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Modeling and Control of High Speed Mechanisms Using Inverse ...

Proceedings of IMECE2006

2006 ASME International Mechanical Engineering Congress and Exposition

November 5-10, 2006, Chicago, Illinois, USA

Proceedings of IMECE2006

2006 ASME International Mechanical Engineering IMECE2006-13920

Congress and Exposition

November 5-10, 2006, Chicago, Illinois, USA

IMECE2006-13920

MODELLING AND CONTROL OF HIGH SPEED MECHANISMS USING INVERSE

DYNAMIC ANALYSIS

Yanzhi Li

Centre for Power Transmission and Motion Control

Department of Mechanical Engineering

University of Bath

Claverton Down, Bath BA2 7AY

Email: enpyl@bath.ac.uk

M. Necip Sahinkaya ∗

Centre for Power Transmission and Motion Control

Department of Mechanical Engineering

University of Bath

Claverton Down, Bath BA2 7AY

Email: ensmns@bath.ac.uk

ABSTRACT

Inverse dynamic analysis can be used in designing controllers,

but the computational requirements may prevent its use in

real time. A combined piecewise linearization and off-line inverse

dynamic analysis approach has been exploited in this paper

to achieve the required computational efficiency with the convenience

of traditional linear controllers, coupled with the accuracy

and adaptability of inverse dynamic analysis.

The paper simulates a three-degree-of-freedom RRR planar

mechanism to demonstrate the techniques. An inverse dynamic

analysis of this mechanism is carried out for different desired

motions and for different operating speeds. A piecewise linearization

approach is introduced to represent the system as a multiinput

multi-output linear system with motion and speed dependent

coefficients.

A separate linearization method is developed to determine

the error dynamics off-line. A standard linear-quadratic regulator

is applied to the linearized model for the design of a feedback

controller. The robustness against external disturbances of

the proposed adaptive linearized feed-forward and feedback controller

is examined by simulated examples. Most of the computationally

demanding inverse dynamic and linearization calculations

are carried out off-line, therefore the technique offers many

potential applications involving highly complex systems.

∗ Address all correspondence to this author.

Nomenclature

(ˆ·) denotes estimate

α Speed parameter

¯v Time average value of the motor voltage

∆R 2 Margin used to detect break points in the piecewise linearization

∆x Movement of x

ˆv Estimated motor voltage value

λ j Lagrange multiplier

0 Matrix or vector of zeros in the appropriate dimension

λ Vector of Lagrange multipliers

θ m Vector of motor angles, their first and second derivatives

A N × N array of a i, j functions

B Control input coefficient matrix

b Vector of parameters in the estimation

C 1 Coefficient matrix of function of q and t

C 2 Matrix of function of q, ˙q and t

D 1 ,D 2 Vectors of q, ˙q and t

F Constraint Jacobian matrix

Q Vector of generalized inputs

q Vector of generalized coordinates

Q,R LQR weighting matrices

U Control input vector

u Control vector

V Vector of voltage values

X Matrix of independent variables in the estimation

y Desired motion specifying coordinates

z State vector

1 Copyright c○ 2006 by ASME


θ c

θ i

a i, j

Absolute angle of the platform.

Angular position of the i th motor

Functions of generalized coordinates and time

b Linearized coefficients

f j Constraint equation

i, j,k Integers

J Cost function

K Number of degrees of freedom of the required motion

L Lagrangian function

M Number of degrees of freedom

N Number of generalized coordinates

n Number of data points in the estimation

P Covariance matrix for the recursive least square estimator

Q i Generalized input

q i Generalized coordinate

R 2 Goodness of fitness measure

t time

T s () Settling time (percentage in parenthesis)

u Normalized time

v i Voltage input to the i th motor

x any coordinate

x 0 Initial condition of x

x c ,y c Cartesian coordinates of the center of platform

Introduction

Parallel manipulators for high speed applications offer many

advantages compared serial manipulators because closed loop

kinematic mechanisms lead to high stiffness and good dynamics

properties [1]. In addition, accuracy and reliability are also

improved by the elimination of cable transmissions and placing

the drive motors on the ground [2, 3]. There is increasing research

activity in the field of three or less degrees-of freedom

(DOF) Planar Parallel Manipulators (PPM). Although kinematic

problems with different combinations of joints and legs can be

successfully tackled [4–6], dynamic analysis is complicated because

of the existing of multiple closed loop chains.

