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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/s00170-003-1860-2ORIGINALARTICLEIntJAdvManufTechnol200525130–139J.SongJ.MouAnear-optimalpartsetupalgorithmfor5-axismachiningusingaparallelkinematicmachineReceived19February2003/Accepted4July2003/Publishedonline28July2004Springer-VerlagLondonLimited2004AbstractAnear-optimalpartsetupmodelNOPSMisde-veloped.Thepurposeofthismodelistofindthenear-optimalpartsetuppositionandorientationbasedontheworkspace,stiff-nessandaccuracycapabilityofaparallelkinematicmachinethatcanbeusedfor5-axismachining.TobuildtheproposedNOPSM,theknowledgeonthehexapodkinematics,workspace,stiffness,structuralimperfection,nonuniformthermalgradientandaccuracyisrequired.Thus,itisacomprehensiveperform-ancecapabilitystudyforaparallelkinematicmachine.Thepro-posedmodelisasoftwaresolutionconcepttoimprovethema-chine’sperformance.Itisverycosteffectiveandcanalsobemodifiedforother5-axismachinetoolapplications.Keywords5-axismachiningParallelkinematicPartsetupPerformanceenhancement1IntroductionManyCAD/CAMalgorithmscansimulatethemachiningpro-cessforslantedandfree-formsurfacesaswellasgeneratethecorrespondingNCcodes;however,thetraditionalthree-degree-of-freedomCNCmachinerestrictsthepotencyofthesealgo-rithms.Oneoftheadvantagesofparallelkinematichexapodmachineoverthetraditional3-axismachinecentreisitsdex-terityandflexibility[1–7].Theoretically,thehexapodmachineunderstudypossessessixdegreesoffreedom.Actually,theplat-formorientationaroundthez-axiscoincideswiththemachinespindlerotation;thus,thishexapodmachinehasfiveeffectivede-greesoffreedomformachining.Manypartsthathaveslantedorfree-formsurfacescanbegeneratedonahexapodmachinewithonesetup.Anotheradvantageofthehexapodmachineisitshigherstiffnesscomparedtotheseriallinkedstructuralma-J.SongJ.Moua117DepartmentofIndustrialEngineering,ArizonaStateUniversity,Tempe,AZ85287-5906,USAE-mailjimoutsmc.comchinesothatthehigh-speedoperationcanbecarriedoutonthismachine[8,9].However,adrawbackthatcomeswiththedexterityofthehexapodmachineisitsrelativelysmallworkspace.Likeallmanufacturingequipment,imperfectstructureandnon-uniformthermal-gradient-relatederrorsalwaysexisttodegradethema-chine’sperformanceinproducingqualityproducts[10–14].Duetotheuniquecharacteristicsofparallelkinematicstructures,themachineinaccuracydistributionwithinitsworkspacewillchangeastheplatformpositionandorientationchange.Mean-while,itsstructuralstiffnessvariesatdifferentplatformpositionsandorientations.Therefore,basedontheinformationonthehexapod’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,thejointrotationangleandmobileplatformpositionandorientation.Twoconstraintsaretakenintoaccountinthisworkspaceanalysis.First,themachine’sstrutlengthlimitationsmaximumlengthdefinethelowerboundoftheworkspace.Fig.1.HexapodmachineworkspaceanalysisflowchartSecond,themachine’ssphericaljointrotationallimitationsde-finetheupperboundoftheworkspace.Althoughthemachine’sminimumstrutlengthlimitationshouldalsobetakenintoconsid-eration,thisconstraintisoverriddenbythesphericaljointrota-tionallimitationindeterminingthemachine’supperworkspace.AnalgorithmforthedeterminationofhexapodworkspaceforaspecificplatformorientationisshowninFig.1.Differentplatformorientationshavediversemachinework-spaceenvelopes.Byupdatingtheorientationinformation,theworkspaceenvelopfordifferentmachineplatformorientationscanbedetermined.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.Sincetheplatformorientationanglearoundthez-axis,γ,coincideswiththespindlerotatingdi-rection,theeffectofγissuperimposedonspindlerotationandthusnottakenintoconsiderationinworkspaceenvelopeanalysis.TheNOPSMadoptstheworkspaceanalysistodeterminewhetherornotthemachiningsurfacesarewithinthehexapodworkspace.Toensuretheefficiencyofthealgorithm,thefollow-ingtwoconstraintsaretestedforalltheselectedpointsonthesurfacetobemachined1.Thehexapodmaximumstrutlengthconstraint.2.Thehexapodmaximumjointanglerotationconstraint.Ifalltheselectedpointsonthemachiningsurfacessatisfytheabovetwoconstraints,structuralstiffnessandmachineaccu-racywillthenbeanalysedtoidentifythenear-optimalpartsetuplocationandorientation.3StiffnessanalysisForaparallelmechanism,thereusuallyisaclosed-formsolutionfortheinversekinematics.Theinversekinematicsforthehexa-podmachinecouldbeusedtocalculatethesix-strutlengthbasedontheplatformpositionandorientationinformation[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.