian matrices often play an important role in relating input/actuator and
output/ end-effector coordinates. With parallel manipulators, it will be shown using some
examples, that these relations differ from those constructed from a serial manipulator. In
Fig. 2.4, a 2 degree-of-freedom serial joint robot is depicted. The end-effector coordinates
x and y can in the serial configuration explicitly be stated as functions of actuated rotating
joints angles q1 and q2 as
And by differentiation
x = l1 cos(q1) +l2 cos(q1 + q2) (2.30)
y = l1 sin(q1) +l2 sin(q1 + q2) (2.31)
_x
_y = ;l1 sin(q1) ; l2 sin(q1 + q2) ;l2 sin(q1 + q2)
l1 cos(q1) +l2 cos(q1 + q2) l2 cos(q1 + q2)
_q1
_q2
= Jx�qq (2.32)
At q2 =0 , the jacobian of this configuration becomes singular. At such configurations,
physically, the actuators can only generate one d.o.f. motion/velocity of the end-effector
locally. Further, it can be noted that, for each nonsingular configuration, two actuator positions
generate the same end-effector state.
The dual parallel configuration of a 2 d.o.f. robot is given in Fig. 2.5. In fully parallel
systems, the sliding actuator lengths can explicitly be stated as functions of the end-effector
coordinates. With xT =[xy] and li(x) =x ; bi
q
klik = lT i li
(2.33)
and by differentiation
k _ lik = l T n�i _x (2.34)
Thus J T l12�x = [ln�1 ln�2]. If the unit actuator direction vectors, ln�i are in parallel, the
configuration becomes singular. In this case, however, the end-effector can still move but

42 2 Mechanics of parallel driven motion systems
Fig. 2.5: A 2-d.o.f. parallel system in a singular configuration
Fig. 2.6: A 3-d.o.f. system with serial and parallel singular configurations
not be forced in 2 d.o.f. and from the speed of the actuators, the end-effector velocity can
not be determined. In this system, in a nonsingular configuration, there are two end-effector
states for each actuator position.
In general, a rigid body mechanical system, will have combinations of serial and parallel
connections as in Fig. 2.6. In this example, the actuator coordinates, q, are no explicit functions
of the end-effector coordinates, x and v.v. Also both kind of singular configurations
can usually occur. An important observation in the following sections will be the fact that
the Stewart platform flight simulator motion system is a purely parallel system.
The parallel manipulator construction of the Stewart platform is first defined. To derive
the equations of motion, the velocity and accelerations should be described w.r.t. a
limited number of generalized variations. This defines the kinematics after which the semiequilibrium
equations of the active and inertial forces can be stated.
φ

2.2 Kinematics 43
2.2.7 Stewart platform definitions and assumptions
The Stewart platform (Fig. 2.7) consists of an end-effector body with mass, m c, and 3x3
inertia matrix I m c w.r.t. the end-effector frame, M, connected to the centre of gravity (c.o.g.)
of the body which has varying coordinates, c, in the inertial frame, G. The end-effector body
or platform is connected by six parallel actuators at the moving upper gimbal points with
fixed coordinates, ai, in the moving frame, M, tobi in the inertial frame, G. The length of
the six actuators can be varied. In describing a specific actuator the subscript i for the i th
actuator will be left away.
The platform position is defined as
sx =
c
s
(2.35)
With the last three euler parameters, ,todefine the rotation matrix, T ( ), asR in (2.20),
from the moving (platform) frame, M, to inertial (ground) frame, G and a scaling factor
s = 2kakmax equal to the maximum distance of the c.o.g. to any of the upper gimbal
points. The platform speed is not defined as the derivative of platform position but as
_x =
_c
! � (2.36)
where ! is the angular velocity of the platform. The platform (generalised) speed and
position are related through the jacobian, J sx�x, using (2.22)
sx _ =
_c
s_
= I 0
0 sG T s =2
_c
! = Jsx�x _x (2.37)
An actuator (Fig. 2.8) will be modelled as 2 bodies, the rotating cylinder and the moving
piston. The rotating body, with mass mb and a constant distance of rb of the c.o.g., bc, to
the connection of a 2-d.o.f.-rotational gimbal joint to the inertial frame at b. The moving
actuator body, i.e. the piston, with mass ma with a constant distance of ra of the c.o.g.,
ac, is connected with a 3-d.o.f.-rotational gimbal joint to the platform at a. Witha1d.o.f.
controlled sliding joint between these two bodies the length of the actuator can be
varied.
It is assumed that the inertia of the actuator bodies can be neglected around the actuator
axis and to be uniform perpendicular to this axis. i a is the inertia of the moving actuator
body at ac and any axis perpendicular to the actuator. i b is the inertia of the rotating part
of the actuator along any axis perpendicular to the actuator w.r.t. the connection point to the
inertial frame (b) .
With these assumptions also the case, often seen in practise, in which the moving part of
the actuator not only slides at the connection with the rotating part but also rotates around
the l-axis , and has only a 2-d.o.f. rotation gimbal in connection to the platform, results in
the same dynamics. Finally, the c.o.g.’s of the actuator parts are assumed to lie on the line
connecting the upper and lower gimbal point.

44 2 Mechanics of parallel driven motion systems
Fig. 2.7: A schematic view of the Stewart platform
2.2.8 Stewart platform kinematics
The kinematics of the Stewart platform will be described by first defining the transformation
of the platform to actuator coordinates. Then by differentiation also velocity and acceleration
of all relevant points can be calculated as a function of the platform motion, whose
velocities will be taken as the generalized speeds.
Almost all vectors can be conveniently described in the inertial frame. Apart from a m i
whose time derivative in the moving frame is 0.
As shown in Fig. 2.7, the vector, li, between the two attachment points of an actuator
can be described by
li =c + T a m i ; bi (2.38)
Now the squared length of the actuator, j li j 2 = l T i li, and the unit vector in direction of
the actuator, ln�i = li
jlij can be calculated from the platform variables c g and the orientation
matrix T = g R m which will be the only rotation matrix used.
The velocity of the length of the actuators can be calculated by projection of the velocity
of the upper gimbal attachment point, v a, in the direction of the actuator, since
d
dt j li j= d
dt
q l T i li = lT i va
j li j = lT n va: (2.39)

2.2 Kinematics 45
Using (2.8), the velocity of the upper gimbal points is given by
vai = _c +! T a m i � (2.40)
where ! is the angular velocity of the moving frame in the inertial frame. By projection of
this velocity with (2.39) and filling in (2.40), the velocity of the actuator is described as a
function of platform velocities and poses.
_li = l T n�ivai = lT n�i _c + l T n�i (! T ami ) (2.41)
With some reordering and written as matrix equation (ln�i, am i and vai stacked in Ln, Am and Va) for all the actuators the jacobian between the actuator and platform velocities is
defined.
_
l = L T n _c +(TA m Ln) T ! = Jl�x _x = L T n Va (2.42)
The semi-jacobian matrix, Jl�x, is one of the most important variables in the Stewart platform,
relating the platform coordinates to be controlled and used as basic model coordinates,
and the actuator lengths, which can be measured. Further, in transposed form, J l�x relates
the actuator input forces with the platform forces as will be shown in deriving the platform
dynamics. Note that with defining _x T =[_c T ! T ], x does not exist.
2.2.9 Interpretation and use of the jacobian matrix, Jl�x
If the system has to be controlled by the actuators, the jacobian specifies how the control
inputs, the actuator forces, influence the platform (accelerations), which are, especially in
flight simulation applications, often the variables to be controlled. Further, the measurable
outputs are often only the actuator lengths. The derivatives (actuator speed) of these outputs
are given by the product of the jacobian and the platform speed.
There are two interpretations to the jacobian. In the force interpretation the rows of J l�x
give the (generalized) forces in the platform coordinates given a unit force in an actuator. In
the velocity interpretation the columns of Jl�x specify the velocity of the actuators required
to have unit velocity of the platform.
In model based control, the inverse information is of interest. The measured variations
of the actuator have to be put in platform variations to calculate corrections in a model
specified in platform coordinates. Each column of the inverse jacobian, J ;1
l�x , specifies what
velocity (angular velocity included) of the platform is necessary to have elongation of just
one actuator while the others only rotate. Given measurable actuator velocities, the platform
velocities can be calculated.
The correction forces in a model based controller are also most conveniently calculated
as a function of platform coordinates. Each row of J ;1
l�x specifies the forces necessary in the
actuators to have unit force correction in platform coordinates.
The inverse jacobian appears in feedback linearising structures (like computed torque,
etc.), which will be dealt with in the forthcoming chapters.
Another problem of a parallel manipulator, with only the link position measured, are
the forward kinematics. It is not known how to analytically calculate the platform position
(without decision making about set of possible solutions) from link measurements.

46 2 Mechanics of parallel driven motion systems
r b
l
bc
ac
b
Lower Gimbal
ra
Upper Gimbal
Fig. 2.8: Stewart platform actuator link construction
The jacobian Jl�x(sx) is an important element in the jacobian Jsx�l(sx) from actuator
variation to the platform position variation.
Jsx�l(sx) =J ;1
;1
l�sx (sx) =Jsx�x(sx)J l�x
(sx)� (2.43)
where Jsx�x is given by (2.37).
This jacobian provides a way to apply a Newton-Raphson iteration to calculate the solution
provided one starts in a point sufficiently close to the solution and away from jacobian
singularities.
sxj+1 = sxj + J ;1
l�sxj (lmeasured ; lj) (2.44)
The condition number of Jl�x also provides a measure for the controllability of the platform
from the actuators which becomes uncontrollable at singularities of this matrix.
Further, most of the constraints of the platform are caused by the characteristics of the
actuators like limited stroke, maximum speed and force. The jacobian plays an important
role in translating these limitations into platform coordinates. E.g. given a maximum velocity,
j v jmax, in extending an actuator either direction, the maximum velocity in the j’th
platform coordinate is given by the 1-norm of the respective row of J ;1
l�x times the maximum
actuator velocity.
_xi�max = kJ ;1
l�x (i� )k1 j v jmax=
6X
j=1
j J ;1
l�x (i� j) jj v jmax (2.45)
a

2.2 Kinematics 47
So, the maximal platform velocity along a specific coordinate can be calculated by adding
all absolute values of the specific row of the inverse jacobian and multiplying with the
maximum actuator velocity. In this case, every other platform coordinate is left unspecified.
In the more interesting case, with the other platform coordinates constraint to zero
_xi�max =j v jmax =kJl�x(i� )k1 =j v jmax = max(j
Jl�x(i� j) j)� (2.46)
j
e.g. the maximum pure surge speed can be calculated by dividing the maximum actuator
velocity by the largest number in the first column of the Jacobian. For position and acceleration,
these formulas can be used as an approximation.
2.2.10 Velocity and acceleration of the actuator joints
If the inertia of the actuator joints is important or gimbal forces have to be calculated, the
kinematical formulas have to be expanded with those to derive the velocities and accelerations
of these joints.
The jacobian between the platform and the upper gimbal point velocity is defined by
vai = I T ( ~ A m i )T T T _x = Jai�x _x (2.47)
To determine the inertial forces of the actuators, the jacobians, from gimbal point to the
c.o.g.’s of the actuators, are also important. The angular velocity of the actuator perpendicular
to the actuator, !a,isdefined by
va
!a = ln
(2.48)
j l j
Now the velocities of the c.o.g.’s of the actuator bodies v ac and vbc can be stated as a function
of va:
and
vac =va +!a
(;raln) =(I ; ra
j l j Pln )va = Jac�ava� (2.49)
vbc =!a rbln = rb
j l j Plnva = Jbc�ava� (2.50)
as becomes more clear by looking at Fig. 2.9. The acceleration of the actuators can be
calculated by differentiating (2.42),
l = Jl�xx + _
Jl�x _x = L T n _ Va + _ L T n Va: (2.51)
The derivative of the unit vectors, ln�i, of each actuator, i, can be calculated with:
_
ln = d l
dt j l j =
l
_ j l j;ld dt j l j
j l j2 = (I ; lnlT n )
va =
j l j
1
j l j Plnva
(2.52)

48 2 Mechanics of parallel driven motion systems
rb ln
vbc
ω a
bc
ω a
ac
Lower Gimbal
va
ω a
vac
ra ln ra l n
x
ω a
ln .
ln
l
Upper Gimbal
Fig. 2.9: Stewart platform actuator construction of link velocities. With the upper gimbal
velocity, va, which can be derived from the platform velocity by (2.40), all other
actuator velocities can be constructed. The actuator speed, _ l, is the projection of
va along the actuator ln (2.41). The figure is best read by considering v a in the
plane of the picture. Then, only the actuator angular velocity, ! a, given by (2.48),
is out of the plane. In fact, it is orthogonal to the plane.

2.2 Kinematics 49
The acceleration of the actuator length consists of a term which is the projection of the
acceleration of the upper gimbal in the direction of the actuator and a positive quadratic
term, which is the centripetal acceleration of the actuator.
l = l T n _va +v T 1
a ( Pln )va
(2.53)
j l j
So the acceleration of the actuator length is always positive if the platform is moving with
constant translational speed in any direction and constant orientation since _va =0in that
case. The acceleration of the upper gimbal can be derived directly with (2.9) and (2.40)
_vai = c + _! ai +! (! ai) =Jai�xx; j! j 2 P!ai� (2.54)
where the projection matrix, P! =(I +!n! T n ), projects the upper gimbal vector, ai, on the
plane orthogonal to the normalised angular velocity direction of the platform, ! n.
The acceleration of the c.o.g. of the moving actuator part also generates inertial forces
and can be written as a function of platform motion.
_vac = d
dt vac = d
dt (Jac�ava) =Jac�a _va + _ Jac�ava
So this jacobian needs to be differentiated. With (2.49) and (2.52),
Now with j Plnva j 2 =v T a Plnva,
Jac�a
_ = d ra
(I ;
dt j l j
=
ra
j l j2 vT ra
a lnPln +
j l j2 (Plnval T n + lnv T a Pln )
(2.55)
Pln ) (2.56)
Jac�ava
_ = ra
j l j2 (j Plnva j 2 ln +2(v T a ln)Plnva)� (2.57)
which shows a quadratic centripetal term in actuator direction and a coriolis term orthogonal
to the actuator. Clearly, the piston acceleration, _vac, only depends on the velocity, va and
acceleration, _va, of the upper gimbal, given the positional coordinates.
The motion of the actuators has now been explicitly stated as a function of the platform
coordinates and its derivatives. The kinematic relations thus provide means to state the
motion of the system as a function of a limited number of variables. Together with the
dynamics i.e. semi-equilibria of active and inertial forces stated in Section 2.4, the equations
of motion result.
2.2.11 The Simona flight simulator motion system kinematics
In this thesis, any system connected to the environment through six parallel joints with five
(passive) rotational and one (active) translational d.o.f. is considered a Stewart platform.
The structure of the Simona flight simulator motion system design is much more specific. It
is schematically depicted from above in Fig. 2.10 and the measures are given in Table 2.1.

50 2 Mechanics of parallel driven motion systems
Fig. 2.10: The Simona flight simulator motion system kinematics represented schematically
in a top view in an ’all the actuators down’ position
Both the upper as the lower gimbal points virtually form a platform, which looks like two
blunt equilateral triangles rotated 180 degrees with respect to each other.
All the upper gimbal points as well as the lower lie on a circle. The circles lie in parallel
planes if the actuator lengths are equal. All the minimal and maximal actuator lengths are
equal. The actuators and gimbal points, which are numbered 1 to 6, are pairwise distributed
clockwise over the circle. (6,1), (2,3) and (4,5) for the moving upper and (1,2), (3,4) and
(5,6) for the inertial lower platform. The distance between the two elements of the pairs are
also specified in the table. The upper pair (6,1) is connected on the positive x-side of the
simulator. The lower pair (3,4) at the negative x-side of the ground.
In most of the derivations, the motion system kinematics is assumed to belong to the
most general class of Stewart platforms. This leaves room to adjust easily to a structure
which differs from its design.
This concludes the kinematical modelling, which is used to derive the equations of motion
in modelling the mechanical system in Section 2.4. The nest section finishes the treatment
of the kinematics with two issues, which will be specifically important for the model
based control structure to be used in Chapter 5.
2.3 Analysis of the Stewart platform kinematics
A vast majority of the literature on parallel robotic systems is devoted to their kinematics.
This is an important subject because in the design of the kinematics of a robotic system,

2.3 Analysis of the Stewart platform kinematics 51
Table 2.1: The Simona system parameters
variable value description
ra 1:60 m upper gimbal radius
rb 1:65 m lower gimbal radius
du 0:20 m upper gimbal spacing
dl 0:60 m lower gimbal spacing
lmin 2:08 m minimal actuator length
qmax 1:25 m actuator stroke
qop 1:15 m operational actuator stroke
_lmax 1 m=s maximum actuator speed
its manipulability, its workspace, etc. is to be fixed. This problem in combination with the
application to flight simulation has been studied by Advani [7].
In this section, two new solutions towards kinematical problems will be presented,
which have a direct relation with the ability to perform real time model based feedback
safely on a specific kinematic design of a Stewart platform motion system.
- Given a specific design of the manipulator, one has to ensure the system can not be driven
into a singular configuration.
- Further, model coordinates have to be calculated from the measurements taken in real
time (e.g. a finite number of calculation steps) without the chance to have divergence
between actual and calculated coordinates.
In parallel robots such as most flight simulation motion platforms, the position of the
system is usually indirectly measured by the length of the actuators. The forward kinematical
problem of calculating the platform coordinates given the actuator lengths of the Stewart
platform is seen to be solved in roughly two ways in literature.
Using analytic techniques, the problem can be transformed to a set of combined polynomial
equations whose roots have to be found to solve the forward kinematics e.g. [57].
Although these equations can provide insight into the structure of the problem, closed form
solutions are only seen to be presented for special classes of platforms e.g. Merlet [97],
which still is an active area of research [68]. Solving the roots of the equations, still leaves
the problem of choosing the actual pose. The complexity of this problem is reflected by the
fact that it has been shown that the general version of the Stewart platform can have up to
40 real solutions [34].
Secondly, for long, the forward kinematics has been tackled numerically by performing
the Newton-Raphson (NR) iteration scheme [35] already given in (2.44).
sxk+1 = sxk + J ;1
l�sx (sxk)(lmeasured ; lk) (2.58)

52 2 Mechanics of parallel driven motion systems
As the actuator lengths, l, are explicit functions of the platform coordinates, sx, the
jacobian, Jl�sx, is a function of platform coordinates.
Jl�sx(x)(i� j) = @li(sx)
@sxj
(2.59)
This latter method is preferred here as it is less involved to be implemented in a realtime
model based controller where inversion of the jacobian is already part of the control
structure. Convergence or convergence to the actual physical pose in this scheme is not
guaranteed in general. As the forward kinematics of a Stewart platform have more than one
solution [57], and since singular points of the jacobian for unconstrained actuator lengths
exist [87], the iteration scheme does not converge globally to the right solution. This problem
has only recently been considered in literature [21], [70].
Chetelat [21] considers the problem of convergence in a general way by considering reconstruction
of coordinates of an implicit (matrix) function from which the image is known
e.g. a Stewart platform pose from length measurements. He points out that if one considers
functions, of possibly interconnected vector equations, of polynomial degree

2.3 Analysis of the Stewart platform kinematics 53
and minimum gain of J and again the Lipschitz condition. These can be calculated for
Stewart platforms. Exclusion of singular points of J is necessary to calculate a radius of the
neighbourhood in which convergence is guaranteed.
From this radius, the maximum gain of J and the maximum speed of the actuator, a
sufficient update frequency of the iteration can be calculated above which (quadratic) convergence
is guaranteed.
The parameters of the new Simona research simulator are used as an example which
shows that reasonable results can be obtained although the conditions derived are rather
conservative.
2.3.1 Convergence NR-iteration
A weak version of the Newton-Kantovorich theorem [109] given by Stoer [141] will be
used. From this theorem convergence of the NR-iteration can be inferred. It is stated as
follows.
Theorem 2.1 Given: a set D IR n , a convex set Do with exterior Do D and a function
f : D ! IR n which is continuous on D and differentiable with derivative Df(x) on Do.
If positive constants r, , , and h can be found for x o 2 Do such that Sr(xo) =
fx jkx ; xok 0
B) limk!1 xk = exists, 2 Sr(xo), and f( )=0
C)
8 k 0� kxk ; k
h 2k;1
1 ; h 2k
With 0

54 2 Mechanics of parallel driven motion systems
The proof of this theorem is given in [141].
Roughly speaking, this theorem states that a solution can be found in the NR-iteration
(B) if the differential Df (x) does not vary too much (a)), is far enough from singularities
(b)) and eventually does not jump too close to the boundary of the defined neighbourhood,
at the first iteration (c)).
It also states that the iteration will not go out of a specified neighbourhood (A) and
converges at a certain speed (C). Conditions for a NR-iteration towards the Stewart platform
coordinates will be derived.
Chetelat [21] uses the fact that if one ensures that the function, f (x), is not of polynomial
degree higher than two, then the jacobian is Lipschitz and in this case the Lipschitz constant
can be calculated using the infinity norm.
2.3.2 NR-convergence Stewart platform
Having stated the kinematical structure of the Stewart platform and given Theorem 2.1 on
convergence of a NR-iteration, it is now possible to investigate under what conditions the
specific NR-iteration of (2.58) will converge to the physical platform pose.
First an appropriate definition of the coordinates has to be given. The defined jacobian,
Jl�x, is the description of platform translational and angular speed/variation to actuator
length variations. To go from orientational parameter variations to angular speed depends
on the parametrization used.
In this case the last three euler parameters, are used to parametrize orientation, sx T =
[cT s T ] T , where s is a scaling factor which can be used to get less conservative results
in specifying a variation of x since c and = sin( 1
2 )n have different dimension (m and
rad). The scaling factor s =2kakmax will shown to be appropriate in the sequel.
It will be assumed that after each iteration will be reset to zero. In that case the
relation between the angular velocity and the euler parameters is very simple, ! j =0= 2I _.
Off course the rotation matrix in the iteration is now the multiplication of all the rotation
matrices calculated, Tk+1 = T ( k+1)Tk. The continuous function, f, in Theorem 2.1, from
which the platform coordinates have to be found can now be given by k l i k;klik = fi(sx)
where kl i k is the measured length of the i th actuator (fixed value for each iteration) and klik
is the length of the i th actuator given sx (li given by (2.38)). Since the measured length is
fixed, the derivative function with scaled euler parameters, is only slightly different from
the jacobian Jl�x given earlier in (2.42).
Df (sx) T = Jl�sx(sx) T =
Ln
2(TA m Ln)=s
(2.62)
Since it is a function of unit direction lengths, it is only defined for k lk 6=0.
To derive the conditions (a,b,c)-st(ewart)pl(atform) for convergence of the NR-iteration
defined in Theorem 2.1 for the Stewart platform, the constants ( � � ) can be specified
using the kinematics.
a-stpl) In this case the 2-norm of the matrix is taken which is equal to the largest singular
value, and in the formulas d means an finitely small difference.

2.3 Analysis of the Stewart platform kinematics 55
kJl�sx(sx1) ; Jl�sx(sx2)k < stplksx1 ; sx2k (2.63)
kJl�sx(sx1) ; Jl�sx(sx2)k =
dLn
2d(TA m Ln)=s
= (dJ) (2.64)
The Frobenius (semi)-norm is an upper bound on this norm and is advantageous in
case of the Stewart platform since specific bounds can be derived on the matrix elements
as will be shown later on.
kdJkF =
vu
u
t 6X
i=1
(dJ) kdJkF (2.65)
(kdln�ik 2 + k2d(T ai ln�i)=sk 2 (2.66)
To derive a constant, stpl, the separate elements of the last equation will be described
as a function of sx. This will be done in the Section 2.3.3.
b-stpl) To state the second condition from which a constant number stpl has to be calculated
also the 2-norm is used which can be upper bounded by one over the minimal
singular value, min, of the jacobian at some pose, xo. By using the maximal
variation of the jacobian which has been calculated for the previous condition, the
constant, , becomes an upper bound for the maximum gain of the inverse jacobian
over a volume of poses, sx.
kJ ;1
l�sx (sx)k =( min(Jl�sx(sx))) ;1
(2.67)
(max( min(Jl�sx(sxo)) ;kdJkF � 0)) ;1 = stpl (2.68)
c-stpl) To calculate stpl, also use can be made of the singular values. Since any point
in the work space is a possible initial start of the iteration, the maximum condition
number of the scaled jacobian over the work space can be taken as the constant, stpl.
kJ ;1
l�sx f (sx)k = min (Jl�sx)kdsxk = stpl (2.69)
In the next part it will be shown that indeed the jacobian of a Stewart platform is Lipschitz,
as a-stpl) requires, as long as the actuators of the platform have minimal stroke strictly
larger than 0.

56 2 Mechanics of parallel driven motion systems
2.3.3 Lipschitz condition on jacobian
By constructively analyzing the kinematics of the Stewart platform the following theorem
can be derived.
Theorem 2.2 The Stewart platform jacobian, Jl�sx, is lipschitz i.e. a can be found such
that
kJl�sx(sx1) ; Jl�sx(sx2)k ksx1 ; sx2k 8 sx1� sx2 2 Do (2.70)
if kli(sx)k > >0 8 sx 2 D� i 2f1�::: �6g: (2.71)
Note that the requirement of an actuator length larger than zero is an implicit constraint
on the set of platform poses.
To prove Theorem 2.2, it is observed that jacobian, J l�sx, in (2.62) consists of two
elements: the unit actuator direction, ln, and the vector product, (ln T a), for each actuator.
By bounding the difference between the unit actuator directions and vector products of two
jacobians with a function linear in kdsxk, Theorem 2.2 can be proven and a can actually
be constructed.
In two steps dln will be bounded first.
1. Difference of an upper gimbal connection coordinate, dx a, will be bounded given the
finite difference between two platform coordinates dsx.
2. Then the difference between two actuator unit vectors, d ln, givendxa, will be considered.
See Fig. 2.11 in which the motion of actuator as a function of the moving
gimbal point xa is schematically depicted.
With
the following bound directly follows
xa�i =c + T a m i
(2.72)
kdxa�ik kdck +2kakkd k p 2kdsxk (2.73)
as the motion of a point due to rotation can be bounded easily with the reseted euler parameter
description as
kd(T a)k =2kakkd k (2.74)
As from Fig. 2.3, it can be observed that the absolute change of a vector due to rotation has
length 2 j sin(1=2 ) j=j 2 j.
The second bound takes into account that as the upper gimbal moves within a ball
(Fig. 2.11), the maximum change of the actuator direction is achieved if the new l, just

2.3 Analysis of the Stewart platform kinematics 57
l n
l
Φ
dl n
dx a
Fig. 2.11: Change of the unit vector in the direction of the actuator, ln, as a function of
change of the upper gimbal vector, dx a.
touches the ball. In that case (ln + dln) ? dxa. With some geometry, sin( )=kdxak=klk
and
cos( )=
s
b
1 ; kdxak
klk
the following monotonous upper bounding function can be derived for d ln.
r
1 ; cos( ) p kdxak
kdlnkmax =2
2
2
klk
2
a
(2.75)
(2.76)
Taking into account (2.73) gives kdlnk 2kdsxk=klk. So, the actuator (minimum) length
directly influences the bound, which can be put on kd lnk. If the length of the actuator is not
strictly larger than zero, one can show that arbitrary small variations of the pose can result
in nondecreasing variations of dln i.e. not satisfying the Lipschitz condition.
Now, bounding the vector product is also possible. In general ka bk kakkbk and
(a +c) (b + d) = (a b) +(a d) +(c b) +(c d). Further, rotation does not
change the 2-norm, kT ak = kak. Change of the moving gimbal due to rotation is bounded
by kd(T a)k =2kakkd k = kds k. Now,
kd(ln T a)=kakmaxk
kdln T ank + kln d(T an)k + kdln d(T an)k

58 2 Mechanics of parallel driven motion systems
p 2
klk kdxak + kds k +
2kds kkdsxk
: (2.77)
klk
So in this way every element of the jacobian matrix is explicitly bounded by the variation
of the platform pose and the (explicitly platform pose dependent) actuator length. If a
minimal actuator length is given, a specific stpl can be calculated.
Adding the bounds, for each of the actuators in the F-norm of the jacobian, results in an
explicit number for stpl. Assuming small platform pose variations i.e. second order effects
in the last equation are relatively small e.g. kdsxk

2.3 Analysis of the Stewart platform kinematics 59
Algorithm
- Choose an expected minimal singular value of the jacobians, over the grid points.
- Take a grid such that the boxes with radius rs�min around the grid points fill the whole
work space.
- Calculate the minimal singular value of the jacobian at every grid point.
- If the minimal singular value is larger than the expected value, there are no singular points
in the work space, the algorithm finishes. If not, start another iteration choosing a
smaller expected minimal singular value.
Lemma 2.3 If the work space of a robotic manipulator having a bounded Lipschitz constant,
does not have any singular point this non singularity will be detected by Algorithm 7.1
in a finite number of iterations.
Of course the boundary of the work space (in six dimensions!) should be known which is a
stand alone problem (treated by Haug et al. [86]). To calculate an upper bound for the gain
of the inverse jacobian ( stpl) afiner grid should be taken. (This will increase calculation
time tremendously, e.g. n 6 grid points extra.)
2.3.5 Sufficient update frequency
The bounds derived are rather conservative in most cases. However, by calculating a convergence
radius for the NR-iteration in the practical example of the new Simona research
simulator shows that it is possible to guarantee that this iteration can be used if a reasonable
frequency is used to update the platform pose. Further, the singular points can really be
excluded from the work space in this case.
To satisfy NR-iteration convergence the following constants were obtained using the
kinematics of the Simona flight simulator motion system given in Table 2.1.
a-simona Using (2.78), assuming a minimal length of the actuator of l min =2:08m, results
in simona =5:2.
b-simona With gridding a minimal radius of rs�min = :09 is found. (About 200000 grid
points had to be calculated to exclude singularity from the work space). The smallest
singular value found is 0:75 and with finer gridding a simona =2can be guaranteed.
c-simona The first value of f(sxo) in any point can not be larger than f(sxo) (J)kdsxk
Together with the smallest gain this gives simona = kdsxk 3:5kdsxk
Now with a bound kdsxk 0:02, hsimona =( )=2 =(3:5 0:02 2 5:2)=2 =0:23 and
rsimona = =(1 ; h) =(3:5 0:02)=(1 ; 0:23) = 0:11. Within the every ball with radius
rsimona, [a-simona] and [b-simona] should be guaranteed which is the case in the work
space given the operational stroke (not including actuator cushioning part). To guarantee
convergence in the whole work space also non-singularity, etc. should be guaranteed further

60 2 Mechanics of parallel driven motion systems
outside the work space which needs lots of calculation (with an extra stroke of 0.15m,
singularities can be obtained so rs�min becomes very small).
With a bound on the maximal speed of the actuator it is possible to calculate a minimal
update frequency which guarantees kdsxk < 0:02. Givenk _ lk < 1m=s, now
simona t _ lmax < kdsxk (2.80)
and t 100 Hz convergence of the
NR-iteration is attained.
Lemma 2.4 The Stewart platform with the Simona motion system parameters has no singular
points in the work space and the NR-iteration with this platform will converge to the
right platform pose if the update frequency is larger than 100 Hz.
Summarising, to satisfy a general convergence theorem on Newton-Raphson iteration,
one of the requirements is the Lipschitz condition on the derivative function, which was
shown to be satisfied for Stewart platforms. Variations of the jacobian can be bounded by
the variations of the platform coordinates.
Next to this requirement, the jacobian should not be singular in any point of the work
space. With the Lipschitz condition on the jacobian and gridding, volumes can be excluded
from singularities.
Although only sufficient (and thereby conservative) conditions could be derived for convergence,
it was shown to be possible to obtain an important convergence result in the practical
example of Simona flight simulator motion system. It will be one of the requirements
for safe application of model based feedback.
2.4 Modelling the mechanical part of the system dynamics
After the thorough analysis of the kinematics of the parallel driven Stewart platforms in the
previous section, this section will focus on the dynamics of such systems. The aim of this
section is to show how to derive a limited number of differential, and possibly also some
algebraic, equations, which describe the motion of mechanical systems such as the motion
system of a flight simulator as a function of the forces acting upon this system. It will
be assumed that these systems consist of rigid bodies. In the extensions these bodies are
possibly interconnected by springs and dampers.
2.4.1 General theory in modelling the dynamics of mechanical systems
Most theory described in this section can be found in many textbooks on mechanics and
robotics, rigid multi body systems like [64, 81, 102, 106, 120, 129, 134]. Based on this
general theory, a rigid body model of the Stewart platform, possibly with actuator joint
inertia or linear ground flexibility, will be derived.
All mechanics treated here are based on the assumption of a semi-equilibrium given by
Newton’s second law f ; mp =0in an inertial frame for any mass particle. To describe the

2.4 Modelling the mechanical part of the system dynamics 61
acceleration, p, of all parts in a system as a function of a limited number of variables some
kinematics has been introduced.
Having defined the kinematics of a mechanical system, its dynamics can be specified. In
the equations of motion the semi-equilibria are described in a compact form as a function of
generalized velocities or coordinates. The integrals over the mass-particles of a body result
in inertia matrices and the active forces are projected along the velocities by a virtual work
argument.
The equations of motion can be stated in several ways. First the Lagrange equations will
be given. Then it will be shown that from these equations the more simple Newton-euler
equations follow if one rigid body is considered. Finally the method based on Kane [64]
will be introduced. This method in which all forces, active and inertial, are projected along
a limited set of generalized velocities, will be central in the derivation of the motion system
mechanical model.
Considering generalized velocities or coordinates ( _z� z), with the method of Lagrange
the difference between the kinetic energy, K(z� _z), and potential energy, P(z), of a system
is called the Lagrangian, L(z� _z): L = K ; P. For a system, which can be described by a
minimal number of generalized coordinates z, the Lagrange equations are given by:
d
dt
@L
@ _z
; @L
@z
= (2.81)
The driving moments/forces, , are all the working forces (non inertial or conservative) projected
along the variations of the generalized coordinates. These are called the generalized
forces.
The kinetic energy can in general be described by
K = 1
2 _z T M (z) _z� (2.82)
where the mass matrix, M (z), is a symmetric positive semi definite matrix.
It can be appropriate to apply a coordinate transformation by introducing the generalized
momenta (assuming a positive definite and thus invertible mass matrix)
m = @L
@ _z = M (z) _z: (2.83)
The implicit second order differential Lagrange equations (2.81) can now be transformed
to a (nonlinear) state space description, i.e. the Hamiltonian equations of motion following
Van der Schaft [147]. A key role is played by the hamiltonian or total energy function, H,
consisting of the addition of kinetic and potential energy,
H(m�z) = 1
2 mT M ;1 (z)m + P(z) =m T _z ; L(z� _z): (2.84)
Taking the partial derivative of (2.84) to m for the first set of state equations and filling in
(2.84) into (2.81) eliminating L for the second, results in
_z = @H
(m� z)
@m
(2.85)
_m = ; @H
(m� z) +
@z
: (2.86)

62 2 Mechanics of parallel driven motion systems
In this non energy dissipating system the increase in energy can be shown to be equal to the
amount of work delivered,
d
dt H = _z T
(2.87)
By observing that the work W, done by forces , does not change if a change of coordinates
z, is applied, it is easy to show that projecting the forces along other coordinates is equal to
multiplying by the transpose jacobian.
W = T z1 = T Jz1�z2 z2 =(J T z1�z2 )T z2 (2.88)
Considering such collocated pairs of forces and velocities as inputs ( = J T (z)u) and outputs
y = J(z) _z, a hamiltonian system results. The Stewart platform rigid body mechanics
will have this structure. The hamiltonian system structure does not require that the number
of input/output pairs and the number of generalized momenta/velocity pairs is equal. This
allows to take into account parasitic resonant effects without losing the structural properties
of a hamiltonian system.
Further, manipulation of the second order differential equation (2.81) can lead to
M (z)z +
Compactly written as
d
dt (M (z) _z ; _z T
@
@zi
M (z) _z + @
P(z) = : (2.89)
@z
M (z)z + C(z� _z) _z + G(z) = : (2.90)
with a mass matrix, M, a non linear coriolis/centripetal matrix, C, and a gravity vector, G.
If a rigid body (with mass, m, and inertia, Ic) is considered at its centre of gravity c, the
Newton-euler equations result. The mass matrix is block diagonal in this case.
Kbody = 1
2
_c !
mI
0
0
Ic
_c
!
(2.91)
From the two blocks, two independent equations follow from (2.81). The impulse law:
f = d d
(m (c)) = mIc (2.92)
dt dt
And the impulse moment law with the generalized forces f g c (moments in this case) is
derived similarly from (2.81) taking the time derivative of the Lagrangians partial derivative
to the second set of (angular) velocities. The Lagrangian, in this case, only consists of the
kinetic energy, which does not depend on the position.
f g c
= d
dt (g R m I m c ! m )= g R m I m c _! m + g R m ~ m I m c ! m
= I g c _! g + ~ g I g c ! g = I g c _! g +! g
I g c ! g
(2.93)
In this way, the Newton-euler equations are easily stated for each rigid body in a system.
However, extra equations with unknown internal forces result, in using this method to state

2.4 Modelling the mechanical part of the system dynamics 63
the equations of motion for a multi body system. Applying Lagrange, results in taking
partial derivatives of complex energy functions if the whole system is considered. These are
disadvantages which can be circumvented by using Kane’s method.
With Kane also generalized variations or velocities have to be specified. The general
formula
f + f = 0 (2.94)
states the (semi-)equilibrium of the active forces, f, and inertial forces, f , projected along
the directions of the generalized velocities. Generalized velocities do not have to be derivatives
of some positional coordinate e.g. in principle the angular velocity can be used as
such.
To calculate the over-all inertial forces, as in the Newton-euler approach, the specific
inertial forces generated in the frame of each body can be stated. As in the Lagrangian
approach, a minimal number of equations results by writing the motion of the bodies as a
function of the generalized velocities and projecting each specific force from its local coordinates
to the generalized ones. Also the active forces can first be stated in an appropriate
frame after which projection follows. The projection in general consists of a change of coordinates
i.e. a multiplication with a jacobian. This procedure can also be automated [65].
Especially in fast modelling with the aim of simulation, this can be useful.
With this method it is possible to start with a strongly simplified system by calculating
part of the (inertial) forces and separately adding other forces if a more accurate model
has to be taken into account. Many of the specific merits of the method will become clear
considering the modelling of the mechanical part of the Stewart platform including actuator
inertia.
The amount of generalized coordinates can exceed the number of d.o.f. of a system. In
that case, next to the differential equations given by the semi-equilibria of the forces (2.94)
constraint equations of velocities and position, generally stated as
A(x� t) _x + b(x� t) =0� (2.95)
should be added. With mechanical systems having so called kinematic chains i.e. parallel
manipulators, such additional equations easily appear if the motion of some parts of the
system can not explicitly be written as functions of the minimal number of coordinates
(amounting to the number of d.o.f.). This will not be the case with the Stewart platform apart
from the additional coordinate necessary to attain a global description of the orientation.
Systems where the constraint equations cannot be integrated to constraint position equations
are called non-holonomic. A parallel manipulator, like the Stewart platform, will occur
to be a holonomic system. Since there are kinematic chains, it is often easier to state the
equations of motion of a parallel manipulator by using constraint equations. In that case the
manipulator is described as a serial system (with some of the joints disconnected). The parallel
connections are incorporated by adding constraints. A combined differential/algebraic
description results. This kind of description causes difficulties (index problems etc., [19])
in simulation and model based control. With these goals in mind during modelling, it is
more convenient to state the model in explicit differential equations if possible. Starting
with Section 2.4.3 in the second part of this section it is shown that this can be done with

64 2 Mechanics of parallel driven motion systems
φ
r
x
y
m
Fig. 2.12: Example system consisting of one mass rotating in the plane on the left. On the
right also a variable length, r, exists to the centre of rotation.
the Stewart platform. But first a simplified example will be given in Section 2.4.2 to clarify
the modelling method used.
2.4.2 Example in modelling using Kane’s method
To clarify the method of Kane, the dynamics of a simple example system will be modelled.
The system is given in Fig. 2.12. A simple mass, m, can be rotated in the plane around
some point, which is taken as the origin in the inertial frame. It is assumed we do not know
how to define the mass properties in rotation but only know the translational inertial force
equations:
f c =
f x
f y
= ;
m 0
0 m
φ
x
y
r
x
y
m
= ;Mc (2.96)
We, however, want to choose the angular velocity, _ ,defined in Fig. 2.12 as the generalised
velocity for this 1-d.o.f.-system. The coordinates, x and y, and their derivatives can be
formulated as a function of and _ with
and by differentiation
_c =
c =
_x
_y
x
y
= r cos( )
r sin( )
= ;r sin( )
r cos( )
_ = Jc�
(2.97)
_ : (2.98)
The local inertial forces, f c can be projected along the generalised velocity by
f = J T c� f c (2.99)
Further differentiation of (2.98), provides us with the acceleration
c = Jc� + _
Jc�
_ (2.100)

2.4 Modelling the mechanical part of the system dynamics 65
with
_
Jc� =
;r cos( )
;r sin( )
_ : (2.101)
This equation can be filled into (2.96), which can be filled into (2.99) to arrive at the simple
f = ;mr 2 � (2.102)
since J T c� Jc�
_ =0and J T c� Jc� = r2 in this example. The equilibrium equation of motion,
considering an active force already projected along _ , i.e. a moment fa, will be
f + f = fa ; mr 2 =0 (2.103)
In principle, also _c could have been defined as the generalised velocity. The only constraint
in this choice is the ability to describe the motion of any body with this set of velocities.
In that case (2.96) would give the inertial forces and an additional constraint equation
_c T c =0 (2.104)
would have to be satisfied. I.e. a combined algebraic and differential set of equations would
result. Simulating such a system is not trivial. E.g. in the example any finitely small step
satisfying (2.104), a velocity orthogonal to c, would enlarge the radius r.
In the case sketched, it is relatively easy to choose a minimal number of coordinates
and end up with differential equations only. But already with planar systems such as the
manipulator given in Fig. 2.6, it is not clear how to choose three coordinates (minimal local
degrees of freedom) which globally describe this system.
The method presented seems somewhat overdone if applied on such an example but it
can easily be extended to the more complex cases. Consider for example the system of
Fig. 2.12 with an variable radius (e.g. due to a non rigid spring/damper) connection.
If the force of the spring always acts along the radial direction it could be appropriate to
choose the polar generalised velocities, _p, consisting of _r and _ .
Now
_c = Jxp _p =
and the inertial forces follow from
and using
cos( ) ;r sin( )
sin( ) r cos( )
_r
_
(2.105)
f p = ;J T xpm(Jxpp + _
Jxp _p) (2.106)
Jxp
_ = ; sin( ) _ ;r cos( ) _ ; _r sin( )
cos( ) _ ;r sin( ) _ +_r cos( )
gives the inertial forces more explicitly as
f xp = ;
m 0
0 mr 2
r ; ;mr _2
2mr _ _r
(2.107)
: (2.108)

66 2 Mechanics of parallel driven motion systems
In this case nonlinear velocity forces like the quadratic centripetal force, mr _2 , and the
velocity product coriolis force, 2mr _ _r, naturally show up.
Including an additional body is relatively easy. E.g. the inertia, i tube of the tube drawn
in Fig. 2.12, can be taken into account by adding the inertial force, f tube = itube .
In the calculations, the projections using the jacobians often enable geometric simplifications.
This will also be the case in using this method in modelling the Stewart platform
including the actuator inertial properties.
With this example, the method of Kane has been made explicit and will be used in the
next sections to derive a model of the mechanical part of the Stewart platform motion system
dynamics.
2.4.3 Stewart platform dynamics
The general procedure of stating the equations of motion has now been given. In this part,
this will be applied in modelling the Stewart platform. Kane’s method of projecting local
semi-equilibria (equations of motion) will be used to arrive at a compact description. To
state the local equations, both Lagrange and Newton-Euler are applied wherever either one
is most appropriate.
With the choice of platform position/orientation as the generalized coordinates, all equilibria
can be written as explicit functions of these coordinates (and its derivatives). The
modelling approach, however, still enables a clear assignment of the contribution each part
of the system has on the over all dynamics.
First a simplified model, with the platform as the only (rigid) body, will be derived.
Then, the influence of the actuator inertial forces is quantified.
2.4.4 Basic Stewart platform mechanical model dynamics
The basic structure of the Stewart platform model results if one considers the platform
alone, not taking into account the inertial forces of the actuators. Since the system in this
case consists of only one body, the equations of motion are easily derived with Newton-
Euler taking the velocity of the platform coordinates as the generalized speed. With the
impulse law, (2.92), the translational motion is described
Lnfa + mcg = mcc� (2.109)
where fa are the active forces generated by the actuators and g is the gravity vector.
And with the impulse moment law, (2.93), the rotational motion is described by
TA m Ln fa = Ic _! + ~ Ic!� (2.110)
with Ic = TI m c T T . Combining these two results in the simplified model of the Stewart
platform.
Ln
TA m Ln
fa = mcI 0
0 Ic
c
_!
+ 0 0
0 ~ Ic
_c
! ; mcg
0
(2.111)

