Multiple Simultaneous Control Problem Its Application to ...Multiple Simultaneous Specification Control Problem and Its Application to Robot Trajectory Tracking Systems Ph.D. Thesis,

Multiple Simultaneous Specificat ion Control Problem and Its Application to

Robot Trajectory Tracking Systems

Hugh Hong-Tao Liu

A Thesis submitted in confonnity with the requirements for the Degree of Doctor of Philosophy

Department of kIechanical and Industrial Engineering University of Toronto

@Copyright by Hugh H.T. Liu, 1998

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Multiple Simultaneous Specification Control Problem and Its Application to Robot

Trajectory Tracking Systems

Ph.D. Thesis, 1998

Augh Hong-Tao Liu

Depart ment of Mechanical and Indust rial Engineering

University of Toronto

Abstract

A practical robot control design problern typically consists of several performance require-

ments, The control designer may need to take into account a few of different design specifications

together. Therefore, there exists a need for a rnethodology to address the design of a controller

to satisfy multiple specifications. The objective of this thesis is to develop a controller design

method to meet muItiple simultaneous specifications (MSS), and apply it to robot trajectory

tracking systems. -4 so called convex combination design method is proposed to solve the MSS

control problern. Through a linear system framework, the design specifications are described and

considered uniformly. W hen the specifications have t his convex property. the proposed met hod

offers a two-stage strategy that can effectively design the controiler to meet multiple specifica-

tions a t the same time. The design strategy is straightforward and easily implemented. Further,

through feedback linearization, this proposed linear design method is applied to nonlinear robot

tracking systems to solve the MSS problern. Under imperfect linearization conditions due to sys-

tem uncertainties, system robustness is viewed as one of multiple specifications to be met, and

the convex combination method can also be validated. The application is conducted through

simulation and experirnentation on one commercial robot, and the proposed design method is

verified to be effective in practice. %y arriving at the thesis objective, industry will have at its

disposal, a performance oriented automated control design procedure. The furt her impact of this

thesis is that new tasks that have never been done before may be automated.

Acknowledgement

1 wodd like to thank Professor J.K. Mills for giving me the opportunity to explore the very

exciting robot control field, and for his guidance with the work presented in this thesis.

As a friend, 1 would also like to th& him for his support and inspiration throughout my

study at the University of Toronto.

I wish to thank the people in the Laboratory for 'lonlinear Systems Control. Matthieu.

Andrew, Cris. Kate, Mingwei, Donald, Weihua, and others. They made my lab life much

more funt and 1 have become richer for the experience through various discussions with

them.

Finally, 1 wish to express my gratitude to my wife Hong and our new boni baby Man.

for their continuing encouragement and patience.

Contents

Abstract

Acknowledgement

Table of Contents

List of Figures

List of Tables

Nomenclature

1 Introduction

1.1 Thesis Scope . . . . . . . . . . . . . . .

1.1.1 Performance oriented procedure

1.1.2 Multiple specifications . . . . .

1.3 Literature Review . . . . . . . . . . . .

1.2.1 Tuning approaches . . . . . . .

1.22 Analyt ical approaches . . . . .

1.3 Thesis Objectives . . . . . . . . . . . .

1.4 Thesis Contributions . . . . . . . . . .

1.5 Thesis Overview . . . . . . . . . . . . .

vii

2 MSS Control Problem 10

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Control System Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Convex Performance Specifications . . . . . . . . . . . . . . . . . . . . . . 12

2.4 MSS Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3

2 5 Convex Combination Method . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Simulation and Experimentation Test Bed 22

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 -- 3.2 Modeling of CRS 4460 Robot . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.5 Experimental Set Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Simulation Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :32

3.5 Controller Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Robot Trajectory Control: Joint Space 38

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3S

3.2 M S S Control Problem: Simulation . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 MSS Control Problem: E-xperimentation . . . . . . . . . . . . . . . . . . 44

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4 Summary 54

5 Robot Trajectory Control: Task Space 59

.5 .1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5 Tracking in Cartesian Space . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.3 Performance Transformation Approach . . . . . . . . . . . . . . . . . . . . 60

5.3.1 Robot control system framework . . . . . . . . . . . . . . . . . . . 62

5.3.2 Convex specifications . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.3.3 Experimental resdts . . . . . . . . . . . . . . . . . . . . . . . . . . 67

. . . . . . . . . . . . . . . . . . . . . . . 5.4 Control Transformation Approach 71

. . . . . . . . . . . . . . . . . . . 5.4.1 Robot cootrol in Cartesian space 72

. . . . . . . . . . . . 5.4.2 Performance specifications in Cartesian space 73

5.4.3 Experimental resdts . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Robot Trajectory Control: Robustness 86

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.2 Robot Control Under Systern Uncertainty . . . . . . . . . . . . . . . . . . 86

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Robust Specifications 9'1

6.3.1 Discussion of condition (C-1 ) . . . . . . . . . . . . . . . . . . . . . 96

6.3.2 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.5 Summ- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7 Convex Optimization Application 107

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.2 blSS Problem Application . . . . . . . . . . . . . . . . . . . . 108

7.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11s

. . . . . . . . . . . . . . . . . . . . 7.5 Initialization of Cutting-plane Method 119

. . . . . . . . . . . . . . . . 7.6 Cornparison of Performance Oriented Met hods 1%

7.7 Sumrnary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

8 Conclusions and Discussions 134

. . . . . . . . . . . . . . . . . . . . . . 5.1 Convex Combination hlethodology 134

8.2 Application to Robot Tracking Systems . . . . . . . . . . . . . . . . . . . . 135

List of Figures

1.1 (a ) Traditional control design procedure . ( b ) Performance oriented control

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . design procedure

2 . 1 Linear system framework . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 Performance oriented control design flow chart . . . . . . . . . . . . . . . .

3.1 CRS A460 Robot mode1 illustration . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Major Frames of Reference

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 CRS '4460 industrial robot

3.4 ROBOTOOL interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.0 Robot control schematics . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Cascade realization

3.7 ParalIel realïzation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1 Output response of unit step input and noise input: (a) under sample con-

troller Gi . (b) under sample controller Gz. (c) under sample controller G3.

. . . . . . . . . . . . . . . . . . and (dj under convex designed controuer G

4.2 Actuator response of unit step input and noise input: (a) under sample

controller Gi, (b) under sample controller G2? ( c ) under sample controller

G3? and (d) under convex designed controller G . . . . . . . . . . . . . . . .

. . . . . . 4.3 Joint positions el, 02, d3 vs time with convex designed controller

vii

. . . . . . 4.4 Joint velocities & ? gZr ë3 VÇ time with convex designed controller 58

. . . . . . 5.1 Trajectory in Cartesian space with performance transformation 82

5.2 Trajectory in Cartesian space with performance transfomat ion: 3-D grap hics 53

. . . . . . . . 5.3 Trajectory in Cartesian space with controller transformation 84

5.4 Trajectory in Cartesian space with controller transformation: 3-D graphics 55

. . . . . . . 6.1 Joint positions O3 vs time with convex designed controller 105

. . . . . . 6.2 Joint velocities el. f&? & vs time with convex designed controller 106

7.1 Subgradient of convex function . . . . . . . . . . . . . . . . . . . . . . . . 121

. . . . . . . . . . . . . . 7.2 Response: xl. x2 vs time~vithdisturbanceiÇI= 1 131

7.3 Response: x1 . x2 vs time with disturbance &1 = 10 . . . . . . . . . . . . . . 13'2

. . . . . . . . . . . . . . . . . . . . 7.4 H, noms under different control gains 1:3:3

viii

List of Tables

3.1 CRS A460 Robot mode1 parameters . . . . . . . . . . . . . . . . . . . . . . 25

. . . . . . . . . . . . . . . . . . . . . . . 3.2 Analysis of Controiler Realization 36

. . . . . . . . . . . . . . . . . . . . . . 4.1 Robot simulation mode1 parameters 39

. . . . . . . . . . . . . . . . . . . . . 4.2 Joint specificat ions simulation results 43

. . . . . . . . . . . . . . . . . . 4.3 Joint specificat ion test: combination vector 52

. . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Joint specification test results 53

. . . . . . . 5 Cartesian specifications test with performance transformation 69

. . . . . . . . . 5.2 Cartesian specification test with controuer transformation 79

. . . . . . . . . . . . . . . . . . . . . . 6.1 Robust specification test: trajectory 100

. . . . . . . . . . . . . . 6.2 Robust specification test: sample controller results 102

7.1 Cutt hg-plane optimization results with diflerent initialization approaches . 12'7

Nornenclat ure

Hi- i .i

link 2 length of CRS A460 Robot

cos(Bi). where Bi is the joint angle of the i-th link of a robot

mani pulator

COS( oi + e, )

COS(^^) where ai is the twist of the i-th link of a robot manip-

ulator

the symrnetric. positive definite inertia matrix of an n DOF

robot manipulator dynamic equat ion

link 3 offset of CRS -4460 Robot

the error vector of the joint angles

the error vector of the joint velocities

subgradient of a conver function

one t e m of an n DOF robot manipulator dynamic equation.

which includes the centripetal. Coriolis. and gravitationai t e m s

closed-loop transfer matrix from external input w to s p t e m

output z in the control system frame (2.1)

a set whose element is a closed-loop transfer matrix H in the

control system frarne (2.1)

transformation matrix that t ransforms the coordinates and ori-

entation of the link i reference frame into the i - 1 reference

frame coordinates and orientation

Hos transformation matrix t hat transforms the coordinates and ori-

entation of the link (6) reference frame into the {O) reference

frame coordinates and orientation in CRS ,4460 Robot

H& = H& - H06 error transformation matrix. which is the differece between the

desired transformation rnatrix H& and the actuaI one !Iolos

1 transformation matrix that transforms the coordinates and ori-

entation of the tool {TI) reference frarne into the (6) reference

frarne coordinates and orientation in CRS A160 Robot

HLvo transformation matrix t hat t ransforms the coordinates and ori-

entation of the link {O) reference frame into the {CC') reference

frame coordinates and orientation in CRS A460 Robot

H L ~ ~ ~ ~ transformation matrix t bat transforms the coordinates and ori-

entation of the tool reference frame {Tl) into the {CC') reference

frame coordinates and orientation in CRS A360 Robot

H = H l - K T error transformation rnatrix. which is the differece between the

desired t ransfocmat ion mat rix H&-l and the act ual one HcyTi

Z a set

1 . . Inn identity matrix with dimension n x n

J Jacobian matrix

h- E Rn" X n ~ controller transfer matrix from actuator y to controller u in

the control system frame (2. L)

K p = diag(kPi) diagonal proportional control matrices! whose elements kP*. i =

l. . . . . n7 are positive gains

heV = diag(k&) diagonal deribative control matrices. whose elements k,;. i =

I l . . . , n, are positive gains

diagonal integral cootrol matrices. whose elements /tri. i = 1. . . . . n.

are positive gains

the vector which results by combiniog the centrifugai. Coriolis

forces, gravit ational forces. and viscous and Coulomb friction

terms. with respect to the Cartesian space

the inertia mat rix descri bed in Cart isesian space

plant. system t o be controlled

transfer matrix from external input w to system output r in

the control system frame (2.1)

transfer matrix from controller u to system output r in the

control system frame (2.1)

transfer matrix from external input w to actuator y in the

control system frame (2.1)

transfer matrix from controIIer u to actuator y in the control

system frame (2.1)

position vector that transforrns the origin coordinates of link i

frame into link i - 1 frame origin

position vector that transforms the origin coordinates of link

(6) frarne into link {O) frame origin

position vector of H$

position vector that transforms the origin coordinates of the

tool { T l ) frame into link (6) frame origin. where d E = 13.97

inches

Pivo = (O O d B ) = position vector that transforms the origin coordinates of link

{O) frarne into the world {CI;) frame origin. where d B = 1 3

inches

xii

position vector that transforms the origin coordinates of tool

{Tl ) frame into the ivorld { W ) frame origin

position vector of H&Tl

rotation matrix that transforms three axes of the link i coordi-

nate frame into the link i - I frame axes

rotation matrix that transforms three axes of the Link 6 coor-

dinate frame into the link O frame axes in CRS A460 Robot

rotation t ransforrnat ion matrix of H$

rotation matrix that transforrns three axes of the tool coordi-

nate frame {TI) into the world { W ) frarne axes in CRS A460

Robot

rotation transformation matrix of H&-TI

compact notation of matrix K( I - P,,K)-' in the closed-loop

cont rol system t r a d e r matrix (2.2)

a set whose element is a real number

a set whose element is a positive real number

a convex set

sin(Oi). where Bi is the joint angle of the i-th Link of a robot

rnani pulat or

sin(& + 8,) sin(ai). mhere ai is the twist of the i-th link of a robot manip-

ulator

controller output vector in the control system framework: Fig-

ure '2.1

external input vector in the control s-stem framemork: Figure

r = ( t y i)=€R3 thevectorofend-effectorpositions

= ( I y f )' E R3 the vector of end-effector velocities

y controller input vector in the control system framework: Figure

z system output vector in the control systern framework: Figure

a. expected design specification value

! \ = (X I ..... x,)~ cornbinationvector.whereXi zO.C:=,Ai= 1

30 set of subgradients of the convex function o

o functional performance specificat ion

baaj objective function

op-4 specificat ion: pat h accuracy

oiv;l specificat ion: velocity accuracy

o s s ~ p specification: steady state error of path

OROBL:S+ specificat ion: robust ness

@ = {O,} combination rnatrix. where oij is specification ai value under

cont roller Ii,

Q = ( a . a ) expected specification vector. where ai is the expected value of

specificaiton oi

rL. functional performance specificat ion: constraint function

9.6 . ë E Rn the robot joint angle vector. the joint velocity vector. and the

joint acceleration vector respectively

9d?éd?ëd E Rn the desired robot joint angle vector, joint velocity vector. and

joint acceleration vector respectively

T , the vector of applied generalized forces (torque or forces) acting

at the local joint torque side.

~ e r l the externai disturbance described in joint space

T, the vector of Cartesian force input

T,a the external disturbance in ta& space

r, the calculated torque based on the known s-s tem dynamics

model parameters

Ï the external disturbance on t h e torque side

Notations

defined by. denoted by

end of proof

norm

for a vector z E RnI it is an Euclidian norm: I I s ~ ~ ~ = /=: for a rnatrix

-4 E RnXm. it is d e h e d by: 11-4112 = omoz(--l). where a(A) represents the

singular d u e of rnatrix -4

H, norm

for ail. any

convolu tion operation

for vector r. y E Rn, x 2 y represents that each element of t h e vector

satisfies the inequality: xi 2 yi. 1 5 i 5 n.

Acronyms

DDC Direct Digital Control

DOF Degree Of Freedom

LTI Linear Time Invariant (System)

MSS Multiple Simult aneous Specificat ions

PD Proport ional-Derivat ive (cont rol)

PI Proport ional-Integral (control)

PID Proportional-Integral-Derivative (control)

RMS Root Mean Square

ZOH Zerc-Order Hold

Chapter 1

Introduction

1.1 Thesis Scope

f he robotics industry has experienced a sustained growth surge for the past few J-ears. By

1997. more than 70.000 robots urere estimated to be at work in L7.S. factories (RIA. 1997

[ill]). The robotics industry in North America is now a 'j: 1 billion business. With 10.000 or

more new robot orders annually received by U S . based robotics companies. and enormous

untapped opportunities still amilable throughout the rvorld. the robotics industry will be

one of the most important global industries of the '1st Century. In order to stay on the

leading edge in today's global market and competit ion. the robotics indust ry wants to offer

better performance and to ~ r o v i d e more reliable technology to suit a mide-range of tasks

that cannot be performed as effectively by manual labor or fixed automation. Among those

many different engineering techniques, the robot controller design procedure is of interest

to the author of this thesis1. The control problem for a robotic system is the problem of

deterrnining the time history of joint inputs required to cause the end-effector to execute a

commanded task (Spong and Vidyasagar, 1989 [XI]). The purpose of a controller or control

'In the rest of this thesis, unless otherwise specified, "the author of this thesis" is abbreviated as "the

authot" .

Chapter 1 Introduction -7 -

law design is to improve. or in some cases enable the performance of a system. rvhere the

term performance broadly describes how the robotic system is required or desired to behave.

1.1.1 Performance orient ed procedure

Since the invention of the programmable robot in 1950's. proportional-clerivative (PD).

proportional-integral ( PI). and proportional-integral-derivat ive ( PID ) control have been

the dominant control technology in practice. Such a traditional control design procedure

as shown in Figure l . l (a) operates as follows: Set the control structure (e.g.. PD. PI. or

PID). the control designer tries to tune the gains to meet the performance specifications.

This design strategy is easily implemented and cost-effective. due to simple control struc-

tures. However. when the performance requirements are changed. the control has very

little flexibility to be customized to meet a wide range of different specifications. due to

the structure and the control gain limits. Moreover. i t takes time and requires experience

in finding proper gains.

COIcTROL DESIGN

COhTROL DESIGN

Figure 1 .l: (a) Traditional control design procedure. (b) Performance oriented control

design procedure

A natural thought of improvement is to offer a custorner based. performance oriented

cont rol design procedure, as shown in Figure 1.1 (b ) : When the performance specificat ions

Chapter 1 Introduction 3 - -

are formulated. a controiler can be systematically derived to meet the design specifications,

as apposed to empiricai gain tuning.

1.1.2 Multiple specificat ions

The control design procedure is to meet system performance. A practical robot control de-

sign problem typically consists of several performance requirements. For example. in order

to complete a trajectory tracking task, the control designer may need to take into account

a few factors at the same t ime, such as tracking error, speed of tracking, and robustness to

mode1 uncertainty or extemal disturbances. Therefore, there exists a need for a method-

ology to address the design of a controller to satisfy multiple performance specifications

simult aneously.

Based on information of the above two subsections, this thesis is concerned with develop

ment of a performance oriented controller design rnethod to address multiple simultaneous

specifications, and its application to robot trajectory tracking systems.

1.2 Literature Review

Many controller design approaches have been proposed to improve performance in the robot

trajectory control area. These approaches are loosely divided into two categories: tuning

approaches and analytical approaches.

1.2.1 Tuning approaches

In a tuning approach, we imply that for a given control structure, the controller param-

eters rnust be somehow chosen or "tunedn. The design is based on a particular choice

of a control law with adjustable gains (parameters). These approaches include the well

Chapter 1 Introduction 4 - - - - -- -

known PD control, PID control; the so called computed torque control that is a PD control

through the feedback linearization technique: adaptive control where the parameters are

variables adjusted on-line by a self-tuning mechanism; and parameter optimization meth-

ods, amongst others. For instance, Arimoto and Miyazaki (1984 proved the asymptotic

stability under local PID control. While PID/PD control is adequate in most position

control applications such as pick and place tasks, it is not expected to deal effectivel- with

the dynarnic path-following demands in trajectory tracking tasks. Perfect tracking in the

sense of zero tracking error can be achieved through computed torque control. which has

been developed for many years by various researchers. and given an exposition in Baines

and Mills (1995 [4]). Recently, Qu (1996 [Ml) showed that under bounded s-stem mode1

uncertainty, the computed torque control can make the system robust if the proportionai

derivat ive gains are properly selected. Lihen t here exists system parameter uncert â i n t - the

trajectory tracking problem can be addressed by using adaptive control (e.g.. Slotine and

Li. 1991 [dg]. Colbaugh and Seraji. 1994 [9]. and Misawa. 1992 [XI). where an adaptation

law is constructed to learn explicitly unknown parameters. By using nonlinear parameter

optimization programming techniques. Park and h a d a (1993 [39]) and Rai and Asada

(1993 [12]) search for the optimal structural variables and control panmeters (PD control

gains) to opt imize the settling time.

Tuning approacheç make it possible to treat a few diffeïent design specifications. such as

stability, tracking error, and robustneçs in a trajectory tracking problem? if the control gains

are properly chosen. In many cases? the gains are adjusted (tuned) empirically. Even when

the parameten are designed through optimization rnethods. it is a hard task to design an

iterative optimization to converge quickly. and it may also be hard to guess a "reasonablen

initial value to start the optirnizat ion algorithm. Another significant disadvant age of tuning

approaches is the following: Even if a satisfactory controller of the selected structure is

found, the designer does not know whether a dramatic improvement in performance woüld

Chapter 1 introduction ., 3

--

be obtained by using some other controuer structures. In other words. controllers ore not

effectively designed through tuning approaches to meet performance criteria.

1.2.2 Analytical approaches

By an andytical approach, we imply that the controller is derived as an analytical solution

to a sort of control equation. These approaches include: siiding control, L yapunou 'S direct

method, H, theo y, linear quadratic (LQ) optimai control. and so on. For example. Çlotine

and Li (1991 (481) addressed the trajectory tracking problem that tolerates disturbances or

dynamic uncertainties. Hanmancllu and Pandian (1993 (131) proposed a generalized mode1

based sliding controller, which incorporated the fdl-order, nonlinear uncertain actuator

dynamics in control law design. One drawback of this control approach is that it exhibits

chattering when implemented in pract ice (Su and Leung, 1993 [Ml). L yapunov's direct

method has become one of the most important tools for nonlinear system andysis and

design. Some robus t nonlinear controllers are directly derived from the Lyapunov stability

theory, and it is usually applied to the stability robustness problem in the face of parameter

uncertainties (e.g., Spong, 1992 [19]. Koo and Kim, 1994 [KI). 4 controller can also be

derived to achieve the robustness requirement by using H , theory. such as in Rotstein

(1997 [G]).

Generdly speaking, analwytical approaches try to derive a controller based on perfor-

mance. However, they only address very specific performance niteria due to specific control

equations. For example, the controller derived from the equation of the sliding condition

keeps the trajectory on the sliding surface, but it does not contribute to other performance

specifications like overshcot. Another popular approach belonging to this category is to

cast the control problem as a classical optimal control question: minimization of a cost

function with or without constraints. Many approaches have been developed by Linear-

Quadratic (LQ) optimai control theory to improve performance. For exâmple, time-optimal

Chapter 1 Introduction 6 -- -

control (Bobrow 1985 [7]) minimized the time needed to perform a given task, subject to

the constraints imposed by the actuators. A time/energy optimal control (Shiller. 1994

[4'ij) rninimized a scalar function of many but finite variables with equality and inequâlity

constraints. These approaches minimize an LQR-Like cost function. Unfortunately, the cost

functions may not be directly related to the performance specifications. In that case. even

when an optimal control law has been designed, a better performance using some other

control law may be possible.

In conclusion, the developmen t o f a performance orien ted con troller des& met hodology

to solve the multiple sim ul taneous specification (MSS) con trol problem is an open research

topic. The reason could be due to any of the foflowing: 1) The tuning approaches are

not performânce oriented. They involve empiricd tuning procedures. or have to face the

dificulties associated with nonlinear parameter optimization problems. 2) The ê n e i c d

approaches only derive the controller for specific performance criteria. not generaJy for

multiple specifications.

1.3 Thesis Objectives

From an industrial point of view, a successful control design m e t h o d o l o ~ to address the

multiple simultaneous specifications (MSS) may operate as follows: Whenever the multiple

design specificat ions are defined by the system engineer based on the custorners require-

ments, the control engineer will either directly derive a controller to achieve these design

specifications simultaneously, or tell that the required specifications are too tight to be met

at the same time at the control design side. In the latter case. this information of the lirnit

of control engineering is very usefd to the system design. -4 modification on the structure

(mechanical, electrical, or chernical, etc.) may be considered a t this stage as one solution, or

Chapter 1 Introduction 7

the relaxation of some specifications may be suggested if the whole system performance can

be maintained. With the help of such a performance oriented control design methodology,

the industry wilI have at its disposal, an automated system design procedure.

Recently, a new design approach, called convex optimization theory, has been discussed

in the control Literature (Boyd and Barratt , 1991 [S]). This theory shows that many design

specifications have a simple geometric convex property. Under this condition, a linear

controller can be analyt i c d y derived using the convex theories ( Rockafellar . 1970 [44] ) .

This research work is of interest to us due to the fact that under a linear system framework,

many different design specifications are described uniformly. Therefore, we may be able to

benefit from it to address multiple specifications.

On the other hand, we dso note that in the robotics literature, i t is well known that

the state equation of an n degree of freedom (DOF) rigid robot can be globally linearized

and decoupled by the method of feedback linearization (as an example, refer to Spong

and Vidyasagar, 1989 [SO]). That makes it possible to apply linear design methods to

the robotic system. The computed torque control (linear local PD controller) has some

successful applications (e.g., Baines and Mills, 1995 [4]). Spong (1957 [XI) proposed a

linear controller to address the robustness problem by using the factorization approach

(Vidyasagax, 1985 [%I).

To the best of our knowledge, no research work has been done on automated perfor-

mance oriented design method for the MSS control problem, especially in the robot control

area. This lack of work in this area and the advantages inherent in the convex optimization

theory and feedback linearization technique motivate this P h.D. research.

The objectives of this thesis are:

1. To develop a generic performance oriented design methodology to solve the MSS

control problem (i.e., to meet multiple control specifications sirn~ltaneously)~ or to

Chapter 1 Introduction 8

determine that the specifications are too stringent to be achieved (we cd1 it an MSS

feasibility problem). By adopting the convex concept from the convex optimization

theory, we propose a so called convex combination method to solve the MSS problem

effectively. Furthemore? this rnet hod offers a simple judging criterion to address the

MSS feasibility problem. The design strategy is straightforward and easily imple-

mented.

2. To apply the proposed performance oriented control design rnet hodolog?; to robotic

systems to address the MSS problem in trajectory tracking tasks. The feedback

Linearization technique makes it possible in the application of such a linear control

method to the nonlinear robotic systems. under the assumption of perfect lineariza-

tion. However. there always exist unnoticeable and unknotvn aspects of the real

system in the mathematical model, which are accounted for by the notion of uncer-

tainty. We further address the MSS control problem of the robot trajectory tracking

systems under uncertainty. The robustness issue can be viewed as one of multiple

specifications to be met simultaneously As a result. it validates the application of

our proposed control method for the MSS problem.

3. To verify the effectiveness of the proposed design methodology through simulation

and experimentation on a commercial CRS Robotics Corporation A460 robot.

1.4 Thesis Contributions

By arriving at its objectives, this thesis will rnake contributions as follows:

The MSS control of a robot trajectory tracking system is a practical design problem,

and as far as we know, is a new resexch topic in this area.

Chapter 1 1ntrod.uction 9

0 A generic performance oriented control design methodology is developed to solve the

MSS problem. It offers a practicd design procedure and simple criterion to address

the feasibility issue. The design strategy is straightforward and easily implemented.

0 The developed rnethod can offer better performance than current methods used to

control robots. Further, this methodology even permits new tasks to be automated

that had never been done before.

The developed method has strong potential in industrial applications. The impact

on industry is that industry will have at its disposal, an automated control system

design procedure.

1.5 Thesis Overview

The outline of the remaining of the thesis is as follows. Chapter 2 presents the definition of

the MSS problem and the developrnent of the proposed performance oriented control design

method, the so called convex combinat ion theory. Chapter 3 introduces the simulation and

experimentation test bed. In chapters 4 and 5. the proposed convex combination method is

applied to solve the MSS problern for a six (6) degree of freedom (DOF) robot manipulator.

The problem is described in local joint space, and ta& space respectively. Chapter 6

discusses the robustness problem under system uncertainty. As a cornparison. Chapter 7

illustrâtes the application of convex optimization theory to a simple example. Through

discussions of its limitations and difficulties, it is further shoivn that our proposed design

method perforrns better in the MSS control problern. Finally. Chapter 8 offers conclusions

and discussions.

Chapter 2

MSS Control Problem

Introduction

In this chapter. we formulate the multiple simultaneous specification (RilSS) control prob-

lem, and develop a new performance oriented controller design method to solve this prob-

lem. The rest of this chapter is organized as foilows. Section 2 establishes a generic

framework, where the multiple specifications are described uniformly. In Section 3. the

concept of convex design specificat ions is int roduced. Section 4 present s the formulation

of the MSS problem. The main research done by the author, the development of a perfor-

mance oriented design met hod, the so c d e d conuez combination method. is presented in

Section 5. Finally, Section 6 offers a summary of the author's research work in this chapter.

2.2 Control System Framework

It is well known that any linear tirneinvariant (LTI) system can be formulated in a uniform

frarnework (e.g.. Stoorvogel, 1992 [53]) as s h o w in Figure 2.1.

Actuator input u E RnY denotes the signals generated by the controller. and al1 other

inputs are denoted by the vector w E Wu. The measured signai vector? denoted by

Chapter 2 rWS Control Problem I l

Controiier K(s)

Figure 2.1 : Linear sys t em framework

y E Rny. consists of those output signals that are accessible to the controller. The output

signals of interest to the designer are lumped into the vector z E Rn=. In the frequency

domain, the system framework is represented by the following t ransfer mat rices:

cont roller U ( S ) = K ( s ) =*y(s) (2.1)

closed - loop Z ( S ) = H ( s ) --w(s)

where Pm E Rnzxnw is the transfer matrix from w to z; Pz, E RnZXnu is the transfer

rnatrix from u to z : P,, E Rny Xnw is the transfer rnatrix from w to y; P, E R n y Xnu is

the transfer matrix from u to y; K E RnuXny is the controller transfer matrix from y to

U; and H E 7 P X n w is the closed-loop tramfer matrix from w to z.

The mathematical mode1 of the system to be controlled (plant P) includes information

about the system. Assume that the controiler to be designed is ais0 LTI, it is generally

represented by the transfer matrix K ( s ) . With this uniform description, the system input-

output relationship is described by the closed-loop transfer matrix H ( s )

Chapter 2 MSS Control Problem 12 - - --

t hen the closed-loop t ransfer mat rix H can be rewri t t en compact iy as

H = Pz, + PzuRPy, (2.4)

Rernark: For a nonlinear system plant, it is possible to mite its 1inea.r part into the uniform

framework, while considering the nonlinear part as an input signal W. Therefore. the effects

of plant nonlinearities c m be accounted for in this framework. The details will be given in

the following chapters when we discuss applications to (nonlinear) robotic systems.

2.3 Convex Performance Specifications

The term performance broadly describes how the control system is required or desired to

behave. The design (performance) specifiations are mathematical functions that techni-

cally represent the performance requirements. Given the mathematical mode1 of the control

system, the functional specifications are associated with system information.

Definition 2.1 (Specifications) A performance specification or design specification is a

function mapping from a system set 2 to a real value, denoted by 4 : 1 -t R+.

Note that the system set Z may have different choices. as long as it contains the con-

trol system specification information. Given the uniform framework description (2.1): one

obvious choice of Z is the set of the closed-loop transfer matrices in (2.2):

Unless particularly mentionedl, we assume Z = 7l in the rest of this thesis.

'In some cases, for example, when the controller K has a fiaed structure, such as PD control and PID

control, the specifications can be described as functions of the controller parameters. In such a case, the

set Z can be chosen as the set of those parameters (gains). An example will be given in Chapter 7.

Chapter 2 MSS Control Problem 13

Many control design specifications have a cornmon convex geometric property (Salcud-

ean, 1986 [46])*. In this thesis. we will focus on designing controllers for convex performance

specifications.

Definition 2.2 (Convexity) A specifiation function 4 : 1 -+ R+. is called a convex

function, if for a n y x, I E Z, and constant X (0 5 X 5 1), it always satisfies:

2.4 MSS Problem Formulation

The functional specification descriptions (2.1) and (2.2) also imply that a.li performance

specifications can be considered simultaneously as functions of H, which are evaluated

under every different controller hp(s).

Assume n convex specificat ions are to be sat isfied simultaneously

where a; ( i = 1,. . . , n) denotes the expected specification value, which quantifies the

satisfactory performance. Then the MSS control problem can be defined as follows.

Definition 2.3 The ~~ control problern is fonalired as: design a controller K ( s ) such

that the specifications in (2.7) hold sirnultaneously. CVe cal1 such a controller a satisfactory

controller.

'In the following chapters, we wiii also show Chat many design specifications are convex in the robot

control systerns.

2.5 Convex Combination Method

This research develops a new design method. to address the MSS problem. -4 so called

conuer com6ination method is proposed to derive a satisfacto- controller by cornbining

exist ing cont rollers (or cont rol techniques).

Definition 2.4 (Sarnple controller) Consider the design probfern roith n simultaneous

specifiations (2.7). -4 controller Ii(s) in (2.2) is cafled a sumple controller if there exists

one specification that can be met under this controller: O, < a, (j E [ l . n ] ) .

The objective of the convex combination method is to find n different sample con-

trollers. each of which satisfies a t least one specification respectivelx then design a con-

troller through the convex combination of the ciosed-loop transfer matrices corresponding

to each of n sarnpliog controllers such that it satisfies (2.7) simultaneously. This objective

can be achieved thro~lgh the following proposed two-stage design s t r a t e s :

1: Sample controller. Select n sample cont rollers: Ki. h;. . . . . K, such t hat each con-

trouer h;. satisfies at least one specification di. Such sample controllers can be selected bq-

using any existing linear control design approach (e-g. PID control) for one specification

each time. 'iote that how to select the sarnple controller seems quite challenging. In fact. it

lessens much burden on the controller designer when he/she only needs to place emphasis

upon one specification each time. When the sample controllers are obtained. calculate the

n specificat ion values under every sample controller: 4, = oi( H, ) is the specificat ion

value under the control structure Hj, where H j is the closed-loop transfer matrix with the

controller 4. Therefore: if 4, 5 ai. where ai is the required specification value. it implies

that controller ICj meets the specification di. Otherwise (di j > ai) . the specification di is

not met by this controller.


II: Combination. Find the combination vector A through solving a linear prograrnming

problem

@ - A 5 9 (2-8)

T where B = {dij},A= [Xi, . . 3 O,C:='=l Xi = 1,9 = [al ,..., a,] , (1 5 i t j 5 n).

If the inequality (2.8) is solvable, a satisfactory controller is derived through the convex

combination of n closed-loop transfer matrices with the combination vector A. The main

result is given by the follow ing t heorem.

Theorern 2.1 Assume A' = [A; A; . . . is one solution O/ (2.8). then there exists a

controller h" such that the corresponding closed-loop transfer matriz H' satisfies:

Such a satisfactory controller K*(s) could be derived through the following constructive

proof of this theorem.

Proof Under each sample controller h;., the closed-loop transfer matrix Hi can be

calculated from (3.2),

Combine the n sample closed-loop t r a d e r matrices

H a = X;Hl +A;H2 + ...+ XiH,

Since A' is a solution of ( Z S ) , it implies that for every i (i = 1,. . . , n)? 4iiX; i @ i 2 X ; + . . . + 5 ai, and the specifications is a convex function. This leads to

Chapter 2 MSS Control Problern 17

Proof Combining the constraints together. we c m rewrite Inequality (2.8) as follows:

where eT = [ 1 1 . . - 1 1. From Lemma 2.1. we have the conclusion that this inequality is

inconsistent if and only if there exists a vector y = [ uT vT k1 k2 IT 3 0. U, v E Rn. kl, kz E

R, such that

It follows that

= O and

From - kl > u T ~ > 0, we set k2 - kl = A > O. Substitute back to the above inequality.

we have

uT@ < A 5 A + Uj = U T p j j - l c . . . n (2.19)

It leads to the conclusion that the inequality (2.8) is inconsistent if and only if there

exists a u E Rn 3 O such

In other words, the inequality (2.8) is consistent if and only if there exists no such

vector u to satisfy (2.15).


Therefore, the convex combined closed-loop transfer matrix H' satisfies the specifica-

tions (2.7). Note from (2.3) that the correspondence between R and K is one-to-one. the

satisfactory controller K' c m be derived from its closed-loop transfer matrix H' as follows:

Sheorem 2.1 shows that the convex combination method is successful as long as the

inequality (2.8) is solvable. Nowl the only question left is when this inequality has a

solution. The following lemrna is one classical conclusion in linear programming theory.

Lemma 2.1 (Consistency) For any A E RmXnl b E Rn, The linear inequality

is consistent Le., it has solution x E Rn i f and only if for any y E Rm 1 O , if y T . ~ = O,

then I.J'~ > O . On the other hand, this inequality is inconsistent (has no solution) i f and

oniy i f there ezists a y E Rm 2 0 , such that yT.4 = O and yTb < O (Webster, 1994 [56]).

