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附录A:英文资料Thespeed-controlledinterpolatorformachiningparametriccurvesS.-S.Yeh,P.-L.HsuAbstract:ModernCNCsystemsaredesignedwiththefunctionofmachiningarbitraryparametriccurvestosavemassivedatacommunicationbetweenCAD/CAMandCNCsystemsandimprovetheirmachiningquality.AlthoughavailableCNCinterpolatorsforparametriccurvesgenerallyachievecontouringpositionaccuracy,thespecifiedfederate,withdominatesthequalityofthemachiningprocesses,isnotguaranteedduringmotion.Recently,someapproximationresultsconcerningmotionspeedhavebeenreported[ShpitalniM,KorenY,LoCC.Computer-AidedDesign1994;26(11):832-838;BediS,AliI,QuanN.TransASMEJEngngIndustry1993;115:329-336;ChouJJ,YangDCH.Trans.ASMEJEngngIndustry1991;113:305-310;ChouJJ,YangDCH.TransASMEJEngngIndustry1992;114:15-22].Inthispaper,byderivingacompensatoryparameter,theproposedinterpolationalgorithmhassignificantlyimprovedcurvespeedaccuracy.Inrealapplications,theproposedalgorithmresultsin:(1)constantspeed;and(2)specifiedaccelerationanddecelerationanddeceleration(ACC/DEC)tomeetthefeedratecommands.Themotionaccuracyandfeasibilityofthepresentinterpolatorhavebeenverifiedwithaprovidednon-uniformrationalB-spline(NURBS)example.©1999ElsevierScienceLtd.Allrightsreserved.Keywords:Parametriccurve;CNCinterpolators;Speedaccuracy;NURBS1.Introduction:InmodernCAD/CAMsystems,profilesforpartslikedies,vanes,aircraftmodelsareusuallyrepresentedinparametricforms.AsconventionalCNCmachinesonlyprovidelinearandcirculararcinterpolators,theCAD/CAMsystemshavetosegmentacurveintoahugenumberofsmall,linearizedsegmentsandsendthemtoCNCsystems.Suchlinearized-segmentedcontoursprocessedontraditionalCNCsystemsareundesirableinrealapplicationsasfollows:●thetransmissionerrorbetweenCAD/CAMandCNCsystemsforahugenumberofdatamayeasilyoccur,i.e.lostdataandnoiseperturbation;●thediscontinuityofsegmentationdeterioratessurfaceaccuracy;●themotionspeedbecomesunsmoothbecauseofthelinearizationofthecurveineachsegment,especiallyinaccelerationanddeceleration.Asthegeneratedcurvesorprofilesmaybeinaparametricform,onlyparametriccurveinformationisrequiredtobeefficientlytransferredamongCAD/CAM/CNCsystemsasshowninFig.I.Shpitalnietal.[1]proposedthecurvesegmentstransferbetweenCADandCNCsystemsandBedietal.[2]proposedtheB-splinecurveandB-splinesurfaceinterpolationalgorithmtoobtainbothaccuratecurvesandgougefreesurface.HuangandYang[3]developedageneralizedinterpolationalgorithmfordifferentparametriccurveswithimprovedspeedfluctuation.Moreover,YangandKong[4]studiedbothlinearandparametricinterpolatorsformachiningprocesses.IntheseCNCsystems,parametriccurvesareprofilesindifferentformatsliketheBeziercurve,B-spline,cubicspline,andNURBS(non-uniformrationalB-spline).Thegeneralparameteriterationmethodusedis错误!未指定书签。ui+1=ui+△(ui)whereuiisthepresentparameter,ui+1isthenextparameter,and△(ui)istheincrementalvalue.Theinterpolatedpointsarecalculatedbysubstitutinguiintothecorrespondingmathematicalmodeltorecovertheoriginallydesignedcurves.Asthecuttermovesstraightbetweencontiguousinterpolatedpoints,twopositionerrorsmayoccuras:(a)radialerror;and(b)chorderror[5]duringmotionforaparametriccurveasshowninFig.2.CADCNCFig.1.ThemachiningsystemswithparameterstransmissionParametriccurveCutterpath●DRadialerrorAChorderrorInterpolatedB●CSegmentationParametriccurverepresentedmodelParametriccurveinterpolatorToolmotionMan-MachineinterfacedeviceFig.2.TheradialerrorandthechorderrorTheradialerroristheperpendiculardistancebetweentheinterpolatedpointsandtheparametriccurve.Basically,theradialerroriscausedbytheroundingerrorofcomputersysterms.Withtherapiddevelopmentofmicroprocessorswithhigherprecision,theradialerrorisnolongeramajorconcerninpresentapplications.Inaddition,thechorderroristhemaximumdistancebetweenthesecantCDandthesecantarcAB.Asmallcurvatureradiuswithafastfeedratecommandmaycausethechorderror.Thus,anadaptivefeedrateisrequiredtokeepthechorderrorwithinanacceptablerange.However,cuttingspeeddominatesthequalityofthemachiningprocess.Toachievethespecifiedfeedrateforparametriccurvesisusuallydifficult.Theundesirablemotionspeedmaydeterioratethequalityofthemachinedsurface.Severalresearchershavedevelopeddifferentinterpolationalgorithmsforparametriccurvestoimprovemotionspeedaccuracy.Bedietal.