Robust 3C,- Based Control of Flexible Joint

Robots with Harmonic Drive Transmission

Majid M. Moghaddam, B.Sc., M.Eng.

A thesis submitted in conformity with the requirements

for the Degree of Doctor of Philosophy

Graduate Department of Mechanical and Industrial Engineering

University of Toronto

@ Majid M . Moghaddani. 1997

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To my wife

and

to our parents

Acknowledgments

I woiild like to thank niy advisor, Professor -4ndrew A. Goldenberg, for his guidance

and assistance in this research. He provided me with freedom and support to pursue

11iy own rwearch ideas. My education has been enhanced because of him.

Than ks to the people in Kobotics ancl Automation Laboratory (RAL) of the

I 'niversity of Toronto for their support. Special thanks are due to Professor i l enad

1iirr.anski for his help particularly during the experirnental phase of the work. 1 would

l i ke to than k al1 ~ i iy friends in the Departnient of Mechanical Engineering for their

generoiis support ancl friendship. Thanks to kirs, B. Fung, the secretary of the

( ;radiiatc Stuclies of the Departnient. for being always kind and cooperative during

riiy stiidies in the tlepart~iient.

I wo~iItl ais0 like to than k Profèssor B. Francis for being always availahle for

any dis(-iissions or questions that 1 hacl dtiring the course of this work.

I wisli to express riiy gratitude to iiiy faniily for their encouragement. under-

standing and support over the course of iiiy eclucation that eventually culminated with

t h i s thesis.

Finally, 1 would like to gratefully acknowledge the financial support of Nat-

1 1 r d Srienc-e and Engineering Research CounciI (NSERC) , Institute of Robotics and

Intelligent Systeiris (IRIS) and Ministry of Culture and Higher Education of Iran

(kl('HE1).

MAJID M. MOGHADDXM

Robust X,- Based Control of Flexible Joint

Robots with Harmonic Drive Transmission

Majid M. Moghaddani, Ph.D.

Departinent of Mechanical and lndust ria1 Engineering

( :niversity of Toronto. 1997

Abstract The probleiii of the control of flesible joint robots is considered. Motion and torqtre

(-ont roi of robot joints, especially tliose equipped with harriionic drive (HD) traris-

i~iission. is a challenging task due to t h e inherent nonlinear characteristics and joint

f l w i bility of such systeiris. Two issues relatecl to the control of flexible joint robots

are investigated. The first is the developnient of a systematic schenie for selecting

un(-ertainty bounds of robot joint nonlinearity and fiexibility for control design piir- .

poses. The second is the development of motion and torque control schemes that

guarantw robiist perforniance in the presence of ttioclel uncertainties.

\VP propose a twofold robust control design for flexible joint robots with H U :

an a r t iiator-level torque control and a lin k-levd motion con trol. We u tilize iiiiil-

tivarialile R, - basecl optiirial controi laws supported entirely by frequency doriiain

riirasures at both leveis. The proposed method provides a unified franiework for

a(-liieving the desired performance requirements and for preserving robust stability in

the presence of iriodel uncertainties. lJsing simulation i t is shown that the proposed

iiiethocl is more robust than conventional niethods because the uncertainties due to

the actuator-t ransrnission nonlinearity are explicitly considered in the controI design

prowrlu re.

In the design of the actuator level torque contro1, we analyze nonlinear har-

riionic clrive phenomena that have been widely observed in experiments. The focus is

on incorporating into the control design process a knowledge of the niismatch betweon

the physical systerii and i ts tiiatheri~atical niodeis. The describing function and corric-

scctor-bounded rrodirrearity riiethods are used to build into the controt design pro(-ess

the effec~5 of tiiisriiatch between hysteresis, friction and nonlinear stiffness of H D

transrtiission and their mathematical models. In the design of the link level motion

control. a nonlinear conipensator based on the cornputed torque technique is de-

riveci. It i~ shown that the closed-loop systeni achieves robust performance using the

prc~pos~d c'ont rol design technique.

! 'si~ig the Srriali Gain theorerii and the Lyapuriou Furrctiorz method, the sta-

hiiity of the proposed control scheme is verified. Finally, in order to illcstrate the

propos~cl tech nique two con t rol designs are presented for the IRIS-facility experi-

riieri ta1 test bed (a versatile. rriocl uiar, and reconfigurable prototype robot developecl

at the Koho tics and .i\utor~iation Laboratory of the University of Toronto) together

tvith siiiitilation and experiirientaI results.

Contents

Abstract

List of Figures

List of Tables xiv

Chapter 1 Introduction I

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 blotivation 1

. . . . . . . . . . . . . . 1.2 Background on Flexible Joint Robot Control :3

1 ('ontributions of this Research Work - ; > . . . . . . . . . . . . . . . . . . -

1.4 Oiitline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . I

Chapter 2 Modeling and Uncertainty Description of Robots with Har-

monic Drive Transmission 8

. . . . . . . . . . 2.1 .L-\ctuator and Haririonic Drive Transrnision Models 9

. . . . . . . . . . . . . . . . . . . 1 .1 .1 Slotor-Ariiplifier Subsysteiii 10

. . . . . . . . . . . . . . . . . . . 2.1.2 Wave-Cimerator Subsysterii 10

. . . . . . . . . . . . . . . . . . . . . . 2 . 1 Flex-splin~ Subsysteni 1 1

. . . . . . . . . . . . . . . . . . . . 2.1.4 Circular-Spline Subsysterii 14

. . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Harnionic Drive 14

. . . . . . . . . . . . . . . . . . . 2.1.6 Flexiblejoint RobotModel 1.5

. . . . . . . . . . . 2.2 hloclel Verification Based on Experiniental Results 15

. . . . . . . . . . . . . . . . . 2.2. I Tinie Response Characteristics 16

. . . . . . . . . . . . . . . 2 - 2 2 Freqiiency Response Characteristics 16

. . . 2 3 t'ni-ertainty Description: The Frequency Doiiiain-Based Method 19

2.4 I :nc~rtainty Description: The Descri bing Function-Based Method . . 22

. . . . . . . . . . . . . . 2.4.1 Coiiiputing Describing Frinction (DF) 24

. - . . . . . . . . . . . . . . . . . . . . . . 2 ..5 I Yrirertainty Weights Selection 14

. . . . . . . . . . 2.5.1 Motor Friction Uncertainty Weight Selection 'LI

. . . . . . . . . . . . 2.5.2 Hysteresis Uncertainty Weight Selection 30

. . . . . . . . 2.5.3 Nonlinear Stiffness Uncertainty Weight Selection 31

. . . . . . . . . . . . . . . . . . . . . . . . 2.6 Siiiii~iiary and Discussion 94

Chapter 3 A New Robust Motion and Torque Control Design Method 35

. . . . . . . . . . . . . . . . . . . . . . . . . 1 Kigid Body Error Mode1 36

. . . . . . . . . . . . . . . . . . . . . ij.2 Kobiist Motion Cont rol Design 38

. . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Design Method 38

. . . . . . . . . . . . . . . . . . . . . . . . . 2 . 2 Design Exa~iiple -10

3 -4ctuator-Level Torqiie C'orit roi Design . . . . . . . . . . . . . . . . . -U

. . . . . . . . . . . . . . . . . . . . . . . . . . 3 . 1 Design .M ethod 4:J

. . . . . . . . . . . . . . . . . . . . . . . . . . 2 Design Exam ple 46

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Sii~riiriary 53

Ch~pter 4 Experimental Evaluation and Robustness and Performance

'Ikade-offs in Uncertaint y Mo deling 54 . P . . . . . . . . . . . . . . . . . . . . . . . 4 . i Expeririiental Setup Facility ;ln

F . . . . . . . . . 4.1.1 Expeririiental Identification of Transfer Function .>i

. . . . . . . . . . . . . . . . 4.2 Kobiistness and Performance Trade-ORS 60

. . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 ( lontrol Objectives 60

vii

. . . . . . . . . . . . . . . . . . . . . 4 . 2 Uncertainty Descriptions 60

. . . . . . . . . . . . . . . . . . -4.2.: Ciontrol Probleiii Forriiulation 61

. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Expeririiental Results 62

. . . . . . . . . . . . . . . . . . . . 4.3.1 Mode1 iincertainty Weight 62

. . . . . . . . . . . . . . . . . . . . 4 . . Actiiator-Saturation Lirni t 64

. Chapter 5 Stability Analysis of the Proposed Robust Torque Control

Design 73

. . . . . . . . . . . . . . . . . . . . . . . 3 .1 Sector Bounded Conditions 74

. . . . . . . . . . . . . . . . . . . . 5.2 Linear Fractionai Tramfornation 76 .. . . . . . . . . . . . . 5 .il Input-Output Stability and Srnall Gain Theorerii I I

. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 LoopTransforriiation 130 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ., i Stability Kesults Hl

. . . . . . . . . 5 . 1 Sniall-gain T heoreiri-Based S tabili ty .A nalysis S l

. . . . . . . . . . . . . . . 5 5 2 Lyapunov-Based S tability Analysis H.3

Chapter 6 Conclusions and Future Directions 88

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 1 ('onclusions S 8

. . . . . . . . . . . . . . . . . . . . . (5.2 Siiggestions for Future Research 89 O

References 91

Appendix A N,,. Design and Structured Singular Value (p) F'rarne-

work 98

. . . . . . . . . . . . . . . . . . . . . . . A . 1 .M athematical Preliniinaries 9s

. . . . . . . . . . . . . . . . . . . .A 1 . 1 The Signal Spaces Cz and X2 9%

A.1.2 The Function Spaces Cm and 31, . . . . . . . . . . . . . . . . 99

. . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 R, C'ontrolTheory 101

viii

. . . . . . . . . . . . . . . . . . . . . . . 4 2 . 1 Nominal Performance 102

. . . . . . . . . . . . . . . . . . . . . . . . . . h.2.2 Kobust Stability 104

. . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 p -AnaIysis Methods 105

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 R., Synthesis 108

List of Figures

2.1 Harrrionic drive systerii . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Hysteresis iiiocfel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 . 3 lclentified paranieters y and R vs . load history T; . . . . . . . . . . . 13

2.4 Iclentified and ~iieasured hysteresis . TI, verses q,. . . . . . . . . . . . . 13

2 ... Theoretical and experi~nental wave-generator angle . . . . . . . . . . 17

2.6 Theoretical and experiniental wave-generator velocity . . . . . . . . . 17

2.1 blagnitiide Bode plot of the iriodel and experinient . . . . . . . . . . . 18

2.X Phase Bode plot of the rnodel and expeririient . . . . . . . . . . . . . 19

. . . . . . . . . . . . . . . . . 2.9 Block tliagratii of additive uncertainty 20

. . . . . . . . . . . . . . . . . . 2-10 Nycluist lot of additive uncertainty 21

2.11 Hlock diagrarii of output riiuhiplicative uncertainty . . . . . . . . . . 22

2 . 12 .A nonlinear eleriient and its describing fiinction representation . . . . . 24

2-13 Saturatiori nonlinearity ancl corresponding input-output relationstiip 25

2.14 ( 'oiilorth friction type noniinearity . . . . . . . . . . . . . . . . . . . 2li

2.15 Hysteresis type nonlinearity . . . . . . . . . . . . . . . . . . . . . . . 'LI

2.16 ('oiiiplete nonIinear triode1 of the robot HD drive system . . . . . . . 23

2-17 Motor and DF of C~oulomb friction . . . . . . . . . . . . . . . . . . . 28

2.1s Freqiiency response of the niotor-friction niodel . . . . . . . . . . . . 30

2 . 19 .Cl irltipliçative uncertainty rriodel of the niotor . . . . . . . . . . . . . 3 1

2.20 Frequency response of the variation between the nominal and per-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t il r bec1 inotlel 32

. . . . . . . . . 2.2 1 ( 'onir. sertor representation of the nonlinear stiffness 32

. . . . . . . . . . . . . . . . . . 2.22 Liriearization and r-onir-sector bound 33

. . . . . . . 2-23 Aclclitiv~ irnçertainty representation of nonlinear stiffness 34

. . . . . . . . . . . . . . . . . . . . . . 11.1 Error tlynairiics block diagram 38

. . . . . . . . 3.2 Error iiiotlel c-ontrol problern formulation block diagratii 39

. . . . . . . . . . .... LFT of error niodel control probleni block diagrani 10

. . . . . . . . . . . . 3.4 Frequency response of weighi ng transfer function 42

:)J Step response error of rlosed-loop systeiii . ( e = q,d - g, ) . ( 6 = qC - 4) 43

3.6 ('Iosetf-loop interconnection of the complete flexible joint robots . . . 44

i3.7 Freqiiency response of weighing transfer functions . . . . . . . . . . . 47

. . . . . . . . . . . .I.S Norriinal performance plots of cont rollers: KI, . Kp 4s

. . . . . . . . . . . . . . 3.9 Hobust stability plots of controllers: /ih. /Cp 19

. . . . . . . .$.IO LFT block diagrani of robiist perfortiiance control design 50

. . . . . . . . . . . . i1.11 Kobust perforiiianze plots of cont rollers: Kl,. Ii, 50

. . . . . . . . . . . . . . . .I 12 Yoiiiirial i ~ j step response fih.solitl Ii, . c las hed .il

. . . . . . . . .( 1 3 Noitiirial cc and t l - step responses . /<,,.solicl. K , .dashed 52

.1.14 FCrttirbncl cta, step response . lilL.solid. li, .dashed . . . . . . . . . . 52

.I . 1.5 Pert u rbecl e, and è, step responses . K I solid. K, .dashed . . . . . . . 53

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 IRIS arriis 56

. . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Kestrained motion setup 56 .. . . . . . . . . . . . . . 4.3 Real tiine controi iniplenientation organization a I

. . . . . . . . . . . 4.4 input-output torque magnitude plot of IRIS-joint 5s

. . . . . . . . . . . . . . 4.5 Iriptit-outpiit torqiie phase plot of IRIS-joint 5s

. . . . . . . . . . . . . 4.6 Estiiriated and ineasured input-output torque 59

4.7 Magnitiide plot of error torque and multiplicative uncertainty weight . 6 1

. . . . . . . . 4.H Rlock diagrarn of trade-off control probleni formulation 62

4 9 L FT of tracle-off control problerii forniulation . . . . . . . . . . . . . . 6:3

. . . . . . . . . 4.10 S t ~ p response of the IRIS-joint for controller MU-K1 61

. . . . . . . . . 4.11 Step response of the IRIS-joint for controller MU-K2 67

. . . . . . . . . 4.12 Step response of the IRIS-joint for controller MU-K3 68

. . . . . . 4.13 Sintisoidal response of the t RIS-joint for controller MU-KI 68

. . . . . . 4.14 Sintisoiclal response of the IRIS-joint for controller MU-K2 69

. . . . . . 4.15 Sinitsoicial response of the IRIS-joint for controller MU-K3 69

. . . . . . . . . 4.16 Step response of the IRIS-joint for controller AC-KI 70

4.17 Step response of the IRIS-joint for controller A G K 2 . . . . . . . . . 10 4.1 S tep response of the IRIS-joint for controller .4 C-K3 . . . . . . . . . 71

1.19 Sintisoidal response of the IRIS-joint for controuer A G K 3 . . . . . . 71

4-20 Sinusoiclal responseofthe IRIS-jointforcontrollerAC-K5 . . . . . . 72

4.2 1 Sintisoidal response of the IRIS-joint for controller AC-KG . . . . . . 72

1 ('losmi-loop nonlinear iiiodel of flexible joint robot . . . . . . . . . . . 74

-. 2 S~ctor-bounded nonlinearity . . . . . . . . . . . . . . . . . . . . . . . r.1

5.3 Nyqtiist plot of G ( s ) = (s+2jps+5) insicle sector [-0.4. 11 . . . . . . . 76

.- 5.4 Linear fractional transformation . . . . . . . . . . . . . . . . . . . . ( , . . ...a Feedback connection . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

.>.Ci .4bsolu te stability frarriework . . . . . . . . . . . . . . . . . . . . . . 4q0

. - 1 . 1 Loop t ransfoririation . . . . . . . . . . . . . . . . . . . . . . . . . . . SI

5 . H Li~iear fractional transfoririation . . . . . . . . . . . . . . . . . . . . $2

9 Staldi ty block diagrairi . . . . . . . . . . . . . . . . . . . . . . . . . #V:3

.F.lO Sertor noniinearity transformation . . . . . . . . . . . . . . . . . . . 8.5

5.1 1 Transforriied block diagram . . . . . . . . . . . . . . . . . . . . . . . 86

