1

,

Adaptive Damping Control for Robotic Teleoperation

MiLhel Pelletier

B Ing

Department of Electrical Engmeering

McGill University

Montreal

A thesis submitted to the Faculty of Graduate Studies and Researcn

ln partIal fulfillment of the requircments for the degree of

Master of Engineering

July 1989

© Michel Pelletier

l

l

Abstract

This thesls mvestlgates the design and Implementation of a compilant control

scheme based on dampmg control, that faClhtates position and force control ln robotle tele­

operation The dynamlc performance of such il system ln constratned situations IS very

dependent on the environ ment parameters e g thf' force response at low stlffnesses IS slug­

glsh, wh,le hlgh stlffnesses glve mc to bounîlng and Instabdlty An adJptlve controller based

on the model reference approach (MRAC) 15 developed and analyzed for a single-axIs maOlp­

ulator model, and 15 further extended to the multl-aXls case uSlng the concept of dlrectlOnal

adaptation I-he MRAC <lnd the compllance are active only ln the direction of the force con­

stralnts, thus allowlng near-perfect veloclty and force tracklng ln ail directions Simulation

results show that the adaptlve scheme can slgmflcantly Improve the performance ln force

tracklng and enhance 5tablltty at high stlffneS5es, by rendertng the behavlor Independent of

the envlronment parameters

1\

Résumé

La présente thèse porte sur le développement d'un con:l,tolleur d'accomodatlon

(dampmg control) qUi facilite le contrôle de position et de force pour [j'n robot commandé en

téléoperatlOn Les performances dynamiques d'un tel controlleur sont ',rès dépendantes des

paramètres de l'enVironnement l,) réponse er, force est lente pour une ng .. dlté faible, et oscil­

latOire et/ou Instable pour une ngldlté élevée Un controlleur adaptatif b1Sé sur un modèle

de référence est développé pour un modèle de manipulateur à un seul ax,,,. et étendu pour

l'utilisation à plUSieurs axes gr<Îce au concept d'adaptation dlrectlonelle L accomodatlon à

la force est active seulement dans la dtrectlon de la force normale, permetL3nt ainsI un ex­

cellent SUIVI de la vitesse et de la force dans toute,> les directions Les résultats de Simulation

montrent comment le controlleur adaptatif peut améliorer les performances et la stabilité

du controlleur traditionnel, en rendant la réponse en force Indépendante des paramètres de

l'environnement

'"

1

1

Ack nowiedgeillents

The author wishes to express hls gratitude fHst of ail, ta hls thesls supervlsor

Laeeque Oaneshmend for hls support and the great mterest he has shawn for thls proJect

from the be~lnnlng 1 \'\Iould also Ilke to thank Dean Pierre Bél.:lnger for Ills understandlng

and encouragements. Jnd my supenors at CAE Electronlcs for thclr ~upport <Jnd for glvlng

me the opportunlty to work on the force-feedback proJcct /\ very special thanks goes to

Yvan Lagacé of dept 24 for his comprehension and support throughout thl5 proJect

Je tiens aussI à remercier tous les gens du groupe de robotique de l'IREQ,

spéCialement Pierre Girard, Martin Boyer et George Osmolskl, pour m'avoir rendu la tâche

de rédaction SI bcile J'JlmeralS aussi remercier mon anw~ FranCIne pour sa patience et son

soutien moral lors de la rédaction, ainSI que mes parents pour leur soutien constant durant

toutes mes études sans lequel, Je ne serais Jamais rendu ICI

1he flnanclal support for thls work was provlded by the National Science and

Englneerrng Research (ouncr! of Canada. through a graduate studles scholarshlP, and by

CAE Electronlcs Ltd of Ville St-Laurent

iv

C

List of Figures . ..

Nomenclature ..

Chapter 1 Introduction

Contents

1.1 Control of Teleoperated Robots.

1 1 1 Bilateral Force-Reflexion

Contents

VIII

1

4

5

5

1.1.2 Position and Rùte-Resolved Control Modes.. ............. ...... 6

1.2 (omphance and Force Control . . . . . ~ .............. 7

1.2.1 Compilant Control Schemes .. . . . . . ......... , .. 7

122 Compilant Control ln TeleoperatlOn .. •••••••••• t. 8

1 2.3 Force Control ln Constralned Situations . , ..... " ., ........ 9

1.24 State-of-the-Art ln Comphance and Force Control ... , ....... 10

1.3 An Adaptlve Dampmg Controller for Teleoperatlon ............. 11

1.4 Thesls Overvlew . .. . . . . . ...... . . ... 11

Chapter 2 The Damping Control Scheme ........................ 13

2.1 General Description

2.2 Behavlor ln Contact

13

. ... .. ....................... 15

2.2.1 Relation to a PI Force Controller ... ......................... 15

2.2.2 Stabliity . ........................... 16

2 2.3 Step Response .. .......................... 18

2.3 Why use Adaptlve Control7 . ....... . . ......... ... 22

Chapter 3 Adaptive Control with M RAC ..... . . . . . . . . . . . . . . . . . .. 23

3.1 The MRAC Scheme . . . .. .................... 23

3.2 Dampmg Control wlth MRAC .......... .......................... 28

v

1

1

Contents

3.3 Implementation Concerns . ...... ............... 33

3.3.1 Range of Use of the MRAC Controller ..................... 34

3.3 2 Estlmate Llfnlts ......... ........... 34

3.3.3 EstimatIOn Aigonthm . . . ................ .. . . . . . 36

3.34 Cholee of the Iteration Rate .... , .............. 36

33.5 Cholee of the Reference Model ....... '" ....... 37

3 3 6 Cholee of the Inltlill Estlmates . . .. .. ..... 37

Chapter 4 Mliiti-axis Adaptive Damping Control 39

39 4 1 Dampmg Controller fv10dcl

42 The OlreetlonJI MRI\C Hie.) ........ . .. 44

43 Adaptlve (ontroller Model . ... . . . . . . .... . . . . . . ... . . . . .. 45

44 ImplementatIOn Concerns . .... ................. ......... 49

4.5 Wall Orientation Estimation

Chapter 5 Simulation Tests

50

52

52

52

53

54

5.1 1 AXIS Slnlulatlon

5.11 Model

5.1.2 Results

5.1.3 DIScussion.

5.2 3 Axes Simulation.

5.2.1 Model

5.2.2 Results

5 2.3 DISCUSSion

'" •• ,. o •••••••••••••••••••• 1 •••• , •

' ................................. , ........ .

•••••••••••••••••••••• 1 ••••• 'O ............ .

•••• 1 •••••••••••••••••••• o ••••••••••••

57

57

59

74

5.3 Preltmmary Experimentai Tests. . . . . . . . . . . . . . . . . . . .. . . . . . . . ... . . . . .. 76

Chapter fi Conclusion .........................................

6.1 ContributIons • • • • • 1 • • .. .. • ~ • • • • • • • • • • • • • • • • • • • • • • • • .. • • • • • • • • • ..

77

77

VI

Contents

6.2 Future Work ...........•. ,...................................... 78

References . ... .... ~ .. ~ ~ ...... , , f • f •••• 1 l , r , , ••• f • t • t ••••• ~ ...... t ••• of ••• • , 80

c

vii

a

List of Figures

2.1 Single-axIs damping controller

2.2 PI force cO:1troller black dlagram

2.3 P controller root locus

2.4 1 controller root locus

2.5 P controller force step response

2.6 1 controller force step response

2.7 Step rf'sponse behavlor of the damplng controller

3 1 MRAC loslde the dampmg controller

4 1 Multl-aXls damplng controller

42 Multl-axls adaptlve damplng controller

5.1 Force response at low stlffness

5.2 Force response at medium stlffness

5.3 Force response at hlgh stlffness .

5.4 Response wlth MRAC at low stlffness

5.5 Response wlth M RAC at hlgh stlffness

56 Robot and environ ment Initiai configuration

57 Force response at low strffness (100Hz. no Iv1RAC)

5.8 Force response at medium stlffness (100Hz. no [,.t1RAC) ..

List of Figures

14

15

17

17

19

19

21

30

40

46

54

55

55

56

56

58

63

63

5.9 Force response at hlgh stlffness (100Hz, no MRAC) ......... '......... 64

VIII

1

List of Figures

5.10 Force response at low stiffness (100Hz. with MRAC) 64

5 11 Normal position response at low stlffness (100Hz, wlth MRAC) 65

5.12 Position response ln " at low stlffness (100Hz. wlth MRAC) 65

5.13 Response wlth RLS estimation (100Hz, wlth MRAC) 66

514 Parameter convergence for RLS estimatIon (100Hz. wlth MRAC) ....... 66

5.15 Response wlth CG estimation (100Hz. wlth MRAC) 67

5.16 Parameter convergence for CG estImatIon (100Hz, wlth MRAC). .. .. . . .. 67

5.17 Response wlth 20% jOlOt friction (100Hz. wlth MRAC)

518 Force response for hlghest stlffness (1000Hz. no MRAC)

5.19 Force response wlth MRAC (1000Hz)

520 Force response wlth C) polynomial (1000Hz. wlth MRAC)

5.21 Force response wlth dynamlc compensation (1000Hz, wlth

68

68

69

69

MRAC) ............ 70

5.22 Force response at hlghest stlffness (1000Hz, wlth M RAC and

comp) . . . . .. ....... 70

5.23 Force response wlth tangentlal motion (1000Hz, no MRAC) . . . . . . . . . . .. 71

524 Tangentlal position demand (lOOOHz, no MRAC) · .... " ..... 71

5 25 Force response at Impact (1000Hz. no MRAC) · ... ....... 72

526 Force response at Impact (1000Hz. wlth MRAC) · . . . . . . .. . . ~ . 72

5.27 Force response wlth dlr estlm (1000Hz, wlth MRAC. 45° wall) ......... 73

5.28 Tangentlal position response (1000Hz. wlth MRAC, 45° wall) ........... 73

IX

Nomenclature

Nomenclature

(Ii, bj Plant model parameters

.1.(:: ) Denomlnator of plant model

h Motor damplng (.Y ,)

il Damplng term ln Impedance model

il{::) Numerator of plant model

c(t) Force controller step response (S)

Cl Plant model numerator

('2 Regulation dynamlcs polynomial

Co,C~ Robot cOriolis and centnfugal terms

dr Damplng controller model bJas term

D7,D:: Plarll model denomlnators

~ Jf::1 Environment friction forces (tangential directions) (N)

'il' F,F., Contact force ln task frame (J"n

Fo Nominal force Input (X) 1 Fez Environment force feedback ln task space (N)

FeO Environment force feedback ln world frame (N)

Fuz Disturbance and JOint fnctlon forces ln task frame (N)

Fez Controller forces ln task frame (N)

Fk Adaptation gain matrlx

Y/oad Loau and Wrlst welght (X)

90,9;; Robot gravit y terms (Includlng payload) (N)

Gp Plant transfer functlon

Gzoh Zero order !laid Laplace transfer functlon

Ii Robot links moments of Inertla (Kg· m,2)

.l Motor and link lOertla for single-axIs model (1\g' m 2)

.lOO Manlpulator Jacoblan miltnx

... .lOz Manlpulator Jacoblan matnx ln task frame

~ .lOdz Manlpulator Jacoblan of deSlred positions ln task frame

1

1\

Ka

Ke

1\,

K" Ki JO; .

3

1\]1

Kv

Ka

[. t

mi

ml

1 M~,AI:::

p r .. ri

n,s T

Ta:

u

u M

uf

V(k)

X,X

1

Stiffness term in impedance model

Actuators armature constants (i\" . rnlV)

Envi ronment stlffness ( SI ln )

Force controller proportlonal gain

Position servas veloclty feedforward gains (V . s Irad)

Force controller Integral gain

Damplng term (X .. ' / m)

Position servas position errer gaIns (F/I'ad)

Position servas veloClty gain (1 r ,1 rad)

Discrete-tlme plant model gain

Robot links lengths (1/1)

Robot links positions of center of gravit y (711)

Robot links masses (l':y)

Load and WfiSt mass (I\ y)

Nomenclature

Robot Inertla matnx ln JOInt and task coardinates (Kg· m,2, Kg)

Estimated parameter vector

Robot IInk length for single aXIs model (m)

Plant model unknown parameters

Plant model polynomlals

Sampllng penod (.,)

Tra nsforrnatlon matnx from world to task frames

Plant Input

Model Input

Wall Coulomb friction coefficient

Lyapunov stabtlity candidate functlon

Carteslan position and velocity of robot end-point

in world frame (1/1, Iii J.,)

Desi red Cartesla n position (117)

Velocity demand (/11;'~)

Nominal veloclty Input (mis)

2

t

",

1t"

y

yM

zd, zd

ze

zi . ;.o. -1

fi f(k)

€*(k)

(

0,8, ë ae

ad À 1, ;\2

/\

"00

"Oz

rc

Tf

rt!

<P

W

wd

Wn

Plant output

Model output

Desired position and veloclty in task frame (m, ml8')

Environment equllibrium positIOn ln task frame (m)

Robot posItion ln task frame coordmates (m)

Robot velocity ln task fra me coordi nates (In j s)

Discrete-tlme plant model parameter

Plant model error

Filtered plant model error

System dampmg factor

Nomenclature

Joint positions, velocltles and accelerations (l'ad, 7'ad/8, md182)

Envlronment equdlbrium position in joint frame (7'ad)

Desired JOint positions (J'url)

Estlmator convergence parameters

System tlme constant (." -1 )

Manlpulator forward kmematlcs operator

Mampulator forward klnematlcs in task frame

Control torques at output shaft (N . ln)

JOint torques created by contact with the enVironment (N . m)

Dlsturbance and Jomt fnctlon torques (.Y . III)

RegreSSion (measurement) vector

System frequency (l'fUij...,)

System resonant frequency (/'wlj.s)

System natural frequency (radis)

G'''' C'* }"* ;". 1"· j". J". J"* J"* T::, ::' \ 1" \ J J' \ 1'0' \ 1l' \ 1) l' \ pd' \ pc

Linear gains from multl-aXls modellineanzatlon

3

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1

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Chapter 1 Introduction

Teleoperation is an important subdivision of robotlcs ln whlch manlpulators are

controlled remotely by human operators. Teleoperated master-slave systems orlglnate from

the very beginnmg of lobotlcs, where they were used ln the nuc\ear Industry to manipulate

dangerous substances behrnd protectlve walls. Today, telemanlpulators are belng considered

for use ln many new applications such as assembly of space structures or maintenance of

hlgh-voltage power lines

It IS clear that the actual state of the technology does not allow for a {ully au­

tonomous solution to these complex manipulations Much work remalns to be done ln the

under5tandtng of tasks and the Integration of sensory Information ln intelligent controllers

For many applications, teleoperatlon 15 the only Viable solution, It closes the gap between

manual and fully autonomous manipulations Furthermore, the study of sophlstlcated teler­

obotic control schemes can provlde InSlght5 which aid ln understandlng tasks and how they

should be automated.