2.4 Modelling the mechanical part of the system dynamics 67
In short
J T l�x (sx)fa = Mc(sx)x + Cc( _x� sx) _x + Gc� (2.112)
from which the mass matrix, Mc, the coriolis/centripetal effects, Cc, and gravity, G, are
clearly separated and the structure given in (2.90) is visible. Further, the jacobian, J l�x,
already playing an important part in the sections on kinematics, also occurs in the equations
of motion directing the actuator driving forces.
In many cases, especially with large and heavy motion systems, these equations will sufficiently
describe the rigid body dynamics of the mechanical part of the system. In practice,
these dynamics will interact foremost with the hydraulic actuators, which will be modelled
in the next chapter. This chapter concludes with some sections on extending the mechanical
model.
2.4.5 Influence of the actuator inertial forces
In some cases, the inertial forces (i.e. mass properties) of the actuators can not be neglected.
With many heavy flight simulation systems they can, but with the light weight motion systems
requiring high performance motion this has to be reevaluated. Explicitly taking into
account the actuator inertial forces will complicate the model. However, this model can be
used to attain an appropriate approximate version as will be shown in the next chapter.
Most of the assumptions considering the actuators have already been given considering
their kinematical properties in Section 2.2.11. See e.g. Fig. 2.8. The gimbals are assumed to
rotate frictionless. The inertia of both actuator parts rotating around the actuator direction
is neglected and further assumed symmetric w.r.t. this axis. The mass properties of each
actuator part can therefore be described by two parameters, the mass and the inertia around
an axis orthogonal to the actuator direction. Mass m a and inertia ia are taken for the upper
moving part of the actuator (mainly the piston), and (m b, ib) for the lower rotating part, the
cylinder.
The inertial forces of the actuators can be split up in three parts: the gravitational forces,
the inertial mass forces and the influence of its inertia. These forces will first be projected
on the upper gimbal points. Any force generated at this point is easily projected at the
generalized platform velocities with (2.47) by
f x a = J T a�xf a a : (2.113)
The gravitational forces are easily projected along the platform velocities. The lower part
of the actuator using (2.50),
f a bg = J T bc�ambg = mbrb
j l j Plng = Gmb : (2.114)
Clearly, in a position where the gravity vector is directed along the actuator ( ln) this force
will not contribute. The contribution of the moving part is in that case maximal at a as is
shown by equivalently using (2.49),
f a ag = J T ac�amag = ma(I ; ra
Pln )g = Gma
(2.115)
j l j

68 2 Mechanics of parallel driven motion systems
The inertial force generated by the mass at the c.o.g. of the moving part of the actuator is
easily described w.r.t. a frame at its c.o.g.
fac = ma _vac
(2.116)
Projection of this force at the upper gimbal point using (2.49) and writing _vac as a
function of _va using (2.55) and (2.57), results in
f a ma = ma(I ; Pln
+ ma
ra
2
(j l j;ra)
+
j l j2 Pln ) _va ::: (2.117)
j l j 2 j Plnva j 2 ln +2ma
= Ma _va + Cmava
(j l j;ra)ra
j l j3 (l T n va)Plnva
The first term consists of a part in the direction of the actuator (I ; P ln) where the mass
directly acts on the gimbal point. Perpendicular to this direction the influence gets smaller
with the squared ratio of distances to the lower gimbal point. The second and third term are
the quadratic velocity dependent coriolis and centripetal force.
The inertial forces generated by the inertias of the lower and upper part of the actuator
can be taken together considering Lagrange and observing that their contribution to the
kinetic energy is equivalent. With (2.48)
Kia�ib
= 1
2 !T a !a(ia + ib) (2.118)
= 1
2 vT a
(ia + ib)
j l j2 Plnva = 1
2 vT a M a ia�ibva The derivation of the following two equations can be found in Appendix B together with
a proper choice for factoring the quadratic velocity terms. p a is the position of the upper
gimbal point.
d
dt (Mia�ib )=; (ia + ib)
j l j3 Kia�ib
pa
(2v T a lnPln + Plnval T n + lnv T a Pln )� (2.119)
= ; (ia + ib)
j l j3 ((l T n va)Plnva + lnv T a Plnva)� (2.120)
With Lagrange (d=dt(M ) _v ; @K=@p g a ), (2.119) and (2.120) the inertial forces at p a result
f a ia�ib = (ia + ib)
j l j 2
Pln _va ; 2(ia + ib)
j l j3 (l T n va)Plnva = Mia�ib _va + Cia�ibva: (2.121)
The contribution to the mass matrix only exists at motion perpendicular to the direction of
the actuator. Next to this, only coriolis and no centripetal terms appear. The coriolis force
is generated as a result of the inertia points in the opposite direction as the one generated by

2.4 Modelling the mechanical part of the system dynamics 69
the mass. This is due to the fact that the influence of the inertia decreases while that of the
mass increases as the actuator gets longer.
All the separate inertial force contributions of each actuator represented by f a bg , f a ag ,
f a ma and f a ia�ib together with the active actuator force generated by the hydraulics, result in
total forces at the upper gimbals. Projection to the platform coordinates via the jacobian,
Ja�x, puts these forces into place for the equations of motion.
2.4.6 The Stewart platform model including actuator inertia
The equation of motion of the Stewart platform including the inertia of the actuators can
still be put in form of
Where Mt, Ct and Gt are given by
Ct _x = Cc _x +
6X
i=1
J T l�x (sx)fa = Mt(sx)x + Ct( _x� sx) _x + Gt(sx) (2.122)
Mt = Mc +
6X
i=1
J T ai�x (Mma�i + Mia�i�ib�i )Jai�x
(2.123)
(J T ai�x (Cma�i + Cia�i�ib�i )Jai�x _x; j! j 2 (Mma�i
Gt = Gc +
6X
i=1
+ Mia�i�ib�i )P!ai)
(2.124)
J T ai�x (Gma�i + Gmb�i ) (2.125)
In this model, the platform coordinates, velocities and acceleration are the only variables
and the effect of each term (mass, coriolis, centripetal, gravity, driving forces) and parameter
(mass, inertia, centres of gravity, gimbal point) can clearly be distinguished. The model is
well defined in each state except for those in which any of the actuator lengths becomes
zero. Note that this puts an implicit limitation to the validity of the model.
Adding the inertial influence of the actuators to the simplified model did not change the
compact form of six coupled second order differential equations. The equations, however,
became much more complex which is not favourable in model based control in which the
model has to be calculated at high speed.
Ji claims in [61] that the actuator inertial effects can be seen as a change of the platform
mass, inertia and c.o.g. This claim should be carefully interpreted as this change is not only
dependent on the position, but also on the direction of the motion i.e. not even valid at one
operating point. E.g. in case of the Simona flight simulator, the mass of the actuators add
more to the mass matrix of simulator in heave than in the lateral directions of surge and
sway.
With the equations given, it is possible to give bounds on the forces not taken into account
if the inertial forces of the actuators would be neglected. Although with conventional

70 2 Mechanics of parallel driven motion systems
motion systems this is often justified, the tendency towards light weight platforms makes
the actuator inertial forces more evidently come into play. Total neglection would result
in a too rough approximation in that case. Approximation by a constant additive term will
shown to be more convenient.
2.4.7 Parasitic mechanical aspects: modelled as linear flexibility
The models presented only take into account the mechanical system as the rigid bodies.
Together with the hydraulic actuators, this forms the most relevant part of the system dynamics.
However, it appeared that two kinds of parasitic mechanical effects due to flexibility
could not be neglected.
First, the foundation on which the Simona motion system was tested in the Central
Workshop was not entirely rigidly attached to the inertial world. Though relatively heavy
(45 tons), the foundation formed by a block of concrete appeared to be in motion in the
plane orthogonal to z if the simulator motion contained too much energy in the frequency
area above > 5 Hz [41]. The foundation was not ideal for these tests, but in countries
like the Netherlands, where buildings are often build on large piles in weak ground, similar
effects can be expected if no special precautions are being taken.
Flexible effects like this, can be modelled by considering additional mass/ spring/ damper
systems in connection with the rigid body model. In Fig. 2.13, a representation of a foundation
is given, which has two translational and one torsional spring/damper pot connecting
the mass/inertia of the foundation to the inertial world. Further, the motion system introduces
forces through the lower gimbal points.
If the foundation is given, the three degrees of freedom: x f along the surge (x-)direction,
yf along the sway (y-)direction and the angle f around the z-axis (yaw-motion), three
additional equations of motion will result. At first instance, the interaction of the foundation
with the inertial world will be modelled by decoupled spring/damper connections with
spring stiffnesses cxf, cyf, c f, and damping coefficients bxf , byf, b f, which are used for
the respective directions. Now, the equations of motion of the foundation with mass m f and
inertia If can be modelled by using the additional matrix equation:
2
4 mf 0 0
0 mf 0
0 0 If
32
5
4 xf
yf
f
3
5 +
2
4 bxf 0 0
0 byf 0
0 0 b f
2
4 cxf 0 0
0 cyf 0
0 0 c f
32
5
32
5
4 _xf
_yf
_ f
4 xf
yf
f
3
5 + (2.126)
3
5 = J T l�xf fa
The jacobian of the general (six) foundation body parameters, x fg to actuator lengths
has a similar structure as Jl�x.
Jl�xfg = ;LT n ;(B Ln) T (2.127)
In case only the aforementioned three degrees of freedom in the plane are possible a second
projection is required through the selection matrix P f with unit vectors selecting x, y and

2.4 Modelling the mechanical part of the system dynamics 71
Fig. 2.13: Physical representation of the model used for the flexible attached foundation
.
P T f =
2
4
1 0 0 0 0 0
0 1 0 0 0 0
0 0 0 0 0 1
Jl�xf = Jl�xfgPf (2.129)
3
5 (2.128)
Finally, the equation for the platform velocity has to be adapted since b will start to vary.
_
l = L T n (Va ; Vb) (2.130)
With the lower gimbal point velocities, _vb stacked in Vb. If the model with the platform
alone (or a constant mass matrix with respect to the moving simulator) is taken, the model
equation (2.111) does not have to be adapted. All together these equations form a model of
the system with a platform attached to a non-inertial ground, which has the same validity as
the simple model (for all simulator poses except those for which (klk i =0). This problem
has not seen to be considered in literature apart from cases in space technology [93, 108] in
which the moving platform has very limited manoeuvrability and the actuators are used to
supress vibrations.
At first sight, the foundation, though also moving if the actuators force the simulator
to move, does not have too much direct interaction with simulator motion, since the basic
mechanical model equation (2.112) looks the same. Indirectly, the following dependencies
can be observed.

72 2 Mechanics of parallel driven motion systems
- The motion of the foundation has influence on the actuator velocities. The hydraulic
actuators form a port connection with the mechanical system through the active forces
and velocity. Generally, the active force will be influenced by the velocity due to a
coupling in the dynamics of the (controlled) hydraulic system. This influence will
therefore affect the simulator motion. Note that this influence will be much larger if
the actuators are position/velocity controlled than with force control in the relevant
frequency area.
- The platform pose can not be calculated from the actuator lengths anymore without correcting
for the moving foundation.
- The jacobian, Jl�x, depends on the motion of the foundation since it depends on the unit
direction vectors of the actuators. These directions will change as the foundation
moves resulting in change of direction of the active forces acting on the simulator.
This is a second order effect and is mostly compensated for by reconstructing the
Jacobian from the actuator length measurements. The motion in the plane of the
foundation will not influence the forces necessary to compensate for gravitational
effects of the gross moving platform.
- The motion of the actuator (inertias) is directly influenced by the foundation. The equations
of motion for these parts have to be adapted and become somewhat more complex.
If the actuator inertia can be neglected or taken as part of the constant mass
matrix w.r.t. moving platform, this effect can also be neglected.
Given influences, both in the choice of the control strategy as in identifying the dynamics
of the system, the parasitic behaviour of the foundation should be taken into account.
The second important parasitic mechanical effect to be analyzed are the structural modes
of the simulator itself. Especially in optimising the ability to realistically simulate the high
frequency vibrations in a vehicle, the unwanted resonances of the simulator should not be
hit up to noticeable level. In the design of the shuttle of the Simona simulator, the ratio
of stiffness and mass has been optimized [8]. However, finite element models pointed out
that with this system one will not be able to extend the lowest structural vibrations above
15-20 Hz
As a rule of thumb the flexible modes should only appear in a frequency area well
distinct from the area where the motion to be simulated has relevant energy and thus usually
form an upper bound on the achievable bandwidths. Since there are only a limited number
of actuators, the moving platform loses functional controllability if structural deformation
takes place. In theory, one could well be able to apply a desired acceleration profile to the
pilots head but one would in these cases structurally have to settle for vibration in other parts
of the platform e.g. the large visual projection screen. Additional active damping devices
can probably be of help in making this restriction less tight.
The unneglectable deformations in the moving platform drastically change the structure
of the system. Modelling using the projection methods like Kane’s is still possible if one can
approximate the modes by splitting up the previously rigid bodies in a limited number mass/
spring/ damper systems i.e. smaller rigid parts connected by springs and dampers ([146]).
Deriving these equations by hand, however, becomes very tedious and does in general not

2.5 Chapter Resume 73
result in a model from which analysis can take place more easily than from automatically
derived models. In a master’s project as part of this research Rijnten [119] modelled the
system, including a structural vibration, using the symbolic equation modelling software,
Autolev [65]. Though simulation with such models can clarify systems behaviour, not much
structural understanding of systems like a deformed Stewart platform can be gained since
the number of variables in the model well goes above a thousand.
Other, more advanced, methods in modelling flexible structures [62, 63, 66, 129] often
used in space applications with large flexible deformations, usually result in models, which,
due to their complexity, can not directly be used in model based control. It was decided to
do the actual analysis of the structural deformations in the Simona motion system using the
actual data measured at the system. This will be discussed in Chapter 4. Further analysis
and also simulation of the models derived will take place after introducing and integrating
the models of the hydraulic system to be discussed in the next chapter. This will be more
realistic since the integration of the hydraulics and the mechanics forms the most relevant
part of the dynamics considering the whole flight simulator motion system.
2.5 Chapter Resume
Since the kinematic model of the Stewart platform construction is fully parallel, the platform
(end effector) coordinates and velocities can globally be used to describe the state of
the mechanical system. It appeared most convenient to state orientation applying an euler
parameter (quaternion) like parametrization. Apart from the numerical advantages, an algorithm
could be defined to evaluate the appearance of singular points in the work space of a
parallel manipulator like the Stewart platform.
With this method, it finally became possible to proof that the SRS kinematical construction
has no singular points. This is not trivial, since it would have been possible to reach
such a point with only 15 % more stroke of the actuators. Moreover, it could be proven that
the iterative Newton-Raphson procedure to calculate the platform pose from the actuator
lengths converges anywhere in the SRS work space during any kind of possible (velocity
limited) motion. It converges sufficiently fast for safe use in a real time digital feedback
loop as will be done in the model based controller discussed in Chapter 5.
The dynamics of the Stewart platform was stated as a set of differential equations without
algebraic constraints resulting from the kinematic chains in the system. This is possible
by writing the actuator motion explicitly as a function of platform motion. By using Kane’s
method of projecting forces onto the generalized velocities, the contribution of each inertial
or active force e.g. stemming from the actuators piston mass, etc., can be quantified separately
using local (simple) Newton-Euler like equations and (by projection) ending up with
a limited number of equations of motion.
The models derived only describe the dynamics of the mechanical system. The observed
dynamics of a flight simulator motion system will shown to be much better approximated by
models which incorporate both the hydraulic and mechanical characteristics and their (two
sided) interaction. This chapter provided for the mechanical modules for such models. The
next chapter will treat the modelling of the hydraulic system itself and the interconnected
structure of hydraulically driven mechanical systems.

Chapter 3
Hydraulically driven motion
systems
In the previous chapter, the mechanical system was modelled with input actuator forces
from which accelerations, velocities and positions of the motion system and possibly its
parasitic dynamics could be calculated. The energy necessary to move is supplied by the
actuators. In robotics both electrically as hydraulically driven systems are seen to be applied
and the mechanical model can be merged easily to models of either kind of systems.
With the larger flight simulator motion systems almost invariably use is made of hydraulic
actuators as is also the case with the Simona motion system. Since both the energy
and the cooling with fluid power technology can be supplied at an exterior hydraulic pump,
a favourable high ratio of force delivered over the weight and size of the actuator can be
sustained well into the high power area. But even more importantly, the use of hydraulics
enables smoothly running (low friction, low wear) long linear actuators, which is necessary
in simulation requiring an unnoticeable change of sign in the direction of motion.
An important aspect of the motion systems under consideration is the fact that the dynamics
of the actuators heavily interact with the mechanics. So modelling either one of
these phenomena does not provide much insight in the dynamics of such systems. In most
textbooks treating the modelling and control of mechanical systems the dynamics of the
actuators are at best considered as parasitic influences to be taken into account at high frequencies
[11, 31, 113]. However, with (heavy) systems requiring a high amount of power,
especially if driven hydraulically, the interaction between the actuators and mechanical system
determines the ’rigid’ resonant modes. By using an integrated model, which will be
presented in Section 3.3, the most relevant dynamics is captured as will be shown in the
subsequent Chapter 4 in which evaluation with the experimental data takes place.
The basic integrated model can still have a relative simple structure although the response
of such systems may already seem quite complex. The basic structure of the hydraulically
driven mechanical system allows the derivation of some important properties
like passivity, discussed in Section 3.3.1, which can be of use in control and experimental
identification. In this chapter, the dynamics of the hydraulic actuators will be modelled with
a bilateral coupling to the mechanics through the energetic pairs of force and velocity at the
mechanical side and flow and pressure in the hydraulical part of the system.
75

76 3 Hydraulically driven motion systems
Fig. 3.1: Schematic drawing of a symmetrical hydraulic actuator.
The long stroke hydraulic actuators, which are typical for flight simulators, have been
modelled in detail for motion control design by Van Schothorst in [124]. In Section 3.1, an
overview will be given of the parts of those models which proved to be essential as either
being an explicit part in the model based controller considered in this research or in analysis
exploring the limitations of such controllers. This last part usually requires an additional
degree of detail in modelling discussed in Section 3.2. In this case, the dynamics of the
hydraulic servo valve together with the transmission lines had to be taken into account.
For a more fundamental introduction into hydraulic systems, one is referred to textbooks
like [99] and [153]. Most of the notation used here follows Van Schothorst [124]. The
modelling for control approach in hydraulic systems can also be found in references like
[48] and [146]. In these references, however, the dynamics of the transmission lines was
assumed to be neglectable. In this research, this was not the case due to the long stroke
and high performance requirements, and therefore a modelling for control approach with
the interplay of transmission lines dynamics and servo valve as the limiting factor, had to be
chosen.
3.1 The basic structure of hydraulic actuators
The basic notion of the functionality of a hydraulic actuator can be observed by looking
at Fig. 3.1, although the actually used actuator looks far more complicated as shown in
Fig. 3.3. In Fig. 3.1, the piston connected to the load can be made to move by letting oil
flow into the compartment of the cylinder on the left, o1, and out of the compartment on
the right, o2, or v.v. This flow will result in a pressure build up, P o1 ; Po2, which through
the reflected areas, Ap, will force the load to move. In a symmetrical actuator, this area,
Ap, is equal on each side. In motion, the oil flows will also have to compensate for the
change in volume in the respective actuator chambers. This effect accounts for the twosided
coupling of flow/ pressure and/ or velocity/ force. Usually one also has to account for
some dissipation resulting from leakage flows.
The dynamics due to the hydraulic part of the system is caused by the compressibility
of the oil. The relative decrease in volume, ; _ V=V, causes a pressure rise, P _ , linear with

3.1 The basic structure of hydraulic actuators 77
the effective bulk modulus of the oil, E.
P _ = ;E _ V
(3.1)
V
By taking into account the oil flow into and out of the compartments, the following equations
can be obtained, which describe the pressure dynamics of each chamber.
Po1
_ = E
( o1 ; l1 ; Ap _q)
V1
Po2
_ = E
( l2 ; 02 + Ap _q) (3.2)
V2
The sign of the oil flows due to leakage, l1, l2, are taken negative w.r.t. the main flows,
o1, o2, usually controlled by a valve. The position, q, of the piston is set to zero at the
point where the volumes of the actuator chambers are equal, V m = V1 = V2. This will
typically be at or near half stroke position as drawn and will be referred to as mid position.
As a result
V1 = Vm + Apq
V2 = Vm ; Apq (3.3)
At this point it is most appropriate to apply a change of coordinates by taking mean and
difference pressures, flows and oil stiffnesses (Cm and dC), which are defined as follows
Pm m lm Cm
dP d d l dC =
1
2
1
2
1 ;1
The pressure dynamics of (3.2) can now be transformed to
Po1 o1 l1 E=V1
Po2 o2 l2 E=V2
_
dP = 2Cm( m ; lm ; Ap _q) +dC=2(d ; d l)
(3.4)
_
Pm = Cm=2(d ; d l) +dC=2( m ; lm ; Ap _q) (3.5)
The change of coordinates can be motivated by the fact that only the pressure difference,
dP , directly affects the mechanics through the actuator force, f a = ApdP . Further analysis
reveals that even more advantage can be taken of this transformation. First of all, dC =0
in the mid position, which fully decouples the two equations. Also in non mid position,
the term d ; d l is small or even zero which makes the second equation not only badly
observable but also almost uncontrollable. So, in most cases a sufficiently accurate model
can be obtained by only taking into account the structure given by
_
dP =2Cm(q)( m ; LlmdP ; Ap _q)� (3.6)
assuming a laminar leakage flow proportional to the pressure difference, lm = LlmdP .
Equation (3.6) provides one of the two main equations to describe the basic hydraulically
driven mechanical system as given in the block scheme of Fig. 3.2.
Removing the dynamics of Pm not only simplifies the model but also reduces problems
(e.g. numerical) in model based control of a hydraulic actuator due to the minor effect on

78 3 Hydraulically driven motion systems
Fig. 3.2: Basic block scheme of a hydraulic actuator connected to a mechanical system.
the input/ output relation of this part of the dynamics. This model reduction step is often
performed without consistently showing what is to be neglected i.e. not considering the
change of coordinates.
The first order differential equation is nonlinear due to the position dependence of the
oil stiffness, using (3.3) and (3.4),
Cm(q) =
EVm
V 2 m ; A2 p
q2 = E
Vm
which attains the minimum at the mid position (q =0).
3.1.1 Leakage
1
1 ; A 2 p q2 =V 2 m
� (3.7)
The leakage flows are considered laminar, which results in flows lineary dependent on the
pressure drops. Assuming two reference pressures: P s as the higher level supply pressure
and Pt as the lower level tank pressure (which can chosen to be the 0-level), the leakage
flows can depend on five different parameters, L i, given in the following equations.
l1 = L1t(Po1 ; Pt) ; L1s(Ps ; Po1) +Ln(Po1 ; Po2) (3.8)
l2 = ;L2t(Po2 ; Pt) +L2s(Ps ; Po2) +Ln(Po1 ; Po2) (3.9)
Not all terms will appear in every construction. E.g. in the double concentric actuator used
for the SRS, schematically given in Fig. 3.3, the terms Ln and L1s do not exist.
Also in this case it is appropriate to consider a change in coordinates
Ltm Lsm
dLt dLs =
1
2
1
2
1 ;1
L1t L1s
L2t L2s
(3.10)

3.1 The basic structure of hydraulic actuators 79
The leakage can now be described as function of the new coordinates
lm
d l
=
+
Ln +(Lsm + Ltm)=2 dLt + dLs
dLt + dLs 4(Ltm + Lsm)
;dLs
;dLt
;4Lsm ;4Ltm
Ps
Pt
dP
Pm
(3.11)
It can be analyzed at what value the mean pressure, P m, will stabilize, using these equations
together with (3.5) with _ dP = _ Pm =0.
In dropping Pm, only the main leakage parameter, Llm, influencing the resulting basic
hydraulic equation (3.6), is relevant. It can be constructed as
Llm = Ln +(Lsm + Ltm)=2 (3.12)
3.1.2 Basic hydraulically driven mechanical system model
In Fig. 3.2, a 2-port block scheme of (3.6) is given in connection with a mechanical system.
This mechanical system with mass, M, viscous/ damping force constant, B p, and possibly
some additional disturbance forces, Fext, can be described by the following equation
Mq = fa ; Bp _q ; Fext
(3.13)
As all the parameters given are strictly positive, the hydraulic system can viewed as a velocity
to force feedback with a first order system. The loop gain over theqtwo integrators,
which is usually dominating, determines the undamped eigenfrequency 2CmA2 pM ;1 .
Both leakage, Llm, and viscous friction/ damping, Bp will dissipate energy.
It is not advisable to model the physical hydraulic actuator as a fully separate module.
E.g. some parts of the hydraulic system to be modelled like the mass of the piston/ cylinder
and the viscous friction should be made part of the mechanical system model to be
connected.
In case of the multivariable flight simulator, basically Fig. 3.2 holds, considering vector
paths and matrices. To include platform coordinates, a mapping using the jacobian has to
be applied. The actuator forces fa = ApdP will effect the platform coordinates through
the transpose jacobian J T l�x and the platform velocities will have to be projected along the
actuators by using _q = _ l = Jl�x _x. The undamped eigenfrequencies of the hydraulically
driven mechanical system can be predicted by calculating the square roots of the eigenvalues
of the loop matrix gain, p (2CmA2 pJl�xM ;1J T l�x ), assuming equal Ap and Cm for all
actuators.
If this is not the case one should consider taking matrices with the specific actuator
values for Ap and Cm on the diagonal. All these variables, except for A p, depend on the
simulator pose. These mass and oil stiffness dependent eigenvalues will be shown to reflect
the most prominent part of the dynamics of a hydraulically driven flight simulator motion
system.
The structure given in Fig. 3.1 is a very rough simplification of an actual hydraulic
actuator although the model derived will describe the main dynamical effects which can be

80 3 Hydraulically driven motion systems
Fig. 3.3: Schematic drawing double concentric symmetric hydraulic actuator.
observed in practice. However, the way the controlled oil flow, m (d ), behaves, has not
yet been given and should also be studied to arrive at an appropriate model. This requires
the modelling of the servo valve and transmission lines as will be discussed in the next
section.
3.2 Extensions
In Fig. 3.3, a schematic drawing is given of the so-called double concentric symmetric
hydraulic actuator, which is used in case of the SRS. This construction enables a symmetric
actuator without the disadvantage of a piston at both sides. The arrows are drawn in
the direction where the flows and velocity take a positive sign. Note that e.g. the leakage
flow, l2;, is always negative. Next to the cylinder, the picture shows two other parts of a
hydraulic servo actuator, which haven’t been discussed yet, the servo valve and the transmission
lines.
3.2.1 Servo valve
The input flows, i1 and i2, given in Fig. 3.3 are determined by the servo valve. Modelling
the relevant dynamics between our control input signal (voltage), u, to the servo valve and
these oil flows is the subject of this section. The valve consists of an pilot valve and main
spool as given in Fig. 3.4. The main spool is controlled electronically using a control scheme
given in Fig. 3.5.
With the external connection to two reference oil compartments to the servo valve, the
oil flow can be controlled by the valve opening. One external reference is the hydraulic
pump supply pressure, Ps, which is 160 Bar higher than the second reference in case of the
SRS. This second reference, the tank pressure, Pt, can be considered zero. Both references
are assumed to be constant in the sequel.

3.2 Extensions 81
Fig. 3.4: Schematic drawing of a three stage valve. The main spool position, X m, is measured
electronically and controlled by feedback to the flapper positioning current,
ia. The flapper/ nozzle mechanism internally stabilizes the positioning of the pilot
valve with position, Xs, with oil flows ni. This is used to enable large oil flows
at the main spool, which is positioned with oil flows mi. For a valve model,
describing the most relevant high frequent dynamics as given in (3.17), only a
limited number of variables given in the figure have to be taken into account.

82 3 Hydraulically driven motion systems
Fig. 3.5: Valve main spool control scheme
Considering a turbulent oil flow through the servo valve, the main behaviour of this
flow can be described by a nonlinear algebraic relation depending on spool position and the
square root of the pressure drop. In the higher frequency area (> 100 Hz in case of the SRS
servo valves), the dynamics of the spool positioning mechanism have to be considered too.
The spool is an important part of the servo valve. By positioning the spool, the valve
opening can be set. The valve opening and pressure drop over the valve determine the
oil flow through the valve to the actuator cylinder compartment itself. If large oil flows are
required (typically larger than 100 l=min), the spool positioning itself requires an additional
hydraulic circuit, in itself an extra servo actuator controlled by a pilot spool. This multistage
concept is applied in the SRS as flows up to 150 l=min are needed.
A schematic drawing of the valve is given in Fig. 3.4. The pilot spool (with position
xs), is positioned by an internal hydromechanical feedback mechanism called the flapper/
nozzle. As already noted, the positioning of the main spool is performed by an electronic
feedback given in Fig. 3.5. This feedback uses the main spool position x m, or positioning
error m to the current ia which forces the flapper position, xf , aside.
Over a broad frequency area, usually including the area of interest in positioning and/
or forcing a mechanical system (servo-valve bandwidth in case of the SRS is 150 Hz), the
transfer function from desired to actual spool position, x m, can be considered a static unity
gain.
So the main servo valve phenomena can be captured by considering the turbulent flow
equations of the main four-way valve. With symmetric hydraulic actuators, one usually
applies symmetric 4-way valves. The main spool position, x m, together with the pressure
drops determine the flows, i1 and i2, to and from the actuator chambers from P s and to
Pt, as is drawn in Fig. 3.4.
The equation describing a turbulent flow restriction is given by Merrit [99]
=CdAm(x)
s 2(Pin ; Pout)
(3.14)
In this case the discharge coefficient, Cd, and the oil density, , can be considered constant.
The geometric properties, Am(x) of the flow restriction depend (almost linearly) on the
spool position. The flow gain evaluated w.r.t. the spool position varies with the square root
of the pressure drop and, moreover, this results in a hard non-linear change in gain if the
flow changes sign and Pi1 or Pi2 are not both equal to 1=2Ps.

3.2 Extensions 83
x_s, P_n, f_p, f_v
2
1.5
1
0.5
0
P_n
Step response pilot valve
−0.5
0
f_v
0.001 0.002 0.003 0.004 0.005
time (s)
0.006 0.007 0.008 0.009 0.01
Fig. 3.6: A state step response of the model (3.17) of the pilot valve with the identified
parameters given in Table 3.1.
As observed earlier, a more compact description can be arrived at at the actuator side in
applying a change of coordinates, using difference and mean flows and pressures. This is
also the case with the valve. With the important additional assumptions that
x_s
Pi1 + Pi2 = Ps� (3.15)
and unaltered geometric properties A(xm), and considering normalised pressures and flows,
^P = P=Ps, ^xm = xm=xx�max, ^ = = max, the maximum flow can be found by calculating
or measuring max := f (xm�dPi�nom) =f (xm�max� 0). Now, one equation describes
the oil flow through the four-way valve
q
^
im = f2(^xm) 1 ; sgn(^xm)d ^ Pi� (3.16)
where the mean normalised valve flow is defined by ^ im =( ^ i1 + ^ i2)=2. Again note that,
due to the sign convention taken, this mean valve flow, im, means the average amount of
oil going into transmission line 1 and out of transmission line 2.
With this equation, as proposed in [124] by Van Schorhorst, static geometric nonlinearities
can be captured by the function f 2(^xm), which have been identified experimentally for
all the six SRS servo valves. The sign-function (sgn) accounts for the switch in ports at
the four-way valve. Note that a reduction step by omitting one state in taking away mean
pressure and difference flow at the actuator is also motivated at the input, the valve, since
these variables are not involved in (3.16).
The main limitation in only using (3.16) in describing a valve lies in the high frequency
area, as the spool position, xm, can not be positioned infinitely fast. The positioning mechanism,
as depicted in Fig. 3.4, can be modelled following Van Schothorst [124] by using the

84 3 Hydraulically driven motion systems
variable value variable value
c1 4:29e3 c6 1e4
c2 2:96e7 c7 9:93e-1
c3 2:3e10 c8 4e3
c4 1:13e11 c9 2:68e10
c5 1e2 c10 1:27e2
Kp 8 Kd 4:9e-4
Table 3.1: The Simona three stage valve parameters identified by Van Schothorst [124]
state space equation with normalized states given for the pilot valve:
2
6
4
^xf
_^xf
_^xs
d _ ^ P n
3
7
5 =
2
6
4
;c1 ;c2 ;c3 ;c4
1 0 0 0
0 ;1 ;c5 c6
0 c7 1 ;c8
3 2
7
6
5
6
4
_^xf
^xf
^xs
d ^ Pn
3
7
5 +
2
6
4
c9
0
0
0
3
7
5 ia
(3.17)
All the coefficients, c1:::9 are positive and the numerical values, identified on the actual
actuators of the SRS by Van Schothorst [124], can be found in Table 3.1. The model takes
into account the force balance and acceleration of the flapper, x f , the limited stiffness of
the oil at Pn and a reduced force balance (velocity only) of the pilot spool, x s. The model
incorporates a right half plane zero as the flapper forces the pilot spool to move in negative
direction, which can be compensated only after pressure build up has taken place.
Further, the dynamics of the SRS pilot valve has four poles, among which is one lightly
damped resonance, at ca. 900 Hz and one pole at ca. 150 Hz. The step response of the pilot
valve is given in Fig. 3.6. Next to each other the flapper velocity (f v), position, xf , and
pressure difference peak, Pn, in resonating at 880 Hz. The pilot spool moves slightly non
minimum phase and resonating along a first order response in approx. 6 ms to its final value.
In this simulation the four states have been scaled to achieve a numerically more favourable
description. xsc = diag(1e ; 7� 1e ; 3� 1� 10): The pilot spool lets oil flow ( m) tomove
the main spool. Considering a dPm 0, this can be modelled by a additional equation for
the third stage adding an integrator.
_^xm = c10f1(^xs) (3.18)
Also the quasi static functions f1(^xs) and constants c10, have all been experimentally identified
[124].
As already noted, the position of the main spool is controlled with an electronic PDcontroller
(Fig. 3.5) with proportional (K p) plus derivative (Kd) feedback given by
ia = Kp ^xm ; Kd _ ^xm = Kp(u ; ^xm) ; Kdc10f1(^xs) (3.19)
where u is the input signal sent to the valve.
The additional state of this system in feedback results in a slightly resonating behaviour
( =0:3) at 150 Hz as opposed to the mainly first order like response of the pilot valve.

3.2 Extensions 85
x_m, x_s, P_n, f_p, f_v
1.4
1.2
1
0.8
0.6
0.4
0.2
0
x_s
Step response three stage valve
x_m
−0.2
f_v
−0.4
P_n
−0.6
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
time (s)
Fig. 3.7: Model state step response of the three stage servo valve.
As can be seen from the state step response of the valve in Fig. 3.7, the high frequency
behaviour can hardly be noted in the input/ output relation. Up till 200 Hz, the valve can
also well be approximated by a second order system (e.g. via balanced reduction), which
will lack physical interpretation, however. Taking a bode plot of the linearised version of
the valve shows the close correspondence as shown in Fig. 3.8.
In hydraulic servo actuators, the valve characteristics are often limiting the achievable
performance of the system. In this case, the high bandwidth valves still introduce a phase
lag of approx. 10 at 25 Hz and double this at 50 Hz. The combination with the long
transmission lines, which will be described next, form a risk to stability.
3.2.2 Transmission lines
A disadvantage of requiring long stroke actuators is the unavoidable use of relatively long
transmission lines. Transmission lines often show a lightly damped resonating behaviour
and if the transmission lines are longer, the eigenfrequencies tend to be lower which is a
disadvantage in striving towards a smooth high bandwidth control by the servo valve of the
flow to and from the actuator. Moreover, if actuator pressure control (dP o) is intended to
be applied measuring the pressure, dPi, at the valve, the transmission line dynamics are in
between. It will be shown that the pressure at the other side of a long transmission line, the
output pressure, Po1�2, of the actuator, can deviate considerably from the measured input
pressure, Pi1�2, at frequencies ten times lower than the transmission line eigenfrequencies.
The transmission line dynamics can often be neglected if properly designed and with
moderate performance requirements. In [48, 146] practical examples can be found of hydraulically
driven mechanical systems (Brick laying robot, RRR-robot) in which the transmission
lines can be omitted in modelling and control. In that case, the bandwidth of the
servo-valve can be made reasonably lower than the lowest resonance resulting from the

86 3 Hydraulically driven motion systems
Amplitude
Phase (deg)
10 1
10 0
10 −1
10 1
10 −2
0
−100
−200
−300
10 1
−400
Bodeplot of full and reduced (2nd order) model valve
10 2
10 2
Frequency (Hz)
Fig. 3.8: Bode plot of 5 th -order linearised servo valve model and 2 nd -order reduced version.
transmission lines. With [153], the lowest transmission line eigenfrequency, ! t can roughly
be approximated by
Lt
!t =
co 2
10 3
10 3
(3.20)
With line length, Lt =1:4m in case of the SRS and the wave propagation velocity, c o
given by
s r
E 109 co = =
1000� (3.21)
850
where E is the bulk modulus of the oil and the density, this results in an estimated eigenfrequency
of 180Hz.
In [124] it has been shown that these dynamics can not be neglected if one wants to apply
high performance motion control on long stroke hydraulic actuators typically used in flight
simulation. The lightly damped resonances of the transmission lines together with the phase
lag introduced by the limited bandwidth of the servo valve easily result in stability problems.
By taking these phenomena into account in modelling, proper model based control can be
applied.
The main steps in modelling the transmission lines will now be described, following Van
Schothorst [124] and Yang and Tobler [158]. Details can be found in these references. The
main conclusion will be that transmission lines can be modelled as a (theoretically infinite)
set of parallel connected linear second order systems. Each second order system appoints a

3.2 Extensions 87
φ i
P i
Fig. 3.9: A causal two-port configuration for a transmission line.
specific vibrational mode with increasing eigenfrequency and, in approximating, all modes
of interest are easily picked.
In choosing the input flow and output pressure as input and the input pressure and output
flow as output, a proper causal solution will result as discussed by Van Schothorst in
[124]. This two-port representation of the transmission line is given in Fig. 3.9. The exact
causal solution to a one dimensional distributed parameter model of a uniform rigid fluid
transmission line with laminar flow in the Laplace domain, is given by Yang and Tobler in
[158] by
Pi(s)
o(s)
=
1
cosh(;(s))
1 ;Zc(s) sinh(;(s))
; sinh(;(s))=Zc(s) 1
φ o
oP
Po(s)
i(s)
(3.22)
with the characteristic impedance, Zc(s), and wave propagation operator, ;(s), of a single
transmission line, as in [158]. One can approximate this system by a finite dimensional state
space model by using the first n terms of an exact infinite modal sequence, which can be
derived from (3.22).
This modal approximation technique leads to
Pi(s)
o(s)
=
=
n
k=1
Pik(s)
ok(s)
n 1
k=1
s2 + cks + dk
acks + bck ;azks + bzk
ask ; bsk acks + bck
Po(s)
i(s)
(3.23)
Considering each second order system, an even more compact state space representation
can be derived [158], requiring only four parameters for the first mode and two additional
(due to parameter dependencies) for each higher harmonic to be taken into account.
Each harmonic can then separately be described by
_
Pik
_ ok
=
0 (;1) k+1 1k
(;1) k 2k ; 3k
Pbk
ak
+
0 4k
; 2k
1k 4k 0
Pok
ik
:
(3.24)
These parameters still have a clear physical interpretation, as given in [124]. A block scheme
representing this model is given in Fig. 3.10 with for k =1, t j = j j =1�::: �4� t5 =
; 4 2= 1. Only one mode is given in this block scheme, as the structure of the others is

88 3 Hydraulically driven motion systems
Fig. 3.10: Basic block scheme of the parallel modal transmission line model. Note that each
additional mode, i, can be accounted for by an equivalent scheme connected in
parallel through the sigma signs.
equivalent. The sigma-terms show the connection of these additional harmonics, which can
be taken into account by a parallel connection of similar block schemes.
To maintain approximate steady state gain for n =1, which will be lost due to simple
neglection of all higher order terms, 4 can be set to ; 1 if pressure and flow have been
normalised. The term p 1 2 is equal to the undamped eigenfrequency of each mode and
3 is the dissipating term, damping the resonance. In the hydraulic actuators of the SRS
the lowest eigenfrequencies of the transmission lines could be found in the range of 160 ;
220 Hz and the first higher harmonic at a factor three higher. Given the bandwidth of the
valve of 150Hz, in this research each transmission line was modelled by taking only one
mode into account
Apart from the resonating behaviour, the transmission lines result in a difference between
the pressure measured over the valve, dP i and the pressure dPo, over the actuator
chambers, which actually drive the mechanical system. This difference can become a problem
if one wants to apply a well defined force to accelerate with high precision. As will be
shown, these differences occur at much lower frequencies than those where the transmission
line resonances occur.
In [124], it was argued that the state describing the mean pressure in the hydraulic actuator,
Pm, should be retained if transmission lines are to be taken into account and the
model is to be used over the whole stroke including the position dependent stiffness of the
hydraulic actuator. But in applying model based control, it is unfavourable to include an
almost uncontrollable and unobservable state. Therefore it is proposed to use an alternative
model.
There are two separate transmission lines as shown in Fig. 3.3. The model of transmission
line 1 has inputs i1 and Po1 and outputs Pi1 and o1. The model of transmission line
2 has inputs i2 and Po2 and outputs Pi2 and o2. The transmission line inputs Po1 and
Po2 are constructed from the reduced hydraulic actuator model, (3.6), by
Po1 = 1=2(Ps + dPo)
Po2 = 1=2(Ps ; dPo)� (3.25)

3.3 Integrated system 89
which is a similar assumption as given in (3.15). Further, the other connection of this two
sided coupling between the transmission lines and the hydraulic actuator is given by
m =( o1 + o2)=2 (3.26)
At the other side of the transmission lines, the servo valve is connected. The models of the
two can be merged by retaining (3.16) and using i1 = i2 = im and dPi = Pi1 ; Pi2.
By evaluating the differences between the reduced and full order model after connecting
the building blocks, this step can and will be validated. Integrating the set of submodels
introduced for the mechanical and hydraulical part of a flight simulator motion system will
be the subject of the next section.
In this section two relevant extensions, the valve characteristics and transmission line
dynamics, to the basic hydraulically driven mechanical system model were introduced. Of
course many more effects could be investigated. One, which should be mentioned here,
but will not be discussed much further, are the hydraulic actuator cushioning areas at both
ends of the stroke. By gradually disabling oil flow to and from the actuator, the cushioning
prevents from high accelerations (max. 2g) occurring if an actuator for some reason runs out
of stroke. This mechanism, however, also influences the system dynamics for at least the
first and last 0:15m stroke of all the actuators. Design and evaluation of the SRS cushioning
is discussed in [124].
3.3 Integrated system
Having introduced models of the mechanical part of a flight simulator motion system in
Chapter 2 and of the hydraulic servo actuators which drive the system in Section 3.1 and
Section 3.2, the two can be connected. This results in a nonlinear model describing the
hydraulically driven mechanical system.
An important system analytic tool, passivity analysis, will be used to derive some interesting
properties of these non-linear models. E.g. it will be shown that the full multivariable
nonlinear system model is passive from input oil flows to input valve pressure differences
irrespective of parasitic effects like transmission line resonances and/ or flexible mechanical
modes. As a result proportional (or either other passive) input pressure feedback to valve
oil flow can be applied without destabilising the system, as is of course already long known
heuristically.
In the sequel, the system characteristics due to the hydraulic system are best clarified by
first considering a one degree-of-freedom actuator driving a single mass. Also the effect of
a parasitic mass or a non rigidly connected foundation is best understood by using such a
simplified model first. The complexity of the fully non-linear six degree-of-freedom Stewart
platform model driven by its hydraulic actuators is considerable since it easily includes over
fifty states. Most of the system properties seen in practise are, however, well understood
by using these kinds of models because the physical structure of the whole system has been
maintained.

