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

DOI10.1007/s00170-003-1860-2ORIGINALARTICLEIntJAdvManufTechnol200525130–139J.Song·J.MouAnear-optimalpartsetupalgorithmfor5-axismachiningusingaparallelkinematicmachineReceived19February2003/Accepted4July2003/Publishedonline28July2004Springer-VerlagLondonLimited2004AbstractAnear-optimalpartsetupmodelNOPSMisde-veloped.Thepurposeofthismodelistofindthenear-optimalpartsetuppositionandorientationbasedontheworkspace,stiff-nessandaccuracycapabilityofaparallelkinematicmachinethatcanbeusedfor5-axismachining.TobuildtheproposedNOPSM,theknowledgeonthehexapodkinematics,workspace,stiffness,structuralimperfection,nonunithermalgradientandaccuracyisrequired.Thus,itisacomprehensiveper-ancecapabilitystudyforaparallelkinematicmachine.Thepro-posedmodelisasoftwaresolutionconcepttoimprovethema-chine’sperance.Itisverycosteffectiveandcanalsobemodifiedforother5-axismachinetoolapplications.Keywords5-axismachining·Parallelkinematic·Partsetup·Peranceenhancement1IntroductionManyCAD/CAMalgorithmscansimulatethemachiningpro-cessforslantedandfree-surfacesaswellasgeneratethecorrespondingNCcodes；however,thetraditionalthree-degree-of-freedomCNCmachinerestrictsthepotencyofthesealgo-rithms.Oneoftheadvantagesofparallelkinematichexapodmachineoverthetraditional3-axismachinecentreisitsdex-terityandflexibility[1–7].Theoretically,thehexapodmachineunderstudypossessessixdegreesoffreedom.Actually,theplat-orientationaroundthez-axiscoincideswiththemachinespindlerotation；thus,thishexapodmachinehasfiveeffectivede-greesoffreedomachining.Manypartsthathaveslantedorfree-surfacescanbegeneratedonahexapodmachinewithonesetup.Anotheradvantageofthehexapodmachineisitshigherstiffnesscomparedtotheseriallinkedstructuralma-J.Song·J.Moua117DepartmentofIndustrialEngineering,ArizonaStateUniversity,Tempe,AZ85287-5906,USAE-mailjimoutsmc.comchinesothatthehigh-speedoperationcanbecarriedoutonthismachine[8,9].However,adrawbackthatcomeswiththedexterityofthehexapodmachineisitsrelativelysmallworkspace.Likeallmanufacturingequipment,imperfectstructureandnon-unithermal-gradient-relatederrorsalwaysexisttodegradethema-chine’speranceinproducingqualityproducts[10–14].Duetotheuniquecharacteristicsofparallelkinematicstructures,themachineinaccuracydistributionwithinitsworkspacewillchangeastheplatpositionandorientationchange.Mean-while,itsstructuralstiffnessvariesatdifferentplatpositionsandorientations.Therefore,basedontheinationonthehexapod’snom-inalkinematicstructure,structuralerrors,thermalerrors,andworkspaceandstiffnessanalyses,anear-optimalpartsetupmodelNOPSMcouldbedevelopedtosub-optimallysetupapartwithintheworkspaceofahexapodmachine.Thecon-ceptofthisalgorithmisgenericandcanbeeasilyintegratedwithexistingkinematicandthermalmodelsofanyotherparallelkine-maticmachineswithsimplemodifications.Thisapproachcouldalsobeemployedintheapplicationofseriallylinkedrobotsandmachinetools.ForNOPSM,thefirstconstraintisthatallsurfacestobema-chinedneedtobelocatedwithinthehexapod’sworkspace.Oncetheworkspaceconditionissatisfied,thenextcriterionappliedtofindthenear-optimalpartsetupisthehexapodmachine’sstiff-nessanalysis.Togeneratehigh-qualityproduct,thepartneedstobeplacedatthemostdesirablepositionsothatthemachinecanpossessthehigheststiffnessandaccuracywhilegeneratingthepart.Thealgorithmsderivedin[15]tofindtherelationshipbetweenthemachine’sstructural/thermalerrorsanditsaccuracydistributionbasedonthemachine’sstructuralcharacteristicsandmachine’stemperaturegradientprofilesareadoptedforsearch-ingthenear-optimalpartpositioningandorientation.Inpractice,thehexapodmachine’sdynamicsandcontrolsys-temshouldalsobeconsideredfornear-optimalpartpositioningsearches.However,duetotheproblem’scomplexityandtolim-itationsonthescopeofthisresearch,wewillnotdiscussthosetopicshere.1312WorkspaceanalysisTheworkspaceistheworkingvolumeofamachinewithspe-cifictoolsandfeasiblespindlepositionandorientation.Inordertodeterminetheusableworkspaceofthehexapodmachine,ade-rivedkinematicmodel[15]canbeappliedtodeterminethestrutlength,thejointrotationangleandmobileplatpositionandorientation.Twoconstraintsaretakenintoaccountinthisworkspaceanalysis.First,themachine’sstrutlengthlimitationsmaximumlengthdefinethelowerboundoftheworkspace.Fig.1.HexapodmachineworkspaceanalysisflowchartSecond,themachine’ssphericaljointrotationallimitationsde-finetheupperboundoftheworkspace.Althoughthemachine’sminimumstrutlengthlimitationshouldalsobetakenintoconsid-eration,thisconstraintisoverriddenbythesphericaljointrota-tionallimitationindeterminingthemachine’supperworkspace.AnalgorithmforthedeterminationofhexapodworkspaceforaspecificplatorientationisshowninFig.1.Differentplatorientationshavediversemachinework-spaceenvelopes.Byupdatingtheorientationination,theworkspaceenvelopfordifferentmachineplatorientationscanbedetermined.InFigs.2and3,theorientationsaroundthe132Fig.2.Workspaceenvelopewithspindleorientationangle[0˚0˚0˚]Fig.3.Workspaceenvelopewithspindleorientationangle[−30˚0˚0˚]y-andz-axess,βandγ,arekeptconstant；onlytheorientationanglearoundthex-axis,α,ischanged.Astheorientationanglearoundthex-axisincreases,theworkspaceistiltedandthez-dimensionoftheworkspaceenvelopeisdecreased.Thelargertheorientationangle,themoreseveretheworkspacetilting.Theworkspaceanalysisresultsshowthatasimilarphenomenonoc-curswhentheorientationanglearoundthey-axis,β,ischanged,butwithdifferenttiltingdirection.Sincetheplatorientationanglearoundthez-axis,γ,coincideswiththespindlerotatingdi-rection,theeffectofγissuperimposedonspindlerotationandthusnottakenintoconsiderationinworkspaceenvelopeanalysis.TheNOPSMadoptstheworkspaceanalysistodeterminewhetherornotthemachiningsurfacesarewithinthehexapodworkspace.Toensuretheefficiencyofthealgorithm,thefollow-ingtwoconstraintsaretestedforalltheselectedpointsonthesurfacetobemachined1.Thehexapodmaximumstrutlengthconstraint.2.Thehexapodmaximumjointanglerotationconstraint.Ifalltheselectedpointsonthemachiningsurfacessatisfytheabovetwoconstraints,structuralstiffnessandmachineaccu-racywillthenbeanalysedtoidentifythenear-optimalpartsetuplocationandorientation.3StiffnessanalysisForaparallelmechanism,thereusuallyisaclosed-solutionfortheinversekinematics.Theinversekinematicsforthehexa-podmachinecouldbeusedtocalculatethesix-strutlengthbasedontheplatpositionandorientationination[16].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.1bythedifferentialtimeelementδtandexpressingitinmatrixatyields˙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¯vintothestrutdisplace-mentratevector.Applyingtheprincipleofvirtualworktoanarbitrarymech-anismallowsonetoequateworkdoneinCartesianspacetermstoworkdoneinconfigurationspaceterms.Specifically,workinCartesiantermsisassociatedwithaCartesianforce/torquevec-tor,[F],appliedatamechanism’stoolframeandactingthroughaninfinitesimalCartesiandisplacement,δ¯v.Workinconfig-urationspacetermsisassociatedwithaconfigurationspaceforce/torquevector,[f],appliedatamechanism’sjointsandact-ingthroughinfinitesimaljointdisplacements,δ¯l.Thestiffnessofthehexapodcanbedeterminedusingmatrixstructuralanalysis,wherethestructureisconsideredtobeacom-binationofelementsandnodes.Thederivationofthehexapodstiffnessmodelisbasedonthefollowingassumptions1.Theonlydeationofthelinksisintheaxialdirection.2.Thereisnobendingortwistingofthelinks.3.Thereisnodeationofthejoints.Workiscalculatedasthedotproductofaforce/torquevectorwithadisplacementvector,[F]T·δ¯v[f]Tδ¯l,where[f]T[f1,f2,f3,f4,f5,f6]aretheforcesrtedoneachofthesix133strutsand[F]T[Fx,Fy,Fz,Mx,My,Mz]aretheforcesandmomentsactingatthecentreofgravityoftheplat.Notethatδ¯vJδ¯l,so[F]T·Jδ¯l[f]Tδ¯l⇒[F]TJ[f]T.Trans-ingbothsidesoftheequationyields[F]TJT[f]⇒[f]JT[F].Onecouldconcludethat‘actuating’amechanismwithaforce/torquevector,[F],appliedatthetoolisequivalenttoac-tuatingthatmechanismwithaforce/torquevector,[f],appliedatthejoints,whenthesameamountofvirtualworkisdoneineithercase.Therelationshipbetweenanappliedforce[F]atthetoolandtheresultingaxialforcesinthestrut[f]canbedefinedas[F]J−T[f].Givenpureaxialloading,εδli/liσ/Efi/AE⇒fiAE/liδli,whereEistheelasticmodulusofthestrutmaterialandAisthecross-sectionalareaofthestrut.Inmatrixat,[f]AE/l1000000AE/l2000000AE/l3000000AE/l4000000AE/l5000000AE/l6.