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A.... LICATION OF NEURAL NETWORKSIN ROBOTICCONTROL AND DESIGNOF MECHANISMS

By

eRAGH UBALASUBRAM AN1AN.D.E.

to.thesissubmitte d 10theSchoo l of Graduate Stud ies in partial fulfillmentof(he requirementsfor thedegreeof

Masterof Engineer ing amiAppliedScience

Faculty ofEngineeringandAppliedSciences MemorialUniversi tyofNewfoundlan d

December 1993

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Abstract

Neuralnetwork has been widely usedinvarious rich,isIll'robotics.luthiswork, the neuralnetworkanalysisusing.hackpmpagalilll1algllrilhm isappliedIIItheinverse velocity analysisofrobotic manipulators ncarthe sin);!,u l;lrity POilllS ;lcClIlllllinl!torrlw trac king erro randfeasibilityof jointvelocities, Theinverse computationsIIsill!!11Il' pseudo-inverseoftheJacobianmatrixarccomparedwiththoseIIht:lilWdh~the neural networ k analysis . The resultsillustrurcd usingexample s ofIwu wellknownnmmpul.uors showtheadvantagesofusingthepresent work.i\new1c;lrningalglll'ilhmcallcd 1.1' neuromethodisthendevelopedtnsolveneural network problems lnrhisalgl,rilhlll .thL' weigh tsareobtainedby a combinationofLinea r I'rngral1lll1illghaving;1sjxlrs,':llCffit:il'l1t matrix anda singlevariable non-linearnpthuizatiuumethod.Theresultsarcdlustral\'d by solving threediffe rent problems.twoofwhicharcusefulin theon-fine\olllr"lIII roboticmanipulators.

Thedesigns ofa functiongenenuor and afour-bar mechani smwhose l,:oupkl curvepassesthroughninespe cifiedpoints,have been carried outusillgneu ralnetwOl'k methods. Thedesignproblemhasbeen solvedusingnon-linear techniqueswhichyield aweightmatrix ineachofthecases. The:ll.;CUr<ICYofthemcthudsisalsodiscussed Finally,gain parameters requiredforthe trajectoryuuurul arcevaluatedusi ngrumlinear optimizationmethod.Neuralnetworkisthen trainedtoevaluate uicga inpnuunctcrs basedon error historyofdifferentrrajcctoncs

Acknowledgements

Iwould like 10 expressmyapprccauon and profound gratitude 10 Ill)' adv isor Prof. A.M.Shamn forhispuncmguidan ceand constantsupport. Iam grateful 10 Profs.~1 .J.Hinchey.A.SJ .SwnmidasandK.Munaswamy fortheir valuablesuggestions :1IIl.!guidanceuuringthecourse.Iamalsothankful10theDean.Faculty ofEng i~e ring anllAppliL,jScicrceforthefinancialsupport duringtheprogram.Iwouldalsoliketo IhanklIurAS~'lCiateDeans.Dr.1.J.Sha!'JIand Dr.T.R.Chari for theirenco urage ment eXlcl1~ktldur ingmystay inthe campus.Finally,Ithankallmyfe llow grad uatestude nts

;Ilk!frie ndsfortheir help aoo mo ral supportduringthecourse of this work.

iii

Contents

Abstract Acknowledgemen ts

List ofFigures List ofTllblcs ListofSymbols

IIntroductionand Literatu reSurw )' 1.1Introduction

1.2LiteratureSurvey

1.2.1Artificial NeuralNetworkMethods 1.2.2 SingularityProblems in Robotics 1.2 ,3 MechanismSynthesis. 1.2.4 NeuralNetwork Contro lin Robotics 1.3 Thesis Object ives

2NeuralNet workMeth od s z.tImroduction.

2.2 BackpropagationMethod. 2.2.1Multilayer NeuralNetwork

2.2.2Feed forwnrd Recalland Error Buckpropagatiun Algorithm

2.2.3Propertiesand its Slgnificance

_-I

III

10 II II

. . . .16 21

2.2.4 Applicannn- Singularity Problemsin Velocity

Analysis ofRobots 2-1

2.2.4.1 Velocit yAnalys isUsingPsucdo-lnv erse Method 25 2.2.4. 2velocityAnalysisUsing theDamped

Least Squares Method 26

2.2.4.3VelocityAnalysisUsing NeuralNetwo rk

Method 29

2.2.4.4CaseStudy 31

2.2 .4.5Resultsand Discussion 37

2.3 LP-NeuroMe thod -16

2.3.1A NewApproach-De velopmentofLp-Ncuro Method . 46

2.3 .1.1Lp- Ne uroMethod -Type1 46

2.3.1.2 LP- NeuroMe thod- Type 2: 51

2.4 ApplicationsuftheLP-NeuroMethod. 5-1

2.-1- .1 Function Genenuion 54

2.4.2 Accele ration AnalysisofaTwo-linkPlanarManipulator 56 2.4.3 SolutionofTorqueand ReactionForces ofthe

Two-linkManipulator 62

2.5Conclusions 74

JNcur;llNetworksin MechanismDesign 76

3.1 Introduction 76

3.2 ImplementationofNeuralNetworkinMechanism Design 76

3.3.1Nine-PointPathProblem 76

3.-1- Cnncluslons :\7

-INcu rnl :'\elwor:.Cont rolillRoh ul ics

ss

.t.tImroducnon .xx

-1.2 TrajectoryControl. XII

-1.2. 1InverseDynamics of an-LinkManipulator .XII 4.3Evalu ation of Gain Parnnnncrx for Trajectory Comrot III

4.3.1Proc edure to EvaluateGain values IIJ

-1,..1. Neural NetworksinTrajecto ry Comml

or

Two-Linkf\.'lallipulaw,'S lllll -1.5Conclusions

5Con clusions 5.1Conclusions.

5,2Fut u reRecomm e ndatio nsofthe Work. Referen ces

Ap pendix AProgramListin gs

vl

.. ...113

115 115

IlfJ

us

A.I 1\.2

List of Figures

1.1 ApplictuionofNeuralNetwork inRnhotil.:s.

2.1 AppJit:<lliomofNeuralNetworks I:!

2.2 !I.TypicalNeu ra lNet wor k ....13

2.3 ActivationFunctions. . ....IS

2.4 RepresentationofNeuralNetwork Layers-ForwardComput.uious .17 2.5

2.6 2.7 2.M 2.1) 1.10

Rcpr-senrationofNeuralNetwork Lavers -Backprcpagauon of Erro rs.

Flo wChan-BackpropugmionMethod

Mo vementof Wl.:ightVector(2-D)ontheErrorSurface. TrajectoryUsed forPUMA·560Manipulator. 1\Planar Two-Link Manipulator. PUMI\·560Manipulato r.

... . ..19

"

.. . .. . .. .

^~-

..23 32 .33 35

z.t

t Varia tionIll'theNorrnof theAngularVelocityVector,~

e !.

^Along

theTrajectoryof aPUMA-560Manipulater 38

1.12 VariationoftheAngularVelocity.

0:.

Alongthe Trajector yofa

PtrMA-5 60Manipulator 39

2.13 VnrfntkmIll'the AngularVelocity,

e. ..

AlongtheTrajectory ofa

PUMA·560Manipulmor. ..40

2.1-t Variationnfthe Norm oftheCartesianVelocityVector.

I I

x

I I.

AIlingtheTraj ectory ofa PUMA-5 60Manipulator .41 2.IS VariationortheNormoftheAngula r Veloci ty Vector.

li eI.

AIlingthe Traicctoryora Two-Link Manipulator ..42

2.16 Variationofthe Angularvelocity.

0

1,Along theTrajectoryofa

Two -LinkMan ipulator. ..43

~.17 Variationpfmel\ lIl!ulal'\·c1l1.:iIY.0:.:\I" ll~therraje':I 'ry"f.1

Two-Link~j;mirul~I1,;r. ..1_1

~.'8 Vuriruiou

o f

the Norm \11'the Canc slan Yc1llei,y.

Ilx

1\.:\1I'11~me

Trajectoryof a Two-LinkManipulator .1:'

2.19 DiagrurruuaticRcprcsenuuiou oftheNetwork-l.I'cNcuro~k1hlld 4:-:

2.20 FlowChart·Lp-NeuroMethod :':'

2.21 ComparisonofValues for theSine Curve(til.I.P-Neu1'll~leth,',1.

HI'Method and theDesiredValues) 57

A PlanarTwo-LinkManipulator andthe Trajectoryused for AccelerationAnalysis

2,23 2.24

2.25 2.16 2.27 2,28

2.29 2.30 3.1 3.2

•.1 4.2

•. 3

...

