IDENTIFICATION OF PARAMETERS AND CONTROL OF ROBOTIC MANIPULATORS USING ARTIFICIAL NEURAL NETWORKS

IDENTIFICATION OF PARAMETERS AND CONTROL OF ROBOTIC MANIPULATORS USING

ARTIFICIAL NEURAL NETWORKS

By

©Rajn ishAggarwal (B.E.)

A Thesissubmitted to the School of Graduate StudiesIn partial fulfilment of the

requirements

tor

the degree of Master of Engineer ing

Faculty of Engineering and Applied Science Memorial University of Newfoundland

August 1995

SI. John's Newfoundland Canada

1+1

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ABSTRACT

Inthepresentwor1l.the estimation of dynamic parametersandthetrajectory

control01atwolink planar manipulatoriscarried out.Thederivationof system equationsinvolve thekinema ticparam eters such asjoin tpositions.velocitiesWld accelerations.Forboth thedynamicparameterestimationandthetrajectorycontrol an ArtificialNeuralNc-tworksmethodcalled.UnaarProgramming(l PI-Neuro Method.i'iused.In this algori thm, theweigh ts are obtainedbya combination of linearprogramming having asparse coeHicienlmatrixandasingle variablenon- linearoptimizationroutine.

The training setvaluesrequiredforparame terestimation aregenerated by the IterativeNewton-Euler DynamicsAlgorithm.TheArtificialNeuralNetworkis trained10prediC1lhedynamicparameters.The valuesof the forces andtorques arerecomputed basedonthe estimated dynamicparameters.Thismethod is usefulfortheon lineparameter estimation ofthemanipulatorsnavingoddmass distribution,which isthecasein actualpractice.

Forthecontrol problem non-linearoptimizationmethodis usedtoevaluate thegain param etersrequired for the manipulator tofollow the desiredtrajectory.

Thenthe UnearProgramming(lP)·Ne uroM~thodIsusedtoobtaintheweight matrix whichrerates theInput(Joint positions andvelocitIes) andtheoutput(gain parameters).Thisweight matri x,for each point along the trajecto ry,canbeused on-line toevaluate the gainparametersthereby elimi nati ng the lime consuming

calculationsat eachpoint alongthetrajectory.Finally,the effect of thevariation01 the maximumtangentialvelocityandgeneralcontrollaw are studied.

ACKNOWLEDGE MENTS

With sincereappreeialiOflandgratitud eI wouldliketothank my supervisor, Dr. Anand M.Sharan.lorhisinvaluable direction. support.patientguida nceand constantencou ragement during thecourse ofthis work.Workingwith him wasa valuable experiencebothpersonallyandprofess ionally.

Iwouldalsolike 10 extendmythanks10Dr.John Malpa e,De an of$chool of Graduate Studiesforsupporting mefinancially throug h thisperiod.Iamalso thankful to Or.R.Seshadri,Dean01EngIneering an dOr.J.J.Sharp,Associate DeanofEngineering fortheircoopera tionandnecessary helpduringthis graduate program,

Finally,Iwouldlike tothank allmy fnendslortheirconstantencourage ment and moral support.

Thiswork isdedicatedtomyparents.

TABLEOFCONTENTS

ABSTRACT ACKNOWLEDGEMENTS

TABLE OF CONTENTS LIST OF FIGURES LIST OFTABLES NOMENCLATURE

CHAPTER1 INTRODUCTIONAND LITERATURE SURVEY

1.1 INTRODUCTION 1.2 LITERATURE SURVEY

1.2.1 ARTIFICIALNEURALNETWORKS 1.2.2ESTIMATION OF PARAMETERS 1.2.3 ARTIFICIALNEURALNETWORK

CONTROL IN ROBOTICS 1.3 OBJECTIVESOFTHE THESIS

CHAPTER2 ESTIMATIONOF DYNAMIC PARAMETERS

2.1 INTRODUCTION

vi

lv vi

"

xiii

10

2.2 ROBOT KINEMATICS 11

2,2.1 INVE RSE KINEMATICS 11

2.3 ROBOT DYNAMICS 14

2,3,1GENERAT IONOF INERTIAIII- MATRIX 14

2.4 PARAMETERIDENTIFICATION 19

2.4.1 THE LINEARPROGRAMMING-NEURO

(LPN)METHOD 19

2.4 .2 ARTIFICIAL NEUR AL NETWORK

(ANN)INPARAMETERESTIMATION 23

2,5 RESULTS ANDDISCUSSIONS 26

2,5 CONCLUSIONS 29

CHAPTER 3 CONTROL OFROBOTIC MANIPULATORS

3,1 INTRODUCTION 40

3,2 DYNAMICEQUATIONS OF APLANAR

TWO LINK MANIPULATOR 41

3,3 TRAJECTORY CONTRO L 43

3,3,1INDEPENDE NT POCONTROL 43

3,3,2EVALUATI ON OFGAIN PARAMETER S 44 3,3,3HOOKE"; ANDJEEVESNON-LINEAR

OPTIMIZATIONMETHOD 46

vii

3.3.4NUMERICAL INTEGRATION TECHNIOUE 3.3.4.1 RUNGE·KUTTAMETHOD FOR

A SECOND ORDER DIFFERENTIALEOUATION 3.3.4.2 RUNGE·KUTTAMETHODFOR

THE CONTROLPROBLEM 3.4 TRAJECTORY GENERATION

3.5 OPTIMALCONTROLMETHOD (NON·LINEAROPTIMIZATION) 3.6 ARTIFICIAL NEURAL NETWORKSIN

TRAJECTORY CONTROL

3.7 THE EFFECT OFVARIATIONOF MAXIMUM TANGENTIALVELOCITY

3.8 THE GENERAL CONTROL LAW 3.9 CONCLUSIONS

CHAPTER 4 CONCLUSIONS ANORECOMMENDATIONS 4.1 DISCUSSIONAND CONCLUSIONS 4.2 RECOMMENDATIONSFOR F' 'TUREWORK

REFERENCES

viii

49

49

52 54

62

64

77

87 92

93 95

96

APPENDIXA EXPRESSIONSFOR INERTIA TENSOR 101

APPENDIXB FORMULAFORROTATIONABOUT THE

PRINCIPALAXIS BY

e

104

APPENDIXC EXPRE SSIONSFORTHEDERIVAT ION

OF THE ITERAT IVENEWTON ·EULER

OYNAMICALGORITHM lOS

APPENDIX D PROG RAMLISTIN GS 110

LIST OF FIGURES

tiQ DESCRIPTIQ!:!. PAGE

t.t THEUNIMATIONPUMA 500 IN OPERATION 1.2 THE PUMA·56DMANIPULATOR

2.1 PLANARTWO·L1NKMANIPULATO R 12

2.2 ALINK WITHREGULARMASS DISTRIBUTION 16 2.3 PLANARTWO-LINKMANIPULATOR

WITHIRREGULA RMASS DISTRIB UTION 18

2.4 DIAGRAMMATICREPRESENTATIONOFTHE

NETWO RK(LP·NEURO METHOD) 20

2.5 DESIREDTRAJECTOR Y 30

2.6 VARIAT IONOF TANGENTIAL VELOCITYALONG

THE TRAJECTOR Y 31

2.7 FLOWCHART :DYNAMIC PARAMETER ESTIMATI ON 32 2.6 FLOW CHART:COMPARISONOF FORCES AND

TORQU ESCORRESPONDINGTOESTIMATED

PARAMETERS 33

2.9 ERROR VALUES IN F.FORLINKS 1 AND2 34 2.10 ERROR VALUES INF, FOR LINKS 1 AND2 35 2.11 ERROR VALUES INt,FORLINKS 1 AND2 36

2,12 ERROR VALUES INtvFORLINKS1AND2 37

2.13 ERROR VALUESIN'. FOR LINKS1AND2 38

2.14 PERCENTAGEERRORINt_lFOR LINKStAND 2 39 3.1 SPECIFICATIONSOFTHE DESIREDAND

ACTUALTRAJECTORY 45

3.2 DESIRED TRAJECTORY 55

3.3 DESIRED TANGENTIALVELOCITY PROFILE 58 3.4 DESIRED CARTESIANTRAJECTORY ACCORDING

TOTABLE 3.3 61

3.5 FLOWCHART:OPTIMAL CONTROLMETHOD

(NON·L1NEAROPTIM IZATION) 63

3.6 FLOW CHART:LINEARPROGRAMMING(LP -NEURD)

METHOD(ANN METHOD) 66

3.7 VARIATION OF8, ALONGTHE TRAJECTORY 68 3.8 VARIATIONOF

e ,

^{ALONG THETRAJECTORY} 69 3.9 VARIATIONOF

e ,

^{ALONG THETRAJECTORY 70}

3.10 COMPARISONOF

1<"

VALUESOBTAIN EDUSING OPTIMALCDNT ROLAND ANNMETHOD 72 3.11 COMPAR ISONOF

1<"

VALUESOBTAINED USING

OPTIMALCONTROLAN'OANN METHOD 73

~.1 2 COMPARISONOF

1<..,

VALUES OBTAINEDUSING

OPTIMALCONTROLAND ANNMETHOD 7.

