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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

JOURNALOFLNTELLGENTANDROBOTICSYSTEMS4393,02,1991,39391991KLUWERACADEMICPUBLISHERSPRINTEDINTHENETHERLANDSAPPLICATIONOFUNIFORMCUBICBSPLINECURVESTOMACHINETOOLCONTROLPANDRE,MCHADDADANDCMORLECLABORATOIREDAUTOMATIQUEDEBESANRURACNRS822,ENSMM,LNSTITUTDEPRODUCTIQUE,IMPASSEDESSAINTMARTIN,25000BESANFON,FRANCERECEIVED19JUNE1989;INFINALFORM5NOVEMBER1990ABSTRACTTHEAPPLICATIONOFBSPLINEFUNCTIONSINTHEDOMAINOFMACHINETOOLCONTROLCANBEEFFICIENTONLYIFITISPOSSIBLETOSIMULTANEOUSLYCONTROLTHESHAPEOFACONTOURANDTHESPEEDALONGTHISCONTOURTHEPAPERDESCRIBESAMETHODALLOWINGUSTOCONTROLTHESPEEDOFATOOLMOVINGALONGACONTOURDEFINEDBYANUNIFORMCUBICBSPLINETHISMETHODISBASEDUPONTHEDISCRETIZATIONOFTHEBSPLINEPARAMETERWITHRESPECTTOBOTHAGIVENGEOMETRICALERRORATTOTHEDESIREDSPEEDLAW,KEYWORDSMACHINETOOL,NUMERICALCONTROL,INTERPOLATION,BSPLINEOINTRODUCTIONTHANKSTOTHEIRSPECIFICPROPERTIES,BSPLINEFUNCTIONSHAVEBECOME,ININTERACTIVECADSYSTEMS,THEMOSTCOMMONLYUSEDMATHEMATICALTOOLFORDEFINITIONANDREPRESENTATIONOFFREESHAPECURVESANDSURFACESINAMACHINETOOLFIELD,THECONTOURSOFTHEPARTSTOBEREALIZEDAREGENERALLYMADEOFASEQUENCEOFSTRAIGHTSEGMENTSANDCIRCLEPORTIONSNEVERTHELESS,INSPECIFICFIELDSLIKETHELASERORWATERCUTTINGORSPARKMACHINING,SHAPESAREVERYOFTENDEFINEDBYIMPOSEDPASSPOINTSINSUCHCASES,ITSEEMSTOUSTOBEINTERESTINGTODEFINE,ATTHEMACHINETOOLCONTROLLEVEL,THECONTOURSBYMEANSOFBSPLINECURVESWEWILLSEELATERTHATTHEBSPLINETECHNIQUEALLOWSUSTOREALIZEANYCURRENTLYUSEDINTERPOLATIONSTRAIGHTLINESANDANGLESCANBEEXACTLYACHIEVED,FORCIRCLESITISALWAYSPOSSIBLETOLIMIT,ATAGIVENVALUE,THEGEOMETRICALERROR1BSPLINEINTERPOLATIONBYDEFINITION,THEBSPLINESAREPARAMETRICPLECEWISEPOLYNOMIALCURVESOFORDERK,WHOSENEIGHBOURINGSEGMENTSMEETATTHEPARAMETERKNOTSWITHAPARAMETRICCONTINUITYOFK2FIRSTDERIVATIVESTHEIRMATHEMATICALTHEORYHASBEENTHOROUGHLYREPORTEDIN\1,3,4,6\11CUBICBSPLINECURVESITHASTOBENOTICEDTHATAMONGALLVARIOUSMETHODSUSEDTODEFINEANDREPRESENTFREESHAPECURVESANDSURFACESINCADSYSTEMS,THETWOMOSTWIDELYUSEDARETHOSEOFB6ZIER\2\ANDTHEBSPLINEMETHOD\3\394PANDREETALTHEFIRSTMETHOD,BASEDUPONBERNSTEINSPOLYNOMIALSCONCEPT,DOESNTSEEMTOUSTOBEADEQUATEFORREALTIMECONTROLAPPLICATIONSINDEED,THEHIGHDEGREEOFPRACTICALLYUSEDB6ZIERSPOLYNOMIALS8TO10DOESNTLEADTOASUFFICIENTLYFASTCOMPUTATIONALGORITHMTHESECONDMETHODLEADSTOACURVEDEFINEDINAPIECEWISEMANNER;EACHPIECEISASPLINECURVESEGMENTHAVINGTHESAMEDEGREEKORDERK1;THEOVERALLRESULTINGCURVEISK1TIMESDIFFERENTIABLEATSIMPLEDEFINITIONPOINTSTHEBSPLINEDEGREEONLYDEPENDSUPONTHETYPEOFAPPLICATIONITMUSTALLOWUSTOREALIZEACOMPROMISEBETWEENTHEWEALTHOFTHEREPRESENTATIONANDTHESIMPLICITYOFTHECALCULUSLEADINGTOAFASTCOMPUTATIONFORAPPLICATIONSINMACHINETOOLCONTROL,THEMAINREQUIREMENTSAREPOSITION,SPEED,ANDACCELERATIONCONTINUITYITHASTOBENOTICEDTHATONEOFTHEMOSTINTERESTINGPROPERTIESOFTHESPLINECURVES,INTHEMACHINETOOLFIELD,ISTHATTHEYMINIMIZETHEENERGYNEEDEDTOFOLLOWTHECOURSECONTOURAND,THUS,THEMECHANICALSOLICITATIONOFTHEMACHINESSTRUCTUREPRACTICALLY,WECANACHIEVEACONVENIENTTRADEOFFBETWEENAGOODSMOOTHINGOFTHECONTOURIMPOSINGHIGHORDERSPLINESANDAGOODABILITYTOREALTIMECONTROLLEADINGTOAREDUCTIONINTHEORDERBYCHOOSINGCUBICBSPLINECURVES,SINCETHEYACHIEVEC2CONTINUITYANDMINIMIZETHECONTOURSCOURSEENERGY12UNIFORMCUBICBSPLINESIF,INADDITION,WECHOOSEAUNIFORMNODEVECTORSUCHASUJJ,WEDEFINECUBICBSPLINESASHAVINGEQUIDISTANTNODESCALLEDUNIFORMCUBICBSPLINESWHICHENHANCETHEREALTIMEPROCESSINGABILITYINORDERTOSIMPLIFYTHEWRITING,WEUSETHEABBREVIATIONUCTODESIGNATEUNIFORMCUBICINTHEENSUINGANALYSISUCBSPLINESFORASETOFCONTROLVERTICES\P0,P,99,P,2,PN3\,THECOORDINATESOFAPOINTONTHEITHUCBSPLINECURVESSEGMENTAREGIVENBYSEE\4,5,6\3Q,UNUPGL1R0WHEREUISTHEPARAMETERVARYINGFROM0TO1ALONGTHEITHCURVESEGMENT,AND\NOU,NLU,N2U,N3U\914103030\L,U,U2,U3\1/63630133112UNIFORMCUBICBSPLINECURVESANDMACHINETOOLCONTROL395121RELATIONSHIPBETWEENCONTROLPOINTSANDPASSPOINTSDEVELOPINGTHEABOVERELATIONSHIPLEADSTOP,4/\,/\23P,3/\2Q,UU663/\6P/L32U26G3/\13/\2/\3U3136WITH0U1THESTARTINGPOINTOFTHEITHCURVESEGMENTCORRESPONDSTOU0/\4/\1PI2Q,0146THEFINALPOINTOFTHISSAMECURVESEGMENTCORRESPONDSTOU1/\L4/\2/\3Q,1156THISPOINTISALSOTHESTARTINGPOINTOFTHEI1THCURVESEGMENTWRITINGQI0MLEADSTOASETOFLINEAREQUATIONSLINKINGTHECONTROLPOINTSP,ANDTHEPASSPOINTSMOFTHEUCBSPLINECURVESEE\5\P4/\1/\26MIII,,,N16AORINTHEMATRIXFORM\P\\A\6\M\L6B\A\BEINGAMATRIXLINKINGTHE/\SANDTHEMISIF,DEFININGTHEPASSPOINTSMI,WEWANTTOFINDTHECONTROLPOINTS,THEPROBLEMHASNDATAMIANDN2UNKNOWNS/\TWOCASESMUSTBECONSIDEREDITHECURVEISCLOSED;IITHECURVEISOPEN1211CLOSEDCURVEINORDERTOHAVEACONTINUOUSCLOSEDCURVE,ITISNECESSARYTOIMPOSECONTINUOUSFIRSTANDSECONDDERIVATIVESNOTED,RESPECTIVELYBYMANDM2ATTHEJUNCTIONPOINT,IEATTHEENDOFTHELASTCURVESEGMENTANDTHEBEGINNINGOFTHENEIGHBOURINGCURVESEGMENTOFTHEBSPLINETHATCANBEEXPRESSEDBYM1M,,M\1MNLLMI2,MN2IWHICHIMPLIESSEE\5\P,PIP2PZPP3396PANDREETALTHESENEWCONSTRAINTSADDTWOEQUATIONSTOTHEPREVIOUSSYSTEM16ASO,WENOWHAVEASETOFNEQUATIONSWITHNUNKNOWNSTHATCANBESOLVEDBYCOMPUTINGTHEINVERSE\A\MATRIXINORDERTODERIVETHEVALUESOFTHEPCONTROLPOINTSFROMTHEMIPASSPOINTSITHASTOBENOTICEDTHATTHEAMATRIXISNOTASINGULARONE1212OPENCURVEWEMUSTADDTWOCONDITIONSTOTHESEOF16AINORDERTOBEABLETOSOLVETHEPROBLEMTHESETWOCONDITIONSMUSTBECHOSENACCORDINGTOEITHERTHEGEOMETRICALORKINEMATICALCONSIDERATIONSREQUIREDBYTHEMACHININGCONDITIONSTHECONDITIONSPROPOSEDBYBOUJON\5\,IMPOSINGTHEVALUESOFTHEFIRSTDERIVATIVEATTHEENDPOINTSOFTHEOVERALLBSPLINECURVE,AREVERYINTERESTING,SINCETHEYCORRESPOND,INTHECASEOFMACHINETOOLCONTROL,TOIMPOSETHEINITIALANDFINALSPEEDSALONGTHECONTOURONECANALSOIMPOSETHESECONDDERIVATIVEATTHEENDPOINTSOFTHECURVEIFWEWANTTOIMPOSEAVALUEOFTHECURVATUREATTHESEENDPOINTSINMOSTCASES,ANULLCURVATUREWILLBECHOSENNATURALENDOTHERCONDITIONSAREPROPOSEDBYBARSKY\4\SUCHAS,FOREXAMPLE,ADDINGPHANTOMVERTICESATBOTHENDPOINTSOFTHECURVEDEPENDINGUPONTHEREQUIREMENTSCONCERNINGEITHERTHESHAPEOFTHECONTOURORTHESPEEDLAWALONGIT,TWOOFTHESECONDITIONSCANBECHOSENINORDERTOMEETTHEREQUIREMENTS13USEOFUCBSPLINEFORMACHINETOOLCONTROLINORDERTOBEABLETODESCRIBETHEGEOMETRICALCONTOURSCOMMONLYUSEDINCONVENTIONALMACHINING,ASFREESHAPECONTOURS,WEMUSTBEABLETOACHEVELINEAR,CIRCULAR,ANDEVENTUALLYPARABOLICINTERPOLATIONITHASTOBENOTICEDTHATTHEELLIPTICINTERPOLATIONISDERIVEDFROMTHECIRCULARONE,SINCEANELLIPSEISTHERESULTOFTHETRANSFORMATIONOFACIRCLEBYANORTHOGONALAFFINITYREFERINGTOEITHERTHEBIGAFFINITYCOEFFICIENT1ORTHESMALLAFFINITYCOEFFICIENT1AXISOFTHEELLIPSEALLTHETANGENCYPROPERTIESARECONSERVEDBYTHISTRANSFORMATION131LINEARINTERPOLATIONTHEFOLLOWINGSECTIONPROVIDESABRIEFREVIEWOFTHEMETHODSUSEDTOACHIEVETHESEINTERPOLATIONSANDPRESENTSSOMEOFTHERESULTSOBTAINEDMOREDETAILSABOUTLINEARINTERPOLATIONCANBEFOUNDIN\3,5\1311STRAIGHTSEGMENTSTHETECHIQUEOFUCBSPLINESALLOWSUSTOEASILYANDEXACTLYDEFINESTRAIGHTSEGMENTSONEWAYCOSISTSOFUSINGTRIPLECONTROLPOINTSIF,FORINSTANCE,WEASSIGNPLP2P3,P4BEINGANYOTHERPOINTDIFFERENTOFP,ACCORDINGTO14WECANWRITEMJQ0PJANDM2AL1PTP4PJ/6

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