This thesls focuses on the deSign of a new control 5cheme for robot manrpulators

whlch 15 sUitable for use ln teleoperation This controller will support the control of positions

and forces ln Carteslan space, and should allow easy rnteractlon wlth an external envlronment

to perform complex manipulations The lnltlal motivation for thls research arose from the

author's work on teleoperatlon at CAE Electronlcs Ltd This flrm developed the three

degrees-of-freedom hand controller for the Shuttle Remote Manlpulator System (Canadarm),

and was rnterested ln Implementrng force feedback in the system to expand the control

1

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l 1 Control of T eleoperated Robots

posslbilities. The flrst step in the desIgn process will' be ta revlew the prevlous work and

state-of-the-art ln teleoperatlon and compilant control

1.1 Control of Teleoperated Robots

1.1.1 Bilateral Force-Reflexion

The flrst and most natural method of controlling posItion and force ln teleoper­

ation IS bilateral force reflexlon, where the Inputs are Interpreted as positIon demands and

the forces felt at the robot end pOint are reflected ta the user through the master robot or

the force-reflectlng Joystlck This glves the operator a great feeling of what 15 happenrng

at the remote sIte, It actually feel5 Itke he IS holdmg the end-effector ln hls hands When

correctly Implemented, thl5 method can allow a task to be completed two to tE"n tlmes

faster than wlthout the force reflexlon [Shendan78]. and 15 generally agreed ta be the best

method of controllmg a teleoperated robot Force reflexlon, however, IS not always possIble

It requlres a lot of space and power ta actuate the Joystlck or masteT robot, whlch may

not be avatlable ln certain applicatIons Furthermore. the behavlor and performance of a

force-reflectlng system 15 very dependent on the tlme delay between the Joystlck and remote

site

Ferrell [Ferrell66} studled thls problem extenslvely and showed that when tlme

delays are Introduced ln a teleoperatlon control system, the operator adopts a move-and-walt

strategy ta complete th. tasks and when force feedback 15 present, the lag ln the Joystlck

response can cause the manlpulator to go unstable Direct vlsual feedback of the remote

site or the use of predlctor dlsplays [Shendan86] can help the human operator to aVOId the

unstable movements The Instabllity problem ln force reflexlon IS also very dependent on

the Impedances of the environ ment and the human operator [Shendan86} [HJnnaford88j

The two-port network model representatlon [RaJu89], the forward-flow archItecture [Han­

naford89]. and the passIve Hilbert networks [Anderson89] were used rec.ently ta descrrbe tt- ..

III put-out put relations between the effort (force) and flow (veloclty) va na bles ln the pres­

ence of tlme delays ln the transmiSSion These representatlons have slgnlflcantly Increased

1 1 Control of Teleoperated Robots

the understanding of force-reflectlng systems and have allowed the development of many

Improvements and new control laws that can assure stabllity of the control The stablhty

problem can now be solved, however the tlme delay IS always detnmental to the system

performance as It constralns the cholce of Impedances As the tlme delay Increases. the

system response will become slower and will feel slugglsh to the operator, thus Increaslng

the task completlon tlmes

1.1.2 Position and Rate-Resolved Control Modes

Bilateral force reflexlon requlres that the inputs from the Joystick must be Inter­

preted as position demands. Jnd that the Joystlck must be active to reslst human Inputs

When force reflexlon IS not Implemented, there 15 a greater flexlbdlty ln the choice of the

Input devlce, Joystlck or master robot, and type of Input Smaller passive Joystlcks whlch

requlre much less space and power than force reflectlng devlces can be used, and the in­

puts can be Interpreted as position, rate or force demands. An excellent survey of the

dlfferent hand-controller configurations and control modes for teleoperatlon can be found

in [Brooks85j

Ta perform position wntrol 10 (arteslan space, It 15 generally agreed that the

best method to use IS the position resolved mode Compared ta the rate resolved mode.

the positIOn resolved method allows the same tasks to be performed 1.3 to 5 tlmes faster,

dependmg on the conditions [Sherldan78] [Klm87] This IS especlally true when the manlp­

ulator workspace IS small or comparable ta the operator's control spac.e The advantage of

position control over rate control dlsappears for large and slow telemanlpulatlon systems

ln order ta obtaln good posltlonal accurJcy from a positIOn re!>olved system,

the Input hand controller must have good pOSitIOn resolutlon ln Its control space and allow

sufflclently large movements If thls IS not the Célse, variable scallng between the Joystlck and

remote site workspaces may be necessary to allow fast gross motions to reach an obJect, and

fine motions to perform precise manipulatIOns near the target ThiS adds some complexlty

for the user whlch IS unnecessary ln the rate resolved mode.

6

1

1 2 (ompllance and Force Control

1.2 Compliance and Force Control

1.2.1 Compliant Control Schemes

Compllance and force control capabdities are an Important part of any robot

controller that has ta deal wlth external forces and interact wlth the envlronment The

bilateral force reflexlon scheme IS one example of such a controller ln which the human

operator closes the feedback loop ln recent years, many new control schemes th3t provlde

compllance and force control have been developed An excellent survey of these methods

can be found ln [Whltney87} The most Interestlng ones for teleoperatlon applications are

those that can provlde both position and force control for free and constralned motion

These are stlffness control, dampmg control, Impedance control and hybnd pOSItion/force

control Compared ta tradition al position or rate control methods. these schemes have force

feedback signais from force or torque sensors that allow the campllance Instead of reslstlng

external forces, they comply to them and apply the deslred amount of force

The hybTld control method IS 'luite dlfferent from the others It uses expllcit force

control 10 the constralned directions, and Simple position control for the others [Ralbert81].

ThiS allows easy monitoring of the force and posItion wlth tradltlonal servo controllers, Ilke

PlO controllers for example The Implementation of tl1l5 scheme usually requlres exphclt

knowledge of the envlronment constramts to declde ln wl1lCh dlrectlOn(s) to apply force and

position control T yplcally, force IS controlled ln the directions normal to the constramt

surfaces, whde tangentlal motion IS controlled ln position To achleve thls, the hybnd

controller has two separate control lcops whlCh contam the pOSitIOn and force controllers

that accept position and force Inputs. The force feedbJck 15 performed only ln the force

control1eù directions

The other methods, stlffness control, damplng control and Impedance control.

use a dlfferent approach The Input IS always position and/or rate When contact forces

arise from the envlronment. the force feedback IS used ta modlfy the Input demand , thus

7

t

,

l 2 Compllance and Force Control

creating the deslred Impedance [Kazeroom86j [Hogan85j. ln fact, stlffness control and

damping control can be seen as partlcular reallzatlons of Impedance control [Whltney87]

Ideally, the target (deslred) mechanlca! Impedance determlnes the dynamlcal

behavior of the system when external forces are applled The slmplest example IS stlffness

control [Sahsbury80jln whlch the desned behavlor IS an Ideal sprlng for whlch the st.ffness

can be deflned 5eparately ln ail Carteslan axes ln thl5 case, the Impedance 15 equal to the

deslred spnng constant F ln the general c.ase, the target Impedance can be speclfled by.

!(oS) = .1:,2 + ils + 1\. (1.1 )

An ideal impedance controller will make the dynamlcal system behave acwrding

to:

(1.2)

ln this type of controller, there IS only one controlloop whlch handles both con­

strained and free motion The envlronment constramts do not need to be known ln general,

unless different Impedances are deslred ln dlfferent constralnt directions [Kazeroonl86j The

force feedback IS performed ln ail directions and the system will comply to ail external forces

1.2.2 Compliant Control in Teleoperation

ln teleoperatlon, the mampulator 15 controlled dlrectly from manual Inputs in

an environment that IS ln general consldered unknown From a control pomt of Vlew, thls

imphes that no model of the envlronment IS avallable, and also that the Input signai from the

joystick has to make sense to the user It should be somethlng simple and easlly controllable,

IIke force, position or rate for example

From thls pOint of Vlew, hybnd control should perform weil since It tracks POSI­

tion and force Inputs However slOce the envlronment configuration IS unknown, It is often

dlfflcult ln practlce ta determme the constramt frame and the pOSition and force direc­

tIons. A few sImple on-Ime direction estimation schemes have been used recently [Merlet87]

8

1

1

l 2 Comphance and Force Control

[Kazanzides89] [Yoshikawa89j and have glven good results for low frictIon and hlgh stlffness

conditIons. The effects of high frictIon and error5 ln the dIrection estImatIon on the hybnd

controller have not been Investlgated.

Another problem asso\::lated wlth hybnd control 15 that when the control mode

changes from positIon ta force, or VIce-versa, ln one dIrectIon, or If the constrarnt directIon

changes suddenly, there 15 a sWltchlng between the force and position control loops. This

may create unwanted tran51ent5 ln the response whlch do not make sense for the user since

they do not correspond ta any Input. ThIs sWltchlng may occur at any tlme and 15 hard

ta predlct and therefore. "bumpless transfer" schemes need ta be developed ta smooth the

t ra nSltlons

The other control schemes based on Impedance control seem eaSH.'r ta Implement

for control ln teleoperatlon It 15 unnecessary ta have knowledge about the environ ment ,:lnd

there IS no control 5wltchlng For the user, the system will react accordlng ta the deslred

Impedance whlch can be a stlffness, a damper, an mertla, or a combmatlon of these These

methods deal wlth ail the Intermedlate cases between free space motion and pure force

control on a ngld surface. For example, wh en pushlng a slldlng abject wlth hlgh fnctlon, it

15 unneces5ary ta declde If the position or force mode should be used as ln the hybnd control

case

1.2.3 Force Control in Constrained Situations

The behavlor of ail compilant and force control schemes when ln Interaction with

an external envlronment IS dependent on bath the manlpulator and envlronrnent dynamlcs,

as these two systems become coupled through the contact surface Many authors have rn­

vestlgated the problem of force control ln constréllned situations Results show that stabrllty

and performance are very dependent on the stlffnesses of the emllronment, the force sensor,

and the robot Itself [Epprnger861IAn87] [Epprnger87] [Kazeroom88] As the effective ex­

ternal stlffness (force sensor + envlronment) Increases, the dynam,c response becomes less

and less damped and wlth very hlgh natural frequenCles, eventually becomlng unstable The

9

«

(

1 3 An Adaptlve Dampmg Controller for Teleoperatlon

bandwldth of such systems has to be hmited for the sake of stabrllty by addlng compllance

in the system. This can be done by decreasJng the controller gains, whlch 15 equlvalent to

detumng the controller, or by addlng passive or active compltance ln the wnst [KazeroonI87]

ln the case of paS51ve compltance, thls decreases the posltlonal accuracy of the end-effector

and it may be necessary to compensate the position error actlvely IRoberts85]

1.2.4 State-of-the-Art in Compliance and Force Control

To deal wlth these problems and obtaln acceptable behavlor for dlfferent external

enVlronments, many sophlstlcated control schemes have been presented ln the Iiterature.

Most of these schemes use complex dynamlc compensation or some sort of adaptation to

preserve stabthty and achleve good performances

Yoshlkawa [Y05hlkawa87] [Yoshlkawa88] and Khatlb [Khatlb86] extended the hy­

bnd controlldea to Inc\ude full compensation of the manlpulator dynamlcs and constralnlng

forces A hybrld Impedance controller was developed by Anderson dnd Spong IAnderson87]

in whlch the Impedance 15 monltored ln the force and position controlled directIOns by us­

Ing a complete Inverse model of the manlpulator These two methods bath need expliclt

knowledge of the envlronment constralnts and a complete dynamlc model of the robot

Slotlne [Slotlne87} developed and tested an adaptlve controller ln Carteslan space

that can track position and force trajectones by Includlng in the adaptlve model, both the

robot and environ ment parameters ThiS control 1er needs only Jornt position and veloclty

measurements to perform the adaptation, and assumes that the robot and environ ment

are Infrnltely 5tlff ThiS Idea was further extended ta adaptlve Impedance control [Kelly89j

These approaches do not requlTe exact models of the robot and enVlronment, but slnce the

adaptation IS complex, convergence IS relatlvely long to achleve and requlles more than one

task cycle The scheme IS best sUlted for repetltlve tasks where the robot and environ ment

parameters are unknown but relatlvely flxed rn tlme

10

1

1

1 4 Thesis Overvlew

1.3 An Adaptive Damping Controller for Teleoperation

The work presented here concerns the design of a new compltant control scheme

for the control of robot rnampulators ln teleoperatlon This scheme does not support btlateral

force-reflexlon and uses the resolved-rate mode for Input cornmands The basIc controller

Idea 15 based on Whltney's damplng control scheme [Whltney77] ThiS IS a partlcular case

of irnpedance control where the Impedance IS chosen ta react Ilke a generaltzed damper

F = n(Xo - Y) ( 1.3)

The Input 15 veloclty and the force applled IS proportlonal to the veloClty error ln a con­

~.ralned situation, the veloclty 15 zero ln the direction of the surface normal and the force

applted IS dlrectly proportlonal to the Input at the Joystlck Because of the compltance

ta external forces, th,s method can slmpllfy the planning ot aS5embly and 1 nsertlon tasks

[Lozano-Perez84] and therefore. 15 a good cholce for compltant teleoperatlon

ln arder to deal wlth the stablilty and performance problems assoclated wlth the

control of forces ln constralned Situations, an adaptation mechanlsm will be used whlch

will assure the tunlng of the response accordlng to the deslred behavlOr Unltke the control

schemes mentloned ln the prevlous section. thls controller does not use complex adaptation

or dynamlc models to perform compensation It uses simple unl-dlmenslonal adaptation

to the envlronment and robot parameter5, ln the direction of the external constralnts. ta

tune the force response and render It Independent of the parameters Such an adaptlve

compilant motion controller IS weil sUlted to use ln teleoperatlon, ~Ince It would allow the

human operator to use the sarne Inputs to a klnesthetlc Interface to perform tasks in wldely

dlffertng constralned envlronments

1.4 Thesis Overview

The followlng chapter discusses the baSIC behavlOr and stablhty of a simple

single-axIs damplng controller ln free and constralned motion, and also compares It to

other force control schemes. Chapter three concerns the deSign and Implementation of an

11

t

(

1 4 ThesÎs Overvlew

adaptation scheme based on the model reference approach, which allows automatlc tunlng

of the controller ta achieve good performances under varying conditions This adaptlve

controller 15 then extended to the general multl-axls case ln the next chapter, UStng the

concept of directlonal adaptation The results and discussion of the simulatIon tests for

bath the single and multi-axls cases are presented ln Chapter flve, whtle concludtng remarks

and recommendatlons for future work can be found ln the final chapter

12

----------------............

-

• ..

Chapter 2 The Damping Control Scheme

2.1 General Description

The basIc block dlagram for the damprng control scheme apphed ta manlpulator

control can be found ln IWhltney771 ln thls chapter. the single aXIs Implementation of

thls controller IS Investlgated ta understand the behavlor. and ta analyze the stabillty and

performance The black dlagram of the smgle-axls damplng controller model used ln the

analyses IS shawn ln Fig 2 1

The damptng control 1er consists of a simple position control system that uses

rate Inputs. ta whlch force feedback IS ddded ln free motion, there IS no force feedback and

the robot's position 15 controlled ln resolved rate mode dtrectly from a hand controller. When

the robot hlts a wall. the dlfference between the actual position H and the wall position HI'

glves nse ta a contact force whlch IS fcd back ta modlfy the veloclty Input At equtllbnum,

the steady-state force appiled F on the wall surface will be equal to I\r\Q. where .f\) IS

the damplng term whlch determlnes the cornpltance of the system The motor dynamlcs

are simphfled ta a Simple gain X". whde the robot IS modeled with a second arder system

of Inertla .J and damplng b The closed-Ioop transfer functlon when ln contact (HI' = 0) is

given by:

2 2 Behavior in Contact

(2.1)

where

(2.2)

. Xo

~--~j ~----------------------------------------------~

Force Control 1er Position Servo Robot + Envlronment

Figure 2.1 Single-axIs dampmg controller

When movmg ln free space, the stlffness term becomes nul! and the transfer

function becames

(2.3)

According to this relation, the steady-state position error in free space is zero,

and if the feed-forward term IS set ta

1- _ li + [\-,,!{(, \ fJ - J-

\11 (2.4 )

the position error to an Input ramp will also be null The performance ln free space should

therefore be adequate for accurate position tracklng, espeCially If the position servo gains

Kr and 1\1 are set to allow a hlgh bandwldth wlth a damplng factor ( rv 1 0

•

11.

-.

-

2,2 Behavior in Contact

2.2 Behavior in Contact

2.2.1 Relation ta a PI Force Controller

The behavior of the damptng contro"er when in contact wlth a surface, as given

by equation (2.1), can be studled easlly If It 15 compared ta an explicit PI force controller.

