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UncertaintypropagationincalibrationofparallelkinematicmachinesBernhardJokielJr.a,JohnC.Ziegertb,LotharBiegcaSandiaNationalLaboratories,Albuquerque,NM871850958,USAbUniversityofFloridaMachineToolResearchCenter,237MEB,Gainesville,FL32611,USAcSandiaNationalLaboratories,1Albuquerque,NM871850958,USAAbstractThispaperoutlinesindetailamethodfordeterminingtheuncertaintypresentinthekinematicparametersjointlocations,initialstrutlengths,andspindlelocationandorientationforparallelkinematicdevicesaftercalibration.TheuncertaintyestimationmethodusingMonteCarlosimulationswasappliedtoasequentialmethodfordeterminingthekinematicparametersoffullyassembledHexelTornado2000a6–3Stewartplatformmillingmachine.ResultsfortheuncertaintypresentinthekinematicparametersofaHexelTornado2000millingKeywordsStewartplatformCalibrationKinematicparametersUncertainty.1.IntroductionOverthelastdecade,multiaxismachinetoolsbasedonparallelkinematicmechanismsPKMshavebeendevelopedandmarketedworldwideasalternativestotraditionalserialstackedslide,orthogonalmachinearchitectures.ThegeneraldifferencebetweenPKMandorthogonalserialmechanismsisthearrangementoftheactuators.Inorthogonalserialmechanisms,individualactuatorsresponsibleformotioninindividualCartesiandegreesoffreedomDOFarejoinedendtoendinaserialchainconnectingthestationarygroundframetothemovingframe.InPKMarchitectures,individualactuatorsarenottypicallyarrangedtoindependentlycontrolasingleCartesianDOF.Insteadtheactuatorsarearrangedsothateachactuatorisconnectedbetweenthestationarygroundframeandthemovingframe,sothatacombinationofactuatormotionscontrolsthemotionofthemovingframeinCartesianspace.ApopulartypeofPKMistheStewartplatformFigure1.TheStewartplatformconnectsamoveableplatformtoastationarygroundframebysixextensiblelinksorstruts,allowingforcontrolledmotionoftheplatforminallsixDOF.GenerallyforPKMmachines,theCartesianpositionandorientationofthetoolpointcarriedontheplatformisobtainedfromakinematicmodeloftheparticularmachine.Accuratepositioningofthesemachinesreliesontheaccurateknowledgeoftheparametersofthekinematicmodeluniquetotheparticularmachine.Theparametersinthekinematicmodelincludethespatiallocationsofthejointcentersonthemachinebaseandmovingplatform,theinitialstrutlengths,andthestrutdisplacements.Thestrutdisplacementsarereadilyobtainedfromsensorsonthemachine.However,theremainingkinematicparametersjointcenterlocations,andinitialstrutlengthsaredifficulttodeterminewhenthesemachinesareintheirfullyassembledstate.Thesizeandcomplexityofthesemachinegenerallymakesitdifficultandsomewhatundesirabletodeterminetheremainingkinematicparametersbydirectinspectionsuchasinacoordinatemeasuringmachine.InorderforPKMstobeusefulforprecisionpositioningapplications,techniquesmustbedevelopedtoquicklycalibratethemachinebydeterminingthekinematicparameterswithoutdisassemblyofthemachine.AnumberofauthorshavereportedtechniquesforcalibrationofPKMs.{Soons1,2,Masory,3,Zhuanget.al.4,5,Ropponen6}Inanotherpaper7,theauthorshavereportedonworkrecentlycompletedbytheUniversityofFloridaandSandiaNationalLaboratoriesoncalibrationofPKMs,whichdescribesanewtechniquetosequentiallydeterminethekinematicparametersofanassembledparallelkinematicdevice.CalibrationofPKMs,oranyothertypeofmachine,beginsbycollectingsetsofmeasurements.Thecollectedmeasurementdataissuppliedtoanalgorithm,whichcomputesthemodelparameters.Aswithanymetrologytask,1SandiaisamultiprogramlaboratoryoperatedbySandiaCorporation,aLockheedMartinCompany,fortheUnitedStatesDepartmentofEnergyunderContractDEAC04–94AL85000.PrecisionEngineeringJournaloftheInternationalSocietiesforPrecisionEngineeringandNanotechnology25200148–55machineaftercalibrationusingaSMX4,000lasertrackerareshown.thereisuncertaintypresentinthecollectedmeasurementdata.Theuncertaintypresentinthemeasurementdatawillpropagatethroughtheparameteridentificationalgorithmandresultinerrorsinthecomputedkinematicparameters.Theerrorsintherecoveredkinematicparameterswillthencreatepositioningerrorswhenthekinematicmodelisusedformachinecontrol.ThepurposeofthispaperistoexploretheissueofpropagationofuncertaintyinthecalibrationofPKMdevices.Wewilldescribeamethodologyforevaluatingtheuncertaintyintherecoveredparameters,andtheresultingerrorboundsinmachinepositioning.Wewilldemonstratethismethodologyforthenewcalibrationmethodwehavedevelopedandreportedinanotherpaper7.Thispaperisorganizedinthefollowingmanner.Section2describesthemethodologyforevaluatinguncertaintypropagationinPKMcalibration.Section3containsabriefsynopsisofthecalibrationmethodologyusedinthiswork.InSection4,theerrorbudgetforthecalibrationmeasurementsisdeveloped.Section5givestheresultsoftheuncertaintypropagationsimulation.Section6comparesthesimulationresultstoexperimentalresultsperformedonthemachine.2.MethodologyforevaluatinguncertaintypropagationInanymachinecalibrationtask,uncertaintyisintroducedintwoways.First,thecalibrationmethodinvolvesperformingmeasurementswithaspecifiedsetofinstruments.Thesemeasurementinstrumentswillhaveanassociateduncertaintyintheiroutput,whichisafunctionofthephysicalprincipalsandconstructionoftheinstrumentsthemselves,aswellastheenvironmentinwhichthemeasurementstakeplace.Inconjunctionwiththeuncertaintyassociatedwiththeexternalcalibrationinstruments,theremaybeadditionaluncertaintyintroducedbythemachineitself.ThefeedbackdevicesonthestrutsofPKMmachinestypicallycannotdirectlymeasuretheabsolutedistancebetweencorrespondingpairsofjointcenters,onlythechangeinlengthofthestrutfromsomehomepositionatwhichtheabsolutejointcenterdistanceisassumedtobeknown.Changesinthethermalstateofthemachinemaycausethisassumedvaluetobeincorrectand/ortofluctuateovertime.Thestrutdisplacementfeedbackdevicesalsohaveafiniteresolutionandsomeuncertaintyassociatedwiththeiroutput.Geometricaleffectsnotincludedinthekinematicmodel,suchasnonsphericaljointmotion,alsocreateaneffectivestrutlengtherrorwhichisnotsensedbythestrutfeedbackdevices.Therefore,themachinesrepeatabilityandabsolutepositioningaccuracyduringthecalibrationmeasurementscontributestotheoveralluncertaintyofthebasicdatausedbythecalibrationalgorithm.Theuncertaintiespresentintheexternalmeasurementinstrumentcombinewiththeuncertaintiesinthemachinemotionstocreateerrorsinthedatausedbythecalibrationalgorithmtocomputetheparametersofthemachineskinematicmodel.Sinceerrorispresentinthecollecteddata,therewillbeerrorpresentinthecomputedmodelparametersaswell.Sincethesecomputedmodelparametersdonotidenticallymatchtheactualphysicalparametersoftherealmachine,positioningerrorswillresultwhentherecoveredparametersareusedduringpositionalcontrolofthemachine.Thesepositioningerrorsduetoincorrectparameterizationofthemodelwillbecompoundedwiththeuncertaintiesinherenttothemachineitself,resultinginthefinalvolumetricaccuracyofthemachine.Thisisanimportantpointthatshouldnotbeoverlooked.Theuncertaintiesarisingfromthemachineitselfi.e.,repeatability,sensorresolution,unmodeledthermalandgeometriceffectsaffecttheoverallpositioningperformanceofthemachinetwice.First,theyadduncertaintytothebasicmeasurementdatathatisusedbythecalibrationalgorithm.Second,afterthecalibrationiscomplete,andthenewlyparameterizedkinematicmodelisusedformachinecontrol,thesesamemachineuncertaintiesstillexistandcontributetopositioningerrors.WeproposethefollowingmethodologyforevaluationofuncertaintypropagationthroughthePKMcalibrationprocess1.Constructanerrorbudgetforthemeasurementdevicesusedduringcalibrationtopredicttheiruncertaintycontributions.2.Constructanerrorbudgetforthemachineitselftoestimatethemagnitudeofstrutlengtherrorwhichcanoccurduetothermaleffects,sensorresolutionanduncertainty,nonsphericityofthejointmotions,andothereffectsinherenttotheparticularmachinewhichmaycausenonrepeatabilityofpositioning.3.Combinethesetwouncertaintysourcestoestimatetheoveralluncertaintypresentinthecollectedmeasurementdatausedbythecalibrationalgorithmtoobtainmodelparameters.Fig.1.A12jointsixsixStewartplatformdevice.49B.Jokieletal./PrecisionEngineering25200148–554.Constructaperfectdatasetfromthemachinemodel,whichassumesnouncertaintiesexistinthemeasuringinstrumentsormachine.5.PerformaMonteCarlosimulationofthecalibrationprocessbyrunningthecalibrationalgorithmmultipletimes,eachtimecorruptingtheperfectdatasetwithrandomlygeneratederrorsassumingauniformdistributionovertheuncertaintyboundspredictedbythecombinederrorbudgetanalyses.6.AnalyzethedistributionoferrorsinthemodelparametersobtainedfromtheMonteCarlosimulation.7.Usingthemachinekinematicmodelwitheachsetofmodelparametersobtainedfromthecalibrationsimulation,simulatethemachinemotionsduringaparticularmachineaccuracyperformancetesti.e.,circularballbartests,laserinterferometricdisplacementmeasurements,etc.,beingsuretoincludetheuncertaintyassociatedwiththemachineitselfinthesimulation.8.Analyzethesimulatedperformancedata.Theresultsofthissimulationwillshowtherangeofresultsoneislikelytoachieveonthefinalmachinepositioningperformancetestsfromaparticularcalibrationprocedure.ThisprocedureisoutlinedintheflowchartinFigure2.Itisalsopossibletomodifythesesimulationstoexaminetheeffectofvariouscontributorstotheoverallpositioningperformanceofthemachine.Forexample,onemightassumethatthemachineitselfisperfecttodeterminehowuncertaintiesintheexternalcalibrationmeasurementinstrumentscontributetomachinepositioningerrors.Conversely,onecanassumethattheexternalmeasurementsareperfectandexaminethepropagationofvarioussourcesofmachineerror,suchasthermaleffectsornonsphericityofjointmotions,tothefinalpositioningaccuracyperformance.3.Calibrationmethodology–sequentialdeterminationtechniqueTheapproachoutlinedinSection2wasusedtoevaluateuncertaintypropagationincalibrationofPKMdevicesforanewmethodforsequentialdeterminationofkinematicparametersinPKMsdevelopedbytheauthorsandreportedelsewhere10.Asaconveniencetothereader,thismethodisbrieflydescribedhere.Theparameteridentificationmethodusesaspatialcoordinatemeasuringdevicesuchasalasertrackeroralaserballbar.Thecalibrationisperformedinfoursteps1.LocationofacentralreferenceframeRandthemachineframeM.2.Identificationofthespatiallocationsofthejointscentersofrotation.3.Determinationofthespindleorientationandnoselocation.4.Determinationoftheinitialstrutlengths.3.1.LocatingthecentralreferenceandmachinereferenceframesIngeneral,severallocationsofthespatialcoordinatemeasuringdevicemayberequiredtocompleteallofthenecessarymeasurements.Therefore,asingle,stablereferFig.2.Uncertaintypropagationalgorithmflowchart.Fig.3.Locationsofcoordinatereferenceframesusedduringcalibration.50B.Jokieletal./PrecisionEngineering25200148–55enceframe,R,isnecessarytowhichallofthecoordinatedatacollectedinvariousframesmaybetransformedFigure3.TheRframeconsistsofthreegaugepointssecuredtotheworktable.Utilizingthespatialcoordinatemeasuringdevice,thelocationsoftheRgaugepointsaremeasuredrelativetothemeasuringdevicescoordinatesystem,andthehomogeneouscoordinatetransformationHTMbetweenthemeasurementdevicescoordinatesystemandtheRframeMeasTRiscomputedusingthefollowingequations.TheHTMbetweentheRsystemandmachinesreferenceframeMmayalsobedeterminedatthistime.Usingthespatialmeasurementdevice,theplaneoftheworktable,thedesiredlocationoftheXaxisandthemachineoriginaremeasured.Usingtheunitnormalvectorofthebestfitplane,theunitvectoroftheXaxisdirectionvectorprojectedintothebestfitplane,andthecoordinatesofthedesiredmachineoriginprojectedintothebestfitplanetheHTMoftheMframerelativetothemeasurementdeviceframe,MeasTM,isconstructed.TheHTMrelatingtheRandMframesMTRisthencalculatedbymatrixmultiplication.3.2.JointcenterlocationidentificationStewartplatformmachinesrequiretheuseofsphericalorHookejointstoconnectthestrutstothemachinebaseandtothemoveableplatform.Thesejointsallowthestrutendstorotateaboutfixedpointsonthemachinebaseandplatformasthemachinegeometrychanges.Ifastrutisheldatafixedlengthandrotatedaboutoneofitsjoints,allofthepointsonthelinkmovewithsphericalmotionaboutacommoncenterofrotation.Thisfactcanbeleveragedtodeterminethelocationofthejointscenterofrotation.Twogaugepointsareaffixedtoeithersideofonestrutsothatthegaugepointcentersandthestrutcenterlinelieinthesameplane.Threegaugepointsareaffixedtotheplatform,whichdefineaplatformreferencecoordinatesystemPR.Theplatformisthencommandedtomovealonganarbitrary,predeterminedpathdesignedtoholdthestrutinquestionatanarbitraryfixedlength.Astheplatformmoves,thisfixedlengthstrutrotatesaboutitsjointcenters,andpointsonthestruttraceasphericalpathinspaceFigure4.Atseverallocationsalongthepath,themachinemotionispaused,andthespatialcoordinatesofthetwolinkandthreeplatformgaugepointsaremeasured.Thismovepausemeasuresequenceisrepeateduntilthedesirednumberofplatformposeshasbeenreached.Thegaugepointcoordinatesmaybeexpressedrelativetotwodifferentcoordinatesystems,theMsystem,orthePRsystem.GaugepointcoordinatesexpressedrelativetotheMsystemlieonthesurfaceofaspherewhosecenteristhecenterofrotationofthebasejointoftheexaminedstrut.ThesamegaugepointcoordinatesexpressedrelativetothePRsystemlieonthesurfaceofaspherewhosecenteristhecenterofrotationoftheplatformjointfortheexaminedstrut.Thelocationsofthebaseandplatformjointcentersmaybedeterminedbyfittingtheappropriatelytransformedstrutgaugepointcoordinatestotheequationofasphere.Assumingthejointsproducesphericalmotion,thecalculatedcoordinatesofthecenterofthebestfitsphereisthecenterofrotationofthejointinquestion.Thismethodisrepeatedsequentiallyforeachofthesixstrutstorecoverall12ofthebaseandplatformjointcenterlocations.MeasTR5MeasxˆRMeasyˆRMeaszˆRMeasr1000MeasxˆR5Measr22Measr1uMeasr22Measr1uMeaszˆR5MeasxˆR3Measr32Measr1uMeasr32Measr1uMeasyˆR5MeaszˆR3MeasxˆR1T5MeasxˆMMeasyˆMMeasnˆplaneMeaspO0001MeasxˆM5Measuˆx2MeasuˆxzMeasnˆplaneMeasnˆplaneuMeasuˆx2MeasuˆxzMeasnˆplaneMeasnˆplaneuMeasyˆM5Measnˆplane3MeasxˆMMeaspO5Measp2MeaspzMeasnˆplaneMeasnˆplaneMTR5MeasTM21MeasTR251B.Jokieletal./PrecisionEngineering25200148–55
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