Achieving accurate motion control at high speeds with

high dynamic performance requires accurate dynamic modeling

of multi-physics systems and nonlinear controllers to handle

complex dynamic interactions between system components and

other connected systems. Various modeling techniques can be

used [7–9], but the analytical development of equations of motion

is tedious and prone to errors even with the help of symbolic

programming tools such as MATLAB [10], and MAPLE [11]. A

constraint Lagrangian based approach [12] is used in this paper,

where the equations are automatically developed by the software

package Dysim [13, 14] from topological data. This is an object

oriented approach suitable for multi-phsyics systems. The model

developed can be used as part of other simulation environments

such as MATLAB and SIMULINK.

Controllers based on linear system theory do not always

function well, and various adaptive control algorithms have been

developed to improve performance. Due to the highly nonlinear

behavior of these systems, it is difficult to design a controller

to perform well for different motions and at different speeds.

Inverse dynamic analysis tools do however offer high performance

and take into account nonlinear system dynamics [14–17].

They have been used to predict the controller inputs in order to

achieve a required motion, and various feed-forward controllers

have been suggested in the literature. Inverse dynamic analysis

can also be utilized in feedback controllers, but the computational

requirements usually prevent real-time implementation.

Previous work [18] introduced an linearization approach for

the system and error dynamics of a three DOF PPM by using the

inverse dynamics analysis off-line. It was shown that accurate

linear models could be obtained for a specified motion that can

be used in real time. It resulted in three decoupled second order

equations, and the robustness under different motion speeds was

tested. Control issues using such linear models were discussed

in [19] and simulation results were presented. In this paper,

multistage recursive estimation is introduced by utilizing statistical

data to obtain more reliable and accurate piecewise linear

models for a specified motion. The same method is also applied

for the error model. The linearized error model is implemented

with a Linear Quadratic Gaussian (LQR) control algorithm

to ensure stability and improve performance.

System Dynamics

The Lagrangian equations of motion are [12]

( )

d ∂L

− ∂L N−M

+

dt ∂ ˙q i ∂q i


j=1

λ j

∂f j

∂q i

= Q i , i = 1···N (1)

The symbols are defined in the nomenclature. The Lagrangian

function represents the total kinetic energy minus the total potential

energy of the system, and can be written in the following

general form in terms of the generalized coordinates:

L = 1 2

N


i=1

N


j=1

a i, j ˙q i ˙q j +

N


i=1

a i,0 ˙q i + a 0,0 (2)

The functions a i, j s are functions of generalized coordinates and

time. Due to the use of superfluous coordinates, (N − M) constraint

equations are needed.

f j = 0 , j = 1···(N−M) (3)

2 Copyright c○ 2006 by ASME


Inserting (2) into (1) and double differentiating (3) gives the following

2N − M algebraic differential equations:

A 3

−θ 3

−θ c

[

AF

T

F 0

[ ] [ ]

][¨q D1 Q

= +

λ]

D 2 0

(4)

C 3

This can be solved for 2N − M unknowns, namely the second

derivatives of generalized coordinates and Lagrange multipliers,

and then the second derivatives can be integrated twice to obtain

the system forward dynamic response. The time history of Lagrange

multipliers is automatically calculated and can be used to

calculate the forces of constraints.

In inverse dynamics analysis, control inputs are calculated in

order to achieve a desired motion, which is specified in terms of

the second derivatives of the generalized coordinates. Assuming

that the required motion has K (≤ M) degrees of freedom, it can

be represented as follows:

A 1

B 3

C

B 1 C 2

C 1

B

θ 2

2

θ 1

Figure 1. Three degrees of freedom PPM

Table 1. Manipulator Data

A2

ÿ = C 1 ¨q − C 2 (5)

The control input vector U of dimension K can be added to the

generalized input vector with a coefficient matrix of B specifying

the location of the control action. There are various ways

of formulating an inverse dynamics model as discussed in [14],

but the general formulation can be written as follows by moving

the unknown control input vector to the left hand side in (4) and

using the desired motion as additional constraint equations:


⎣ A ⎤ ⎡ ⎤ ⎡

FT B ¨q

F 0 0⎦

⎣λ⎦=

⎣ D ⎤

1+Q

D 2

⎦ (6)

C 1 0 0 U ÿ+C 2

The above algebraic differential equations can be solved for the

second derivatives of the generalized coordinates, which can then

integrated twice to obtain the motion of the system. The required

control input vector and the Lagrange multipliers are automatically

calculated during this process.

