HIN - Hovedoppgave - Høgskolen i Narvik - hovedside

HIN - Hovedoppgave
Master i teknologi
Lodve Langesgt. 2, Postboks 385
8505 NARVIK
Telefon: 76 96 60 00
Telefax: 76 96 68 10
Tittel
Styring av orientering med bruk av
reaksjonsthrustere i en mikrosatellitt
Attitude control of a micro satellite with the use of reaction
control thrusters
Forfattere
Jøran Antonsen
Institutt for
Data-, elektro- og romteknologi
Veileder
Raymond Kristiansen
Oppdragsgiver
HiN
Studieretning
Romteknologi
Dato
14. juli 2004
Gradering
Åpen
Antall sider
89
Vedlegg
3 stk
Oppdragsgivers kontaktperson
Raymond Kristiansen
Sammendrag
Denne hovedoppgaven er et studie av styresystemet til den tre-akse stabiliserte mikro
satellitten European Student Earth Orbiter (ESEO). Styring av orientering skjer ved bruk
av fire reaksjonsthrustere og et reaksjonshjul. Det er utviklet en matematisk modell av
ESEO som det er utført stabilitetsanalyse på. To lineære og to ulineære kontrollere er
foreslått til styring av orientering.
Abstract
This thesis is a study of the attitude control system for the three-axis stabilized micro
satellite European Student Earth Orbiter (ESEO). The attitude is controlled by four
attitude control thrusters and one reaction wheel. A mathematical model of ESEO is
developed and stability analysis is performed. Two linear and two nonlinear controllers is
suggested for attitude control.
Norske stikkord
Styring av orientering
Stabilitetsanalyse
Thruster
Ulineær kontroll
ESEO
Keywords
Attitude control
Stability analysis
Thruster
Nonlinear control
ESEO

Hovedoppgave vår 2004
for
Stud. Techn. Jøran Antonsen
Styring av orientering med bruk av reaksjonsthrustere i en mikrosatellitt
Den europeiske romfartsorganisasjonen ESA har satt i gang et samarbeidsprosjekt med
universiteter over hele Europa. Prosjektet går under navnet Student Space Exploration &
Technology Initiative (SSETI), og har som hensikt å opprette et nettverk av studenter og
utdanningsinstitusjoner for distribuert design, konstruksjon og oppskyting av mikrosatellitter
og andre romfartøy.
Den første mikrosatellitten som utvikles i programmet er European Student Earth Orbiter
(ESEO). Denne delen av prosjektet befinner seg på nåværende tidspunkt i fase B av utvikling.
ESEO skal plasseres direkte i en geostasjonær bane av en Ariane 5 bærerakett i 2004.
Oppgaven går i hovedsak ut på teoretiske analyser av stabilitet av ESEO, samt utvikling og
simulering av robuste regulatorer for styring i tre akser. Konklusjoner skal underbygges med
teoretisk analyse og simuleringer av dynamikk utført i MATLAB.
Tekniske spesifikasjoner og krav for ESEO finnes i rapporten ESEO Phase-A Study Report
utgitt av ESA.
Deloppgaver:
1. Gjør et litteraturstudium innen styring av orientering av mikrosatellitter og forstyrrelser.
2. Utled en matematisk modell av ESEO med pådragsorganer, ut fra en spesifisert
konfigurasjon av reaksjonsthrustere.
3. Studer aktuatordynamikk for reaksjonsthrustere, og hvordan denne kan innvirke på styring
av orientering.
4. Foreslå regulatoralgoritmer for styring av satellitten i tre akser, og benytt simuleringer for
å vise at regulatoralgoritmene overholder krav til nøyaktighet.
5. Gjør vurderinger av størrelsesorden på forstyrrelser, og vurder de foreslåtte regulatorers
evne til å undertrykke disse.
6. Gi en algoritme for dumping av overflødig spinn i reaksjonshjul, og vis med simuleringer
at denne fungerer tilfredstillende.

Som bakgrunn for oppgaven anbefales følgende litteratur (utgangspunkt):
R. Kristiansen (2000): Attitude control of a micro satellite. Diplomoppgave, Institutt for
Teknisk kybernetikk, NTNU
M. J. Sidi (1997): Spacecraft Dynamics & Control. Cambridge University Press
J. R. Wertz (1978): Spacecraft Attitude Determination and Control. D. Reidel Publishing
Company, Dordrecht, Holland.
Utleveringsdato: 28.01.2004
Innleveringsdato: 14.07.2004
Hovedveileder: Høgskolelektor Raymond Kristiansen, M.Sc
Medveileder: Førsteamanuensis, Dr. Ing. Per J. Nicklasson
Høgskolen i Narvik
Institutt for data, elektro og romteknologi
Veileder
(sign)

Preface
The thesis is the concluding work of the Master of Science education provided by Narvik University College. It
has been carried out at the Department of Computer Science, Electrical Engineering, and Space Technology. I
would like to thank my advisor M.Sc Raymond Kristiansen and co-advisor Associate Professor Dr.ing. Per J.
Nicklasson for their support, valuable advice, and interesting discussions during this work.
Most of all I would like to thank my girl Gry for her love and support during the last five years. I would
also like to thank my fellow students M.Sc Frank Robert Blindheim, M.Sc Aleksander L. Marthinussen and
M.Sc Morten Topland for the many discussions during these last months.
Narvik July 14 th 2004
Jøran Antonsen
iv

Abstract
This thesis is a study of the attitude control system for the three-axis stabilized micro satellite European
Student Earth Orbiter (ESEO). The satellite is regarded as a rigid body. The thesis introduces the satellite
body dynamics relative to the orbit frame, where the angular velocity is represented relative to the orbit frame
in stead of the inertial frame. This provides a model dependant of the same representation of angular velocity
for both kinematics and dynamics.
The attitude is controlled by four attitude control thrusters and one reaction wheel. A mathematical model of
ESEO is developed and stability analysis is performed on both the linearized and the nonlinear model. The
stability analysis of the linearized unactuated model concludes with an unstable satellite in the point where
the body frame coincides with the orbit frame. Four equilibrium points is found with stability analysis on the
nonlinear model, but none of them are where the body frame coincides with the orbit frame. Simulations of
the unactuated satellite show that is unstable in the equilibrium point where the body frame coincides with the
orbit frame.
Four controllers is deduced for attitude control in this thesis, a PD controller, a Linear Quadratic Gaussian
controller, a nonlinear controller based on feedback linearization, and a nonlinear controller based on Lyapunov.
The controllers are compared against each other with respect to step response, power consumption and how
robust they appear against disturbances. The controller based on feedback linearization is shown to have the
fastest settling time whereas the PD controller has the second fastest settling time. The PD controller and the
controller based on Lyapunov is shown to be most robust against disturbances and where the controller based
on Lyapunov has the lowest power consumption. The LQG controller seems to be unstable with disturbances
at several occasions.
A Bang Bang controller with deadzone is used for control of the thrusters dynamic, but is not used when
the attitude controllers is compared. A momentum dumper is deduced to prevent wheel saturation and also a
suggestion for how to prevent wheel disturbance for the PD controller is shown. A reference trajectory is shown
to be useful to keep the angular velocities low and to reduce the power consumption.
v

CONTENTS 1
Preface iv
Abstract v
Contents
1 Introduction 4
1.1 StudentSpaceExploration&TechnologyInitiative,SSETI ..................... 4
1.1.1 EuropeanStudentEarthOrbiter(ESEO) ........................... 4
1.1.2 SSETIExpress .......................................... 5
1.1.3 EuropeanStudentMoonOrbiter(ESMO)........................... 5
1.1.4 EuropeanStudentMoonRover(ESMR)............................ 5
1.2 Other satellite missions ......................................... 5
1.2.1 NCUBE.............................................. 5
1.2.2 NSAT-1.............................................. 5
1.2.3 Ørsted............................................... 6
1.2.4 Rømer............................................... 6
1.2.5 Proba1&2............................................ 6
1.3 Outlineofthethesis ........................................... 7
2 Mathematical background and notations 8
2.1 The satellite ................................................ 8
2.2 GeostationaryTransferOrbit(GTO).................................. 8
2.3 Referenceframes ............................................. 11
2.3.1 EarthCentredInertialframe(ECI)............................... 11
2.3.2 Orbitframe(O) ......................................... 11
2.3.3 Bodyframe(B).......................................... 12
2.4 Notations ................................................. 12
2.4.1 Vectors .............................................. 12
2.4.2 Matrices.............................................. 12
2.4.3 Pseudoinversematrices ..................................... 14
2.5 RotationmatrixrepresentedinEulerangles.............................. 15
2.6 RotationmatrixrepresentedinEulerparameters ........................... 16
2.7 Stability definitions............................................ 17
2.8 Liederivative ............................................... 19
3 Mathematical modelling 20
3.1 Dynamical satellite model . . . ..................................... 20
3.2 Gravitationaltorque ........................................... 20
3.3 Dynamicalmodelofthereactionwheel................................. 23
3.4 Thrustertorque.............................................. 23
3.5 Actuators ................................................. 26
3.5.1 Fourthrustersandnoreactionwheel.............................. 27
3.6 Disturbancetorque............................................ 28
3.6.1 Gravitygradient ......................................... 28

CONTENTS 2
3.6.2 Atmosphericdrag ........................................ 28
3.6.3 SolarRadiationandSolarWind ................................ 29
3.6.4 Internalnoise........................................... 29
3.6.5 Thermal flexibility . . . ..................................... 29
3.6.6 Sloshing.............................................. 29
3.7 SolarPanels................................................ 30
3.8 Dynamical model summary and simplifications ............................ 30
3.9 Implementation.............................................. 31
3.9.1 Eulerintegration......................................... 32
3.9.2 ExplicitRungeKutta ...................................... 32
4 Theoretical analysis 34
4.1 Linearizationofthedynamicmodel................................... 34
4.1.1 Linearizationofthekinematicequations............................ 34
4.1.2 Linearizationoftherotationmatrix .............................. 34
4.1.3 Linearizationoftheangularvelocity .............................. 34
4.1.4 Linearizationofthegravitationaltorque............................ 35
4.1.5 Linearizationofthereactionwheeltorque........................... 35
4.1.6 Linearizationofthethrustertorque .............................. 35
4.2 Completelinearizedmodel........................................ 36
4.3 Stability of the linearized satellite model . . . ............................. 38
4.4 Stability of the satellite - the Lyapunov approach ........................... 39
4.5 Referencetrajectory ........................................... 42
5 Controllers 44
5.1 Thrusterallocation............................................ 44
5.2 Pulsemodulation............................................. 44
5.2.1 BangBangController ...................................... 44
5.2.2 Pulse-WidthPulse-FrequencyModulator(PWPFM)..................... 44
5.2.3 Pulse-WidthModulator(PWM) ................................ 45
5.3 LinearControl .............................................. 45
5.3.1 PDcontroller........................................... 45
5.3.2 PDcontrollerwithwheelcompensation ............................ 47
5.3.3 LinearQuadraticGaussian(LQG)controller ......................... 47
5.4 NonlinearControl ............................................ 48
5.4.1 NonlinearcontrolbyLyapunov................................. 49
5.4.2 NonlinearcontrolbyFeedbackLinearization.......................... 50
5.5 Momentumdumpingcontroller ..................................... 53
6 Simulation and results 55
6.1 Unactuated satellite ........................................... 55
6.2 Stepresponse............................................... 56
6.3 Disturbance................................................ 60
6.3.1 Disturbanceonthemeasurements ............................... 60
6.3.2 Disturbanceontheactuators .................................. 64
6.3.3 Disturbanceonthemeasurementsandtheactuators ..................... 68

CONTENTS 3
6.4 Summaryofresults............................................ 72
6.5 BangBangcontroller........................................... 75
6.6 Momentumdumpingcontroller ..................................... 76
6.7 Wheeldisturbancecompensation .................................... 76
6.8 Powerconsumption............................................ 77
6.9 Referencetrajectory ........................................... 81
7 Discussion 82
7.1 Theoreticalanalysis ........................................... 82
7.2 Simulations ................................................ 82
7.2.1 Unactuated satellite . . ..................................... 82
7.2.2 Stepresponse........................................... 83
7.2.3 Disturbance............................................ 83
7.2.4 BangBangcontroller ...................................... 84
7.2.5 Momentumdumpingcontroller................................. 84
7.2.6 Wheeldisturbancecompensation................................ 84
7.2.7 Powerconsumption........................................ 84
7.2.8 Referencetrajectory ....................................... 85
8 Concluding Remarks and Recommendations 86
8.1 Conclusion ................................................ 86
8.2 Recommendationsforfuturework ................................... 87
9 Bibliography 88
A Notations 90
A.1 Vectricesanddyadics........................................... 90
B Simulations 91
C Matlab sourcecode 98

1 INTRODUCTION 4
1 Introduction
This thesis is a study of the attitude control system for the micro satellite European Student Earth Orbiter
(ESEO). The satellite is three-axis stabilized by the use of four attitude control thrusters and one reaction
wheel. ESEO is currently being deigned and constructed by European students, members of the Student Space
Exploration & Technology Initiative (SSETI) and is planned to be launched late 2005. The satellites main
objectiveistotakepicturesoftheearthandthemoon,and requires an attitude control system to be able to
rotate to desired positions.
1.1 Student Space Exploration & Technology Initiative, SSETI
Student Space Exploration & Technology Initiative (SSETI) was founded by the support of the European Space
Agency (ESA) in the year 2000. The main objective of the SSETI is stated as (ESEO Phase A Study Report,
2001)
To create a network of students, educational institutions and organizations (on the Internet) to perform the
distributed design, construction and launch of (micro)-satellites and other spacecraft (s/c).
The objectives are reached when a spacecraft is design, build and launched by a significant number of European
students in a highly distributed way. The completion of this project objective is independent of a mission
success or failure. More about SSETI ongoing and future missions can be found at the SSETI homepage,
http://www.sseti.org/.
1.1.1 European Student Earth Orbiter (ESEO)
• The SSETI micro satellite (ESEO), is a micro satellite designed, built and operated as the first SSETI
mission by the European students working in the network of participating universities. The ESEO is the
first SSETI project which started late 2000 and defined as mission one. The main mission objectives for
the ESEO is as stated (SSETI Phase A Study Report, 2001)
• Take picture of impact on public order to increase the interest of European students in space missions.
• Take one picture of the moon to increase the enthusiasm of students to reach the Moon on future missions.
• Test and qualify a star-tracker developed from a commercial device.
• Deploy integrated radiation dosimeters within OBDH nodes and central "PC box" to monitor radiation
dosage during the mission.
• Test and qualify a propulsion system for orbit maneuvers and future moon missions, by putting the spacecraft
in a 12-hour orbit.
• No life critical failures should occur in the first 28 days of the mission. End of mission is when it is not
possible to communicate with the spacecraft anymore.
These objectives shall be reached by means of a spacecraft system and related instruments and payload. The
mission is defined as successful by maintaining the satellite in GTO for at least 28 days, testing the essential
subsystems and performing the simple science payload experiments. The satellites is set to launched from Korou
late 2005.

1 INTRODUCTION 5
1.1.2 SSETI Express
SSETI Express is a micro satellite where the work started in December 2003 and defined by SSETI as mission
zero. This is a satellite which uses some of the already designed subsystems developed for the ESEO. The
satellite is planned to be launched early 2005.
1.1.3 European Student Moon Orbiter (ESMO)
The European Student Moon Orbiter (ESMO) is the future mission two of SSETI. A feasibility study of the
ESMO started in 2004, but no final objectives have been defined yet. The main idea for ESMO is to use if
possible some of the knowledge and hardware from the ESEO mission.
1.1.4 European Student Moon Rover (ESMR)
The European Student Moon Rover (ESMR) is another future mission three of the SSETI which not yet has
started, but where the idea is to land the ESMR on the moon.
1.2 Other satellite missions
1.2.1 NCUBE
The Norwegian student satellite NCUBE is an experimental spacecraft which is developed and built by students
from four Norwegian universities. The project was initiated by the Norwegian Space Centre with support from
Andøya Rocket Range, Norway. The main mission of the satellite is to demonstrate ship traffic surveillance
from a satellite in LEO using the maritime Automatic Identification System (AIS). The AIS system is based
on VHF transponders located onboard ships. These transponders broadcast the position, speed, heading and
other relevant information from the ships at regular time intervals. The main objective of the satellite is to
receive, store and retransmit at least one AIS-message from a ship. Another objective of the satellite project is
to demonstrate reindeer herd monitoring from space by equipping a reindeer with an AIS transponder during
a limited experimental period. This part of the project is conducted by the Norwegian Agriculture University
(NLH). A third objective is to demonstrate efficient attitude control system using a combination of passive
gravity gradient stabilization and active magnetic torquers. It is possible to control the attitude of the satellite
such that the nadir surface points towards the earth within limits of ±10 ◦ ,whichissufficient for antenna
pointing. For more information, see e.g. Rise et al(2003) and the NCUBE hompage at http://128.39.102.180/.
1.2.2 NSAT-1
The development of the proposed three-axis stabilized Norwegian micro satellite NSAT-1 has been ongoing for
several years. The satellite is intended to fly in a low earth orbit at 600 km altitude with the purpose of locating
ships at sea. The satellite should be able to locate a ship within a few hundred meters, and high demands are
placed on the attitude control system. The satellite control system will consist of a composition of a gravity
boom, reaction wheels and magnetic coils.
Several people have performed studies on NSAT-1. More information about the NSAT-1 can be found in
(Kristiansen, 2000) and (Kyrkjebø, 2000). Kristiansen (2000) performed several simulations of the attitude
control system of the NSAT-1. Disturbances on the actuators where added to be 20%, and showed that the
proposed controllers where robust and handled the disturbances quite well.

1 INTRODUCTION 6
1.2.3 Ørsted
The Ørsted satellite is a 62kg Danish micro satellite, and was launched from California on January 8, 1999. The
main purpose of the Ørsted satellite has been to provide a precise global mapping of the Earth’s magnetic field.
The satellite evolves the Earth in an elliptic orbit of heights between 600 and 850 km. The attitude controls
system consists of three orthogonal magnetic coils and a gravity boom. Flight results and experience from the
Ørsted attitude control system can be found in (Bak, T. et al, 1999). For more information about the Ørsted
mission, see the Ørsted homepage on the internet at
http://dmiweb.dmi.dk/fsweb/projects/oersted/homepage.html.
1.2.4 Rømer
Rømer is the next Danish micro satellite mission scheduled to be launched in 2005 - 2006 from Plesetsk Cosmodrome
in Russia. It has the same dimensions as the ESEO but the weight is 85 kg. The satellite is scheduled
to be launched into a Molniya orbit in which one orbit takes approximately 12 hours, the same as for the ESEO
GTO orbit. The scientific experiments implemented are Measuring Oscillations in Nearby Stars (MONS) with
the primary science objective of observing 25 nearby, solar like stars by measuring tiny oscillations in intensity
and color by precision photometry.
For attitude control the Rømer satellite is equipped with a tetrahedron configuration of Wide Angle Telescopes
for Cosmic Hard x-rays (WATCH), that serves the dual purpose of x-ray detectors and momentum
wheels for short term attitude control (seconds to hours). Also a set of magnetic coils or rods for long term
control (days) and momentum dumping is used in addition to the WATCH for attitude control. A proposed
nonlinear attitude controller for the Rømer satellite based on feedback linearization can be found in (Jensen and
Wi´sniewski, 2001). Another passivity based nonlinear attitude controller suggested for the Rømer satellite can
be found in (Quottrup et al, 2000). More information and documents about the Rømer mission can be found
at the Danish Space Research Institute (DSRI) at http://www.dsri.dk/.
1.2.5 Proba 1 & 2
The Proba-1 satellite was the first micro satellite launched by ESA. It has a weight of 94kg and has the same
dimensions as ESEO, 60x60x80 cm. The name Proba comes from the Project for On-board Autonomy. Proba-1
was launched from Antrix (India) 22 October, 2001, initially for a one-year mission. The satellite was injected
directly into its final polar, sun-synchronous orbit at an altitude of 817km, 98.7 degrees inclination. There is
no on-board propulsion because of only 2 degrees orbital drift per year. The satellite is three-axis stabilized
by means of attitude provided by a star tracker and by on-board control through a set of reaction wheels and
magnetic coils.
ESA’s second micro satellite, Proba-2, is under development for launch early in 2006. It will have the same
dimensions as Proba-1. Proba 2 will offer increased miniaturization and integration of avionics, improved performance,
and with more resources allocated to payloads. The satellite uses its main instrument called SWAP
to watch the Sun and make detailed images of the solar atmosphere once a minute, by the light of energetic
ultraviolet rays. Proba-2 will give early warnings of eruptions on the Sun that provoke stormy weather throughout
the solar system.
The Proba missions are a part of an overall effort to promote technological missions using small spacecraft.

1 INTRODUCTION 7
For more information about the Proba missions, see the ESA website at
http://www.esa.int/SPECIALS/Proba_web_site/index.html.
1.3 Outline of the thesis
This M.Sc thesis is organized as follows:
Mathematical background information and notation used in this thesis is described in Chapter 2. This includes
parameters and definitions of the space environment, in addition to reference frames and conversion between
these and stability definitions.
A complete mathematical model of the satellite with all its components is deduced in Chapter 3. A description
of the actuator configuration and different disturbance torques is mentioned.
Chapter 4 consists of a theoretical analysis where the dynamical model is linearized and stability analysis
is performed both on the linear and nonlinear model. The principal of a reference model is also shown in this
chapter.
Chapter 5 is dedicated to control strategies for the satellite where different attitude controllers are deduced
including a momentum dumping controller and a Bang Bang controller.
Simulation of the theoretical control strategies is shown in Chapter 6, for the main purpose of illustrating
the satellite step response and the robustness of the controllers and to substantiate the result found in the
theoretical analysis. The power consumption of each controller is investigated. The motivation for using a
reference model is illustrated and the need for angular momentum dumping from the reaction wheel by using
thrusters.
A discussion of the theoretical analysis and the results from simulations is done in Chapter 7.
Finally Chapter 8 includes conclusions made from the work presented in this thesis, together with some
recommendations for future work.

2 MATHEMATICAL BACKGROUND AND NOTATIONS 8
2 Mathematical background and notations
2.1 The satellite
The Student Space Exploration & Technology Initiative, SSETI is building a micro satellite called the European
Student Earth Orbiter, ESEO with the dimension 60 × 60 × 80 cm3 , and a total mass of approximately 120
kg. For Attitude and Orbit Control System (AOCS), the ESEO uses one reaction wheel in y-axis controlling
the pitch movement, four thrusters for attitude control (from here on called Attitude Control System thruster
or ACS thruster), one main Orbit Control thruster (from here on called Orbit Control System thrusters or
OCS thrusters) for orbital manoeuvres, and additional four Reaction Control Thrusters (from here on called
Reaction Control System thruster or RCS thruster) used to correct orbital manoeuvres since the OCS thrust
vector might not go through the centre of mass. The RCS thrusters are also used as a redundancy for the ACS
thrusters. This thesis is about the attitude control system (ACS) which includes the ACS thrusters and the
reaction wheel. The required attitude accuracy for ESEO is 1◦ in x-axis and y-axis, and 5◦ in the z-axis (ESEO
phase B document). The satellite in this thesis is assumed to be a rigid body, with an inertia matrix
⎡
I = ⎣ Ix
0
0
Iy
0
0
⎤ ⎡
⎦ = ⎣
0 0 Iz
2.2 Geostationary Transfer Orbit (GTO)
4.3500 0 0
0 4.3370 0
0 0 3.6640
⎤
⎦ kgm 2
The ESEO is expected to be launched by Ariane 5 from Kourou into a Geostationary Transfer Orbit (GTO).
However, the final orbital parameters are not yet defined. Estimated orbital data for the ESEO is listed in
Table 2.1 (ESEO phase B document)
Table 2.1. Orbital parameters
Apogee 35843.369 km
Perigee 252.560 km
Semimajor axis 24421.104 km
Eccentricity 0.7285
Inclination 7 ◦
A graphical description of the GTO is shown in Figure 2.1where perigee is the point in the orbit where the
apogee
GTO
equator
Figure 2.1: Geostationary Transfer Orbit.
perigee
distance to earth centre is minimum and apogee is the point in the orbit with the maximum distance. In any
(2.1)

2 MATHEMATICAL BACKGROUND AND NOTATIONS 9
given orbit the intersection of the orbit plane and the reference plane through the centre of mass of the Earth is
the line of nodes. The ascending node of the satellite is the point where it crosses the equatorial plane travelling
from south to north. Similar, the descending node is where it crosses the equatorial plane travelling from north
to south. The nodes are shown in Figure 2.2. The eccentricity e defines the shape of the orbit ellipse as the
ratio
r
a2 − b2 e =
(2.2)
a
where a and b is the semimajor and semiminor axis, which is half the long and short axis in an ellipse, respectively.
For circular orbits, the eccentricity is e =0because the semimajor and the semiminor axis is of the same length.
Similar, the apogee and perigee of the orbit will be the same in a circular orbit.
Earth orbit
direction
North
South
Descending
node
Ascending
node
Satellite orbit
direction
Equatorial
plane
Figure 2.2: Ascending and descending nodes in an orbit.
The nominal duration of this mission is 28 days. During this period the ESEO will spend 10 days in GTO
phase, and then ESEO will perform manoeuvres in order to reach a suitable orbit for suffering a natural de-orbit.
The angular velocity in the orbit, is given as (Sidi, 1997)
r
µ
ωo =
(2.3)
R3 C
where µ = G·mearth =3.986·1014Nm2 /kg and G =6.669·10−11m3 /kg2 is the universal constant of gravitation,
mearth =5.97·1024 ¯
kg is the total Earth mass and R = ¯ ¯
R¯
is the distance from the centre of Earth to the centre
of the satellite. Since the ESEO is in a GTO, the distance from Earth centre to the satellite centre will not be
constant. The distance RC is given as (Sidi, 1997)
RC = a(1 − e2 )
(2.4)
1+e cos θ
where a is the semimajor axis, e is the eccentricity, and θ is the true anomaly defined as the angle between the
major axis pointing to the perigee and the radius vector from the centre of earth to the satellite. True and
eccentric anomaly θ and ψ is shown in Figure 2.3. It can be shown that (Sidi , 1997)
cos ψ =
e +cosθ
1+cosθ
(2.5)

2 MATHEMATICAL BACKGROUND AND NOTATIONS 10
and by substituting cos θ from (2.5) into (2.4),
and by using Kepler’s time equation (Sidi, 1997)
apogee
orbit
RC = a (1 − e cos ψ) (2.6)
b
a
O
ψ
r
θ
F
perigee
Figure 2.3: True and eccentric anomalies.
2π
tM
T =(t− tp) n = M = ψ − e sin ψ (2.7)
where tM = t − tp is the time elapsed from the last passage at perigee, n = 2π
T is the mean motion, and M is the
mean anomaly. Equation (2.7) is not solvable in closed form with respect to ψ, itmustbesolvednumerically.
A solution of (2.7) in the form of trigonometric series was developed by Lagrange as
ψ = M +2
∞X
n=1
1
n Jn (ne)sin(nM) (2.8)
where Jn is a Bessel function of the first kind of order n 1 . A simple numeric procedure based on successive
approximations may be used to find ψ in an elliptic orbit (e

2 MATHEMATICAL BACKGROUND AND NOTATIONS 11
Rc [m]
wo [rad/s]
x 107
4.5
4
3.5
3
2.5
2
1.5
1
Distance from center of earth to center of satellite
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10 4
0.5
x 10-3
1.2
1
0.8
0.6
0.4
0.2
Satellite angular velocity relative to earth
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10 4
0
Figure 2.4: Satellite angular velocity and distance from the center of earth to the satellite.
2.3 Reference frames
Without reference points and frames it is not possible to model positions or motions of a spacecraft. Only the
frames that are of importance for the modelling of a satellite like the ESEO is presented. For earth orbiting
satellites the rotation of the earth can often be neglected. Under these circumstances the earth itself becomes
the inertial reference.
2.3.1 Earth Centred Inertial frame (ECI)
For satellites it is common to define an inertial coordinate system with the centre of mass of the earth as its
origin (Earth Centred Inertial, ECI frame) and whose direction in space is fixed relative to the solar system 2 .
The Zi axis is directed from the centre of earth to the celestial north pole. The Xi axis is directed to the vernal
equinox, the point where the ecliptic, or the plane of the Earth’s orbit round the Sun, crosses the Equator from
south to north. Last the Yi axis together with the Xi axis make the equatorial plane of the Earth, which is
perpendicular to the earth’s axis of rotation. Figure 2.5 shows a graphical description of the ECI frame.
2.3.2 Orbit frame (O)
For earth orbiting satellites, it is most convenient to define a reference frame called the orbit frame. The
orbit frame has its base point at the centre of mass and has the vectors XO, YO, andZO. The XO-axis is
directed forward in the travelling direction of the satellite, ZB is directed to the centre of earth, and YO- axis
is perpendicular to the orbit plane and completes a right-hand orthogonal system.
2 This is actually not a inertial frame because the ECI system moves slowly relative to the stars. Today, it is common to classify
the stars relative to the ECI for the year 2000 (Sidi, 1997).

