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IntJAdvManufTechnol19938344191993SpringerVerlagLondonLimitedTheInternationalJournalofRdvancedmanufacturingTechnoloouFullPoseCalibrationofaRobotManipulatorUsingaCoordinateMeasuringMachineMorrisR.Driels,LtW.SwayzeUSNandLtS.PotterUSNDepartmentofMechanicalEngineering,NavalPostgraduateSchool,Monterey,California,USATheworkreportedinthisarticleaddressesthekinematiccalibrationofarobotmanipulatorusingacoordinatemeasuringmachineCMMwhichisabletoobtainthefullposeoftheendeffector.Akinematicmodelisdevelopedforthemanipulator,itsrelationshiptotheworldcoordinateframeandthetool.Thederivationofthetoolposefromexperimentalmeasurementsisdiscussed,asistheidentificationmethodology.Acompletesimulationoftheexperimentisperformed,allowingtheobservationstrategytobedefined.Theexperimentalworkisdescribedtogetherwiththeparameteridentificationandaccuracyverification.Theprincipalconclusionisthatthemethodisabletocalibratetherobotsuccessfully,witharesultingaccuracyapproachingthatofitsrepeatability.KeywordsRobotcalibrationCoordinatemeasurementParameteridentificationSimulationstudyAccuracyenhancement1.IntroductionItiswellknownthatrobotmanipulatorstypicallyhavereasonablerepeatability0.3ram,yetexhibitpooraccuracy10.0mm.Theprocessbywhichrobotsmaybecalibratedinordertoachieveaccuraciesapproachingthatofthemanipulatorisalsowellunderstood\1\.Inthecalibrationprocess,severalsequentialstepsenabletheprecisekinematicparametersofthemanipulatortobeidentified,leadingtoimprovedaccuracy.Thesestepsmaybedescribedasfollows1.Akinematicmodelofthemanipulatorandthecalibrationprocessitselfisdevelopedandisusuallyaccomplishedwithstandardkinematicmodellingtools\2\.Theresultingmodelisusedtodefineanerrorquantitybasedonanominalmanufacturerskinematicparameterset,andanunknown,actualparametersetwhichistobeidentified.Acceptedforpublication21October1991CorrespondenceandoffprintrequeststoProf.MorrisR.Driels,DepartmentofMechanicalEngineering.NavalPostgraduateSchool,Monterey,California93943,USA.2.3.Experimentalmeasurementsoftherobotposepartialorcompletearetakeninordertoobtaindatarelatingtotheactualparametersetfortherobot.Theactualkinematicparametersareidentifiedbysystematicallychangingthenominalparametersetsoastoreducetheerrorquantitydefinedinthemodellingphase.OneapproachtoachievingthisidentificationisdeterminingtheanalyticaldifferentialrelationshipbetweentheposevariablesPandthekinematicparametersKintheformofaJacobian,PJK1andtheninvertingtheequationtocalculatethedeviationofthekinematicparametersfromtheirnominalvalues8K\jrj\,jrp2Alternatively,theproblemcanbeviewedasamultidimensionaloptimisationtask,inwhichthekinematicparametersetischangedinordertoreducesomedefinederrorfunctiontozero.Thisisastandardoptimisationproblemandmaybesolvedusingwellknown\3\methods.4.Thefinalstepinvolvestheincorporationoftheidentifiedkinematicparametersinthecontrolleroftherobotarm,thedetailsofwhichareratherspecifictothehardwareofthesystemunderstudy.Thispaperaddressestheissueofgatheringtheexperimentaldatausedinthecalibrationprocess.Severalmethodsareavailabletoperformthistask,althoughtheyvaryincomplexity,costandthetimetakentoacquirethedata.Examplesofsuchtechniquesincludetheuseofvisualandautomatictheodolites\4,5,6\,servocontrolledlaserinterferometers\7\,acousticsensors\8\andvidualsensors\9\.Anidealmeasuringsystemwouldacquirethefullposeofthemanipulatorpositionandorientation,becausethiswouldincorporatethemaximuminformationforeachpositionofthearm.Allofthemethodsmentionedaboveuseonlythepartialpose,requiringmoredatatobetakenforthecalibrationprocesstoproceed.FullPoseCalibrationofaRobotManipulator352.TheoryInthemethoddescribedinthispaper,foreachpositioninwhichthemanipulatorisplaced,thefullposeismeasured,althoughseveralintermediatemeasurementshavetobetakeninordertoarriveatthepose.ThedeviceusedfortheposemeasurementisacoordinatemeasuringmachineCMM,whichisathreeaxis,prismaticmeasuringsystemwithaquotedaccuracyof0.01ram.Therobotmanipulatortobecalibrated,aPUMA560,isplacedclosetotheCMM,andaspecialendeffectorisattachedtotheflange.Fig.1showsthearrangementofthevariouspartsofthesystem.Inthissectionthekinematicmodelwillbedeveloped,theposeestimationalgorithmsexplained,andtheparameteridentificationmethodologyoutlined.2.1KinematicParametersInthissection,thebasickinematicstructureofthemanipulatorwillbespecified,itsrelationtoauserdefinedworldcoordinatesystemdiscussed,andtheendpointtoilmodelled.Fromthesemodels,thekinematicparameterswhichmaybeidentifiedusingtheproposedtechniquewillbespecified,andamethodfordeterminingthoseparametersdescribed.ThefundamentalmodellingtoolusedtodescribethespatialrelationshipbetweenthevariousobjectsandlocationsinthemanipulatorworkspaceistheDenavitHartenbergmethod2,withmodificationsproposedbyHayati10,Mooring11andWu12toaccountfordisproportionalmodels13whentwoconsecutivejointaxesarenominallyparallel.AsshowninFig.2,thismethodplacesacoordinateframeonJointn//Joint.n1L/Linkn1..L/ZnnX.Y/Yn1X,IFig.2.Linkcoordinateframeallocation.eachobjectormanipulatorlinkofinterest,andthekinematicsaredefinedbythehomogeneoustransformationrequiredtochangeonecoordinateframeintothenext.ThistransformationtakesthefamiliarformA.rotz,O.transz,d,transx,a.rotx,t.roty,fl.3Theaboveequationmaybeinterpretedasameanstotransformframen1intoframenbymeansoffouroutofthefiveoperationsindicated.Itisknownthatonlyfourtransformationsareneededtolocateacoordinateframewithrespecttothepreviousone.Whenconsecutiveaxesarenotparallel,thevalueof/3.isdefinedtobezero,whileforthecasewhenconsecutiveaxesareparallel,d.isthevariablechosentobezero.WhencoordinateframesareplacedinconformancewiththemodifiedDenavitHartenbergmethod,thetransformationsgivenintheaboveequationwillapplytoalltransformsofoneframeintothenext,andthesemaybewritteninagenericmatrixform,wheretheelementsofthematrixarefunctionsofthekinematicparameters.Theseparametersaresimplythevariablesofthetransformationsthejointangle0.,thecommonnormaloffsetd.,thelinklengtha.,theangleoftwista.,andtheangle/3..ThematrixformisusuallyexpressedasfollowsAnCe.CO.SO.Sa.SO.Se.Ca.CO.SO.Se.Sa.CO.a.CO.SO.CO.CO.Sa.SO.CO.Ca.SO.SO.CO.Sa.CO.a.Se.4Ca.SO.Sot.Ca.CO.d.0001Fig.1.Calibrationequipment.Foraseriallinkage,suchasarobotmanipulator,acoordinateframeisattachedtoeachconsecutivelinksothatboththeinstantaneouspositiontogetherwiththeinvariantgeometryaredescribedbythepreviousmatrixtransformation.The36M.R.DrielsetaL982/4.Z41,...X4Fig.3.PUMAframeallocation.transformationfromthebaselinktothenthlinkwillthereforebegivenbyTnA1A2...An5Fig.3showsthePUMAmanipulatorwiththeDenavitHartenbergframesattachedtoeachlink,togetherwithworldcoordinateframeandatoolframe.Thetransformationfromtheworldframetothebaseframeofthemanipulatorneedstobeconsideredcarefully,sincetherearepotentialparameterdependenciesifcertaintypesoftransformsarechosen.ConsiderFig.4,whichshowstheworldframexw,y,,z,,theframeXo,Yo,z0whichisdefinedbyaDHtransformfromtheworldframetothefirstjointaxisofthemanipulator,frameXb,Yb,Zb,whichisthePUMAJe4ZlxbYb/1/do9.Yw.xwZwFig.4.Basetransformations.manufacturersdefinedbaseframe,andframexl,Yl,zlwhichisthesecondDHframeofthemanipulator.Weareinterestedindeterminingtheminimumnumberofparametersrequiredtomovefromtheworldframetotheframex,Yl,z.TherearetwotransformationpathsthatwillaccomplishthisgoalPath1ADHtransformfromx,,y,,z,,tox0,Yo,zoinvolvingfourparameters,followedbyanothertransformfromxo,Yo,z0toXb,Yb,ZbwhichwillinvolveonlytwoparametersbanddinthetransformTobrotz0,4transzo,d6Finally,anotherDHtransformfromxb,Yb,ZbtoXt,y,ZwhichinvolvesfourparametersexceptthatA01and4arebothabouttheaxiszoandcannotthereforebeidentifiedindependently,andAdlanddarebothalongtheaxiszoandalsocannotbeidentifiedindependently.Itrequires,therefore,onlyeightindependentkinematicparameterstogofromtheworldframetothefirstframeofthePUMAusingthispath.Path2Asanalternative,atransformmaybedefineddirectlyfromtheworldframetothebaseframeXb,Yb,Zb.Sincethisisaframetoframetransformitrequiressixparameters,suchastheEulerformAbrotz,C\broty,0brotx,bbtransPxb,Pyb,Pzb7ThefollowingDHtransformfromxb,Yb,zbtOXl,Yl,zlwouldinvolvefourparameters,butA0mayberesolvedinto4,,0b,,andAdresolvedintoPxb,Pyb,Pzb,reducingtheparametercounttotwo.Itisseenthatthispathalsorequireseightparametersasinpathi,butadifferentset.EitheroftheabovemethodsmaybeusedtomovefromtheworldframetothesecondframeofthePUMA.Inthiswork,thesecondpathischosen.ThetooltransformisanEulertransformwhichrequiresthespecificationofsixparametersAsrotz,b6roty,04rotx,/68transPx6,Py6,P,6Thetotalnumberofparametersusedinthekinematicmodelbecomes30,andtheirnominalvaluesaredefinedinTable1.2.2IdentificationMethodologyThekinematicparameteridentificationwillbeperformedasamultidimensionalminimisationprocess,sincethisavoidsthecalculationofthesystemJacobian.Theprocessisasfollows1.Beginwithaguesssetofkinematicparameters,suchasthenominalset.2.SelectanarbitrarysetofjointanglesforthePUMA.3.CalculatetheposeofthePUMAendeffector.4.MeasuretheactualposeofthePUMAendeffectorforthesamesetofjointangles.Ingeneral,themeasuredandpredictedposewillbedifferent.5.Modifythekinematicparametersinanorderlymannerinordertobestfitinaleastsquaressensethemeasuredposetothepredictedpose.Table1.NominalparametersforthePUMArobot.0bPbP,,bPzbooommmmmm180.00.090.0394.0383.0474.0Link50,d,a,/3,ommmm1000.090.0020.00431.850.00.030.0149.0920.3390.0040.0433.00.090.0050.00.00.090.00mmmmram90.00.00.00.00.0134.0Theprocessisappliednottoasinglesetofjointanglesbuttoanumberofjointangles.Thetotalnumberofjointanglesetsrequired,whichalsoequalsthenumberofphysicalmeasurementmade,mustsatisfyKp.NDt9whereKpisthenumberofkinematicparameterstobeidentifiedNisthenumberofmeasurementsposestakenDrrepresentsthenumberofdegreesoffreedompresentineachmeasurementInthesystemdescribedinthispaper,thenumberofdegreesoffreedomisgivenbyDt610sincefullposeismeasured.Inpractice,manymoremeasurementsshouldbetakentooffsettheeffectofnoiseintheexperimentalmeasurements.TheoptimisationprocedureusedisknownasZXSSO,andisastandardlibraryfunctionintheIMSLpackage\14\.2.3PoseMeasurementItisapparentfromtheabovethatameanstodeterminethefullposeofthePUMAisrequiredinordertoperformthecalibration.Thismethodwillnowbedescribedindetail.TheendeffectorconsistsofanarrangementoffiveprecisiontoolingballsasshowninFig.5.ConsiderthecoordinatesofthecentreofeachballexpressedintermsofthetoolframeFig.5andtheworldcoordinateframe,asshowninFig.6.TherelationshipbetweenthesecoordinatesmaybewrittenasPTPI11wherePiisthe4x1columnvectorofthecoordinatesofFullPoseCalibrationofaRobotManipulator37X6Iz6Fig.5.Tooltransform.z//p,XwF6/X6Fig.6.Ballcentroidinthetoolandworldframes.theithballexpressedwithrespecttotheworldframe,Pisthe4x1columnvectorofthecoordinatesoftheithballexpressedwithrespecttothetoolframe,andTisthe44homogenioustransformfromtheworldframetothetoolframe.IfPisknownbyprecalibratingthetool,andPismeasured,thenTmaybefound,andusedasthemeasuredposeinthecalibrationprocess.Itisnotquitethatsimple,however,sinceitisnotpossibletoinvertequation11toobtainT.Theaboveprocessisperformedforthefourballs,A,B,CandD,andthepositionsorderedase,P{3P/PbT\PAesPcPo\12orintheformPTP13SinceP,TandPareallnowsquare,theposematrixmaybeobtainedbyinversionTPp114InpracticeitmaybedifficultfortheCMMtoaccessfourbailstodeterminePwhenthePUMAisplacedincertainconfigurations.ThreeballsareactuallymeasuredandafourthballisfictitiouslylocatedaccordingtothevectorcrossproductP4P3PIP2PI15Regardingthedeterminationofthecoordinatesofthe
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