3.13 COMPARISONOF

K"

VALUES OBTAIN EDUSING

OPTIMALCONTRO LAND ANN METHOD 75

xi

3.14 TRAJECTORYCONTROLOBTAINEDUSING

OPTIMAL CONTROLAND ANNMETHOD 76

3.15 VELOCITYVARIATIONACCOR DING TO TABLE3.S 79 3.16 DES IRED TRAJECTORY AND THE TRAJECTORY

OBTAINED USING ANN METHOD WHEN

THE MAXIMUMVELOCITY IS 0.15mi. 60 3.17 DESIRED TRAJECTORYANDTHE TRAJECTORY

OBTAINED USING ANNMETHODWHEN THE MAXIMUMVELOCITYIS 0.15mi.

(SMOOTH CORNER) 81

3.18 DESIREDTRAJECTORYANDTHETRAJECTO RY OBTA INED USING ANNMETHOD WHEN

THE MAXIMUM VELOCITY IS 0.'m1s 82

3.19 DESIR EDTRAJECTORY AND THETRAJECTORY OBTAINED USINGANN METHOD WHEN

THE MAXIMUMVELOCITY IS0.05mi. 83

3.20 DESIRED~RAJECTORYANDTHE TRAJECTORY OBTAINED USING ANN METHOD WHEN

THEMAXIMUMVELOCITYIS 0.001

mi.

84

3.21 DESIREDTRAJECTORYAND THE TRAJECTORY OBTAINED USINGANN METHODWHEN THE MAXIMUMVELOC ITY IS 2inch/min

xli

(0.0008466mls) 65 3.22 DESIREDTRAJECTORY ANDTHE TRAJECTORY

OBTAINED USING ANN METHODWHEN TH EMAXIMUMVELOCITYIS 2inch/min

(0.0008466mls : SMOOTHCORNER) 66

3.23 VARIATIONOF THENORMALIZED ERROR BETWEEN

THEOBJECTIVE FUNCTIONS 91

A.l RIG IDBODYWITHAN ATIACHEDFRAME 102

c.i FREEBODYDIAGRAMOFA LINK 108

xiii

LISTOFTABLES

1JQ DESCRIPTIO!!. PAGE

2.1 THE ITERATIVE NEWTON-EULERDYNAMICS

ALGORIT HM 15

2.2 INPUTVECTOR,OUTPUTVECTORANDWEIGHT

MATRIXUSED FOR TRAINING 25

2.3 VARIOUSPARAMETERS USEDFOR A TWO-LINK

MANIPULATOR 27

2.' COMPARISONOF ESTIMATED DYNAMIC

PARAMETER S 28

3,1 RUNGE·KUTIA METHODINVARIABLE FOAM 51 3.2 VARIOUSPARAMETERSUSED FORA TWO-LINK

MANIPULATOR 57

3,3 VARIATIO NOFJOINTDISPLACEM ENTVALUES 60 3.4 COMPARISONOFGAINVALUES OBTAINED

USINGOPTIMALCONTROL(NON-LINEAR

OPTIM IZATION)ANDANN METHOD 71

3.5 VARIOUSMAXIMUMTANGENTIALVELOCITYVALUES 78 3,6 GAINVAL UESOBTAINEDUSINGOPT IMAL

CONTROLMETHODWITH4 DESIGNVARIABLES 88 3,7 GAINVALUESOBTAINEDUSINGOPTIMALCONTROL

MET HODFORTHEGENERALCONTROLLAW 89

3.B COMPA R ISON OFOBJECTIVE FUNCTIO N VALUES OBTAINEDUS ING 4DESIGN VARIABLEANO

GENERALCONT ROLLAW 90

{}

II

10

(Wi,[VJ {II,{D},(D}

NOMENCLATURE

vec tororcolumnmatrix matrix

activationfunction weight matr ices

input,desiredandoutput vectors respectively link lengths

link masses radiu s of thecircular arc

a,b,c length. height andwidthof linkrespectively

ell! inert ia matrixina framelocated atthecentreof mass ofthe body

'[IJ '[II

A{PC}

'{PeJ [C,I X,Y,

X , Y, X, Y

ine rtiamatrix inanarbitrarily translat ed frame finalinertia matrix involving massmoment ofinertiaand mass product ofinertiaterms

positionvector ofthe

e.G.

of thelinkinafram elocat ed altha centreof mass ofthebody

positionvectorof the e,G.of thelink inan arbitra rilytranslated frame

finalpositionvectorof the

e.G.

ofthelink orthogo naltransforma tion mat rix,X-Y·Zllxed angle matrix cartesiandisplacements,velocitiesandaccelerations

xvi

nx1vectorcontainingcentrifugalandcoriollsterms nx1vectorofgravity terms

accelerationdue to gravity timeincrement for Hunqe-Kutta method

exponents 01the jointdisplacementerror andthejointvelocity error in the generalcontrol law

h, .6T A"A, [M(e)]

(V(e,a)}

(G(e)}

IK,J IK,]

u

e, 6" e

j displacement.angular velocityand angularaccele ration 01the

ith joint respectively

c.

p,

y rotation about Z,Y andX axes respectively

F.t forceandtorque

edt6~ desired joint displacementand Jointvelocity

ea.

6. actual jointdisplacementand joint velocity errorinjoint position

error injointvelocity gain parameters proportionalgainmatrix velocitygain matrix objective function radialandtangentialvelocity radialand tangential acceleration n x n mass matrix ofthe manipulator

xvii

CHAPTER 1

INTRODUCTION AND LITERATURE SURVEY

1.1 INTRODUCTION

In the recentyears therobot;have foundan everincre asing use inthe field of industrial automation.Industrial robotsare alreadyassumingmanyhazardous, unpleasant or monotonoustasks, while simultaneously improvingthe productivity of factoriesinthe industrializedworld.Robots potentiallyfindapplicationsin the hazardousorinaccessible environments,such as,nuclearreactors,furnace operations,mines,deep sea andouterspace.Fig. 1.1showsthe UnimalionPUMA 500 (Programm ableUniversalManipulatorfor Assembly) inop eration andFig. 1.2 showsthe rigid body modelofthe 6Degree-ot-Freedom(OOF) PUMA-S6D manipulator.Thestudy of mechanicsand controlofmanipulatorsis not a new science, but merelyacollection oftopics takenfromclassicalfields.Controltheory providestools for designing and evaluatingalgorithmsto realize desiredmotions orforce application.

Althoughcomputersoutperformboth biologicalandartificialneuralsystems for tasksbasedon preciseand fast arilhmelicoperations, artificialneuralsystems representthepromising new generationof informationprocessingnetworks.

Advanceshave been madein applyingsuch systemsfor problems found

Fig.1 .1 THE UNIMATIONPUMA 500 IN OPERATION (SpongandV1dyasagar,1989]

Flg.l.2 THEPUMA·560MANIPULATOR

intractableor difficult for traditional computation.Since,the area ofrobotics requires numerous andcomplicatedon-linecalculations50, ArtificialNeural Networks (ANN) findanimmediateapplicationinthis area.

1.2 LITERATURESURVEY

1.2.1 ARTIFICIALNEURAL NETWORKS

The useofArtificialNeu ralNetworks hasincreasedtremendously In recent yea rs because of the availabilityoffasterandparallel processors andthebasic learningalgorithms(Grossbe rg, 1982;Hopfield, 1982;Rumelhart andMcClelland, 1986;Kohonen,1988).Neuralnetwo rk can learnmappingbetween theinputand outputspace andsynthesizeanassociativememorythatretrieves theappropriate outputwhenpresentedwithaninput,andhasthe ability to generalizewithnew inputs,ANNsare being used to accomplish complexfunctions such as generalizat ion,errorcorrection,info rmationreconstruction, pattern analysisand learning.Because of their massively parallel nature,neuralnetworkscanperform computationsatvery high speeds (FukudaandShibata,1992).

Neuralnetworkshave alsobeen usedto successlullysolve complex problemsliketheTravellingSalesman problem.Ithas beenobservedthat neural netwo rks have often beenopportunistic,i.e.thenetworkmodel iscustomizedto serve theneeds ofthe taskat hand(Kulkarni, 1991).They represent anew app roachthatisrobu stand fault-tole rant.

Neural networksrequire basic algorithms loraccomplishingthe learning

task.Severalalgorithmsarefunctionalat present.One suchalgorithmwhich is widely used is backpropagalio n(BP)algorithm.In backpropagation algorithm, during theleamingphase, the observedoutputsare comparedwiththe desired outputs, andtheweightsare optimized10 minimizeIheerro rfunction. In compe titivelearning, the weightsare updatedwitheachnew input(Aumelhartand McClell and,t986).BannannandBiegler-Kong(1992)discuss efficient learning algorithmslorneural networks.