The in-house simulation package Dysim [14] is used in this

paper. The program automatically develops the equation of motions

for the forward and inverse dynamic analysis from physical

data. It is suitable for multi-physics system, and its object oriented

properties allow the models to be used by other simulation

environments. Matlab/Simulink [10] environment is used to generate

the results in this paper.

Simulated System

The PPM shown in Fig. 1 is consist of two equilateral triangles,

and a moving platform B 1 B 2 B 3 connected to the ground

Object Length Diameter Mass Inertia

(i = 1···3) (m) (m) (kg) (gm 2 )

A i C i 0.17 0.02 0.144 0.3617

C i B i 0.15 0.02 0.127 0.2513

B 1 B 2 B 3 0.13 √ 3 - 0.331 0.1663

Center Load - 0.03 1.000 0.09

by three independent kinematic chains A i B i C i , i = 1···3. Each

chain has three independent revolute joints, and the actuators

are placed on the ground and connected to the manipulator at

A 1 ,A 2 ,andA 3 . The origin of the earth fixed coordinate system

is defined as the center of the triangle A 1 A 2 A 3 . The positions

of A 1 ,A 2 ,andA 3 are (−0.15 √ 3,−0.15) m, (0.15 √ 3,−0.15) m,

and (0,0.3) m respectively. Other relevant data for the manipulator

are shown in Table 1. The three angles θ 1 , θ 2 , and θ 3 are

driven by three similar dc-motors. Data for each motor are given

in Table 2.

The mechanical manipulator consists of 7 objects, namely

six connecting rods and the central platform. The properties of

each object and the connection information are fed to Dyism. The

program selects 21 generalized coordinates (three for each object)

for the mechanism and 3 generalized coordinates (charge

for each motor) for the motors. The total number of degrees of

freedom is 6; 3 mechanical and 3 electrical. The Lagrangian

function and the dynamic equations of motion including the 18

constraint equations are automatically developed. The program

also calculates the initial conditions of the superfluous coordin-

3 Copyright c○ 2006 by ASME


Table 2.

Motor Data

Parameter Value Unit

Power 50 Watt

Gear ratio 66 -

Resistance 0.4 Ohm

Induction 5 × 10 −5 mH

Torque Const. 8 × 10 −3 Nm/A

Back emf 8 × 10 −3 Nm/A

ates from the initial conditions of the user-selected independent

coordinates.

As a test trajectory, a point-to-point motion is used in this

paper. In order to minimize the vibration for high speed motions,

a third order exponential function is used to specify the motion.

The properties of this function were reported in [20, 21]. For the

motion of any coordinate x(t):

x = x 0 + ∆x (1 − e −(αt)3 ) (7)

This is a smooth and twice differentiable continuous function for

all values of t ≥ 0. Its first and second derivatives are zero at

the start and at the end of the motion. In addition, the motion is

controlled by a single parameter which gives complete control of

the motion speed. The speed parameter α determines the motion

speed, and related to the 1% and 5% settling times as follows:

T s (0.01)= 1.66

α , T s(0.05)= 1.44

α

The velocity and acceleration can be written as

(8)

ẋ = α∆x (3u 2 ) e −u3 (9)

ẍ = α 2 ∆x (6u − 9u 4 ) e −u3 (10)

where u = α t is the normalized time.

A straight line point-to-point motion of the platform is selected

as the desired motion. Only the acceleration-time history

is required for the inverse dynamic analysis. The Cartesian coordinates

of the center of gravity of the platform is initially set to

(-0.025, -0.025) m with an angle of θ c = 0deg. The final position

is specified as (0.025, 0.025) m with an angle of 10deg. The

motion duration is selected as 0.25 s. Equation (7) is applied to

ẍ c ,ÿ c , ¨θ c

Inverse

Dynamics

Figure 2.