Variation of0).AlongtheTrajcctury Variationof8:.AI(lngtheTrajectory Error ValuesufI~ActingonLinkI Error Valuesofl~ActingonLinkI ErrorValuesof¹1ActingonLink I ErrorValuesoff,Acting on Link 2 ErrorValuesoffrActing onLink2 ErrorValuesof1)ActingonLink 2 A Four-Bar!'.'ech,·nism- Nine-P ointPa th Generation AFour-BarFunctionGenerator

Specificationsof the Desiredand the ActualTrajectory, DesiredTrajectoryandtheTrajectoryObtainedUsingNon-Linear Optimization Method.

Desired TangentialVelocityProfile

Variationof G1ande~Alongthe DesiredTrajector y. viii

.hX

fI"

..•7(J

.71 72

..77

"1

, n

94

..9K

4.6 How Chart-Trnjcuury Control Using Non-Linear OpnmizuiionMethod 100

4.7 vannuon orC 1ande!Along 'heTrajectory 101

4.X Vari'l lio nori:1andi::AlongtheTrajector y IO~

4.9 vnri.uionof kl,1and kr;AlongtheTrajecto ry 107

4.10 Variationofk;, andk.:Along theTrajectory 1U8 4.11 Ho wCburt-Evalu.uionIll'WeightMatrix[WI forTrajec toryControl

lb ing Ll'<Ncum M,:tho(I ...110

4.12 Compa rison Ill'Gain Values krlandkr~ObtainedUsing Non-Li near Optimiza tionMethod ,U1d Lt' -NcuroMethod III 4.13 Comparisontil'Gainvaluesk'iand k,!Obtained UsingNon-Linear

Opti m izationMethodandLlt-NcuroMethod 112 4.14 DesiredTrajector y lindthe'l'rejcctory Obtaine dUsing LP-Neu roMethod 114

lx

List of Tables

2.1 LinkParametersofPll ~ I :\-; 60Mnuipulnuu ;h

")") Thelterntivc Newton-EulerDyl1:1111icsAl!-=llrilhm 1l.1

2.3 Link Parameters

or

theTwo-Link Manipulanu ./1-1

3.2 Coordinates of theNmc-PoinrPalh Problem ComparisonbetweenIhe

LP-NcuroMethod and Back-propagationMethod Xl

3,3 Link Lengths forthePuncrion-Gcncmrc rMcchautsm. ThreeI'rct.isio l1

Points. .X.'i

3.4 ComparisonnfyValues( Theoreticaland1.1'NCUfOMethod) Three

PrecisionPoints. .K'i

3.5 Link Lcnglus fo r theFuncuon -Gcnc nuorMccb.mivm-Eil-:hl

PrecisionPl1illlS . . Xlt

3.6 Comparison ofyvalues(Theuretic,,'andLlt-NeuruMclhlltll-Ei~hl

PrecisionPoints.

*1

4.1 Various Parametersused forthe TrajectoryControl

List of Symbols

r

, ,

II

accelerationof the endeffectorin theradialdirection errorin jointposition

error injointveloc ity n.)

f,'.l~'

\',.a,

l'l 111 ·IDI.ID}

III WI

11<,.1 IK.I

IWI.IVI

nctlvauon function forcesacting onlinki link len gths

Cartesian velocityand accelerationofillljointrespective ly veloc ity ofthe endeffector inthetangent ialdirection precisionpoints

Cartesianveloci ty vecto r

inpu t.desired.•mdou tputvectorsre spectively Jacobianmatrix

psuedo-mversc of Jacobian matrix gainparame ters

propo rt ionalgainmatrix velocitygain1ll,IITh linklengthsoffour-barmechanism wcigfu nrurtccs

xi

IWd IW,I IW"I

X"Y,

(Ji'U, andv,

e,

{S}

&,.,

w.:ighluuurtxIoraccclcnuiou ;lll:llysis wciglu rnnn-ixfortl'l"lilleanalysis weightmatrixconnec ting itllundjrh layers coordinates ofthe nine-point path problem errorinthe illllayer

learning Factor displacementnfuhjoint

angular velocityandacceleration{I filll jointrespectively damping factor

componentsofsingularvaluedccolilpo scdnuurtx torque actingon linki

displacement ofithjoint angular velocity vector

upperfeasible limitof angularvelocity joint acceleration

xfi

Chapter 1

Introduction and Literature Survey

1.1 Intr oduction

Aniflcia lIntelligence(AI) isapplied in diversified fie lds to achieve faster and betterresults. They areusefulfor achievingcomputat ionallyfastandapproxi mate solutions of certaindecisio n problemsthat arcbased oninform ation of diversecriteria.

Expe rt syste ms, ArtificialNeural Networ ks(ANN),Kno wledge-based representa tions CIC ..arcexamplesofulffere nr toolsusedin the applicat ionofAI. Roboticsisafieldthat requires such techniques becauserobots are oftenemployed toworkinhazardous

environments impossiblefor huma ninteractions, andwhere the calculations are numerous andcomplicate d.Intherecentpast, ANN havepro ved quite useful inrobo tics.Fig.1.1 showsthevar ious field sinrobo ticsin whichANNis beingwidelyused,

Singula rityavoidance. synthesisof mechanisms,finercontro lofthetrajectories

or

roboticmanipulatorsare still the topics thatrequire furtherresearch,A newtechniq ue whichoptimizestheefficiency andspeedwould beofgreathelpbecauseofon-line cnm putariunal requirementsinthe roboticsarea.

ANN IN ROBOTICS

t.fnverse xlnemaues- nonlinearmapping 2.Velocity Calculations

at every instant offimeete.,

I.PosilionITrajcetoryConlrol 2. Controlof Galnparameters 3.ForceControl

I.PatternRecognition :20CollisionAvoidance 3. Sensing and Perception

Figure1.1 App licationofNeura lNetworkinRobolicl:

J. 2 Liter atu re Survey

1.2.1ArtificialNeur alNetworkMethods

ArtificialNeural Network s (A NN) have beenstudiedformoretha n 30yea rs.Its uschasincreasedtremendou sly inrecent years becauseoftheavailabilityof fasterand paralk:1processorsanrJthe basic learning algorithm s (Grossberg . 1982;Hopfield.1982:

Rumclhartand McClelland.1986;Kchonen.1988 ).ANNs also referredas neural networksinthisthesis arcbeingusedto accomplishcomplex funct ions suchas generalization.errorcorrection.informationreconstruction,patternanalysis andlearning.

Neuralnetwor k can learnma pping between theinputandoutputspace andsynthesize anassociativememory thatretrievesthe appropriate outputwhenpresented withan input.

and11:15the abilityto genera lizewithnewinJK!IS.Because ofthe ir mass ivelypa rallel nature, neuralnctwmks tan pcrfonncomputationsal veryhigh speed(Fukudaand Shibata.1992 ).

Neuralnetworkshavealsobeen used10successfully solvecomplexproble ms like theTravellingsalesmanproblem.Ithas beenobservedthat neu ra lnetworkshaveofle n beenuppnnunistic.i.e.thenetworkmodel is customizedto servetheneeds ofthetask alhard(Kulkarni.19(1).They representanew approachthatisrobustandfaulr-tole mm.

Neuralnetworksrequirebasicalgorithmsfor accomplishingthelearningtask.

Se VCTll1algo rithms arc functional in the present. Onesuchalgorithmwhichiswidely used isbackpropagarton (OP)algorithm.Inbackpropagaricn algorithm.during thelearn ing

phase.tbeobserved outputsarc cumparcd withthe desiredouquus.and the weightsarc optimized tominimizetheerror function. Incomrcunvc leaming.the wciglus arc updatedwitheach new input(Rumelhnn,mdt\kClelland..191\(1).Hanuannand Biegler·

Kong (1992)discussefficientlearningalgorithmsfurneuralnetworks.

Neuralnetworkscanperfor mfunctiona l approxinnuiousIhiltMCbeyondthescope- of optimallinear techniques.Gulatictul..(1990)have introducedneuralformalismIII

efficientlylearnnon-linearmap pingusmg a marhcmauculconstructcalledterminal attractc rs.

Neuralnetwor ks havebee nfoundusefu linthe fieldofroboucs illthe rece nt times.Forward andinve rse displacem entanalysesofroboricnuunpututors have hecn doue byNyugenct al. (1990 )andGu lat ietal..(1990).Neu ralnetwork s seem In he:1 promisingapproach to solve non-linear controlproblem sas wen [TabaryaudSnlnnn, 1992).Someother lnteres ung applications in the comrotofmhutic manipulatorscanhe seenin Fukudaer at..1991;andAkio ctaI.,1992.

1.2.2 Singularity Problems inRobotics

Inver sekinematicsproblem s of roboticmanipulato rsarc always difficulthisolve becauseof(a) the multiple solutions inthedisp lacementanalystsproblems.or (bJ the occurre nceof singularitypointsalo ngthetrajectoriesin the case ofvelocit y nnalysis. The singu la r ity problem s , whic hinvolvetherank de fici ency inthe Jacobia nMatrix,have been

de,thwithhy Chiaverni(]992,.In thisregard.general discussion s on pseudo-inver se soluuonscan he seeninLawsonand Hanson. 1974.The pseudo-inversesolutions do not lead to satisfactoryper formancencarthe points ofsingularity because of abrupt changes in the clementsof the joint velocityvector.

