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外文资料--A near-optimal part setup algorithm for 5-axis machining using a parallel.pdf

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外文资料--A near-optimal part setup algorithm for 5-axis machining using a parallel.pdf

DOI10.1007/s0017000318602ORIGINALARTICLEIntJAdvManufTechnol200525130–139J.SongJ.MouAnearoptimalpartsetupalgorithmfor5axismachiningusingaparallelkinematicmachineReceived19February2003/Accepted4July2003/Publishedonline28July2004SpringerVerlagLondonLimited2004AbstractAnearoptimalpartsetupmodelNOPSMisdeveloped.Thepurposeofthismodelistofindthenearoptimalpartsetuppositionandorientationbasedontheworkspace,stiffnessandaccuracycapabilityofaparallelkinematicmachinethatcanbeusedfor5axismachining.TobuildtheproposedNOPSM,theknowledgeonthehexapodkinematics,workspace,stiffness,structuralimperfection,nonuniformthermalgradientandaccuracyisrequired.Thus,itisacomprehensiveperformancecapabilitystudyforaparallelkinematicmachine.Theproposedmodelisasoftwaresolutionconcepttoimprovethemachinesperformance.Itisverycosteffectiveandcanalsobemodifiedforother5axismachinetoolapplications.Keywords5axismachiningParallelkinematicPartsetupPerformanceenhancement1IntroductionManyCAD/CAMalgorithmscansimulatethemachiningprocessforslantedandfreeformsurfacesaswellasgeneratethecorrespondingNCcodeshowever,thetraditionalthreedegreeoffreedomCNCmachinerestrictsthepotencyofthesealgorithms.Oneoftheadvantagesofparallelkinematichexapodmachineoverthetraditional3axismachinecentreisitsdexterityandflexibility1–7.Theoretically,thehexapodmachineunderstudypossessessixdegreesoffreedom.Actually,theplatformorientationaroundthezaxiscoincideswiththemachinespindlerotationthus,thishexapodmachinehasfiveeffectivedegreesoffreedomformachining.Manypartsthathaveslantedorfreeformsurfacescanbegeneratedonahexapodmachinewithonesetup.AnotheradvantageofthehexapodmachineisitshigherstiffnesscomparedtotheseriallinkedstructuralmaJ.SongJ.Moua117DepartmentofIndustrialEngineering,ArizonaStateUniversity,Tempe,AZ852875906,USAEmailjimoutsmc.comchinesothatthehighspeedoperationcanbecarriedoutonthismachine8,9.However,adrawbackthatcomeswiththedexterityofthehexapodmachineisitsrelativelysmallworkspace.Likeallmanufacturingequipment,imperfectstructureandnonuniformthermalgradientrelatederrorsalwaysexisttodegradethemachinesperformanceinproducingqualityproducts10–14.Duetotheuniquecharacteristicsofparallelkinematicstructures,themachineinaccuracydistributionwithinitsworkspacewillchangeastheplatformpositionandorientationchange.Meanwhile,itsstructuralstiffnessvariesatdifferentplatformpositionsandorientations.Therefore,basedontheinformationonthehexapodsnominalkinematicstructure,structuralerrors,thermalerrors,andworkspaceandstiffnessanalyses,anearoptimalpartsetupmodelNOPSMcouldbedevelopedtosuboptimallysetupapartwithintheworkspaceofahexapodmachine.Theconceptofthisalgorithmisgenericandcanbeeasilyintegratedwithexistingkinematicandthermalmodelsofanyotherparallelkinematicmachineswithsimplemodifications.Thisapproachcouldalsobeemployedintheapplicationofseriallylinkedrobotsandmachinetools.ForNOPSM,thefirstconstraintisthatallsurfacestobemachinedneedtobelocatedwithinthehexapodsworkspace.Oncetheworkspaceconditionissatisfied,thenextcriterionappliedtofindthenearoptimalpartsetupisthehexapodmachinesstiffnessanalysis.Togeneratehighqualityproduct,thepartneedstobeplacedatthemostdesirablepositionsothatthemachinecanpossessthehigheststiffnessandaccuracywhilegeneratingthepart.Thealgorithmsderivedin15tofindtherelationshipbetweenthemachinesstructural/thermalerrorsanditsaccuracydistributionbasedonthemachinesstructuralcharacteristicsandmachinestemperaturegradientprofilesareadoptedforsearchingthenearoptimalpartpositioningandorientation.Inpractice,thehexapodmachinesdynamicsandcontrolsystemshouldalsobeconsideredfornearoptimalpartpositioningsearches.However,duetotheproblemscomplexityandtolimitationsonthescopeofthisresearch,wewillnotdiscussthosetopicshere.1312WorkspaceanalysisTheworkspaceistheworkingvolumeofamachinewithspecifictoolsandfeasiblespindlepositionandorientation.Inordertodeterminetheusableworkspaceofthehexapodmachine,aderivedkinematicmodel15canbeappliedtodeterminethestrutlength,thejointrotationangleandmobileplatformpositionandorientation.Twoconstraintsaretakenintoaccountinthisworkspaceanalysis.First,themachinesstrutlengthlimitationsmaximumlengthdefinethelowerboundoftheworkspace.Fig.1.HexapodmachineworkspaceanalysisflowchartSecond,themachinessphericaljointrotationallimitationsdefinetheupperboundoftheworkspace.Althoughthemachinesminimumstrutlengthlimitationshouldalsobetakenintoconsideration,thisconstraintisoverriddenbythesphericaljointrotationallimitationindeterminingthemachinesupperworkspace.AnalgorithmforthedeterminationofhexapodworkspaceforaspecificplatformorientationisshowninFig.1.Differentplatformorientationshavediversemachineworkspaceenvelopes.Byupdatingtheorientationinformation,theworkspaceenvelopfordifferentmachineplatformorientationscanbedetermined.InFigs.2and3,theorientationsaroundthe132Fig.2.Workspaceenvelopewithspindleorientationangle0˚0˚0˚Fig.3.Workspaceenvelopewithspindleorientationangle−30˚0˚0˚yandzaxess,βandγ,arekeptconstantonlytheorientationanglearoundthexaxis,α,ischanged.Astheorientationanglearoundthexaxisincreases,theworkspaceistiltedandthezdimensionoftheworkspaceenvelopeisdecreased.Thelargertheorientationangle,themoreseveretheworkspacetilting.Theworkspaceanalysisresultsshowthatasimilarphenomenonoccurswhentheorientationanglearoundtheyaxis,β,ischanged,butwithdifferenttiltingdirection.Sincetheplatformorientationanglearoundthezaxis,γ,coincideswiththespindlerotatingdirection,theeffectofγissuperimposedonspindlerotationandthusnottakenintoconsiderationinworkspaceenvelopeanalysis.TheNOPSMadoptstheworkspaceanalysistodeterminewhetherornotthemachiningsurfacesarewithinthehexapodworkspace.Toensuretheefficiencyofthealgorithm,thefollowingtwoconstraintsaretestedforalltheselectedpointsonthesurfacetobemachined1.Thehexapodmaximumstrutlengthconstraint.2.Thehexapodmaximumjointanglerotationconstraint.Ifalltheselectedpointsonthemachiningsurfacessatisfytheabovetwoconstraints,structuralstiffnessandmachineaccuracywillthenbeanalysedtoidentifythenearoptimalpartsetuplocationandorientation.3StiffnessanalysisForaparallelmechanism,thereusuallyisaclosedformsolutionfortheinversekinematics.Theinversekinematicsforthehexapodmachinecouldbeusedtocalculatethesixstrutlengthbasedontheplatformpositionandorientationinformation16.ThiscanbeexpressedasfollowsLifix,y,z,α,β,γ.1Theapplicationofthechainruleyieldsdifferentialsoflii1,2,...,6asfunctionsofthedifferentialsofx,y,z,α,β,γ.δli∂fi∂xδx∂fi∂yδy∂fi∂zδz∂fi∂αδα∂fi∂βδβ∂fi∂γδγ.2DividingbothsidesofEq.1bythedifferentialtimeelementδtandexpressingitinmatrixformatyields˙l1˙l2˙l3˙l4˙l5˙l6∂f1∂x∂f1∂y∂f1∂z∂f1∂α∂f1∂β∂f1∂γ∂f2∂x∂f2∂y∂f2∂z∂f2∂α∂f2∂β∂f2∂γ..................∂f6∂x∂f6∂y∂f6∂z∂f6∂α∂f6∂β∂f6∂γ˙x˙y˙z˙α˙β˙γ.3NotethestandardJocobianexpression,˙¯vJ˙¯l.Byletting˙¯lJ−1˙¯v,theinverseJocobianmatrix,J−1,facilitatesthemappingoftheCartesianspacevelocityvector¯vintothestrutdisplacementratevector.ApplyingtheprincipleofvirtualworktoanarbitrarymechanismallowsonetoequateworkdoneinCartesianspacetermstoworkdoneinconfigurationspaceterms.Specifically,workinCartesiantermsisassociatedwithaCartesianforce/torquevector,F,appliedatamechanismstoolframeandactingthroughaninfinitesimalCartesiandisplacement,δ¯v.Workinconfigurationspacetermsisassociatedwithaconfigurationspaceforce/torquevector,f,appliedatamechanismsjointsandactingthroughinfinitesimaljointdisplacements,δ¯l.Thestiffnessofthehexapodcanbedeterminedusingmatrixstructuralanalysis,wherethestructureisconsideredtobeacombinationofelementsandnodes.Thederivationofthehexapodstiffnessmodelisbasedonthefollowingassumptions1.Theonlydeformationofthelinksisintheaxialdirection.2.Thereisnobendingortwistingofthelinks.3.Thereisnodeformationofthejoints.Workiscalculatedasthedotproductofaforce/torquevectorwithadisplacementvector,FTδ¯vfTδ¯l,wherefTf1,f2,f3,f4,f5,f6aretheforcesexertedoneachofthesix133strutsandFTFx,Fy,Fz,Mx,My,Mzaretheforcesandmomentsactingatthecentreofgravityoftheplatform.Notethatδ¯vJδ¯l,soFTJδ¯lfTδ¯l⇒FTJfT.TransformingbothsidesoftheequationyieldsFTJTf⇒fJTF.Onecouldconcludethatactuatingamechanismwithaforce/torquevector,F,appliedatthetoolisequivalenttoactuatingthatmechanismwithaforce/torquevector,f,appliedatthejoints,whenthesameamountofvirtualworkisdoneineithercase.TherelationshipbetweenanappliedforceFatthetoolandtheresultingaxialforcesinthestrutfcanbedefinedasFJ−Tf.Givenpureaxialloading,εδli/liσ/Efi/AE⇒fiAE/liδli,whereEistheelasticmodulusofthestrutmaterialandAisthecrosssectionalareaofthestrut.Inmatrixformat,fAE/l1000000AE/l2000000AE/l3000000AE/l4000000AE/l5000000AE/l6.