翻译英文.docx
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翻译英文.docx
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翻译英文
Full-PoseCalibrationofaRobotManipulatorUsingaCoordinate-
MeasuringMachine
MorrisR.Driels,LtW.SwayzeUSNandLtS.PotterUSN
DepartmentofMechanicalEngineering,NavalPostgraduateSchool,Monterey,California,US
Theworkreportedinthisarticleaddressesthekinematic
calibrationofarobotmanipulatorusingacoordinatemeasuring
machine(CMM)whichisabletoobtainthefullposeof
theend-effector.Akinematicmodelisdevelopedforthe
manipulator,itsrelationshiptotheworldcoordinateframeand
thetool.Thederivationofthetoolposefromexperimental
measurementsisdiscussed,asistheidentificationmethodology.
Acompletesimulationoftheexperimentisperformed,allowing
theobservationstrategytobedefined.Theexperimentalwork
isdescribedtogetherwiththeparameteridentificationand
accuracyverification.Theprincipalconclusionisthatthe
methodisabletocalibratetherobotsuccessfully,witha
resultingaccuracyapproachingthatofitsrepeatability.
Keywords:
Robotcalibration;Coordinatemeasurement;Parameter
identification;Simulationstudy;Accuracyenhancement
1.Introduction
Itiswellknownthatrobotmanipulatorstypicallyhave
reasonablerepeatability(0.3ram),yetexhibitpooraccuracy
(10.0mm).Theprocessbywhichrobotsmaybecalibrated
inordertoachieveaccuraciesapproachingthatofthe
manipulatorisalsowellunderstood[1].Inthecalibration
process,severalsequentialstepsenabletheprecisekinematic
parametersofthemanipulatortobeidentified,leadingto
improvedaccuracy.Thesestepsmaybedescribedasfollows:
1.Akinematicmodelofthemanipulatorandthecalibration
processitselfisdevelopedandisusuallyaccomplishedwith
standardkinematicmodellingtools[2].Theresultingmodel
isusedtodefineanerrorquantitybasedonanominal
(manufacturer's)kinematicparameterset,andanunknown,
actualparametersetwhichistobeidentified.
2.Experimentalmeasurementsoftherobotpose(partialor
complete)aretakeninordertoobtaindatarelatingtothe
actualparametersetfortherobot.
3.
Theactualkinematicparametersareidentifiedbysystematically
changingthenominalparametersetsoastoreduce
theerrorquantitydefinedinthemodellingphase.One
approachtoachievingthisidentificationisdetermining
theanalyticaldifferentialrelationshipbetweenthepose
variablesPandthekinematicparametersKintheform
ofaJacobian,
andtheninvertingtheequationtocalculatethedeviationof
thekinematicparametersfromtheirnominalvalues
Alternatively,theproblemcanbeviewedasamultidimensional
optimisationtask,inwhichthekinematicparameter
setischangedinordertoreducesomedefinederrorfunction
tozero.Thisisastandardoptimisationproblemandmay
besolvedusingwell-known[3]methods.
4.Thefinalstepinvolvestheincorporationoftheidentified
kinematicparametersinthecontrolleroftherobotarm,
thedetailsofwhichareratherspecifictothehardwareof
thesystemunderstudy.
Thispaperaddressestheissueofgatheringtheexperimental
datausedinthecalibrationprocess.Severalmethodsare
availabletoperformthistask,althoughtheyvaryincomplexity,
costandthetimetakentoacquirethedata.Examplesof
suchtechniquesincludetheuseofvisualandautomatic
theodolites[4,5,6],servocontrolledlaserinterferometers[7],
acousticsensors[8]andvidualsensors[9].Anidealmeasuring
systemwouldacquirethefullposeofthemanipulator(position
andorientation),becausethiswouldincorporatethemaximum
informationforeachpositionofthearm.Allofthemethods
mentionedaboveuseonlythepartialpose,requiringmore
datatobetakenforthecalibrationprocesstoproceed.
2.Theory
Inthemethoddescribedinthispaper,foreachpositionin
whichthemanipulatorisplaced,thefullposeismeasured,
althoughseveralintermediatemeasurementshavetobetaken
inordertoarriveatthepose.Thedeviceusedforthepose
measurementisacoordinate-measuringmachine(CMM),
whichisathree-axis,prismaticmeasuringsystemwitha
quotedaccuracyof0.01ram.Therobotmanipulatortobe
calibrated,aPUMA560,isplacedclosetotheCMM,anda
specialend-effectorisattachedtotheflange.Fig.1shows
thearrangementofthevariouspartsofthesystem.Inthis
sectionthekinematicmodelwillbedeveloped,thepose
estimationalgorithmsexplained,andtheparameteridentification
methodologyoutlined.
2.1KinematicParameters
Inthissection,thebasickinematicstructureofthemanipulator
willbespecified,itsrelationtoauser-definedworldcoordinate
systemdiscussed,andtheend-pointtoilmodelled.Fromthese
models,thekinematicparameterswhichmaybeidentified
usingtheproposedtechniquewillbespecified,andamethod
fordeterminingthoseparametersdescribed.
