緊密輪廓控制器的設(shè)計(jì)

BurakSencerYusufAltintasPh.D.CandidateModelingandControlofContouringErrorsforFive-AxisMachineTools—PartII:PrecisionContourControllerDesignTheaccuratetrackingoftool-pathson?ve-axisCNCmachinetoolsisessentialinachiev-inghighspeedmachiningofdies,molds,andaerospacepartswithsculpturedsurfaces.BecausetraditionalCNCscontrolthetrackingerrorsofindividualdrivesofthemachine,thismaynotleadtodesiredcontouringaccuracyalongtool-paths,whichrequirecoor-dinatedactionofallthe?vedrives.Thispaperproposesanewcontrolapproachwherethetooltipandtoolorientationerrors,i.e.,thecontouringerrors,areminimizedalongthe?ve-axistool-paths.Thecontouringerrorandkinematicmodelofthemachine,whicharepresentedinPartIofthepaper,areusedinde?ningtheplant.Amulti-input–multi-outputslidingmodecontroller,whichtriestominimizepathtrackingandpathfollowingvelocityerrors,isintroduced.Thestabilityofthesystemisensured,andtheproposedmodelisexperimentallydemonstratedona?ve-axismachinetool.Thepatherrorsorigi-natingfromthedynamicsof?vesimultaneouslyactivedrivesaresigni?cantlyreduced.?DOI:10.1115/1.3123336?ProfessorFellowASMEManufacturingAutomationLaboratory,TheUniversityofBritishColumbia,Vancouver,BC,V6T1Z4,Canada1IntroductionContouringerrors,de?nedasthenormaldeviationfromthedesiredreferencetool-path,occurinmulti-axismotioncontrolsystemsduetotrackingerrorsofindividualservodrives.InPartIofthepaper,twotypesofcontourerrors,whicharedetrimentaltotheparttolerancesduringsimultaneous?ve-axismachining,arede?ned.The?rstoneisthenormaldeviationofthetooltipfromthedesiredtool-path,calledthetooltipcontourerror.Consider-ingthekinematicsofthe?ve-axismachinetools,tooltipcontourerrorsariseasanonlinearfunctionofthetrackingerrorsofalltheaxis.Orientationcontourerror,ontheotherhand,isde?nedasdeviationfromthedesiredtoolorientationinsphericalcoordinatesandcontrolledonlybytherotarydrivesofa?ve-axismachinetool.ModelsthataccuratelyestimatebothofthosecontourerrorsinrealtimearepresentedinPartI.Thedesignofasimultaneousmulti-axisslidingmodecontroller?SMC?forminimizingcontourerrorsisintroduced.Twomajorapproacheshavebeenadoptedtoreducecontouringerrors.Inthe?rstapproach,contourerrorsarereducedindirectly,byattemptingtoreduceaxistrackingerrors.Traditionalalgo-rithmssuchasP,PI,andPIDarebasedonthefeedbackprinciple?1,2?.Tomizuka?3?developedazerophaseerrortrackingcontrol-ler?ZPETC?bycancelingthestabledynamicsoftheservodriveinafeed-forwardfashion.Thebandwidthoftheoverallsystem,hencethetrackingaccuracyofthedrive,increaseswithZPETC,providedthatthedrivemodelisaccurateanddoesnotvarywithtime?4,5?.Recenteffortsaredirectedtowardimprovingtheband-widthofthedrivesusingslidingmodecontrollersthataremorerobusttothechangesinthedrivedynamics.ThegeneralSMCwas?rstintroducedbyUtkin?6?,whichrequiredswitchingaroundtheslidingsurfaceresultinginadiscontinuouscontrollaw.Toovercomethisproblem,SlotineandLi?7?proposedanadap-tiveslidingmodecontroller,whichestimatesandcancelsvariousuncertaintiesthatdonotvanishattheequilibriumpoint.Inparal-lel,Stepanenkoetal.?8?notedthatthetransientresponseoftheContributedbytheManufacturingEngineeringDivisionofASMEforpublicationintheJOURNALOFMANUFACTURINGSCIENCEANDENGINEERING.ManuscriptreceivedJuly14,2008;?nalmanuscriptreceivedFebruary27,2009;publishedonlineMay1,2009.ReviewconductedbyEricR.Marsh.discontinuousSMCscanbeimprovedbyincludingtheintegral,position,andderivativeofthetrackingerrorsintheslidingsur-facedesign.Later,Altintasetal.?9?proposedacontinuousslidingmodecontrollerforaccuratetrackingcontrolofaxisbyconsider-ingonlytherigidbodydynamicsoftheballscrewdrive.TheirSMCdesignwasbasedonpanelizingposition,aswellasthevelocityerrorsoftheaxes,anddemonstratedsimilarperformancetotheZPETCbutwithimprovedrobustnesstouncertaintiesinthedrivedynamics.Animprovedmethodofcontrollingthemachinetoolfeeddrivesystemistointroducecouplingactionsinthecontrollertomain-taincoordinationalongthedesiredcontour.Byintroducingcou-plingeffectsamongmultipleaxes,coordinatedmotionisachievedbyeitherthe“equalstatus”orthe“master-slaveapproach”?10?.Whenthedynamicsaresigni?cantlydifferentamongmultipleaxis,thecontrollerdesignedbytheequalstatusapproachmaysaturatethesloweraxisactuator.Inordertoovercomethisdraw-back,themaster-slaveapproachisfavored,whichassignstheslowaxisasthemasterofthefasterdrive.Suetal.?11?developedanadaptivecoordinationcontrollerforpositionsynchronizationofmultipleaxis.Theyde?nedthesynchronizationerrorasthedif-ferentialpositionerroramongmultiplemotionaxesandpenalizeditinthefeedbackloop?12?.Thesecondmajorapproachinreducingcontourerrorsistoattempttoestimatecontourerrorsinrealtimeandgeneratecon-trolactionagainstit.KorenandLo?13?estimatedthecontouringerrorintwoaxismachinesasafunctionofaxistrackingerrorsandlinearfeeddirectionbetweenconsecutivepathpoints.Theyreducedthecontouringerrorbyinjectingcorrespondingcouplingcommandtotheservosinordertopushtheactualtoolpositiononthedesiredtool-path,knownasthecrosscoupledcontroller?CCC?.Later,KorenandLo?14?usedtimevaryingcouplinggainstoimplementCCCalongcircularpaths.Whentheaxiscon-trollersminimizethetrackingerrorsalongacurvedpath,thecon-tourerrorisincreased,whichwillforcethecontourcontrollertoresistit.Asaresult,itisdif?culttodistinguishwhichcontrolelementdominatesthe?nalcontouringresult.Theanalysishin-deredimplementationoftheCCCschemeinnonorthogonalma-chinetools.ErkorkmazandAltintas?15?developedanumericalmethodtoestimatethecontourerrorforarbitrarilyshapedtool-JUNE2009,Vol.131/031007-1JournalofManufacturingScienceandEngineeringCopyright?2009byASMEDownloaded17Nov2009to06.RedistributionsubjecttoASMElicenseorcopyright;see/terms/Terms_Use.cfmTable1Driveparametersx-axis?Vs2/mm?m=J/KaKtrgc=B/KaKtrg0.001620.00681y-axis?Vs2/mm?0.001740.00863z-axis?Vs2/mm?0.002960.01518a-axis?Vs2/rad?0.006820.01964c-axis?Vs2/rad?0.000540.00368paths.TheyimplementeditinaCCCschemetogetherwithfeed-forwardaxisdynamicscompensationanddemonstratedimprovedcontouringperformanceforCartesianmachining.ChiuandTomi-zuka?16?usedthecoordinatetransformationapproachtodirectlydesignforthedesiredcontourerrordynamics.UsinglineartimevaryingPDregulators,thedecouplederrordynamicsintangentialandnormaldirectionsarestabilized.Thisapproachdemonstratedthatthecontouringperformanceisimprovedbyincreasingtheclosedloopbandwidthinthenormaldirection.However,thePDcontrollerisnotsuf?cientlyrobustenough,andthecontourerrorapproximationmaybecomeinaccurate.Therefore,thecontouringaccuracydegradesinspeedmachiningofpathswithsharpcurva-tures.PengandChen?17?de?nedageometriccontouringindex?CI?thatimprovestheerrorestimationoncirculartool-pathsanddesignedabacksteppingslidingmodecontrollerinthenormaldirectiontointroducerobustnessagainstfriction.Chenetal.?18?laterdesignedrobustcontrollersinpolarcoordinatestoestablishcontouringcontrolforverysimplenoncirculartool-paths.Acompletelydifferentphilosophyisintroducedinthispaper.Ratherthentrackingofindividualdrives,thetool-pathfollowingtheaccuracyof?ve-axismachinetools,i.e.,theminimizationofcontouringerror,isconsideredastheprimeobjectiveofthecon-trollaw.ByutilizingthekinematicandcontourerrorestimationmodelspresentedinPartI,amulti-input–multi-output?MIMO?continuoustimeintegralslidingmodecontourcontrollerisintro-duced,whichismorerobustagainstdisturbancesandmodelingerrors.Theeffectivenessofthecontrolstrategyisdemonstratedexperimentallyonthein-housecontrolled?ve-axisCNCmachinetool.Thispaperisorganizedasfollows:Section2describesthede-signofslidingmodecontrollersforminimizingthetooltipaswellasthetoolorientationcontourerrors.Section3investigatestheeffectivenessofthecontrolalgorithmsthroughcontouringtests.2.1DesignofSlidingModeControllerfortheToolTipContourErrors.InPartI,theactualtooltipcontourerrorvector???ismodeledbyre?ectingthetooltrackingtiperrors?e?onthedelaycompensatedFrenetframe??F?asfollows:?eF=?FTe+?hF˙˙˙F=?˙+??eFTeFTe+?hF˙˙?¨¨?¨F=?¨+2F?eFTehFe+FTe+???3?Thecontourerrorsleftonthepartsurfaceareestimatedfromthe?Х?eF,n+?eF,b?oftheerrorsnormalandbinormalcomponents???Fig.1?,whichareminimizedbytheslidingmodecontroller.ThecontourerrordynamicsarederivedbysubstitutingEq.?2?intoEq.?3?:˙T˙?¨¨˙q?¨F=?¨b=?¨ref?J˙?JM?1?u+d?Cq˙??+2F?FT?xe+FTe+?hFee¨a?e¨c?e??¨t?e¨n?e?4?2ContouringControlThesimpli?edlineardynamicsofatypicalfeeddrivesystemona?ve-axismachinetoolwasmodeledinPartIas˙?t??¨?t?=M?1?u?t??d?t??Cqq?1?whereq?t?=?x?t?,y?t?,z?t?,?a?t?,?c?t??Tcontainsthedriveposi-˙?t?andq¨?t?containthedrivevelocitiesandaccelerations,tions.qrespectively.u?t?isthecontrolinputtotheampli?ersandd?t?istheexternaldisturbancere?ectedattheampli?er’sinput.M?R5?5andC?R5?5arethediagonalmatricesthatcontaindrives’equivalentinertiaandviscousdampingtermsgiveninTable1.TheJacobianmatrix,J?t??R5?5,relatesdrivevelocities˙?t?,P˙?t?,P˙?t?,˙?t??tothetoolposevelocity?x˙?t?=?P?qxyzT˙˙a?t?,c?t???andisobtainedfromthekinematicsofthe?ve-axismachine?19?.ThedrivedynamicsaremappedtothetoolposeusingtheJacobian,andthetrackingerrordynamicsofthetoolintheworkpiececoordinatesystemareexpressedby˙?t?q¨?t?=x¨ref?t??J˙?t??J?t?M?1?u?t??d?t??Cq˙?t??e?2?Notethatthedynamicsofthe?rstthreecomponents,namely,theen,?eb?errors,arenonlinearandtangent,normal,andbinormal??et,?timevaryingduetotheJacobianandFrenetframetransforma-tions.Thelasttwocomponentsareregulartrackingerrordynam-icsoftherotarydrives.Asecondorderslidingsurface?S?R5?1?isdesignedtocontaintheproportional,integral,andde-rivativeofthetimevaryingerrorsaccordingtoa)DesiredContourbFtnb)zzeFh,Petzreefε,Py,Pz,PActualToolTipLocationεFPxrefyrefPx¨ref?t?isthereferencetoolposeacceleration