. A i ( ;meral LFT fratiiework of p design . . . . . . . . . . . . . . . . . . . 101

A.? K., disturbance attenuation problerii . . . . . . . . . . . . . . . . . . 103

sii

Robtist stability control problem forniulation 104 A. . . . . . . . . . . . . . . -4 -4 ( ;mers1 analysis franiework . . . . . . . . . . . . . . . . . . . . . . . 106

. . . . . . . . . . . . . . . . . . . . . . . . . . . . A ..ï ( ieneral frarriework 101

. . . . . . . . . . . . . . . . . . . . . . . . . . . . A -6 ( ;erieral frariiework 109

... X l l l

List of Tables

3.1 Flexible joint robot parameters . . . . . . . . . . . . . . . . . . . . . 41

4.1 Paranleters for control design with fixed performance weight . . . . . 64

4.2 F'arariieters for control design with varying performance weight . . . . 6.3

4 . Parameters for controi design with varying performance and uncer-

tainty weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7

Nomenclature Fields, Norms and Spaces

Corii plex nu~iibers

:= ess siiptél+ 11 f ( t ) l l , .Cs nortn of f : R+ -t Rr

Space of Fourier transforiii of signais in C2+

Space of Fourier t ransforrris of signals in C2-

Hardy space of functions analytic in C+ and bounded

on the imaginary axis, with norm (IQI(, := s u p W E ~ I IQ(jw) 11 Linear 31, cont roller transfer function

Yorrried tinear space

A n extendecl linear space

Space of Lebesgue rneasurable functions on R+ with finite C2 norni

Space of Lebesgue measurable functions on R+ with finite L, norni

Space of signals defined for positive tirne and zero for negative tiiiie

Space of signals ciefined for negative time. and zero for positive tiriie

Real niim bers

Space of real-rational fiinctions in 3C,

.r Fourier transforrii of J

flmui [.] Maxi triuni singular value of a coiri plex niatrix

11-41 := (X'Z)~. Euclidean noriii of x E C"

Il41LL .- .- niaxi=l,..., I Z ; ~ , OC-norm of L E Rr

X Maxiiiiuni eigenvalue of the niat rix

Parameters

n j l and a f 2 Constant coefficients

&(iw) Total friction torque generated a t the bearings

H,"(G~, + Friction terni of the wave-generator bearing

e Angular position and velocity errors

iipper-bound of the c-iirrent clefind by the an1 plifier electronics

Motor current

c-oriibined inertia of the rotor. shaft and the wave-generator

rt x 71 diagonal niatrix of actuator inertia

Soft windup correction factor

constants

Back e.ni.f. constant of the motor

blotor torclue constant

An1 pli fier gain

IL x n link inertia niatrix

Output signal amplitude

Matheiiiatical niodel of the :LI

Matheniatical tirodel of the iV

IL x 1 vector of centrifugai, gravity, and Coriolis(genera1ized) for(-es

Gear ratio

'i uiii ber of teeth on the Hex-spline ou ter circumference

Descri bing function

Soiilinal plant riioclel

Perturbeci plant transfer function

A n unknown perturbation

Motor angular veloci ty

Flex-spline angular vdocity

C'-end angular clisplaceiiient

an unknown nonlinearity

IL x 1 link position

11 x 1 vector of desired link position trajectory

xvi

Motor armature resistance

C'-end load torque

Driving torque

Motor Torque

Torque sensor output located at the link side

rr x 1 veïtor of control torque input

Torque exerted on the wave-generator

New control input

Applied torque

Amplifier in put corri niancl

TL x 1 vector of fictitious control input

Maxiniurii output voltage of the amplifier

Frequency points

A real-rational, stable minimum phase transfer function

Frequency dependent weighting mat rices

Noise weighting function

Slope at the transition from loading to unloading

.4 norrri bol nded coriiplex nurriber

Phase shif't

.An arbitrary signal

xvii

Chapter 1

Introduction

1.1 Motivation

The use of gearing systenls is a well established means for lighter manipulators to

hnnclle heavier payloads. However. gear systenis suffer froni nonlinearities such as

l x d h . s h . nonlinear stiffness. and various types of friction. Correspondingly. the in-

I i ~ r ~ n t c-o~ri plianc-P between the input and output sliafts of a transmission systeni

i ~ ~ t rurlui-PS iinclesirable resonant behavior du ring robot motion. The design of con t rol

laws that overcoriie t ransrii ission elastici ty is qui te challenging from both practical

and theoretic-al stand points. The highly nonlinear nature of robot dynamic nlotlels.

c-oiiplecl with the joint flexibility probleni, is indeed an open control problem. As

the operating bandwidth of the robot increases, the joint flexibility becomes more

iiiiportant (47. 46. 521. The joint flexibility introduces a fast dynaniic characteristic

iii robots [do]. Tlie in herent interaction of the fast dynamics (joint flexibility dynain-

ics) atid the slow varying robot dynaniics (rigid body dynamics) can destabilize the

systetii. Moreover, high frequency present in the input signal niay excite the fast

c1ynatriic.s i n troduced by the joint flexibility.

Most of the developed control schenies for flexible joint robots assunie that

torqiie coriitiiancls can be arbitrarily applied to each robot joint. In reality. these

torqties are applied through the use of joint actuators. Unfortunately, the actuators

possess non-negligible mechanical dynamics such as inertia, friction and cornpliance.

For the design of high perforriiance motion and torque control of robotic applications

it is necessary to inodel accurately the drives involved. Motion and torque control

o f iiianipulators relies inostly on the abitity of the actuation systems to provide d e

sirerl joint torques. B u t , for robotic manipulators, particularly those equippecl wi th

hartiionic clrives (HD). the ability of the actiiation system to provide desired joint

torqi~es is consiclerably restrirted by the inherent nonlinearity, friction and flexi bility

of t h e drive involved. Therefore, riiore advanced controllers must be employetl to

giiaran tee robiist stability and perforniance of such robotic systenis.

There are various approaches in the literature dealing with the control of

fl~xihle joint robots. Such approaches suffer froni two major drawbacks. First. t h e

joint flexi bility is iisually tiiodeled as a pure Iinear spring, whereas the joint stiffness

is rionliriear and there is hysteresis and nonlinear friction i n the joints. Second. the

perforixianïe is sensitive to mode1 uncertainty and rneasurement noise. The existing

c-ontrol iriethods fail to integrate into a cornmon framework explicit knowledge of

iiioclel errors. robustness bounds and performance specifications. These deficiencies

oftteri degraile the effeçtiveness of the control schenies available in the literature.

To overc-ortie these deficiencies. this thesis forniulates a robust cont rol scheiii~

1 ~ ; t c i ~ ~ l o n t h e N, - optinial control approach. The X, - optiriial control design teçh-

n i c l w . hasecl ~ntirely on frequency dorriain tiieasirres, has been developed recentiy

[S. I l ] . It provicies a iinified franiework for atldressing simultaneously uncertainty

iiimsilres and performance requirernents. It has also the advantage of allowing the

designer to use experiiriental results to yield uncertainty descriptions and correspond-

irig levtels (i-e.. weights) for control design purposes. We propose a twofold robiist

i.otitrol design of flexible joint robots with HD: an actuator-level torque control ancl

a lin k-lwel motion control. U'e propose riiultivariable ?Lm - optinial control laws at

110th levels. This approach allows us to focus on lin k and actuator control problems

separately. It is shown that the closed-loop systeni achieves robust performance using

the proposecl control design technique. Experiirients are also perforined to test the

proposecl con t rol design schetne.

1.2 Background on Flexible Joint Robot Control

As the operating bandwidth of a robot increases, joint fiexibility becomes niore irri-

portant. High freqiiency preseni in the input signal niay excite the fast dynaiiiics

i~itrocluced by the flexibility of the drive and tiiay destabilize the system. The source

of joint flexibility may be torque transducers [57, 31, drive shaft stiffness r2], or har-

iiionic drives [47]. The problem of robot nianipulation in experiments, and sources

of the perforriiance limitation of a robot were investigated by Sweet and Good [4T].

Kivin [4 11 deteriiiined many sources of joint flexibility such as harmonic drives. belt.

chains. torsional shafts, and gear reducers.

Mtich effort ha5 been devoted to control of flexible joint robots. Different

tpc-hniqiics Iiaw been proposed in the literature to deal with modeling and control

issues. Iri general. the control schenies suitable for flexible joint robots can be rat-

egorizecl into the foilowing techniques: siniple PID, feedback linearization, in tegral

tiianifolcl. singiilar perturbation, passivity approach, torque feedback and aciaptiv~

<-on trol rlesigns. -4 simple PD controller for robots with elastic joints was introdiiced

hy P. Toiti~i [49]. .\ singular perturbation technique to control flexible joint robots

WFLS proposcd by Marino and Nicosia [30]. Khorasani [1] constructed an adaptive

[-ontroller based on the integral tnanifold and singular perturbation techniques, b u t

h is design assumes high joint flexibility. Several other researchers proposed the use

of joint torque feedback for control of robot nianipulators [15, 22, 27, 291. Spong in-

trodiireci a feedback linearization approach to the control of elastic joint robots [46].

A class of output feedback globally stabilizing controller for flexible joint robots was

proposed by R. Ortega et al. [:38]. Readman [39] showed that when the actuator drive

iriertia tiiatri'c is sirfficiently small, there exists a decentralized velocity control law

t;)r flexible joint robots which asyiiiptotically stabilizes the flexible joint dynaii~ics

. An adaptive controller for flexible joint robots is designed in [44]. J. Yuan and

Y. Stepanenko proposed a composite adaptive control of flexible joint robots [Bx].

An adaptive cont rol of robot nianipulators with flexible joints was also presented

I,v K. Lozano and B. Brogliato ['26]. Furtherniore. since the control law containecl

variable striirtiire-like ternis, the rontrol input exhibited an undesirable chattering

ph~nomenon.

roiiiprehensive study of adaptive control of flexible joint robots was given in

[-Ri]. ( ; hortiel and Spong showed that an asyniptotic link trajectory could be achieved

t q inoclifying an adaptation law, assurning that the desired trajectory tracking error

approaclies zpro as tiiiie approaches infinity [l'LJ. I n [.II, Benallegue and M7Sirdi

designer1 an aclaptive controller baseci on the pasivity approach. However. their

approacti rrqiiired iiieasurenients of lin k acceleration and jerk. In [23], Chen and Fu

iised an nclaptive approach which required riieasurenient of Iink acceleration. Dawsori

et al. designed a hybrid adaptive flexible joint robot controller that achieved (;.AS

position tracking [6]. Using joint torque feedback, Lin and Goldenberg designed an

aclaptive control of flexible joint robot [25]. However. it was required to con ipu t~

1111 to the s~çoncl clerivative of the torque signal. A global dynamic output feedback

t rai-king (-ont roller for robots having elastic joints was presented in [Xi]. O bservrrs

for robots wi th elastic joints were also reported in [36, 481. In [4X] an observer

wi tti global convergence property was O btained assuniing that the link position and

spred ivere available froiii iiieasu reiiients. In [36] only the lin k position was assuiriecl

to I>e neecled. Tracking control of flexible joint robots with uncertain paranieters

and tlistiirbanr-es were considered in [50]. Control of robots with elastic joints via

noiilinpar dynaniic feed back tvas also reported in ['LX].

At the sanie tiiiie, extensive research was carried out to deal with the uncer-

tainty ~.oiiipensation of robots. Adaptive ancl robust control scherries are major areas

of this res~ari-h work. However. there is very liniited aiiiount of published work in the

area, of flexible joint robot control that niade use of 31, - based robust controllers.

The kasillility of Xcw, - based control of a rigid robot rnanipulator was investigated

i n [XI. but no assuniption r~garding the existent-e of joint flexibility was niade. It

tvas assumecl that the parariietric uncertainty was the only type of uncertainty to be

c.i~tisiderecI. Then, a very simple uncertainty niodel was proposed for the robot under

stiicty. AISO. there was no experiniental validation of t h i s work.

.A rohust i-ontrol schenie based on R, theory was presented i n [IO]. The

cari t roller (.onsistecl of a riiodel- basecl ron t roller (inverse-dynaniics) followed Ily a

liriear dynariiic controller (designed by using Z, - control tlieory). There was no

c-onsideration of joint ff exibility or of any form of friction in the design. Uncertainty

iiiodels basecl on un niodeleci d y namics assum ptions tvere proposed . Using the experi-

riiental result of the resolved acceleration control iniplemented on a two D.O.F. robot.

weighting ftinc-tions were chosen. Then. the suggested controller performance was

c-o~iipared witli the aforementioneci resolved acceleration controller implemented on

the sanie robot. This method did not guarantee a consistent performance in practice. - .A (.ont roi st rategy w hich consisted of feedforward and feed back compensation loops

f i ~ r a ningl~lin k iiianipiilator wi th joint flexi bility was investigated in [!XI. A state-

spwe foriii of' 'H,, - controller was used. The theoretical development was validated

1.y iiiipleiiienting the proposed controIler on a single-link robot. Then. it was arguecl

that the proposed SIS0 niethod couid be extended to the M I M O case too. However.

frirtlier investigation is necessary to conclude whether or not this assertion is feasible.

-4 nonlinear U,, cont rol design in ro botic syste~iis under parameter perturbation and

cxternal disturbance was proposed by B.S. Chen et al. [5].

1.3 Contributions of this Research Work

The (-on tri b i ~ tions of this research work are as follows:

Flexible Joint Robots Mode1 and Model/Data Consistency Problem

It was observecl that the H D flex-spline subsysteni xnodel coulcl not represent the

~'cperiiiiental data adequately. Hence, the model was fiirther extended and coniparetl

with existing niodels. An extended mode1 that estiniates the hysteresis phenoirienon

i r i itiore detail was proposed. The extended niodel along with other subsystem niodels

were iisecl and a set of paranieters were identifiecl in order to verify the niodel/data

wiisistency. This issue wns acldressed in the çontext of experiniental data O btainecl bv

riin nirig open-loop experiments on the systeni. It was shown that with the extendecl

~iioclel. the open loop time and frequency domain responses based on the mathematicai

tr:otlel are i-onsistent with the expeririiental data.

Robust Control Design of Flexible Joint Robots with HD

A twofolcl robiist control design schenie of flexible joint robots with HD is proposed:

a n a(-tiiator-Ievel torqiie çontrol and a lin k-level motion control. IJsing R, - op-

t.iriial c-ontrol. a systeniatic rriethod of designing robust controt laws at each levei is

introdtiçed. Two niultivariable 3.1, - controI designs based on the model of the IRIS

(K.AL) testbed are presented together with simulation results to illustrate the tech-

~iiqtie. It is shown that, because the uncertainties due to the actuator-transniission

rioti l i n ~ a r i ty are explici tly considered i n the cont rol design procedure, the proposed

iiietiiotl is riiore robiist than conventional ~tiethocls.

Uncertainty Description of Flexible Joint Robots *

[ri the propo~ecl iiiethod the selection of uncertainty bounds plays a major role in

the t rade-off5 between robustness and performance requirements . A systematic ap-

proach to select uncertainty bounds of flexible joint robots with HD is presenteti.

The dtsrribing furzction iiiethod is introduced to select friction and hysteresis un(-er-

tainty weighting functions (bounds). The conic-sector-bouirded rronlirtearity rtietliod

is proposed to select the nonlinear stiffness unçertainty weights.

Experimental Implementation of the Proposed Controller

.A siic-c.essfiil impleinentation of the proposed control design technique is presented.

The cont rol laws are syn thesizeci for different uncertain ty bounds. 1 t is experimentally - verifiecl ttiat the perforniançe of the closed loop systeni is sensitive to uncertainty

(les(-riptions and bounds, and high perfor~iiançecan be achieved by tuning the bounds.

The espeririiental results have aIso shown that the stability of the control system

is [nain tainecl. Moreover, the robustnes of the proposed control schenie to niodel

ri rirertain ty and riieasureriien t noise is denionst rated experimen tally.

1.4 Outline of the Thesis

In the previous sections the motivation, background on flexible joint robot cont rol

and the contribution of the research have been presented. The renlainder of the

thesis is organized as follows. Chapter 2 considers flexible joint robot models. Actu-

ator and HI) transriiission systeni rnodels are presented. Chapter 2 ais0 discusses the

motivation for the need of uncertainty modeling of flexible joint robots with H D t rans-

riiission. Different approaches to uncertainty riiodeling are then presented. Chapter :{

cletails robiist [notion and torqw c-ontrol design of flexible joint robots. C'hapter 4

explains i n tletail the experimental setup (IRIS faciiity) developed at the Robotics

ancl Xiitoniation Laboratory (RAL) of the University of Toronto. Robustness and

p e r f o r ~ m n c ~ t rade-offs due to uncertain ty niodeting are also investigated. Chapter 5

addresses the stability analysis of the proposed control design of flexible joint robots.