90 3 Hydraulically driven motion systems
3.3.1 Passive input/output pairs in flight simulator motion systems
Passivity is a strong input/ output systems property, which can be derived for a large class
of systems also inhibiting nonlinearities. It roughly tells that a system does not generate
energy by itself. From this property, stability is often easily derived, also in combination
with feedback.
There exist a large amount of control strategies based on the passivity property e.g.
found in [11, 66, 128, 147]. Mathematically it is defined here following Van der Schaft
[147]. The inner product for the vector signals, f and g in the L n 2 -space, is taken as
Z 1
< f�g>=
0
n
i=1 fi(t)gi(t)dt: (3.27)
Definition 3.1 Let the causal input/ output system G map L n 2 ! L n 2 (thus stable). Then G
is passive if there exists some constant such that
� 8 u 2 L n 2 : (3.28)
By introducing the extended L2e-space, the definition can be generalised to a larger signal
set and noncausal/ nonstable systems. A causal or nonanticipating system means that
G(u)T = G(uT ) 8 u 2 Ln 2e where the subscript T gives the truncation at time T . In the
extended space, integration of (3.27) is required only up till time T.
Starting with small subsystems from which passivity can be derived, complex system
passivity can be evaluated by using the fact that feedback of a passive system with a passive
system results in a over all passive system and two parallel passive systems together form
one passive system. This does not hold for passive systems in a series connection.
Consider the feedback system of Fig. 3.11. With the assumption that (e 1� e2) 2 Ln 2 !
(u1� u2) 2 Ln 2 and passive G1 and G2 i.e. < yi� ui > i for i = 1� 2, the feedback
system with inputs (e1� e2) and outputs (y1� y2) is also passive since
< y1� e1 > + < y2� e2 > = < y1� e1 +y2 > + < y2� e2 ; y1 >=
< y1� u1 > + < y2� u2 > 1 + 2 8u1�2 2 L n 2 (3.29)
Notice that since this holds for a multivariable system, any input/ output combination, for
which a system is passive, can be fed back by a passive system without influencing the
passivity of some passive input/ output combination somewhere else. Passivity of a parallel
connection of some passive G1 and G2 with output y =y1 +y2 and u =u1 =u2 is proven
by
< y� u>= < y1� u1 > + < y2� u2 > 1 + 2 8 u 2 L n 2 (3.30)
Passivity can be proven easily for some basic system structures. For the time varying
positive gain ’system’ described by y(t) =k(t)u(t) with k(t) 0, it can be proven by
Z 1 Z 1
y(t)u(t)dt = u(t)k(t)u(t)dt 0 8 u 2 L2 (3.31)
0
0

3.3 Integrated system 91
e1
y2
+
- d
- 6
u1
-
G1
G2
u2
Fig. 3.11: A general feedback structure
For the stable linear first order system described by _y =(u;ky) with k 0,oru = _y +ky,
it follows from
Z 1 Z 1
y(t)u(t)dt = y(t)ky(t) +
0
0
1
Z 1
2 0
d
dt y2 (t)dt > ; 1
2 y2 (0) 8 u 2 L2 (3.32)
For linear systems passivity means that the real part of the transfer function in the frequency
domain will be positive. For passive linear single input single output systems the phase will
thus vary between =2.
The collocated input torque - output velocity pairs of a mechanical system, which can be
described in hamiltonian form, are passive for every initial condition. From the definition of
the Hamiltonian (2.84) and assuming that the potential energy function is chosen as P(z)
0 follows H 0. With the mechanical system described by the generalised coordinates,
z, consider an output mapping y = J(z) _z, and the input, u, defined by the dual map,
= J T (z)u. Now the system from input u to output y will shown to be passive. Consider
(2.87), then
< y� u>T =< _z� >T =
0
d
H(m(t)� z(t))dt =
dt
H(m(T )� z(T )) ; H(m(0)� z(0)) > ;H(m(0)� z(0)) 8 u 2 L2e (3.33)
Take y = _q, u = fa, z = sx and J(z) = Jl�sx(sx) and this establishes passivity of the
rigid body Stewart platform (2.122) for the system considered from input actuator force,
fa to output actuator velocity, _q. Moreover, it tells that passivity will be preserved in case
of connecting a large class of parasitic mass- spring- damper systems connected e.g. a non
rigidly attached foundation.
Having established passivity of the mechanical system, the system can be extended to
the hydraulics retaining the passivity property.
-Viscous friction/ damping Every Rayleigh function, R satisfying _z T (@R)=(@ _z)( _z) 0
is a passive system. Rayleigh functions often occur as dissipative feedback over passive
mechanical (flow) systems retaining passivity. A simple example of a Rayleigh
function is the viscous friction, bi _q i in each of the actuators:
Rvact( _x� sx) =J T l�x (sx)diag(bi _qi) =J T l�x (sx)diag(bi)Jl�x(sx) _x: (3.34)
Z T
+
d?
+
y1
e2
-

92 3 Hydraulically driven motion systems
Amplitude
Phase (deg)
10 2
10 1
10 0
10 −1
10 0
10 −2
100
0
−100
dPo−dPi
−200
10 0
−300
10 1
10 1
Bodeplot Phi_im−>dP_i,o
dPo−dPi
Frequency (Hz)
Fig. 3.12: Bode plot of 7 th -order linearised hydraulically driven mechanical system model
including transmission lines. Input valve flow, im, outputs, dPi (full), dPo
(dashed), dPo ; dP i (dotted).
Equivalently damping induced through resonances can also be included without destroying
passivity.
-The hydraulic actuator The model of the hydraulic part of an actuator, given in (3.6)
can be considered as a stable first order system with a varying gain C m(q). Since
Cm(q) > 0 each linearisation of the hydraulics described by (3.6) results in a passive
system for every position q. Further, as shown as a part in the block scheme of
Fig. 3.2, it is connected to the mechanical system as a negative feedback over a passive
system. So each linearised hydraulic actuator connected to the nonlinear mechanical
system preserves passivity. The passive input/ output system of the actuator can be
chosen from m to dPo respectively. With the varying gain, the situation is somewhat
different:
< m�dPo > =
+
Z 1
0 Z 1
0
d
dt dPo(t)
10 2
10 2
dPi
dPo
dPi
dPo
dPo(t) Llm ; d
dt
1
4Cm(q(t)
1
4Cm(q(t))
10 3
10 3
dPo(t) dt
dPo(t)dt (3.35)
Due to the time derivative of the inverse actuator stiffness, this system is not always
passive. With an upper bound on the actuator velocity of e.g. 1 m=s and with a
polynomial fit Cm(q) = b2q 2 + b1q + b0 =99:2q 2 ; 9:8q + 104 [124], a leakage
term of 0.002 (0.01 is normal) will passify this system.
-The transmission lines The structure of the transmission lines models in the block scheme
given in Fig. 3.10 is clearly a parallel connection of passive systems which can be considered
as two stable linear first order systems in feedback. The actuator side of the

3.3 Integrated system 93
10 2
10 0
10 −2
10 2
10 0
10 −2
10 2
10 0
10 0
10 −2
10 1
10 2
Frequency (Hz)
10 3
100
0
−100
100
0
−100
100
0
10 0
−100
10 1
10 2
Frequency (Hz)
Fig. 3.13: Bode plots from input valve flow ( im) to pressure difference (dPi) of linearised
hydraulically driven mechanical system models including transmission lines in
upper/ mid/ lower position (q = :47� 0� ;:47m) for full (full), reduced (n =6,
dashed) order models and difference (dotted).
port feeds back over the passive input/ output pair dP o and m respectively and at
the valve side the input/ output pair is given by im and dPi, considering the parallel
transmission lines over the pairs Pi1 and i1 and Pi2 and i2.
-The valve port dynamics The valve port dynamics modelled as turbulent flow port can
be considered as a positive time varying gain. If j dP i j 1,
p
1 ; sgn(xm(t))dPi(t) =k(t)xm� (3.36)
im(t) =xm(t)
implies k(t) 0. This is a passive system which preserves passivity if used in a feedback
connection with e.g. constant gain from dP i to xm. As shown in Section 3.2.1,
the valve spool positioning system cannot be considered a constant gain over an infinitely
large frequency area but inhibits dynamics which is clearly not passive from
input voltage (u) to output spool position (x m). At this point passivity based control
strategies may cause possible (stability) problems.
Summarising, it has been established that the hydraulically driven nonlinear mechanical
system is passive with respect to a large number of input/ output pairs. Under moderate
conditions, which are typically satisfied in practice, constant feedback of the pair (valve
spool position, pressure difference measured at the valve) leaves the passivity properties of
the nonlinear system unaltered. In fact, this kind of feedback will dissipate energy, which
drives the system with badly damped resonances faster to its stable equilibrium.
10 3

94 3 Hydraulically driven motion systems
Variable Description Value Unit
n Max. valve flow 25e ;4 [m3 =s]
Ps Supply Pressure 160e5 [N=m2 ]
Fmax = Ps Ap Max. actuator force 40e3 [N ]
_qmax = n=Ap Max. velocity 1 [m=s]
Normalised Operators Description Value Unit
An = Ap _qmax= n Piston area 1 [;]
C1 = E n=(V1Ps) Stiffness chamber 1 101 [1=s]
C2 = E n=(V2Ps)
L1 = L1tPs= n
Stiffness chamber 2
Leakage 1
104
4:32e
[1=s]
;2
L2 =(L2s + L2t)Ps= n Leakage 2 5:06e
[;]
;2
Wpn = wp=Fmax Visc. friction 3:5e
[;]
;4
M
[;]
;1
n = Fmax=(Mp _qmax) Inv. Mass 10 [1/s]
Table 3.2: Parameters taken for a 1-d.o.f. hydraulic mechanical system. First the four normalising
quantities are given. Then all the normalised values for of the operators
are specified.
Variable Value Variable Value
t11 = 1737 t12 = 1254
t21 = 650 t22 = 1184
t31 = 30 t32 = 54
Table 3.3: Parameters taken for the two transmission lines.
3.3.2 The 1-d.o.f. hydraulically driven mechanical system
Further insight into the characteristics of the hydraulic mechanical models, derived in the
previous sections, can be obtained by filling in realistic values into the parameters, e.g. SRS
design numbers, and performing an evaluation in the frequency (linearised models) and
time domain. First a one-degree-of-freedom hydraulically driven mechanical system with
transmission line dynamics will be considered. For these models, the difference in the
dynamics of the actual driving pressure in the actuator and the measured pressure over the
valve will be evaluated. Further, the implications of the reduction step proposed by taking
(3.6) instead of (3.2), assuming (3.25) and (3.26), can be quantified.
It is important to normalise the variables such as pressures (typically 10 7 in SI-units)
and flows (typical values of 10 ;3 in SI) when building numerical models which contain
hydraulics. In Table 3.2, first the normalising values of these variables are given. Then the
normalised values of the operators taken for the models consisting of (3.13) together with
(3.6) or (3.2) are specified at neutral position (at full stroke q =0:625). With the mass taken
of 4000kg and the typical values of the hydraulic actuators of the SRS, an undamped rigid

3.3 Integrated system 95
body mode eigenfrequency, f rm of approx. 7Hz results.
frm = 1
2
r
2Cm
= 1
2
r
2 4:1 106 4000
= 1 p
(101 + 104)10 = 7 Hz
2
(3.37)
Mn
This corresponds with the lowest eigenfrequencies found for the SRS. Due to the normalising
velocity of one, the normalised inverse mass can also be interpreted as the maximum
acceleration to be attained, which in this case approx. amounts to gravity. The leakage term
related to the damping in the actuator driving direction amounts to L lm =2:35%, which is
equal to the value experimentally found by Van Schothorst [124]. This is also the case for
the viscous friction term.
The model is completed by connecting the transmission lines using (3.24) and assuming
an ideal valve input, i1 = i2 = im. The parameters taken for the transmission lines
correspond to the values derived in [125] adapted with 10 % additional line length to arrive
at the eigenfrequencies found in practice. The three resulting parameters, t 1k:::3k, for the
two lines (k = 1� 2) are given in Table 3.3. Eigenfrequencies of the unconnected transmission
lines therefore amount to 169 Hz and 194 Hz with damping of t1 = 0:01 and
t2 =0:02 respectively.
Influence of the transmission lines on input and output pressure dynamics
Bode plots of this system from input m to the outputs dPi and dPo are given in Fig. 3.12.
Connection of the impedance of the hydraulic actuator to the transmission lines shifted
the transmission line eigenfrequencies approx. 10 Hz higher. This shift of approx. 5%
is also observed for the lower (rigid body) mode and in that case to lower values for the
eigenfrequency resulting from the load together with the oil spring stiffness. This effect
of interaction between the transmission lines and the actuator should not be neglected. It
means e.g. that the effective oil stiffness with the SRS-actuators will be 12% lower than
the theoretically found value in case one identifies this hydraulically driven system with a
model structure not accounting for the transmission lines.
The phase of the passive input/ output pair ( im�dPi) clearly remains within 90 .It
is important to notice that although the pressures have a one-to-one correspondence in the
lower frequency area, they start to deviate much earlier than the transmission line eigenfrequencies.
This is due to the anti-resonance in the transfer function from im to dPi at
approx. 60 Hz.At20 Hz this amounts to 16% and at 30 Hz already accounts for a difference
of 25%. Further, the resonating behaviour of the transmission lines is expected to be
much (10 times) heavier observed in the measurable output, dP i, than it will be felt by the
mechanical system (through dPo).
Influence of the varying actuator stifness and model reduction step
Since the stiffness of the oil column depends on the actuator position, the eigenvalues, also
those related to the transmission lines, will shift if the actuator moves. As argued in [124],
this shift will be slightly different if the actuator model is reduced to (3.6). However, the
reduced model describes the system reasonably well as can be observed in Fig. 3.13. Bode
plots of the system from input, im, to the measurable output dPi in an upper/ mid and
lower position are taken for the reduced and nonreduced models. Also the differences are

96 3 Hydraulically driven motion systems
Amplitude
Phase (deg)
10 2
10 1
10 0
10 −1
10 0
10 −2
0
−200
−400
10 0
10 1
10 1
Bodeplot u−>dPi,dPo
Frequency (Hz)
Fig. 3.14: Bode plot of 8 th -order linearised hydraulically driven mechanical system model
including transmission line and valve dynamics. Input valve input voltage, u,
outputs dP i (full), dPo (dashed).
shown. The upper and lower positions are just within the operational area where the cushioning
does not influence the actual system. Since the eigenvalues are relatively lightly
damped, very small differences in the eigenvalue shift locally result in larger deviations but
the general behaviour is described fairly well by the reduced order model. Both for the
upper as the lower position the anti-resonance between the hydromechanical mode at approx.
7 Hz and the first transmission line related mode moves up from 60 Hz to maximally
90 Hz.
Influence of the valve dynamics
The model can be extended by adding the reduced linear second order model of the three
stage valve dynamics derived from (3.19) and (3.17). In Fig. 3.14 the Bode plot of this
model is given. The main effect of the valve is an additional lag in phase, which becomes
prominent for frequencies > 100 Hz. Together with the high gains introduced with the
transmission lines resonances, the valve phase lag forms an important stability risc in feedback
design, which has to be taken into account.
In systems with moderate requirements, the valve bandwidth can be decreased in order
to filter most of the transmission line resonances. But also in that case, the phase lag of
the valve will be dangerous in some frequency area e.g. in combination with resonances
due to structural deformation. In the lower frequency region, the nonlinearities of the valve,
e.g. due to (3.16), will form the most important factor.
In the time domain, the step response of this model is given in Fig. 3.15 on a 200 ms
range. The steady state response to the input and output pressures is neglectable. This
means it is not very feasible to induce sustained accelerations with hydraulically driven
10 2
10 2
10 3
10 3

3.3 Integrated system 97
5
4
3
2
1
0
−1
−2
−3
Step response u−>dPi,dPo,dPo−dPi
−4
0 0.02 0.04 0.06 0.08 0.1
time (s)
0.12 0.14 0.16 0.18 0.2
Fig. 3.15: Step response of 8 th -order linearised hydraulically driven mechanical system
model including transmission line and valve dynamics. Input valve input voltage,
u, outputs dPi (full), dPo (dashed). dPo ; dPi (dash-dot). The high frequency
resonances result from the transmission lines, the slower vibration with
approx. 0:16s period time is the rigid body mode.
mechanical systems. A quasi static gain of a moving actuator is attained from valve input
voltage to actuator velocity. Of course in practice, the limited stroke is the main limiting
factor for the low frequency accelerations. In control design for acceleration (simulation),
however, the virtually zero gain from valve input to force/ acceleration will be an important
factor to be taken into account.
In both pressures the 160 ms time period (7Hz) of the hydromechanical mode is most
prominent. The difference in response is mainly due to the transmission line resonances to
which the pressure measured over the valve is much more sensitive. Due to the interference
of the two transmission line modes, the high vibrational amplitude comes up and falls within
a 40 ms time range.
3.3.3 Additional mechanical modes
The models presented in the previous paragraphs include the dynamics of the hydraulic actuator(s)
into the higher frequency areas of approx. 200 Hz. In deriving accurate models of a
hydraulically driven mechanical systems over this broad frequency area, one can usually not
assume the mechanics to be entirely rigid. In this section, two of such cases are examined
for a 1-d.o.f. system. First, the possible nonrigidness of a foundation is taken into account.
In the second case, the driven system consists of a main mass connected to a parasitic mass/
spring/ damper.
A schematic drawing of hydraulically driven mass, m s, connected to a nonrigid foundation
is given in Fig. 3.16. The foundation with mass, m f is fastened to the ground with a

98 3 Hydraulically driven motion systems
Fig. 3.16: Schematic drawing of a hydraulic actuator moving a consisting of a single mass
ms in 1-d.o.f. pushing from a non rigidly attached foundation, acting as a mass/
spring/ damper system mf �bf �cf .
spring of stiffness cf and a damper, bf . The equations of motion become
msxs = AdPo (3.38)
mf xf = ;AdPo ; bf _xf ; cf xf
Although the actual force fa = ApdPo necessary to accelerate the mass, ms does not
change with the nonrigid foundation, the feedback path relating _q = _x s ; _xf changes. In
case of the structure with simple mass given in Fig. 3.2, the feedback path becomes _q =
ApdPo=(mss), assuming no viscous friction. Every pole of the feedback path becomes a
transmission zero in the transfer function from input (valve flow) to output pressure, dP o.
With the foundation the feedback path changes to
considering the additional equation of motion,
_q = (ms + mf )s 2 + bf s + cf
mss(mf s 2 + bf s + cf ) ApdPo� (3.39)
mf xf + bf _xf + cf xf + fa =0: (3.40)
p
cf =mf ,
A complex pair of transmission zeros with the undamped eigenfrequency, ! f =
of the foundation alone results. The structure of the poles of this interconnected system is
somewhat more complex. The foundation will add two complex poles and the transmission
zeros will always lie in between, since the passivity properties will be preserved. Usually the
mass of the foundation will be much higher than the mass to be moved. In this case, although
the respective eigenfrequencies can be close, there will not be a large shift in connecting the
models of the hydraulically driven mass, ms and the foundation. In Fig. 3.17 the Bode plots
of this model structure with three different parameter values are given. In all cases (1f,2f,3f)
the model of the hydraulically driven mass is as specified in the previous paragraphs. The
parameters concerning the foundation are given in Table 3.4. f is the relative damping
given by 2 f cf = !fbf .
In both cases 1f and 2f, in which the mass of the foundation is relatively large (the
realistic cases), the nonrigid foundation results in a relatively small distortion of the open

3.3 Integrated system 99
Amplitude
Phase (deg)
10 2
10 1
10 0
10 −1
10 0
10 −2
100
0
−100
−200
10 0
−300
Case mf =ms !f f
1f 10 2 10 0:05
2f 10 2 4 0:05
3f 1 2 10 0:05
Table 3.4: Parameters taken for the foundation.
10 1
10 1
Bodeplot Phi_im−>dP_i
Frequency (Hz)
Fig. 3.17: Bode plots of input valve flow, im, to pressure difference over the valve, dP i,
with 1-d.o.f. hydraulically driven mass connected to a nonrigid foundation. Case
1f (full), 2f (dashed), 3f (dash-dotted).
loop characteristics of the original system. The additional resonance and anti-resonance are
clearly visible and also the, mostly oil spring stiffness-moveable mass related resonance is
seen to be shifted slightly. In the third case, where the mass of the foundation and moveable
mass are equal, the interaction between the modes is shown to be much higher. Although
this example is highly hypothetical in case of a foundation, these typical effects can occur
if one would place two platforms on top of each other like McInroy et al. [93]. In all
cases the transfer functions from im to dPi clearly remain strict positive real i.e. passive.
For the high and low frequencies, the response is not sensitive to the characteristics of the
foundation.
The second example of a possible parasitic resonating behaviour is schematically drawn
in Fig. 3.18. The construction to be moved itself, e.g. the simulator, is nonrigid. In practice
such a situation can occur if one connects a large projection screen in front of and heavy
projectors on top of the cockpit of a simulator. In this 1-d.o.f. case the parasitic behaviour
is modelled as two masses connected through a spring/ damper. Although this schematic
example will not catch the nonlinear behaviour of a flexible distributed mass with gross
spatial movement, some typical, linear parts of the characteristics can be investigated.
10 2
10 2
10 3
10 3

100 3 Hydraulically driven motion systems
Fig. 3.18: Schematic drawing of a hydraulic actuator moving a mass m s in 1-d.o.f., which
is connected to a parasitic mass/ spring/ damper system mp�bp�cp.
Case mp=ms !p p
1p 1 2 10= p 2 0:05= p 2
2p 1 2 10 p 2 0:05= p 2
3p 3 2 10 p 2 0:05= p 2
Table 3.5: Parameters taken for the parasitic mass.
The equations of motion for this system become
msxs = cp(xp ; xs) +bp(_xp ; _xs) +ApdPo (3.41)
mpxf = cp(xs ; xp) +bp(_xs ; _xp)
In this case the velocity measured at the actuator is equal to the inertial velocity of the
mass, ms. The transfer function from applied actuator force, f a = ApdPo, to the velocity,
_q = _xs changes to
_q =
mps 2 + bps + cp
mss(mps 2 + rmbps + rmcp) ApdPo� (3.42)
with the mass ratio, rm = (ms + mp)=ms, since the actuator force is not the only force
on ms as also the spring and damper force are taken into account. Again, the zero at zero
frequency is retained and an additional complex pair is introduced. This time predicted at
p cprm=ms. In principle, the structure of (3.39) and (3.42) is equal. As a result, one can not
discriminate between the case of a nonrigid foundation or a nonrigid structure to be moved,
given the measurement from valve input to a pressure difference (dP i or dPo) or actuator
velocity, _q. To determine what causes the parasitic resonance, additional measurements
have to be taken e.g. from accelerometers attached to the foundation or simulator. This
identification problem will be considered in the next chapter.
In Table 3.5 the parameters of three cases are given for the flexible structure of Fig. 3.18.
All the parameters of the hydraulic actuator are also taken equal to the ones presented in
the previous section (Table 3.2). The total moveable mass (m s + mp) is taken equal to

3.4 Hydraulically driven Stewart platform 101
Amplitude
Phase (deg)
10 2
10 1
10 0
10 −1
10 0
10 −2
100
0
−100
−200
10 0
−300
10 1
10 1
Bodeplot Phi_im−>dP_i
Frequency (Hz)
Fig. 3.19: Bode plots of input valve flow, i, to pressure difference over the valve, dP i, with
1-d.o.f. hydraulically driven structure consisting of two masses interconnected by
spring and damper. Case 1p (full), 2p (dashed), 3p (dash-dotted).
4000 kg in all cases. The first case (1p) results in exactly the same model parameters
as case 3f, considering (3.39) and (3.42). The transfer function from i to dPi for these
parameters, plotted in Fig. 3.19, is almost the same (the viscous friction of the actuator
results in very small differences). The other cases shown have somewhat more realistic
behaviour considering a resonant frequency twice as high (2f) and further taking 25% of
the part of the moving object nonrigidly connected (3f). In reducing this part, the zeros
and poles mostly related to the parasitic resonance shift back to lower values towards the
original eigenfrequency of the hydraulically driven load since there is less interaction.
The effect of interaction will be similar to the parasitic resonating spring/ damper/
masses treated in this section if one measures the flow to pressure/ velocity transfer function
of one actuator in case of a hydraulically driven Stewart platform where more than one
actuator is connected to the system. This structure will be treated in the next section.
The parasitic resonance shows a moderate peak in the frequency domain in the more
or less realistic setting of case 3p. In open loop the characteristics of the standard hydromechanical
mode will dominate. With feedback this will change and will change more
drastically if performance specifications require a bandwidth close to these parasitic effects.
This problem will be more closely looked at in Chapter 5 where the closed loop system is
validated.
3.4 Hydraulically driven Stewart platform
Having considered the hydraulic actuator and its most relevant properties to be taken into
account in modelling in a 1-d.o.f. setting and considering the description of the 6-d.o.f. me-
10 2
10 2
10 3
10 3

102 3 Hydraulically driven motion systems
dP o
R
b J T fa - A - - - b
l�x
Mt
?
-L 6
C
-B
6
R?
?
6
-A
_q _x
- b - b
Jl�x -
om
Fig. 3.20: Basic structure hydraulically driven motion system.
chanical system model of the Stewart platform introduced in Chapter 2, the two can be
combined. As interaction will be the most prominent additional effect to be taken into account,
the structure of this phenomenon will be analyzed first, using a simplified model. The
inertial effect of the actuators will be treated in the hydraulically driven setting. Finally, the
full model will be considered performing analysis in the frequency domain (through local
linearisation) for several platform poses using numerical design values of the SRS for the
model parameters.
3.4.1 Basic model structure hydraulically driven systems
As a resume, the basic structure of a hydraulically driven parallel motion system is given
in Fig. 3.20. Through the servo valves and transmission lines, oil flows to the actuator
compartments, om, can be regulated. Strong coupling with the mechanics results through
actuator velocities, _q, requiring a in/decreasing amount of oil in the compartments. Together
with a small leakage term, ;L, which provides for some damping, the nett flow will result
in in/decreasing pressure rises, dP
_
o, through the oil stiffness, C. Actuator pressure times
piston area, A, will force the system to accelerate. With parallel systems the actuator forces,
fa, have to be transformed to platform forces though the transposed jacobian, J T l�x . Some
additional damping force results from the viscous friction along the actuators, B. The acceleration
of the actuators can be calculated as the nett sum of forces times the inverse mass
matrix, M ;1
t , considered along the platform coordinates. Coulomb, coriolis and centripetal
forces are not shown in this system and neither are the gravity terms.
The mass matrix as seen from the actuators consists of the simulator pose, sx, dependent
jacobian, Jl�x(sx) as defined by (2.42), and an almost constant simulator mass matrix, M t,
6
M ;1
;1
act = Jl�xM t J T l�x : (3.43)
This matrix determines the coupling between the actuators. Given similar hydraulic actuators
in equal position, all other matrix operators can be considered as one operator times
the identity matrix. So, the eigenvectors of the linearised version of this system can be calculated
from the mass matrix. This still holds if decentralized feedback (diagonal feedback
structure) is applied. As the mass matrix is symmetric the eigenvector matrix can be made
?
-1

3.4 Hydraulically driven Stewart platform 103
unitary. This results from the singular value decomposition
M ;1 T
act = U U (3.44)
With the basic model for the hydraulically driven Stewart platform, it will be shown that the
matrix, U, can be used in a state transformation matrix, which decouples the 12 th order system
into 6 second order systems each having the same structure as the 1-d.o.f. hydraulically
driven mechanical system.
Considering for a moment dP o and _q as state1 , the state space equations for Fig. 3.20
become
dP
_
o
q
=
;CL
M
;CA
;1
actA ;M ;1
actB
The state transition matrix can be decomposed as
dP o
_q
+ C
0
om
(3.45)
;CL
M
;CA
;1
actA ;1
;M actB =
U
0
0
U
;CL
A
;CA
; B
U T
0
0
U T : (3.46)
This can be used to decouple the system using the assumption that all matrices involved,
apart from the mass matrix, have a scalar times identity structure, I. From this observation,
a decoupled system results if the system is described w.r.t. a new set of inputs ^ om
U
=
T ^
om and outputs ^ dP o = U T dP o, _ ^q = U T _q.
If lower captions are used for the diagonal elements of the respective operators, e.g. b
from B = bI, the six decoupled second order systems can be described by the state space
equations
"
_^
dP o�i
^q i
#
= ;cl ;ca
ia ; ib
dP ^
o�i
_^q i
+
c
0
^
om�i� (3.47)
with i =1�::: �6. So the system can be described with six second order systems in parallel.
Moreover, exciting the system with an oil flow, om, directed along one of the columns, ui,
of U will only result in response of the pressure, dP o, and velocity, _q along this very same
direction, ui. Therefore the following definition makes sense.
Definition 3.2 Each column of the matrix, U, is defined as the rigid body modal direction
of the hydraulically driven Stewart platform. The mode, eigenfrequency/ damping, of
each subsystem, (3.47), is defined as a rigid body mode of the hydraulically driven Stewart
platform.
The eigenvalues of the subsystems can be calculated from the respective determinants,
i = ; p ic( l p
c= i +
2
bp
i=c
2
ja)� (3.48)
1 The most appropriate choice for the state equations in modelling e.g. in numerical simulation, explicit model
based control, are of course the platform pose, sx and velocity, _x as state instead of q and its derivative and require
variational actuator stiffness. The model which results from that choice will be given with equation (3.66) in
Section 3.4.4

104 3 Hydraulically driven motion systems
where the product of l and b is assumed to be neglectable. The eigenfrequencies can be
considered the rigid modes and are mainly determined by the hydraulic stiffness, c, and
the singular values of the inverse mass matrix, i, taken w.r.t the actuators. These singular
values can be interpreted as the inverse of the generalised masses, which the actuators will
have to accelerate moving along the six orthogonal directions described by the columns of
the unitary decoupling matrix, U.
The characterisation with U can already be found by Hoffman [51], who uses this decoupling
to analyse a flight simulator motion system, a Stewart platform, in combination
with a decentralised feedback. It has not seen to be applied after that time. However, as
will be shown in the sequel, both in model analysis, identification and decoupling control it
can be very appropriate, since the decoupling is seen to hold in practice even in the face of
moderate additional variations/ dynamics.
The parallel structure makes the calculation of the directions in U simpler. It can be
calculated from a ’scaled’ jacobian, Jm without inversion by assuming M ;1
t does not vary
and can be calculated in advance
Jl�x(sx)M ;1
t J T l�x (sx) = Jm(sx)J T m (sx)
= U (sx) 1=2 (sx)V T (sx)V (sx) 1=2 (sx)U T (sx) (3.49)
As shown in the previous sections, the stiffness of the actuators, C, also varies with
the position of the actuator. With different stiffness values, the decoupling with U will
not be exact anymore. However, in many practical cases, as will be analyzed by taking
numerical design values in the next sections and experimentally found parameters in the next
chapter, a strongly decoupled system results calculating U and only from the platform
pose dependent Jacobian and a constant inverted platform mass matrix M t.
In the next section, first, the specific properties of the SRS motion system will be discussed.
Then, possible neglection of the variation of M t and appropriate choice of its value
will be treated.
3.4.2 Dynamical and kinematical properties of the SRS-motion system;
Design aspects
Having introduced the generic formulas to describe the mechanics and hydraulics of a (flight
simulator) motion system, additional insight can be gained by considering a specific design
example of such a system and analyzing the Simona Research Simulator characteristics.
The analysis presented in this section can be done before actually building a motion system
and can/ should therefore be taken into account in the process of design. The parameters for
the kinematics of the SRS were already given in Table 2.1.
The inertial properties of the SRS-motion system discussed here will be the values calculated
in the design of a fully operational flight simulator. In that case the centre of gravity
(cog) of the moving body excluding the actuators is predicted 1 m above the upper centre
gimbal point (cgb). The cog will be taken as the origin of the moving reference frame. The
vector from cog to cgb will thus be given by
x m T
cgb = 0 0 1
(3.50)

3.4 Hydraulically driven Stewart platform 105
It is not required to take the c.o.g. as the origin but usually specific design values, such as
the mass matrix are more easily interpreted. Further, an important point in simulation, the
design eye point (DEP) of the pilot is predicted very close to this point, only :2 m higher.
x m T
DEP = 0 0 ;0:207
(3.51)
As discussed, flight simulation usually requires a neutral position of the motion system. The
high pass wash-out filter characteristics should keep the motion system not to far away from
this point to prevent the actuators from running out of stroke. The most straight forward
choice of the neutral position is at half stroke of all the actuators. The moving reference
frame of the SRS is in that case 3:38 m above the lower centre gimbal point, which is taken
as the origin of the inertial grounded frame (or its nominal position in case of a flexibly
attached foundation).
x g
T
neutral = 0 0 ;3:38 (3.52)
Since the motion system will normally be close to this point 2 the (dynamic) characteristics
are often specified at this point. The jacobian in neutral position, sx n, is given by:
Jl�x(sxn) =
2
6
4
0:1943
;0:4668
0:2725
0:2725
;0:4668
;0:4269
;0:0452
0:3817
;0:3817
0:0452
;0:8832
;0:8832
;0:8832
;0:8832
;0:8832
0:3386
;1:2204
;1:5589
1:5589
1:2204
1:6047
;1:0955
;0:5091
;0:5091
;1:0955
;0:7011
0:7011
;0:7011
0:7011
;0:7011
7
5
0:1943 0:4269 ;0:8832 ;0:3386 1:6047 0:7011
(3.53)
Each column of this matrix specifies at what velocity the actuator should run in case only
one of the platform states is to be moved with unit velocity ( _ l = Jl�x _x). The columns,
in this specific pose, are orthogonal except for the pitch/ surge and roll/ sway direction.
This strongly influences the hydromechanical modes of the system evaluated along the platform
directions. The rows of the matrix specify the platform forces felt in case of a unit
force of one of the actuators. These rows are not orthogonal showing strong coupling of
neighbouring actuators and strongly influence the mass matrix evaluated along the actuator
coordinates.
Recalling the discussion on the jacobian in Section 2.2.9, it is possible to calculate what
can be the maximum velocity in a certain platform direction given a maximum actuator
velocity, given the jacobian. If the infinity norm (max) of the actuator velocity vector is
given, the 1-norm (added absolute values) of the rows of the inverse jacobian specify the
maximum attainable platform velocity.
k _xik1 = kJ ;1
i� k1k _ l1 (3.54)
Given a maximal applicable actuator force, a similar formula can be used to calculate the
maximal attainable platform force (take 1-norm of each row of the jacobian). In both cases
2Directing the gravitational vector properly often results in a certain off set from neutral as an airplane is usually
operated slightly pitched.
3

106 3 Hydraulically driven motion systems
Acceleration limits [m/s2,100 deg/s2]
10 2
10 1
10 0
10 −1
10 −1
10 −2
x,y z m_z m_x,m_y
10 0
Frequency [Hz]
Fig. 3.21: Approximate accelerational limits for sinusoidal platform motion considered at
the neutral position, along the platfrom directions, surge (x), sway (y), heave (z),
roll (mx), pitch (my) and yaw (mz).
it should be noted that the other platform variables are allowed to vary. A more appropriate
bound is therefore given by the maximal attainable velocity/ force in a specific direction not
moving/ forcing the others.
For the velocity this bound can be found by
And for the force
_xi�max = _ lmax=kJ �ik1
10 1
(3.55)
fxi�max = fa�max=kJ ;1
i� k1 (3.56)
The maximal actuator force and velocity are parameters, which are determined by the specific
design of the hydraulic system. The maximum valve flow, max, together with the
hydraulic actuator operational area, Ap, determine the maximum velocity
_qmax = max=Ap =SRS 1 m=s (3.57)
The maximal applicable actuator force can be found by multiplication of the supply pressure,
Ps, with this area, Ap,
fa�max = PsAp =SRS 40 kN (3.58)
The operational area is clearly a trade-off parameter in design for velocity vs. force given a
specific available amount of energy. There are, however, also other constraints in choosing
this area e.g. radial stiffness, and of course also the stroke of the actuator limits motion.
A first estimate of this last limit can be found by using (3.55) from which the jacobian to

3.4 Hydraulically driven Stewart platform 107
Pos. displacement x y z x y z
[m� deg] 1.34 1.10 0.74 22.3 20.9 45.2
Neg. displacement x y z x y z
[m� deg] -1.05 -1.10 -0.69 -22.3 -30.0 -45.2
Est. displacement (1 it.) x y z x y z
[m� deg] 1.34 1.46 0.71 23.1 22.5 52.9
Max. velocities _x _y _z !x !y !z
[m=s� deg=s] 2.14 2.34 1.13 36.8 35.7 81.7
Max. force fx fy fz mx my mz
[kN,kNm] 47.7 46.3 212.0 159.1 174.1 168.1
Max. acceleration x y z _!x _!y _!z
[m=s 2 �kdeg=s 2 ] 13.5 12.9 55.4 1.39 1.24 1.01
Freq. of pos. to vel. constr. x y z x y z
[Hz] 0.32 0.39 0.29 0.31 0.32 0.35
Freq. of vel. to acc. constr. x y z x y z
[Hz] 0.99 0.87 7.71 5.85 5.56 2.01
Table 3.6: SRS-properties considered at neutral position and c.o.g.
euler parameter velocities for the rotational variation can be calculated. Iteratively using this
formula (Newton-Raphson) and lmin�max, the least available stroke, a precise limit can
be found very fast as opposed to a bilinear search. This Newton-Raphson iteration usually
requires three or four steps, ending up far below measurable error in a well designed system.
With a given mass matrix, it is also possible to calculate the maximal attainable acceleration.
The specific mass matrix (design values) of the SRS is discussed in the next section. All
these values calculated for the SRS in the neutral position are specified in Table 3.6.
Given sinusoidal inputs, the velocity and positional constraints limit the maximal attainable
acceleration for lower frequencies. For the 1-d.o.f. hydraulic actuator, Viersma [153]
specifies exact performance limit diagrams in the velocity domain. In Fig. 3.21 approximate
performance diagrams for the 6-d.o.f. motion system are given in the platform coordinates
using the limits specified in Table 3.6. The precision of the approximate velocity and acceleration
constraints for the sinusoidal motion is favourably influenced since the maximum
velocity of the sinusoid is attained at neutral position where the velocity constraint is exact
and the maximal acceleration is attained (with limited positional offset at these high frequencies)
at zero velocity with no centripetal and coriolis accelerations. Gravity, however,
results in a more serious offset to be taken into account. Given the force which needs to
be applied to compensate gravity, the least available maximal force required to move in a
specific direction determines the maximal attainable acceleration.
The frequencies where the positional constraint, p max, and velocity constraint, vmax,
meet, fpv, and velocity and accelerational constraint meet, f av, are given by fpv = vmax
2 pmax
and fpv = amax
2 vmax .Afirst estimate of fpv is equal for all platform directions and does only
depend on the actuator specifications: ~ fpv = vact�max
2 pact�max =SRS
1
2 0:625
=0:25Hz. From

108 3 Hydraulically driven motion systems
−z [m]
3.5
3
2.5
2
1.5
1
0.5
0
Upper Gimbal
Lower Gimbal 1 Lower Gimbal 2 (6)
−1.5 −1 −0.5 0
−y [m]
0.5 1 1.5
Fig. 3.22: Envelope and largest circle within this envelope of a 2-d.o.f. parallel planar manipulator.
Parameters taken as for actuator 1 and 6 of the SRS except for merging
of the upper gimbals.
Fig. 3.21 it can be observed that the actual constraint will be attained at a frequency which
is somewhat higher. It is interesting to observe that already from :3 Hz and higher it is not
so much the finite stroke of the actuators which limits the (flight) simulation but the velocity
constraint. For these frequencies it is of no use to build a larger motion system. One should
first choose a larger valve.
Only looking at heave, roll and pitch, the operational area of the actuators could have
been chosen somewhat smaller as velocity is the main constraint for these directions. For
surge and sway, however, the force constraint comes in at much lower frequencies. In this
respect the jacobian (kinematical structure) of the Stewart platform is also a trade-off design
parameter where e.g. with the SRS neutral position high surge velocity can be attained
(twice as high as the maximal actuator velocity since the actuators are mainly rotating)
compromised with limited applicable force in this direction.
The specific values given in Table 3.6 should be treated with extreme care since they
can vary considerably over the motion envelope. Especially the motion envelope itself has
a complex non convex structure. Consider a simplified (planar) system with two actuators
having limited stroke manipulating a point. In Fig. 3.22 this structure is drawn with the
lower gimbals the same distance (2:5 m) apart as actuator 1 (2,4) and 6 (3,5) in the SRS
and the same constraints on the minimal and maximal length (2:08 m,3:33 m). The upper
gimbals have been merged (are :2 m apart in the SRS). Choosing the neutral point somewhat
lower in this construction, would lead to a considerable growth of the maximal attainable
sway moving straight away from this point. But the constraint towards the attainable heave
would become even more asymmetric in doing so. Thus changing the neutral point in this
way will probably not be a appropriate decision.
Appropriately describing the motion envelope of the general 6-d.o.f. Stewart platform

3.4 Hydraulically driven Stewart platform 109
in task space is still an open problem [27]. The inverse problem of checking whether a
specific task space envelope fits in a specific kinematical structure can be interesting for
(flight) simulation and can sometimes be solved. Consider e.g. a normed task space for the
2-d.o.f manipulator of Fig. 3.22. With the 2-norm, ( p x T x), this space describes a ball and
the maximal radius (norm) of the circle can be found in the half stroke neutral position and
is exactly equal to this half stroke (0:625 m).
With the general Stewart platform, exactly the same bound can be found if the platform
is not allowed to rotate since the upper gimbals in that case are also only allowed to move in
this same ball. So the maximal ball of translational motion of the SRS which is applicable
can be found in the all actuator half stroke neutral position and has radius r =0:625 m. For
any ball larger than this, translations can be found which run (some of) the actuators out of
stroke.
In case of orientational variations together with translation, a radius can be found which
satisfies the constraints but is not necessarily maximal. Recall the motion of any upper
gimbal
which can be bounded by
xa�i(sx) =c + T ( 13)ai
kxak2
(3.59)
p 2ksxk (3.60)
In case of the SRS kak = 1:6m when using the centre upper gimbal point as origin (and
gets larger/worse in using DEP or cog). So in case of the SRS r = :44m and this means a
maximal rotation of =5:7 . It would be interesting to know whether it is possible to use
more structure concerned with the interaction between rotation and translation in order to
arrive at possibly less conservative results.
In some cases it is favourable to define a (hyper) ellipsoid instead of a ball in which
the task space states are allowed to move. E.g. in case of Fig. 3.22 it would be appropriate
to choose a larger allowable variation in y-direction. Also this problem is in some cases
tractable in optimising a norm given quadratic constraints, which can be transformed to
linear matrix inequalities. The problem has to made convex, however. This can be attained
by replacing the constraints w.r.t. the minimal actuator lengths by convex approximations
in the task space e.g. a planes in the tangent space of the l min balls. The ellipsoidal form is
attained by multiplying xa by a (positive definite) scaling matrix S.
E.g. maximize =x T x such that
(Sx ; b) T (Sx ; b) 0 8kxk (3.62)
By use of the S-procedure [18], the ’uncertainty’ ball of x’s can be removed from the equations,
leaving a tractable problem.

110 3 Hydraulically driven motion systems
Parameter Description Value Unit
ma Mass upper part actuator (piston) 120 [kg]
mb Mass lower part actuator (cylinder) 135 [kg]
ra Distance upper gimbal to cog piston 0:7 [m]
rb Distance lower gimbal to cog cylinder 0:5 [m]
ia Inertia upper part actuator wrt cog 20 [kgm 2 ]
ib Inertia lower part actuator wrt lower gimbal 36 + mbr 2
b
Table 3.7: Inertial parameters SRS-actuators.
[kgm 2 ]
Limitations of a hydraulically driven motion system with specific kinematical and dynamical
properties have been quantified in this section. This knowledge can be used in
design of the system itself but also in the design of experiments in which it is often important
not to approach such limits to closely.
3.4.3 Actuator inertial effects
In model based control choice of an appropriate model is important. The model has to
describe the control relevant aspects of the system. On the other hand it should not be too
complex since it has to be calculated on a real-time control computer. W.r.t. modelling, the
question arises whether the mass properties of the actuators have to be taken into account.
And if so, how this should be done in a convenient way.
With the currently used simulator motion platforms the weight of the actuators is relatively
low (ca. 250 kg) w.r.t. the platform (up to 12:5 tons). In these cases the weights do
not contribute significantly to the motion of the total system. As the designed weight of the
simulator was reduced drastically (to ca. 3:2 tons) with the Simona research simulator, one
has to reevaluate this assumption.
Let’s look into the mass contribution of the system. The one body simulator has according
to the design (finite element modelling) roughly the following mass matrix, M c,
in platform coordinates w.r.t the c.o.g. which is calculated 1 meter above the centre upper
gimbal point.
Mc =
2
6
4
3200 0 0 0 0 0
0 3200 0 0 0 0
0 0 3200 0 0 0
0 0 0 7000 0 500
0 0 0 0 7000 0
0 0 0 500 0 8000
3
7
5
(3.63)
The inertia of the simulator is somewhat higher than the weight but this does not necessarily
mean that the impact is higher since the actuators are connected on a circle with radius
larger than 1m. The main axes of inertia do not exactly correspond to the platform body
axes. The y-axis does, as the system can be mirrored w.r.t the xz-plane. The projectors on
the upper back of the simulator are the main reason for the xz cross term.