Damped-LeastSquaresmethod(DLS) approach hasheen used by many researchers(Wampler.1986;Nakamura andHanatusa,1986; Maciejewskiand Klein.

1989; WamplerandLeifer.1988; Mayorgactal.,1992). The additionaladvantagewith this method is thatonecan set the limit(achievable limit)onthe normofthejoint velocit y vectorandli ndthe correspondingdamping factm, A.which yieldsthe minimum Maciejewski and Klein(1989) also proposeda truncatedSingularValue Decom position(SVO) solutionmethodwhichcouldbeusedfor on-linecomputat ions.

Howevertheresultingerrors could be more in thismethod.So far.therehas notbeen anymethod which takes intoaccount factors such as the errors aswellas the computationalefficiency. Neuralnetwork sareknownto performwellinthoseareas provideda relationshipis establishedbetweenthejoint velocity vectors andCartesian Velocityvectors onanoff-linebasis. Thiscircumventsthe on-linecomputational requirements of the jointvelocity vector,aswasdone by researchersmentionedearlier (Maciej ewski andKlein,1989).

1.2.3Mechanism Synthesis

Synthesisof amechanismis;1means of finding thelinkage that willproducethe

specifie dmotion. The problemofap prox imatesymhe sls ofafour-hal'mechanismII"h,,,,,' couplercurve isa planartrajectory wassolvedbyWamplercr:11..lll)t)2).SOhLli"1L.,1 such problems dateascattyas1923and someoftheimportantworksarcgiven ill Preudensretnand Sandor.1959:Shigleyand Vicker.1980;Erdmanand Sall\lm .l'm-l;

Morganand Wampler. 1989:Subbinn lind Flugrud.1989. Theusc IlftlJltillliz;llil'lI techniquehas been madebySuh and Radcliffe(1978).Angelesct :11..(198M),Ill'Akhr,ls and Angeles(1990)have applied a variable-separationtechnique and uon-Hncar optimization scheme to solve thefour-barpathgeneration problem.TsniantiI.utllJl-NI have solved the nine-pointpathproblemusinga newconnuuauon method.Wampler cr al.,(1992)have solvedthisproblem usinga cnmblnationor unnlyticalandmnncrtcut too ls.Problemswhere thenumberofpointsisgreater IIm11nineresultin allover- determined systemwhoseexactsolurlons an:notpossible.

Thefour-barmechanisms have also beenusedinthe designofIunctnm-gcncrarnrs.

Preuuenstetn(1955)proposedan algebraicIormulanonfor thenppruxhuurc symncs!s01 suc h a mechanism. Wilde(982)applied errorlinearizationtechniques 10solve rhts prob lem.Otherinteresting reference s onsuch problemscnn he seenill(Mnlmn 1{,lllcl al.,1973;TinubuandGupta,1984;andLiu and Angeles.1992).

1.2.4NeuralNetworkControlin Robotics

There hasbeenrecenttrendwithintheroboticscontrolliteratureto applyneural networksforthecontrol of robotic systems. in manyapplications reported in the

literature(Gu anti Chan.1989; Fukudaand Sh ibata,1990 :Helferty and Biswas.1990:

Jamshidi ctal..1990;Karakasoglu antiSundareshan.1990:Yamamura eral.,1990)lilt:

pw<:cssofneuralnetwork learningisconducted on-line(i.e.thedynamics of the neural networkisembedded inthe closed-loop with the dynamicsofthe robotic system),yet there appearstobe alackofstudies focussingon the dynamic behaviorofthe neural network during learningand/orcontrol whenthe neuralnetwork isusedin suchcontext.

Kawaiu(1990)used feedback error learning to computethe feedforwardtorques requiredfo ramanipulatorto followa path. The neural networkimplemented inthis method usesthedesiredjointpositions.velocitiesand acce lerationsasinputs and adjusts thenetwork weights using thefeedbac k torque astheerror signalto a backpropagation parameter optimizingalgorithm.Yuh(1992)alsouseda neural network formanipulat or control.He used a "critic"equation,which is afunction ofthe manipulator outputerror, 10trainthe networkto directly computethe manipulator inputtorques.

Asada (l990) useda multilayered feed fcrwardnetwork tolearna no n-linear m:lppingfo r compliancecontrol.Fromthemeasuredforces and torques inan assembly lask he usedthenetworkto computetherequiredvelocities, whichwouldaIJowthe assembly task10be completed.

1.3 Thesis Objectives

Wehave see n in the lastfewsectionsthat the neuralnetworks arequitever satile

tools tosolve problems in a widevaricty\11'arc;ls.With thisinmind.i,W;IStl" ,u~h lh' applythis10011(\solveproblems inIheareasofmechanismu.:si~nanllnllll,t;el;nntn'l.

Basedonthis,tbefollowingarcthe:objec tives,Ifthisthesis:

I)DevelopmentofOJnewneuralnl:lw(lfL:learning.atgoruhmlLlI-I\l.·Utll tncth<lo.1l which isIasr andaccurate.

2)Applicationof neuralretworksforinverse kinem.llK.:S til"tllhtllielIl;lllil'ul;lltirs nearsingularconfiguralion.sand comparison withdamrell-Icasl SIIU;lrcS;1[\\1 pseudo-inversemelhods.

3) Velocity.acceleration anlltorque analysis of ruhlllicnumipuhuors Llsing.nctnul networks.

4)Synthesisofmechanismsusing neuralnetworks

5)Trajectory controlof lhe roboticmanipulators usingneuralnetworks.

Chapter2.brieflyreviewsthebasicsof1ll.'Ur.J1nclWtJl"ks.Ib :kpmpag;llillll algorithmisintroduced hereandvariousfactorsinllueneinganeural networ karc discussedin this chapter.Thesignificanceofsolving forweighlmatrixinneuralnetwurk problems using combinationof LPandasingle variablenon-linearuptimi/.atiu!I mUline isidentified here.Thevalidityoftheapplication of backpropagationalgorithm ischecked byusingthemnearsingularconfigurationsofrobotic manipulators.An inversekinematic relationshipis establishedbetween the Cartesianand jointveloc itieson off-linebasis whichreduces on-linecomputationtime.The relativemenu afkldemeritsofthis mcthud over conventionalpseudo-inverseand damped-least squaresmethod arc discussedinthis

chapter.1\new algorithmcalledLr.ncurumethodis developedtosolveproblems using neuralnetworks.

InChapter .",thebackpropagatlonmethod andthenewalgorithmcalled theLP- Ileummethodarc thenapplied to solvevariousmechanism synthesis problems.

Chapter4dealswithsolutionofnon-linear oradaptive controlproblems.Here the non-linearcontrol problem is solved usingLP-neu ro method develo pedin Chapter2.

Next.thegainvaluesobtainedbythenon-linea rmethodarethenusedinthe neural controlmethodwherethe methodologydeveloped insections 2.4.1 to2.4. 3 arc used . In this way,thenumber of trainingsetsrequiredis a101 lessthan whatmanyothe r researchershave used.

Finally .inChapter5.thecontributions ofthethesis and recommendations for futureresearch arcoutlined.

Chapter 2

Neur al Network Meth ods

2.1 Introdu ction

Neuralnetworkmethodsarc widelyusedin manycl1gillLocrillg;lpplic;lIillllS.'lhcy canbethoughtof asa mathcuuuicultooltosolveCOm 1l10 1lcnl,\lnccr inJ;pwhkmsSUd1:IS optim ization.pattern recognitionere.Them!lIml ,,('tworkinuic:llcs,hl,:slm ilo,ritylit modelli ng.networkofneuro nsinthe brain. Many{incar:lUdlIUnlillC<lrneuruumlll,lds are connectedinthenetworkandinfo nnallOlI isproc essedin 11 parallel JislrihutcJ manner.This grea tlyn:tluccs thecomputation lime.NeuralnclwtJrb haveIe:.m ill),::llIJ self-orga nizatio n capa bilities.TheyadapttochangesinUilt".kamingthedWlu,;lcrislK:s oftheinput signa l.

Neuralnetwork scanbe broad lyclcsslfl..xlinmtWIItypes:

l}The ncuralnetworksthai learn and,.d aptto cha nges :1I"e c:llk Jrecurrent networksor backprcpagationnetworks. Muhiluyer pcrccpeunneuralnets.Ilupfid unels, Adapt iveResona nce Theory(AR T )networksfallunderthis category.

2) Those thatdu notinvolvelearn ingandsometimes calledfcedtorwa rd nets.Outer-product associativememories and multilayer nelswithoutbackwarderror

10

~orr~'t;li llll~belong10thistype.The mostpopular neuralnetworksusedtmlay arethe llupfielt.!netsKohoncn'ssell-organizing maps.multilayerpcrceptronsan~ARTnets.

