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附錄,文獻(xiàn)原文及翻譯(1)原文Atool-pathcontrolschemefor?ve-axismachinetoolsChih-ChingLo*DepartmentofMechanicalEngineering,FengChiaUniversity,Taichung407,TaiwanReceived16November2000;accepted8May2001AbstractThispaperpresentsanewservocontrolmethodfor?ve-axismachiningapplications.Theproposedmethodconductsadirecteliminationofthedeviationerror,theorientationerror,andthetracking-lagerrorthatarethemainconcernsfor?ve-axistool-pathcontrol.Toachievethispurpose,theproposed?ve-axiscontrolsystemisbasedonareal-timetransformationbetweenthedrive-coordinatebasis,inwhichthe?vedrivesareoperated,andtheworkpiece-coordinatebasis,inwhichthedeviationerroretc.,arede?ned.2001ElsevierScienceLtd.Allrightsreserved.Keywords:Five-axismachinetool;Servocontroller;Tool-pathtrackingcontrol1.IntroductionToachievehighprecisioninmodernCNC(computernumericalcontrol)machiningapplications,designofservocontrolsystemsthatgenerateaccuratecoordinatedmulti-axismotionisofgreatimportance.Tosynchronizethemotionsofthedifferentaxeswhenmachiningacomplexsurface,aconventionalmulti-axisservocontrolsystemconsistsofaninterpolatorandseveralaxialcontrollers.Theinterpolatorgeneratesthedesiredtoolmotionthatisrelativetotheworkpiece,andthen,decomposesthedesiredmotionintothereferencepositioncommandsfortheseparatedrivingaxes[1–4].Thepracticalmotionisrealizedbythedrivingaxes.Eachaxisiscontrolledbyanaxialcontroller,whoseobjectiveistotracktheaxialpositioncommand(i.e.,toeliminatethepositionerroralongeachdrivingaxis).Manyresearchershavedevelopedcontrolalgorithmsthatimprovethetrackingaccuracyforanindividualaxis.Traditionalalgorithmsarebasedonthefeedbackprin-ciple[5,6].Inaddition,feedforwardcontrolalgorithmscanbeimplementedtoaugmentthetrackingperformance.Currently,asigni?cantcontributionhasbeenmadebyTomizuka[7],whoproposedazerophaseerrortrackingcontroller(ZPETC).OnthebasisoftheZPETCmethod,somevariationaloraccessoryalgorithms(e.g.,adaptiveZPETC)havebeenproposed[8,9].Althoughthetrackingperformanceforeachindividualaxiscanbesigni?cantlyimprovedbytheabovemethods,theoverallcontrolperformanceforthemulti-axismachinetoolisnotalwaysguaranteed[6].Atypicalperformanceindexforevaluationofthemulti-axisservocontrolisthecontourerror,whichdenotesthedeviationfromthedesiredtoolpath.Toconductaneffectivereductionofthecontourerror,Koren[10]proposedacross-couplingcontroller(CCC)thatisconstructedbetweenandparalleltotheaxialcontrollers.Atypicalcross-couplingcontrollerconsistsofareal-timecalculationofthecontourerrorandacontrollawtoeliminatethecontourerror.BasedontheconceptoftheCCCmethod,numerouscross-couplingcontrollers(withdifferentcontour-errormodelsand/orcontrollerlaws)havebeenproposed[11–13].Thecontourerror,however,isnottheonlyconcernformulti-axistool-pathtrackingcontrol.Forinstance,thepositionlagalongthetrackingdirectionisanotherconcern[6].Besides,theorientationerror,whichdenotesthedeviationanglebetweenthepracticaltoolaxisandthedesiredtooldirection,isalsoanimportantconcerninfour-or?ve-axismachine-toolcontrol[14].Inthispaper,themainconcernsfora?ve-axistool-pathcontrolarediscussed?rst.Then,aconventional?ve-axiscontrolsystemisdiscussed,anditsdrawbackisaddressed.Finally,a?ve-axiscontrolsystemthatconductsadirecteliminationoftheseconcernsisproposedandcomparedwiththeconventionalone.Fig.1.ThetoolpathalongthesculpturedsurfaceFig.2. Machininginaccuracyduetoimperfecttool-pathtrackingcontrol.2.Mainconcernsin?ve-axistool-pathcontrolLet’sconsiderthefollowing?ve-axismachiningcase(asreferredtoinFig.1):utilizingacylindricaltooltocutasurface.Attheplanningstage,thetoolpaththatcomprisesthetool-centerlocation(L)andthetoolorien-tation(O)isscheduledsothatthecutteredge(S)canpassoverthesculpturedsurface[15,16].Here,welet