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