Neuralnetworkscan performfunctionalapproximatio nsthatarebeyond the scopeofoptimal linear techniques.Gulatl etal.,(1990)have lntrccscednaural formalismto efficientry leamnon -linearmappingusing amathemat ical construct caned terminal anractcrs.

Neural networkshavebee nfounduseful in the fieldofroboticsIn therecent times.Forward andinversedisp lacementanalyses01robotic manip ulatorshave beendon ebyNyugan at al.(1990) and Gulallet al.,(1990).Neuralnetworks seem 10 bea promising approachto solvenon-linearcontrolproblemsas well(TabslY andSalaun,1992).Someotherinterestingapplicationsinthe co ntrolofrobotic manipulat ors ca n beseenIn Fukuda atal.,1991; andAkioetal.,1992.

1.2.2ESTIMATIONOF PARAMETERS

The kinematicparamete ridentificationtechniques were reviewed by Hollerbach(1989). Many research ers (Ma"ednat at 1984;Khosla and Kandate 1985;Kawasakiand Nishimura1986;An at.al.1986; Atkans onet.al.1986]have

worke dtoestimate thedynam ic param e ters.Themeth o d 01calculationof thebase Inertialparametersofclosed-looprobotsthroughsymbolic computation sis given byKhalil andBennis(1995).A metho dofestimatingthemass propertiesofa manipulatorby measuringthereaction momentsat thebasehas been discussed byDubowsky and Cheah (19 89). The se estimates arenot accurateenough for dyna miccontrol. Amethod ofgenerati ngthevariableforgettinglactorfor on-line parameterestimation,hasbeenstudie d by WeipingLi andStotlne(19 88).These estimatesareapplicable tolinearlyparameterizedmo delsonly. One ofthe ways of ascertainingthese (the dynamicparameters)wouldbebyusin gArtifi'~ialNeu ral Networks(A NN).The applicationof NeuralNetworks lor the identificationand controloflo w order nonlinea rdynamica lsystemsis studied by Narendra and Parthasarathy(1990 ). In the presentinvestigation,an ANN method is usedto estima te the dynamicparam et ers and is discussed in detailin Chapter2 ofthis thesis.This would be usinga fasteralgorithmdeveloped byBalasubramanlanand Sharan(199 3).

1.2.3ARTIFICIALNE URAL NETWORKCONTROL IN ROBOTICS Therehas been recanttrenet within theroboticscontrolliteratureto ap ply neuralnetworksforthecontrolofrobotic systems.In manyapplicationsreported inthe literatu re(Guand Chan,1969;Fukud a andShIbata,1990;He lferty and Biswas,199 0;Jamshidiet at,1990;Karakasoglu and Sundaresha n,19 9 0:

Yamamuraatal.,19 9 0) theprocess ofneura l network learning is conducte don- line (La.the dynamics01theneuralne twork is embeddedInthe closed -loopwith

the dynamicsof therobot icsystem),yet thereappearstobelac kofstudies focussing onthedynamicbehavioroftheneuralnetworkduringlea rning and'or controlwh enthene ural networkisused insuch conten.

Kaw ato(1990) used feedback error leaming10comp utethe feedforward lorq ues re q uired for a manipulator10fo llowa path.The neural netwo rk imp lemente d inthis methoduses the desiredjoi ntpositions. velocitiesand accelerationsasinputsandadjuststhe nefwor1(weights usingthefeedbacktorque as theerro rsignalto a backpropagallonparameteroptimizing algo rithm. Yuh (1992)alsouseda neural networkfor manipulatorcontrol. Heuseda"critic' equation,which is a functionc,fthemanipula tor outp uterror, totrainthenetw ork todir ectlycomputethe mar.ipu lalor In puttor ques,

Asada(1990) useda multilayeredleed forward network toleama non-linear mappi ng forcomplianc econ t rol.Fromthemeasur e dforees and torquesin an asse mbly taskheusedthe network 10 compotethe requiredveloc iti es,which wou ldallowtheass embly task10 be completed,Oza kiatal,(t991)presentsthe method01trajectorycootrolofrcbotcrnancuratcrsusin g a non linear compensator, Theadaptivecapabi lity of neuratne MO ""controller tocompensate unstruc1ured uncertaintiesisdiscussed.A method01uSIngtwoneuralnetworks for the control of a robotic armis suggested by Snonametal.(1992),The neu ra lnetwork weights whichareus ually chosenarb Itrarily have beendeterminedIn asystematic waybased ona geometricjnte rprerancnoltha neuron function.Balasubraman ian (199 3 )usedANNover alimite d range cttrarectorycontrol,

1.3 OBJECTIVES OF THE THESIS

We have seen in the tast few sectionsthatthe neural networks are quite versatiletools to solve problemsina widevarietyof areas.With this in mind.II was thoughtto applythis 1001 to solveproblemsIn theareas of parameter estimationand controlofrobolicmanipulat ors. Also. basedon thereview ofthe literature ,the followingare the objectivesof this thesis:

1. The identificationof thedyn amic parameters01rob otic manipulators.

2. Theoptimalcontrolof the robotic manipulatorsusing the Hooke s and JeevesDirectSearchnon-linearoptimizationmethodoverapractical range of a trajectory.

3. Use ofANN method for the trajecto ry controlover a practicalrange.

4. Studytheeffectofthe variationofth e maximumtangentialvelocity over a tra jectory on the controllability(erro r) of amanipul ator.

5. Study the generalconlrol lawin the trajectorycontrol.

InChapter2the inversekine maticrelations hipis establishedbetweenthe Cartesian andjointdisplacement,velocityand accelerationvalues onoff-Unebasis whichred uceson·linecomputationtime. The ANNmetho d,linear programm ing · neurn (L PN) me thod,is discussedin thischapte r. This method is a non-linea r method wheremost,but not all ,01the comp utallonsaredoneusinglinear program m ing.This methodisfa stconv erging as compared tomany other methods.Inthis chapterthegenera tionofInertia Matrix[I]for an irre gularsh aped

link is discussed.Finally,in thischapter theresults of estimationof dynamic parametersare reported.

In Chapter 3 the various controlissues fora twolinkplanar manipulatorare studied.Herethe optimalcontrolmethod(non-Hnearoptimization) and linear programming -neuro (LPN)methodisusedin theon-line roboticcontrol.The effectof variation of the maximumvelocityovera trajectory and generalcontrol laware alsoincluded in thischapter.

Finally, conclusions and recommendationsfor futureworkare presented in Chapter4.

CHAPTER 2

ESTIMATION OF DYNAMIC PARAMETERS

2.1 INTRODUCTION

A manipulatorconsistingofconcatenatedrigid linkshas highly non-linear dynamics.Thejointtorques are relatedto thejoint angles.velocitiesand accelerationsby asetof non-linearequations involving various constant parameters suchaslinklengths, masses,inertias andso on,Theseparameters areseparatedintotwogroups:the kinematic parameters, and the dynamic parameters.The kinematicparameters do notincludeanyphysicalquantitiessuch as massorinertia, while the dynamicparametersInclude them.To control the roboticmechanismsitisrequired 10knowbolh groupsof parameters.

Inactual pract ice themassdistribution alonganylinkisquite complexdue tothe various hydrauli cdrives orotheraccessoriesattachedto these links. The theoreticalvalues(dynamic parameters) are notaccu ratelyknown asaresult 01 this.Therefore , forprecisioncontrolone has toincorporate these quantities in the model.

In thischapter,at first.the unknowndynamicparame ters areidentified usingANNmethod. Thenetwork istrained by conside ringthe forces and torques acting on links asinputparametersandposition01thecentreofgravity of the links

10

aswellasthe various moments ofinertias as the outputparameters. Then, the training the unknown outputparametersofthe links areobtainedusingthe trained ANN.

2.2 ROBOT KINEMATICS

2.2.1INVERSE KINEMAnCS

Fig.2.1showsa planar two-linkmanipulatorhavingrevolute Joints. Inthis figure, (Xo' Yo)representstheglobalcoordinatesystemand(XI'Y_I)representthe localcoordinateframeoflink'i'(i=1,2).Thejointvariables are given by el•

Inthis figure thepositionof theendeffectorIsrepresentedbythe coordinatesX andY.Theinversesolullon at this pointcan becarriedout by the use of followingequations,insequence:

=sin6z:we startwith

(2.1)

(2.2)

From this, we get62as

(2.3) IIwehave two intenne diale variables k1and k2as

11

Fig.2.1 PLANAR TWo-UNK MANIPULATOR

12

(2.4)

(2.5)

then we can obtain

(2.6) Theaboveinverse kinematicequations can be written invectorform as

(2.7)

where I,. 1₂are the lengthsof link1 and 2respec tively.