Someofthe opcrauorsrhm neuralnetworks perform arc shown inFig.2.1. They arca~valll"g~'t)USinthefollowing situations:

I)Dccision-nnklngfrom amassiveamount ofdata 2)NOli-linearmapping

3)Obtainingnear-optimalsolutionsto optimizationprobleminI~time.

2.2 B ackpropagation Method

2.2.1Multilayer NeuralNetwork

A typic,11neuralnetworkisshownin Fig.2.2.Basic componentsofa neural network arc:

I)luput aliI.!11lIlpuldamsets

3)Processing Etemens (PE)orneurons 4)Activationfunction

'lhcncural networksthatneedtobe trainedare supplied withpredefinedinputandoutput t.!;lla sctsin a vectorfurru.Eachlayer ofa neuralretworkconssrsofseveralprocessing clements.EachPE inaneuralnetworksumsall ofits inputvaluesand performsa rn.:tldim:d operationand produces a single outputvalue.rE'sarc connectedwith

II

NEURAL NETWORK APPLICAnONS

CLASSIFICATION

NOISE REMOVAL

PATTERN RECOGNlTlON

Figure2.1 Applicationsor NeuralNetworks

12

.l<;tivat ion(unctionhidde n layers

processing element(PEl

weighed connections

Figure2.2 A TypicalNeuralNetwork

13

weighedconnecti ons. lnfunuationisstoredin a networkillrbc tonn nl" II'cighlS.In neural networkmethod the weight matrixis obtained basedI'lltill· karning.l'fllccssi.c ..

based on the inputand output informationused forthat purpose.

Activation functions,alsoknownas squashing.functkms.pcrlilTll1l1laplling of 1'1':\ infinite domain intoa prespccined ra nge.CommontyUSl.'1J activation

runcuons

(shown inFig.2,3)are:

l)Linear activation [unction 2)Stepactivation function 3)Ramp activationfunction

4) Sigmoidalactivationfunction orsquashingfunction 5) Gaussian function

Neural networksareorganized into severallayers ofPE'swhichinclude inputlayer.

hiddenlayers andoutput layeras shown in Fig.2.2.I\.fccdforward networkis one llmt has connections whichfeedinformationin om: direction withoutanyfeedback path.II"a network has feedbackpaths,thenitis called feedback network. Thetraiuing III multilayerneuralnetworksdependonthe followingfactors:

1)The numberoflayers 2)The number ofPEineachlayer

3) The amountofdataneededfor sufficienttraining.

Therearenopredefined setofrules availablefor (Jclcrminingthe ab oveIac tors.Several techniques areavailablefor the multilayer neuralnetworks tohavetheir connection

14

(a) Linear activation function _(b)Hardlimiting function

f(nel,) (netJ

f~

_{(c)Thresholdfunction} (d) Sigmoidal

f

activationfunction Figure2.3 ActivationFunctions

15

weightsadjustedtolearn mapping.Till' most populartechniqueIs till'bal.:kpr\II'a~;lli\ll1 algorithmtwertos.197.J:Parker.198:!;Rumelhan.lIill1011,a11l1WilliamsIlJS6).

Learningprocesscan beclnssif'iedinto two categories :supervised learning :lnd unsupervisedlearning. Supervisedlearningmonitorsthedurathm ufthetrainingandthe errorperformanceetc..Unsupervisedlearningincorporates numtlllitorin~pTllI:essand relies only uponlocalinformationduring the entirelearning proce ss, Mostlearning techniquesarecarried outoff-line.

2.2.2Feedforward Recall andError BackpropagationAlgorithm.

In neuralnetworkmethod,oneestablishes a relationshipbetweentheinputandthe desiredoutput parameters. The matrixrelationshipbetween these two Vl'l.:l!1 rsarc approximated byusingseveral hiddenlayers as shown in rig,2A. Inthisfigure,the relationshipbetweentheinputvector and thefirsthidden layervectoris,IItlrstcxrrcs.~l.:d involvinga weightmatrixwhose elementsvary between -IandIand arcrandomly generated.Similar procedure isadoptedfor therelationshipbetweentwo adjacenthidden layers or the last hidden layer and theoutput layer. Mathematically.one ofthesetypical relationshipscanbewrittenas.

(2.1)

where {I}istheinput vectorand {H},isthefirst hiddenlayer.

16

HID DEN

LAY ERS

[W);

Figure 2.4 Representation of Neural Network Layers -ForwardComputations

17

vector{H},arecomputedand an: symbolicallyrepresented by a slluan: (0)illFi~.~A.

For example.fora typicaldementitwllulJ he written as

whereaisthesteepness factor andh, is one otthc clementsorvector(II},.This process is continueduntilthelasthiddenlayeri.c.. each layer is related to titherhyamanix containing weights.andalso. thereis a similarrelationshipwrittenbetween thelasl hiddenlayer and theoutput layer.

Defining twovector- {a}and {d}<ISthe vectorof(lUirut.~ i!lmn i d alfuucrlons and desiredvaluesrespectively.wewishtominimizetheerrorE definedhy

N

E "

~ ~

^(d1-

oi

Eachofthe summation terms (E,) isrepresented by triangular Itt.) symholinFig.2.4 Thiserror has tohe backpropagatcdusingthesamewcightsmentioned above. Todo this.we firstwrite theequation

18

(2.4,

HIDDEN LAYERS

I ^{( 6 ,.)}

Figure 2.5 Representationor NeuralNetworkLayers.Back-propagation of Errors

19

whichisrepresentedl\y

, I

dial\lllndsymbol(,» inFi ~,2.5.Fhcerror inthelaslhidden layerelement wiseis compute d us ing

,

I).v

v,

(I-Yj)

f;

^{8,,1I"V j} ~I•...~I

whereYjisthe sig moidalelementaloutputofthelasthidde n layerin Fig,2.-1and\\'~,is anelementof thecorrespo nding (to therightofy,) weightmatrix. This processi., repeat ed untilonecomputesall the clements of thefirsthiddenlayer.Theweight matrix betw een the output layerandthelasthidden layerIIIhellSL1..Iillthenext cycleis recomputedas

(2.6)

where thesuperscriptsrefer10 thecycle numberandIIis thelearningIacnrrwhichis normallyassumed between10-)to10.The relationshipfurtheweigh tmatrixill other layersis give n by

(2.7 )

Finallythewe ightmatrix betwee nthe input and the firsthidden layeris calc ulated using

(2.K)

20

r.rnccthcflCweightmatricesan:obtained,then for any input '"eCIOrone:has togo lhrnugh the forward c!.Imputalionsas shownin Fig.2.410obtaintheOUtputvector.This pml.:e~~iscontinueduntil the final set of weightmatricesare obtainedwhichyieldthe tlesirt.-dnutputvalueswithin the:accuracyspecifietl. Flowchartfor the backpropagation methodis shown in Fig.2.6.

2.2.3 Proper ties and itsSignificance

Backpropagation algorithm usesgradientdescent techniquetoadjust the weights so,IStominimizetheerror

(2,9)

where'Iis the stepvalue.Themovementof the weightvectorintwo-dimensionalspace canIlCobservedon the error surface shown in Fig..2.7.Theweigbts ofthenetwork 10 helraiOl.'lIan: typically initializedatsmallrandom values. Theinitialization:tro ngly affectstheultimatesolution.Al10lherfactor thataffects the convergenceisthe steepness factorQ. in the sigmoidalactivationfunctiongivenin Eq.(2.2). Theeffectiveness and convergenceoftheerrorbackpropagationlearning algorithm depend sign ifica ntlyonthe valueof thelearning constant'I.Ingeneral, however.theoptimumvalue of II depends upon thepr oblembt'ing solvedand thereisnosingle learningconstantsuitable for different training cases.Acti vation functionswithlargersteepnessfactorproducesthe same effect asincreas ing thelear ningfactor.So. the steepness factor isusuallytaken as

21

ls trair-ng NO patternover?

Isnumber of

NO

~:2/

YES Isconvergence

achieved? NO

Figure 2.6 now Chart -Back-propagationMethod 22

Error(E)

l!litialError

Errorminima

w2

wI

wl,w2 -weigbtJ

Figure2.7 MovementorWeightvector(2-D)on Ihe Error Surface

23

tand the learningfactoris adjustedtocontrol the convergence.llnwcver,gl';l,lkll1 descent t:gorithm surfers

nom

localminimum probrcm which is :1c\1111nl<l1lIlrllpcrl~'of any nonlinearoptimization algorithm.

2.2.4 Appllcaticn>Singular ity Problem sin VelocityAlIld)'sis01' Robots

Whena manipulator is in singular configuration.it loscs oneor more degrees of freedom inthe Cartesianspace.Singularities inrobotic manipulatorsmayurtscdueIII the geometrical limitations (constraintsin thc connectinglinks) of the manipulators. This problemcan behandled by the useor redundantmanlpulnrors.There nrc two kimlsof singularities:

I)Boundary singularities arise due to the gccmctricnl limitations.