Applying the above lemma, we find a simple criterion for judging the consistency of the

inequality (2.8):

Theorem 2.2 Inequality (2.8) is consistent i f and only ifthere ezists no uector u E Rn 2

0 , such that

zuhere 9, = [ [ l j #2j ... & j ]*.

From the consistency conclusion, we can further address another important aspect of

the MSS problem: feasibility.

Definition 2.5 (Feasibility) The MSS problem defined in Definition 2.3 is feasible, if

there exists a controller K to satisfy the design spec i f i t ions (2.7) simultaneously. On the

other hund, the MSS problem is called infeasible, if such a satisfactory controller does not

exist.

ControiIer design. Lke aU engineering design, involves trade-offs. When a wide range of

performance specificat ions are considered simult aneously, the coexist ing specificat ions may

confikt with each other. For examplel generaily speaking, larger actuator signals lead to

smaller output signal errors. However, with limits on the actuator output (constraints).

regdation performance may oot be achieved. Therefore, whether or not the performance

specifications can be achieved simultaneously is also important information to the designer.

When the MSS problem is found to be infeasible, the designer's option is to suggest that

the specifications be relaxed.

From the consistency conclusion (Theorem 2.2). we have a result to address the feasi-

bility problem.

Theorem 2.3 If there ex2sts a vector u E R; such that

where 9 = [ Oi . . . , , 1 , and 4; = b i ( H ) is the i-th specification with respect to the

closed-loop transfer rnatrix H, then the MSS problem shoen in Def i i t i on 2 3 is infeasibh.

The proof of this theorem is a natural outcome of Theorem 2.2.

Proof Suppose the MSS problem is feasible, then there exists a controller K', its

corresponding closed-loop transfer matrix is H', and the performance specifications (2.7)

Chapter 2 MSS Control Probiern 19

are satisfied sirnultaneously: df 5 ai. i = 1.2.. . . .n. Let 9' = [ Q; os . . . O;, lT. and

V = [ al, a* .. . a, ]*. it follows that uTg- 5 uT*. for any v E R;. Set c = u. then we

have minuT9 5 uTV' 5 uTQ. contradictory to (2.21). This conflict shows that the SISS

problem is infeasible.

Remark: Consider (2.21). Ive notice that the feasibility problem bas a similar form to a

typical optimization problem. Mathematically. when the objective function is convex. the

minimum value of (2.21) constructs a Pareto optimal boundaq (Rockafellar. 1970 [+il). I t

states t hat exactly one of tw-O alternat ive assert ions must hold: the performance specifica-

tions are eit her achievable, or unachievable. divided by t his boundary. Thus identification

of this boundary can be cast as convex optimization problem. Therefore. rve show that the

feasibility conclusion obtained here (Theorem 2.3) is t h e same as the related conclusions in

(Boyd and Barrat t. 199 1 [SI). Severt heless. discussion of the feasibility problem completes

our proposed convex combinat ion t heory3.

T hrough oiir proposed convex combinat ion met hod. the performance oriented cont rol

design procedure is shown in Figure 2.2: Establish a control system framework. so t hat

a wide range of different design specifications c m be described uniformly and considered

simultaneously. Whenever a set of requirernents is specified. we can a p p l ~ the convex

combination rnethod to meet these specifications at the same time. mhere in the sample

controller stage, it lessens much burden on the designer when hefshe onl - needs CO focus

on one specification each time, and takes ad~antage of many existing control approaches:

3Unfortunately. to solve the inequality (2.21) one bas to face the difficulties of the conves optimization

methods as discussed in details in Chapter 7. Therefore, in the following research, wve will focus o n

applications of the conves combination theory to derive the controller to solve the 3ISS problern. assurning

it is feasibte.

and in the combinat ion stage the satisfactory controller is systemat ically derived and it

guarantees t hat the multiple specifications are sirnultaneously satisfied. Furt her. if the

required specifications are too stringent to be achieved, the designer determines that the

specifications be relaxed.

Figure 2.2: Performance oriented control design Born chart

2.6 Summary

In this chapter we addressed how to solve the MSS control and its feasibility problern. Our

research work in this chapter is summarized as follows.

We developed a new performance oriented control design method. the convex corn-

bination method, to solve the MSS problem and its feasibility problem (Liu. 1996

[29]).

Our proposed convex combination method is generic and easily implemented. It has

strong potential application in the robot industry and other areas.


By applying the proposed convex combination method. industry will have an auto-

mated design procedure as shown in Figure 2.2.

Chapter 3

Simulation and Experiment at ion

Test Bed

Introduction

The Laboratory for Nonlinear Systems Control (LXSC) at the University of Toronto is

equipped with a CRS Robotics Corporation -4460 industrial robot (see Figure 3.3). C\ë

will apply our proposed convex combination control design method to this robot to address

some b1SS problems in the hrajectory tracking taslis. In this chapter. we ivill present this

re-engineered commercial robot test bed. on rvhich our simulation and experimentation is

conducted.

The rest of this chapter is organized as follows. Section 2 briefly introduces the kine-

matic and dynarnic mode1 of CRS A460 Robot that kvas developed at LXSC (Baines. 1995

[.j]). and presents some derivations that is prepared by the author for the esperiments to

be conducted. Section 3 presents the e'rperimental set up that kvas also developed at LXSC

(Mills, 1995 [36]). Section 4 introduces a simulation software that is developed by the au-

ITrademark of CRS Robotics Corporation, Burlington, Ontario, Canada.

Chapter 3 Simulation and Expe rimentution Test Bed 23

thor, to evaluate any new and cornplicated control algorithm in an environment simulating

the CRS A460 Robot. In section 5 . the aut hor presents some of his research and develop

ment work during the control law implementation. Finally. section 6 offers a summary of

the aut hor's research work in this chapter.

3.2 Modeling of CRS A460 Robot

The CRS A460 Robot is a six degree of freedom ( DOF) mechanical manipulator configured

ivith six revolute joints as illustrated in Figure 3.1.

Figure 3.1: CRS A460 Robot model illustration

The arroivs shown in this figure indicates the positive sense of joint rotations. This

robot model possesses a spherical wrist defined by intersect ing mrist axes at a single point.

The major frames of reference employed by the CRS Robotics Inc. is presented in Figure

i3.2.

These frames include the World Frame {W): the Base Frame {B). the Zeroth Frame

{O) to the Sixth Frame (61, the Flange Frarne {T f) , and the Tool Frame {Tl} defining

the tip of the end-effector (CRS, 1990 [IO]).

G h a ~ t e r 3 Si.mulation and Ezperimentation Test Bed 24

Figure 3.2: Major Frames of Reference

Chapter 3 Simulation and Ezperimentation Test Bed 25

Kinematics deals with the geometry of robot Link motion with respect to a fixed refer-

ence coordinate systern frame { W ) as a function of time without regard to the forces that

cause the motion. The kinematics of any manipulator c m be formulated by defming a Ho-

mogeneous transformation matrix through the weil known Denavit-Hartenberg convention.

The matrix transforms the coordinates and orientation of the link i reference frame

into the i - 1 reference frame coordinates and orientation:

and link offset di of the CRS A460 Robot is shown in Table 3.1.

Table 3.1: CRS A460 Robot mode1 parameters

Link Twist Length Offset

a (deg) a (in) d (in)

1 +90 O O

The transformation matrix Hi-ls can also be denoted by a rotation matrix and

Chapter 3 Simulation and Ezperimentation Test Bed 26

a position vector

&-i.i Pi-1.i (3.2)

where transforms three axes of the Link i coordinate fiame into the link i - 1 frame

axes, and transforms the origin coordinates of iink i frarne into link i - I frame

origin.

In our test bed, we set the World Frame {W) coincident with the Base Frame {B) and

has a fixed relationship with the Zeroth Frame {O). W e dso set the Flange Frame {Tf)

and the Tool Frarne {Tl) such that they have a h e d relationship with the Sixth Frame

( 6 ) . The transformation matrices become:

where dB = 13 (in) and d E = 9.5 (in).

The transformation matrix Hm has the following formula (Baines. 1995 [SI):

Chapter 3 Simulation and Experimentation Test Bed 27

P& = azclcz - d 4 ~ 1 ç 2 3

Po6 = C I ~ S L C - - d 4 ~ 1 ~ 2 3

Pi6 = a2s2 + d4c23

where = COS(@^). c;j = COS(^^ + 0,)- si = sin(Bi). sij = sin(Oi + 8,).

In our experiments. ive also need to calculate the .Jacobian matrix. the inverse .Jacobiân

matrix. and its deriwttive matris. We derive these matrices as follows.

Frorn = ~ ( 6 ) 6 . the Jacobian matrix J ( 0 ) becomes:

-aîslc2 + d4slsz3 -a2cl.s2 - & c ~ c ~ ~ - ( & c ~ c ~ ~

J = a2clc2 - d4cLss3 - a z s p 2 - d4s 1c2:3 -&s1cZ3

1 O

The inverse Jacobian becomes

where

Chapter 3 Simulation and Expen'mentation Test Bed 28

The derivative of J, J

The dynamic behavior of manipulator Links is described in terms of the time rate of

change of the link configuration in relation to the joint torques exerted by the actuaton.

This relationship is expressed by a set of digerential equations. called equations of motion.

that describe the dynamic response of the manipulator to forcing functions applied at the

joints. The equation of motion of our CRS A460 Robot Test Bed is developed by the

Lagrangian formulation, which is a generic dynamic mode1 for rigid manipulators (Luh.

1983 [33]). In free motiont the dpamic equation for an n degree-of- freedom manipulators

is given as:

~ ( e ) j + h(0: O) = ro (3.2'7)

where O, &, 6 E 'R" ore the joint angle vector, the joint velocity vector, and the joint accel-

eration vector respectively. D E RnXn represents the symmetric, positive definite inert ia

matrix; and the vector h(0 ,8) E Rn includes the centripetal? Corioliso and gravitational

terms. rd is the vector of applied generalized forces (torque or forces) acting at the local

joint torque side. The detailed derivations refer to Baines (1995 [j]).


Based on this model, we will conduct simulation and e-xperimentation CO solve some

MÇS control problems by applying the proposed convex combination method.

3.3 Experimental Set Up

The Laboratory for Nonlinear Systems Control at the University of Toronto is equipped

with a CRS Robotic Corporation -4460 industrial robot, with six degrees of freedom (see

Figure 3.3).

Figure 3.3: CRS A460 industrial robot

The robot is of duminun construction and has six revolute joints including a spherical

wrist with a payload of 3 kg. Each joint is actuated by a DC servo motor through a

H m o n i c Drive, which provides a gear transmission ratio of 100:l. A spherical wis t is a

wrist configuration whose last three axes of rotation intersect at a single point defined as

the wrist center. Such a coniiguration allows a decomposition of the six DOF knematic

computation into two sets, three DOF each, of kinematic computations. It separates the

orientational coordinates, controlled by the last three joints' local PID controllers, from the

positional coordinates, controlled by the first three joints' local user defined control laws.

Chapter 3 Simulation and Eqerimentation Test Bed 130

In this experimentation, we focus on the positioning trajectory, therefore. the controuer is

designed and implemented on the first three joints, keeping the local controllers on the last

t hree joints as commercid PID controllers.

The CRS robotic system includes the commercid (2500 robot controller. It is a multi-

processor real time controller that makes use of an IXTEL 80286 processor and two INMOS

805 transputers. The tramputers are capable of hosting multiple processes simultaneously.

working on one process at a time according to the specified priority of the processes. Each

transputer has four communication links that allow easy expansion of the system by adding

more transputers to the network.

The industrial robotic system provided by CRS has undergone many modifications to

facilitate the implementation of different control laws. and help with the research performed

at the Laboratory for Xonlinear Systems Controi. Tramputers have been added to the con-

trollers. as well as a Windows based interface that runs on a 486-66 Mhz personal computer

(PC). Another transputer was implemented in the PC circuitry to handle the communica-

tion between the PC and the controller. This e-xpansion of the network was necessary to

handle the large number of computations penerdly required when implementing complex

control laws used in research. control laws developed in areas such as neural network and

force cont rol.

The fist three joints have been equipped with torque sensors constructed of two pain

of strain gauges mounted on the outer perimeter of the flexsplines of the harmonic drives.

The torque sensors measure the actual torque output fiom the harmonic drive which is used

in an inner torque feeciback loop. This work is dso reported in MLlills e t a1 (1993 [:36]). The

torque sensors feed their signais to a torque signal conditioner, and thus: each controller

network has been equipped with an A/D module to sample the joint torque data.

The detailed description of this systern can also be found in Bines (1995 [SI). Nguyen

(1995 [38]), and Laliberte (1996 [Hl).

Chapter 3 Simvlation and Experimentation Test Bed 31

3.4 Simulation Software

The robot simulation is an essential step towards test and verification of robot control

laivs and performance ~~ecifications. In this section, we discuss the development of a

simulation software, ROBOTOOL. which is compatible with our experimental test bed

setup. Such a sophisticated software system is necessary for the user to evaluate any new

and complicated control algorit hm before experimentation. Addit ionally, it gives the tester

better understanding and confidence, which makes the test procedure smoother during the

experimental implementation. The development consists of software architecture, robot

kinematic and dynamic model, and control law strategies (Liu and Mills, 1997 [XI). This

LVindowsT" based software, as s h o w in Figure 3.4. offers user-friendly interface: and is

also associateci with M A T L A B * ~ and S I M U L I N K ~ ~ for simulation. For details refer to

Liu (1998 [32]).

R O B O T O O L Hugh T. Uu and J.K. Mllls

Labirraîoiy for Nonlinear Systems Control University of Toronto

Figure 3.4: ROBOT00 L interface

Choptez 3 Simulation and Expen'mentation Test Bed 32

3.5 Controller Implementation

In the following chapters, we will present, in detail, how to apply the (linear) convex

combination met hod to (nonlinear) robotic systems to address the MSS control problem.

The whole control schematic is shown in Figure 3.5.

Figure 3.5: Robot control schematics

Xote that even when the controller is derived based on the performance specifi~ations~

our work isn't done yet. As shown in Figure 2.2, we need to irnplement the designed

controller into a digital control code to control the robot in real time. In t his section. we

discuss the implementation of a controller into our CRS A460 test bed.

A real-time control system reads input from the plant and sends control s ipals to the

plant to produce the correct response within a definite time Lirnit, which is determined by

plant operational considerations (Bennett, 1994 [6]). When the digital computer is utilized

as a control device in the feedback loop to interface directly with the plant, the system is

referred to as a direct digital control (DDC) system (Houpis and Lamont, 199'2 [16]). In

this section, we discuss how to implement the designed controller into a digital control code

(Liu, 1997 [30]).

In most cases, as we will show in the following chapters, the derived controller through

the proposed convex combination theory to solve the MSS problem is a continuous high-


order linear transfer function. To irnplement it into a digital control law. it oeeds to be

discretized. There is a very extensive range of methods for time discretization. such as

forward, backward. or bilinear approximation, pole-mapping approach, and substitution

approach, amongst others. The most popular and widely applied approach is the s ~ c a l l e d

zeroorder-hold (ZOH) approach: Given a trmsfer function of the controller G ( s ) , it can

be discretized by using the r transformation through a zeroorder-hold (ZOH) represented

by its transfer function (1 - e -ST) / s

where 2 denotes the z-transformat ion.

When the controller is discretized, it has to be realized? i.e.. it is converted into a

computer algorithm (DDC code). In the following, several techniques for realization are

discussed ( Leigh. 1992 [19]).

Direct Method 1. The controller transfer function can be expressed as the ratio of two

It is converted into a difference equation, which c m be directly implemented in computer

code

Direct Method II. Assume, as before, that the transfer function c m be expressed as

in equation (3.29), then in Direct Method II, the difference equation is formulated by

introducing an auxiliary variable P(z) such that

and


Therefore, two difference equations are obtained:

Cascade Realization. If the transfer function is expressed as the zerwpole-gain fom,

it can be expressed as the product of simple block elements of k s t and second order as

shown in Fig. 3.6.

Figure 3.6: Cascade realization

Each elernent can be converted to a difference equation using Direct hlethod 1:

and the overd algorithm is a set of difference equations.

ParaIlel Realization. If the transfer function is expressed as the residue form

it can be expressed as the summation of simple block elements of first and second order as

shown in Fig. 3.7.

Each element caa be converted to a difference equation using Direct Method 1:

and the overd algorithm is the set of difference equations.


Figure 3.7: Parallel realization

State-Space Equation. If the controller G(z) is represented as the state-space equation:

t his difference equation can be direct ly implemented.

Exarnple: CVe are concerned with the implementation of the controller using the proposed

convex combinat ion t heory. For example. the control equation on the fist joint is ( refer to

Chapter 4):

v = h',(t)el + &(t)él

where el is the joint 1 position error: el = 4:-BI; è1 is the joint 1 velocity error: é1 = jf-6,;

fi,, hTV are the derived controllers by using the proposed convex combination theory, which

are represented by the following transfer functions:

Chapter 3 Simdation and Experimentation Test Bed 36

Realization I

Parallel

Realization

Cascade

Realizaiion

State-Space

Equation

which are described

Table 3 -2: Anaiysis of Cont roller Realizat ion

controlfe r position

by using ZOH approach:

uelocit y 1 controller

In order to compâre different realization approaches. we monitor the controller signal

v to e d u a t e how weil the controller algorithrns are implemented. Using the same ex-

perimental joint position and velocity data, the off-line computation shows the theoretical

controller. The norm of the error between the theoretical and actual signal v values is set

as one evaluation index. .&O, to complete the algorithm at each sarnpling tirne. the count

of floating point calculations involved is estimated. This floating point operatioos (flops)

is set as another evaluation index. We choose our sampling time as Ts = 3ms or T, = 5ms.

From the implementation test, we conclude that:

Chapter 3 Simulation and Experirnentation Test Bed 37

O The selection of sampling time depends on the floating point operations of the real-

ization algorithm. The larger the number of flops, the longer should be the sampling

interval. For example, the state-space equation realization algorithm requires 45 flops

to finish the computation. When the sampling time is not long enough (T' = 2rn.s):

it leads to incornplete calculations of the cont roller. which means the correct control

signal cannot be fed back to the robot within the sampling interval. Therefore. the

control system does not meet the tracking requirements as expected.

0 When the sampling time is selected to guarantee the cornpletion of the controller

realization algorithm, the smaller the sampling i n t e rd . the better is the achieved

performance.

a When the sampling time is fixed, the fewer the floating point operations. the better

is the achieved performance.