DifferentiatingEq. (2.7)withrespectto time,we gel

(2.8)

Finally, the equations fortheaccelerationofthe end-effectorcanbe expressed in terms of both thefirstan dsecondderivatives of9₁and9₂as

(2.9)

13

2.3 ROBOT DYNAMICS

2.3.1 GENERATION OFINERTIA [1]-MATRIX

As discussedinthe introduction,theinertia matricesof therobotic

man ipulators are notknown in most practicalsituations.Inactua lcontrolofrobotic manip ulatorsonehas 10 computeforces andtorques whichcan becalculated usingthe IterativeNewt on-Eu lerDynamics Algorithmshown in Table 2.1.Asone canseein thistable,oneneedstopreciselyknow the numericalvaluesof the matrices(

1 1

shown inSleps6·8inorder tocalculate the joint forces and torques.

Fig.2.2shows alinkof regular geome trywhose inertiamatrices,in the framec() aregiven by[Rothbart,1973]

'[I]·

Thismatrix inframe...{ )willbegivenby[Craig,1994]

(2.10)

11] • 11) -

^m['(P,}''(p,)II']-'(P,}

'(P, } ']

(2.11) where"{Pc}""

{x".

y~,

z , t

^{locate sthecentre of}mass relativeto~{} andl(]Isthe 3 x 3 identity matrix.

In thisfigure we also have aframe9( }which isobtained byrotatingthe frame~{} abo utX"by anangle1,thenaboutY"by an angle~and finallyabout Z~by anangleCt.Hence,the transfonn ationmatrix,[e,l,relatin gthe frames"{}

14

TA BLE 2.1 THEITERATIV E NEWTON·EULER DYNA MICS ALGORITHM [CRAIG,1994)

FORWARD RECURS ION

Step 1: fro}, '"

[RJJ

{ro},., {z}

8 ,

Step 2: {a}I "

[RJ;

{a},., {z}

a, ..

(Rj J{ro}1x{z}

0,

Step 3: {a)1'"

tRJJ

({a},_1 ...{a}/-l x{P)I_' ...{roll-!x(p}/-l) ...(z)

a, ...

2x

[RJ;

fro}"~x{z}

9/

Step 4: {aIel'"{a}, ...{a}, x(5), ...fro},x fro}, xIs}, Step 5: {F}, . mj{a}eI

Slep 6: {N}, ' [~,{a}, • (oo),x([~,(oo),)

BACKWARDRECURSION

Step7: {~" (F),• [RI,.,{~,.,

Step 8: (n), '{RI,.,{n),., • (N), .(s),x(F), • {PI,x {[RI,.,(~,.,) Step 9: 'tj .. (z){nIT .. nil

15

Fig.2.2 A LINK WITH REGULAR MASS DISTRIBUTION

16

ands{ } canbewrittenas

:(C,I '

:[R ...

ly,~,a)J⁼

[R , (a)!

[R,I~)!

IR,ln!

.f t : : _{~s: ~][c: ~ 5:] r~} :y _ : y]

o 0 1

-5 ~

⁰

c~ l ~

^5ycy(2.12)

" [ca c~ ca s~sy-sacy cas~cy.sasy]

sac13 sa s13s1"cacysasl!CY-casy

·sp CPSl clky

whichcan beused 10 transform avectorInI{ }spaceintoA{ }space.

To obtain theinertia matrix in frame

Sf }

^{one hastousethe}equation [D'Souza and Garg,1988;Greenwoo d,19701

'[ I)"[C,I' '(I )[C, ) (2.13)

Applyingthe same transformationmalnll. (C.I.tothe position vectorof the centroid ofthelink we get

~Pc '•[C.J' '(Pc) ^(2.14)

Forthislinkthe InertiamatnxIIIInboth c{)andA{ }frameswillbea diagonalmatrix.Onthe otherhand.on1M8{Ilrameitwillbea symmetricbutfull matrix (all elementsnon-zero).Acomptell mass distributio nofthe robotic links wouldalso have its Inertia metnx.alull ".,alnllas expresse din8[1]. Fig.2.3shows IhtJ III} frameandirregular massdlsl nbuhon ofa link.Thus,if the matrix8{1}Is known thenonecancompu telorees.lndtorquesfromTable 2.1.

17

Flg .2.3 PLANAR TWO·L1NKMANIPULATOR WITH IRREGULAR MASS DISTRIBUTION

18

2.4 PARAMETER IDENTIFICATION

2.4.1 THE LINEAR PROGRAMMING·NEURO(LPN) METHOD Beforedescribingthe detailsof the parameteridentificationbyusingANN methods,letus discuss brieflyone suchmethod calledtheLP-Neuro method(LPN method).The basic details aboutthe ArtificialNeuralNetworks canbe seenin any standardtext such as [Hertzatal.,1991;Zurada,1992J.This,theLPNmethod, is a non-linearmethodwheremost,but nol all,ofthe computationsaredone using linearprogramming. This methodutilizesthe faster convergencepropertyof linear programmingwhichresults Inbetter errorminimization.

TheLPNmethodis diagrammaticallyexplained inFig. 2.4.Here the weight matrix

(WJ

whichrelates {IJ.'lrtj{H}is givenby [BalasubramanianandSharan, 1993J

h, w" w" ^••.W,k (H)

.

^{h, w" w" ...W2k}

h, w" wI'

. . W .

(2.15)

The activatio n functionusedinthis caseIs

(2.16)

In Eq.(2.16), each variable h_Jhas a correspondingscalar

m ,

^{anda constantoJ'}

The relationshipbetween {OJ (output)and {I}(input)is thereforegivenby

19

Cj

~dl

n 9'

~ d'

92

~dl

9j DESIRED

OUTPUT

._/J~L. __

OUTPUT

LAYER

\ (O}

-- -iiii··---·

INPUT

LAYER

f-_ L :

: i1

i,O~-"'=,.L-~~~::':!"{

:i.a"t---:.~>_!!!!i.(

... . . .. .1

Flg.2.4 DIAGRAMMATIC REPRESENTATIONOF THE NETWORK (Lp·NEUROMETHOD)

20

{OJ • 1M ) [W ] {I) - IC) (2.17) where,

{M)is the diagonalslope matrix.whichhasscalar"\ asits diagonalelements,and

(elistheintercept vector.

A similaractivalionfunction for thedesired outputsido(shownbythe box on the extreme rightin Fig.2.4) canbewritten as

IS)-[NJlD) • (G) (2.18)

where (N]is a diagonalmatrix.The matrices [N]and {G}are analogous10[M]and {e lin Eq.(2.17).Oneof the ways10obtainthesetofweight matrices with minimumerror wouldbeby writing a costfunctlonE,inthefollowing form:

Minimiz.E,=[NIlD]+IG)-(M)[WJ{I)•{C)

subject 10

[M)[WJ(J)+IC) : [NIlD)+{Gl

orinthe scalarfonn,itcan berewritten as

Minimize E, ::n,d,+":ld2+..,+f\~

-

m,{wlIl,+

w , *

^{+•..+wu.iJ• rn2(w211,}

+

w ni2

+...•W2lo~}~...~~{W.lil+W~2+...+w..\}+9,+92+..•+9_1•C,- e,-...-~

subjecllO

21

whered_jarethe elements ofthe desired outputvector and w,..are theweights .The weights wI<can be expressedinasingledimensio nalarrayor avecto r as

(2.19)

The weightsW1_.

l1

andQ₁when replaced bytwo positivenumbers (inthelinear programming,thevariablesare required to have non-negative values only)can be writte nas

WI"' Y,·v.•

c;==Yo-V,.. 'and i==1,2,..•

After these substitutions,onearnvesat

I: 1.3.... (2.20)

MinimizeEI==n.d,+Oz~+ nd.m.II.V, -i,vz⁺izv,-Izv.+..)-mz(i,vzo..,

-I,v_2II•2+.•.)-m₃(i.v , / l-m_l(n.+\,v21<"•\.v~+V,II-

VOl+...-vel+Va• subjectto

[AllV)=

to}

whe rethecoefficient matrix [A]isgIventy

(2.21)

I:

-mJ,-111,10-m,1"1-I ·1 I 0 0 00 0 000 0_0 0 00 0 00 00]

o 0 0 0 0 00 0Owy,-mJ.,-m,J"...'· , -11 -0 0 00 0 00 00 , • • • • •' • • • • • . . •• ' • • • , • • • I . I

o 0 0 0 - 00000 0 00 _ 0 0 00_"\ ...-m., ""'"'\I.,-I1

(2.2 1.) Here,the coefficient matrix(AI.is asparse matrix.Intheabove equation,~are thedesired values and~aretheinputvalues;VI_velandvQlare the unknown variables andsoarern_jand oJ' The problemshownin Eq.(2.21) isnon-linear becauseofthe occurrenceofproduct termssuchasm.v,...etc.However,ifthis problemis combinedwith anothe r multi-variabl eoptimization problemcontaining allffilonly, thentheprob leminvolvingtherem ainingvariables canbesolvedby thelinearprogramming method.Since.thenumberofvariables far exceeds the number01constraints,itwouldbebetter tosolvefor

n;

usingnon-linea r optimization andtheremaining variableswhichinclude weights.by linearmethod.