RandPdenotethereferencepositionvectorandthepracticalpositionvector,respectively.BothRandParepositionvectorswith?vecomponents(threeforthetool-centerlocationL andtwoforthetoolorientation O).Thedifference(orerror)betweenthereferenceandthepracticalpositionvectors(i.e., ER-P)isaconcernin?ve-axismachining.However, Eisnotthemainconcern,becauseasmall Edoesnotnecessarilyguaranteeanegligiblemachininginaccuracy.AsillustratedinFig.2,although P(2)ismuchcloserto RthanP(1)(i.e.,|E(2)| |E(1)|),itresultsinmoremachininginaccuracy.TwomaincausesforpartinaccuracyareillustratedinFig.3.AsshowninFig.3(a),thedeviationerror(ed),whichdenotesthedistancebetweenthepracticaltoollocation(P)andtheclosestlocation(C)onthedesiredtoolpath(ratherthantheinstantaneousreference R),isanimportantconcern.AsshowninFig.3(b),theorien-tationerror( ),whichdenotestheanglebetweenthepracticaltoolaxisandthedesiredtooldirectioncorre-spondingtoC,isanotherimportantconcern.AscanbeseeninFig.3,thedeviationerrorandtheorientationerroraremaincausesformachininginaccuracy.Inadditiontothedeviationerrorandtheorientationerror,atracking-lagFig.3.ThedeviationerrorandtheorientationerrorFig4Twoconsecutivepathsthatthein?ve-axismachiningcontroltoolpathpassesover.(a)deviationerror;(b)orientationerror.error(d)thatdenotesthecomponentof E alongthetrackingdirectionisalsoanimportantconcern.AsshowninFig.4,asigni?canttrackinglagwillalsocauseanunacceptablemachininginaccuracybetweentwoconsecutivesurfaces.3.Conventional?ve-axiscontrolsystemTypically,a?ve-axismachinetoolconsistsofthreetranslationalaxes(x,y,z)andtworotationalaxes(a,b).Theblockdiagramforaconventionalcontrolsystemfor?ve-axismachinetoolsisshowninFig.5.Inthsystemtheinterpolator,whichconsistsofapath-plan-ningmoduleandaninverse-kinematicstransformation[2,14],generatesinrealtimethedesiredreferencepositioncommandstothe?veseparatecontrolloops(respectivelyforthex-,y-,z-,a-,andb-axis).Thepath-planningmodulegeneratesthedesiredtoolmotionrela-tivelytotheworkpiece.Inotherwords,thetoolpathisde?nedintheworkpiece-coordinatebasis(WCB)forwhichtheaxesare?xedontheworkpiece.Inthefollowing,thedesired,practical,anderrorpositionvectorsthatarede?nedintheWCBaredenotedasRw,Pw,andEw,respectively.Incontrasttothepath-planningmodule,theaxialcontrolloopsfocustheireffortontrackingtheindi-vidualmotionsalongthe?vedrivingaxes.Thesemotions,however,arede?nedinthe drive-coordinatebasis(DCB).Inthefollowing,thedesired,practical,anderrorpositionvectorsthatarede?nedintheDCBaredenotedasRd,Pd,andEd,respectively.TotransformthereferencepositionvectorfromWCBtoDCB,aninverse-kinematicstransformationalgorithmisrequiredtobeimplementedintheinterpolator.Notethatinpracticethemechanicalstructureofthe?ve-axismachinetoolplaysaroleasadirect-kinematicstransformationthatconvertsPdto Pw.Let K(·)and K1(·)representthedirect-andinverse-kinematics transformations, respectively. Inotherwords,wehaveRwK(Rd),PwK(Pd),RdK?1(Rw)andPdK?1(Pw).