The block diagram of such a contro"er is shawn ln Fig. 2 2. The closed-loop transfer

function is given by:

F(s) _ ____ l_+_(..:...-!t-_~i~) _s _--,-__

FO{s) - (1+1" ) ( J ) ,3 + ( b ) ,2 + ~ , + 1

l\iI\e c~ /\;1\'('" l\i"

(2.5)

Figure 2.2 PI force controller black dlagram

As can be seen from the block dlagrams, the basic structure of the two controllers

is roughly the same and the transfer functlons show some slmdarities. In fact, If it is assumed

that the environment stlffness IS reasonably hlgh compared to the dampmg control 1er gain,

i.e. r 2J(e » 1{flI\.JI! then the two systems become completely identical under the fo"owlng

correspondences'

(2.6)

(2.7)

(2.8)

J J" " 'Ps\

1

1

t

2 2 Behavlor ln Contact

Ta allow easy analysls and aVOId unnecessary long expreSSions, the PI controller

nomenclature IS used ln the followmg diScussions Now that the dampmg controller mode: I~,

equlvalent ta a simple force controller, It cou Id be mterestlng ta study the effect of varyln,~

the gains and addlng a denvatlve term or lead-Iag compensators ln the PI controller, and

posslbly Implement then equlvalent ln the dampmg controller For thls application though,

the fact that the damplng controller IS used also ln free space motion Imposes certain

constralnts on the controller ln unconstramed situations, the Input has to be Interpreted as

a veloClty demand and therefore, the Integral gain should be kept constant ln ail situations to

always have the same veloClty output for a glven Input The proportlonal gain J.: frelates to

the veloclty feed-forward term ln free space motIOn and therefore, should not be hlgher than

the value glven by equatlon (2 4) The etfect of the lead/lag and denvatlve compensators are

not mvestlgated here, although the latter could be Interpreted as acceleratlon feed-forward

ln free spa ce motion

2.2.2 Stability

The baSIC behavlor and stabihty of P and PI force controllers was studied exten­

slvely ln [EpPlnger87], [An87] and [Eppmger86]. uSlng Simple single-aXIS controllN models.

ln contlnuous tlme, If It is assumed that the robot and envlronment are Inflnltely stlff, the

proportlonal controller will remaln stable for ail values of gains ThiS IS nc.- longer true for

1 or PI controllers, or for other compilant control schemes It IS shawn ln [lshlkawa89J that

the stabdlty reglon depends on the environ ment stltfness and the controller gains, even If

a stlff robot model IS assumed The root-locus diagrams for the proportlonal and Integral

force controllers, when Irnk flexlbdlty is Included 10 the model, are shawn ln Flgs. 2.3 and

2.4. 80th controllers eventually become unstable as the loop gain Increases

ln [ChIOU89], the root-ioci for stlffness control and damptng control are plotted

for a two-Itnk robot model wlth Joint and Itnk flexlbility. Although the pole configuratIon

IS more c.omplex ln thls case, the root-ioci diagrams show simllar behavlors as Figs. 2 3

and 2.4. ThiS was expected slnce damplng control 15 equlvalent ta PI force control at high

stlffnesses, and slmtiarly, stlffness control IS equlvalent to a proportional force controller

16

1

1

lO

20

10

J 0 r--

J

·10

·zo

0 .] .2,5

15

10

.5

0

·s

·10

·15 ·8

2 2 Behavlor 10 Contact

x / C--- x

(;

~- x

)(

~ ·20 ·IS ·10 o 5 10 1.5

Real

Figure 2.3 P controller root locus

X

·6 0 2 4 6

Real

Figure 2.4 1 controller root locus

It is interestrng to note that the Integral controller, and dampmg controller,

have an addltlonal pole at the on gin WhlCh 15 marglnally stable Clearly, thls pole has a

destabliizmg effect at low gains, 1 e. at low stiffnesses the response becomes more and more

17

1

1

2 2 Behavlor 10 Contact

oscillatory. The margmal stability condition IS reached only when the stlffneS5 15 zero, I.e

in free space, when the force loop is no longer closed

For discrete tlme force controllers, the stabliity IS also dependent on the samphng

frequency Faster Iteration rates usually allow stabliity at hlgher stlffnesses, but thls effect IS

IImlted The speed of response of the system IS dependent on the Iteration rdte, but IS also

hmlted by the mechanlcal tlme constant of the robot and actuators [Whltney87] A faster

Iteration rate may be useless If the response IS Ilmlted by the Inertla and compliance of thf'

system.

2.2.3 Step Response

The force step responses of the P and 1 controllers for three dlfferent values of the

environ ment stlffness are shown ln Figs. 2 5 and 2 6 The proportlonal controller response

is good for low stlffnesses, although there IS a steady-state error As stlffness Increases, the

natural frequency rncreases and the damplng factor decreases, thus creatlng large overshoots

and nngmg ln the response This scheme IS equlvalent to hlgh gain positron control and

the overall gam, whlch IS related to the environ ment stlffness, has ta be hmlted to preserve

stabllity

The Integral controller behaves qUite differently For low values of sttffness,

the response IS very slow and underdamped, due to the dominant pole at the ongln For

hrgh values of the stlffness the response IS rnuch faster and the overshoot IS completely

elrminated The response IS much better tha n the P controller, the addltlonal pole clearly

has a stablltzrng effect at hlgh stlffnesses, as noted ln [An87j

These step responses glve good Insights on how PI force controllers behave at

dlfferent stlffnesses It would be useful to have a general expression that clearly shows how

the behavlor 15 affected by the other parameters as weil A good apprOXimatIon of thls

behavlor can be found by decomposlng the thlrd arder denommator of equatlon (2.5) Into

two slmpler terms. The approximate transfer functlon IS glven by.

18

1

1

} '2 ~ehav.or 10 ( ontact

~-------------------------------------------------

08

06

04

18

16

14

Il

08

06

04

01 ,

Ke=lef Ke=lef

- Ke=lfP

0.2 04 06 08 12 14 16

Figure 2.5 P controller force step response

. '. " .................... ...

- Ke = 1 çz?

18 2

0' 0 02

Ke=l02 i Ke=lC2f _J

:'-"""':'-:---"----'--~-~-~~.- ~ -_. 04 06 OS 12 14 16 1 Il

Fi gu re 2.6 1 con troller force step response

!(.<;L Fb{·" )

2

(2.10)

lY

t

,

2 2 8ehavior in Contact

where

(2.11)

(2.12)

The decomposltlon will be valld If the followlng Inequallty holds

(2.13)

This will be true if !{p is hlgh, whlch was already assumed, or If J{ f IS hlgh. With equation

(2.10), the analysis of the system behavlor tS ~tralghtforward The effect of the envlronment

and controller parameters on the system's natural frequency, dampmg and tlme constants

can be seen ln the decamposed denomtnator Usmg the Inverse Laplace transform, the step

response to a force Input can be determmed as a functlon of the system parameters

(2.14)

where

(2.15)

(2.16)

The effects of the flrst and second order terms DI(S) and D2(.s) are clearly

separated in the step response. The effect of Dl ($) is related to the integral term 1\:1 and

is dominant when the proportlonal gain J{f IS low. The response IS charactenzed by a

decaying exponentlal wlth a relatlvely slow tlme constant The second arder term D2(.~) Îs

responslble for the hlgh frequency nnglng Its effect IS dosely related to the proportlonal

gain: as Kf mcreases, the amplitude and frequency of the OSCillation becomes more and

more Important

Fig. 2.7 IIlustrates the shape of the response wlth respect ta the parameters

Wh en companng thls figure to Figs 2.5 and 2.6, the effect of the- approximation becomes

20

1

J 2 Hehavlor ln ( onta,!

_____ ~w l

C (t )

.~, ,,\'" e .' Cos (wt )

••.. \ -~t

e t

'---------------------------- - ---- -

Figure 2.7 Step response behavlor of the damplng controller

visible. According to (2 14 )-(2 16), the stlffness should Influence only tl](' r.nglllg frc'lLJCllcy,

wh Ile It 15 clear that It has a much more Important cffeet, the behJvlor 15 very d,fferent olt

lowand hlgh stlffnesses The approxllnJte decomposltloll should therefore be llsed only for

hlgh stlffness situations

Accordlng ta these results, there appears to be no Ideal <;olutlon for the tunlng

of the controller for a wlde range ot stlffnesses A.t hlgh stlffn('<;.,et, .1 pure Integr<11 1',.1111

performs better and avolds the rlnglllg. but the system becorTw<; VNY ,.Iow .lnd r,llIgglsh .1t

low stlffnesses IncreJslng the proportlOnal gain <;pc€'d<; IIp tlH' respoll,>e but the "y<;tcrn

becomes osctllatory and eventually IInstable at much lower ~tdfn('''5(''' 11)(' l)l'st 'lolutlon

will always be a compromise between speed of rpsponsc and stabdlty at IHglt "tdfnesses One

possIble method of Improvlng the performance IS to .1dd ;}ctlve or p".,<;lve cornpli,lnC(1 III the

system. thlS allows hlgher gains and faster re5ponc;es but decr{,d<;e~ tllf' pOC,ltl(HlJI .Jleur<lcy

These concerns Jpply (le; w(·11 to the d.nnplllg (I)ntrfJllI'r ()f "quatlOll (/ 1) ac­

cordmg to the equlvalences (26) to (20) Ac, nwntlorH'd f',HItf'r. Il)1' ff/'p '.paf(· motion

performance and damplng specdlcdtICJn<, (1\) term) IInpO'>f~ l!llpOlldnt tfm'itr,lInl, on the

chOice of the contraller gains and parameters 0115 rnpanc, that th(· performJf)u' ln contact

)1

(

2 3 Why use Adaptlve Control?

has to be compromlzed even more to allow an acceptable behavior in ail situations

2.3 Why use Adaptive Control?

It IS clear from the above discussions that any non-adaptlve damplng controller

will have very IImlted performancE- specIfications. slugglshness will always be pr~sent and

compliance will have to be added to aVOId IOstablilty at hlgh stlffnesses

For many teleoperatlon applicatIons. thls poor behavlor can be Insufflelent to

allow effiCient replacement of human functlons The Ideal response of such a system should

allow near-perfect posItion tracklng wlth no lag ln free space, and a fast and stable force

response ln contact for a Il possible stlffnesses

The use of an adaptlve controller seems weil sUited for thls applIcation' when

contact IS detected, an adaptation mechanlsm could be started ta adJust the system pa­

rameters to tune the response accordlng ta the stlffness ThiS way, the controller gains can

imtlally be set accordmg to the optimal performance ln free space only, thus allowmg the

perfect poslt.on trackmg The slugglshness ln the force response will be ehmmated by the

adaptlve controller and stabllity will be preserved at hlgher stlffnesses

Such a controller makes the force response IOdependent of the environ ment

parameters The percelved Impedance between the robot and surface IS always the sarne,

whlch allows easy operatIon for the liser at the Joystlck The baSIC equatlons and model of

this adaptlve controller are developed ln the next chapter

22

l

Chapter 3 Adaptive Control with MRAC

3.1 The MRAC Schenle

The maIn purpose of havlIlg adaptatIon ln the controller 15 ta Illlprove the forc/'

tracking capabtlltles, 1 e to render the behJvlOr of tht' ,>ystem Independpl1t of the ('IlVI

ronment parameters fhe model reference control ,1ppro,lch ,>eclm th!' 1)4'~t ~ulh,d for tll1'>

application ThiS method perforrns tr0ckmg of the deslred r~'spons!' ,1C(Qfdln~ to .l referel111~

model, the parameters Jre automatlcJlly Jdj1lst!'d to obt,lIr1 th(' 5m,ille,>t output ('rror 1(1 .. )

Unlrke the self-tumng approach, It 15 unneceSSJry to ')olvl' the linder/Y/fig d('sign probfem

IAstrom83]. 1 e how to adJust the parameters to obt,ltrl thr c!e<>!rt'd !){'havlOr

landau and Lozano [landau81l developed .1 lllllflpd dl5crete tllllC explrClt model

reference ada ptlve controller ( 1'./1 RAC) th a t perforll1s r {'gula tlon lf1 ,I<l<htlon t 0 mode! foilowi n~

for Single-Input sIngle-output (5150) plants This appro,)(h has alrcady bf'f'n slJ(ce~,>flJlly

applted to simple force control .Jlgor/thms [f).1ncshmclld86l!DdllPshrnprl<lH81 .ind Will be lIset!

here to perform the adaptatIOn rhe <ksign 1<, only ,lppllc.abl(· to ITllnlflHHlI phase pl,\I)\s,

smce It relies on cancellatlon of the plant 1/:rOI';5 If) d(hl('V/! pHi'!( t f11(Jd(·1 foll(Jwlnp, fOI

purposes of provmg 'itJbllrty. It Jssumes that tf)(! tllm' dd.Jy and lJpperbollllds cA thl' plant

polynomlals are known

The Irnear 5150 plant to be controlled 1'> r(:pre')~nt('d sn the: domaln by

,

1

• y(l.:) :;-<iB(:::-l) Gp(z) = Il(!.') = .-1(z-l)

where z-<l is a pure time delay of ri sample perlods. and

/1 .-1 ..1(z-l) = 1 + L (!,:;-I

1=1 /lB

B(Z-l) = î-:=b1z- t

!=o (bo ~ 0)

3 1 The MRAC Scheme

(3.1)

(3.2)

(3.3)

It is assumed that B( ;:-1) has ItS zeroes mSlde the unit circle so their cancellatlon

will not lead to an unstable control Input

The tracklng obJec.tive whlch relates the Input and output IS defined with the

equation

y(k) ::;-dD(=-l)

wU ( k) - Cl ( ;:; - 1 ) (3.4)

where '!lM (1.:) is a reference input and Cl and D are the polynomidls describing the deslred

behavior:

ar1

G't(z-l) = 1 + L c: Z-I

1=1 uD

D(z-1) = L d,Z-1

1=0

(3.5)

(3.6)

The regulation design objective Imposes that an initial disturbance y(O) 1- 0

(uA! (k) = 0) is eliminated accordtng to the dynamlcs defined by

(3.7)

24

1 : , , ,

1

J

tU 4 44 «4",**, HIQ' .... WIl

1 1 1 he MHA< ~,hemt'

wnere

(3.8)

is an asymptotlcJlly stable polynomllll

Ideally ('A:: 1) should equJI lInlty. ~Ince thl5 would cnc;ure the I)('.,t p(w,lblp

performance ln regulatlon However . .1s shawn by slmul,Hlcn rpsults III IL,lIHIJu8lj. tilis

polynomial 15 crucial to the perform.1f\(e of the !vlRAL ~chern(' It Slllooth., Ollt tht' fl'<,ponw

of the adaptation mech~nl5m to bath p,HJllleter chJnp;e':> ,111<1 to dl,>tll,b.HIU·<'

The solution to thls problelll. 1 (' ,1 If'ferellfl' Illodd th.lt ~.ltlstHH) both th('

trading Jnd regulatlon objectives. 15 found by 1I5111g the expltctl ret{'fc>nn! mode!