.BasedonthewherexcontourerroranddrivedynamicsmodelspresentedinPartI?19?,aslidingmodecontrollerthatminimizesboththetooltipandorientationcontourerrorsisintroducedinSec.2.1.031007-2/Vol.131,JUNE2009en+ebxyen+ebActualToolTipLocationyWorkpieceCoordinateSystemxFig.1TooltipcontourerrorestimationTransactionsoftheASMEDownloaded17Nov2009to06.RedistributionsubjecttoASMElicenseorcopyright;see/terms/Terms_Use.cfmS=SbSaScwhereCP?R5?5,CI?R5?5,andCD?R5?5arethediagonaldesignmatricesthatrepresentthedesiredachievabledynamicsoftheerrorsontheslidingsurface.Thecontrolinput?u?R5?1?mustbemanipulatedinsuchawaythattheerrorsandthetimederivativeconvergeasymptoticallytothestableslidingsurface,so˙F→0.ethattheyeventuallyslidetooriginS→0as?eF→0and?TheerrordynamicsontheslidingsurfacecanberepresentedinLaplacedomainas˙=?Cs+Cs+C??SDPIeF?s?=02??StSn=CP?eF+CI?˙=STM?C???T?JT?˙F+CD?¨???d?dVeeFSIeF+CP?t?12?˙F?eFd?+CD?e?5?andEq.?4?isusedtoexpandthederivativeoftheLyapunovfunction,˙q˙=STM?C?˙F+FTx¨ref?FTJ˙?FTJM?1u+FTJM?1Cq˙eVIeF+CP?˙T˙?¨¨???e+FTe+?h?+STFTJd?STFTJ?d?d+2F?13????1????0.Asaresult,thefollow-Eq.?9?becomesSTFTJ?d?d˙?t?ingcriterionissuf?cienttoensuretheasymptoticstability?V?0?ofEq.?13?:˙=STM?C?˙q˙F+FTx¨ref?FTJ˙?FTJM?1u+FTJM?1Cq˙VeIe+CP?˙T˙?¨¨?=?STKS?+2Fe+FTe+?h?+STFTJds?14???inEq.?13?canberewrittenasNotethatST?FTJd?STFTJ?d?d???1???andduetothedisturbancelimitsimposedinSTFTJ?d?d?6?andstablesecondordererrordynamicscanbedesignedbychoos-ingconstantdesignmatricesCD=I5?5CP=diag?2?t?n,t,2?n?n,n,2?b?n,b,2?a?n,a,2?c?n,c?22222CI=diag??n,t,?n,n,?n,b,?n,a,?n,c?whereKs=diag?Ks,t,Ks,n,...,Ks,a?isapositivede?nitivediago-nalfeedbackgainmatrixtopushindividualerrorcomponentsontotheslidingsurface.Hence,thecontrollawisobtainedfromEq.?14?as?7?where?iand?n,i,i=t,n,b,a,caretheindividualdampingandnaturalfrequenciesforeachtrackingerrorcomponent.Thevaria-tionintheexternaldisturbancecausedbythecuttingprocessandfrictionisconsideredtobeconstantandremainsbetweentheup-per?d+?R5?1?andlower?d??R5?1?limitsduringshorttimeintervals?9?.Theexternaldisturbancesforcecontouringerrorstodeviatefromtheslidingsurface,buttheycanbetrackedusingthefollowingobserverthatintegratestheslidingsurface:˙??k?=d??k?1?+T??JT??FS→dFSd=??JT??8?u=ubuauc¨?˙+d+?hF+M?1KsS?+Cq??utun˙T˙?¨˙q?˙F+FTx¨ref?FTJ˙+2F=J?1FM?CI?eF+CP?ee+FTe?15?whereTisthecontrolsamplingperiod,?=diag??t,?n,...,?c?istheobservergainmatrix,JTisthetransposeoftheJacobian,and?=diag??t,?n,...,?c?isamatrixwithpositiveentriesusedtoimposelimitsontheintegralcontrolactionagainstthedistur-bancessothattheobservationsarewithinthegivenbounds?d???d+?:?d??isevaluatedfromEq.?8?.wherethedisturbance?dTheasymptoticstabilityofthetooltipcontourcontroller?Eq.?15??isguaranteedasfollows:TheLyapunovfunction?V?postu-˙?isforcedlatedinEq.?10?islowerbounded,anditsderivative?VtodecreasebytheconditionimposedinEq.?14?.Asaresult,the˙F?,andeFd?,?eF,?eslidingsurface?S?,thecontourerrorstates??t0???areallbounded.theestimateddisturbancesactingonthetool?dThestableslidingsurface?S?designedinEq.?15?issubstitutedintothecontourerrordynamics?Eq.?4??toobtainthefollowing:??˙+MKS=FTJM-1?d-dSs?16??