3.6 Summary

In this chapter. we introduced the simulation aod experimentation test bed. which is equip

ment in the Laboratory for Nodinear Systems Control at the Eniversity of Toronto. The

contributions of the Ph-D. research discussed in this chapter are the following:

a W e developed the simulation software, such that the controller c m be tested and

verified even before experimentation, which saves implementation time (Liu and Mills

(211). The avâilable simulation data will dso be very useful during test malysis.

We discussed the implementation of high-order linear controller transfer functions into

direct digital control codes. Some practical issues were addressed. The compaxison of

different implementation approaches helps the designer to make better choices (Liu,

1997 [30]).

Chapter 4

Robot Trajectory Control: Joint

Space

4.1 Introduction

In this chapter, we discuss how ta apply the proposed convex combination theon;. to robotic

systems to solve the MSS control problem in trajectory tracking tasks. When the tracking

specifications are described in local robot joint coordinates or joint space. the controller is

designed and implemented in joint space. W e wiil discuss such cases in this chapter. and

address the case of tracking in Cartesian space in the following chapter.

The rest of this chapter is organized as follows. Sections 2 and 3 present the robot

control framework through feedback linearization? which makes it possible to apply the

linear convex combination rnethod to nonlineu robotic systemç. In section 2. a specific

MSS control probiem is solved and verified by simulation results: which are obtained using

our developed ROBOT00 L simulation software. Section 3 addresses another MSS control

problem? which is illustrated by the experimentation on the CRS A460 Robot Test Bed,

introduced in Chapter 3. Findy, Section 4 offers a summary of our research work in this

chapter.

Chapter 4 Robot Trujectory Control: Joint Space 39

4.2 MSS Control Problem: Simulation

Consider the dynamics of a 3 Degree-of-Freedom (DOF) robot manipulatorl

where 8.&8 are the joint angle vector. the joint velocity vector. and the joint acceleration

vector respectively. D represents the symmetric, positive definite inertia matrix: and the

vector h(0. e ) includes the centripetal' Coriolis. and gravitational terms. rd is the vector

of applied generalized forces (torque or forces) acting at the local joint torque side. The

system parameters are listed in Table 4.1.

Table 4.1: Robot simulation mode1 parameters

1 Twist (degree) 1 90 / O 1 -90 /

For illustration of the proposed convex combination method, this robotic system is sirn-

plified through application of the feedback linearization technique (Spong and Vidyasagar.

1989 [SOI), by assuming that perfect decoupling and linearization is obtained2.

lThe reason for considering a 3 DOF robot dymamics equation for simulation is to follow the experi-

mental robot setup that has 6 degrees of Çeedorn including a spherical wrist, where the computation of the

rnanipulator position can be simpiified through the first three (3) joints. For details refer to Chapter 3. 'As a matter of fact, perfect linearization cannot be achieved. There always exist system uncertainties.

To validate the proposed hear method, robustness is an important issue. We d l address it in Chapter 6.

Chapter 4 Robot Trajectory Control: Joint Space 40

where v is the linear controiler to be designed, and d represents a Gaussian white noise

with intensity o2 = 10, then the linearized system3 h a . the transfer function for each joint

with zero initial values Oi(0) = O i ( ~ ) = O. The systern plant is decoupled.

where

Therefore, the controller design on the three joints c m be independent. For simplicity.

we only consider the design problem on one joint d i . Rewrite this Linear system in the

uniform framework as shown in Figure 2.1 and equation (2.1) with

where r = d t is the desired trajectory, and v = u + d. The open-loop transfer matrix wiU

O P P

0 o : i

and the controller is

3The reasoo using the term, v - 2e - 6, is that the plant transfer Function P,, in the d o r m framework

(2.1) will be stable, which will be used to address the internai stability later in this section.


The closed-loop transfer rnatrix H can be obtained

Consider the following four (4) performance specifications:

(1) The step response (relative) overshoot from the reference signal r to y, must be less

Q1(H) P sup y,(t) 5 1.2

t > o

(2) The output to the process noise must be regulated within -5.48%

(3) The actuator effect u to the process noise must be limited within 20%

(4) The closed-loop system (4.9) must be internaily stable, Le.. Hw7 Hgd . Hure and Hud

must be al1 stable transfer functions. Since the transfer function P is stable (4.5): the

systern is internally stable, if and only if R gf G/(l + PG) is stable. Note that since

any convex combination of stable R keeps its interna1 stability. Ive focus on designing a

stabilizing controller to meet the above three design specifications &. &. simultaneousl~

through the convex combination method.

Theorem 4.1 Performance specifieations dl , & are concez funetions in ternzs of the

closed-loop transfer matriz H .

Proof Assume r is a step input, the time response of y, is

Chapter 4 Robot Trajectory Control: Joint Space 42 - - --

where h,, is the tirne-domain map of Hy-. For any H a . H ~ , denote the step response by

y," and respectively. Let H be the convex cornbination of Ha and H~ wit h O < X < 1 :

H = AHa + AH'. where h = 1 - A? the corresponding step response is

because of the linearity. Therefore, we have

which proves the convexity of specification di according to Definition 2.2. On the other

hand. we know that the system root rnean square (RMS) specification can be cornputed in

frequency domain by Parseval's Theorem (Francis, 1987 [12]), the regdation specification

& then becomes

1 O0

p?(H) = [ R M S ( ~ ) ] ~ = -/ SdlH,d(ju)12d~ Ln m

where Sd represents the power spectral density of noise d. For any Ha, H b . denote the

transfer function between y, and d by H> and H:, respectively. Let H be the convex

combination of Ha and H~ with O < X < 1: H = AHa + X H ~ . ive have

which proves the convexity of specification &, according to Dehi t ion 2.2. Identically,

specification & is a1so a convex function.

Chapter 4 Robot Trajectory Control: Joint Space 43 - - -

Now a e can apply the proposed convex combination rnethod to solve this kfSS control

problem. First, select the stabilizing sarnple controllers to satisfy one specification at one

t ime:

(1) Select sample controller to satisfy d l : Gi = 5 + i. (2) Select sarnple controller to satisfy &: Gz = 5 + $.

( 3 ) Select sample controller to satisfy Q3: G3 = 2 + y. The calculated specifications and the expected values are listed in Table 4.2.

Table 4.2: Joint specifications simulation results

Note that each sarnple controller satisfies one specification. However. no sample con-

troller c m satisfi d the three specifications at the same time. Based on these calculated

data? the convex combination vector is obtained through linear programming:

Specification

functions

0 1

d2

d3

and the designed controller becomes:

where a0 = 3.68, al = 30.27, a2 = 138.03, a3 = 426.40, a4 = 945.38, as = 1533.92, a6 =

1'794.06,a7 = 1452.02,ae = 766.26,ag = 241.73,alo = 39.42,all = 2.5, and bo = l.O,bl =

8.01b2 = 36.32, b3 = 111.59,b4 = 245.46, b5 = 394.08, b6 = 450.0, bî = 314.51,b8 =

161.93, b9 = 4.01, bto = 3.58: bli = 0.0.

Controller

Gi

1.1301

0.0017

0.0509

Controller

G2

Controller

G3

Expected

specijicat ions

1.4563

0.0015

0.0856

1.2

0.003

0.04

0.9.512

0.0046

0.0146

t

1 . -


The corresponding specification values are

The simulation results verif'y the effect iveness of the proposed convex combinat ion

method (Liu and Millsl 1997 [25]). The output response y, and actuator response u due to

the unit step input and white noise input are shown in Fig. 4.1 and Fig. 4.2 respectively.

4.3 MSS Control Problem: Experimentation

The proposed convex combination theory is verified through simulation. In this section:

we further apply it to a robot trajectory system in an experimental environment: the CRS

Robot Test Bed.

Consider the dynamics of a n degree-of-freedom (DOF) rigid manipulator:

where 0 E Rn is the vector of joint positions; 6 E Rn is the vector of joint velocities;

D(6) E RnXn is the inertia rnatrix; h(8 ,8) E Rn is the vector which results by adding

the centrifuga1 forces, Coriolis forces, and gravitational forces expressed in the robot joint

space; Te is the vector of joint torque input; Tda represents the disturbance or uncertaint-

such as noise and friction.

Through application of the feedback linearization technique (Spong and Vidyasagar.

1989 [SOI):

7 , = o(e)[ëd + VI + h(e, e ) (4.18)

where Bd is the vector of desired joint positions, and the vector v represents for the linear

controller to be designed, the complex highly coupled nonlinear dynamics of the manipu-


lator is replaced by a simple set of second-order linear difFerentia1 equations:

where q = D - ' ( B ) T ~ ~ is considered as an externd disturbance input. Therefore? it is highly

advantageous from a control viewpoint to consider the linearized system and to quantify

the performance.

Assume that the expected trajectory is generated by its acceleration function r ( t ) =

ëd(t) . Let p = [pl = [O elT. q = [q, q2]T = [Od é d l T Then the linearized system can

be described as the state space equation

where

W e can equivaiently describe this linear system in the frequency domain

where

When the linear control Iaw utilizes feedback of the joint position and velocity errors,

it has the following form:

Describe this linearized control system in the uniform framework (2.1). By defining

Chapter 4 Robot Trajecto y Control: Joint Space 46

we have

The experimental testbed: CRS A460 Robot, at the Laboratory for Nonlinear Systems

Control, University of Toronto, is a 6 DOF robot including a spherical(3 DOF) wrist. where

the computation of the manipulator position can be simplified through the first three (3)

joints. For details refer to Chapter 3.

Define point O with O1 = 0: e2 = 90, 83 = -90 (deg); point 1 with O1 = 20. = 70. 83 =

-70 (deg): and point 2 with Ol = 40, O2 = 50, B3 = -50 (deg). The desired robot trajectory

is the joint movement as follows: stop at each point, move from point O to point 1, then

to point 2, and then back to point O through point 1. The performance criteria are chosen

according to the American National Standard (1992 [II) as follows. Path accuracy (PA) is

a measurement of the distance between a reference path (desired trajectory) and any given

attained path. Velocity accuracy (VA) is defined as the error between the programmed

speed and the attained speed. The performance of â robot under motion conditions is

important in understanding the "steady state of error (SSEP) performance. While trying

to meet the above input-output functional specifications, the designer needs to keep the

system internally stable. For the closed loop transfer matrix H as in equation (2.2), the

Chaptet 4 Robot Trajectory Control: Joint Space 47

system internal stability (1s) requires that Pz, K ( I - P,,. K)- l , K( I - P, K)-' Pm, K( I -

P,, h')-l, ( I - P, K) -' are a.Il stable. We also call the linear controller K stabilizing if t his

controller makes that the closed-loop system internally stable.

Through the feedback linearization technique, the robot dynamics can be globally lin-

earized and decoupled. Assume the linear control law v utilizes feedback of the joint position

and velocity errors, the linearized system with inputs r, r ) , measured signals B . & & control

o? and outputs O, O is rewritten in the uniform framework (Figure 2.1).

With the uniform description of the linear system. we define the design specifications as

the functions of the closed-loop transfer matrix H (4.30) mapping from input w to output

z (4.27). The functional description of path accuracy (PA) is defined as the square of the

l2 n o m (Desoer and Vidyasagar, 1975 [Il]) of the joint position error

In the sarne way, the velocity accuracy (VA) is dehed as

The steady state error of position is given by the following function

The specification of internal stability (1s) will be addressed later.

We show that the above four specifications are convex functions of the closed-loop

transfer matrix H.

Theorem 4.2 Specifiations $pa, 4va, &SEP, and IS are convex functions.

Proof Suppose Ki, & are two stabilizing controllers. i.e.


are both stable. For any X E R, obviously

is stable. When P, (5.17) is stable, the closed-loop system is internal stable if R is stable.

It turns out that internal stability (1s) is an &ne (convex) specification.

From (-5.19), uve know that p(s) = H(s)w(s) . Suppose

It can be transformed into time domain

where * denotes convolution. Accordingiy, the vector of positions B. the first part of the

vector p. becomes

For any X E [O, 11, and A = 1 - A, the convex combination of the closed loop transfer

rnatrix H is

H = x H " + X H ~

0 ( t ) = h l ( t ) * w ( t )

= (Xhf ( t ) + Ah!(t)) * w ( t )

= A(h:(t) * w ( t ) ) + h ( h l ( t ) * w ( t ) )

= xea(t) + Xeb(t)


Therefore, the path accuracy (PA)

#PA = IlW) - od(t)ll2

= IIP~W - q1(t)Il2

= IPPW + X d ( t ) - q M I 2

= I I ~ P W - q i w + J(~b,(t) - d t ) ) 1 I 2

5 ~ I P W - q1(t)1I2 + Xllp;(t) - qdt) I l2

= A 4 , + X O ~ , ,

is a convex specification in terms of the closed loop transfer matrix H. Identicall- the

velocity accuracy (VA) can be proven to be a convex specification as weil.

On the other hand. equations (4.41.4.42) have the following f o m in s-plane

PW = H f ( s ) w ( s ) (4.46)

pbi(4 = H:(s)zL~(s) (4.47)

then, B(s) = p&) = Xpy(s) + Xpl(s). Therefore. the steady state error of path (SSEP)

QSSEP = I t+= W W ) - o d W I

is a convex specification in terms of the closed-loop transfer matrix H.

Now, the convex combination design method c m be applied to solve this MSS control

problem. We set the expected specification values in Table 4.4.

Chapter 4 Robot Trajectory Controlr Joint Space 5 O - - -

Sample Controller. Each ~ a m p l e controller is selected to meet one specification. -4ny

Linear control law can be applied. as long as one specification can be satisfied. In this

experimental test, we select the computed torque control as the sample controller. The

control structure of the cornputed torque is

where Kp = diag[kpii] and Kv = diûg[kGi] are diagonal matrices whose elements are

positive constants (gains). Comparing wit h equation (4.18) the lineâr sample controller u

is the local PD control. For each joint i = l7 2.3, the closed-loop transfer matrix (4.30) has

the form

Hi =

and the characteristic equation is

Since the gain values of kpii and kGi are positive. the roots of this equation axe con-

tained in the left hâlf of the s-plane. Therefore- the system is ioternally stable. When

selecting al1 sample controllers as cornputed torque structure (with different gain values).

the sampled closed-loop systern is internally stable. Select the sample controller to satisfy

one specification at each time:

(1) Select sample controller a to satisfy dPa: kiïi = 800. kz i = 60. A&) = s2(s2 + ktiis + k k i ) .

2 2 (2) Select sample controuer 6 to satisfy +VA: ci = 600, kLi = 80, & ( s ) = s ( s + kLis + k:,i).

(3) Select sample controller c to satidy $ssEp: kiii = 800, k z i = 40, A&) = s2(s2 + kLis + k;ii).

Chapter 4 Robot Trajectory Control: Joint Space 51 - - - - -

Knowing that interna1 stability (1s) is an affine specification. it turns out that after the

convex combination. the find designed closed-loop system is:

with the characteristic equation 4(s) = &(s) &(s) il,(s)? which is also internally

stable. Thus. in this experimentation. IS is always guaranteed. Now we only need to

design controllers to meet <ep-4, @SSEP-

From experimental results. calculate the specification values under each sample con-

troller. The results are also shown in Table 4.4. Note that each sample controller satisfies

one specification (srnaller than the expected value). However, no sample controLler can

satisfjr ail the three specifications at the same time (compare the results of columns "Con-

troller a,b.cY with the contents of column "E-xpected specifications' in Table 4.4). In fact.

it is not hard to adjust the PD gains to treat one specification. but it is difficult and te-

dious to tune the PD gains to try and meet the simultaneous specifications. because they

appear to codlict each other in ferms of the control gains (Liu. 1997 [:31]). CVe apply

the coovex combination method to design a controller to meet these three specifications

simult aneously

Combination. Based on the collected data, run the Lineu programming algorithm to find

a feasible combination vector X as in (2.8) (see Table 4.3).

When the combination vector is found? a satisfactory controller for each joint is derived

t hrough the convex combinat ion (2.13) :

Chapter 4 Robot Trajectory Control: Joint Space -5 2

Table 4.3: Joint specification test: combinat ion vector

The solutions show the fact that the designed controllers have new control structures

(here each controller is a 4th order transfer function)? which camot be achieved by any

-.

conventional design approach.

Implement ation The CRS robotic system includes a commercial CJOO robot cont rolleler ,

which is a multi-processor real time controller that makes use of an INTEL 80286 proces-

Controller 6)

sor and four INMOS 505 transputers. as well as a Windows based interface that runs on a

Joint Controller c

486-66 MHz personal computer (PC). This network can hândle the Large number of corn-

putations required when the cornplex complex control laws (4.53 - 4.58) are implemented

and executed at a rate of 500 Hz.

The experimental results tested under the new controllers are shown in Figures 4.3.

4.4. The calculated specifications are listed in Table 4.4: in which each column of 'Con-

troller a,b,cV shows the specification values under sarnple controller a, b. or c: the column

Controller a

"Expected specifications" lists the desired specification values; and the column "Actud

designed specifications" represents the specification d u e s under the convex combined con-

When focusing on one specification, it is easier for the designer to find a (sample)

Chupter 4 Robot Trujectory Control: Joint Space 5 3

Table 4.4: Joint specification test results

.- Specification 1 Controller a 1 Controller b 1 Controlkr c 1 Expected 1 Joint

1

Actual

designed

s pecificat ions

I- n controller to meet it. Compare the contents of columns -Controller a.b.cZ with the column

"Expected specifications". ive note that each sample controiler meets one specification each

time, but no one c m meet the four specifications simultaneously. By combining the different

sample controllers, a nerv control structure is generated, which cannot be designed by any

convent ional tuning approach, t O meet the four simultaneous specifications. In conclusion,

the convex combination method is effective in treating multiple simultaneous specifications,

and the proposed controller design procedure is automated. The method also offers an

opportunity to combine existing control techniques to treat the MSS problem, and even

achieve better performance.

Chapter 4 Robot Trujectory Control: Joint Space 54

4.4 Summary

We discussed how to apply the proposed (linear) convex combination method. to a (nonlin-

ear) robotic system. to solve the MSS problem in trajectory tracking tasks. Our research

work in this chapter is summarized as follows.