This methodclearly differsfromothers becaus e,for themajorityofthevariables (otherthan

"\>'

thelinea rmethod yieldsfaste r convergence as compared 10a totallynon- linear method.The otherdetailscanbeseenin[Balasubrama nian.

'994).

2.4.2ARTIFICIAL NEURAL NETWORK (ANN)IN PARAMETER ESTIMATION In theArtificialNeural Network method one ob tains theweightmatrix[W]

mentionedin Eq.(2.15) byconsidering differentset ofvectors {H},and[I}Iandby carrying outvariouscalculationsas givenbyEq.(2.21).One mustremem ber that

23

the weightsareexpressedas thedifferenceof twopositive variablesVIandViolin Eq. (2.20). The trainingoftheneuralnetw ork canbe carriedout by observing Table 2.2where the parametersto beidentified areshownas the outputvector.

To trainthenetwork,onecan generateanirregularlinkinertia valuesfroma regularlinkvalues usingEq.(2.13).Itshould be pointedout here thatfor different numberof trainin gsets.I,lor example ifi=3,Eq. (2.21)canbeexpressedas [SharanandAggarwal,19 94}

(D}, [A],

(D), • [Ak Iv)

^(2.22)

n x

t

(D), [A !,

3mx 1 3mxn

For exampte,here the dimension of the vec tor{Ohis mx1, thematr ix{Allism xn and the vecto r{V}is nx 1.If we vary c,~,yincertainsequential mannerwe would obtaindifferentsetsofinertiamatricesinB{}space.For eachof those sets, fora givensetofdisplacement,velocityand accelerationvalues of the end effector,onecancompute theforces and torquesforthatparticularposition of the end effecto r.IIshould be clarifiedherethat th ese forcesor torquesare correspondingto a given point ofthetraversingendeffecto rata givenposition, Therefore, the forcesortorques are thedynamic values at agivenpointonthe

24

TABLE2.2 INPUTVECTOR,OUTPUTVECTORANDWEIGHT MATRIX USED FORTRAINING

t"

t••

W,.,

Wl.2

W,.

W',12

W2.1

W "

^{W. . WZ-}12

W" W, . W, .

-

^W3,12

W,.. , W'U W,'"

..

_{W, ,, 'Z}

t.,

Here.thesubscripts 1 and2 referto thelinknumber

25

trajectory.

By varyingc,]3,yone can generate the output vectorshownin Table 2.2 . whereas the inputvectorin thistable can be calculated usingTable 2.1(fora givenset of displacement,velocityandacceleratio nvalues).These vectorsare then used toobtainthe weightsLe.{W]matrix shownin Eq. (2.17).Theseweights can be used to identify the outputvectorofa manipulato rwhose dynamic quantitiesareunknownby measuring the forces and torq ues an dusing Table 2.2. It must be cautioned her e that the kinematic par ametersi.e.linklengths.velocity, acceleration,position,etc . must remainthe samefor boththe knownand unknown manipulators,otherwise the traininghas tobedone diffe rently.

2.5 RESULTS AND DISCUSSIONS

To illustratethe theorydeveloped,a two-linkman ipulatorwasusedand the variousparameters for this manipulatoraregiveninTa ble 2.3.Atfirst , diHerent values ofthe Inertiamatricesweregeneratedby varyinga,~andy.Thereafter, the LPNmethod wasusedto identifythe dynamic parameter s.The comparisonofthe values ofthedynamicparameters correspondin gto oneunknown setof values of a,j3andyand theLPNMetho daredepictedinTable2.4.Inthisfigure,the calculationsusingtheknownkinematicand dyn amic paramete rsare doneand it includesthedetermination of[WImatrix. Afterthetraining ,the inputvecto r mentione din Table2.2fora manipulator whose [I] matrixwas not known,was fed totheANN. Then,theunknown (output)parametersare predict ed bythese

26

TABLE 2.3 VARIOUS PARAMETERSUSED FOR ATWO-LINK MANIPULATOR

PROPERTY NAME LINK 1

I

LINK2 Link Lengths(m) 0.3

I

^0.2

Mass(kg) 4.0

I

^3.0

Height of Link,b (m) 0.1 Widthof Link.c(m) 0.1 Radius ofCircle(m) 0.2 MaximumTangential 0.15

Velocity,VI(mls)

27

TABLE2.4 COMPARISON OF ESTIMATEOOYNAMICPARAM ETERS

KinematicValues:X = 0.08m, Y=0.08m.X=0.6mis,Y=0.6 mis,X=0.4 m²/s,

Y

= 0.6mlls

S.No. OYNAMIC UNKNOWN SET LPN METHOO

PARAMETERS a.~. y=8· VALUES LINK1 LINK2 LINK1 LINK2 1 I."(kg_m^l) 0.011 0.00644 0.0097 ·2.88 2 I (kg·m²) 0.121 0.0419 0.095 0.0431 3 1.. (kg-m²) 0.12 0,0416 0.122 0.042 4 I (kg_m2) 0.0135 0,00436 0.0156 0.002 5 I (kg-m²) 0.00215 0.000693 0.000957 0.00067 1 6 I:.(kg·m²) -0.0178 -0.005 73 -0.0177 ·0.00572

7 Pc, lm) 0.147 0.098 0.146 0.0977

6 Pc.(m) -0. 0178 -0.0118 -0.0178 ·0.0116 9 Pc, 1m) 0.0233 0.0156 0.0236 0.0157

26

weights, usingTable 2.2.Its predictedinertia valueswere thenused forthe force and torque calculations alongthe trajectoryshownin Figs.2.5 and 2.6,Figs.2.7 and 2.8show the variousprogrammingdetails.Theresults of these calculations are shown in Figs.2.9 to 2.14.These resultsclearlyshow the usefulness ofthe ANN method.It is quite obviousthat as theendeffector is madeto travelalong the trajectory,the errorsin forces or torquesarevery small. The maximumerror is less than2%for aUpoints having significantand comparabletorque values.The onlyexception was the torque at point3forlink 2 which was 0.Q15 N-m,andthe errorwas 0.0041N-m. In thiscase the torque itselfwas negligible.

2.6 CONCLUSIONS

Inthischapter theproblemoftheidentification of thedynamic parameters was carried outbyusing a linkof regulargeometry and associatedinertiavalues.

The Inertiamatrixof this link was thentransformed to anotherset of axes to obtain a full matrixwhichalink havingcompli catedmassdistribution is normally expected to have.These matriceswerethen usedto trainan ANN.After the training,the ANN was usedto predictthe dynamicparameters of anunknownmanipulator.The results obtained showthat thistechnique of parameter estimatJon can be successfullyused.

29

0.18 10 0.16

0.14

0.12

~

^0.1

0.08

0.0 6

0.04

0.02 0.12 0.14

X(m)

0.16 0.18 0.2

Fig.2.5 DESIRED TRAJ ECTORY

30

0.181

!

0.15m1s 5

, ^,

0.14 i ¹⁰

!

I CONSTANT

0.12 VELOCITY

i ~

^0.1

"

~

^{" ACCELERATING}

~ s

^0.08

.

z

~ 0.02

01

1

•

⁵

,

7 '0

POINTSALONGTHETRAJECTORY

Fig.2.6 VARIATIONOF TANGENTIAL VELO CITY ALONGTHE TRAJECTORY

31

Inpulunknown setofdata

' ,

I

GenerateI•matrixandPc veclor;

~ I

~

Tra,n,ng sets lorItleLPN ·method

IJsllefaled

" -,

NO

cc;:=~

r

^{C.llCula t.}

-1

1

ou tpu tparamet.rs

0 '

VES Slop

Fig. 2.7 FLOW CHART:OVNAMIC PARAMETER ESTIMATION

J2

~ .---Actual

I

, dynamic

! parameters

I

rs;;;;;-': _{. ' - - - '} i~

Iterative Ne'Nlon-Euler - ._~Ulal;on

f~~amiCS

~ ~ NO

< ^{Are ~~?}

LPNmethod dynamic parameters

perf°

new

fl

training J

Fig. 2.8 FLOW CHART:COMPARISONOF FORCES AND TORDUES CORRESPONDING TO ESTIMATED PARAMETERS

33

l

3 4 5 7 8

POINTSALONGTHETRAJECTORV 10

-0-ERRORVA LUESFORLINK1 -6-ERRORVA LUESFORLINK 2

Fig.2.9 ERRORVALUESIN F.FOR LINKS1ANO 2

34

-'0-EAROR VALUESFO R LINK1

"""6-ERROAVAL UESFaA LINK2

0.00 4

1

1

'OO: ~ : :

1 2 3 <I 5 6 7. 8 10

POINTSALONGTHE TRAJECTORY 0.02

,

0.018

r

'01. r

0.014 0.012

g

~- 0.01 .S

I

^0.0080.006

Flg.2.10 ERROR VALUES IN F,FORLINKS1ANO 2

35

--o-EAAOR VALUES FORLlNK 2

5 8 10

POINTSALONGTHE TRAJECTORY 0.012

0.002'

o I

^{'-~- ~-}

--'

1 0.01

0·" 1

0.018

0.Ot6I

0 .014 i

: .:i

l

0.004.