=01???d?ifd??d+ifdotherwiseandandS?0S?0?9?Theboundeddisturbanceobserveravoidschatteringofthecon-trollerontheslidingsurface.Inordertopushthetrackingerrorsontotheslidingsurface,thefollowingLyapunovvectorfunctionispostulatedas˙isboundedsince?d-d??isbounded,Equation?16?provesthatS¨FiseandJ,FTarebothboundedbythereferencetrajectory.?boundedfromEqs.?6?and?16?,thesecondderivativeofLyapunovfunction?Eq.?14??,¨=?C?VPeF+CIV=VbVaVcThisfunctionpanelizesthedeviationfromthesurfaceandthesquareofthedisturbanceestimationerror.ThederivativeoftheLyapunovfunction˙˙=STMS˙??d?d??T??1d?V?11???VtVn??t˙F?CP?˙F+CI?¨F??eFd?+CD?eeeF+CD?e?17??1??T??1?d?d???=?STMS+?d?d2?10?mustbenegativeforasymptoticstability,whichalsopushesthestatestowardtheslidingsurfacewheretheyfollowthedesiredsecondorderdynamics.Equations?6?and?8?aresubstitutedintoEq.?11?toyieldJournalofManufacturingScienceandEngineeringisalsobounded.Toconclude,Vispositivede?nitelower˙ispositivede?nitedecreasing,andV¨isboundedandbounded,V˙isauniformlycontinuousfunction.Hence,thisprovesthatVBarbalat’slemma?20?dictatesthatV→0,S→0ast→?,andallthecontourerrorsontheslidingsurfaceconvergetoorigin,?eF˙F→0.Consideringthesimpli?edhomogenoustransfor-→0,and?emation??eF=?FTe?betweenthecontourerrors??eF?andtrackingerrors?e?fromEq.?3?,ityieldsthatbothcontourandthedrivetrackingerrorsconvergeasymptoticallytotheorigin:e→0as?eF→0.TheoverallcontrollerstructureisillustratedinFig.2.Unlikeinsingleinputsingleoutput?SISO?controlstructures,theinterpola-tordoesnotusetheinversekinematicstogeneratereferencecom-JUNE2009,Vol.131/031007-3Downloaded17Nov2009to06.RedistributionsubjecttoASMElicenseorcopyright;see/terms/Terms_Use.cfmFig.2Tooltipslidingmodecontourcontrollerblockdiagrammandstothedrivesintheproposedcontouringcontroller.Instead,theinterpolatorgeneratesthereferencetooltippositioncommandintheP?workpiececoordinates?systemandtherotarydrivepo-sitionsintheM?machinecoordinates?system.Theproposedcon-trollerusestheposetrackingerrorsandutilizesFrenetframetransformationstocomputethecontourerrordynamics.Thecon-trolactionisthensenttothe?vephysicaldrivesofthemachinetool.AsexplainedinPartI,duetothekinematicsofthemachinetool,trackingerrorsofall?veaxisin?uencethetooltippositionintheP-system.However,thekinematicsofthemachinetoolallowscompensationforthetooltippositioningerrorsbyutilizingonlytheCartesianaxisofthemachinetool.ThetoolpositioninP-systemisusedinthecontourerrorestimationandCartesianaxesarecoupledtogether.Sincethetooltipcontourerrorsthataredirectlyin?uencedbytheerrorsinnormalandbinormaldirections?en,eb?areemphasized,higherbandwidths??n,n,?n,b?andfeed-backgainscanbeusedtoachievemoreaccuratecontouringper-formance.Dynamicsinthetangentdirectionaresetslowersothatthedrivesarenotsaturated.Incontrast,therotarydrivesarenotcoupledduringcomputationofthecontrolaction,ratherregulatedindividually.Hence,theaboveslidingmodecontrollerensuresthestabilityofthecompletemachinetoolduringsimultaneous?ve-axismachining.2.2DesignofSlidingModeControllerfortheToolOrien-tationContourErrors.Theorientationcontourerrorisde?nedasthenormaldeviationfromtheorientationtrajectoryinPartI?19?,whichisbrie?ysummarizedheretoderivethecontrollaw.Theorientationerrorsde?nedinsphericalcoordinates?eo=?eo,i,eo,j,eo,k?T?arere?ectedinthenormaldirectiontothede-siredtool-path,andthecontourerrorsareestimatedas?ea,ec?.ByconsideringonlythemotionoftherotarydrivesfromEq.?1?,thedecoupledtrackingerrordynamicsforrotarydrivescanbeexpressedas¨R=e??