(1)Fig.5. Aconventionalcontrolsystemfor?ve-axismachinetoolsThefunctionofthe?vecontrolloopsistotrackthereferencepositioncommandsthataregeneratedbytheinterpolator.Foreachloop,thecontrollerobjectiveistominimizethepositionerroralongthedrivingaxis.Let[Hd]and[Gd]berespectivelythetransferfunctionmatricesforthecontrollersandthedrives.Inthematrices,[Hd]and[Gd],thenon-diagonaltermsarezerosandthediagonaltermsarethetransferfunctionsfortheaxialcontrollersandtheaxialdrives,respectively.Notethatinacomputer-controlledsystem,[Hd]and[Gd]arefunctionsof z-variable(indiscrete-timedomain).Thestrat-egyoftheconventional?ve-axiscontrolsystemistoreducethepositionerrorsalongthedrivingaxes(i.e.,EdRdPd),andthen,toexpectaquality?ve-axistool-pathcontrolthatfocusesontheeliminationofthedeviationerror,theorientationerror,andthetracking-lagerror.However,theexpectationisindoubt.Ashasbeenillustratedintheabovesection(refertoFig.2),thereductionof Eddoesnotnecessarilycorrespondtothereductionofthedeviationerror,etc.4.Proposed?ve-axiscontrolsystemTheproposed?ve-axiscontrolsystemisdepictedinFig.6.Incontrasttotheconventionalsystemthatconstructs?velocalandseparatecontrolloops(refertoFig.5),theproposedcontrolsystemconstructsaglobalandcoupledlooptoachieveaneffectivecontroloftheover-allperformancethatisintermsofthedeviationerror,theorientationerror,andthetrackinglag.Thedeviationerror,etc.,whichwillbederivedlaterinthefollowing,areerrorcomponentsde?nedintheWCB.Incontrast,thefedbackpositionsignals(Pd)andthecontrolsignals(Ud)senttotheaxialdrivesarebothde?nedintheDCB.Consequently,coordinatetransformationsareintroducedtotheproposedcontrolsystem.AsdepictedinFig.6,theservocontrollerconsistsoffourparts:(1)adirect-kinematicstransformationalgorithmthatcalculatesthepracticaltoolpositioninWCB,i.e., PwK(Pd);(2)anerrormodelforcalculationofthedeviationerrorantheorientationerror(thatarerepresentedbye),andthetracking-lagerror(d);(3)acontrollawthateliminateseandd;(4)aninverse-JacobianmatrixthattransformsthecontrolintheWCB(i.e., Uw)tothatintheDCB(i.e., Ud).Throughtheaboveprocedure,theproposedcontrolsystemfocusesitscontroleffortintheWCBandconductsadirecteliminationofeandd.ThelastthreepartsoftheservocontrolleraredescribedindetailinthefollowingdFig.6. Theproposedcontrolsystemfor?ve-axismachinetools.4.1.ErrormodelAsstatedabove,thedeviationerror,theorientationerror,andthetracking-lagerrorarethemainconcernsfor?ve-axistool-pathtrackingcontrol.Therefore,themodelfortheseconcernederrorsisthecoreoftheproposedservocontroller.AschematicillustrationfortheseconcernederrorsisshowninFig.7.InFig.7,Cwdenotesthepositionthatislocatedonthedesiredtoolpathandistheclosesttothepracticaltoolposition,Pw.LetthedifferencebetweenCwandPwbedenotedase,i.e.,eCwPw. (2)Note that e consists of ?vecomponents, i.e.,e(ex,ey,ez,ea,eb).CwfromPwtothedesiredtoolpath.Fig.7. ThereferencepositionRw,thepracticalpositionPw,andtheclosestpositionAshasbeende?nedabove,thedeviationerroristhedistancebetweenthepracticaltoollocationandtheclosestpointonthedesiredtoolpath.Consequently,the?rstthreecomponentsofeareinpracticethecomponentsofthedeviationerror(ed),i.e.,ed(ex)2+(ey)2+(ez)2. (3)Theorientationerroristheanglebetweenthetoolaxisandthetoolorientationfortheclosestpointonthedesiredtoolpath.Inthispaper,thetworotationalanglesarede?nedsothatthetoolwithrespecttotheWCBisoriginallyinthez-direction,thenrotateswithaalongthe