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

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JournaloflntellgentandRoboticSystems4393,02,1991,39391991KluwerAcademicPublishers.PrintedintheNetherlands.ApplicationofUniformCubicB-splineCurvestoMachine-ToolControlP.ANDRE,M.C.HADDADandC.MORLECLaboratoired AutomatiquedeBesanrURACNRS822,ENSMM,lnstitutdeProductique,ImpassedesSaint-Martin,25000Besanfon,FranceReceived19June1989;infinalform5November1990Abstract.TheapplicationofB-splinefunctionsinthedomainofmachine-toolcontrolcanbeefficientonlyifitispossibletosimultaneouslycontroltheshapeofacontourandthespeedalongthiscontour.ThepaperdescribesamethodallowingustocontrolthespeedofatoolmovingalongacontourdefinedbyanuniformcubicB-spline.ThismethodisbaseduponthediscretizationoftheB-splineparameterwithrespecttobothagivengeometricalerrorattothedesiredspeedlaw,Keywords.Machinetool,numericalcontrol,interpolation,B-spline.O.IntroductionThankstotheirspecificproperties,B-splinefunctionshavebecome,ininteractiveCADsystems,themostcommonlyusedmathematicaltoolfordefinitionandrepre-sentationoffree-shapecurvesandsurfaces.Inamachine-toolfield,thecontoursofthepartstoberealizedaregenerallymadeofasequenceofstraightsegmentsandcircleportions.Nevertheless,inspecificfieldslikethelaserorwater-cuttingorsparkmachining,shapesareveryoftendefinedbyimposedpasspoints.Insuchcases,itseemstoustobeinterestingtodefine,atthemachine-toolcontrollevel,thecontoursbymeansofB-splinecurves.WewillseelaterthattheB-splinetechniqueallowsustorealizeanycurrentlyusedinterpolationstraightlinesandanglescanbeexactlyachieved,forcirclesitisalwayspossibletolimit,atagivenvalue,thegeometricalerror.1.B-SplineInterpolationBydefinition,theB-splinesareparametricplecewisepolynomialcurvesoforderk,whoseneighbouringsegmentsmeetattheparameterknotswithaparametriccontinuityofk-2firstderivatives.Theirmathematicaltheoryhasbeenthoroughlyreportedin\[1,3,4,6\].1.1.CUBICB-SPLINECURVESIthastobenoticedthatamongallvariousmethodsusedtodefineandrepresentfree-shapecurvesandsurfacesinCADsystems,thetwomostwidelyusedarethoseofB6zier\[2\]andtheB-splinemethod\[3\].394P.ANDREETAL.Thefirstmethod,baseduponBernstein spolynomialsconcept,doesn tseemtoustobeadequateforreal-timecontrolapplications.Indeed,thehighdegreeofpracticallyusedB6zier spolynomials8to10doesn tleadtoasufficientlyfastcomputationalgorithm.Thesecondmethodleadstoacurvedefinedinapiecewisemanner;eachpieceisasplinecurvesegmenthavingthesamedegreekorderk1;theoverallresultingcurveisk-1timesdifferentiableatsimpledefinitionpoints.TheB-splinedegreeonlydependsuponthetypeofapplication.Itmustallowustorealizeacompromisebetweenthewealthoftherepresentationandthesimplicityofthecalculusleadingtoafastcomputation.Forapplicationsinmachine-toolcontrol,themainrequirementsareposition,speed,andaccelerationcontinuity.Ithastobenoticedthatoneofthemostinterestingpropertiesofthesplinecurves,inthemachine-toolfield,isthattheyminimizetheenergyneededtofollowthecoursecontourand,thus,themechanicalsolicitationofthemachine sstructure.Practically,wecanachieveaconvenienttradeoffbetweenagoodsmoothingofthecontourimposinghigh-ordersplinesandagoodabilitytoreal-timecontrolleadingtoareductionintheorderbychoosingcubicB-splinecurves,sincetheyachieveC2continuityandminimizethecontour scourseenergy.1.2.UNIFORMCUBICB-SPLINESIf,inaddition,wechooseauniformnodevectorsuchasujj,wedefinecubicB-splinesashavingequidistantnodescalleduniformcubicB-splineswhichenhancethereal-timeprocessingability.Inordertosimplifythewriting,weusetheabbreviationUCtodesignateuniformcubicintheensuinganalysisUCB-splines.Forasetofcontrolvertices\[P0,P,9.9,P,2,PN3\],thecoordinatesofapointontheithUCB-splinecurve