(3.9)

where Il If (/;) and /1.\1 (1. ) are the model output .1nd Input If the pl,ll1t model

error is defined as

(3.10 )

then the two control objectives will hold If the followlng equallon IS ~atlsfled

J .. / 0 (111 )

This can be solved uSlng the Identlty

(3 12)

t

1

where

and

ns S(z-1) == 1 + ).SiZ-i

.:.--J

;=1

llR

R(z-l) == L lï Z - 1

1=0

3.1 The MRAC Scheme

(3.13)

(3.14)

It can be shown that 5(.::-1) and R(=-1) are uniquely defined by (3.12) (see

Appendix A of [Landau81]), with the degree of the polynomlals glven by

Ils=d-l

With this identity, equatlon (3.11) ca n now be wntten as

(3.15)

(3.16)

C2(Z-1)f(k + cl) = (.-1(.:;-1)5(z-1) + =-dR(.::-l») !J(" + ri) - C'2(=-1)y"/(J. + li)

= A(z-l)S(z-l)y(k + li) + R(=-l)V(k) - C'2(.:-1)VM(J.. + d)

where

== B(=-l)S(.:-l )1I(k) + H(=-l )V(!.:) - ('2(=--1 )!I.ll (/." + fi)

= bOue "-) + PÔ 1> 0 {I.: ) - C2(': -1 )v,U (J, + ri)

= pT rf;(k) - C'2(.::-1)U·\!(I,· + cl) (3.17)

rfJ6 (k) = [u( k - 1), ... , tt( k - d - n B + 1), y( k), ... , y( k - n R )] (3.18)

rf;T(k) = [u(k); QÔ(!.:)]

pT = !bo;pôl (3.20)

(3.21)

26

" . . '

1 1 1 ht M~A( ~chemt'

ln these equatlOns, l'l' and <,'J Jre the estlmated p.1r.lIlH'tN and regrt'<;!>lon (/lw.,­

surement) vectors respectlvely Equatn.g the rtght hand side of equiltlon (3 17) 10 lNO. Ihl'

control objectives can be achleved wlth the followlng IIlpllt

( 3.22)

The parameter estImatIon IS performed ll!>lng the genNilllzed parameter ('~tl-

mation algonthm for dlscrete-tlme 5150 plants presented Ifl !L.lnd,1Il81! T 11<' adaptlve

controller adJustment illgonthm IS glven by

(3_ 23)

where the adaptatIon gain matnx is glven by

(3 24)

Fa ,0

with the a posteflori flltered plant-model l'rror glven by

• r(l .. ) ( = 1 + (:/(i-- --:,-) . j.j. 1 ~ l, ~ -(1) (3.25 )

where

/' Il (l, tI) (3.26 )

This scheme IS a generillililtion of the rer lHc,lve least-squares (RLS) estlrn.1tlon

method WhlCh salves il set of recurslve IIrwM f'qu.ltlonc, Wllh the obJcctlvl' of nllnlllllllng thp

squares of the output estimation errorc,

(3 /7)

32 Dampmg Control wlth MRAC

ln the RLS scheme, equal welght 15 glven ta each measurement cP( Ii,), 50 the

adaptation gain matnx Fk has tlme-decreasing gains given by

(3.28)

ln the general case, two parameters ;\1 and /\2 are introduced ta allow flexlbility ln

the estimation scheme The flrst parameter /\1 is a forgettlng factor whlch allows exponential

discountlng of past data, whlle '\2 determines the varratlon of the adaptation gain in tlme.

The most common algorrthms are constant gain adaptation (/\1 = 1, '\2 = 0), recursive

least-squares ('\1 == 1, '\2 = 1), and exponentlally welghted least-squares (r\l < l, /\2 = 1).

Wlth these parameters, equation (3 28) IS rewntten as

(3.29)

The expression of F" (3.24) IS found by using the matrix Inversion lemma on

this relation

3.2 Damping Control with MRAC

The adaptlve scheme developed ln the prevlous section can be applied ta the

single aXIs ddmplOg controller model of Fig 2.1 ln arder to achleve the deslfed performance

Improvements, It IS Important to choose carefully how the MRAC will be used, I.e. what

Input and output wlliit mOnitor and modlfy

Since the contact force 15 the parameter that nas to be controlled wlth the

adaptation, It 15 natural to use It as the plant output. The choice of the Input 15 not 50

obvious The torque sIgnai ta the actuator cannot be u5ed, 5tnCe the position servo uses

posItion and veloclty outputs from the robot The model would not be 5150 The velocity

Input from the user 15 al50 a bad cholce, It 15 beneflclal to have a feedback loop around the

MRAC for robustness and stabtllty

This leaves us wlth three cholces, the velocity demand Sil, the Carteslan posItIon

demand Xd, and the Joint position demand Bd The best one to use IS the velocity demand

28

1

1

{'2 Damplog ( ont roi 'Nlth MIM(

because it is an unblased term, 1 e It IS mdependent of the position of the robot. It will

also be shown further that for the multi-axis Implementation, fiel IS a bad (hOlee bec.llIse It

is not ln the same reference frame JS the force

The MRAC configuration mSlde the dJmplng controller as deswbed above IS

shown in Fig 3 1 Accordlng to Flgs 2 1 Jnd 3.1, the plant model equJtlon .)'i seen by the

MRAC IS glven by'

(3.30)

where

This plant model is 5150 wlth the exception of the constant b,as tl'rm due to

the enVHonment position The effeet of tlm offset can be removcd \I~\I)g data pletrl',)tlTlf'nt

methods JS descnbed ln 1 LJung88J The 1110St n.Jtur.J1 appro.Jch 1'1 \0 dl'tl.'rrmllP t Ill' v,lllIe

of the offset and subtract It from the data entnes ln thls case, tl1l'; V.lllIP 1" IJnkllOWII ,Hlel

It 15 not possIble elther ta deterrnlne Its value from rneasurement me,HlS. '-.11)( P III (onl.1ct

the pOSition IS always greJter ttlan the surface equdlbnurn positron I\t vpry 11Igh "ttffne.,.,(,s

however. the JctuJI pOSition IS very c\oc,e ta the wall pO~ltton ilnd ,f li t l , I~ .J.,.,urncd. the

blas term ln cquatlon (3 30) dlsJppcars If X" .\ IS u!.E'd as the plant Input 1 hls 1<11:.1

worked reasonably weil for very 11Igh ..,tlffnesses wh!'n trlrd ln prelllrllllary t{'~ts. but dc~radcd

rapldly for medium and low sttffnesses. as expectcd

Another possible solution 15 to f'/pllCltly e..,t,mate the I1la'- Lerm .1'- ,\ part of thf~

plant model The estlmated v<lluc tht::n h;ss to be mcluded ln the reff'rcll("(! Illodr.' I!quatton

(3 9) whlch becomes

(3.31 )

ï\)

----------------.........

1

t

3 2 Damping Control wltn MRAC

F

'----t.< J to-----------,-------'

Figure 3.1 M RAC Inslde the dampmg controller

where J f IS the estimated b,as t€'rm When trred ln prellmmary tests, the estlmator did not

converge properly and the eontroller was unstable It seems that the reference output IS

too sensitive to the vanatlons ln ;/~ As seen from equatlon (330), the value of the b,as IS

non-neghglble compared ta the Input term, small errors ln the blas estlmate will therefore

result ln Important vanatlons ln the model output The posItion errors generJted will glve

ri se ta large forces or 1055 of contact, thus cre<1tlng the Instabllity

The method that glves the best results for the b,as term ellmmation IS the use

of a nOise model wlth Integration [LJung88] This scheme IS equivalent to prefdtenng the

data wlth il (1 -- _ -1) fllter, 1 e dtfferencmg the Input and output signais (Iearly, th,s

eliminates the blJS term, but It also has the effeet of pushlng the fit of the of the transfer

functlOns IOta a high-frequency reglon t, whlch can be undes.rable for certaIn applicatIOns

ln thls case, th,s effeet should Improve the performance of the controller at hlgh stlffnesses

(i.e. high natural frequenCles), but may detenorate the low-frequency response e g to slow

varying Inputs.

The effeet of the dlfferencmg fllter IS eqUivalent ta a dlfferentlating term .'l ln the

The fIt is the dlfference between the actual and estlmated transfer functlons as seen on a Bede plot 1 e ln the frequency domaln Pushlng the fit at hlgher frequp.ncles means that the mapptng at hlgh frequency will be Improved, wn,cn usually means that the low-frequency fit will deterlorate For more delads see Chapter 13 of [LJung88]

10

1

,

Laplace domain. Multlplying by 8 bath sides of equatlon (3 30). the pl.lnt model becol1ws

(332)

The constant blas term 15 ehmrnated and the plant IS now SISO It becomes c!eJr

from this equation why .\d was chosen as the plant Input IflsteJd of '\,{I It would h.we been

necessary ta differentlate It anyway

ln the:: domaln, the plant model is glven by

where

(3.33)

(3.34)

(3.35)

and T is the samphng penod Usmg the partial fractions expansIOn and the;; transform

theorems, the dlscrete-tlme plant model can be solved

where

and

(lI = -2e->'TC().~(wT)

(l2 == c-2>·T

bo = Ko (1- (,-XI'('o::.(wT) -,Jr:->''J'SI1I(wT»)

1 1· ( - nT - ,\T fI ( 'l') l ,J ,\) L' ( 'J')) )1=\'0(- -t t-,().~W Tut .7/1/"'-'

(3.36)

(3.37)

(3.38)

(3.39)

(3.40)

31

, 32 Damping Control wlth MRAC

and

Wn = KpJ\a + 1'2 Ke

J

The plant model has the same form as equation (3.1). i.e.

11 B = 1, cl=1

y(z) == (1 - .::-l)F(z) - contact force difference

'U( z) == .-tct( z) == velocity demand

(3.41)

(3.42)

(3.43)

(3.44)

(3.45)

(3.46 )

(3.47)

(3.48)

Similarly, the desned performance of the system can be represented by an exphcit

second-order reference model with the sa me form, 50 that

and

nD = 1, d=1

vM (z) == (1- z -1 )pM (z) = desired force output

ll M(Z):::: .-tj.f(z) == model input (velocity demand)

(3.49)

(3.50)

(3.51)

ln the same manner, the regulation dynamlcs can be deflOed wlth a second-order polynomial:

(3.52)

The order of 5(z-1) and R(z-l) can now be determined usmg (3.15) and (3.16):

32

a

1

t

l j Irnplemt'ntatlon ( onCl'tns

Il.') = Q.

The identity (3.12) can now be easlly solved for R(::: -1) :

Ï'o ;:::- ri - Iq

- ') 1'1 =(2- H 2

(3 53)

(3.54 )

(3.55 )

Hence, the general MRAC scherne for damping control in the presence of varylng

environment and robot parameters, uSlng secol,d-order process, re-ference model. and second­

order regulatlon dynamlcs. IS deftned by

(1) Regression vector

1> T (k) = [Il( k ). 1/ (1.' - 1), y( ' .. ), y( /.. - 1)] (3.56)

(ii) Estimated parameter vector

(3.57)

(iii) Adaptive control law

(3.58)

(iv) Adaptive controller adJustment (parameter estl matlon) algonthm glven by (3.23),

with the adaptation gam, F, belng a 4/4 rnatrrx

(v) Adaptation gain ca\culatlon algortthm glven by (3.24) and (3.25) wlth li ~ 1.

3.3 Implementation COl1cerns

n

1

1

3 3 Implementation (oncerns

3.3.1 Range of Use of the MRAC Controller

The use of the MRAC controller should be supervlsed carefully to avold insta­

bilities and Incoherent responses The MRAC should be active when the robot IS ln contact

with a stlff wall, but inactIve when III free space The behavlor ln the intermedlate cases,

I.e. soft surfaces, IS up ta the designer and depends on the applIcation TYPlcally, If the

non-adaptlve robot hlts a soft wall at a certain Initiai veloclty, It wlilloose speed slowly tlll

it reaches the deslred force (It may very weil overshoot) This behavlor IS slugglsh from a

force control pOint of vlew but reaets accordlng ta the deslred Impedance On the other

hand, If the MRAC IS active, the controller will speed up the force response This Will cause

the robot to Increase ItS speed to reach the deslred force faster

From the user's pOInt of Vlew, the perception of the envlronment's mechanlcal

Impedance IS lost, stnce the rapld acceleratlon ta reach the deslred force was not commanded

at the Joystlck On the other hand, the klnesthetlc feeling of the task, 1 e. the input-output

relation between demand and force, 15 always the same The user can command his Inputs

in the same way tndependently of the environment parametels

For very low stlffnesses, the acceleratlon tnduced by the MRAC may be hlgh and

can be dangerous in free space for example, where the stlffness 15 null, the MRAC would

tncrease the commanded torques Indeflnltely to butlcl up a force whlch Will never occur

The system designer must therefore choose an acceptable level of Input Increase whlch,

wh de Improvmg the force response, does not generate dangerou5 speeds and acceleratlons

This IS do ne by ltmltlng the MRAC Input to the piant 1/ k to an acceptable value. The

most eonservatlve approach woulcl be ta set the Itmlts on liA- 50 that the M RAC can only

decrease Its value, thls way, there 15 no veloclty Increase The perceptIon of the environ ment

Impedance IS preserved at low stlffnesse5 and the MRAC Îs used only for Improvlng stabillty

at very hlgh stlffnesses

3.3.2 Estimate limits

ln order to have robustness ln the estlmator and help the c.nnvergence. the

14

1 {t Implemt'nt oJtlon ( \)I\cern~

estlmator values have to be hmlted accordtng to thclr physlCJI rne.lllillp; The v,\lu{'s ot

1'0 and h are related to the plant denomtnator parameters lit .1Ild Il,' .w .. ordlng to (3 54)

and (355) As seen from equatlons (337) and (338), these pJrameters ~hoLJld veIlly the

followlng Inequahtles

(3.59)

(3.60)

For the numerator terms, It IS clear that ho should be positive smce il positive Input should

glve a positive output, 1 e an tncreastng value ot conta~l force The upper "mit on 1'0 I!>

related ta [\0 and should be determmed usmg the vJlues of the controller p . .uametcrs and

the maximum effective stlffness to be encountered

ln arder ta preserve stabdlty of the MRAC, unstable zeroes tn the plilnt model

should be avolded Therefore the absolute value of hl must be smaller than {'o to stay Inslde

the Unlt ClYcie. ThiS ylelds the foliowlOg relations

(3.61)

(3.62)

The estlmated parameter vector j/' IS th us bounded bya hyper .. cube (' dellrnlted

by the four parameter "mlts The most nJtural method of hmltmg the po;tlmJtes IS Slrnply

to limlt thelr respective values accordlng ta the mJXlmum and minimum values glven Jbove

This IS eqUivalent to orthogonally proJectmg thetr values onto (' ln the space of the parJm­

eters. ThiS scheme IS used ln the simulatIon progrJm ta perform the IlInltln~, bllt ilccordmg

to the estimation scheme (323) and (324), thl,) 15 not compldely ilCCIHJte l3eCJll,>e the

convergence of the estimatIOn Jlgorlthm I~ based on tlte Lydpunov funrtlOn

0.63)

It is necessary to perform the projection ln the ':.pace (' d,storted under the ImeJr tran':> ..