Fig. 2.11 ERRORVALUES INt.FORLINKS 1 AND 2

36

0.02• 0.0181 0.016 0.014 11.012

I

^0.01

. "

.5 0.008

!

^0.006

000 4

t

0.0021

i

0 1

~E RRO RVALUES FORLINK 1 - b -ERRORVALUESFOR LINK 2

3 4 5 6 il

POINTS ALONGTHE TRAJECTORY

Fig.2.12 ERROR VALUESINt,FOR LINKS 1AND 2

J7

0.1- - - -

009

!

:: j l

I

^0.05

- "

.S 0.04

t

I

^0.03

^t _:

~ERROR VALUE SFORLINK 1 ....rERRORVALUES FOR LINK2

o ~

.. _~

3 4 5 6 7 8 10

POINTSALONGTHETRAJECTORY 0.02~~O---~~>-^...

...,-l

0.01

t

o...

~~·'===:=~=:::::::_;::==1

Flg.2.13 ERRORVALUESIN'.FOR LINKS 1 AND2

38

: r

ALLTHE POINTSHAVETHE ERRORINTORQUEALONGZ·

AXISBELOW2%

-o-ERROA VALUES FOR LINK 1

!

-tr-EARQR VALUES FOR LINK 2

:~ O~

^{, , •}

^{! • •} '=H

5 6 7 8 10

POINTSALONGTHETRAJECTORY

Flg.2.14 PERCENTAGEERROR IN'.FOR LINKS 1 AND 2

39

CHAPTER 3

CONTROL OF ROBOTIC MANIPULATORS

3.1 INTRODUCTION

Controllinga robot manipulator along a given tralectory can be dividedinto two subtasks. The firstone is an inverse kinematic calculation which transforms the trajectory,usually specified in a Cartesian coordinate,inlo a sequenceof requiredjointpositions.The second one is the generationof thejointtorques from required joint angles and their derivatives. For the intelligentrobots the inverse kinematic calculations have tobedoneinreal-time whichis a limeconsuming computationalprocedure. Also,any attemptto upgradethe existing industrial robots controllerimposes a much grealercomputationalload on thecontrol computer.

Recently, significanteffortsweremade indevelopingANNcontrollers capable of controlling robotic manipulators. The ANN canbe quiteuseful in this respect because the trainingof thenetwork can be done off-line,inmost ofthe situations whereextensive computationscan be done on a main-frameor fast computers.Once thetraining is completethenthesenetw orks can enable the manipulatorto perform various tasks on the on-linebasis.

In thischapter the ANN Is successfully used by training thenetworkby

40

obtainingthe gainvalues,namelypositiongainvalues(kp)andvelocity gainvalues (1<.,),correspondi ngto jointdisplacement(0),jointvelocity(6),error in joint position values(e) anderrorin jointvelocity values(e) byusing ncn-uneer optimal control techniqueonoff-linebasis first.Then.thetraining of the network is performed using anArtificialNeuralNetwo rkMethodcalledtheLinear Programming-Neuro (LPN)Method,whichis fasterand more accurateas comparedto many other methods.Theresults obtainedare tested ona twolinkplanar manipulatorwhose endeffecto r moves alonga circulartrajectory. The study includesthe effect of the variation of the maximum velocityalongthe trajectory.Theeffects of thesharp chang esinthe velocity profile andthe general con trollaw arealso includedinthis work.

3.2 DYNAMIC EQUATIONS OF A PLANAR TWO LINK MANIPULATOR

Whenthe Newton-Eulerequationsareevaluate dsymbo lically for any manipula tor, theyyield the dynam ic equationwh ichcan be writteninthe form

(, )•[M{a)J{e}•(V{a,a}) •(G(a) } ^(3,1) Inothe rwords, this isthedynamic equa tionwritte nin matrix fonnfor ann-link manipulator.Here,(M(O)J is the nx n inertia matrixof themanipu lator, (V(O,en is ann x 1 vectorcontainingcentrif ugaland coriolistermsand(G(O)}is ann x1 vectorhavinggravityterms.Theterm (V(8,9)},appea ringin Eq.(3.1) hasboth the

41

positio nandvelocity terms.Inthe caseof a simple planar two·link manipulator shown in Fig.2.1,the matricesinvolv edare:

Any manipula tormass matrix is symme tricandpositivedefinite,andis,therefore, always inve rtible [Craig,1989).Similarly,in the vector

.

l-mzlllzs~t:I:

^-

2mzlll~z919Z}

^(3.3)

{V(8,8)}..

m11,lzsze~

^{, and}

the term-mzlllzs~IVis causedby acentnfugaJforce, andisrecognized as such becauseitdependsonthesquare ofthe JOlnlvelocity.The next term, -2mzl,lzsz916z is dueto theCorio/isforce andalways containsthe prod uct of twodiffer ent joint velocities. Finally,

(3.4)

contai nsall thosetermsin whichlhe gravItationalconstant,9,appears.

Inallof Ihese equations,Eqs.(3 21.(3 3)and(3.4)

These equatio nsarederived usingtheNewto n-Euleralgorit h m(shown in Table 2.1)based on the followingassumpnoea

1.All mass existsas a pointmass atthe distal endof the link.

2. Inertial tensorwritten atthe centerof masslor each link is thenull matrix.

3.There areno forces acting on the end-effector.

The idea01Inverse dynamicsisto seek a non-linearfeedbackcontrollaw

t • f(9,9), ^(3.5)

whichwhensubstitutedInEq. (3.1),results inalinear closed loop system. It is quitedifficult,if notimpossibleto findcontrolparameters for general non-linear systems.Since (M(9)J is invertible, wemay solve lor jointacceleration(a)of the manipulatoras

(O) • [M(9)]·' {{,}-(V(9,9)j-IG{9))} (3,6)

We thenapplyany of theseveralknownnumericsllntegrationtechniques,where {O}and {a} areexpressed in terms of

fa}

mentionedabove, to integratethe accelerationto compute thefuture positions andvelocities.

3.3 TRAJECTORY CONTROL

3.3.1INDEPENDENT PO CONTROL

An independent Proportional-Plus-Derivative (PO)Controlscheme(classical controlscheme)isusedto control themovement ofthemanipulator. While PO schemes are adequate In most controlapplications, there is overshootingl.e.the end-effectorcould gobeyond the specified positionbefore actuallysettling down.

Overshootingis quite undesi rable, becausein order to eliminateovershooting,an

43

integratorisused whichintroduces damping andcausesthe end-effectortomove slowly throughanumber of intermediateset points, thus considerably delayingthe completionof the task, andthe quality of the displacementetc.The controller designcanthusbecomemore sophisticatedonaccountoftheinvolvementof non- linear system dynamics.This type ofcontrolalgorithm isoneclassof computed- torque-likecontrollers.

3.3.2EVALUATION OF GAIN PARAMETERS In aPOcontrol scheme,the torqueequationIsgivenby

It} • [K,J{e} • [K.JIe} ^(3.7)

where{e}={6,,(t+1)} -(6.(t)} and {e}=[a,II'I)}- (a.ll)}

FromFig.3.1,thefollowingcbservattonscanbe drawn:

1. Thekinematicparameters with the subscript d represent the desired valuesonthetrajectory. These values are computed based on Table 2.1.The subscripta referstothe actualvalues of thekinematicparametersobtained by solving the control equationswhich involvethe finite differenceschemediscussed inSection 3.3.4.

2.Itshould be noted thatthe desired valuesobtainedusing Table 2.1can be compu tedIn anoff-line based trajectory planning.Ontheother hand, theactual valuesandthe errorsetc.,have to be computedon-line. Oneshould try10 minimizethe on-linecompU1alionstoincrease the speedwith whichthe lask Is

44

o DI:SIIlI:DTR.UJ:croRY X AcruAL TR.UJ:croRY

'.(1+3), ;.('+3) 8,(t+3) ,i4f.t+3)

Fig.3.1 SPECIFICATIONSOF THE DESIRED AND ACTUAL TRAJECTORY

45

performed.

