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外文资料--Application of Uniform Cubic B-spline Curves to Machine-Tool Control.pdf

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外文资料--Application of Uniform Cubic B-spline Curves to Machine-Tool Control.pdf

JournaloflntellgentandRoboticSystems4393,02,1991,39391991KluwerAcademicPublishers.PrintedintheNetherlands.ApplicationofUniformCubicBsplineCurvestoMachineToolControlP.ANDRE,M.C.HADDADandC.MORLECLaboratoiredAutomatiquedeBesanrURACNRS822,ENSMM,lnstitutdeProductique,ImpassedesSaintMartin,25000Besanfon,FranceReceived19June1989infinalform5November1990Abstract.TheapplicationofBsplinefunctionsinthedomainofmachinetoolcontrolcanbeefficientonlyifitispossibletosimultaneouslycontroltheshapeofacontourandthespeedalongthiscontour.ThepaperdescribesamethodallowingustocontrolthespeedofatoolmovingalongacontourdefinedbyanuniformcubicBspline.ThismethodisbaseduponthediscretizationoftheBsplineparameterwithrespecttobothagivengeometricalerrorattothedesiredspeedlaw,Keywords.Machinetool,numericalcontrol,interpolation,Bspline.O.IntroductionThankstotheirspecificproperties,Bsplinefunctionshavebecome,ininteractiveCADsystems,themostcommonlyusedmathematicaltoolfordefinitionandrepresentationoffreeshapecurvesandsurfaces.Inamachinetoolfield,thecontoursofthepartstoberealizedaregenerallymadeofasequenceofstraightsegmentsandcircleportions.Nevertheless,inspecificfieldslikethelaserorwatercuttingorsparkmachining,shapesareveryoftendefinedbyimposedpasspoints.Insuchcases,itseemstoustobeinterestingtodefine,atthemachinetoolcontrollevel,thecontoursbymeansofBsplinecurves.WewillseelaterthattheBsplinetechniqueallowsustorealizeanycurrentlyusedinterpolationstraightlinesandanglescanbeexactlyachieved,forcirclesitisalwayspossibletolimit,atagivenvalue,thegeometricalerror.1.BSplineInterpolationBydefinition,theBsplinesareparametricplecewisepolynomialcurvesoforderk,whoseneighbouringsegmentsmeetattheparameterknotswithaparametriccontinuityofk2firstderivatives.Theirmathematicaltheoryhasbeenthoroughlyreportedin\1,3,4,6\.1.1.CUBICBSPLINECURVESIthastobenoticedthatamongallvariousmethodsusedtodefineandrepresentfreeshapecurvesandsurfacesinCADsystems,thetwomostwidelyusedarethoseofB6zier\2\andtheBsplinemethod\3\.394P.ANDREETAL.Thefirstmethod,baseduponBernsteinspolynomialsconcept,doesntseemtoustobeadequateforrealtimecontrolapplications.Indeed,thehighdegreeofpracticallyusedB6zierspolynomials8to10doesntleadtoasufficientlyfastcomputationalgorithm.Thesecondmethodleadstoacurvedefinedinapiecewisemannereachpieceisasplinecurvesegmenthavingthesamedegreekorderk1theoverallresultingcurveisk1timesdifferentiableatsimpledefinitionpoints.TheBsplinedegreeonlydependsuponthetypeofapplication.Itmustallowustorealizeacompromisebetweenthewealthoftherepresentationandthesimplicityofthecalculusleadingtoafastcomputation.Forapplicationsinmachinetoolcontrol,themainrequirementsareposition,speed,andaccelerationcontinuity.Ithastobenoticedthatoneofthemostinterestingpropertiesofthesplinecurves,inthemachinetoolfield,isthattheyminimizetheenergyneededtofollowthecoursecontourand,thus,themechanicalsolicitationofthemachinesstructure.Practically,wecanachieveaconvenienttradeoffbetweenagoodsmoothingofthecontourimposinghighordersplinesandagoodabilitytorealtimecontrolleadingtoareductionintheorderbychoosingcubicBsplinecurves,sincetheyachieveC2continuityandminimizethecontourscourseenergy.1.2.UNIFORMCUBICBSPLINESIf,inaddition,wechooseauniformnodevectorsuchasujj,wedefinecubicBsplinesashavingequidistantnodescalleduniformcubicBsplineswhichenhancetherealtimeprocessingability.Inordertosimplifythewriting,weusetheabbreviationUCtodesignateuniformcubicintheensuinganalysisUCBsplines.Forasetofcontrolvertices\P0,P,9.9,P,2,PN3\,thecoordinatesofapointontheithUCBsplinecurvessegmentaregivenbysee\4,5,6\3Q,uNuPgl.1r0whereuistheparametervaryingfrom0to1alongtheithcurvesegment,and\Nou,Nlu,N2u,N3u\914103030\l,u,u2,u3\1/6363013311.2UNIFORMCUBICBSPLINECURVESANDMACHINETOOLCONTROL3951.2.1.RelationshipBetweenControlPointsandPassPointsDevelopingtheaboverelationshipleadstoP,4/\,/\23.P,3./\2Q,uu663/\6P/l32u26g3./\13./\2/\3u31.36with01axisoftheellipse.Allthetangencypropertiesareconservedbythistransformation.1.3.1.LinearInterpolationThefollowingsectionprovidesabriefreviewofthemethodsusedtoachievetheseinterpolationsandpresentssomeoftheresultsobtained.Moredetailsaboutlinearinterpolationcanbefoundin\3,5\.1.3.1.1.StraightSegments.ThetechiqueofUCBsplinesallowsustoeasilyandexactlydefinestraightsegments.Onewaycosistsofusingtriplecontrolpoints.If,forinstance,weassignPlP2P3,P4beinganyotherpointdifferentofP,accordingto1.4wecanwriteMjQ0PjandM2al1PtP4Pj/6.

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