A siiiiiiiiary of resiilts aiid a clisciission of future research directions are presented in

( 'liapter 6 . .A review of X, design and structured singular value (p) framework is

given i n Appendix -4.

Chapter 2

Modeling and Uncertainty .

Description of Robots with

Harrnonic Drive Transmission

Chie to its unique gearing configuration. w hich offers conipactness. light weight , high

ratio. effici~nc-y and virtually zero backlash. harrnonic drive transmission hns gainecl

witle açceptance in industrial applications. However, the sources of soriie nonlinear

pti~noiii~ria ancl their iiiipact on the actiial dynatiiic response have not been fiilly

;~( l ( l ress~d tliris far. The design of high perfortiiance motion and torque i:ontrolters

fur robotic applications requires an accurate knowledge of the characteristics of the

drive involved. Previoiisly published niodels characterize the nonlinear kineniatic a n d - kin~tir- behavior of harinonic drives. The t ransrnission corn pliance, resulting froiti

gear-tooth interaction anci wave-generator deformation due to high radial forces. is

coiiiiiionly approxiiiiated by a nonlinear piecewise iinear stiffness curve [Li, 24.21. 491.

However. hysteresis losses caused by nieshing gears are mostly ignored. Extensive

~xpr i i i i en ts fciçusing on kineinatic error ancl frictional effects have been conciucted

Ily Tii t t l~ (5 11.

~ V P toc.tis on the aspects of both irioclel developnient and parameter identifir-

ations [31. 33, 34, 321. We review the previously published models of HD ['LI]. Ex-

per iriien ta1 ~lx;~-rvations illust rate that the torque t ransrnission characteristics show

hysteresis depending on the operating conditions. Afso. the deviation in the output

rotation dile to hysteresis effects cannot be ignored. In ['LI] these effects are repres-

entecl by a siiiiple C'ouloirib friction that does not match the HD behavior cornpletely.

These tiieixiory-depenclent properties and hysteresis effects are captured in our new

irioclel. ~ V P then identify a set of paraineters to coniplete the niode1 develop~rient, and

the pr~posed niodels are used for motion and torque control designs.

In prartice, there is a trade-off between niodel compIexity and perforniance.

in that a sinipler model is easier to simulate and use in the design. ,4t the same tinie,

iincertainties due to unniodeted dynamics, neglected nonlinearities, the conipliance

effect of gear transniission and rneasurement noise rnust be considered.

This c-hapter is organized as follows. First, we present the flexible joint ro-

bot ciiodel. Second, we propose tiriie and frequency doniain experiments to verify

the çonsistency of the proposed niodel against the experimental data. Then, we pro-

pose freqiiency doriiain- based uncertainty descriptions to accoun t for unmodeled high

freqilenry cfynaniics. Finally, we introduce the describing function and conic-sector-

bouridt-d-desci-iptiotr niethods to derive flexible joint robot uncertainty bounds.

2.1 Actuator and Harmonic Drive 'Pransmission Models

In tliis section, a brief introduction of the niatheniatical model of the IRIS-facility

joint is provided. The entire system including the actuator and HD can be dividecl

into five subsystenis: 1) motor-amplifier, 2) wave-generator, 3) flex-spline, 4) circular-

spline and 5 ) load (Fig. 2.1). The detailed description and mathematical formulation

of each sutxystein can be found in p2 11.

f load

Figure 2.1 : Harnionic drive systeni

2.1.1 Motor- Amplifier Subsystem

The ixiotor sliaft is clriven by the riiotor input torque r,,, that is proportional to the

in p l i t cwrren t i,,, in the non-saturatecl region of the niotor amplifier s u bsysterri, i.e..

wi th kt heirig the niotor torque constant. On the other hand. the relationship between

the aiilplifier input çoriiniancl u. niotor ciment i,,, and rotor velocity q , can be es-

tatdishd as:

- w h ~ r e k, i s the amplifier gain, I , is the upper-bound of the current defined by the

aitiplitiw ~l~i-tronics. ïJs is the iiiaxiiiiuiii output voltage of the amplifier, kb is th^

ha(-k e.i~i.f. ronstant of the niotor. ancl r,,, is the riiotor armature resistance.

2.1.2 Wave-Generator Subsystem

The torqice r,,,. cleveloped by the DC rriotor, drives the niotor armature and the

liarnion ic-cl rive wave-generator. The torque exerted on the wave-generator will be

w~isidered as the subsystei~i output. The followirig riiodel describes the wave-generator

subsysteui iiiodel. It invoives C:oulortib and viscous friction at the bearings, and the

wave-generatorballbearing:

tvtier~ il is t h e flex-splinr angiilar v~locity with the reference direction opposit~ to

the wave-grri~rator velocity (q,), J,,,, is the coixibined inertia of the rotor, shaft and

tlw waveg~nerator, ~ ( 4 , ) is the total frit-tion torque generated at the bearitigs ancl

R,(tjW. il) is the friction terni of the wave-generator bearing, and rW is the t o r q i i ~

exertecl on the wave-generator.

2.1.3 Flex-spline Subsystem

The experiiiiental observations reveal three physical phenornena related to the b ~ h a -

vivr of the flex-spline: nonlinear stiffness, hysteresis, and quasi-backlash due to a soft

windiip effeït. The transniision conipliance acting between input and output shaft is

xpprosiitiat~ci by a rubic plus a linear function of the torsion angle, and a correction

teriri . i.e..

G ( q ) = aliq3 + alzq 4- KSW (2.4)

tvhere and n ~ 2 are constant coefficients. q is the angular displacenient of the

fies-spline and /<,, is the soft windup correction factor that can be niodeled as a

sacidleshape fiinction

where k,,, and a,, are constants. Furtherniore, the flex-spline hysteresis is modelecl

as a siiiiple Coulonib friction (i.e., BQ) [2 11. However, it does not completeiy represent

tliis phenottienon when the systerii is in motion. An extended mode1 of the hysteresis

is presented. The shape of the hysteresis is estiniated as a combination of C:ouloitib

friction ancl a weighted friction function, which is represented by a hyperbolic function - I l

Figure 2.2: Hysteresis niodel

with 1 7 - i = TL - (2.7)

w liwe r, lias been defined in (2.4) and the Coulonib friction parameter Bf defines

the liitiits r, - BI 5 TI& <_ T, + BI of the torque transinission function. whereas

the factor 7 deterniines the dope at the transition froni loading to unloading.i.e..

4 5 O -i q > O. or vice versa. The hyperbolic function in (2.6) intplies that the friction

torclue asy iiiptotically approaches a constant friction torque B for large arguments

-, ((1-(lm). so that the paranieter y determines the contribution of the actual friction. as

sliown in Fig. 2.2. The superscript ( )' in (2 .6 ) relates to quantities a t the reversal

point froni loading to unloading, that can occur a t any point (q', Th), so that the

S ~ I J P of the torque function (2.6) also depends on the load history.

I -sing (2.6). the rorresponding paranieters(i.e., 7, B I) niust be estimated lin k-

irig ~xpr i i i i en ta l phenoiiiena to niodeling a~suiliptions. The estimated paranieters for

7 and BI as functions of the load Iiistory ri of the test-joint are shown in Fig. 2.3.

l :sirig the ~stiiiiated paranieters y and B values. the identified hysteresis has k e n

sliown i n Fig. 2.4, which shows good agreement with the nieasured hysteresis.

Octput Toque mm1

Paranieter -, O u p i Toque [kirni

Paralneter BI

Figure 2.3: ldentified paranieters 7 and BI vs. load history ri.

mur mgleigear ratto (deg 1

Iden tified Hysteresis Measu red Hysteresis

Figure 2.4: ldentified and nieasured hysteresis. rh verses qm

*

2.1.4 Circular-Spline Subsystern

T tie oiitput link to the joint can be attached either to the fiex-spline or to the circirlar

sp l in~ . Therefore. the foliowing two equations will be used to describe C-end/F-end

lin k clynarriics:

where y, is the C'-end angular dispIacen~ent. is the C-end load torque, w h i l ~ r,

is the driving torque transmitted to the circular spline at the engagement zone cross

sertion. The paranieters Jc and those in 8, are inertia and friction paranieters a t the

( '-mcl. The F-end paraiileters are also analogous to the C-end (2.9).

*

2-1.5 Harmonic Drive

Tt ip Hm-spline h a s two fewer teeth than the circular spline. and thus each fiill turn cd

the ivave generator inoves the ftex-spline two teeth in the opposite direction re!atiw

tu the (sir(-iilar spline. This directly iiriplies the kineniatic constraint

ivher~ .\' is the gear ratio defined as iV = ,Vt/2, and Xt is the nuinber of teeth on the

flpx-splin~ oiiter (-irciiriiference. Applying the power conservation law on the ;3-port

clevic-e descri bed by kineniatic constraint (2.1 O), or more precisely on i t s derivative

forrii:

yi~lcls the input/output torque relationships: - TI = .Vrw,

Tc = (iV + l )r ,

2.1.6 Flexible Joint Robot Model

With reference to ['LI] and section 1.1.3. the mode1 of a n-DOF flexible joint robot

tvith H t) and a n actuator can be written as follows:

wtiere qc and q,,, are the n x 1 vectors of shaft displacement on the joint side and

actuator sicle respectively, M(qc) is a n x n link inertia matrix, iV(q,, Q,) is a 11 x I

vet-tor of centrifugai, gravity, and C:oriolis(generalized) forces. r, is the torque sensor

outpitt located a t the link side, r is the harrnonic drive gear ratio, J , is the TL x n

diagonal oratr ir of actuator inertia, B,,a, is a 72 x I vector of daniping and (BZ,, B;,)

are friction terms associated witti the actuator and H.D. bearings. r,,, is the ri x L

v ~ c t o r of control torque input. The rest of the paranieters have been previously

- clefinetl. It is obvious that in the following model if we ignore viscous and Couloitib

frictions ancl represent the jcrint flexibiiity by a simple linear spring, then the proposecl

iiio(1el wiIl rerlrice to ri. cotiinionly iisetl nioclel of flexible joint robots, for exaniple in

[-KI .

2.2 Model Verification Based on Experimental Results

Ir1 the previous section a niathetnatical niodel of a flexible joint robot was in t roduc~d.

This section addresses the problem of checking the consistency of the model with

open-loop experiniental tirrie and frequency response data for the experimental setup

(IRIS fa(-ility) tleveloped at the Kobotics ancl Automation Laboratory ( M L ) of the

I-~iiversity of Toronto. The tiriie domain experiitients are performed with the output

l ink of the joint heing allowed to move without any constraint. By contrast. the Iink is

cmnstrainecl against the ground in the frequency domain experiments. This prevents

t h interat-tion of the fast dynaii~ic motion of the joint and the slow dynaniic niotion

of the oiitpiit I ink so that the joint is not daniaged.

2.2.1 Tirne Response Characteristics

i n order to verify the mode1 and its parameters, a set of tests was performed that

involved the clynariiic excitation of the systeni and the cornparison of actual output

signals with niinierically derived data. Fig. 2.5 and Fig. 2.6 contrast experirriental

wavegenerator angle and velocity responses for five different current aniplitudes wi th

resii lts o b tai ned by siriiulation. The current coriiiriand amplitude covers the full range

of riiotor r~ t r r en t froni O to 3 ainps. Fig. 2.5 shows the theoretical and expeririiental

waveg~nerator angle response to these sets of ciirrent coni~iiand inputs. Obvious

froiri the figure, there is a good consistency between the rnodel and the real systeni

responses. Fi%. 2.6 shows the wave-generator velocity response For the rnodel and the

~speriiiiental setup. A coniparison of the experiniental and siniulated tinie responses

stiows gooc1 agreement between experinient and siinulation for different C U rrent inpu t

ariiplitiitles. For ari input airiplitude of 3 amp. there is a slight increase i n the siiiiii-

lation response coniparecl with the experiniental response which suggests a variation

iri the niotor armature and amplifier paranieters.

2.2.2 F'requency Response Characteristics

The freqiiency response was predicted via simulation and compared with the response

of the physical systeni. Du ring testing, the output shaft was restrained against the . groiind and the riiotor was driven by a sinusoidal current input with varying frequen-

(mies. .Joint torqiies in simulation ancl expeririient were tiieasu red for a sinusoidal iiiotor

.;mi -*--..-- -.-. t i a I U? 0 3 ;XI 95 a i 3;

O 1 O 2 O 3 O 7 i k s tri 0 4 0 5 O 6

Timc [sefi

Fip i re 2 3 : T heoretical and experi inen ta1 wave-generator angle for step in put of r u r- ren t

Figure 2.6: Theoretical and experiniental wave-generator velocity

Figure 2.7: Magnitude Bode plot of the 11iodel and experinient.

utirrent with a sweeping frequency ranging froni O to 500Hz. The Bode magnitude

ancl phase plots of the input/output torque transfer functions are shown in Fig. 2.1

ancl Fig. 2.8. .A corriparison of experimental and simulated transfer functions shows a

slight redtwtion in the breakpoint frequency, indicating a small variation of the effect-

ive stiffness. For excitation freqiiencies of f < 30 Hz, the systeni response riiatc-hes

the 1ttod4 response perfectly. However, due to the inaccuracy of the torque sensor

r~arl inp aric1 ~rieasureirient noise for f > 100 Hz. the recorded data were unreliable.

Fur c-oatrol design purposes. the working range of the systein is below 30 Hz, so the

tiiotlel is applicable.

U;P use the niodels introduced for control design purposes. The following sec-

tioris cl~scribe iiow to approxiniate the mode1 uncertainty bounds between the exper-

iiiimtal data and the inoctel. The firquericy dotnain-basai uncertainty description is

proposecl to iiiodel the effect of uniiiodeled high frequency dynaniics. The describitiy

functiotl-based uncertain ty description niethod is used to approxirnate the effects of

hysteresis ancl friction at robot joints. The conic-sector-bounded rlonlinean'ty niethod

is proposecl to approximate the cl rive nonlinear stiffness bounds.

Figure '2.3: Phase Bode plot of the triodel and experiment.

2.3 Uncertainty Description: The Frequency Domain-

Based Method -

The fwquctrcy dorriain-based niethod is proposed to descri be frequency dependent

variations betweeri ttie experiirien ta! data and the iiioclel via uncertainty bounds

(iwigliting fiinctions). These allosv ils to accoiint for the variation in experiiriental

(lata at specific frequency points. .A frequency response experinient is perforiried to

~stal-ilisti t i p p r and lower bounds on both the magnitude and phase of the real sys-

terii as a funçtion of frequency. Variations in the data are then approxiniated by disk

s t i apd regions in the cotiiplex plane, leading to either a multiplicative or additive

uni-ertain ty description of the bounds [a]. The plant transfer function can be described by P(s ) + AP(s ) , where P ( s )

is the noiiiinal plant iiiodel and AP(s ) is an unknown perturbation [35]. Consider

a SIS0 systprii witti A P bounded across frequency by a weighting function M/,: a

real-rational. stable iiliniiiiiini phase t ransfer function. and a nomi bounded A, w here

[AI < 1, sirch that

( w ) < ( w ) ( w for al1 O 5 i~ 5 m, (2.17)

where ru represents individual frequency poinLs. Equation (2.17) is referred to as

a n additive uncertainty description and defines the bound on the allowable additive

un(-ertain ty.

-4s sliown in Fig. 2.9, the additive uncertainty weighting is wrapped around

the plant and is often used to account for additive plant errors and uncertain right

tialf plane zeros. 4 variety of additive uncertainty weights can be developed for a

M I M 0 systerii. each adding states to the control probleni. For instance, the additive

iinc-~rtainty weights can be wrapped around the plant from each input channel to

- eac-h output channel with different weights. Low order weighting functions are usually

~iriployecl to liriiit the number of states added to the problem formulation. - 4 ~ 1 0 t h ~

reason for low order weights is t h a t knowledge of the exact size of the uncertainty

is often liriiited. Therefore describine; the variation by a cornplex, high-order weight

r-an iiot be j ustifiecl.

Figure 2.9: Block diagram of additive uncertainty

( 'onsicler the following SIS0 systetti

This clescribes a set of plant niodels within

with additive plant uncertainty:

O.lA)u, ~~&, 5 1. ('2.18)

which the "real" systern lies. A Nyquist

plot of this [incertain systeiii is shown in Fig. 2. I O . The plant is described at each

frqiienry point w by a circle centered at P(jur) of radius /W. ( j w ) 1.

