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FullPoseCalibrationofaRobotManipulatorUsingaCoordinateMeasuringMachineMorrisR.Driels,LtW.SwayzeUSNandLtS.PotterUSNDepartmentofMechanicalEngineering,NavalPostgraduateSchool,Monterey,California,USTheworkreportedinthisarticleaddressesthekinematiccalibrationofarobotmanipulatorusingacoordinatemeasuringmachineCMMwhichisabletoobtainthefullposeoftheendeffector.Akinematicmodelisdevelopedforthemanipulator,itsrelationshiptotheworldcoordinateframeandthetool.Thederivationofthetoolposefromexperimentalmeasurementsisdiscussed,asistheidentificationmethodology.Acompletesimulationoftheexperimentisperformed,allowingtheobservationstrategytobedefined.Theexperimentalworkisdescribedtogetherwiththeparameteridentificationandaccuracyverification.Theprincipalconclusionisthatthemethodisabletocalibratetherobotsuccessfully,witharesultingaccuracyapproachingthatofitsrepeatability.KeywordsRobotcalibrationCoordinatemeasurementParameteridentificationSimulationstudyAccuracyenhancement1.IntroductionItiswellknownthatrobotmanipulatorstypicallyhavereasonablerepeatability0.3ram,yetexhibitpooraccuracy10.0mm.Theprocessbywhichrobotsmaybecalibratedinordertoachieveaccuraciesapproachingthatofthemanipulatorisalsowellunderstood1.Inthecalibrationprocess,severalsequentialstepsenabletheprecisekinematicparametersofthemanipulatortobeidentified,leadingtoimprovedaccuracy.Thesestepsmaybedescribedasfollows1.Akinematicmodelofthemanipulatorandthecalibrationprocessitselfisdevelopedandisusuallyaccomplishedwithstandardkinematicmodellingtools2.Theresultingmodelisusedtodefineanerrorquantitybasedonanominalmanufacturerskinematicparameterset,andanunknown,actualparametersetwhichistobeidentified.2.Experimentalmeasurementsoftherobotposepartialorcompletearetakeninordertoobtaindatarelatingtotheactualparametersetfortherobot.3.Theactualkinematicparametersareidentifiedbysystematicallychangingthenominalparametersetsoastoreducetheerrorquantitydefinedinthemodellingphase.OneapproachtoachievingthisidentificationisdeterminingtheanalyticaldifferentialrelationshipbetweentheposevariablesPandthekinematicparametersKintheformofaJacobian,andtheninvertingtheequationtocalculatethedeviationofthekinematicparametersfromtheirnominalvaluesAlternatively,theproblemcanbeviewedasamultidimensionaloptimisationtask,inwhichthekinematicparametersetischangedinordertoreducesomedefinederrorfunctiontozero.Thisisastandardoptimisationproblemandmaybesolvedusingwellknown3methods.4.Thefinalstepinvolvestheincorporationoftheidentifiedkinematicparametersinthecontrolleroftherobotarm,thedetailsofwhichareratherspecifictothehardwareofthesystemunderstudy.Thispaperaddressestheissueofgatheringtheexperimentaldatausedinthecalibrationprocess.Severalmethodsareavailabletoperformthistask,althoughtheyvaryincomplexity,costandthetimetakentoacquirethedata.Examplesofsuchtechniquesincludetheuseofvisualandautomatictheodolites4,5,6,servocontrolledlaserinterferometers7,acousticsensors8andvidualsensors9.Anidealmeasuringsystemwouldacquirethefullposeofthemanipulatorpositionandorientation,becausethiswouldincorporatethemaximuminformationforeachpositionofthearm.Allofthemethodsmentionedaboveuseonlythepartialpose,requiringmoredatatobetakenforthecalibrationprocesstoproceed.2.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,thismethodplacesacoordinateframeoneachobjectormanipulatorlinkofinterest,andthekinematicsaredefinedbythehomogeneoustransformationrequiredtochangeonecoordinateframeintothenext.ThistransformationtakesthefamiliarformTheaboveequationmaybeinterpretedasameanstotransformframen1intoframenbymeansoffouroutofthefiveoperationsindicated.Itisknownthatonlyfourtransformationsareneededtolocateacoordinateframewithrespecttothepreviousone.Whenconsecutiveaxesarenotparallel,thevalueof/3.isdefinedtobezero,whileforthecasewhenconsecutiveaxesareparallel,d.isthevariablechosentobezero.WhencoordinateframesareplacedinconformancewiththemodifiedDenavitHartenbergmethod,thetransformationsgivenintheaboveequationwillapplytoalltransformsofoneframeintothenext,andthesemaybewritteninagenericmatrixform,wheretheelementsofthematrixarefunctionsofthekinematicparameters.Theseparametersaresimplythevariablesofthetransformationsthejointangle0.,thecommonnormaloffsetd.,thelinklengtha.,theangleoftwista.,andtheangle/3..Thematrixformisusuallyexpressedasfollows
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