x-axis,and?nallyrotateswith b alongthey-axis.Basedonthede?nitionsofthetworotationalangles(a,b),theorientationerror(f)canbecalculatedbyf cos?1[cos(Pwa)sin(Pwb),sin(Pwa),cos(Pwa)cos(Pwb)]·cos(Pwa+ea)sin(Pwb+eb)(4),?sin(Pwa+ea)cos(Pwa+ea)cos(Pwb+eb)AsshowninEq.theorientationerrorisdeterminedbyeaandeb,whicharethelasttwocomponentsofe.Therefore,intheproposedcontrolsystem,theeliminationofthedeviationerrorandtheorientationerrorcanbeconductedthroughthecontrolofe.Thetracking-lagerror(d)istheprojectionvectorofEw(RwPw)alongthetoolpath.AccordingtoFig.7,wehaved RwCw.(5)Notethatdconsistsof?ve components, i.e.,d(dx,dy,dz,da,db),andthe?rstthreecomponentsofdareforthetracking-lagdistance(dd),i.e.,dd(dx)2+(dy)2+(dz)2(6)SubstitutingEq.(5)intoEq.(2)yieldse(Rwd)PwEwd(7)AscanbeseeninEq.(7),ifdisdetermined,eisalsoobtained.However,ananalyticalsolutionof d isnotavailableforgeneraltrajectories.Anumericaliterativemethodistime-consumingandisnotsuggestedforreal-timecontrol.Therefore,anapproximatedmodelforcalculationofdisrecommendedhere.AsshowninFig.7,Cwisapositionvectorthatislocatedonthedesiredtoolpathandlagsbehindtheinstantaneousreferencepositionvector,Rw.Thedistance(dd)thatCwlagsbehindRwcanbeapproximatedbytheprojectionofthepositionerrorvector(Ewx,Ewy,Ewz)onthetangentialdirectionofthepathonRw,i.e.,ddEwxtxEwytyEwztz,(8)where(tx,ty,tz)isthetangentialvectortothetoolpathonRw.Thetracking-lagerror(d)canberegardedasavariationofthereferencepositionvector Rwaccordingtothetracking-lagdistance,dd.Consequently, d canbeapproximatedbyWhereddiscalculatedbyEq.(8);thevariable l isthepathlengthalongthedesiredtoolpath dRw/dl andd2Rw/dl2arethe?rstandsecondderivativesofthereferencepositionvectorwithrespecttothepathlength(l).InaCNCsystem,dRw/dlandd2Rw/dl2canbeapproxi-matedbywherefisthefeedrate,Tisthesamplingperiod,andkTdenotesthesamplinginstant.4.2.ControllerlawLet[He]and[Hd]bethetransferfunctionmatricesforthecontrollawsforeandd,respectively.Notethatboth[He]and[Hd]are55diagonalmatrices,forwhicheachdiagonaltermrepresentsthecorrespondingcontrollawforeachcomponentofeandd.Withthecontrollaws,thecontrollercommandsareUw[He]·e[Hd]·d (12)4.3.Inverse-JacobianmatrixInthe?nalstep,thecontrolsignalsintheWCBaretransformedtothoseintheDCB,andthen,fedtothe?veaxialdrives.ThistransformationisconductedbymultiplyingthecontrollercommandsbytheinverseoftheJacobianmatrix,i.e.,Ud[J]?1Uw,wheretheinverse-Jacobianmatrixarede?nedas?Pdj?Pwi;i,j x,y,z,a,b(14)5.StabilityconsiderationForsimplicity,weadoptthesamecontrollawfor eand d (i.e.,letting[He][Hd][Hw])inthefollowing.Consequently,basedonEqs.(7)and(12),wecanhaveUw[Hw]Ew[Hw](RwPw). (15)Consequently,theoutputsofthedrivesarePd[Gd][J]?1[Hw]Ew.(16)Inpractice,EdandEwareincrementalpositionvectorsforPdandPw,respectively.Accordingtothede?nitionoftheinverse-JacobianmatrixreferredtoEq.(14),wecanhaveEd[J]?1Ew,(17)andconsequently,getPdRdEdK?1(Rw) [J]?1Ew.(18)ThecombinationofEqs.(16)and(18)yieldsEw([J]?1[Gd][J]?1[Hw])?1·K?1(Rw). (19)Consequently,thecharacteristicequationfortheproposedcontrolsystemis(z) ||[J]?1[Gd][J]?1[Hw]|| 0. (20)Itiswellknownthat,tosatisfythestabilityrequirementforacomputer-controlledsystem,therootsoftheaboveequationmustbelocatedinsidetheunitcircle.BecauseEq.