V

θ m

Least Square

Estimation

Linearization process

Linearized

Model

Statistical

Data

all three coordinates of the platform, x c , y c and θ c to specify this

desired motion to the inverse dynamic modeling.

Piecewise Linearization

If the motor input voltages are selected as control inputs,

inverse dynamic analysis will generate the time history of control

voltages and the motor output coordinates (angles and their first

derivatives). Considering a decoupled linear model for one of the

motors, say the i th motor, the control voltage can be written in

the following form as described in [18, 19]:

v i (t)=b 0,i +b 1,i¨θ i +b 2,i˙θ i +b 3,i θ i (11)

If PPM is moving on a horizontal plane, the required motor

voltages at the end of the motion should approach to zero giving

the following constraint on the linearized parameters:

b 0,i = −b 3,i θ i (T s ) (12)

if the data from the simulation are stored in a discrete form, they

can be arranged in a matrix form as follows:


V = ⎢


v i (1)

v i (2)

···

v i (n)




⎦ , X= ⎢


¨θ i (1) ˙θ i (1) θ i (1)

¨θ i (2) ˙θ i (2) θ i (2)

··· ··· ···

¨θ i (n) ˙θ i (n) θ i (n)



⎦ (13)

This gives the following Least Square Estimator (LSE) of the

parameter vector b [22]:

ˆb =(X T X) −1 X T V (14)

The estimation process is illustrated in Fig. 2, where the vector

θ m contains motor angles, and their first and second derivatives.

Also the statistical data from the LSE, such as the goodness of

fit measure R 2 and the standard errors of each estimated coefficients

can be utilized to assess the accuracy of the linearized

model and significance of the estimated parameters. The goodness

of fit measure R 2 , corrected by the degrees of freedom of

4 Copyright c○ 2006 by ASME


the estimation, is defined as follows:

R 2 = 1 − n − 1 (1 − ∑n k=1 (v i(k) − ˆv i ) 2 )

n − 3 ∑ n k=1 (v i(k) − ¯v i ) 2

(15)

where n is the number of data points. If a high R 2 value (close

to 1) is obtained then the linearized model can be used in real

time implementations instead of the nonlinear inverse dynamic

analysis [18, 19] for the specified motion.

This paper extends the method to cases where the estimation

produces poor R 2 by breaking the motion duration into two or

more linearized sections. For this a recursive version of the LSE

is needed so that the estimates are updated at each time interval.

The recursive estimation of coefficient matrix at the k th time

interval, b(k) can be formulated as [23]:

LSE results (the numbers in parenthesis denote standard devi-

Parameters Motor 1 Motor 2 Motor 3

b i,1 -0.0312 0.0262 -0.0350

(0.0001) (0.0002) (0.0006)

b i,2 0.6645 0.6554 -1.2045

(0.0015) (0.0045) (0.0174)

b i,3 0.0299 0.3714 -22.358

(0.0155) (0.0495) (0.4521)

R 2 i 0.9989 0.9894 0.9251

Table 3.

ations)

ˆb(k) = ˆb(k−1)+P(k−1)X(k)[1 + X T (k)P(k − 1)X(k)] −1

×[V(k) − X T (k)ˆb(k − 1)] (16)

where the covariance matrix P is updated as follows:

V 1

(V)

1

0

−1

P(k)=[P(k−1) −1 +X(k)X T (k)] −1 (17)

The initial values of the coefficient estimates are taken as zero,

and the initial value of the P matrix is set to a very large diagonal

matrix.

The piecewise linearization break points are calculated by

an algorithm based on R 2 measurements at each time step. The

R 2 value should initially be increasing with each sampled data

until a peak value is achieved. If the subsequent measurements

of R 2 drops by a predefined margin ∆R 2 below the current maximum

value of R 2 , then the algorithm defines this point as the

break point and stops the estimation. The recursive estimator is

then initialized and started again until the next break point or end

of the motion is reached. This enables the linearized controllers

to be used in a wide range of applications involving highly nonlinear

systems and high speed motions. This process is demonstrated

by using a numerical example.

Numerical example: A sampling time interval of 0.0005 s

(sampling frequency of 2 kHz) is selected, which gives 501

sampling times including the initial values for the duration of the

motion. Straight LSE results by using the complete data set for

the simulated motion are shown in Table 3. Although a good fit is

obtain for all three motors, the R 2 value for Motor 3 is relatively

low. The calculated and linearized motor control voltages are

shown in Fig. 3. The lower R 2 for the Motor 3 can be observed

graphically from the figure.