δ¯l.4OrfKSδ¯l,wherethematrixisidenticaltothestrutspacestiffnessmatrix,Ks.Notethatδ¯lJ−1δ¯v,sofKSJ−1δ¯vandFJ−TKSJ−1δ¯v.LetKCJ−TKSJ−1thenFKCδ¯v,whereKcistheCartesianspacestiffnessmatrix.Bysettingupaneigenvalueproblem,theprinciplestiffnessaxes,¯ηi,andprinciplestiffness,κi,canbefoundasfollowsFKCδ¯vκiδ¯v5KC−κiI6δ¯v06|KC−κiI6|0.7Here,¯ηiisinthedirectionofδ¯vwheretheaboveconditionholds.Theprinciplestiffnessκiwillchangeasplatformorientationandpositionchange.Thehigherthemachinesstructuralstiffness,thebetterthepartsqualityandaccuracy.4StructuralerrordetectionmodelAsmentionedearlier,thehexapodmachinestructureisnotperfect,andstructuralimperfectionandassemblyerrorsexist.Thestructuralandassemblyerrorsarenotdistributedevenlyamongthehexapodjointsandstruts.Thisunevennesscausesdiverseaccuracylevelsatdifferentplatformpositionsandorientations.Afteramachineisassembled,itisdifficulttomeasurethemachinestructuralandassemblyerrorbyusinginstrumentsorsensorsdirectly.However,themachineplatformsorientationandpositioncanbepreciselymeasuredbyusinganexternalinstrumentsuchasa5Dlaserinterferometersystemoralasertrackersystem.Amodelisthenneededtoreverseidentifythemachinestructuralerrorsbasedonthemeasuredplatformpositionandorientationerrors.Thehexapodnominalinversekinematicsisderivedas15λTmlmTpTRPPbn−Tsmm1,2,...,6nintm1/2.8Differentiatingtheequation,sinceallthevectorsarewithrespecttothetablecoordinatesystem,thesuperscriptofTcanbeomittedδλmlmλmδlmδpδRPPbnRPPδbn−δm.9Tosimplifythecalculation,therotationerrormatrixcanbewrittenasδRPδ˜ΩRP,whereδΩδα,δβ,δγTistheorientationerrorvector,RPisthenominalorientationmatrix,andδ˜Ωisdefinedasδ˜Ω1−δγδβδγ1−δα−δβδα1.Equation9cannowbeexpressedasδλmlmλmδlmδpδ˜ΩRPPbnRPPδbn−δsm.10Sincelmisaunitvector,lmTlm1lmTδlm0.MultiplyingEq.10bylmTresultsinδλmlTmδplTmδ˜ΩRPPbnlTmRPPδbn−lTmδsmm1,2,...,6nintm1/2,11whereδλmisthestrutlengtherror,δsmisthetopplatformsphericaljointpositionalerror,δbnisthemobileplatformballjointpositionalerror,δpandδΩarethemobileplatformpositionandorientationerrors,respectively,lTmisthestrutvector,andRPisthetransformationmatrixbetweenthemobileplatformandthetablecoordinatesystem.Nowtherelationshipbetweenthehexapodstructuralerrorsandthemobileplatformpositionandorientationerrorsisestablishedforanyspecificpointwithinthehexapodworkspace.Sincethehexapodmachinehas6strutlengtherrors,18basejointpositionalerrorsand9platformjointpositionalerrors,thetotalstructuralerrortermsaddupto33.Bothanalyticalandexperimentalsensitivityanalyseswereconductedtodeterminethesignificanceofeachstructuralerroronmachineaccuracy15.Theresultsshowthatall33structuralerrortermshavesignificantimpactonplatformpositionandorientationaccuracy.Thusthecompleteerrormodelincludesatotalof33structuralerrorterms.Asimulationisusedtovalidatethederivedstructuralerrordetectionmodel.Theperturbationtothepositionandorientationof6testingpointsiscreatedbyrandomlyassigningasetofoffset∆X,∆Y,∆Z,∆α,and∆βtothemobileplatformcoordinate.ThestructuralerrordetectionmodelcanbewritteninmatrixformasECM−1,whereEisthevectorconsistingof33hexapodstructuralerrorterms,Cisa33elementvectordeterminedbythehexapodstructuralparametersandmeasuredplatformpositionandorientationerrors,Misa3333matrixdeterminedbythehexapodstructuralparametersandtheselectedsixpositionsandorientations.

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