Yyquist plot of additive uncertainty

Figure 2.10: Nyquist plot of additive iincertainty

Another approach to rnodeling errors involves niultiplicative uncertainty de-

scriptions. blultiplicative uncertainty descriptions are used to account for relative

variations in input or output signais. Input multiplicative uncertainty is usefui in de-

scribing actuator errors at high frequency and unmodeled actuator dynamics. Output

~iiiiltiplicative uncertainty is used to niodel similar quantities o n output signals and

tiriie delays. Sensor noise attenuation and output response to output cornmands are

prforrriarice ineasures that can be specified with such weighting. Typically, testing of

wtuators and sensors involves inputting signais i n to the components and riieasuring

their response (i-e.. force, displacerrient, torque). The output response is rneasurecl

with a percvntage error froiii a nominal plant riiodel that may Vary across freqiiency.

Fig. 2.11 shows the biock diagrani of a n output iiiultiplicative uncertainty, represen-

Figure 2.1 1: Block diagram of output muttiplicative uncertainty

2.4 Uncertainty Description: The Describing F'unction-

Based Method

* I I I rincertainty riiodeling, one needs a niethociology for dealing with nonlinear ph*

riomena i n the systeni (e.g., friction. hysteresis). For sonie nonlinear systenis and

iintler 'ertain conclitions. an extendeci version of the frequency response method. the

dcsc-ribirrg /urictioti rnetliod [45. 531, can be used to analyze and predict nonlinear

I)ohavior appro~irrrately. The niain use of the describirig function rnethod is the pre-

clic-tiori of litnit cycles in nonlinear systerxis, although the rnethod ha5 a nurnber of

o t l i ~ r applications. LVe propose to use it for uncertainty weight description.

Befor~ addressing uncertainty weight description, let u s briefly d iscus how

to represent a nonlinear coniponent using a describing function, which is critical

for our purposes [45]. As shown in Fig. 'L.I'La, let us consider a sinusoida1 input

( ~ ( t ) = .4s in(wt)) to the nonlinear elenient of amplitude A and frequency u. The

u i i t l ~ t of the nonlinear coniponent c ( t ) is often a periodic function though generally

non-siniisoidal. Note that this is always the case if the nonlinearity f ( e ) is single-

- v a l i i ~ c l because the output is f [As in(w( t + 2~/w))]. Using a Fourier series, this

p~rioclic- fiin(-tion can be expantled as

w l i ~ r e the Fourier coefficients a,,'s and 6,'s are generally functions of -4 and i ~ . de-

tertriiri~rl hy

tf w~ assriiiie the nonlinearity is odd, one has a0 = O. Furthermore, if we consider the

fiinclaillental coniponent cl ( t ) in the oiitpiit c ( t ) then,

cl ( t ) = :VI s i n ( d + oj

h I ( . l . i ~ ) = + 6: and o ( . ~ . J ) = a r c t a n ( a l / b i ) ( 2 . 2 5 )

~ v h e r e .LI is the output signal amplitude. and Q is the phase shift. Expression (2 .24)

ititlicates that the fundanlental coniponent corresponding to a sinusoidal input is a

siniisoirls a t the sariie frequency. We define the describing functiorr of the nonlin-

par eleiiieiit to he the conrplcx ratio of tllc fundanientai conrponent of the rronlirrrrrr.

t l t r ~ r t . r t f by titc iîlput sintrsoids, i. c ..

With a describing function representing the nonlinear coinponent, the nonlinear el+

~iient . in the presence of sinusoidal input, can be treated as if it were a linear elenient

witli a freqaency response function N(A, w), as shown in Fig. 2.12b.

Figure 2.12: A nonlinear elenient and i t s describing function representation.

2.4.1 Computing Describing Function (DF)

In this section, we take a closer look a t the nonlinearities found in HD transmission

systeriii;. The describing fitnctions for a few COIII Inon nonlinearities are coniputed.

a. Nonlinear Eieriient b. Describing Function

N ( A , w ) CM A sin(ult)

- - + -4 sin(wt) -

Saturation(Friction) DF:

!Ms in (w ' t - + O ) - !V.L.

- The input-output relationship for a saturation nonlinearity is plotted in Fig. 2-13.

with n and k denoting the range and slope of the linearity. Since this nonlinearity

is singk-valiiecl. we expect the describing furiction to be a real function of the input

am pli tutle.

('onsicler the input ~ ( t ) = Asin(ut). If -4 < a. then the input remains in the

I in~ar range. antl therefore the output is y ( t ) = k.,4 s i n ( d ) . Hence. the describing

fiirii-tiuri is siiiiply a i-onstant k.

SOW (.onsider the case .4 > a. The input and the output are plotted in Fig.

2.!:3. The output can be expressed as

w h e r ~ w.tl = arcsin(al.4). The odd nature of c( t ) iniplies that al = 0, and the

syiii iiietry over the four quarters of a period itnplies that

Ther~fore. the describing function is 7

Figure 2-13: Saturation nonlinearity and corresponding input-output relationship

.As a special case, one can obtain the describing function for the relay-type

( ( 'oiilorii b friction) nonlinearity shown in Fig. 2.14. This case corresponds to shrin k-

ing the l in~arity range in the saturation function to zero, i.e., a + O, k + x. but

krr = .II. Thotigh bl can be obtained from (2.23) by taking the liniit, it is more easily

c~l~tain~dclirectlya.;

Ther~fore. the clescribing function of the CoiiIoirib friction nonlinearity is

Hysteresis DF:

( ionsider the hysteresis nonlinear operator N shown in Fig. 2.15. In the steady-state,

the output (Yz) ( t ) follows the upper straight line when the input is increasing i.e..

Figure 2-14: Couloinb friction type nonlinearity

i ( t ) > O and the lower straight line when the inpu t is decreasing. The value a in Fig. *

2.1.5 tlepentls on the amplitude of the input and is not a characteristic of iV itself.

rrtrc(t) + b if t ( t ) > O [SJ) ( t ) =

7 w ( t ) - b if i ( t ) < O

and "jiiriips" when i ( t ) goes through zero.

Çiipposç. a sinusoidai input r is applied to LV. The resulting steady-state output

is shown in Fig. 2.15. It is ciear that the first harmonic of the steady-state part of

.Vr is

Note that !V is independent of i;. because time scaling does not affect the output of

.v *

Figri re 2.15: Hysteresis type nonlinearity

2.5 Uncertainty Weights Selection

Once the describing function representation of a nonlinear systeni is known, it can be

iisd to rharacterize th- ~!ncertainty involved in the system. The describing functions

( ~ f ' the ( 'oiiloiiib friction and the hysteresis phenornenon were introduced in the previ- - 1 oiis se(-~ions. i iiia section addresses t h e problem of finding uncertainty weights of HD

- frirtioii . hysteresis and nonlinear stiffne.~. The descri bing function method is tised

t o îind th^ rioriri boiinclecl weighting funcrtion for riiotor friction and H D hysteresis.

2.5.1 Motor Friction Uncertainty Weight Selection

Fig. 2.16 shows the coniplete nonlinear block diagram of flexible joint robots with

HI) transiiiission. The Couloriib frictioq at the riiotor side can be replaced by its

eqiiivalent DF introduced in (2.31).

Figure 2.16: CompIete nonlinear mode1 of the robot H D drive system

Figure 2.17: Motor and DF of Couloinb friction

Let the noiiiinal iriodel of the niotor in Fig. 2.17 b~ as

where there is no Coulomb friction and where J , is the motor inertia and B, is the

viwoiis tlaiiiping coefficient of niotor. .41so assunie the perturbed mode1 of the niotor

1 J P

- i P(.) =

JTri s + B,, + N (s) w h ~ r e .V(s) is the DF of the CouIoinb friction (Fig. 2.17). Now if a muItiplicativ~

iinwrtainty description is used to accoiint for relative variations between nominal and

pertiirhed plants. one can write.

w t i ~ r ~ P ( s ) is tlie noniinal plant transfer function (2.35). ~ ( s ) is the perturbed plant

trarisfer function (2.36), Wfs) is the weighting function to be found and l ( s ) is the

n o m borinded coniplex uncertainty transfer function such that IIAllw 5 1. Taking

t h x-norirl of both sides of (2.37). and using the commutative property of oc-nortri.

WP (-an wri te

tliat is.

Frequenc y [rad/sec 1

Figure 2.1s: Frequency response of the motor-friction niodel for different amplitude of inputs

Fig. 2 . l x shows the Bode magnitude freqiiency response plot of the for dif- P ( s ) - P ( , )

h ~ n t inpiit signal amplitudes. Using MATLAB [18], one can find a suitable weight-

ing fun(-tion W ( s ) to satisfy (2.40). The niuItiplicative uncertainty model of the

riiotor+fric.tion is shown in Fig. 2.19.

2.5.2 Hysteresis Uncertainty Weight Selection

The H D hysteresis uncertainty weight can be selected following the sanie procedure as

i n the previous section. The noniinal H D model is considered aç a pure linear spring - ( i . ~ . . rio hysteresis). The perturbed H D model is selected when the hysteresis effect is

presen t. A tiililtiplicative uncertain ty weight is used to describe the variation betw~e11

tfiç. rioininal and pertiirbed iiiodels. Moreover, the DF of the hysteresis introdiiced

in (2.34) is iisecl for uncertain ty weight selection. The frequency doniain hysteresis

iin(w-tainty weight is obtained using a set of swept sine input signals with varying

aiiiplitiicies. Fig. 2.20 shows the freqiiency response of the variation between the

Figure 2-19: Multiplicative uncertainty niodel of the motor

no~iiinal ancl perturbed niodels (i-e., A). and the corresponding weighing function. 1 n

- the following section. the so called niethod of the conic-sector-bounded nonlinear-ity is

i isrd to represmt nonlinear stiffness nonlinearity [59].

2.5.3 Nonlinear Stiffness Uncertainty Weight Selection

1 ri t h is tliesis, we use the so called conic-sector-bonded n ~ n l i t r e a ~ t y niethod, w hose

(Ietiiiition is given below, to find an uncertainty boiind for the nonlinear stiffness [-Ki].

Definition: A continuoiis funrtion ~ ( y ) is sait1 to belong to the sector [k l . k 2 ] .

if tliere wist two non-negative nunibers k l and k2 such that

(;eonietrically, condition (2.41) implies that the nonlinearity function always lies

Iwtween two straight lines rCil y and kzy, as shown in Fig. 2.21.

The uncertainty weight description for the nonlinear stiffness is derived using

tliis ~~ie thod . In section 2.1.3, the torque-torsion relation is defined as follows:

Frequency [mci/sec]

Figure 2-20: Frequency response of the variation between the nominal and perturhed *

t~iodei of liys teresis and the corresponding weigh ting function

Figure 2.2 1 : Conic-sector representation of the nonlinear stiffness

Figure 2-22: Linearization and conic-sector bound

w l i e r ~ the parameters are defined in section 2. l.:]. This nonlinear function can h~

lin~arizecl nlmiit an operating point of the systeni:

w h e r ~ r = TI = (T'- - rd), and qr = ( q - qo). rdl qo denote the values of

variables nt the operating point. If we wish to have bounds on the error involved in

the linearization, we can eniploy a "conic-sector" description of the nonlinearity by

nrriting

This procedure is depicted in Fig. 2.22. Equation (2.44) iniplies the charac-

- teristic- falls in the cone ci rawn in Fig. 2.22 ( t h i s of course is valid for a limited range

of r l .q l ) .

The previous bound is a static constraint, but we can generalize this by writing

w h ~ r ~ d ( q l ) is an iinknown nonlinearity operator such that llS(ql)[J, 5 1. We inight

ttiink of it '*c:overing" dynatiiic effects which are not describeci in Our static equations.

Figure 2-23: Additive uncertainty representation of nonlinear stiffness

Fig. 2.23 shows the additive type uncertainty representation of (2.45). 6 is the

weighting t'iinction which can be selected using experimental resiilts(e.g. Fig. 2 .3 ; .

2.6 Summary and Discussion

I n this çhapter, the nonlinear properties of H D transmission are analyzed, and a

riiodel is presen ted that takes into account hysteresis, friction and nonlinear stiffness

(Section 2.1). Moreover, the HD flex-spline hysteresis mode1 is further developecl

(S~r t ion 2.1.3). Using the proposed niodel, a coniplete niodel of an n-DOF flexible

joint robot witti H D is introduced (Section 2.1.3). The niodel and its paranieters

are vwifiwl ~xperiitientally against the actiial data (Section 2.2). The rrlodel is then

I I S P ~ as a tmsis for uncertainty houncl selection. Frequency-based doniain iincertainty

d~srription is proposetl to describe frequency dependent variations between the ex-

pririien ta1 data and the iiiodel. In addition, a systeniatic approach is introdiiced

fur selecting uncertain ty bounds for the actuator-transmission stiffness nonlinearity.

fric-tiori. and hysteresis (Section 2.;1). The describing futzction and the conic-scctor-

bocrrrdcd tzonlincan'ty niethods are used to incorporate the effects of hysteresis, friction

and nonlinear stiffness into the control design. Using these methods, appropriate un-

wrtainty weighting functions are selected. In the following chapter, a control design

tech nique is proposed which. in conjunction with the proposed uncertainty bound

ci~si-riptions. yielcls robust perforniance for flexible joint robots.

Chapter 3

A New Robust Motion and

Torque Control Design Method

This chapter proposes a new niotion and torque control design schenie to achicvo

rohiist perfor~iiance in robots with harriionic drive transmission. It describes the

steps one neecls to follow for control design irsing the X,, - optimal control and

p-analysis and synthesis riiethod outlined in Appendix A.

The design procedure proposed involves the following steps:

0 ( knerating the error mode1 of the rigid body dynaniics of a flexible joint robot

(Serti011 3.1)

0 h f in ing perforrrlance specifications and iincertainty bounds of the error 111ocle1

(Section 3.2)

0 C.'onst ruïting open-loop in terconnections

a Oesigning a controller for the error iiiodel to satisfy the performance require-

~iieri ts (Section 3.2)

0 Closing the feedback loop with the robust niotion controller and exaniining the

behavior of the systerri (Section 3.2.2)

a (ienerating uncertainty mode1 of the actuation system

0 Clefin ing perforriiance specifications and uncertainty bounds (Section 3.3.1 )

0 ( 'onstrticting the open-loop interconnection of the overali systeni

a Cl~signing an actuator-level t o q u e rontrol law based on R, (Section 3.3)

rn F1~rforriiing a variety of tests on the closed-loop system. and exploring the ilri-

pac-t of the uncertainty iriodels on the robust stabiIity and robust perfornianc-e

rey iiireiiients (Section 3.3)

3.1 Rigid Body Error Mode1

The first step in motion control design of flexible joint robots is the forniulation of

the error systeril. Let the link position error be clefined as

\ v \ . l i~ r~ (1, is th^ rl x 1 link position and q,d is the n x 1 vector of desired link position

t raj~ctory. CVe assui~ie that and its derivatives up to the third order are boiincled.

.\:on+ NP (-an write rigicl-body dynaiiiics (1.14) in teriiis of (3.1) as:

I n the statespace forrri (3.2) can be written as

Since t h e r ~ is no çoiitroi input in ( 3 . 3 ) . let add and subtract the BoM-' ur on

the RHS of (3.3) yield

w here U I is a rt x 1 vector of fictitious control input defined later. ur can be designecl

so ttiat the trading error e approacties zero in spite of external disturbances. H e r ~ i r i .

we design U I as

u/ = .ir(fCd - U,) +;Y (:3.5)

rvherr :cl and :q are the niatheniatical triodels of the iM and N respectively, and u,

is the nrw (-ont rol input designed iising X, and p -synthesis design methods definecl

in Appendix A. Substituting (3.5) for only the first ul in RHS of (3.4) yieids.

r + 1 e = Aoe + Bo(q + u,) + ~ ~ A 4 - l [UI - (3.6)

r

w here

q = A(&-" + u,,) + 5.

let iletine

thm. ive c'an write (3.6) as:

Yow i n (3.8) if we niake the last term (w) disappear, then it reduces to a simple

state-space error equation where by appropriate design of u, the error stably van-

ishes. Henre. our goal is to force % to zero. This requirement can be satisfied if

dynar~iics of the actuator and transmission systeni is known. In other words. an

a-triator-IPVPI torqlie control law is required to be designed to provide the desirecl

Figure 3.1 : Error dynariiics block diagram.

torqiie of the error-niodel iiiotion control ( 3 . 5 ) . The proposed niethod in this work

reqiiires only the iiieasu renient of lin k position, velocity and output torque. Section

ij.2 pr~sentç the robust ~iiotion control design procedure. Then, in section 3.2.2 a

riiiirierica1 ~xarnple of the n~otion cont rol design without considering the actuation

syst~tii dy nairiics is given. Section 3.3 presents the design procedure for the actuator-

lwel torque feecl back cont roi law. 1 t starts with uncertain ty weights selection (section

3.3.1 ) and continues with the robust U, control design procedure. Two different

wntrollers are designeci, one ignoring the effects of actuator uncertainty (iii,). and

the next one consiclering uncertainty weight levels (K, ) . The noriiinal perforriianr'e.

robust stability and robust performance characteristics of the closed-loop system with

two controllers are coriipared. Finally, the step response characteristics of the systeni

are investigated.