¨ae¨ce¨ref?M?1?u?d?Cq˙?=x?21?whereM=diag?ma,mc?andC=diag?ca,cc?containdriveinertiasanddampingvalues.Trackingerrorsoftherotarydrivesarethenrede?nedastheweightedsumoftheregularrotarydrivetrackingerrors?eR=?ea,ec??andtheintegralofthecontourerrorcompo-nents??R=??a,?c??fromEq.?18?as?22?whereW=diag?wa,wc?isthepositivede?nitivediagonalweight-ingmatrix.Notethatwhentheweights?wa,wc?aresettozero,thenewerrorstate??e?isbasedonlyonthetrackingerrorsofrotarydrives.Astheweightsareincreased,theeffectsofcontourerrorsareraisedin?e.Consequently,minimizingthenewerrorstatewithnonzeroweightwillintroducecouplingandsynchronizationbe-tweentherotarydrivestoimprovethecontouringperformance.SimilartotheslidingmodecontrollerdesignprocedurepresentedinSec.2.1,aslidingsurface?S?R2?1?thatspinsoverthepro-posederrorstatesisselectedas˙=0S=??e+?e?23??o=eo?eo·??18?where=?/???isthenormalizedangularvelocityofthetoolaxis.UsingtheorientationJacobian?JO?thetoolorientationtrackingerror?eo?isevaluatedfromrotarydrivetrackingerrors?eR=?ea,ec?T?aseo?JOeR?19?TherelationshipbetweenthetoolorientationcontourerrorvectorintheP-system??o?andthecorrespondingrotarydriveerrors??R=??a,?c??intheM-systemwasobtainedaswhere?=diag??a,?c?isthedesiredbandwidthoftheerrorsontheslidingsurface.Thecontrolinput?u?R2?1?ismanipulatedinsuchawaythattheerrorsandderivativesconvergeasymptoticallytothestableslidingsurfacewheretheyeventuallyslidetotheorigin.Thedisturbances?d?R2?1?areestimatedfromthefol-lowingobserver:˙??k?=d??k?1?+T??S?d=??S→d?24??R=T???a?c=eR?·eR??20?where??R2?2containsparameteradaptationgains,and??R2?2isusedtokeeptheobserveddisturbancewithinthegivenboundariessimilartotheexpressionpresentedinEq.?9?.Inordertopushtheerrorsontotheslidingsurface,apositivede?nitelowerboundedLyapunovfunctionisused,1??T??1?d?d???V=?STMS+?d?d2˙?t??0?isensuredbysettingandtheasymptoticstability?V˙+x¨ref?M?1?u?d?Cq˙?+W?˙R?=?STKsS?26?eSTM???whereKs=diag?Ks,a,Ks,c?isthefeedbackgainmatrix.Thecon-trollawisobtainedfromEqs.?21?,?22?,and?26?asTransactionsoftheASME?25?where=?ˉva,ˉvc?containsnormalizedrotarydrivevelocities.TheslidingmodecontourcontrollerdesignedforthetooltipcontourcontrolinSec.2.1alsosendscontrolsignals?ua,uc?tominimizeindividualtrackingerrors?ea,ec?oftherotarydrives.However,aseparatecontrollerisredesignedinthissectionfortherotarydrivesinordertominimizetheorientationcontourerrorcomponents??a,?c?inparalleltotheregulartrackingerrors031007-4/Vol.131,JUNE2009Downloaded17Nov2009to06.RedistributionsubjecttoASMElicenseorcopyright;see/terms/Terms_Use.cfmFig.3Toolorientationcontourcontrolleru=??uauc?+KS˙+Mx˙+MW?¨ref+Cq˙R+d=M??es?27?eR?s?=?W?R?s?Is?31?Theproposedcontrollawminimizestheerrorstatethatcontainsweightedsumoftrackingandthecontourcomponents,?e=eRt+W?0?Rd?.Theblockdiagramoftheorientationcontourcontrol-lerispresentedinFig.3.Stabilityoftheorientationcontroller?Eq.?27??isstudiedsimi-lartotheonepresentedforthetooltippositioncontroller.SincethederivativeofLyapunovfunctionpostulatedinEq.?25?isde-?areallbounded,thisimpliesthatthe˙,anddecreasingandS,?e,?