2 MATHEMATICAL BACKGROUND AND NOTATIONS 12
2.3.3 Body frame (B)
Figure 2.5: Earth Centered Inertial frame.
A rigid body can be represented by the so-called body frame. This frame is important for modelling location
and orientation of a satellite under study. Its base point also coincides with the centre of mass, and the base
vectors XB, YB, andZB are aligned with the principal axes of the moment of inertia tensor to simplify the
kinetic equations. Rotation about the axis XB, YB, andZB is defined as roll, pitch, andyaw respectively. The
body frame together with the orbit frame is shown in Figure 2.6.
2.4 Notations
2.4.1 Vectors
A vector independent from any frame will be denoted v, and coordinate vectors represented in a frame will be
denoted with a bold font and a superscript to indicate the frame. Then vB is a vector represented in the body
frame. Angular velocity vectors are denoted similar to other vectors, but with additional subscript to state
which frame that has angular velocity and relative to what frame. The vector ωB BI denotes the angular velocity
of the body frame relative to the inertial frame expressed in body frame.
2.4.2 Matrices
Matrices will be denoted similar to vectors with bold typing and a superscript to indicate the frame that matrices
are represented in for dependent matrices and with an arrow above the matrix for independent ones. The unity
matrix will be denoted as 1. Normally it is denoted as I, butI is in this case the notation for the moment of
inertia for the satellite. The rotation matrix, which transforms a coordinate vector from a system A to system

2 MATHEMATICAL BACKGROUND AND NOTATIONS 13
B, is denoted R A B
Orbit
YO
YB
Satellite
ZO
To Earth
Center
ZB
Figure 2.6: Body and orbit frame.
v A = R A Bv B
This can be used to describe the orientation of a satellite relative to the orbit frame, RB O
R B O =
⎡
⎤
⎦ (2.12)
⎣ c11 c12 c13
c21 c22 c23
c31 c32 c33
where the elements cij is called directional cosines, and as vectors they are named c1, c2, andc3
⎡ ⎤
⎡ ⎤
⎡ ⎤
c1 = ⎦ c2 = ⎦ c3 = ⎦ (2.13)
⎣ c11
c21
c31
⎣ c12
c22
c32
XB
⎣ c13
c23
The rotation matrix RB O is orthogonal, which makes the vectors orthonormal, and
c1 × c2 = c3 c2 × c3 = c1 c1 × c3 = c2 (2.14)
The rotation matrix RB A has the property
R A ¡ ¢T B
B RA = 1 ⇐⇒
A
RB = ¡ R B¢−1 ¡ ¢T B
A = RA this means that its inverse equals its transpose. This can be derived as (Egeland and Gravdahl, 2003)
˙R A B
c33
XO
(2.15)
¡ ¢ ³
BT A
RA + R ˙R B
B ´ T
A =0 (2.16)

2 MATHEMATICAL BACKGROUND AND NOTATIONS 14
By defining a matrix S as
S , ˙R A ¡ ¢ B
B RA and inserting this into (2.16), the derived expression turns into
S + S T =0 ⇐⇒ S = −S T
(2.17)
(2.18)
which implies that the matrix S is skew-symmetrical, and therefore it must exist a vector given as ωA AB =
£ ¤ T
ω1 ω2 ω3 in such a way that
S ¡ ω A ¢
AB =
⎡
⎣ 0 −ω3 ω2
ω3 0 −ω1
−ω2 ω1 0
⎤
⎦ (2.19)
where the coordinate vector ωA AB represents the angular velocity of frame A relative to frame B expressed in
frame A. The matrix S ¡ ωA ¢
AB expresses the cross product of two vectors as
S ¡ ω A ¢ A
AB RB = ω A AB × R A B
(2.20)
This relation implies that
S ¡ ω A ¢ A
AB RB = −R A BS ¡ ω A ¢
AB
The derived rotation matrix ˙R A B can now be solved using (2.17) as
˙R A B = S ¡ ω A ¢ A
AB RB = R A BS ¡ ω B ¢
AB
2.4.3 Pseudo inverse matrices
(2.21)
(2.22)
Pseudo inverse is a method to find the inverse of a non square matrix. A system matrix has to have either
existence or uniqueness properties to be invertible, defined as (Strang, 1988)
Definition 2.1
Existence: The system Ax = b has at least one solution x for every b if and only if the columns span R m ;then
r = m. In this case there exists an n by m right-inverse A ` such that AA ` =1m, the identity matrix of order
m. This is possible only if m ≤ n.
Definition 2.2
Uniqueness: The system Ax = b has at most one solution x for every b if and only if the columns are linearly
independent; then r = n. In this case there exists an n by m left-inverse A a such that A a A =1n, theunity
matrix of order n. Thisispossibleonlyif m ≥ n.
There are two simple formulas of pseudo inverse, left and right inverses, if they exist
A a = ¡ A T A ¢−1 T
A and A ` ³
T
= A AA T ´ −1
(2.23)
Then A a A = 1 and AA ` = 1. It is however not always that A T A,andAA T are actually invertible. A T A
has an inverse only if the rank is n, andAA T has an inverse when the rank is m.

2 MATHEMATICAL BACKGROUND AND NOTATIONS 15
2.5 Rotation matrix represented in Euler angles
The Euler angle rotation is defined as successive angular rotation about three orthogonal frame axes. The
rotation from the orbit frame to the body frame can be expressed as a rotation first about the z axis then about
y, and x axis with the angles ψ, θ, andφrespectively. This rotation is shown in Figure 2.7, and can be expressed
as (Egeland, 1993)
(2.24)
R O B = Rz,ψRy,θRx,φ
where Rz,ψ, Ry,θ, andRx,φ is the unity rotation
⎡
Rz,ψ = ⎣
cψ −sψ 0
sψ cψ 0
0 0 1
⎤ ⎡
⎦ Ry,θ = ⎣
cθ 0 sθ
0 1 0
−sθ 0 cθ
⎤ ⎡
⎦ Rx,φ = ⎣
1 0 0
0 cφ −sφ
0 sφ cφ
⎤
⎦ (2.25)
where s(·)and c(·) is the same as sin(·) and cos(·) respectively just to simplify the notation. With the help of
the unity rotations, RO B can be written as
⎡
R O B = ⎣
cψcθ cψsθsφ − sψcφ cψcφsθ + sψsφ
sψcθ sψsθsφ + cψcφ sψsθcφ − cψsφ
−sθ cθsφ cθcφ
⎤
⎦ (2.26)
The rotation matrix RO B becomes the unity matrix 1 if the angles φ = θ = ψ =0, which implies that no rotation
has occurred. The rotation matrix RO B can be described as a rotation by an angle θ about a unit vector k where
RO B is given by (Egeland, 1993)
R O B =cosθ1 + S ¡ k B¢ sin θ + k B ¡ k B¢T (1 − cos θ) (2.27)
and inserting
S 2 ¡ k B¢ = k B ¡ k B¢T ¡ B
− k ¢T B B
k 1 = k ¡ k B¢T − 1 (2.28)
gives the equation
R O B = 1 + S ¡ k B¢ sin θ + S 2 ¡ k B¢ (1 − cos θ) (2.29)
This implies that the directional cosines in (2.13) enters the form
⎡
c1 = ⎣ cψcθ
⎤
cψsθsφ − sψcφ ⎦
⎡
c2 = ⎣
cψcφsθ + sψsφ
sψcθ
⎤
sψsθsφ + cψcφ ⎦
⎡
c3 = ⎣
sψsθcφ − cψsφ
−sθ
⎤
cθsφ ⎦ (2.30)
cθcφ
This parameter representation has its disadvantage in the singularity problem. Consider for instance that,
θ = ± π
2 , which gives the rotational matrix RO B
⎡
R O B = ⎣
0 cψsφ − sψcφ cψcφ + sψsφ
0 sψsθsφ + cψcφ sψsθcφ − cψsφ
−1 0 0
⎤
⎦ (2.31)
which is singular. This implies that there will be no inverse solution to this problem, and the rotation can not
be found.

2 MATHEMATICAL BACKGROUND AND NOTATIONS 16
YB
θ
ZB
Satellite
cm
ψ
Figure 2.7: Description of a rotation roll, pitch, and yaw with the angles φ, θ, andψ respectively.
2.6 Rotation matrix represented in Euler parameters
Euler parameters are a four parameter representation and have no singularity. The Euler parameters are defined
in terms of the angle-axis parameter θ and k, andaregivenbythescalarηand the vector ε defined by (Egeland
and Gravdahl, 2003)
q , £ ¤ T
η ε1 ε2 ε3
(2.32)
where
µ
θ
η =cos
2
and ε = £ ε1 ε2 ε3
where ε and k is represented in the body frame. Note that
η 2 + ε T ε =cos 2
µ θ
2
+sin 2
φ
¤ T = k sin
XB
µ
θ
2
(2.33)
µ
θ
=1 (2.34)
2
The Euler parameter product between vectors q1 = £ η1 εT 1
¤ T
and q2 = £ η2 εT (Egeland and Gravdahl, 2003)
2
∙
η1
q , q1q2 =
ε1
¸ ∙
η2
ε2
¸ ∙
η
=
1η2 − εT 1 ε2
η1ε2 + η2ε1 + S (ε1) ε2
¸
¤ T can be found as
(2.35)

2 MATHEMATICAL BACKGROUND AND NOTATIONS 17
The rotation matrix RO B can be written as (Egeland, 1993)
R O B = Rk,θ = Rη,ε , ¡ η 2 − ε T ε ¢ 1 +2εε T +2ηS (ε) (2.36)
= ¡ 2η 2 − 1 ¢ 1 +2εε T +2ηS (ε) (2.37)
= ¡ 1 − 2ε T ε ¢ 1 +2εε T +2ηS (ε) (2.38)
and by using (2.28) and (2.36) gives
R O B = 1 +2ηS (ε)+2S 2 (ε) (2.39)
whichcanbeexpressedincomponentsas
R O ⎡
B = ⎣ 1 − 2 ¡ ε2 2 + ε2 ¢
3
2(ε1ε2 + ηε3)
2(ε1ε2 − ηε3)
1−2 2(ε1ε3 + ηε2)
¡ ε2 1 + ε2 2(ε1ε3 − ηε2)
¢
3
2(ε2ε3 + ηε1)
2(ε2ε3 − ηε1)
1−2 ¡ ε2 1 + ε2 ⎤
⎦
¢
2
(2.40)
The inverse rotation RO B can now be found as
R B O = 1 − 2ηS (ε)+2S2 (ε) (2.41)
and from (2.15) the inverse rotation matrix is the same as the transpose rotation matrix, hence
R B O = ¡ R O ⎡
¢T
B = ⎣ 1 − 2 ¡ ε2 2 + ε2 ¢
3
2(ε1ε2 − ηε3)
2(ε1ε2 + ηε3)
1−2 2(ε1ε3−ηε2) ¡ ε2 1 + ε2 2(ε1ε3 + ηε2)
¢
3
2(ε2ε3−ηε1) 2(ε2ε3 + ηε1)
1−2 ¡ ε2 1 + ε2 ¢
2
⎤
⎦ (2.42)
The directional cosines from (2.30) can now be expressed in Euler parameters as
⎡
c1 = ⎣ 1 − 2 ¡ ε2 2 + ε2 ¢ ⎤
3
2(ε1ε2 − ηε3) ⎦
2(ε1ε3 + ηε2)
⎡
c2 = ⎣ 2(ε1ε2 + ηε3)
1 − 2 ¡ ε2 1 + ε2 ⎤
¢
⎦
3
2(ε2ε3 − ηε1)
⎡
c3 = ⎣ 2(ε1ε3 − ηε2)
2(ε2ε3 + ηε1)
1 − 2 ¡ ε2 1 + ε2 ⎤
⎦
¢
2
(2.43)
The kinematic equation for the Euler parameters can be expressed as (Egeland and Gravdahl, 2003)
˙η = − 1
2 εT ω B BO
2.7 Stability definitions
˙ε = 1
2 [η1 + S (ε)] ωB BO
(2.44)
(2.45)
For stability analysis of the satellite system, Lyapunov theory is a suitable choice. Definitions and theorems in
this section is inspired by (Slotine and Li, 1991). Lyapunovs method is often used to investigate the stability in
nonlinear systems. Lyapunovs method is used in this thesis where the idea is to define a continuous derivable,
positive definite function which reflects the energy in the system. This function is called the Lyapunov Candidate
Function (LCF). If the initial energy in the system dissipates, the system can be regarded as stable. In Lyapunov
analysis of non-autonomous nonlinear systems of the form
˙x = f (x, t) (2.46)

2 MATHEMATICAL BACKGROUND AND NOTATIONS 18
where f is a n × 1 nonlinear vector function, and x is the n × 1 state vector, some different concepts of functions
has to be defined. The first is the concept of equilibrium points, which are definedaspointswherethesystem
states can stay forever. This implies that the state derivative is equal to zero, ˙x = 0. Many of the stability
problems are naturally formulated with respect to equilibrium points.
Definition 2.3
A state x ∗ is an equilibrium state or equilibrium point of the system if once x is equal to x ∗ , it remain equal to
x ∗ for all future time.
Mathematically, this means that the constant vector x ∗ satisfies
0 = f (x ∗ ) (2.47)
Definition 2.4
AscalarfunctionV (x,t) is said to be decresent if V (0,t)=0, and there exists a time invariant positive definite
function V1 (x) such that
V (x,t) ≤ V1 (x) , Vt ≥ (0)
this means that a scalar function V (x, t) is decresent if it is dominated by a time invariant positive definite
function.
Definition 2.5
If, in a spherical region (ball) BR0 defined by kxk

2 MATHEMATICAL BACKGROUND AND NOTATIONS 19
Global uniform asymptotic stability: If the ball BR0 is replaced by the whole state space, and condition 1,
the strengthened condition 2, condition 3 and the condition
4. V(x, t) is radially unbounded
are all satisfied, then the equilibrium point at 0 is globally uniformly asymptotically stable.
2.8 Lie derivative
This section is inspired by (Khalil, 2002) and (Vidyasagar, 1993). Given a single input single output system as
x = f (x)+g (x) u (2.49)
y = h (x) (2.50)
where f, g, andh are sufficiently smooth in a domain D ⊂ R n . The derivative ˙y is given by
˙y = ∂h
def
[f (x + g (x) u)] = Lf h (x)+Lgh (x) u (2.51)
∂x
where
Lfh (x) = ∂h
f (x) (2.52)
∂x
is called the Lie derivative of h with respect to f or along f. Tis is the familiar notion of the derivative of h
along the trajectories of the system x = f (x). The new notation is convenient when repeating the calculation
ofthederivativewithrespecttothesamevectorfield or a new one, such as
LgLfh (x) =
∂ (Lfh)
g (x)
∂x
(2.53)
L 2 fh (x) = LfLf h (x) = ∂Lfh
f (x)
∂x
(2.54)
L k fh (x) = LfL k−1
f
f h (x) =∂Lk−1
h
f (x)
∂x
(2.55)
L 0 fh (x) = h (x) (2.56)
If Lgh (x) =0,then ˙y = Lfh (x), independent of u. When calculating the second derivative of y, denotedby
y (2) leaves
y (2) = ∂Lfh
∂x [f (x)+g (x) u] =L2 fh (x)+LgLfh (x) u (2.57)
Definition 2.6
The nonlinear system in (2.49) and (2.50) is said to have relative degree ρ, 1 ≤ ρ ≤ n, inaregionD0 ⊂ D if
for all xD0.
LgL i−1
f
ρ−1
h (x) =0, i =1, 2, ..., ρ − 1; LgL h (x) 6= 0 (2.58)
f

3 MATHEMATICAL MODELLING 20
3 Mathematical modelling
3.1 Dynamical satellite model
The angular motion of a satellite can be modelled as an ideal rigid body. However, most satellites have
flexible parts like solar panels and antennas. Internal effects like fuel sloshing and thermal deformations are
not accounted for using a rigid body model. The rigid body model is nevertheless a good approximation,
especially for small satellites. Eulers moment equation (Sidi, 1997), which is the dynamic equation for its
angular momentum change related to applied torque, is given as
T B = ˙ hI = ˙ h B BI + ω B BI × h B BI (3.1)
= I ˙ω B BI + ω B BI × Iω B BI (3.2)
where h is the angular momentum, I is the satellite moment of inertia, ωB BI is the angular velocity between
the body frame and the inertial frame represented in the body frame. TB is the total torque working on the
satellite,
T B ⎡
= ⎣ Tx
Ty
⎤
⎦ = T B g + T B a = T B g + T B w + T B T + T B d
(3.3)
Tz
where T B g is the gravitational torque working on the satellite body and T B a istheactuatortorquethatconsist
of the torque resulting from the reaction wheel T B w, the thruster torque T B T , and the disturbance torque TB d .
3.2 Gravitational torque
Any non symmetrical satellite of finite dimension in orbit around the earth is subject to a gravitational torque
because of the variation in the earth’s gravitational force over the satellite. In a uniform gravitational field,
there would be no gravitational torque and the centre of mass would become the centre of gravity.
Controlling roll, pitch and yaw of a satellite in such a way that pitch is maintained parallel to the orbit normal,
and yaw is nadir pointing, has been called stabilite. When a satellite is nadir pointed, it means that its x-axis
is pointing directly to the earth’s centre of mass. This means that the roll and yaw axes rotate once about
the orbit normal per orbit. A few assumptions are made for modelling of the gravitational torque. Only the
gravitational field from the earth is considered and the earth is considered to have a spherical symmetrical mass
distribution. The spacecraft is small compared to the distance from the spacecraft’s centre of mass of the earth’s
centre of mass. The spacecraft consists of a single rigid body. The force that a mass element is influenced by,
is given by Newton’s gravitational law (Larson & Wertz, 1992)
d f = µ R
dm (3.4)
R3 where µ = G·mearth =3.986·10 14 Nm 2 /kg and G =6.669·10 −11 m 3 /kg 2 is the universal constant of gravitation,
mearth =5.97 · 1024 ¯
kg is the total earth mass and R = ¯ ¯
R¯
is the distance from the center of Earth to the mass
element. The total gravitational torque is given as
T B Z
g = r × d Z
r ×
f = −µ g
R
dm (3.5)
R3 B
B

3 MATHEMATICAL MODELLING 21
Body
O
r
dm
Rc
R
Earth
Figure 3.1: Spacecraft with an mass element and mass center O in body, orbiting earth.
where r is the location of the mass element dm relative to the mass centre of the satellite (See Figure 3.1). The
terms R−1 and R−3 can be expressed as binomial series as
R −1 = R −1
Ã
C 1 − r · RC
R2 C
− 1 r
2
2
R2 !
+ ....
C
(3.6)
R −3 = R −3
⎛ ³
3 r ·
⎝1 C −
´ ⎞
RC
+ .... ⎠ (3.7)
Inserting (3.7) into (3.5) gives
T B g = µ g
Z
B
R 3 C
R 2 C
⎡ ³
r × RC
3 r ·
· ⎣1 −
´ ⎤
RC
⎦ dm (3.8)
T B µ
µg
g =
R3 Z µ
3µg
RC × rdm −
C
B R3 Z
· RC × rrdm
C
B
RC
(3.9)
T B g = − 3µ g
R3 Z
RC × rrdm
C B
R (3.10)
By definition of the centre of mass it is known that R
R
rdm =0. The expression rrdm is a part of the
B B
expression of the inertial torque of the body, represented by the inertial dyadic
Z
Ī ,
¡ ¢ 2
r 1 − rr dm (3.11)
where 1 represents the unity dyadic. By defining
σO = RC
RC
the gravitational torque for the satellite becomes
T B µ
µg
g =3 σO × IσO =3ω 2 oσO × ĪσO
R 3 C
B
R 2 C
(3.12)
(3.13)

3 MATHEMATICAL MODELLING 22
where
ω 2 µ
µg
o =
R3 C
(3.14)
By using some of the properties of vectrices and dyadics given in Appendix A, the gravitational torque can be
expressed in body frame coordinates as
where c3 is given by the rotation matrix
as
and the inertia matrix is generally denoted as
T B g = FB · Tg =3ω 2 CFBσO ×FBIF T BσO =3ω 2 oc3 × Ic3 (3.15)
R B O = Rz,ψ Ry,θ Rx,φ
⎡
c3 = ⎣
I =
⎡
− sin θ
cos θ sin φ
cos θ cos φ
⎣ I11 I12 I13
I21 I22 I23
I31 I32 I33
(3.16)
⎤
⎦ (3.17)
⎤
⎦ (3.18)
which gives the resulting gravitational torque
T B g =3ω 2 ⎡
⎣
o
(I33
¡
2 − I22) r23r33 + I23 r23 − r2 ¢
33 + I31r13r23 − I12r33r13
¡
2
(I11 − I33) r33r13 + I31 r33 − r2 ¢
13 + I12r23r33 − I23r13r23
¡
2
(I22 − I11) r13r23 + I21 r13 − r2 ¢
23 + I23r33r13 − I31r23r33
⎤
⎦ (3.19)
If the axes of the body-frame are chosen to coincide with the principal axes of the satellite, the inertial matrix
would be denoted as
⎡
I = ⎣ Ix
0
0
Iy
0
0
⎤
⎦ (3.20)
and the gravitational torque becomes
0 0 Iz
T B g =3ω 2 ⎡
⎣
o
(Ix − Iy) r23r33
(Ix − Iz) r33r13
(Iy − Ix) r13r23
⎤
⎦ =3ω 2 ⎡
⎣
o
(Iz − Iy)sinφcos φ cos2 θ
(Iz − Ix)cosφsin θ cos θ
(Ix − Iy)sinφsin θ cos θ
⎤
⎦ (3.21)
and expressed with the use of Euler parameters, described in previous chapter, with the directional cosines c3
given as
⎡
2(ε1ε3 − ηε2)
c3 = ⎣ 2(ε2ε3 + ηε1)
1 − 2 ¡ ε2 1 + ε2 ⎤
⎦
¢
2
gives the gravitational torque
T B g =3ω 2 ⎡
⎣
o
(Ix
⎤
− Iy) c23c33
(Ix − Iz) c33c13
(Iy − Ix) c13c23
⎦ =3ω 2 o
⎡
⎣ 2(Iz − Iy)(ε2ε3 + ηε1) £ 1 − 2 ¡ ε2 1 − ε2 ¢¤
2
2(Ix − Iz)(ε1ε3 − ηε2) £ 1 − 2 ¡ ε2 1 − ε2 ⎤
¢¤
⎦
2
(3.22)
4(Iy − Ix)(ε1ε3 − ηε2)(ε2ε3 − ηε1)

3 MATHEMATICAL MODELLING 23
3.3 Dynamical model of the reaction wheel
An angular torque on the satellite is generated by an accelerated rotor or gyro in the opposite direction as
Iwωw = −I ˙ω B BI
(3.23)
where Iw is the moment of inertia of the reaction wheel, ωw is the angular velocity of the reaction wheel, I is
the satellite moment of inertia, and ˙ω B BI is the derived angular velocity between the body and inertial frame
represented in the body frame. The ESEO has only one reaction wheel mounted on the y-body axis which
controls the pitch motion. The resulting angular momentum is then (Sidi, 1997)
hw = Iwωw = £ 0 hwy 0 ¤ T
and the reaction wheel torque expressed in the body frame is
T B w =
µ B
dhw
+ ω
dt
B BI × hw − T B friction
(3.24)
(3.25)
hw is the angular momentum of the wheels, and TB frictionis the friction characteristics on the bearings. The
friction characteristic can be expressed as (Kristiansen, 2000)
T B friction = Ncsgn(s)+fs (3.26)
where Nc is the Coulombs friction coefficient (Nc =7.06 · 10−4 −6 Nm
Nm), f =1.21 · 10 rpm is the viscous friction
coefficient, and s is the wheel speed in rounds per minute (rpm). The wheel friction is thus only dependent
of the speed of the wheel. If the reaction wheel is chosen to be large, it will limit the speed and hence, less
friction. The ESEO has an output torque of 0.4 Nm and a maximum wheel speed of 5000 rpm (ESEO phase B
document). The resulting friction torque becomes
T B friction = Ncsgn(s)+fs ≈ 6.76 · 10 −3
(3.27)
which only represents a maximum friction of a little over 1,5% of the output torque. Since the friction torque is
so small it will neglected in this thesis. The simplified equation for the reaction wheel around the body frame
therefore becomes
T B µ B
dhw
w = + ω
dt
B ⎡
BI × hw = ⎣ Twx
⎤ ⎡ ⎤
−hwyωz
Twy ⎦ = ⎣ ˙hwy ⎦ (3.28)
3.4 Thruster torque
Twz
hwyωx
The ESEO has one main thruster for orbit control (OCS thruster), and four reaction thrusters (RCS thrusters)
are used to correct orbital manoeuvres since the OCS thrust vector might not go through the centre of mass.
Additional four thrusters (ACS thrusters) are used for attitude control and momentum dumping of the reaction
wheel. The RCS thrusters are also a redundancy for the ACS thrusters. The thruster torque is not only
dependent on the thrusters force but also its orientation and distance from the centre of mass. The centre of
mass for the ESEO is not yet calculated, so in this thesis it is assumed to coincide with the geometric centre.
This thesis will only consider the ACS thrusters together with the reaction wheel as actuators for attitude

3 MATHEMATICAL MODELLING 24
control. The attitude is controlled by activating one set of the ACS thrusters, which gives a rotation in the
desired direction. Given a thrusters location
r = irx + jry + krz
(3.29)
where r is the vector from the centre of mass to the thrusters location, then the thrusters torque is given by
(Sidi, 1997)
T B ⎡
t = ⎣ Ttx
⎤
⎡
⎤
cos α cos β
Tty ⎦ = r × F = r × ⎣ sin α ⎦ F (3.30)
Ttz
cos α sin β
T B ⎡
t = ⎣ ry
⎤ ⎡
sin β cos α − rz sin α
rz cos α cos β − rx cos α sin β ⎦ F = ⎣
rx sin α − ry cos (α)cosβ
∆x
⎤
∆y ⎦ F (3.31)
∆z
where F is the thrust force in Newton, β is the elevation angle from the thrust vector to the XB axis in the
XBZB plane, and α is the azimuth angle from the thrust vector to XB axis in XBYB plane shown in Figure
Figure 3.2. The thrust vector is in the opposite direction of the nozzle which is illustrated in the figure. This
means that the angles β and α for thrusters Th1, Th2, Th3, andTh4respectively is β1,2,3,4 =0◦ because all
thrusters are directed along the XB axis, and α1,2 = ±135◦ and α3,4 = ±45◦ . With the ACS thrusters, there
YB
Th3
Th4
r
cm
ZB
YB
Th2
Th1
X
B
YB
ZB
α
ZB
β
Thrust
vector
XB
Thrust
vector
Figure 3.2: Thruster placement with the angles β, α, andthevectorr.
exists six different ways to rotate the spacecraft in positive and negative X, Y, and Z directions. This means
that two thrusters must be fired to give one rotation in roll, pitch or yaw. By using (3.31) on the configuration
XB

3 MATHEMATICAL MODELLING 25
given in Figure 3.2 gives the total ACS thrusters torque
T B t =
⎡
⎣ r1y
⎤ ⎡
sin β1 cos α1 − r1z sin α1
r1z cos α1 cos β1 − r1x cos α1 sin β ⎦
1 F1 + ⎣
r1x sin α1 − r1y cos α1 cos β1 r2y
⎤
sin β2 cos α2 − r2z sin α2
r2z cos α2 cos β2 − r2x cos α2 sin β ⎦
2 F2
r2x sin α2 − r2y cos α2 cos β2 ⎡
+ ⎣ r3y
⎤ ⎡
sin β3 cos α3 − r3z sin α3
r3z cos α3 cos β3 − r3x cos α3 sin β ⎦
3 F3 + ⎣
r3x sin α3 − r3y cos α3 cos β3 r4y
⎤
sin β4 cos α4 − r4z sin α4
r4z cos α4 cos β4 − r4x cos α4 sin β ⎦
4 F4(3.32)
r4x sin α4 − r4y cos α4 cos β4 Since β 1,2,3,4 =0 ◦ the torque becomes
T B t
=
⎡
⎣ −r1z sin α1
r1z cos α1
r1x sin α1 − r1y cos α1
⎡
+ ⎣ −r3z sin α3
r3z cos α3
r3x sin α3 − r3y cos α3
⎤ ⎡
⎦ F1 + ⎣ −r2z sin α2
r2z cos α2
r2x sin α2 − r2y cos α2
⎤ ⎡
⎦ F3 + ⎣ −r4z sin α4
r4z cos α4
Because of the angles α1,2 = ∓135◦ and α3,4 = ∓45◦ gives the relations
sin α1 =cosα1 = − 1
√
2 2
sin α2 = − cos α2 = 1
√
2 2
sin α3 = − cos α3 = − 1
√
√ 2 2
2
which leads to
sin α4 =cosα4 = 1
2
r4x sin α4 − r4y cos α4
T B t = 1
⎡
⎡
⎤
√ r1z −r2z r3z −r4z ⎢
2 ⎣ −r1z −r2z r3z r4z ⎦ ⎢
2
⎣
−r1x + r1y r2x + r2y −r3x − r3y r4x − r4y
⎤
⎦ F2
F1
F2
F3
F4
⎤
⎦ F4
⎤
⎥
⎦
(3.33)
(3.34)
(3.35)
With the assumption that the centre of mass coincides with the geometric centre makes it possible to make
some simplifications of the vector r
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
r = ⎦ = ⎦ = ⎦ = ⎦ = ⎦ (3.36)
and T B t takes the form
T B t
⎣ rx
ry
rz
⎡
1√
= 2 ⎣
2
⎣ r1x
r1y
−r1z
⎣ r2x
−r2y
−r2z
⎣ −r3x
r3y
−r3z
−rz rz −rz rz
rz rz −rz −rz
− (rx − ry) (rx − ry) (rx − ry) − (rx − ry)
⎣ −r4x
−r4y
From (3.37) it can be shown that by using a combination of two thrusters, gives the following rotations
−r4z
⎡
⎤
⎢
⎦ ⎢
⎣
F1
F2
F3
F4
⎤
⎥
⎦
(3.37)

