1、IntJAdvManufTechnol(1993)8/34-4191993Springer-VerlagLondonLimitedTheInternationalJournalofRdvancedmanufacturingTechnoloouFull-PoseCalibrationofaRobotManipulatorUsingaCoordinate-MeasuringMachineMorrisR/Driels/LtW/SwayzeUSNandLtS/PotterUSNDepartmentofMechanicalEngineering/NavalPostgraduateSchool/Monte
2、rey/California/USATheworkreportedinthisarticleaddressesthekinematiccalibrationofarobotmanipulatorusingacoordinatemeasuringmachine(CMM)whichisabletoobtainthefullposeoftheend-effector/Akinematicmodelisdevelopedforthemanipulator/itsrelationshiptotheworldcoordinateframeandthetool/Thederivationofthetoolp
3、osefromexperimentalmeasurementsisdiscussed/asistheidentificationmethodology/Acompletesimulationoftheexperimentisperformed/allowingtheobservationstrategytobedefined/Theexperimentalworkisdescribedtogetherwiththeparameteridentificationandaccuracyverification/Theprincipalconclusionisthatthemethodisablet
4、ocalibratetherobotsuccessfully/witharesultingaccuracyapproachingthatofitsrepeatability/Keywords/Robotcalibration；Coordinatemeasurement；Par-ameteridentification；Simulationstudy；Accuracyenhance-ment1/IntroductionItiswellknownthatrobotmanipulatorstypicallyhavereasonablerepeatability(0/3ram)/yetexhibitp
5、ooraccuracy(10/0mm)/Theprocessbywhichrobotsmaybecalibratedinordertoachieveaccuraciesapproachingthatofthemanipulatorisalsowellunderstood1/Inthecalibrationprocess/severalsequentialstepsenabletheprecisekinematicparametersofthemanipulatortobeidentified/leadingtoimprovedaccuracy/Thesestepsmaybedescribeda
6、sfollows/1/Akinematicmodelofthemanipulatorandthecalibrationprocessitselfisdevelopedandisusuallyaccomplishedwithstandardkinematicmodellingtools2/Theresultingmodelisusedtodefineanerrorquantitybasedonanominal(manufacturers)kinematicparameterset/andanunknown/actualparametersetwhichistobeidentified/Accep
7、tedforpublication/21October1991Correspondenceandoffprintrequeststo/Prof/MorrisR/Driels/DepartmentofMechanicalEngineering/NavalPostgraduateSchool/Monterey/California93943/USA/2/3/Experimentalmeasurementsoftherobotpose(partialorcomplete)aretakeninordertoobtaindatarelatingtotheactualparametersetforther
8、obot/Theactualkinematicparametersareidentifiedbysystemati-callychangingthenominalparametersetsoastoreducetheerrorquantitydefinedinthemodellingphase/OneapproachtoachievingthisidentificationisdeterminingtheanalyticaldifferentialrelationshipbetweentheposevariablesPandthekinematicparametersKintheformofa
9、Jacobian/P=JK(1)andtheninvertingtheequationtocalculatethedeviationofthekinematicparametersfromtheirnominalvalues8K=jrj-/jrp(2)Alternatively/theproblemcanbeviewedasamultidimen-sionaloptimisationtask/inwhichthekinematicparametersetischangedinordertoreducesomedefinederrorfunctiontozero/Thisisastandardo
10、ptimisationproblemandmaybesolvedusingwell-known3methods/4/Thefinalstepinvolvestheincorporationoftheidentifiedkinematicparametersinthecontrolleroftherobotarm/thedetailsofwhichareratherspecifictothehardwareofthesystemunderstudy/Thispaperaddressestheissueofgatheringtheexperimentaldatausedinthecalibrati
11、onprocess/Severalmethodsareavailabletoperformthistask/althoughtheyvaryincomplexity/costandthetimetakentoacquirethedata/Examplesofsuchtechniquesincludetheuseofvisualandautomatictheodolites4/5/6/servocontrolledlaserinterferometers7/acousticsensors8andvidualsensors9/Anidealmeasuringsystemwouldacquireth
12、efullposeofthemanipulator(positionandorientation)/becausethiswouldincorporatethemaximuminformationforeachpositionofthearm/Allofthemethodsmentionedaboveuseonlythepartialpose/requiringmoredatatobetakenforthecalibrationprocesstoproceed/Full-PoseCalibrationofaRobotManipulator352/TheoryInthemethoddescrib
13、edinthispaper/foreachpositioninwhichthemanipulatorisplaced/thefullposeismeasured/althoughseveralintermediatemeasurementshavetobetakeninordertoarriveatthepose/Thedeviceusedfortheposemeasurementisacoordinate-measuringmachine(CMM)/whichisathree-axis/prismaticmeasuringsystemwithaquotedaccuracyof0/01ram/
