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JournalofZhejiangUnivem时SCIENCEAISSN1673565XPrintISSN18621775Onlinewww.zju.educn/Izuswww.springerlink.cornEmailjzuszju.edu.cnLengeta1./JZhejiangUnivSciA2008913358365ResearchoncubicpolynomialaccelerationanddecelerationcontrolmodelforhighspeedNCmachining★HongbinLENGt,YijieWU竹,XiaohongPANInstituteofModernManufactureEngineering,ZhefiangUnivers魄Hangzhou31002LChinaEmaillenghb2002zju.edu.cnw埘1116zju.edu.cnReceivedJuly2,2007revisionacceptedNov.3,2007publishedonlmeDec.30,2007AbstractTosatisfytheneedofhighspeedNCnumericalcontr01machining,anaccelerationanddecelerationace/deccontrolmodelisproposed。andthespeedcurveisalsoconstructedbythecubicpolynomial.Theproposedcon仃olmodelprovidescontinuityofacceleration.whichavoidstheintensevibrationinhighspeedNCmachining.Basedonthediscretecharacteristicofthedatasamplinginterpolation.theace/deccontroldiscretemathematicalmodelisalsosetupandthediscreteexpressionofthetheoreticaldecelerationlengthisobtainedfurthermore.Aimingatthequestionofhardlypredeterminingthedecelerationpointinacc/deccontrolbeforeinterpolation.theadaptiveace/deccontrolalgorithmisdeducedfromtheexpressionsofthetheoreticaldecelerationlength.Theexperimentalresultprovesthattheace/deccontr01modelhasthecharacteristicofeasyimplementation,stablemovementandlowimpact.ThemodelhasbeenappliedinmultiaxeshighspeedmicrofabricationmachiningSUCCESSfUlly.KeywordsHi曲speedNCmachining,Accelerationanddecelerationace/deccontrolmodel,Cubicspeedcurve,Discretemathematicalmodel,Adaptiveaccelerationanddecelerationcontrolalgorithmdni10.1631/jzus.A071351DocumentcodeACLCnumberU441.4INTRODUCTIONNCnumericalcontr01machiningisnowde.velopingtowardshighspeedandhighefficiency.Inhighspeedmachining。eachmotionaxismustaccel.erateintomovingstateandrealizeprecisestopinfewseconds.So,researchingonanefficientaccelerationanddecelerationace/deccontrolmethodtomeetthedemandofhighspeedmachiningisoneofthecriticalproblemsinmodemhighperformanceNCsystem.ThecommonlyusedmethodsinmostdomesticeconomicalCNCcomputernumericalcontr01systernarelinearacc/decmodeandexponentialace/decmode.Butvibrationiseasilycausedbydiscontinuityofacceleration,whichaffectsmachiningqualityandequipmentlifeHuetaL,1999Zhang,2002.TolCorrespondingauthorProjectsupportedbytheHi.1echResearchandDevelopmentProgram863ofChinaNo.2006AA042233,theNationalNaturalScienceFoundationofChinaNo.50575205andtheNaturaIScienceFoundationofZhejiangProvmceⅢos.Y104243andY105686,Chinadecreasethevibration,thescurveace/decmotionplanningmethodisadoptedinadvancedCNCsystem.Theaccelerationordecelerationstageinthescurveacc/decmotionplanningmethodiscomposedofin.creasingaccelerationphase,constantaccelerationphaseanddecreasingaccelerationphaseorincreas.ingdecelerationphase,constantdecelerationphaseanddecreasingdecelerationphase.Throughgradedcontrolofaccelerationineachstage.machiningfeedratecanbechangedsmoothly.HoweveLthea1.gorithmistoocomplexKaanandYusuf,200lNamandYang.2004.Thetrigonometricfunctionace/decmethodismoreflexible,butthealgorithmisalsocomputationextensiveandmorecomplex,whichisrelativelydi币culttosatisfyrealtimerequirementGunandLi,2003.Themethodselectingpolynomialfunctionscangeneratesomanykindsoface/deccharacteristicsand,furthermore,canmakethecharacteristicsofdecelerationbeindependentfromthoseofacceleration.Inordertoachievehighperformancemotioncontrol,themotionprofilesmustbematched堡万方数据Lengeta1./JZhejiangUnivSciA200893358365tothesystemlimitssuchasthemaximumaccelerationandthemaximumvelocity.Ifpositiontrajectoriesofwhichthevelocityprofilesaresmootharegeneratedbythemethodselectingpolynomialfunctions.itrequiresalotofcomputationsInabaandSakakibara.1985Park.1996.Thedigitalconvolutionmethodismuchmoreemcienlthanthemethodselectingpolynomialfunctionsandiseasilyimple.mentedbyhardware.But,inthevelocityprofilesgeneratedbythemethod.theaccelerationintervalisalwaysthesameasthedecelerationintervalandthecharacteristicsofthedecelerationaredependentonthoseoftheaccelerationKhalsaandMahoney,1990ChenandLee.1998.Asimpletoefficientsstoredmethodforgeneratingvelocityprofilesisproposed.Accordingtothedesiredcharacteristicsofacc/dec,eachsetofcoefficientsiscalculatedandstored.Givenamovingdistanceandace/decintervals,avelocityprofilehavingthedesiredcharacteristicsofacc/deccanbee币cientlygeneratedbyusingthesetoem.cients.Butforlongandshortdistances.thesameace/decintervalsareselectedthee币cientvelocityprofilecannotbecalculatedforthevarieddistancemovementsJeonandHa,2000.Thispaperisorganizedasfollows.Section2proposesacubicpolynomialace/deccontrolmodelforhighspeedNCmachining.Basedonthediscretecharacteristicofthedatasamplinginterpolation,thecubicpolynomialacc/deccontroldiscretemathematicalmodelissetupinSection3.Theadaptiveace/deccontrolalgorithmforpredeterminingthedecelerationpointofarbitraryroutesegmentisdeducedinSection4.ExperimentalresultsarepresentedinSection5.andconclusionsaresummarizedinSection6.struttedbythecubicpolynomial,359y甜q2a2u3a3u24a4u3/fm.1Itisassumedthattmistheacceleratingordeceleratingdurationtime.whichistakenassynchronizedmotionaxestoaccelerateordeceleratefromthebeginningspeedtotheendspeed,妒t/tm,f∈O,tm.Theotherkineticcharacteristiccurvesofaccelerationandierkcanbeobtainedbydifferentiatingthefeedratecurve,.㈣Au撕2a26asU12a.46as24a4u/t铲∥。2【,.Again,integratingEq.1withrespecttotimeyieldsthedisplacementcurvefunctionas,Szfa0aI.a2u2a3u3a4u4.3Theboundaryconditionsare,s00,voK,矿1圪,AOA10,whereKand圪standforthebeginningspeedinacestageordecstageandtheendspeed.Thenthecurvilinearfunctionsarederivedas,,6圪一K12/4,彳u6KV,3KV,一uK甜u2/t2m,VuK一圪材,,4K3KK咖22K一圪弦,stmKU圪一V。tmU30.5KVotmU4.CUBICPOLYNOMIALACC/DECCONTROLequalFtoromtheE孟qimumuac0ce5lerathteioanc彳ceml瓤.eraTtihoennfmains.4,let.,彳MODELbededucedas,Tomeettheneedofhighspeedmachining,thefeedratemustbechangedsmoothlyandtheaccelerationmustbecontinuous.Theboundaryconditionsare1thedisplacementatthebeginningtimeis02boththebeginningspeedandtheendspeedarethesameasrequired3theaccelerationsbothatthebeginningtimeandtheendtimeare0.Theacc/decfeedratecurve如nctioniscontm3KKI24。。nT.5玎isarealnumber,denotingthetheoreticaltimesofthetheoreticalrunningtimetmtointerpolationperiodTwhendeceleratingoracceleratingfromKto%.Whenttm,thetheoreticalaccelerationordecelerationlengthSlisobtainedfromEq.4as,万方数据Lengeta1./JZhejiangUnivSciA200893358365墨圪V。tm/23lK2一曙II4A.。.Therelationofspeed叻,acceleration0andjerk∽isexpressedasshowninFig.1.FromFig.1,inthecubicpolynomialace/deccontrolmodel.theac.celerationiSconsecutive。whichavoidstheOccurrenceofintensevibrationinhighspeedmachining.Thecalculationsofjerk,acceleration,speedanddis.placementinvaryingspeedprocessaresimpleandeasytorealize.becauseoffeWfourfundamentalop.erations.矿匕二上二金.