3.2 Robust Motion Control Design

3.2.1 Design Method

Let us represerit the transfer matrix of the error dynamics in (3.8) (without terril)

by F', (s). The systetn can be represented in Fig. : 3 . 1 , where Ii = 7 + u, is the

appliecf torqtie and e is the resultant angutar position and velocity errors.

('onsider X,- optirrial design of a controller for the error dynaniics. Let A-,

denote the riiotion controller for the error dyna~nics niodel. The design specifications

arP taken to be as follows:

1. The artri position and vehcity errors should vanish (e + 0).

2. The control torque, u,, should not exceed a pre-specified saturation liriiit.

pert

Figure 3.2: Error riiodel cont rol probleni formulation block diagrani.

3. The ~inrxiodeIed high-frequency dynamics should be cornpensated for.

4. The effects of ~iieasurenient noises should be cancel out.

Therefore. t h e ?-f, design of K , niay be carried out with reference to Fig.

3.2. wliere W L l ,Wr,2,kli,3 and LV,,, are frequency dependent weighting matrices. used

tu reflect aforeiiientioned perforiiiance specifications. The tiiultiplicative uncertainty

weight (kC',,,) açcounts for the neglected high frequency modes and sorne lorv frequency

Prrors. It is tiiocleled as an unstructureci full block uncertainty, A,, after the error

dyriaii~ics transfer niatrix (P,). To limit the actuator power in control design bVa2 hns

I w n corisitlered. Also, to reflect the noise propagation in different frequency range

aiiother weighting mat rix ha3 been included ( WrrS).

The block diagrarii is reformulatecl into the LFT framework to design control

laws iisiiig the p-synthesis methodology (Fig. 3.3) . A series of control law syntheses

for the error niodel using the p synthesis approach are designed. Robustness and

perforniance of the control designs are traded off in the design process, as one is

increased the other is decreased. Each design is iterated on until it achieves a p 0

value of approxiniately 1. A control law with a p value of 2.0 indicates that for the

iinc~rtain ty and perfortiiance criteria prescribed, the control laws achieves f or 50%

r!J con t rol

Figure 3.3: LFT of error tiioclel control problerii block diagrain.

cil the perforiiiance for f or 50% of the unîertainty level.

3.2.2 Design Example

This section demonst rate a design exam ple of the con trol design procedure in trodiicecl

in previoiis section. It consists of the selection of nominal plant parameters. the

weighting matrices, and clarification of the control design objectives.

.A one DOF flexible joint robot ha5 been considered for siniplicity. The nom-

i r i a l paraiiiet~rs of the robot and the actuator+H.D. are given in Table 3.1. For this

exam ple. the design specifications are:

1. settling tiriie 5 10 sec.

2. Iluw[l, < 2 N.rri.

The weigtiting ~iiatrices are chosen as

for the weiglit on the error signal,

r mani~ulator paranieter I value

Table 3.1 : Flexi ble joint robot parameters.

t

for the weight on the control signal u,.

tu i r i o c l ~ l the noise and

fi)r the weight on the unniodeled high-frequency dynamics.

Figure 3.4 shows the frequency response of these weighting matrices. lising

SI AT L.4 B p-Analysis and Synthesis Toolbox a -1'" order controller is obtained (after

Figure 3.3: Frequency response of weighing t ransfer function.

I!sing this controller for the closed loop system, one can examine the response

of the systeiii. The ciosed loop error response for the step input (gC = 1 rad) are

illustrated in Fig. 3.5.

Figure 3.5: Step response error of closed-loop systeni.(e = q,d - qc) .(è = 4: - 9,)

3.3 Actuator-Level Torque Control Design

The niotion control design procedure for the error dynamics of the flexible joint . rol>ots w a s presented in section 3.2. This section presents a procedure to design

a n act tiator-level torque cont rol law. It consists of selecting uncertainty bounds for

the a(-tiiator ancl HD transniission. To design uncertainty weighting functions of

rtonliriear stiffness. hysteresis and friction, the established method in Chapter 2 is

iised. Fig. 2.16 shows the open-loop interconneîtion of the overail systetii including

the actiratiori systerii, HD. ancl the a m . A new mode1 replacing the nonlinearities

by ttieir PCI uivalent linear riiodels plusuncertain ty weights is presented in the next

sec-tion. The riiodel is then used for the actuator-level torque control design purpose.

3.3.1 Design Method

Fig. 3.6 shows the closed-loop interconnection structure of flexible joint robots with

weighting fiinctions. In Fig. 3 .6 , W,,, is the weighting function of the niotor friction

. M;, is the nonlinear stiffness weighting function, and Wh is the hysteresis weight- - ing function. A's are un known distrirbances of the aforementioned nonlinearities

K,, Pr WJ ~2

t f t t

Yltl

JLctuator w H.D.

4

LI Closed-Loop [ :] Arm Error

Mode1 P

Figure 3.6: Open-loop interconnection of the complete flexible joint robots with weighti ng fiinctions.

Sensor Noise: *

E x h irieasiirerrien t is (-orruptecl with sensor noise w hich becoriies more severe wi th

in(-reasing fr~quency. Since TI, is iiieasiired with torque sensors, their sensor noise

w~ights vari be tliodeIed as

This weighting function irriplies a low frequency nieasurement error in TI, of 0.025

N..LI.. ancl a high frequency error of 0.05 Nm. The niodel of the nieasurecl value of

TI, ,. derioteci r ~ e r a 3 is given by

where 11, is a n arbitrary signal, with (177pjI2 <_ 1. The noise weighting functions are

(lenotecl by W,,,;,, i n the coritrol block diagrairi.

Errors

There are several variables which are to be kept -s~iiall" in the face of the exogenoiis

sigrials. In this con text. these variables wi11 be considered errors.

0 Actuator signal Ievels: the amplifier cii rrent conimand (i,,) should reniai11

rw..sonal>ly ..siiiall" in the face of the exogenous signals. The signals are weighted to

give a desi r d actuator Ievel using WcZct.

0 Performance variables: the actuator torque response(rh) should follow

the desired torque (&ui ) . In other words. the torque error signal should vanish

as titi~es goes to infinity. These c m be ensured if they are weighted by frequency

dependent weights, to give required performance. In the open-loop interconnection

(Fig. 3.6) WPerj is used for this purpose.

Uncertainty Weights

Appropriate weigh ting functions can be selected following the methods descri bed in

( 'hapter 2 for the nonlinear stiffness, friction, and hysteresis effects. It results.

hir the nonIinear stiffness weighting fùnçtion

for the hysteresis weighting function

for the friction weighting function. The frequency response of the weighting functions

are shown iri Fig. 3.7.

3.3.2 Design Exarnple

In t h i s section, the robustness properti O different controllers for a one-DOF

flexible joint robot are analyzed using p. The controllers receive 1 sensor rneasiire-

nient (TI,) dong with the qd cornniand signal which are required for feedback and

proc-liic-e one r:oritrol signal for the actuator (i,,,). In this section. each controller hzw

cl ifferen t c-haracteristics:

O is designeci to optiniize 3C, perforniance, uncler the assumption that there

is no riiode[ iincertainty;

IC, is clesigned with the help of p-synthesis, and it is assumed that moclel

unwrtainty is present.

10'

W . . 0 h a i o n

10-'1 . - . . . . . . * - - - - - * . . . . . . . y 10- 1 O-' 1 on 1 O' 1 O=

Frequency [rad/sec]

Figure 3.7: Freqiiency response of weighing t ransfer functions for acttiator side con t rol ( l~sign.

Frequency Response

Nominal Performance

In the rlosed-loop systeiri. there are -i exogenous signals (1 disturbance signal ( I I ) .

1 sensor noise. and 2 conimand signals), and 2 errors ( weighted performance error.

weighted actiiator error). The noniinal performance objective is that the transfer

fiinc-tion ~iiatrix frorii exogenous signals to errors should have a n 31, norni less than

L. l:sirig, p-Analysis and Synthesis Toolbox it is easy to evaluate this perforniance

c.rit~rion. Fig. 3.8 shows the frequency response of the closed loop system for two

(liffrr~nt i-ontrollers (l ih, Ii,). Note that the best nominal performance is achiwed

hy i.ontroller XI,.

Notuinal Perforitiance: h',,:sotid, K,:dashed

Figii rr 3.8: Noniinal performance plots of controllers: Kh, K,.

Robust Stability

I -sirtg p. the robust stability characteristics of ea th of closecl-loop systeni can be eval-

i i a t ~ d . The uncertain paranieters can be treated as cor~iplex. -4ccording to Fig. 3.9.

the ri, i-ontroller has the best robust stability properties. when the perturbations are

t r ~ a t e d as i*oinplex (dynaiiiic). The Peak of the p. 0.98, implies that there is a diag-

onal. romplex perturbation of size & that causes instability. Similar interpretation

is possible for tlie closed-loop systeni with controller KA, though the p plots have

Iarger peaks. tlie bounds on allowable perturbations is siualler. Hence the closed-loop

systerii with the controller K , achieves robust stability t o cornplex perturbations.

w hereas the other c-ontroller cloes not.

Robust Performance

I.:sirig p. the robust perforniance characteristics of each of closed-loop systetns c-an

he evaliiaterl.

Robust Stabili ty: K,,:soIid, K,:dashed

Figure 3.9: Robust stability plots of controllers: K h , K,.

The appropriate biock structure for the robust performance test is

w tiic.11 is si111 ply an augmentation of the original real robitst stability uncertain ty set.

with a ( W I I I ~ I P X 4 x 'L fu l l block to inïlude the perforriiance objectives.

The p calculations are performed on the entire X x 6 closed-loop niatrix. which

irir-liirle the perturbation channels and the exogenous signals and errors (Fig. :3.10). . Fig. 3.1 t shows the closed-loop response of the system for two different con-

trollers. At very low frequency, the closed-loop robust performance with both of the

c-ontroll~rs açhieved, but a t riiecliuin and high frequency with [ch iniplemented the p

gpts as bac1 as 1.3. The closecl-Ioop system achieves a robust performance p vattie of

0.8 with c-ontroller I<,.

corn niand

Figii re 3 . L O: LFT block diagrain of robust perforniance control

Ro bust Perforn~ance: :solid? K,:dashed

<r-----l design.

Figure 3.1 1: Robust perforniance plots of controllers: Kh, A-,'.

Figure S. 12: Yoriiinal '11 step response. Kb:solid, K,:dashed

Time Response

1x1 this s t i bsection noiiiinal and perturbeci tinie-domain simulations will be demon-

stratecl. For each of the closeci-loop controllers a simulation mode1 is producecl to

rl~inonst rate the tiirie response characteristics of the closed-loop systerii.

Nominal Step Responses

The main performance objective is link angle tracking, so the response to a 1 radian

step input of g'f is investigated. The output signals of interest are 7,. ec and 6,. Fig.

:Li i! shows the 111 response for nominal system without any perturbation present.

Fig. 3.13 also shows the l ink angle and angular error velocity. It is obvious that the . perfor~iiarire of the error closed-loop systern with i<, is much siiperior than controller

K I , .

.As it (-an be seen froiii the noriiinal step response, the controller Ku perfortiis

rriiwh I~etter ttian I ï h .

Figure 3.i3: Noniinal cc and 6, step responses. KtL:solid, K,:dashed

Perturbed Step Responses - The tiltle-cioriiain analysis is rcp~ated on the perturbed niodels. Fig. 3.14 and

shoiv th^ response of the systerii for this case. As expected, the time domain sirriiila-

tioris reinforw the conclusions that were reached in the frequency domain analysis.

: Perturbecl Case 2

C

1::

Figure 3-14: Pertur bed ta^ step response. KIL:soiid, K,:dashed

È = q: - q, : Perturbed

Figure 4.15: Perturbed cc and é, step respoiises. h ' h :~~ l id . K,:dashecl

3.4 Surnmary

.-\ new technique is proposed for designing motion and torque control of flexible joint

r u h t s . It is based on a careful study of the characteristics of HD transmission i n

robotir syst~iiis. A closed-Ioop ff exi ble joint robot built according to this niethod

is ~ ~ p w t ~ c l to bp more robust. because the uncertainties due to the actriator systeiri

nonlinearities are explicitly considered in the control design procedure. The tectiniqri~

iisw tlw perturbed iiiodel of flexible joint robots introducod in Chapter 2. I t involves

two steps: a robust arm motion control is designed; and the desired torqiie for thp

i~iotiorr (-ontrol design is provided by an actuator-level torque feedback control law.

This allow the designer to focus on ariii and actuator control problenis separately. .4

c-oiiildete exaniple deriionstrates the syste~natic procedure for designing cont rol laws.

Different rontroliers are designed to demonstrate the effect of uncertainty bounds on

(*ont rol design. I t is shown that considering uncertainty bounds in control problciii

foriiiiilation will create closed-loop systeiiis which perform robustly in the face of

iincertainties.

Chapter 4

- Experimental Evaluation and

Robustness and Performance

Trade-offs in Uncertainty

Modeling

In the previous çhapter. a twofoId control design strategy was presented. In this

c-hapter. the focus is on the expeririiental evaliiation of the proposed control design.

Hu t becaiise the experimental setup ( IRIS-facility) was itself under development , and

1)ecaiise of the lack of link side measurement instrumentation, we were not able to

wnluate the entire control strategy experinientally. Therefore, we limited our exper-

iiiimts to only the acttiator-level torque control . In spite of this, the results were

quite proiiiising, and it is expected that the proposed theoretical control design c m

IJP iiiipleiiiented siiccessfully when the setiip is ready. For the purpose of torque con-

trol design. a notriinal linear niodel of the joint is identified from the input-output

expeririiental tests. Subsequently, by varying the input signal amplitude level, a

set of iriodels incorporating the effect of nonlinearities is extracted. The diflerences

0

hetween t h e noniinal rtiodel and the set are formulated a s uncertainty bounds for

t-oiitrol design purposes. lsing the uncertainty bounds, an %, - based optimal i-on-

trol1t.r is designed. Experitiients are performed for different uncertainty levels o n the

1 K IS robot f a d i ty test betl .

4.1 Experimental Setup Facility

This section briefly presents the IRIS-facility experiniental setup introduced in [16.

20. I l ] (Fig. 4.1). The IRIS-facility is a versatile and reconfigurable robot arni. It is

clesigned to be easily disassembled and assembled as required. It provides a multitude

of i-onfigurations. Each joint-module is coiiiposed of a frameless DC-motor. an H D

p a r . an optical rotary encocler to rrieasure the i~iotor displacement, and a çustoiii-

designet1 torqrie sensor to riieasure the load torque.

For the purpose of the experinientai tests. the joint-module (this joint will

t ~ e referrecl to as the **test-joint" in the text to fotlow) is constituted by a n RBE-

O 1202 iriotor coriiponents set (Inland Motor Corp.). capable of delivering up to 1.12

Siii. The iiiotor is couplecl to a harnionic drive with LOO: L speed reduction [l'il. The

rat~cl torqiie for this unit is 40 Xni, the niaxirriuni average torque is 49Nrri, ancl the

iiioiiieritary peak torqiie is 108.4 Yrri. The cirstotii-designed torque sensor, which ha.;

a stiffness c-oefficient 10 tiiiies tiigher than that of the harriionic drive itself. is ilsecl

to riieasiire the loacl torque.

For the purpose of torque-feedback controI law designs, the stator of the joint-

test motor is fixeci to a stationary franie to prevent it from rotation. Hence, the

iiieasiirecl torque is proportional to the torque experienced by the rotor. (Fig. 4.2).

The IRIS-facility is controlied by a distributed computer system baseci on

KISC' processor nodes able to provide up to 80 MFLOPS per node, and a fast 1/0

systerii associateci witli each node allowing up to 5 KHz sampling rate.