3 MATHEMATICAL MODELLING 26
Xpositive Th2 and Th4
Xnegative Th1 and Th3
Ypositive Th1 and Th2
Ynegative Th3 and Th4
Zpositive
Th2 and Th3
Znegative Th1 and Th4
Equation (3.37) holds for both the ACS and the RCS thrusters. The only difference will be the vector rz and
the thrust force F. Considering that the centre of mass is not in the geometric centre, the thrusters torque will
change. However (3.35) will not change since the elevation and azimuth angles β and α respectively will be
thesame.Theonlychangewillbethevalueofthevectorr. It should be mentioned that a rotation about the
XB axis will result in lateral motion along the YB axis, and similar a rotation about the YB axis will result in
lateral motion along the XB axis.
3.5 Actuators
From (3.1) the dynamical model with gravitational and actuator torque is given as
I ˙ω B BI = −ω B ³
BI × Iω B ´
BI + T B g + T B a
(3.38)
The actuator torque applied to a fully actuated satellite with one reaction wheel and four thrusters, can be
obtained by adding the torques T B w from the reaction wheel in (3.28) and T B t from the thrusters in (3.37), as
T B a = T B t + T B w
= 1
⎡
√ −rz
2 ⎣ rz
2
− (rx − ry)
rz
rz
(rx−ry) −rz
−rz
(rx−ry) rz
−rz
− (rx − ry)
⎡
⎤
⎢
⎦ ⎢
⎣
F1
F2
F3
F4
⎤
⎥
⎦ +
⎡
⎣
−hwyωz
˙hwy
hwyωx
⎤
⎦ (3.39)
This actuator torque can be expressed in terms of two matrices an actuator matrix Ba and a disturbance matrix
Da as
T B a = Bau + Daω B BI
(3.40)
where u is the vector of actuators given as
u = £ F1 F2 F3 F4
˙hwy
¤T
(3.41)
The actuator matrix Ba contains elements from reaction wheel and thrusters torques, and the disturbance
matrix Da contains dynamic terms added from the velocity in the reaction wheel. For the configuration of
ESEO with the actuator torque given in (??), the matrices Ba and Da can be written as
Ba = 1
⎡
⎤
−rz rz −rz rz √ √
0
2 ⎣ rz rz −rz −rz 2 ⎦ (3.42)
2
− (rx − ry) (rx−ry) (rx−ry) − (rx − ry) 0

3 MATHEMATICAL MODELLING 27
and
⎡
Da = ⎣
0 0 −hwy
0 0 0
hwy 0 0
⎤
⎦ (3.43)
which together makes the total actuator matrix
T B a = Bau + Daω B =
BI (3.44)
1
⎡
−rz √
2 ⎣ rz
2
− (rx − ry)
rz
rz
(rx−ry) −rz
−rz
(rx−ry) rz
−rz
− (rx − ry)
⎡
⎤
√
0 ⎢
2 ⎦ ⎢
0 ⎣
F1
F2
F3
⎤
⎥
⎦
=
⎡
+ ⎣
⎡
⎣
1
2
0 0 −hwy
0 0 0
hwy 0 0
⎤ ⎡
⎦
⎣ ωx
ωy
ωz
F4
˙hwy
⎤
⎦ (3.45)
√
1
2 2rz (−F1 + F2 − F3 + F4) − hwyωz
√
1
2 2rz (F1 + F2 − F3 − F4)+ ˙ ⎤
hwy ⎦
√
(3.46)
2(rx − ry)(−F1 + F2 + F3 − F4)+hwyωx
By inserting (3.40) into (3.38), gives the dynamical model of the satellite
I ˙ω B BI = −ω B ³
BI × Iω B ´
BI + T B g + Bau + Daω B BI
Equation (3.47) can be written in component form as
˙ωx =
¡
σx ωyωz − 3ω 2 ˙ωy =
∙ ¸
¢ 1 1√
oc23c33 + 2rz (−F1 + F2 − F3 + F4) − hwyωz
Ix 2
¡
−σy ωxωz − 3ω 2 ∙
¢ 1 1√
oc13c33 + 2rz (F1 + F2 − F3 − F4)+
Iy 2
˙ ˙ωz =
¸
hwy
¡
−σz ωxωy − 3ω 2 ∙ ¸
¢ 1 1√
oc13c23 + 2(rx − ry)(−F1 + F2 + F3 − F4)+hwyωx
2
Iz
(3.47)
(3.48)
(3.49)
(3.50)
3.5.1 Four thrusters and no reaction wheel
Aconfiguration of four thrusters is used later on in the development of a momentum dumping controller, the
resulting actuator torque with four thrusters becomes
T B a = T B t = 1
⎡
⎤
√ rz (−F1 + F2 − F3 + F4)
2 ⎣ rz (F1 + F2 − F3 − F4) ⎦ (3.51)
2
(rx − ry)(−F1 + F2 + F3 − F4)
and since there is no reaction wheel TB w =0. This means that the disturbance matrix Da equals zero and the
actuator torque can be written as
T B a = Bau = 1
⎡ ⎤
⎡
⎤ F1
√ −rz rz −rz rz ⎢ F2 ⎥
2 ⎣ rz rz −rz −rz ⎦ ⎢ ⎥
2
⎣ F3 ⎦ (3.52)
− (rx − ry) (rx−ry) (rx−ry) − (rx − ry)
F4

3 MATHEMATICAL MODELLING 28
The dynamical model now becomes
I ˙ω B BI = −ω B ³
BI × Iω B ´
BI + T B g + Bau (3.53)
and in component form
˙ωx =
¡
σx ωyωz − 3ω 2 ˙ωy =
∙ ¸
¢ 1 1√
oc23c33 + 2rz (−F1 + F2 − F3 + F4)
Ix 2
¡
−σy ωxωz − 3ω 2 oc13c33 ˙ωz =
∙ ¸
¢ 1 1√
+ 2rz (F1 + F2 − F3 − F4)
Iy 2
¡
−σz ωxωy − 3ω 2 ∙ ¸
¢ 1 1√
oc13c23 + 2(rx − ry)(−F1 + F2 + F3 − F4)
2
3.6 Disturbance torque
Iz
(3.54)
(3.55)
(3.56)
The satellite will be exposed to several different internal and external disturbance torques. Atmospheric drag
is the most dominating external disturbance in altitudes lower than 500 km (Wertz, 1978). There exist other
external disturbances like solar radiation and wind, and variation in the gravitational field. Internal disturbance
torques might come from the electronics inside the satellite, which can result in magnetic torque from current
loops in the satellite, moving of internal parts such as the reaction wheel and thermal flexibility.
3.6.1 Gravity gradient
An asymmetric body subjected to a gravitational field will experience a torque tending to align the axis of least
inertia with the field direction. This is known as the gravity gradient or by some called tidal forces. The gravity
gradient is a good thing for a stable unactuated satellite but not so good for an unstable satellite because of
the satellites mass distribution. The gravitational torque will contribute to keep a stable satellite around the
equilibrium point RB BO = 1, which is not the case for an unstable satellite.
3.6.2 Atmospheric drag
The principal non gravitational force acting on satellites in low-Earth orbits is atmospheric drag. Drag acts
in the opposite direction of the velocity vector and removes energy from the orbit. This reduction of energy
causes a decrease in the orbit altitude, leading to further increases in drag. Eventually, the altitude of the orbit
becomes so small that the satellite re-enters the atmosphere. The perturbation force Fp due to atmospheric
drag is (Larson and Wertz, 1992)
Fp = 1
2 ρCD
A 2
V (3.57)
m
where ρ is the atmospheric density, CD is the drag coefficient, A is the satellite cross-section area, m is the
satellite mass, V is the satellite velocity with respect to the atmosphere. Computing the drag force on a satellite
is not as simple as (3.57) might lead on to believe. The density of the atmosphere at very high altitudes varies
unpredictably, and is strongly dependent on the level of solar activity. Surface’s, such as solar panels, may
actually cause the satellite to experience lift forces. Therefore, (3.57) should be viewed as a first approximation
and a starting point for more detailed modelling.
Because ESEO will be in altitudes lower then 500 km for a short period of time per orbit, and the lifetime of
ESEO is set to be only 28 days, atmospheric drag will be neglected in the further analysis.

3 MATHEMATICAL MODELLING 29
3.6.3 Solar Radiation and Solar Wind
Solar radiation consists of all the electromagnetic waves radiated by the sun with wavelengths from X-rays to
radio waves. The solar wind consists mainly of ionized nuclei and electrons. Both solar wind and radiation
may produce a physical pressure when acting on any surface of a body. The pressure is proportional to the
momentum flux (momentum per unit area per unit time) of the radiation. The solar radiation momentum flux
is greater than the solar wind by a factor of 100 - 1,000 (Sidi, 1997).
The mean solar energy flux of the solar radiation is proportional to the inverse square of the distance from
the sun. The mean integrated flux at the earth’s position is given by (Sidi, 1997)
Fe =
1, 358
1.0004 + 0.0334 cos(D) W/m2
(3.58)
where D is the ”phase” of the year, which is calculated as starting on July 4, the day of the earth aphelion 3 .
The mean momentum flux is given by
P = Fe
c =4.5 · 10−6kgm −1 s −2
(3.59)
where c is the speed of light. The solar radiation pressure |FR| is proportional to P, to the cross-sectional area
A of the satellite perpendicular to the sun line, and to a coefficient CP that is dependent on the absorption
characteristic of the spacecraft
|FR| = PACP
(3.60)
0

3 MATHEMATICAL MODELLING 30
All these environmental disturbance torques is small relative to the control torques of the satellite. The environmental
disturbance torques except the gravity gradient will be neglected in the further study for simplicity
reasons.
3.7 Solar Panels
The satellite is equipped with solar arrays to generate energy from the sun. The satellite has two symmetrical
solar panels. If the angular position of the two solar panels relative to the sun is not symmetrical, a torque will
be generated, which means that a torque about the line connecting the sun and the satellite will be created.
The solar panels are nominally directed toward the sun for maximum energy absorption. The angular motion
of the solar panels will then produce a torque. The solar panels and its resulting torques are only mentioned,
but not included in this thesis.
3.8 Dynamical model summary and simplifications
As stated in (3.1), the dynamic model of the satellite can be expressed as
T B = I ˙ω B BI + ω B BI × Iω B BI
where ωB BI can be expressed with the help of rotation matrices given in (2.15) as
ω O OI
ω B BI = ω B BO + ω B OI = ω B BO + R B Oω O OI
(3.61)
(3.62)
is the angular velocity of the orbit frame relative to the inertial frame expressed in orbit frame components,
and can be expressed as ωO OI = £ 0 −ωo 0 ¤ q
µg
, where ωo = . By applying this to (3.63) leaves
c23
ω B BI =
⎡
⎣ ωx
ωy
ωz
R 3 c
⎤
⎦ = ω B BO − ωoc2
(3.63)
where c2 is given from (2.30) and (2.43) as
⎡
c2 = ⎣ c21
c22
⎤ ⎡
⎦ = ⎣ sψcθ
⎤ ⎡
sψsθsφ + cψcφ ⎦ = ⎣
sψsθcφ − cψsφ
2(ε1ε2 + ηε3)
1 − 2 ¡ ε2 1 + ε2 ⎤
¢
⎦
3
2(ε2ε3 − ηε1)
(3.64)
The dynamical model can now be written in component form as
For ease of notation the ratio between the moments of inertia is defined as
Ix ˙ωx = (Iy−Iz) ωyωz + Tx (3.65)
Iy ˙ωy = (Iz−Ix) ωxωz + Ty (3.66)
Iz ˙ωz = (Ix−Iy) ωxωy + Tz (3.67)
σx, = Iy − Iz
Ix
σy, Ix − Iz
Iy
σz, Iy − Ix
Iz
(3.68)

3 MATHEMATICAL MODELLING 31
Then the dynamic model can be written as
˙ωx = σx (Iy − Iz) ωyωz + Tx
Ix
˙ωy = −σy (Iz − Ix) ωxωz + Ty
Iy
˙ωz = −σx (Ix − Iy) ωxωy + Tz
Iz
(3.69)
(3.70)
(3.71)
The actuator torque from (3.3) is given as TB = TB g + TB a , and by inserting this together with (3.22) into the
equations above leaves
¡
˙ωx = σx ωyωz − 3ω 2 ¢ Tax
oc23c33 + (3.72)
Ix
¡
˙ωy = −σy ωxωz − 3ω 2 ¢ Tay
oc13c33 + (3.73)
Iy
¡
˙ωz = −σx ωxωy − 3ω 2 ¢ Taz
oc13c23 + (3.74)
Iz
The total model of the satellite consists of the dynamic equations given by (3.61) and the kinematic differential
equations stated in (2.44) and (2.45),
I ˙ω B BI = −ω B BI × Iω B BI + T B
˙η = − 1
2 εT ω B BO
˙ε = 1
2 [η1 + S (ε)] ωB BO
and the angular velocity between the body frame and orbit frame as from (3.63)
3.9 Implementation
ω B BO = ω B BI + ωoc2
(3.75)
(3.76)
(3.77)
(3.78)
The satellite dynamics and kinematics in (3.75) and (3.76) is continuous in nature, and has to be implemented
to a digital computer. The continuous equation can be transformed into discrete system by using some kind of
integration method. In this thesis, two such methods are presented.

3 MATHEMATICAL MODELLING 32
3.9.1 Euler integration
∆ tλ1
∆ tλ2
-1
Im
Re
Figure 3.3: Euler stability region.
Euler integration is based on the simple formula (Iversen, 1997)
xk = xk−1 + ∆tfk−1 (x, t) (3.79)
where k is the discrete time index, ∆t is the integration step size and f (·) is the function integrated. The Euler
integration has a stability region inside the circle described in Figure 3.3. Equation (3.79) can be written as
xk =(1+∆tλ) xk−1
when evaluated on the scalar equation ˙x = λx. The stability criteria for the discrete system is then
(3.80)
DEUL = {∆tλC||1+∆t| < 1} (3.81)
where C is the complex numbers. The step size for a stable integration is determined by the eigenvalue λ. The
eigenvalue λ will move along the lines in Figure 3.3 for different time steps, and an undamped system with
purely imaginary eigenvalues integrated by Euler integration will not be stable even for small time steps.
3.9.2 Explicit Runge Kutta
The Explicit Runge Kutta (ERK) integration methods are nonlinear one-step integration methods which uses
a weighted mean of several computed values for f (x, t) between tk−1 and tk. The number of computed values
is identical to the order of the method, and the stability region of an ERK is dependent on the order. The
stability regions up to order four is shown in Figure 3.4.

3 MATHEMATICAL MODELLING 33
-1
ERK1
ERK2
ERK3
ERK4
Im
Figure 3.4: ERK stability region.
The stability region can be expressed for orders up to four as (Iversen, 1997)
( ¯ sX
¯ )
¯ (∆tλ)
¯
DERK = ∆tλC| ¯ ¯ < 1
¯ i! ¯
s =1, 2, 3, 4 (3.82)
i=0
wheresistheorderoftheERK,andthescalarequationof ˙x = λx is evaluated. As seen from Figure (3.4) the
ERK methods is stable for small eigenvalues on the imaginary axis.
Re

4 THEORETICAL ANALYSIS 34
4 Theoretical analysis
4.1 Linearization of the dynamic model
By linearizing the dynamical model and writing the linearized system as a state space model, linear controllers
can be developed.
4.1.1 Linearization of the kinematic equations
The kinematic equations described in (2.44) and (2.45)
˙η = 1
2 εT ω B BO
˙ε = 1
2 [η1 + S (ε)] ωB BO
When linearizing these around the point ε =0, η =1,andωB BO =0we get
4.1.2 Linearization of the rotation matrix
(4.1)
(4.2)
˙η = 0 (4.3)
˙ε = 1
2 ωB BO
The rotation matrix was presented previously in (2.39) as
and by linearizing around the point ε =0and η =1gives
and
(4.4)
R O B = 1 +2ηS (ε)+2S 2 (ε) (4.5)
S 2 (ε) =0 (4.6)
R O B = 1 +2S (ε) (4.7)
4.1.3 Linearization of the angular velocity
The angular velocity in the body frame relative to the inertial frame represented in the body frame is from
(3.62)
ω B BI = ω B BO + R B Oω O OI
(4.8)
By using the inverse rotation from previous Chapter 2.6 together with the linearized rotation matrix, the
linearized inverse rotation can be found as
R B O = ¡ R O¢T B = 1 − 2S (ε) (4.9)
which gives
R B ⎡
⎤
1 2ε3−2ε2 O = ⎣ −2ε3 1 2ε1⎦
(4.10)
2ε2 −2ε1 1

4 THEORETICAL ANALYSIS 35
This leads to the linearized directional cosines as the three column vectors of RB O = £ ¤
c1 c2 c3 . By
inserting ωB BO =2˙ε from (4.4) and (4.9) into (4.8) where ωO OI = £ 0 −ωo o ¤ T
gives the linearized angular
velocity
ω B BI = ω B BO + R B Oω O ⎡
OI = ⎣ 2˙ε1
⎤
− 2ωoε3
2˙ε2 + ωo ⎦ (4.11)
2˙ε3 +2ωoε1
and this can be derived as
˙ω B ⎡
BI = ⎣ 2¨ε1 − 2ωo ˙ε3
2¨ε2
2¨ε3 +2ωo˙ε1 ⎤
⎦ (4.12)
4.1.4 Linearization of the gravitational torque
The gravitational torque is given from (3.22) as
T B g =3ω 2 ⎡
⎣
o
(Ix
⎤
− Iy) c23c33
(Ix − Iz) c33c13
(Iy − Ix) c13c23
⎦ =3ω 2 o
⎡
⎣ 2(Iz − Iy)(ε2ε3 + ηε1) £ 1 − 2 ¡ ε2 1 − ε2 ¢¤
2
2(Ix − Iz)(ε1ε3 − ηε2) £ 1 − 2 ¡ ε2 1 − ε2 ⎤
¢¤
⎦
2
(4.13)
4(Iy − Ix)(ε1ε3 − ηε2)(ε2ε3 − ηε1)
and by linearizing in the point ε = 0 and η =1gives
T B g =3ω 2 ⎡
⎣
o
2(Iz − Iy) ε1
2(Ix − Iz) ε2
0
⎤
⎦ (4.14)
4.1.5 Linearization of the reaction wheel torque
The torque from the reaction wheel was defined in (3.28) as
T B µ B
dhw
w = + ω
dt
B ⎡
BI × hw = ⎣
−hwyωz
˙hwy
hwyωx
⎤
⎦ (4.15)
This equation can be linearized in the operation point where h =0and ω B BI =0,whichgives
T B w =
4.1.6 Linearization of the thruster torque
⎡
µ B
dhw
= ⎣
dt
0
˙hwy
0
The torque from the thrusters can be expressed according to (3.37) as
T B t = Bau = 1
⎡
√
2 ⎣
2
−rz rz −rz rz
rz rz −rz −rz
− (rx − ry) (rx − ry) (rx − ry) − (rx − ry)
⎤
⎦ (4.16)
⎡
⎤
⎢
⎦ ⎢
⎣
F1
F2
F3
F4
⎤
⎥
⎦
(4.17)

4 THEORETICAL ANALYSIS 36
and is already linear and as from (3.51) gives
T B t = 1
⎡
√ rz (−F1 + F2 − F3 + F4)
2 ⎣ rz (F1 + F2 − F3 − F4)
2
(rx − ry)(−F1 + F2 + F3 − F4)
4.2 Complete linearized model
The dynamic model of the satellite was presented in (3.38)
I ˙ω B BI = −ω B ³
BI × Iω B ´
BI + T B g + T B a
where the actuator torque T B a is
The linearized dynamic model is obtained as
⎤
⎦ (4.18)
(4.19)
T B a = TB t + TB w (4.20)
I ˙ω B BI = −ω B ³
BI × Iω B ´
BI + T B g + Bau +
µ B
dhw
dt
(4.21)
By inserting the linearized torque equations (4.14), (4.16) and (4.18) into(4.19) the model can be written in
component form by using c3 from (4.10) as
¡
˙ωx = σx ωyωz − 3ω 2 oc23c33 ∙ ¸
¢ 1 1√
+ 2rz (−F1 + F2 − F3 + F4)
(4.22)
Ix 2
¡
˙ωy = −σy ωxωz − 3ω 2 ∙
¢ 1 1√
oc13c33 + 2rz (F1 + F2 − F3 − F4)+
Iy 2
˙ ¸
hwy
(4.23)
¡
˙ωz = −σz ωxωy − 3ω 2 ∙ ¸
¢ 1 1√
oc13c23 + 2(rx − ry)(−F1 + F2 + F3 − F4)
(4.24)
2
Inserting the angular velocities ω B BI and their derivatives ˙ωB BI from (4.11) and (4.12) respectively gives
¡
2¨ε1 − 2ωo ˙ε3 = −σx 2ωo ˙ε3 +8ω 2 ¢ 1
oε1 +
Ix
∙
1
Iz
Iz
∙ ¸
1√
2rz (−F1 + F2 − F3 + F4)
2
2¨ε2 = −6σyω 2 oε2 + 1 √
2rz (F1 + F2 − F3 − F4)+
Iy 2
˙ hwy
¡
2¨ε3 +2ωo˙ε1 = σz 2ωo ˙ε1 − 2ω 2 ∙ ¸
¢ 1 1√
oε3 + 2(rx − ry)(−F1 + F2 + F3 − F4)
2
and solving for ¨ε gives the equation
¨ε1 =(1−σx) ωo ˙ε3 − 4σxω 2 oε1 + 1
∙ ¸
1√
2rz (−F1 + F2 − F3 + F4)
2Ix 2
¨ε2 = −3σyω 2 oε2 + 1
∙
1√
2rz (F1 + F2 − F3 − F4)+
2Iy 2
˙ ¸
hwy
¸
(4.25)
(4.26)
(4.27)
(4.28)
(4.29)

4 THEORETICAL ANALYSIS 37
¨ε3 = − (1 − σz) ωo ˙ε1 − σzω 2 oε3 + 1
∙ ¸
1√
2(rx − ry)(−F1 + F2 + F3 − F4)
2Iz 2
By defining the state vector x
x = £ ε1 ˙ε1 ε2 ˙ε2 ε3 ˙ε3
¤ T
and the torque vector u
u = £ F1 F2 F3 F4
˙hwy
¤T
the dynamic model equations can be written as a state space model
where
and
⎡
⎢
A = ⎢
⎣
(4.30)
(4.31)
(4.32)
˙x = Ax + Bu (4.33)
0 1 0 0 0 0
−4σxω 2 o 0 0 0 0 (1− σx) ωo
0 0 0 1 0 0
0 0 −3σyω 2 o 0 0 0
0 0 0 0 0 1
0 − (1 − σz) ωo 0 0 −σzω 2 o 0
⎡
0 0 0 0 0
⎢ −
⎢
B = ⎢
⎣
√ 2
4Ix rz
√
2
4Ix rz − √ 2
4Ix rz
√
2
4Ix rz 0
0 0 0 0 0
√
2
4Iy rz
√
2
4Iy rz − √ 2
4Iy rz − √ 2
4Iy rz 1
0 0 0 0 0
− √ 2
4Iz (rx − ry) √ 2
4Iz (rx − ry) √ 2
4Iz (rx − ry) − √ 2
4Iz (rx − ry) 0
⎤
⎥
⎦
⎤
⎥
⎦
(4.34)
(4.35)

4 THEORETICAL ANALYSIS 38
4.3 Stability of the linearized satellite model
To study the stability of the satellite, the actuator torque is equated to zero, T B a = 0, andfromthelinearized
dynamic model of the satellite in (4.28) - (4.30)
¨ε1 = (1−σx) ωo ˙ε3 − 4σxω 2 oε1 (4.36)
¨ε2 = −3σyω 2 oε2 (4.37)
¨ε3 = − (1 − σz) ωo ˙ε1 − σzω 2 oε3 (4.38)
The stability analysis is performed in two stages, due to the coupling between the roll and yaw movement.
Pitch motion: By Laplace transforming (4.37), the characteristic equation for the motion about the YB
axis becomes
s 2 +3σyω 2 o = s 2 +3ω 2 o (Ix − Iz) /Iy =0 (4.39)
which has one unstable root if Iz >Ix, so the conditions for stability becomes Ix >Iz. Since there are no
damping factor in this second-order equation, the satellite will oscillate in a stable motion about the YB axis
with an amplitude proportional to the initial condition θ(0).
Roll/yaw motion: Performing the Laplace transformation on (4.36) and (4.38) leaves
¡ 2
s +4σxω 2¢ o ε1
¡ 2
s + σzω
= (1−σx) ωosε3 (4.40)
2¢ o ε3 = − (1 − σz) ωoε1 (4.41)
Solving for ε1 in (4.40) and inserting the result into (4.41) gives
µ 4
µ 2
s
s
+(1+σxσz +3σx) +4σxσz =0
ωo
ωo
(4.42)
which is a stable oscillator for σx > 0 and σz > 0, and a small region where σx < 0 and σz < 0. Remembering
that
σx = Iy − Iz
Ix
> 0 (4.43)
gives the condition Iy >Iz, andfrom
σz = Iy − Ix
Iz
> 0 (4.44)
gives Iy >Ix. By combining these results with the stability for pitch motion, gives the required relation for
stability,
Iy >Ix >Iz
(4.45)
The moments of inertia for the ESEO is given as
Ix =4.3500kgm 2
Iy =4.3370kgm 2
Iz =3.6640kgm 2
which means that the linearized satellite system is unstable around the point RB BO = 1 without any actuators.

4 THEORETICAL ANALYSIS 39
4.4 Stability of the satellite - the Lyapunov approach
A stability analysis on the nonlinear system can be performed to see if there exits any other rotation in which
the satellite is stable4 . The energy of the satellite can be divided into kinetic and potential energy. The kinetic
energy is mainly due to rotation in the inertial and orbit frame, while the most important source to potential
energy is the gravity gradient and rotational effect due to revolution about the Earth. The kinetic energy in an
orbit can be written as (Hughes, 1986)
Ekin = 1 ¡ ¢T B B
ωBO IωBO (4.46)
2
The potential energy due to the gravity gradient
Egg = 3
2 ω2 ³
o (c3) T ´
Ic3 − Iz
and the potential energy due to the revolution of the satellite about the Earth is given by
Egyro = 1
2 ω2 ³
o Iy − (c2) T ´
Ic2
The Lyapunov Candidate Function (LCF) V (x) can now be written as the total energy
(4.47)
(4.48)
V (x) = Ekin + Egg + Egyro (4.49)
V (x) = 1
2
¡ ¢T B B
ωBO IωBO + 3
2 ω2o (c3) T Ic3 − 1
2 ω2o (c2) T Ic2 + 1
2 ω2o (Iy − 3Iz) (4.50)
The LCF has to be checked if it satisfies the demands for indeed being a proper LCF. By using
(c1i) 2 +(c2i) 2 +(c3i) 2 =1 (4.51)
V (x) can be written in component form as
V (x) = 1 ¡ ¢T B B
ωBO IωBO 2
+ 3
³
(Ix − Iz)(c13) 2 +(Iy − Iz)(c23) 2´
wherethestatevectorisdefined as
x =
2 ω2 o
+ 1
2 ω2 o
h
¡ω ¢
B T
BO
³
(Iy − Ix)(c12) 2 +(Iy − Iz)(c32) 2´
c12 c32 c13 c23
For (4.52) to be a proper LCF, then V (0) = 0 and V>0 for x 6= 0.Thisisthecasewhen
Iy >Ix >Iz
4 A similar stability analysis of the NCUBE satellite can be found in (Gravdahl, J. T. et al, 2003)
i T
(4.52)
(4.53)
(4.54)

4 THEORETICAL ANALYSIS 40
but for the ESEO the moments of inertia is
Ix >Iy >Iz
(4.55)
There exists equilibrium points where the satellites relative motion in the orbit plane becomes zero, ωB BO = 0.
These equilibrium points are often called relative points of equilibrium because the satellite has a rotation
relative the inertial frame, but has no rotation in the orbit frame. There exists a total of 24 equilibrium points
(Hughes, 1986). There are more then just one orientation that might be stable. To find these equilibrium
points, the LCF is rewritten so it better fulfill the demands of a proper LCF. The new LCF can be rewritten
so that it satisfies the demands of (4.55) for the ESEO as
V (x) = 1 ¡ ¢T B B
ωBO IωBO +
2
3
2 ω2o (c3) T Ic3 − 1
2 ω2 o (c2) T Ic2 + 1
2 ω2o (Ix − 3Iz)
and in component form
(4.56)
V (x) = 1 ¡ ¢T B B
ωBO IωBO 2
+ 3
³
(Ix − Iz)(c13) 2 +(Iy − Iz)(c23) 2´
The new state vector is defined as
x =
2 ω2 o
+ 1
2 ω2 o
h
¡ω ¢
B T
BO
³
(Ix − Iy)(c22) 2 +(Ix − Iz)(c32) 2´
c22 c32 c13 c23
Now V (0) = 0 and V>0 for x 6= 0when
Ix >Iy >Iz
which is the case for the ESEO. Differentiation of (4.56) becomes
The dynamical equation was found in (3.75) as
˙V (x) = ¡ ω B ¢T B
BO I ˙ω BO +3ω 2 o (c3) T I˙c3 − ω 2 o (c2) T I˙c2
I ˙ω B BI = −ω B BI ×
i T
³
Iω B ´
BI +3ω 2 oc3 × Ic3 + T B a
The angular velocity between the body frame and orbit frame is given from (3.78) as
and the differentiated expression of this relation gives
ω B BI = ω B BO − ωoc2
˙ω B BI = ˙ω B BO − ωo ˙c2
Inserting these results into the dynamical model in (4.61), leaves the expression
I ˙ω B BO = ωoI˙c2 − ¡ ω B ¢ ¡ ¢ B 2
BO − ωoc2 × I ωBO − ωoc2 +3ωoS (c3) Ic3 + T B I ˙ω
a
B BO = ωoI˙c2 − S ¡ ω B ¢ B
BO IωBO + ωoS ¡ ω B ¢
BO Ic2 + ωoS (c2) Iω B BO
−ω 2 o S (c2) Ic2 +3ω 2 o S (c3) Ic3 + T B a
(4.57)
(4.58)
(4.59)
(4.60)
(4.61)
(4.62)
(4.63)
(4.64)
(4.65)