2) Interiorsingularitiesaredue to two or more joint axes liningup.

Redundantmanipulatorsalsohavesingularcontigurarionxwhichhavetil he either avoidedor handled.Near singularpoints.very highjoint velocities resultif the Cartesian velocities have components in the direction in which{he armlosesmuhlfiry.I'lu:se arc the points at which theJacobian matrix becomes rank-deficient.

While thisproblemcan be handh..'<!using mnthcmnticnltechniques likepseudo- inversemethods.yet i{ has certainlimitations. The problems of singularitiescanhe tackled atthe task planning levelitselfhycarefullydesigningthe trajectorywhich'lvoids singularconfiguration.Onthe otherhand, if due to wrong taskplanning or insituations

24

whereon-lineI:lIll1pUlatiCinsan:made andjhesingula rityappearsinthetrajectory.Ih..:

ruhllll: llntmisystemmustheableIIIpass throughthemsafely.Multiplesolutionsexrsr atsingularitypoints.

2.2A.1VelocityAnalystsUsingPsuedo-tnverseMethod

Theinverse kinematicsfor-oboucmanipulators is give n by(Craig.1986)

Iii • [1 ) lei

^(2.10)

where10 }represe ntsthejointvelocityvectorand

Ix}

isthe end-effectorvelocityvector H/uJPIis theJacobianmatrix.The refore , thejointvelocitycorresponding(0a given {x}

i.~give n hy

Ie ) 1 1r' Ii i

zs

[./'] I·q ^l~.l~l

where[1'Iisc:.l~ kdthepseudo-inverseof theJac obianmatrix.'1'111:b:ISi..: ideaish>

minimizethe norm

II

{x}-[ll{a}~since

ul'

docsnotexist:1'siugularpoims.IJ'I~i\c's an approximate solution satisfyingthe condit ion

min i\81i and min tte!-[ J IlSII

12l.ll

Nearthe singular points,[J"]isequivalentto[11¹ami psuudn-inverse findsIll llthe cx;"':l solution.Thoughpseudo-inversegivesexactsolutionncarsingularpoints.theyarl'Ilul feasiblebecause of very high valuesof{O}.lienee " compromise isrequiredhC [WCCIl

feasibility !lOUexactnessincaseof inverse kinematic solutionncarsingularpoints.

Otherwise, pseudo-inverse solutions resultin undesirablecontinuityIc:tllingIIIhi~hjoint velocitywhiehresults invery high oscitlations.

2.2.4.2 VelocityAnalysis Using the Damped LeastSllUllres:'\letlwll DampedLeastSquares (DLS) methodhasbeenproposedhy severalresean:hers

tosolveinverse kinematicsproblems. Inthis mcrhrd.nnewritestherelationbetween {e} and{x}as

[[J)'

II I ' . '

[I]

I

JIJI^T

I.il

26

12.141

In order Inre<llislic<lll yachieve the desired jointvelocity values. onemust modifythe abuveequationto suit thehighestachievable limit ofthemanipulatorinterms of angular Velocities.In other words,we haw 10 minimizetheexpress ion

Min

Ili l - IlJ le ll ' •

1-'

I lei "

(2.15)

when:h is knownas the damping factor.~

Ie}

~is the norm of the joint velocityandthe

term~{x}•PH0 }Uaccountsfor the minimizat ionof the trackingerror orexactness of

the solution,lnl!h~~

Ie}

II!lakes care of the feasibilityof the solution.Itis equivalent 10 solvingaminimization proble m.

Mi' Qlx/

- 1 11 1 8 / '

subjectto constraint (2.16)

whereOn....ispractical limiton manipulatorsjointvelocity . An appropriatevalue of damping rector.h, willgivethe desired solution.Dampingfactor.>..,is computedusing lMaciejcwski;ll1u Klcin(l989»

(2.17)

where x,"=={U,}T{x] andr isthe rank of thematrixandai'{v,} and{u;}are obtained

27

fromSingula rValue DecompositionlSVO)lit"theJacobianmatrix[J]. 1\, express Eq.(2.l7)in asimple manneronecall write

e;

I

em..

I"

e~

e;

whe resuperscnpr»representsthemaximum ullownhlcvutuclorthatpuruculnrjoint.AI first,oneevaluates

l2.1lJ) andthen us ingEqs.(2.18)and(2.19)and using a nonlinear optimizationtechnique .Ilndx the valueof>' whichwouldminimizethefunction

C!.20 ) The optimalvalueof~is thensubstituted intheloll owing equat ion to gCIthe damped join!ve loc ityvector

Ie''',

^f2.21J

Unfctrunarely,boththese method s.i.e.. the pseudo-inve rseas wellas the IJLS nre,

28

expensiveinterms ofcomputations, andnotsuitable foron-linetasks.Itisimportantto selectnn appropriate valueof dampingractcr.A. A low value of A minimizesthe tracking crror and gives riseto undesirablehighjoin!velocit ies. A highvalueof~ accounts forthe robustness but leadstolowtracking accuracy(Chiaverni,1992).The term0,I(o,~+AI) far awayfrom singularpoints.becomes(as~-.0)

_ _0,_, _ •

0; +),.2 OJ

(2,22)

DLS sotutfonovercomestwo main limitations of pseudo-inversesolutionnear singularconfigurationsnamely thediscontinuityandinfeasible high jointvelocities.But SVDcalcuhuions arecomputationally expensiveand errorcompromise ishigh. In theory, itis possible10calculatethe damping factorA ateachofthe points alongthe trajectory (ncar singularpoints) but anoptimalvalueof~,ifchosenforallthe points wouldminimizethecomputational burden.

2.2.4.3Velocity AnalysisUsingNeuralNetworkMethod

Asingle layer neural network iscapableenough10[earnthe relationship between the Cartesianandjoint velocities nearsingularconfigurations.This is ahighly non-linear m:lpr ingwherejointvelocitiesincreaseat ahigher rate.

Consideringthe factthatinthe real-time controlproblems onehas to keepinmind

29

both,the errors (displacement. vclocit y.force ere..l.aswell;IStheI.:l llll lll 11:l titl ll ' i1 efficiency(real-limecomputations):therefore.in the presentwork,therelationship betweenthe Cartesianvelocityandthe joint velocityvectors wasestablished 11n1I11-liue basisusing the neuralnetworksover a segment

or

a trajectory.ThiscircumventstheOll- line computationalrequirementsor the jointveloc ity vector.as was done byresearchers (MaciejewskiandKlein.19 89)mentionedearlierin Chapter1. In theirmetluxl.thl' calculationswererequiredto be doneon<Ipoint hypoint basis hut whichresults in the slow ing down ofthe actual task.The additionalbenefitofthe neuralnetwor k method is thatone canachievebetteraccuracyalso ,

The inputvectoristhe Cartesianvelocityvector :llldthe outputvector isthejoint velocityvector.Thetrainingis performedon eithersideofthesiugulnrttypointtShar;ll1 and Balasubramanian.1993).The followingpointsarckeptillmindwhileperforming the train ing:

1)Maintain the jointvelocitiescloseto theupperfensiblc limitncarthesingular point.

2)A smooth transitioncurveofjointvelocitiesisrequiredoneithe rsideof singularitypoints.

3) Minimizethe errors betweenthe actualand achievablejoint veloci ties.

4)Haveoptimal numher of trai ningtasks\0achievethe non-linearmapping.

30

2.2.4 .4CaseStudy

To illustrate thetheorydeveloped so far.thetaskof moving the end effectoralong a traj ectory consistingof asegmentof a circleand a radial lineisshown inFig.2.8.

The pointof singularitywas the pointBinthis figure. While performingthe taska constantt.mge ntial velocityalongtheradialpathwasdesired.Thistask was performed using (a) Aplanar two degrees of freedom(ooF)manipulator (b) PUMA-S60 manipulator.These aretypicalmanipulators widelyusedby various researche rs inthe licldafrobotics.

APlana rTwo-!ink Manipulator

A simnle two-linkmanipulatorisshown in Fig .2.9.The velocity relationships betwee njointvelocityandtheCart esian velocityforthismanipulatoris givenby

(2.23)

where xandy arc coordinatesofthepathfollowed bythe end-effectorexpressed in universal frame.Theinverse of theJacobian is writtenas

[1]" (2. 24)

31

z

-- - -- - --- -- --- -- - --- --- -,

Singular Point

x

.:!india!

[''Trajec tory

Circular TrajectoryA

y

Figure2.8 Trajectory UsedforPUMA~S60Manipulator

32

Figure2.9 APlanar Two-LinkManipulator

33

Here. 61and0larethe joint anglesof the manipulntnrand IIandI:an: the link lengths.