˙Rand??R,?RarealsoboundedwithrespecttoEq.signalseR,e?22?.Control?u?fromEq.?27?issubstitutedintoEq.?21?toobtain??˙+M-1KS=M-1?d-dSs?28?whereI2?2isidentitymatrix.Whenassumingthatthereisnocontourerror?W=diag?0,0??addedtotherotarydrivetracking?errors,eRisdenotedasthepurerotarydrivetrackingerrorswith-outtheeffectofcontourcoupling.Equation?31?canbere-expressedtorelatethedrivetrackingerrorsbetweenthecoupled??eR?andtheuncoupled?eR?cases,?eR?s?=eR?s??W?R?s?Is?32?Similarly,Eq.?18?canberewrittenforthecoupled??R?andthe?uncoupled??R?casesas?R=eR??·eR?????R=eR??·eR?˙and?¨areboundedfunctions.UsingEq.?21?whichshowsthatSe¨isandtheconclusionsdrawnfromEq.?28?,theboundnessofV˙isuniformlycontinuous,andBarbalat’slemmaproven.Thus,V?20?impliesthatast→?,V→0,andS→0provingthatthepro-˙→0andtheeposederrorstateconvergestoorigin,?e→0,and?controllerisstable.SincetheCartesianaxispositionsdonotin?uencethetoolori-entation,thetoolaxisorientationcontrollerworksasanun-coupledsystem.Thecontourerrorweight?W?speci?estheamountofcouplingintroducedbetweentherotaryaxes.IfW=diag?0,0?,theMIMOslidingmodecontrollawinEq.?27?isreduceddowntotwoseparatestableSISOSMClawsfortrackingcontrolofaandcrotarydrives.Ontheotherhand,theeffectofnonzeroweightsfortheorientationcontouringperformanceisin-vestigatedasfollows:Whenthestatesareawayfromtheslidingsurface?S?thefeedbackterm?KsS?isnonzero,whichpushesthestatesbacktothesurface.Whentheerrorstatesareontheslidingsurface?KsS=0?duringsteadystate,thedisturbanceestimationtermcanbeneglected,andtheequivalentcontrolisobtainedfromEq.?27?as˙+Mx˙+MW?¨ref+Cq˙Ru=M??e?29??33?Hence,substitutingEq.?33?intoEq.?32?leadsto???R+?·?eR?eR??=?R?W?RIs?34???s??eR?s?=W??R?s?/s?fromEq.?32?intoEq.andsubstitutingeR?34?yields??R?s?+?·?R?s??=?R?s??W?R?s?Is?35?Notethatthecontourerrorvectorandtheangularvelocityvectorareperpendicular?19?,?·?R=0→v·?R=0;therefore,Eq.?35?canbesimpli?edas?R?s?=Is??R?s?Is+W?36?Hence,Eq.?36?showsthatforpositivede?nitiveweightmatrix?W?,thecontrollerminimizesthecontourerrorsinthecoupled??es-mode??R?ascomparedwiththeuncoupledinitialcase??Rpeciallyatlowfrequencyrange.Inparallel,thestabilityofthetrackingerrors?eR?forthecoupledcasecanbestudiedbysub-stitutingEq.?33?intoEq.?32?:??s??eR?s?=eRLetussubstitutetheequivalentcontrol?u?fromEq.?29?intotheerrordynamicspresentedinEq.?21?toobtainthesteadystatetrackingerrors?eR?ofthedrives˙+Mx¨ref?M?M??¨ref+MW?˙R?¨R=xee?1W?eR?s??·eR?s??Is?37?whichissimpli?edas??s??eR?s?=eR?30?1?ˉva?ˉvaˉvcW??·eR?s??←?=Is?ˉvaˉvc1?ˉvc???38?whichcanbefurthersimpli?edbysubstitutingtheexpressionfor˙fromEq.?22?,andrewritingitinLaplace?s?domainyields?es?s+??IeR?s?=??s+??IW?R?s?JournalofManufacturingScienceandEngineeringwhere?haspositivesemide?niteeigenvalues.Equation?38?isreorganizedtoobtainthetransferfunctionoftrackingerrors?eR?inthecoupledcase:JUNE2009,Vol.131/031007-5Downloaded17Nov2009to06.RedistributionsubjecttoASMElicenseorcopyright;see/terms/Terms_Use.cfmFig.4Complete?ve-axiscontourcontrollerblockdiagrameR?s?