Full-PoseCalibrationofaRobotManipulatorUsingaCoordinate-MeasuringMachineMorrisR.Driels,LtW.SwayzeUSNandLtS.PotterUSNDepartmentofMechanicalEngineering,NavalPostgraduateSchool,Monterey,California,USTheworkreportedinthisarticleaddressesthekinematiccalibrationofarobotmanipulatorusingacoordinatemeasuringmachine(CMM)whichisabletoobtainthefullposeoftheend-effector.Akinematicmodelisdevelopedforthemanipulator,itsrelationshiptotheworldcoordinateframeandthetool.Thederivationofthetoolposefromexperimentalmeasurementsisdiscussed,asistheidentificationmethodology.Acompletesimulationoftheexperimentisperformed,allowingtheobservationstrategytobedefined.Theexperimentalworkisdescribedtogetherwiththeparameteridentificationandaccuracyverification.Theprincipalconclusionisthatthemethodisabletocalibratetherobotsuccessfully,witharesultingaccuracyapproachingthatofitsrepeatability.Keywords:Robotcalibration；Coordinatemeasurement；Parameteridentification；Simulationstudy；Accuracyenhancement1.IntroductionItiswellknownthatrobotmanipulatorstypicallyhavereasonablerepeatability(0.3ram),yetexhibitpooraccuracy(10.0mm).Theprocessbywhichrobotsmaybecalibratedinordertoachieveaccuraciesapproachingthatofthemanipulatorisalsowellunderstood1.Inthecalibrationprocess,severalsequentialstepsenabletheprecisekinematicparametersofthemanipulatortobeidentified,leadingtoimprovedaccuracy.Thesestepsmaybedescribedasfollows:1.Akinematicmodelofthemanipulatorandthecalibrationprocessitselfisdevelopedandisusuallyaccomplishedwithstandardkinematicmodellingtools2.Theresultingmodelisusedtodefineanerrorquantitybasedonanominal(manufacturers)kinematicparameterset,andanunknown,actualparametersetwhichistobeidentified.2.Experimentalmeasurementsoftherobotpose(partialorcomplete)aretakeninordertoobtaindatarelatingtotheactualparametersetfortherobot.3.Theactualkinematicparametersareidentifiedbysystematicallychangingthenominalparametersetsoastoreducetheerrorquantitydefinedinthemodellingphase.OneapproachtoachievingthisidentificationisdeterminingtheanalyticaldifferentialrelationshipbetweentheposevariablesPandthekinematicparametersKintheformofaJacobian,andtheninvertingtheequationtocalculatethedeviationofthekinematicparametersfromtheirnominalvaluesAlternatively,theproblemcanbeviewedasamultidimensionaloptimisationtask,inwhichthekinematicparametersetischangedinordertoreducesomedefinederrorfunctiontozero.Thisisastandardoptimisationproblemandmaybesolvedusingwell-known3methods.4.Thefinalstepinvolvestheincorporationoftheidentifiedkinematicparametersinthecontrolleroftherobotarm,thedetailsofwhichareratherspecifictothehardwareofthesystemunderstudy.Thispaperaddressestheissueofgatheringtheexperimentaldatausedinthecalibrationprocess.Severalmethodsareavailabletoperformthistask,althoughtheyvaryincomplexity,costandthetimetakentoacquirethedata.Examplesofsuchtechniquesincludetheuseofvisualandautomatictheodolites4,5,6,servocontrolledlaserinterferometers7,acousticsensors8andvidualsensors9.Anidealmeasuringsystemwouldacquirethefullposeofthemanipulator(positionandorientation),becausethiswouldincorporatethemaximuminformationforeachpositionofthearm.Allofthemethodsmentionedaboveuseonlythepartialpose,requiringmoredatatobetakenforthecalibrationprocesstoproceed.2.TheoryInthemethoddescribedinthispaper,foreachpositioninwhichthemanipulatorisplaced,thefullposeismeasured,althoughseveralintermediatemeasurementshavetobetakeninordertoarriveatthepose.Thedeviceusedfortheposemeasurementisacoordinate-measuringmachine(CMM),whichisathree-axis,prismaticmeasuringsystemwithaquotedaccuracyof0.01ram.Therobotmanipulatortobecalibrated,aPUMA560,isplacedclosetotheCMM,andaspecialend-effectorisattachedtotheflange.Fig.1showsthearrangementofthevariouspartsofthesystem.Inthissectionthekinematicmodelwillbedeveloped,theposeestimationalgorithmsexplained,andtheparameteridentificationmethodologyoutlined.2.1KinematicParametersInthissection,thebasickinematicstructureofthemanipulatorwillbespecified,itsrelationtoauser-definedworldcoordinatesystemdiscussed,andtheend-pointtoilmodelled.Fromthesemodels,thekinematicparameterswhichmaybeidentifiedusingtheproposedtechniquewillbespecified,andamethodfordeterminingthoseparametersdescribed.ThefundamentalmodellingtoolusedtodescribethespatialrelationshipbetweenthevariousobjectsandlocationsinthemanipulatorworkspaceistheDenavit-Hartenbergmethod2,withmodificationsproposedbyHayati10,Mooring11andWu12toaccountfordisproportionalmodels13whentwoconsecutivejointaxesarenominallyparallel.AsshowninFig.2,thismethodplacesacoordinateframeoneachobjectormanipulatorlinkofinterest,andthekinematicsaredefinedbythehomogeneoustransformationrequiredtochangeonecoordinateframeintothenext.ThistransformationtakesthefamiliarformTheaboveequationmaybeinterpretedasameanstotransformframen-1intoframenbymeansoffouroutofthefiveoperationsindicated.Itisknownthatonlyfourtransformationsareneededtolocateacoordinateframewithrespecttothepreviousone.Whenconsecutiveaxesarenotparallel,thevalueof/3.isdefinedtobezero,whileforthecasewhenconsecutiveaxesareparallel,d.isthevariablechosentobezero.WhencoordinateframesareplacedinconformancewiththemodifiedDenavit-Hartenbergmethod,thetransformationsgivenintheaboveequationwillapplytoalltransformsofoneframeintothenext,andthesemaybewrit