One can find fromthe aboveequation. the singularityllriSCSwhens:=(l(1/:=0) t.c.

when asthe ann stretchesoutwardand bothjoint ratesgotoinfinity.The two-link manipulatoris movingits tip ata constant tangential velocityof 0.03m/s.Thelink lengths used wereII=0.4 m and11":0.2m:the radius ofthe circle was 0.07IIIaml the damping facto rhobtainedfromnonlinearoptimization routineWOlSn.0077.

PUMA-560Manipulator

TheforwardkinematicrelationshipbetweenCartesiancourdiuatcs andjoint coordinatesfor a PUMA·560manipulator(shown inFig. 2.10)is given hy

x.

^a3ctcZ3

-

_{d~CIS23 a 2clc2}

-

d] St

Y. ^a]slc21

-

_d~sls23

.

^{a2s l e2}

^-

^{d3c l a.lSI}

Z. -a~2J

-

_d4c 2J a~2

The link parameters for thismanipulatorarcshowninTable2.1.While perfor mingthe task .the desiredtangential velocity alongthe circularpath forPUMA-560 was 0.5 m/s andit was the samevelocity alongthe radialpathalso.The maximum achievableHnur

o,....

for eachof themanipulatorswastaken10 be 25radts.

34

Figure1.10 PUMA· S60 Manipulator

35

Table2.1LinkParametersof PUMA·560Manipulator

Linki

a ,

B, H, D,

(degrees) (degrees) (m) (111 )

1 0 B. 0 0

2 -90

e,

0.4318 0

3 0

e,

0.02032 0.127

4 -90

e.

0 0.43 18

36

2.2.4.5Result.. andDi'iCU~'iion

AtFlrstaI· UMA·560manipulatoris conside red.The{€I}vectorwas obtained u'iin~Eqs. r2.J J) or(2.12)depending upon the proximityofthepoint to!hepointof singularity.Theresultsobtainedarcsbownin Figs.2.11to2.14.Similarly.theresults furdampedleast squaresmethodusing F..q.(2 .2t) are also shownin thesefigures.II is quiteclearherethattherequired valuesnearthepointofsingularity arehighandnOI achievablebecausethis manipulator has amaximum

I €I I

equalto 25 rad!s. For the neumlnetworkanalysis.the input:1Or.!the outputvaluesfor thelearningphase were specifiedinaccordance withEqs.(2 .11) or the maxim umlimitsoverthe traject ory.

Al'ler this,the weightmatrix(W Iwhich relates [x]and{€I}as

Iii ' (WI lSI ^(2.26)

was tlhtairk.'tl usingEqs.(2.1)to (2.8).Theresults areshewninFigs.2.1tto2.14.In atttbe scfigures.theresultsoh(ainedbyneuralnetworkanalysis arefarmoreaccurate thanuose obtained hy till:DL.Smethodi.e .the neuralnetwork method givesthenoon values muchclosertothe values given byEqs.(2.11)and(2.12)than the DLS method.

Sr..ocondly.the error in

I

x

I

(to theright ofpointB) inFig.2.14in the case of neural nelWtm.: method.is duetoI~maximum achievablelimitand notdue tothemethod ilsclf.In adduio n.asmentionedearlier.the DL.Smethod requiresmuchmore on-line computations.Thesefacts were further confirmedin the caseof two-linkmanipulator asshownin Fig.2.9 .Theresultsinthis case areshown in Figs.2.15to2.18.The tmj ~l'luryin this casewastilt:sameasusedearlier.

37

s iii

en

0

~

"'

'"

0

~

r- 0

~

'"

0

'"

rn io 0

g

~ I: 0

a: _~ 0

Z N

m0

m

l'j N

;:;

W

0

•

0

,;

EQS.(2.1l)&(2.12) NEURAL DAMPED

TRAJECTQAYA

TRAJECTORY C

•

0. 0 0 0.50 1.00 1.50 2.00 2.5 0 3.00 3.50 4.00 DISTRNCE ALONG THETRAJ ECTORY(rnm)

Figure2.11 Variation of theNorm of theAngularVelocity Vector,

I 0

~.Alonl the Trajectory ofaPUMA·560Manipulator

38

s ,,;

'"

0

:il

,

TRAJ ECTORYA 0

'- ⁱⁿ

"C0

.. ,

0r-

"

.""~

~

0

>-

~ t:

u

..

a..J

~

UJ>

"

a:a:

~

...J::>

'"

⁰

z ~

a:

..;

~

ui

':'

0.0 0

£05 . (2.11)&(2.12 NEURAL DAMPED TRAJECTORY C

4.00 01STANCE ALONG THE TAAJE CTOAYImmJ

Figure2.12 Variationof the AngularVelocity,9,.AlongtheTrajectoryofa PUMA-S60 Manipulator

39

0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 a>0

cO '"

N<D

~ .,. _.,.

;;

<D

~ N

~ ^~

a>

"

⁰

'"

..,;

^N

'"

0

~ ^NN

U N

0 r-

~

..JW

>

.,.

a:a:..J

'" ~

::>

<!l <D

Z

'"

a:

..

N

'" _r s

'" ^,

0.00

EQS.(2.1I)&(2.12) NEURAL DAMPED

TAAJECTDAYA TAAJECTOAYC

DISTAN CERLDNGTHE TAAJECTORY(mm)

Figure2.13 Variation

o r

^theAngularVelocity, 9).Along the Trajectory01.

PUMA·560Manipulatur 40

0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 aa

,..:

~

'"

~

~

^.;

~ a

<D

.;

a

.~ rn

'"

a

:E a

a: '"

0z

~ N

~.;

0

'"

0

Na a

I

is r

0.00

- - EQS.(2.11)&(2.12) - NEURAL -)(- DAMPED

TRAJECTORYA TRAJECTORY C

DISTANCEALONG THE TAAJECTORY ImmJ

Figure 2.14 Vari ation oftheNor-m ofthe Cartesian VelocityVector.ijx

I.

AlongtheTrajector yofaPUMA·560Manipula tor 4\

'"

'"

.;

iii

'"

coN

'"

mr- N

il

N

"'

co

>:

..:

II:a

z

_~

.;

s

ci r-0

ci

~

? s

r

0.00

- - E0 5.(2.II)&(2.12) - NEURAL -)(- DAMPED

TRAJECTOHYC TRAJECTORYR

B

0. 50 1.00 1.50 2.00 2. 50 3. 00 3. 50 4. 00 DISTANCE ALONG THETRAJECTORYLmrnl

Figure2.15 VariationoftheNormofthe AngularVelocity Vector.

1 91.

AlongtheTrajectory of a Two-Link:\fanipulator 42

0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

g ui g

~ .. ^{..; g}

N

0

'"

:;; 0

0

:;2

L

.,

. ... _:;2

>-

'"

t: ^,

u 0

0 r-

--'

"'

UJ>

,

a:

:;2 a:

r-,

--'::>

,

CO

:;2 z

a:

i

r-

g

ei

.,

0.00

TRAJECTORYA

EQS.('.l1l&('.1') NEURAL DAMPE D

TRAJECTORYC

DISTANCE ALONG THETiAJ ECTDAYLmrn)

Figure2.16 Variationofthe AngularVelocity,

e

l•AlongtheTrajectory ofa Two-LinkManipulator

43

0 0

0

'"

,;

~ '"

0

M CD

. g ^g

~ ^.n

^N

.:

^~

0

-.;:

^N₀

"'

>-

~

>-

U S'

0^...J

~

W>

a:

s

a:^...J

.n

l!l::JZ 0N

a:

ci

0

'"

~

0

,

0. 00

_ EQS.(2. 1l)&(2.12)

- NEURAL

- OAMPED

TRAJECTOAYC

TRAJECTORY A

0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 DISTANCE ALONG THE TRAJ ECTORY[mmJ

Figure2.17 Variation

or

the AngularVelocity,

o,

AlongtheTrajectory

o r

a Two-LinkManipulator

44

a

i\i

a on

;;;

aa

gj

_TRRJECTORYR

~

^aon

~ -e-

N

aa

.~

N

iii

>:

c:

0

a:

a

z

a

., iii

o!

