外文翻译--校准并联机床的不确定参数 英文版.pdf

UncertaintypropagationincalibrationofparallelkinematicmachinesBernhardJokielJr.a,JohnC.Ziegertb,LotharBiegcaSandiaNationalLaboratories,Albuquerque,NM87185-0958,USAbUniversityofFloridaMachineToolResearchCenter,237MEB,Gainesville,FL32611,USAcSandiaNationalLaboratories,1Albuquerque,NM87185-0958,USAAbstractThispaperoutlinesindetailamethodfordeterminingtheuncertaintypresentinthekinematicparameters(jointlocations,initialstrutlengths,andspindlelocationandorientation)forparallelkinematicdevicesaftercalibration.TheuncertaintyestimationmethodusingMonteCarlosimulationswasappliedtoasequentialmethodfordeterminingthekinematicparametersoffullyassembledHexelTornado2000(a63Stewartplatform)millingmachine.ResultsfortheuncertaintypresentinthekinematicparametersofaHexelTornado2000millingKeywords:Stewartplatform；Calibration；Kinematicparameters；Uncertainty.1.IntroductionOverthelastdecade,multi-axismachinetoolsbasedonparallelkinematicmechanisms(PKMs)havebeendevel-opedandmarketedworldwideasalternativestotraditionalserialstacked-slide,orthogonalmachinearchitectures.ThegeneraldifferencebetweenPKMandorthogonalserialmechanismsisthearrangementoftheactuators.Inorthog-onalserialmechanisms,individualactuatorsresponsibleformotioninindividualCartesiandegreesoffreedom(DOF)arejoinedendtoendinaserialchainconnectingthestationarygroundframetothemovingframe.InPKMarchitectures,individualactuatorsarenottypicallyarrangedtoindependentlycontrolasingleCartesianDOF.Insteadtheactuatorsarearrangedsothateachactuatorisconnectedbetweenthestationarygroundframeandthemovingframe,sothatacombinationofactuatormotionscontrolsthemo-tionofthemovingframeinCartesianspace.ApopulartypeofPKMistheStewartplatform(Figure1).TheStewartplatformconnectsamoveableplatformtoastationarygroundframebysixextensiblelinksorstruts,allowingforcontrolledmotionoftheplatforminallsixDOF.GenerallyforPKMmachines,theCartesianpositionandorientationofthetoolpointcarriedontheplatformisobtainedfromakinematicmodeloftheparticularmachine.Accuratepositioningofthesemachinesreliesontheaccu-rateknowledgeoftheparametersofthekinematicmodeluniquetotheparticularmachine.Theparametersinthekinematicmodelincludethespatiallocationsofthejointcentersonthemachinebaseandmovingplatform,theinitialstrutlengths,andthestrutdisplacements.Thestrutdisplace-mentsarereadilyobtainedfromsensorsonthemachine.However,theremainingkinematicparameters(jointcenterlocations,andinitialstrutlengths)aredifficulttodeterminewhenthesemachinesareintheirfullyassembledstate.Thesizeandcomplexityofthesemachinegenerallymakesitdifficultandsomewhatundesirabletodeterminetheremain-ingkinematicparametersbydirectinspectionsuchasinacoordinatemeasuringmachine.InorderforPKMstobeusefulforprecisionpositioningapplications,techniquesmustbedevelopedtoquicklycalibratethemachinebydeterminingthekinematicparameterswithoutdisassemblyofthemachine.Anumberofauthorshavereportedtech-niquesforcalibrationofPKMs.Soons1,2,Masory,3,Zhuanget.al.4,5,Ropponen6Inanotherpaper7,theauthorshavereportedonworkrecentlycompletedbytheUniversityofFloridaandSandiaNationalLaboratoriesoncalibrationofPKMs,whichdescribesanewtechniquetosequentiallydeterminethekinematicparametersofanas-sembledparallelkinematicdevice.CalibrationofPKMs,oranyothertypeofmachine,beginsbycollectingsetsofmeasurements.Thecollectedmeasurementdataissuppliedtoanalgorithm,whichcom-putesthemodelparameters.Aswithanymetrologytask,1SandiaisamultiprogramlaboratoryoperatedbySandiaCorporation