-1/2 formation 1'~ IGoodwtn84j l hls hils no effect when J 1. \':. dugond\' but III the general

case, when the recurslve alJ?;orlthm (3 24) 1<; u,>ed. the d,':.tortlOl1 may affect the conver~en(e

1

3 3 Implementation Concerns

3.3.3 Estimation Aigorithm

The best estimation algonthm for robustness and speed of convergence is usually

the recursive least-squares scheme The drawback of thls scheme IS that It is incapable of

trackrng vanatlons of the system parameters ln tlme ln the present cas", It IS clear that the

plant parameters (st,ffness, tnertla etc) can change ln tlme, this will be especlally true in

the multl-axls case. For thls reason It IS useful ta add a forgettlng factor '\1 = 0.95 rv 0 99

If the Input IS not sufflelently rleh, thls seheme will rapldly bnng the adaptation gains to zero

and prevent the Fk matnx inverSion from bemg performed suceessfully It IS thus necessary

ta perform a test on the values of Fl' ta avold the updatlng If It beeomes slngular or badly

condltloned

The constant gain scherne 15 the slmplest of ail because there 15 no gain adap­

tation. ThiS method responds ta changes faster than the two prevlous methods because of

the high garns, but It 15 also the wor5e from the pOint of vlew of robustne5s and speed of

convergence, as will be seen ln the results

3.3.4 Choice of the Iteration Rate

The cholce of the Iteration rate of the MRAC IS dlrectly related ta the natural

frequency ofthe plant ta be eontrolled, whlch ln thls case 15 related ta pI\e + !\aI";p)/J

Usually, the samplmg frequency should be at least tWlee as fast as the natural frequency

ta avold ahaslng [Franklln80], but for the MRAC scheme, It IS reeommended ta sample flve

times faster than the natural frequency [Goodwln84]

The cholce of a very fast samplmg rate 15 Itmlted by the avarlable computlng

power, but there are also other problems ta conslder

• Llmltlng the sampltng rate Ilmlts the eontroller bandwidth thus preventing unmodeled

high-frequency dynamlcs to be exclted

36

1

y

li

l 3 Implementa lion ( oncerns

• The speed of response of the system IS Ilmlted by the .lctuator b;}ndwlth wlllch 15

dependent on physlcal parameters such as mertl;}, torque II/nits Jnd ':Hlll (ompll;}nce

The use of a faster samphng rate can be totally llseless

• Accordlng to the plant model of equatlons (3 36) to (3 40), a very small '/' moves the

pales and zeroes doser to the boundary of the unit wcle, thus decreasmg robu!>tness

ta Instability

The effect of dlfferent samphng rates IS studled ln the multl-axls simulation and

IS discussed ln Chapter flve

3.3.5 Choice of the Reference Model

The best method to flnd an appropnate reference modells to descnbe the deslred

behavlor wlth a second-order tranc;fer funetlon (natural frequency, dampmg factor etc ), and

determme the correspondlng parameters ln the ~ dom':lIl1 uSlng (3 37)-(340)

A simpleT method eonsists of uSlng the parameters found by the least squares

estlmator dUTIng a test that gave good results, 1 e eopy the behavlOr of the system when It

IS weil tuned

3.3.6 Choice of the Initial Estimates

The choiee of the initiai estlmates can slgnlflcdntly Influence stabdlty and con­

vergence of the results The most Crltlc;ll parameter from thls pOint of vlew IS 1/0 becJuse

It is directly related to the environ ment stlffn('ss It 15 Import<lnt to chaast:' ïl hlgh vdlue

for thls parameter, thls way, the IARAC wdl II1ItlJlly (lct ;l,) If the wall WilS very stlff and

Input small eommands ta the plant ThiS wdl prevent hounClng dnd 10'>5 of contact w~lIch

could cause the IV1RAC to gc unstable r hl":! Initiai v.llup,> for the other p.H<lmeters 'ihould

satlsfy the Ilmlts deflned earller and should correspond ta poles and 7eroes th;)t Jre Inslde

,

t

3 3 Implementation Concerns

the unit circle. In the simulation tests, these values were found by observing the estimated

parameters ln different environments. and then choosmg mtermedlate values

The results of the single-axIs simulation tests are presented ln Chapter five. The

next chapter dlscusses the multi-axis Implementation of the MRAC controller

18

•

1

J

Chapter 4 Multi-axis Adaptive Damping Control

4.1 Oamping Controller Madel

This chapter IS concerned wlth the extension of the present ~tudy of the .1dJptlve

damptng control scheme to the general multl-axls case ThIS step 15 necessary for tills SCheITH."

be useful for real telemantpulatlan operations The generJI black dlagram of th!' IllUlthlXI'i

damptng controller can be found ln [Whltney87]. and 15 .1150 shawn III morr detad ln fig

4.1.

ln the followlng diSCUSSions Jnd tests, pOlllt contact 15 a~sunH'd between the

robot end-effector ;md envlronment surface, 1 e the enVlronnlf'nt (allnot I!,en('rate torqlles

The robot Wrlst IS also consldered frozen 50 that orientation term<; are Ignor('(1 ~In(e pOint

contact 15 assumed . .111 the vectors and matrIces ln the figure Jf(' rec,pf'( tlv('ly ()I dlmrnSlon

3 x 1 and 31-3 The environ ment 15 modeled as J srnooth. I(m (urV.1ture c,urtJU> whlch (Jn

be consldered to be locally plane

The major dlfferences between the single and multl il/I'> ca.,,-,~ ,H(' Introduced by

the non-Itnear klnemJtlc and dynamlc transformations ÎII<,o, tlH' !ciree fl'f'dbJck "I~nal trom

the Wrtst contalns more than the f.'xternal contact forcp f,c., .,hown III tlH' dl,IIJ,rJIl1, the

Wrtst sensor Plcks up the gravit y forces on the grlprrr .Incl p,wlo,l<! 1 tlls dt"ct hac, to be

compensated cornpletelv to avole! the falllnj!, ot the IOJe! h"C<1u"r' ot th,. forr f' j",·db,Hk to

the veloClty demand the controller will try ta cornply ln thl> pdvlo.H! w1'IJ;ht ,IIII! t Iv' robot

----------------.........

III N

4.1 Damping Controller Model

" • .!! Q'I

L 0 • c .. N ~ '1-0 • u L 0 ...

0

u..QI

f-0 CO 0 cr

Figure 4.1 Multi-axis dampmg controller

+' C III E c 0 l

> C w

+ +' 0 .0 0 cr

\11 o > 1.

~ C o

\1\ o

a..

l QI

o l +' C o

U

QI U L o

u...

ru QI E QI E 11) E lU l ro l

u... L u... u....

lJ +' .j[. C

l VI 0 ro 0

:3 l- l

.. ~ NŒ

•

1

,

1\ l Damptng Controller Model

will start to fall downwards tillit reaches the ground To compensate thls ('Hect, the liser

can Input a force upwards trom the joystlck however, thls would slgnlflcantly Increase the

user stress and fatigue

This fallmg problem IS present ln ail compilant and force control methods It

can be solved by compensatlng the force feedback wlth the Jctual or estlrnated value of

the wnst and payload welght Small errors ln the welght value can be overcome wlth the

addition of a deadband ln the feedbJck loop ln ail future diSCUSSions .lnd ln the simulatIOn

program used further, It 15 assumed that thls wnst gravit y signaiis pedectly cornpensated

The mertlal forces sensed at the wrrst caused by the Wrlst and payload welght

are also neglected These terms are related to Jcceleratlon Jnd should have no slgnlhcant

effect on the behavlor, wh en perforrnlng typlcal teleoperatlon tJsks

The only transformations needed for control purposes are the rnampulator Inverse

kinematlcs and the force sensor matrlx that solves the forces ln CarteSIJIl c;pace If It IS

preferred ar computatlonally slmpler. the deSigner can choose ta lise the Inverse Jacablan

matnx Instead of Inverse kmematlcs ln thls case, the Integration terrn 1/., will have to be

performed after the Jacoblan multlpltcatlOn

The multl-aXls dampmg controller behaves baslCally the S,WH> WJy d':> the ~lIlgle

aXIs controller The major dlfferences Jre Introduced by the coupllng of the ,lxes Jnd the

Introduction of the gravit y forces ln the feedback The dynamlcal rf'sponse of the system

can be found usmg Lagrange's dynalllicai equatlon

-- , ( t- T'L t- T, (4 1)

A better understandlng ot the <'yc:,tem dynamlcs IS achlf'vNi by \l'Jing t ~H~ t.lsk

frame formulation The task coardlnJtts Jr" d~flnf>d JS . (J.) l). Whfl[f' thf' J .!YI" cOlllcldes wlth the ~nvlronrnent SlJrfd( ,. norrwl dnd p()lntc:, tow.H(h th,· Jrl'>ld" (Jt tflf' ',)Hl.l' l'

• 4 1 Damping Controller Model

the task frame state variables to the joint space are summanzed here:

. . z = JOze = TozJeoB

ï == iozÉJ + .10::8

Fz = JizT r

(4.2)

(4.3)

(4.4)

( 4.5)

MultiplYlng both sides of equatlon 4.1 by JizT and replacing the joint variables

by their task frame equlv~lents, the dynamics of the manipulator can be expressed directly

in task coordlnates:

where

Mzz + Cz + gz = Fez + Fuz + Fez

Mz == Jo'/1\-10Jo/ C:: = '''0/ { Co(8, 8) - ~u(}Jô"/ iozÉJ}

g:; == .lizTg(B)

The controller force is given by:

(4.6)

(4.7)

(4.8)

(4.9)

The envlronment surface is modeled as a pure stlffness of value 1\(' wlth fnction ln the

tangentlal directions --------------.........

•

1

4 '2 The Dlrectlonal MRAr Id~a

where

(

KI' 0 1\1' = 0 0

o 0 (4.12)

::,J := ( ::~t') (4.13)

(4.l4 )

An explicit expression for the transfer functlon of thls system is dlfflcult ta derive

because of the non-Itnear inverse klnematlcs operator Jnd the coupltng of JII the axes The

basic structure should still be the sarne as equatlon (2 1), wlth the dlfference that ail terms

are multl-dlmenslonal and tlme varylng

The basIc behavlor wtll be the same as for the single-axIs case Recause of the

varymg terms though, the best tunlng will change wlth the robot configuratIon and veloClty

This requlres a worst case tunmg ta assure stabdity for .111 robot r,tates ThiS will requrre

sufflclent damplng in the control and the robot may feel sllJgglsh Jnd slow to the operJtor

One possIble Improvement ta the controllér 15 to use dynalnlc compensation <lnd

feed-forward terms to Improve the decoupling of the .lxes Jnd ellfntn,lte non-Itncéli effects

The computed torque method [Asada861 whlch uses a complete dynJrmc model can be used

to compensate C;;, y;; , and decouple the Inertltl rnJtnx Irl (46) lorque fe(>dback at the

JOint level can also be used ta compen'iJte for the unwanted d,c,tllrb,lIH es F". n1tS would

reduce (4 6) to a simple decoupled system

(4.15 )

ThiS approach should slgnrflt:antly Irnprov(~ th~ pf!rformélncr:, thf! system r,hould

behave as weil JS the 'ilngle-Jl,ls controll('r dr·'/:.rlhf'd ln ( h,q)tl'r two Ac, 11\ tl)(· <'lllr,k ,IXI~

-(

1

4 2 The Dlrectlonal MRAC Idea

4.2 The Oirectional MRAC idea

At flrst glance, It IS not obvlOus how the MRAC scheme should appll(.d ta the

multi-axis case Wh en the robot hlts a wall, the forces are felt ln ail the JOints and the

compllance should also occ.ur ln ail the JOints The flrst Idea that cornes to mlnd would

be to Implement a MRAC .Jt each actuator, thus creatlng adaptlve compllance ln ail JOints

This method 15 very computatlonally demandlng, but thls 15 not Its blggest problem' the

adaptlve scheme developed ln Chapter three IS designed for force control, It expects that

the veloclty Inputs will create force outputs Jccordlng ta a stlffness model ln a robot

jOint, the dynamlcal behavlor does not depend on the stlffness only, there may very weil be

îomponents of the response due ta tJngentlal motion on the wall and re.:octlon forces due

ta fflctlon or 5tlctlOn ThiS Implles that the stlffness model Inslde the M RAC IS no longer

vahd, the controllN wtll try to trJck forces ln ail directions and the veloclty tracklng ln the

unconstralned dIrections Will be lost

To solve thls problem, the concept of dlrectlonal adaptation IS used, 1 e. the

MRAC IS set up ta work ln the SJme directIOn JS the normal contact force ThiS way, the

adaptive controller sees the envlronment as a pure stlffness and will perform tracklng of the

normal contact force, whlch IS what IS deslred ThiS approach ellmlnates the fr/ctlon and

stlctlon forces problem, since these forces Jre not seen by the MRAC The adaptation IS

Independent of what IS happenIng ln the tangentlal directions

ThiS method has another advantage since the MRAC performs the compllance

ln the normal direction, It IS pOSSible ta shut-off the force feedback ln the other directions.

thus allowJng near-perfect velo cIty trackmg ln the tangentlal directIOns Instead of complymg

ta friction and stlctlon forces, the controller wdl see them as dlsturbJnces TI1IS resembles

much more what a hybnd force controller would do, rather than a damplng controller,

It performs position and force trackmg rather thJn systematlcally comr1ylng to ail forces

encountered

ThiS approach has the same major drawback as hybnd control' It needs knowl­

edge about the surface normal onentatlon. Since the system IS designed for teleoperatJon,

1 1\ '1 l\dJptlve ( ootrollN Model

the envlronment IS unknown and It IS necessary to estlmate the wall orlentat,on (see sectIon

4.5) This IS not a very dlfhcult probletn and ln thls CJse, the accur,ley IS not too ultlcal

The actIon of the MRAC 15 Just h) modlfy the Inputs ta the plant ln one dltectlOn to .,dlleve

a better performanc.e, so the basIc structure of the ad.lptlve controller I!) the same ,15 the

non-adaptlve one, damplng control IS perforrned ln <111 dilections 1 he eHect of an error ,n

the direction estimation should be less harmful than III hybfld control where posItIon Jnd

force control are 111 completely separate controlloops ln the estllnated norm<11 duectlOn, the

MRAC can compensate for small components of the force due to the tJngentl.11 directions

This new approach should therefore work reasonably weil even for ,Ill IInperfect surface

mode\. The method IS also computatlonally simple slI1ce It hJS only one SISO MRAC

The block dlagram of the dlrectlOnal MRAC ImplementatIOn Idea IS shown ln

FIg 42 The l'o~ matnx IS the estlmated world ta task trames transformation matflx

As seen on the figure, the MRAC 15 apphed only ln the :"1 directIon whlch cOlllCldes wlth

the surface normal The structure of the damplng controller 15 mamtallled and the sIIl1pler

non-adaptlve scheme of Fig 4 1 can be obtamed simply by 5wltchlng off the MRAC

4.3 Adaptive Controller Madel

ln order to develop the adaptlve controller equatlol1s and ta determme the MRAC

plant mode\. It IS assumed that the wall Orientation, Jnd thu5 the worlel ta task transforma­

tion matnx, IS known The ta5k frame dynélmlcal equZltlOn le; used .15 J startmg pomt.

(4.6 )

The controller torques ln task space can be wntten as

}~ ,-l'J" J" /\ -l( ), (':: = . {}:; \ 1/ \" li _ :,{ t

, _. j' /" /- /\ -1 ( ) - 'fI- \,/ \" fi -. - (4.16 )

These equatlOns can be IlI1eanzed élfound an equdlbnum position defll1ed by

- = zOo :; = 0, zr! = ':dO and z,( --;:. 0, correspondlng to the JOint positions Ho, H"O' ùnd

--------...... ------------------~--------------~

::!! crq

= ~

l'D J::a

IV

:::: 0:..

t.J X ln

'"' 0-eu

-0

<: ... 0-llJ

3 -0

:::J ()q

()

0 ::l

2. ~

;. 0'

X 0

- . .

1 oz

0: Z: 8: f:

oz

t,.Jar l d Frame Task Frame

Joint Frame

ROBOT

Ze

Feo

For-c .. S.n .. or

Estlmated world/task transformation matrtx

fez

.J!» <,.)

» 0-... .., r'O

<: .. f"'. o ::;, .... ... ~ !f s:: o 0-!!..

1

l

,,) i\dapllve ( onlroller Model

velocltles 60 :::: ti"O = 0 This can be done uSlng the T.lylor senes

f(::,::d':';;'{)-= r(;;O·~dO·O.O). f~(-o.-,fO'O,O)(: 0)1-

t~)-o.:"o.O.O)(,j ,/0) 1- f~(-o, ,10,0,0).: !

t~./-O.:dO'O.O) d 1 (417)

where the hlgher order terms are dropped ln arder to allow easy re,ldtng, the followlng IlIleJr

operators are defmed under the assumptlon thJt the Jac.oblan matrices <He slowly v.Hymg

The terms that reduce to zero have been omltted.