=?Is??s?eRIs+W??39?whichisstableforpositivecontourweights?W?.Thecomplete?ve-axiscontourcontrollerisobtainedbycom-biningthetooltipcontrollerpresentedinSec.2.1withthetoolorientationcontourcontroller,asshownintheblockdiagramgiveninFig.4.AspresentedinSec.2.2andalsoillustratedinFig.4,theorientationcontourcontrollerisstrictlyuncoupledfromtheCartesiancontourcontrollerandrequiresfeedbackonlyfromtherotarydrives.Thestabilitystudyhasshownthattheorientationcontrollerisstableandbothcontourandtrackingerrorsignalsarebounded.ThetooltipcontourcontrollerusespositionandvelocityfeedbackfromtherotarydrivesinthecomputationoftheJacobianmatrix?J?andintheinversekinematicstransformation.Thetooltipcontrollerisinternallyboundedinput-outputstable,andthesignalsrequiredfromtherotarydrivesareallboundedallowingthetooltipcontrollertogenerateboundedcontroltotheCartesiandrives.Asaresult,theoverall?ve-axiscontourcontrolsystemisstable.3ImplementationandExperimentalResultsTheopenarchitecturecontrolledexperimental?ve-axisma-chinetoolispresentedinPartI?19?,anditsdynamicparametersaregiveninTable1.Thissectioncomparesthe?ve-axiscontour-ingperformanceoftheproposedcontourcontrolleragainsttheSISOaxisbasedslidingmodetrackingcontroller?9?.TheSMCofeachdriveistunedseparately,andthebandwidthsoftheCartesianaxisarematchedinordertoobtainbettercon-touringperformance?seeTable2?.Previouslypresented?ve-axiscontourtool-path?seeFig.5?isusedinair-cuttingtests.Asmoothtrajectoryisgeneratedwithamaximumcruisespeedof50mm/sandcubicacceleration/decelerationsof1500mm/s2.Correspond-ingaxisreferencepositionandvelocitycommandsareshowninFig.6.Itshouldbenotedthatalthoughthetangentialfeedrateisselectedconservatively,higherreferenceaxisvelocitiesandaccel-erationsareobservedbecauseofthecontinuouslyvaryingpathcurvatureandthekinematicsofthemachinetool.AsshowninFig.6?c?,yandzaxisarecommandedashighas200mm/sand90mm/s,respectively.Experimentalair-cuttingresultsunderSISOslidingmodecontrolispresentedareFig.7.Trackingerrorsaremeasuredbycomparingthereferenceaxistrajectoryagainsttheactualpositionsmeasuredfromrotaryencodersmountedatthemotorsideofeachdrive.AsshownonFigs.7?b?–7?f?,dedicatedSMCswithdisturbanceobserversprovideeachaxistoaccuratelytracktheirreferencepositioncommands.Highestaxistrackingerrorsareobservedasmax?ex?=15?m,max?ey?=13?m,max?ez?=7?m,max?ea?=2.2mrad,andmax?ec?=1.7mrad.ByinspectingthevelocitytrajectoryofthedrivesfromFigs.6?c?and6?d?,itcanbeobservedthatpeaktrackingerrorsoccurespeciallyatthevelocityreversaloftheaxiswhereCoulombfrictionactsasastepdisturbancetothedrives.ThecontourerrorestimationmethodpresentedbyErkorkmazandAltintas?15?isthenusedtomeasurethetooltipcontourerroralongthetool-pathandpre-sentedinFig.7?a?.Preciseaxistrackingperformancedoesnotalwaysguaranteethedesiredcontouringaccuracyduringsimulta-neous?ve-axiscontourmachining.Sincetrackingerrorsofalldrivesaffectthetoolposition,themeanabsolutetooltipcontourerroriscalculatedasmean???=45?mandthemaximumismax???=317?m.Itshouldalsobenotedthatthemaincontribu-tiontothecontourerrorsareoriginatedfromthetrackingerrorsofaandcrotarydrives.Bothaxesrotatetheworkpiece,andduetotheoffsetbetweentherotarydrives’axisofrotationsandtheplacementoftheworkpiececoordinatesystem,smallrotationsmaycauselargedevi