(20)includestheinverse-Jacobianmatrix[J]1,thecon?gurationofthemachinetoolisintroducedtothedesignofthecontrollaw.Thiswillbringadif?cultytothedesignofthecontrollerlaw,becausethemachine-toolcon?gurationnotonlydependsonthemachine-toolstructurebutalsoisvaryingduringthemachiningprocess.However,iftheparametersofthecontrollawsarethesameforthedifferentcomponentsof[Hw](i.e.,[He]and[Hd]),wecanhave[Hw] hw[I],(21)where[I]isa55identitymatrixandhwisthetransferfunctionfortheindividualcontrollaw.BasedonEq.(21),thecharacteristicequationcanbesimpli?edas(z) ||[I] hw[Gd]||(1 hwgi) (22)where giisthetransferfunctionforanaxialdrive.AsillustratedbyEq.(22),ifhwcanseparatelystabilizethe?vedrives,thesystem’sstabilityisassured.Notethattheabovesimpli?edresultistrueonlywhenadoptingthesamecontrollerparametersforthedifferentaxialcomponents.Ifdifferentcontrollerparametersareutilizedforthedifferentaxialcomponents,weshouldcheckthesystem’sstabilitybasedonEq.(20).6.SimulationexamplesAschematicillustrationforthe?ve-axismachinetoolusedinthesimulationisshowninFig.8,wherethecorrespondingdrivingaxesarede?ned.Themachinetoolincludesthreeslidingaxes(forx,yandz)andtwotiltingaxes(foraandb)sothatthetoolis?xedandtheworkpieceisdriven.InFig.8,L(lxx?+lyy?+lzz?)denotesthecutterlocationandisa?xedpointontheFig.8. Themachinetoolstructureusedinthesimulation.machine.Misapointdrivenbythethreeslidingaxesandisthepivotoftherotationalangle a. C isthepivotoftherotationalangle b andistheoriginoftheworkpiececoordinate frame. Besides, we de?ne C Mmxx?+myy?+mzz?.Thecontinuous-timemodel(inLaplace-ors-domain)forthemotor-drivenservomechanismischosenas[5,6]wherekiandti(ix,y,z,a,b)arerespectivelythespeedgainsandthetimeconstants.Precededbyazero-order-hold(ZOH),thedrive’sdigitaltransferfunctionisrepresentedbyInthesimulation,atypicalPIDcontrollawisutilizedforboththeconventionalmethod(forcontrolofEd)andtheproposedcontrolmethod(forcontrolofeandd).ThetransferfunctionofthePIDcontrollerisdescribedbywhere hp, hi,and hdareproportional,integral,andderivativegains,respectively.Notethatinordertogetafaircomparison,thesamesetofPIDgainsarechosenforthetwodifferentmethods.ThemachinegeometryandthesystemparametersusedinthesimulationarelistedinTable1.Notethatweintroducesomedifferences(ormismatches)betweenthedrivedynamicsofthedifferentaxes(5%forthespeedgainsand20%forthetimeconstants).ThePIDgainsarechosensothatthestabilityrequirementissatis?ed.Fortheconventionalcontrolmethod,theinverse-kin-ematicsalgorithmthattransforms Rwto RdcanbedescribedbyRdx?Rwxcos(Rwb)?Rwzsin(Rwb)(27)Rdymzsin(Rwa)?Rwxsin(Rwa)sin(Rwb)?Rwycos(Rwa)+Rwzsin(Rwa)cos(Rw)Rdzlz?mzcos(Rwa)?Rwxcos(Rwa)sin(Rwb)?Rwysin(Rwa)?Rwzcos(Rwa)cos(Rwb)RdaRwaRdbRwbFortheproposedcontrolmethod,thedirect-kinemat-icsalgorithmthattransforms Pdto PwcanbedescribedbyPwxmzsin(Pdb)?Pdxcos(Pdb)?Pdysin(Pda)sin(Pdb)?(lz?Pdz)cos(Pda)sin(Pdb)Pwy?Pdycos(Pda)+(lz?Pdz)sin(Pda)Pwx?mzcos(Pdb)?Pdxsin(Pdb)+Pdysin(Pda)cos(Pdb)?(lz?Pdz)cos(Pda)cos(Pdb)PwaPdaPwbPdb(28)BasedonEqs.(14)and(27),theinverse-JacobianmatrixthattransformsUwtoUdcanbeformulatedasInthe?rstsimulationexample,acylindricalsurfaceiscutbyacylindricaltool.Thecylindricalsurfaceisexpressedas(x+y)2+2z2800.AsillustratedinFig.9,Sisthecutter–contactlocationthatfollowsacircularpathonthesurface,whileListhecutter–centerlocationthatisobtainedthroughcutter–offsettingFig.[15,16].Toachievepivotoftherotationalangle a. C isthepivotoftherotationalangle b andistheoriginoftheworkpiececoordinate frame. Fig.9. Five-axismachiningofacylindricalsurfaceBesides,we de?ne C Mmxx?+myy?+mzz?.Thecontinuous-timemodel(inLaplace-ors-domain)forthemotor-drivenservomechanismischosenas[5,6]wherekiandti(ix,y,z,a,b)arerespectivelythespeedgainsandthetimeconstants.Precededbyazero-order-hold(ZOH),thedrive’sdigitaltransferfunctionisrepresentedbyInthesimulation,atypicalPIDcontrollawisutilizedforboththeconventionalmethod(forcontrolofEd)andtheproposedcontrolmethod(forcontrolofeandd).ThetransferfunctionofthePIDcontrollerisdescribedbywhere hp, hi,and hdareproportional,integral,andderivativegains,respectively.Notethatinordertogetafaircomparison,thesamesetofPIDgainsarechosenforthetwodifferentmethods.ThemachinegeometryandthesystemparametersusedinthesimulationarelistedinTable1.Notethatweintroducesomedifferences(ormismatches)betweenthedrivedynamicsofthedifferentaxes(5%forthespeedgainsand20%forthetimeconstants).ThePIDgainsarechosensothatthestabilityrequirementissatis-?ed.Fortheconventionalcontrolmethod,theinverse-kin-ematicsalgorithmthattransforms。Rdx?Rwxcos(Rwb)?Rwzsin(Rwb)Rdymzsin(Rwa)?Rwxsin(Rwa)sin(Rwb)?Rwycos(Rwa)+Rwzsin(Rwa)cos(Rwb)(27)Rdzlz?mzcos(Rwa)?Rwxcos(Rwa)sin(Rwb)?Rwysin(Rwa)?Rwzcos(Rwa)cos(Rwb)RdaRwaRdbRwbFortheproposedcontrolmethod,thedirect-kinematicsalgorithmthattransformsPdtoPwcanbedescribedbyPwxmzsin(Pdb)?Pdxcos(Pdb)?Pdysin(Pda)sin(Pdb)?(lz?Pdz)cos(Pda)sin(Pdb)Pwy?Pdycos(Pda)+(lz?Pdz)sin(Pda)Pwx?mzcos(Pdb)?Pdxsin(Pdb)+Pdysin(Pda)cos(Pdb)?(lz?Pdz)cos(Pda)cos(Pdb)PwaPdaPwbPdb(28)BasedonEqs.(14)and(27),theinverse-JacobianmatrixthattransformsUwtoUdcanbeformulatedasInthe?rstsimulationexample,acylindricalsurfaceiscutbyacylindricaltool.Thecylindricalsurfaceisexpressedas(x+y)2+2z2800.AsillustratedinFig.9,Sisthecutter–contactlocationthatfollowsacircularpathonthesurface,whileListhecutter–centerlocationthatisobtainedthroughcutter–offsetting[15,16].Toachievehighef?cientmachiningofconvexsurface[17],wemaysettheinclinationandthetwistinganglesabout S aszero. Consequently, thetool-orientation vector,OO(a,b),isequaltothenormalvectortothesurfaceonS.Basedontheaboveassignment,aspeci?ctoolpath(Rw)isscheduledasRwz20sin(q)+rtcos(q))Rwa?sin?1cos(q)(31)Rwbtan?12tan(q)Wherert(5mm)isthetoolradiusand q isthepathparameterthatstartsfrom90to0(inCW).The?ve-axismachiningisconductedatafeedrateof450mm/min.Inthesecondexample,aconicalsurfaceiscut.Theconicalsurfaceisexpressedasx2+y2(45 2z)2.Asillus-tratedinFig.10,thecutter–contactFig.10. Five-axismachiningofaconicalsurfacelocation(S)followsacircularpathonthesurfaceandthetoolisneitherinclinednortwistedabout S.Consequently,aspeci?ctoolpath(Rw)isscheduledasRwx15cos(u)+rtsin(u)(32)where rt(5mm)isthetoolradiusanduisthepara-meteralongthetoolpath.Fig.11. Five-axismachiningofaruledsurface.Inthesimulation,ustartsfrom0to2p(inCCW).The?ve-axismachiningisconductedatafeedrateof600m/min.Inthethirdexample,aruledsurfaceiscut.Theruledsurfaceisexpressedax 2(20u3?30u2+30u+5)+20vy 2(20u3?30u2+30u+5)?20v(33)whereuandvarethesurfaceparameters.AsillustratedinFig.11,thecutter–contactlocation(S)followsacubicsplinepath(intheu-direction).Inordertoavoidthereargaugingattheconcavepart,thecutterisinclinedat13.Notethatananalyticalsolutionforthetoolpath(suchasEqs.