EQS.(2.11)&(l.U) NEURAL DRMPED aa

,.:

aon

'"

aa

o+-_~_~_~

__

~_~_~_~_---I 0.00 0.50 1.00 1.50 2.00 2.5 0 3.00 3.50 4.00

DISTANCE ALONGTHE TAAJE CTOAY Ernm J

Figur e 2.l8 Variationofthe Normof the CartesianVelocity,II:KIAlongtbe Traje-rtoryof a Two-Link Manlpul alor

4S

2.3 LP-Neuro Method

2.3.1 .-\New Approach- Development of U'-Neu roMelhod

Asdtscussed earlier.3.mClhtlllroIr.k,-:-,' IT Ilk: a":I.1.Ir:Jcy;1Ik! C' llllr U1.'t i" lIa l effic iency . is sought. A1l\:Wtnt:lhl'l1calledLP-n,:u runtetht-.l tlb!:tsuhr. llll;mi;m;11-.1 Sharan.1993)is dcvckl(x'''din thissecuonwhil:h utilizes theIasrer convergencepn'l""'r1~­

oflinear progra mming:lhis resultin bcucremirminimiza tion.Thearchitecture

tlr

this methodissimilarto thefl,.ocJforward errorhm:kpmp;lgatil1l1ncurnt uct work cxcc111thaI asingle layer isenough.'111eactivationfunction uscl1illthisC;ISl.'is;.Iinc.rractivation functionwithslopemand intercept c. 1\nonlinear curve is"pprnxilllatcllhyscvcra!

linearcu rvesof differentslopesand tracrccprs.The errorminimizutiouobjec tivefUlll:liUII hasweightsandinterceptsaslinearvariablesandtheslUflCilSunn-Iiucarv;lri;l hlc~which is solvedusingHookesand Jccvesmethod.

2.3 .1. 1 LP-Neu r oMeth od-TypeI

Intbeneuralnelworkmel.hod(asUSl.'l1inSec.2.2.4. 3l.lhc::input

III

illxllk....ircu output{D)vectorsarerel atedbythe euuatio n

ID} ~

I W IIII

^(2.271

Ho wever.due to errors. oneobtains avector{Olinsteaduf{D}. Thewergln matrixIWl whichrelatestheinputamioutp u tvectorintll:ll cnscis givenhy

46

hI ^wil w₂₁ w_kJ

IHI

^{h, w21w22}

W"

ⁱ² (2,28 )

h_j w_jl w_{j 2} w_{j i} i,

TheLP-I1I.:uromethodisdiagrammatically explained inFig.2.19.The functional relationsh ipbetween{H} andID}canbe writtenas

101• [Willi

(2.29)

(2.30)

The elementOJisshownbyasquaresymbol(0 )inFig.2.19and hjaretheclements of vector{II}. Thisissimilar to the sigmoidalfunctionalrelationshipusedin backpropagation method,whereone uses theequation

0; " f(hJ" 1

s , 1+exp(- h) ^{(2. 31)}

One ofthewaystoobtai n thesciof weightmatriceswithminimumerrorwould beby 47

i,

IN PU T LAYER

f-- L _!

, .

: It

, ,

L ,

O UTPUT LAYER

\ {O}

-- -{Hi---

: Cj •

L. •• •

DESIRED O UTPUT

./ ~~L. _

~ _~ T

dj 9i

figure1.19 DiagrammaticRepresentation of the Network-LP-NeuroMethod

48

writing;I,o~tfunctionEinthe foll()wingterm:

Subjectto

(2.32) where<.I,arctheclements ofthe desiredoutputvector {D}inEq.(2.27 ),andw_Jlare the weights.Eq.(2.32)has(j*k) weights ami they can be collectedin a single dimensional arrayoravectorOlS

"" ",

{WI ^IVl2

",

1Vj.\ 1V"l-

(2 .33)

This vector{W~containsjs kunknowns.Sincethesecan take positiveor negative values , cuch{)fthese can he replacedby twopositive variables( arequir ement for solvinglinearprogramming).For examp le.one can writeWI

=

VI•v~•w~

=

VJ-V4etc.

SuhstitulingWiin terms of v, one can rewrite the Eq. (2.32)as

49

SuhjCCIlO

IAIIV} • (D}

Thedetails ofthecoefficient nuurtxIAI canheshown,IS

(AI

=

t~.J.J1

-II;~ -;1^...j, -;l 0 0 0 0 00 0

..

0 0 0 0 0 0 0 0 0 0 0 0 0 0 j, -i_{li~ -}i~.. j, -i_l

. .

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...j, -i._i~ -j~...i,

.',

1:!.J5l Justlike IW).one canalsowrite

"

(2.3ft)

50

ThelIliHri.\1/1.1clmlaim.anumber orzeroesin a givcnro...lien: thenon-zcroelemcnu 1I1:t:Urtogerberandnnlyonce inagtvenrow. II is wellknownthat theopumaicost fuoclionfursuchprublcms, involving sparsity.canbe oolail"ll.'\Jmuch more quickly CMI.'Cnrmick.1990,as compared10a casewherefA Iisadensematrix.

2.3.1.2I.P-i'lcur oMeth od~Type 2:

Further rcflrc me ntsontheabovemethod ca nbemadebyreplacingthe acuvauc n functiongiveninEq.(2.29)byanother function givenby

(2.37)

In1:4.(2 .37)eachvariahlehihasacorrespondingscalarm,andaconstantcr Thenew relationship correspondingtoEq.(2.30)willbe

101 • {MI(IV](II • ICi (2.38)

where

IMI =

diago nalslopematrix.which has scalar'"Jas its diagonal etemems. /I.similaractivationfurct ion fortheoutputsidecanbewritten3S

lSI-INII D I •

IGI

^(2.39)

where

INI

isadlugnnalmatrix.Thematrices

INI

and{G}areanalogousto 1M)and{C}

in E4. (2.38),ThenewFormulationusingE4S.(2.38 ) and(2,39)willbe

51

MinimizeE,

=

[NI{D}+{G}-[MlIWHI}-{C]

subjectttl

[MlIWI{I}+{C}

=

INI{D}+{G}

or in the scalarform.itcanbe rewritten as

MinimizeE,= nidi+nf!)+... +n,d, 'md\\'lIit+w,:i:+ ...+w,.i ,}-1Il.•{

w)li ,+w::i:

+ ...

+w:~i.}·...-11I1{W,li1+w,:i:+...+w,1i.}-I-g\I- g:+... +g,"

subjectto

Againhere.theweightswJI..cJand gJarc replacedbytwo positive numbers asbeforeill thefollowingmanner:

gi=V~I•V~I

Afterthesesubstitutions.onearrivesat

i=1,2•..•

1=1,3, ..

(2.41)

MinimizeE1=n,tl,+n~2+...+n,tll -mJ(i1v,"i,vl+ilYl·i2Y~+...J+mii.v1k,,- i1y,-,.:+ ..)

+

m)(i,Y4..1 -i'Y~I02+..)+rol( ...+il.v2J!' _1-i, vl,,'+V~ I-V~l+... .

52

Suhjecrm

(2.42)

This cl{ui.llioncanhewritteninthestandard LP notationas.

SUbjl"t:IIO IAIlV } • ID}

whcn:thectlcflicic nlmatrixIA Iisgivenby

[

.... .... ....

II II II

",i.\·1·1 1II

n " ...!, -",i.I-I-11

: :: : :]

... _" ....

-~

^{...I-"I-I I}

(2.43)

Inthe allowequation,U, an:thedesired valuesandil.are (heinput values;

v,.

V<I andv~,aretheunknow n variablesand soaren~and oJ' The problem showninEq.

53

(2.4:!)isnon-linearbecaus e oftheoccurrcnc cororooucricnu'slI.::h,IS 1lI1'"1 However. if this problem is comhincdwithanothe rmuhi-vnnnblcIlptimil.;llillllpTl1hk m containingallm,only,thentheproblemiuvolviugtherem,lini ngv:lri;lbkscanbeSllh'cd by thelinear programmingmethod.Since.thenumberIll"vartablcs far c:\l;e\\1 the numberof constraints.itwouldhe beucr10 solvefor 111,lIs ing non-linearllptimiwtinn and the remaini ngvariableswhichincludeweights. hy linearmethod. ThislI\elh'ld clearlydiffers fromothersbecause.for themajority(Ifthevariables(otherthan11\1.the linear method yieldsfasterconve rgenceascompared IIItnta llynon-linearIIIcthod."1"11\' additionaladvantage inthelinear methodisnunonecan cxplourhc xpursityrn]I\llIIalrl.\

in Eq.(2.35) .Forexample.iftheite rative valuesIll'111,arc ulunincdInnuthcnOIl-lil1car method. andsubstituted inEq. (2,42).lhenthe resultingproblem hecnl\leslinearand \';111 be solvedusing the Revised SimplexMethod(Sidd ul. 19K2).The actual Flow chartlit the combined methodis showninFig.2.20,Infact,one can nnempr to solveusing;1 singlem valueinsteadof}differe ntIll)values andcheck ferconverg e nce.It"results arc satisfactory.thentheproblem can be reduc edttlsingle variable non- linearuptilllil.alillll problem follo wed bylinearprogramming.