3.4 Hydraulically driven Stewart platform 111
Case/Actuator 1 2 3 4 5 6
Neutral position lmid lmid lmid lmid lmid lmid
Extreme pitch/surge lmin lmin lmax lmax lmin lmin
Extreme twist (yaw) lmax lmin lmax lmin lmax lmin
Extreme roll/sway lmid lmin lmin lmax lmax lmid
Table 3.8: Actuator lengths for platform poses considered.
Case/Platform pose x y z 1 2 3
Neutral position 0 0 -3.38 0 0 0
Extreme pitch/surge 2.00 0 -2.43 0 -0.32 0
Extreme twist (yaw) 0 0 -3.02 0 0 -0.52
Extreme roll/sway 0.14 0.41 -3.25 0.20 -0.01 0
Table 3.9: Platform poses considered.
Recalling the additive mass matrix term, Ma�n, in (2.123), given by
Ma�n =
6X
i=1
J T ai�x (Mma�i + Mia�i�ib�i )Jai�x� (3.64)
which consists of the contribution of all six actuators. In neutral position it becomes the following,
using the relations determining (2.122), and the inertial parameters for the actuator
given in Table 3.7.
Ma�n =
2
6
4
500 0 0 0 ;525 0
0 500 0 525 0 0
0 0 675 0 0 0
0 525 0 1400 0 0
;525 0 0 0 1400 0
0 0 0 0 0 1325
3
7
5
(3.65)
The actuators contribute considerably to the total mass matrix (about 15 %). The terms in
the upper right and lower left block contribute to a lower cog of approx. .1 m. Remarkable
is the fact that the weight experienced in moving in z-direction will be higher than in x
or y-direction, due to the actuators. The actuator mass matrix will vary as a function of
platform position since the actuators move w.r.t. the platform. These variations, however,
are rather small. By taking a constant mass matrix in platform coordinates into the model
based controller, a relatively simple but accurate choice is made.
If a Stewart Platform with even less relative weight is used, the full model should be
taken into account. Apart from extra calculational effort this does not result in additional
problems if the modelling steps described, are taken.
With respect to the nonlinear quadratic velocity terms in the model describing the coriolis
and centripetal terms, the constraints on the velocities are important. The most restrictive

112 3 Hydraulically driven motion systems
10 2
10 0
−500
−1000
10 2
10 0
0
100
0
−100
10 0
10 2
10 0
10 2
Fig. 3.23: Bode plots from input valve voltage (u) to pressure difference (dP i) of linearised
multivariable hydraulically driven mechanical system models. Masses: Simona
design values. Rows 1,2 gives response of input 1 to outputs 1 (full), 2-6 (dotted).
Rows 3,4 give responses of the diagonal elements along the decoupled rigid body
modal directions (1,1:full), (2,2-6,6:dotted). Column 1: mid position, Column
2: extreme pitch/ surge, Column 3: extreme twist, Column 4: extreme roll/
sway. X-axis: frequency (Hz).
constraint, which holds for both pitch and roll of 35 =s =0:61rad=s in the neutral position,
was taken as the rotational velocity bound in calculating the influence of these nonlinear
terms in [119] and it was found that these only contribute up to ca. 500 N for the SRS.
3.4.4 Analyzing the SRS hydraulically driven motion system model
Having introduced the specific parameters of the SRS, the SRS-model can be analyzed considering
the interaction between the hydraulics and mechanics in this 6-d.o.f. system. It will
be evaluated whether the hydromechanical rigid body modes describe the main characteristics
of the system. Along the rigid body modal directions, it is expected that the system
locally behaves as six independent 1-d.o.f. hydraulic actuators moving a mass. This has to
be checked also as is the predictability of these directions. The influence of the main mechanical
nonlinearity, the platform pose dependent variation of the jacobian, J lx, has to be
quantified.
The state equations, constructed from Fig. 3.20, describing the nonlinear dynamical
10 0
10 2
10 0
10 2

3.4 Hydraulically driven Stewart platform 113
frb 1 2 3 4 5 6
1: (y� x) (x� - y) ( z) ( y� -x) ( x�y) (z)
[Hz] 4.5 4.5 7.6 13.3 13.5 14.8
2: (x� z� - y) (y� x� - z) ( z� - x) ( y� -x� z) ( x�y) (z)
[Hz] 4.1 5.8 9.1 15.5 16.1 17.4
3: ( z� -z ) (y� x� y) (x� - y� x) ( y�x) ( x� z� -y) (z� z� y)
[Hz] 3.6 4.5 4.5 16.1 16.3 17.2
4: (x� -y� - y) (y� x� x� - y) ( z) ( x�z�-y) ( y�x) (z� - x�x)
[Hz] 4.9 5.4 7.7 12.0 14.3 18.0
Table 3.10: Model eigenfrequencies, frb, in Hz of the rigid body modes and roughly the
rigid body modal directions. 1: Neutral position, 2: Extreme pitch/ surge, 3:
Extreme twist (yaw), 4: Extreme roll/ sway. i is the rotation around the i th
axis.
model from the output oil flows, om, are given by
x
_
dP o
;bI aI
;aI ;lI
=
;1
M
t J T lx (sx) 0
Jlx(sx) 0
0 I
:::+ M ;1
t
0 Ch(sx)
_x
dP o
+ 0
I
G(sx) +C(sx� _x)
0
::: (3.66)
om + :::
In this case it is assumed that there are similar actuators (operational area, a, viscous friction,
b and leakage, l) easily replaced by the respective matrices, A� B� L otherwise. By taking
the platform pose, sx� as state variable, the model is valid for singular poses as well. Only
if actuators would reach zero length, the jacobian is not defined.
The two-sided coupling with all ( om,dPo) pairs to the transmission lines and valves can
readily be made. For a reduced but fairly good model, each transmission line has four states
and each valve has two. Further, the six platform pose coordinates, sx, have to be calculated
through integration of the velocities. The model thus consists of 12 + 6(2 + 4) + 6 = 54
states.
This model with the SRS-parameters has been analyzed for several poses. Together
with the neutral position, three extreme poses were taken into account. Actuator lengths
are given in Table 3.8. The extreme pitch/ surge and twist only consist of extreme actuator
positions. As the oil stiffness varies parabolically with the position, the extreme roll/ sway
pose is important since it contains actuators with maximal (extreme position) and minimal
(mid position) stiffness.
The platform poses, which are given in Table 3.9, do result from these actuator lengths.
The euler parameters can be translated to angles by taking 12 per :1 up to :6 with 1

114 3 Hydraulically driven motion systems
Case/mode 1 2 3 4 5 6
Neutral position 8.8 8.8 3.0 1.0 1.0 0.8
Extreme pitch/surge 14.8 7.4 3.0 1.0 0.9 0.7
Extreme twist (yaw) 19.0 12.2 12.2 0.9 0.9 0.8
Extreme roll/sway 9.0 7.8 3.6 1.8 1.0 0.6
Table 3.11: Generalised masses (tons). The large differences in (generalised) mass easily
result in interaction. E.g. consider a force along the 8.8 ton direction in neutral
position, which is only 1 misaligned into the 0.8 ton direction. This will result
in 20% interaction. In more extreme positions e.g. pitch/ surge this will even
amount to more than 40%!
accuracy. In the first two rows of Fig. 3.23 the Bode plots of the linearised models in these
positions are given from the first valve input voltage to the various pressures, dP i, taken
over the valves. The hydromechanical modes are visible and are highly interacting in the
frequency area between 1 Hz and 30 Hz. The phase remains within 90 for the (1� 1)
transfer function up till the point where the valve itself starts to be relevant (> 70 Hz).
Above 30 Hz the measurable interaction strongly fades as the nondiagonal terms lack zeros
in the transfer functions, which can be interpreted as springs in between input/ cause and
output/ point of measurement.
In the last two rows the transfer functions have been considered from the other inputs and
outputs (only diagonal terms) resulting from the approximate decoupling matrix U defined
in (3.44). This matrix was calculated from the approximate mass matrix (not taking into
account the variation due to the specific inertial properties of the actuators) and does not
take into account the varying actuator stiffness. Still the modes in the system can clearly
be distinguished. In the hydromechanical frequency interaction area, only one mode per
transfer function results in a single shift in phase of 180 . All the nondiagonal transfer
functions in these coordinates are small over the full frequency area considered.
Considering stability from a control point of view, the connection of a hydraulic circuit
still enables a simple passivity preserving dissipating feedback from pressure/ force supplied
to the mechanical system to the oil flow through the valve instead of velocity feedback to
force in a mechanical system.
The eigenfrequencies related to the hydromechanical rigid body modes of the system are
given in Table 3.10. There is considerable variation for the different rigid body modes both
for a specific pose as for the different poses considered. As each extreme position considered
was only given a specific variation of the platform pose w.r.t the neutral position, the rigid
body modal directions in platform coordinates, given by J ;1
lx U, can still be interpreted.
E.g. in extreme pitch/ surge, symmetric involving x� z� y and asymmetric modes involving
y� x� z. If all platform coordinates are varied this structure will be lost.
Looking at the Bode plots in Fig. 3.23, it can be observed that building up a unit pressure
in a specific direction will take very different input values (more than a factor 10 variation).
Very slight misalignment (wrong input) will easily result in large coupling. This is roughly
linear with the generalised masses, which affect the eigenfrequencies through a square root.

3.4 Hydraulically driven Stewart platform 115
Case/actuator 1 2 3 4 5 6
Neutral position 18 18 18 18 18 18
Retraction gain -1.08 -1.08 -1.08 -1.08 -1.08 -1.08
Extraction gain +0.90 +0.90 +0.90 +0.90 +0.90 +0.90
Extreme pitch/surge 26 67 -42 -42 67 26
Retraction gain -1.12 -0.57 -1.19 -1.19 -0.57 -1.12
Extraction gain +0.86 +1.29 +0.76 +0.76 +1.29 +0.86
Extreme twist (yaw) -24 48 -24 -24 48 -24
Retraction gain -0.87 -1.21 -0.87 -1.21 -0.87 -1.21
Extraction gain +1.11 +0.72 +1.11 +0.72 +1.11 +0.72
Extreme roll/sway 21 32 25 6 11 18
Retraction gain -1.10 -1.15 -1.12 -1.03 -1.05 -1.09
Extraction gain +0.89 +0.83 +0.86 +0.97 +0.95 +0.90
Table 3.12: Static load as percentage of maximal load F max = PsA and the resulting normalized
valve gain discontinuity.
The generalised masses, 1= i from (3.44), are given in Table 3.11.
Already in the neutral position, the SRS is expected to have very different characteristics
along the rigid body modal directions. In the sway/ roll and surge/ neg. pitch directions the
system will experience approx. 9 tons where the sway/ neg. roll and surge/ pitch directions
only actuator forces to accelerate 1 ton of load have to be applied. In the extreme positions
the largest mass ratio (mass matrix conditioning) along the different modes grows from nine
to more than twentyfour in extreme twist. Although the jacobian, J lx, has a main influence
on this variation, its conditioning is not a very well predictor for this factor. It grows from
4:5 to 5:2 going from the neutral position to extreme twist.
With the SRS design parameters, gravity requires approx. 18 % of supply pressure in
the neutral position. The platform contributes 15 % and the moving parts of the actuators
3 %. The contribution of the lower parts is almost neglectable. This percentages can rise
extensively in other platform poses. In the extreme surge/ pitch position considered, actuator
2 and 5 will have to carry two third of the theoretically maximal weight. In practice, two
third is relatively high to take care of the static forces only. This means that the SRS design
weight and height of the c.o.g. are in fact maximal for this motion system to operate safely.
Another important aspect of the static loads on the actuators is the nonlinear characteristic
of the valve as given in (3.16). A preload results in a discontinuous valve voltage to flow
gain as im = u p (1 ; sgn(u)dPi) for normalized variables. In Table 3.12 the required
preload for the four platform poses considered is given with the resulting valve gains for
actuator extension and retraction. Negative percentages for the load mean that retraction
preload has to be applied to statically balance the platform. The all actuator minimal length
position is not the global minimum for the potential gravitational energy. As a result, the
system will not always go to this pose if for some reason the preload disappears e.g. pump
supply pressure drop to tank pressure in shut down.
All together, the hydraulically driven Stewart platform model can be transformed to six

116 3 Hydraulically driven motion systems
Direction Mfii [tons� tons m 2 ] !fi [Hz] fi
xf 45 2 7 0:05
yf 45 2 7:5 0:05
!f 80 2 6 0:10
Table 3.13: Parameters taken for the foundation.
independent 1-d.o.f. hydraulically driven dynamic and static masses for each pose using
the variable unitary decoupling matrix, U (sx), describing the rigid body modal directions.
Analysis of the SRS motion system design showed considerable differences in the rigid
body eigenfrequencies among these 1 d.o.f.-systems for the SRS.
3.4.5 Connection of a flexible foundation to the SRS system model
The hydraulically driven Stewart platform model of (3.66) can be extended with a model
including parasitic dynamics such as a flexible foundation. This model is used to predict
in what way the dynamics of the system is influenced by such distortion. A model of a
flexible foundation was introduced in Section 2.4.7 described by (2.126). Combination of
these models leads to the following state equation
22
6
6
4
6
4
2
6
4
_xf
xf
x
_
dP o
3
7
5 =
2
6
4
I 0 0 0
0 M ;1
f 0 0
0 0 M ;1
t 0
0 0 0 Ch
0 I 0 0
Cf ;Bf ; J T f BJf ;J T f BJlx J T f A
0 ;J T lx BJf ;J T lx BJlx J T lx A
0 ;AJf ;AJlx ;L
32
7
6
5
6
4
3
7
5
xf
_xf
_x
dP o
::: (3.67)
3
7
5 +
2
6
4
0
0
0
I
3
7
5 om
The foundation is allowed to move in the horizontal plane and introduces three additional
modes, foundation surge, xf ,sway,yf , and yaw, !f , described by six states (xf ,
_xf ). As an example, the parameters given in Table 3.13, are filled into the equations. These
describe roughly the characteristics of the foundation on which the system was tested in the
central workshop. The mass matrix of the foundation has numbers on the diagonal, which
are ten times higher than those of the platform. The stiffness matrix and damping matrix
are chosen such that the undamped eigenfrequencies and damping factors specified for the
foundation alone result. The high mass of the foundation in relation to the platform lets
these frequencies only slightly altered in connecting the motion system.
In Fig. 3.24 of Appendix A, the system is evaluated in the frequency domain from 1 Hz
to 50 Hz. The upper left bode amplitude plots shows the model actuator pressure difference
responses to the oil flow into the first actuator (not all 36 responses as the other 30 are
3
7
5

3.4 Hydraulically driven Stewart platform 117
10 1
10 0
10 −1
10 1
10 0
10 −1
10 0
10 0
10 1
10 1
Frequency (Hz)
10 1
10 0
10 −1
10 1
10 0
10 −1
10 0
10 0
10 1
10 1
Frequency (Hz)
Fig. 3.24: Bode amplitude plots of linearised multivariable hydraulically driven mechanical
system models without transmission lines but with flexible foundation. Upper
left: neutral position from first input actuator flow om1 to all output pressures
(dP o). Upper right: neutral position along the (original) rigid body modal direc-
tions from all ^ om to ^
dP o, nondiagonal terms dashdotted. Lower left: neutral
position along the (original) rigid body modal directions from all ^ om to the
accelerations (m=s 2 )(xf � yf� _!f ) of the foundation, nondiagonal terms dashed.
Lower right: extreme pitch/ surge along the (original) rigid body modal directions
from all ^ om to ^ dP o, nondiagonal terms dashdotted.
similar) in the neutral position. The complex dynamics of the hydromechanical modes
prevents one from recognising the parasitic dynamics. Using the decoupling transformation
U calculated from the platform mass matrix the bode amplitude plots in the upper right part
of the figure results. In the diagonal responses the fact that there are parasitic modes is
directly visible. Three additional modes show up separately along the eigenvectors of the
lowest three original platform modes. Further, the nondiagonal terms are not zero anymore
though still quite small.
As discussed in Section 3.3.3, both a flexible foundation as a mass/ spring/ dampersystem
connected to the platform result in similar characteristics evaluated at (combinations
of) actuator coordinates. To evaluate whether the foundation is causing parasitic dynamics,
one can measure accelerations at the foundation itself. The model predicts the acceleration
frequency response at the neutral position given in the lower left part of Fig. 3.24 of Appendix
A. This was calculated by multiplying the velocity frequency response of the model
by j!. Remarkable is the fact that the peaks in the response, which are most prominent, are

118 3 Hydraulically driven motion systems
not in all cases mainly the original modes of the foundation. Input along the higher platform
eigenfrequencies are expected to result in relatively high (approx. 15 Hz) frequency vibrations
and of the three main transfer functions the yaw platform direction to yaw foundation
modal response peaks are maximal at approx. the original platform yaw mode.
The fact that the parasitic modes influence only three platform modes is not generally
true. In the lower right part of Fig. 3.24, the system is evaluated again along the original
platform eigendirections. But now the platform has been put into the extreme surge/ pitch
mode. The fact that the platforms c.o.g. is now approx. 2 m in front of the c.o.g. of the
foundation makes the sway mode of the platform interact with both the foundation yaw
and sway mode and through the foundation yaw mode with the platform yaw mode. The
response along this direction shows four resonances. Also the decoupling is almost completely
lost between 4 Hz and 15 Hz.
So, parasitic dynamics such as a flexible foundation can have an important influence
on the motion system characteristics. This is not always directly visible in the measurable
responses (inputs and outputs of the actuators).
3.5 Chapter Resume
The motion systems rigid bodies interacting with the hydraulic actuator stiffnesses, the
hydromechanical rigid body modes, describe the main characteristics of the hydraulically
driven motion system dynamics. The eigendirections of these modes can be quite accurately
predicted by taking the eigenvectors of the approximate actuator mass matrix consisting of
the variable jacobian, Jlx(sx), and the platform mass matrix. Along the rigid body modal
directions the system locally behaves as six independent 1-d.o.f. hydraulic actuators moving
a mass in every platform pose.
Under mild conditions, these systems are passive evaluating pairs of the oil flow and
pressure difference at the valve or at the actuator of the hydraulic servos or (collocated)
combinations of these pairs, no matter what parasitic flexible mechanical systems are to be
included. Any passive feedback over these pairs will result in stability of the full nonlinear
system.
The valve dynamics causes loss of passivity mostly relevant in combination with the
high frequency (e.g. transmission line) modes. The transmission lines, though mainly resulting
in high frequency dynamics, cause a considerable shift of the hydromechanical rigid
body modes and introduce large differences at relative low frequencies (10 times smaller
than the transmission line modes) between the measurable pressure at the valve and the
equivalent force (pressure difference times operational area) the actuator induces into the
motion system.
Parasatic dynamics such as a flexible foundation or an additional mass/ spring/ damper
system connected to the moving platform is easily included in the models introduced. This
dynamics can have considerable influence on the systems characteristics and is most easily
recognized evaluating along the hydromechanical rigid body modes. Only measuring at the
actuators, the cause of the parasitic dynamics is not identifiable.

Chapter 4
Parameter identification and
model validation
The models presented in the two previous sections were derived from the laws of physics.
Where numerical examples were given, the parameter values were taken from design values
or rough approximations of what could be expected in practice. These models can be
valuable for the structural analysis of (parallel, hydraulically driven) motion systems. The
strengths of these models should, however, be proven by experimental validation. Proper
choice of model parameter values is required in using the models to predict the characteristics
of the actual system and enable direct comparison of model and experimental response.
Usually, the parameter choice is also to be based on experiments. Experiments from
which the parameters can be identified. Small differences between model and practice,
i.e. a well validated model structure with proper parameter values is also important in the
use of a model based motion controller.
Identification can also be performed with more general model structures which are especially
suited for parameter identification. In nonlinear systems it is very hard to find
convenient model structures. The physically motivated structure of the nonlinear models
presented in this thesis, however, enables one to investigate what influence changes in the
design of the system will have on its dynamics and helps in pointing at the relevant parts
of the statics and dynamics in a complex system as the flight simulation motion systems at
hand. This also helps in the design of proper experiments.
As already discussed, the motion system (model) consists of kinematics and dynamics
related to the mechanics i.e. the Stewart platform, and the hydraulic actuators which drive
the system. Further some additional parasitic dynamics can be expected from a nonrigid
foundation and a flexible platform/simulator. The identification of the dynamics specifically
related to the hydraulic actuators has already been discussed in detail by Van Schothorst
[124] and will not form an explicit part of this thesis though the hydraulic model structure
presented does. Of course the dynamics resulting from the interaction between the
hydraulics and mechanics will be, since the models presented in the previous chapter appointed
this effect as the most relevant.
This chapter will have to provide a good starting point for model based control on the
actual system. The gap between the theoretically derived model structures in the previous
119

120 4 Parameter identification and model validation
chapter and the actual motion system to be controlled will be bridged in three steps.
Parameter estimation. The calibration method for the Stewart platform kinematical
model will be presented, implemented and evaluated in Section 4.1. Next, in Section
4.2, the experiments to infer the dynamic model parameters for the mechanics
are set up, implemented and evaluated. Dynamic experiments to infer the position
and orientation of measurement equipment consisting of accelerometers and rate gyros
will be treated in Section 4.3.2.
Model validation. The model characteristics with the parameters estimated will be
validated in the frequency domain in Section 4.3. Especially the conclusion in the
previous chapter that the hydromechanic rigid body modes are most relevant and that
the system can be considered locally as six independent 1-d.o.f.-hydromechanic systems
along the rigid body modal directions, will have to be evaluated experimentally.
If this conclusion is justified in practise, the models, which describe this dynamics,
are expected to provide a good starting point for model based control.
Quantification of parasitic dynamics. The cause of the main parasitic dynamics
observed in Section 4.3 will be clarified in the final part of this chapter, Section 4.3.2,
by additional measurements showing vibration of the foundation and non rigid shuttle.
Together with the hydraulically generated parasitic dynamics stemming from the
valve and transmission lines, these parasitic dynamic effects are to be considered the
limitations to the control scheme based on the basic model.
All experiments discussed in this chapter are open loop based i.e. no feedback is involved
apart from a very low bandwidth controller, which is to stabilise the system without
relevant influence on the information inferred from measurements for calibration/identification
(higher frequencies or static relations).
4.1 Calibration
In this section the parameters of the kinematical model of the simulator motion system will
be identified (calibrated) and the kinematical model itself will be validated. In robot calibration
[15], parameter identification of both the kinematics and dynamics of the mechanical
system is referred to if calibration is considered. In this thesis, calibration will assume to be
concerned with the identification of the kinematical model parameters or of specific measurement
equipment. Parameter identification will generally refer to the model parameters.
It is most convenient to perform calibration first as this step can be made independent of the
system dynamics.
Calibration is a very important topic in robotics. Especially if positional accuracy is
relevant since this can often be improved by an order of a magnitude [15]. Further, one
can expect that most experimental effort will have to be put into calibration as related to
the time spend in testing the motion control strategies in practice [52]. In flight simulator
motion systems positional accuracy is clearly not the main issue but indirectly it affects the
accelerational accuracy and is therefore considered in this thesis.

4.1 Calibration 121
The basics of calibration can be found in textbooks such as [101] and books containing
selections of articles published such as [15]. The framework adapted in this thesis is based
on an article of Hollerbach and Wampler [52]. In this article, a large set of robot calibration
methods is unified considering joints and selecting closed kinematic loops. Further, a number
of numerical issues is discussed, which provide a theoretical basis on the identifiability
problems of kinematical parameters in practice. This method easily integrates the parallel
driven structures but does not explicitly do so by presenting equations which are explicit
in the end effector coordinates. The approach of Zhuang in [161] generalizes the idea of
making use of residuals in redundant equations for parallel structures in identifying the unknown
parameters i.e. calibrating such systems and these ideas will also be incorporated in
the calibration procedure.
4.1.1 Calibrating the Stewart platform
In this research no specific new theory will presented considering the calibration of a Stewart
platform. The contribution will be a fully worked out applicable method consisting of a
combination of known only slightly modified steps, which, not often seen in literature, is
brought into practice and is put into a general framework. A large part of the procedure
followed has been worked out as a part of two master projects [16] [151] in calibrating the
SRS motion system with the dummy platform of Fig. 1.5. Here the results for the practical
part of the calibration will involve the calibration of the motion system with the SRS-shuttle
of Fig. 1.4.
The calibration procedure proposed consists of only two basic steps.
A redundant measurement procedure for platform pose reconstruction.
Identification of the kinematical parameters of each platform leg separately using the
redundancy of the actuator length measurement.
Only a small part of the literature on calibration is devoted to the parallel manipulators.
These systems differ from most serial systems considering the number of kinematical
parameters involved, which is high. For the Stewart platform, a kinematical model,
schematically given by Fig. 2.7, was presented by (2.38) from which
klmi + loik 2 =(T ai +c ; bi) T (T ai +c ; bi) i =1�::: �6 (4.1)
follows, which involves 42 unknown kinematical parameters. Seven for each equation consisting
of an unknown offset length, loi, which together with the measured length, lmi represents
the total length of the i’th leg. The unknown upper and lower gimbal vectors, a i
and bi, each consist of three parameters. These parameters will be identified in this section
after redundantly measuring the platform pose consisting of the rotation matrix, T and the
translation, c.
This kinematical model assumes that all the gimbals are perfect ball and u-joints as
schematically given by Fig. 2.8 and the prismatic joint axis of the sliding actuator exactly
intersects the two gimbal points. In [154, 155] a more general kinematical structure is
considered, allowing each axis of rotation in one gimbal to cross the other at a specific

122 4 Parameter identification and model validation
distance not perfectly orthogonal. This model involves 132 unknown kinematic parameters,
which would heavily complicate the calibration problem. Validation of the simple model
after calibration can point out whether this is necessary.
Reducing the number of unknown parameters to be identified in one step usually helps
the conditioning of the problem to be solved and therefore improves the accuracy of the
results. In [162] by Zhuang and Roth, the separate calibration of each leg of the Stewart
platform is made possible by assuming a redundant measurement of the platform pose. This
will leave one redundant actuator length variation measurement per actuator from which
the seven unknown kinematical parameters in a i, bi and loi will have to be identified. In
[162] also the calibration of the offset length, l oi, is decoupled from the others, but as this
strongly reduces the number of measurements, which can be taken into account, this part of
the procedure has not been adopted.
The solution method to identify the kinematical parameters is iterative with only local
convergence, like the forward kinematical problem discussed in Section 2.3. In [59, 60] a
method is presented to calculate every solution to the calibration of a i and bi nonredundantly
taking only six measurements into account and using a seventh measurement in deciding
which solution is the right one. As it involves rootfinding of a twentieths order polynomial
equation, this method is usually numerically not suitable for full (noisy) calibration in
practice but it can be used for validation of the calibration procedure used here.
The mobility of a system, roughly its degrees of freedom, can be specified with the
mobility index, M as done by Hollerbach and Wampler [52]. It can be calculated given the
number of links, L, including the base link and the number of constraints, D i, for each joint.
Putting the method of Zhuang and Roth [162] for the Stewart platform actuator of Fig. 2.8,
in the framework given in [52], the mobility index, M, of a separate leg of the platform is
M =6(L ; 1) ; J i=1 Di = 3(4 ; 1) ; (4+5+6)=6 (4.2)
The four links consist of base, lower and upper actuator part and the platform. There is
a u-, prismatic and a ball joint. The calibration index, C, a redundancy measure given by
Hollerbach and Wampler [52], can also be calculated as the total number of sensors, S, is
seven, assuming an independent pose measurement, with
C = S ; M =1: (4.3)
This means there is only one redundant measurement involved from which all seven kinematical
parameters have to be constructed. As in system identification of a dynamic miso
system with several parameters, this is only possible if more than one, here at least seven,
measurements are taken and if the inputs, the platform poses chosen, can be made sufficiently
rich to make the parameters identifiable.
As opposed to the discussion of serial manipulator examples given in [52] for systems
with C =1, the number of measurements in the joints in a parallel system is one instead
of the number of measurements for the end-effector. An advantage of C = 1 is that all
equations involved and the residuals resulting, are of the same type. This is favourable in
conditioning.
In applying a calibration method in practice, the desired precision of the result should
be known since as a rule of thumb the measurement equipment should be an order of a

4.1 Calibration 123
−z
−z
2
1
0
−1
−2
3
2
1
0
View in perspective
0
x
Side view
2 −2
0
−y
−1
−2 −1 0
x
1 2
2
−z
−y
3
2
1
0
Front view
−1
−2 −1 0
−y
1 2
2
1
0
−1
Top view
−2
−2 −1 0
x
1 2
Fig. 4.1: Two calibration measurement frames for the case of the Stewart platform. The
fixed frame consists of three points. With the SRS they were chosen somewhere
between the lower gimbals. The moving frame consists of three points chosen
somewhere on the upper gimbal blocks. Knowledge on the exact positioning of
the measurement frame points is not necessary for this method.
magnitude more precise [15]. For the measurement of the relative position of the prismatic
joint a linear position transducer of the Temposonic type is used. The accuracy of these
sensors is 0:1 mm (resolution with [55] 0:009 mm). A calibrated motion system with
an positional accuracy of 1 mm will be strived for. The independent measurement of the
platform pose discussed next will have to be done with equipment of :0:1 mm accuracy
also.
4.1.2 Redundant measurement of the platform pose
The redundant measurement of the platform pose is based on the method presented by Geng
and Haynes [43]. One defines two measurement frames e.g. as in Fig. 4.1. One fixed on
the ground and one on the moving platform/simulator using three points for each frame.
The motion of a point with respect to the fixed frame can be determined by measuring the
distance of this point to the three points fixed to the ground if the distance between the
fixed points is known and if the moving point remains in a known area w.r.t. plane spanned
by the three points. With three additional length measurements (two to the second moving
point and one to the third) the orientation can also be determined if the distance between the
moving points is known.
In case of the calibration of the shuttle of the SRS the distance between the moving
points fixed to the shuttle are not precisely known and can not easily be measured directly
since the shuttle shape does not allow so. This problem is solved by taking three distance
measurements for each moving point (nine in total) from which the relative position of the
moving points can be calculated. The relative distance of these points is redundant if more

124 4 Parameter identification and model validation
Fig. 4.2: Determination of the position of a moving point in space using three length measurements.
In case of the SRS, the three fixed points, Gi, can be thought in between
the lower gimbal blocks. Each moving point, P i, was taken as part of the
upper gimbal blocks.
than one pose measurement is taken. This redundancy can and will be used to decide on
a confidence level for the measurements in the weighted least squares calibration of the
kinematical parameters. As only one distance to a specific point can be measured at the
time this redundancy does not take significant additional effort to be measured since the
platform has to be moved to a specific pose three times at least (using three pairs of three
measurements instead of three, two and one measurement respectively).
In Fig. 4.2 a top view of a three point distance measurement with three lengths, l i,
i = 1�::: �3 is given to determine the position of a moving point P i with respect to the
frame defined by the three points Gi, i =1�::: �3. Point, G1, is taken as the origin and the
line through G1 and G2 as the y-axis. Using the cosine formula:
l 2
2 = l2
1 + b2
2 ; 2l1b2cos( )=l 2
1 + b2
2 ; 2b2y1� (4.4)
y1 can be calculated. By letting the third fixed point, G 3, define the fixed measurement
xy-plane, similar application of the cosine rule will lead to a parameter, s. This parameter,
s, is the distance and direction of the origin to the point S(s x�sy� 0) in Fig. 4.2. So also the
following holds
s
p 2 a3 + b2
3
= sx
=
a3
sy
b3
(4.5)
Further, the line in the xy-plane through S, orthogonal to the line through G 1 and G3, runs
through (x1�y1� 0), i.e. direction @y=@x = ;a3=b3 can also be given by
y1 ; sy
= ;
x1 ; sx
a3
b3
Combining these equations to get rid of sx and sy leads to
q
2
b3y1 + a3x1 = s a
3
: (4.6)
+ b2
3
(4.7)

4.1 Calibration 125
So in applying the cosine rule, s can be replaced by x 1 and y1, from which x1 can be solved
l 2
3 = l2
1 + a2
3 + b2
3 ; 2l1
=
q
2
cos( ) a3 + b2
3
l 2
q
2
1 + a2
3 + b2
3 ; 2s a3 + b2 3
Finally j z1 j can be determined from
= l 2
1 + a2
3 + b2
3 ; 2(b3y1 + a3x1) (4.8)
l 2
1 = x21
+ y2 1 + z2 1
(4.9)
The sign of z1, above or below the plane of measurement, should be known in advance. In
this way the three vectors from the fixed frame to the moving points, p i, can be calculated.
The frame defined through Gi given in Fig. 4.2 is convenient in determining the vectors to
the moving reference points pi. In identifying the kinematical parameters it will be more
suitable to define a slightly different fixed and moving frame from G i and Pi respectively,
which are almost equal to the ground frame and moving frame defined for the SRS (Fig. 2.7).
The new fixed reference measurement frame origin, G m, is given by a translation
oGm =(g1 +g2 +g3)=3 (4.10)
with respect to the frame given in Fig. 4.2. The orientation remains unchanged. The moving
reference measurement frame origin, Pm, is specified w.r.t Gm accordingly with an origin
(translation cpg)
oPm =(p1 +p2 +p3)=3 ; oGm =cpg
(4.11)
and the orientation, with the line through P 1 and P2 chosen parallel to the y-axis, consists
of three unit vectors along the new axes. With pij = pj ; pi, npy = p12= p p T 12 p12,
npz = (npy p13)= p p T 13 p13 and npx = npy npz. These together form the rotation
matrix from Pm to Gm
TGm�Pm =[npx npy npz ]: (4.12)
And this is the pose measurement we were looking for. In short it is given by (c pg, Tpg),
which basically can be interpreted along the same lines as the pose defined for the Stewart
platform.
All together the used platform pose measurement requires nine absolute length measurements
and the knowledge of the absolute distance between the fixed reference points.
The quality of the calibration procedure heavily depends on these measurements. The positioning
of the reference points themself is not that important. A rough idea will help in
selecting a proper initial estimate in the calibration and the conditioning of the optimization
problem to be solved is improved if all the distances to be measured (e.g. in Fig. 4.2) are
about the same. Further the platform reference points have to reside at a predefined side of
the plane defined by the fixed reference points.

126 4 Parameter identification and model validation
Fig. 4.3: In practise, the measurement device could not be used to take direct measurement
of the length between two points at an angle ( 6= 0). A funnel (a cone shaped
utensil) had to be placed in between. The picture shows that the measurement
cable will not run directly from Gi to Pi but will follow the rounded edges of the
funnel from Gi to D. This requires funnel length measurement compensation.
4.1.3 Stewart platform pose measurement in practice
The length measurements to reconstruct an independent, redundant platform pose in case of
the SRS-motion system have been performed with relatively low cost string encoders made
by AMS Gmbh. A measurement cable winds onto an accurately machined cable drum. A
coil spring delivers a constant pull in force to maintain cable tension. An incremental rotary
encoder on the drum gives a digital output. The resolution of these encoders is 0.03 mm and
the calibrated relative precision 0.1 mm. The standard version of this equipment does not
allow the cable to be used at different angles w.r.t. the output point. This would damage the
device.
To allow calibration for an absolute length measurement, also for measurements taken
at an angle, a funnel was designed. As shown in Fig. 4.3, this slightly affects the length
measurement between Gi, which is chosen at the bottom of the funnel and P j. In practice
this difference can take values up till 4 mm (R=15 mm) and should be corrected.
Considering Fig. 4.3, lpg, the length between Pj and Gi is equal to
p
lpg = (X0 ) 2 + Z2 (4.13)
the measured length will be equal to
d1 + R
p
= (X0 ) 2 + Z2 ; 2RX0 + R (4.14)
By using the measured lengths to reconstruct a first estimate of p i the required correction

4.1 Calibration 127
can be approximated by taking
0 = tan X 0
Z
(4.15)
or more precise (in this case not necessary since R=lpg was small, only used for evaluation)
reconstructing from
Z tan( )=X 0 + R
1 ; cos( )
cos( )
(4.16)
It is assumed that the direction (Z) of the funnels (in the frame G m) is known.
The choice for the approximate positions of the reference measurement points is given
in Fig. 4.1. Considering Fig. 2.10, the fixed reference points (the encoders) have been
placed in between the lower gimbal pairs (4,5), (2,3), (6,1) respectively. This allows the
funnels to point along the z-axis of the measurement (and ground) frame. The upper, moving
reference points have been taken in between (slightly lower: 80 mm) the same pairs of
upper gimbal points. In most platform poses, the lengths l pg will therefor not be the same
(slightly less favourable conditioning) but in practice prevention from actuators running into
the measurement cables is more important. As a result, the fixed ground frame and platform
frame almost coincide with the measurement reference frames. This helps in interpretation
of numerical results but does not improve the calibration itself.
To allow measurement in between a favourable maximum and minimum length the limited
(2 m) range string encoders were enlarged with steel cables (as used for fishing!) which
were calibrated with the string encoders themselves. Two sets of these additional offset cables
were used (for Gi to Pj with i = j approx. 1.5 m and 2.0 m in case of i 6= j). The offset
of the string encoders themselves is also necessary to allow absolute measurement and was
measured at a small, approx. 60 mm, retracted up right position with a sliding calipers to
the funnels. The distance ( 1:8 m) between the string encoders could be measured with a
large (2 m range) sliding calipers.
Careful attention in performing these (offset) measurements largely improved the calibration.
By calibration before and after a measurement session also some drift compensation
(1.4 mm for one string encoder) could be applied. Another important point of consideration
was the dry cable friction in the funnels. Before each measurement, the cables had to be
vibrated a little bit to get rid of play.
To allow proper choice of the sensor range to be used, the set of platform poses, approximate
since the actual kinematics is not known in advance, used for calibration can be
simulated. In Fig. 4.4 all the positions of upper gimbal 1 are given for all 64 combinations
of setting the six actuators in a minimum length position of 20 cm above minimal and below
maximal stroke. These positions are sufficiently far away from the cushioning area of the
actuators to be able to hold the system stable at these positions in taking a measurement.
It is assumed that this set of positions will give relatively high variation in order to
enhance the identifiability of the kinematical parameters. From the front view it can be observed
that the positions are roughly the extremes of the 2-d.o.f. structure given in Fig. 3.22
which are rotated if considered from the side view. The fact that some points are relatively

128 4 Parameter identification and model validation
−z
3
2
1
0
−2
3
2.5
2
1.5
1
0.5
View in perspective
0
Side view
2 −2
0
−2 0 2 4
x
0
2
−z
−y
3
2.5
2
1.5
1
0.5
Front view
0
−2 −1 0
−y
1 2
2
1
0
−1
Top view
−2
−2 0 2 4
x
Fig. 4.4: Positions of upper gimbal 1 for calibration.
close to each other in this picture does not imply that they will contain the same information
since the orientation of the platform, which will be different, is not shown.
If compared to the unconstraint actuator, it can be observed that especially the range of
angles to be reached by the platform actuators is relatively limited. E.g. the angles of the line
between the lower gimbal point 1 and the dots in Fig. 4.4. This will affect the conditioning
of the calibration as will be discussed in the next sections.
4.1.4 Identification of the kinematical parameters
After the determination of the platform poses, this information plus the actuator displacement
measurements can be used to identify the kinematical parameters. Taking an initial
estimate of these parameters in (4.1), each measurement regarding the j’th actuator, will
have a residual. One can try to change the parameters in order to minimize these residuals.
Using the quadratic form of (4.1), the jacobian of kinematical parameter variations, k, to
residual variation, r, has a favourable linear structure
Jrk = @r
@k = Xo + 7
i=1 Xiki
with the kinematical parameters, k, of one actuator defined by
and the i’th row (measurement) of the jacobian
(4.17)
k T =[a T b T loj] (4.18)
Jrk(i� ) = 2[(lm(i) +lo(i)) (c T i Ti + b T Ti +a T ) (c T i +aT T T
i + bT )]: (4.19)

4.1 Calibration 129
It is immediately clear from the last equation that every row can be split in a non kinematical
parameter dependent row
Xo(i� )=2[lm(i) c T i Ti c T i ] (4.20)
from which the matrix, Xo, can be constructed and parts which are linearly dependent on
only one kinematical parameter e.g.
X1(i� )k(1) = 2[0 1 0 0 T T
i ( � 1)]a(1): (4.21)
A weighted least squared solution to the linearised problem of minimising the residuals
is given by
kwls =(J T rk WJrk) ;1 JrkW r (4.22)
In robot calibration, this is usually effective since the equations are only mildly nonlinear
[52]. The weighting matrix, W , can be used to discriminate between the measurements.
This can be appropriate since the confidence level for the different platform pose measurements
can differ. If one of the string encoders showed e.g. a little hysteresis due to friction
once in a while, the redundancy in the platform pose measurement can be used to detect
this. With
Wii =
1
0:5 + 1000 j lPP ; mlpp j
(4.23)
the confidence is upper bounded by 2 and goes to zero if high difference is detected between
the measured distance, lpp, and the mean distance, mlpp, between two points, Pi.
Iteration on (4.22) is similar to a Newton Raphson iteration. For convergence, it helps if
the jacobian is Lipschitz, far from singular, well conditioned and starts with a well chosen
initial estimate. Due to the linearity in the parameters, the jacobian is clearly Lipschitz.
Further the structure of each parameter (Xi) is almost equal. The choice of platform poses
can be used for conditioning. E.g. not varying the orientation, T , will result in a singular
jacobian, since all solutions with only a translational vector added to the actual solutions of
both a and b will have equal residuals.
The singular values of the jacobian, Jrk, are also called the observability indices of the
calibration problem. In a comparative study cited by Hollerbach and Wampler in [52] it was
concluded that optimal conditioning of these indices or maximising the minimal index gave
good quantative results.
The calibration problem of the actuator kinematics was idealised by an unconstrained
simulation i.e. no constrains of platform poses to be chosen and all actual parameters zero.
Choosing all (729) combinations of the tri state (-1,0,1) for translation and (-.5,0,.5) for the
euler parameters results in a jacobian with condition number two at the solution, which is
almost optimal. By Hollerback and Wampler [52] it is referred to a research concluding the
condition number should be better (lower) than 100 to have reasonable results. With the
SRS Stewart Platform to be calibrated with constrains on the actuator lengths the condition
number increases to sixty using the 64-state set given in Fig. 4.4.

130 4 Parameter identification and model validation
−z
0.03
0.02
0.01
0
−0.01
0.03
0.02
0.01
0
view in perspective
0
0.01
side view
0.01
0
−0.01
−0.01
−0.02 −0.01 0
x
0.01 0.02
−z
−y
0.03
0.02
0.01
0
front view
−0.01
−0.02 −0.01 0
−y
0.01 0.02
0.02
0.01
0
−0.01
top view
−0.02
−0.02 −0.01 0
x
0.01 0.02
Fig. 4.5: SRS calibration result with the design structure on a 1:100 scale and the differences
1:1.
e_length [mm]
e_trans [mm]
e_rot [meps]
5
0
−5
0 20 40 60
5
0
−5
0 20 40 60
5
0
−5
0 20 40 60
non calibrated system
5
0
−5
0 20 40 60
5
0
−5
0 20 40 60
5
0
−5
0 20 40 60
calibrated system
Fig. 4.6: Positional errors for all 64 platform poses used for calibration made by the noncalibrated
and the calibrated system. The first row shows length measurement
prediction errors of the actuators length measurement given the redundantly measured
pose. The next rows give the transformation of these errors to translational
(2 nd row) and rotational (3 rd row) prediction errors of the platform pose.

4.1 Calibration 131
4.1.5 Results in calibrating the SRS motion system
The calibration procedure leads to kinematical parameter (vectors) defined in the reference
measurement frames. By construction of the forward kinematics of the newly found parameters
for the actuators in half stroke, the transformation, sx gp, between the two measurement
frames can be calculated. All upper gimbal positions can now also be referred to the ground
measurement frame, Gm.
a Gm =cgp + Tgpa Pm (4.24)
As the parameters were not constrained to a plane, the simulator ground and moving
frame are not trivially defined anymore. As for the reference measurement frames, the
ground frame origin is chosen as the mean of the lower gimbal points. The ground frame xdirection
is chosen as the addition of the vectors between the origin and gimbal point 1 and
6 and the xy-plane is formed by making the vector from the origin to the mean of gimbals 2
and 3 part of it. This defines the transformation to a ’design’ ground frame.
Both the upper as the lower gimbals can now be referred to the design frames. In Fig. 4.5
the gimbal points found through calibration are given in a design frame which is a 100
times compressed making the differences, which are less than 0.1 %, visible. As opposed
to the dummy platform [16] [151], no large deviations are found in the construction of the
shuttle. It is now interesting to see what the differences are in predicting the actuator length
measurements i.e. using the design values and the calibrated values. These deviations for
all six actuators can also be transformed to the deviations in platform pose using
sx = J ;1
l�sx (sx) l (4.25)
In Fig. 4.6 it is shown that the positional accuracy of the calibrated platform has almost
improved an order of a magnitude.
The accuracy of the noncalibrated system is not better than 5 mm in actuator lengths,
2 mm in translation and 10 mrad in orientation. With the calibrated system this improves
as shown in detail in Fig. 4.7 to .75 mm in actuator length predictability, apart from one
outlier also predicted with upper reference point distance measurement. In the more important
platform pose translational and orientational accuracy the errors are reduced to less
than 0.5 mm and 2 mrad respectively.
The initial estimate in the calibration procedure was more than another magnitude worse
(> 100 times) since the pose of the measurement reference frame was only roughly (within
100 mm) taken into account. Still the iteration on the kinematical parameters converged
quickly within 4 steps for each actuator leg. Convergence is not guaranteed with such
initial errors, but since the procedure can be done off line, as opposed to the platform pose
reconstruction treated in Section 2.2.11, this is not so important.
One important aspect, the direction of the gravitational field has not been identified yet.
This can be done using an additional measurement device (e.g. an inclinometer) as has been
done in [16] [151]. But the pressure (force) measurements in the actuators can also be used
to detect this as shown in the next section.

132 4 Parameter identification and model validation
residual [mm]
error [mm]
Variation upper triangle measurement
3
2
1
0
−1
−2
0 20 40 60
0.5
−0.5
Translational pose prediction errors
1
0
−1
0 20 40
pose no.
60
residuals in [mm]
residual [meps]
0.5
0
−0.5
−1
−1.5
Errors in actuator length prediction
1
−2
0 20 40 60
Orientational pose prediction errors
1.5
1
0.5
0
−0.5
−1
−1.5
0 20 40
pose no.
60
Fig. 4.7: Positional errors for all 64 platform poses used for calibration made by the calibrated
system in detail. Also showing the variation in upper reference point distance
measurement, which is used to calculate the confidence levels.
Resume
The following conclusions can be drawn w.r.t. the calibration procedure described.
It has been shown that the kinematic mechanical parameters of a Stewart platform
used as a flight simulator motion system can be calibrated resulting in a system with a
positional accuracy better than 1 mm (for DEP). This is an order of a magnitude more
precise than before calibration.
The fact that the kinematical model itself is accurate within these error bounds for the
SRS is another important conclusion.
Further, it means that with calibration such motion systems can be made more accurate
without putting more extreme tolerances on the fabrication and constructional
process.
The two step calibration procedure has been shown to work in practice.
For the first step, measuring relative platform pose variations w.r.t. unknown, but
platform and inertial world fixed, reference frames is sufficient redundant information
necessary for calibration.
The parallelism of the platform can be used in the second step to identify the 42
kinematical parameters in six separate actuator groups. This enhances accuracy.
It is important to realise that the result strongly depends on the accuracy of some
of the measurements taken. These are the absolute length measurement of, and the
distances in between, the redundant sensors.

4.2 Stewart platform model parameter identification 133
As the calibration is concerned with sub mm precision on a sup m motion system, it is
recommended that the procedure is rerun after each reconstruction of a platform e.g. in
moving the SRS motion system to another site.
4.2 Stewart platform model parameter identification
Now the kinematic model has been validated and its parameters have been calibrated, the
parameters concerning the dynamics of the motion system can be identified. The main
unknowns in the Stewart platform model are its mass properties.
First, the experiments to reconstruct the parameters related to the gravity forces, the
location of the centre of mass and the weight itself will be discussed. Then, the identification
of the inertial parameters is treated. These can be determined experimentally without much
concern about the precise dynamics of the hydraulics. In this way, a reasonably accurate
approximate model, i.e. (2.112) with a general mass matrix, of the mechanics presented in
Chapter 2 taking into account the actuator intertia as in (2.122) can be identified.
In the model based control structure to be presented in the next chapter, this part of the
model will be important since the forces required for the accelerations to be applied for
simulation will for most be generated by feed forward which requires an accurate model.
Identification of the hydraulics, considering the separate actuators, has been discussed in
[124]. The model parameters concerning the hydraulics are therefore assumed to be known.
Validation of the dynamics resulting from the interaction in the hydraulically driven Stewart
platform will be discussed in the next section.
The set up of the hydraulically driven motion system supplied with the six valve inputs
and measurement equipment to determine both pressure and positional information of the
actuators, the legs, provides a complex but sufficiently rich environment by itself to perform
almost all experiments required. The only additional measurement equipment which is
used in this research are accelerometers and rate gyros which do more directly reflect the
simulation performance on the platform and with which one can more accurately appoint
the cause of parasitic dynamics resulting from flexible modes in the shuttle or foundation.
4.2.1 Gravitational force determination
The parameters to be identified, the static mass experienced and the position of the centre of
gravity, can be used to incorporate a gravity compensation in a controller. This is important
since gravity forces form a considerable part of the total force (up to 70 % of full pressure
required as specified in Table 3.12). Further, a reasonable mass matrix results from taking
the centre of gravity as the origin of the moving body axes in the dynamic model.
Considering the model (2.122) in a static pose, leaves
Gt =
fg
mg =(fg co) = J T l�x (sx)fa (4.26)
with the gravity force, fg = Tgg[0 0 mg] T , transformed to the fixed lower gimbal frame by
the rotation matrix, Tgg. Usually the matrix Tgg is approximately the identity matrix.