,aLockheedMartinCompany,fortheUnitedStatesDepartmentofEnergyunderContractDE-AC0494AL85000.PrecisionEngineeringJournaloftheInternationalSocietiesforPrecisionEngineeringandNanotechnology25(2001)4855machineaftercalibrationusingaSMX4,000lasertrackerareshown.thereisuncertaintypresentinthecollectedmeasurementdata.Theuncertaintypresentinthemeasurementdatawillpropagatethroughtheparameteridentificationalgorithmandresultinerrorsinthecomputedkinematicparameters.Theerrorsintherecoveredkinematicparameterswillthencreatepositioningerrorswhenthekinematicmodelisusedformachinecontrol.ThepurposeofthispaperistoexploretheissueofpropagationofuncertaintyinthecalibrationofPKMdevices.Wewilldescribeamethodologyforevalu-atingtheuncertaintyintherecoveredparameters,andtheresultingerrorboundsinmachinepositioning.Wewilldemonstratethismethodologyforthenewcalibrationmethodwehavedevelopedandreportedinanotherpaper7.Thispaperisorganizedinthefollowingmanner.Section2describesthemethodologyforevaluatinguncertaintypropagationinPKMcalibration.Section3containsabriefsynopsisofthecalibrationmethodologyusedinthiswork.InSection4,theerrorbudgetforthecalibrationmeasure-mentsisdeveloped.Section5givestheresultsoftheun-certaintypropagationsimulation.Section6comparesthesimulationresultstoexperimentalresultsperformedonthemachine.2.MethodologyforevaluatinguncertaintypropagationInanymachinecalibrationtask,uncertaintyisintroducedintwoways.First,thecalibrationmethodinvolvesperform-ingmeasurementswithaspecifiedsetofinstruments.Thesemeasurementinstrumentswillhaveanassociateduncer-taintyintheiroutput,whichisafunctionofthephysicalprincipalsandconstructionoftheinstrumentsthemselves,aswellastheenvironmentinwhichthemeasurementstakeplace.Inconjunctionwiththeuncertaintyassociatedwiththeexternalcalibrationinstruments,theremaybeadditionaluncertaintyintroducedbythemachineitself.ThefeedbackdevicesonthestrutsofPKMmachinestypicallycannotdirectlymeasuretheabsolutedistancebetweencorrespond-ingpairsofjointcenters,onlythechangeinlengthofthestrutfromsome“home”positionatwhichtheabsolutejointcenterdistanceisassumedtobeknown.Changesinthethermalstateofthemachinemaycausethisassumedvaluetobeincorrectand/ortofluctuateovertime.Thestrutdisplacementfeedbackdevicesalsohaveafiniteresolutionandsomeuncertaintyassociatedwiththeiroutput.Geomet-ricaleffectsnotincludedinthekinematicmodel,suchasnon-sphericaljointmotion,alsocreateaneffectivestrutlengtherrorwhichisnotsensedbythestrutfeedbackdevices.Therefore,themachinesrepeatabilityandabsolutepositioningaccuracyduringthecalibrationmeasurementscontributestotheoveralluncertaintyofthebasicdatausedbythecalibrationalgorithm.Theuncertaintiespresentintheexternalmeasurementinstrumentcombinewiththeuncertaintiesinthemachinemotionstocreateerrorsinthedatausedbythecalibrationalgorithmtocomputetheparametersofthemachineski-nematicmodel.Sinceerrorispresentinthecollecteddata,therewillbeerrorpresentinthecomputedmodelparame-tersaswell.Sincethesecomputedmodelparametersdonotidenticallymatchtheactualphysicalparametersoftherealmachine,positioningerrorswillresultwhentherecoveredparametersareusedduringpositionalcontrolofthema-chine.Thesepositioningerrorsduetoincorrectparameter-izationofthemodelwillbecompoundedwiththeuncer-taintiesinherenttothemachineitself,resultinginthefinalvolumetricaccuracyofthemachine.Thisisanimportantpointthatshouldnotbeoverlooked.Theuncertaintiesarisingfromthemachineitself(i.e.,re-peatability,sensorresolution,unmodeledthermalandgeo-