JournaloflntellgentandRoboticSystems4:393,02,1991,39391991KluwerAcademicPublishers.PrintedintheNetherlands.ApplicationofUniformCubicB-splineCurvestoMachine-ToolControlP.ANDRE,M.C.HADDADandC.MORLECLaboratoiredAutomatiquedeBesanr(URACNRS822,ENSMM),lnstitutdeProductique,ImpassedesSaint-Martin,25000Besanfon,France(Received:19June1989；infinalform:5November1990)Abstract.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-splinetechniqueallowsustorealizeanycurrentlyusedinterpolation:straightlinesandanglescanbeexactlyachieved,forcirclesitisalwayspossibletolimit,atagivenvalue,thegeometricalerror.1.B-SplineInterpolationBydefinition,theB-splinesareparametricplecewisepolynomialcurvesoforderk,whoseneighbouringsegmentsmeetattheparameterknotswithaparametriccontinuityof(k-2)firstderivatives.Theirmathematicaltheoryhasbeenthoroughlyreportedin1,3,4,6.1.1.CUBICB-SPLINECURVESIthastobenoticedthatamongallvariousmethodsusedtodefineandrepresentfree-shapecurvesandsurfacesinCADsystems,thetwomostwidelyusedarethoseofB6zier2andtheB-splinemethod3.394P.ANDREETAL.Thefirstmethod,baseduponBernsteinspolynomialsconcept,doesntseemtoustobeadequateforreal-timecontrolapplications.Indeed,thehighdegreeofpracticallyusedB6zierspolynomials(8to10)doesntleadtoasufficientlyfastcomputationalgorithm.Thesecondmethodleadstoacurvedefinedinapiecewisemanner；eachpieceisasplinecurvesegmenthavingthesamedegreek(orderk+1)；theoverallresultingcurveis(k-1)timesdifferentiableatsimpledefinitionpoints.TheB-splinedegreeonlydependsuponthetypeofapplication.Itmustallowustorealizeacompromisebetweenthewealthoftherepresentationandthesimplicityofthecalculusleadingtoafastcomputation.Forapplicationsinmachine-toolcontrol,themainrequirementsareposition,speed,andaccelerationcontinuity.Ithastobenoticedthatoneofthemostinterestingpropertiesofthesplinecurves,inthemachine-toolfield,isthattheyminimizetheenergyneededtofollowthecoursecontourand,thus,themechanicalsolicitationofthemachinesstructure.Practically,wecanachieveaconvenienttradeoffbetweenagoodsmoothingofthecontour(imposinghigh-ordersplines)andagoodabilitytoreal-timecontrol(leadingtoareductionintheorder)bychoosingcubicB-splinecurves,sincetheyachieveC2continuityandminimizethecontourscourseenergy.1.2.UNIFORMCUBICB-SPLINESIf,inaddition,wechooseauniformnodevectorsuchasuj=j,wedefinecubicB-splinesashavingequidistantnodescalleduniformcubicB-splineswhichenhancethereal-timeprocessingability.Inordertosimplifythewriting,weusetheabbreviationUCtodesignateuniformcubicintheensuinganalysis(UCB-splines).ForasetofcontrolverticesP0,P,9.9,P,+2,PN+3,thecoordinatesofapointontheithUCB-splinecurvessegmentaregivenby(see4,5,6):3Q,(u)=N(u)Pg+(l.1)r=0whereuistheparametervaryingfrom0to1alongtheithcurvesegment,andNo(u),Nl(u),N2(u),N3(u)91410-3030l,u,u2,u3-(1/6)3-63013-31(1.2)UNIFORMCUBICB-SPLINECURVESANDMACHINE-TOOLCONTROL3951.2.1.RelationshipBetweenControlPointsandPass-PointsDevelopingtheaboverelationshipleadstoP,+4-/+,+/+2-3.P,+3./+2Q,(u)=+u+663/-6P/+l+3+2u2+6+-g+3./+1-3./+2+/+3u3(1.3)6with0u1.Thestartingpointoftheithcurvesegmentcorrespondstou=0:/+4/+1+Pi+2Q,(0)=(1.4)6Thefinalpointofthissamecurvesegmentcorrespondstou=1:/+l+4/+2+/+3Q,(1)=(1.5)6Thispointisalsothestartingpointofthe(i+1)thcurvesegment.WritingQi(0)=MleadstoasetoflinearequationslinkingthecontrolpointsP,andthepasspointsMoftheUCB-splinecurve(see5)P+4./+1+/+2=6Mi(i=I,.,N)(1.6a)orinthematrixformP.A=6M.(l.6b)Abeingamatrixlinkingthe/sandtheMis.If,definingthepasspointsMi,wewanttofindthecontrolpoints,theproblemhasNdata(Mi)and(N+2)unknowns(/).Twocasesmustbeconsidered:(i)thecurveisclosed；(ii)thecurveisopen..ClosedCurve.Inordertohaveacontinuousclosedcurve,itisnecessarytoimposecontinuousfirstandsecondderivatives(noted,respectivelybyMandM(2)atthejunctionpoint,i.e.attheendofthelastcurvesegmentandthebeginningoftheneighbouringcurvesegmentoftheB-spline.ThatcanbeexpressedbyM1=M.+,(M1)=Mn(l+)l.MI2,=Mn(2+)Iwhichimplies(see5)P,=P.+I.P2=P.+z.P=P.+3-396P.ANDREETAL.Thesenewconstraintsaddtwoequationstotheprevioussystem(1.6a).So,wenowhaveasetofNequationswithNunknownsthatcanbesolvedbycomputingtheinverseA-matrixinordertoderivethevaluesofthePcontrolpointsfromtheMipasspoints.(IthastobenoticedthattheAmatrixisnotasingularone.).OpenCurve.Wemustaddtwoconditionstotheseof(1.6a)inordertobeabletosolvetheproblem.Thesetwoconditionsmustbechosenaccordingtoeitherthegeometricalorkinematicalconsiderationsrequiredbythemachiningconditions.TheconditionsproposedbyBoujon5,imposingthevaluesofthefirstderivativeattheendpointsoftheoverallB-splinecurve,areveryinteresting,sincetheycorre-spond,inthecaseofmachine-toolcontrol,toimposetheinitialandfinalspeedsalongthecontour.Onecanalsoimposethesecondderivativeattheendpointsofthecurveifwewanttoimposeavalueofthecurvatureattheseendpoints.(Inmostcases,anullcurvaturewillbechosen:naturalend.)OtherconditionsareproposedbyBarsky4suchas,forexample,addingphantomverticesatbothendpointsofthecurve.Dependingupontherequirementsconcerningeithertheshapeofthecontourorthespeedlawalongit,twooftheseconditionscanbechoseninordertomeettherequirements.1.3.USEOFUCB-SPLINEFORMACHINETOOLCONTROLInordertobeabletodescribethegeometricalcontourscommonlyusedinconven-tionalmachining,asfree-shapecontours,wemustbeabletoachevelinear,circular,andeventuallyparabolicinterpolation.Ithastobenoticedthattheelliptici