134 4 Parameter identification and model validation
z
f 1
co
f 2
fg l1 l2
c1
f 3
l
3
X
Fig. 4.8: Simplified planar example of identifying mass and centre of gravity. Through the
three actuator forces of a parallel 3-d.o.f. system the gravity vector (length and
direction) plus line of action can be identified with two force equilibria and one
moment equation. By rotating the system and identifying another (rotated) line
of action, the intersection of both lines provides for the position of the centre of
gravity. In a spatial 6-d.o.f.-system, also two measurements are sufficient but the
two lines of action can cross in practice. By identifying the point, which is closest
to a set of lines of action, a more robust estimation can be attained.
Further, the moments, mg, are induced by the vector product of this force with the vector
of the upper centre gimbal point to the centre of gravity, c o. The vectors, fg, the gravity force
i.e. platform mass times gravity and the centre of gravity, c o, have to be identified.
From Fig. 4.8, it can be observed that one can only identify the part of c o, which is
orthogonal to fg considering one pose as the moments result from a vector product i.e. a
line of action can be identified on which the centre of gravity point resides. By measuring
the forces required in other static poses in which the platform has been rotated (around a
vector which is at least partly orthogonal to the gravitational direction), a set of lines can
be determined from which co can be obtained. In theory all these lines will intersect at this
point but in practice they will cross. The point which is closest to all lines in some sense
can be taken as best guess for the centre of gravity. Here the least squares optimal solution
of all (Euler) distances will be taken.
Consider a specific static pose measurement of the pressure differences at the valves,
[f T g mT T
g ]=AdP i J(sx), assuming the input and output pressures are equal statically. A
valid point of the line containing the centre of gravity, o, can be found by
with ng = fg=(mg) and
c1
z
l
1
f 1
o = ; 1
f T g fg
fg Pngmg� (4.27)
q
f T
g fg = mg. Now, the line of action can be described by
o + ng. Every line, ( o + ng)i, found for a pose (ci�Ti), can be transformed back to the
platform coordinates in one specific pose e.g. the neutral pose.
( on + nngn)i = T T
i ( o + ng) ; ci (4.28)
f g
co
l
2
c2
f 2
f 3
l
3
X

4.2 Stewart platform model parameter identification 135
Configuration A B C
m [tons] 2.60 2.10 4.25
co(x) [m] 0.00 -0.12 0.00
co(y) [m] 0.00 0.00 0.00
co(z) [m] -0.23 -0.26 -0.45
Table 4.1: The Simona Research Simulator gravitational parameters
The vector, dlp, with smallest distance of some point, xp, to such a line is given by
dlp = Pnng (xp ; on) (4.29)
The vector, co, to the point, which minimizes the sum of squared distances to n of such
lines, is given by
co = 1
n
n
i=0 Pnng�i
;1 1
n
n
i=0 Pnng�i oni (4.30)
The conditioning of this optimization problem greatly depends on the ability to perform a
measurement in which the summation of the projection matrices, P nng�i, equally span the
3D-space i.e. whether the platform has been rotated sufficiently.
Due to the limited stroke of the actuators of the Stewart platform, rotation can not be
performed freely. In fact, performing valid pressure measurements can only be done if
the actuators are out of the cushioning area (approx. 15 cm from max or min). Further,
incorporating the mass of the actuators in the total mass is less tolerable with large rotations.
With the SRS, the fast gravitational force identification procedure consisted of measuring
pressures in five poses: neutral, +roll, -roll, +pitch and -pitch from neutral considering the
limits: (lmin + :15,lmax ; :15).
The more elaborate procedure takes into account all 64 poses used in the kinematic
calibration. Over time, the mass and position of the centre of gravity of a simulator under
construction, as the SRS, is likely to change. Therefore this procedure is to be repeated
more often than full kinematic calibration.
The conditioning of the matrix to be inverted in (4.30) is 12 and as a result the x- and
y-position of the centre of gravity can be found approx. 10 times as accurate as the zposition.
Fortunately, as this point is mainly used for gravitational compensation, the x- and
y-position are also much more important.
The gravitational forces of the three SRS system configurations evaluated in this research,
A) The dummy platform alone,
B) The shuttle and
C) The dummy platform with additional load

136 4 Parameter identification and model validation
have been identified. The results are given in Table 4.1.
In the poses considered, the observed gravity forces are varying related to a mass of
approx. 20 kg due to the relative motion of actuators w.r.t. the platform. In all cases the
c.o.g. is found above the centre gimbal plane. As the dummy platform has symmetricity, it
is natural to find the c.o.g. in the origin of the xy-plane. The c.o.g. of the shuttle, which is
somewhat to the back, is expected to move to the front if the visual system will be added.
Measurements are best taken with a very slowly moving platform (e.g. sinusoidal motion
in z-direction < :5Hz, amplitude < 2mm over a whole number of periods) to eliminate
most of the friction forces. Coulombic friction forces are small for the prismatic joint, the
actuator with hydrostatically bearings, [124] but the rotation of the gimbal joint also has
friction. These friction forces together are seen to be as large as 250 N.
Both the calibration procedure as the gravitational force determination require a simple
controller to stabilize the system and moving the system to a specific platform pose and hold
it there. The standard local (actuator per actuator) pressure feedback [153] in combination
with positional PI-feedback with low bandwidth (1/10 of lowest eigenfrequency in neutral
position i.e. approx. 0.5 Hz) will usually be sufficient.
To determine the inertial properties of the system, discussed in the next section, experiments
preferably require vibrations of higher frequency, e.g. approx. 4 Hz. This will still
be attainable with a local control structure as no harsh requirements will be stated towards
cross talk in the identifying procedure.
4.2.2 Identification of the inertial properties
Basic concept
The main unknown parameters left to be determined are those contained in the mass matrix,
the inertial properties of the system. In short, these parameters will be estimated by
considering force and acceleration (through position) measurements and evaluating these
signals in a domain where the basic equation force is equal to mass times acceleration in
matrix form, f = Mx, holds. The mass matrix, M, can be deduced by performing six
approximately sinusoidal vibrations which span the 6-d.o.f. coordinate space with sufficient
acceleration (position) and force amplitude to be measured accurately and sufficiently small
positional amplitude to have a system with only moderate nonlinear effects. It is assumed
that no relevant parasitic flexible modes are hit at this harmonic.
Basically from calculation of the platform forces fp = J T l�x (sx)AdP i and position
(sx = f ;1 (l)), the amplitudes of the ground harmonic signal parts can be determined. The
amplitudes of the accelerations, xa, resulting from the ground harmonic with frequency ! s
are easily obtained from the positional as j xa j=j ! 2
s xa j. Now, taking all experiments
together in a matrix equation,
Mt = FapX ;1
a � (4.31)
Mt can be identified, with for the i’th experiment M txai fapi and the force basic harmonic
amplitudes, fap, stacked in the columns of Fap and stacking the accelerational amplitudes,
xa, into Xa.

4.2 Stewart platform model parameter identification 137
mpp (40kNm,kN)
pitch (3eps,m)
6
4
2
0
−2
−4
−6
0.2
0.1
0
−0.1
x 10 −3
0 0.5 1
−0.2
0 0.5
time (s)
1
1
0.5
0
−0.5
−1
0.03
0.02
0.01
0
−0.01
−0.02
x 10 −4
0 0.5 1
−0.03
0 0.5
time (s)
1
Fig. 4.9: Decomposition of platform forces and position into ground harmonic and residual
signals in a vibrational (pitch) test of the dummy platform without additional load
(configuration A.) for mass matrix determination. Upper left plot gives positional
ground harmonics for the second euler parameter (pitch in three times the euler
parameter, eps) and the ground harmonics for the other d.o.f.’s. Upper right are
the residual positional measurements during the test. Lower left are the platform
forces/moments ground harmonics, normalised to the maximum force of one actuator
(times 1m for the moments). The lower right plot provides the measured
residual forces and moments.

138 4 Parameter identification and model validation
Estimation in the presence of nonlinear terms
To evaluate under what conditions a reasonable estimate of the SRS motion system mass
matrix, Mt, results, the nonlinear model will be considered. As proposed in the previous
chapter, the mass matrix is assumed constant in the platform frame i.e. the relative motion of
the actuator parts is not taken into account. Referring to (2.122) and including a dissipating
velocity term, Bt(sx) _x,
Mt(sx)x + Ct(sx� _x) _x + Bt(sx) _x + Gt(sx) =J T l�x (sx)fa
it is immediately clear that even in order to identify the mass matrix, M t, locally at some
sx, dynamic experiments (x 6= 0) have to be performed in which nonlinear terms are
unavoidable. Fortunately, as the term, Ct, is quadratic in the velocity it does for most
have second and higher harmonic signal components since 2 sin( ) cos( ) = sin(2 ) and
2cos 2 ( ) = (1 + cos(2 ))). So in convolution with the ground harmonic, C t almost disappears.
Further, the ground harmonic function can be split up into two parts. One in phase
with the position and one shifted over =2 rad, or the part which is in phase with velocity.
By only considering the part of the signal (force, position) which is in phase with position,
also the dissipating forces drop out.
In Fig. 4.9 the necessary measurements are provided for one typical experiment out of
six in estimating the mass matrix. Only taking the ground harmonic into account allows
accurate determination of the ground harmonic accelerational amplitude. As the ground
harmonics along all degrees of freedom are taken into account, no severe requirements hold
for the controller during the experiment.
Another cause left of calculating a biased estimate of the mass matrix will be the dry
friction terms. Also these are velocity (sign) related and will also mainly influence the
ground harmonic which is =2 out of phase.
In practice, the system will not move in one platform coordinate only. Often the parasitic
motion will also be out of phase. The system will therefore move along an ellipsoidal
trajectory instead of a line as can be observed from Fig. 4.9, which shows phase shifted
positional ground harmonics. This will be a possible cause of still having friction like terms
in phase with the ground harmonic used in the estimation procedure.
Another nonlinear term to be considered is the position dependent jacobian, J l�x(sx).
Linearising in a static pose, sxo, with static gravitational actuator forces, fao results in
Mt x = J T (sxo) fa + 6 @Jl�sx(sxo)
i=0
(sxo) sxifao
(4.32)
@sxi
With sxi sufficiently small, these terms can be neglected. To check what is sufficiently
small, the partial derivative of the jacobian, Jl�sx to the states sx is to be evaluated. As
given in (2.42), this jacobian consists of two kind of vectors. All the unit direction vectors,
lni of the actuators and the vector products, T a i lni. Using the time derivative of ln given
in (2.52), and decomposition with (2.47) the partial derivative to sx is given by
@ln 1
=
@sx klk PlnJai�x
I 0
0 J!
(4.33)

4.2 Stewart platform model parameter identification 139
with the transform from euler parameter variations to angular velocity given by J ! =
2G( ) (2.23). Now
@(T a ln)
@sx
= T a
@ln @T
+
@sx @sx a ln (4.34)
where @T=@sx follows from _
T = ~!T. Validation is easily done by calculation of the
difference between two jacobians for which the pose is slightly altered.
Experiment design
Calculation of each @Jl�sx=@sxi-matrix in the neutral position gives matrices with roughly
the same gains as Jl�sx itself. With the gravity forces at the same magnitude as the dynamics
forces, the relative effect of the variation (to compensate for gravity) in the linearisation
is nearly the same as the ratio of positional over accelerational amplitudes. As the test
frequency becomes higher, this ratio decreases quadratically.
In practice the test frequency of the ground harmonic, ! t is chosen somewhat below
the lowest open loop ’rigid’ eigenfrequency. In this manner, it possible to successfully
perform the test with a simple non model based controller. In case of configuration A.
and B., !t = 2 4 and for the dummy with additional load it was taken somewhat lower
!t =2 =0:3. At these frequencies it is possible to perform the measurements with reasonable
pressure (up till 25 % of full load) and velocity (up till 20 % of maximum velocity)
amplitudes by requiring harmonic reference signals for each platform coordinate in a separate
test of positional amplitude of 10mm in the translational and :003 for k 13k in
rotational directions. Also an equivalent ratio of measurement errors in position (0.1 mm)
and pressure ( 25 mV noise on 10V scale) in relation to the amplitudes is attained in this
way (both 1%). Typical measurement signals, for the forces, S f , and positions, Sp, for such
a test are given in Fig. 4.9. The actual test length was tl =5s and the measurements were
sampled at 500 Hz.
Determination of the ground harmonics
The harmonic coefficients of the positional and force (pressure) measurements contain the
information of the mass properties. They are calculated by using a matrix, H, with a basis
function in each column over the time span 0�::: �tl sampled at the same frequency
H =[1t ; :5tl sin(!tt) cos(!tt) sin(2!tt) cos(2!tt) :::] (4.35)
Now the coefficient matrices for the platform forces C f and positions Cp are the least square
estimates found by
[Cf Cp] =(H T H) ;1 H T [Sf Sp] (4.36)
For the ground harmonic (third, c 3, and fourth, c4 rows of the coefficient matrices), the
phase, p, of the platform coordinate, which is tested, is calculated. The columns fap and
xap of the matrices Fap and X of (4.31) are now found by
[f T ap x T
ap ]=real(e;j p ([c3f + jc4f c3p + jc4p])) (4.37)

140 4 Parameter identification and model validation
Configuration Place in 6x6 A B C
Mass matrix Mt
mx [tons] (1,1) 2.6 2.0 4.3
my [tons] (2,2) 2.6 2.0 4.3
mz [tons] (3,3) 2.7 2.3 4.3
Ixx [tons m 2 ] (4,4) 2.3 2.6 4.1
Iyy [tons m 2 ] (5,5) 2.3 2.8 4.0
Izz [tons m 2 ] (6,6) 3.8 3.5 6.7
Ixz�zx [tons m 2 ] (4,5),(5,4) 0.0 0.3 0.0
M15�51 [tons m] (1,5),(5,1) 0.0 0.0 0.1
Table 4.2: The Simona Research Simulator identified nonzero inertial parameters. These
are mainly appearing on the diagonal of the mass matrix, M t, apart from
the Ixz�zx term in case of the shuttle (configuration B.). At least 1 % error
(25 ; 50 kg) can be expected since this is the maximum singular value of the
nonsymmetric part of the estimated mass matrix.
In this way, a maximal part of the ground harmonic for each test direction is taken into
account and only the part of the force and positional signal which are in phase with this
motion.
In Fig. 4.9 it is shown that a significant part of the forcing signal does not belong to the
ground harmonic and will not be taken into account in the mass matrix estimation.
Experimental results of the mass matrix estimation
The estimated mass matrices for the three configurations are given in Table 4.2. An advantage
of taking the identified centre of gravity as origin in the moving body frame is that the
mass matrix becomes almost diagonal. The dummy platform has the advantage of having
its main axes of inertia along the platform frame axes. For the shuttle, configuration B, the
xz-plane is a plane of symmetry (making the y-axis a main axis of inertia, running I yz�zy
and Ixy�yx zero) but also has a cross term Ixz�zx of approx. 250 kg in case of an empty
shuttle.
In principle, the mass matrix should be symmetric. The nonsymmetricity of the identified
mass matrix can give some impression of the errors made due to measurement inaccuracy,
etc. In case of configuration A., the maximum singular value of the non symmetric
part is 26 kg(m 2 ) and in B. and C. it amounts to 50 kg i.e. 1 % of the estimate itself for the
dummy platform and somewhat higher for the shuttle. This means no more accurate results
were obtained than specified in Table 4.2.
Resume
A very compact test sequence was presented, which allows estimation of the relevant
dynamic mechanical parameters of the Stewart platform in the presence of hydraulic

4.3 Frequency response model validation 141
actuators.
In practice five static poses (at least two in theory) measuring pressure and position,
allowed identification of the total mass experienced and the position of the centre of
gravity.
With six persistently exciting periodic motion tests, the inertial properties of the motion
system could be identified through a (6 by 6) mass matrix.
By only taking the ground harmonic in phase with positional motion, velocity related
(friction) and higher order terms (nonlinearities) have minimal influence on the
procedure.
Together with the earlier identified hydraulic actuator parameters, these mechanical
parameters allow a reasonably tight fit to practice of the complex multivariable nonlinear
hydraulically driven mechanical system model.
In the next section, this will be validated in comparing the responses of the models derived
with the parameters determined with ’open loop’ motion system frequency response
measurements.
4.3 Frequency response model validation
At this point, the model structures have been formulated and the model parameters have
been estimated. With this information, the calibrated models can be evaluated by comparison
of model and actual responses. In this section the model open loop system characteristics
will be validated in the frequency domain. Especially with mechanical systems, also
hydraulically driven, many important properties are revealed considering responses to sinusoidal
inputs.
Characterising nonlinear systems by describing functions
With frequency response measurements not only the input-output relation of a linear system
can uniquely be described but also many nonlinear system structures can be characterized
by performing harmonic excitation. E.g. a sinusoidal input describing function identification
procedure approximately characterises a nonlinear system by considering amplitude
and phase of the response as a function of frequency and input sinusoidal amplitude, as
extensively discussed in Van Schothorst [124]. This only requires a filter hypothesis, which
says that the non-linear system has low-pass characteristics.
Further, the linear dynamics of a non-linear system (for some operating point) can be
defined as the describing function of the system as the amplitude approaches zero i.e. is
sufficiently small. The describing function can be found measuring a frequency response
such that the amplitude of the response is flat. This requires the design of an input amplitude
filter found e.g. by iteratively measuring the frequency response.
Appropriate experiments characterising hydraulically driven mechanical systems
Systems experiencing structural vibrations such as mechanical systems are often charac-

142 4 Parameter identification and model validation
terised experimentally using modal analysis techniques. A large part of the modal analysis
also heavily relies on measuring frequency responses [39, 90]. With modal analysis, the velocity
response of a mechanical system to force excitation is called mobility. With a single
degree of freedom (one mode, vibration) e.g. a mass-spring-damper system, the mobility
frequency response has a zero at zero and at infinite frequency and a complex pole pair
assuming reasonably low damping with an undamped eigenfrequency at the square root of
the mass vs. spring stiffness ratio.
Similar responses can be expected measuring the response of our motion system from
input valve voltage to the pressure difference over the valve. As discussed in the previous
chapter, the hydraulically driven mechanical system has roughly a mobility like characteristic
if the response of the pressure difference over the actuator compartments is considered
from input actuator valve flow. Measuring from input valve voltage sets a low pass filter
in series with this system thereby fulfilling the filter hypothesis. Viscous damping not only
damps the resonance but also shifts the zero at zero frequency to the left in the complex
plane. As shown in Section 3.3.1 considering the models, the response of the multi degree
of freedom multi input valve/output pressure difference of the motion system at hand is expected
to have a mobility like characteristic with multiple modes. Also with parasitic effects
from transmission lines or mechanical flexibility, this does not change, apart from inducing
additional anti-resonance and resonance pairs.
4.3.1 Frequency response dummy platform
The models of the previous chapter appointed the rigid body modes as the main characteristics
of the hydraulically driven mechanical system. In this section, this will be evaluated by
the actual frequency responses of the dummy platform (configuration A. with no additional
load). In this configuration relatively little influence of parasitic effects, e.g. stemming from
flexible deformation, is expected to enter the frequency area of interest i.e. the frequency
area up till 50 Hz where the rigid body modes reside.
With the inertial parameters found in the previous section and the parameters for the hydraulic
actuators given in Table 3.2, the theoretic basic model structure of the hydraulically
driven motion system given in Section 3.4.1 is specified. Calculation of the eigenfrequencies
of the rigid body modes in the neutral position for the dummy platform, configuration A.,
results in 6:8 Hz for a surge/pitch and sway/roll mode, 13:6 Hz for a yaw mode, 17:7 Hz
for heave and 23:4 Hz for the different sign surge/pitch and sway/roll modes.
With a HP frequency analyzer ([91]) the relation at the neutral pose from each valve
steering voltage to each actuator pressure difference was measured between 5 Hz and
50 Hz by a swept sine at 1:5% of full input voltage. With the analyzer 201 logarithmically
spaced datapoints per input output relation were obtained. As this analyzer only resolves
one input output relation at the time, 36 separate experiments were performed. Most of the
frequency domain measurements are given in 6x6 form in Appendix A.
The systems pose was stabilized by a moderate position feedback of gain k p = 0:2
resulting in a bandwidth of 0:05 Hz. At 5 Hz the influence of this feedback can be
1% and grows at the first resonance to maximally 2%. At the frequencies larger than
10 Hz it will be as little as 0:1% and decrease quadratically. This influence is estimated

4.3 Frequency response model validation 143
Amplitude
Phase (deg)
10 1
10 0
0
−200
−400
−600
10 1
10 1
Frequency (Hz)
10 1
10 0
0
−200
−400
−600
10 1
10 1
Frequency (Hz)
Fig. 4.10: Bode plots of the highly interacting responses from the first valve input voltage
to the six pressure input differences (dPi), measured at the dummy platform
(configuration A.) at the left. On the right side the model response (18 th order)
is given incorporating the rigid body modes and three additional modes for the
foundation. Six by six separated Bode plots can be found in Fig. A.1 and Fig. A.2
of Appendix A
by taking the measured ’closed loop’ gain, k m, and recalling the open loop response gain,
kg, to be measured is found through kg = km(1 ; kckm) -1 with kc, the position loop gain
including the transfer function from (normalized) pressure to position. For small (complex)
values, kckm is the relative gain variation.
j kckm j = kpAdpnormkm(m! 2 ) ;1
= 0:2 25 10 -4 200 10 5 7=(8:8 10 3 (2 5) 2 )
= 0:008
(4.38)
The measured Bode plot of the measurement from the first valve input to the six pressure
differences is given in the left column of Fig. 4.10. The predicted rigid body modes are
present and additionally some parasitic resonances are present between 10 ; 13 Hz and at

144 4 Parameter identification and model validation
Direction Mfii [tons� tons m 2 ] !fi [Hz] fi
xf 45 2 10:5 0:05
yf 45 2 10:0 0:035
f 140 2 12:5 0:05
Table 4.3: Parameters taken for the foundation.
approx. 30 Hz.
Additional measurements at the floor, which will be discussed in the next section,
pointed out that the vibrations at 10 ; 13 Hz could be assigned to the foundation. By modelling
the foundation as a 3-d.o.f mass/spring/damper system as modelled in Section 3.4.5
with the parameters of Table 4.3, an 18 th -order model results.
In general the damping structure of mechanical systems is hard to predict theoretically.
The damping of the first three rigid body modes is well described by the experimentally
found values for leakage and viscous friction of the hydraulic actuators. For the three highest
frequency modes some additional damping had to be introduced.
This can be explained by the fact that the gimbal rotation is probably also dissipating
energy and not explicitly taken into account. Further, it is observed in modal analysis that
damping does not always act as the linear viscous effect, which is part of the model used
here. Alternatively, hysteretic damping structures are proposed in [39, 90], in which the
frequency dependence of the dissipation is changed. A drawback of that structure is that it
can not be incorporated in a state space (time-domain) model. Therefore hysteretic damping
has not been included in the models proposed in this thesis for motion systems.
The Bode plot from the first valve input to the six pressure differences of the (6 6)
18 th -order model is given in the right column of Fig. 4.10. The measured transfer functions
are fully shown in Fig. A.1 and Fig. A.2 of Appendix A. The model gives a reasonably
close description of the characteristics observed from the measurements. Both the phase as
amplitude frequency responses are very much alike. Note that only the phase of the first
input to the first output remains within 90 necessary for passivity.
The largest discrepancy can be observed at 30 Hz where the measurements point out
an additional resonance, which is not incorporated in the model. Limited stiffness of the
actuators in the radial direction are a possible cause of this vibration but this could not be
confirmed by some additional measurements of an accelerometer waxed at the actuators.
Further, the additional phase lag coming in at frequencies 40 Hz can be assigned to
the valve characteristic. The model including this effect will be evaluated with the shuttle
(configuration B.). The frequency responses of that system were measured up till 500 Hz.
Decoupled response by coordinate transformation
In the frequency responses of Fig. 4.10 is it not immediately clear how many modes are to be
used to the describe the 6 6 system. Now we can use the basic model structure we obtained
for hydraulically driven mechanical systems in Section 3.4.1. An important system property
of the model is the coordinate transformation along the rigid body modal directions, which

4.3 Frequency response model validation 145
Amplitude
Phase (deg)
10 1
10 0
100
50
0
−50
−100
10 1
10 1
Frequency (Hz)
10 1
10 0
100
50
0
−50
−100
10 1
10 1
Frequency (Hz)
Fig. 4.11: Bode plots along the six estimated decoupling rigid body modal directions (without
parasitic vibrations) from redirecting the input valve voltages and valve pressure
differences measured (left) and modelled (right). In the upper left, the
largest singular value over the frequency of the error system regarding all the
nondiagonal elements is given in the dashed plot. Six by six separated Bode
plots can be found in Fig. A.3 and Fig. A.4 of Appendix A
decouples the system into six independent systems. Each of which incorporates one of the
rigid body modes.
With additional parasitic modes this decoupling is not exact anymore but in the previous
chapter it was found that at least in the neutral position, this transformation still puts the
main modes on the diagonal. This transformation with an unitary matrix, U, can be derived
from the mass matrix evaluated from the actuators as defined in (3.44). A quick impression
for these modes can attained by evaluating the singular values for each separate frequency
measured.
In Fig. 4.11, the frequency responses of the diagonal elements of the 6 6 transformed
transfer function are plotted. (Full transfer functions are shown in Fig. A.3 and Fig. A.4
of Appendix A. In the left column, the measured response is given and on the right, the
model response is drawn. Still, the model and the measured response have very much the
same characteristics. Having the rigid body modes decoupled on the diagonal of the transfer
function enables a much easier interpretation. E.g. the phases of the system are now clearly
90 from which the conclusion can be drawn that the linearised system is passive at this
point of operation as was predicted from the non-linear system. As predicted, the dynamics

146 4 Parameter identification and model validation
of the foundation and other additional mechanical resonances do not influence this property.
Of course, the valve characteristics will make the system lose its property of being strict
positive real and cause the additional phase shift at the higher frequencies of the measured
response.
In this decoupled form, the two frequency responses, the measured response and the
one derived from the model, can be compared in more detail. Further, it is easier to adjust
the model in this form in order to let the model more tightly fit reality. For example the
damping structure is most conveniently changed in these coordinates since each rigid mode
can be varied independently. Also parameters like massmodes (eigenvalues massmatrix)
can be adjusted. In the model, the mass viewed along the z-direction can be changed to
a 13% higher value to have higher correspondence of the modelled and measured z-mode.
Probably the actuator masses are somewhat higher than expected.
Important is the fact that the non diagonal part of the responses is almost neglectable
as the largest singular value of this part of the transfer is much smaller than the rigid body
modes, which appear separated on the diagonal transfer functions. In the measured responses
it is clear that there is some distortion at the resonance frequencies coming into the
low gain parts of the other modes.
Resuming, the model which describes
the rigid body mechanical system
together with the basic hydraulic system,
and in this environment some additional dynamics from the non rigid foundation,
gives a reasonable description of the system over the frequency range, which will be important
for control (5 ; 50 Hz). Both the rigid modal eigenfrequencies and directions can
be predicted from the model structure together with the kinematic and inertial model parameters
identified. The damping structure had to be adapted somewhat using the measured
frequency responses. Something which is not unusual since the damping structure of a
mechanical system is hard to predict.
Frequency domain identification
A more tightly fitted local description of the system at the neutral pose can be obtained by
frequency domain identification. A flexible tool was used to identify a multivariable model
from frequency domain data with the method of De Callafon et al [28] as already reported
in [75]. No reasonable results could be obtained, however, from directly fitting the original
6 by 6 by 201 datapoints. Results improved drastically by separately identifying the rows of
the data transformed by the coordinate transformation, U. Rows instead of columns were
taken as the singular directions could be predicted more closely at the directly measured
output than at the input through the valve.
As these models will usually contain redundant modes from non exact decoupling, especially
at the dominant lowest rigid modes at 6:8 Hz, the full multivariable model can be
reduced. By using the block balancing approach of Wortelboer [157], the reduction problem
can be transformed to finding a good multivariable representation of each observed mode.

4.3 Frequency response model validation 147
Amplitude
10 0
10 −1
10 1
Input filter shuttle frequency response measurements
Frequency (Hz)
Fig. 4.12: Bode amplitude plot of the 6 th -order experiment notch filter.
The resulting 20 th -order model describes both the six rigid modes as four additional
parasitic modes, including the modes at 30 Hz which were not incorporated in the more
global model with physical interpretation. The largest singular value of the difference between
model and data at each frequency at which the data was measured is equivalent with
the dashed line in the upper left of Fig. 4.11. Thus the error remains well below the relevant
dynamics.
With the inverse coordinate transformation, U T , this model can be transformed to the
actuator coordinates from which the original measurements were taken. The error is approximately
10%.
More research into identification of the flight simulator motion system has to point out
in what way models or model parameters can be identified which provide a tight description
of the system on a global scale, e.g. over all platform positions.
4.3.2 Additional dynamics into the higher frequency area
With the shuttle on top of the motion system, Configuration B., it was decided to measure
the frequency responses over the wide frequency area between 5 Hz and 500 Hz again
with a sine (up) sweep of 201 logarithmically spaced frequencies. In this way the possible
flexible modes of the shuttle and the influence of the transmission lines on the pressure
dynamics, could be observed and evaluated with the system characteristics predicted by the
models obtained in the previous chapter.
In performing the experiments, more care w.r.t. the dummy platform measurements had
to be taken not to hit resonances to badly. Especially because the pilot seats, experiencing
some play in the mounting, appeared to have their resonant behaviour close to the main
rigid modes of the shuttle. Moreover, the high frequency area, containing the transmission
line dynamics, did not allow, already for the audible noise level alone, input voltage levels
necessary to measure the low frequecy rigid mode area with sufficient signal/noise ratio.
A 6 th -order experiment filter was designed to obtain a sufficiently flat response level,
which is also favourable in spreading the relative effect of possible nonlinearities. The measured
bode amplitude plot of this filter is drawn in Fig. 4.12. Having the poles of the filter
10 2

148 4 Parameter identification and model validation
!p1 !p2 !p3 !n1 !n2
2 5 2 35 2 100 2 7:8 2 42
p1 p2 p3 n1 n2
0:7 0:7 0:7 0:25 0:25
Table 4.4: The parameters of the input frequency response measurement filter.
appear just before the lightly damped (notching) zeros on 7:8Hz and 42Hz, gradually reduces
the amplitude level and above 100Hz filters more rigorously. The filter was designed
in the continuous time domain and converted into a digital version with a sample frequency
of 5kHz by a bilinear transformation (in Matlab) for implementation on the motion control
computer. By measuring the response of the filter implemented on the computer, the system
frequency responses can be adequately compensated for. The input level of the experiments
could be set on 2 % but the filter already starts to reduce this level at 6Hz.
The continuous version of the experiment filter, U f (s) is given by
Uf (s) = (s2 =! 2 n1 +2 n1s=!n1 +1)
(s 2 =! 2
p1 +2 p1s=!p1 +1)
(s2 =! 2
n2 +2 n2s=!n2 +1)
(s2 =! 2
p2 +2 p2s=!p2 + 1)(s2 =! 2
p3 +2 p3s=!p3 +1)
(4.39)
with the parameters specified in Table 4.4.
The direct measurements of the system with the shuttle from the actuator inputs to the
input pressure differences show an even higher number of resonances than those with the
dummy platform. However, the structure of the system characteristics is revealed by decomposition
along the eigen directions of the rigid modes predicted through the (rigid body)
model of the shuttle using the mass properties identified by the experiments discussed in the
previous section. This requires pre and post multiplication of the measured 6 by 6 transfer
function by a constant unitary matrix Ush�n and its transpose.
Also with the shuttle and over the larger frequency area, this transformation puts the
dominant responses on the diagonal, which show only one rigid mode per direction, while
all 36 directly measured responses contain every mode in principal. This result and the
model predicted frequency responses are given in Fig. 4.13 and in Fig. A.7 and Fig. A.8
of appendix A all plots are given seperately, while Fig. A.5 and Fig. A.6 of Appendix A
provide the Bode plots for the untransformed frequency data.
Due to the nondiagonal inertia matrix, there are more platform directions involved in
each mode considering the neutral pose than in case of the dummy platform. Still, in this
pose, one can discriminate between symmetric (x� z� -related) and non-symmetric (y� � -
related) modes. Premultiplying the Ush�n-matrix, whose columns describe actuator displacements,
by the inverse jacobian results in a column wise description of the modal direction
in the platform coordinates. Rescaling the columns to a (2-)norm of one, by postmulti-

4.3 Frequency response model validation 149
Amplitude
Phase (deg)
10 1
10 0
10 −1
100
0
−100
−200
−300
10 1
10 1
10 2
10 2
Frequency (Hz)
10 1
10 0
10 −1
100
0
−100
−200
−300
10 1
10 1
10 2
10 2
Frequency (Hz)
Fig. 4.13: Bode plots of the shuttle response redirected along the six estimated decoupling
rigid body modal directions considering the valve input voltages and the valve
pressure difference responses measured (left) and modelled (right). Six by six
separated Bode plots can be found in Fig. A.7 and Fig. A.8 of Appendix A
plying with Ncsc gives:
J ;1
lx Ush�nNcsc =
2
6
4
0:987 0:000 0:000 0:083 ;0:078 0:000
0:000 0:988 ;0:032 0:000 0:000 ;0:064
;0:026 0:000 0:000 0:828 0:676 0:000
0:000 0:153 0:089 0:000 0:000 0:988
;0:159 0:000 0:000 0:554 ;0:733 0:000
0:000 ;0:022 0:996 0:000 0:000 ;0:139
3
7
5
(4.40)
The eigenfrequencies, which belong to these modal directions, are 7:3, 7:3, 14:7, 20:3, 22:0,
24:2 Hz. The (60 th -order) model predicts these frequencies (and its directions) appropriately
as are the transmission lines anti-resonances at approximately 70Hz and the characteristic
due to the (anti)-resonances at 200Hz. This does not include the exact behaviour at
the resonant transmission line frequencies since only the actuator cylinder pressure difference
and not the absolute pressures were taken into account as was discussed in the previous
chapter.

150 4 Parameter identification and model validation
The transfer function from valves flow to input pressure differences is passive despite
the transmission lines, flexible shuttle modes and moving foundation. In Fig. 4.13 this
clearly shows by the phase moving between =2 rad. The valve voltage to valve flow
characteristic with two poles modelled at 150Hz results in the additional phase shift, which
(together with the transmission line resonances and flexible shuttle modes) can result in
stability problems when using pressure feedback.
Another property, which clearly shows, is that the hydraulic actuators are velocity generators
at low frequencies below the rigid body resonance. The Bode plots show that this
requires different pressure levels since the mass experienced along the modal directions
varies. Above the resonant frequencies, the actuators become pressure difference derivative
generators and in this respect the fact that all actuators are similar causes indifference for
direction considering the amplitudes. A dual effect will show in considering the velocity
(accelerating) behaviour over the frequency using constant valve input amplitude.
The characteristics of the foundation having resonant frequencies at 10:2, 10:7 and
12:7Hz do show the same characteristics in the model and measurements. The model parameters
and the evaluation of the foundation causing these resonances were obtained by
performing measurements at the foundation itself, which will be discussed in the next part
of this section.
The main discrepancy between the model and the actual system is the omission of the
flexible modes which show at 33Hz, again, as with the dummy platform probably due to
the radial stiffness of the actuators. Further, new resonances show up most prominently at
43� 57 and 65Hz and are still visible around 100Hz. These resonances were most probably
caused by the flexibility of the shuttle. To confirm this, accelerometers measurements were
performed at several points of the shuttle.
Although the characteristics of the pressure measurements in Fig. 4.13 are for most
well contained in the model, the model does not predict the measurements very accurately
considering the numerical values. This probably requires a more thorough identification
procedure and/or more research into the nonlinear behaviour of the system. The Bode response
is based on a linearised version of the model. E.g. the nonlinear valve characteristics
could be a cause of some of the discrepancies.
The vibrating foundation
As already shown in several Bode plots, the vibrating foundation is part of the model according
to the equations presented in Section 2.4.7. In principle, the parasitic resonance,
which shows up in e.g. Fig. 4.13, can be caused by any flexibly attached mass at either
side of the platform. In experiments and demonstrations it was readily verifiable that also
the floor was moving if the motion system was made to vibrate at frequencies approaching
10Hz.
To confirm that it was actually the concrete foundation, which is approx. 20 times as
heavy as the shuttle of Configuration B., which causes the specific behaviour, and to help
identifying the model parameters and structure, experiments were set up measuring the
acceleration at the floor itself. Creating the experimental environment to perform the measurements
for closer analysis of parasitic mechanical behaviour of the system (so also those

4.3 Frequency response model validation 151
10 0
10 −1
10 −2
10 0
10 −1
10 −2
10 1
10 1
Fig. 4.14: Bode plot amplitude responses of measured (dashed) and modelled accelerations
along the xf (upper left), yf (upper right), x2f at (0,-2) (lower right) and constructed
angular acceleration around the z-axis, ! (lower left) due to input valve
vibration along the first three ’rigid modes’.
100
0
−100
−200
−300
100
0
−100
−200
−300
10 1
10 1
Fig. 4.15: Bode plot phase responses of measured (dashed) and modelled accelerations
along the x (upper left), y (upper right), x 2 at (0,2) (lower right) and constructed
angular acceleration around the z-axis, ! (lower left) due to input valve vibration
along the first three ’rigid modes’.
10 0
10 −1
10 −2
10 0
10 −1
10 −2
100
0
−100
−200
−300
100
0
−100
−200
−300
10 1
10 1
10 1
10 1

152 4 Parameter identification and model validation
in the shuttle itself) have been part of a masters’ project described in [41].
Use was made of the (at that time) new compact accelerometers of Analog Devices, the
ADXL05 1 , which evolved from airbag system initiators and now measure accelerations at a
range of 5g at a resolution of 5mg from DC into reasonable high frequencies (10 kHz).
Three of these sensors were mounted orthogonally in a small package, such that acceleration
measurement in all three directions, xf , yf and zf was possible.
This package was mounted on the floor in between the motion system on three places
performing 6(x6) experiments in total assuming that the floor itself was rigid but flexibly
attached to the inertial frame. Right in between the lower gimbal points as the frame given
in Fig. 2.13, all three accelerations were measured, again while performing a sine sweep
at the valve input voltage for each actuator at the time (between 5Hz and 500Hz usually
stopping the experiment if no motion could be observed anymore above 50Hz). Again, the
motion system provides a nice 6-d.o.f. force inductor to perform the required experiment 2 .
Recording both accelerations and the forces induced is required in identifying the mass
and spring stiffness, which can hardly be identified if only the resonant behaviour in the
e.g. pressure dynamics is taken into account.
At the coordinates (0� -2)[m] and (2� 0)[m] taken in the frame of Fig. 2.13, measurements
were performed in x and z and z direction respectively. All measurements in z-direction
did not show any acceleration (apart from gravitation), allowing to take only three directions
(xf ,yf, f ) into account.
In Fig. 4.14 and Fig. 4.15, the bode amplitude and phase plots of the model and measurements
in the xf and yf direction are given as well as for the x2f and reconstructed f
direction due to the inputs along the rigid modal directions most concerned with the motion
to be observed. Clearly, in Fig. 4.14 and Fig. 4.15 both the rigid shuttle modes can be observed
along with the mode due to the fact that the foundation is not rigidly attached to the
floor. In the neutral state, there is no interaction between the rigid shuttle modes due to the
floor but in other platform poses it can.
Although the accelerometers were relatively noisy, the characteristics observed in the
models can be recognized in the measurements. The model fails to identify a secondary
mode in x-direction (at approx 12 Hz) which is hardly affecting the yaw mode (as it does
with the model). The most relevant behaviour is, however, well described by the model. No
additional effort has been done to more accurately describe the dynamics of the foundation
as the system had to be moved to another building in the end.
The parameters used in the model of the foundation are given in Table 4.5. The total system
eigenvalues, which are due to this part of the mechanics, are shifted somewhat w.r.t. the
values given here for the foundation alone. The direction of this shift naturally depends on
the eigenfrequencies of the rigid body modes of the platform (up in case of x and y, down
in case of ).
The experiments point at the importance of taking into account the dynamic behaviour
of the construction of the building, foundation, best already in design, in order to prevent
1This sensor is not available anymore and has been replaced by the ADXL105 having higher resolution and
temperature compensation
2As already put forward, the hydraulic actuators are no force generators by themselves. Probably, the experiments
could be enhanced by using some sort of force control loop.

4.3 Frequency response model validation 153
Phase (deg)
Amplitude (m/s2/10V)
Direction Mfii [tons� tons m 2 ] !fi [Hz] fi
xf 45 2 10:5 0:035
10 2
10 1
10 0
0
−200
−400
−600
yf 45 2 10:0 0:03
f 182 2 12:5 0:07
Table 4.5: Parameters taken for the foundation.
10 1
10 1
Frequency (Hz)
Fig. 4.16: A typical frequency response with a Bode plot of measured and reconstructed
(from the other measurements) acceleration approx. along the y-axis in the back
left floor position due to a sinusoidal sweep of the input voltage at the 4 th actuator.
this part of the system influence the characteristics of a flight simulator motion system in a
relevant frequency area as it did with the experiments performed at the Central Workshop
of mechanical engineering.
Accelerating a flexible shuttle
At this point only measurements were presented, which indirectly provide insight in the behaviour
of acceleration in the shuttle. By mounting a (minimal) number (of six) accelerometers
onto the shuttle, the rigid body motion, felt by a person sitting in the simulator, can be
observed. Additional sensors can provide insight into the parasitic flexibility resulting from
the shuttle itself.
10 2
10 2

154 4 Parameter identification and model validation
10 2
10 1
10 0
10 2
10 1
10 0
10 2
10 1
10 0
10 1
10 1
10 1
10 2
10 2
10 2
Frequency (Hz)
10 2
10 1
10 0
10 2
10 1
10 0
10 2
10 1
10 0
10 1
10 1
10 1
10 2
10 2
10 2
Frequency (Hz)
Fig. 4.17: Bode amplitude plots of the platform acceleration response due to the valve inputs
along the rigid body modal input directions. Left column x� y� z, and on the
right _!x� _!y� _!z. For each plot one of the modal direction is not dashed. It is
clear that strong interaction and hard to analyze responses result from evaluation
along the platform directions. Much stronger decoupling will result evaluating
along the rigid body modal directions at the output also.
Reconstruction of platform acceleration
To reconstruct the platform acceleration, first the gain, direction and positional parameters
of the accelerometers used, have to be identified. By using a method in which the procedure
in identifying the platform mass matrix and centre of gravity are combined, the sensors can
be characterised. In general, an accelerometer with gain k, mounted at position, r, w.r.t. a
reference frame and directed along the unit vector, n, will output a signal, s, determined by
s = kn T (c ; g + _! r +! (! r)) (4.41)
where c and _! provide the reference frame translational and rotational acceleration.
By performing sinusoidal motion, the accelerational amplitudes, ( c� _!)i, in six directions,
which span the motion space, can robustly be derived from the position measurements.
By correlating with the basic sinusoidal frequency, ! t, noise and the quadratic

4.3 Frequency response model validation 155
velocity term will be very small as in (4.36) and will be neglected. (4.41) also holds for
the ground harmonic amplitudes. With some manipulation using (4.41), the accelerometer
ground harmonic output amplitudes of all the experiments, s, can be written as
s = Am� (4.42)
the multiplication of a known matrix, A, of platform accelerational ground harmonic amplitudes
of which each i’th row is given by
h
Ai� = aT c�i aT i
_!�i � (4.43)
and the measurement device vector, m, with the yet unknown accelerometer parameters,
direction, n, gain, k, and line of action, r, given by
h
m = knT (kn ( g
! 2 t ; r))T i T
(4.44)
Gravity gives an offset to the line of action of the sensor which is inversely proportional with
the squared frequency, ! 2 t , at which the test has been performed. This offset results from
the fact that gravity is constant in the inertial frame and the direction of the accelerometers,
n, moves (and thus rotates) with the platform. If e.g. pitch rotation occurs, gravity will
show at the accelerometer measuring in x-direction proportionally to the sine of the angle
rotated. For small angles the sine operation can be dropped and the angle decreases with
the squared frequency with constant rotational acceleration. By inversion of A, the gain, k,
and direction, n, of the accelerometers can be determined.
m = A ;1 s (4.45)
The position, r, can not, just as in the determination of the centre of gravity. Using only
the linear terms (not the quadratic velocity term), the sensor can be anywhere on a line
parametrized by the free variable , described by
with
r( )= o + n (4.46)
1
o = ;
mT 11m11 (m11 Pnm21) (4.47)
where [m T 11 mT 21 ]T = m T .
With at least two accelerometers, not measuring in parallel directions, in a measurement
device, the position of this device can be defined by the point closest to both lines identified
as r( ). As in the identification of the centre of gravity (4.30), incorporation of more
measurements (accelerometers) can easily be done. In case of the shuttle measurements, a
device with three accelerometers was attached in several positions, p o, as given in Fig. 4.19.
For each position, the accelerometer lines of measurement were identified and from these
lines po.