一p毒三}\/7l\/一\/一Fig.1Speed,acceleration似andjerkJofthecubicpolynomialacc/deccontrolmodelCUBICPOLYNOMIALACC/DECCONTROLDISCRETEMODELCharacteristicsofdatasamplinginterpolationThecontrollingwithdatasamplinginterpolationisatypeofdiscretecontrollingmode.Thedatasam.piinginterpolationisbasedonapproximatingacurvewithstraightlinesegments,whoselengthsareproportionaltothelocalaxialplannedvelocities.Theace/deccon仃OlalgorithmiSperformedonthedesiredfeedratecommand矿firstly.Thenthefeedratecorn.mandVafterace/deccontrolalgorithmissenttotheinterpolatortocomputethetravelingdistaneecom.ponentforeachaxisintheCartesiancoordinates△K△Yand△刁.Thetravelingdistancecomponents△Z厶Yand觇alongtheX.YandZaxesaretransmittedtothemotioncontrolroutine邪positioncommandstothepositioncontrolloopforthedesiredmachiningfinally.Whendatasamplinginterpolationisusedinpositioncontr01.itiStheconditiontOmakesurethateverysynchronizedmotionaxisreachesthedestinationsimultaneouslyandtheirmovementiScontinuousSOthateachaxisrunningtimetisjustintegraltimesthatoftheinterpolationperiod互i.e.timepartitioningrule.Thisconditioncanberealizedbyadjustingthecommandfeedrateofeachsynchronizedmotionaxisi.e.acceleratingordeceleratingGuoeta1.,2003.Throughtheaboveanalyses.nshouldbeanin.teger.SetNastheminimumintegernotsmallerthan玎.Nceiln.N≥n.Replacingtherealnumber,1withtheintegerNinEq.5yields,3IKKI/24。。NT.6ConstructionofdiscretemodelGivenanycurvegivenbythefunctionyflt,theinterval【to,纠isdividedintoNdivisions.PickingtheleftendpointstodeterminetheheightsisshowninFig.2atheapproximateareaunder少钡Dontheintervalto,明is,r儿灿嘻小娟棚孚孚.Ototlt2r卜l,,l肛ltMtOtotlt2缸lntltutFig.2ImplementcurveofJft.aLeftendpointsusedfortheheightsofrectanglesbRightendpointsusedfortheheightsofrectangles万方数据Lengeta1./JZhejiangUnivSciA20089f3J358365Thiswayofapproximatingtheareaunderthecurveiscalledtheleftsummethod.ChoosingthefightendpointstodeterminetheheightsisshowninFig.2btheapproximateareaunder尸砜fontheintervalto,明is,IjⅣ删,≈喜小z可tNto警.Thiswayofapproximatingtheareaunderthecurveiscalledtherightsummethod.SupposethattheslopeofierkiSKsandinitialvaluesofjerk,acceleration,speedanddisplacementareJo,Ao,goandSo,respectively.Thediscretemodelofjerkcurve,implementedbyIeftsummethod,iS。JiT厶0f一1T,1≤i≤N.7Thediscretemodelsofacceleration,speedanddisplacementcurve,realizedbyrightsummethod,are,iAiT.40∑,/7r,IiNI,/1I矿f丁Vo∑AjTT,jl,sfrso∑VjTT,jl1≤i≤N.1≤f≤Ⅳ.361州耻器一嘉NN斋‰、Ⅳ1Ⅳ2r21Ⅳ2r2l≤f≤NThediscretemodelsofacceleration,speedanddisplacementcurve,implementedbyrightsummethod,are,彳fr坠竺二竺f一篁兰二竺2f,、NN2TNN1Ⅳ2丁71≤i≤N1yczr,K孑÷糕zi揣r2一j噪f3,liNNN1Ⅳ2SfnN33N21V.2N1Ve乃、7NN1Ⅳ2竺二竺堕丝±12Ti24.竺二竺21丝二乃,2NN1Ⅳ2NN1Ⅳ2一旦熹忑巩liN.2NN1Ⅳ29c8,me。罢等蠹慧i酱竺罴舞嚣篡。警Tosatisfythattheaccelerationattheendtimeis0andconsideringcharacteristicsofimplementingtheaccelerationcurvebyfightsummethodtheheightofcorrespondingrectangleareaattime矗is以廿1,theboundaryconditionsoftheaccelerationcurveare,40,彳Ⅳ1T0.Theboundaryconditionsofspeedanddisplacementcurveare,N一1Vff一2210Fortheaccelerationcurve,when圪刮7mx,thediscreteexpressionofthetheoreticalaccelerationlengthis,只MT1一V,T2Forthedecelerationcurve,when圪屯,thediscreteexpressionofthetheoreticaldecelerationlengthSdis,蜀生笋TN1Vff,VoK,VNT圪,so0.whereRmisthemaximumallowedfeedrate,MisSothediscretemodelofthejerkcurve,impletotalinterpolationperiodsinaccelerationprocess,Nmentedbyleftsummethod,is,istotalinterpolationperiodsindecelerationprocess.万方数据
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