We show that rnmy control design specification in the robot trajectory tracking

systems have the property of convexit. Therefore. the proposed convex combination

theory is applicable to a wide class of problems.

a The simulation (Liu and Mills, 1997 [25]). and experimentation (Liu and Mius. 1997

r24]) both show the effectiveness of this theory.

Chapter 4 Robot Trajectory Controir Joint Space 55

f me (sec)

O 2 4 6 8 10 tirne (sec)

4 6 time (sec)

O 2 4 6 8 10 time (sec)

Figure 4.1: Output response of unit step input and noise input: (a) under sâmple controller

G1, (b) under sample controller GÎ , (c) under sample controller G3, and (d) under convex

designed controller G.

Chapter 4 Robot Traiectory Control: Joint Space 56

tirne (sec)

-2 O 2 4 6 8 10

tirne (sec)

6

4

ci-

; 2 w

3

O

-3

fme (sec)

2 4 6 8 1 O tirne (sec)

Figure 4.2: Actuator response of unit step input and noise input: (a) under sample con-

troller Gi, (b) under çample controller GÎ, (c) mder sarnple controller Gj, and (d) under

convex designed controller G.

Chapter 4 Robot Traiecto~l Control: Joint Space W C .

3 l

time (sec)

Figure 4.3: Joint positions 0?, O3 vs time with convex designed controller


tirne (sec)

üme (sec)

tirne (sec)

Figure 4.4: Joint velocities ëi: ê2, ë3 YS tirne with convex designed controller

Chapter 5

Robot Trajectory Control: Task

Space

5.1 Introduction

In this chapter, we discuss how to apply the proposed convex combination theory to robotic

systems to solve the MSS control problem in trajectory tracking tasks. When the tracking

specifications are described in Cartesian coordinates or task space. the controuer is designed

in ta& space. In this case. we either transfer the performance specificâtions described in

task space into the local joint space. and design the controller in local joint space: or

we t r a d e r the derived controller in task space into the local joint space. at the t h e of

implementation. PVe address both approaches in this chapter.

The rest of this chapter is orgonized as follows. Seaion 2 presents the robot control in

tosk space. In section 3, the so c d e d performance transformation approach. which trans-

forms the performance specifications fkom task space into the local joint space, is discussed.

and a specific MSS control problem is solved and verified through experimentation con-

ducted on the CRS A460 Robot test bed. Section 4 addresses the control transformation

approach, which transforms the controller from ta& space into the local joint space, and

Chapter 5 Robot Trajectory Control: Task Space 60

another MSS control problem is illustrated by experimentation on the CRS A460 Robot

test bed. Findy, Section 5 offers a summary of our research work in this chapter.

5.2 Tracking in Cartesian Space

Ln the previous chapter, we discussed how to solve the MSS problem of the robot trajectory

control systemsl when the dynamic mode1 of the robot manipulator is described in the locd

joint space, and the performance specifications are specified in the joint space as weH. In

many cases, however, the designer may need to describe the performance in Cartesian

space, which gives a more direct evaluation of the trajectory tracking property. Depending

on the sensor configuration on the robot, the controller feeds back either the robot joint

signals in the case of joint motor sensing, or directly the end-efFector signals in the case

of endpoint sensing. For the first case: the designer needs to translate the performance

specifications described in Cartesian space back into the local joint space, theo apply the

convex combination theory to solve the transformed MSS control problern. Here. it is

referred to as the performance transformation approach. For the latter case. the designer

can directly solve the hlSS control problem, by transforming the feedback linearization

control law. Here, it is referred to as the control transformation approach. In this chapter.

we discuss both cases. We will use the kinematic and dynamic mode1 of the CRS A460

Test Bed, to illustrate how to solve the MSS control problem when the system performance

is required in Cartesian space.

5.3 Performance Transformation Approach

The position and orientation of the end-effector of the CRS A460 Robot can be described

by the following vect or


where the coordinate frame on the end-effector is called the tool frame. denoted by {Tl).

the reference frame is cded world frame, denoted by {W). For details refer to Chapter

3. The homogeneous transfomat ion matrix describing the orientation and position of the

tool frame {Tl) with respect to the world frame {W) is denoted by:

where the rotation matrix Rwrl is calculated from the Euler angles OwTr. TwTr: and

the position vector Pivn is represented by [ XwTl YwTl Zwn 1- and is also denoted by

[ Pz P, Pz ] = [ x y z 1. When HWTl is obtained. the position and orientation of the wrist

center (frame ( 6 ) ) with respect to the zeroth frame {O) can be calcuiated:

From (3.3) to (3.5).

When there is error between the desired and actual end-efFector positions and orienta-

tions, the error c m also be described as a homogeneous transform matrix

L hen the error transform matrix of the wrist center will be

where

Chapter 5 Robot Trajedory Control: Task Space 62

This homogeneous transform matrix shows the position and orientation relationship

between the wrist center and the end-effector. A spherical wrist is a wrist configuration

whose last three axes of rotation intersect at a single point defined as the wrist center. Such

a configuration dlows a decomposition of the six DOF inverse kinernatic computation

into two, t hree DOF kinematic computatioos. through a separation of the orientation

specification, cont rolled by the last t hree joints' local PID controllers. from the position

specification, controlled by the first three joints' local user defined control laws.

Assume that the PID control of last three joints can adjust the orientation satisfactorily

such that the rotation matrix of the wrist center frame (6) is almost the same as the desired

rotation matrix of the tool frame {TI)

& = RWTl = R & ~ ~ =

Based on this assumption: we conclude that the position error of the end-effector is the

same as the position error of the wrist center

pST1 = ~k (5.9)

Since the position of the wrist center is only related to the first three joints. it is obvious

that in this simplified case. the designer only needs to design a controller on the first three

joints such that the desired trajectory (position in Cartesian space) is achieved.

5.3.1 Robot control system framework

Consider the dynamics of a n n degree-of-freedom (DOF) rigid manipdator

where 8,é, 8 E Rn are the joint angle vector, the joint velocity vector, and the joint accel-

eration vector respectively. D E RnXn represents the symmetric, positive definite inertia

Chapter 5 Robot T~ajectory Control: Task Space 63

matrix; and the vector h ( 8 , b ) E En includes the centripetal. Coriolis, and gravitational

terms. ~g is the vector of applied generalized forces (torque or forces) acting at the local

joint torque side, and QA represents the extemal disturbance? such as noise coming from

the rnotor brushes.

Through application of the feedback Linearization technique (Spong and Vidyasagar.

1989 [SOI):

Te = D ( B ) [ ~ ~ + +] + h(d. 4) (5.11)

where Bd is the vector of desired joint positions. and the vector v represents for the linear

controller to be designed.

The cornplex, highly coupled nonlinear dpamics of the manipulator (5.10) is then

replaced by a simple set of second-order linear differentid equations:

ë = i d + o f q ( 5 - 12)

where 7 = D-' (B)rea is considered as an extemal disturbance input. Therefore. it is highly

advantageous from a control viewpoint to consider the linearized system and to quanti-

the performance.

Assume that the expected trajectory is generated by its acceleration function r ( t ) =

bid(t)? this linear system can be eguivdently written in the frequency dornain

where

When the linear control law utilizes feedback of the joint position and velocity errors? it

has the following fom:

Chapter 5 Robot Trajectory Control: Task Space 64 - -

Define r = e, y = e, u = v . w = [r the linearized robot control system (5.10,5.11.5.12)

can be described in the uniform frarnework, as in (2.1):

5.3*2 Convex specifications

Many robot control design performance requirements can be technically represented as

functions rnapping input signals to output signals (American Xational Standard Institution,

1992 [Il). The performance specificat ions tested in t his experiment at ion are chosen as

follows. Path accuracy (PA) is a measurement of the distance between a reference path

(desired trajectory) and any given attained path. Whiie trying to meet the above input-

output funct ional specifications, the designer needs to keep the system interna& s table.

For the closed loop transfer matrix H as in equation (2.2)- the system internal stability

(1s) requires that P,,K(I - P,K)-',K(l- P&)-'P,,K(I - P,h')-'JI - PwK)-'

are all stable. We also c d the linear controller K stabilizing if this controuer makes the

closed-loop syst em interndy stable.

With the uniform description of the linear system, we define the design specifications

as functions of the closed-loop transfer matrix H mapping input w to output z (2.2). The

functional description of path accuracy (PA) is defined as the square of the l2 n o m (Desoer


and Vidyasagar, 1915 [Il]) of the joint position error

W h e n we translate the above specifications in Cartesian space into Local joint space,

Ive show that all the above design specifications are convex in terms of closed-loop transfer

matrix H.

Theorem 5.1 Specijications PA and LS are convez.

Proof Suppose Kl ? h; are two stabilizing controllers. Le.

are bot h stable. For any X E R, obviously

is stable. When P, (5.17) is stable, the closed-loop system is internal stable if R is stable.

It t u r n s out that the internal stability (1s) is an convex specification.

We know that the position of the wrist center is only related to the h s t three joints

where = COS(B~), S j = Sij = sin(di + e j ) ? and az, d4 are constants.

Expressed with the Tayior series for the position functions about the desired position,


where * denotes the convolution operation. For any X E [O, II, and 1 = 1 - XI the convex

combinat ion of the closed loop transfer matrix N is

Therefore, the pat h accuracy (PA)

is a convex specification in terms of the closed loop transfer matrix H . Ident ically. & and

<j, are proven to be conver specifications as well.

5.3.3 Experimental results

Since the design specifications are convex after transformation, we can apply the convex

combinat ion t heory to solve the following MSS problern:

def 4 < QI - 4.1

def & 5 ~3 - 3.5

def & < QQ - 1.8

subject to IS

Chnpter 5 Robot Trajector-y Control: Task Space 68

where ai. CI*. a3 are the expected specification values.

1. Sample Controller. Each sample controller is selected to meet one specification. . h y

linear control law can be applied, as Long as one specification can be satisfied. In this

example. we select the local PD and PID control as the sample controller. The control

structure is

where rip = diag[ltpii]. KI.. = diag[kCii] and Kr = diûg[kiii] are diagonal matrices whose

elements are positive constants (gains). For each joint i = 1.2.3. the closed-loop transfer

matrix (3.19) has the characteristic equation

Since the gain values of kpii , Li and krii are positive. the roots of these equations are

contained in the Left half of the s-plane when kpii > kr i i /k , i i . Therefore. the system is

internally stable. Note that IS is a conves specification. then a n - convex combination of

internally stable system becomes

with the characteristic equation A(s) = il.($) - &(s) - A,(s) which is also internally stable.

Thus, in this example, the designer only needs to focus on designing a controller to meet

+i, Q2: and d3 simu1taneously.

Select the sample controller to satisfy one specification at one time:

Chapter 5 Robot Trajectory Control: Task Space 69 .

(1) Select sample controller a to satisfy &: k>i = 2000, kb = 70, &(s) = s2 + kz is + k;ii .

(2) Select sample controller b to satisfy &: k:ii = 2500, kLi = 50, A&) = s2+ktiis+k:ii.

(3) Select sample controller c to satisfq. &: k;ii = 2000, k z i = 60, k;ii = 10- &(s) =

s3 + ktiis2 + k&s + kiii

From experimental results, calculate the specificat ion values under each sample con-

troller. The results are also shown in Table 5.1. Note that each sample controller satisfies

one specification (smaller than the expected value). However, no sample controller can

satisfy d the three specifications at the same time (compare the results of coiumns "Con-

troller a,b,c' with the contents of column "Expected specifications" in Table 5.1). In fact.

it is not hard to adjust the PD gains (Liu, 1997 [XI) to treat one specification. but it is

difficult and tedious to tune the PD,PID gains to try and meet the simultaneous specifica-

tions, because specifications conflict with each other in terms of the control gains.

Table 5.1: Cartesian specifications test with performance transformat ion

11. Combination. Now we can apply the convex combination method to design a con-

troller to meet these three specifications simultaneously. Based on the expected and actual

specification values, we conclude from Theorem 2.2 that the necessary and sufficient condi-

I

J

Expected

specifications

-- - -

4.1000

2.5000

1 .SO00

Specification

functions

Controller b Controller a

d2

@3

Controller c

k

2500

Ic," 2000

k;

ZOO0

k

50

k:

70

5.5870

2.1130

k:

60

0.4815

2.5600

k:

10

0.2083

0.6760


tion for an inconsistent solution of (2.8) is that there exists a vector u = [ul u* u3IT. Ui 2 O

such t hat :

It cas be equivalently mitten as

-1.1070 3.0870 0.3130

0.7470 -3.0185 0.7600 (5 .45 j

0.7520 -2,2915 -1.1240

Denote this inequality as Au > O. If such a u exists? then for any v = Lul v2 v3IT > 0.

we have vTAu > 0. In fact, we can find a o = r2.5 1.0 %.51T such that V'AU = [-0.0655 -

0.0297 - 1.26751~ < O. This codict illustrates that the vector u 3 O? Au > O cannot exist.

It implies that the inequality (2.8) has a feasible solution. Using the linear programming

software (M-4TLAB Optimizat ion Toolbox [35]), we find a feasible combinat ion vector

Al = 0.4087, X2 = 0.2S19J3 = 0.3064. Through a convex combination of closed-loop

transfer funct ions, the final designed controller transfer funct ion is

Cm = [G + Fs]

where G = diag[gij] F = diag[f i i ] are diagonal matrices:

Chapter 5 Robot Trajectory Controk Task Space 71

We conducted an experiment on the CRS A460 Robot test bed (refer to Chapter 3)

under the final designed controller. The specifications were found to be are found to be:

The experimental results are shown in Figures 5.1 and 5.2. m e n focusing on one spec-

ification, it is easier for the designer to find a (sample) controller to meet it. Comparing

the contents of columnç ' Controller a.b.cF with the column A E.xpected specifications".

we note that each sample controller meets one specification each time. but no single con-

troller can meet d l four specificat ions simultaneously. By combining the different sample

controllers, a new control stmcture is generated. which cannot be designed by an- conven-

t ional tuning approach. to meet the four simult aneous specificat ions. In conclusion. the

convex combinat ion met hod is effective in treating the multiple simultaneous specifications.

5.4 Control Transformation Approach

In the previous section ive discussed the case when the performance specifications are

described in Cartesian space. and the controuer only accesses the s ipa l fiom the robot

joints. Hence we needed to transform the specifications from Cartesian space into joint

space. On the other hand, if we have direct endpoint sensing, Le.? the position of the end-

effector of the robot can be directly measured. we can address the MSS problem directly

in Cartesian space. In this case, the closed-loop system is desnibed in Cartesian space.

therefore, the controller acting on robot joints needs to be transformed from the Cartesian

space into the local joint space.

Chapter 5 Robot Tmjectory Control: Task Space 72

5.4.1 Robot control in Cartesian Space

The d p a m i c mode1 in Cartesian space is given by:

where x = [ x y z IT E 7S3 is the vector of end-effector positions; x = [ i y 5 IT E R3 is the

vector of end-effector velocities: M ( x ) E R3x3 is the inertia matrix: L ( x . x ) E R3 is the

vector which results by combining the centrifugal, Coriolis forces. gravitational forces? and

viscous and Coulomb friction terms. with respect to the Cartesian space; T, is the vector

of Cartesian force input; r , ~ is the external disturbance.

Through the feedback linearization technique in Cartesian space:

where u generally represents a linear controller. Then the nonlinear robot dynamics equa-

tion (5.52) is replaced by a simple set of second-order linear differential equations:

where = M - ' ( x ) T , ~ is viewed as an external disturbance input. Therefore. it is highly

advantageous from a control viewpoint to consider the Linearized system and to quantify

the performance. Assume that the expected trajectory is generated by its acceleration

function r(t ) = zd(t ) ? this linear systern (5.54) can be equivalently written in the frequency

where


When the linear control 1aw utilizes feedback of the joint position and velocity errors. it

has the following fom:

Define r = e? y = e: u = v w = [r this linexized robot control system can be described

in the uniform framework. as in (2.1 ):

5.4.2 Performance specifkations in Cartesian space

The performance characteristics are chosen according to the Arnerican National Standard

(ANSI/RIA R15.05-2-1992 [Il) as follows. Path accuracy (PA) is a measurement of the

distance between a reference path (desired trajectory) and any given attained path. A

functional description of PA is defined as the square of the l2 norm (Desoer and Vidyasagar,

1973 [Il]) of the end- effector position error on the x-buis

Velocity accuracy (VA) is defined as the error between the programmed speed and the

attained speed. A functional description of VA is defined as the square of the l2 norm of


the end-effector velocity error on the x-axis

The performance of a robot under motion conditions is important in understanding the

"steady staten performance. The steady state error of position (SSEP) on the z-axis is

described by the following function

While the designer tries to meet the above input-out put functional specifications. he/she

needs to keep the system internally stable. For the closed loop transfer matrix N as in

equation (2.2), the system intemol stability (1s) requires t hat Pz, hp( 1 - P, K)-' . K( l -

P,,K)-' Pp, K(1- P,, K)-': (1 - P,K)-' are all stable. iVe also cd1 the linear controller

K stabilizing if t his controller makes the closed-loop system internally stable.

We show that ail the above performance specifications are convex functions.

Theorem 5.2 Design specifications PPA bYA, Q ~ ~ ~ ~ , and the intemal stabifity (1s) are

convex functions in termv of closed-loop trunsfer matriz H in (5.61).

Proof Suppose ATi , h; are two stabilizing controllers i-e.?

are both stable. For any X E R, obviously

is stable. Since P, (5.59) is stable, it turns out that after the convex combination, the

closed-loop system is still in temdy stable. Therefore, IS is a convex specification.

Chapter 5 Robot Trajectory ControZ: Task Space CI la

From (5.61), we know that e(s) = H(s)r)(s) . Suppose

It can be transformed into the time domain

For any X E [O. 11. and X = 1 - A, the convex combination of the closed loop transfer

matrix H is

H = x H ~ + ) ~ H ~ (5.66)

The corresponding output B(t )

Therefore

is a convex specification in terms of the closed loop transfer matrix H. Identicdy , VA is

a convex specification as well.


On the other hand,

is a convex specification in t e m s of the closed loop transfer matrix H.

5.4.3 Experimental results

Since the CRS A460 robot is actuated through DC motors on al1 six joints. the Cartesian

space control needs to be transformed back to the joint space. -4ssurne that the locd

dynarnic model is

D ( o ) ~ + h(0' é) = r, (5-69)

and the dynarnic model in Cartesian space is written as in (3.52). From 5 = J0 + J&,

where J is the Jacobian matrix, we have

Multiply JT to the both side of the above equation

Compare the above equation with (5.69), we obtain

Chapter 5 Robot Trajectory Control: Task Space -- I I

Therefore. the control law in Cartesian space can be transformed to joint space

Since the design specifications are convex after transformation. we can apply the convex

combinat ion t heory to solve the following MSS problem:

mhere al. n2. a3 are the espected specification values.

1. Sample Controller. Each sample controller is selected to meet one specification. Any

linear control law can be applied. as long as one specification can be satisfied. In this

example. we select the PD and PID control as the sample controller. The control structure

of PD control with feedback linearization is

where Kp = d i a g [ b i ] and Ku = d i a g [ L i ] are diagonal matrices whose elements are positive

constants (gains). Comparing wit h the uniform framework. the controller t ransfer matri':

has the form

C = [Kp+sh;]

The control structure of PID (feedback linearization + PID) control is


where Kp = diag[kPjjll Ku = diag[kVjj] and Ki = diag[kijj] are diagonal matrices whose

elements are positive constants (gains). Comparing with the uniform framework. the con-

trouer transfer matrix has the form

Since the gain values of kpii, kuii and klii are positive, the roots of t hese equations are

contained in the left half of the S-plane when kpii > krii /kGi. Therefore. the system is

internally stable. Note that IS is a convex specification. Then any convex combination of

internally stable systems becomes

with the characteristic equation A(s) = A&) .Ab(s) -4,(s), which is also internally stable.

Thus. in this example. the designer only needs to focus on designing a controller to meet

bPA dvA. and bSSEP simultaneously.

(1) Select a sample PD controller a to satisfy

(2) Select a sample PD controller b to satisfy dvA.

(3) Select a sample PID controller c to satise +sSEP.

Based on the experimentation data, calculate the specification values under each sample

controller. The cdculation results with the expected specifications are listed in Table 5.2.

Note that each sample controller satisfies one specification (smder than the expected

value). However, no sample controller can satisf'y all the three specifications at the same


Table 5.2: Cartesian specification test wit h controller transformation

Specification Controller a ControZler b

functions Ic," 4 kP k:

2500 50 3000 70

Controller c 1 Ezpected

tirne. Apply the convex combination method to design a controller to meet these three

specifications simult aneously-

II: Combination. Using the linear prograrnming algorithm LP (MATLAB Toolbox), the

combination vector is obtained

Through convex combinat ion of the closed-loop t ransfer funct ions, the final designed

controller transfer function is given by

C'= [ G f Fs] (5.86)

where G = d iag[g j j ] , F = diag[fii] are diagonal matriices:

fii(s) = 59-98 (5.88)

Upon conducting experimentation on the CRS A460 robot test bed (refer to Chapter 3

G a p t e r 5 Robot Trajectory Control: Task Space 80

for details)? under the final designed controller, the specifications were found to be:

The experimental results are shown Figure 5.3 and Figure 5.4.

Summary

We have addressed the solution of the MSS control problem of robot trajectory tracking

systems in Cartesian space. There are two approaches to deal with the MSS problem

when the performance and controller are described in difFerent coordinate sys tems: ei t her

transform the performance, or transform the controiler, such that the control and the design

specifications can be considered in the same coordinate system. Our research work in this

chapter is summarized as follows.

0 Performance transformation approach t ransforms the design specificat ions descri bed

in task space into joint space, therefore the MSS control problem can be solved in

joint space. Our experimental example shows that this approach is effective. under the

condition that the transformed performance still sa~isfactorily represents the original

requirements. The benefit is that it is easier to implement. The example dso shows

how to verify the feasibility problern.

r Control transformation approach solves the MSS problem directly in ta& space, t hen

transforms the designed controller fiom task space to joint space. The advantage is

that we can derive the controller to directly meet the performance without transfor-

mation. However, the trade-off is that the control stmcture (5.73) is more complicated

and time consuming, thus the sacrifice in time may lose performance as well. The


experimentd results show that this approach is acceptable, but a little worse than

the results obtained by using the performance transformation approach. It implies

that the proposed convex combination method can be extended to treat multiple

performance specifications, even in different coordinate systems.


üme (sec)

tirne (sec)

1.5 2 2.5 Sme (sec)

Figure 5.1 : Trajectory in Cartesian space with performance transformation

Chapter 5 Robot Trajectory Controlr Task Space $3

actual . . . . . . desired

x (in)

Figure 5 . 2 Trajectory in Cartesian space with performance transformation: 3-D graphics

Chavter 5 Robot Tra-jectorg C'ont rd: Task Space SA

lime (sec)

Figure 5.3: Trajectory* in Cartesian space with controller transformation

Chapter 5 Robot Tr~jectorv Cont rol: Task S p m e 85

actual 1 - desireci 1

x (in)

Figure 5.4: Trajectory in Cartesian space wit h contrder transfomat ion: 3-D graphics

Chapter 6

Robot Trajectory Control:

Robustness

6.1 Introduction

In this chapter. we address the MSS control problem of robot trajectory tracking systems

in the presence of systern mode1 uncertainty and disturbances. Le.. the robustness is one of

performance specifications to be satisfied. The rest of this chapter is organized as follows.

Section 2 presents robot control under system uncertainty. In Section 3 , the robustness

conclusion in the sense of boundness is derived by the author. In Section 4, an e-perimental

example is used to solve an MSS pmblem, including the robustness specification. Finally,

Section 5 offers a summary of the author's reseorch work in this chapter.

6.2 Robot Control Under System Uncertainty

In the previous two chapters, we discussed how to solve the MSS problem of robot trajectory

control systems, by using our proposed convex combination design method. This method

requires that the nonlinear robotic system be Iinearized, through the feedback Linearization

Chapter 6 Robot Trajectory Control: Robustness 87

technique. This linearization depends on knowledge of the robot manipulator dynamics.

Unfortunately, uncert ainty always exists in the mat hematical model. for which are counted

by the notion of uncertainty. For a robotic system, systern parameters. unknown payload,

friction. disturbance, and unmodelecl dynamics can be viewed as uncertainty. Robust

control design implies the design of a deterministic or k e d control in the presence of

significant system uncertainties.

Consider the nonlinear dpamics equation of an n DOF rigid robot manipulator:

where D(B), h ( 0 , 8 ) denote the real model of the systern dynarnicso TA represents the extemal

disturbance on the torque side, such as the noise generated by the joint motor. and r, is

the calculated torque based on the known system dynamics mode1 parameters:

Dc is the calculated (known) inertia matrix. h, is the calculated (knoivn) part of h.

and u is the output of a linear controller. .Assume that the linear controller feeds back

information of the joint position error vector e = Bd - 0 and the joint velocity error vector

é = gd - 9. It can be generdy represented in the frequency-domain:

while it can also be generally mitten as a state-space equation in the time-domain:

where t E Rm is the state miable; A, E RmXm is the state matrix: B, E FXn is the input

matrix; C, E RnXm is the output matrix; Kp = d a ' t ~ g [ k ~ ~ ] ~ K,, = diag[kuj] ( j = 1: 2,. . . , n)

are the diagonal direct feedthrough matrices with the joint position gain kpj and the joint

velocity gain le,j respectively.

Chapter 6 Robot Trajectory Contd: Robustness SS

Obviously we have

Define A/ and BI as

With the notation of f = [ zT eT 1. we alço d e h e a vector uo that directly feeds back

the position error e; velocity error e' and the state variable z by proper gains:

where R is a constant matrix: I? = [ BJ., al,, ] with cu > 0. 3 > O. identity n x n rnatrix

Inn, and a constant matrix J,, E RnXm.

Therefore. we d e h e the linear controiler o as o = u + uo with the generic formulation

of (6.4) and (6.7) .

Remark: W e briefly elrpiain that a generic Lnear controller v in (6.3) can be written in the

form of (6.4) and ( 6 . 7 ) . Based on the assumption that the linear controller u feeds back the

position error and velocity error. the generic form of u c m be written in transfer function

fom:

(L(s) = & ( s ) e ( s ) + &(s)se(s ) = ( h ; ( s ) + s K 2 ( s ) ) e ( s ) (6-8)

where Kl ( s ) and &(s) are proper transfer matrices (the degree of the numerator poly-

nomial is not greater than that of the denominator polynomial). Equation (6.8) can be

remit ten by taking out the constant part,

where E(s) is strictly proper, in the sense that the degree of the numerator polynomial is

less than that of the denominator polynomid. Translate h'(s) into the state-space equation,


then we obtain the general form of the linear controller u in a state-space representation

(6.4). Finally note that uo directly feeds back e, è, and z, which can also be viewed as

part of u. Without loss of generality, we specify the term Kp + KVs separately for the

sake of convenience when we address the robustness by using Lyapunov's direct method,

as discussed in the following section.

Note t hat

The Lnear controller v can be rewritten as

Substituting the linear controller (6.10) and the caiculated torque (6.2) into the d ~ a m -

ics equation (6.1) , leads to the closed-loop system:

where E = D-ID, - Inn. This leads to

Chapter 6 Robot Trajectory Control: Roliustness 90

which is denoted by

z j = A y - BA (6.13)

The following assumptions (Spong and Vidyasagar. 1987 [51]) account for the system

uncert aint ies.

Assumption 6.1 Intertia matriz D is bounded by known values r n ~ and mu

Remark Boundedness of the inertia matrix D is based on the property t hat D is symmetric,

positive definite. Positive definiteness of the inertia matrix reflects the fact that most

systems in motion always have nonzero kinetic energy. Note that there are special cases

in which the inertia matrix is mathematicdy only positive semidefinite, which may cause

unboundness. These cases involve only robots equipped wit h prismat ic joints and, unless

their physical structure allows Bi for prismatic joints in motion, the inertia matrix remains

positive definite. That is, the positive definite property holds except for, at most. finite

isolated points in the space of motion (Qu. 1996 [do]).

Assumption 6.2 External disturbance TA is bounded by a known value E ,

Assumption 6.3 Centripetal/Coriolis term is bounded by known values e,, E,, and

Assurnption 6.4 Desired acceleration signai $d is bovnded by a k n o m value e,

Assurnption 6.5 E is bounded by a knovn value o

Gàapter 6 Robot Trajectory Contra[: Robvstness 91

Note that although this assumption seems restrictive, it is dways possible to satisS

this assumption by a suitable choice of D,. This conclusion can be shown in the following

lemma (Spong, 1987 [51]).

Lemma 6.1 When the calculated inertia matriz D, is chosen as '

where mu and m L are the bounding values of D-L as shozun in (6.14). Thhen 11 E 11 satisfies

Proof Define F = LI-' - DE'. From E = D-'DC - Inn = (D-' - D;')D, = F D,. we

have

On the other hand, matrix n o m of F is defined as

and

'The author of paper [fil] rnistakedy selected Dc (denoted by k in this paper) as In, - (mt + mu )/2.

We correct it here.


It follows that

Therefore. we have the conclusion

With the above assumptions, we can address the robustness specification.

6.3 Robust Specifications

We address the robustness specifications in the sense of boundness ( Lagrange stability )

under system uncertainties, by using Lyapunovos direct method.

Definition 6.1 A function V : Rn -L R+ is called a Lyapunou function candidate i f there

exists a positive constant and a scalar function X: Rn -t R+ such that

By defining a Lyapunov function candidate. we have the weU-knom stability conclu-

sions, often called the Lyapunov's direct method.

Theorem 6.1 (Lyapunov's Direct Method) Consider a nonlinear system

Let a Lyapunou function candidate be V , and let v be the derivative of V along the trajectoq

of the system (6.22). This system is unïformly stable, if V 5 O. Tt is asymptotically stable,

if v is negative definite. Tt is exponentially stable i f v satisfies 5 -cX(1~11~ for sorne

c > O . System (6.22) is un i fomly ultirnately bounded i f v satisfies i.' -cXll~11~ + c for

some c > O and r > 0.

CIhapter 6 Robot Tmjectory Control: Robustness 93

By constructing a Lyapunov function candidate for the system (6.13). we obtain the

robustness conclusion. i.e.. ultimate boundness. Construct a function

w here

and .CI is a positive definite matrix. which will be deterrnined later. LVe show that this

function Le' is a Lyapunov function candidate.

Lemma 6.2 V(y) i.9 a L yaprnoc junction candidate.

Proof For any vector wT = [ uT cT 1. where u E Rmçn and c E Rn. we have

w T P t ~ = LuTJ[u 2 + f - ( r + Q u ) ~ ( L ' + Q U ) > O. when .II is positive definite. It follows that

P is a positive definite rnatrix. For a quadratic function V ( S ) = -YT P X for symrnetric

matris P. it always satisfies (Stewart. 1973 [52] )

where Amin and A,,, are the minimum and maximum eigendues of P respectivel-. There-

fore. V(y) is a quadratic function satisfjring AminllY (1' 5 \,-(y) 5 ~ ~ ~ ~ l l ~ l l < Positive definite

matrix P has positive eigendues. It leads to the conclusion that C.-(y) is a Lyapunov func-

t ion candidate.

With this Lyapunov function candidate L'(y). the robustness conclusion is gik-en by the

follow ing t heorem.

Theorem 6.2 If the follouing conditions are satiqîed:

Chapter 6 Robot Trajectory Control: Robwtness 94

( C I ) there ezists a positive definite matriz M such that

and

(ATM + MA,) - (DTQ + f i T @ ) + d n = -9, (6.26)

where QF is positive definite.

(C-2) -2h', + In, = -Qu, where Q, is positive definite.

(C-3) A is bounded

1 1 ~ 1 1 s P

Then the system (6.13) is uniformly ultimately bounded.

Proof .A Lyapunov function candidate is given by (6.23). Differentiate (6.23) along the

state trajectory of system (6.13). This leads to

Xote that

Tt follows that

Chapter 6 Robot Trajectory Control: Ro bustness 95

where

Therefore. we have

Substituting for P' Po. and A. we have

where:


When the conditions (C-1,C-2,C-3) hold, it leads to LI2 = LZl = O(,+n),, Li i = -QF,

and LZ2 = -Qu- It follows that

From Lyapunov?~ direct met hod, Theorem 6.1 system (6.13) is ultimately bounded.

Remark: When there is measurement noise, i.e.? the controller feeds back the measured

joint position and velocity errors, which include the noise on the measurement side: v =

K f & where = é + é A 7 2 = r+za , ê = e+ea . -4s long as the noise is bounded; we

have a similar conclusion.

Obviously, in Theorem 6.2: condition (C-2) is easily satisfied by selecting &. Condition

(C-3) will lead to our robust specification, which is to be addressed later. We now show

that condition (C-l ) c m hold in many cases.

6.3.1 Discussion of condition (C-1)

Assume t hat

From MBr = DF + QT& - RT, we have

It leads to

Chapter 6 Robot Trajectory Control: Robusf ness 97

Obviousl- k122 is positive definite. And we can always select J such that iWlz = O. In

this case, we have

CT = - P J ~ ( K , , - I,,) (6.41)

From (AFlLl + M.4/) - ( R T D I + DFR) + RTfl = -QF7 and assume

After some algebraic manipulation, we have the following conclusions:

1) QE = a ( X p - al,,) is obviously positive definite.

2) iWl1 B, = ,8JT(h', - a&)

3) -4: LM,, + ibf11-4Z = -Qm

4) Q, = Qz + (0 J)=Q@J) is also positive defmite when Q, is positive definite.

If we can find a positive definite matrix satisfying 2) and 3)- then condition (C-1)

in Theorem 6.2 will hold.

Now we conclude that when condition (C-3) in Theorem 6.2 holds. the system (6.13) is

robust. We specify condition (C-3) as our robust specification. which also has the convex

property

6.3.2 Convexity

Now that ne have the robustness conclusion under the condition that is bounded (con-

dition (C-3) in Theorem 6.2), a robust specification can be considered as: hom to design

a controller C in (6.3) such that 6 iç bounded. We follow the same approach as that in

Spong (1987 [SI]) to address this problem.

From (6.13), we have

Chapter 6 Robot Trajectory Control: Robvstness 98 -- - - ..

denoted by

? = A l - B U - B A (6.14)

Written in the frequency domain, we have r = -Gu - GA. Assume that the controller

in the frequency domain is v = Cr, we c m put the system in the framework ( 2 4 , where

t =x ,Y = r, w = a, u = v. The closed-bop transfer matrix becornes r = H&:

From the assumptions we made in Iast section, we c m easily derive the following con-

clusion:

where y,, y, can be calculated by t hose bounded values.

Therefore, we obtain

and

Ml = l lC4 5 IICFII - 1141 (6-45)

Let c& = IIC F11, J2 = II FI[: and put the above inequâlities together, tve have

When y,d' + cdl < 1, it leads to (Spong and Vidyasagar, 1987 [XI)

m a p t e r 6 Robot Trajectory Control: Robustness 99

That is. L. and are bounded:

11a11 5 Tc dcf

1 - ~ / 1 6 2 - = P

Therefore? we consider this inequality as our robust specification.

Theorem 6.3 Robust specifiation

is conuez in tenns of elosed-loop transfer matriz H.

Proof Obviously&(H) = IIF(I = 11-HI1 isconvex. Xotethat GCF = GC(I+G'C)-'G =

H + G1 thus Ji(H) = IlCFll is dso convex. Therefore. for any Ha. Hh. O < X < 1. and

define 1 = I - A. n.e have

It leads to the conclusion that dRoso-sT is convex with respect to the closed-loop transfer

rnatrix N.

6.4 Experimental Results

W e conducted the experimentation on the CRS A460 robot test bed as introduced in

Chapter 3. The desired robot trajectory was specified as foilows: joint movement bom


the straightup position (position O) to position 2 , and back to position O. These three

positions are listed in Table 6.1.