156 4 Parameter identification and model validation
Amplitude (160Bar/10V)
Phase (deg)
10 1
10 0
0
−100
−200
−300
−400
10 1
10 1
Frequency (Hz)
Fig. 4.18: Bode plot of the diagonal elements of the input valve voltage to output pressure
transfer function along the rigid body modal directions reconstructed from the
accelerometer measurements. As in Fig. 4.13 the system is strongly decoupled
and each plot shows only one rigid body mode. The transfer function to output
(actuator) pressure difference lacks the transmission zeros at approx. 70Hz and
shows phase shift much earlier in comparison with input pressure difference as
output plotted in Fig. 4.13.
Three positions were chosen at the shuttle floor, the part of the shuttle which was expected
to be the most rigid (highest flexible eigenfrequencies). Just behind the pilot’s chairs
in the middle, in the back at port and starboard. At the front side, the ’chin’ of the shuttle,
just below the place where the window is to be placed, in the middle and at the starboard
side, the redundant measurements were taken. The identified positions, using 4Hz sinusoidal
motion, are given in Table 4.6. These positions give some idea of the placing of the
accelerometers. In further calculation also the relative positioning of the measurement lines,
r( ), and their direction should be taken into account.
Assuming a rigid body shuttle floor, the rigid body frequency response can be identified
by measuring at least six accelerometer responses at the floor. To include some redundancy
for evaluation, seven sensors were used in measuring the response to the sine sweep of all
the actuators independently. x, y and z for the mid and y and z for the (port and star) back
10 2
10 2

4.3 Frequency response model validation 157
Position 1:mid 2:port back 3:star back 4:front mid 5:front star
x [m] -0.44 -1.33 -1.32 1.21 1.11
y [m] -0.18 -0.40 0.43 0.07 0.89
z [m] -0.11 -0.11 -0.12 -0.39 -0.39
Table 4.6: Positions identified in the shuttle for the accelerometer device relative to the
centre of gravity in platform coordinates.
4
5
x
y
Fig. 4.19: Approximate positions of the accelerometer device in the shuttle xy plane as also
given by Table 4.6.
positions. Port y is used for evaluation. The other six are used to construct the platform
response by
1
3
2
x = M ;1
p s (4.48)
where each row of Mp is given by m T of (4.45). With the platform response, x, it is possible
to reconstruct all the (linear) parts of the acceleration resulting from the rigid body motion
at all positions of the shuttle.
Reconstructing rigid body acceleration, pressure and velocity dynamics using accelerometers.
Sine sweeps from 5Hz up to 500Hz with 201 logarithmically spaced frequency points were
made with 150mV amplitude send through the experiment filter of Fig. 4.12. All measurements
presented have been compensated for this filter.
Reconstruction of rigid body redundant accelerometer measurement.
In Fig. 4.16, a typical accelerometer frequency response to a sinusoidal valve voltage input
is given. It is the response of the accelerometer measuring in y-direction at the port back
position. In dashed form the reconstructed response is plotted.
Clearly visible are the rigid body resonance at 8Hz (sway/roll), 15Hz (yaw) and 24Hz
(roll/sway). At 10Hz the resonance due to the foundation can be observed. Modes, which
are not identified yet, are visible at 35� 57 and 80Hz. Irrespective of the high number of
resonances, the two responses correspond quite well. Apart from the valve phase drop, the

158 4 Parameter identification and model validation
Amplitude ("m"/s/10V)
10 1
10 0
10 −1
10 1
Frequency (Hz)
Fig. 4.20: Bode amplitude plot along the rigid body modal directions of valve input to the
velocities along these directions reconstructed by the accelerometer measurements.
Again these plots are decoupled showing only one rigid mode per plot
and further the convergence to the typical hydraulic actuator static velocity gain
going to frequency zero shows.
transfer function phase start to drop below =2 around 57Hz and further at 80Hz possibly
due to flexibility between the input (actuator) and the sensor.
Reconstructed dynamics of input valve voltage to platform accelerations.
Step by step the responses can be manipulated such that a structured response results which
can be analyzed more easily. In a first step, Fig. 4.17 provides the responses of the platform
pose accelerations to the inputs along the rigid body directions. Almost all platform
directions (apart from the yaw-direction) are influenced by more than one rigid body mode.
In the neutral pose, where the measurements where taken, it is possible to discriminate between
symmetric and non-symmetric modes (x� z� ) and (y� � ).
Output pressure dynamics.
More fully decoupled responses (up till the flexible resonances) can be achieved by considering
the reconstructed output pressure differences or velocities along the rigid body
directions. The output pressures are reconstructed using a constant premultiplication given
by
dP
^
o = U T A ;1 J ;T
l�x Mx (4.49)
containing the unitary modal direction matrix, U, the actuator area matrix, A, the jacobian,
Jl�x, and the identified mass matrix, M.
10 2

4.3 Frequency response model validation 159
actuator 1 2 3 4 5 6
yaw 1 -1 1 -1 1 -1
chin up 1 0 0 0 0 1
torsional x-axis 1 -1 -1 1 1 -1
Table 4.7: Input directions flexible mode measurements.
In Fig. 4.18 it is shown that all the rigid modes have been decoupled in this way. It
looks like the six accelerometer measurements allow a reconstruction of the (linear part
of the) output pressure difference characteristics. The rigid modes appear on the diagonal
elements of the 6x6-transfer function. For the lower frequencies, there is a considerable
difference in gain between the different directions due to the variation in mass (and thus the
pressure necessary to move with the same speed). Above the rigid mode eigenfrequencies
the gains are expected to be equal apart from the effects due to the parasitic modes.
The three assumably flexible modes at 43, 57 and 65Hz mainly appear along the direction
of the most stiff two rigid modes. Along the pitch/surge mode direction the 43
and 65Hz resonances are experienced. The 57Hz vibration comes in along the roll/sway
direction.
The pressure transfer function constructed from the accelerometers assuming a rigid
construction should be passive apart from the influence of the transmission lines (and of
course the valve characteristics). Passive behaviour is certainly lost at approx. 60Hz where
the phase along the direction of the pitch/surge mode (6) drops another rad.
Velocity dynamics.
Another alternative to reconstruct a decoupled transfer function is to use the velocities along
the rigid body modal directions. These frequency responses are plotted in Fig. 4.20. These
plots show the fact that for low frequencies, the hydraulic actuator is a velocity generator.
This enables easy evaluation of the differences in the rigid mode resonance gains. It also
shows that for the most stiff modes, the acceleration at the frequencies above the rigid body
modes is the highest. Probably the (high frequency) flexible modes appear mainly along
these directions due to the fact that they are best measured along these directions.
Flexible phenomena.
A closer inspection of these flexible modes was done by evaluation of the additional accelerometer
measurements taken at the front of the shuttle. From visual inspection of the
shuttle, it can be expected that the weakest part of the shuttle appears at this point where
a large hole has been provided for the front window. Possibly torsional or chin up like
deformation could be induced. By comparison of the actual accelerations measured at the
front and those reconstructed from the (back) floor assuming rigid motion, this has been
investigated.
In Fig. 4.21 three examples of actually measured and reconstructed frequency responses
from inputs along some specific directions are given, some of which are expected to induce
energy for the flexible modes. These directions are given in Table 4.7.

160 4 Parameter identification and model validation
10 1
10 0
10 −1
10 1
10 0
10 −1
10 1
10 0
10 −1
10 1
Frequency (Hz)
10 2
0
−200
−400
0
−200
−400
−600
−800
0
−200
−400
−600
−800
10 1
Frequency (Hz)
Fig. 4.21: Bode plots of measured (dotted) and reconstructed rigid part of the acceleration at
front middle in approx. y (upper row), x (middle row) and y (lower row) direction
due to input vibration along the yaw, ’chin up’ and ’torsional x-ax’ directions
respectively. Left column provides amplitude and right column provides phase
responses. Differences in measured and reconstructed plots can point at flexible
modes.
10 2

4.4 Chapter Resume 161
In the yaw direction it can be observed that the accelerometer output at front can be predicted
fairly well by the accelerometers at the back of shuttle on the floor. However, none
of the flexible modes to be investigated (43,57,65 Hz) is very pronounced in this direction.
For the chin up input, the reconstructed and measured output for the accelerometer mounted
for the x-direction result in totally different frequency responses. In the reconstructed frequency
response, it is not expected that the modes at 43 and 65Hz will be observed while
they appeared to be very pronounced in the measurements at this point. Finally, with the
torsional x-axis input it is shown that around 57Hz the phase of the bode plots starts to
differ 180 . The measurements have a complex zero pair like characteristic at 50Hz while
the reconstructed response has not.
In many ways, the reconstructed and measured responses differ. Too much to closely
predict the characteristics of the flexible modes very accurately at this point. In some directions,
it seemed the accelerometers were not sufficiently excited by the experiments performed.
If exact knowledge of the flexibilities is required, more thorough modelling together
with further experiments is necessary. In this research the parasitic resonances were
mainly investigated in order to give a proper view on the extent to which the models not
including these effects are valid.
Resume
Through frequency response model validation it was shown that the model structure
proposed fairly well describes the characteristics of the actual system.
Especially the rigid body modes and directions resulting from the interaction of hydraulics
and mechanics, which were predicted rightly to dominate the response, are
well caught by the model.
Low frequency ( 10Hz) parasitic behaviour could be pinpointed to the dynamics of
the floor, also well described by the model including the floor.
Accelerometers could be used to reconstruct rigid body platform accelerations, velocity
and pressure dynamics and point at possible flexibilities, after careful identification
of accelerometer line of action and gain.
The shuttle flexible modes could be assigned to part of the high frequent parasitic
resonances but could not yet be included in the model structure. This will also require
more rigorous measurement to allow accurate mode shape identification.
4.4 Chapter Resume
In this chapter it has been shown that the parameters of the models derived on physical laws
could be identified by performing a very limited number of experiments. The responses
of the kinematical model of the motion system did have a very close match with the measurements
taken. The characteristics of the dynamics modelled for the hydraulically driven

162 4 Parameter identification and model validation
mechanical system does correspond fairly well to the actually observed dynamics up till
reasonable high frequencies ( 30Hz) if the dynamics of the foundation is taken into account.
The properties due to the valve and transmission lines in the hydraulic system are
still important for the over all motion system. But at the frequencies where these effect have
their main influence ( 30Hz) additional parasitic resonances due to the mechanics have
to be taken into account, especially in case of the light weight shuttle.

Chapter 5
Model based control of the flight
simulator motion system
5.1 Introduction
In previous chapters it was shown through theoretical modelling and experimental verification
that in systems like the Simona flight simulator motion system, the dynamics due to the
hydraulic actuators and the parallel driven mechanical system with six degrees of freedom
form an integrated structure with typical characteristics, which can be modelled with reasonable
complex equations. This is in practice the most elaborate part of design in using a
model based control strategy as laid down in this chapter.
Referring to the discussion in Chapter 1, the eventual task of high performance simulator
motion control is improved motion realism. With flight simulator motion systems one wants
to provide the pilot in the simulator with appropriate generalized specific forces i.e. both
rotational and translational accelerations plus gravity [92]. Particularly in case of tight pilot
control, e.g. landing, phase lag of realized versus desired accelerations should be minimal.
In this way smaller differences between simulator and in-the-air flight conditions will exist.
In flight simulation, the pilot relies on the perception of self-motion through several
stimuli, and uses this perception to exercise control over the aircraft. Stimuli cueing systems
include the visual system, the motion system, the audio system, control loading and the
aircraft instruments stimulation. In the control task, the pilot uses his visual perception to
make a good estimation of the aircraft’s attitude and velocity. The approximate frequency
response of visual perception can be modelled as a first order low-pass filter with a break
frequency of 0.1 Hz [92], which is rather slow.
Motion in the higher frequency area is, however, mostly perceived by the pilot’s vestibular
and tactile sensors, which are sensitive to specific forces and angular accelerations.
These signals are processed rapidly by the central nervous system and, therefore, give the pilot
lead information. With this lead information, the pilot can react more quickly to changes
in the vehicle motion state. This information is most important at the higher frequencies
(above the bandwidth of visual perception). These high-frequency motions are, in simulation,
often called onset-cues.
163

164 5 Model based control of the flight simulator motion system
Because of the high-frequency nature of these onsets, it is important that the motion
system has sufficient bandwidth. Moreover, it is important that the time-delay in the motion
simulation is kept as small as possible, since an onset is also time-critical. If the onset is
simulated noticeably late, simulator training quality will be decreased considerably. Time
delays, furthermore, lower the pilot-vehicle crossover frequency and may require the pilot
to adapt by applying lead compensation. The reduction of the adaptation required is
precisely one of the goals of future flight training [7]. Therefore, a primary objective of
a flight simulator motion system is to provide the pilot in the simulator with appropriate
representations of, what one could call, the ”generalized forces” i.e. both translational and
rotational accelerations and gravity. This has to be attained by driving the simulator with
six parallel hydraulic servo actuators. With the models of the SRS, an important basis for
high performance motion has been attained further helped by construction design for control
including low-mass, low centre-of-gravity, and a high-stiffness structure. Its potential
can only be fully exploited by advanced controller design techniques such as used in the
area of robotics.
Another necessary requirement is the application of appropriate hardware components,
like fast multi-processor Digital Signal Processor (DSP) boards, and software which performs
automatic DSP-code generation from higher level programs e.g. Matlab/Simulink.
Thereby, it is possible to use highly structured complex control strategies which take model
knowledge into account. This includes fast iteration on controller design methods consisting
of setting the specifications, analyzing the system, synthesising the controller and
implementing on the actual experimental set up.
5.2 Another look at the control problem
Since the motion controller to be designed has to be implemented on a complex system, the
previous chapters dealt with extensive modelling, which took place with the control objective
in mind, even before construction [74], [124]. In the earlier stages measurements could
be done on an experimental set up in which each hydraulic actuator was tested separately.
Further, tests were performed with a dummy platform and the empty shuttle replacing the
eventual simulator on top of the motion system [75].
Model based control strategies such as the computed torque like methods are directly
applicable for mechanical systems modelled in the standard structure of explicit differential
equations whose states can be measured and whose control inputs are the torques which are
aligned to the model coordinates. So, in applying a computed torque like technique, at least
the following issues have to be regarded.
- Model equations: A model-based method requires modelling. Modelling parallel systems
typically can result in combined constraining algebraic and differential equations.
For instance, the motion of an actuator mass in the Stewart Platform, will be
fully dependent on the motion of the platform itself. Preferably the model equations
will have to be parametrized in such a way that an explicit differential structure results.

5.2 Another look at the control problem 165
- Model coordinates: Parametrization of the model also includes choice of the model coordinates.
In the Stewart Platform one can choose end-effector/ platform coordinates,
input coordinate/actuator length or any other coordinate system to describe the 6 degrees
of freedom. Parallel systems are usually and most conveniently modelled in
end-effector coordinates. In general these are not the coordinates which can be directly
measured or steered.
- Extent of modelling: Every system can be modelled in several ways and in various degrees
of accuracy and complexity. In case of the mechanics of the Stewart Platform
the simplified model of (2.112) could be taken. But next to the simulator body also
the actuators themselves have mass properties which can not always be neglected.
- Measurements: A full state measurement is assumed. In mechanical systems this means
that a number of independent positions and velocities equal to the number of degreesof-freedom
of the system has to be measured. On top of this, not only position and
velocity measurement are assumed to be taken, but it is also assumed that these quantities
directly represent the model coordinates.
With the Simona motion system only actuator lengths (next to actuator pressure) are
measured. So not only the velocity but also the appropriate model position or platform
pose have to be constructed somehow.
- Control inputs: The same goes for the control inputs. These are assumed to be the generalised
forces along the model coordinates. These would be the platform forces in
case of the Stewart Platform. Since the actuator forces have several (position dependent)
components in platform coordinates, these can not be considered to be the
required control inputs using a standard feedback linearisation scheme without any
further manipulation.
Also the forces will have to be generated by an actuator which in practice cannot
apply required forces instantaneously. Actuator dynamics has to be looked into. In
case of the Simona motion system, the actuators are hydraulic servo systems. This
kind of actuators can not be considered force generators as such.
Direct application of the feedback linearising control on the motion system requires
consideration of the various problems observed. Other aspects also need to be looked into.
In the sequel several issues will be discussed, and where necessary the control strategy will
be modified.
By analysis of both theoretic models derived from basic physical laws as experimental
models based on measurements taken, an inventory of the relevant control problems can be
put together.
Control objectives
Flight simulation or fooling a pilots motion awareness basically forms a control problem
with mixed objectives. The system should provide for the accelerations being
simulated without running out of stroke. This problem is mainly left to a host which
has to come up with feasible trajectories but the motion controller still has to both
track reference accelerations as to stabilize platform pose.

166 5 Model based control of the flight simulator motion system
Hydraulics.
Control of the long-stroke hydraulic actuators used in flight simulator motion systems
is not easy since the phase lag introduced by the servo valve together with the nonnegligible
high frequent transmission line resonances form a stability problem [125].
Hydraulics/Mechanics
Further, the bilateral coupling of the hydraulic mechanical system with strong energy
exchange via pressure/flow and force/velocity introduces mechanical pose dependent
resonances with interaction over the actuators. As was shown, the dynamics resulting
from this interaction forms the most relevant part of the system.
Mechanics
With the actuators mounted in parallel to the simulator, the construction forms a Stewart
platform. Due to the resulting kinematic loops, care should be taken to model this
system with only explicit differential equations [74]. This required modelling in appropriate
coordinates i.e. the platform pose. As only the actuators lengths are being
measured, a transformation to platform pose was required to be able to apply model
based control. This transformation is, dual to the actuator trajectory generation of serial
robots, explicitly known from platform pose to actuator length but not injective.
Trajectory generation
The reference acceleration and pose will be calculated at a low sample rate host computer
which incorporates a complex airplane model. References have to be introduced
smoothly to the motion controller. But smoothing the signal should not result in responses
with too much phase lag since the timing of on-set motion is an important
part of simulation quality.
In the next section, a control strategy will be described, which takes into account the
afore mentioned problems and resulted in a controller which could be implemented on a
real-time control computer connected to the motion system.
5.3 Control Strategy
The control strategy will be targeted at generating the appropriate accelerations for the simulation
and at the same time stabilising the platform pose. Taking into account the reference
generating system, e.g. the airplane, enables sufficiently smooth accelerating trajectories.
Now, acceleration of the simulator mechanical system results from the forces applied.
Through a model of the mechanical system the necessary forces can be calculated. Minimal
invasive but stabilising position feedback is attained through use of a decoupled coordinate
system. The hydraulic actuators can generate the required forces but in the previous
chapters we observed that they face strong interaction from the mechanics. By supply of
appropriate oil flow the hydro-mechanical coupling can be minimized and local pressure
feedback enables the desired force generation.
By design of a control structure which has four levels, each of which have their own
specifications in close relation with the other levels, one can circumvent having to solve

5.3 Control Strategy 167
Fig. 5.1: Multi level control of the Simona Motion System.
1. Inner loop feedback: Each hydraulic actuator is feedback transformed into a
force generator by the inner loop feedback using the input pressure, dp i, and estimated
velocity, ^ _q i, of an actuator. Input to these subsystems are the desired input
pressures, dpi�d.
2. Feedback linearisation: The desired pressures are calculated using a model (2a)
of the multivariable mechanics, which calculates gravity, centripetal and coriolis
force compensation and uses the mass matrix to calculate the required forces for
acceleration. This block uses the measured platform state (pose, sx, and estimated
velocity, ^ _x) to do this feedback linearisation step. This state is calculated on line
from the measured actuator position and velocity in an iterative manner (2b).
3. Outer loop feedback: This is outer loop state feedback of the platform coordinate
and velocity errors, the difference between the desired xs dr and _xdr and
reconstructed coordinates. Corrective accelerations, xc, result from this block and
are fed to to feedback linearisation block (2.).
4. Reference model-based feedforward: Smooth and possibly predictive acceleration,
velocity and position, xdr� _xdr� xs, are reconstructed from the desired accelerations
supplied by the host, xd. They are output to the feedback linearisation
block to be used for feed forward.
a too complex set of problems at once. Further, it is shown that this structure leads to an
implementable controller.
General idea
Looking at Fig. 5.1, in which the control structure is schematically depicted and described,
we consider the following control levels with reference to the applied control theory [76].
Level 1. Local hydraulic pressure control loops [49], [124], [127].
Level 2. Multivariable feedback linearisation [71], [133].
Level 3. Outer loop position stabilisation [113].

168 5 Model based control of the flight simulator motion system
-Llm -B
R R
- V - ? d- - d?
- M - -
6
1
C
- Ap J -
u dp
f
_x
T -
-Ap
Fig. 5.2: Basic structure hydraulically driven motion system.
Level 4. Reference model based feed-forward [72], [111], [144].
In short, the actuators are turned into pressure generators by local controllers. These
controllers receive their reference pressure from a feedback linearisation loop in which pressures
can be calculated necessary to track desired accelerations. Desired accelerations are
partly corrections which are required to stabilise the pose of the simulator and for most
the cues generated to provide the pilot with reasonable motion awareness. As these cues
have to be smoothed but not delayed, a reference model based controller has to calculate
appropriate cues for the feedback linearisation controller.
The next sections will describe the different control levels more closely.
5.4 Inner loop pressure control
To turn the hydraulic actuators into nice force generators two of the afore mentioned control
problems have to be solved at this level. Feedback of the pressure can result in stability problems
since the relatively long transmission lines cause badly damped resonances together
with phase lag of the valve. Further, the coupling between the mechanics and hydraulics
results in the pose and load dependent rigid modes of the system.
As shown in Fig. 4.13, for the shuttle the rigid modes can be observed in the frequency
area between 7Hzand 25 Hz. At200 Hz the transmission lines cause peaking and at 75 Hz
also a notch results. The valve has a bandwidth of 150 Hz which can clearly be observed by
looking at the phase. Flexibility in the mechanics caused some additional parasitic modes
between 40 Hz and 80 Hz.
The coupling from which the rigid modes result, can be dealt with using the control
method introduced by Sepehri [127] and successfully applied by Heintze [49]. A hydraulically
driven motion system basically has the structure given in Fig. 5.2. Through the valves,
V , the oil flows, , can be steered by the inputs, u. The required oil flows are mainly determined
by the speed at which the volumes in the actuators have to be filled. These are
equal to the velocities, _q, of the actuators times the area of the piston, Ap. Together with
the oil loss due to the leakage Llm, the net oil flow difference cause the pressures dp to rise
through the hydraulic oil stiffness, C.
fa
_q
J
_
dp = C(V u ; Llmdp ; Ap _q) (5.1)

5.4 Inner loop pressure control 169
dp d
?
-1
V ;1
d
d?-
- -
6
- Kdp
Inner loop
controller
A ;1
p
6
-
u
-Llm
V - ? d-
C -
6
Fig. 5.3: Basic structure inner loop controllers. On the right, the hydraulic part of the system
structure of Fig. 5.2 is given. On the left, the inner loop controller, the cascade
dp structure, compensates for the influence of the mechanics and controls the
pressure, dp, which results in a theoretically arbitrarily fast first order response dependent
on the controller feedback gains in K dp. In case of the SRS, _q can not be
measured and has to be reconstructed.
The acceleration of the actuators is determined by the inverse mass matrix, M ;1 , which
causes the interaction, times the forces supplied by the actuators minus the viscous friction
e.g. b due to each hydraulic bearing, B = J T l�x bIJl�x, with Jl�x as the pose dependent
jacobian.
R
dp
-
Ap
-Ap
Mx = J T l�x Apdp ; B _x (5.2)
The actuator and platform velocities are related through _q = Jl�x _x. Gravity forces and the
less relevant coriolis and centripetal forces are assumed to be dealt with at the higher levels.
In Fig. 5.3, the basic structure of an inner loop pressure controller, which decouples the
mechanics from the hydraulics, is given together with the hydraulic part of the system of
Fig. 5.2. By compensation of the oil flow due to actuator velocity, the hydraulics can be
decoupled from the mechanics.
A smooth 50 Hz bandwidth pressure generator was obtained by filtering the pressure
feedback signal properly for a one degree of freedom set up in which the hydraulic actuator
were all tested separately. As the inner loop controller should not interact with the mechanics
its characteristics could be designed and tested in this setting at first instance [124]. In
this way the hydraulic servo actuators, which usually are considered velocity engines, are
approximately turned into force generators. However, as already put forward, the measured
pressure difference (at the valve) does not reflect the applied force over the full relevant
frequency area due to the transmission lines. Already at 30 Hz, this difference amounts to
25%.
The structure of the implemented inner loop controller as given in Fig. 5.4 is somewhat
different from the basic structure given in Fig. 5.3. The structure implemented on the single
hydraulic actuator as was used by Van Schothorst [124] is taken. As the velocity of the
hydraulic actuator is not measured directly it has to be reconstructed and three filters, C 1,
C2 and C3 have been added to deal with the transmission lines and the valve dynamics.
-
fa
_q

170 5 Model based control of the flight simulator motion system
Roughly the filters remove signal content from the frequency area above 100Hz where
the valve and transmission line dynamics are most prominent and only mildly change the
characteristics up to 30Hz where the rigid body dynamics resides.
The linear dynamic filters, C1�C2�C3, used for the platform are given in the frequency
domain by
and for the shuttle
C1(s) =
C2(s) =
C3(s) =
1
1
(2 95) 2 s 2 +
1
1
2 120
C2s(s) =
s +1
1
(2 200) 2 s 2 +
1
(2 400) 2 s 2 +
1
2 0:3 s +1
2 95
2 0:2
2 200
1 s +1
2 120
s +1
2 0:2
s +1
2 400
1
(2 65) 2 s 2 +
1
(2 65) 2 s 2 +
1
1
(2 150) 2 s 2 +
2 0:015
2 65
2 0:13
2 65
2 0:6
s +1
2 150
s +1
s +1
(5.3)
(5.4)
(5.5)
(5.6)
a notch had to be added due to the parasitic resonance of the mechanical system at this
frequency, which should not be hit. Implementation was done through the standard bilinear
transformation to digital filters running at 5kHz.
The flexibility of the shuttle, which enters well above 30 Hz has still to be taken into
account. An illustrative picture is given in Fig. 5.5. Nyquist plots of the characteristic
loci are given reconstructed from the 6x6 frequency domain measurements taken for the
shuttle by taking the eigenvalues of the measured transfer function, T , (T (j!)), for each
frequency, !. Recall that since the multivariable system can approximately be decoupled
by a unitary matrix at each operating point, Nyquist plots can be constructed, which can be
analyzed as in the SISO case as given in (3.47).
In the upper left part of Fig. 5.5, no feedback filter is applied and a pressure feedback
gain of kdp = 0:35 is chosen. All rigid body modes and flexible resonances remain
’circling’ in the right half plane as our passivity analysis of Section 3.3.1 predicted. The
additional phase lag of the valve rotates the transmission line dynamics into the left half
plane encircling the stability point, ;1.
The gain of kdp =0:35, is well below the achievable gain of kdp =0:75 reported on
in the SISO case [124]. For the SISO case this gain could safely be used in combination
with the dynamic filters. The parasitic dynamics due to the flexibility of the shuttle, however,
complicates the separation of low (rigid body modes, foundation) and high frequent
phenomena (valve and transmission lines). The upper right plot of Fig. 5.5 still shows a
resonance (at 65Hz) encircling the stability point. An additional notch solves the stability
problem as shown in the lower left and right plots of Fig. 5.5 although the resonances at 43
and 57Hz become more prominent.
In theory, measured velocity compensation should remove all influences, also the flexibilities,
from the pressure dynamics. As velocity could only be estimated with limited
accuracy, a considerable amount of this compensation was done with the required velocity
instead. This leaves the pressure feedback dynamics for a large part unchanged.

5.4 Inner loop pressure control 171
dpi�d
C1
- -+ ? - -+ -
+
6
C3
e Kdp
e
C2
-
Fig. 5.4: Implemented structure inner loop controllers with velocity observer and additional
filters, Ci. The output, ^ i, of this controller part is still an intermediate control
variable, which is input to the valve flow compensation module of (5.7).
IMAG
IMAG
5
0
−5
−10
−10 0 10 20
REAL
5
0
−5
−10
−5 0 5 10
REAL
IMAG
5
0
−5
K _q
6
_qobs
−10
−5 0 5 10
REAL
−1.5
−1 −0.5 0 0.5
REAL
Fig. 5.5: Largest characteristic locus for pressure feedback with just feedback gain (k dp =
:35) (upper left), feedback filter applied (upper right), feedback filter with notch
for the flexible platform (lower left and right).
IMAG
0.5
0
−0.5
−1
dpi
- ^ i
_qref
dp i
q

172 5 Model based control of the flight simulator motion system
10 0
10 −1
0
−100
−200
−300
10 1
10 1
Fig. 5.6: Bode plot of pressure feedback filter (C 2KdpC1) with flexible platform.
Given the normalized stiffness, C =2Cm 220, similar as in Table 3.2, this results in
a bandwidth of 2Cmkdp
=12:2Hz in case only the first order pressure system is considered.
2
A Bode plot of the resulting feedback controller is shown in Fig. 5.6.
As discussed in Section 3.2.1, the valve has a nonlinear characteristic given by (3.16),
especially in the face of load variations. Van Schothorst [124] proposes a control structure
u =
10 2
10 2
f ;1
2 (^ i)
q 1 ; sgn( ^ i)dpi
to compensate for these effects. As this was shown to be successful experimentally, it was
made part of the inner loop controller for the SMS.
(5.7)
Velocity estimation
The positive velocity compensation loop requires knowledge of the actuator velocity. Position
and (indirectly through pressure difference) applied force are measured. Further, it
is known at what velocity the actuators should run. Again, in the thesis of Van Schothorst
[124] a range of velocity observers is presented. The most simple version consists of differentiating
the position signal (with some low pass filtering). With the application of the
increased resolution position measurement [55], this version seemed feasible. However, the
position measurement appeared to have a (complex deterministic) error of 100 m in amplitude
(and 4:3mm period length) which could result in severe vibrations running at higher
velocities.
Therefore it was decided to use the other sources of information as well, i.e. the measured
pressures and the required actuator velocities, which can be calculated by v des =
Jl�x _xdr. A simple second order filter was used to attain cut off at both low (no bias) and

5.5 Multivariable feedback linearisation 173
high (less noise) frequencies for all three inputs of the velocity reconstructor.
1
2 0:7
2 6 svdes + sq
(2 6) ^_qobs =(1; )
2 s^q dpi +
1
(2 6) 2 s2 + vdes (5.8)
2 0:7
+ s +1
2 6
The acceleration of the actuator, q, had to be reconstructed at the higher level using the
information on the mass properties of the system and the measured pressure differences
at the valves. The velocity compensation gain was set at k _q = 0:9, just safely below the
theoretic value of 1 as the velocity valve gain is still slightly nonlinear. Further, in practice
the feedforward part, vdes, had to be set to 50% , i.e. =0:5.
The parametrization of the inner loop control, which was implemented, has more or less
been attained through loop shaping. A more structural approach has been taken in [150] by
doing inner loop control design using robust control techniques. This unfortunately did not
lead to controllers which could be implemented safely yet. Additional effort in this direction
is required.
5.5 Multivariable feedback linearisation
Considering the actuators as smooth force generators, a task of the upper level is to come
up with proper reference forces. These can be calculated using the model of the mechanical
part of the system. A feedback linearisation structure results.
5.5.1 Feedback linearising robotic manipulators
General feedback linearisation as described in [105] can for rigid body robotic manipulator
models be given in a computed torque control structure [133] or an inverse dynamics control
structure [31]. In [77] it is shown that many of such differently named control strategies in
fact lead to the same control structure.
For a rigid body serial robotic manipulator, the controller is constructed as follows. See
Fig. 5.7. Given the general equations of a mechanical model of such a system
M (q)q + C(q� _q) +G(q) = (5.9)
in which similar terms appear as in (2.112). The torques/forces, are assumed to drive the
system. In the computed torque controller all terms in (5.9) are compensated for according
to
= M (q)(q d + c) +C(q� _q) +G(q) (5.10)
which leaves decoupled double integrators.
This linear system is achieved by compensation with terms, which depend on the measured
state and is therefore called feedback linearisation.
The desired accelerations, q d, can be steered directly, and the double integrators can be
stabilised independently by the outer loop, e.g. a PD position feedback controller for each
input-output channel

174 5 Model based control of the flight simulator motion system
Fig. 5.7: Feedback linearising control of a robotic manipulator
c = K(s)(qd ; q): (5.11)
As the feedback linearising control strategy provides a suitable feedforward path for
desired accelerations and turns the non-linear multivariable structure into an easy to control
decoupled double integrator system, the method should be able to deal with most of the
specifications set for the flight simulator motion system [83].
However, in the theory presented quite some assumptions are implicitly made. These
have to be checked in applying the method. In the next section it will be put forward that
with parallel robotic systems the assumed model structure does not fit in general. The
control method has to be modified, which requires additional analysis.
5.5.2 Feedback linearisation of a Stewart platform
Also with parallel motion systems, the force generators should be used to control the nonlinear
and multivariable mechanics. With model based calculation of the required forces to
accelerate along the desired path, xd, given a measured pose and velocity, the system is to
be provided with both feed forward and decoupled feedback linearised correction paths to
be used by the higher level controllers.
The proposed control structure is given in Fig. 5.8. This structure differs from the standard
computed torque controller of a mechanical system. In modelling for control the parallel
Stewart platform configuration, one has to take care of generating an explicit set of
differential equations [74]. Since this is only possible taking the platform pose as the generalised
coordinates, the controller has to incorporate an algorithm which calculates these

5.5 Multivariable feedback linearisation 175
Fig. 5.8: Modified feedback linearising control structure
coordinates from the measured actuator lengths, l, and translates desired platform forces
into required actuator pressures.
The actuator lengths, l can be calculated from the platform pose sx.
l = f(sx) (5.12)
As discussed in Chapter 2, in measuring the actuator lengths, the platform pose has to be
reconstructed iteratively e.g. by (2.58).
With the jacobian, Jl�sx, which was defined by
sxk+1 = sxk + J ;1
l�sx (sxk)(lmeasured ; lk) (5.13)
Jl�sx(x) = @l
@sx
(5.14)
In Chapter 2 it was shown that this iteration converges sufficiently fast if some requirements
are fulfilled.
The desired actuator pressures can be constructed by calculation of a platform mass
matrix M, coriolis and centripetal forces C _x and gravity G and filling in the desired platform
accelerations, xd for the simulation plus the corrective accelerations, xc, from the outer loop
position control. These are functions of the reconstructed platform pose and velocity. The
simplified model of the Stewart platform taking into account the actuator inertia in addition
to the platform mass matrix given in (2.112) will be considered and further the actuator
forces are approximated by fa = ApdP o ApdP i, where Ap is the normalised operational
area of the hydraulic actuator. Multiplying (2.112) by J ;T
l�x , the desired pressure is given by
dP i�d =(ApJl�x) ;T (Mt(xd + xc) +Cc( _x� sx) _x + Gc): (5.15)

176 5 Model based control of the flight simulator motion system
In the first approach, filling in design values for these parameters like masses, inertias and
centres-of-gravity already results in reasonable performance by a considerable reduction in
interaction. This was further improved by taking identified model parameters. With the
parameters of the kinematical model of (4.1) calibrated in Section 4.1.4 the jacobian, J l�x,
can be calculated, as follows from (2.42). The identified platform mass matrix M t of (4.31),
including actuator intertia constant w.r.t. the platform, is taken instead of the more limited
one body mass matrix Mc of (2.112). By using the identified center of gravity in (4.30) as
origin of the platform coordinates, the model parameters become relatively simple. E.g. as
seen in (4.26), the gravity vector force, G c only consists of the identified mass, mg, times
gravitational acceleration g in z-direction.
The inverse jacobian calculation was performed at 1kHz twice as required for the implicit
state measurement. Inversion was implemented through a LU-factorisation with full
pivoting as given in [112].
Gravity compensation is easily made explicit for the actuators by use of (the third column
of) the inverse jacobian information as
G = ;J ;T
l�x ( � 3)mg� (5.16)
Easily derived from (2.111).
For coriolis and centripetal forces, only the nonlinear term given in (2.111) at platform
force level by
fc =! (Ic!) = ~ Ic! (5.17)
was taken into account. Thereby neglecting the (small) actuator related nonlinear inertial
terms.
5.5.3 Implicit state measurement requirements
As the iteration (5.13) reconstructing the state is part of the feedback loop, it has to converge
at all times and moreover sufficiently fast in order to prevent the system from going
unstable. This problem has been considered in detail in Section 2.3. Summarising, the convergence
properties of the implicit state measurement by the NR-scheme can be looked into
by considering the weak Newton-Kantovorich theorem given by Stoer [141], stating that a
NR-scheme results in a well-defined sequence in a limited area with a solution (limit point)
to which it converges with a guaranteed speed if the following three properties hold.
1. The Lipschitz condition on the jacobian This condition implies that a limited difference
between two platform poses should result in a limited difference between the
two jacobians at those points.
kJl�sx(sx1) ; Jl�sx(sx2)k ksx1 ; sx2k (5.18)
2. ’Away’ from singular positions This means that the smallest gain of the jacobian should
be sufficiently far away from zero:

5.5 Multivariable feedback linearisation 177
kJ ;1
l�sx (sx)k (5.19)
With Stewart Platforms this is not trivially true. Singular points exist in the construction
if the actuator lengths are not constrained by limited stroke [97]. For example, if
a platform is moving down until all legs are in the XY-plane, no forces in Z direction
or moments around X and Y axis are possible.
Singularity can be excluded by considering a nonsingular point and the Lipschitz
condition. This excludes a (possibly small) volume from singularity, and the whole
workspace can be proved to be free of singularities by gridding. In this way the
workspace of the Simona motion system was proved to be free of, and even far enough
from singular points.
3. ’Near’ to solution The third and last condition requires an initial guess which is not too
far away from the real solution:
kJ ;1
l�sx (lmeasured ; lk=0)k : (5.20)
With the described control structure implemented on a digital computer with given
sample rate (> 100 Hz) and limited speed of the system, the previous solution can
be used as an initial guess which can be proved to be close enough. The hydraulic
actuators are physically limited in speed.
Since the three conditions can be proved to be true for the Simona motion system, as was
shown in detail in Chapter 2, the proposed control structure can be used without stability
problem concerning the iterative part of the controller. Given the speed of convergence, the
solution is close enough (within sensor accuracy) after two steps. The update frequency is
easily attained by calculating two iterations at 1 kHz.
5.5.4 Outer loop control
Considering Fig. 5.8, the transfer function from our input, the desired accelerations, x d, to
the platform pose without the feedback path K(s), does contain unstabilized double integrator
paths. The outer loop controller, K(s), will have to stabilise the simulator pose to prevent
the actuators from running out-of-stroke. As the feedback linearising controller decouples
the mechanics into separate double integrators, the outer loop can generate correction accelerations
resulting from a PD-structure (kpos + kvels) to stabilise these integrators.
The output of the outer loop controller, K(s), can be considered as an extra desired
acceleration, additional to xd, to correct the errors in platform pose. These correction accelerations,
xc, should not exceed human sensory thresholds [54] i.e. generate no noticeable
false cues. Therefore the correction should ideally be sufficiently smooth (filtered) and
only requires limited bandwidth (well below 1Hz). In practice, a bandwidth of 2Hz was
chosen necessary to achieve sufficient suppression of the disturbances and for most the unmodelled
dynamics. With the normalised signals, the feedback gains are easily chosen at

178 5 Model based control of the flight simulator motion system
kpos =158=(2 2) 2 and kvel =17:5 for a damping of =0:7 for each platform direction.
The measured platform velocity is constructed from the actuator velocity estimation
multiplied by the inverse jacobian. The angular velocity, which follows from this multiplication
can then be used to calculate euler parameter velocities.
The outer loop controller are split up into translational and rotational platform coordinate
controllers. The outer loop for the translational directions is given by
cc = kpos(cd ; c) +kvel(_cd ; _c) (5.21)
For the rotational directions, the nonlinear structure for the parameters used for the description
of the rotation should be used. In this case the euler parameters are taken and as
= G T s _!=2, equivalent to (2.22), a double integrator structure of the euler parameters can
also be considered in the platform structure of Fig. 5.8. The appropriate corrective angular
acceleration to generate a corrective euler parameter acceleration can be found through
where the corrective euler parameter acceleration, T
control structure on the reduced euler parameters
and the reconstruction of 0�c is found by
_!c =2G( e) e�c� (5.22)
e�c = [ 0�c
T
c ], is taken as the P(I)D
c = kpos( d ; )+kvel(_ ; _) (5.23)
0�c = ;( T c + _ T
e _ e)= 0� (5.24)
twice differentiating (2.19). Of course, using this strategy, rotations, , should satisfy
< . A disadvantage of this structure is the nonlinear structure of the euler parameters.
This results in different response on similar rotational errors in different poses. In [45]
an alternative method is proposed, which controls difference rotation matrices but it is not
clear yet how to incorporate integral control in this structure.
As the pressure generators could not be made an order of a magnitude faster than the
outer loop, the references for the outer loop (the desired velocity and position) were made
to lag by using a second order filter with the estimated bandwidth of the acceleration generation
(12Hz). Otherwise the outer loop would start to compensate errors, which were
already taken into account.
Although a P(I)D-structure of the outer loop controller robustifies the system [113],
explicit robust control will have to be used to more accurately deal with the varying system
conditions one encounters working within a real-time environment. A first step towards
such a controller was done as part of a master thesis research [122], but this did not lead to
controllers, which could be implemented safely. So also at this point many research aspects
are still open.
5.6 Reference model based feed forward
The motion control computer is provided with desired simulator accelerations ( xd) and
poses (sxd) by a host, which controls the over all simulation (including visual, instrumental

5.6 Reference model based feed forward 179
and acoustic stimuli). These signals are generated by a model of the vehicle to be simulated
and the subsystem with the wash-out filters which translate vehicle motion into feasible
simulator motion. Although the host comes up with new set points at a relative low update
frequency (ca. 60 Hz), the fact that the systems being simulated are known, will enable a
reasonable prediction of the next set point.
The reference model based control has the task to deal with the set points and future
predictions in a proper way. Using knowledge of the set points supplied (mainly the fact
that the signals do not contain information at frequencies higher than 30 Hz) a smooth
interpolation filter provides a suitable reference acceleration to the feedback linearisation
level together with a smooth jerk (derivative acceleration) signal which can be used as lead
signal in the same feed forward channel.
With the reference model based control considerable improvement can be achieved
w.r.t. phase lag or delay in simulating on-set of abrupt (e.g. landing bump) and fast varying
motion (e.g. turbulence) [111].
The objective of predictive reference model based feed-forward control is to reduce
latencies to effectively zero. This is achieved by using knowledge of the vehicle to be simulated,
known as the reference model, to guide the simulator by feed-forward and feedback
control, which also accounts for the motion system dynamics.
Providing the motion controller solely with the desired system’s output results in latencies,
since pure feedback has limited bandwidth. This observation led to the key idea
that predictive knowledge from the fully available simulation model should be obtained and
used in an appropriate way.
To obtain a solution that can be implemented on and properly integrated with the available
subsystems, namely the host and motion computer, the problem was split into the
following:
reference model prediction
reference model based feed forward control design
systems interconnection
These problems will be treated subsequently now.
5.6.1 Reference model based predictors
The simulation model can be used as a reference signal generator. The only uncertain factor
is the pilot. However, due to the relatively slow response by the human operator, it is quite
easy to predict with reasonable accuracy the future accelerations over a short period of time
(30-50 ms). The major problem is due to the complexity of the vehicle model, making it
difficult to calculate future values within the real-time environment. Approximate models
can however be used, and several methods have been studied and evaluated [29].

180 5 Model based control of the flight simulator motion system
Approaches
The model approximation methods ranged from the very general method of series expansion
(with little knowledge of the simulation model), to the fast-time modelling approach, in
which the structure of the simulation model is taken explicitly into account.
Taylor series expansion.
With the most recent data points of the simulated signals, a polynomial is fitted. Extrapolation
of this polynomial results in a prediction of the future values of the signal.
Identified linear model predictor.
A linear state-space model is identified from the non-linear time-variant vehicle model.
With the identified model and its state, a future prediction can be made.
Self tuning predictor.
The previous model parameters can be updated by an adaptation using a stochastic
model and correlation methods.
Fast time modelling (FTM).
The simplified model of the equations of motion of the aircraft can be updated by the
coefficients calculated by the original (and more complex) simulation model. In the
construction of the predicted motion, these coefficients are assumed to be constant.
The advantage of the first three methods is that they can be used more independent of
the specific vehicle to be simulated. However, the expansion method and self tuning predictor
appeared to suffer heavily from high frequent distortion like a turbulence model. One
constant linear approximation model was not able to come up with high quality predictions
at all operation points. The FTM method proved to be best suited for prediction over limited
time horizons. It revealed constant performance under different test conditions.
The parameters transferred to a linear model of the equations of motion of the aircraft
are:
Aerodynamic coefficients,
Thrust,
Mass parameters,
Aircraft state vector.
With these parameters together with the wash out filter parameters (which do not have much
effect on the short horizon as they merely act as high pass filters) the required acceleration
of the simulator can be obtained
The total of forces and moments considered acting on the aircraft is a summation of
those generated by the engine model, aerodynamic model, landing gear model and turbulence
model. As forces immediately result in accelerations considering the equations of
motion, the prediction horizon should be obtained by the ability to predict the future forces

5.6 Reference model based feed forward 181
smoothly over some time frame. As thrust, aileron, elevation and rudder steering and landing
gear model have reasonable time constants, this can be done with respect to the engine,
aerodynamic and landing gear forces. The turbulence related forces vary considerably faster
but as the dependence on the pilots actions can be assumed constant over short horizons, the
stochastic nature can be simulated in advance to deliver future predictions.
5.6.2 Construction of the reference model based controller
An important aspect with respect to the reference model based control is that there is no
direct response of the acceleration (or force generation) resulting from the control inputs,
u. This is due to the presence of the valve dynamics and the finite oil stiffness in the
column. With the application of model-based control, the desired motion outputs (namely,
the specific forces and angular accelerations) can be driven more directly. Therefore the
design of a feed forward signal becomes more straight forward.
First, the requirements for the reference model based control task will be given. Then,
the different approaches will be outlined.
Requirements
In the ideal case, the multiple-level controller would result in a transfer function from the desired
platform accelerations, xd to actual accelerations x equal to a fast first-order response,
determined by the time constant of the pressure feedback.
In practice, however, several other aspects lead to deficiencies in the system. First of all,
the velocity compensation is not perfect and, as a result, the pressure feedback gains cannot
be increased to arbitrarily high values. Nonetheless, the hydraulic actuators can be turned
into ’force generators’ with a bandwidth of about 2-3 times the lowest natural frequency of
the rigid-mass system with finite oil spring stiffness. In the case of the SRS, this natural
frequency, with a design load of 4000 kg, is 4-7 Hz. Finally, the limited bandwidth of the
servo valves accounts for an additional latency of about 5 to 10 ms as was shown in Fig. 3.7.
Furthermore, the reference accelerations should not have a frequency content higher
than 20-30 Hz: Undesirable deformations of the simulator result in parasitic resonances
slightly above this frequency range. The pilot’s visual system is highly sensitive to vibrations
of the visual display system optics.
Approach
A study of literature resulted in three options for a trajectory tracking control approach.
Servo Control. If little use can be made of knowledge of a reference model or of the
system to be controlled, the servo control of Desoer [33] still results in the robust
asymptotic tracking of a reference signal. Transient response quality, which is considerably
relevant in motion simulation where onsets play an important role, cannot
be guaranteed however. The hydraulically driven motion system has tracking quality
w.r.t. step signals of the position if position feedback is applied due to the physical
structure given in Fig. 5.2.