(31)and(32)fortheprevioustwoexamples)isnotavailable.However,wecanobtainthenumericalsolutionthroughthetooloffsetting.Inthesimulationexample,aspeci?ctoolpathisscheduledalongv0.Theproposedtool-pathcontrolmethodisevaluatedandcomparedwiththeconventionalmethod.Thecomparisonisbasedontheircapabilityineliminationofthedeviationerror(ed),theorientationerror(f),andthetracking-lagerror(dd).ThesimulationresultsforthethreeexamplesareshowninFigs.12–14,andaresum-marizedinTable1.Basedonthesimulationresults,wecanhavethefollowingobservations.1.Thedeviationerror(ed)issigni?cantlyreducedbytheproposedmethod(forboththemaximumandthemeanvalues).2.Theorientationerror(f)issigni?cantlyreducedbytheproposedmethodforthe?rstexample.Forthesecondandthethirdexamples,onlyalittleimprovementcanbeachieved.AsillustratedinFig.3(a),thetoolorientationerrorcancauseasurfaceerror(overcutorundercut)nearthecuttercontactlocation(S).Geometrically,thiserrorcanbeapproximatedbyrtf where rtisthetoolradius.Fortheabovethreeexamples,themaximumorientationerrorislessthan0.06deg(0.001rad).Accordingly,thesurfaceerrorduetothetoolorientationerrorislessthan0.005mm.Ascomparedwiththetooldeviationerror(ed),thisisnotimportant.3.Thetracking-lagerror(dd)isnotsigni?cantlyreducedbytheproposedmethod(especiallyatthetransientstate).However,ashasbeenillustratedinFig.4,thetracking-lagerrordoesnotcausemachiningerrorduringthetracking.Itplaysanimportantroleonlyattheendofthepathoronthecorneroftwoconsecutivepaths.AscanbeseeninFigs.12(c),13(c)and14(c),theproposedmethodcandrivethetracking-lagerrortoaverysmallvalueatthesteadystate.Thismeansthatthemachiningerrorattheendofthepath(orthecorner)canbewellcontrolledbytheproposedmethod(ascomparedwiththeconventionalmethod).However,wemaynotreallywantddtogotozero,especiallywhenasigni?cantovershooterroratthecornerisnotallowed.Toavoidasigni?cantovershootatthecorner,wecanutilizePDlaw(byremovingtheintegralpart)tocontroldd.However,PDcontrolmayresultinalargedd,andconsequentlycauseanunder-cutatthecorner(pleaserefertoFig.4).Therefore,whenchoosingthePIDgainsfor dd,thereisatrade-offbetweentheovercutandtheundercutatthecorner.Fig.12. SimulationresultsformachiningofFig.13. Simulationresultsformachiningacylindricalsurface;solidline:conventionalofaconicalsurface;solidline:conventionalmethod,dashedline:proposedmethod.method,dashedline:proposedmethod.表14.Inordertoobtainafaircomparisonbetweentheconventionalandtheproposedmethods,weutilizethesamePIDgains.Inpractice,toaugmenttheFig.14. Simulationresultsformachiningofarulesurface;solidline:conventionalmethod,dashedline:proposedmethodperformanceoftheproposedmethod,wemayutilizedifferentPIDgains(orevendifferentcontrollaws)forcontrollinged,fanddd.Inotherwords,[He]isdifferentto[Hd].Forinstance,wecanutilizehigh-gainPIDcontrolfor[He](becauseourpurposeistoeliminateedand f),andutilizemedium-gainPDcontrolfor[Hd](sothatddisneithertoolargenortoosmall).7.ConclusionThemainconcernsfor?ve-axistool-pathtrackingcontrolarethedeviationerror,theorientationerror,andthetracking-lagerror.Consequently,theconven

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