The dynamic model of CRS A460 Robot is developed in Chapter 3. From equation

(6.14) one can calculate the bounding values of the inertia matrix D: r n ~ = 0.2165. mr; =

0.7935. The calculated model of D, is chosen as (6.19). Then from Lemma 6.1. a = 0.5713.

Assume that the calculated term h, does not contain viscous friction and Coulomb friction

terrns, then the bounding values of (1 h, - hl1 can be found: 2h = 0.547, a, = 1.3. From the

deçired trajectory, one can also obtain the bounding values of l ( jd 11: e. = 1.439. With t hese

calculated bounding values, we can derive the controller to satisfy the robust specification

(DRO BUST-

The performance specifications tested in this example are: path accuracy (PA). velocity

accuracy (VA), steady state error of path (SSEP), and robustness (ROBUST). As discussed

before, we know that the specifications dPa, qjVA, and +ssEp are convex specificat ions. We

also know that QROBLiST iç a convex specification as well. Therefore. we can follow the

proposed convex conbination design procedure to derive the controller to satisfy the mul-

tiple specificat ions simultaneously. Note t hat the robust specification cont ains informat ion

of system stability. which is a basic requirement of the control system. LVe formulate the

Table 6.1: Robust specification test: trajectory

Position

O

dl (deg)

O

O1 (deg) 1 BI (deg)

90 -90


MSS control problem as foilows:

where cri, 02, a3 cr4 are the expected design specificat ion values.

Step 1: Sample controller. Select cornputed torque control law as the sample controller:

where e = gd - 9, é = Bd - 6, and K p = diag[kp], f i = diag[kv] are diagonal matrices with

positive gains k, and It, respectively. Comparing with (6 .2 ) , the linear controller c is chosen

as local PD control. As the convex combination theory requires. the sample controller is

selected to meet each specificat ion:

(1) Select sample controller a on each joint to satisfy <bpA subject to o ~ o ~ c i f l : k; =

800. kt = 60.

(2) Select sample controller 6 on each joint to satisfy dvl subject to o ~ o s c . s ~ : k; =

600, k: = 80.

(3) Select sarnple controller c on each joint to satisfy 4 S S E p subject to P R ~ B ~ ~ ~ ~ : ki =

800, Ic," = 40.

Knowing that &osun is a convex specification. it turns out that after the convex corn-

bination with any combination vector, the designed controller must also satisfy 4ROBUST.

Now we only need to focus on designing controllers to meet Qpa, 4va, QSSEp by using the

convex combination theory. From experimental results, calculate the specification values

under each sample controller. The calculation results with the expected specifications are

Listed Table 6.2. Note that each sample controller satifies one specification ( s m d e r than

Chapter 6 Robot Trajectory Control: Robvstness 102

the expected value). However. no sample controller can satisfy d the three specifications

at the same time (compare the results of columns 'controller a,b,cE with the contents of

column uExpected specifications" in Table 6.2).

Table 6.2: Robust specification test: sample controller results

Step II: Combination

Using the linear programming algorithm LP (MATLAB Optimization Toolbos [Xi] ).

L

the combination vector is obtained as:

Through t the convex combinat ion of the closed-loop transfer func t ion. the final designed

Specification

functions

OPA

VA

, OSSEP

~ R O B U S T

Controller 6

controller transfer function on each joint is v = Cz(s) e + kp e + li, ë:

ki 600

Transforrning the new controller transfer funct ion into its state-space representat ion,

k:

80

Controller a Controller c Expected

specifications

1 .OS

5

1.10

< 1

1.1077

3.3'190

1 .S790

0.6204

k;

800

k;

800

t

4 60

k

40

0.5420

6.0530

0.9994

0.6662

0.8183

5.5440

O -9905

0.7498

Chapter 6 Robot Txjectory Control: Ro bustness 103

we will have the same equations as (6.4) and (6.7): where

It turns out that there exist a positive definite rnatrix M

such that condition (C-1) in Theorem 6.2 is satisfied. Obviously (C-2) is satisfied. When

the robust specification o ~ o s u n < 1. from (6.52). condition (C-3) holdç as well. Therefore.

from Theorem 6.2. we have the conclusion of robustness-

The designed control code is implemented on a CRS A460 robot system and executed at

a rote of 5OOHz. The experimental results are shown in Figure 6.1 and 6.2. The performance

specifications are calculated as:

Chapter 6 Robot Trajectory Control: Robvstness 104 -

We note that d e r convex combination, d the four specifications are satisfied sirnul-

taneously. Furthermoret the final derived controller (6.39) has a totally different control

structure from the sample PD control. It shows that convex combination theory is a

performance oriented met hod, i.e., the controller is derived based on the performance spec-

ifications, and the result may have not only new control parameters. but also new control

structures. Such controllers cannot be found by current gain tuning approaches.

6.5 Summary

We have addressed the MSS control problem in the robot trajectory systems. including the

robustness issue. Our research work in this chapter is sumrnarized as follows.

a We showed that robust performance can be considered as one of expected multiple

specificatioo funct ions. Therefore: the proposed convex combinat ion method can

be applied (Liu and Mills, 1997 [-21). As mentioned before. convex combination is

based on the feedback Linearization technique. which has the advantage that such

a control structure makes it possible to treat a wide range of different performance

specificat ions simultaneously~ The obvious disadvant age is t hat t his met hod depends

on our knowledge of the robot dynamics model, which is never exactly the same as

the real dynarnics. Therefore. the main challenge is how to solve the 41SS probiem in

the presence of system uncertainties. Discussion of robustneos malies it clear that the

proposed convex combination method also works well under this system uncertainty.

Furthermore, Theorem 6.2 offers a robustness conclusion under a generic Linear control

structure (Liu and Mills, 1997 r23]). Such a generic conclusion, to our best knowledge,

is new.

Chapter 6 Robot Trajectory Control: Robusfness 10.5

time (sec)

time (sec)

time (sec)

Figure 6.1: Joint positions O1 O*, O3 vs time with convex designed controller

Chapter 6 Robot Trajectory Control: Robvstness 106

time (sec) 1 I 1 1

Figure 6.2: Joint velocities &, &, e3 vs tirne with convex designed controller

Chapter 7

Convex Opt imizat ion Application

7.1 Introduction

As we mentioned in Chapter 1. the development of our proposed convex combination

rnethod adopts the convex design specification concepts frorn the convex optimizat ion the-

ory (Boyd and Barratt. 1991 181). This theory offers a performance oriented method to

derive a controller under a uniform Linear system frarnework. As a matter of fact. before

we developed the convex combination method. we first tried CO use the convex optimization

theory to address the MSS control problem. In this chapter. this application work is illus-

trated through a simple example. The result will show the limitations and difficulties of this

theory when it is applied to solved the MSS problem. In other words, the primary purpose

of this chapter is to further explain why a new design methodology (convex combination

method) is needed to solve the MSS control problem.

The rest of this chapter is organized as follows. Section 2 formulates the WSS control

problem into an optimization question that needs to be used by the convex optimization

theory. In Section 3, the optimization algorithms are applied to a simple example for

illustration. The limitations and difficulties are addressed in Section 4. Further, an im-

provement on the initialkation approach of one convex optirnization dgorithm, proposed

Cliapter 7 Convex Optirnization Appiication 1 O8 - -

by the author, is presented in Section 5. Section 6 compares current performance ~ r i -

ented methods with our proposed one. Finally, Section 7 offers a summary of the author's

research work in this chapter.

MSS Problem Application

As we described in Chapter 2, the system performance can be technically represented by

a functional specification: 4 : Z -t R+ (Definition 2.1). For a single design specifica-

tion, it can be cast into a typical optirnization problem: min #. When the optimal value

6 = mind 5 a. where a denotes the desired specification value, then the specification is

satisfied,

For multiple specifications. the design specifications can be classified into t wo categories:

constraints and objectives, based on their degrees of priority. A constraint is a specification

that is firmly required. For example. the stability of a control system must be considered

as a constraint since it is the very basic requirement in control system design. and any

unstable situation is not tolerated. On the other hand, an objective is a specification that

is expected to be improved, and any slight change may affect the system performance, but

will not cause serious problems.

Therefore, the MSS control problem c m be modified to a classical optimization problem:

minimization (or mâximization) of the cost function (objective) with constraints:

where &Pi generally represents some sort of combination of the many objective functionals

into one single funct ional.

Two common met hods of combinat ion are weighted-sum met hod:

Chapter 7 Convex Optirniration Application 109

and weighted-max met hod ( also called rninirnax met hod) :

When the design specifications defined in Definition 2.1 are convex in a finitedimensional

system set 1: the optimization problern (7.1) can be solved effectively by using convex o p

timization met hods. such as the cut ting-plane met hod and ellipsoid met hod ( Boyd and

Barratt, 1991 [SI). Convex opt imization methods have the advantage of global convergence

over many nonlinear optimizat ion problerns (Horst. 1990 [Ml ). In the lollowing section.

the convex optimization algorithms are applied to a simple exarnple.

7.3 Exarnple

The convex opt imizat ion t heory offers effective algorit hms to derive cont rollers when the

performance specifications are convex in a finite-dimensional set. Consider the following

linear system:

where x is the position and velocity error vector: x = [et ezIT = [xf - xi x i - r2IT: 1

represents an identity matrix; the controller feeds back Cx: Kp = dia&,], KV = diag[k,]

are diagonal matrices- whose elements are positive gains; d denotes an unknown bounded

external disturbance.

Define the controller pararneters P sr A - BC in (7.1). The goal of the designer is

to tune the controller parameters (entries of the matrices of Kp and hi.) to irnprove the

system performance. When the expected performance can be described as convex design

specifications with respect to the gains, the specifications are convex in a finite-dimensional

set: Z = RP, p is the dimension of the controller pararneters (gains).

Chapter 7 Conuex Optirnizution Application 110 - - . . -

Assume that the following specifications Si and S2 are to be satisfied together. The

system is required to be stable: the H, norm (Francis, 1987 (121) of the transfer matrix of

(7.4) is h i t e :

SI kf pl(P) = ll(sI - P)-lIlm 5 a (7.5)

Assuming that the input d is unknown but bounded by

the dist ur bance reject ion specification is defined as:

where 5 is the tolerance error under the case of disturbance.

Consider the MSS control problem: Find a controller C in (7.4) to rneet the foilowing

two specifications sim ultaneoudy:

s = si A s* ( 7.8)

Convex optimization theory requires that the MSS problem be modified into an opti-

mization problem in the form of (7.1 ). Since the first specification is associated wit h system

stability. we modify this MÇS problem (7.8) into an optimization problem with objective

function $ and constraints Si:

f = min f vJ1 sa

If the optimal value f' 5 E / M , t hen there exist proper control parameters to meet the

MSS (7.8) . On the other hand, if the optimal vdue > - /Mt or in any case constraints

y1 5 a cannot be met, then no controuer with structure (7.4) c m meet the specifications

simultaneously. Therefore, the optimization problem description (7.9) not only solves the

MSS problem, but also answers the feasibility question (refer to Definition 2.5 and related

discussions).

Chapter 7 Convez Optirniration Application 11 1 - - - - - . .. -- . . -

As we discussed before, nonlinear parameter optimization methods generally cannot

treat the problem (7.9) satisfactorily. However, if the objective function f and the con-

straint functions pl have convex geometric property, then the optimization problem (7.9)

can be effectively solved by convex optimization methods. W e show that SI. S2 are convex

specifications in a finite-dimensional set.

When the Hm norm in Si and Sz is achieved at the zero frequency: uo = O (The case

of w # O will be addressed later)'.

obviously it is convex in terms of the control gains, represented by PL.

Two types of powerful algorithms are specificdy designed for convex optimization prob-

lems: the cutting-plane and the ellipsoid methods (Boyd and Barratt. 1991 [SI). Both of

them require that the subgradients of the convex specifications be calculated, as in Defi-

nition 7.1 (Refer to Section 5 for details). Consider the constrained optimization problem

( K g ) , we show how to construct one subgradient of objective h E i?f(P-') (Liu and MilIso

1996 [26])? where f is II P-L I l2 in Our case.

Theorem 7.1 For V P ~ ' , suppose a singufar value decomposition of Po1 is

where Uo. Vo are unitary matrices, and diagonal rnatrix C contains the singdar ual,ues 01

pol .

Let uoluo be the cofumns of Uo and that correspond to the maximum ualue of S . A

subgradient of$(P-l) ut Po1 k given by a linear function

- -

'Through calculations, we have empirical criteria: when k: < 2kp , system transfer matrix has H, norm

at w # O.

Chapter 7 Conuex Optimiiation Application 112

The linear function f(uo, vo) can be found through the following constructive proof of

t his t heorem.

Proof CVe know that (Stewart, 1973 [52])

Since the singulâr d u e decomposition of Po1 is Po' = UoCK, and uo, vo correspond

the maximum value of C, which is the maximum singular d u e O,,,. it can be represented

Therefore, for VP-'. we have

[IP;iI(, = O r n a ( ) = u;PôlvB

= up-'?Io + U ~ ( P ; ' - P-')vo

5 m=l,ull=l#i=l IuTP-'VI + u,T( PoL - p-' )vo

= I I p-y2 + zl,T(P;' - P-')?Io

Note that U;(P-' - P~-')V~ is a linear function. It c m be reorganized to have the form

of fT(p-' - Po')' where f = f (uo, v o ) Therefore,

T p-i IIP-'112 2 IIPo1ll2 - uo ( 0 - P-')v*

= II P;'II* + uT(P-' - P[')vo 2 IIPo-'II* + fT - (P-' - Po')

It shows that f (uo, va) is a subgradient of +(P-') at P;'.

We can apply the ellipsoid or cutting-plane algorithm to h d the gains (matrices) of

KP a~ld h'v.

Assume the required specifications are the same as we discussed before, (i.e.. 4, and

Sz), which are rnodified into an optimization problern (7.9), where the tolerance is i = 2,

Chapter 7 Conuez Optimizatation Application 113

and a = 1000. Suppose the desired trajectory is defined as

and apply either the cutting-plane algorithm (Liu and Mills. 1996 PT]) or the ellipsoid

algorithm (Liu and Mills. 1996 ['16]). Wihen the external disturbance is bounded by A1 = 1.

then the design result is -

min f = 1.618 < 1 = 2 31 <a Ad

It shows that under the constraint of SI. the optimal (minimal in this case) specifica-

tion value of specification S2 is 1.618. within the expected range. Therefore. t h e multiple

specifications Si. S2 cm be achieved simultaneously. The conesponding controller C is

r 7

The simulation results. as in Figure 7.2 show the satisfactory tracking propert- under the

maximum disturbance !;1f = 1. On the other hand. if the external disturbance is bounded

by k1 = 10, the design result given by the algorithm is

It implies that with the same constraints. the best specification that con be achieved esceeds

the expected one. Therefore. there exists no controller with the structure (7.4) that can

meet the specifications sirnultaneously. The simulation results. as in Figure 7.3. show that

the tracking error is significant under the rnaxinium M = 10 disturbance. The H, n o m

of the system transfer matrix (SI - P);' is dso shown in Figure 7.4.

When the H, norm of the system matrix in SI and S2 is reached at nonzero frequency

wo # O, we also show that the specifications are convex in tems of the control gains under

sorne constraints.

Chapter 7 Convex Qptimization Application 114

Lemrna 7.1 If a constant matriz A satisjk

where wo # O denotes the frequency at which H, n o m is reached, then we hane

Proof Obviously ( j u J - A)(juOI - A)-' = 1. Then from jwo(jUoI - = +

A(jwo1- A)-': we have

It follows that

( d o - ~ ~ A ~ ~ 2 ) ~ ~ ( j ~ ~ o I - -4)-'Il2 5 1

When (7.17) holds. it leads to the conclusion of (7.18)-

According to Lemma 7.1. we can transform the functional specification Si into

subject to

llPll2 - Uo 5 o.

From Lemma 7.1, we also know that in the frequency domain. if

then distubame rejection specification S2 (7.7) cm be met. We then substitute the ob-

jective function f (P) by a new objective Q ( P ) in the frequency domain. Therefore: with a

Chapter 7 Conoez Optimization Application Il5

new constraint (7.20) and objective (7.21), the optimization problem (7.9) can be rewritten

as follows:

It is obvious to see that constraint function tL(P) is a convex function with respect to

matrix P, which is finite-dimensional. We show that &( P ) is aiso a convex function of P.

Theorem 7.2 d(P) is a conaex junction with respect to P.

Proof For simplicity, denote II - I l z by II - 11. and wo by w. For any P. P and VO c X < 1.

we have 1

Let x = I(Pll,y = 1 1 PII. Since for any O < z, y < w. we have

1 'd - [Xy + ( 1 - X)z] X 1 - X =-+- iJ - p z + (1 - ~ ) y ] b - 4 b ~ - Y) - Yt-y

It shows that

On the other hand

Thus, rve have the conclusion

From Definition 2.2, objective 6 is a convex function of P.

Chapter 7 Convez Optimization Application

Before we construct a subgradient of the objective do we oeed the following inequality.

Lemma 7.2 I f x , y < w , then

1 1 -2- + x - Y ..'-z w - y ( w - y ) 2

Proof Formula (7.23) is equal to

If x = y, (7.24) obviously holds.

Lf x > y, we have (x - y) > O and O < (w - x ) < (a - y). then (7.24) holds.

If x < y, we have (x - y ) < O and O < (w - y) < (w - x). then (7.24) still holds.

Therefore (7.23) always holds- if x , y < d.

With the help of Lemma 7.2, we can construct a subgradient for the objective function

O( P) .

Theorern 7.3 For YPo,

is a subgradient of d(P) a t Po . where h is a subgradient of &( P ) u t Po, as shown in

T h e o r e m 7.1.

Chapter 7 Convez Optirnization Application 217 - - - -

Since h is a subgradient of 11 P II y we know t hat

IlPl1 - IlPoIl 2 hT(P - Po)

T herefore

Thus, we can say that g is subgradient of 4 ( P ) at Po.

Apply the ellipsoid and cutting-plane algorithms to the convex optimization problem

(7.22). with = 1.5, and sirniliar tolermce E = 2 under disturbance boundedness :CI = 1.

It turns out that the constraint leads to a point KP = diag[ll 1] and f i = diag[O,O],

where i ~ o = 1, however. the Hm n o m is unbounded (refer to Figure 7.4). That means the

constraint is too tight to meet the objective function. It shows that no matter what the

parameters of C are. the specifications (7.92) cannot be achieved sirnultaneously. Once

this fact is determined,

controller structures, or

gains.

it is also very valuable to the designers. They must select other

relax the design specifications. This will Save time spent tuning

Remark: Note that the conclusion of non-existence of the controiler only refers to the fact

that there exists no control parameter K p and h> to meet the multiple specifications?

under the fixed control structure (7.4). We cannot exclude the possibility of existence of

some other control structures to satisfy the specifications. Therefore, strictly speaking, this

convex opt imization application does not solve the feasibility problern as in Definition 2.5.

Chapter 7 Convet Optirnirat ion Application 118

7.4 Discussion

When the performance specifications are convex in a fuiite-dimensional set. such as the

set of controller gains in the computed torque control. the MSS control problem can be

effectively solved by using convex opt imization algorit hms. Compared wi t h many exist ing

controller design approaches proposed in robotic systems, convex optimization cao be used

to find the controller gains systernatically without empirical tuning approaches. It also

has the advantage of global optirnization over many nonlinear parameter optimization

methods. Furthermore, the convex optimization theory addresses the feasibility problern,

which cannot be treated by current existing approaches.

The limit of the convex optimization application is that it assumes that the design

specifications are convex in a finite-dimensional set, which. unfortunately cannot hold in

many cases. The situation of a fixed control structure (such that the control parameters

can be optimized) dso limits its capability of satisfying a wider range of performance spec-

ifications. Many design specificat ions are convex functions wit h respect to the closed-loop

transfer matrix H (Salcudean. 1986 [46]). In this case. the M S S problem leads to a con-

vex optimization problem over the indefinite-dimensional set Rfl. Boyd and Barratt ( L99 1

[8]) suggest a solution to this problem by using the secolled Ritz approximation method:

form a finite-dimensional subset (approximation) first and solve this finite-dimensional op-

timization problem: then enlarge the subset and operate the optimization in the larger

(finite-dimensional) set; when the subset becomes larger and larger, the optimal value wilI

tend to converge to the actual optimal value over the indefinite-dimensional set. Unfor-

tunately, the convergence of the finite-dimensional subsets is not always paranteed, and

computational problems may arise with the increase of the dimensionality of the approxi-

mation. Furthermore, it is required to construct a sequence of controllers as the starting

subset. How to pick these controllers properly is an open research problem.

The most difficult part is that the original MSS problem must be modified into as

Chapter 7 Convex Optimization Application 119 - -

optimization problem (7.1). Through the application to the robot trajectory system to

address three specifications, we experienced that this modification is not an easy task,

and the trade-offs arnong the different specifications may lose the original goal of treating

multiple simultaneous specifications directly. Another shortcoming is that the objective

funct ion in Equat ion (7.1 ) is constructed t hrough weighted summat ion (7.2) or weighted

maximization (7.3). It is obvious that selection of the weight coefficients has no direct

relat ionship with the multiple specifications themselves. and the maximization. as we did

in the above example, constrains the solvability of the MSS problem.

In conclusion, the application of convex optimization theory to the MSS control problem

is technicdy challenging. Our proposed new convex combination method. as we have

shown, performs better in solving this type of control problem.

Both the cutting-plane method and the ellipsoid method require that an -initial boxz

be guessed to start the optimization procedure. which is a hard task. almost as difficult

as solving the optirnization problem itself. In the following section, we develop a new

init ializat ion approach to overcome t his drawback.

Initialization of Cutting-plane Met hod

The convex cutting plane method was developed to solve convex programming problem of

the form

min (j(z) zES

where S c Rn is a closed convex set and the function 8(-) is convex over S. The general

form of a cutting plane algorithm for problem (7.26) is as follows (Luenberger. 1984 [34]):

Given an initial polytope (a set defined by the intersection of a finite number of closed

haif spaces) P contâining S, conduct minimization (by linear programming) over P. If the

optimal point x E S, then it is the solution of problern (7.26). Otherwise, find a hyperplane

Chapter 7 Conuex Optirnizotion Application 120

H separating r from S (such a hyperplane always exists because of the convexity of the set

S) (Rockafellar. 1970 [44]), and update the polytope to be a new (outer) approximation

polytope including a half space defined by H . Repeat this procedure until an optimal

point is located in S. Xote that the cutting plane method is based on the assumption

that an initial polytope P is given. If the convex set S is open, Le.. the unconstrained

case S = Rn, it is impossible to find such a P. Boyd and Barratt (1991 [8]) suggest a

practical approach: guess an *initial box" i.e.? a polytope that contains the optimal point.

However. to correctly guess such a poIytope is a hard task. almost as difficult as solving

the optimization problem (7.26) itself.

ÇVe propose a new initialization approach to overcorne this drawback. Furthemore.

with this new initialization approach, faster convergence speed may be achieved. Consider

an unconst rained convex opt imization problem

min ~ ( z ) ,TER"

where ci(.) is a convex function. Denote the optimal point as x'. at which

Note that specific algorithms differ rnainly in the mamer in which the hyperplane H

is selected. When the function p(-) is convex? we c m choose such a hyperplaoe through

subgradients (Boyd and Barratt. 1991 [SI).

Definition 7.1 (Subgradient) If 4 : Rn R" R+ is conuex . we will say that g E 'Rn iu a

subgradient of 4 at x if

o ( z ) 2 &(x) + gT(r - x) for al1 3

Denote the set of al1 subgradients of c$ at x as &(x).

Chapter 7 Convez Optimization Application 121

Figure 7.1: Subgradient of convex function

Assume t hat a subgradient of the coovex function b( r ) is obtained. which always exists

(Rockafellar, 198% [43]). and function values at k points X I . . . , xk are also computed. For

ony xi among those k points, we know that

since r' is the optimal point and gi is the subgradient of Q at point xi. It follows that the

optimal point x' must be located in the half space

which is dehed by a h-yperplane

Therefore, with the computed k points and their corresponding h-perplanes. we know

that the optimal point x x is located in the polytope

Defme a piecewise linear function

at which polytope Pk is bounded. It follows that a lower bound c m be found


where Lk can be cdculated by linear programming methods (Boyd and Barratt , 1991 [8] ).

Denote X ~ + I as the optimal point of that linear programming problem (7.35).

We dso can find an obvious upper bound

4' 5 U k = r l i ~ k min 4 ( x i )

such that the optimal point xm is located between the lower and upper bounds. If the Lr

and Lik are close enough. the algorithm stops and xk+i is considered as x-. Otherwise, by

selecting a new hyperplane

a new polytope can be constructed

Thus a new lower bound L k f l and upper bound Crk+l can be computed in the same w a -

as (7.33) and (7.36) respectively. Notice that

which guarantees the convergence of the cutting plane method.

Now the on- problem left is how to select the initial polytope Pl containing z'. -4

starting point is required to begin the cutting plane method. We c m arbitrarily select it

because of the global convergence of this convex method. However, the lower bound of this

starting bound (7.35) is negative infinity, which is derived by the author and s h o w in the

following lemma.

Lemma 7.3 The starting lower bound Li is negative infinity:

Chapter 7 Convex Op ti mizut ion Application 123 - - -

Proof For any nonzero subgradient vector g, there always exists nonzero solution to the

equation

T g x = c (7.41)

where c is an arbitrary number. For any N > O, select

where x l is the starting point and 6 is any positive number: 6 > O. It is concluded that

there exists a nonzero solution to the equation (7.41 ) with the selected value c. It shows

that for any !V > O? there dways exists 5 such that

It Ieads to the conclusion

In this case, the second point x2 cannot be determined such that the algorit hm has to

stop without success. It is suggested in Boyd and Barratt (1991 [SI) to guess an -initial

box"

Bo = {Z E Rn 1 x,in 5 r 5 x m a r ) (7.12)

If this initial box contâins the optimal point x', it can be found by iterating the cutting

plane algorithm. If the point rk stays on the box boundary for too many iterations.

then "increase the box size and continuen. However, as mentioned before. such an ad hoc

procedure which requires the initial box to be guessed is empirical. This task may be as

difficult as solving the op timizat ion problem (7.27) itself, because sometimes a successfd

guess of an initial box depends on the successfd guess of the optimal point. Even if a "safe"

guess contains the optimal point, which u s u d y requires a very big initial box, the algorithm

Chapter 7 Conuez Optimization Application 124

will actually start from the boundary of that big box so that the number of iterations will

increase, leading to an increase of the size of the linear prograrnrning problem (Luenberger;

1984 (341).

By properly selecting two starting pointsl instead of one point, we c m avoid guessing

the initid box. Furthermorel faster convergence speed c m be achieved. The main resuit is

given by the following theorem (Liu and Mills, 1997 PSI).

Theorem 7.4 Given g(-) E R, the function of a subgradient of the function O ( - ) . select

two starting points xo, x l y such that for their subgradients go: 91: go f 91. Further. denotr

u = (go +9i)/Y

v = (go -gi)/2

X = U - u T - V - U T

Y = t p u T - v - u T

If matrix X E RnXn is semi-negatioe definite (s.n.d. j . then the louer bound L is jînite:

where rl zs the cornmon point of two linear functions

and 21 can be set as the nem point x*.

Proof First of all, since go # gl, there always exists a common point rl such t hat (7.51)

is obtained. Obviously, the matrix Y is skew-symmetric. It follows that for any A E R"

If the matrix X is s.n.d., we have

A ~ X A 5 O for dl A E Rn

Chapter 7 Convex Optimization Application 125 - - -- - . --

It foLlows that for any A E R"

aTg,$~ = A*(U + V ) ( U - V)*A = A ~ ( X - Y ) A 5 O

which means that

From (7.34) we know that

There are two cases:

(1) For sorne s, +:b(=) = & ( I I ) + 935 - XI)- (II) For some other x. bib:b(x) = ~ ( x o ) + &(" - IO).

Case 1: It shows that

From (7.51) and (7.56) it leads to

T herefore

È 4(xd+g:(zL -4 It follows that the lower bound Ll is finite:

Chapter 7 Conuex Optirniration Application 126

Case II: Identically: it c m be shown that

Therefore. in either case the lower bound LI is the value a t common point ZI, which can

be set as the new point 1 2 to construct a new polytope. Thus. the convex cutting plane

method can be continued. Further, in most cases such a new point xz is much closer to the

optimal point than the boundary of an initial box, which will enable faster convergence.

Example: Consider a parameter optimization problem

w here

P = A + B C -a=

1 1 Pllz e-presses the Euclidean 2- matrix norm (Stewart, 1973 [Z])

where a,.,(P) is the maximum singular value of matrix P.

For any 2': x2 E R4 and X E [O, 11, we know that

Therefore function 4(x) is convex over R4. A subgradient c m be computed by a method

in (Liu and Mills, 1996 [26] ) . The optimization results with different initialkation ap-

proaches are shown in Table 7.1.

Chapter 7 Convex Optimisation Application 127

Table 7.1: Cutt ing-plane optirnization results wit h different init ializat ion approaches

Initiaiizat ion Approach -- -

Initial Box - -

No initial box

Without initial box

Resul t kerat ions

The solution is unbounded I

and at infinity 1 S/A

O" = 1.0000

Subgradients go, gl satisfies

the condition of Theorem 3.1

Chapter 7 Conuex Optimization Application 128

the optimal point can be found at x = 5 = O mathematically with the value

From the optimization results. it is verified that only one s tx t ing point does not lead

to a successful optimization procedure (Case 1). By guessing an initial box, the algorithm

may stop at the boundary after some iteration (Case 2). It is therefore required to increase

the size of the box. However, how to reset this box effectively depends on how good the

guess of where the optimal point 5 is. which is as difficult as finding the point itself. If a

big initial box is selected and fortunately the optimal point is contained in this box (Case

3). this point can be found alter the optimization procedure. On the other hand. if we

properly select two starting points (Case 4)? the optimization problem can be effective-

solved. and with faster convergence.

7.6 Cornparison of Performance Oriented Methods

There are many controller design approaches proposed in the control field. S tarting from

establishing a control structure. some approaches address very specific performance criteria.

or try to meet a few more requirements by tuning the controller parameters. On the other

hand. some approaches try to derive the controller (structure and parameters) based on the

performance specifications. Such approaches are called performance orien ted methods. [n

this sense. the Hm theory (e.g. Francis, 1987 [1%]), the convex optimization theory (Boyd

and Banatt, 1991 [g]), and our proposed convex combination theory dl belong to the latter

class. A brief comparison of these three methods shows the novelty of our proposed convex

corn bination t heory.

Clrapter 7 Conuez Op t imitation Application 129

Based on the uniform framework of control systems, all these three methods describe

the design specifications as functions of the closed-loop transfer rnatrix, Therefore, it

is possible for them to address the MSS control problern. Further, the H, norm is

also a convex function. Thus the three methods all treat convex specifications.

Hm theory focuses on the control problem in the presence of system uncertainty or

external disturbances. Since the H, norm is defined in the frequency-domain. it also

iimits its application to the MSS problem, which often involve specifications both in

frequency domain and in t ime domain.

0 The convex optimization method solves the MSS problem through an optimization

procedure. However. it is difficult to rnodify the standard MSS problem defined in

Defmit ion 2.3 to the optimization problem formalized in (7.1). because combining the

multiple objectives into one single cost function loses the aspect of satiseing multiple

specifications. Further, this method has to face sorne difficulties associated with the

infini te-dimensional op timizat ion pro blem.

Our proposed convex combination method directly solves the 4ISS problem (Defi-

nition 2.3), without going through the modification or any optimization procedure.

The two-stage design strategy is straightfonvard and easily irnplernented. . h o t her

benefit of this method is that the designer only needs to focus on one design spec-

ification each time, when he or she can t&e advaotage of existing controller design

approaches. After the combination. the derived controller wiU have new control struc-

ture and parameters, which cannot be found by other existing approaches. The limit

of this method is that the feasibility problern is not as obvious as that by using the

convex optimization method. Wevert heless. the discussion of the feasibility problem

of the convex combination method makes it clear that it leads to the same conclusion

as that in convex optimization method.

Chapter 7 Convex Optimization Application 130

Summary

In this chapter we addressed the application of the convex optimization theory to solve the

MSS problem. Our research work in this chapter is summarized as foUows.

a We conclude t hat it is possible to apply the convex optimization method to solve the

MSS problem, especially when the performance specifications are convex in terms

of a finite-dimensional set of parameters under a fixed control structure. A simple

example is applied for illustration. The simulation results veriS the effect iveness of

the eilipsoid (Liu and Mills, 1996 1261) and the cutting-plane algorithms (Liu and

Mills, 1996 PZ?]).

O We dso note that the application of convex optimization has limitations and difficul-

ties in its capability t o address the MSS problem. This situation generates the need

for a better design method to solve the MSS problem more etticientl- practically. and

in a wider range. We developed the convex combination rnethod to meet this need.

a During Our study of the convex optimization theoq introduced by Boyd and B ~ r a t t

(1991 [8]), we proposed a new initialization approach to start the convex cutt ing-plane

optimization algorithm (Liu and Mills, 1997 [281).

O Through cornparison of different performance oriected rnethods. we show that our

proposed convex combination method is more generic and easily to implement. It

has a strong potential application in the robot industry and other areas.


time (sec)

time (sec)

Figure 7.2: Response: XI, 22 vs tirne with disturbance i\.I = 1

Chapter 7 Convex Op ti mizat ion Application 132

1.5 2 time (sec)

time (sec)

Figure 7.3: Response: x i 7 x2 vs time with disturbance M = 10

Chapter 7 Conuez Optimization Application 133

Figure 7.4: Hm noms under different control gains

Chapter 8

Conclusions and Discussions

This thesis is concerned with development of a performance oriented controller design

method to address multiple sirnultaneous specificat ions, and its application to robot t ra-

jectory tracking systems.

8.1 Convex Combination Methodology

We proposed a so called convez com6ination method to solve the multiple simultaneous

specification ( MSS) cont rol problem.

Based on a linear system framework, a system input-output relationship is u n i f o d y

described by the closed-loop transfer matrix, and the design specifications are for-

mulated as mathematical funct ions of t his closed loop t ransfer matrix. Therefore.

multiple specifications are associated with the sarne variable (transfer matrix H as

shown in equation 3 2 ) , and hence they are considered together.

O When the uniform linear system frarnework has been established and a set of multiple

design specifications has been specified, the MSS control problem is formulated (2.7).

A controller that can satisfy these multiple specifications simultaneously is called a

sat isfact ory cont rouer.

Chapter 8 Conclusions and Discussions 133 - - - -

r When the design specifications have a convex property, the MSS control problem

can be solved by applying the proposed convex combination method. The convex

combination method is achieved through a two-stage design strategy: First, fmd dif-

ferent sample controllers, each of which satisfies at least one specification. Then,

design a cont roller t hrough a (convex) combinat ion of the closed-loop transfer matri-

ces corresponding to each of the sample controllers such that it satisfies the multiple

specifications simultaneously. Such a design strategy lessens much of the burden on

the controuer designer, when helshe only needs to focus on finding the sample con-

troller to meet one specification at each time (first design stage). The controller can

also take advantage of existing control techniques to find sample controllers. Fur-

ther. the convex designed controller has a new control stmcture that cannot be found

by many current control a~proaches (second design stage). Therefore, the convex

combinat ion method is straightforward and easily implemented.

0 When the design specifications are too tight to be met simultaneously, the convex

combination method has the ability to determine whether the MSS control problem

is solvable (feasibility problem).

8.2 Application to Robot Tracking Systems

We applied the convex combination to robot trajectory tracking systems to solve the MSS

control problem.

The feedback linearization technique rnakes it possible to apply the linear convex corn-

bination method to nonlinear robotic systems. Through feedback linearization, the

nonlinear dynamic equation of a robot rnanipulator c m be linearized and described

in the uniform system framework. Therefore, the convex combination rnethod can be

applied to solve the MSS control problem.

Chapter 8 Conclusions and Discussions 136

0 The application is conducted on a commercial CRS A460 robot. The MSS control

problem is addressed both in joint space and task space. When the design specifi-

cations are described in joint space, the controiler is designed and implemented in

joint space. When the design specifications are described in task space. we either

transform the specifications into the joint space, and design the controller in joint

space; or we transform the designed controiler from task space into the Local joint

space, to be implemented.

The application of convex combination method to robot trajectory tracking systems

is achieved through feedback linearization. However, uncertainty always exists in the

mathematical model. Robust control design implies the design of a deterministic or

fixed control in the presence of significant system uncertainties. \Ve show that the

robustness in the sense of system boundness also has a convex property and can be

considered as one of multiple specifications that are to be met together. Therefore.

the convex combination method designs a cont roller to meet t hose specificat ions.

including the robust specification. It implies that the proposeci design method is also

valid under system uncertainty.

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