Thefundamentalmodellingtoolusedtodescribethespatial
relationshipbetweenthevariousobjectsandlocationsinthe
manipulatorworkspaceistheDenavit-Hartenbergmethod
[2],withmodificationsproposedbyHayati[10],Mooring
[11]andWu[12]toaccountfordisproportionalmodels[13]
whentwoconsecutivejointaxesarenominallyparallel.As
showninFig.2,thismethodplacesacoordinateframeon
eachobjectormanipulatorlinkofinterest,andthekinematics
aredefinedbythehomogeneoustransformationrequiredto
changeonecoordinateframeintothenext.Thistransformation
takesthefamiliarform
Theaboveequationmaybeinterpretedasameansto
transformframen-1intoframenbymeansoffouroutof
thefiveoperationsindicated.Itisknownthatonlyfour
transformationsareneededtolocateacoordinateframewith
respecttothepreviousone.Whenconsecutiveaxesarenot
parallel,thevalueof/3.isdefinedtobezero,whileforthe
casewhenconsecutiveaxesareparallel,d.isthevariable
chosentobezero.
Whencoordinateframesareplacedinconformancewith
themodifiedDenavit-Hartenbergmethod,thetransformations
givenintheaboveequationwillapplytoalltransformsof
oneframeintothenext,andthesemaybewrittenina
genericmatrixform,wheretheelementsofthematrixare
functionsofthekinematicparameters.Theseparametersare
simplythevariablesofthetransformations:
thejointangle
0.,thecommonnormaloffsetd.,thelinklengtha.,theangle
oftwista.,andtheangle/3..Thematrixformisusually
expressedasfollows:
Foraseriallinkage,suchasarobotmanipulator,acoordinate
frameisattachedtoeachconsecutivelinksothatboththe
instantaneouspositiontogetherwiththeinvariantgeometry
aredescribedbythepreviousmatrixtransformation.'The
transformationfromthebaselinktothenthlinkwilltherefore
begivenby
Fig.3showsthePUMAmanipulatorwiththe
Denavit-Hartenbergframesattachedtoeachlink,together
withworldcoordinateframeandatoolframe.Thetransformation
fromtheworldframetothebaseframeofthe
manipulatorneedstobeconsideredcarefully,sincethereare
potentialparameterdependenciesifcertaintypesoftransforms
arechosen.ConsiderFig.4,whichshowstheworldframe
xw,y,,z,,theframeXo,Yo,z0whichisdefinedbyaDH
transformfromtheworldframetothefirstjointaxisof
themanipulator,frameXb,Yb,Zb,whichisthePUMA
manufacturer'sdefinedbaseframe,andframexl,Yl,zlwhich
isthesecondDHframeofthemanipulator.Weareinterested
indeterminingtheminimumnumberofparametersrequired
tomovefromtheworldframetotheframex~,Yl,z~.There
aretwotransformationpathsthatwillaccomplishthisgoal:
Path1:
ADHtransformfromx,,y,,z,,tox0,Yo,zo
involvingfourparameters,followedbyanothertransform
fromxo,Yo,z0toXb,Yb,Zbwhichwillinvolveonlytwo
parameters~b'andd'inthetransform
Finally,anotherDHtransformfromxb,Yb,ZbtoXt,y~,Z~
whichinvolvesfourparametersexceptthatA01and4~'are
bothabouttheaxiszoandcannotthereforebeidentified
independently,andAdlandd'arebothalongtheaxiszoand
alsocannotbeidentifiedindependently.Itrequires,therefore,
onlyeightindependentkinematicparameterstogofromthe
worldframetothefirstframeofthePUMAusingthispath.
Path2:
Asanalternative,atransformmaybedefineddirectly
fromtheworldframetothebaseframeXb,Yb,Zb.Sincethis
isaframe-to-frametransformitrequiressixparameters,such
astheEulerform:
ThefollowingDHtransformfromxb,Yb,zbtOXl,Yl,zl
wouldinvolvefourparameters,butA0~mayberesolvedinto
4~,,0b,~,andAd~resolvedintoPxb,Pyb,Pzb,reducingthe
parametercounttotwo.Itisseenthatthispathalsorequires
eightparametersasinpathi,butadifferentset.
Eitheroftheabovemethodsmaybeusedtomovefrom
theworldframetothesecondframeofthePUMA.Inthis
work,thesecondpathischosen.Thetooltransformisan
Eulertransformwhichrequiresthespecificationofsix
parameters:
Thetotalnumberofparametersusedinthekinematicmodel
becomes30,andtheirnominalvaluesaredefinedinTable
1
2.2IdentificationMethodology
Thekinematicparameteridentificationwillbeperformedas
amultidimensionalminimisationprocess,sincethisavoidsthe
calculationofthesystemJacobian.Theprocessisasfollows:
1.Beginwithaguesssetofkinematicparameters,suchas
thenominalset.
2.SelectanarbitrarysetofjointanglesforthePUMA.
3.CalculatetheposeofthePUMAend-effector.
4.MeasuretheactualposeofthePUMAend-effectorfor
thesamesetofjointangles.Ingeneral,themeasuredand
predictedposewillbedifferent.
5.Modifythekinematicparameters
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