The control designs are iniplenientecl on the [RIS-joint via a computer node

hiiilt aroiincl a 50 MHz EISA bus-based IBM-PC co~iipatible host computer. A RISC'

Figure 4.1: IRIS amis

(-oprocessor board (ATM-29050) is attached to the PC bus. The RISC board repres-

ents a whole computer system except for 1/0 peripherals. It has a powerfuI RISC'

processor wi t h a built-in floating-poin t unit and i t s own rnemory(severa1 M By tes).

The iiieiriory access tirne that is very critical to the execution speed of standard coiii-

piiters (typicalIy Tons), preventing theni froni crossing over the speed of a couple of

MFLOPS. is effectively shortened by a factor of 2 in this RISC board. The details

about the RISC board can be found in the YARC Systems reference manual. The

Figure 4.2: Restrained motion setup

r-ontrol systeiri hardware includes the following I/O boards attached to the Host bus:

a. .ICI(' boards, b. DAC boards, and c. Digital 1/0 boards.

The 1/0 boards are connected to the low-level con trol system that includes in-

terfaces to the optical encoders, signal-conditioning circuits and the power amplifiers.

The IRIS hardware organization is shown in Fig. 3.3.

Figure 4.3: Real tirne cont rol irriplenientation organization

4.1.1 Experimental Identification of Transfer F'unction

The frequency response method is used expeririientally to provide an input-out put

tortlue rriodel of the test-joint. Such experitnents can be dangerous if the arrri is

u n rest rained. due to the elastic coupling between high-frequency "internal" (joint)

anil low-freqiiency "external" ( a m ) clynaniics. In contrast, the expeririients can be

p~rforiiied siiccessfully if the artii is restrained against a stiff environment. The stiff

environnient prevents the external dynamics from being excited such that no effect on - the internal tlynaniics takes place. The experiments were carried out for the test-joint

iising a n in prit signal frequency ran90 of 0- 150 Hz. The same set of experiments were

r~peatecl for varying levels of input signal aitiplitude. Fig. 4.4 and Fig. 4.5 show

the magnitude and phase plot of input-output torque for three different amplitudes

of s i n iisoidal signal.

Since the highest rvorking bandwidth of the systein is below 100 Hz, a trans-

Frequency [Hz!

Figure 1.4: Input-output torque magnitude plot of IRIS-joint for three different input signal a111 plitude

Frequency [Hz1

Figure -t..',: Input-output torque phase plot of IRIS-joint for three difkrent input signal aiiipli tiicle

10 - c. O - m 3 - y -10- -- 'O 7 .'=c -20-

2' T a -

Figure 4.6: Estiniated and iiierrsured input-output torque transfer function of the 1 KIS-join t

fPT furii:tion of a riiodel of the input-output torque baçed on the experimental data

between O- 1OOHz is identifieci. More precise analysis using Signal Processing und

Id~ntijication Toolbox of MATLAB [LW] shows that the transfer function is better

approximated by P7,,7,,(s) = 3, where b(s) and a(s) are second and fourth ortlcr

poly notitials respectively given as:

The coiiiparison of the real and the identifieci input-output torque signals in Fig.

4.6 shows very good match between the iiieasured and the estirnateci signals for the

o p r a t i rig range of 0- 100 Hz.

[ i tilizing the 31, control design fratnework, the aforenientioned identifid

rioiitinal triodel and uncertainty levels are used for control design purposes.

- 4.2 Robustness and Performance Trade-offs

The seh-tion of iincertainty description plays a major role in the trade-off between

rol~iist ness and perforruance requirenients in the cont rol design process. Uncertainty

des(-riptions are introdiiced tu account for variations between models and the actual

systerii ancl provide a quantitative rneasure of the differences.

This section investigates this trade-off in the selection of uncertainty desrrip-

tions and levels. -4 set of control laws is designed by changing the iincertainty Ievels

fur the test-joint experiiiient. A frequency dotiiain uncertainty description of the

variation between the model and the "real" system is used for the design of control

Iaws. These designs make use of a multiplicative uncertainty model to account for

high-freqiienry iin~iiodeled dynarriics (Fig. 4.8). It is shown that the closed loop

systerii iiiay main tain the perfortnance or even becorne unstabIe as the uncertain ty

lev~ls are çhanged. These clearly indicate the iriiportance of uncertainty descriptions

i n the control design process.

4.2.1 Control Objectives

The çontrol objective is to track a desired coniniand torque in spite of nonlinearities,

friction and Aexibility in the actuator-transmission system. This is formulatecl as

rtiiiiiriiizing the 1) .))., norni between the input disturbances and sensor outputs.

4.2.2 Uncertainty Descriptions

-4 freqiiency doniain clescription of uncertainty is ernployed to account for the vari-

ation between the mode1 and the "real" systeni. A multiplicative uncertainty weight

is iised for the low frequency (below 0.2 Hz) and for the unmodeled high frequency

dynarriics (above 50Hz). The noininal linear mode1 of the joint (4.1-4.2) is used.

Siil>sequentIy, by varying the input signal aniplitude level, a set of niodels incorpor-

ating the effect of nonlinearities in the systeni are ext racted. A plot of the differenrw

lretween the noruinal triode1 ancl the set along with the ~riultiplicative uncertainty

Figure 4.7: !Magnitude plot of error torque and rriultiplicative uncertainty weight

w ~ i g h t a r p shown in Fig. 4.7. The ~nagnitude of the multiplicative uncertainty weight

a t high frequency is selected to envelop the uniiiodeled inodes of the systeni (Fig.

4.7).

.klditive uncertainty weight is aiso included to represent the torque serisor

noise ~rieasiirenient (Fig. 4.8).

4.2.3 Control Problem Formulation

The icientified SIS0 nominal iiiodel of the test-joint. P,,,,,, , is used to describe the

test-joint esperinient. It serves as a baseline rriodel to which uncertainty niodels are

appencled. A block diagratri of the probleni forriiulation is shown in Fig. 4.8 The

tiiiiltipliïntive uncertainty weight accotints for the neglected high-frequency niocles

arirl sotiie Iow frequency error. It i s inodeleci a5 an unstructured ful l block uncertaintÿ,

A,,,uII, after the test-joint nominal riiodel as seen in the block diagram. Perforniance

weigh t. Wp, and actuator-satu ration liniit weigh t, Wn, are the paranieters variecl to

examine trade-off between the robustness and performance of the control designs.

The block diagram is reforrnulated into the LFT general framework to design

pert. dist u r bance

noise ++f

Figure 4.8: Block cliagrani of tracle-off control problern forniulation

cmntrol laws using the p-synthesis methodology. The block diagram is shown in Fig.

-4.9 The dirtiensions of the A blocks are: 1 x 1 for A l , and 3 x 2 for Al. A I is

a.sso(-iated wi th the multiplicative uncertainty, l2 with the performance block.

-4 pure ' H , control design would synthesize a control law for one full block of

size -4 x 3. neglecting the inherent structure associateci with the two blocks. Ignoring

the strwture of the uncertainty block leads to overly conservative control laws. The p-

synthesis iiiethodology incorporates knowledge of this structure in the control design

prowss, reducing the conservatisrn.

4.3 Experimental Results

4.3.1 Model Uncertainty Weight a

Thwe control laws are synthesized based on the biock diagram in Fig. 4.8 with

varying Ievels of iricdel uncertainty weight (i.e. WmUrt varied), while the rest of the

1 Generalized Plant

l noise

dist. + pert. setpoint

I 1 controls

Figure 4.9: LFT of tracle-off control probleiri formulation

weighting functions reniain fixeci. The set of weighting functions that are used for

control design purposes are as follows:

I . Mode1 iincertainty weight:

500s + 5000 W,,rult (s) = r.

s + 10000

:{. F'erfoririance weight 0.005s + LOO

wp(s) = '*o.ols + 0.005 18 4. Ac+tiiator saturation weight

Table 1 çontains a list of the control parairieters used in the design and the results of

itii plenien tation on the test-joint experirnen t.

r p Figure

0.01 9.4 Fin. 4.10

1.0 1 2.7 1 Fig. 4.12

Table 4.1: Parariteters for control design with fixed performance weight

The closed-loop torque responses of the test-joint experinient for implementing

these controllers are shown in Fig. 1.10 to Fig. 4.15. M U-KI achieves a p v a h e of 9.4.

and the step response settles down in about 1 second. But the step response shows

an overshoot as high as two tiriies the desired step coniniand. .MU-K:! is designed by

iricr~asing the iriodel uncertainty weight by a factor of 10 coiiipared to the previoris

case. However, the step response doesn't show any overshoot, but a steady-state error

- of 15% ran be observed in Fig. 4-Ll. The level of uncertainty weight is increasetl

tu 1 i n the .LI I V - I i : 3 controller and it can be observed that the step response of the

s y s t ~ ~ i i g ~ t s hetter cotiipard to the two previoiis (.ases.

Fig. 4.13 to Fig. 4.15 show the sinusoiclal response of the systeni for the

saiiie sets of (.ontrollers. In this set of expeririients. controller hl U-IC3 shows superior

performatiw coriipared to the two other controllers.

4.3-2 Actuator-Saturation Cimit

.A series of con t rol laws is synthesized using varying tevels of actuator-satu ration

l i i i i i ts (i.e.. Lt:, varied), whiIe multiplicative uncertainty and noise weights reniai ri

tired. Table 1. shows the control parameters used in the design and the results of

iiiiplenientation on the test-joint experinient.

Fig. 4.16 shows the step torque responses of the test-joint for controller .AC-

- I< 1. The actiiator-saturation level is chosen as 10 in this case. It shows a n steady-

Table 4.2: Paratiieters for control design with varying performance weight.

control

AGKl

a 1 71 10 1 IO-''

input

step L r

2.0 1.0

AC-KP AC-Ki3

controll input

'AC-K4Is in

T a l h 4.3: Yarariieters for control design with varying performance and uncertain ty iveight .

P

step s t e ~

AC-K5 4 ' - 6

state error of iip to 1.5% which continues to increase. If the actuator-saturation l i ~ i i i t

dm-reas~s to 2 (i.e. AC'-K2 controller)! the perforriiance of the systeni does not change

coiilpareci to the first case (Fig. 3.17. But if the actuator-saturation limit decrease,.;

to 1. the steacly-state error gets smaller values (Fig. 4.18).

Three new control laws are synthesized. using sinusoidal input torque c o w

iiianrl. Table ;3 shows the pararrieters useci in the design as well as the result of

iiiipleriientation on the test-joint expeririient. .As can be observed, the performance

of th^ syste~n for the case of AC-Ii4 controller is considerably better thân two other

. c.ontrollers for the best selection of uncertainty and actuator-saturation levels.

, -

a

2.0

Represen ting the actuator-transmission systeni in flexible joint robots with a nom-

i d mode1 and an uncertainty description provicies an excellent design model for

ilse in robiist torque cont rol techniques. The theoretical and experirnental results

inclicate that uncertainty iriodeling plays a major role in the trade-off between the

r

Z 1 0 - ~ 1 0 - ~

n

s in sin

p 2.26

10-~ 1 0 - ~

L O - ~ 1 0 - ~

1.0 0.5

Figure

Fia. 4.16

P I r 1 0 - ~ 1 1 0 - ~

1.0 1.0

L O - ~

- -

p ' Figure

2.27 2.7

1.55 0.01

10-"-001

Fig. 4.17 Fia. 4.18

Fig .4 .19 9.4 0.73

- Fig. 4.20 Fie. 4.21

req:iireriients and the robustness properties of synthesized control laws. A series of

- i-ontrot laws are synthesized with varying levels of niodei uncertainty and different

artiiator-saturation lirriits. It is experinientally verified that the performance of the

s y s t ~ n i is sensitive to the uncertainty description and the actuator saturation level

i r i the wntrol design. Increasing the level of multiplicative uncertainty while the

a(-tuator-satiiration level is fixed leads t o better performance of the system when the

cwritrol law is applied experirrientally. On the other hand, choosing a high actuator-

sattiration liiiiit while the iiiodel uncertainty weight reniains fixed causes perfornianc~

to deteriorate and large steady-state errors to appear in the step response. Finally.

it is verified that siniultaneously varying the actuator-saturation ievel and the triode1

un(-ertairity weights in con trol design leads to high performance closed-loop systmis

ici ~xp&rriental irripleirientation.

Figure 4.10: S t e p response of the IRIS-joint for controller MU-K 1

1 .sr

1

- f 0.5 Z - Q> 3

g O

-0.5

- 1

-1 .5 O

T h e [sec]

I

\ I

-

-

-dV

2 4 6 8 10 12

Figure 3.1 1: Step response of the IRIS-joint for controller M U-K2

Time [sec]

Time [sec]

Figure 4.1'2: Step response of the IRIS-joint for controller MU-K3

2.5

2

1.5 - E z 1 w a

g- 0.5 + O

-0.5

- 1

-1.5 O 2 4 6 8 10 12 14

Time [sec]

Figure 3.1:3: Sinusoicial response of the IRIS-joint for controller MU-Kl

Figure 4.14: Sinusoidal response of the IRIS-joint for controller M U-K'L

- - --

Tirne [sec]

Figure 4.15: Sinusoidal response of the IRIS-joint for controller MU-K3

Figure 4.16: Step rosponsc of the IRIS-joint for controller ..\C-1<1

2

1.5 - - C c Z O 1 - a z

E=

O 1 I I 1 1 O 2 4 6 8 10 12

Tirne [sec]

0.5

Figure 4.17: Step response of the IRIS-joint for controller A G K 2

-

O

-0.5 , 1 , 1

O 1 2 3 4 5 6 7 8 9 10

Time [sec]

O 1 1 I I

O 1 2 3 4 5 6 7 8

Tirne [sec j

Figure 3.18: Step response of the IRIS-joint for controller AC-K3

-

-0.5- * O 5 10 15

Time [sec]

Figure 4.19: Sinusoidal response of the IRIS-joint for controller AC-K4

Figrire 4.20: Sinusoidal response of the IRIS-joint for controller .4C-K5

~ & e [sec] 8

Figure 4.2 1: Sinusoidal response of the IRIS-joint for controller AC-KG

Chapter 5

Stability Analysis of the Proposed

Robust Torque Control Design

Thr prohleiii of absolute stability of a feedback systerii which h a s a linear tiriie in-

variant part and a nonlinear elernent in the feedback loop is known as the classical

Lur .c*s Pr-oblciri [53, il. Many systenis whi îh are priinarily characterized as linpar

ti rile-invariant systetris except for a few nonlinear cornponents such as saturating ar-

tuators can be represented as Lure systenis. Therefore, traditionally, there h a s been

significari t in terest in the stability of such systerns. The stability results are available

priiiiarily for sector bounded nonlinearity [,59. 71.

This chapter presents a stability analysis of the feedback control of flexible

joint robots with sector-bounded perturbations, as shown in Fig. 5.1. The controller

is designecl following the procedure rnentioned in Chapter l3. The focus is on the stabil-

ity of the c-losed-Ioop torque feedback control of flexibIe joint robots with harnionic

cl rive t raiisiliission. I t is asstinied that the t ransrnission systeni exhibits nonlinear

stiffness. WP transforni the closed-loop systeiii in to a Linear Tinie lnvarian t (LTI)

part in the feedforwarcl loop and a nonlinear elenient (i.e., noniinear stiffness) which

is insicle sector [a, b] (i-e., having bounded m-norni) in the feedback loop. Iitilizing

t l i ~ sr~sull-gain theorem, we denionstrate that this guarantees the closed-loop syste~ii

Fig i i r~ 5.1 : C'losed-loop nonlinear tiiodel of flexible joint robot with nonlinear stiffness pertiir bation

stability providecl that the cm-norni of the interconnection of the LTI system and the

nonlin~arity is less than one. Linear fractional transforniation and loop transforiiiation

terhniqiies are erripIoyed to prove the niain results of our stability analysis. Finally.

as an alternative to the srnall-gain-thcoretr~ approach, a Lyaputtou Jirtlctiorr is derivecl

tliat guarantees that the feed baçk interconnection of the LTI and the nonlinearity arp

stable.

-4 review of a sector bounded perturbation condition and a linear fractional

transforrmtion of the systeni is given in the next two sections. The small-gairi theorcru

is tlien presented, and the probleiri of robust stability of torque feedback control of a

flexible joint robot is fortiiulated. Finally, the stability of the system is investigated . iising the sntall-gain tlreoreirt and the Lyapunov function.

5.1 Sector Bounded Conditions

Seirtor conditions for the stability of the feedback interconnection of general input-

output systeiiis were introduced in [59], and further expanded in [55, 421.