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,facilitatesthemappingoftheCartesianspacevelocityvectorvintothestrutdisplace-mentratevector.Applyingtheprincipleofvirtualworktoanarbitrarymech-anismallowsonetoequateworkdoneinCartesianspacetermstoworkdoneinconfigurationspaceterms.Specifically,workinCartesiantermsisassociatedwithaCartesianforce/torquevec-tor,[F],appliedatamechanism’stoolframeandactingthroughaninfinitesimalCartesiandisplacement,δv.Workinconfig-urationspacetermsisassociatedwithaconfigurationspaceforce/torquevector,[f],appliedatamechanism’sjointsandact-ingthroughinfinitesimaljointdisplacements,δl.Thestiffnessofthehexapodcanbedeterminedusingmatrixstructuralanalysis,wherethestructureisconsideredtobeacom-binationofelementsandnodes.Thederivationofthehexapodstiffnessmodelisbasedonthefollowingassumptions1.Theonlydeformationofthelinksisintheaxialdirection.2.Thereisnobendingortwistingofthelinks.3.Thereisnodeformationofthejoints.Workiscalculatedasthedotproductofaforce/torquevectorwithadisplacementvector,[F]Tδv[f]Tδl,where[f]T[f1,f2,f3,f4,f5,f6]aretheforcesexertedoneachofthesix133strutsand[F]T[Fx,Fy,Fz,Mx,My,Mz]aretheforcesandmomentsactingatthecentreofgravityoftheplatform.NotethatδvJδl,so[F]TJδl[f]Tδl⇒[F]TJ[f]T.Transform-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.Inmatrixformat,[f]AE/l1000000AE/l2000000AE/l3000000AE/l4000000AE/l5000000AE/l6.δl.4Or[f]KSδl,wherethematrixisidenticaltothestrutspacestiffnessmatrix,Ks.NotethatδlJ−1δv,so[f]KSJ−1δvand[F]J−TKSJ−1δv.LetKCJ−TKSJ−1;then[F]KCδv,whereKcistheCartesianspacestiffnessmatrix.Byset-tingupaneigenvalueproblem,theprinciplestiffnessaxes,ηi,andprinciplestiffness,κi,canbefoundasfollows[F]KCδvκiδv5KC−κiI6δv06|KC−κiI6|0.7Here,ηiisinthedirectionofδvwheretheaboveconditionholds.Theprinciplestiffnessκiwillchangeasplatformorien-tationandpositionchange.Thehigherthemachine’sstructuralstiffness,thebetterthepart’squalityandaccuracy.4StructuralerrordetectionmodelAsmentionedearlier,thehexapodmachinestructureisnotper-fect,andstructuralimperfectionandassemblyerrorsexist.Thestructuralandassemblyerrorsarenotdistributedevenlyamongthehexapodjointsandstruts.Thisunevennesscausesdiverseaccuracylevelsatdifferentplatformpositionsandorientations.Afteramachineisassembled,itisdifficulttomeasurethema-chinestructuralandassemblyerrorbyusinginstrumentsorsen-sorsdirectly.However,themachineplatform’sorientationandpositioncanbepreciselymeasuredbyusinganexternalinstru-mentsuchasa5Dlaserinterferometersystemoralasertrackersystem.Amodelisthenneededtoreverseidentifythemachinestructuralerrorsbasedonthemeasuredplatformpositionandorientationerrors.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,δsmisthetopplatformspher-icaljointpositionalerror,δbnisthemobileplatformballjointpositionalerror,δpandδΩarethemobileplatformpositionandorientationerrors,respectively,lTmisthestrutvector,andRPisthetransformationmatrixbetweenthemobileplatformandthetablecoordinatesystem.Nowtherelationshipbetweenthehexapodstructuralerrorsandthemobileplatformpositionandorientationerrorsises-tablishedforanyspecificpointwithinthehexapodworkspace.Sincethehexapodmachinehas6strutlengtherrors,18basejointpositionalerrorsand9platformjointpositionalerrors,thetotalstructuralerrortermsaddupto33.Bothanalyticalandex-perimentalsensitivityanalyseswereconductedtodeterminethesignificanceofeachstructuralerroronmachineaccuracy[15].Theresultsshowthatall33structuralerrortermshavesignificantimpactonplatformpositionandorientationaccuracy.Thusthecompleteerrormodelincludesatotalof33structuralerrorterms.Asimulationisusedtovalidatethederivedstructuralerrordetectionmodel.Theperturbationtothepositionandorienta-tionof6testingpointsiscreatedbyrandomlyassigningasetofoffset∆X,∆Y,∆Z,∆α,and∆βtothemobileplatformcoordinate.ThestructuralerrordetectionmodelcanbewritteninmatrixformasECM−1,whereEisthevectorconsistingof33hexapodstructuralerrorterms,Cisa33-elementvectordeterminedbythehexapodstructuralparametersandmeasuredplatformpositionandorientationerrors,Misa3333matrixdeterminedbythehexapodstructuralparametersandtheselectedsixpositionsandorientations.

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