182 5 Model based control of the flight simulator motion system
Model Matching. Vehicle simulation can also be seen as trying to match the dynamics
of the ’washed out’ vehicle model by the dynamics of the simulator motion system.
Model matching [156] can be applied with feed-forward, feed back, or both.
Preview Control. With preview control [144], predicted references can be used in a
controller in order to yield limited phase lags for systems not having stable inverses.
The scheme, given in Fig. 5.1, results in tracking for the long term of the platform pose
which is stabilized by the outer loop position feedback. Further, as already pointed out, the
inner loop feedback and feedback linearising control result in first order responses of the
system, from desired to actual accelerations nominally. This system can be described by
Gnom(s) =
1
I� (5.25)
s +1
where the time constant is determined by one over the product of pressure feedback and
the oil stiffness times valve gain, CV . The oil stiffness, C, varies slightly with the position
of the actuator[124]. The valve gain also shows some non-linearity.
The immediate unit (step) response of a system can be obtained by precompensation
with an inverse system. Because the system Gnom(s) is strictly proper, the inverse is not a
proper stable system. Two approaches can be taken to overcome this problem.
First, knowledge of the reference system can be used to determine the derivative acceleration,
the ’jerk’ signal d=dt(xd). If
xr = d
dt (xd) +xd
(5.26)
is used as feed forward, both the phase lag and high frequent amplitude decrease are
compensated for. As high frequency components tend to be amplified in this way, one
needs to be sure that the reference signal does not contain high frequent components
which could excite the parasitic resonances of the motion system. This could be
reduced somewhat by filtering the reference acceleration with first order filter with
a shorter time constant than , which results in a faster but still lagging response.
However, as is not even exactly known and is not constant, the compensation can
be wrong in the mid-frequency area i.e. the bandwidth area of the motion system of
10-20 Hz.
Alternatively, using a prediction of the future acceleration over a horizon equal to
the time constant of the controlled motion system can be used to compensate for the
phase lag over the relevant frequency area without unnecessarily amplifying the high
frequency area. The reference to the feedback linearising controller would then be
xr(t) =xd(t + ): (5.27)
Also with this approach high frequency reference signal will not be compensated for
properly, but at least will not be amplified.

5.6 Reference model based feed forward 183
In either case, the connection of the host to the motion computer has to be such that
the reference signal does not have a significant high frequency content. This is especially
relevant since the host is operating at a relative low sample rate w.r.t. the motion computer.
Caution has to be taken to prevent aliasing effects from mapping into this area. This will be
discussed next.
Host/Motion computer connection
The host computer generates the simulator feasible trajectories on a sample frequency which
is usually set at 60 Hz. In earlier days even 30 Hz was seen to be used. In near future, one
is likely to be able to set this frequency at 120 Hz.
According to Shannon’s theorem, reference signal frequencies higher than half the sample
frequency, > 0:5fn, can not be discriminated from the area 0 :::0:5fn. In practice,
aircraft model time constants should be well below this frequency. This alone motivates a
higher sampling frequency as also flexible effects of large aircraft or rotorcraft need to be
taken into account in simulation models in the future.
Extensive literature [26] towards signal processing and filtering in the area of multirate
signal processing exists. Since the motion system control computer requires (smooth)
signals at 1-5 kHz, the lower-frequency sampled signals of the host computer will have to
be interpolated. Ideally, if the signal x(k) to be interpolated would be known from time, t,
in the interval ;1 to 1, filtering with the well known anti-causal filter h(t),
xc(t) =
=
1X
h(t ; k=fn)x(k)
k=;1
1X
k=;1
sin( (fnt ; k))
x(k)� (5.28)
(fnt ; k)
would exactly reconstruct a continuous signal x c(t) with only frequency content in relevant
area without any deformation in this area. In practice none (or at best only part) of the future
is known; different techniques exist to approximate h(t). Every filtering technique then
typically not only reduces high frequency signal contents, but also introduces phase lags.
There are always shortcomings, which the designer must be aware of.
Choice of polynomial interpolation technique
In this research, several filtering techniques like Finite Impulse Response (FIR) filters, Infinite
Impulse Response (IIR) filters and Cubic Polynomial Reconstruction (CPR), have been
considered. Unlike FIR and IIR filters which are designed in the frequency domain, polynomial
interpolation techniques can be designed in time domain. This has some advantages:
time domain criteria can be used,
non-linear design is possible,
interpolation functions are continuous time functions, so they are independent of the
interpolation factor.

184 5 Model based control of the flight simulator motion system
The last point is considered important because in case of inter-connecting systems with
large (> 50) ratios between their sampling rate (as in the case of host-motion computer
integration), CPR interpolation techniques can still be implemented as low order filters.
This is unlike the FIR-filters. Further, no redesign has to done as the interpolation factor
changes, unlike the IIR-filter design.
In polynomial interpolation on the interval t k to tk+1 of order p, a function
f ( )=
pX
i=0
ai i
(5.29)
is defined on the local time interval from 0 (tk) to1(tk+1). The coefficients ai of the
polynomial are chosen such that the value of the signal at the sample time points, and
possibly the time derivatives thereof, satisfy certain constraints. With a third-order CPR,
which was used to filter the acceleration signal, the four coefficients can for example be
specified by defining the value f (0) = x(k ; 1) and f (1) = x(k) and their first time
derivative _
f(1) = x(k) ; x(k ; 1) and _
f(0) = x(k ; 1) ; x(k ; 2). If a prediction of the
signal x(k +1)is known. the interpolation can be shifted.
With the position signal interpolation of a fifth-order function, (C)PR can be used to
allow the accelerations (second order time derivatives) to correspond to the desired values
also. In case signals from the host would arrive late, or not at all, precaution should be taken
not to extrapolate the polynomials. This could run the signal off-line very quickly. Here,
by taking the next sample equal to the previous one for the position the system gradually
comes to a stop.
With CPR, the host and motion computer communicate the reference signals with a simple
low-order method, which has relatively little high-frequency contents. The method can
easily be combined with the use of the time-derivative of the acceleration for feed forward.
This ’jerk’-signal can be extracted explicitly at every time point. Predicted time points can
be incorporated directly to reduce the phase lag.
Resume concerning the reference model based control
In this section a number of approaches to reduce time delays or lags in the response of
simulation motion systems were presented. A synergistic procedure was proposed in which
system knowledge, of both the vehicle model to be simulated and the motion system to
be accelerated, is used to have virtually immediate response to the desired motion. By a
division of sub-tasks, the modular simulation structure can be maintained.
Further research with a full operational simulator will be needed in order to point out
exactly what delays are allowed while having no influence on the perceived realism of simulation.
Having a simulator motion system as the SRS, which is able to operate nominally
with low latencies, will help enabling this research.
Resume concerning the multi level control structure
Motion control of a complex non-linear multivariable system like the Simona Research
Simulator can be performed by using a control strategy in which the controller is structured
in multiple levels. Each level has its own specifications in close relation with the level it

5.7 Implementational issues 185
communicates with. In this way a set of various control problems can be solved more or
less separately. Further, the control structure enables implementation on a multi-processor
dsp-system.
5.7 Implementational issues
The multiple level controller has been implemented on a real-time multi-processor DSP
motion computer [37] connected to the motion system with configuration C., a temporary
dummy platform (ca. 4tons). In this set-up, as shown in Fig. 5.9, one C40-processor has to
perform all communication with the outside world (bottle-neck w.r.t. sampling frequency)
and could be run at 5 kHz. Also the coprocessor, which calculates the inner loop control,
runs at 5 kHz, necessary to deal with the relevant fast actuator dynamics.
In this respect the multiple level structure pays-off since the other levels, especially the
feedback linearising control ran on yet another coprocessor at 1 kHz which is just sufficient
to go through all the algorithms involved.
Design of the control structure was performed in the user friendly environment of Matlab/Simulink
1 from which c-code can be generated automatically and connected with userwritten
code in so called S-functions. The four processor c-code for the full model based
controller consisted of about 9500 lines automatically generated code from the simulink
model and 1850 lines of user written code (of which 600 lines came from the standard templates).
Though probably not optimal in efficiency, a very user friendly environment has
been enabled, which allows reasonable complex control structures to be implemented. In
this way, rapid prototyping of complex controllers, as presented in this research, becomes
feasible. Going from a Simulink model to a controller running in a real time environment
takes about 10 minutes.
5.8 Performance quantification
The evaluation procedure of the multiple level controller implemented on the flight simulator
motion system SRS is presented in this section. The results, as partly also described
in [73], were obtained through experimental measurements with the dummy platform with
additional weights added, which amounted to the predicted final weight of 4 tons as in
Fig. 1.5.
The motion-base must be able to guarantee a high level of performance throughout its
workspace. To do this, first a test procedure and a framework was defined whereby the
dynamic characteristics could be quantified. A basis for the procedure is the long existing
AGARD Advisory Report AR-144 [1], which quantifies several independent tests including
the measurement of describing functions, dynamic thresholds, noise levels and the hysteresis.
This set of tests gives insight into several linear and non-linear properties of a controlled
motion system in both the time and frequency domains. The AGARD tests had to be modified
to include the high-frequency dynamics, and extended to enable characterisation of
1 Matlab and Simulink are registered trademarks of the MathWorks, Inc.

186 5 Model based control of the flight simulator motion system
Fig. 5.9: Motion controller hardware setup: The model based controller is implemented on
a multi-C40-processor board. A master processor has to deal with all the I/O and
forms the bottle-neck w.r.t. the necessary 5 kHz sampling time for the hydraulic
loops. Apart from I/O the master only performs some safety checks on the outgoing
signals. The two slave processors calculate through the multi level control
structure given in Fig. 5.1 On the first coprocessor (slave 1), the six inner loop
controllers (module 1 of Fig. 5.1) are calculated at 5 kHz. They receive their reference
(pressure) from the second coprocessor (slave 2), whose task is to process
the mechanical model based control (modules 2 to 4 of Fig. 5.1). Finally, a third
co-processor (slave 3), was used to generate the host signal (at 100 Hz).
the system throughout its operating area. In order to arrive at a standardised approach, it is
proposed that the latter be achieved by performing a set of path tracking tests. Furthermore,
it was noticed that in order to enable an accurate characterisation of a high-performance
motion system, both the experimental set up and test method have to be critically observed.
Since the goal of the SRS simulation facility is to develop and validate new simulation
technologies and to investigate human-machine interface design concepts, the performance
demands on the motion cueing capability are higher than current devices. The high level of
performance has to be guaranteed throughout the workspace of the motion system.
Qualifying or even quantifying the performance of a non-linear, multivariable, physical
system is complex, especially if one wishes to compare the results with other configurations
or simulators. Ideally, first a model structure is chosen which describes the characteristics
of such systems. Then, the parameters of this model structure are identified, which requires
experiments or testing. Finally, the models are classified by assignment of a measure or cost
function. The performance classification will at best produce results as good as the method
and devices used to measure it. The identification of rigorous models, which describe all

5.8 Performance quantification 187
systems characteristics, generally does not lead to accurate results [84]. Therefore, motion
systems are often classified by performing several tests to identify specific parameters,
sometimes assigned within independent simplified model structures.
In case of motion simulation, the objective cueing performance should preferably be
measured by the ability to properly stimulate the human vestibular system [54]. This consists
of two organs; the otoliths and the semicircular canals. Consider the perception models
given in Section 1.2 and the parameters provided in Table 1.1. The otholiths are primarily
and almost proportional sensitive to linear specific forces, the so-called ”regular” unit. The
time derivatives of the specific forces, called the ”irregular” unit or ”jerk”, are also detected.
The semicircular canals are proportionally sensitive to rotational acceleration in some frequency
area (1:44 ; 30Hz) but are also proportional to rotational speed at lower frequency
(0:026;1:44Hz). These sensitivities form the basis of the use of acceleration as the primary
metric to measure the performance of a motion system.
5.8.1 Motion system evaluation methods and requirements
At present, only one standard method to characterise the performance of a motion system
is known to exist. This is described in the AGARD Advisory Report 144 [1]. Although this
report dates back to the end of the seventies, the technique is still up to date. Additional
aviation requirements, like JAR-STD-1A8 [118] and FAA 120-40C5 [3] generally, overlap
AR-144 [1] and are basically less stringent. In robotics, even though rapid developments are
underway in design and testing parallel robots, these have not led to a generally accepted
characterisation standard. The AGARD report defines tests to measure
- the describing function as a frequency domain evaluation,
- dynamic threshold as the time domain response,
- noise levels to characterise parasitic motion,
- and hysteresis to identify hard non-linearities.
It uses these parameters to describe the performance in one prescribed point within the
workspace. As this is an international standard, which has been applied to several motion
systems [46], [123], it will be used as a basis in further expansion to evaluate the performance
of our mechanism.
The AGARD method does have some limitations.
- First of all, it operates in only one point of the workspace, the neutral point. The dexterity
in this point is often minimal [7], implying that the excursion forces are well within
the operational limits of the motion system, and the limited non-linearity in this point
implies that simple controllers can still perform reasonably well.
- Secondly, the simulator upper-gimbal centroid is taken as the point of reference, but this
point is almost never the pilot’s head reference location.
- Furthermore, only a limited bandwidth of up to 10 Hz is tested in the AR-144 procedures.

188 5 Model based control of the flight simulator motion system
- No evaluation of the limitations of the experimental set up is included. Especially in
measuring the noise levels and hysteresis in high-performance motion systems, the
discrimination between measurement noise and system-generated parasitic motion
should be possible.
- Finally, the separate tests do not guarantee a level of performance of the system being
operated during actual flight motion simulation. This evaluation is lacking in AR-
144.
To improve the performance characterisation this section will therefore propose some
modifications to the original AGARD method in order to enable the calibration of highperformance
motion systems. The newly proposed test should not be unnecessary extensive
to remain economically feasible.
The objective of this section on performance quantification is to characterise the performance
of a high-fidelity flight simulation motion system throughout its workspace in an
efficient manner. This can not be achieved with existing methods. Therefore, a first attempt
is made to modify the test procedure in order to overcome the current limitations. For the
sake of practicality and standardisation purposes, the experimental set up is considered to be
part of the procedure. Application to the Simona motion system will lead to a preliminary
characterisation and an evaluation of the procedure.
Although tests exist which are independent of the system at hand, it is preferred to
take the specific characteristics of a simulator motion system into account in designing a
procedure.
- A motion system is a hydraulically or electrically-driven mechanism in which the uncontrolled
dynamics usually have resonance frequencies with a low level of damping.
A test procedure should point out whether the degree of damping of the controlled
system is sufficient.
- A purely mechanical system is considered to have immediate force generation and, as
a result, direct acceleration. If the dynamics of the actuators is relevant, and this is
certainly the case with hydraulics (no direct feed-through), then there will be a limited
bandwidth to be quantified.
- The use of a synergistic Stewart Platform implies that all actuators have to move to perform
pure motion of one platform degree-of-freedom. Interaction in moving along
the different degrees of freedom easily results if not compensated for. This should be
taken into account in testing. Further, the interaction and reflected masses depend on
the platform pose.
- The hydraulic actuators are symmetric, enabling similar force generation in both directions.
Asymmetricity can still occur if there is a pre-load, as is usually the case if
gravitational forces exist. Hydrostatic bearings are applied to reduce friction, thereby
reducing hysteresis, turn-around bump and parasitic motion. As a result, there are
several sources of possible non-linear behaviour of the system, which should be considered.

5.8 Performance quantification 189
In the design of a modernised test procedure an attempt will be made to take these
characteristics into account.
5.8.2 New test procedure
In setting up the test procedure to describe the dynamic properties of a controlled flight
simulator motion system, the AGARD Advisory Report 144 [1] is taken as a basis. In
the following it is discussed, which part of AR-144 is considered important, what relevant
modifications were made, and what extensions were taken into account.
Describing function
With the describing function method, the response of a system is given along two axes, in
frequency and amplitude. As extensively discussed in Van Schothorst [124], the describing
function method can be used to characterise a large class of non-linear systems in the
frequency domain. It is stated there that hydraulically-driven mechanical systems belong to
this class even if hard non-linearities such as Coulomb friction are relevant. If no effects
like Coulomb friction are present, as should be the case with high performance motion systems
with hydrostatic bearings, then the sinusoidal input describing function converges to
the frequency response of the linear dynamics for small input signals. This assumption has
to be checked, e.g. by the hysteresis and threshold tests described below.
Considering the frequency response of a linear system, several properties can be deduced.
Bandwidth can be calculated, stated as the -3 dB and/or the ;45 point. The ”peaking”
or degree of damping of the controlled system can be found, and cross talk can be
analyzed. By taking a sinusoidal input amplitude of about 10 percent of the system’s positional,
speed or acceleration limits, the non linear effects are observed to have a relatively
minor influence.
By using an analog frequency analyzer, a broad frequency spectrum can be taken into
account and highly accurate frequency responses can be measured. This is considered to be
preferable to the method described in AGARD in which an extensive number of discretefrequency
measurements have to be taken. The limitations of this technique are however
that one should measure within the (amplitude) range where the system can be considered
linear, and that no higher harmonic response is taken into account.
Threshold response
With step response measurements, the response of the system can be analyzed in the time
domain. It is considered an easily applicable test to infer the non-linearity of the system
with respect to different amplitudes. Originally, these kinds of tests were used to measure
the lowest amplitude or threshold acceleration to which the system still responded. With
modern motion systems having virtually no friction, the motion-base will respond to any
amplitude and the problem at very low amplitude responses mainly results from the accuracy
of the measurement apparatus.
By considering a very simple first-order linear system response model with time delay,

190 5 Model based control of the flight simulator motion system
acceleration (m/s2)
position (m)
1
0.5
0
−0.5
−1
0.1
0.05
0
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5
time (s)
2 2.5 3
Fig. 5.10: Representative input signal, acceleration and position, for the dynamic threshold
test.
the following structure results:
G(s) =
K
s +1 e- ds
(5.30)
This is often a convenient structure to approximate the motion system accelerational response.
If e.g. a feedback linearisation and inner loop cascade-dp is applied, the motion
system acceleration roughly follows the first order presssure controlled dynamics. The dynamics
at higher frequencies and computational implementation mostly influences phase
and can therefore often appropriately be described by one time delay.
The parameters of this model are identified from the different step responses. The time
delay, d, is the time that the system takes to respond, and the time constant, ,isgivenby
the time taken from this point to reach 63% of the final value.
With flight simulator motion systems, the acceleration step response is considered important
for good cueing. To measure this response, a trajectory as given in Fig. 5.10 can
be applied. One should notice that this cycled trajectory actually consists of eight different
acceleration step responses. In this research, a response starting with zero velocity is
considered.
Path tracking
To evaluate the system in its normal operating mode, and to analyze whether the properties
measured at one operating point can be extended to a relevant part of the workspace, a new

5.9 Experimentally evaluating performance 191
test is introduced. Advani [7] already introduced a set of benchmark manoeuvres to check
the design of the kinematics of a motion system. These include standard manoeuvres during
flight simulation training which are considered critical in the utilisation of the simulator.
Some 30 manoeuvres, responses to heavy turbulence, rejected take-off, landing, etc., are
included. The ability of the controlled system to track these manoeuvres is considered. At
this moment, the mean and standard deviation of the acceleration error, after being compensated
by the time delay, and the time delay itself are taken as the parameters to be measured.
With this test, two aforementioned limitations of the AGARD procedure are accounted for.
Further tests
The AGARD procedure also includes noise levels and hysteresis tests. With high performance
systems, the acceleration noise levels are very close to perceivable thresholds of 1%
of gravity, and become hard to measure. This is because the measurement apparatus also
introduces measurement noise, which should be discriminated from system-generated parasitic
motion. With the gyro measurement taken with respect to rotation, the situation even
becomes more severe as this signal has to be differentiated. With the hysteresis test, in
which the system is made to track very low-frequency sinusoids, the situation is the same.
It was concluded that at this moment these tests can (and should) only be used to check
whether the system’s noise and hysteresis are below measurable thresholds e.g. to investigate
if turn-around-bumps can be noticed.
5.9 Experimentally evaluating performance
In this section the results of the model based controller will be evaluated using the proposed
test method and with the reference of a conventional controller with a decentralised
pressure/position feedback for each actuator.
5.9.1 Experimental set up
As explained in the previous sections, the test procedure consists of several trials in which
the controlled motion system is made to perform a series of manoeuvres. During these
tests, the response of the system has to be measured. First, the design and calibration of the
experimental set up will be considered.
Performance test set up
The experimental configuration is given in Fig. 5.11. It consists of the Simona motion
system, which is controlled by the motion computer. With the actuator extensions being
measured, together with the kinematics of the system, the platform pose (position and orientation)
can be calculated on-line. In fact the system’s response with respect to the accelerations
can also be calculated from these measurements. However, as the signals then
need to be differentiated twice, the high-frequency component of the reconstructed acceleration
becomes corrupted by measurement noise. Therefore, an independent data acquisition
system is used to measure the systems accelerating response.

192 5 Model based control of the flight simulator motion system
Fig. 5.11: Motion control evaluation test setup: Independent of the motion control system
given in Fig. 5.9 implemented on the motion computer, the accelerations and
angular velocities of the system are measured at the motion platform by another
data acquisition system (also a dSpace computer). The motion controller and
data acquisition system are synchronised by a frequency generator which is used
to trigger the start of each run and detect possible discrepancies between the
system clocks.
With a reference measurement package consisting of three accelerometers and three
rate gyros attached to the dummy platform, the local specific forces and angular velocities
are measured. This still requires one differentiation step in order to calculate the angular
acceleration.
As measurements of the motion computer and the data acquisition computer will be
used in the evaluation of the test procedures, the time scale of both systems has to be synchronised.
To achieve this, an external reference signal from a frequency generator was fed
to both systems. This sinusoid was used to trigger the start of each run. The test setup is
depicted in Fig. 5.11.
Calibration measurement system
Reconstruction of the actual acceleration of the platform in all six degrees-of-freedom cannot
be done with the measurements taken by the reference package alone. Some additional
information has to be known in order to process the data properly. The gain and orientation
of the measurement devices, and also their relative position, are required. Some (but not all)
of these parameters can be calibrated by off-line measurements. As part of this research,

5.9 Experimentally evaluating performance 193
another approach was taken.
In the low-frequency region, the accelerations of the platform can be reconstructed with
the actuator extensions and the kinematics of the platform as also done in Section 4.2.2.
By performing a manoeuvre at a specific frequency (e.g. at f=4 Hz), a high signal-to-noise
ratio can be obtained. Especially if only this component of the signal is taken into account,
the non-linear components can be neglected. This frequency should be chosen such that
both the accelerometers and the position transducers have an approximately equal signalto-noise
ratio, taking into account the accuracy and resolution of the measurement devices.
Manoeuvring in six independent platform directions assures spanning the whole motion
space. Here, we assume only rigid-body motion of the platform, and a rigidly-attached
base. This assumption is not valid in the case of the current placement of the Simona
motion system (on the floating concrete floor plate) at frequencies higher than 8 Hz.
The position and orientation of the reference package in platform coordinates can be reconstructed
taking the procedure presented in the previous chapter (4.46) for the accelerometers
used in identifying the flexible dynamics. The identification was improved further by
incorporating the dynamics of the anti-aliasing filters.
This procedure was applied to the reference package attached to the dummy platform
moved by the Simona motion system. The accelerometers and gyros appeared almost perfectly
aligned with the axes of the moving reference frame. The gains also corresponded
to the earlier off-line calibrated values (approx. 3.6V/g). The position of the lines of measurement
for the accelerometers were also very much the same with respect to their relative
design positions. The absolute position of the reference package was identified at
683mm 1mm below the centre of gravity (which is 233mm below centre upper gimbal
point). The x-y coordinates (forward and to the side) depend on the accelerometer at hand
which should be compensated for in the following test procedure. As this easy to apply
procedure was successful, it could be used in the future to reconstruct the relative position
in the interior of the simulator.
5.9.2 Characteristics of the Simona motion system
Given the description of the test procedure and the experimental set up, the resulting performance
characteristics of the Simona motion system will be outlined.
Frequency response
First, the application of the describing function test will be discussed. In Fig. 5.12, both
the amplitude and phase frequency characteristics of the Simona motion system with the
multiple level controller from 0.5 to 50 Hz are shown. All plots are separately shown in
Fig. A.9 and Fig. A.10 of Appendix A with direct comparison of the model based controller
to a conventional controller and in Fig. A.11 of Appendix A also the error response of both
controllers to reference signals is given.
It can be seen that the -3 dB point can be found at 13 to 15 Hz for the ’non horizontal’
directions of pitch (omy), roll (omx) and heave (z). In surge (x), sway (y) and yaw (om z),
the bandwidth is lower due to the motion of the current floor foundation. Except for yaw,

194 5 Model based control of the flight simulator motion system
Direction Response time (ms) Direction Response time (ms)
Surge 32 Roll 35
Sway 32 Pitch 35
Heave 33 Yaw 34
Table 5.1: The Simona Research Simulator latency with a model based controller.
the ;45 bandwidth can be found at approx. 5 Hz. The peak amplitudes remain well below
2 dB. The most relevant parasitic motion is measured between pitch due to surge and vice
versa, and roll due to sway and vice versa.
For reference purposes, a conventional approach was used to control the same motion
system. The frequency response is given in Fig. 5.13. The same pressure feedback gain
is applied, however, since there is no compensation for the different natural frequencies,
some responses, like surge and sway, demonstrate peaks much more than others (like pitch
and roll, which are over damped). The bandwidth is twice as low, and the peaking and
interaction are twice as high with this control approach. The describing function enables
the characterisation of some of the parameters, which are considered most important in
control, like bandwidth, damping and interaction in a multivariable system.
Threshold response
Threshold tests have been performed for all platform directions for four different amplitudes
in acceleration (1, 0.4, 0.1, 0.05 m/s2) for both negative as positive directions. The response
in the surge direction is shown in Fig. 5.14. Note that the signal amplitudes have been scaled
to a desired final value of 1. As can be seen the response becomes less regular for the lower
amplitudes, but the time delay (approx. 10 ms) and time constant (approx. 22 ms) do not
vary too much and correspond with the ;45 bandwidth ( =1=(2 fbandwidth)) measured
with the frequency response. Also, the smooth response without too much peaking was
predicted by the frequency response.
The characteristic parameters for the other platform directions are given in Table 5.1. In
all cases, a time delay of about 10 ms can be observed. Given the sample frequency of the
host processor of 100 Hz, such values can be expected. It should be noted that the responses
in the rotational directions were much more corrupted by measurement noise, as the signal
had to be differentiated. Especially for the low amplitude responses of 0.05 and 0.1 rad/s2,
no exact time constant or delay could be extracted.
Path tracking
To check whether the characteristics as measured in one operating point would still be preserved
while performing relevant flight manoeuvres, the path tracking test was introduced.
Testing Characteristic Manoeuvres
At the KLM flight crew training centre, 31 simulator training-critical manoeuvres were
flown in a Boeing 747-400 simulator and the aircraft model responses registered. These

5.9 Experimentally evaluating performance 195
Amplitude m/s2−>m/s2
Phase (deg)
10 0
10 −1
0
−50
−100
−150
10 0
10 0
om_z
z
x y
Frequency (Hz)
om_y
om_x
om_z
Fig. 5.12: Actual model based control motion system frequency response. The Bode
plot of the closed loop system from required to actual (translational and rotational)
accelerations applying the model based controller in the neutral position.
The most important interactions are also given (dashed) and rotations are multiplied
by the upper gimbal radius. Highest bandwidths are attained with pitch
(omy), roll (omx) and somewhat lower with heave (z). For the directions surge
(x), sway (y) and yaw (omz), where the foundation is flexible and not included
in the model used to design the controller, some energy does not accelerate the
platform above 4 Hz. The phase of the yaw direction does lag the others. Maybe
the gyro filter was not compensated for properly. All responses are reasonably
flat without peaking more than a few percent above 10 0 . Interaction, especially
between pitch to surge and roll to sway and vice versa is unexpectedly high up
till 30 % above 4Hz. The six by six Bode response plots directly comparing the
model based and conventional controller can be found in Fig. A.9 and Fig. A.10
of Appendix A
10 1
10 1

196 5 Model based control of the flight simulator motion system
Amplitude m/s2−>m/s2
Phase (deg)
10 0
10 −1
0
−50
−100
−150
−200
10 0
10 0
y
x
x
y om_z
z
om_z
Frequency (Hz)
10 1
10 1
om_y
om_x
Fig. 5.13: Actual conventionally controlled motion system frequency response. The
Bode plot of the closed loop system from required to actual (translational and rotational)
accelerations applying the basically conventional control strategy in the
neutral position. The most severe interactions, between pitch (om y) and surge (x)
and roll (omx) and sway (y) are given (dashed) and the rotations are multiplied
by the upper gimbal radius. As expected, with the conventional controller it is not
possible to tune the bandwidths of the platform accelerational loops w.r.t. each
other. Further, a compromise has to be found between peaking (up till 6 dB)
of the lowest bandwidth loops of surge (x) and sway (y) and the overdamped
response of pitch (omy) and roll (omx). The 4 Hz bandwidth of surge (x) and
sway (y) is also a maximum attainable bandwidth using this control structure, the
amount of peaking given and the lowest eigenfrequency of the platform. The six
by six Bode response plots directly comparing the model based and conventional
controller can be found in Fig. A.9 and Fig. A.10 of Appendix A

5.9 Experimentally evaluating performance 197
1.5
1
0.5
0
−0.5
−1
−1.5
+/−1.0 m/s2
+/−0.4 m/s2
+/−0.1 m/s2
0.05 m/s2
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Fig. 5.14: Short response times of the model based controller of a large range of accelerational
amplitudes. The normalised threshold or acceleration step response
measurements are given over a range of amplitudes for the most difficult surge
direction.
manoeuvres included a wide range of dynamic conditions, including take-offs, normal and
hard landings, engine seizures, and response to heavy turbulence. These manoeuvres were
used for research into kinematical design of motion platforms by Advani [7] and reference
model based feedforward control within a Masters project as part of this research by Piatkiewitz
[111].
It was decided to use a set of these manoeuvres, which were dynamically most challenging,
as a benchmark for the SRS motion platform. Five of the thirty-one manoeuvres were
selected, since these represented the most motion-demanding conditions.
These were:
1. Response to maximum clear air turbulence
The motion system is made to move in a stochastic manner experiencing vibrations
up to 10 Hz. Also an airpocket is included which requires almost full speed of the
actuators (:9 m/s with vmax=1 m/s).
2. Taxiing
Fast vibrations introduced through the landing gear have to be experienced in simulator
during which it has to move gradually through surge, sway and yaw motion.
3. Landing with cross wind
In the air, the simulator has to experience the stochastic nature of cross wind. At

198 5 Model based control of the flight simulator motion system
landing three instances of landing gear touch down bumps require fast acceleration
peak generation after which the landing slide introduces even faster vibrations up to
15 Hz.
4. Rejected take off
Immediate negative surge together with tilt coordination (in which gravity is made
to introduce long term breaking force) move the simulator through a wide area of
operating conditions w.r.t the platform pose.
5. Rotation during take off roll
Limited amount of motion also has to be introduced smoothly in the asymmetric
(roll) motion. Faster resonances (before take off) during the first seconds have to be
replaced by the more smooth vibrations in flying.
To test the system with the load conditions of a full operational SRS the control strategy
proposed was evaluated with a dummy platform of 4000 kg on top of the SRS motion system
depicted in Fig. 1.4. The host computer was replaced by an additional processor on the
motion computer running at a lower sampling rate of 100 Hz. In the ’host’-processor the
tracks for platform pose and acceleration are made available and send to the motion computer
control processors at demand. The original tracks were aircraft model output. By a
plain wash out these trajectories were made fit for the SRS. Actuator stroke had to be within
1.25 m and velocity within 1 m/s. To test the motion system with the control strategy these
signals were considered the feasible trajectories which had to be tracked as close as possible.
Evaluation of the path tracking
Each manoeuvre has a duration of 16 seconds, however, due to the limited hydraulic power
supply (temporarily) available at the Central Workshop of Mechanical Engineering this duration
is just manageable. In Fig. 5.15, the measured heave response together with the
reference acceleration, are depicted.
The measured heave signals correspond very closely to the desired acceleration. If it
is considered on a short time frame an almost constant lag of approximately 30 m can be
observed. The main contribution of the lag in response is due to the smoothing filter at 30 Hz
(63 % rise time of 20 ms). This motivates the use of a prediction horizon of 20-30 ms, in
which case the lag in response could very well be reduced to a situation of ’virtual zero time
delay’. In this chapter a number of approaches to achieve this were presented.
In Fig. 5.15 part of the response during a landing with cross wind is shown. Acceleration
peaks can be observed as the three landing bumps occur. These are well represented by the
controlled motion system. The frequency content of the signal is clearly changing as the
wheels start to roll over the landing track (more high frequency rumble required). The
errors are well below 2% of gravity. As the high-frequency part of the error signal can be
significant (due to measurement noise and also anti-alias filtering, starting at 30 Hz), it is
hard to draw conclusions with respect to this frequency region.
The figure shows that the motion system reacts without gain decrease on the stochastic
nature of wheel landing track contact at least up to vibrations of ca. 8 Hz (which can be seen
during the time span 8.05 s:::8.45 s). The ’noise’ in the measured signal is mainly caused

5.9 Experimentally evaluating performance 199
acceleration (m/s2)
2
1
0
−1
−2
−3
−4
landing with cross wind, desired/actual heave
−5
0 2 4 6 8 10
time (s)
12 14 16 18 20
Fig. 5.15: Path tracking in landing with cross wind manoeuvre with the model based controller
considering accelerations in the platform heave (z-direction). Upper plot
gives the desired accelerations in heave direction and the lower one gives the
actually measured accelerations (off-set 2m=s 2 brought in on purpose for comparison)
by the application of servo valve dither. As said, the top-top values remain however well
below 0.02 g, which almost unnoticeable.
The other manoeuvres, not depicted here, show comparable characteristics. This proves
that this motion system control and design strategy resulted in a high-quality system. The
use of a predictive reference model based control will even enable a synergistic connection
between host and motion computer, which will considerably reduce latencies.
The two aforementioned controllers, (1) the multiple level controller and (2) the conventional
controller, were compared using the manoeuvres. The characteristic parameters,
given in Table 5.2, show again that the multiple level controller has higher performance with
respect to the conventional one.
Further tests
Further, as already pointed out, noise level and hysteresis tests are also part of this procedure.
As both were near or below measurable or noticeable levels, no exact values can be
given here. The hysteresis tests showed values less than 0.3 mm. This value was measured
going through a sinusoid with 0.4 m of amplitude and period time of 5 minutes, and was
still decreasing as the period time was increased. It was observed that higher values of hys-

200 5 Model based control of the flight simulator motion system
Manoeuvre Control Delay error Mean error Std error
(ms) (m=s 2 ) (m=s 2 )
Landing Model based 32 0.032 0.037
Conventional 62 0.042 0.041
Taxiing Model based 34 0.022 0.019
Conventional 62 0.041 0.033
Table 5.2: Comparison path tracking results in heave (delay, mean error, standard deviation
(std)).
teresis appear if the dither - a high-frequency vibration in the pilot valve of the actuator - is
not properly tuned. Therefore, this test should be part of the procedure to guarantee some
upper bound.
The noise levels measured with respect to the translational acceleration are acceptable,
though close to measurable levels of 0.01 to 0.02 g. It should be noted that with the experimental
set-up used, no predictions with respect to high-frequencies ( 50 Hz) can be
given. The noise levels in the rotational directions could not be measured as there was too
much measurement noise caused by differentiating the angular velocity. Therefore, more
accurate gyros should be used here. The frequency responses measured in Fig. 5.12 do not,
however, predict higher sensitivity in the rotational directions compared to the translations.
5.10 Chapter Resume
Almost invariantly, in every test performed, the model based controller did a far better job
than the reference conventional controller. Higher bandwidths (8 Hz for 90 phase lag)
were measured in the frequency domain along with less peaking (max: 2 dB). Further,
higher suppression of interaction was also attained, though somewhat less than expected. In
the time domain, fast responses (response time between 30 and 35 ms) were observed over
a large range of amplitudes including 5 mg.
In the conventional controller, a compromise had to be settled for in which some loops
have high bandwidth (still not more than 4 to 5 Hz) and too severely damped response
and other loops show already relatively high peaking up till 6 dB. Only in heave direction,
in the neutral position, performance is almost equal to the model based control approach.
Fortunately, for the conventional control approach, most accelerations in the simulated manoeuvres
were required along the heave direction.
In this chapter also a test procedure was proposed to characterise the performance of a
high-fidelity flight simulation motion system throughout its workspace in an efficient manner.
The procedure was evaluated with the Simona motion system, which, although in preliminary
form, already shows favourable properties due to its design and the model based
control approach. The main parameters of such a system appear to be given by its linear
dynamics. Measuring the frequency response is an efficient way to characterise this. Other
tests such as threshold step responses, path tracking and hysteresis, can be used to quantify

5.10 Chapter Resume 201
to what extent these properties are maintained throughout the full operating area. Further
research has to be done to identify the noise levels with higher accuracy.
Some modifications to the current standard outlined in AGARD-144 can yield a better
quantification of high-performance motion systems.
Performance of the motion system was defined by the degree of motion realism attained.
There are no measures known which exactly quantify this. More research into human perception
has to point out how this has to be done. The Simona Research Simulator could
play a role in attaining this goal.

Chapter 6
Review and discussion on the
results
The previous chapters went over the full system/control design process of flight simulator
motion systems. Before drawing the final conclusions towards this research, the main
considerations in this process will be discussed.
It has been investigated to what extent the use of relevant system knowledge in the motion
control strategy improves the controlled dynamics of a flight simulator motion system.
This required structuring the research in several subproblems. Defining quality i.e. setting
specifications, obtaining system knowledge through theoretical modelling, quantifying and
verifying these models and the relevant dynamics by experiments. Definition of a modelbased
but implementable control strategy and a test to validate and compare the obtained
closed loop dynamics with conventionally controlled motion systems. In literature various
solutions to many of these subproblems are proposed. Integratibility in the full design
scheme and actual applicability have been the main arguments in this research in the choice
for the most suitable available alternative or, in some cases, newly proposed variant.
6.1 Flight simulation
Flight simulation is about creating a complex environment, which can be used to train pilots,
evaluate flight system characteristics, etc. in a system which is not actually flying.
Using a flight simulator is cheaper, safer, environmentally less harmful than flying an aircraft
and training or evaluation can more easily be defined, standardised, modularised and
repeated. Since these advantages become more and more important, there is a continuing
market driven tendency towards the use of flight simulators. However, requiring higher standards
in simulation quality. Major breakthrough in technology, mainly due to the increase
in computer power, concern the visual system and the ability to take complex flight characteristics
into account in the aircraft (environment) modelling. Motion is one the stimuli
which can never be perfect given the limited dexterity of motion systems kinematics. This
foremost constrains the ability to duplicate the low frequency part of manoeuvres. This can
partly be compensated for by current visual systems. The attainable quality of simulating
203

204 6 Review and discussion on the results
high frequent vibration or onset motion is determined by the controlled dynamics of the
motion system together with the simulation model. By design of the Simona motion system
construction, expectedly favourable properties such as low mass and inertia, low centre of
gravity and high rigidity were strived for. Design of a suitable controller to exploit this, is
the next step. This has been investigated in this research.
6.2 Motion system specifications
Perceived motion or realism of motion simulation is a subjective measure which can not
exactly be specified. Research into motion perception models is one of the arguments in
trying to attain a relatively high performance motion system. Flight simulator users explicitly
ask for quantification of required motion cues and accuracy necessary to attain a certain
training quality. Literature provides for quite some indications in what range specifications
should be tighter. First of all, predictable characteristics of the controlled motion system
over its full working area should be strived for to be able to perform well defined simulator
training experiments. This alone motivates the use of as much relevant system knowledge
as possible.
Control specifications should result from the requirements set, the internal structure of
the system to be controlled and the properties of the external signals which are expected
to perturb this system. A fundamental difference in simulator motion systems w.r.t. usual
robotics, is the fact that acceleration instead of positional accuracy is most important. In the
low frequency range (preferably below :1 Hz) position stabilisation to prevent the actuators
running out of stroke should prevail, but as from this frequency on, acceleration should
be tracked. Since the simulator is running free, tracking of acceleration reference signals
is the major task, which has to be compromised with unmodelled dynamics, uncertainties
and measurement noise. Disturbance rejection is less evidently necessary. If the system is
known, feed forward tracking can be used. As uncertainty grows, feedback will have to be
applied.
Although there is no difference in the ratio between desired and attained position or
acceleration, errors or sensitivity to external signals are to be weighted with the squared
frequency (! 2 ) if one evaluates accelerational w.r.t. positional accuracy in the frequency
domain. This means sensitivity to noise is much more important in simulators than it is in
robotics. As a system is typically sensitive to noise around the bandwidth, it will be more
difficult to obtain a high bandwidth simulator motion system. Use of system knowledge
through model-based control, can prevent from unnecessary amplification of noise.
In the usual setting, the controlled motion system is in serial closed loop connection
with the aircraft model and pilot. Both the aircraft and pilot are not exactly specified, however,
an aircraft typically has its highest relevant frequency modes at 2-5Hz and pilots will
close the loop in order to obtain a bandwidth of .3 to 1Hz. This is attained by providing
lead over a decade around this frequency (so up till 3 Hz). To have minimal influence on
the pilot-aircraft(-model) loop, phase lag of the controlled simulator motion system should
be virtually zero in this frequency area, which otherwise would influence relative stability
of this loop. This requires a bandwidth of an order of a magnitude higher ( 10 Hz).
The frequency response of an aircraft evaluated w.r.t. position can be misleading as this

6.3 Theoretical modelling 205
response quickly drops above the highest mode while acceleration response remains constant.
Apart from these loop characteristics, realism might also be improved by simulation
of vibrations well above these frequencies such as turbulence, rotorcraft resonance, taxiing
wheel to ground rumble and impact on landing bump. Amplitude characteristics should be
unchanged in simulation of these kind of manoeuvres.
Finally the multivariable system should not introduce parasitic coupling in the other
degrees of freedom and as this depends on the relative position, the response in each degree
of freedom should be similar.
To summarise, a controlled motion system should preferably be able to simulate references
with frequency content up to 1 Hz with virtually zero time delay and an order of
a magnitude higher without amplitude attenuation. Use of system knowledge can help in
attaining a predictable response without unnecessary noise amplification.
6.3 Theoretical modelling
Modelling is the first step towards model based control but also helps in analysing the characteristics
of the open loop system, the response of the conventionally controlled system
and picking the relevant dynamics. Theoretic physical modelling enables analysis before
actually building a system and helps in attaining an experimentally fitted model of limited
complexity describing the relevant dynamical features and pointing at the physics from
which these features result. In describing the dynamics of a flight simulator motion system,
the most relevant dynamics can be found in the mechanical system, hydraulic drives and the
connection between the two.
The mechanical system is, unlike the usual robotic manipulators, a fully parallel system.
This leads to some dual properties if compared with the serial connected systems. In
singular points, the manipulator becomes unsupported in one or more degrees of freedom,
while in serial connected systems these become fixed. Kinematically, the parallelism poses
problems in describing the position as an explicit function of the actuated coordinates, while
in series connections these problems can occur using end effector coordinates. In describing
the dynamics, the equations of motion of a mechanical system, not choosing appropriate coordinates
then leads to combined differential algebraic equations which results in additional
problems in simulation or model based control. In the most general mechanical systems
both problems occur together with possible nonholonomy, in which case it is not clear if
or which coordinates can be chosen to have a global description in explicit form. In the
Stewart Platforms, i.e. the flight simulator motion systems, only parallelism has to be taken
care off and its dynamics can be described explicitly as a function of platform coordinates.
Of course still choices have to be made in describing the spatial rotation. The use of quaternions
or Euler-parameters leads to a computationally simple description also favourable in
system analysis although an additional but easy to handle, constraint equation results.
Symbolic multibody equation of motion solvers are suitable to generate a simulation
model in short time and without to much difficulties. These solvers, however, as they handle
the most general form of mechanical systems, do not easily recognise special properties, like
the possible explicit form of certain systems like the Stewart Platforms. Large numbers of
scalar variables result with systems of limited complexity like the motion system. They are

206 6 Review and discussion on the results
not suited to represent automatically appropriate vector and matrix equations or to take into
account equivalence between the different actuator legs. This makes system analysis more
difficult.
Using a projection method, the equations of motion of hexapod flight simulator motion
system can be written down preserving insight. Further, this shows that in this case both
Euler-Newton as Lagrange, whatever is appropriate, methods can be used in stating the
local equations of motion.
System analysis of multibody mechanical system shows the following. The strong coupling
in the system evaluated along the platform coordinates is mainly due to the coupled
transposed jacobian w.r.t. the centre of gravity mapping the actuator forces onto the platform
with specific mass and inertial properties. The jacobian varies with the platform position
and forms the main non-linearity in the mechanical system. Choice of appropriate kinematical
parameters, mass and inertial properties in the design of the construction should result in
reasonable conditioning of the mass matrix evaluated at the actuators. Inertial properties of
actuators are not very relevant in conventional systems due to high mass payload. With the
Simona motion system the one body mass matrix structure is not enough. Varying relative
actuator inertia is, however, still neglectable. The centripetal and coriolis forces will remain
relatively small due to the limited velocity of the actuators and reasonably low centre of
gravity.
Connection of hydraulics and mechanics results in the most relevant modes, the platform
pose dependent rigid modes. Direction of eigenvectors of these modes only depend
on the mass matrix of mechanical system in which the jacobian plays an important part.
Eigenfrequencies typically vary three to four times in magnitude as the eigenvalues of the
mass matrix vary ten to twenty times.
The collocated input/output pair of valve position and valve pressure difference of a
basic set of hydraulic actuator models connected to a general non-linear mechanical system
is passive. This is not the case with taking resulting actuator output pressure or platform
acceleration as an output. The property is also lost if valve dynamics can not be neglected
as is the case with the range where transmission line dynamics plays a role.
Passive feedback of pressure difference will not cause stability problems with any flexible
mode (e.g. position dependent rigid modes, fundament or simulator parasitic dynamics,
(transmission lines dynamics)). Direct feedback of pressure to flow e.g. leakage provides
for an additional dissipative term like viscous friction as feedback from velocity to force
does in purely mechanical systems. Total energy consists of both the mechanical kinetic
part as the hydraulic pressure oil stiffness part.
6.4 Experimental modelling and validation
The kinematical model forms a basic part of the dynamics. Validation of the kinematics
and identification of the parameters is the first step in identifying a full model. Redundant
measurements are required to identify the parameters. It is important to distinguish measurements
in calibration, which should be accurate vs. those which do not matter. Combining
two existing methods with limited adaptation, first redundant measurement of platform pose