'/:() == '1 \ __ _ ,-- '0

,* nlh \ (,. -= -.-• 0:: ::=':0

C~ == aGz 1 __ _

- D::· '-=-.0° .-

The hneanzatlon allows (4 6) to be rewnttp.n d'i

GrouPlng the terms

1-• 4- \ T"

{ .. \ l' -

/ )

(4 18)

(4 19)

( 4.20)

( 4.21)

( 4.22)

( 4.23)

(4.24 )

(4 25)

(4 26)

(4.27 )

(4 28)

"'

4 3 Adaptlve Controller Model

} "* ~"* l"· F -J" - G'* t' \péd + J\. f f::r/ + \1'(' + IIZ + \.",::1' t 7:.:'0 - fJ:O - . f= ( 4.29)

( '1 2 (l"* C'*) -1" l"t G'*) 1\ Z S + 1\ V + :: li + \. (' + \ l' + 7 = .:: =

(430)

D::(.,)::: =

(J "* J"*) 1"* F -/" - G'* f \pd + \fr ::" + \.1'(' +- 1/:: + \,-;:;1' + 7:.:'0 - !J:o - fz (4.31 )

where

( 4.32)

Equation (431) descrtbes the behavlor of the plant ln ta!.k coordlnates As seen

from thls equatlon. the system 15 coupled in general L10tlons and Inputs ln a partlcular

direction Influence motion ln ail directions, because matrices D:, I\l~d and l":j f are not

diagonal ln order to be able to apply the MRAC controller ln the normal contact force

direction, It IS necessary ta assume that the system IS decoupled, 1 e that tangentlal motion

does not affect the normal contact force This IS physlcZllly Justifiable the motIOn ln ,)

partlcular direction should depend mostly on Inputs ln that direction TI1I5 apprOXimatIOn

should be valtd for manlpulators wlth simple configuratIOns, especlally If the posItion servos

are weil tuned and the velocltles and Jcceleratlons are 10\'11 ThiS hypothesis IS later venfled

ln simulation for the PUMA 560 manlpulator

Wlth thls assumptlon, the matnces mentloned above became diagonal From

equatlOn (4 31), the flrst Ilnf> IS extracted to get a relation between the Input and the cantact

force ln the surface normal direction ThiS ylelds

D:(l.l){.'l)Zl = (I\l~" + !{Îr)(l.l) :,/1 -1\I~r(l) l- FlI:(l) + 1\', ::, -r- «(1.1)-01 - '/:0(1) ( 4.33)

The model now depends only on the normal direction terms The contact force

Îs found by applymg equatlon (4 11) and as for the Single aXI~ case (equatlans 3 30 - 3 32).

the constant blJS terms are ellmlnated by multlplylng bath sides bv .'

1

1

h", FI/.(I)

f):(I.l)(')

1 4 lmpl~mtnt;)t,on (enceins

, j ( 4,34)

" 1\, ~, ( 4,31))

(4 Jo)

If It IS assumed that the dl5turbiwce term FI/_ l.Hl br (ornpensJted, or that Ils

effect 15 negllglble because It IS slowly varytng, the SISO plant model uc,ed for the MRAC IS

glven by

,I! ( 4.37)

ThiS glves the expression tI·latlng the VI·IOCltV InplJt .lnd the <.ont<1et force III

the normal direction The lv1 RAC schernp le, ,lppll('d to t 111'; pl,lnt ,1., ln the '>1I1!!,le aXIs CJ~e

(equatlons 332 - 336) The estlmator c,hOlJld 1)(' "blp !o rqrnçwn,>atp for tlH' V.HI.ltlon of

the klnematlc and dynarnlc terms, and for th!' (Duplnl!', Il rm', thal ~"JI'r(' rll'l',lcLted 1 he

model IS vahd only for locally pl,wc, IIxf>d ~lIrf,I(('<" IIIP /'/tf'I1'>IOfl (li thl'> Idea to morC'

general envlronments (an not be don/> f',lsllv .1<; Il wOlJld r/'qUtrf' /'I!f>n'>lv(' knowlcdge of the

envlronment configuration .lnd ')t .. tf~C,

4.4 Implementation COllccrns

r he sa me 1 rnplernen ta tlon (Of)c/:'rn<; (j,.v r dll'd tr\ <,1'( t Ion ~ ~ for t IH' .. mgle aXIS

case apply to the multi-axis case fhe (hol((~ 01 Ihf' Ilmlt on fi 1 Wflldl (if·termtn('s the ranp,e

,1'1

,

f

4 5 Wall Orientation f:.stlmatlon

of use of the MRAC is IImited by an addttional factor' the value has ta be suff,clently low ta

aVOId actuator saturation If th,s occurs, the decoupling between the axes will be lost and

the system will behave erratlcally, e g at low stlffnesses, an Input ln the J" direction may

result 10 motion or force ln any direction

Because stabdlty and convergence are more fragile ln the multl-aXls case, malnly

because of the coupllng and non-hneantles, the zero of the d,screte-tlme plant model IS

ehmlnated and replaced by a simple gam, tnstead of constantly estlmatlng and hmltmg the

hO and b1 parameters Therefore. equatlOn (336) becomes

(438)

This Simplification lmproves the robustness and convergence of the !v1RAC wlth­

out compromlslng the performance of the controller ThiS problem would have been aVOIded

by the use of a dlfferent adaptation scheme based on pole placement or self-tunlng ap­

proaches wh,ch do not rely on the cancellatlon of zeroes, but the slmpllclty of the MRAC

design would have been lost

ln the three-axis Simulation, a 5upervIsmg module for the MRAC 15 al50 Imple­

mented to start/stop the adaptlve controller when contact torces are detected/lost.

4.5 Wall Orientation Estimation

The flrst Idea that cames to msnd to flnd the wall orientation IS to use the force

measurements, as ln [Merlet87] and [Yoshlkawa89] If there 15 no friction, the resultant

contact force 15 ln the same direction as the surface normal ln reJI applications, friction

and stlctlon forces are present and may be Iligh Furthermore, force signais are very nOlsy

ln general For these reasons, tht> accuracy of th,s scheme IS poor

Another method that IS very Simple, IS ta use the veloCity rneasurements If a

manlpulator IS rnovmg along a surface, the veloclty of Its end-point IS always perpendlcular

50

1

1

,

4'> \VJII Oll~nlatlon l sllmatlon

ta the surface normal and It IS possible, wlth t'nough measurements, ta extract the w,lll

Orientation The problem wlth thls method IS that for low stlffness w,llls, there m.ly be some

motion ln the normal direction If thls 15 the only motloo ,lt one tlme, the .llgorlthrn will

thtnk the surface 15 perpendlcular to tlm direction, whlch 15 tompletely wrong Tll/S tlwthod

should only be used when veloCltl€S are sufftclently large and present for J certain amount

of tlme

A combmatlon of the two ml:'thod5 mentloned above should glve 5Jtlsfactory

results An algorrthm could be developed that uses the force, or velouty, or both '>Igoal",

dependlng on the conditions Such ,Hl estlm.ltlOn ~cheme IS d, veloped ln !KazJnzldps89!

There are ev en more sophlstlc.lted nlpthod'i 01 ftndlng the er vlfonment p;H.lnwterc; rlle

Extended KalnlJn filter approJch [BIJlIcr871 Jnd robot VISion are two f'xJmples wll/ch would

slgnlflcantly IncreJ5e t~e system cornplcxlty

ln the followmg simulation tests, oolv the simple force based rnethod IS tested

The wnst force signai IS fdtered to el1l11lnate 11lgh-frequency nOise and the n'sultant I!> Ilsl'd

as the surface norm.Jl directIOn The vcctor IS then sCOlled clown to lJllit length .1nd the

dlrectlonal coslnes are fed III the fox I1ldtrlX

1

t

Chapter 5 Simulation Tests

This c.hapter presents the simulation results of the adaptlve damplng control

scheme The single-axIs tests are presented flrst. followed by the three-axls simulations The

last section dlscusses prehmtnary expenmental results presented at the F,rst International

SymposIUm on Experimental Robot/cs

The simulation programs whlch rely on simple numeTicallntegratlon were run on

a VAX computer and are wntten ln Fortran7ï language The position servas run at 3 KHz

and the robot dynamlcs are updated at 30 KHz The forcr:. cnntroller and Jdaptlve scheme

run at 100 Hz unless otherwlse speclfled

5.1 1 Axis Simulation

5.1.1 Model

Simulation of the adaptlve damplng c.ontroller was perfarmed flrst for the slngle­

axis case ta achleve famtllanty wlth the system behavlor Jnd tuntng The equatlons far thls

controller are developed ln Chapters two and tlHee and correspond to the model of Fig 2 1

The damptng term h-) 15 set ta 250. so that the maximum r .:lte II1put of ::0 ( III / '

corresponds ta cl force of 50.\ The controller gain adJustments are tuned for a medium

stiffness (h-p == 2 y 10~ .\" / Ill) The system parameters are glven by

t Controller: 1\1' -= 30

{\I -.::; 10

/\'/ f .- 0,00833

{\" =- 80

System Parameters' ,,-:: 2 .\' , .... /111

Soft EnVironment

5tlff EnVlronment

.1 oc: 1 10;1/

/' -= 05111

.)

1/1"

{.;, _:: 1)(10~ .Y/III

'> 1 l ,\X'S ~Imulat'on

)

It should be noteu thJt the effective r,tlffness c,e{'11 by tll!' .lI tlJ.ltor IS l/I" ttmes

smaller t han the Jet UJI ~tlffness The reterenee model lI~ed W.IS derlvl·d from t hf' ft'sponse of

thls eontroller wlth the medium stlffness envltonment. for whl<n It 1'> wdl tlln('d The Hlltlal

estlmates veetor ,/(0) was found aeeordlng to the rnethod~ descrl!J('d ln c,cctlOn 536. The

parameters have the followtng v,llues

(' 1-= 1.0, ·0694. o 2991

('/:::::' 1.0, 0.0, 00

/Je.::: 41 74. 1408

ji J (0)= 170.0. 00. 08, 0.0

5.1.2 Results

Flgs 5 1. 5 2 and 1) 3 show the responsps of th,. non Jd;tptIV1' d.lfnptng controliN

to a varytng demand Input force for tlHet' dlHerent ('nvlronnwnt "tdfn"',<,r'c, I\c, expect!'d,

the low stlffness response 1'; qUite slugglsh whdr· dt hlgh c,tlffnp,,<,(,<, th,. rp<,pOflc,f' becomf's

osctllatory

The effect of the 'I1RAC (,ln b~ seen tn h~ .. 1) 4 .Ind 1) ,) At low !>ttflnesse ...

the response IS much better than ln the non adaptlve caC,e (h~ ':J 1) 1 he MHAC follows

5 1 l Axis Simulation

the desired force very weil, and the estimator converged to the correct values ln only elght

iterations.

Fig. 55 shows the response of the same adaptlve control 1er for the very stlff

envlronment Once agaln, the MRAC slgntflcantly Improves the controller performance, the

oscIllation and bounclng are completely ellm.nated The adaptlve controller glves stable

results for stlffnesses up ta 8.5 x lO'-\ï'l/, whtle the non-adaptlve controller shows unac­

ceptable responses above 5 x 107 SI III ln ail cases, the recurslve least squares algonthm

was used for the estlmator ('\1 :::: \2 .::: l.0)

f DES

bù 00

1 \)

48.00

36.00

24.00

12 00

il DO

o 00

5.1.3 Discussion

.', e = l 00l~

o 50 l 00 1 ." 1 00

Figure 5.1 force response at low stlffness

It IS clear from these results that the M RAC scheme can slgnlflcantly Improve the

performance of the damptng controller ln force tracktng over il wlde range of envlronment

parameters Stabillty can be preserved at hlgher stlffnesses, Jnd the tracklng response at

low stlffnesses IS slgnlflcantly Improved The best results were obtillned 'Nlth a reterence

model tuned for a relatlvely stlff envlronment A controller tunecJ for lower stlffnesses will

1

1

.----------------------------------""------- - l F_OES

5S 00

(N)

44.00

33 00

n 00

11 00

00

-VS- Time

Figure 5.2 loree response at medium stdfness

,------------------------.--- --_. __ ._-----------------..., f _DES ..." 1 .: l', r ~ m, l

l sa 00-, 1

(.V) 1 V 1

1 l~~ OO-!

1

1

1 "1) oo-l

:

:

1\ 1

1 1

\ 1 dl Il 1

1~ 1 1 \ \

~o 00-1

'0 00 .... ,

\1 ~\I !'-'" ,

J 00

.J {JO ! r f,

( " 1 , tif)

'------------------ -

mltlally have hlgher loop gaIne; and rnily bf~cotne un,>tablf' bdore the estlmùtor converges at

hlgh stlffnesses

These good results are not very surpflsang slnce a Ilnear Simulation model WëlS

,

1

F_DES

60,00

(,V)

46,00

36,00

24,00

12.00

o 00

a 00

F_OES 55.00

(X)

44 .00

33 00

22.00

11,00

0.00

a 00

5,2 3 Axes Simulation

-vs- Time

Ke=1000

0.50 l 00 l 50 2.00

Figure 5.4 Response wlth MRAC at low stlffness

-vS- Tlme

o 50 l 00 l 50 ~ 00

Figure 5.5 Response wlth MRAC at hlgh stiffness

used, ln multi-axls tests, the complex dynamlcs and interactIon wlth the environ ment should

significantly i nfl uence the behavlor.

----------------..........

1

", •

,2 J Axes ~imul3tlon

5.2 3 Axes Simulation

5.2.1 Model

ln thls section, the simulatIOn model used to test the dampmg controller and the

directional adaptation scheme in the multl-aXIS (<Ise IS descnbed The dyn'Hllicai equatlons

for this system were developed ln Chapter four and correspond ta the black dlagramc; ot

Figs. 4.1 and 42

ln thls simulation model, pOint contJct wlth the envlfonment IS assumed, 1 e

reaction force~ can not generate torques Jt the contJct pOll1t l'he robot mode! Il!>ed for

simulation has the SJrne bJSIC conflgurJtton JS J PUMA aTm, Wlth tilt' exceptions that <111

offsets h,lVe been ellmlnJted, Jnd the two Jrrn links hJve the SJIlW I('n~th The three WTISt

JOints are also consldered to be frozen, 1 e the Orientation of the WfiSt has no effec.t, only

the Cartestan pOSItIOn ln r, 1/, Jnd ,~ IS consldered

The dynanllc model was developed wlth LJgrJnge's eqUJtlons Jnd was taken

from [pelletler87j The Inertlas and motor c.onstJnts were tJken frorn the PUMA 1)60 robot

speclflC<ltlons IArmstrong86j The envlTonment le:; modeled JS J pure c,tlffner,c" It le; defllH'O

as an InfinIte plane whlch could have .1ny position ,md OflentJtlOl1 tn c,p;\ce A friction model

based on Coulomb's equatlons IS also IInplemented for rnovernent'i on thl:' WJII surface ln

ail tests, the friction coefficient 1/ f IS 0 3

The Inttlal configuration of the robot as shawn ln Fig ':> 6, 1'> glven by

o} =..: 0,0 1 (ft!