14、Therobotmanipulatortobecalibrated/aPUMA560/isplacedclosetotheCMM/andaspecialend-effectorisattachedtotheflange/Fig/1showsthearrangementofthevariouspartsofthesystem/Inthissectionthekinematicmodelwillbedeveloped/theposeestimationalgorithmsexplained/andtheparameteridentifi-cationmethodologyoutlined/2/1K
15、inematicParametersInthissection/thebasickinematicstructureofthemanipulatorwillbespecified/itsrelationtoauser-definedworldcoordinatesystemdiscussed/andtheend-pointtoilmodelled/Fromthesemodels/thekinematicparameterswhichmaybeidentifiedusingtheproposedtechniquewillbespecified/andamethodfordeterminingth
16、oseparametersdescribed/ThefundamentalmodellingtoolusedtodescribethespatialrelationshipbetweenthevariousobjectsandlocationsinthemanipulatorworkspaceistheDenavit-Hartenbergmethod2/withmodificationsproposedbyHayati10/Mooring11andWu12toaccountfordisproportionalmodels13whentwoconsecutivejointaxesarenomin
17、allyparallel/AsshowninFig/2/thismethodplacesacoordinateframeonJointn/Joint/n+1L/Linkn+1/L-/ZnnX/-Y/Yn-1X/-IFig/2/Linkcoordinateframeallocation/eachobjectormanipulatorlinkofinterest/andthekinematicsaredefinedbythehomogeneoustransformationrequiredtochangeonecoordinateframeintothenext/Thistransformatio
18、ntakesthefamiliarformA/=rot(z/O/)trans(z/d/)trans(x/a/)rot(x/t/)rot(y/fl/)(3)Theaboveequationmaybeinterpretedasameanstotransformframen-1intoframenbymeansoffouroutofthefiveoperationsindicated/Itisknownthatonlyfourtransformationsareneededtolocateacoordinateframewithrespecttothepreviousone/Whenconsecut
19、iveaxesarenotparallel/thevalueof/3/isdefinedtobezero/whileforthecasewhenconsecutiveaxesareparallel/d/isthevariablechosentobezero/WhencoordinateframesareplacedinconformancewiththemodifiedDenavit-Hartenbergmethod/thetransformationsgivenintheaboveequationwillapplytoalltransformsofoneframeintothenext/an
20、dthesemaybewritteninagenericmatrixform/wheretheelementsofthematrixarefunctionsofthekinematicparameters/Theseparametersaresimplythevariablesofthetransformations/thejointangle0//thecommonnormaloffsetd//thelinklengtha//theangleoftwista//andtheangle/3/Thematrixformisusuallyexpressedasfollows/An=Ce/CO/-S
21、O/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/(4)-Ca/SO/Sot/Ca/CO/d/0001Fig/1/Calibrationequipment/Foraseriallinkage/suchasarobotmanipulator/acoordinateframeisattachedtoeachconsecutivelinksothatboththeinstantaneouspositiontogetherwiththeinvariantgeometryaredescribedb
22、ythepreviousmatrixtransformation/The36M/R/DrielsetaL9%82)/4/Z41-//X4Fig/3/PUMAframeallocation/transformationfromthebaselinktothenthlinkwillthereforebegivenbyTn=A1A2/An(5)Fig/3showsthePUMAmanipulatorwiththeDenavit-Hartenbergframesattachedtoeachlink/togetherwithworldcoordinateframeandatoolframe/Thetra
23、nsform-ationfromtheworldframetothebaseframeofthemanipulatorneedstobeconsideredcarefully/sincetherearepotentialparameterdependenciesifcertaintypesoftransformsarechosen/ConsiderFig/4/whichshowstheworldframexw/y/z/theframeXo/Yo/z0whichisdefinedbyaDHtransformfromtheworldframetothefirstjointaxisofthemani
24、pulator/frameXb/Yb/Zb/whichisthePUMAJe4ZlxbYb/1/do9/Yw/xwZwFig/4/Basetransformations/manufacturersdefinedbaseframe/andframexl/Yl/zlwhichisthesecondDHframeofthemanipulator/Weareinterestedindeterminingtheminimumnumberofparametersrequiredtomovefromtheworldframetotheframex/Yl/z/Therearetwotransformation
25、pathsthatwillaccomplishthisgoal/Path1/ADHtransformfromx/y/z/tox0/Yo/zoinvolvingfourparameters/followedbyanothertransformfromxo/Yo/z0toXb/Yb/ZbwhichwillinvolveonlytwoparametersbanddinthetransformTob=rot(z0/4)trans(zo/d)(6)Finally/anotherDHtransformfromxb/Yb/ZbtoXt/y/Zwhichinvolvesfourparametersexcept
26、thatA01and4arebothabouttheaxiszoandcannotthereforebeidentifiedindependently/andAdlanddarebothalongtheaxiszoandalsocannotbeidentifiedindependently/Itrequires/therefore/onlyeightindependentkinematicparameterstogofromtheworldframetothefirstframeofthePUMAusingthispath/Path2/Asanalternative/atransformmay