4 THEORETICAL ANALYSIS 41
Inserting (4.65) in (4.60), gives
By using the facts that
˙V (x) = ¡ ω B BO
¢T
[ωoI˙c2 − S ¡ ω B ¢ B
BO IBO + ωoS ¡ ω B ¢ B
BO IBOc2 + ωoS (c2) Iω B BO
−ω 2 oS (c2) Ic2 +3ω 2 oS (c3) Ic3 + T B a ]+3ω 2 oc3I˙c3 − ω 2 oc T 2 I˙c2
˙ci = S (ci) ω B BO
(4.66)
(4.67)
and ¡ ¢T ¡ ¢ B B
ωBO S ωBO =0 (4.68)
equation (4.66) can be written as
˙V (x) = ¡ ω B BO
¢T [ωoIS (c2) ω B BO + ωoS (c2) Iω B BO
−ω 2 oS (c2) Ic2 +3ω 2 S (c3) Ic3 + T B a ]+3ω 2 oc3IS (c3) ω B BO − ω 2 oc T 2 IS (c2) ω B BO
(4.69)
Since
S T (x) =−S (x) (4.70)
the last term in (4.69) can be written as
−ω 2 o (c2) T IS (c2) ω B BO = ω 2 ¡ ¢T B
o ωBO S (c2) T Ic2
(4.71)
Similarly the second last term in (4.69) can be expressed as
3ω 2 o (c3) T IS (c3) ω B BO = −3ω 2 ¡ ¢T B
o ωBO S (c3) Ic3
which simplifies (4.66) to
The term ωoIS (c2) ωB BO can be written as
which again simplifies (4.73) to
(4.72)
˙V (x) = ¡ ω B ¢T
BO [ωoIS (c2) ω B BO + ωoS (c2) Iω B BO + T B a ] (4.73)
ωoIS (c2) ω B BO = −ωoS (c2) Iω B BO
˙V (x) = ¡ ω B ¢T B
BO Ta (4.74)
(4.75)
h
¡ω ¢
iT If the actuator torque Ta = 0, then the equilibrium point B T
BO c22 c32 c13 c23 = 0 is uniformly
stable according to Lyapunov (Theorem 2.5). The equilibrium points can be found by investigating the rotation
matrix RB BO . The rotations can be found by using c22 = c32 = c13 = c23 =0,
where i 6= j, and the fact that
(c1i) 2 +(c2i) 2 +(c3i) 2 = 1 , i =1, 2, 3 (4.76)
(cj1) 2 +(cj2) 2 +(cj3) 2 = 1 , j =1, 2, 3 (4.77)
c1 = c2 × c3
(4.78)

4 THEORETICAL ANALYSIS 42
This gives a total of 4 rotation matrix
R O ⎡
0
B = ⎣ −1
0
1
0
0
⎤
0
0 ⎦
1
R O ⎡
0
B = ⎣ −1
0
−1
0
0
0
0
−1
⎤
⎦ R O ⎡
B = ⎣
0 −1 0
1 0 0
0 0 1
⎤
⎦ R O ⎡
B = ⎣
0 1 0
1 0 0
0 0 −1
⎤
⎦ (4.79)
As mentioned earlier, there exists a total of 24 equilibrium points, and where the four stated above are stable.
It is desirable to have the body frame together with the orbit frame such that RO B = 1, the identity matrix. As
seen this is not the case for the unactuated satellite. A controller is needed such that RO B = 1, andthismust
be done by controlling ε → 0 and ωB BO → 0.
4.5 Reference trajectory
A reference model filters the reference signal so that a step in the reference will be filtered into a smooth
transition curve. A second order filter is appropriate for this system. The reference model can be expressed as
(Fossen, 1994)
¨ψ r +2ζωn ˙ ψr = ω 2 nψd (4.80)
where ψd is the desired state, ψr is the reference signal given to the system, ωn is the cross-over frequency and
ζ is the relative damping factor. It should be denoted that when filter enters a steady state, ¨ ψd = ˙ ψd =0,and
accordingly
ψd = ψr (4.81)
By altering the parameter ωn and ζ, the response from the model can be adjusted. The relative damping factor
is usually chosen to be in the interval 0.8

4 THEORETICAL ANALYSIS 43
Choosing ζ 1 = ζ 2 = ζ 3 =0.8 and ωn1 = ωn2 = ωn3 =0.01 rad/s leaves a slightly overdamped model, which
expresses the reference trajectory for roll, pitch and yaw angles. A simulation of the reference trajectory is
shown in Figure 4.1.
Roll angle [deg]
Pitch angle [deg]
Yaw angle [deg]
10
5
0
Reference trajectories in roll, pitch and yaw
-5
0 100 200 300 400 500 600 700 800 900
10
5
0
-5
0 100 200 300 400 500 600 700 800 900
10
5
0
-5
0 100 200 300 400 500 600 700 800 900
sec
Figure 4.1: Reference trajectories for roll, pitch and yaw from initial angles of 10 ◦ on all axis to the reference
angles 0 ◦ .
φ ref
θ ref
ψ ref

5 CONTROLLERS 44
5 Controllers
5.1 Thruster allocation
A controller normally consists of an actuator vector u which may have both positive and negative components.
The ESEO uses four thrusters and one reaction wheel as attitude actuators in which only the reaction wheel can
produce both negative and positive torque. A thruster alone can only produce a positive torque. The mounted
position of the thrusters may result in a negative torque. Therefore, if a negative value in one or more of the
first four elements of the actuator vector u is detected, the negative element is set to null and the absolute
value of the element is added to a thrusters which is mounted in the opposite direction. A new actuator vector
is formed where u1 − u4 only has positive values and where u5, the reaction wheel can be either positive or
negative.
5.2 Pulse modulation
Reaction wheels can provide continuous control action according to the desired torque profile for attitude
control. Thrusters on the other hand are on—of devices and are normally capable of providing only fixed
torque. Therefore, achieving high attitude control performance using thrusters is a challenging task. The task
becomes even more complicated for flexible spacecraft where thruster firings could excite modes resulting in
attitude control instability 5 . The two major approaches for thruster control are bang bang control and pulse
modulation.
5.2.1 Bang Bang Controller
Bang Bang control is simple in formulation, but results in excessive thruster action. Its discontinuous control
actions may interact with the flexible modes of the satellite and result in limit cycles.
An attitude controller outputs the continuous u = £ ¤
u1 u2 u3 u4 u5 where u1 - u4 are the thruster
F1 - F4, andu5is the reaction wheel ˙ hw. By using a bang bang controller with deadzone the continuous control
u1 − u4 can be transformed into on of control (Song and Agrawal, 2000)
Fi = { Fmax, ui ≥ D
0, ui

5 CONTROLLERS 45
straightforward because continuous or analogue electronics technology is the natural medium for realizing such
modulators. However, PWPFM causes some practical problems with today’s onboard microprocessors. Digital
microprocessors work in a synchronous timing created by an onboard electronic clock. The onboard computer
sends control commands at equal time intervals, and pulse frequency modulation cannot be easily implemented.
5.2.3 Pulse-Width Modulator (PWM)
Modulators based solely on pulse width modulation are simpler then PWPFM to apply in a microprocessorbased
onboard computer (Sidi, 1997). There are no fundamental difference between PWPFM and PWM. Both
are based on the Schmidt trigger schemes. In attitude feedback control loops based on PWM, the sampling
frequency is constant and the thrust pulses are applied at equal time intervals. If the dynamics of the plant
contains additional structural dynamics with low damping coefficients and eigenfrequencies equal to the sampling
frequency of the control loop, then the structural dynamics might be excited, thus degrading the quality of the
feedback control loop. Special precautions, such as adequate structural filters, must be incorporated to account
for this phenomenon.
5.3 Linear Control
Linear controllers are based upon the linearized state space model of the satellite from chapter 4.2.
5.3.1 PD controller
The actuator torque for the satellite was deduced to be
where
T B a = Bau + Daω B BI
u = £ F1 F2 F3 F4
is the actuator vector, ωB BI is the angular velocity of the satellite body relative to the inertial system referenced
in the body system,
Ba = 1
⎡
−rz √
2 ⎣ rz
2
− (rx − ry)
rz
rz
(rx−ry) −rz
−rz
(rx−ry) rz
−rz
− (rx − ry)
⎤
√
0
2 ⎦
0
(5.3)
is the actuator matrix, and
⎡
Da = ⎣
0 0 −hwy
0 0 0
hwy 0 0
˙hwy
¤T
(5.2)
⎤
⎦ (5.4)
is the wheel disturbance matrix. A proportional derivative (PD) controller is proposed given as
u = −Kp˜ε − Kd ˙ε (5.5)
where Kp and Kd is the state feedback matrix calculated from desired pole placement for the system and ˜ε
denotes the deviation from the current to the desired Euler parameter vector, ε and εd, respectively, found from
the Euler product (Egeland and Gravdahl, 2003)
∙ ¸ ∙ ¸ ∙ ¸ ∙
˜η ηd η
=
=
˜ε
−ε
εd
η dη + εdε
η dε − ηεd − S(εd)ε
¸ ∙
=
η dη + εdε
η dε − (ηεd + ε × εd)
¸
(5.6)

5 CONTROLLERS 46
The total actuator torque can be expressed as
T B a = Bau + Daω B BI (5.7)
= Ba(−Kp˜ε − Kd ˙ε)+Daω B BI (5.8)
The desired poles p in the system is set to be in the stable region p = ¡ −1, − 1
¢
1
1
2 , −1, − 2 , −1, − 2 .
The mathematical calculation of KP and Kd can be difficult with a large A matrix. KP and Kd can be found
by comparing the characteristic polynomial
det (s1 − A + BK) (5.9)
where A and B was stated in (4.34) and (4.35) respectively as
⎡
0
⎢ −4σxω
⎢
A = ⎢
⎣
1 0 0 0 0
2 o
0
0
0
0
0
0
0
−3σyω
0
1
0
0
(1−σx) ωo
0
2 0 0
o
0
0
0
0
0
0
1
0 − (1 − σz) ωo 0 0 −σzω2 ⎤
⎥
⎦
o 0
and
⎡
0 0 0 0 0
⎢ −
⎢
B = ⎢
⎣
√ 2
4Ix rz
√
2
4Ix rz − √ 2
4Ix rz
√
2
4Ix rz 0
0 0 0 0 0
√
2
4Iy rz
√
2
4Iy rz − √ 2
4Iy rz − √ 2
4Iy rz 1
0 0 0 0 0
− √ 2
4Iz (rx − ry) √ 2
4Iz (rx − ry) √ 2
4Iz (rx − ry) − √ 2
4Iz (rx − ry) 0
and where K is 5 × 6 gain matrix consisting of the vectors
h
K = k1 k2 k3 k4 k5 i
k6
in which Kp and Kd
with the wanted polynomial
Kp =
Kd =
h
k1 k3 i
k5
h
k2 k4 i
k6
µ
(p +1) p + 1
µ
(p +1) p +
2
1
µ
(p +1) p +
2
1
2
⎤
⎥
⎦
(5.10)
(5.11)
(5.12)
However there does not exits only one solution of the K matrix. By using the command place(A, B, p), Matlab
uses a solver based on least square solution to find a suitable K matrix to be used with the PD controller. The
state feedback matrix now has the eigenvalues of (A − BK) which is the new system matrix and has the desired
poles p. The orbit angular velocity ωo in matrix A is set to be the constant angular velocity at perigee so that
the K matrix is constant.

5 CONTROLLERS 47
5.3.2 PD controller with wheel compensation
By introducing an extra term to the PD controller deduced in chapter 5.3.1 it is possible to compensate for the
disturbance Da from the reaction wheel. The PD controller from (5.5) was deduced as
and adding
to (5.13) as
gives the resulting torque
u = −Kp˜ε − Kd ˙ε (5.13)
−B àDaω B BI
u = −Kp˜ε − Kd ˙ε − B ` Daω B BI
(5.14)
(5.15)
T B a =
¡
Ba −Kp˜ε − Kd ˙ε − BàDaω B ¢
BI + Daω B BI
(5.16)
= Ba (−Kp˜ε − Kd ˙ε) (5.17)
5.3.3 Linear Quadratic Gaussian (LQG) controller
Theory about the LQG controller can be found in (Fossen, 1994), (Anderson and Moore, 1989). An LQG
controller can be design by using the linearized state space model
˙x = Ax + Bu (5.18)
The error vector of the system can be defined as
˜x = x − xd
where xd is the desired state. The feedback control law is found by minimizing the performance index
Z T
1 ¡ ¢
T T
J =min ˜x Q˜x + u Pu
u 2 0
(5.19)
(5.20)
where P = PT > 0 and Q = QT ≥ 0 are weighting matrices. The system Hamiltonian can be written as
(Fossen, 1994)
H = 1 ¡ ¢ T T T
˜x Q˜x + u Pu + P (Ax + Bu)
2
(5.21)
Deriving H with respect to u yields
∂H
∂u = Pu + BT p =0 (5.22)
and
u = −P −1 B T p
Assuming that p can be expressed as a linear combination
(5.23)
p = Rx + h1
where R and h1 are unknown quantities to be determined. Inserting (5.24) into (5.23) leaves
u = −P −1 B T Rx + P −1 B T h1
(5.24)

5 CONTROLLERS 48
Deriving p leaves the result
From optimal control it is known that
˙p = ˙Rx + R ˙x + ˙h1
(5.25)
˙p = − ∂H
∂x = Qx − AT p = Qxd−Qx − A T Rx − A T h1
Eliminating ˙p from (5.25) and (5.26) yields
(5.26)
˙Rx + R ˙x + ˙ h1 − Qxd + Qx + A T Rx + A T h1 =0 (5.27)
By inserting (5.18) and (5.23) into (5.27) gives the equation
³
˙R + RA+A T R − RBP −1 B T ´
R + Q x =0 (5.28)
and
h1 + ¡ A − BP −1 B T R ¢ h1 − Qxd =0 (5.29)
This implies that R must be solved from the algebraic Ricatti equation given as
˙R + RA + A T R − RBP −1 B T R + Q =0 (5.30)
The boundary conditions for (5.29) and (5.30) are given by
h1(T )=0 and R(T )=0 (5.31)
The weighting matrices P and Q are first formed as diagonal matrices defined as
P = diag ([p1,p2,..., pna]) (5.32)
Q = diag ([q1,q2,..., qns]) (5.33)
where na is the number of actuators on the control system and ns is the number of interesting states in the
system. According to the linearized system shown in (4.33), na =5and ns =6in this case. The elements can
be chosen according to a suggestion from (Balchen and Mummé, 1988) as
pi =
1
(∆xi) 2
qi =
1
(∆ui) 2
(5.34)
where ∆xi and ∆ui is the nominally acceptable variation to a reference of the states and actuator torques,
respectively. The eigenvalues of the final system matrix (A − BG) will lie in the left half-plane, and the system
will be stable.
5.4 Nonlinear Control
When the required operation range is large, a linear controller is likely to perform very poorly or to be unstable,
because the nonlinearity in the system cannot be properly compensated for. Nonlinear controllers, on the other
hand, may handle the nonlinearity in large range operation directly. Nonlinear controllers might be designed
to guarantee global stability, improved efficiency and increased control performance for such systems. This is
especially interesting for satellites, since they are nonlinear and efficiency is an important factor as the energy
used on board the satellite is self-obtained.

5 CONTROLLERS 49
5.4.1 Nonlinear control by Lyapunov
As mentioned earlier it is desired to coincide the body and orbit frame such that RO B = 1. By using (4.56) and
adding a term with the Euler parameters, the new LCF can be written as (Soglo, 1994)
h
Vc (x) =V (x)+k ε T ε +(1−η) 2i
(5.35)
where k>0 is a constant. When the satellite is in the correct position, ε = 0 and η =1. By using the fact that
ε T ε + η 2 =1, the last term in (5.35) can be written as 2k(1 − η). Differentiating this expression gives −2k ˙η,
and with the use of (3.76), the relationship
−2k ˙η = kε T ω B BO
(5.36)
is found. The differentiated equation ˙ V (x) without the last term of Euler parameters was found in (4.75) as
˙V (x) = ¡ ω B ¢T B
BO Ta hence the differentiated expression of (5.35) with the Euler parameters becomes
(5.37)
˙Vc (x) = ¡ ω B ¢T ¡ ¢ B
BO kε + Ta A controller can now be proposed as
(5.38)
T B a = −Dω B BO − kε (5.39)
where D>0. This is a similar controller found for the Norwegian satellite project NISSE by (Soglo, 1995).
The constant k has to be investigated further. By using the dynamical model from (4.65), given as
I ˙ω B BO = ωoI˙c2 − S ¡ ω B ¢ B
BO IωBO + ωoS ¡ ω B ¢
BO Ic2 + ωoS (c2) Iω B BO − ω 2 oS (c2) Ic2 +3ω 2 oS (c3) Ic3 + T B a
and inserting (5.39) and the fact that ˙ci = S (ci) ω
(5.40)
B BO gives
−Dω B BO − kε = I ˙ω B BO − ωoIS (c2) ω B BO + S ¡ ω B ¢ B
BO IωBO − ωoS ¡ ω B ¢
BO Ic2 (5.41)
−ωoS (c2) Iω B BO + ω 2 oS (c2) Ic2 − 3ω 2 oS (c3) Ic3
(5.42)
When ωB BO = 0 and ˙ωB BO = 0
ω 2 oS (c2) Ic2 − 3ω 2 oS (c3) Ic3 = −kε (5.43)
k has to be large enough such that ε = 0 is the only solution. Writing (5.43) in component form gives
ω 2 o (Iz − Iy)(c22c32 − 3c23c33) = −kε1 (5.44)
ω 2 o (Ix − Iz)(c12c32 − 3c13c33) = −kε2 (5.45)
ω 2 o (Iy − Ix)(c12c22 − 3c13c23) = −kε3 (5.46)
The product of the directional cosine has a value in the range ±0.5, which makes the values of (c22c32 − 3c23c33),
(c12c32 − 3c13c33), and(c12c22 − 3c13c23) in the range ±2. The left side of (5.45) has the highest value since
Ix >Iy >Iz. By investigating the maximum value of the directional cosine of (c12c32 − 3c13c33), leavesthe
rotation about the y axis as the maximum value. A rotation about the y axis gives c12 = c32 =0, c13 =sinθ,
c33 =cosθ and ε2 =sin θ
2
. Equation (5.45) can now be written as
3sinθ cos θω 2 o (Ix − Iz) =k sin θ
2
(5.47)

5 CONTROLLERS 50
so k now has to be chosen so that
k> 3 sin (2θ) ω
2
2 o (Ix − Iz)
sin θ
2
The maximum value of the right side of (5.48) is when θ =0, hence the expression sin(2θ)
sin θ 2
(5.48)
=1which leads to
k> 3
2 ω2o (Ix − Iz) (5.49)
Equation (5.39) is the torque provided by the controller, so the right side pseudo inverse actuator matrix has
to be used in order find the actuator vector u. Multiplying the right side pseudo inverse of Bà matrix with the
torque from the controller, gives
⎡
F1
⎢ F2 ⎢
u = ⎢
⎤
⎥
(5.50)
⎢
⎣
F3
F4
˙hy
⎥
⎦ = BàT B a
And finally the actual torque using the thrusters and reaction wheel as actuator becomes
T B a = Bau + Daω B BI
5.4.2 Nonlinear control by Feedback Linearization
(5.51)
The basic principle of feedback linearization is to make the nonlinear system act as a linear system and then
design a simple tracking controller that stabilizes the system. In other words exact linearization is used to
change the appearance or behavior of a nonlinear system into a linear one, which is controllable, and then
design a stabilizing controller, that tracks the references. The Feedback Linearization in this thesis is based
upon (Jensen and Wi´sniewski, 2001) and (Khalil, 2002) 6 .
The dynamical model was defined in (3.75) as
I ˙ω B BI = −ω B BI × Iω B BI + T B
˙η = − 1
2 εT ω B0
BO
˙ε = 1
2 [η1 + S (ε)] ωB BO
(5.52)
(5.53)
(5.54)
Where TB = Tg + Tcontrol. The gravitational torques is neglected in the design of the controller which makes
it easier to find the Lie derivatives. The derived ˙ω B BI with Tcontrol = £ ¤ T
Tx Ty Tz can be written on
component form as
˙ωx = σx (ωyωz)+ Tx
˙ωy =
Ix
¡
−σy ωxωz − 3ω 2 ¢ Ty
oc13c33 +
Iy
˙ωz =
¡
−σz ωxωy − 3ω 2 ¢ Tz
oc13c23 +
Iz
6 Another nonlinear controller by feedback linearization can be found in (Bang et al, 2004)
(5.55)
(5.56)
(5.57)

5 CONTROLLERS 51
or
˙ω B BI =
⎡
⎣ σx
⎤ ⎡
(ωyωz)
−σy (ωxωz) ⎦ + ⎣
−σz (ωxωy)
1
Ix
0
0 0
0
1
Iy
0 0 1
Iz
The derived Euler parameters can be written as
⎡
˙ε1
⎢ ˙ε2 ⎢
⎣ ˙ε3
⎤ ⎡
η
⎥
1 ⎢ ε3
⎦ = ⎢
2 ⎣ −ε2
−ε3
η
ε1
ε2
−ε1
η
ε1
ε2
ε3
⎤ ⎡
⎥ ⎢
⎥ ⎢
⎦ ⎣
˙η −ε1 −ε2 −ε3 η
The orthogonal matrix Q has the property
since
Q −1 =
ωx
ωy
ωz
0
QT ε2 1 + ε22 + ε2 = QT
3 + η2
⎤ ⎡
⎦
⎣ Tx
Ty
Tz
⎤
⎥
1
⎦ =
2 Q
⎡
⎢
⎣
⎤
⎦ (5.58)
ωx
ωy
ωz
0
⎤
⎥
⎦
(5.59)
(5.60)
ε 2 1 + ε 2 2 + ε 2 3 + η 2 =1 (5.61)
Before feedback linearization can be done, the complete system has to be defined as
Using the models derived in (5.58) and (5.59), the system takes the form
⎡
˙ωx
⎢ ˙ωy ⎢ ˙ωz ⎢ ˙ε1 ⎢ ˙ε2 ⎢
⎣ ˙ε3
⎤ ⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ = ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
σx (ωyωz)
−σy (ωxωz)
−σz (ωxωy)
1
2
˙η
Q
⎤ ⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎡ ⎤ ⎥ ⎢
ωx ⎥ + ⎢
⎢ ωy ⎥ ⎥ ⎢
⎢ ⎥ ⎥ ⎢
⎣ ωz ⎦ ⎦ ⎣
1
Ix
0
0
1
Iy
0
0
⎤
⎥ ⎡
⎥
⎦
0
˙x = f (x)+g + u (5.62)
0 0 1
Iz
0 0 0
0 0 0
0 0 0
0 0 0
⎣ Tx
Ty
Tz
⎤
⎦ (5.63)
The control characteristic indices are derived using the Lie derivatives for multi-variable systems as from definition
2.6 in Chapter 2.8. The new system is based upon the Lie derivatives L 2 f h and LgLf h as
L 2 f h = Q
2
LgLf h = Q
2
⎡
⎢
⎣
⎡
⎢
⎣
σx (ωyωz)
−σy (ωxωz)
−σz (ωxωy)
− 1
¡
2
2 ωx + ω2 y + ω2 z
1
⎤
0 0 0
Ix
1 0 0 0 ⎥
Iy ⎥
1 0 0 0 ⎦
Iz
0 0 0 0
⎤
⎥
⎦
¢
(5.64)
(5.65)

5 CONTROLLERS 52
The system which is to be feedback linearized has the following structure
d2 ⎡
ε1
⎢ ε2 ⎢
dt ⎣
⎤
⎥
⎦ = L2f h+LgLf hu
ε3
η
= 1 ¡ 2
2Lf h
2
¢ + 1
2 (2LgLf hu)
= 1
2 f2 (x)+ 1
U (x, u)
2
(5.66)
Using feedback linearization theory, the new input vector U is then calculated, so that it exactly linearizes the
system in (5.66)
U (x, u) =−f2 (x)+2vt
Combining the feedback linearization loop (5.66) with (5.67), results in the linear system
(5.67)
⎡ ⎤
d 2
dt 2
⎢
⎣
ε1
ε2
ε3
η
⎥
1
⎦ =
2 f2 (x)+ 1
2 (−f2 (x)+2vt) =vt
The relationship between the real input vector u and the feedback linearizing vector U is
⎡
⎢
U = U (x, u) =LgLf hu = Q ⎢
⎣
Tx
Ix
Ty
⎤
⎥
⎦
Isolating u in (5.69), gives
⎡
⎢
⎣
Tx
Ty
Tz
0
Iy
Tz
Iz
0
(5.68)
(5.69)
⎤
⎥
⎦ =
⎡ 1
⎤
0 0 0
Ix
⎢ 1
⎢
0 0 0 ⎥
Iy ⎥
⎣
1 0 0 0 ⎦
Iz
0 0 0 0
QT U (5.70)
The problem now is that the feedback linearizing input vector U has four components that may change from zero,
but the real input vector u has three components. The three upper components of U are calculated according
to (5.67). In order for the fourth component of u to be zero, £ ε1 ε2 ε3 η ¤ T has to be perpendicular to U.
In order to do this, the forth component, U4 in U is changed as
By doing this, gives the extra term
U4 = − ε1U1 + ε2U2 + ε3U3
η
ve =
⎡
⎢
⎣
¡ ε1 2
2η ωx + ω2 y + ω2 ¡ z
ε 2
2η ωx + ω2 y + ω2 ¡ z
ε 2
2η ωx + ω2 y + ω2 z
0
¢ ⎤
¢
⎥
¢ ⎥
⎦
(5.71)
(5.72)

5 CONTROLLERS 53
to the input vector u which changes the ideal feedback linearized system in (5.68) to
⎡ ⎤
ε1
⎢ ε2 ⎥
⎢ ⎥
⎣ ⎦ = vt − ve
d 2
dt 2
The feedback form has to be chosen to get asymptotic tracking (Marino and Tomei, 1995)
⎡
⎤
−k1ε1 − k2 ˙ε1
⎢ −k1ε2
vt = ⎢ − k2 ˙ε2 ⎥
⎣ −k1ε3 − k2 ˙ε3 ⎦
0
ε3
η
(5.73)
(5.74)
where vt is a simple multi variable proportional controller, which gives asymptotic tracking on the linearized
system. Now Tcontrol can be written as
⎡ ⎤
Tx
⎢ Ty ⎥
⎢ ⎥
⎣ Tz ⎦
0
=
⎡ 1
⎤
0 0 0
Ix
⎢ 1
⎢
0 0 0 ⎥
Iy ⎥
⎣
1 0 0 0 ⎦
Iz
0 0 0 0
QT
⎛
⎜
⎝−Q ⎡
σx (ωyωz)
⎢ −σy ⎢ (ωxωz)
⎣ −σz ¡
(ωxωy)
2 ωx + ω2 y + ω2 ⎤
⎥
⎦
¢
z
+2vt
⎞
⎟
⎠ − ve (5.75)
All the elements in the last row will be zero, hence the torques Tx, Ty, andTzcan be converted to find the
actual actuator vector based on the thrusters and reaction wheel. This is done by using the right hand pseudo
inverse Bà ⎡ ⎤
F1 ⎡ ⎤
The actual actuator torque, T B a is given as
⎢
⎣
F2
F3
F4
˙hy
T B a = Ba
5.5 Momentum dumping controller
⎥ = Bà
⎦
⎡
⎢
⎣
F1
F2
F3
F4
˙hy
− 1
2
⎣ Tx
Ty
Tz
⎤
⎥
⎦ + Daω B BI
⎦ (5.76)
(5.77)
Reaction wheels used for attitude control of a satellite tends to suffer from saturation, due to the fact that
there is an upper limit for the velocity of the wheel. Hence, a momentum controller for the removal of angular
momentum from the reaction wheels is needed. The total angular momentum of the reaction wheel can be
written as
hw = £ 0 ˙ hwy 0 ¤T = Iw ˙ωw (5.78)
where Iw is the moment of inertia for the reaction wheel, and ωw is the angular velocity of the reaction wheel.
Since the wheel is only controlling the pitch angle, the problem can be solved by only looking at the y-axis.

5 CONTROLLERS 54
The approach to this problem is to apply proportional momentum feedback by using the thrusters which result
in a positive or negative torque in roll, or
T B yt = Bayu = ˙ hwy
where
Bay = 1√
£
2 rz
2
rz −rz −rz
¤
is the actuator matrix in y-axis and
u = £ F1 F2 F3 F4
¤ T
is the thruster actuator. By using the right sided pseudo inverse of the matrix B` y to find u as
u = B ày ˙ hwy
(5.79)
(5.80)
(5.81)
(5.82)
The thruster torque is then
T B t = Bayu = BayBày ˙ hwy = ˙ hwy (5.83)
which will compensate for the acceleration or deceleration of the wheel.