02 :..: 1.090-' 1 (fi}

'h = 2.1813 /'(fI/

The wall 1') Inltlally rosltlont·cl .1t 041/1 dlong; t~lP 1 dtle, ,lnd (.In IH' rotated ln

.1nv direction r he sy<,tem pilTanH:ters (pOc,ltlon <,crvoc" f(,,'d·fnrwdrd t"rm (·t. ) w(~rp tllned

t

,

5 2 3 Axes Simulation

Figure 5.6 Robot and enVIYonment initiai configuration

to reproduce the behavlor of a PUMA 560 robot. The gain values and system parameters

are given by

Controller Gains: Kp= (30 30 30)

K lI = (10 10 10)

Kf! = (0.33 0.33 0.33)

l':] = (250 250 250)

Actuator constants Ka= (80 160 80)

T('max = (97.6 186.4 89.4)

Tu = (20.0 40.0 20.0)

Robot parameters: Inl= 17.4 1112 =: 4.8

Il = 0.068 l2 = 0.070

Il =: 0.53 12 = 0.076

ml= 2.24 1-' - 0.433

The MRAC was tuned accordmg to the methods descflbed ln sections 3.3.5 and

3.3.6. Two reference models are used, one for the 100 Hz Iteration rate, and the other for

58

1

•

J

5 2 3 Axes Simulation

1000 Hz. The parameters have the followlng values:

100 Hz mode!. Cl = 1 0, -0 104, 0.0042

D= 180 O. 00

î/(O)=500 0, 0 0, a l, 0.0

1000 Hz model Cl = 1 O. -1.47,0.74

D= 7.31, 0.0

~1'(0)=55 0, 0 O. 08, 0.0

To allow good position tracklng along the environment surface, the force feed­

back ln the tangentldl directions IS shut-off as shown ln Fig 42. and unless otherwise

speclfled. the wall Orientation IS a5sumed to be perfectlv known

5.2.2 Results

The flfSt three plots, Flgs 57,58. and 59, show the force response of the robot

wh en Inltlally ln contact wlth a wall at 0° for three dlfferent stlffneS5es, !{/ -= 103, 25 x 104

and 5 /104 .YI III ln these tests, the adaptlve controller IS not active and there IS no motion

on the wall The force controller IS runnlng at 100 II:: There IS no compensation for the

dynamlcs (conolls, gravit y etc) but the fnctlon at the JOints IS cancelled

The effect of the fV1RAC controller can be seen ln the next plots Flgs 5 10,

5.11 and 5 12 show the behavlOr dt low stlffnesses ([\-(' -= 10 3) The m;Jnlpu!ator 15 Inltlally

movmg at 20 ('/II J.' ln the 1 direction, and then hlts the wall surface perpendlcularly after

0.07 seconds Fig 5 10 shows the force response. whde flgs 5 Il and 5 12 show the deslred

and actual positions \0 the normal (1 JXIS) and tangentlal (.:, aXIs) directions As seen ftom

Fig 5 10, the speed of the force response Jt low ~tlffnesses IS lirnlted to Jvold saturation

of the actuators (see 5cctlon 44) ln thls test III. 15 Ilmlted to 0 5/{{/~ The ITlCre3Se ln

the contrai Input produced bv the M RAC can be seen by the chJnge of slope ln the normal

position demand (Fig 5 1l) The talhng th;]t occurs 111 the -: direction 1') due ta the welght

of the robot ln non-Jdaptlve tests, the total droop 15 Jpproxlmately jill/II. whde Irl thlS

test It falls 6 6,,111/ ThiS effect IS due to the Imperfect decoupilng of the MRAC action at

,9

5 2 3 Axes SImulatIon

low stlffnesses, i.e wh en the controlmput Iq. IS hlgh ln the case where there 15 saturation,

I.e. If Uk is not limlted, the droop can reach 20mm or more

Figs 5 13 ta 5 17 show the force response ot the MRAC controller at the hlgnest

possible stlffness whde preservlng stabdlty (I{/ -= 2 " 10~ S/III) when tnltlally ln contact

wlth the wall surface Figs 5 13 and 5 14 show the torce response and convergence of the

rO and rI parameters for the recurslve least-squares estimation algonthm ( \ l ~ \2 = 1 0)

The force response IS oscJlIJtory but stable, and the convergence IS smooth but rather slow

Figs 5 15 and 5 16 show the same parameters obtalned by uSlng a constant adaptation gain

estimatIOn scheme ln thls case the force response 15 still stable but parameter convergence

IS not achleved, the estlmates constantly oscdlate between thelf Ilmlts ln genera\. the RLS

algonthm performed best Jnd show€'d rnuch better robustness The constant gain algonthm

often went unstable ln the mlddle of a test when the Input was not sufflclently nch The

results obtalned wlth the exponentlally welghted leastsquJres ( \1 --- 095"" 0 99) algoTithm

are not shown because they are very sirndar ta Figs 5 13 and 5 14 It seems th;)t the addition

of a forgettlng factor ln the RLS scheme does not Influence the convergence slgnlflcantly tor

these short test~ ln real-life Implementations, the use of a forgettlng factor will be necessary

to cope wlth VJrlatlons ln the envlronment and manlpulator parameters

ln Fig S 17. the friction compensation IS shut-off and the sarne hlgh stlffness

test 15 run wlth thE' RLS estlm;:lt/On The d,sturbJnce torques created by fflctlOn are ap­

proxlmatlvely 20~o of the maXlrTllIrn output torques rhe effect of thls hlgh frictIOn on the

force response 15 slgnltrcar1t The response IS still stdble Jnd the oscillations are ehmmated

completelv The steJdy-state accuracy 15 lost thougb, especlally for slow Vdrlatlons IrI the

demand, the eontroller CJn not track the force as weil as before At lower stlffnesses, the

effeet of fnctlon becornes less Irnportant The SJme test was performed at !\-,'" 5 '" 104

and K, -= 10:l .\ 111/ and the results show th3t the effeet of friction decrrJses wlth stlffness

At 1\·, = 10:l .\/111 there was no slgolflcaot dlfference between the friction and non·trlctlon

tests ThiS effeet IS due ta the tact thJt ;)t lower stlffnesses, the control torques are hlgher

and the relative Impo~tJnce of the fnctlon IS decreased

ln order to Irnprove the results at hlgh stlffnesses, the Iteration rate of the adap-

hO

1 5 2 3 Axes Simulation

tive force controller was Increased from 100 H:: to 1000 H -:, whtle preservmg the JOint

fnctlon compensation Results for the hlghest possible stlffness wlth and wlthout the adap­

tlve controller are shawn ln Figs 5 18 Jnd 5 19 The non-adaptlve controller performs much

better ln thls ca~e The response 15 good up ta ]\-/ ::- 8 x lO~ S/III, whde the adaptlve

controller behaves poorly at !\-/ = 1 v 10' SIIII ThiS response can be Improved wlth the

addition of a smoothtng polynomial C'? ~ (1 O. -1 2l3. 0 368) The response for the same

stlffness 15 shown 111 Fig 520 Although the response 15 rnuch better wlth the smoothlng

polynomial, It IS still not possible ta Increase the stlffness slgmflcantly Jnd the non-adaptlve

controller can still cope wlth hlgher stlffnesses

The behavlor of the M RAC ilt 1000 IL CJn be slgmflc.Jntly Improved by uSlOg

dynamlc compensation for the gravit y ;lnd veloclty dependent terms (coriolis and centflpetal

forces) The mertlal terms are still configuration dependent but ')lnce they vilry slowly, the

effeet IS negllglble The response of thls adaptlve controller wlth feed-forwJrd dynamlcs

IS shawn ln Fig 5.) 1 for a stlffness of 106 -\i II/ The trJcklllg oi force 15 achleved and

stabdlty can be m,llntalned for stlffnesses up to 5 x 10b _\) lIl, .JS shawn ln hg 5 22. whlch IS a

slgntftcant Ifnprovement over the non-adaptlve (<:Ise The reason why dyn;lmlc compensation

makes suc.h .J dlfference dt 1000 ff ,: IS that the controller excites bst dynaml(s whlch were

~,ot slgmflGlnt at 100 II: At the hlgher Iter;:ltlon rate. the controller becomes much more

sensitive ta the couplln~ ot the axes and the nOn-lll1earitlec; rhe dyn.1n1lc compens<ltlon has

no slgnlflcant effect on the non-adaptlve controller force response

ln the next test, the assumptlon that the system CJn be consldered decoupled

(see sectIon 4 3) Hl the normal force directIon IS verlfled by observlng the effect of tangentlal

motIOn wlthout the dvnamlc compensation The same Input force demand as before IS

applied III the direction of the surfilce normal, whde fast motIon 15 commanded ln the

tangtntlal directions rlg ') 23 shows the force response whtle hg ':J]4 shows the deslred

and actual positions ln the tangenttJI direction rhe speed vallee; bptween ~ 201/11 lM ( ln

the:: directIOn III thls case Other than the uSllJI droap of )111111 of the WlISt, the tracktng

IS perfect and the effect on the force response tS h,lrdly notlceable Th,s same test was also

performed wlth motion ln the 1/ direction, wlth or wtthout the JdZlptatlon rnechanlsm, and

61

1

5 2 3 Axes Simulation

also wlth the 100 H:: controller ln ail cases, the effect of the couphng 15 neghglble The

assumptlons about the decouphng are thus val Id ln practlce and the MRAC controller model

of equatlon 439 can be applted ln th,s confIguration

The effect of an initiai Impact IS shawn ln Figs 5 25 and 526, wlth and wlthout

the adaptlve controller The wallis posltloned further 50 that the robot Inltlally hlts the

surface perpendlcularly at 20('11/ /,' ln general, the adaptlve control 1er response IS less

oscillatory but It creates larger force overshoots The response 15 also very dependent on the

smoothlng polynomial (''} A slow ('2 renders the controller Incapable of deahng wlth the

rapld force variations assoclated wlth the Impact On the other hand, If the polynomIal IS

slmply set to unit y, convergence 15 longer to achleve and the response mJy oever stabdlze

The best solution 15 a compromise between these two effects At even hlgher stltfnesses,

nelther the adaptlve or nan-adaptlve controllers could recover from J vIolent Impact

The last test shows the behavlor of the M RAC contrai 1er when the wall orlenta-

tlon 15 Imperfectly known 1he e~tlmatlon algonthm simply uses a fdtered force measurement

from the wnst sensor ta determlne the normal direction Slnc€' there 15 considerable surface

friction (II! ::-:: 03), the orientatIOn estlmate error 15 Important (17°) The wallis r:.>tated

45° around the:: ;)XIS and the same Input as before 15 commanded ln the direction of the ,1'

aXIs As can be seen from Flgs 5 27 Jnd 5 28, the force trackmg 15 still Jchleved at medium

stlffnesses and the position tracklng along the wallis good also For hlgher stlffnesses, the

effect of the Imperfect direction estimatIon does not Influence the position trdcklng but the

force response degrades gradually A more accurLlte direction estlmator should be used for

better results

02

5 2 3 Axes Simulation

F_DES_~(l)F_~(l) -YS- rime

54.00~------------~-----,-------------

(N) Ke=1000

43.00

32.00 --~---+--------+-----

21.00~--------~--~----~~----~----.H

-1.00

0.00 a la o 20 0.30 0.40

Figure 5.7 rorce response at low stlffness (100Hz, no MRAC)

-ys- rime F_nES_~(l)F_Z(l)

54.00-r----------------------------r1-----

(N) Ke=25000

43.00~r---------_+----------------

32. 00 -H--~- --4----------~------

21.00~-----------~------~---------~

lO.OO~----~----+-----~--~-~~+-~

-1.00 1 ~------r-----~----~------r_----~ 0.00 0.10 0.20 0.30 0.40 O. 5C\:~ec)

Figure 5.8 Force respon<e at medium stlffness (100Hz. no MRAC)

63

5 2 3 A:(I!S Simulation

F_DES_Z(l)F_Z(l) -YS- Tl.me S9.00-r------------------~----~------

( ."1) Ke=50000

3S.00~~--_+------44------------------

2).00~----_+------~~~~~--------~

11.00~----------·--~--------~----·~--

-1 00

0.00 o 10 0.20 0.30 o 40 0.50(.w:)

Figure 5.9 Force response at hlgh stlffness ~ 100Hz, no MRAC)

( F_DES_Z( l)F _Z(l) -YS- Time 54.00~----------·--__ ----------------~

(N) Ke=lOOO

.3.00~~------~-~~----~-----------

32.00~~----~~---~-~---------------~

21. 00 -+t----++-----+-----+~--__.. -----.1-4

10.00~-----+4------4_----~--~~--4_~

-1. 00

0.00 0.10 0.211 0.30 0.40 O.50(sec)

Figure 5.10 Force response at low sttffness 1100Hz. wlth MRAC)

(

64

, •

CTSX_Z(l) CTSX_DEZ(l ~Vs' Time 62977.00~----~----~r-----~------~---

(lO-lmm)

50374 . 00 +------'- -I----r\---j.-----+----.....,

37771.00~---~~-----~~FA~---__ ----~

25168 . 00 -+--.--Ir--

12565.00~--J~-------~~----~--<L-~

"38.00

5 2 3 Axes Simulation

0.00 0.10 0.20 o 30 0.40 0.50(sec)

Figure 5.11 Normal pOSition response at low stlffness (100Hz, wlth MRAC)

CTSX_Z(J) CTSX_DEZ(3 -VS- Time 2.00~----~------~-----r----------~

(10- lmm) 1

Ke=lOOO

-1328.004-----~-~~----~-----~-----

-2658 OO~---~---+_r------------

-3'88.00-r----~~----~-----~----~---~

! l , 1 1

l ' .-~--'r+-._.~-- ----1 -5318. 00 4----+-

-6648.00

0.00 0.10 o 20 0.30 0.40

Figure 5.12 Position response ln: at low stlffness (lOOllz, wlth MRAC)

65

1

5 2 3 Axes Simulation

F_DES_Z(l)F_Z(l) -vs- Time

S4.00~------------~----~-----'----~

(N) Ke='200000

1

43.00~~---4--ïO~~-----------+------;

3l.00~~----------·~~----------~-----4

Figure 5.13 Response wlth RlS estimatIon (100Hz. wlth MRAC)

o O)Ofl

" Dun 11' o~Ut

03 oou. /1 OOUI

-- -------~---------------

--- --- ------------------c=;:.~==:::=::::===+ " Ual4

-------- _~~---____ _\..I -, "u,

OOOOlO L---~-~------~------r-~----~-----~------T-------~:=~~ -, n."

•• u o 000 1) Ut , 1" 1) JOI o lU out

Tl_

Figure 5.14 Parameter convergence for RlS estimation (100Hz. wltn MRAC)

66

1

5 2 3 Axes SimulatIon

-vs- rime F_DES_Z(l)F_Z(l) 54.00~----~------~----~-----,----~

(N) 1

Ke=200000

43 . 00 -HI~---~.

0.00 0.10 0.20 0.30 0.40 0.50(.see)

Figure 5.15 Response wlth CG estImatIon (100Hz, wlth MRAC)

P'IIEC (3l. i''1EC l' 1

l U'U

J. G?ut

1 OJ:UI

o ,.U4

o U.,a 1) .tUI

1 ......

o JUJI

o nOI"

o lOlU

" I!.'"

1) 'lOQl

OUlU

'3 'HU)

o Uotll

o ualt

o ln.' 1) U'U

o lUt)

OJJ4ll

o U"I

o 140"

~ 0' If.

--1 --~ -~----

11)!)!)

1 - ,

'" '} JI.

-- -- -- ---.-,,-r-..,--

.. -- --+ 1 .. .--/

-l-

i r +-+ - .j -

tt .- i i

.., J lOI

j •• 1It

- ~~ :..-----f1-- : ::::: --- ........ - - 0 OOIU

_._--~- - - ·0 OU,O

-- ~ ---+-. 0 OU))

! -0 UUI

..\ - ~ • -0 11~OO --- \ t- -.1 0 .,. ..

\ : ~-r -1 -: :::::

, 16'

\ • l t 1- 0 u.))

.-~ ---t o ..... l- - 1) H'" ~. - -0 iHI)

,. -~t -0 Hl ..

t 0 UH'

- + 0 UJJl

t- 0 "tU

- t- ·0 nn' ~- t- 1) 110U

.,.

l J _: ::::: - ~ + 0 ".u

... --r 0 'Uit

o "t" l ,,'Ut

Figure 5.16 Parameter convergence for CG estimation (100Hz. wlth MRAC)

67

----------------..........