27、bedefineddirectlyfromtheworldframetothebaseframeXb/Yb/Zb/Sincethisisaframe-to-frametransformitrequiressixparameters/suchastheEulerform/Ab=rot(z/Cb)rot(y/0b)rot(x/bb)trans(Pxb/Pyb/Pzb)(7)ThefollowingDHtransformfromxb/Yb/zbtOXl/Yl/zlwouldinvolvefourparameters/butA0mayberesolvedinto4/0b/andAdresolvedin
28、toPxb/Pyb/Pzb/reducingtheparametercounttotwo/Itisseenthatthispathalsorequireseightparametersasinpathi/butadifferentset/EitheroftheabovemethodsmaybeusedtomovefromtheworldframetothesecondframeofthePUMA/Inthiswork/thesecondpathischosen/ThetooltransformisanEulertransformwhichrequiresthespecificationofsi
29、xparameters/As=rot(z/b6)rot(y/04)rot(x//6)(8)trans(Px6/Py6/P/6)Thetotalnumberofparametersusedinthekinematicmodelbecomes30/andtheirnominalvaluesaredefinedinTable1/2/2IdentificationMethodologyThekinematicparameteridentificationwillbeperformedasamultidimensionalminimisationprocess/sincethisavoidsthecal
30、culationofthesystemJacobian/Theprocessisasfollows/1/Beginwithaguesssetofkinematicparameters/suchasthenominalset/2/SelectanarbitrarysetofjointanglesforthePUMA/3/CalculatetheposeofthePUMAend-effector/4/MeasuretheactualposeofthePUMAend-effectorforthesamesetofjointangles/Ingeneral/themeasuredandpredicte
31、dposewillbedifferent/5/Modifythekinematicparametersinanorderlymannerinordertobestfit(inaleast-squaressense)themeasuredposetothepredictedpose/Table1/NominalparametersforthePUMArobot/0bP*bP/bPzb(o)(o)(o)(mm)(mm)(mm)180/00/090/0-394/0-383/0474/0Link(50/d/a//3/(o)(mm)(mm)(1000/0-90/0020/00431/850/00/030
32、/0149/09-20/3390/0040/0433/00/0-90/0050/00/00/090/00(mm)(mm)(ram)90/00/00/00/00/0134/0Theprocessisappliednottoasinglesetofjointanglesbuttoanumberofjointangles/Thetotalnumberofjointanglesetsrequired/whichalsoequalsthenumberofphysicalmeasurementmade/mustsatisfyKp/NDt(9)whereKpisthenumberofkinematicpar
33、ameterstobeidentifiedNisthenumberofmeasurements(poses)takenDrrepresentsthenumberofdegreesoffreedompresentineachmeasurementInthesystemdescribedinthispaper/thenumberofdegreesoffreedomisgivenbyDt=6(10)sincefullposeismeasured/Inpractice/manymoremeasure-mentsshouldbetakentooffsettheeffectofnoiseintheexpe
34、rimentalmeasurements/TheoptimisationprocedureusedisknownasZXSSO/andisastandardlibraryfunctionintheIMSLpackage14/2/3PoseMeasurementItisapparentfromtheabovethatameanstodeterminethefullposeofthePUMAisrequiredinordertoperformthecalibration/Thismethodwillnowbedescribedindetail/Theend-effectorconsistsofan
35、arrangementoffiveprecision-toolingballsasshowninFig/5/Considerthecoordinatesofthecentreofeachballexpressedintermsofthetoolframe(Fig/5)andtheworldcoordinateframe/asshowninFig/6/TherelationshipbetweenthesecoordinatesmaybewrittenasP；=TPI(11)wherePiisthe4x1columnvectorofthecoordinatesofFull-PoseCalibrat
36、ionofaRobotManipulator37X6Iz6Fig/5/Tooltransform/z/p/XwF6/X6Fig/6/Ballcentroidinthetoolandworldframes/theithballexpressedwithrespecttotheworldframe/Pisthe4x1columnvectorofthecoordinatesoftheithballexpressedwithrespecttothetoolframe/andTisthe44homogenioustransformfromtheworldframetothetoolframe/IfPis
37、knownbyprecalibratingthetool/andPismeasured/thenTmaybefound/andusedasthemeasuredposeinthecalibrationprocess/Itisnotquitethatsimple/however/sinceitisnotpossibletoinvertequation(11)toobtainT/Theaboveprocessisperformedforthefourballs/A/B/CandD/andthepositionsorderedase/P3P//Pb=TPAesPcPo(12)orintheformP
38、=TP(13)SinceP/TandPareallnowsquare/theposematrixmaybeobtainedbyinversionT=Pp-1(14)InpracticeitmaybedifficultfortheCMMtoaccessfourbailstodeterminePwhenthePUMAisplacedincertainconfigurations/ThreeballsareactuallymeasuredandafourthballisfictitiouslylocatedaccordingtothevectorcrossproductP4=(P3-PI)(P2-PI)(15)Regardingthedeterminationofthecoordinatesofthe
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