6 SIMULATION AND RESULTS 55
6 Simulation and results
This chapter contains some of the results of the satellite step response and settling time. The attitude accuracy
demands is 1 ◦ in x-axis and y-axis and 5 ◦ in the z-axis as mentioned in Chapter 2. All simulation with the
satellite starts in perigee where the gravitational torque is at its highest. Euler parameters are used in the
simulations to prevent singularity. All Euler parameters are converted to Euler angles and plotted after simulations.
The nonlinear controller based on feedback linearization will from her on be referred to as the Feedback
controller and the nonlinear control by Lyapunov will be referred to as the Lyapunov controller.
The simulations of the controllers are done with continuous thrust levels, but the thrusters allocation is used.
Continuous thrust level will be more close to a simulation with the use of PWM mentioned in chapter 5.2.3 than
with the use of the Bang Bang controller. By using continuous thrust, the settling time for each controller is
easier to investigate. A simulation where the PD controller together with the Bang Bang controller is performed
just to illustrate the difference between the Bang Bang controller and continuous thrusters.
1
Several simulation over 800 orbit with different inertial and reference angles are performed to evaluate the
step response for each controller, but only one set of angles is plotted. Other rotations can be found in appendix
B. There does not exist any demands for the response time for the ESEO, therefore the main focus will be on
finding the best response time for the system. No large angle manoeuvres are performed.
6.1 Unactuated satellite
The stability analysis on the linearized dynamical model of the satellite without actuators showed that the
satellite is unstable around the equilibrium point RO B = 1. A simulation of the unactuated satellite during ten
orbits and released with the initial angles 0◦ which is the rotational matrix RO B = 1 isshowninFigure6.1. The
roll and yaw axis remain stable, but the pitch axis is spinning and does not stabilize.
RPY angles [deg]
Angular velocity [rad/s]
200
150
100
50
0
-50
-100
-150
Attitude angles [deg]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10 5
-200
sec
-0.5
-1
-1.5
x 10-3
0
Angular velocities without reference trajectories [rad/s]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10 5
-2
sec
Figure 6.1: Unactuated satellite relased at perigee with the initial Euler angles 0 ◦ .
φ
θ
ψ
ωx ωy ωz

6 SIMULATION AND RESULTS 56
6.2 Step response
The resulting step response from the initial Euler angles Φi = £ 5◦ 10◦ 15◦ ¤ T
to the reference angles
Φr = £ 10◦ 15◦ 5◦ ¤ T
using the PD controller is shown in Figure 6.2. The response of the pitch angle
overshoots the reference with approximately 3.3◦ , but all the Euler angles are settled within 13.10 sec. The
reason for the overshot might be because in addition to the thrusters the pitch angle has an extra actuator, the
reaction wheel, which contributes to the pitch rotation. The small overshot in the yaw angle is insignificant
because it is well within the attitude requirements of 5◦ . The highest angular velocity that occur in roll, pitch
and yaw is 0.0350, 0.0231 and 0.0385 rad/s respectively. The control torque exceeds the maximum available
torque from the thrusters and the reaction wheel as seen in the actuator plot in Figure 6.2.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
ωx 0.02
ωy 0
ωz -0.02
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.2: Step response using the PD controller with the initial angles Φi = £ 5◦ 10◦ 15◦ ¤ T
and the
reference angles Φr = £ 10◦ 15◦ 5◦ ¤ T
.
φ
θ
ψ

6 SIMULATION AND RESULTS 57
The LQG controller is used with the same Euler angles as the PD controller as shown in Figure 6.3. The
pitch angle does not overshoot as it did with the PD controller, but a small overshoot in the roll angle occur, in
which the settling time is nearly the same as for the PD controller, 13.15 sec. The maximum angular velocity in
roll increases by a factor of 44.5% and is reduced by a factor of 44.3% and 59.6% in pitch and yaw respectively
compared to the PD controller. All the actuators except thruster Th1 and the reaction wheel reaches the
maximum actuator torque, and the overall usage of actuator seems to be lower than for the PD controller.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
0.02
ωx ωy ωz 0
-0.02
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.3: Step response using the LQG controller with the initial angles Φi = £ 5◦ 10◦ 15◦ ¤ T
and the
reference angles Φr = £ 10◦ 15◦ 5◦ ¤ T
.
φ
θ
ψ

6 SIMULATION AND RESULTS 58
When using the Feedback controller the step response in all the axis is has a settling time of 8.62 sec. Only
a very small overshot in the pitch angle occur but it is does not pass the limit of 1 ◦ , as seen in Figure 6.4.
The small overshot might also be a result of the extra actuator in form of the reaction wheel. The response
of the yaw angle is much slower than both roll and pitch angle, but since the attitude accuracy demand for
the yaw axis is 5 ◦ it does not matter. The response of the angular velocity for the Feedback controller looks
quite the same as for the PD controller, but with a reduction of 6.5%, 24.6% and 16.4% in roll, pitch and yaw
respectively. It is worth noting that the actuators are activated much more rapidly as seen in Figure 6.4, than
with the PD and LQG controller and it is only thruster Th3 that does not reach the maximum torque limit.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
ωx 0.02
ωy 0
ωz -0.02
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.4: Step response using the Feedback controller with the initial angles Φi = £ 5◦ 10◦ 15◦ ¤ T
and the
reference angles Φr = £ 10◦ 15◦ 5◦ ¤ T
.
φ
θ
ψ

6 SIMULATION AND RESULTS 59
The step response of using the Lyapunov controller is shown in Figure 6.5, where a small overshoot in the
pitch angle of approximately 1.5 ◦ occur. This is as mentioned earlier probably due to the reaction wheel. The
settling time for the Lyapunov controller is as high as 15 sec due to the overshoot. The first thing to notice
about the angular velocity is that they appear like smooth curves with no sudden change. This is due to the
fact that none of the actuators reaches the maximum torque limit and therefore also appears like smooth curves
as shown in the actuator plot in Figure 6.5. The angular velocity is reduced by a factor of 18.6 %, 97.6 % and
30.1 % compared to the PD controller which therefore has the lowest angular velocity in roll and pitch, but not
inyaw,wheretheLQGcontrollerhasthelowest.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.02
0.01
ωx ωy 0
ωz -0.01
-0.02
-0.03
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.5: Step response using the Lyapunov controller with the initial angles Φi = £ 5◦ 10◦ 15◦ ¤ T
and
the reference angles Φr = £ 10◦ 15◦ 5◦ ¤ T
.
The settling time for the simulation with the controllers performed in Figure 6.2 to Figure 6.5 can be found
in Table 6.1
Table 6.1. Settling time [sec] Maximum angular velocity, [rad/s]
Controller No disturbance roll pitch yaw
PD 13.10 0.0231 0.0350 0.0385
LQG 13,15 0.0334 0.0195 0.0156
Feedback 8,62 0.0216 0.0264 0.0322
Lyapunov 15,00 0.0188 0.0164 0.0269
The Feedback controller has the fastest settling time which actually is 34.2 % faster than the second fastest
controller, the PD controller. Compared to the Lyapunov controller, the Feedback controller is 42.5 % faster.
The absolute value of the angular velocity with the different controllers is also shown in Table 6.1. The Lyapunov
controller has the lowest angular velocities in roll and pitch,buttheLQGcontrollerhasthelowestangular
velocity in yaw. The Lyapunov controller is also the only controller that does not produce a commanded control
φ
θ
ψ

6 SIMULATION AND RESULTS 60
torque larger than the maximum available torque from the thrusters and the reaction wheel as mentioned earlier.
The LQG controller has been tuned so the yaw angle stays within the reference angle by a margin within ±5 ◦
which again results in a low angular velocity as seen both in Table 6.1 and also from the plot in Figure 6.3.
6.3 Disturbance
For the actuators, a proposed 10 % disturbance is typical, and is also used in (Kristiansen, 2000) for the
NSAT-1 and also estimated by the AOCS team for the ESEO (ESEO phase B document). To compensate
for uncertainties in the mounting of the thrusters and the reaction wheel, and other mechanical disturbance
factors, 20 % disturbance is added on the commanded control torque in the following simulations. Another
20 % disturbance is added to the measurements. The noise is set quite high to evaluate the robustness in the
different control systems. During simulations a random noise maker is used, where the result can be can be
between ±20 %. Several simulations have been perform to see which of the controllers are most robust against
disturbance.
First 20 % disturbance is added on the measurements, second 20 % disturbance is added on the actuators,
and third 20 % disturbance on both the measurements and the actuators. The same simulation time and Euler
angles as the one without disturbances is plotted in this chapter.
6.3.1 Disturbance on the measurements
The step response using the PD controller is shown in Figure 6.6 where again a overshoot in the pitch angle of
2.95 ◦ above the reference occur. It means that an overshot of 1.95 ◦ above the attitude requirements has occurred
which actually is 0.35 ◦ less than without noise in Figure 6.2, resulting in a faster settling by a factor of 2.1%.
The angular velocities and the actuator plot in nearly the same as without disturbance and the measurements.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
0.02
ωx ωy 0
ωz -0.02
-0.04
-0.06
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.6: Step response with 20 % disturbance on the measurements using the PD controller with the initial
angles Φi = £ 5 ◦ 10 ◦ 15 ◦ ¤ T and the reference angles Φr = £ 10 ◦ 15 ◦ 5 ◦ ¤ T .
φ
θ
ψ

6 SIMULATION AND RESULTS 61
The resulting plot using the LQG controller with 20 % disturbance on the measurements is shown in Figure
6.7, where both roll and pitch angles does not stay within the attitude requirements mentioned in Chapter
2. The actuator is used extensively and they reach the maximum limit several times during the simulations
because of the disturbance on the measurement as seen in Figure 6.7.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
ωx 0.02
ωy 0
ωz -0.02
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.7: Step response with 20 % disturbance on the measurements using the LQG controller with the initial
angles Φi = £ 5 ◦ 10 ◦ 15 ◦ ¤ T and the reference angles Φr = £ 10 ◦ 15 ◦ 5 ◦ ¤ T .
φ
θ
ψ

6 SIMULATION AND RESULTS 62
The step response using the Feedback controller is shown in Figure 6.8, where the overshoot in the pitch
angle has increased by a factor of 15 % compared to no disturbance in Figure 6.4. The overshoot results in a
increase of the settling time by a factor of 18.6 % compared to the same controller without disturbances. It can
also be seen from Figure 6.8 that the actuators is used more rapidly trying to counter for the disturbance on
the measurements.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
ωx 0.02
ωy 0
ωz -0.02
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.8: Step response with 20 % disturbance on the measurements using the Feedback controller with the
initial angles Φi = £ 5 ◦ 10 ◦ 15 ◦ ¤ T and the reference angles Φr = £ 10 ◦ 15 ◦ 5 ◦ ¤ T .
φ
θ
ψ

6 SIMULATION AND RESULTS 63
The step response of the Lyapunov controller which is shown in Figure 6.9, shows actual a little decrease in
the pitch overshoot. This results in a faster settling time of 14.55 sec which is 3 % less than without disturbance
on the measurements. The angular velocities is reduced by a factor of 18.1 %, 44,7 % and 16.8 % in roll pitch
and yaw respectively compared to no disturbance in Figure 6.5. The actuator plot is not so smooth as it was
without disturbance in Figure 6.5. Sudden changes in the actuator torque appear during the step response and
as a gives small changes in the angular velocities as seen in Figure 6.9.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.02
0.01
ωx ωy 0
ωz -0.01
-0.02
-0.03
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.9: Step response with 20 % disturbance on the measurements using the Lyapunov controller with the
initial angles Φi = £ 5 ◦ 10 ◦ 15 ◦ ¤ T and the reference angles Φr = £ 10 ◦ 15 ◦ 5 ◦ ¤ T .
The settling time for the simulation with the controllers performed in Figure 6.6 to Figure 6.9 can be found
in Table 6.2
Table 6.2. Settling time [sec]
Controller Measurements disturbance
PD 12.82 (-2.2 %)
LQG -
Feedback 10.60 (+18.6 %)
Lyapunov 14.55 (-3.7 %)
The LQG controller does not settle within the required attitude demands seen in Figure 6.7, hence no settling
time is noted in Table 6.2. The same table also shows the difference in settling time noted in (_) by comparing
the settling time with disturbance on the measurements in Table 6.2 with the settling time without disturbance
in Table 6.1 where ± denotes increased and decreased settling time respectively. The change in settling time is
smallest for the PD controller and largest for the Feedback controller when the LQG controller is not considered
as shown in Table 6.2. The deviation for the PD and Lyapunov controller is very low and nearly the same as
seen in Table 6.2. The absolute value of the angular velocity with disturbances on the measurements is shown
φ
θ
ψ

6 SIMULATION AND RESULTS 64
in Table 6.3.
Table 6.3. Maximum angular velocity, [rad/s]
Controller roll pitch yaw
PD 0.0231 (0 %) 0.0322 (-8 %) 0.0412 (+7 %)
LQG 0.0372 (+11.4 %) 0.0285 (+46 %) 0.0204 (+30.8 %)
Feedback 0.0210 (-2.8 %) 0.0258 (-2.3 %) 0.0353 (+9.6 %)
Lyapunov 0.0177 (-5.9 %) 0.0145 (-11.6 %) 0.0268 (-0.4 %)
The Lyapunov controller has the lowest angular velocities in roll and pitch and the LQG controller in yaw. The
change in maximum angular velocities from the controllers without disturbance to the controller with disturbance
on the measurements is shown as (_) in Table 6.3 where ± denotes increased and decreased angular velocity
respectively. The Feedback controller has the smallest change in angular velocities in roll and pitch and the
Lyapunov controller in yaw. The LQG controller has large changes in the angular velocities because of the
disturbance.
6.3.2 Disturbance on the actuators
The disturbance from the measurements is removed, and instead a 20 % disturbance is added to the actuators.
This might result in either an increase or a decrease of the calculated actuator torque. As seen in Figure 6.10 a
small increase in the use of the actuators has happened compared to the PD controller without any disturbance
in Figure 6.2. This can be confirmed by investigating the power consumption, which is done later in this thesis.
Still all actuators reaches the maximum actuator limit. Because of this increase, the settling time is 0.28 sec
faster than without any noise which is an improvement of 5.5 %. The overshoot in the pitch axis has also
improved and is 9.1 % lower than without disturbance.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
ωx 0.02
ωy 0
ωz -0.02
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.10: Step response with 20 % disturbance on the actuators using the PD controller with the initial
angles Φi = £ 5 ◦ 10 ◦ 15 ◦ ¤ T and the reference angles Φr = £ 10 ◦ 15 ◦ 5 ◦ ¤ T .
φ
θ
ψ

6 SIMULATION AND RESULTS 65
ThestepresponseusingtheLQGcontrollerwithdisturbance on the actuators is much better than when the
disturbance where added to the measurements as seen in Figure 6.11. The attitude settles down within 13.27
seconds which is a small increase of 0.9 % from the response without any disturbance, which might be caused
by the small overshot in the pitch angle. The angular velocities is nearly the same as for the controller without
any disturbances.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
0.02
ωx ωy ωz 0
-0.02
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.11: Step response with 20 % disturbance on the actuators using the LQG controller with the initial
angles Φi = £ 5 ◦ 10 ◦ 15 ◦ ¤ T and the reference angles Φr = £ 10 ◦ 15 ◦ 5 ◦ ¤ T .
φ
θ
ψ

6 SIMULATION AND RESULTS 66
The step response using the Feedback controller shown in Figure 6.12 does not seem to be affected much by
the disturbance on the actuators, only a small decrease in the settling time by a factor of 4.3 % compared to
the plot with no disturbance in Figure 6.4.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
ωx 0.02
ωy 0
ωz -0.02
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.12: Step response with 20 % disturbance on the actuators using the Feedback controller with the initial
angles Φi = £ 5 ◦ 10 ◦ 15 ◦ ¤ T and the reference angles Φr = £ 10 ◦ 15 ◦ 5 ◦ ¤ T .
φ
θ
ψ

6 SIMULATION AND RESULTS 67
The disturbance on the actuators using the Lyapunov controller as shown in Figure 6.13, results in a increase
in the overshoot in the pitch angle. This results in a increase of the settling time of 4.7 % compared to the same
controller with no disturbance as shown in Figure 6.5. The angular velocities is just a little higher than without
any disturbance. The disturbance on the actuators causes the actuators to reach the maximum torque level for
the thrusters Th1 and Th4 which together results in a rotation in yaw. The result of the increased torque from
the two thrusters can be seen for as the overshot in angular velocity about the z-axis which has increased by
9.6 % compared to the same simulation without disturbance.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.02
0.01
ωx ωy 0
ωz -0.01
-0.02
-0.03
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.13: Step response with 20 % disturbance on the actuators using the Lyapunov controller with the
initial angles Φi = £ 5 ◦ 10 ◦ 15 ◦ ¤ T and the reference angles Φr = £ 10 ◦ 15 ◦ 5 ◦ ¤ T .
The settling time for the simulation with the controllers performed in Figure 6.10 to Figure 6.13 can be
found in Table 6.4
Table 6.4. Settling time [sec]
Controller Actuator disturbance
PD 12.38 (-5.5 %)
LQG 13.27 (+0.9 %)
Feedback 8.27 (-4.1 %)
Lyapunov 15.75 (+5 %)
The Feedback controller has the fastest settling time and the Lyapunov has the slowest as shown in Table 6.4.
The settling time in Table 6.4 is compared to the settling time without disturbances in Table 6.1 and the result
is presented in Table 6.4 as (_) where ± denotes increased and decreased settling time respectively. The change
in settling time is small for all the controllers, but the LQG controller has the smallest change as seen in Table
6.4. The disturbance on the actuators results in a faster settling time of 5.5 % and 4.1 % for the PD and
Lyapunov controller respectively. The absolute value of the angular velocity with disturbances on the actuators
φ
θ
ψ

6 SIMULATION AND RESULTS 68
is shown in Table 6.5.
Table 6.5. Maximum angular velocity [rad/s]
Controller roll pitch yaw
PD 0.0231 (0 %) 0.0352 (+0.6 %) 0.0384 (-0.3 %)
LQG 0.0321 (+3.9 %) 0.0205 (+5.1 %) 0.0156 (0 %)
Feedback 0.0230 (+6.5 %) 0.0280 (+6.1 %) 0.0309 (-9.6 %)
Lyapunov 0.0181 (-3.7 %) 0.0160 (-2.4 %) 0.0270 (-0.4 %)
The Lyapunov controller has the lowest angular velocity in roll and pitch and the LQG controller in yaw, the same
case as for no disturbance and disturbance on the measurements. The change in maximum angular velocities
from the LQG controllers without disturbance on the actuators to the LQG controller with disturbance on the
actuators is shown in Table 6.5 as (_), where ± denotes increased and decreased angular velocity respectively.
The change in angular velocity is not very high, and the highest change is for the Feedback controller which
initially had the lowest angular velocity both with and without disturbance on the actuators as seen in Table
6.5 and 6.1 respectively.
6.3.3 Disturbance on the measurements and the actuators
In this section 20 % disturbance is added to both the measurements and the actuators, and the same angles as
previously is used in the plots. The step response has increased in all the Euler angles compared to the response
with no noise (see Figure 6.2) when using the PD controller as shown in Figure 6.14. The overshoot in pitch
angle is approximately 3.6 ◦ over the reference angle of 15 ◦ , which increases the settling time to 5 % more than
without any disturbance.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
ωx 0.02
ωy 0
ωz -0.02
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.14: Step response with 20 % disturbance on the measurements and the actuators using the PD controller
with the initial angles Φi = £ 5 ◦ 10 ◦ 15 ◦ ¤ T and the reference angles Φr = £ 10 ◦ 15 ◦ 5 ◦ ¤ T .
φ
θ
ψ

6 SIMULATION AND RESULTS 69
The attitude using the LQG controller behaves similar to when disturbance where added to the measurements,
where the attitude travels outside the accuracy limit of Φ = £ 1 ◦ 1 ◦ 5 ◦ ¤ T as shown in Figure 6.15.
The commanded output torque is higher than the available torque from the actuators as seen from the actuator
plot in Figure 6.15. This result in high angular velocities and sudden changes as seen in Figure 6.15.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
0
Attitude angles [deg]
-5
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
ωx 0.02
ωy 0
ωz -0.02
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.15: Step response with 20 % disturbance on the measurements and the actuators using the LQG
controller with the initial angles Φi = £ 5 ◦ 10 ◦ 15 ◦ ¤ T and the reference angles Φr = £ 10 ◦ 15 ◦ 5 ◦ ¤ T .
φ
θ
ψ

6 SIMULATION AND RESULTS 70
The step response using the Feedback controller is shown in Figure 6.16, where there is only a small overshot
of 0.4 ◦ in the pitch angle. This results in an improvement of 4.3 % in the settling time compared to the attitude
with no noise shown in Figure 6.4. There is more variation in the angular velocities using the Feedback controller
with disturbance on the measurements and the actuators in Figure 6.16 than it is without disturbances as shown
in Figure 6.4. This variation is probably due to the actuators is used more as seen in Figure 6.16.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
ωx 0.02
ωy 0
ωz -0.02
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.16: Step response with 20 % disturbance on the measurements and the actuators using the Feedback
controller with the initial angles Φi = £ 5 ◦ 10 ◦ 15 ◦ ¤ T and the reference angles Φr = £ 10 ◦ 15 ◦ 5 ◦ ¤ T .
φ
θ
ψ

6 SIMULATION AND RESULTS 71
The settling time when using the Lyapunov controller as shown in Figure 6.17 is approximately the same as
the settling time without noise as seen in Figure 2.49. The output of the Euler angles is looks almost the same
as for the angles with no noise. There is only small changes in the actuator plot where thruster Th1 reaches the
maximum allowed torque for a short period of time.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
ωx 0.02
ωy 0
ωz -0.02
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.17: Step response with 20 % disturbance on the measurements and the actuators using the Lyapunov
controller with the initial angles Φi = £ 5 ◦ 10 ◦ 15 ◦ ¤ T and the reference angles Φr = £ 10 ◦ 15 ◦ 5 ◦ ¤ T .
The settling time for the simulation with the controllers performed in Figure 6.14 to Figure 6.17 can be
found in Table 6.6
Table 6.6. Settling time [sec]
Controller No disturbance
PD 13.79 (+5.3 %)
LQG -
Feedback 8.25 (-4.3 %)
Lyapunov 15.00 (0 %)
The Feedback controller still has the fastest settling time and the Lyapunov controller has the slowest as seen in
Table 6.6. The settling time in Table 6.6 is compared to the settling time without disturbances in Table 6.1 and
the result is presented as (_) in Table 6.6 where ± denotes increased and decreased settling time respectively.
There is no major change in the settling time except the LQG controller which does not settle within the
required attitude demands. The settling time for the Lyapunov controller is exactly the same with disturbance
on the measurements and actuators as it is without any disturbances as in Table 6.6, 15 sec. The absolute value
of the maximum angular velocity with disturbances on the measurements and the actuators is shown in Table
φ
θ
ψ

6 SIMULATION AND RESULTS 72
6.7.
Table 6.7. Maximum angular velocity [rad/s]
Controller roll pitch yaw
PD 0.0236 (+2.2 %) 0.0325 (-7.1 %) 0.0393 (+2.1 %)
LQG 0.0282 (+15.6 %) 0.0247 (+26.7 %) 0.0170 (+9 %)
Feedback 0.0221 (+2.3 %) 0.0250 (-3.8 %) 0.0280 (-13 %)
Lyapunov 0.0202 (+7.4 %) 0.0167 (+1.8 %) 0.0240 (-10.8 %)
The Lyapunov controller has the lowest angular velocity in roll and pitch and the LQG controller in yaw as seen
in Table 6.7, the same case as from Table 6.3 and Table 6.5. The change in maximum angular velocities from
the controllers without disturbance on the actuators to the controller with disturbance on the measurements is
shown as (_) in Table 6.14, where ± denotes increased and decreased angular velocity respectively. The LQG
controller has the most change in angular velocity due to the fact that the attitude does not stay within the
accuracy demands because of the disturbance. The Feedback controller has a rather big change of -13 % of the
angular velocity in yaw, but this is not so crucial since the angular velocity is lower than without noise. The
Lyapunov controller has some change of the angular velocity in roll and yaw as seen in Table 6.7, but still the
maximum absolute value of angular velocity is lower than for any other of the controllers as seen in Table 6.7
except the LQG controller in yaw.
6.4 Summary of results
This chapter summarizes some of the simulation results in Tables in addition to some other simulations which
can be found in appendix B. The change in settling time from previous chapter with initial and reference angles
Φi = £ 5 ◦ 10 ◦ 15 ◦ ¤ T and Φr = £ 10 ◦ 15 ◦ 5 ◦ ¤ T respectively, is shown in Table 6.8. The table makes it
is easier to compare the robustness of the controllers with respect to settling time.
Table 6.8. Change in settling time, Φi = £ 5 ◦ 10 ◦ 15 ◦ ¤ T ,Φr = £ 10 ◦ 15 ◦ 5 ◦ ¤ T [%]
Controller No noise [sec] Measurement noise Actuator noise Angle and Actuator noise
PD 13.10 -2.2 -5.5 +5.3
LQG 13.15 - +0.9 -
Feedback 8.62 +18.6 -4.1 -4.3
Lyapunov 15.00 -3.7 +5 0
The Feedback controller has the fastest settling time without disturbances as seen in Table 6.8, but the settling
time increases with 18.6 % when the controller is suppressed with disturbances on the measurements. In spite
of the increased settling time, the Feedback controller is still the fastest controller with disturbances on the
measurements. There is only a small change in the settling time when using the Lyapunov controller as seen in
Table 6.8, where the maximum deviation is 5 %. The LQG controller is quite influenced by the disturbances
and therefore does not stabilize around the reference angles. In Table 6.9 to Table 6.11 is a summary of the
change in settling time for the same controllers but with other initial and reference angles. The other plots of
the settling time without noise performed in this chapter can be found in appendix B.
Table 6.9. Change in settling time, Φi = £ 10 ◦ 10 ◦ 10 ◦ ¤ T ,Φr = £ 0 ◦ 0 ◦ 0 ◦ ¤ T [%]
Controller No noise [sec] Measurement noise Actuator noise Measurement and actuator noise
PD 14.39 +8.6 -5.2 -2.2
LQG 17.48 - -2.7 -
Feedback 11.50 -7 -0.3 +10.8
Lyapunov 16.05 +4.1 +2.8 -2.9

6 SIMULATION AND RESULTS 73
In Table 6.9 where the initial and reference angles is Φi = £ 10 ◦ 10 ◦ 10 ◦ ¤ T and Φr = £ 0 ◦ 0 ◦ 0 ◦ ¤ T
respectively, the Feedback control has the fastest settling time. The disturbance has the same effect on the
LQG controller as before where the attitude does not stabilize round the reference angles as seen in Table 6.9.
The Lyapunov controller has the second longest settling time and it still has the smallest change in settling
time, in fact an improvement of 7 % and 0.5 % occur with disturbance on the measurements and the actuators
respectively as seen in Table 6.9. The maximum deviation in settling time for the Feedback controller is 10.8
% when disturbance is added to the measurements and the actuators. The change in settling time in Table 6.9
for the PD controller is almost the same as it is in Table 6.8.
Table 6.10. Change in settling time, Φi = £ 8 ◦ −6 ◦ 9 ◦ ¤ T ,Φr = £ 3 ◦ 1 ◦ 5 ◦ ¤ T [%]
Controller No noise [sec] Measurement noise Actuator noise Measurement and actuator noise
PD 7.61 -2.5 -2 +8.7
LQG 19.02 - -31.1 -
Feedback 9.72 -1 -0.6 +1.4
Lyapunov 8.15 -2.8 +2.5 +2.3
In Table 6.10 the initial and reference angles is changed to Φi = £ 8 ◦ −6 ◦ 9 ◦ ¤ T and Φr = £ 3 ◦ 1 ◦ 5 ◦ ¤ T
respectively, and where the PD controller has the fastest settling time without any disturbances. The deviation
in settling time when adding noise is generally very low and the maximum deviation when not considering the
LQG controller occur with the use of the PD controller with 8.7 % increased settling with disturbance on the
measurements and the actuators as shown in Table 6.10.
Table 6.11. Change in settling time, Φi = £ 5 ◦ −5 ◦ 0 ◦ ¤ T ,Φr = £ −5 ◦ 5 ◦ 10 ◦ ¤ T [%]
Controller No noise [sec] Measurement noise Actuator noise Measurement and actuator noise
PD 15.47 -4.6 +1.7 0
LQG 21.00 - - -
Feedback 12.00 -6.7 -2.3 -3.9
Lyapunov 16.81 +1.7 +0.5 -1
In Table 6.11 the Feedback controller has the fastest settling time with the initial and reference angles Φi =
£ 5 ◦ −5 ◦ 0 ◦ ¤ T and Φr = £ −5 ◦ 5 ◦ 10 ◦ ¤ T respectively. The same occur as before where the LQG controller
has problems with reaching the reference angles with disturbances with the initial and reference angles
Φi = £ 5 ◦ −5 ◦ 0 ◦ ¤ T and Φr = £ −5 ◦ 5 ◦ 10 ◦ ¤ T as shown in Table 6.11. The increase in settling time
for the different controllers except the LQG controller is quite small as they also where in Table 6.10. The
Lyapunov controller only has a maximum increase of the settling time of 1.7 %.
An average settling time for the simulations performed in this chapter without any disturbances is shown
in Table 6.12 where the Feedback controller has the fastest settling time and the LQG controller the slowest
settling time.
Table 6.12. Average settling time [sec]
Controller No noise
PD 12.64
LQG 17.66
Feedback 10.46
Lyapunov 14.00

6 SIMULATION AND RESULTS 74
The Feedback controller has the fastest average settling time as seen in Table 6.12. The average settling time
increases when using the PD, LQG and Lyapunov controller with 17.2 %, 40.8 %, and 25.3 % respectively
compared to the Feedback controller.
It is possible to see how the settling time change for each of the controllers with respect to the inertial and
reference angles. In Table 6.13 is the variation of the settling time with four different initial and reference angles
shown for the PD controller. The settling time without any disturbance is nearly the same except for the initial
and reference angles Φi = £ 8 ◦ −6 ◦ 9 ◦ ¤ T and Φr = £ 3 ◦ 1 ◦ 5 ◦ ¤ T which has a much faster settling time.
The variation in settling time when adding noise is not very high and does not get higher than 8.7 % compared
to the settling time without disturbance.
Table 6.13. Variation in settling time using the PD controller [%]
Initial and reference angles [ ◦ ] No noise [sec] Measurement noise Actuator noise M. & A noise
(5, 10, 15) − (10, 15, 5) 13.10 -2.2 -5.5 +5.3
(10, 10, 10) − (0, 0, 0) 14.39 +8.6 -5.2 -2.2
(8, −6, 9) − (3, 1, 5) 7.61 -2.5 -2 +8.7
(5, −5, 0) − (−1, 5, 10) 15.47 -4.6 +1.7 0
In Table 6.14 is the variation of the settling time with four different initial and reference angles shown for the
LQG controller. It is very hard to do any analyses the settling time with respect to disturbances because the
LQG controller has problems with stabilizing round the reference angles. But without noise the LQG controller
has some variation in the settling time as seen in Table 6.14.
Table 6.14. Variation in settling time using the LQG controller [%]
Initial and reference angles [ ◦ ] No noise [sec] Measurement noise Actuator noise M. & A noise
(5, 10, 15) − (10, 15, 5) 13.15 - +0.9 -
(10, 10, 10) − (0, 0, 0) 17.48 - -2.7 -
(8, −6, 9) − (3, 1, 5) 19.02 - -31.1 -
(5, −5, 0) − (−1, 5, 10) 21.00 - - -
In Table 6.15 is the variation of the settling time with four different initial and reference angles shown for
the Feedback controller. The settling time using the Feedback controller with the initial and reference angles
Φi = £ 5 ◦ 10 ◦ 15 ◦ ¤ T and Φr = £ 10 ◦ 15 ◦ 5 ◦ ¤ T results in a increased settling time of 18.6 % with
disturbance on the measurements as shown in Table 6.15. Also a large increase in settling time occur with
disturbance on the measurements and the actuators of 10.8 % when using the initial and reference angles is
Φi = £ 10 ◦ 10 ◦ 10 ◦ ¤ T and Φr = £ 0 ◦ 0 ◦ 0 ◦ ¤ T respectively. Except for these two deviations, there is
only some small changes in settling time as seen in Table 6.15.
Table 6.15. Variation in settling time using the Feedback controller [%]
Initial and reference angles [ ◦ ] No noise [sec] Measurement noise Actuator noise M. & A noise
(5, 10, 15) − (10, 15, 5) 8.62 +18.6 -4.1 -4.3
(10, 10, 10) − (0, 0, 0) 11.50 -7 -0.3 +10.8
(8, −6, 9) − (3, 1, 5) 9.72 -1 -0.6 +1.4
(5, −5, 0) − (−1, 5, 10) 12.00 -6.7 -2.3 -3.9
In Table 6.16 is the variation of the settling time with four different initial and reference angles shown for the
Lyapunov controller. The Lyapunov controller is the controller with least variation in settling time when adding