[2]set△(ui)asaconstanttoformtheuniforminterpolationalgorithmwhichisthesimplestmethodanditschorderrorandcurvespeedhoweverarenotguaranteed.AccordingtotheanalysisofCNCmachinekinematicsandcutterpathgeometry,animprovedinterpolationalgorithminposition,velocity,andaccelerationwasproposedbyChouandYang[6]iftheCNCmachinekinematicsmodelisknownexactly.Further,HoungandYang[3]developedthecubicsplineparametriccurveinterpolatorbyusingtheEulermethod.Shpitalnietal.[1]derivedthesameinterpolatoralgorithmbyusingTaylor’sexpansion.LoandChung[7]proposedthecontouringerrorcompensationinterpolationalgorithmwhichcontainsreal-timecontouringerrorcalculationsandasimplifiedparameteriterationmethodtoachievesatisfactorymotionaccuracy.Inaccelerationanddeceleration,Kim[8]obtainedasimplemethodforparametriccurveswhileitspositionandspeederrorsaresignificant.Thepresentspeed-controlledinterpolationalgorithmisproposedbyderivingasuitablecompensatoryparameterforthefirst-orderapproximation[1,6,8]toobtaindesirablemotionspeed.Then,theproposedinterpolatorisappliedtotheconstant-speedmodeandtheacceleration/decelerationmodetoachieveconstantfeedrateandthespecifiedspeedprofiles,respectively.Thus,thepresentCNCinterpolatorsresultinstablemotionandsmoothlychangedspeedtoavoidmechanicalshockorvibrationinmachiningprocesses.Theproposedspeed-controlledinterpolatorshavealsobeensuccessfullyappliedtoaNURBScommandonapersonalcomputertoachievehighmotionprecision.2.Parametriccurveformulation:SupposeC(u)istheparametriccurverepresentationfunctionandthetimefunctionuisthecurveparameterasu(ti)=uiandu(ti+1)=ui+1ByusingTaylor’sexpansion,theapproximationuptothesecondderivativeisui+1=ui+(du/dt│t=ti)(ti+1–ti)+1/2(d2u/dt2│t=ti)(ti+1–ti)2+HOT(1)AsthecurvespeedV(ui)canberepresentedasV(ui)=‖dC(u)/dt‖u=ui=(‖dC(u)/du‖u=ui)(du/dt│t=ti)thefirstderivativeofuwithtisobtainedasdu/dt│t=ti=V(ui)/(‖dC(u)/dt‖u=ui)(2)BytakingthederivativeofEq.(2),thesecondderivativeofuwithtisd²u/dt²│t=ti=–1/(‖dC(u)/dt‖2u=ui){V(ui)[d(‖dC(u)/dt‖)│u=ui]/dt}(3)where[d(‖dC(u)/dt‖)│u=ui]/dt={[d(‖dC(u)/du‖)│u=ui]/dt}(du/dt│t=ti)={[d(‖dC(u)/dt‖)│u=ui]/dt}{V(ui)/[‖dC(u)/du‖u=ui]}(4)BysubstitutingEq.(4)intoEq.(3),thesecondderivativeofuisrewrittenasd2u/dt2│t=ti=–V2(ui)/(‖dC(u)/du‖3u=ui){[d(‖dC(u)/du‖)│u=ui]/du}(5)where[d(‖dC(u)/du‖)│u=ui]/du=(dC(u)/du)(d2C(u)/du2)/(‖dC(u)/du‖u=ui)(6)BysubstitutingEq.(6)intoEq.(5),thesecondderivativeofubecamesd2u/dt2│t=ti=–[V2(ui)(dC(u)/du)(d2C(u)/du2)]/(‖dC(u)/du‖4u=ui)(7)LetthesamplingtimeininterpolationbeTssecondsandti+1-ti=TsThefirst-andsecond-orderapproximationinterpolationalgorithmsareobtainedbysubstitutingEqs.(2)and(7)intoEq.(1),respectively.Byneglectingthehigherorderterm,theinterpolationalgorithmsinEq.(1)canbeprocessedasfollows:Thefirst-orderapproximationinterpolationalgorithm[1,6,9]ui+1=ui+V(ui)Ts/(‖dC(u)/du‖u=ui)(8)Thesecond-orderapproximationinterpolationalgorithm[6,9]ui+1=ui+V(ui)Ts/(‖dC(u)/du‖u=ui)–{V2(ui)T2s[(dC(u)/du)(d2C(u)/du2)]│u=ui}/[2(‖dC(u)/du‖u=ui)](9)whereV(ui)canbeeitherthefeedratecommand,thespecifiedspeedprofilesofACC/DEC,oranydesiredspeedinageneralmachiningprocess.3.Thespeed-controlledinterpolationalgorithm:3.1.ThecompensatoryparameterInEqs.(8)and(9),thefirst-orderandthesecond-orderinterpolationalgorithmsareapproximatedresultsbyneglectingthehigherordertermasinEq.(1).Therefore,the
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