3.{Kpland [K..lare diagonal matriceswithdiagonalelementsconsisting01 positiongaink,andvelocitygaink,valuesrespectively.Mathematically. theyare writtenas

[K.J

' [~' ~]

(3.8)

(3.9)

Theuse of a singlevalue respectively fortheentiretralectory lork,and

k..

maynot beable 10produce torquestofollowthe desiredtraiectory.The traiectory controlcan be achieved byevaluatingthe setofgainparametersforthe entire trajectory usingnon-linear optimization method(theoptimalcontrolmethod)as describedinthe Secticn3.5on apoint by pointbasis.Thisrequires the gain valuestobe differentfor eachpoint alongthetrajectory ofthe manipulator.The objectiveof theoptimalcontrolis tominimize the errorsin joint positionsandjoint velocitiesbetween theactualvalues andthedesiredvalues, based on the gain variables(I<,]and[K..].

3.3.3 HOOKES AND JEEVES NDN·LINEAROPTIMIZATION METHOD The gainvalues areevaluated usingnon-linear optimizatio n routine that wouldminimize16.(1+1)-

e

d(t+1)1and1'8.(1+1)-

e

d(I+1)1.

[KJ

and[K.lare obtained usingtheHookesandJeeves method(Aao,1978).Thepattemsearch

46

method 01 Hookeand Jeevesis a sequentialtechnique each step 01 which consistsoftwokinds01moves,onecalledthe exploratorymove andtheOlh~r calledthepattern move.Thefirst kindofmoveisincludedtoexp(orethekxal behaviouroftheobjectivefunction andthe secondkindofmoveis included10take advantage ofthepatterndirection.

Let usdefine .

and

lk,.

(X}<0

ik:~

k.,

k .,

(3.10)

F((X))•(O, (" ')- .,11.1 1)'•(e. (I'l)- e.(I.l )}' (3.11)

Theobjectivefunctioncan bemathematicallywritten as

U({X)) • FI{XII •

t. ^P.g~H(g.)

^(3.12)

where {X}is the design vector,kconsua--rsarerepresented as9.,andH(gJIs the Heavys ide unit step functiondefinedsothat

H(g ,)

=

^1forg..~0 or,

H(g~)

=

⁰for g_ < 0 (3.13)

InEq.(3,12),p~arelarge penaltyconstantswhich arepositivebecause the present problemisaminimizat ionproblem Next,one needsto solve forthe

"'

minimumof U({X}) llsing the Hookesand Jeevesmethod and the step-by-step procedure foradesign vector {X}havingn componentsis mentioned below;

(I)Start withan Initialestimate of the design vector

x,

{X}· : (3.14)

and choosedxl,i=1,2,...

.n

as steplengths in each of the coordinatedirections u"

i=1,2,,,.,0.

(Ii)Set temporary base point{V~.o}={X~}

(iii)Start the exploratory moveby perturbing one design variableat alimeinorder to find the improvedvalueof the objective function.Set:

{YkJ-1}...dX,{U,} if V'..U({Y~H}...1X/fuJ)

<V"_U({Y~.H})

if V- "U({Y~.i-I}-.1X,{uJ)

< U..U({Y~.H})

<U· •U({Y".,}''''',(u) 1 (Y~J-,l ifV..U({ Y~,'_I})<min(V',V')

(3.15)

In this way,all the designvariablesx_nare perturbed and the improved position {Y~.n}found .

(Iv) If the point{Vl<,n}is notdifferent from{X_l, reducethe step lengths ox_{l :}set I= 1and go to step (iii).If{Y~.n}is different from{X~}obtain the new base pointas

48

(3.16)

(v)Fin d the patterndirectio n{5} using

{S}.{X••,}-IX,}

Findthe point{Yhl.o}as

(Y••,.,)• {X,.,}•AIS}

(3.17)

(3.18)

Find'A.",the opt im umsteplengthinthe directio nfS)anduse

r

in Eq.(3.1 8).

(Vi)Setk=k+ 1tUll==U({Yllol)andi=1;repeatstep(Iii).Ifatthe endofslep(iii), U({Yk•n} )<U(X\c)usethe newbas epoint as {XII• I}={VII,"}and go tostep (v).If U({YIl,n})>=U( X'Il

H .

set {XII.'}={XII}and reduce step lengths; set k=k+1 and go to step(Ii).

(vii)Theprocess is terminatedifthestep lengthsbecomelessthane,averysmall quantity.

3.3.4 NUMERICAL INTEGRATION TECHNIQUE

3.3.4.1 RUNGE·KUTTA METHOD FOR A SECOND ORDER DIFFERENTIAL EQUATION

TheAung e-Kutlamethod Is self-startingand givesquiteacc urate results.

Ifwehave a singlesecond orderdifferentialequationas

x •

~

[F(I)-cx-.,].I(,.X.I) Bydefiningx=

v.

Thiscan bewrittenastwo firstorder equations:

49

(3.19)

x,=y (3.20)

Y.1(x.y.l) (3.2')

Bydefining

(X('»)•

lX

_y(l)

( I I)

^(3.22)

(F(')} .l,(/Y .,>!

^(3.23)

thefollowing recurrenceformulais used tofindthe values ofX{t)at differentgrid points~accordingto the fourthorder Runge·Kutlamethod.

where

I K, } • h F( (X .j.,,)

(K ,l ' h F( (X.}.(K, }. \., )

Inthevariableformthe Aunge·Kulta Method is showninTable3.1.

(3.25)

(3.26)

(3.27)

(3.28)

Thesequantities evaluatedinthe tableare thenusedin thefollowingrecurrence

50

TABLE 3.1RUNGE-KUTTA METHOD IN VARiABLEFORM

t x

y=x

f

= Y =

ⁱ

T,=ti X,

=

^Xi V,

=v.

^F1:=f(Tl'

x,

Y,)

Tz^:=~+hI2 Xz^:=~+Y,.(h/2) Vz=YI+F1·(hJ2) Fz=f(T2•X2•Yz)

T3= ~ + h12 X₃=~+Yz·^(hJ2) V3

=

Y,+Fz·(hI2) F3=f(T3•X3•Y3)

T.= t,+h X.:=~+Y3.(h/2) Y.

=

YI+F3·(h/2) F.=f(T••X.,Y.)

51

formula to calculate thevalues of

x

and

x

at the next limestepas:

Y,., • Y, •

~ [F,

^{·2F,·2F,.F.I}

3.3.4.2RUNGE-KUTIAMETHOD FOR THE CONTROL PROBLEM (3.29)

(3.30)

RewritingEq.(3.6) , the diffe rentialequationfor thepresent problemisgiven by

{i} • [M{e}}'IItl.(Vle,e)}-IGle)}) (3.6)

herethe quantities[M(8)],{V(8,6n and{G(ij )} are givenbyEqs. (3.2), (3.3 ) and (3.4) respectively.Aftersu bslilul ingEq.(3 7)IntoEq. (3.6) onecanwriteEq.(3 .6) forthe planartwo link manipulatoras [Aggarv.31et.aI.,19 9 5)

(3. 31)

Where (3,32)

and[M(8)t canalsobewrittenas

[MI')}" '•_ _,_ .[

r Im,

de'(M(al)

-{ lIm , •

1,1,m,cJ

.(IIm , - 1 ,1,m,e,) ]

~2mll•21,llImzCz ..1,2(m,+m,) (3.33)

Eq. (3.31) is a com binatio nof two secondorderdifferen tialequations so,wewill use thefourth Order Aunge-KuttaMethoddiscussedinSection3.3.4.1.Defining:

(3.34) thetwosecondorderdifferential equationsin Eq.(3.3 1) cannowbewrittenas four firstorderdifferentialequa tions as

(3.35)

8

₂•'1'2

tV

_z..

9

₂~G(61'92.'I',.'l-'z}

Therefore.one candefine

53

{

'I',(t)

e '(ll l

(X(III · e,(I) ' and '1',(11

(F(I)}

·!IJ

(3,36)

(3,37)

along thesimilar lines asin Eqs.(3.22)and (3.23). Finally,using Eq, (3 .24)with Eqs.(3.25),(3.26),(3.2 7)and (3.28)we numericall y integrale10 calculate {X(I)Jfor the next tlmestep.

3.4TRAJECTORYGENE RATION

In robotic applica tions. a desiredtaskIsusuallyspecnecin the wo rkspace orCartesian space,asthisis where themotion of themanipulator is easily describedin relationtothe externalenvironment and workpiace.HoweverI

trajectory-following control J5 easilyperformedintheJointspaceas this iswhere the arm dynami cs Ismo re easil yformulated. Therefore.itIsimpo rtanllobeable to findthe desired jointspacetrajecto rygiven the Ca rt esian trajectory .Thisis accomplished using theinverse kinematics.

Fig. 2.1showstheplana rtwolin k manip ulator.For this investig ationwe havechosen acircular arc asthedesiredtrajectory(Fig.3.2). In Fig.3.2the

54

25 0,16

21

0.14 20

19 1.

17

0.12 1.

15 1.

0.1 13

12 11

~

^{0.0 8 109}

•

0.06 7

,

5

0.0 4

•

₃

0.02

0

0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

X(m)

Fig.3.2 DESIREDTRAJECT(·.r··.¹f

55

manipulatorendeffector starts atpoint1and goes to point25onthe trajectory withthevariationof anglefrom5°to53°withincrements of2°.The radiusof the circulararcis0.2m.^Table3.2shows thevarious link parametersused.Fig .3.3 showsthevariatio n oftangentialvelocity alongthetrajectory.

Ife(5"to53°)is the angle withwhich the end effectormovesalongthe trajectorythen the Cartesiancoordinatesofthe endeffectorof the twolink manipulatoris given by

x•rcos(O) y,. rsin(9)

where ris the radiusof thecirculararc.

(3.38)

Knowingthe tangentia lveloc ityVIthe Cartesian velocitiesoftheendeffector are given by

The rad ialand tangentialaccelerationsaregivenby

A,·

-+ v'

V 2_V2

A , . _ '_ "_ '_

2 , 6(8,. .-8 ,)

(3.39)

(3.40)

whereVI,.Iis Ihetangentia lvelocity atlhe next timeslep andV~isthe tangential velocity at theprevious value of tim e.Theref ore,the Cartesianaccele rationvalues

56

Table3.2 VARIOUSPARAMETERSUSEDFORA TWO·L1NK MANIPULATOR

PROPERTY NAME LINK 1

I

^{LINK 2}

Linklengths (m) 0.3

I

^0.2

Mass (kg) 4.0

I

^3.0

HeightofLink ,b(m) 0.1

Widthof Link,e(m) 0.1

Radius ofCircle(m) 0.2 Maximum Tan gential 0.15

Velocity,VI(m/s)

Initial Position Gain K:t

=

150.0:~2

=

^150.0

Values

InitialVelocity Gain K.,::r150.0 ;K,.z

=

150.0 Values

StepSize InTime,.1T 0.001 (s) forRKMethod

57

:345 67 B 9101112131415 16 17 18 19 20212223 242 5 POINTSALONGTHE TRAJECTORY

CONSTANTVELOCITY ACCELERATING

\

O.15m1s 0.16

I

0.14

t.

! 0 " 1

~ 0.1i-

~ ^{o oa l}

~

^o.ce

1

~ j

~ 0,04 1 00'

I

od--+-~_---~~~~

_ _

~..-J

1a

Fig.3.3 DESIRED TANGENTIAL VELOCITY PROFILE

56

aregivenby

{ x ).[ cose- sine lk)

9 ema COS9]A.

(3,41)

Finally,knowing the Cartesian displace ment,velocity and acceleration values. we use the inversekinematic catculanons as describedbelowto calculate the jointdisplacement.velocity and accelerationvalueswhich are10beused in Eq.

(3.31).The variation ofJointdisplacementvaluesfor links 1and2Is shownin Table 3.3.Fig.3.4shows theorientatio noftheendeffector of theplana rtwolink manipu lator following thedesired trajectory in accordance with the Joint displacementvalues givenin Table 3.3,

Abriefreview ofthe computationprocedurewhich for the sake of claritycan be stated in thestep-by-stepfonnsas:

STEP 1:Compute thelefthand sideterms Inaqs.(3.38)to (3,41)Ina sequential

STEP2:Compute9₁and9₂us ingEqs .(2,1)to (2,6),

STEP3:Use

X

and

Y

fromEq , (3.39)and compute

El,

and

9

zfromEq,(2.8)which isasetof two simultaneouslinea ralgebraicequations.

STEP 4:Similarly use

X

and

Y

from Eq,(3,41) and compute

ii,

^and

9

₂using Eq.

(2.9).One has to remember that

a t

^and

9

2have alreadybeen calculatedinstep 3above.

59

Table3.3 VARIATIONOF JOINT DISPLACEMENTVALUES

POINTS 9,(rad ) I 9,ld. g ) I81(rad) e1(deg)

1 -0.64 -36.4 1 2.42 136.59

2 -0.60 ·34,4 1 2.42 138.59

3 -0 .57 -32.41 2.42 138.59

•

^{·0 .53 ·30.4 1 2.42 138.59}

•

6 ^-0.50-0. 46 ^-28,41·26.41 ^2.422.42 ^138.59136.59

7 -0 .4 3 ·24.41 2.42 13B.59

9 -0. 3 9 -22.4 1 2.42 138.59

•

^{-0.36 ·20.4 1 2.42 138.59}

10 -0.32 ·18.4 1 2.42 138.59

11 I -0.2 9 ·16.41 2.42 138.59

12 -0.25 I ·14.41 2.42

It

13 -0.22 ·12,41 2.42

" I.

^-0.18-0.15 ^·10.41-8.41 ^2.422.42

16 -0.11 -6.41 2.42 138.59

17 -0.08 I ·4.41 2.42 138.59

ts

-0.04 -2.41 2.42 138.59

"

^{-0.01 .().41 2.42 138.59}

20 0.03 1.59 2.42 138.59

21 0.06 3.59 2.42 138.59

22 0.10 5.59 2.42 138.59

23 0.13 7.59 2.42 138.59

2. 0.17

,

_{9.59 2.42 138.59}

2. 0.20

,

_{11.59 I 2.42 138.59}

60

Y

24

x

Fig.3.4 DESIRED CARTESIAN TRAJECTORY ACCORDING TOTAB LE3.3

61

3.5 OPTIMAL CONTROL METHOD (N ON • LINEAR OPTIMIZATION)

Optim alControl(non·linearoptimization)methodisamethodused10 achievethetrajectorycontrolandob t ain mepositionand ve locitygainvalueson the off-linebasisusingthe Hook esand Jeevesdirect search non-Iinear optim ization meihod.Thismethodminimizes theerrorin!heJoint displacementand Joint velocityvaluesInaccord ancewit h EQ.(3.7).It makesus e01thefourthorder Aunge-Kur.amethod(section3.3.4) 10caeutatetheactualjoint displa cementand velo c ityvalues atthe nexttlme step.Thecalc ulated gain valuesarethe nusedlor trainingthe ANNasdiscussedinsecnot 3.6 .The variousstepsinv o lved inthe

optimalcont rolmethodareshownIn Fig.3.5 .Thevarious etepeinvolvedin this methodare asfollow s:

STEP1:Readthe linkparameters(T able 3.2)andlettheend-effectorbe atsome initia lpositio nhavingcoordinatesIx.YI.

STEP2:USI3theinversekinemanc equations(Section2.2.1)tocalculateIhe actua l jointposition andjointvelocityvanes.

STEP3:Calculate thejointtorQue uSingEq. (3.7 ) based on the errorinthe desire d and actual jointdisp lacement,mo10lnlve locity va lues andthere a fter comp ute thejoinlacceeratlcn lromEq 13 6)IIshouldbenotedherethaiwe use the actualvaluesof

a ,6,

etc.in Eq. (3 6\

STEP4:Numericall y integrate lorwall)Insteps of6T andcalculate the joint positi o n andjoInt velocityva luesatth eeesttimeste p.

r - - -

.~

,

Read linllparametefS

Fig . 3.5 FLOW CHART:OPTIMAL CONTROL METHOD(NON- U NEAR OPTIMIZATION)

63

STEP5 :FinallyusetheHookes andJe eveedirectsearch (non -linear optimiza tion)methodto minimize the errorin theactual(obtained inStep 4)and the desiredJointposition andjoin tvelocit yvalue s.Record thegainvalues. Here , weuse[Kpland[K.Ias designvariablesasshow nInEq.(3.10),an dthe objective functionfor lhepresent case is givenbyEqs.(3.11)and(3.12).

These gai nvalue s whensubstitutedInEq.(3.7)wouldres ultinthedesired torque.The Steps2 to5 werere pealed for theentiretrajecloryandasetofgain valueswas obta ined.The objectiveherewastohave theposition controlpr imarily , whichwas successfullyattained.The gainvalue s foratrajectoryare shown in Figs,3,10to3.13(Sec t ion3.6).Here,thegain valueschangequite slgnif icanUy along the trajec tory.ThisIs becausethe matr ic es{M(El)],/V(E>, 0 ))andIG(On undergo contin uous changefro mpositiontopositionwhichalsoincludesthe changes inthe velocityvector.

3.6 ARTIFICIAL NEURAL NETWORKS IN TRAJECTORY CONTROL

Neuralne tworkmethodha s beenwidelyusedin many con trol applications.

Inthepresentcasewe areusing theLP-neu romethod whichwasfoundtobe very etfecliveintheparamet er estima tionprob lemdiscussed inChapter2,Itwouldbe quitebeneficia ltohaveaweigh t matrixwhich relates lheInput ve ctor

{I}

=

{6"62,8,,62,e,,e2, 61,62}T (3.42)

and theoutput vector

64