6 SIMULATION AND RESULTS 75
noise to eater the measurements or the actuators where the maximum deviation is with noise on the actuators
using the initial and reference angles Φi = £ 5 ◦ 10 ◦ 15 ◦ ¤ T and Φr = £ 10 ◦ 15 ◦ 5 ◦ ¤ T respectively as
shown in Table 6.16.
Table 6.16. Variation in settling time using the Lyapunov controller [%]
Initial and reference angles [ ◦ ] No noise [sec] Measurement noise Actuator noise M. & A noise
(5, 10, 15) − (10, 15, 5) 15.00 -3.7 +5 0
(10, 10, 10) − (0, 0, 0) 16.05 +4.1 +2.8 -2.9
(8, −6, 9) − (3, 1, 5) 8.15 -2.8 +2.5 +2.3
(5, −5, 0) − (−1, 5, 10) 16.81 +1.7 +0.5 -1
6.5 Bang Bang controller
This chapter illustrates the use of a Bang Bang controller and the resulting behavior of the actuators. The step
response using the PD controller together with the Bang Bang controller is shown in Figure 6.18, where the
simulation time is 1
200 orbit to illustrate the deviation of the Euler angles7 . The thrusters act as on of devices
while the reaction wheel is continuous as seen in the actuator plot in Figure 6.18. The settling time is 12.03
sec which actually is 8.2 % faster than without the Bang Bang controller. The attitude is oscillating around
the reference but does not exceed the attitude accuracy demands. The value of the angular velocities is nearly
the same except there is much higher sudden changes compared to the PD controller without the Bang Bang
controller.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 50 100 150 200 250
Angular velocities [rad/s]
0.04
ωx 0.02
ωy 0
ωz -0.02
-0.04
0 50 100 150 200 250
Thrusters [N], Reaction wheel [Nm]
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 50 100 150 200 250
sec
Figure 6.18: Step response using the PD controller together with the Bang Bang controller with the initial
angles Φi = £ 5 10 15 ¤ T and the reference angles Φr = £ 10 15 5 ¤ T .
7 More extensive use of a similar Bang Bang controller for the ESEO can be found in (Blindheim, 2004).
φ
θ
ψ

6 SIMULATION AND RESULTS 76
6.6 Momentum dumping controller
A simulation of the momentum dumper used together with the PD controller is shown in Figure 6.19 with the
initial and reference angles 0 ◦ in roll, pitch, and yaw. The PD controller only uses the thrusters as actuators
but the reaction wheel is used as a fifth actuator when adding the momentum dumper. The initial speed of
the reaction wheel is set to the maximum speed of 5000 rpm and after 20 sec the momentum dumper controller
is activated so that the speed of the reaction wheel decreases to zero as seen in Figure 6.19. The momentum
dumping controller is tuned so it dumps 50 % of the angular momentum of the reaction wheel pr second, so it
is easier to see the decreasing wheel speed in Figure 6.19. The sudden change in the actuator plot is easily seen
where the thrusters is compensating for the torque produced by the decreasing wheel speed.
Velocity
RPY angles [deg]
Angular velocity [rad/s]
1
0
Attitude angles [deg]
-1
0 10 20 30
Angular velocities [rad/s]
40 50 60
1
ωx ωy 0
ωz -1
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.05
0
Th1
Th2
Th3
Th4
h
Actuators
-0.05
0 10 20 30 40 50 60
Reaction wheel velocivy [rpm]
6000
w
4000
2000
0
0 10 20 30
sec
40 50 60
Figure 6.19: PD controller together with the momentum dumping controller with the initial and reference angles
0 ◦ in roll, pitch, and yaw.
6.7 Wheel disturbance compensation
The simulations done previously did not compensate for the disturbance Da from the reaction wheel. The step
response using the PD controller with wheel compensation deduced in (5.15) is shown in Figure 6.20. The
settling time is the same as for the PD controller without wheel compensation, 12.82 sec, hence the disturbance
Da from the reaction wheel has little or no effect on the step response.
φ
θ
ψ

6 SIMULATION AND RESULTS 77
Angular velocity [rad/s]
Actuators
RPY angles [deg]
20
15
10
5
Attitude angles [deg]
0
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
ωx 0.02
ωy 0
ωz -0.02
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure 6.20: Step response with using the PD controller with wheel disturbance compensation, initial angles
Φi = £ 5 10 15 ¤ T and reference angles Φr = £ 10 15 5 ¤ T .
6.8 Power consumption
The power consumption during simulation is logged, and is used to investigate the different controllers. The
input power to the satellite can be obtained as
³ ¡TB P = abs
¢T
´
B
ωBO (6.1)
where TB is the actuator torque, and ωB BO is the angular velocity of the satellite between the body and the
orbit frame represented in the body frame. The input power can be a indication of how much the actuators are
being used.
The nonlinear controllers have a very low power consumption compared to the linear controllers, in which
the Lyapunov has the lowest power consumption as seen in Table 6.17 where the reference and initial angles is
Φi = £ 5 10 15 ¤ T and Φr = £ 10 15 5 ¤ T respectively. As mentioned previously, the LQG controller did
not stay within the required attitude demands with disturbances on the measurements, hence the controller also
has high power consumption as seen in Table 6.17. The variation in power consumption in shown as (_) where
± denotes increased and decreased power consumption respectively in Table 6.17. The power consumption
for the LQG controller in Table 6.17 is very high with disturbances because the controller has problems with
tracing the reference angles as seen in Figure 6.7 and Figure 6.15. The Lyapunov controller has rather small
variations in the power consumption and the small variation that occur is in fact a reduction of up to 12 % from
the initial power consumption without disturbances as shown in Table 6.17. Except for the LQG controller, the
Feedback controller has the highest variation in power consumption (30.3 %), but as shown in Figure 6.12, 6.8
and 6.16 the actuators do not reach the maximum torque available for the actuators in spite of the increased
φ
θ
ψ

6 SIMULATION AND RESULTS 78
power consumption.
Table 6.17. Power consumption, Φi = £ 5 10 15 ¤ T ,Φr = £ 10 15 5 ¤ T [W]
Controller No noise Measurement noise Actuator noise Measurement and Actuator noise
PD 0.0100 0.0091 (-9.0 %) 0.0111 (+11.0 %) 0.0117 (+17.0 %)
LQG 0.0080 0.0235 (+194.0 %) 0.0078 (-2.5 %) 0.0278 (+228.0 %)
Feedback 0.0066 0.0074 (+12.1 %) 0.0086 +30.3 %) 0.0078 (+18.2 %)
Lyapunov 0.0046 0.0044 (-4.3 %) 0.0046 (0.0 %) 0.0040 (-13.0 %)
With the initial and reference angles Φi = £ 10 10 10 ¤ T and Φr = £ 0 0 0 ¤ T respectively as used in
Table 6.18, the Lyapunov controller again has the lowest power consumption with small variation when adding
disturbance. The Feedback controller also uses a small amount of power compared to the linear controllers. The
variation in power consumption with disturbances is shown as (_) where ± denotes increased and decreased
power consumption respectively in Table 6.18. The Feedback controller has increased the power consumption
with 30.6 % with disturbance on the measurements compared to no disturbance as shown in Table 6.18.
Table 6.18. Power consumption, Φi = £ 10 10 10 ¤ T ,Φr = £ 0 0 0 ¤ T [W]
Controller No noise Measurement noise Actuator noise Measurement and Actuator noise
PD 0.0210 0.0212 (+1.0 %) 0.0179 (-14.8 %) 0.0189 (-10 %)
LQG 0.0234 0.0196 (-16.2 %) 0.0249 (+6.4 %) 0.0227 (-3.0 %)
Feedback 0.0124 0.0162 (+30.6 %) 0.0128 (+3.2 %) 0.0136 (+9.7 %)
Lyapunov 0.0087 0.0083 (-4.6 %) 0.0089 (+2.3 %) 0.0085 (-2.3 %)
All the controllers’ except the LQG controller has very low power consumption in Table 6.19 with the initial
and reference angles Φi = £ 8 −6 9 ¤ T and Φr = £ 3 1 5 ¤ T respectively. The Lyapunov controller has
the lowest power consumption, closely followed by the Feedback and PD controller respectively. The variation
in power consumption shown as (_) where ± denotes increased and decreased power consumption respectively
in Table 6.19. The smallest overall variation is with the Lyapunov controller except when disturbances is added
to the measurements. The other controllers has larger variations in the power consumption where the PD
controller has variations with disturbances from -17.3 % to 21.2 % compared to the same controller without
disturbances.
Table 6.19. Power consumption, Φi = £ 8 −6 9 ¤ T ,Φr = £ 3 1 5 ¤ T [W]
Controller No noise Measurement noise Actuator noise Measurement and Actuator noise
PD 0.0052 0.0056 (+7.7 %) 0.0063 (+21.2 %) 0.0043 (-17.3 %)
LQG 0.0107 0.0123 (+15.0 %) 0.0115 (+7.5 %) 0.0114 (+6.5 %)
FL 0.0039 0.0041 (+5.1 %) 0.0044 (+12.8 %) 0.0046 (+18.0 %)
Lyapunov 0.0034 0.0038 (+11.8 %) 0.0034 (0.0 %) 0.0035 (+2.9 %)
An increased power consumption has occurred in Table 6.20 where the initial and reference angels is Φi =
£ ¤ T
5 −5 0 and Φr = £ −5 5 10 ¤ T
respectively compared to Table 6.19 where the initial and reference
angels is Φi = £ 8 −6 9 ¤ T
and Φr = £ −3 1 5 ¤ T
respectively. The Lyapunov controller is still the
controller with lowest power consumption, followed by the Feedback, PD and LQG controller respectively. It is
easier to see the changes in power consumption shown as (_) where ± denotes increased and decreased power
consumption respectively in Table 6.20. For both the PD and Lyapunov controller there occur an decrease in the
power consumption when adding disturbances whereas the Lyapunov controller has the highest reduction. The

6 SIMULATION AND RESULTS 79
LQG controller has a huge increase in the power consumption because the LQG controller has problems tracing
the reference with disturbances. The Feedback controller has a 33.1 % increase in the power consumption with
disturbances on the measurements and 15.7 % increase with disturbance on the actuators, but when disturbance
is added to the measurements and the actuators at the same time there is no change as seen tin Table 6.20.
Table 6.20. Power consumption, Φi = £ 5 −5 0 ¤ T ,Φr = £ −5 5 10 ¤ T [W]
Controller No noise Measurement noise Actuator noise Measurement and Actuator noise
PD 0.0236 0.0229 (-3.0 %) 0.0218 (-7.6 %) 0.0226 (-4.2 %)
LQG 0.0236 0.0358 (+52 %) 0.0242 (+2.5 %) 0.0316 (133.9 %)
Feedback 0.0127 0.0169 (+33.1 %) 0.0147 (+15.7 %) 0.0127 (0.0 %)
Lyapunov 0.0091 0.0086 (-5.5 %) 0.0085 (-6.6 %) 0.0082 (-9.9 %)
The variation of the power consumption with four different initial and reference angles for the all controller is
showninTable6.2. When looking at Table 6.21 it is quite clear that the Lyapunov controller has the lowest
power consumption and the variations in the power consumption is overall much lower than for any of the
other controllers. The LQG controller has the overall highest power consumption and the variation in power
consumption is very high due to the disturbances. It is basically only the Lyapunov controller that show a
pattern in the power consumption with different initial and reference angles. This is very easy seen in Table
6.22 where the average power consumption actually is lower with disturbances than without disturbances for
the Lyapunov controller. This is also the case for the PD controller which has an average power consumption
with disturbances that is lower than without, but the average power consumption without disturbance decreases

6 SIMULATION AND RESULTS 80
with 56.7 % when using the Lyapunov controller instead of the PD controller.
Table 6.21. Variation in power consumption [%]
Initial and reference angles No noise Measurement noise Actuator noise M. & A noise
PD controller
(5 ◦ , 10 ◦ , 15) − (10 ◦ , 15 ◦ , 5 ◦ ) 0.0100 -9.0 +11.0 +17.0
(10 ◦ , 10 ◦ , 10 ◦ ) − (0 ◦ , 0 ◦ , 0 ◦ ) 0.0210 +1.0 -14.8 -10
(8 ◦ , −6 ◦ , 9 ◦ ) − (3 ◦ , 1 ◦ , 5 ◦ ) 0.0052 +7.7 +21.2 -17.3
(5 ◦ , −5 ◦ , 0 ◦ ) − (−1 ◦ , 5 ◦ , 10 ◦ ) 0.0236 -3.0 -7.6 -4.2
LQG controller
(5 ◦ , 10 ◦ , 15) − (10 ◦ , 15 ◦ , 5 ◦ ) 0.0080 +194.0 -2.5 +228.0
(10 ◦ , 10 ◦ , 10 ◦ ) − (0 ◦ , 0 ◦ , 0 ◦ ) 0.0234 -16.2 +6.4 -3.0
(8 ◦ , −6 ◦ , 9 ◦ ) − (3 ◦ , 1 ◦ , 5 ◦ ) 0.0107 +15.0 +7.5 +6.5
(5 ◦ , −5 ◦ , 0 ◦ ) − (−1 ◦ , 5 ◦ , 10 ◦ ) 0.0236 +52 +2.5 133.9
Feedback controller
(5 ◦ , 10 ◦ , 15) − (10 ◦ , 15 ◦ , 5 ◦ ) 0.0066 +12.1 +30.3 +18.2
(10 ◦ , 10 ◦ , 10 ◦ ) − (0 ◦ , 0 ◦ , 0 ◦ ) 0.0124 +30.6 +3.2 +9.7
(8 ◦ , −6 ◦ , 9 ◦ ) − (3 ◦ , 1 ◦ , 5 ◦ ) 0.0039 +5.1 +12.8 +18.0
(5 ◦ , −5 ◦ , 0 ◦ ) − (−1 ◦ , 5 ◦ , 10 ◦ ) 0.0127 +33.1 +15.7 0.0
Lyapunov controller
(5 ◦ , 10 ◦ , 15) − (10 ◦ , 15 ◦ , 5 ◦ ) 0.0046 -4.3 0.0 -13.0
(10 ◦ , 10 ◦ , 10 ◦ ) − (0 ◦ , 0 ◦ , 0 ◦ ) 0.0087 -4.6 +2.3 -2.3
(8 ◦ , −6 ◦ , 9 ◦ ) − (3 ◦ , 1 ◦ , 5 ◦ ) 0.0034 +11.8 0.0 +2.9
(5 ◦ , −5 ◦ , 0 ◦ ) − (−1 ◦ , 5 ◦ , 10 ◦ ) 0.0091 -5.5 -6.6 -9.9
The average and change in power consumption for all the controllers can be summarized as shown in Table
6.22 where the same pattern as before occur where the Lyapunov controller has the lowest power consumption
and also the lowest variation.
Table 6.22. Average power consumption, [W]
Controller No noise Measurement noise Actuator noise Angle and Actuator noise
PD 0.0150 0.0147 (-2.0 %) 0.0143 (-4.7 %) 0.0144 (-4.0 %)
LQG 0.0203 0.0228 (+12.3 %) 0.0171 (-15.8 %) 0.0234 (+15.3 %)
Feedback 0.0089 0.0112 (+25.4 %) 0.0101 (+13.5 %) 0.0097 (+9.0 %)
Lyapunov 0,0065 0,0063 (-3.1 %) 0,0064 (-1.5 %) 0,0061 (-6.2 %)

6 SIMULATION AND RESULTS 81
6.9 Reference trajectory
Some of the controllers in this chapter suffer from large angular velocities. The sensor for attitude determination
will not be applicable for such high angular velocities (Kyrkjebø, 2000). It is therefore desirable to keep the
angular velocities of the satellite as low as possible. A reference trajectory as deduced in chapter 4.5, can be
used to slow down the response and angular velocity of the satellite. A plot of the PD controller where a
reference trajectory is used is shown in Figure 6.21. The settling time is approximately 304 sec and the power
consumption during a simulation period of 1
50 orbitisaslowas6.5785E−6, which is reduction by a factor of
3192 compared to without the reference trajectory.
RPY angles [deg]
Angular velocity [rad/s]
Actuators
15
10
5
0
Attitude angles [deg]
-5
0 100 200 300 400 500 600 700 800 900
x 10-4
5
Angular velocities [rad/s]
0
ωx ωy ωz -5
-10
0 100 200 300 400 500 600 700 800 900
x 10-4
4
2
0
Thrusters [N], Reaction wheel [Nm]
-2
0 100 200 300 400 500 600 700 800 900
sec
Figure 6.21: Step response using the PD controller with the reference trajectory, initial angles 10 degree and
reference angles 0 degree .
φ
θ
ψ
Th1
Th2
Th3
Th4
h

7 DISCUSSION 82
7 Discussion
7.1 Theoretical analysis
The satellite is regarded as a rigid body and where the dynamical model of the satellite is deduced in Chapter
3.8. Two different approaches for stability analysis is performed. A stability analysis on the linearized satellite
is performed and showed that the satellite is unstable in the desired equilibrium point RO B = 1 because of the
relations between the satellite moments of inertia. The stability analysis of the nonlinear model encountered
aprobleminfinding a proper Lyapunov candidate function where the equilibrium point RO B = 1 could be
investigated. However a new Lyapunov candidate function where deduced to satisfy the moments of inertia and
in which the stability analysis of the nonlinear model resulted in four equilibrium points, where non of them
where the desired equilibrium point RO B = 1 as shown in (4.79). From this it is clear that the linearized model
is unstable in the desired equilibrium point RO B = 1, but no conclusion regarding this equilibrium point can be
drawn from the stability analysis of the nonlinear model which only found four other equilibrium points. The
satellite will not settle during simulation at one of the equilibrium point, but would probably stay within close
range of an equilibrium point.
Since the satellite it self is nonlinear, the nonlinear controllers will generally handle such a system better
than the linear controllers. This is because nonlinear controllers can handle nonlinearity in large range operation
directly while linear controllers are likely to perform very poorly or to be unstable. Nonlinear controller
might be design to guarantee global stability, improved efficiency and increased control performance for such
systems.
7.2 Simulations
The discretization method where suppose to be Runge Kutta or in Matlab ODE45, but during simulations with
ODE45, the thrusters were firing rapidly up to 20 times pr second. This is of course not possible with the
thrusters, but due to the lack of time, this problem was never solved. Therefore all simulations where done with
the less stable Euler integration.
A worst case scenario that might happen when using the Euler integration is that a simulation performed
appears unstable, but when performing the same simulation with Runge Kutta the simulation is in fact stable.
This is something that might occur as mentioned in Chapter 3.9.2. By introducing the Matlab Runge Kutta
method ODE45 of fourth order one can eliminate the uncertainties of using the Euler integration. In this thesis
it would probably not be any problem when using the Euler integration because the sampling frequency is not
so high. The simulation of the unactuated satellite shown in Figure 6.1 is in fact simulated with ODE45 because
it did not result in any problems since the thrusters is not used in the unactuated satellite.
7.2.1 Unactuated satellite
The simulation of the unactuated satellite in Figure 6.1 showed that the satellite is unstable about the pitch
axis and stable about the roll and yaw axis. The results from the simulations substantiates the results from
the theoretical analysis of the linearized satellite model which also concluded with an unstable satellite in the
equilibrium point RO B = 1. From the simulation results and the theoretical analysis the satellite is most likely
unstable in the equilibrium point RO B = 1. The stability analysis of the nonlinear satellite found four equilibrium
points where the satellite is uniformly stable, but these points were not found during simulations.

7 DISCUSSION 83
7.2.2 Step response
The results from the step response without any disturbances showed that the Feedback controller had the
fastest settling time for all rotations except with the initial and reference angles Φi = £ 8 −6 9 ¤ T
and
Φr = £ 3 1 5 ¤ T
respectively where the PD controller is the fastest as shown in Table 6.8 to Table 6.11.
From the average settling time without disturbances in Table 6.12 it is shown below the table that the Feedback
controller is 17.2 %, 40.8 %, and 25.3 % faster than the PD, LQG and Lyapunov controller respectively.
The angular velocity is much lower for the Lyapunov controller than for any of the other controllers and it
also has a longer settling time as seen in Table 6.1. The Lyapunov controller is the only controller where the
commanded output torques does not get higher than the available torque from the actuators as seen in Figure
6.5. This is an advantage for the controller which means that there is still available torque if any major disturbance
should occur.
All of controllers except the LQG controller had either a small or larger overshot in the pitch angle as seen in
Figures 6.2, 6.4, and 6.5 which might be caused by the extra actuator in form of the reaction wheel. The LQG
controller does not have the same overshoot as the other controllers because of tuning of the LQG controller.
The tuning of the LQG controller also results in a slow settling time for the yaw angle as seen in Figure 6.3
because the attitude demands for the yaw axis is lower than for the roll and pitch axis.
7.2.3 Disturbance
The LQG controller is not very robust against disturbances and is in fact unstable for several different rotations
as shown in Table 6.14. The PD and Lyapunov controller seems to be the controllers which are most robust
against disturbances because they have least change in settling time with disturbances as shown in Table 6.13
to Table 6.16. There are only small differences in the variation in settling time between the PD and Lyapunov
controllers with the use of different initial and reference angles as shown in Table 6.13 to Table 6.16. It is hard
to say which of the two most robust controllers is the most robust because of the small changes in settling
time. The PD controller has surly the fastest settling time, but then again the Lyapunov controller seldom
has a output torque that exceeds the maximum torque limit in which might be one of the reason for that the
Lyapunov controller is so robust and still capable of tracking the reference with disturbances. It is likely to
believe that the other controllers probably can be adjusted so the output torque does not exceed the actual
torque limit, but it would probably entail a slower response.
The Feedback controller still has the fastest settling time even with disturbances, but the change in settling time
with disturbances is generally higher than for the PD and Lyapunov controller except for one rotation as seen
in Table 6.15. It is when using the initial and reference angles Φi = £ 8 −6 9 ¤ T and Φr = £ 3 1 5 ¤ T
respectively that the Feedback controller is the most robust controller as seen in Table 6.15.
All the controllers have been tuned to have the best possible settling time, but it might be possible to tune them
even better then they already are. An LQG controller is normally a very good control strategy to use, but it
should be mentioned that such a controller requires heavy computations over a finite period of time, hence an
LQG controller is normally not used to control a satellite. This is not the case for any of the other controllers.

7 DISCUSSION 84
7.2.4 Bang Bang controller
A simulation of the Bang Bang controller together with the PD controller illustrated the use of the Bang Bang
controller. The attitude stays within the attitude accuracy demands as shown in Figure 6.18. The actuators are
used rapidly to track the reference and the power consumption is naturally much higher than with continuous
thrust. It was initially intended to use the Bang Bang controller during all simulations, but due to different
deadzones required by the attitude controllers it was decided not to use the Bang Bang controller. If it had been
possible to use the same deadzones for all attitude controller the Bang Bang controller would have been used.
If PWPFM have been used in stead of the Bang Bang controller, it would result in a smoother response of the
attitude by modulating the width of the activated pulse proportional to the level of torque from the attitude
controller. The PWPFM control will also take care of the problem that arises with odd or multiple thruster
firing which may result in vibrations of the satellite.
7.2.5 Momentum dumping controller
The momentum dumping controller insures that the attitude does not change when an acceleration or deceleration
of the reaction wheel is performed as seen in Figure 6.19. The controller removes the initial momentum
of 0.0209 Nm without disturbing the attitude controller, which in this case is the PD controller. It should be
mentioned that the attitude controller, in this case the PD controller used together with the momentum dumping
controller only uses the thrusters as actuators and therefore the response might be different than when the
reaction wheel is included. The result might be increasing settling time or higher power consumption compared
to when using both the reaction wheel and the thrusters. Then again by not using the reaction wheel as an
actuator for the attitude controller the overshot in the pitch angle that has occurred for most of the simulations
performed in Chapter 6.2 might be reduced. Wheel saturation is a phenomenon that occurs when the satellite
is exposed to constant disturbances over a long period of time, and it is natural to assume that the angular
velocities of the reaction wheel will stay in the vicinity of zero.
7.2.6 Wheel disturbance compensation
The wheel disturbance compensation developed for the PD controller compensates for the disturbance Da from
(3.43) caused by the reaction wheel. The simulations showed that the wheel compensator did not have any effect
on the settling time as shown in Figure 6.20. The disturbance from the reaction wheel will have a stabilizing
effect on the x-axis and z-axis which might be useful since the ESEO is mainly suppose to be a nadir pointing
satellite. This stabilizing effect will make the satellite more robust against disturbances in the x-axis and z-axis
and therefore result in lower power consumption when it is desired to keep the satellite at the equilibrium point
RO B = 1.
7.2.7 Power consumption
Analyses of the average power consumption showed that the Lyapunov controller has the lowest consumption
both with and without disturbances and where actually the average power consumption with disturbances is
lower than without disturbances as seen in Table 6.21. This also confirms that the Lyapunov controller is one
of the most robust controller since the variation in average power consumption is very low. The fact that the
power consumption is lower with disturbances than without is a special case where the disturbance contributes
in a positive way for the controller so it uses the actuators less. Normally the power consumption would increase
and does increase for some rotations as for example with the initial and reference angles Φi = £ 8 −6 9 ¤ T
and Φr = £ 3 1 5 ¤ T respectively as shown in Table 6.21.

7 DISCUSSION 85
The PD controller has a power consumption which is 56.7 % higher than for the Lyapunov controller without
disturbances, but the variation in power consumption for the PD controller is as low as the Lyapunov
controller and is also lower than without disturbances as seen in Table 6.21. The LQG controller naturally has
a very high power consumption with disturbances as seen in Table 6.21, since it loses track of the reference.
The Feedback controller has lower power consumption than the PD controller but not as low as the Lyapunov
controller without disturbances as seen in Table 6.21. The Feedback controller also has lower power consumption
than the PD controller with disturbances except for disturbances on the measurements and the actuators
with the initial and reference angles Φi = £ 8 −6 9 ¤ T and Φr = £ 3 1 5 ¤ T respectively where the PD
controller has a 7 % lower power consumption. When looking at Table 6.21 one can see that the variation in
power consumption with disturbances is generally much higher for the PD controller.
7.2.8 Reference trajectory
The simulation of the PD controller with the reference trajectory showed that the settling time increases by a
factor of 23.1 and the power consumption is reduced by a factor of 3192 compared to when not using the PD
controller. The angular velocities are kept low so that any sensors can be able to measure the attitude. The
control system has no problem following the reference trajectory since the bandwidth of the reference model is
lower than the bandwidth of the system. It is possible to control the power consumption by tuning the reference
trajectory, and by using the reference trajectory together with the Feedback controller makes the total system
fast and low on power consumption.

8 CONCLUDING REMARKS AND RECOMMENDATIONS 86
8 Concluding Remarks and Recommendations
This section contains the concluding remarks from the study performed in this thesis and recommendations for
future work.
8.1 Conclusion
This thesis deals with the attitude control system of the three-axis stabilized satellite, ESEO where the attitude
is controlled by four attitude control thrusters and one reaction wheel. The satellite is modelled as a rigid
body. Since the satellite environment is very complex and because is not easy to perform simulation of such
a complex system, the satellite environment is simplified. Stability analysis is performed on the unactuated
linearized satellite model and showed that the satellite is unstable in the equilibrium point where the body frame
coincides with the orbit frame. Another stability analysis is performed on the nonlinear model witch resulted
in a uniformly stable satellite in four equilibrium point, but none of them where the equilibrium point where
the body frame coincides with the orbit frame. The simulations of the unactuated satellite showed an unstable
satellite which substantiates the theoretical analyses on the linearized satellite model. The four equilibrium
points found in the theoretical analysis was not found during simulation.
The attitude control system for the ESEO has the accuracy demands of 1 ◦ in the x- and y-axis and 5 ◦ in
the z-axis. Four controllers are deduced for attitude control in this thesis, a PD controller, a Linear Quadratic
Gaussian controller, a nonlinear controller based on feedback linearization, and a nonlinear controller based on
Lyapunov. Simulations show that all the controllers stabilize the satellite as long as no disturbances are considered.
The controllers is compared against each other with respect to settling time, how robust they appear
against disturbances, and power consumption.
When only considering settling time the Feedback controller has the fastest settling time regardless of disturbances
except for some small angular rotations in witch the PD controller sometimes is faster. The LQG
controller has the slowest settling time but is also the only controller without any overshoot. The Lyapunov
controller is the only controller with a commanded output torque which does not exceed to maximum available
torque. The LQG controller does not seem to be very robust against disturbances because it is unstable for
several different rotations with disturbances. The Lyapunov controller and the PD controller appear to be the
most robust controllers against disturbances. The Lyapunov controller also has the lowest power consumption
between all the controllers both with and without disturbances. It is not easy to find the best possible controller
for the satellite because there is not one controller that is the best regarding settling time, robustness, and power
consumption.
A suggesting of controllers that might be good given certain circumstances is mentioned next. With respect
to settling time, the Feedback controller would be a good choice since it is the fastest controller and in it also
has the second lowest power consumption after the Lyapunov controller. The Lyapunov controller is the best
controller with respect to power consumption where also the controller is the only one where the commanded
output torque does not exceed to maximum available torque. Both the Lyapunov and PD controller would
be a good choice with respect to robustness, in which the PD controller has the fastest settling time and the
Lyapunov controller has the lowest power consumption. The LQG controller would in this case not be a good
choice because it has the slowest settling time and it appears unstable when exposed to disturbances.