Figure 4 shows R 2 values when RLSE is applied to the same

V 2

(V)

V 3

(V)

Figure 3.

voltages

−2

0 0.05 0.1 0.15 0.2 0.25

5

0

−5

0 0.05 0.1 0.15 0.2 0.25

1

0.5

0

−0.5

0 0.05 0.1 0.15 0.2 0.25

Time (s)

Calculated (solid) and linearized (dashed) motor control

data. As it can clearly be seen from the graphs that the R 2 value

for the Motor 3 is decreasing. A slight decrease for Motor 2 can

also be seen towards the end of the motion. When the piecewise

linearization option is selected with ∆R 2 = 0.01, no break

point was detected for Motor 1, but one breakpoint for Motor 2

at 339th sampling interval, and two break points for Motor 3 at

118th, 317th sampling intervals were detected. The calculated

5 Copyright c○ 2006 by ASME


R 2 R 2 R 2 3

2

1

1

0.8

0.6

0 100 200 300 400 500

1

0.8

0.6

0 100 200 300 400 500

1

0.8

0.6

0 100 200 300 400 500

Sampling numbers

V 1

(V)

V 2

(V)

V 3

(V)

0

−0.5

−1

−1.5

0

−2

−4

0.6

0.4

0.2

0

100 200 300 400 500

100 200 300 400 500

100 200 300 400 500

Sampling numbers

Figure 4.

R 2 values calculated at each sampling interval by the RLSE

Figure 5. Calculated (solid) and piecewise linearized (dashed) motor

control voltages

and piecewise linearized control voltages for all three motors are

shown in Fig. 5. The estimated parameters and the goodness of

fit values for Motor 2 and Motor 3 are given in table 4. A significant

improvement on the Motor 3 results is apparent from this

figure with a change of R 2 from 0.9251 to 1.0.

No attempt has been made to ensure continuity of the linearized

voltages between regions. This can be done by introducing

another constraint to the linearization process, but this would effectively

reduce the number of coefficients in the estimation thus

reducing the goodness of fit. The problem due to the jump discontinuity

could also be handled by the error dynamic loop as

discussed later.

Error Dynamics

After successfully linearizing the inverse dynamic model to

provide the feed-forward control input to the system, it is necessary

to add feedback control to take into account the modeling

errors, unknown or unmeasured external disturbances, and measurement

noise. It has been shown previously that the error dynamics

for a nonlinear system can be significantly different than

the system dynamics [18]. A linearization process is applied to

the simulated errors, and proposed to be used in the feedback for

real-time implementation. A multi-frequency disturbance is introduced

at each motor sequentially, and the required changes

in the motor input voltages are calculated as shown in Fig. 6

and used in a RLSE algorithm to calculate the coefficients of

ẍ c ,ÿ c , ¨θ c

Inverse

Dynamics

Figure 6.

SPHS

V

¨θ 1 , ¨θ 2 , ¨θ 3

d i

Inverse

Dynamics

Least Square

Estimation

Error model linearization process

V

Linearized

Model

Statistical Data

the linearized error model. A Schroeder Phased Harmonic Sequence

(SPHS) [24] is used to generate a low-peak-factor multifrequency

signal with controllable power spectrum to represent

disturbances. A coupled linear model is considered for the error

dynamics. The disturbance levels on motor accelerations were

selected as 10% of the desired acceleration. When a disturbance

is introduced to the motion of each motor one at a time, the inverse

dynamics would produce variations of control voltages in

all three motors. Therefore, a total of 9 coefficients would be estimated

from each test. This would give a total of 27 coefficients

for the error model. Figure 7 shows the calculated voltage variations

in order to compensate for the motion errors introduced to

three motors, and compares these with the results obtained from

a linearized model. The goodness of values R 2 for 3 motors are

0.8667, 0.9626 and 0.8214. These are not as good as the R 2 values

obtained for linearizing the system in the previous section,

however the error models are not used directly in the feedback,

6 Copyright c○ 2006 by ASME


Table 4. Piesewise RLSE results for Motors 2 and 3

Parameters Region 1 Region 2 Region 3

Motor 2:

Data Range 1-339 340-501

b 2,1 0.0252 0.0335

b 2,2 0.6518 -0.0797

b 2,3 0.2569 -38.38

R 2 2 0.9898 0.9999

Motor 3:

Data Range 1-118 119-317 318-501

b 3,1 -0.0682 -0.0295 0.0063

b 3,2 -0.6596 -1.0297 1.0327

b 3,3 -12.982 -19.901 41.328

R 2 3 0.9903 0.9896 0.9969

V 1

(V)

V 2

(V)

V 3

(V)

0.05

0

−0.05

0.2

0

−0.2

0.04

0.02

0

−0.02

−0.04

0 0.05 0.1 0.15 0.2 0.25

Time (s)

but to design a LQR regulator. Although this is a path tracking

problem, the error dynamics can be treated as a regulator problem

as constant zero output is desired. Most of the nonlinearities

are compensated by the feed-forward path.

Linear Quadratic Regulator

A Linear Quadratic regulator is selected to achieve a stable

and optimum performance of the linearized error model. The

LQG regulator consists of an optimal state-feedback gain, which

minimizes a quadratic cost function as follows [25]:

J(z,u)=


0

(

z T Qz + u T Ru ) dt (18)

where the state vector z corresponds to motion errors in angular

displacements, velocities and accelerations of three motors,

and the control vector u corresponds to corrections to the control

voltages. The diagonal weight matrices Q and R are selected by

using Bryson’s rule according to the maximum acceptable values

of the elements of the z and u vectors. The linearized error model

is used to form the LQR.

The overall control system is shown in Fig. 8. The controller

is tested by adding SPHS disturbance forces on the platform

along the x and y directions The results of the simulation are

shown in Fig. 9. The tracking errors in terms of x c , y c and θ c

are shown as a function of time during the motion. The results

Figure 7. Calculated (solid line) and linearized (dashed line) control

voltage variations to compensate for the disturbances in motor motions.

testing the LQR requlator control with disturbances on the plat-

Figure 8.

form

ẍ c ,ÿ c , ¨θ c

Inverse

Dynamics

V

θm

SPHS

LQG

d i

Forward

Dynamics

ẍ c ,ÿ c , ¨θ c

are also compared with the case where there is no LQR feedback

loop. The effectiveness of the suggested LQR control based on

the linearized error dynamics can be clearly seen from the figure.

Various simulation runs (not presented here) with different

disturbance levels were also made, and showed the ability of the

LQR controller to produce stable and accurate results when operating

under varying conditions.

Conclusions

In this paper, a numerical dynamic model for a three DOF

RRR PPM is built by using Lagrangian equations and the software

Dysim, which utilizes a numerical coding system. Both

forward and inverse dynamics models are used in a Simulink environment

to design linear feed-forward and feedback controllers

for real-time implementations. The numerical linearization

θ m

7 Copyright c○ 2006 by ASME


x c

(m)

y c

(m)

θ c

(rad)

4 x 10−5

2

0

−2

0 0.05 0.1 0.15 0.2 0.25

10 x 10−5

5

0

−5

0 0.05 0.1 0.15 0.2 0.25

0

−5

5 x 10−5

−10

0 0.05 0.1 0.15 0.2 0.25

Time (s)

Figure 9. Tracking errors with (solid line) and without (dashed line) LQR

under SPH disturbance forces on the table.

process is for a specified motion. A RLSE approach is used to

develop a piecewise linearized model off-line. An approach to

automatically calculate the break points is developed, and shows

that a very accurate generation of the nonlinear inverse dynamics

can be achieved.

An off-line strategy is also developed to obtain a coupled

linearized error model from the nonlinear simulations. The error

dynamics are significantly different to the system dynamics. The

path following requirements are converted to a regulator problem

in the error model, and a LQR controller is designed to ensure

stability of the feedback loop and to optimize the trajectory

tracking performance. The designed regulator-based controller

is tested by applying external unknown disturbance forces

to the platform. The simulation results demonstrate good control

performance. The suggested linear controller not only deals

with the system nonlinearities, it is also robust against unknown

disturbances for high speed path following problems. This approach

performs all the computationally demanding calculations

off-line, and synthesizes an optimized linear controller to be implemented

online.

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