ssegmentaregivenbysee\[4,5,6\]3Q,uNu Pgl.1r0whereuistheparametervaryingfrom0to1alongtheithcurvesegment,and\[Nou,Nlu,N2u,N3u\]91410--3030\[l,u,u2,u3\]-1/63--63013--311.2UNIFORMCUBICB-SPLINECURVESANDMACHINE-TOOLCONTROL3951.2.1.RelationshipBetweenControlPointsandPass-PointsDevelopingtheaboverelationshipleadstoP,4-/\],/\]2-3.P,3./\]2Q,uu663/\]-6P/l32u26-g3./\]1-3./\]2/\]3u31.36with0u1.Thestartingpointoftheithcurvesegmentcorrespondstou0/\]4/\]1Pi2Q,01.46Thefinalpointofthissamecurvesegmentcorrespondstou1/\]l4/\] 2/\]3Q,11.56Thispointisalsothestartingpointofthei1thcurvesegment.WritingQi0MleadstoasetoflinearequationslinkingthecontrolpointsP,andthepasspointsMoftheUCB-splinecurvesee\[5\]P4./\]1/\]26MiiI,,..,N1.6aorinthematrixform\[P\].\[A\]6\[M\].l.6b\[A\]beingamatrixlinkingthe/\] sandtheMi s.If,definingthepasspointsMi,wewanttofindthecontrolpoints,theproblemhasNdataMiandN2unknowns/\].Twocasesmustbeconsideredithecurveisclosed;iithecurveisopen.1.2.1.1.ClosedCurve.Inordertohaveacontinuousclosedcurve,itisnecessarytoimposecontinuousfirstandsecondderivativesnoted,respectivelybyMandM2atthejunctionpoint,i.e.attheendofthelastcurvesegmentandthebeginningoftheneighbouringcurvesegmentoftheB-spline.ThatcanbeexpressedbyM1M.,,M\[1Mnll.MI2,Mn2Iwhichimpliessee\[5\]P,P.I.P2P.z.PP.3-396P.ANDREETAL.Thesenewconstraintsaddtwoequationstotheprevioussystem1.6a.So,wenowhaveasetofNequationswithNunknownsthatcanbesolvedbycomputingtheinverse\[A-\]matrixinordertoderivethevaluesofthePcontrolpointsfromtheMipasspoints.IthastobenoticedthattheAmatrixisnotasingularone.1.2.1.2.OpenCurve.Wemustaddtwoconditionstotheseof1.6ainordertobeabletosolvetheproblem.Thesetwoconditionsmustbechosenaccordingtoeitherthegeometricalorkinematicalconsiderationsrequiredbythemachiningconditions.TheconditionsproposedbyBoujon\[5\],imposingthevaluesofthefirstderivativeattheendpointsoftheoverallB-splinecurve,areveryinteresting,sincetheycorre-spond,inthecaseofmachine-toolcontrol,toimposetheinitialandfinalspeedsalongthecontour.Onecanalsoimposethesecondderivativeattheendpointsofthecurveifwewanttoimposeavalueofthecurvatureattheseendpoints.Inmostcases,anullcurvaturewillbechosen naturalend .OtherconditionsareproposedbyBarsky\[4\]suchas,forexample,addingphantomverticesatbothendpointsofthecurve.Dependingupontherequirementsconcerningeithertheshapeofthecontourorthespeedlawalongit,twooftheseconditionscanbechoseninordertomeettherequirements.1.3.USEOFUCB-SPLINEFORMACHINETOOLCONTROLInordertobeabletodescribethegeometricalcontourscommonlyusedinconven-tionalmachining,asfree-shapecontours,wemustbeabletoachevelinear,circular,andeventuallyparabolicinterpolation.Ithastobenoticedthattheellipticinterpolationisderivedfromthecircularone,sinceanellipseistheresultofthetransformationofacirclebyanorthogonalaffinityreferingtoeitherthebigaffinitycoefficient1orthesmallaffinitycoefficient1axisoftheellipse.Allthetangencypropertiesareconservedbythistransformation.1.3.1.LinearInterpolationThefollowingsectionprovidesabriefreviewofthemethodsusedtoachievetheseinterpolationsandpresentssomeoftheresultsobtained.Moredetailsaboutlinearinterpolationcanbefoundin\[3,5\].1.3.1.1.StraightSegments.ThetechiqueofUCB-splinesallowsustoeasilyandexactlydefinestraightsegments.Onewaycosistsofusingtriplecontrolpoints.If,forinstance,weassignPlP2P3,P4beinganyotherpointdifferentofP,accordingto1.4wecanwriteMjQ0PjandM2al1PtP4-Pj/6.
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