6.4 Experimental modelling and validation 207
and secondly identification of the kinematic parameters, results in closely fit (within evaluation
accuracy) kinematics. This improves positional accuracy by an order of a magnitude
and can be attained by low cost length measuring devices. The location of the devices and
their connection on moving and rigid base is basically not important though it can influence
the sensitivity of the method. Important are the absolute lengths between the devices and
the absolute measurement of length itself.
In evaluating simulator acceleration response, the location of the accelerometers is highly
relevant. Gain, direction and location can be identified using the kinematical model of the
system with reasonable accuracy. Conditioning filters should also be known and taken into
account in evaluating measurements.
In this research, the motion system has been tested with three different load conditions.
The unloaded dummy platform (2250 kg), the empty shuttle (1680 kg) and finally the
dummy platform with full operational flight simulator load (4000 kg).
Already with a basic hydraulic/mechanical system model, the most relevant open loop
dynamics can quite closely be predicted. This was confirmed by measuring the result of
sine sweeps with a frequency analyser. With the Simona motion system this rigid body
dynamics typically showed around five to thirty Hertz. Dynamics, earlier obtained from a
single actuator and resulting from the valve and transmission lines is still valid, confirmed,
in the multivariable system but appears for most in the high frequency area (larger than fifty
Hertz) .
With the dummy platform additional resonances could be observed in the mid frequency
area, eight to fifteen Hertz, which resulted from a non rigid foundation at the test site as
confirmed by additional measurements with accelerometers attached to the ground. By
assuming a planar mass/spring/damper system in the fundament this additional dynamics
could be modelled, which led to a reasonable fit to the measurements taken at both the
pressure dynamics in the actuator and the acceleration measurements at the ground. With
the loaded dummy platform the dynamics of the foundation changes since a relatively higher
dissipation (possibly relatively more work done by friction) can be observed leading to
overdamped characteristics. Design of the foundation at a simulator motion system test site
should be done in careful consideration and always be evaluated properly after the motion
system has been set up.
Accelerometer measurements in the newly designed shuttle reveal deformation at the
frequencies where resonances occur in the pressure dynamics. Though stiff (> 40 Hz), the
modes of the system can be very lightly damped and occur in a frequency area, which was
previously used to attenuate pressure feedback (introducing phase lag losing the passivity
property). Further, one of the modes of the shuttle (at 65 Hz) appeared to be almost unobservable
in the valve pressure dynamics as the transmission line dynamics block (pair of
complex zeros) at this point. The number of measurements and the ability of the data acquisition
system should be enlarged to identify such mode shapes and other characteristics
of a flexible system more closely. Flexible deformations can be expected to occur at much
lower frequencies with a large projection screen attached. Further, a less exciting experiment
should be designed in case of a full operational simulator to prevent such a system
from any harm done.
If flexibility of the fundament can be modelled precisely, it can be compensated for,

208 6 Review and discussion on the results
but this is doubtful. The flexibility of the simulator results in a principle compromise since
only six coordinates can be functionally controlled. Problems with these kinds of parasitic
modes should for most be dealt with in design of the system. Not afterwards in control.
6.5 Control strategy and evaluation
With a conventional control strategy one uses pressure feedback to introduce extra damping
to the rigid mode resonances. With additional position feedback, a bandwidth close to this
resonance frequency can be obtained. As in this case, decentralised feedback is used, no
explicit compensation for each direction, which typically has a range of the rigid mode
frequencies, is taken into account and worst case compensation, usually overdamping the
system with respect to the variation in platform pose, will have to be settled for. Feed
forward can at best locally provide for adequate amplification of desired high frequency
references. If the system is operated and testing is only performed in or close to any neutral
position, this can be sufficient but still can require quite some tuning. Tuning can be reduced
if a model is taken into account in design.
With a model-based control strategy, one tries to compensate for the directionally different
rigid modes varying as a function of the position by taking into account the varying
jacobian and platform mass matrix. Taking into account the integration of hydraulics and
mechanics will of course form the basic part of this strategy. To attain a controller, which is
structured and can be implemented in different modules, it was split up in the following sub
tasks.
- Inner loop decentralised actuator pressure control
- Partial feedback linearisation
- Coordinate reconstruction
- Reference model based feed forward
- Outer loop position stabilisation
As acceleration in the mid-frequency area is important, the hydraulic actuators should
be able to deliver the required force to attain this acceleration. With the so-called cascadedp
control one compensates for the oil necessary to move at actual velocity and feeds back
the pressure necessary to attain required force. With this principle, a bandwidth of two to
three times the rigid mode frequency can be obtained. Velocity compensation principally
decouples the influence of the mechanics on the hydraulics.
The lowest rigid mode frequency can be found at 5 Hz with the fully loaded Simona
motion system in the neutral position. It will typically change to lower values moving to
other positions.
As the velocity is not measured directly, it has to be reconstructed by an observer. Both
position and pressure measurements and the desired velocity can be used to calculate the
required compensation. Differentiation of position is simple but very limited in frequency
due to measurement errors. Pressure can be used with higher frequencies but requires the

6.5 Control strategy and evaluation 209
full multivariable mechanical model which is not exact, adds multiple uncertainties and is
further very sensitive to friction which is small but existent in the motion system. Even with
exact velocity reconstruction, velocity compensation will not be exact due to uncertainty in
the valve characteristics.
Also platform state reconstruction has to be performed since the model is only explicit
in platform coordinates. An iterative scheme is used to calculate the platform pose from
actuator position measurement. Use in feedback requires convergence at all time for stability
and guaranteed accuracy in a very limited amount of iterations if applied in real time.
Sufficient conditions on the convergence (speed) of a general Newton-Raphson scheme can
be translated to the general Stewart Platform. The jacobian used in the scheme is shown to
be Lipschitz under mild conditions. With the Simona motion system, convergence in two
iterations within measurement device accuracy can be guaranteed given the limited actuator
speed and a sufficiently high update frequency (100 Hz), which was easily attainable.
With partly feedback linearising control of the mechanical system, one strives for two
objectives. First, from feed forward of desired accelerations, required forces are calculated
taking the varying mass matrix, non-linear velocity terms and gravity into account. Further,
feedback of positional and velocity error is performed in a theoretically decoupled set of
coordinates. Translation of desired platform forces to actuator forces requires inversion of
the jacobian, which, however, can also be used with the coordinate reconstruction.
In the conventional structure of motion simulation, the series connection of controller
and motion system ideally requires an infinitely fast controlled motion system. Since the
aircraft model and wash-out filters are incorporated in the simulation program, reference
model knowledge can be used to generate a predicted acceleration over a certain amount of
time. 30-50 ms would be sufficient to attain a situation in which the acceleration onset can
be felt exactly at the required instant of time with a finitely fast motion system. This will be
possible since the model based controlled motion system behaves in a predictable manner.
As positional accuracy is not the main objective of the controller, the position feedback
should only be used in preventing the system from running out of stroke and attaining the
right accelerations in the low frequency area. Especially at the bandwidth of the positional
feedback, care should be taken not to distort the appropriately generated accelerations.
A model based controller is able to enlarge the bandwidth of the accelerations to be simulated
by a factor two w.r.t. the directions of the rigid mode with the lowest frequency and
also compared to the conventional controlled system. This was confirmed in this research.
By limiting the bandwidth in the other directions to attain equal response in any direction,
unnecessary excitation of unmodelled dynamics, amplification of noise and coupling between
rotation and translation is avoided.
The dynamics of the foundation at a test site, which is not explicitly taken into account,
limits the performance of the system. Success of compensation is doubtful since the effect
is seen to vary a unpredictable manner w.r.t. payload characteristics.
System feedback control enforces reasonable linear response. With this assumption,
the characteristics of the system can be tested by measuring the frequency response. The
assumption can be verified by performing some additional tests. In case of a motion system,
the linearity can be tested with respect to a set of different amplitude acceleration
step responses and a standardised number of realistic manoeuvres representative for motion

210 6 Review and discussion on the results
through the workspace of the system. Exact quantification of the noise would require more
sophisticated methods and measurement devices. Moreover, the noise seems to vary over
time.
A modern standardised test method is required to evaluate the performance of a simulator
motion system. In this research some building blocks to set up such a test were provided.
Main aspect in such a test should be the predictability of a system i.e. guaranteed simulation
quality. The model based approach taken in this research helps to attain this.

Chapter 7
Conclusion and recommendations
7.1 Conclusion
Central to this conclusion is the main problem statement to investigate what relevant system
knowledge should be used in a model based control strategy for a flight simulator motion
system and to what extent this results in improved controlled dynamics.
In this thesis, the path of choice became physical modelling through balance equations
of the hydraulically driven mechanical system, identification of the model parameters, experimental
evaluation of the model structure and explicit use of the model in the controller.
An important first observation, considering the mechanical construction of Stewart platform
like motion system, is their limited complexity if compared to general higher degree
of freedom manipulators. The dynamics can explicitly be described as a function of a fixed
set of platform coordinates unlike many combined serial/parallel constructions. This also
limits the complexity of a model based control strategy.
The mapping between the variations along the platform coordinates and actuator coordinates,
the jacobian, is an important parameter in control of multi degree of freedom manipulators
such as the Stewart platform. Most of the calculations in the model based controller
presented for the flight simulator motion system are directed towards this parameter.
In parallel systems, controllability is lost if the jacobian loses rank in some position,
unlike serial robotic systems. In this thesis, a method was presented to exclude these kind
of singular positions from the reachable envelope of a parallel system considering limited
stroke actuators. For the Simona Research Simulator (SRS) it was proven that such points,
although not far away, can just not be reached.
If the position of a parallel system is only measured along the actuator coordinates,
an inverse problem reconstructing the platform coordinates has to be solved in real time
using the model based control strategy. In this thesis a method was presented to prove
convergence of the specific Newton-Rapshon iteration chosen anywhere in the reachable
envelope of the platform. It was proven that it does and it does sufficiently fast for the
SRS. The platform coordinate reconstructor could therefore be made a central module in
the model based controller, which was implemented.
The platform pose dependent jacobian together with the platform coordinate related
mass matrix of the system recover the specific masses as seen through the actuators. In
211

212 7 Conclusion and recommendations
most flight simulator motion systems, and also the SRS, large variations of the specific
masses, more than an order of a magnitude, can be expected. Therefore, only misdirecting
a force 1% easily results in a ten times higher accelerational error of 10%. Feedforward
control therefore requires highly accurate models.
The kinematical model structure of the system was shown to be accurate within tens
of mm. Calibration of the model parameters improved the positional accuracy an order of
a magnitude. The structural model properties of the dynamics of the hydraulically driven
mechanical motion system were also confirmed. However, estimation of the system mass
properties can probably be improved despite the scheme developed to robustly derive the
mass properties of such systems in the presence of dissipation or some nonlinear effects.
The specific mass directions are almost equal to the rigid body modal directions connecting
the hydraulic actuators. In analysis and identification of hydraulically driven motion
systems, knowledge of these directions proved to be very helpful to decouple the, at first
sight, highly interactive dynamics. Uncontrolled hydraulically driven systems are usually
lightly damped and this increases the already large difference along the rigid body modal
directions in the neighbourhood of the resonance frequencies. The rigid body resonance
modes form the main dynamics in the most relevant frequency area for flight simulation up
till 30Hz.
The conventional control structure robustifies the decentralised positional feedback through
dissipating energy by pressure feedback thereby damping the resonance peaks. In this thesis
this strategy proved right by showing that feedback of the input valve pressure difference
to valve flow is feedback of a passive transfer function, also in the presence of parasitic
dynamics e.g. due to transmission line resonances or mechanical flexibilities. However,
no directionality can be accounted for in this structure. It was shown that this leads to
limited bandwidth systems, which still have to compromise overdamped directions with
some peaking of others.
With the model based strategy, the already known cascade dp structure was chosen,
which maintained local decentralised dissipating pressure feedback by inner loops and decoupled
hydraulics from mechanics. This leaves the directionality to be dealt with by the
multivariable feedback of the mechanics. Through implementation of this controller on
the SRS it was shown that this indeed leads to a system with higher bandwidth and more
equalised response in each degree of freedom.
The attainable improvement through the model based controller chosen is limited in several
ways. Not all dynamics was or could be made part of the model used in control. In testing
the SRS, of course, the dynamics of the foundation appeared in the relevant frequency
area due to the non ideal experimental environment. But this should not be a problem in a
specific simulator building. However, it can be concluded that the absence of such dynamics
should be confirmed, tested, in any place where motion systems are to be used or evaluated.
The flexible deformations of the system are a more severe problem. This causes loss of
functional controllability, strived for in simulation. Adding additional objects, e.g. projection
screens, will further limit the attainable performance.
The phase lag of the hydraulic valve usually limits the attainable feedback pressure
bandwidth. With fast valves one can choose this bandwidth in between the rigid body
modes and the transmission lines resonances but exactly in this frequency area the flexible

7.2 Recommendations 213
resonances show up if velocity is not precisely compensated for, as is impossible in practice.
The physical modelling in this research enabled a limited complexity model based control
structure for flight simulator motion systems by choosing only the most relevant dynamical
effects to be incorporated. Also the parameter identification procedure could be made
very compact in this way. The model based controller proved better in almost all respects if
compared to a conventional control structure implemented on the actual motion system.
Summarising, the full design process proposed was shown to be applicable and leads to
more predictable characteristics and higher performance of flight simulator motion systems.
7.2 Recommendations
This thesis systematically went through the full design process of motion control design of
flight simulator motion systems but it still leaves many points of research open for more
thorough investigation. Some at the high abstract level of the over all simulation, some at
the lower level of detail though in practise often the most troublesome.
First of all, specifications towards the design of motion controllers in simulation should
become more precise in stating in what way the fundamentally non exact motion simulation
characteristics should be compromised for. E.g. how much more accelerational noise can
be settled for in attaining one additional Hertz of bandwidth? What class of signals will
have to be tracked by the motion system? Usually there is much information in this area,
structure in the trajectories, but not yet very well specified. Short time spectral properties
of the reference signals is relevant in this sense since otherwise the important aspects, high
frequency content, of e.g. a landing bump, would not be revealed.
These specifications should form the basis for a standardised test to measure the performance
and predictability i.e. robustness of a motion system. In this research a number
of tests were introduced to measure the characteristics of a controlled motion system in a
more modernised manner as compared to the current flight simulation standards. But only
providing more specific guidelines, weighting factors w.r.t. desired properties of a motion
system, will enable the design and analysis of the right test and can lead the way towards
the most effective control strategy.
Information from the flight simulation model can also be used more conveniently in the
control scheme itself. Limited short response time of the required pressure results in lagged
accelerational response of the motion system as compared to what was required in simulation.
As the spectrum of the desired accelerations has almost always limited bandwidth,
predictive information theoretically is sufficient to exactly attain the required response. In
this thesis it was shown how this information can be used. Flight simulation models should
be set up in such a way that this predictional information can be communicated to the motion
system controller. It is the most viable way to attain virtual zero time delay response in
simulation.
Dynamics, not taken into account by the model based controller, occurred in practise.
The most important part was shown to be caused by the flexible deformations of the system,
which in future will become even more relevant. Procedures to identify the specific structure
of these parasitic resonances will have to be set up and applied. With more specifically

214 7 Conclusion and recommendations
directed identification techniques in general, parameter estimation of the system model can
probably be improved further.
The control strategy should be extended in order to deal effectively with the uncertain
or identified flexible dynamics. One could even consider applying additional actuators,
to dissipate energy stemming from local resonant modes, to solve the current problem of
functional controllability in the presence of deformations. Also additional sensors, such
as the accelerometers applied in identification, could be considered in an extended control
structure. After all, acceleration felt in the cockpit is what really matters in motion cues for
flight simulation.
Dealing with uncertainty in general in control requires robust feedback/feed forward
schemes. Models can never be exact. Unfortunately many modern robust control techniques
are not mature enough at the moment to deal efficiently with nonlinear six d.o.f. hydraulically
driven motion systems. In the side line of this research some first steps towards robust
control of such systems were taken but this did not lead to satisfying results yet. Still, attaining
robustly guaranteed properties in simulation is important in flight training and makes
more effort in this direction worthwhile.
W.r.t. the specific model based control structure chosen, the interaction between the inner
and outer loops appeared to be more severe than expected. More exact velocity compensation,
central in the decoupling of mechanics and hydraulics, would directly enhance the
proposed control structure. This requires more accurate velocity estimation first. E.g. due
to a structural deviation in the position measurement the increased resolution could not yet
be used in more precise velocity reconstruction though it can probably be compensated for.
Next, enhanced control over the valve, the oil flow to and from the actuators, would be of
help. Especially the nonlinear characteristics of the valves often appeared to be troublesome
in practise. E.g. each valve has to be activated at all time by a high frequent dither signal to
prevent stick of one of the spools. In setting the dither amplitude one had to be very careful
not to hit transmission line dynamics (too high) or otherwise let sticking effects occur (too
low). An adaptive setting can improve this.
Over all this thesis provides on one hand many leads towards more scientific investigations
and on the other hand laid down an improved design procedure, which can be applied
directly by the practioneer. And that’s what’s research is all about.

Appendix A
Frequency domain measurements
215

hdp_i1
hdp_i2
hdp_i3
hdp_i4
hdp_i5
hdp_i6
216 A Frequency domain measurements
hu_1
hu_2 hu_3 hu_4 hu_5 hu_6
Fig. A.1: Bode amplitude plots dummy platform from valve voltage to valve pressure
measured and modelled (dashed). Frequency runs form 5Hz to 50Hz with
ticks at 10Hz, 20Hz and 30Hz. Amplitude from 0:5 to 50 with ticks at 1 and 10.

hdp_i1
hdp_i2
hdp_i3
hdp_i4
hdp_i5
hdp_i6
hu_1
hu_2 hu_3 hu_4 hu_5 hu_6
217
Fig. A.2: Bode phase plots dummy platform from valve voltage to valve pressure measured
and modelled (dashed). Frequency runs form 5Hz to 50Hz with ticks at
10Hz, 20Hz and 30Hz. Phase from ;700 to 100 with ticks from ;135 to
90 every 45 .

hdp_i1
hdp_i2
hdp_i3
hdp_i4
hdp_i5
hdp_i6
218 A Frequency domain measurements
hu_1
hu_2 hu_3 hu_4 hu_5 hu_6
Fig. A.3: Bode amplitude plots dummy platform from valve voltages to valve pressures
along the rigid body modal directions measured and modelled (dashed). Frequency
runs form 5Hz to 50Hz with ticks at 10Hz, 20Hz and 30Hz. Amplitude
from 0:5 to 50 with ticks at 1 and 10.

hdp_i1
hdp_i2
hdp_i3
hdp_i4
hdp_i5
hdp_i6
hu_1
hu_2 hu_3 hu_4 hu_5 hu_6
219
Fig. A.4: Bode phase plots dummy platform from valve voltages to valve pressures
along the rigid body modal directions measured and modelled (dashed). Frequency
runs form 5Hz to 50Hz with ticks at 10Hz, 20Hz and 30Hz. Phase
from ;700 to 100 with ticks from ;135 to 90 every 45 .

hdp_i1
hdp_i2
hdp_i3
hdp_i4
hdp_i5
hdp_i6
220 A Frequency domain measurements
hu_1
hu_2 hu_3 hu_4 hu_5 hu_6
Fig. A.5: Bode amplitude plots shuttle from valve voltage to valve pressure measured
and modelled (dashed). Frequency runs form 5Hz to 500Hz with ticks at 10Hz
and 100Hz. Amplitude from 0:03 to 30 with ticks at :1, 1 and 10.

hdp_i1
hdp_i2
hdp_i3
hdp_i4
hdp_i5
hdp_i6
hu_1
hu_2 hu_3 hu_4 hu_5 hu_6
221
Fig. A.6: Bode amplitude plots shuttle from valve voltage to valve pressure measured
and modelled (dashed). Frequency runs form 5Hz to 500Hz with ticks at 10Hz
and 100Hz. Phase from ;700 to 100 with ticks from ;180 to 90 every 90 .

hdp_i1
hdp_i2
hdp_i3
hdp_i4
hdp_i5
hdp_i6
222 A Frequency domain measurements
hu_1
hu_2 hu_3 hu_4 hu_5 hu_6
Fig. A.7: Bode phase plots shuttle from valve voltages to valve pressures along the
rigid body modal directions measured and modelled (dashed). Frequency
runs form 5Hz to 500Hz with ticks at 10Hz and 100Hz. Amplitude from 0:03
to 30 with ticks at :1, 1 and 10.

hdp_i1
hdp_i2
hdp_i3
hdp_i4
hdp_i5
hdp_i6
hu_1
hu_2 hu_3 hu_4 hu_5 hu_6
223
Fig. A.8: Bode phase plots shuttle from valve voltages to valve pressures along the
rigid body modal directions measured and modelled (dashed). Frequency
runs form 5Hz to 500Hz with ticks at 10Hz and 100Hz. Phase from ;700 to
100 with ticks from ;180 to 90 every 90 .

ax
ay
az
aex
aey
aez
224 A Frequency domain measurements
ax_d
ay_d az_d aex_d aey_d aez_d
Fig. A.9: Bode amplitude plots closed loop around heavy weight dummy platform
from desired to actual ccelerations model based controller (-) and conventional
controller (dashed). Frequency runs form 0:5Hz to 50Hz with ticks at
1� 2� 3� 5� 10� 20� 30Hz. Amplitude from 0:025 (;32dB)to2:5 (8dB) with ticks
at ;30� ;20� ;6� ;3� 0� 3� 6dB.

ax
ay
az
aex
aey
aez
ax_d
ay_d az_d aex_d aey_d aez_d
Fig. A.10: Bode phase plots closed loop from desired to actual accelerations model
based controller (-) and conventional controller (dashed). Frequency runs
form 0:5Hz to 50Hz with ticks at 1� 2� 3� 5� 10� 20� 30Hz. Phase from ;180
to 180 with ticks from ;180 to 180 every 45 .
225

e_ax
e_ay
e_az
e_aex
e_aey
e_aez
226 A Frequency domain measurements
ax_d
ay_d az_d aex_d aey_d aez_d
Fig. A.11: Bode amplitude plots closed loop from desired to error accelerations model
based controller (-) and conventional controller (dashed). Frequency runs
form 0:5Hz to 50Hz with ticks at 1� 2� 3� 5� 10� 20� 30Hz. Amplitude from
;32dB to 14dB with ticks at ;20� ;6� ;3� 0� 3� 6dB.

Appendix B
Derivation of actuator inertial
properties
In this appendix essentially the derivation of (2.119) and (2.120) is performed. Further, it
is shown how to choose a factor N in the quadratic velocity term to make _ M ; 2N skew
symmetric. This last property is important in passivity based control.
First the derivation of the derivative mass matrix equation (2.119).
and
With
d
dt (Mia�ib )=; (ia + ib)
j l j3 (2v T a lnPln + Plnval T n + lnv T a Pln ) (B.1)
d d p l
j l j= lT l =
dt dt
T va
j l j = lT n va
_
ln = d l
dt j l j =
l
_ j l j;l d dt j l j
j l j2 the required steps can be taken
d
(Mia�ib )
dt
=
d
dt ( (ia + ib)
j l j2 Pln )=(ia + ib)
(B.2)
= (I ; lnlT n )
va =
j l j
1
j l j Plnva: (B.3)
d
dt
1
j l j2 I + d
dt
lnl T n
j l j 2
= (ia + ib) ;2lT n va j l j
j l j4 1
(
jlj
I ; PlnvalT 1
n +
jljlnvT a Pln ) j l j2 ;2lT n va j l j lnlT n
j l j4 = ; (ia + ib)
j l j3 (2v T a lnPln + Plnval T n + lnv T a Pln ): (B.4)
N is defined as a factor in the factorisation of the nonlinear velocity term C(x� _x). But
this factorisation is not uniquely defined by requiring that C(x� _x) =N (x� _x) _x.
227

228 B Derivation of actuator inertial properties
Taking
f a ia�ib = Mia�ib _va ; 2(ia + ib)
j l j3 (l T n va)Plnva: (B.5)
The nonlinear velocity term, ; 2(ia+ib)
jlj3 (lT n va)Plnva can be factored into N (x� va)va in an
infinite number of ways. Two possible factors N are in this case:
and
N1 = ; 2(ia + ib)
j l j3 (l T n va)Pln
(B.6)
N2 = ; 2(ia + ib)
j l j 3 Plnval T n (B.7)
And all factorisations N1 +(1; )N2, with 2 IR are valid. Only one factorisation
results in a skew _ M ; 2N.
Take
Then
N = 1
2 (N1 + N2): (B.8)
_M ; 2N = 2(ia + ib)
j l j3 (Plnval T n ; lnv T a Pln ): (B.9)
Check through _ M ; 2N = ;( _ M ; 2N ) T that the skew symmetry is there (P T
ln
= Pln).
Now the derivation of the partial derivative of the kinetic energy actuator inertia given
by
@Kia�ib
@pa
= ; (ia + ib)
j l j3 Define a partial derivative of a vector, x to a vector, y n 1 as:
@x
@y =
h @x
@y1
@x
@y2
l T n (vT a Plnva) +v T a Pln (lT n va) : (B.10)
:::
@x
@yi
:::
@x
@yn
Kinetic energy of the actuator intertia was derived as the scalar term:
with
Kia�ib
i : (B.11)
1
=
2 vT (ia + ib)
a
j l j2 Plnva (B.12)
= 1
2 vT a
(ia + ib)
j l j2 va ; 1
2 (ia + ib)
v T a l
j l j 2
v T a l
j l j 2
� (B.13)
Pln =(I ; lnl T n )� ln = l= j l j : (B.14)

The upper actuator gimbal point, pa, depends on the variables c and in the same way as l:
Therefore
Now
229
pa =c + T ( )a� (B.15)
l =c + T ( )a ; b: (B.16)
@l
= I: (B.17)
@pa
= 1
2 (ia + ib)v T a va
h
1
@
jlj2 i
; (ia + ib)
@pa
h
vT a l
jlj2 i
@Kia�ib
@pa
vT a l
j l j2 @
@pa
: (B.18)
So two partial derivatives to scalars have be examined further. The first
and the second
@
h v T a l
jlj 2
@pa
i
@
h 1
jlj 2
@pa
= vT a j l j 2 ;2v T a ll T
j l j 4
i
= @ 1
l T l
@pa
= vT a
j l j2 ; 2vT a lnlT n
j l j2 = ;2lT
j l j4 = ;2lT n
� (B.19)
j l j3 Filling in these equations in (B.18) results in, using (x T y)=(y T x),
@Kia�ib
@pa
=vT 1
a Pln
j l j2 ; vT a lnlT n
: (B.20)
j l j2 = ; 1
2 (ia + ib)v T 2l
a va
T n
j l j3 ; (ia + ib) v T 1
a Pln
j l j2 ; vT a lnlT n
j l j2 = ; (ia + ib)
j l j 3
= ; (ia + ib)
j l j 3
And this was what was to be determined.
v T a Ival T n ; v T a lnl T n val T n +vT a Pln (vT a ln)
v T a ln
j l j
l T n (vT a Plnva) +v T a Pln (lT n va) : (B.21)

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Glossary of symbols
241

242 Glossary of symbols
Arabic symbols (large)
Variable Unit Explanation
A [m 2 ] Actuator operational area
[m] 3x6 matrix with upper gimbal (to c.o.g.) vectors stacked
Ap [m 2 ] Actuator operational area
B [m] 3x6 matrix with lower gimbal vectors stacked
B,Bp [Ns=m�Ns] Viscous friction (matrix)
C [kg=s,kgm 2 =s] Nonlinear coriolis and centripetal terms
Ci [N=m 5 � ;] Oil column stifness chamber i
dC [N=m 5 ] Difference oil column stifness chambers 1 and 2
Cm [N=m 5 ] Mean oil column stifness chambers 1 and 2
E [N=m 2 ] Effective bulk modulus of the oil
G [ ] Ground frame or intertial frame
[;] 3x4 transformation as part of rotation R = GL T
G T s [;] 3x3 ! to transformation
I [ ] Identity matrix
Iz [kgm 2 ] 3x3 platform inertia matrix evaluated at c.o.g.
Jy�x [ ] Jacobian matrix between two sets of variables y and x
Jl�x [;�m] Jacobian from platform velocities to actuator velocities
K [ips=( c irc=s 2 �m=s 2 )] Semicircular and vestibular organ model gain
[m=(sA)] Hydraulic actuator model gain
Kd As=m Valve main spool differential feedback gain
Kp A=m Valve main spool proportional feedback gain
Kh [nd] Crossover model pilot gain
L [ ] 4x3 eul. par. transformation as part of rotation R = GL T
Ln [;] 3x6 matrix with all unit actuator directions vectors stacked
Llm [m 5 =N] Oil leakage constant due to pressure difference actuator chambers
Lt [m] Transmission line length
M [kg,kgm 2 ] Mass matrix
[ ] Moving frame
[-] Mobility index
Mact [kg� kgm 2 ] Simulator mass matrix actuator coordinates
Mt [kg� kgm 2 ] Simulator mass matrix platform coordinates

Glossary of symbols 243
Variable Unit Explanation
dP [N=m 2 ] Net generated actuator pressure
Pi1 [N=m 2 ] Oil pressure at valve side transmission line 1
Pi2 [N=m 2 ] Oil pressure at valve side transmission line 2
Pm [N=m 2 ] Mean actuator chamber pressure
Pm1�2 [N=m 2 ] Main valve spool pressures
Pn1�2 [N=m 2 ] Pilot valve spool pressures
Po1 [N=m 2 ] Oil pressure at actuator chamber 1
Po2 [N=m 2 ] Oil pressure at actuator chamber 2
Ps [N=m 2 ] Supply oil pressure
Pt [N=m 2 ] Tank oil pressure
Pxn [;] 3x3 projection matrix onto plane with normal x n
R [;] 3x3 rotation matrix
B R A [;] 3x3 rotation matrix from frame A to B
T [;] 3x3 rotation matrix platform
Tx�y�z [;] 3x3 rotation matrix around one axis of the frame
U [;] Unitary (rigid body modal direction) matrix
V [m 3 ] General volume
V1 [m 3 ] Oil volume actuator chamber 1
V2 [m 3 ] Oil volume actuator chamber 2
Vm [m 3 ] Mean oil volume actuator chambers 1 and 2
Va [m=s] 3x6 matrix with upper gimbal velocity vectors stacked
X general matrix (capital)
~X vector product matrix
Arabic symbols (small)
Variable Unit Explanation
ac [;] Upper actuator body c.o.g.
ai [m] Vector from c.o.g. simulator to i th upper gimbal point
bi [Ns=m] Viscous friction coefficient actuator i
bxf�yf� f [Ns=m� Nms=rad] Viscous friction coefficients foundation
bc [;] Lower actuator body c.o.g.
bi [m] Vector from ground frame origin to i th lower gimbal point
c [N=m] Hydraulic actuator stifness
c1�::: �10 [ ] Valve model parameters
co [m=s] Wave propagation velocity
cxf�yf� f [N=m� Nm=rad] Stifness coefficients foundation
c [m] Vector from ground frame origin to c.o.g. simulator
d [ ] Finitely small variation
dl [m] Lower gimbal spacing
du [m] Upper gimbal spacing
e [ ] Error vector
[ ] General external signal
dp [N=m 2 ] Pressure difference over hydraulic actuator compartments

244 Glossary of symbols
Variable Unit Explanation
f N�Nm Inertial force
fa N 6x1 vector active actuator forces
f (:) [ ] Vector function
g(:) [ ] Vector function
g [N,Nm] Gravitational terms
fd [N] Actuator driving force
frm [1=s] Rigid body mode eigenfrequency
i [A] Valve input current
ia [kgm2 ] Inertia upper actuator body around
ib [kgm2 j
]
[;]
Inertia lower actuator body around around gimbal point
Imaginairy unit, p ;1
kdp [Am2 =N, ] (Normalised) pressure feedback gain
li [m] Length ith joint (serial), actuator (parallel) manipulator
l [m] Vector with all actuator platform lengths
li [m] Vector form the i th lower to upper gimbal
ln [;] Normalised unit direction vector actuator
nG�M x�y�z [m] Unit direction vectors
direction in ground, G, and moving frame, M
m [kg] Mass (platform)
ma [kg] Mass upper actuator body
mb [kg] Mass lower actuator body
mf [kg] Mass of the foundation
m [kgm=s� kgrad=s] Vector of generalised momenta
n [ ] Euler paramter unit vector along axis of rotation
pg [m] Vector to point p in frame G
pm [m] Vector to point p in frame M
q [m] Actuator extension
qi [m] Rotation angle i th joint serial manipulator
q [m,rad] Robotic manipulator positional coordinates
qd [m,rad] Desired positional coordinates
ra [m] Length from c.o.g. upper actuator body to upper gimbal
rb [m] Length from c.o.g. lower actuator body to lower gimbal
s [rad=s] Sloppy Laplace operator as j!
[m] Scaling factor, ka jmax
sx [m] Vector with platform coordinates
t [s] Time
t1�::: �5 [ ] Transmission line model parameters
tg [m] Translational vector in frame G
u [V ] Actuator input voltage
[ ] General system input
va [m=s] Upper gimbal point velocity vector
vm m m=s Velocity vector of a point in a moving frame

Glossary of symbols 245
Variable Unit Explanation
x [ ] General scalar
[m] Surge direction
xf [m] Valve flapper position
[m] Foundation surge displacement
xm [m] Valve main spool position
xs [m] Valve pilot spool position
xa [m] Vector from ground origin to upper gimbal point
xn [ ] General normalized vector (index n) with length one
x [ ] General vector (bar on top)
_x [m=s� rad=s] Vector with translational and angular velocity platform
x g [m] Vector in inertial (ground) frame
x m [m] Vector in platform (moving) frame
xin [(rad� m)=s 2 ] Semicircular and vestibular input (rotational) acceleration
xout [(rad� m)=s 2 ] Noticed semicircular and vestibular (rotational) acceleration
y [m] Sway direction
[ ] General system output
z [m] Heave direction
z [m� rad] Vector with generalised (positional) coordinates
Greek symbols (small)
Variable Unit Explanation
[ ] Bound on first Newton Raphson iteration
[ ] Relative damping factor
[ ] Bound on inverse lowest singular value NR-iteration
[rad] Vector of euler angles
[] Infinitely small variation
0�1�2�3 [;] The specific four euler parameters
[;] Vector with last three euler parameters 1, 2, 3
e [;] Vector with all four euler parameters
m [V ] Valve spool position error
~ [;] Cross product matrix with euler parameters
[ ] Lipschitz constant
[rad] Angle in general
[rad] Last euler angle around airplane roll axis
[rad] Euler parameter angle of rotation
[rad] Second euler angle around intermediate pitch-axis
[kg=m 3 ] Oil density
[ ] Singular value

246 Glossary of symbols
Variable Unit Explanation
[N,Nm] Generalized torques, forces
1 [s] Crossover model pilot lead time constant
[s] First semicircular and vestibular lag time constant
2 [s] Crossover model pilot lag time constant
[s] Second semicircular and vestibular lag time constant
L [s] Semicircular and vestibular lead time constant
d [s] Crossover model pilot pure time delay
Nm�N Generalised torques
! [rad=s] Frequency
!o [rad=s] Hydraulic actuator eigenfrequency
! [rad=s] Angular velocity vector platform
!a [rad=s Angularo velocity vector actuator orth. to actuator length
1�::: �5 [ ] Transmission line model parameters
[rad] First euler angle around ground frame yaw axis
Greek symbols (large)
Variable Unit Explanation
i1 [m 3 =s] Oil flow from valve into transmission line 1
i2 [m 3 =s] Oil flow from valve out of transmission line 2
l1 [m 3 =s] Leakage oil flow outof actuator chamber 1
l2 [m 3 =s] Leakage oil flow into actuator chamber 2
lm [m 3 =s] Leakage oil flow from actuator chamber 2 into 1
m [m 3 =s] Mean oil flow into chamber 1 and out of chamber 2
d [m 3 =s] Difference oil flow o1 ; o2
~ [rad=s] Cross product matrix with angular velocity
Caligraphic symbols (large)
Variable Unit Explanation
G [N�Nm] Generalised gravitational forces
H [kgm 2 =s 2 ] Hamiltonian function (addition kinetic and potential energy)
K [kgm 2 =s 2 ] Kinetic energy function
L [kgm 2 =s 2 ] Langrangian function (difference kinetic and potential energy)
P [kgm 2 =s 2 ] Potential energy function

Samenvatting en CV
De hoge veiligheidseisen in de luchtvaart vereisen een goed begrip van het gedrag van de
piloot in extreme weersomstandigheden. Vernieuwend onderzoek op dit gebied moet gebruik
kunnen maken van vluchtsimulatoren die in staat zijn om met hoge nauwkeurigheid
kritische condities tijdens een vlucht te simuleren. Verschijnselen zoals turbulentie en zijwind
in de lagere delen van de atmosfeer vertonen een snelheidsspectrum dat grote krachten
introduceert over een breed scala aan frequenties. Het reproduceren van de benodigde beweging,
het voelbaar maken van de inertiële versnellingen en het juist uitrichten van de
zwaartekracht in geavanceerde vluchtsimulatoren, is dus een uitdagende regeltaak die het
uiterste uit het systeem moet halen. Voor vergelijkbare systemen, zoals robots, bestaan
moderne modelgebaseerde regelstrategieën, die hogere prestaties kunnen behalen dan minder
gestructureerde methoden, meer inzicht geven in de begrenzingen van het systeem
en wellicht op constructieve eigenschappen van het systeem kunnen wijzen waar gebruik
van kan worden gemaakt in het ontwerp. Deze regelstrategieën worden nog vrijwel niet
toegepast bij bewegingssystemen van vluchtsimulatoren. Eén van de redenen zou kunnen
zijn dat deze systemen zo complex zijn dat exacte of zeer gedetailleerde modellen geen
onderdeel van de regelaar konden en kunnen worden gemaakt. Toepassing ervan in dit
onderzoek vereiste een tussenstap waarin een minder gedetailleerd model moest worden
geëxtraheerd dat toch de meest relevante dynamica moest beschrijven.
De vraag welke relevante systeemkennis gebruikt zou moeten worden in een modelgebaseerde
regelstrategie en in welke mate deze strategie bijdraagt aan het geregelde gedrag
van een bewegingssysteem van een vluchtsimulator, is het centrale onderzoeksthema van
dit onderzoek geweest.
Om hierop een antwoord te geven moest het volledige regelaarontwerpproces worden
gedefinieerd, gestructureerd en geëvalueerd. Veel van de tussenstappen toegepast in dit proces
kunnen al in de literatuur worden gevonden maar een integrale aanpak, waarin ook de
specifieke eisen ten aanzien van een vluchtsimulator werden meegenomen, ontbrak nog. Bij
de keuze van bestaande dan wel nieuw voorgestelde bouwstenen in dit proces van systeemen
regelaarontwerp, vormden integreerbaarheid en praktische toepasbaarheid de voornaamste
argumenten.
Door middel van modelvorming op basis van fysische balansvergelijkingen is eerst een
relevante modelstructuur ten behoeve van systeemanalyse en regeling afgeleid. Daarna is
deze structuur geëvalueerd met behulp van experimenten. Daarbij vond identificatie van de
modelparameters en het in kaart brengen van het geldigheidsgebied van de gekozen structuur
plaats. Vervolgens is de meest geschikte modelgebaseerde regelstrategie gekozen en
aangepast voor toepassing op dit bewegingssysteem met zijn specifieke parallelle struc-
247

248 Samenvatting en CV
tuur. Ontwerp en implementatie van deze regeling op het bewegingssysteem van de Simona
vluchtsimulator was de volgende stap. Tenslotte is een testprocedure gedefinieerd om de
prestatie van dit geregelde systeem te kunnen kwantificeren en te kunnen afzetten tegen de
prestatie van een conventioneel geregeld systeem. Gegeven de specifieke toepassing van
vluchtsimulatie moesten verschillende stappen in dit generieke ontwerpproces nader worden
uitgewerkt.
Opbouwen van vergaand inzicht in de kinematica en dynamica van het systeem was
nodig om zowel de robuustheid van een conventionele regeling te behouden als ook de
prestatie en de voorspelbaarheid te verhogen. Met fysische modelvorming kon kwalitatief
inzicht worden opgebouwd en gelijkertijd gaf deze structuur de mogelijkheid om de parameters
van het zich niet-lineair gedragende bewegingssysteem experimenteel te identificeren
en de te implementeren regelaarstructuur te modulariseren. Met deze identificatie en experimentele
evaluatie kon de vereiste nauwkeurigheid voor het model worden bereikt die
nodig was om de prestatieverhoging met de modelgebaseerde regelaar ook in de praktijk te
kunnen behalen.
Fysische modelvorming van bewegingssystemen begint met de analyse van de systeemkinematica.
In dit onderzoek wordt een nieuwe methode gepresenteerd waarmee het
voor het eerst mogelijk werd om te garanderen dat singuliere punten buiten het werkgebied
van het systeem vallen. Hierop verder bouwend kan ook nagegaan worden of een iteratief
schema om de platformcoördinaten te berekenen, gegeven de gemeten lengte van de actuatoren,
convergeert en snel genoeg convergeert voor implementatie in de werkelijke regeling.
Hiermee kon voor het Simona bewegingsysteem de garantie worden geboden dat dit deel
van de regelstructuur veilig toegepast kon worden. Inzicht in de kinematica gaf verder de
mogelijkheid om door middel van calibratie de positioneringsnauwkeurigheid van het al
nauwkeurig gespecificeerde systeem met een factor tien te verhogen waarmee direct ook
het kinematisch model geëvalueerd was.
Van het bewegingssysteem in de vorm van een Stewart platform is afgeleid dat deze tot
de klasse van volledig parallelle systemen behoort waardoor de dynamica als een stelsel expliciete
differentiaalvergelijkingen beschreven kan worden vanuit de platformcoördinaten.
Zowel in theorie als in de praktijk kon aangetoond worden dat de modes van de stijve
lichamen volgend uit de interactie tussen hydraulica en mechanica de meest relevante systeemdynamica
vormen bij simulatortoepassingen. De relatie tussen platform- en actuatorcoördinaten
beschreven door een jacobiaan vormt hierbij de centrale operator. Volgend
uit deze analyse kan het systeem, met op het eerste gezicht sterke interactie, in elke stand
lokaal vanuit de juiste coördinaten gezien worden als zes hydraulische actuatoren die elk
vrijwel onafhankelijk een massa aandrijven.
Met de voorgestelde modelgebaseerde regeling kunnen een aantal zaken bereikt worden
die conventioneel niet mogelijk zijn. Ten eerste kunnen de krachten benodigd voor de
gewenste versnelling aangepast worden voor de standsafhankelijke richtingen waarin ontkoppeld
de sterk verschillende massa’s gevoeld worden. Vervolgens kunnen de toch nog
optredende positioneringsfouten ook langs deze richtingen zonder interactie teruggekoppeld
worden. Tenslotte wordt de interactie tussen de hydraulica en de mechanica geminimaliseerd
door te compenseren voor de oliestroom die nodig is om de gewenste snelheid
aan te houden. Implementatie van deze structuur heeft aangetoond dat hiermee een hogere

Samenvatting en CV 249
bandbreedte, minder opslingering en een meer over de vrijheidsgraden geëgaliseerde respons
kan worden bereikt dan met de minder gestructureerde conventionele regeling. Verder
is aangegeven hoe nog meer bereikt kan worden als ook voorspellende informatie vanuit het
voertuigsimulatiemodel aan de regeling wordt aangeleverd.
De te behalen prestaties van de voorgestelde regelaarstructuur worden eerst begrensd
door het optreden van parasitaire effecten ten gevolge van mechanische vervormingen, flexibiliteiten.
Ook met de combinatie van de altijd iets vertraagde olietoevoer en zingende
leidingen moet op nog wat hogere frequenties rekening gehouden worden. Door in het
ontwerp van een simulator een laag gewicht en hoge stijfheid na te streven is echter aangetoond
dat met de voorgestelde regelaarstuctuur over een breed spectrum kritische simulatieexperimenten
zeer realistisch uitgevoerd kunnen worden.

250 Samenvatting en CV
Curriculum Vitae
Naam Sjirk Holger Koekebakker
25 oktober 1967 Geboren te Vierhouten (Gemeente Ermelo).
1980 - 1986 VWO aan het Agnes College te Leiden.
1986 - 1993 Werktuigbouwkunde aan de Technische Universiteit Delft.
Afgestudeerd vanuit de vakgroep Systeem- en Regeltechniek.
Titel afstudeerwerk: ”Decoupling by means of H 1�F ;
application to a wafer stepper”. Afstudeerwerk uitgevoerd
op het Philips Natuurkundig Laboratorium te Eindhoven.
1993 - 1994 Onderzoeker bij het TNO Productcentrum i.h.k.v.
de vervangende dienst.
1994 - 1998 Promovendus bij de vakgroep Systeem- en Regeltechniek
van de faculteit Werktuigbouwkunde
aan de Technische Universiteit Delft.
1999 - Onderzoeker bij de Group Research Technology
te R & D Venlo van Océ Technologies B.V.