8 CONCLUDING REMARKS AND RECOMMENDATIONS 87
A reference trajectory is deduced for the purpose of keeping the angular velocities low and where simulations
of the reference trajectory together with the PD controller show that the power consumption was reduced by
a factor of 3192 compared to without the reference trajectory. The reference trajectory could be used together
with the Feedback controller which gives the fastest response and low power consumption. If robustness is the
most important issue together with the lowest possible power consumption it would be vice to use the Lyapunov
controller together with the reference trajectory. The power consumption might not be of such high importance
since the satellites mission flight time is only 28 days, hence the importance of settling time might be of more
importance.
The PD controller was modified to include wheel disturbance compensation and where the simulations showed
no change in settling time or power consumption compared to when only using the PD controller. The disturbance
from the reaction wheel will have a stabilizing effect on the x-axis and z-axis which might be useful since
the satellite is mainly suppose to be nadir pointing. This stabilizing effect will make the satellite more robust
against disturbances in the x-axis and z-axis and therefore might result in lower power consumption when it is
desired to keep the satellite so that the body frame coincides with the orbit frame.
A Bang Bang controller deduced in this thesis together with the PD controller showed that the PD controller
stabilizes the satellite within the accuracy demands. The PD controller together with a momentum dumping
controller insures that no change of the attitude occur when a acceleration or deceleration of the reaction wheel
is performed in order to unload the angular momentum of the reaction wheel.
8.2 Recommendations for future work
It might be useful to develop a more realistic model of the satellite environment by implementing some of the
disturbance torques mentioned in Section 3.6 with respect to orientation and angular velocity. Changes in the
satellite moments of inertia can be performed to see if a stable unactuated satellite would have any effect on the
attitude control system. A PWPFM should be implemented to handle the on—of thrusters and also the PWPFM
can be designed in such a way that the thrusters fires at different frequencies of the multiple resonance frequency
of the structure, hence it would be much better than the existing Bang Bang controller. The controller based
on feedback linearization can be changed to include the gravitational torque in the design to see if the controller
will be more robust against disturbances than it already is.
Asshowninthesimulationswithdifferent controllers, the pitch response tends to overshot the reference angle,
which might be due to the fact that more actuators is used for the control of the pitch angle. By using some
form of algorithm that only uses the ACS thrusters during large angle manoeuvres and then after the manoeuvre
activate the usage of the reaction wheel to compensate for the orbit angular velocity. The gravitational torque
has the most disturbances on the pitch axis, and the reaction wheel can easily compensate for the disturbance
and hopefully the thrusters activation would decrease and as a result this will give lower fuel consumption in
addition there might be a reduction of the overshot in the pitch angle.
The wheel disturbance compensation used with the PD controller removes the disturbance produced by the
reaction wheel. A further investigation of the wheel compensator should be done to find out if it is desired to
use the wheel compensation or not, since the disturbance will have a stabilizing effect on the roll and yaw axis,
which might be desired for a nadir pointing satellite.
The discretization method should be changed to the more stable Runge Kutta, as mentioned in chapter 3.9.2.

9 BIBLIOGRAPHY 88
9 Bibliography
Anderson, B D. O. and More, J. B (1989), Optimal Control - Linear Quadratic Methods, Prentice Hall, New
Jersey.
Bak, T., Blanke, M. and Wi´sniewski, R. (1999), Flight results and lessons learned from the Ørsted attitude
control system. Proceedings of the 4. ESA International Conference on Spacecraft Guidance, Navigation and
Control System, ESTEC, Nordwijk, The Netherlands, 18-21 October 1999, (ESA SP-425, February 2000).
Balchen and Mumme (1988), Process control - Structures ans Applications, Van Nostrand Reinhold, New York.
Bang, H., Lee, J-S., Eun, Y-J. (2004), Nonlinear Attitude Control for a Rigid Spacecraft by Feedback Linearization,
Division of Aerospace Engineering, KAIST. KSME International Journal, Vol, 18, No. 2, 2004
Blindheim, F. R (2004), Attitude Control of a Micro Satellite, Master Thesis, Department of Computer Science,
Electrical Engineering and Space Technology, HiN, Norway.
Egeland, O. (1993), Robotdynamikk. Lecture notes in the course 43411 Robot manipulators. Department
of Engineering Cybernetics, NTNU, Norway.
Egeland and Gravdahl (2003), Modeling and Simulation for Automatic Control, Marine Cybernetics.
ESEO Phase A studie report (2001), ESA, ESTEC.
ESEO Phase B dokuments.
Fossen, T. I. (1994), Guidance and Control of Ocean Vehicles, John Wiley & Sons Ltd, New York.
Gravdahl, J. T, Eide, E., Skavhaug, A., Fauseke, K. M., Svartveit, K. and Idergaard, F. K. (2003), Three
axis attitude determination and control system for a pico-satellite: design and implementation, 54th International
Astronautical Congress of the International Astronautical Federation, the International Academy of
Astronautics, and the International Institute of Space Law 29 September — 3 October 2003, Bremen, Germany
IAC-03-A.5.07.
Hughes, P. C. (1986), Spacecraft Attitude Dynamics, University of Toronto.
Jensen H. B.and Wisniewski R. (2001), Quaternion Feedback Control for Rigid-Body Spacecraft, AIAA Guidance,
Navigation and Control Conference, Montreal, Canada, August 6-9, 2001.
Khalil, H. K (2002), Nonlinear Systems, PrenticeHall.
Kristiansen, R (2000), Attitude control of mini satellite, Master Thesis, Department of Engineering Cybernetics,
NTNU, Norway.
Kreyzig, E. (1999), Advanced Engineering Mathematics, John Wiley & Sons.

9 BIBLIOGRAPHY 89
Kyrkjebø, E. (2000), Satellite Attitude Determination, Master Thesis, Department of Engineering Cybernetics,
NTNU, Norway.
Larson, W. J, and Wertz J. R. (1992), Space Mission Analysis and Design, Kluwer Academic.
Marino, R. and Tomei, P. (1995), Nonlinear Control Design, Prentice Hall.
Riise,Å-R.,Samuelsen,B.,Sokolova,N.,Cederlad,H.,Fasseland,J.,Nordin,C.,Otterstad,J.,Fauske,K.,
Eriksen, O., Indergaard, F., Svartveit, K., Furebotten, P., Sæther, E., and Eide, E., (2003), Ncube: The first
Norwegian Student Satellite, In proceedings of The17th AIAA/USU Conference on small Satellites, 2003.
Quottrup, M. M., Sørensen, J. K., and Wi´sniewski, R. (2000), Passivity based nonlinear atitude control of
the Rømer satellite, Institute of Electronic Systems, Department of Control Engineering, Aalborg University,
Denmark.
Sidi, M. J. (1997), Spacecraft Dynamics & Control, Cambridge University Press, New York.
Slotine, J.-J. E and Li, W. (1991), Applied Nonlinear Control, Prentice-Hall, New Jersey.
Soglo, K. (1994), 3-aksestyring av gravitasjonsstabilisert satellitt ved bruk av magnetspoler, Master Thesis,
Department of Engineering Cybernetics, NTNU, Norway.
Song, G. and Agrawal, B. N. (2000), Vibration suppression of flexible spacecraft during attitude control, Department
of Mechanical Engineering, University of Akron, USA.
Strang, G. (1988), Linear Algebra and its Applications, Harcourt Brace Jovanovich College Publishers.
Vidyasagar, M. (1997), Nonlinear Systems Analysis, Prentice Hall.
Wertz, J. R. (1978), Spacecraft Attitude Determination and Control, D. Reidel Publishing Company, Dordrecht,
Holland.
Wi´sniewski , R. (1996), Satellite Attitude Control Using Only Electromagnetic Actuation, Ph.D thesis, Department
of Engineering, Aalborg University, Denmark.

A NOTATIONS 90
A Notations
A.1 Vectrices and dyadics
A vectrix is a matrix where all the elements are vectors, and it has the characteristics of both matrices and
vectors. The vectrices elements are all base vectors in a coordinate system, or a reference frame. If a coordinate
system A is represented by its base vectors a1, a2, a3 a vectrix Fa for the system can be defined as (Egeland,
1993)
Fa = £ ¤
a1 a2 a3
(A.1)
The relation between a vector vA represented in the system A and an independent vector ¯v canwiththehelp
of the vectrix for system Fa be expressed as
v = ¡ v A¢T Fa = F T a v A
(A.2)
v A ¡ A
v
= Fav = vFa (A.3)
¢T
=
T
Fa v = vF T a (A.4)
The dot product of two vectrices is
FaF T ⎡
a = ⎣ a1a1
a2a1
a3a1
a1a2
a2a2
a3a2
a1a3
a2a3
a3a3
⎤ ⎡
1
⎦ = ⎣ 0
0
0
1
0
⎤
0
0 ⎦
1
(A.5)
and similar the cross product between two vectrices as
Fa ×F T ⎡
a = ⎣ a1 × a1
a2 × a1
a3 × a1
a1 × a2
a2 × a2
a3 × a2
a1 × a3
a2 × a3
a3 × a3
⎤ ⎡
⎦ = ⎣ 0 a3
−a3
a2
0
−a1
−a2
a1
0
⎤
⎦ (A.6)
A dyadic has the property that the dot product of a dyadic and a vector produces a vector. Two vectors placed
side by side can be regarded as a dyadic. Vectrices can be used for transformation between dyadics and the
matrix representation of dyadics. The transformation between a dyadic ¯D and the matrix representation of the
dyadic D can be expressed as
D = F T a DFa (A.7)
D = F T a DFa (A.8)

B SIMULATIONS 91
B Simulations
This section contains some of the simulations performed in this thesis with different attitude controllers.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
15
10
5
0
Attitude angles [deg]
-5
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
0.02
ωx ωy 0
ωz -0.02
-0.04
-0.06
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure B.1: Step response using the PD controller with the initial angles Φi = £ 10◦ 10◦ 10◦ ¤ T
and the
reference angles Φr = £ 0◦ 0◦ 0◦ ¤ T
.
φ
θ
ψ

B SIMULATIONS 92
Angular velocity [rad/s]
Actuators
RPY angles [deg]
15
10
5
0
Attitude angles [deg]
-5
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
0.02
ωx ωy 0
ωz -0.02
-0.04
-0.06
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure B.2: Step response using the LQG controller with the initial angles Φi = £ 10◦ 10◦ 10◦ ¤ T
and the
reference angles Φr = £ 0◦ 0◦ 0◦ ¤ T
.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
15
10
5
0
Attitude angles [deg]
-5
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.02
ωx 0
ωy -0.02
ωz -0.04
-0.06
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure B.3: Step response using the Feedback controller with the initial angles Φi = £ 10◦ 10◦ 10◦ ¤ T
and
the reference angles Φr = £ 0◦ 0◦ 0◦ ¤ T
.
φ
θ
ψ
φ
θ
ψ

B SIMULATIONS 93
Angular velocity [rad/s]
Actuators
RPY angles [deg]
15
10
5
0
Attitude angles [deg]
-5
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.01
0
ωx ωy -0.01
ωz -0.02
-0.03
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure B.4: Step response using the Lyapunov controller with the initial angles Φi = £ 10◦ 10◦ 10◦ ¤ T
and
the reference angles Φr = £ 0◦ 0◦ 0◦ ¤ T
.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
10
5
0
-5
Attitude angles [deg]
-10
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
ωx 0.02
ωy 0
ωz -0.02
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure B.5: Step response using the PD controller with the initial angles Φi = £ 8◦ −6◦ 9◦ ¤ T
and the
reference angles Φr = £ 0◦ 0◦ 0◦ ¤ T
.
φ
θ
ψ
φ
θ
ψ

B SIMULATIONS 94
Angular velocity [rad/s]
Actuators
RPY angles [deg]
10
5
0
-5
Attitude angles [deg]
-10
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
ωx 0.02
ωy 0
ωz -0.02
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure B.6: Step response using the LQG controller with the initial angles Φi = £ 8◦ −6◦ 9◦ ¤ T
and the
reference angles Φr = £ 0◦ 0◦ 0◦ ¤ T
.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
10
5
0
-5
Attitude angles [deg]
-10
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.03
0.02
ωx ωy 0.01
ωz 0
-0.01
-0.02
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure B.7: Step response using the Feedback controller with the initial angles Φi = £ 8◦ −6◦ 9◦ ¤ T
and the
reference angles Φr = £ 0◦ 0◦ 0◦ ¤ T
.
φ
θ
ψ
φ
θ
ψ

B SIMULATIONS 95
Angular velocity [rad/s]
Actuators
RPY angles [deg]
10
5
0
-5
Attitude angles [deg]
-10
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.03
0.02
ωx ωy 0.01
ωz 0
-0.01
-0.02
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure B.8: Step response using the Lyapunov controller with the initial angles Φi = £ 8◦ −6◦ 9◦ ¤ T
and
the reference angles Φr = £ 0◦ 0◦ 0◦ ¤ T
.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
15
10
5
0
-5
Attitude angles [deg]
-10
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.06
0.04
ωx ωy 0.02
ωz 0
-0.02
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure B.9: Step response using the PD controller with the initial angles Φi = £ 5◦ −5◦ 0◦ ¤ T
and the
reference angles Φr = £ 0◦ 0◦ 0◦ ¤ T
.
φ
θ
ψ
φ
θ
ψ

B SIMULATIONS 96
Angular velocity [rad/s]
Actuators
RPY angles [deg]
15
10
5
0
-5
Attitude angles [deg]
-10
0 10 20 30 40 50 60
Angular velocities [rad/s]
0.05
ωx ωy ωz 0
-0.05
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure B.10: Step response using the LQG controller with the initial angles Φi = £ 5◦ −5◦ 0◦ ¤ T
and the
reference angles Φr = £ 0◦ 0◦ 0◦ ¤ T
.
Angular velocity [rad/s]
Actuators
RPY angles [deg]
15
10
5
0
-5
Attitude angles [deg]
-10
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.06
0.04
ωx ωy 0.02
ωz 0
-0.02
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure B.11: Step response using the Feedback controller with the initial angles Φi = £ 5◦ −5◦ 0◦ ¤ T
and
the reference angles Φr = £ 0◦ 0◦ 0◦ ¤ T
.
φ
θ
ψ
φ
θ
ψ

B SIMULATIONS 97
Angular velocity [rad/s]
Actuators
RPY angles [deg]
15
10
5
0
-5
Attitude angles [deg]
-10
0 10 20 30
Angular velocities [rad/s]
40 50 60
0.04
ωx 0.02
ωy 0
ωz -0.02
-0.04
0 10 20 30
Thrusters [N], Reaction wheel [Nm]
40 50 60
0.15
0.1
Th1
Th2
Th3
Th4
0.05
h
0
-0.05
0 10 20 30
sec
40 50 60
Figure B.12: Step response using the Lyapunov controller with the initial angles Φi = £ 5◦ −5◦ 0◦ ¤ T
and
the reference angles Φr = £ 0◦ 0◦ 0◦ ¤ T
.
φ
θ
ψ

C MATLAB SOURCECODE 98
C Matlab sourcecode
This section contains documented Matlab sourcecode produced for simulation purpose of the system.
Runsimulation.m
%**************************************************************************************
% File RunSimulation.m
%
%Startfile.
% Set time parameters, initial and reference angles.
% Plotting results.
%
% Created By Jøran Antonsen, 2004
%*************************************************************************************
clear;
clc;
global F h_wheel_dot_vec thruster_vec h_wheel_vec w_wheel_vec Cont noise eps_noise
tic;
%Chosecontroller
Cont = 1; % PD controller
%Cont = 2; % LQG controller
%Cont = 3; % Nonlinear controller (Feedback Linearization)
%Cont = 4; % Nonlinear controller (Lyapunov)
noise = 1; % No actuator noise
%noise = 2; % Noise on the actuators
%eps_noise = 1;% Noise on the euler angles
eps_noise = 2; % No noise on the angles
%for eps_noise = 1:2
%for noise = 1:2 %used 2 simulate noise on the actuators
%for Cont = 1:4 %used 2 simulate all controllers
% Time parameters for the simulation period
t0=0;
tfinal=43200/100;
sampletime=1;
% Initial angles
phi=5;
theta=10;
psi=15;
% reference angles
phi_d=10;

C MATLAB SOURCECODE 99
theta_d=15;
psi_d=5;
% Performing simulation
%[utab,htab,Pe,e,t,ref]=simulate(phi,theta,psi,phi_d,theta_d,psi_d,t0,tfinal, sampletime);
[Pe, e, t, ref, Rc_vec, wo_vec] = simulate(phi, theta, psi, phi_d, theta_d, psi_d, t0, tfinal, sampletime);
% Plotting results
%t=t/43200;
%figure;
% Plotting attitude, angular velocity without any actuators
% Reference trajectories not used
%subplot(2,1,1), plot(t,e(:,4), ’r-’, t,e(:,5), ’k—’, t,e(:,6), ’b:’),grid, legend(’\phi’, ’\theta’, ’\psi’), ylabel(’RPY
angles [deg]’), title(’Attitude angles with no actuators [deg]’),xlabel(’Orbits’);
%subplot(2,1,2), plot(t,e(:,1), ’r-’, t,e(:,2), ’k—’, t,e(:,3), ’b’),grid, legend(’\omega_x’, ’\omega_y’, ’\omega_z’),
ylabel(’Angular velocity [rad/s]’), title(’Angular velocities with no actuators [rad/s]’),xlabel(’Orbits’);
%figure;
% Plotting attitude, angular velocity and reference trajectory
% Reference trajectories not used
%subplot(2,1,1), plot(t,e(:,4), ’r-’, t,e(:,5), ’k—’, t,e(:,6), ’b:’),grid, legend(’\phi’, ’\theta’, ’\psi’), ylabel(’RPY
angles [deg]’), title(’Attitude angles [deg]’),xlabel(’sec’);
%subplot(2,1,2), plot(t,e(:,1), ’r-’, t,e(:,2), ’k—’, t,e(:,3), ’b’),grid, legend(’\omega_x’, ’\omega_y’, ’\omega_z’),
ylabel(’Angular velocity [rad/s]’), title(’Angular velocities [rad/s]’),xlabel(’sec’);
% Reference trajectories used
%subplot(2,1,1), plot(t,e(:,4), ’r-’, t,e(:,5), ’k—’, t,e(:,6), ’b:’),grid, legend(’\phi’, ’\theta’, ’\psi’), ylabel(’RPY
angles [deg]’), title(’Attitude angles using reference trajectories [deg]’),xlabel(’sec’);
%subplot(2,1,2), plot(t,e(:,1), ’r-’, t,e(:,2), ’k—’, t,e(:,3), ’b’),grid, legend(’\omega_x’, ’\omega_y’, ’\omega_z’),
ylabel(’Angular velocity [rad/s]’), title(’Angular velocities using reference trajectories [rad/s]’),xlabel(’sec’);
%subplot(2,1,1), plot(t,ref(:,4), ’r-’, t,ref(:,5), ’k-’, t,ref(:,6), ’b-’),grid, legend(’\phi ref’, ’\theta ref’, ’\psi ref’),
ylabel(’Roll, pitch and yaw angle [deg]’), xlabel(’sec’), title(’Reference trajectories in roll, pitch and yaw’);
%subplot(2,1,2), plot(t,ref(:,1), ’r-’, t,ref(:,2), ’k—’, t,ref(:,3), ’b’),grid, legend(’\omega_x’, ’\omega_y’, ’\omega_z’),
ylabel(’Angular velocity [rad/s]’), title(’Angular velocities using reference trajectories [rad/s]’),xlabel(’sec’);
%size(t)
%size(Rc_vec)
%size(wo_vec)
%figure;
% Plotting Rc and wo
%subplot(2,1,1), plot(t, Rc_vec),grid, ylabel(’Rc [m]’), title(’Distance from center of earth to center of satellite’);
%subplot(2,1,2), plot(t, wo_vec),grid, ylabel(’wo [rad/s]’), title(’Satellite angular velocity relative to earth’);

C MATLAB SOURCECODE 100
%figure;
% Plotting reference trajectories for roll, pitch and yaw
%subplot(3,1,1), plot(t,ref(:,4), ’r-’),grid, legend(’\phi ref’), ylabel(’Roll angle [deg]’), title(’Reference trajectories
in roll, pitch and yaw’);
%subplot(3,1,2), plot(t,ref(:,5), ’k-’),grid, legend(’\theta ref’), ylabel(’Pitch angle [deg]’);
%subplot(3,1,3), plot(t,ref(:,6), ’b-’),grid, legend(’\psi ref’), ylabel(’Yaw angle [deg]’),xlabel(’sec’);
figure;
% Plotting wheel angular momentum of the reaction wheel, speed of the wheel and the thrust vector subplot(2,1,1),
plot(t,thruster_vec(:,1), ’r-’, t,thruster_vec(:,2), ’k—’,t,thruster_vec(:,3), ’b:’, t,thruster_vec(:,4),
’m’),grid, legend(’Th1’, ’Th2’, ’Th3’, ’Th4’), ylabel(’Torque [N]’), title(’Thruster vector’), xlabel(’sec’);
subplot(2,1,2), plot(t,h_wheel_dot_vec(:,1), ’k-’),grid, legend(’h’), ylabel(’Torque [Nm]’), title(’Changes in angular
momentum’), xlabel(’sec’);
%figure;
%subplot(2,1,1), plot(t,h_wheel_vec(:,1), ’k-’),grid, legend(’h’), ylabel(’Torque [Nm]’), xlabel(’sec’), title(’Angular
momentum vector’);
%subplot(2,1,2), plot(t,w_wheel_vec(:,1), ’k-’),grid, legend(’w’), ylabel(’Velocity [rpm]’), xlabel(’sec’), title(’Reaction
wheel Velocivy’);
Pe % printing power consumption
%end
%end
%end
toc
Simulate.m
%**************************************************************************************
% File simulate.m
%
% Performing simulation, defining orbital parameters.
% Input is th initaial and reference andles, together with time parameters.
%
% Created By Jøran Antonsen, 2004
%*************************************************************************************
function [Pe, e, t, ref, Rc_vec, wo_vec] = simulate(phi_i, theta_i, psi_i, phi_d, theta_d, psi_d, t0, tfinal,
sampletime)
global h xd Pe htab utab my a ecc n_mean ref_table h_wheel_vec h_wheel_dot_vec thruster_vec w_wheel_vec
N ref_temp
%Gravityconstant
my = 3.986e14;
% Earth radius

C MATLAB SOURCECODE 101
Re = 6371e3;
% Perigee distance
rp = 250e3 + Re;
% Apogee distance
ra = 35950e3 + Re;
%Sirkular orbitt
%rp = 24471e3;
%ra = rp;
% Semimajor axis
a = (rp + ra)/2; %24471e3
% Eccentricity
ecc = (ra - rp)/(ra + rp); % 0.729434841241;
% Inertia matrix
Ix = 4.3500;
Iy = 4.3370;
Iz = 3.6640;
I = diag([Ix, Iy, Iz]);
% Angular momentum
h = [0, 0, 0]’;
% Power consumption
Pe=0;
% Insertion at perigee
Rc_per = rp;
wo_per = sqrt(my/((Rc_per)^3));
%Initialvalues
x0=[000phi_itheta_ipsi_i];
% Reference values
xref = [0 0 0 phi_d theta_d psi_d];
% Conversion from [w_bo_x, w_bo_y, w_bo_z, phi, theta, psi];
x_temp = conversion1(x0(1, 1 :6), wo_per);
eta_temp = x_temp(1,4);
xd = x_temp;
% Calculation number of steps

C MATLAB SOURCECODE 102
N = round( (tfinal-t0)/sampletime );
tfinal = N*sampletime;
% Initial condition
x(1,:) = x_temp;
t(1) = t0 + sampletime;
n_mean = 2*pi/43200;
anomaly_grense = 0.00001;
psi_0 = psi_i;
% Calculation of reference model
ref_i = x_temp(1,5:7);
ref_temp = conversion1(xref, wo_per);
xref = ref_temp(1,5:7);
%eta_temp = ref_temp(4);
ref_table = referencemodel(ref_i’,xref’,N,sampletime, anomaly_grense);
% Initial vectors used in Thruster.m
h_wheel_vec = zeros(N,1);
h_wheel_dot_vec = zeros(N,1);
thruster_vec = zeros(N,4);
w_wheel_vec = zeros(N,1);
% initializing the LQG controller matrices when they are needed
[A, P, Q, C] = LQinit(I,wo_per);
% Solving the Ricatti equation when the LQ controller is used
[R,h1]=RicattiSolver(A,C,P,Q,tfinal, N, sampletime, wo_per, I, ref_table, xref);
% Satellite simulation loop
for i = 1:N
Ri = zeros(6);
hi = zeros(6,1);
Ri(1,1) = R(i, 1, 1);
Ri(1,2) = R(i, 1, 2);
Ri(1,3) = R(i, 1, 3);
Ri(1,4) = R(i, 1, 4);
Ri(1,5) = R(i, 1, 5);
Ri(1,6) = R(i, 1, 6);
Ri(2,1) = R(i, 2, 1);
Ri(2,2) = R(i, 2, 2);
Ri(2,3) = R(i, 2, 3);
Ri(2,4) = R(i, 2, 4);

C MATLAB SOURCECODE 103
Ri(2,5) = R(i, 2, 5);
Ri(2,6) = R(i, 2, 6);
Ri(3,1) = R(i, 3, 1);
Ri(3,2) = R(i, 3, 2);
Ri(3,3) = R(i, 3, 3);
Ri(3,4) = R(i, 3, 4);
Ri(3,5) = R(i, 3, 5);
Ri(3,6) = R(i, 3, 6);
Ri(4,1) = R(i, 4, 1);
Ri(4,2) = R(i, 4, 2);
Ri(4,3) = R(i, 4, 3);
Ri(4,4) = R(i, 4, 4);
Ri(4,5) = R(i, 4, 5);
Ri(4,6) = R(i, 4, 6);
Ri(5,1) = R(i, 5, 1);
Ri(5,2) = R(i, 5, 2);
Ri(5,3) = R(i, 5, 3);
Ri(5,4) = R(i, 5, 4);
Ri(5,5) = R(i, 5, 5);
Ri(5,6) = R(i, 5, 6);
Ri(6,1) = R(i, 6, 1);
Ri(6,2) = R(i, 6, 2);
Ri(6,3) = R(i, 6, 3);
Ri(6,4) = R(i, 6, 4);
Ri(6,5) = R(i, 6, 5);
Ri(6,6) = R(i, 6, 6);
hi(1,1) = h1(1, i);
hi(2,1) = h1(2, i);
hi(3,1) = h1(3, i);
hi(4,1) = h1(4, i);
hi(5,1) = h1(5, i);
hi(6,1) = h1(6, i);
% Calculating wo and Rc
[wo, Rc] = EllipticOrbit(i, n_mean, anomaly_grense, ecc, a, my, sampletime);
Rc_vec(i) = Rc;
wo_vec(i) = wo;
eta_sat = sqrt(1 - (x(i,5))^2 - (x(i,6))^2 - (x(i,7))^2);
% Diskretization of the nonlinear model

C MATLAB SOURCECODE 104
x(i+1,:) = x(i,:) + sampletime*nonlinearmodel(t(i), x(i,:), xref, I, wo, xd, sampletime, i, eta_sat, A, P, Q, C,
Ri, hi);
t(i+1) = t(i) + sampletime;
eta_new = sqrt(1 - (ref_table(i,1))^2 - (ref_table(i,3))^2 - (ref_table(i,5))^2);
ref_temp(i,:) = [ 0, 0, 0, eta_new, ref_table(i,1), ref_table(i,3), ref_table(i,5)];
end
t=t(1:N);
x = x(1:N,:);
% Conversion from [wx wy wz eta eps1 eps2 eps3] to
% e = [wbo_x, wbo_y, wbo_z, phi, theta, psi]
% psi_0 = psi_i;
e = conversion2( x(:,1:7), psi_0, sampletime, anomaly_grense);
% Conversion of the reference model for plotting purposes
ref = conversion2(ref_temp, psi_0, sampletime, anomaly_grense);
NonlinearModel.m
%**************************************************************************************
% File NonlinearModel.m
%
% Calculation of the nonlinear model
%
% Created By Jøran Antonsen, 2004
%*************************************************************************************
function[xdot]=nonlinearmodel(t,x,xref,I,wo,xd,sampletime,i,eta_sat,A,P,Q,C,Ri,hi)
global h xd Pe Cont
x1=x(1);
x2=x(2);
x3=x(3);
x4=x(4);
x5=x(5);
x6=x(6);
x7=x(7);
w_B_BI = [x1 x2 x3]’;
eta = x4;
epsilon = [x5 x6 x7]’;
S_epsilon = [0, -epsilon(3), epsilon(2); epsilon(3), 0, -epsilon(1); -epsilon(2), epsilon(1), 0];
%Rotation matrix from O to B, R_B_O
R_B_O = eye(3) - 2*eta*S_epsilon + 2*S_epsilon^2;