.A riie~rioryless, tinie-varying nonlinearity, ti(y, t ) . is said to be inside sector

[n. b] if

for a11 ,y E PL [19. 1-41. Ckoriietrically, these sertor conditions iiiiply that th^ graph

of the nonlinearity lies within a conical region in the Y?'" x %"' input-output space for

a l i n . t . For r t r = 1, Fig. 5.2 shows a nonlinearity. l'?(y? t ) , inside sector [a. b]; the

graph of u?(y, t ) tiiust lie in the shaded region within the two lines of dopes a ancl b.

Figii re 5.2: Sector-boiinded nonlinearity

The Nyquist plot of a n LTI systeiii. inside a sector [a. 61, b > a, lies within

a rirrlp in the frequenry plane, whose center is at [(a + b ) / ' ~ , jO] and lias a radius

of ( b - a)/ 'L. Note that a square, bounded, real systeni, that is, a system satisfying

( s ) 1. is inside sector [-1, LI; and its Nyquist plot lies within a unit circle

(-en tered at the origin. For exaniple, G(s) = lO/(s+2) (sf5) is inside sector [-0.4, 1 .O]

and it,s Xyquist plot lies within the corresponding circle, shown in Fig. 5.3. Note

that Ilc;'(s) I l r n < 1 so that i ts Nyquist plot also lies within the unit circle centered a t

the origin, as shown in Fig. 5.:3.

'O inside sector [-0.4.11 Figure 5.3: Yyquist plot of G ( s ) = (,+2)(,+,)

5.2 Linear F'ractional Transformation

( i i v ~ r i a c.-oriiplex variable ~iiatr ix M. partitioned as

am1 le t Al he another çoir~plex variable tiiatrix:

Then we ran forilially define a lower L i n e a r F r a c t i o n a l Transformation (LFT) of .LI

ivith r~spec-t to Al as the iriap

provicled that the inverse ( I - iç122~1)-1 exists. The terminologies of lower LFT

shoulcl be clear from Fig. 5.3.

We can define the closed loop transfer function from w to 2 in Fig. 5.4(or

L o w r Linpar Fractional Transforriiation [SI) as follows:

Figure 5.4: Linear fractional transfoririation

This transforriiation exists provided that t h e closed-Ioop is well posed, i.e. ( I - .1122&)-1 exists.

5.3 Input-Output Stability and Small Gain Theorem

The foriiialisrii of input-ou t p u t stabiIity is useful in studying the stability of interr-on-

n~ctions of dynaiiiical systenis. This is particularly so for the feedback connection of

Fig. 5.5.

To prepare for the small-gain theorenr, we first give the following definitions.

Definition 1: Let LP denotes the norrned linear space where p E [l. 3c).

Also, rlefine L$ as a n extended linear space defined by .

w h ~ r e u belongs to CP, and u, is the truncation of u , defined by

Figure 5.5: Feedback connection

The extenclecl space Lr is a linear space that contains the unextended space CP a s i1

sul)set.

Definition 2: .A niapping H : ÇT -t Ce is ,Cr-stable(r E [ I l cal), if there exist

finite nonnegative constants y and /3 such that 0

1 1 ( W T l l 5 7 11*4l + 1 1 (5.4)

f i ir al1 u E L$' and T E [O. CZ;). Lp is a space of signals u : [O. x) -+ RP. tz is also an

extendeci space of signals

Now. we are ready to introduce the snrall-gain theorern [7]. The small-yuiri

thcorcrrr gives a sufficient condition for the C-stability of the feedback connection in

Fig. 5,.3.

Theorern 1 C'orisider the systern irc Fig. 5.5. Let Hi, H2 : tz + t:, p [L, CQ], be

two Lf stable opcrutors with Errite gains 71. y.1 arzd associated constants f i l , &. Let

tlrc opcrator Hi H2 be strictly causal. if

The reader iiiay refer to [il for the proof of the previous theoreni. Paraphrasing the

srrrall-gain tlteorem proves that the feedback connection of two input-output stable

systmis. ns in Fig. ri.5. will be input-output stable provided that the product of the

systcwi gain is Iess than one.

Xow. for the interconnection of an LTI systeni in forward loop ancl a non-

linearity in the feedback, the following theoreni can also be applied. Consicler the

systeni of Fig. 5.6. Let the LTI systerri be representecl by

Siippose that G ( s ) is Hurwitz. Let

w h w r <r,,,.,,[.] clenotes the iiiaxiiiiuni singular value of a coniplex niatrix. The constant - 71 is finite since ( i ( s ) is Hurwitz. Suppose that the nonlinearity Îi?(., .) satisfies the

iiieqiiali ty

Figure 5.6: Absolute stability franiework

CVP ran apply the striali-gain theoretrr and conciude that the system is absolutely stable

if -

This a a roliiistness result which shows that closing the Ioop around a Hurwitz transf'er

fiinc-tion witli a nonlinearity satisfying (5.10). with a sufficiently stiiall ~2~ does not

clestroy the stability of the systeni.

5.4 Loop Transformation

Hy iiiaking a suitable transformation of the systeni in Fig. 5.5, one can significantly

expand the range of applicability of theoreni 1. The following theorem presents the

generalization of theorern 1.

Theorem 2 [19] Consider tlic systerri showri in Fig. 5.5, and suppose p E [ 1 , DO) is

sym'fied. Suppose Hz is causal and L,-stable. Under these conditions, the systetir is

Figure 5.7: Loop transforiziation

Ç,-stabk if t h c w exists a causal linrar opcrator K which is Lp-stable such tliat:

(i) H i ( I + I i H I ) - ' is causal and L,-stablc, and

(ii) ?,[Hi ( I + li H l ) - ' ] y , ( H 2 - K ) < 1.

Fig. 5.7 shows an equivalent representation of the system in Fig. 5.5 where

a c-oristant-gain negative feed back I i y is applied around the H 2 of the systent. The

proof of tliis theoreiii can be found in [19], [ 5 3 ] .

- 5.5 Stability Results

5.5.1 Small-gain Theorem-Based Stability Analysis

The stability problem of the torque-feedback control of robots with sector-boundetl

stiffness nonlineari ty is introduced in this chapter. The closed loop toque-feed back

i:ontrol law is t ransfornied into an LTI and a nonlinearity which is inside sector [a. 61,

Figure 5.8: Linear fractional transformation

to prove the stability probleni. The linear fractional transfornied (LFT) forrn of the

c-losecl loop systeiii introduced in Section 5.2 is prirrlarily used for this purposo.

TIIP c.losed loop systerri of Fig. 5.1 is transforriied into an LFT forni as

ititrodii(-cd iri Fig. 5.8 . In Fig. 5.8. G i j . ( i q j = 1.2) are a11 known transfer functions.

For instant-e. G I 1 can be found to be as,

where r is the gear ratio, Pm is the actuator transfer function, S the stiffness coeffi-

cient, and K, is the Iinear 31, controtler transfer function shown in Fig. 5.1. The

rptiiaining of Cijk c m be found similarly froiii Fig. 5.1. w is the norm bounded

pxogenoiis inpiiis and z is the signal to be kept srriall.

Figure 5.9: Stability bIock diagrani

The transfer function from y to u can be defined as,

u = =l,y

y = G'2i w + GZ2u

Srihstitiiting (5.1.5) in (5.17) we obtain,

y = v + GZ2ilr9

n. here

Fig. 5.9 shows the block-diagram of the above forinulation.

In Fig. 0.9, G22 is the LTI &-stable transfer matrix, by design: and A, is

the stiffness nonlinearity designed to be in sector [a, b ] . To state our niain results. the

following I~riiiiia is needed[14].

Lminia 1: Civen a stable LTI systei~i Z:k = .4x + Bu. y = C x + Du, where

the qiiadriiple [A! B. C', U] is a minimal realization of the transfer function niatrix.

C;(s) = ( '(si - -4)-l B + D, the following stateiiients are equivalent.

( i ) 'C is inside sector [a. 61.

( i i ) There exist real matrices P = pT > O, L and W which satisfy

w l i ~ r ~ (1 = ( a + b ) / 2 . The proof of this lenima ran be foiind in [14]. Now. we are

r~at ly to state oiir theoreni.

Theorem 3 I 'onsider the rrcgativc fcedback irzterconnection of a stable LTI systern

(i2? : 2 = -4s + Bu. y = C ' r + Du. wlrerc the quadruple [A, B, C'. Dl is a minirual

rculiralion. arrd a rrrerrro yless , tirrie-uaryirrg, nonlincarity, A, (as sho wn in Fig. 5.9).

/ j the rroirlirtcarity . A,, belongs t o an arbitrary sector not rrecessarily centewd at

or-igin. therr the origirt is a Lyapunov stable equdibrium point of the closed-loop systcnl

1 1 ij Ciz2 is iirside seetor [-i;. - --] ( i . c., the i'Vyquist plot of G22 lies Unthin a circlc in A + L -+' - 1

~ / r t Jtcqircncy plarrc, wlrose ceriter i s at [ j ~ ] and has a rndius 01 (y)).

Proof: Since A, does not necessarily belong to sector [a. b] where 6 > O > a. wp

. ~iiiploy the looptr,znsforii~ation technique introduced in section 5.4 for the probleiii

i ~ i Iiancl. Let us clefine

a + b b - a k . = - . r = - (5 .2 1 ) 2 2

Fig. 5.10 shows the definitions of k and r graphically. Then. the new nonlinearity is

g i v w by

It cari easily be verified that if A, satisfies the sector condition (5.1), then i, satisfies

the sector çonclition

Figure 5.1 O: Sector nonlinearity transformation

Fig. 5.1 1 shows the transforrned block diagram of the system. Now, in Fig. 5.1 1. the

new linear systern. G,, represented by

is a H iirwitz t ransfer function by design. [ T tilizing the strtull-gain tlreorenr. we m i

t-onc.liide that the systerii will be stable if

W ~ P ~ C 7 is defined i n (5.9). Sincc r j ~ belongs to sector [-r . r ] , then by definition

-r ( ( : y ) 5 I.. Therefore. froni (5.25) y (ClT) < l /r . Thus, G, should beiong to sector

[- L I r - . 1 / r ] . 1 t can be easily verified that if d~ E [-r . r] then froni (5.22), A, E [a . 61

and also if Ci, E [- f , SI. then froiii (5.44) C;22[I + kC22]-1 E [-!, !] that results in

( ; 2 2 E [- f ; . - '1. Th i s proves the theoreni. I

.Uternatively, the above theoreni can be also proved utilizing Lemma 1 and

Lyapiinov stability criteria. This is shown in the next section.

5.5.2 Lyapunov-Based Stability Analysis *

Proof: For notational convenience, assume that in the state-space form, the GT

trarisfer fiiric-tion is presented as:

and the noniinearity d? belongs to sector [a, 61. Then the following Lyapunov function

csiidiclnte (-an be chosen to prove the systeiii stability. Let V ( z ) = Z*PX be the

Lyapiinov fiinrtion with P = pr > O being a positive definite inatrix which satisfies

the sec-tor t>ouncledness in lemnia 1. Differentiating the Lyapunov function we get.

I'sing the relations of (5.20) in (5.27) leada to

T T CI = y - Du and zTCT = yT - u D .

86

- I 'siiig the fact that the nonlinearity. dy. is inside sector [a. b]' (,u? - by jT( ,@ - a y ) < 0.

tor al1 y and u = - i i r , results in ,

T T C'z = y - and zT<*T = - t/t D . (5 -30)

Henw rising the last relationship iri (5 .20) and sonie algebraic manipulations results

Thus. frorii (5.31) we have V ( x ) 5 0, and the origin is Lyapunov stable.

5.6 Summary

* Siifficien t r-oridi tions are presen ted for the stability of torque-feed back cont rol laws

r )f Hexi blejoint robots subjecteci to nonlinear stiffness perturbation. The closed-loop

systerii is rearranged to include an LTI part and a sector-bounded nonlinearity. The

lirtt ur fractinrial tra~isforntation tech nique is employed to study the stabili ty of the

int~rronnwtion systerii of the LTI part and the nonlinearity. Moreover. conditions

are presentecl for a nonlinearity to be inside a given sector. The sniall-gain tlreorctrr

and looy-ti.cttrsfor7riatiort technique are eriiployed to prove the stability of a closed-loop

systmi. Finally, the Lyapuno~r-stability theoreni is used as a coniplenientary approach

to the striall-gain ttieoreni to prove the stability of the feedback intercorinection of the

LTI systeiri anci the nonlinearity.

Chapter 6

Conclusions and Future Directions

6.1 Conclusions

Yeu- results are presented concerning niotion and torque control problems of flexible

joint robots. A coinplete niodel of flexible joint robots with HD transmission is

proposed and then used for tnotion and torque control design purposes. The thesis

focuses un the selection of uncertainty bounds and their incorporation into the control

design procpss. -4 systeniatic approach is introduced in order to incorporate act uator-

trarisi~iission uncertainties in to the con trol clesign. The describiny furrction and cotlic*-

.swtor- bo unded rronlincan'ty rnethods are used to niodel the effect of hysteresis, friction

and nonlinear stiffness on the cont rol design. Numerical examples of uncertainty

boir ncl selection are provided. This research shows the need to incorporate accu rat^

cl~scriptions of mode1 errors into the control probiern formulation.

A ttwofdd robust control design of flexible joint robots with HD is introdur.ec1:

it inclrides an actuator-level torque control and a link-level motion control. The

clesign riiethod is based on R, - optimal control and p-synthesis techniques. The

c-ontrol design developed allows u s to focus on link and actuator control problems

separately. Moreover, the performance specifications and stability requirements can

be satisfiecl siiiiultaneously. Accurately describing the flexible joint robot models and

correc-tly identifying iiiodel errors guarantee high perforniance control laws. The ro-

liiistn~ss of the control laws is shown to be directly tied to the selection of uncertainty

lwunds. Inacwirately tiiodeling uncertainty bounds leads to unstable control laws in

expriiiiental itiiplenientation. The X, - optiinal control and p-synthesis frainework

prov~rl to he vwy suitable for applications to the motion anci torque control probleiris

of flexible joint robots. This is rriainly because performance and robustness specifica-

tiuiis (-an he incorporated into the control design process. The results presentecl will

aid wntrol engineers in th^ design of control laws using 3C,- optinial control ancl

p-synthesis riiethods.

This research verifies the closed-loop stability of the proposed control design

with nonlinear stiffness using the Snrall-Gain Theorem and the Lyapunou Furzctiori

riiethotl. The actuator-bel tory ue control design was implernented experimentally.

Kobristness and perforiiiance trade-offs in torque control design were investigatd.

Expeririients were performed for different uncertainty levels on the [RIS-facility test-

I ~ e c l . I t was verified that varying the actuator-saturation level and the iiiodel un(-er-

tain ty weights in control design siniultaneously guaranteed high performance closecl-

loop responses and iiiaintained the stability of the system. Moreover. the robustness

of the proposed control scheizie in nlocieling uncertainty and nieasurernent noises was

cleiriorist ratecl experimentally.

6.2 Suggestions for Future Research

The following research directions are suggested:

a) The role of uncertainty descriptions should be quantified in the design of

riiotion ancl torqiie con trol laws for robots with harmonic drive transmission such as

those introtluceci in this work. It is important to understand how the accuracy of

the design iiioclel and uncertainty descriptions affect the ro bustness and perforniance

prqzerties of the control laws.

11) Systeiiiatic tiiethocIs iiiust be fu rther developed for identifying both notii-

inal riiodels and uncertainty descriptions, Currently, uncertainty descriptions are

tleveloped based on the experience of the control engineer. A systematic approach

to systeiii identification, which provides both a plant and uncertainty models of the

systerii, worild he a ~iiajor step forward in the control design process.

c ) A11 robust control design methods require explicit worst-case bounds on

the existirig plant uncertainty. This research established some worst case bounds:

however. i t is desirable that new systeiri identification niethods be i n t roduced to

irientify explkit bounds for the plant uncertainty.

c l ) The proposed control schenie in this work should be extended to the case

where a robot end-effector is in contact with an environnient (force control).

e) In this work, a continuous tinie controller for a continuous plant model is

tlesigned. Ttien, a bilirrear tinns/onnation technique is used to discretize the control-

ler and to iniplenient it experirnentally. Further investigation is required to evalu-

atP the sariipling effects in control design. Moreover, other cornmon digital control

tlesign iiiethods li ke "discrete-ti nie çont roI design for the discretized plant" and "dir-

ect saiiipletl-data control design for the continuous plant" should be compared witli

the [ilethocl proposed in this work.