metriceffects)affecttheoverallpositioningperformanceofthemachinetwice.First,theyadduncertaintytothebasicmeasurementdatathatisusedbythecalibrationalgorithm.Second,afterthecalibrationiscomplete,andthenewlyparameterizedkinematicmodelisusedformachinecontrol,thesesamemachineuncertaintiesstillexistandcontributetopositioningerrors.WeproposethefollowingmethodologyforevaluationofuncertaintypropagationthroughthePKMcalibrationpro-cess:1.Constructanerrorbudgetforthemeasurementde-vicesusedduringcalibrationtopredicttheiruncer-taintycontributions.2.Constructanerrorbudgetforthemachineitselftoestimatethemagnitudeofstrutlengtherrorwhichcanoccurduetothermaleffects,sensorresolutionanduncertainty,non-sphericityofthejointmotions,andothereffectsinherenttotheparticularmachinewhichmaycausenon-repeatabilityofpositioning.3.Combinethesetwouncertaintysourcestoestimatetheoveralluncertaintypresentinthecollectedmea-surementdatausedbythecalibrationalgorithmtoobtainmodelparameters.Fig.1.A12-joint(six-six)Stewartplatformdevice.49B.Jokieletal./PrecisionEngineering25(2001)48554.Constructa“perfect”datasetfromthemachinemodel,whichassumesnouncertaintiesexistinthemeasuringinstrumentsormachine.5.PerformaMonteCarlosimulationofthecalibrationprocessbyrunningthecalibrationalgorithmmultipletimes,eachtimecorruptingthe“perfect”datasetwithrandomlygeneratederrorsassumingauniformdistri-butionovertheuncertaintyboundspredictedbythecombinederrorbudgetanalyses.6.Analyzethedistributionoferrorsinthemodelpa-rametersobtainedfromtheMonteCarlosimulation.7.Usingthemachinekinematicmodelwitheachsetofmodelparametersobtainedfromthecalibrationsim-ulation,simulatethemachinemotionsduringapar-ticularmachineaccuracyperformancetest(i.e.,-cir-cularballbartests,laserinterferometricdisplacementmeasurements,etc.),beingsuretoincludetheuncer-taintyassociatedwiththemachineitselfinthesimu-lation.8.Analyzethesimulatedperformancedata.Theresultsofthissimulationwillshowtherangeofresultsoneislikelytoachieveonthefinalmachineposi-tioningperformancetestsfromaparticularcalibrationpro-cedure.ThisprocedureisoutlinedintheflowchartinFigure2.Itisalsopossibletomodifythesesimulationstoexaminetheeffectofvariouscontributorstotheoverallpositioningperformanceofthemachine.Forexample,onemightas-sumethatthemachineitselfisperfecttodeterminehowuncertaintiesintheexternalcalibrationmeasurementinstru-mentscontributetomachinepositioningerrors.Conversely,onecanassumethattheexternalmeasurementsareperfectandexaminethepropagationofvarioussourcesofmachineerror,suchasthermaleffectsornon-sphericityofjointmotions,tothefinalpositioningaccuracyperformance.3.CalibrationmethodologysequentialdeterminationtechniqueTheapproachoutlinedinSection2wasusedtoevaluateuncertaintypropagationincalibrationofPKMdevicesforanewmethodforsequentialdeterminationofkinematicpa-rametersinPKMsdevelopedbytheauthorsandreportedelsewhere10.Asaconveniencetothereader,thismethodisbrieflydescribedhere.Theparameteridentificationmethodusesaspatialcoordinatemeasuringdevicesuchasalasertrackeroralaserballbar.Thecalibrationisper-formedinfoursteps:1.Locationofacentralreferenceframe(R)andthemachineframe(M).2.Identificationofthespatiallocationsofthejointscentersofrotation.3.Determinationofthespindleorientationandnoselocation.4.Determinationoftheinitialstrutlengths.3.1.LocatingthecentralreferenceandmachinereferenceframesIngeneral,severallocationsofthespatialcoordinate-measuringdevicemayberequiredtocompleteallofthenecessarymeasurements.Therefore,asingle,stablerefer-Fig.2.Uncertaintypropagationalgorithmflowchart.Fig.3.Locationsofcoordinatereferenceframesusedduringcalibration.50B.Jokieletal./PrecisionEngineering25(2001)4855enceframe,R,isnecessarytowhichallofthecoordinatedatacollectedinvariousframesmaybetransformed(Figure3).TheRframeconsistsofthreegaugepointssecuredtotheworktable.Utilizingthespatialcoordinatemeasuringde-vice,thelocationsoftheRgaugepointsaremeasuredrelativetothemeasuringdevicescoordinatesystem,andthehomogeneouscoordinatetransformation(HTM)be-tweenthemeasurementdevicescoordinatesystemandtheRframe(MeasTR)iscomputedusingthefollowingequa-tions.TheHTMbetweentheRsystemandmachinesreferenceframe(M)mayalsobedeterminedatthistime.Usingthespatialmeasurementdevice,theplaneoftheworktable,thedesiredlocationoftheX-axisandthemachineoriginaremeasured.Usingtheunitnormalvectorofthebest-fitplane,theunitvectoroftheX-axisdirectionvectorprojectedintothebest-fitplane,andthecoordinatesofthedesiredmachineoriginprojectedintothebest-fitplanetheHTMoftheMframerelativetothemeasure-mentdeviceframe,(MeasTM),isconstructed.TheHTMrelatingtheRandMframes(MTR)isthencalculatedbymatrixmultiplication.3.2.JointcenterlocationidentificationStewartplatformmachinesrequiretheuseofsphericalorHookejointstoconnectthestrutstothemachinebaseandtothemoveableplatform.Thesejointsallowthestrutendstorotateaboutfixedpointsonthemachinebaseandplatformasthemachinegeometrychanges.Ifastrutisheldatafixedlengthandrotatedaboutoneofitsjoints,allofthepointsonthelinkmovewithsphericalmotionaboutacommoncenterofrotation.Thisfactcanbeleveragedtodeterminethelocationofthejointscenterofrotation.Twogaugepointsareaffixedtoeithersideofonestrutsothatthegauge-pointcentersandthestrutcenterlinelieinthesameplane.Threegaugepointsareaffixedtotheplatform,whichdefineaplatformreferencecoordinatesystemPR.Theplatformisthencommandedtomovealonganarbi-trary,predeterminedpathdesignedtoholdthestrutinques-tionatanarbitraryfixedlength.Astheplatformmoves,thisfixed-lengthstrutrotatesaboutitsjointcenters,andpointsonthestruttraceasphericalpathinspace(Figure4).Atseverallocationsalongthepath,themachinemotionispaused,andthespatialcoordinatesofthetwolinkandthreeplatformgaugepointsaremeasured.This“move-pause-measure”sequenceisrepeateduntilthedesirednumberofplatformposeshasbeenreached.Thegaugepointcoordinatesmaybeexpressedrelativetotwodifferentcoordinatesystems,theMsystem,orthePRsystem.GaugepointcoordinatesexpressedrelativetotheMsystemlieonthesurfaceofaspherewhosecenteristhecenterofrotationofthebasejointoftheexaminedstrut.ThesamegaugepointcoordinatesexpressedrelativetothePRsystemlieonthesurfaceofaspherewhosecenteristhecenterofrotationoftheplatformjointfortheexaminedstrut.Thelocationsofthebaseandplatformjointcentersmaybedeterminedbyfittingtheappropriatelytransformedstrutgaugepointcoordinatestotheequationofasphere.Assumingthejointsproducesphericalmotion,thecalcu-latedcoordinatesofthecenterofthebest-fitsphereisthecenterofrotationofthejointinquestion.Thismethodisrepeatedsequentiallyforeachofthesixstrutstorecoverall12ofthebaseandplatformjointcenterlocations.MeasTR5MeasxRMeasyRMeaszRMeasr1000#MeasxR5Measr22Measr1uMeasr22Measr1uMeaszR5MeasxR3Measr32Measr1uMeasr32Measr1uMeasyR5MeaszR3MeasxR(1)T5MeasxMMeasyMMeasnplaneMeaspO0001#MeasxM5Measux2MeasuxzMeasnplane!MeasnplaneuMeasux2MeasuxzMeasnplane!MeasnplaneuMeasyM5Measnplane3MeasxMMeaspO5Measp2MeaspzMeasnplane!MeasnplaneMTR5MeasTM!21MeasTR(2)51B.Jokieletal./PrecisionEngineering25(2001)4855