C MATLAB SOURCECODE 105
%Calculating the gravitational torque
c3 = R_B_O(:,3); %Tar ut vektor r3 fra R_B_O
g_c=3*(wo^2)*cross(c3, I*c3);
%angular velocity used to calculate the Euler parameters
w_B_BO = w_B_BI - R_B_O*[0, -wo, 0]’;
%Kinematic differensial equation for the Euler parameters, eta and epsilon
xdot(4) = -(1/2)*epsilon’*w_B_BO;
xdot(5:7) = 1/2*(eta*eye(3) + S_epsilon)*w_B_BO;
% Euler parameters to be used in control.m and ActuatorTorqueLQ
x_dot = 1/2*(eta*eye(3) + S_epsilon)*w_B_BO;
if Cont == 1
% Calculating actuator torque for PD controller
[Ta,u]=PD_control(x,xd,x_dot,xref,w_B_BI,epsilon,I,eta,eta_sat,i);
end
if Cont == 2
% Calculating actuator torque for LQG controller
[Ta] = ActuatorTorqueLQ(x, I, x_dot, A, P, Q, C, Ri, hi, xref, i, xd);
end
if Cont == 3
% Calculating actuator torque for Nonlinear controller 1 (Feeback Linearization)
[Ta, u] = Nonlinear_control_1(x, xd, x_dot, xref, w_B_BO, epsilon, I, eta, eta_sat, i, wo, w_B_BI);
end
if Cont == 4
% Calculating actuator torque for Nonlinear controller 2 (Lyapunov)
[Ta, u] = Nonlinear_control_2(x, xd, x_dot, xref, w_B_BI, epsilon, I, eta, eta_sat, i, wo, w_B_BO);
end
%Adding up the power consumption
Pe = Pe + abs(Ta’*w_B_BO);
%Differensial equations for the rotation, wdot
wdot = inv(I)*[-cross(w_B_BI, (I*w_B_BI)) + g_c + Ta];
xdot(1:3) = wdot(1:3);

C MATLAB SOURCECODE 106
Conversion1.m
%**************************************************************************************
% File conversion1.m
%
% Algorithm for conversion from Euler angles to Euler parameters.
%
% Created By Jøran Antonsen, 2004
%*************************************************************************************
function x=conversion1(ev, wo_per);
%x=[wxwywzetaeps1eps2eps3]’;
%ev = [w_bo_x, w_bo_y, w_bo_z, phi, theta, psi];
%Conversion from degrees to radians
ev(4) = pi*ev(4)/180;
ev(5) = pi*ev(5)/180;
ev(6) = pi*ev(6)/180;
%rotation matrix from O to B
c4 = cos(ev(4));
c5 = cos(ev(5));
c6 = cos(ev(6));
s4 = sin(ev(4));
s5 = sin(ev(5));
s6 = sin(ev(6));
R11 = c6*c5;
R12 = -s6*c4 + c6*s5*s4;
R13 = s6*s4 + c6*s5*c4;
R21 = s6*c5;
R22 = c6*c4 + s4*s5*s6;
R23 = -c6*s4 + s5*s6*c4;
R31 = -s5;
R32 = c5*s4;
R33 = c5*c4;
R = [R11 R12 R13
R21 R22 R23
R31 R32 R33];
%Calculating angular velocity between O and I represented in B
wo_i(1) = -wo_per*R21;
wo_i(2) = -wo_per*R22;
wo_i(3) = -wo_per*R23;
%Calculating angular velocity between B and I represented in B
x(1) = ev(1) + wo_i(1);
x(2) = ev(2) + wo_i(2);
x(3) = ev(3) + wo_i(3);
%Calculating Euler parameters
R44 = trace(R);
storst = max([R11 R22 R33 R44]);

C MATLAB SOURCECODE 107
if R11 == storst
p1 = sqrt(1 + 2*R11-R44);
p2 = (R21 + R12)/p1;
p3 = (R13 + R31)/p1;
p4 = (R32-R23)/p1;
elseif R22 == storst
p2 = sqrt(1 + 2*R22-R44);
p1 = (R21 + R12)/p2;
p3 = (R32 + R23)/p2;
p4 = (R13 - R31)/p2;
elseif R33 == storst
p3 = sqrt(1+2*R33-R44);
p1 = (R13+R31)/p3;
p2 = (R32+R23)/p3;
p4 = (R21-R12)/p3;
else
p4 = sqrt(1 + R44);
p1 = (R32 - R23)/p4;
p2 = (R13 - R31)/p4;
p3 = (R21 - R12)/p4;
end
x(4) = p4/2;
x(5) = p1/2;
x(6) = p2/2;
x(7) = p3/2;
Conversion2.m
%**************************************************************************************
% File conversion2.m
%
% Algorithm for conversion from Euler parameters to Euler angels.
%
% Created By Jøran Antonsen, 2004
%*************************************************************************************
function e=conversion2(y, psi_0, sampletime, anomaly_grense);
%y=[wx wy wz eta eps1 eps2 eps3]
%e=[w1, w2, w3, phi, theta, psi]
global my a ecc n_mean
%Calculating number of vectors
N2 = size(y,1);
%Storing initial angle in radians
psi_gammel = psi_0*pi/180;

C MATLAB SOURCECODE 108
for i = 1:N2,
x = y(i,:)’;
%Calculating wo and Rc
[wo, Rc] = EllipticOrbit(i, n_mean, anomaly_grense, ecc, a, my, sampletime);
%calculating angular velocity between O and I represented in O
woi = [0 -wo 0]’;
%Rotation matrix from O to B
x4 = x(4);
x5 = x(5);
x6 = x(6);
x7 = x(7);
R11 = x4^2 + x5^2 - x6^2 - x7^2;
R12 = 2*( x5*x6 - x4*x7);
R13 = 2*( x5*x7 + x4*x6);
R21 = 2*( x5*x6 + x4*x7);
R22 = x4^2 - x5^2 + x6^2 - x7^2;
R23 = 2*(x6*x7 - x4*x5);
R31 = 2*(x5*x7 - x4*x6);
R32 = 2*(x6*x7 + x4*x5);
R33 = x4^2 - x5^2 - x6^2 + x7^2;
R = [R11 R22 R13
R21 R22 R23
R31 R32 R33];
%Calculating angular velocity between B and I represented in B
wbi = [x(1), x(2), x(3)]’;
%calculating angular velocity between B and O represented in B
wbo = wbi - R’*woi;
%Calculating auler angles
psi = atan2(R21,R11);
if psi < 0
psi + 2*pi;
end
psi_test=pi + psi;
if psi_test > 2*pi

C MATLAB SOURCECODE 109
psi_test = psi_test - 2*pi;
end
%Cheking for discontinuoities in psi
d = min([abs(psi_gammel - psi), abs(psi_gammel - 2*pi - psi), abs(psi_gammel + 2*pi - psi)]);
d_test = min([abs(psi_gammel - psi_test), abs(psi_gammel - 2*pi - psi_test), abs(psi_gammel + 2*pi -
psi_test)]);
if d_test < d
psi=psi_test;
end
theta = atan2(-R31, cos(psi)*R11 + sin(psi)*R21);
phi = atan2(sin(psi)*R13 - cos(psi)*R23, -sin(psi)*R12 + cos(psi)*R22);
psi_gammel = psi;
if psi > pi
psi = psi - 2*pi;
end
e(i,1) = wbo(1);
e(i,2) = wbo(2);
e(i,3) = wbo(3);
e(i,4) = phi*180/pi;
e(i,5) = theta*180/pi;
e(i,6) = psi*180/pi;
end
ReferenceModel.m
%**************************************************************************************
% File ReferenceModel.m
%
% Function for producing a smooth reference trajectory for the satellite.
% Created By Jøran Antonsen, 2004
%*************************************************************************************
function ref=referencemodel(init, desired, N, T, anomaly_grense)
%crossover frequencies and relative damping factors
wnx = 0.01;
zx = 0.8;
wny = 0.01;
zy = 0.8;
wnz = 0.01;
zz = 0.8;

C MATLAB SOURCECODE 110
%Reference model
Ar=[010000;
-wnx^2 -2*zx*wnx 0 0 0 0;
000100;
0 0 -wny^2 -2*zy*wny 0 0;
000001;
0000-wnz^2-2*zz*wnz];
Br=[0wnx^20000;
000wny^200;
00000wnz^2]’;
%Initial conditions
phi_i = init(1);
theta_i = init(2);
psi_i = init(3);
ref = zeros(N,6);
xd = [phi_i,0,theta_i,0,psi_i,0]’;
%Reference trajectory generator loop
for i = 1:1:N
xd_dot = Ar*xd + Br*desired;
xd = xd + T*xd_dot;
ref(i,:) = xd’;
end
RicattiSolver.m
%**************************************************************************************
% File RicattiSolver.m
%
% Solving Ricatti equation backwords.
%
% Created By Jøran Antonsen, 2004
%*************************************************************************************
function [R, h1] = RicattiSolver(A, C, P, Q, tfinal, N, sampletime, wo_per, I, ref_table, xref)
%Initial result matrices
Ri = zeros(6,6);
R_dot = zeros(6,6);
R = zeros(N, 6, 6);
hi = zeros(6,1);
h1_dot = zeros(6,1);
h1 = zeros(6,N);

C MATLAB SOURCECODE 111
% Inertia matrix
Ix = I(1,1);
Iy = I(2,2);
Iz = I(3,3);
x_ref = [xref(1), 0, xref(2), 0, xref(3), 0]’;
%Solving the Ricatti equation and feedforward bakwards
t=tfinal;
for i = N:-1:3
% xref = ref_table(i,:)’;
% Thruster vector
r = [0.25456 0.03787 0.363]’;
rx = r(1);
ry = r(2);
rz = r(3);
B=[0,0,0,0,0;
-sqrt(2)/(4*Ix)*rz, sqrt(2)/(4*Ix)*rz, -sqrt(2)/(4*Ix)*rz, sqrt(2)/(4*Ix)*rz, 0;
0, 0, 0, 0, 0;
sqrt(2)/(4*Iy)*rz, sqrt(2)/(4*Iy)*rz, -sqrt(2)/(4*Iy)*rz, -sqrt(2)/(4*Iy)*rz, 1/(2*Iy);
0, 0, 0, 0, 0;
-sqrt(2)/(4*Iz)*(rx-ry), sqrt(2)/(4*Iz)*(rx-ry), sqrt(2)/(4*Iz)*(rx-ry), -sqrt(2)/(4*Iz)*(rx-ry), 0];
R_dot = -Ri*A - A’*Ri + Ri*B*inv(P)*B’*Ri - Q;
h1_dot = - (A-B*inv(P)*B’*Ri)’*hi + Q*x_ref;
hi = hi - sampletime*h1_dot;
Ri = Ri - sampletime*R_dot;
R(i,:,:) = Ri;
h1(:,i) = hi;
t=t-sampletime;
end
Lqinit.m
%**************************************************************************************
% File Lqinit.m
%
% Initializing LQG controller matrices
%
% Created By Jøran Antonsen, 2004
%*************************************************************************************
function [A, P, Q, C] =LQinit(I, wo_per)

C MATLAB SOURCECODE 112
% Inertia matrix
Ix = I(1,1);
Iy = I(2,2);
Iz = I(3,3);
% Constants for the ratio between moments of inertia
rox=(Iy-Iz)/Ix;
roy=(Ix-Iz)/Iy;
roz = (Iy - Ix)/Iz;
% Using the Perigee vector for calculation of wo
rp_temp = 250e3+6371e3;
my_temp = 3.986e14;
wo_temp = sqrt(my_temp/rp_temp^3);
% Linearized system matrix
A=[0,1,0,0,0,0;
-4*rox*wo_temp^2, 0, 0, 0, 0, (1-rox)*wo_temp;
0, 0, 0, 1, 0, 0;
0, 0, -3*roy*wo_temp^2, 0, 0, 0;
0, 0, 0, 0, 0, 1;
0, -(1-roz)*wo_temp, 0, 0, -roz*wo_temp^2, 0];
%system noise
C=[1,0,0,0,0,0;
0, 0, 0, 0, 0, 0;
0, 0, 1, 0, 0, 0;
0, 0, 0, 0, 0, 0;
0, 0, 0, 0, 1, 0;];
% State weight matrix
Q = diag([1.29 0 1 0 1/4 0])*2*inv(2*pi/180)^2;
P = diag([inv(0.13)^2 inv(0.13)^2 inv(0.13)^2 inv(0.13)^2 inv(0.0209)^2]);
ActuatorTorqueLQ
%**************************************************************************************
% File ActuatorTorqueLQ.m
%
% Calculation of the LQG controller
%
% Created By Jøran Antonsen, 2004
%*************************************************************************************
function [Ta] = ActuatorTorqueLQ(x, I, x_dot, A, P, Q, C, Ri, hi, xref, i, xd);
global h ref_table N eps_noise

C MATLAB SOURCECODE 113
% Position deviation
w_B_BI = [x(1) x(2) x(3)]’;
% Without reference trajectory
epsilon = [x(5) x(6) x(7)]’;
if eps_noise ==1
e_noise = 0.20; %Adding 20% noise on the euler angles
epsilon_noise = [epsilon(1)*e_noise*(2*rand(1)-1);
epsilon(2)*e_noise*(2*rand(1)-1);
epsilon(3)*e_noise*(2*rand(1)-1)];
epsilon = epsilon + epsilon_noise;
end
epsilon_dot = [x_dot(1) x_dot(2) x_dot(3)]’;
% Inertia matrix
Ix = I(1,1);
Iy = I(2,2);
Iz = I(3,3);
% Thruster vector
r = [0.25456 0.03787 0.363]’;
rx = r(1);
ry = r(2);
rz = r(3);
% Actuator matrix with reaction wheel, Ba
B=[0,0,0,0,0;
-sqrt(2)/(4*Ix)*rz, sqrt(2)/(4*Ix)*rz, -sqrt(2)/(4*Ix)*rz, sqrt(2)/(4*Ix)*rz, 0;
0, 0, 0, 0, 0;
sqrt(2)/(4*Iy)*rz, sqrt(2)/(4*Iy)*rz, -sqrt(2)/(4*Iy)*rz, -sqrt(2)/(4*Iy)*rz, 1/(2*Iy);
0, 0, 0, 0, 0;
-sqrt(2)/(4*Iz)*(rx-ry), sqrt(2)/(4*Iz)*(rx-ry), sqrt(2)/(4*Iz)*(rx-ry), -sqrt(2)/(4*Iz)*(rx-ry), 0];
G1 = -inv(P)*B’*Ri;
x1 = [epsilon(1), epsilon_dot(1), epsilon(2), epsilon_dot(2), epsilon(3), epsilon_dot(3)]’;
u = G1*x1 - inv(P)*B’*hi;
[Ta_LQ,F]=Thruster(u,B,r,i,w_B_BI);
% Actual torque matrix
Ba = [-rz, rz, -rz, rz 0;
rz, rz, -rz, -rz, sqrt(2);
-(rx-ry), (rx-ry), (rx-ry), -(rx-ry), 0];
% Actuator torque in roll, pitch and yaw with wheel disturbance, Da
Ta = Ta_LQ;

C MATLAB SOURCECODE 114
PD_control.m
%**************************************************************************************
% File control.m
%
% Calculation of the PD controller
%
% Created By Jøran Antonsen, 2004
%*************************************************************************************
function [Ta, u] = PD_control(x, xd, x_dot, xref,w_B_BI,epsilon,I,eta,eta_sat,i)
global ref_table Kd Kp eps_noise Da
% Position deviation
epsilon_dot = [x_dot(1) x_dot(2) x_dot(3)]’;
%epsilon deviation witout the use of a reference model
%epsilond = [xref(1) xref(2) xref(3)]’;
%epsilon deviation with reference model
epsilond = [ref_table(i,1) ref_table(i,3) ref_table(i,5)]’;
etad = eta_sat;
epsilon_dev = - eta*epsilond + etad*epsilon - cross(epsilon, epsilond);
if eps_noise ==1
e_noise = 0.20; %Adding 20% noise on the euler angles
epsilon_noise = [epsilon_dev(1)*e_noise*(2*rand(1)-1);
epsilon_dev(2)*e_noise*(2*rand(1)-1);
epsilon_dev(3)*e_noise*(2*rand(1)-1)];
epsilon_dev = epsilon_dev + epsilon_noise;
end
% Inertia matrix
Ix = I(1,1);
Iy = I(2,2);
Iz = I(3,3);
rox=(Iy-Iz)/Ix;
roy=(Ix-Iz)/Iy;
roz = (Iy - Ix)/Iz;
% Thruster vector
r = [0.25456 0.03787 0.363]’;
rx = r(1);
ry = r(2);
rz = r(3);
% Disturbance matrix, Da

C MATLAB SOURCECODE 115
%Da = [0 0 -hwy;
%000;
%hwy000];
% Using the Perigee vector for calculation of wo
rp_temp = 250e3+6371e3;
my_temp = 3.986e14;
wo_temp = sqrt(my_temp/rp_temp^3);
% Linearized system matrix
A=[0,1,0,0,0,0;
-4*rox*wo_temp^2, 0, 0, 0, 0, (1-rox)*wo_temp;
0, 0, 0, 1, 0, 0;
0, 0, -3*roy*wo_temp^2, 0, 0, 0;
0, 0, 0, 0, 0, 1;
0, -(1-roz)*wo_temp, 0, 0, -roz*wo_temp^2, 0];
% Actuator matrix with reaction wheel, Ba
B=[0,0,0,0,0;
-sqrt(2)/(4*Ix)*rz, sqrt(2)/(4*Ix)*rz, -sqrt(2)/(4*Ix)*rz, sqrt(2)/(4*Ix)*rz, 0;
0, 0, 0, 0, 0;
sqrt(2)/(4*Iy)*rz, sqrt(2)/(4*Iy)*rz, -sqrt(2)/(4*Iy)*rz, -sqrt(2)/(4*Iy)*rz, 1/(2*Iy);
0, 0, 0, 0, 0;
-sqrt(2)/(4*Iz)*(rx-ry), sqrt(2)/(4*Iz)*(rx-ry), sqrt(2)/(4*Iz)*(rx-ry), -sqrt(2)/(4*Iz)*(rx-ry), 0];
% Actuator matrix with no reaction wheel, Ba
% B = [0, 0, 0, 0, 0;
% -sqrt(2)/(4*Ix)*rz, sqrt(2)/(4*Ix)*rz, -sqrt(2)/(4*Ix)*rz, sqrt(2)/(4*Ix)*rz, 0;
%0,0,0,0,0;
% -sqrt(2)/(4*Iy)*rz, -sqrt(2)/(4*Iy)*rz, sqrt(2)/(4*Iy)*rz, sqrt(2)/(4*Iy)*rz, 1/(2*Iy);
%0,0,0,0,0;
% -sqrt(2)/(4*Iz)*(rx-ry), sqrt(2)/(4*Iz)*(rx-ry), sqrt(2)/(4*Iz)*(rx-ry), -sqrt(2)/(4*Iz)*(rx-ry), 0];
% Weighting matrix
W = diag([1 1 1 1 2]);
% Weighted actuator matrix B*W
Bw = B*W;
%C = [1, 0, 1, 0, 1, 0]’
C=[1,0,0,0,0,0;
0, 0, 0, 0, 0, 0;
0, 0, 1, 0, 0, 0;
0, 0, 0, 0, 0, 0;
0, 0, 0, 0, 1, 0;];

C MATLAB SOURCECODE 116
D=0;
% Cheking the poles of the system matrix A
%eig(A);
% checking the rank
%rank = rank(ctrb(A,B));
% Calculation of the K vector is only done one time at
if i == 1
% Wanted poles
p1 = -1;
p2 = -1/2;
p = [p1; p2; p1; p2; p1; p2];
%Calculation of the state feedback matrix K using the wanted poles. Variation in wo does not have much
influense on K vector, hence wo at %perigee is used.
K = place(A,Bw,p);
% Calculating the new system matrix
%A_new = A - Bw*K;
% checking that the eigenvalues or poles of A_new is the same as the wanted poles p
%eig(A_new)
% Proporsjonal ledd
Kpx = K(:,1);
Kpy = K(:,3);
Kpz = K(:,5);
Kp = [Kpx Kpy Kpz]
%Derivat ledd
Kdx = K(:,2);
Kdy = K(:,4);
Kdz = K(:,6);
Kd = [Kdx Kdy Kdz] % Multiplying a factor of 2 to reduce the overshoot.
% Disturbance matrix, Da=0 from the start
Da=[000;
000;
000];
end
% Actual torque matrix
Ba = 1/2*sqrt(2)*[-rz, rz, -rz, rz 0;
rz, rz, -rz, -rz, sqrt(2);
-(rx-ry), (rx-ry), (rx-ry), -(rx-ry), 0];

C MATLAB SOURCECODE 117
% PD controller
u = -Kp*epsilon_dev - Kd*epsilon_dot;% -pinv(Ba)*Da*w_B_BI;
% Finding the F vector by using the calculated u vector.
[Ta_Da,F]=Thruster(u,B,r,i,w_B_BI);
% Actuator torque in roll, pitch and yaw with wheel disturbance, Da
Ta = Ta_Da;

C MATLAB SOURCECODE 118
nonlinear_control_1.m
%**************************************************************************************
% File Nonlinear_control_1.m
%
% Calculation of the Feedback linearization controller
%
% Created By Jøran Antonsen, 2004
%*************************************************************************************
function [Ta, u] = control(x, xd, x_dot, xref, w_B_BO, epsilon, I, eta, eta_sat, i, wo, w_B_BI)
global ref_table Kd Kp ref_temp eps_noise
%c2 = [2*(epsilon(1)*epsilon(2)+eta*epsilon(3));
% 1 - 2*(epsilon(1)^2 + epsilon(3)^2);
% 2*(epsilon(2)*epsilon(3) - eta*epsilon(1))];
% Inertia matrix
Ix = 4.3500;
Iy = 4.3370;
Iz = 3.6640;
I = diag([Ix, Iy, Iz]);
rox=(Iy-Iz)/Ix;
roy=(Ix-Iz)/Iy;
roz = (Iy - Ix)/Iz;
epsilon = epsilon - xref’;
if eps_noise ==1
e_noise = 0.20; %Adding 20% noise on the euler angles
epsilon_noise = [epsilon(1)*e_noise*(2*rand(1)-1);
epsilon(2)*e_noise*(2*rand(1)-1);
epsilon(3)*e_noise*(2*rand(1)-1)];
epsilon = epsilon + epsilon_noise;
end
k1=0.5;
k2=2.2;
I_temp = [Ix, 0, 0, 0;
0, Iy, 0, 0;
0, 0, Iz, 0;
0, 0, 0, 0];
Q = [eta, -epsilon(3), epsilon(2), epsilon(1);
epsilon(3), eta, -epsilon(1), epsilon(2);
-epsilon(2), epsilon(1), eta, epsilon(3);

C MATLAB SOURCECODE 119
-epsilon(1), -epsilon(2), -epsilon(3), eta];
vt = [(-k1*epsilon(1)-k2*x_dot(1));
-k1*epsilon(2)-k2*x_dot(2);
-k1*epsilon(3)-k2*x_dot(3);
0];
ve = [epsilon(1)/(2*eta)*(w_B_BO(1)^2 + w_B_BO(2)^2 + w_B_BO(3)^2);
epsilon(2)/(2*eta)*(w_B_BO(1)^2 + w_B_BO(2)^2 + w_B_BO(3)^2);
epsilon(3)/(2*eta)*(w_B_BO(1)^2 + w_B_BO(2)^2 + w_B_BO(3)^2);
0];
Q_temp = [rox*w_B_BO(2)*w_B_BO(3);
roy*w_B_BO(1)*w_B_BO(3);
roz*w_B_BO(1)*w_B_BO(2);
-1/2*(w_B_BO(1)^2 + w_B_BO(2)^2 + w_B_BO(3)^2)];
%u_temp = I_temp*Q’*(-Q/2*Q_temp + 2*vt + 2*ve);
u_temp = I_temp*Q’*(-Q*Q_temp + 2*vt) - ve;
% Thruster vector
r = [0.25456 0.03787 0.363]’;
rx = r(1);
ry = r(2);
rz = r(3);
% Actuator matrix with no reaction wheel, Ba
B = 1/2*sqrt(2)*[-rz, rz, -rz, rz, 0;
rz, rz, -rz, -rz, sqrt(2);
-(rx-ry), (rx-ry), (rx-ry), -(rx-ry), 0];
u_temp(1:3);
u = pinv(B)*u_temp(1:3);
[Ta_Da,F]=Thruster(u,B,r,i,w_B_BI);
Ta=Ta_Da;
nonlinear_control_2.m
%**************************************************************************************
% File Nonlinear_control_2.m
%
% Calculation of the Lyapunov controller
%
% Created By Jøran Antonsen, 2004
%*************************************************************************************
function [Ta, u] = Nonlinear_control_2(x, xd, x_dot, xref, w_B_BI, epsilon, I, eta, eta_sat, i, wo, w_B_BO)
global ref_table Kd Kp eps_noise

C MATLAB SOURCECODE 120
% Inertia matrix
Ix = I(1,1);
Iy = I(2,2);
Iz = I(3,3);
rox=(Iy-Iz)/Ix;
roy=(Ix-Iz)/Iy;
roz = (Iy - Ix)/Iz;
% Thruster vector
r = [0.25456 0.03787 0.363]’;
rx = r(1);
ry = r(2);
rz = r(3);
% Disturbance matrix, Da
%Da = [0 0 -hwy;
%000;
%hwy000];
% Using the Perigee vector for calculation of wo
rp_temp = 250e3+6371e3;
my_temp = 3.986e14;
wo_temp = sqrt(my_temp/rp_temp^3);
epsilon_dev = epsilon - xref’;
if eps_noise ==1
e_noise = 0.20; %Adding 20% noise on the euler angles
epsilon_noise = [epsilon_dev(1)*e_noise*(2*rand(1)-1);
epsilon_dev(2)*e_noise*(2*rand(1)-1);
epsilon_dev(3)*e_noise*(2*rand(1)-1)];
epsilon_dev = epsilon_dev + epsilon_noise;
end
D=1.8;
k=0.9;%8*wo^2*(Iy - Iz);
% Lyapunov nonlinear controller
T_controller = -D*w_B_BO - k*epsilon_dev;
B = 1/2*sqrt(2)*[-rz, rz, -rz, rz 0;
rz, rz, -rz, -rz, sqrt(2);
-(rx-ry), (rx-ry), (rx-ry), -(rx-ry), 0];
u = pinv(B)*T_controller(1:3);
[Ta_Da,F]=Thruster(u,B,r,i,w_B_BI);
Ta=Ta_Da;

C MATLAB SOURCECODE 121
Thruster.m
%**************************************************************************************
% File Thruster.m
%
% Thruster allocation.
%Momentumdumper.
% Bang Bang controller.
% Calculating torque from the controllers.
%
% Created By Jøran Antonsen, 2004
%*************************************************************************************
function[Ta_Da,F]=Thruster(u,B,r,i,w_B_BI)
global F h_wheel_vec h_wheel h_wheel_dot_vec thruster_vec w_wheel_vec Cont noise
rx = r(1);
ry = r(2);
rz = r(3);
if noise == 2
%Addingnoise
p = 0.20;
u_noise = [u(1)*p*(2*rand(1)-1);
u(2)*p*(2*rand(1)-1);
u(3)*p*(2*rand(1)-1);
u(4)*p*(2*rand(1)-1);
u(5)*p*(2*rand(1)-1)];
u=u+u_noise;
end
F=u;
% Initial condition of the wheel
if i == 1
h_wheel = 0;
end
if F(1) < 0
F(4) = u(4) - u(1);
F(1) = 0;
end
if F(2) < 0
F(3) = u(3) - u(2);
F(2) = 0;
end

C MATLAB SOURCECODE 122
if F(3) < 0
F(2) = u(2) - u(3);
F(3) = 0;
end
if F(4) < 0
F(1) = u(1) - u(4);
F(4) = 0;
end
%Inertial momentum of the wheel taken from: ESEO_B_AOCS_031103_RW_DJD.doc
I_wheel = 4e-5;
% maximum wheel speed [rpm]
wheel_speed_max = 5000;
% maximum angular momentum h = I_wheel*w
%h_max = I_wheel*wheel_speed_max*2*pi/60;
h_max = 0.0209;
% F_max including reaction wheel
F_max = [0.13 0.13 0.13 0.13 h_max];
% F_max with 4 thrusters only
% F_max = [0.13 0.13 0.13 0.13];
Deadzone=0.13;
if F(1) >= Deadzone
F(1) = F_max(1);
end
if F(2) >= Deadzone
F(2) = F_max(1);
end
if F(3) >= Deadzone
F(3) = F_max(1);
end
if F(4) >= Deadzone
F(4) = F_max(1);
end
% remove when not using reaction wheel
if abs(F(5)) > abs(h_max) %

C MATLAB SOURCECODE 123
F(5) = sign(F(5))*h_max; %
end
%Angularmomentumvector
h_wheel = h_wheel + F(5);
% Momentum dumping controller
% B = 1/2*sqrt(2)*[-rz, rz, -rz, rz;
% rz, rz, -rz, -rz;
% -(rx-ry), (rx-ry), (rx-ry), -(rx-ry)];
%B_psaudo=pinv(B);
% temp = [0; 0.5*h_wheel; 0];
% F(1:4) = F + B_psaudo*temp;
% f5= -0.5*h_wheel;
% h_wheel = h_wheel + f5;
h_wheel_vec(i,1) = h_wheel;
% wheel speed vector
w_wheel_vec(i,1) = (h_wheel/I_wheel)*(60/(2*pi));
% Reaction wheel derived
h_wheel_dot_vec(i,1) = F(5);
thruster_vec(i,1) = F(1);
thruster_vec(i,2) = F(2);
thruster_vec(i,3) = F(3);
thruster_vec(i,4) = F(4);
% Wheel distrbance matrix
Da=[00-h_wheel;
000;
h_wheel 0 0];
% Actual torque matrix
Ba = 1/2*sqrt(2)*[-rz, rz, -rz, rz 0;
rz, rz, -rz, -rz, sqrt(2);
-(rx-ry), (rx-ry), (rx-ry), -(rx-ry), 0];
%Ta_thruster = B*F;
%TorquePD
Ta_Da = Ba*F + Da*w_B_BI;
%Ta_Da = [Ta_thruster(2) Ta_thruster(4) Ta_thruster(6)]’ + Da*w_B_BI;
%TorqueLQ
Ta_LQ = Ba*F + Da*w_B_BI;