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Appendix A

3-lco- Design and Structured

Singular Value ( p ) Framework

A. 1 Mathematical Preliminaries

In t h e vontext of this dissertation, Lz is the space of ali signals, or vectors of signals.

~vith I~oiincled energy. This is a Hilbert space with inner product

1.t.. the square root of the total energy. .Mternatively, C2 Itiay be thought of as a

freqtieric-y dottiain space, with inner product

f w h ~ r e I is the Fourier transforiii of 2. Parseval's theorem states that 11211L, = 11z11>,.

I n fact. we shaII think of these spaces as being one and the same and not use a

separate notation for signals in the tiiiie and frequency domains. Thus, when we say

that x is in C2 we niean tliat rr: in a signal of bounded energy, which can be thought

- of either (2.; a function of tiriie or frequency.

I r i the tinie domain, C2 can be decomposed into &+ and L2,, where L2+ is

t l i ~ space of signals defined for positive tinie and zero for negative tiriie. Siniilarly.

L2- is the space of signals defined for negative tiriie, and zero for positive tirrie. I r i

the freqrienry doniain. L2 can be siitiilarly deconiposed into X 2 and ?121, where Hz

is the space of Fourier transforni of signals in L2+ and 'HzL is the space of Fourier

transfornis of signals in C2-. Thus, when we Say that z is in 7d1 we niean that r

ix a signal of bounded energy which, when considered in the time dornain, is zero

for negative tiiiie. For such signals, the Fourier transforni is identical to the Laplace

transforiri with the Laplace variable s replaced by jw . RE2 is the subspace of R2 - consisting of only rational functions. and rriay be equivalently thought of the rationai

f i l tic- tions of the çoniplex variable s with clenoiiiinator of strictly greater order than

rititIierator ancl with no poles in the closed r h p . Similarly, the Hardy space N2 (-an

h~ [lefinecl as the space of functions of the coniplex variable s that are analytic for al1

s i n the open right half plane (i-e., s : R(s) > O ) , and for which the integrals

are iiniforriily bounded for ali cr > O. with norni as in (2.14).

A.1.2 The F'unction Spaces L, and 31,

W e shall often think of systenis as being operators o n X2. A system P will be calletl

stable if. for any input in X2, the output is also in 'Fi2. That is, a stable system niaps

I~orindori energy inputs ont0 bounded energy outputs. If a systeni is unstable then its

- oiitputcan haveinfinityenergyinresponsetoafiniteenergyinput. 31, is thespac~

o f transtèr fiinctions of stable linear. tinie-invariant, continuous tinie systeriis. This

is a Hardy space with norrxi

Thiis stable systetiis have a finite R, n o m , and this norm measures the niaxiiiiu~ri

energy gain of a systeni. For a constant tnatrix A, its maxiniuni singular value, a(A)

is defiriecl as the niaxiriiuni "gain" of the niatrix when the "inputs" and "outputs" are

constant vertors equipped with the standard Euclidean norm 1 1 ~ 1 1 ~ := JE-. Or

The iiiaxiriiiini singular value is riiore easily calculateci as (Asi l ) , where h denotes

the iiia.uiriiiiiii eigenvalue of the niatrix. A n input that conies arbitrarily close to

attaining the 3 C , norni concentrates its energy nt the frequency w h e r e t h e gain of

t I I P systeni. as r t ieasu red by the riiaxiniiim singi i lar value of its freq uency responsp

iiiatrix. is Iargest. Thus,

Foriiially. R, is defïned as the space of functions of the cornplex variable s analytic

for al1 s in the open right haif plane with finite norni as defined in (-4.6). RN, is

defined as the subspace of U., whose elements are rational functions of s and aris~

as the trarisfér functions of finite diniensional systerns. i.e.. those systerns descrilwti

by orciinary différentiai eqiiations. For such systeiiis. the suprenium is attained on the

I~oiintlary ,s = j zu . (for possibly infinite w ) i.e,. -

prc~vicld FJ(s ) is analytic in the right half plane. L, is the (Lebesgue) space of al1

Fun(-tions essentially bouncled on the iniaginary axis with norrii

Siiriilariy RX, is the subspace of L, whose elenients are rational functions of S. for

w hich the norrri sini plifies to

W'IIRL~ := rnax a ( P ( j w ) ) . WER):I~,

bVe shaI1 regard R, as a subspace of Ç,. Any proper rationa1 transfer function

iiiatrix is in Lm provirled it hm no iniaginary axis poles.

Figure A. 1 : General LFT framework of p design.

A.2 ' H , Control Theory

Ir'

This set-tio n briefly reviews frequency clor liain niethods for analyzing and syn t hesizing

tlie perforriiance and robust ness properties of ked back systeiiis using the strurture(1

singiilar value ( p ) [XI. The general framework, shown in Fig. A. 1, is baseci on lirirar

fia-tional transforinations (LFTs), jlny Iinear interconnection of inputs, outputs, ancl

c.oriiitiands along with perturbations and a controller can be viewed in this context ancl

r~arranged to match this diagrani. C represents the system interconnection structure.

1 the iincertainties, and Ïi the control law. v is a vector of exogenous inputs and

clisttirbanr.~~. r is a vector of errors to be kept siriall, y is a vector of nieasurenient

sigiials proviclecl to the control design. u is a vector of inputs froni the controI law. z

ancl li are outputs to and froni the uncertainty block.

control design is concer ned with meeting frequency-domain perfortriance

r r i t~ r i a [ l i l . The Hardy Space a, consists of all coniplex-valued function F(s) of a

c.oiripl~x variable s w hich are analytic and bounded i n the open right half plane(rhp).

For real rational functions, F f RH,, the infinity norm is given by

(A . f O)

whwe 3 denotes the largest singular value [9]. In single-input/single-output (SISO)

(-4.10) states that llFlloo is the distance froni the origin to the farthest point on the

Nyqtiist plot of F ( s ) .

Definitions

Fotlowing is a list of ternis usecl extensively in this chapter.

Nominal Stability. The nominal plant riiodel achieves the nominal stability

if it is stabilized by the controller. This is a inininiuni requirement.

Nominal Performance. In addition to noniinal stability, the noniinal closeci-

loop response shoulrl satisfy sorne performance requirenients.

Robust Stability. The cfosed-loop systerri iriust reniain stable for al1 possibl~

plants as clefined by the uncertainty descriptions.

Robust Performance. The closed-loop system must satisfy the performance

reqiiir~trients for al1 possible plants as defined by the uncertainty descriptions.

'Iost modern control design tiiethods only address the probleni of noniinal

stability and noniinal performance. Stability niargins used in classical frequency

cioninin iiiet hocls atteiti pt to acld ress the robrist stabili ty proble~ri, but they niay t-ie

tii isleatling for niultivariable systenis. These niargins neglect the interaction and c-ross

cwtipliiig present i n niultivariable systetiis. One iii~thoci tiiat deais with the rol~iist

p~rforiiianc-F) question, in a ~riultivariable fraiiiework, is p-based analysis and synthesis

te(-fi riiqiics.

A.2.1 Nominal Performance

The X,- norrii appears as a petforniance irieasure in tnany input/output systeiris.

('onsider the performance in ternis of bounds on the output e in the presence of

un(-ertain bounded input v. Bounds for both v and e can be expressed in ternis of

power. energy, or iiiagnitude nornis.

Figure A.2: 3.1, disturbance attenuation problem.

F r ~ y i ~ e n c y dependent weighting functions can be used to shape the spec-

tral c-ontent of signals and perforriiance specifications. Often the physical process

is iiiocleled as coniposed of inputs of bounded power rather than perfectly known

signals of fixetl power spectruiii, leading to the 11 .(lm nom. lModeling of uncertainties

[vith the ( 1 - 1 1 % norni is iriotivated by the types of niodel errors. The benefit of trans-

foriiiing perforiiiance iiieasure into the 11 . I I r n n o m becomes apparent when robustness

aricl performance objectives are included in the control problem formulation [59, 81.

The systeiri shown in Fig. A.2 is a n exaniple of the application of the X m

rioriri to the disturbance attenuation probleni. The transfer function from zf to c is

the sensitivity function. S. Suppose t is any signal and v = Wz. The disturbance

attenuation objective is to niininiize the energy of e for the worst input signal v, or *

equivalently, the R, n o m of the weighted sensitivity function, I(WS((, is to be

tiiiniiiiizecl. In the R, synthesis probleni, P and W would be given and K wouId be

(.fiosen to iiiiniiriize I l WSIIm SU bject to in ter na1 stability of the closed-loop systerii.

Wi thoii t any iincertainty in the niodel, the control design optimizes the nontinal

Figure A.3: Robust stability control probleni formulation.

prr furwtn tm of the systeni.

A.2.2 Robust Stability

The infinity norm, ( 1 .Il,, lends itself to analyzing systems for stability in the presence

of iiricertainty. Uncertainty is often described as norm-bounded variations froiri a

rioriiinal iiiodel. The uncertainty can Vary across frequency, which is reflected in a

~ v ~ i g h ti ng fiinc-tion associatecl with the norrri-boi~nct. Plants described by a noniinal

n rio cl el and a perturbation of the iiiodel are very different froni systenis describecl

Iy a rioniinal triode1 ancl an additive noise process. The difference lies in the tact

tliat plant pertiirbations can clestabilize a noiiiinally stable plant whereas an additive

noise process cnn not destabilize a plant. Consider the block diagram in Fig. A.3. I r i

Fig. A.3. treat A P as a norni-bounded perturbation and d as the disturbance to the

systeiri. Depending on the size of the norin bound, A P can destabilize the systeni P.

It is even more apparent when feedbacli is introdiiced.

Mortels of real systeriis are never exact, because there is always soriie variation

Iietween the physical systetii and the rnatheniatical model. Descri bing the rnoclel by

a no~iiinal plant P and norrii-bounded perturbations or uncertainties, AP, one is

able to clefine a rich class of ~iiodels. This type of description allows the incliision

of ~lestabilizing perturbations, therefore, the issue of robust stability must be ad-

clrpssecl. That is, we desire a set of plant riiodels, defined by P + A P. to be stabilizecl

hy a controller K, in the presence of the perturbation AP-

Fro~ii the physics of the probleni, the perturbation of the plant model, A P. can

boiiricled arross freqiiency by a function W E RH,, such that for al1 w E [O. 30)

(A. 11)

L i ' is a weighting function ciescribina the uncertainty in the tiiodei as a function of

freqiiency. This uncertainty description is unstruct ured, because the only assuniption

iriacle about the riiodel error is that it is magnitude bounded.

The 3C, control niethodology provides a common fratiiework in which one is

ahle to incorporate knowledge of the rnodeling limitations, in terms of frequency re-

.;potis~ data. into the control analysis and design probietii. .4 shortconiing of the R.,

~iiethodology is that it uses singular value tests for stability and performance ana-

lysis, w hidi often is inappropriate and leads to conservative results. The st ructurecl

sitigiilar value (p) was developed to address this limitation.

A.3 -Analysis Methods

The structured singular value, p , is used as a nieasure of the robustness of systems

to st ructiiretl and unstructured uncertainties. p is a generalization of 3, analysis

iiiethods litiiit~d to unstructu red uncertainty. R, control design and p analysis

iiiethocls are coiiibined to fortii the p - synthesis technique for control design.

p-analysis tnethods are used i n the analysis of systems with structured ancl

iinst ructured uncertainties. For the purpose of analysis, the controller may be thought

of as jiist another systeni cotnponent. The inclusion of the controller into the plant

r~c i i l c~s the diagrarii in Fig. -4. l to that in Fig. A.4. The analysis probleni involves

(I~teriiiining w hether the error e re~iiains in a desired bound for bounded input v and

I>ouricled perturbation A.

Figure -4.4: General analysis franiework.

(; (-an be partitioned so that the input-output riiap froni v to e is expressecl

as tlw following Iinear fractional transforiiiation

w here

The norilinal performance is siniply

This is the transfer function froni u to e with the uncertainty, A, set to zero. Rol)iist

stability for unstrirctured uncertainty (assuniing 8(A) 5 1 known ) depends only on

11,. L i nfortunately, noriii boiinds are inadequate for dealing with robust perforrri-

ance and realistic models of structured plant uncertainty. To handle these questions.

ttip structurecl singular value, p is used. p analyzes linear fractional transformations

whrn 1 has structure. A more coinplete background on p is found in [RI.

Figure A .5: General framework.

We define the structured singular value, a matrix function denoted by p ( - ) .

defirlpcl as follows. Let :LI E Cnxn, S and F be two non-negative integers antl

1.1. . . . . r s : rn 1 . . . . . r n , ~ be positive integers w here

[lefine a farriily A c C'LX'1 of block-diagonal iiiatrices as

A := {diag [61 Ir,. . . . , 6sIr,. Ai. . . . . LF]}

Thiis. a matrix in A is block diagonal wi th S scalar blocks anci F full blocks. Theri

p ( :Cf ) is clefined as

i rn l~ss no 1 E A riiakes I - MA singular. in which case p A ( M ) := 0.

WP riiay interpret the above definition in the following way. Suppose hl €

- Cib . and ïonsider the loop shown in Figure A.5. Then p ( M ) - l is a measure of

the sitiallest s t ructii retl p ~ r t i i rbation. A, that çaiises instability of the feed back loop.

()bvioiisly, p is a function of M , which depends on the structure of A. The iniportance

o f p in stutlying robustness of feedback systerii is due ta two theoreins that charactwize

i r i terirts of p. the robust stability and robust performance of a system in the presence

of struc.turec1 tincertainty.

Theorem [XI: Robust Stabiiity,

F U f C ; . _ 1 ) i ~ ~ t a b l e V A E A i f f ~ u p , ~ ( G t l ( j ~ ) ) < 1, O < W < O C

This r ~ d u c ~ s to sup, a(<;, ( j i ; ) ) 5 1 when A is a full block, unstructiired

tiric-ertainty. Henr-e the interçonnection with the 31, norni. When A ha . structure.

the Il .If ,provides an upper bound w hich is a more conservative rneasure of robustness.

For control design, one is really interested in robust perfornzarrce. That is.

whievirig the perforriiance required in the presence of uncertainty. We will charac-

terize perfortiiance in terrris of the ll.[l, of the transfer function frorri disturbance

( 1 7 ) to error ( e ) i n Fig. A.5. The robust performance question can be formulatecl as

a robust stability question, by associating a full block uncertainty, Ak+, , with the

perfornianc~ n o m . Ak+, is of size(nun1ber of disturbances v ) , by (number of error

ou tputs c ) . T hus, robust performance is equivalent to robust stability with respect

t o a d i f f ~ r p t i t Mock structure. Forrrially statecl:

Theorem[X]: Robust Performance.

F,, ((;. A ) is stable and II Fu (Ci, A) Ilm < f VA E A i ff su p, p ( G ( ju) ) 5 1 0 5

< x. w h r e p is conlputed with respect to the structure A = {diag(A.Akf1)) .

Ak+, is the perfortrm,nce block and A E A.

A.4 31, Synthesis

For the purpose of syn thesis, the ~tiaxiiriuiii sirigiilar value of the perturbation A cari

be assuniecl to be bounded in magnitude by 1. This results in the synthesis problerii

pr~sentecl in Fig. A.6. Hence, the synthesis problem involves finding a stabilizing

c-ont roller Il' such that the performance requirenient are satisfied with the inclusion of

Figure .4.6: General frariiework.

uncertainties. The interconnection structure P is partitioned so that the input-output

iiiap froiii IJ' to É is expressed as the following Iinear fractional transforniation .

For a 'H., optinlal control probleni. the objective is to find a stabilizing controller

11;. whic-h ~nininiizes I ( f i (P , h.)lloo.

The U,, optiniization problerii has been the subject of an enornious anioiint

of r~searc-li in the past 10 years. New state-space forniulas have becorrie availabk.

iv hich iiialie this probleiii ~iuirierically tractable [9. 131. These algorithms involve a

searc-h iiiethocl and the solution of two Ricatti equations. For control design. a value 2

is selected and checked as to whether a controller. Ii. can be generated which satisfies

I I Fr (P. Ii))l, , < y and the closed-loop systein is internally stable. If either of these

tests Fitils. the y value is increased ancl control design is reforniulated. In the liriiit as

7 + .r. the control law approaches the 11.1(2 optitnal solution. Assuniing the weighting

fiinçtions have been selected to noriiialize the desired ]).II, to 1, then, IJFl(P, h)ll, < 1 . inclicat~s the foriiitilation of a control law [< that satisfies the specified criteria. The

'a,, ron trol design tiiethods are lisecl in the p-synthesis design tnethodology.

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