δ¯l.4Or[f]KSδ¯l,wherethematrixisidenticaltothestrutspacestiffnessmatrix,Ks.Notethatδ¯lJ−1δ¯v,so[f]KSJ−1δ¯vand[F]J−TKSJ−1δ¯v.LetKCJ−TKSJ−1；then[F]KCδ¯v,whereKcistheCartesianspacestiffnessmatrix.Byset-tingupaneigenvalueproblem,theprinciplestiffnessaxes,¯ηi,andprinciplestiffness,κi,canbefoundasfollows[F]KCδ¯vκiδ¯v5KC−κiI6δ¯v06|KC−κiI6|0.7Here,¯ηiisinthedirectionofδ¯vwheretheaboveconditionholds.Theprinciplestiffnessκiwillchangeasplatorien-tationandpositionchange.Thehigherthemachine’sstructuralstiffness,thebetterthepart’squalityandaccuracy.4StructuralerrordetectionmodelAsmentionedearlier,thehexapodmachinestructureisnotper-fect,andstructuralimperfectionandassemblyerrorsexist.Thestructuralandassemblyerrorsarenotdistributedevenlyamongthehexapodjointsandstruts.Thisunevennesscausesdiverseaccuracylevelsatdifferentplatpositionsandorientations.Afteramachineisassembled,itisdifficulttomeasurethema-chinestructuralandassemblyerrorbyusinginstrumentsorsen-sorsdirectly.However,themachineplat’sorientationandpositioncanbepreciselymeasuredbyusinganexternalinstru-mentsuchasa5Dlaserinterferometersystemoralasertrackersystem.Amodelisthenneededtoreverseidentifythemachinestructuralerrorsbasedonthemeasuredplatpositionandorientationerrors.Thehexapodnominalinversekinematicsisderivedas[15]λTmlmTpTRPPbn−Tsm；m1,2,...,6；nintm1/2.8Differentiatingtheequation,sinceallthevectorsarewithre-specttothetablecoordinatesystem,thesuperscriptofTcanbeomittedδλmlmλmδlmδpδRPPbnRPPδbn−δm.9Tosimplifythecalculation,therotationerrormatrixcanbewrit-tenasδRPδ˜ΩRP,whereδΩ[δα,δβ,δγ]Tistheorientationerrorvector,RPisthenominalorientationmatrix,andδ˜Ωisde-finedasδ˜Ω1−δγδβδγ1−δα−δβδα1.Equation9cannowbeexpressedasδλmlmλmδlmδpδ˜ΩRPPbnRPPδbn−δsm.10Sincelmisaunitvector,lmTlm1；lmTδlm0.MultiplyingEq.10bylmTresultsinδλmlTmδplTmδ˜ΩRPPbnlTmRPPδbn−lTmδsm；m1,2,...,6；nintm1/2,11whereδλmisthestrutlengtherror,δsmisthetopplatspher-icaljointpositionalerror,δbnisthemobileplatballjointpositionalerror,δpandδΩarethemobileplatpositionandorientationerrors,respectively,lTmisthestrutvector,andRPisthetransationmatrixbetweenthemobileplatandthetablecoordinatesystem.Nowtherelationshipbetweenthehexapodstructuralerrorsandthemobileplatpositionandorientationerrorsises-tablishedforanyspecificpointwithinthehexapodworkspace.Sincethehexapodmachinehas6strutlengtherrors,18basejointpositionalerrorsand9platjointpositionalerrors,thetotalstructuralerrortermsaddupto33.Bothanalyticalandex-perimentalsensitivityanalyseswereconductedtodeterminethesignificanceofeachstructuralerroronmachineaccuracy[15].Theresultsshowthatall33structuralerrortermshavesignificantimpactonplatpositionandorientationaccuracy.Thusthecompleteerrormodelincludesatotalof33structuralerrorterms.Asimulationisusedtovalidatethederivedstructuralerrordetectionmodel.Theperturbationtothepositionandorienta-tionof6testingpointsiscreatedbyrandomlyassigningasetofoffset∆X,∆Y,∆Z,∆α,and∆βtothemobileplatcoordinate.ThestructuralerrordetectionmodelcanbewritteninmatrixasECM−1,whereEisthevectorconsistingof33hexapodstructuralerrorterms,Cisa33-elementvectordeterminedbythehexapodstructuralparametersandmeasuredplatpositionandorientationerrors,Misa3333matrixdeterminedbythehexapodstructuralparametersandtheselectedsixpositionsandorientations.