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首页 A NEW TOOL PATH GENERATION ALGORITHM BASED ON

A NEW TOOL PATH GENERATION ALGORITHM BASED ON.pdf

A NEW TOOL PATH GENERATION ALGO…

dukeyan
2012-12-27 0人阅读 举报 0 0 暂无简介

简介:本文档为《A NEW TOOL PATH GENERATION ALGORITHM BASED ONpdf》,可适用于IT/计算机领域

InternationalConferenceonInnovativeDesignandManufacturingDECEMBER,,Taipei,TaiwanANEWTOOLPATHGENERATIONALGORITHMBASEDONCOVARIANTFIELDTHEORYANDCOSTFUNCTIONALOPTIMIZATIONChenhanLee*WuhanHuazhongNumericalControlCo,Ltd,HuazhongUniversityofScienceandTechnology,Wuhan,PRChinachenhanleehotmailcomChangyaYanWuhanHuazhongNumericalControlCo,Ltd,Wuhan,PRChinayanchangyagmailcomJianzhongYangHuazhongUniversityofScienceandTechnology,Wuhan,PRChinayangjzmailhusteducnABSTRACTThispaperpresentsaunifiedmathematicframeworktocomputeoptimizedtoolpathwithimprovedsmoothnessandstablestepoverdistributions,whicharedesirablerequirements,especiallyforhighspeedmachiningThetoolpathtrajectoryisderivedfromascalarfielddefinedoverthepartsurfaceThescalarfieldissolvedfromminimizinganintegralfunctionalthatmeasurestheviolationofsmoothnessandstablestepoverdistributionThemaximallyallowedstepoverconstraintthatmustbesatisfiedeverywhereistreatedasaninequalityconstraintThisresearchalsoprovidesaFEMnumericalrecipetosolvethisconstrainednonlinearoptimizationproblemThismathematicframeworkisextensibleandothermachiningrequirementscanbeaddedbyintroducingnewfunctionalconstraintsExamplesshowthattheoptimalfieldscanbeusedtoconstructsmoothtoolpathwithstablestepoverdistributionsKEYWORDSToolPathGeneration,Functionalminimization,Stepover,FiniteElementMethodINTRODUCTIONThetoolpathgeneration(TPG)approachesdeployedincurrentstateofartComputedAidedManufacturing(CAM)softwarearenotfullyoptimized,duetothelackofacomprehensivemathematicframeworkoftoolpathcomputationAconventionalsolutionistodividethemultipleoptimalgoalsintoseparatemathematicalproblemsThesemethodscanonlyoptimizespecificaspectsofmachining,insteadofprovidinganoveralloptimizationsolutionGenerallyacompletetoolpathcomputationalgorithmadoptsatwostageapproach:thetoolpositionP(cuttercontactpointorcutterlocationpoint)isfirstlycomputedandthenthetoolaxisvectorisdealtwithaccordinglyThetoolpathcomputationcanbetransformedintocomputetwosetsoffieldsoverpathsurfaces:ascalarfieldforpathtrajectoryandavectorfieldfortoolaxisorientationThispaperfocusesonthevectorfieldcomputationforpathtrajectoriesManypreviousworkshaveprovidedsolutionstoobtainsmoothtoolpathtrajectoriesbasedonfieldmethods(Fenkl,Probstetal)issuedapatenttoproducespiraltoolpathforblademillingThepatentusesanimaginaryelectricalpotentialfieldtogenerateauxiliarylinesforconstructingaspiraltwodimensionalguildpathThequalityofthegeneratedtoolpathgreatlydependsontheimaginaryelectrodesarrangedontheinnerandouterboundaries(MakhanovandIvanenko),(ChiouandLee),(KimandSarma),(KimandSarma),and(Kim)reportfieldbasedmethodstoconverttoolpathcomputationintoanoptimizationproblemManyofthemusegreedymethodtofindthefieldsolution(BietermanandSandstrom)and(ChuangandYang)usedpathscalarfieldstorepresentthetoolpathpatterntooptimizethesmoothnessofthetoolpath,theoptimizationisachievedbysolvinganEllipticPDEasaboundaryvalueproblemTheformulationofBietermanandSandstrommethodlacksofmetrictensorandisnotcovariantTheChuangandYangmehtodisindependentofthesurfaceparameterization,butislimitedtoDplanarsurfacesonlyEachofthesepreviouscontributionsfocusesondifferentaspectoftheproblemandhasdifferentlimitation,buttheyallsharethecommonthreadofusing“fields”todescribethetoolpathandestablishingcostfunctionalorPDEtoformulatecertainmachiningrequirementsInthispaper,wepresentaunifiedmathematicalframeworktofullyaddresstherequirementsofsmoothnessandstablestepoverdistributionsThesolutionistoestablishanenergybasedcostfunctionalandagradientfieldbasedfunctionalinequalityconstraints,suchthattheoptimaltoolpathcomputationisconvertedintoafunctionalminimizationproblemwithequalityandinequalityconstraints,andsolvedbyFiniteElementMethod(FEM)Usingthismathematicalframework,wefocustheattentiononthecomputationoftoolpathtrajectoriesthatsatisfytheabovementionedrequirements,anddon'tconsiderthetoolaxiscomputationParalleltothisresearch,wearealsoworkingonasimilarframeworkfortoolaxiscomputationandtheresultswillbepublishedseparatelyThepresentedmethodisapplicabletononEuclidean(freeform)surfacesandcoversbothplanarandnonplanarsurfacemillingInthispaperweonlyconsiderstablestepoverrequirement,butnotconstantstepoverortoolengagementThemainreasonisthatwewanttoincludetherequirementofcontinuoustoolpathwithzeroorminimallifts(retractsandengages)anditgenerallyconflictswithconstantloadAchievingbothminimalliftsandconstantloadcallsforspecialandcomplexalgorithmssuchasTrueMillofSurfCAMsoftware(seeSURFWARE)andtheyareavailableforonlyfixedaxisDmilling,whilewearetryingtoconstructageneralmethodthatisnotlimitedtoDmillingWebelieveotherqualitycriterionssuchastoolengagementcanbefurtherincorporatedintotheframeworkasnewconstraints,butwillleaveitasafuturetaskThispaperisarrangedasfollows:SectiongivesthemathematicdescriptionandpropertiesoftoolpathbasedonfieldtheoryanddifferentialgeometryTheninsectiontheoverallmathematicalsolutionisestablishedbasedoncostfunctionalrepresentingmultiplemachiningrequirementsSectiongivestheproblemstatementinDforbetterunderstandingFEMrecipeisprovidedinSectiontonumericallysolvethefunctionalminimizationproblemandtoolpathgenerationexamplesarepresentedinSectionFIELDBASEDDESCRIPTIONOFTOOLPATHInmathematics,aaxistoolpathcanbedescribedasavectorvaluefunction:FRR,whereRdenotesacombinationoftwoRvectorspacesonefortoolpositionPandanotherfortoolaxisA()(),()TFtPtAt()ThetoolpathisparameterizedbytimetInsurfacemachiningthetoolpathP(t)formsadensepatternofrepetitivetrajectories(passes)thatcoverstheentiresurfaceThecutvector()QtdPdtisthetangentvectorofdimensitonalsurface(,)Xuvwithparameters(,)uv,andthecutvectorfieldQisthetangentvectorspaceof(,)Xuv,shownasFigFigIllustrationofpathtrajectoriesandcutvectorfieldThetrajectoriesofthetoolpathareassumednevercrosseachotherandeachtrajectoryiseitheraclosedlooporintersectstheboundaryofthedomainregioneventimesWecanrepresentsuchwellbehavingtoolpathastheisocurves(curveswithconstantvalue)ofapotentialfunctionHence,insteadofdealingwiththecutvectorfields,wecanworkwithatoolpathpotentialfunction(scalarfield)Wewillusedifferentialgeometrytoensuretheformulationsarecovariant,thatis,theresultsareindependentofthesurfaceparameterizationInaddition,itgeneralizestheformulationtocoverbothplanar(Euclidean)andfreeform(nonEuclidean)surfacecasesIsoVectorGradientVector====FigZigzagtoolpathThemetrictensoronthesurface),(vuXisdefinedas:,,uuGXX()TheinverseofthemetrictensorisdenotedbygThemetrictensorisnothingbutGauss'firstfundamentalformThemetricdeterminantiscomputedasdetGGGGG()Andinversemetrictensorgis:detGgG()WhereandGgIThegradient(derivative)ofthepotentialfunctionisanirrotationalform:()du()Weassumeissmoothanddifferentiable(correspondingtosmoothanddifferentiableisocurves)Wealsorequirethatitsgradientdoesn’tvanishanywhere(toavoidanycrossoveroftheisocurves)Fromthegradientform,wecandefinetheGradientVectorwiththehelpfromtheinversemetrictensor:()gu()Thevalueofthepotentialfunctionisuniquelydeterminedbyitsgradientform(fromintegration),uptoaconstantTheisolinesarethetoolpathtrajectories,orthe‘streamlines”WealsodefinetheIsoVectoras(),whichisthecutvectorofthetoolpath,orStreamlineVector,asshowninFig,TheGradientVector()isthevectoralongthestepoverdirectionThepotentialfunctionrepresentsthe“stepover”countImaginethateachtrajectoryofthetoolpathisassignedanumber,startingfromzeroandincrementbyoneforeachstepover,showninFig(Zigzagpattern)WethenmatchthevalueofwiththestepovervalueLet’sdefineStepoverDensityfunctionSasthemaximumrateofthechangeofstepovervalueStepoverDensitySmeasuresthenumberofstepoversperunitdistance(ie,thedensityoftrajectories,ortheinverseofthestepoverdistanceperunitchangeof)andcanbecomputedfromtheGradientVector:SGguu()AUNIFIEDSOLUTION:FUNCTIONALOPTIMIZATIONWITHCONSTRAINTSVaryingstepoverUnevenstepoverIlldefinedstepoverIdealstepoverFigThreebadandonegoodtoolpathshapeLet’slookintothequalityofthetoolpathTogainaninsightofhowtodefinethequalityofthetoolpathdistribution,FigillustratesseveralexamplesofbadtoolpathformationversusoneidealformationTodiscouragethesebadexamples,thesolutionistominimizethevariationofthestepoverfunction,ie,tostabilize(makeituniformtoavoidpeaksandvalleys)theStepoverDensitySItwillensurethatthestepoverdistanceisevenlydistributedTostabilizeStepoverDensity,weminimizethesquareofit,integratedoverthewholearea:detdetSGdudugGduduuu()IftheaboveStepoverDensityfunctionalisminimized,wewillobtainwellbehavingisolinesasexpectedaboveKeepinmindthatwehaveothermachiningrequirementstosatisfyandtheinclusionofothercostfunctionalterms(tobediscussedlater)willcompeteagainstthistermandattheendtheoveralloptimizedsolutionmaynotbetheabsoluteminimumofthisonetermObjectCostFunctional:Tosummarize,inordertoobtaintheoptimallyshapedtoolpath,weneedtominimizethefollowingStepoverCostFunctional:()detdetFSGdudugGduduuu()AtrivialsolutionthatminimizestheaboveStepoverCostFunctionalisthattheStepoverFunctionremainsaconstanteverywhere(sothatitsderivativesalwaysvanish)Inreality,itmeansthatthestepoverdistanceisinfiniteandcouldneverhappenThatisbecausewehaven’tintroducedtheconstraintthatsetthemaximallyallowedvalueofthestepoverdistanceStepoverConstraintsRecallthatwedefinetheStepoverDensity,whichmeasuresthenumberofstepoversperunitdistanceLet’sassumethatapotentialvalueincrementequallingtocorrespondstoanewtoolpassnexttotheformeroneIfthemaximallyallowedstepovervalueforneighbouringpassesisshthentherequirementisthatsheverywhereHence,theconstraintforstepoveris:()()()sddghduduuuu()Forbetterunderstanding,let’sfocusonapracticalusercaseinaDplanarareaembeddedinEuclideanspaceFigshowsthemachiningareaofabladesectionГistheouterboundaryГistheinnerboundarydenotestheopensetdomainincludingallmachiningareaisthestepovervalue,whichmakesapotentialfieldonthemachiningareaFigThemachiningareaofblademillingprocessInplanarcase,themetrictensormatrixisnothingbutanidentitymatrixAndthegradientvectoroffieldiscomputedas,Txy()TheobjectivecostfunctionalofEq()is:()Fdxdyxy()AtrivialsolutionthatminimizestheaboveStepoverCostFunctionalisthatthestepoverfunctionremainsaconstanteverywhereInreality,itmeansthatthestepoverdistanceisinfiniteandcouldneverhappenThatisbecausewehaven’tintroducedconstraintsintothisoptimizationproblemOneofthemachiningrequirementsismaximallyallowedvalueofthestepoverdistancehs,whichusuallyis~ofthetooldiameterFigshowsanisolineiwithandanisolineiwith,whereisaninfinitesimalincrementalofLetxdenotethecorrespondinginfinitesimalincremental(inphysicalspace)ofstepoverbetweentwoisolines,thegradientofisolineicanbecomputedas:ddxx()Weassumethatthefieldvaluedifferencebetweentwoadjacentpassesis,ie=,thestepoverdifferencexshouldbelessorequaltotheallowedvalueofstepoverdistancehs,thatissxh()Thefollowingfunctionalinequalitymakesthestepoverconstraintsatisfiedsxyh()BOUDARYCONDITIONSANDEXAMPLECASESByusingcalculusofvariation,theabovefunctionalcanbeconvertedintoPDETosolvetheproblemitrequirestoassignthevalueofattheboundaryofthedomain,alsoknownastheboundaryconditioni:ϕi:ϕΔϕΔxFigIsolineswithinfinitesimalincrementalTherearetwochoicesoftheboundaryconditionsTheDirichletconditionrequiresthespecificationofthevalueofthefunction,whiletheNewmannconditionrequiresthespecificationofthederivativeddtalongthedirectionperpendiculartotheboundaryInwhateverboundaryconditions,thevaluesofkeepconstantatboundaries,thenwecanassignoneboundaryfromwherethepathstartswithandtheotherboundarywherethepathendswithn,wherenistheexpectednumberofpassesThewholemachiningareaisfilledwithasequenceofisolinesfromtonWecanusedifferentboundaryconditioncombinationstoobtaindifferentoptimizedtoolpathpatternLet’slookupintothefollowingtwoexamples:POCKETMILLINGORAREAMILLINGWITHISLANDSTheobjectivefunctionalisthesameasEq()Boundaryconditionisasfollows:onconstconWherecstandsforaconston,whichmeansthederivativeddtalongthedirectionperpendiculartoboundaryiszero(Newmanncondition,tisthecurveparameterof)Theresultanttoolpathpatternwillbesmooth,aswellaskeepingthestepoverdistributionasstableaspossibleCONTOURPARALLELOFFSETTOOLPATHGENERATIONTheobjectivefunctionalisthesameasEq()Boundaryconditionis:onInthiscase,boundaryissetfreetoachieveparalleloffsetisolinesdistributionofcomputedfiledsolutionDifferentboundaryconditioncombinationscanproducedifferentpathpatterns,suchascontourmorphing,zigzag,airflow,paralleloffset,etcWewilladdressthesecasesinfuturepapersFINITEELEMENTMETHODInthissectionwepresentanumericalmethodtosolvetheabovementionedconstrainedoptimizationproblemThismethodconformstothecovariancerequirementandhenceisapplicabletobothEuclidean(planar)andnonEuclidean(freeform)cases,althoughweareusingaplanarexamplecaseinthenumericalexperimentTheproposednumericalmethodisbasedonFEMoftriangularmesh,thatis,weuseatriangulationofthedomainDtoapproximatetheproblemLet’sassumeatriangularmapisbuiltoverDandJu,wherenJ,aretheverticesofthetriangularmapWeusethetriplet)(IJKtorepresentatriangleboundedbyKJIuuu,,ThecollectionofallvalidtripletsisandthecollectionofallvalidedgesisTheimageofthetripletKJIuuu,,intheDphysicalspaceisatriangleboundedby)(IIuSS,)(JJuSS,and)(KKuSSThesidesofthetriangleinthephysicalspaceareIJIJSSS,etcuI(,)uJ(,)uK(,)SIJ=SJSISI=S(uI)SJ=S(uJ)SK=S(uK)SIK=SKSIdu=uJuIdu=uKuIIJKFigAtriangleinparameterspacewithlocalcoordinate(left)anditsimageinthephysicalspace(right)Let’sconsideralocalcoordinatesystem(usuallynonorthogonal)definedinsideatriangle)(IJKsuchthatIuischosenastheoriginandthebasisvectorsareIKIJduduuuuu,Themetrictensorofthislocalcoordinateis:IJIKIJIKGGGGSSSS()Theareaoftheparallelogram(doublethesizeofthetriangle)inthephysicalspaceis:)(detIJIKIJIKIJKGASSSS()TheentriesoftheinversemetrictensoratIuare:()()()IKIJKIJIJKIJIKIJKgAgAggASSSS()Thescalarfieldatthevertexis()IIuThederivativesofthevectorfieldinthetriangularapproximationare:JIKIdduddu()PutEqs(),(),(),and()into(),wecanderivethefiniteelementapproximationoftheobjectivefunctionalEq()as:()()()()IJKIKJIIJKIIKIJKIJIIJKFASSSS()ThefiniteelementapproximationofthestepoverconstraintfunctioninEq()isthencomputedas:()()()IKJIIJKIIKIJKIJIsIJKhAIJKSSSS()ThefunctionalminimizationproblemisconvertedintoamultivariablenonlinearfunctionoptimizationproblembyFEMapproximationTherearehandfuloptimizationmethodsthatcanbeusedtosolvesuchproblems,likeSequentialQuadraticProgramming(SQP),PenaltyandAugmentedLagrangianmethods,orGeneticAlgorithms,etcNUMERICALEXAMPLESFigTheisolinescomputationTheproposedalgorithmhasbeenimplementedtoablademillingprocessFigshowsthegenerationprocessofhowtoobtaintheisolinesbyusingfieldandfunctionaloptimizationFig(a)isthetriangulationmeshofmachiningareaanditsboundaryconditionchoiceThereisonouterboundaryandnoninnerboundaryncanbeanyvalueexceptanditwillnotchangetheshapeoftheresultantpotentialfieldsurfaceexcepttheheightof

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A NEW TOOL PATH GENERATION ALGORITHM BASED ON

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