1

1

,

'_DES_Z( lIF_Z(l) -VS- Time

59.00~-----r----~------------'-----~

(N) 1

Ke=200000 1

47.004-~-4~~---4------------~----~

35.00~~-+~----~~----------~----~

13.00~~~-+----~~-----~----------~

-1.00

0.00 0.10 0.20 0.30 0.40

5 2 3 Axes Simulation

Figure 5.17 Response wlth 20%jomt frICtion (lOOHz. wlth MRAC)

F_DES_Z(l)F_Z(l) -vs- Time

Si. 00 ...,....-----,------r--------~--~

(N) Ke=800000

32,00-H~---------4_---------------~

ll.OO-K-----+------~----~~--~----~

lO.OO~-----+------~--------~~~--~

-1.00

0.00 0.10 o 20 0.30 0.40 0,50(sec)

Figure 5.18 Force response for hlghest stlffness (1000Hz. no MRAC)

t8

'_DES_Z(llF_Z(ll -vS- Time 74.00

: (N)

59.00 "., :

1 Ke=lOOOOO 1

1 i 1 1 1 1

; 1 1

1 , , 1

1

44.00

29.00

14.00

~ 1

1

~r 1

1

\ 1

1 1

, ~ N '~I -1.00 1 1 1 1 1

0.00 o 10 0.20 0.30 o 40

Figure 5.19 f-orce response wlth MRAC (1000Hz)

F _DES_Z( l)F _Z( 1)

54.00~----~-----r----~----~1~----

(.V) Ke=lOOOOO 1

-VS- Time

43. 00 -H-~~-'

32.00~----------~~-------------~

21. 00 -+t---~---+-_.

10.00~---~-----------+-~r---~~

-1. 00

0.00 o 10 0.20 0.30 0.40

5 2 3 Axes Simulation

Figure 5.20 roree response wlth ('? polynomial (1000Ilz. wlth MRAC)

• 5.2 3 Axes Simulation

F_DES_Z(l)F_Z(l) -vs- Time 54.00~----------~----~----~----~

(N)

43.00~~----~~-+-----+----~----~

32. 00 -++-+----------H-----~----+---4

21.00~~--~----_+-----+~--_4----~

lO.OO~~--------_+--------_4,~--~~

-1.00

0.00 0.10 0.20 0.30 0.40 0.50(sec)

Figure 5.21 Force response wlth dynamlc compensation (1000Hz, wlth MRAC)

F_DES_Z(l)F_Z(l) -VS- Time 64. 00 ~----------r-----""-------'------'

(N)

51. 00 ;-;::=====;;H--------t-----j

3 B . 00 -H----,j

2S. 00 -+t--ttI---------H

12. 00 -I-ll--------+-m------~.---<r__n-__l

-1.00

0.00 0.10 0.20 0.30 0.40 o. SO(sec)

Figure 5.22 force response at hlghest stlffness OOOOHz. wlth MRAC and comp) ,

T

-VS- Time F_DES_Z(l)F_Z(l) S4.00~----------------------~------

(N) Ke=40000

1 43.00~~--------~----~-----+------

3 ~. 00 -Hf--------~\----------------

21. 00 -Ht---------+-------\;---~---_fR

10 • 0 a ""*1---

-1. 00

o 00 o 10 o 20 o 3D o 40

5 2 3 Axes Simulation

Figure 5.23 Force response wlth tangentlal motion (1000Hz. no MRAC)

CTSX_Z(3) CTSX_DEZ(3 -YS- Tlme 20505.00 .,--------------,.-------,--

(lO-lmm) Ke=40000 ,

16209. 00 ;-------------+1---+'\--+---------

11913 • 00 -f--------

7617. 00 -t-----+----+-f---

3321. 00 -t---~__I

-975.00

0.00 o 10 o 20 o 30 o 40

Figure 5.24 T.1ngentlal position demand (1000Hz no MRAC) --------------...........

,

5 2 3 Axes Simulation

F_DeS_Z(l)F_Z(l) -vS- Ume

13 •. 00~--~~------------~·-------------( N)

l01.00~---H~---------------------------

BO.OO~---KHH~----~----------~

53.00

li.OO

-1. 00

0.00 o 10 0.20 0.30 J .. 0

Figure 5.25 Force response at Impact (1000Hz. no MRAC)

F_OES_Z(l)F_Z(l) Ut.OO

(N)

171.00

1l8.00

85.00

4l.00

1 -1.00

0.00

J".

0.10

-vs- T1me

1 1

1 1 1

1

1 1

.~

i 1

0.20

1 1 Ke=500000 1 1

1 1 1

1 1

1 1 1

1

1

1

1 1 1 !

1 1 1

i i

1

1 1

1 : 1 1 1

~~ 1

0.30 O.tO 0.50(sec)

Figure 5.26 Force response at Impact (1000Hz wlth MRAC)

72

1

,

-vs Time r_DES_Z(l)f_Z(l) 44.00-r----~------~----~----_::_--~

(Nl Ke=lOOOOO

l6.004-~----------~-----+----~~--~

17. 00.u--+--·-~~=~-_l--

8.00~----~------+_----~--~~--~

-1.00 1

5 2 3 Axes Simulation

1 o 00 0.10 0.20 o 30 0.40 O. SO(S!!c)

Figure 5.27 Force response wlth dlr estlm (1000Hz, wlth MRAC, 450 wall)

CTSX_Z(2) CTSX_DEZ(2 -VS- Time ~9105.00~----_,------._---~----_,r_--~

(lO-lmm)

23284.00~-----4------4---~~------~--~

17463.00 +----t-----4+-----t-- --------;

11642.00+-----+~----·T_---_+----~----~

S821.00~--~_+----+_---_+----~----__;

0.00

0.00 0.10 0.20 0.30 0.40 O.50(sec) ----------------------------~

Figure 5.28 TJngentlal position response (lOOOHz, wlth MRAC 450 wall)

73

1

t

5 2 '3 Axes Simulation

5.2.3 Discussion

The basIc behavlor of the non-adaptlve multl-axls damptng controller 15 very

simllar to the single-axIs case The response IS slow at low stlffnesses and becomes oscdlatory

at very hlgh stlffnesses The adaptlve controller also has slmtlar effects on the behavlor At

very low stlffness, the MRAC speeds up the response, but because of actu3tor saturation,

thls effect has to be Itrnlted ta avold unwanted motIon ln the other directIOns The test

shown on Figs 5 10 ta 5 12 clearly show the Increase ln the droop due to the effect ot the

MRAC

At hlgh stlffnes5es, the 11/1 RAC has the same stabdlZlng effect .1S ln the single-axIs

tests At 100 II.:.. the hlghest stlffness th;)t allows stable response went trom 5 . 104 to

2 x 105 X/ II/ wlthout Jny other c.ompensatlon 1 he fact that the t;)ngentlal frictIon torces

on the wall Jre treated as dlsturbances and Jre not fed back to the controller seems to

help to preserve stabtllty at hlgher stlffnesses FrictIon forces Jre very nOlsv and thelr etfect

on the control Input seems to rnfluence the MRAC even though they ,He not ln the same

dl rection

ln the tests at 1000 II _, stabliity CJn be obtalned at even fllgher stlffnesses

but at greater cost At thls frequency, the system dynamlcs LIre exclted much more Jnd

the simple fv1RAC plant model IS Incapable of compensatlng the dynamlcal eff(>cts of the

coupling and non-llneantles The dynamlc compensùtlon IS necessary to Jchleve the better

performance at hlgh stlffnesses At 100 lI::. dynamlc compensation has no slgnlflcJnt effect

on thf' behavlor

Even though the reactlon tlme of thE' 1000 JI:: controller IS very fast, It cannat

avold the large ovelshoots created by J direct Impact ln facto the value of the force overshoot

IS larger when the MRAC IS active ln practlce. tl1l5 could be dJngerous Jnd cause damage

ta the end-etfec10r or the envlronment The study of Impact ln comtrJlf1ed motion IS J

complex problem related ta the ImpulSive response of the fJst subsystem dynJmlcs [Mills89j

and IS not Investlgated here The srnoothlng polynomial C? must be tuned carefully to tlnd

the best comprornls~ between parameter convergence and behavlor of the force response

74

t

1

1

5 3 Prelimmary Experimental fests

Overall, the simulation results show that the adaptlve controller performs weil

when carefully tuned and can slgnlflcantly Improve the force response The MRAC Will be

useful ln a real system If very hlgh stlffnesses can be encountered dnd compltance ln the

wnst IS undeslrable The !v1RAC can also help ta standardlze the response under wldely

varymg environ ment vararneters

The computatlonal cost of the adaptlve controller wlth parameter estlmator IS

approxlmatlvely 200 floattng pornt operations (when hl 15 frozen) per cycle, excludlng the

direction estlmator and supervlsor A real-life Implementation will .1lso reqUire a secunty

module to check for fallures and dangerous motl0ns The 1000 JI.:: Implementation 15 much

more costly because üf the bst computation r,1te and teed-forward compensation The

convergence of the Jdaptatlon mav be hard to Jchleve ln thls case because the hlgh-speed

controller m.3y eXCite dynamlcs t'lat were not Included ln the ~Imul,)tlon model (JOint and

Ilnk flexlbtllty etc) The 100 Il: Implementation should pertorm better and be rnuch easler

ta Implement

The decision ta use the adaptlve scheme .3l1d the cholce of the Îteratlon rate

should be made by the system deSigner Jnd depend mostly on the <lppltcatlon .3nd the

quallty of the equlpment The valld frequencv r.lnge of the model ~hould be determlned

before choo!>lng the Itf'ratlon r,1te and sensor 5tlffncss Although the SllnUIJtlon model IS

falrly accu rate, there are many unmodeled effects thJt (Jn dlso Intluence the MRAC behavlor

First of ail, most of the simulatIOn tests were run Wlth .) perfect knowledge of the ~urfJce

onentatlon The expenrntnta l ImplementatIOn will requtre ,1 better direction estlmator than

the one used ln the last test The !>Ifnulatlon abo neglects m<lny other effects such as

geartng and backlash, motor tlme constants, wrlst dynamlcs, posItion sensor <lccuracy and

dlgltlZlng, and force ~ensor nOise The latter effec IS very Importz1I1t because It IS necessary

to dlfferentlate the force signai to run the rv1RAC

Wlth a real expenmental apparatus, It will be possible to II1vestlgate the behavlor

of thls adaptlve controller much further by rllnnlng more complex tests such as tracklng a

movrng or curved surface

75

1

1

5 3 PrelimlOary Experimentai Tests

5.3 Preliminary Experimental Tests

The adaptlve damplng control scheme was Implemented on a KALI [Hayward88}

robot controller ln the laboratones of the McGdl Research Center for Intelligent Machines

(McRCIM) and prellmlnary results were presented ln [Daneshmend89] Tflls controller IS

hooked up to J PUMA 560 robot, eqUipped wilh J Lord Ci d 0 f force sensor whlch has

a maximum sample rate of 104 Il:: The testmg was run between 70 El: and 100 Il:

dependmg on the conditions

The non-adaptlve damplng controller was tested flrst and reacted as expected

The behavlor IS very dependent on the environ ment dynamlcs and must be tuned carefully

by varylng the damplng term h-J

ln these Initiai expertments, the adaptlvp. controller dld not stabdlle, and caused

the closed-Ioop system to become unstable for ail values of environ ment stltfnesses tested

The main cause of thls unsatlstactory pertormance appears to be that stlctlon and fflctlon

ln the JOints were very Important Jnd dommated the system dvnamlcs, and also, It was

not pOSSible to dlfferentléite the force Signai because 0t nOise Further testlng Jnd control

on thls robot \'\Id 1 requlfe at least partial compensatIOn for these JOint eHects, and stable

dlfferentlatlon of the force sigoai

Alternatlvely, experlmentatlon on a hlgh performance manlpulator wlth good

actuator and transmiSSion characterlstlcs would be desHable Such manlpulators are under

design, or have been developed, at .J varlet y of ln":tltutlOns 80th CAf Electronlcs and

McRCIM Jre developiOg such Jrms whlCh should be Jvadable for expenmentatlon ln the

near future

76

1

Chapter 6 Conclusion

6.1 Contributions

ln the work presented here, a new compilant controller based on dampmg control

was developed for use ln teleoperatlOn manipulations T!lIS controller uses an adaptrve

scheme based on thE' modelrelerence approach to tune the behavlor of the normal contac.t

force under wldely varyrng environ ment Jnd manlpulator parameters The adaptatIon IS

performed only ln the surface normal direction, usmg a d,rectlonal controller ln the task

reference frame

This scherne was tested wrth srngle and multl-aXiS simulatrons and results at two

drfferent Iteration rates show the usefulness of the adaptation to Ifnprove the performance

of the damplng controller The effects of frrctlon. Impact. estimation schemes. tangent!al

motion and Imperfect surface direction estimatIOn were rnvestlgated dnd 111 ail cases, wlth

proper tumng, the MRAC can Improve the behav!or of the non-ad<lptil/e controller

The actIOn of thE' b1RAC Jutomatlcally tunes the response ta have a sufflclent

amount of compliance ln the controller, thus ehmlnatmg the slugglsh behavlor at low stlff­

nesses and preservmg stabdrty at much hlgher stlffnesses than ongm<"llly possible The re­

sponse cf the system to torce dernands becomes rndependent of the environ ment parameters,

therefore allowlOg unlform control commands for the user at the Input Interface.

,

1

1

6 2 Future Work

It IS clear that the use of the adaptlve scheme developed here IS not IImited to

teleoperation The Interface to the user. usually some sort of hand controller, can be replaced

with a hlgher level controller or traJectory plan'ler ln Industnal applicatIOns, the adaptlve

dampmg control schem'" could be consldered as an alternative to hybnd position/force

control for example

The ongmaiJty of the work presented here Iles ln the application of adaptlve

control to a damplng controller. and ln the concept of dlrectlOnal adaptation ln constramed

motion ln the thesls, the last part of Chapter three on the adaptlve damplng control, the

dlrectlonal adaptation scheme of Chapter four and the results ln Chapter flve are ail ongl­

nal contributions to knowledge developed by the author ur.der the slJpervlslon of Laeeque

Oaneshmend

6.2 Future Work

The next step ln the development of thl5 teleoperatlOn controller, IS to success­

fully Implement the adaptlve scheme on an expenmental set-up Ta praye that the scheme

15 feaslble and useful for performJnce Improvements, rndny tests will h<lve to be run the see

the effect of the neglected dynarmcs and mechanlcill Imperfections of il reill manlpuliltor

system To perform these tests J better robot wlth low friction ilnd effiCient Jctuators will

have to be used. to Insure V;dldlty of the simple second-order model Another solutIon to

the bldS problem rnay have ta be developed If the force sensor signai cannot be dtfferentlilted

adequately

ln future tests. It wdl be useful to develop a more accurate direction estimation

scheme than the one used here It would abo be Interestmg to study the sensltlvlty of the

adaptlve 5cheme to Imperfect knowledge ot the surface onentatlon, and compare It ta hybrld

control

The performance of thls control scheme also has ta be evaluated from the human­

interfaclng pOint of vlew When the system become~ avallable, many tests should be run

18

1

1

62 Future Work

to measure task completion tlmes and ease of use under dlfferent conditions such as partial

vision or tlme delays ln the response Even though t he robot exerts a force proportlonal to

the force the user IS applYlng on the Joystlck. some sort of feedback of force ta the user will

Ilkely be necessary, probably through the use of vlsual dlsplays

ln the present study, the concept of dlrectlOnal adaptation was developed for use

wlth a damplng controller, to adapt ta the stlffness of a plane, flxed surface This Idea cou Id

be reformulated ln a more general manner to be able to de;)1 wlth any type of envlronment ln

the same way, the dlrectlonal adaptation could be apphed to a generallmpedance controller

mstead of J simple damplng controller This could Jllow the development of adaptlve

Impedance control/ers ln whlch the interactIOn wlth the envlron;nent IS constantly monltored

by the adaptlve controller ta tune the performance c1nd pre'ierve stabliity

Wlth thls more general formulation. the adaptlve scheme developed here can be

useful for a much wlder range of applicatIons thJn urtglnally proposed The scheme works

weil for the resolved-rate teleoperatlon mode tested ln Simulation dnd therefore, should also

work ln dlfferent applications It IS hoped thJt expemnentJI Implementation will verlfy the

practlcallty of thls scheme and provlde Inslghts leadlng to further rehnem~nts

79

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1

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