2.4 Applications of the LP-Neuro Method

2.4.1FunctionGenerati on

Approximating asine curvehasbeen a test fornun-linearmarringcarriedOUIhy severalresear chers, Thenon-linearmappingor a sinecurveusingbackpropagationis

54

figure2.20 HowChart,LP-NeuroMethod 55

aSill(h.tl

wherea

=

0.8andb-1".TIlesame example wastakenhen:tortbccascsIUJ~'.111" lc;k1 of usingseveralbiastenusasJOilCinZurada(1992).aJiITerentapproac hwasI"I1Iltl\\\.-J inthepresen twork.To dothis.21pl1illiSalollgthel<oillecurvein a pcrilll.lwere takcu fortraining.In order10identifythiscurve.thetrainingwas performedUIIdifferentSille curves having diffe rentvalues Ilfaandb.IIIallcases .:11pointswereused . Afh:rthis.

the sa menumberofpointsforthispaniculurcurve W<lSpnwjdcd :ISinputaful corresponding outputwas checkednnthe sinecurve.Theresult susingCq.tL \,Hurul Eq.(2.44)are sho wninFig.2.21. Theres ultsinthisIigurcshuwuuuinthefirl<o1 quarter period .theBPmethod yieldssligh tlybetterresultsthantheU>mcthlld0:.(1.2,:0 1 butnotallthrough. Ontheothe rhand.liteLs-ncuromethodC1~.2.41 )is ,llw:lys accurate and decided lythemethodto beUSl.'\I. In view oftileabo ve, nnly theU'-III.'lH'l I methodandBPmethodWert'usedin theocxtlWOexamples .Furthermore,:lsinglevalue of myieldedresultswhichweresuOicicmly accura te. lienee.the sameprocedurei!>

follo wedinsolvingthenextIWOexamples.

2.4.2 Acceleration Analysis

or

^aTwo-linkPlanar Manipula tor

A two -linkplanarmanipulato rhavingrevenue jointsissho wn inFig.2 .22.The end-effector.P.ismade10follow acirculartraje ctory at aconsta nttange ntialvelocity.

V"ofmagnitudeequalto0.15m/ s.(X,..Y"Jrepresentsthe gfcba lcoordinatesystemand

56

>-

0.5

o

-0.5

\

- LP(Eq.2.34) .. LP·NEURO .. DESIRED

o

BP

20

10

15

5

-11-_ _~_ _~_ _ ~~_ _-'-'

o

POINTNUMBER

Figure2.2 1 ComparisonofValues fortheSineCurve(LP,LP·NeuroMethod and BP Method andtheDesiredValues)

57

VI

i,

r

=

0.05m II=0.3In

/,=0.2m

Figure1.22 AP1anar Two-Link:\tanipulator and the Trajectoryusedror Attel erationAnalysis

58

(X"Y,Jrepresentthe localcoordinateframeofthe linkiandthe jointvariables .01and0:

representthenuariuualdisplacements.

Thejointvariables, 01andO~are relatedtothe positionofthe end effector (XI"Y!"inCartesianspacethroughthefollo wing equations:

12.45)

and

Differe ntiatin g Eqs.(2 .45) and(2.46 )with respect10time,weget

whereII'I~arc the lengthsoflinksJand 2 respectively;C1=cosO_{l ;}and (;1:=eos(Oj+0:)etc.The accelemnon of thetipmovingalong thecircularpath inthe radial direc tionisgivenhy

ii, -w~r

r,

59

-=-2 ,

a

;, '

^(2.48)

- ,.,' I

--;-"os(a)

~Sin(a.)

He re, one can obtainby differentiatingthe Ell.(2.49)

Inrobotic control, thereis a greatneedfor minimizationof'oil-linecomputations. Rather thanperforming numerouscomputationsas shownby E4s.(2.45,to12.50).itisdcsirahlc to finda linearrelationshipbetweenthe vectors givenhy

60

° Il

8:1 t,

e,~

Y "

° u

^{(WI] (2.5 1)}

818

s ;

Db

y~

UIItheorr-line basislirst.Thefirst subsc ript.i,inO.represents angularaccelera t ionof a parucula r link.and rbe seco ndone.j,the: pointalongthetrajec tory.Theaccelcranon clcrucnrs

"Pl'

YI'letc.•arecomputedusin gEq.(2.49).

Inccmrolproblems .one1k:C<.IStoknow{O}.asthe end effector traversesthe trajectory.Usually.on-lineco mputationsarcdone on apointbypointbasisi.e.•one has Incom)'outccmp undonsgivenby E4S.(2.45) to(2. 50 ) atevery point. InEq.(2.5 1) above.ifwe nluain thefWdon anoff-line basis then . onecancompute

to}

on anon-line basisfor anyset of points :llongthetrajec tory.muchmorerapidly.Fortra ining,circles

61

of differentradiiwereused andinallcases20p\,i11ls were sl'lc.·tcdon .Hncrcm concentriccircles.Figs,(2,23)and(l,2·t.)showtheresults"htainedhy!:' IS, \2.451til (2.50 ),The sameproblem was;llsudoneusing theUI'mcthod andsbowuinthese figures ,Thesefiguresclearlyshow thatnnecanverysuccessfullyU.'iI,'n"'lIralnctw :\:

conceptingeneral,andLlt-ncuromethod in particular. in;If:" ving<II a bettercontrol strategy forroboticnunipuhuors.Theconstant velocityrcjulrcmcutoftll...endcff"'Cll'!"

ispresentinmanyindustrialapplicationssuch;ISwelding.Ilaintingetc..

2.4.3 Solution of Torque and Reacti on Forces of the Two-link Manipulator

The iterative Newton-Eulerdynamicsalgorithm(secT;lhh:2,2)hasbeenused veryextensivelyby various researchers.TheJinkparameters used inthisC:l~earc shown in Table2.3.Heretoo ,the numberofcomputationsis quitelargetoheperformed011an on-line basis.Inthis method, kinematicsolctionsare carriedoutnilalinkhyliukhasis startingfrom the base(refer toFig.2.22).Whenallthe kinematiccump uuunmsnrc completed,then thedynamic computationsstanfromtheouterlink10theinnerlink.

The detailscan beseenin(Craig,1986 ) andare notmentionedhere.liven illthiscase itwouldbebetter10have thefollowingrelationshipon an nff-Iincbasis:

62

Table2.2TheIterativeNewton- Euler Dyn amicsAlgorithm

FORWARDREC URSIO N

StepI: {wi, _IRI~{wl'l

{'I e ,

Step 2: {ol, IRJ~{Ctl'_1

{,I e ,

IR(Iw},x1'1

e,

StepJ: {a}, IR]~(lo:},I {~}'_ IX {P}i-l

.

^{{W},_Ix(P}I_I)}

{z}

e ,

2 xtR]~{wL,x{a}

a,

Step 4: {a }" {a}, {ol,x1'1, {w};x{w},X{s}/

Step5: IF}, 1Il,{l/I"

Step6: WI, ^III,{al, {wi,x

(in ,

Iw},)

IIACKWARDRE CURSION

Sic"7,

UJ, " {FI , •

[RI" ,

U} ,.,

StepIt {il L IN]",{1l}"1 {N}, "{J }, x{F}, ~ {P},x«RI;.]{f},.]) Stcp'); T, ~ {::l{I/},T~ ",,.

63

Table2.3 linkParametersortheTwo-Linki\lanipul:llnr

DETAILS LINKI I.INK2 UNITS

LINK LENGTH 0.25 0.16 m

LINK CENTER 0.20 O.I~ m

OFGRAVITY

MASS 9.50 5.00 kg

64

.<'

1.2

]

~

.. ^-

z

0

0.8

~

0:

...

~_u

'.t 0.6

0:

:5

:J

0.4

^LP-NEURO

'"

z «

BP

0.2

^DESIRED

0

0 8 10 12

POINTNUMBER

Figure2.23 Variationoff)\.Along theTrajectory

65

-2 -"!

g

^{-2.2 LP- NEURO}

BP

:q)' -2.4 DESIRED

z

~

0 ^-2.6 w-' w

() -2.8

'.:i

'"

:5

-3

::>

o z -c

-3.2 -3.4

-3.6

a

8 10 12

POINT NUMBER

Figure 2.24 Variation of

ell

Along the Trajectory 66

[I: ^e,

I ; .,

" ^B,

^(2.52)

f.

^[W,J

^B,

I, a

l

"

^ti~

InFil;.2.22.'ISthe end effector Pmoves alongthe trajectory,due10 theappliedtorques (11

andr¹)hythe motorson therespect ive links,thereact ion s forces(t~I,

f/

etc.,)aTC prod uced.Onehastoknownot onlythe torquesbutalsothes e reactions forcesat every pointalungthetrajectory.The computationswerecarriedoutfor thetwo-link manipulatoralongthe show ntrajectory.Twentypointswereused here to obtai n[W:I.

Result s;ITeshewn inFigs.2.25 102.30. The results dearlyshow thattheoverallerror isquitesmatt.hisless than0.2'kinalltheCOlSl:Sobtainedby LP-neuromethod.Such allaccurate relationshipwould be of greathelp for on-line controlofsuchsystems.One

~';1l1alsu secinthese figuresthaiLlvncurc methodyieldsbetter resultsthan DP method.

67