JournalofZhejiangUnivem時SCIENCEAISSN1673-565X(Print)；ISSN1862-1775(Online)wwwzjueducnIzus；wwwspringerlinkcornE-mail：jzuszjueducnLengeta1JZhejiangUnivSciA2008913)：358-365ResearchoncubicpolynomialaccelerationanddecelerationcontrolmodelforhighspeedNCmachiningHongbinLENGt,Yi-jieWU竹，XiaohongPAN(InstituteofModernManufactureEngineering,ZhefiangUnivers魄Hangzhou31002LChina)E-mail：lenghb2002zjueducn；w塒1116zjueducnReceivedJuly2，2007；revisionacceptedNov3，2007；publishedonlmeDec30，2007Abstract：TosatisfytheneedofhighspeedNC(numericalcontr01)machining，anaccelerationanddeceleration(acedec)controlmodelisproposed。andthespeedcurveisalsoconstructedbythecubicpolynomialTheproposedcon仃olmodelprovidescontinuityofaccelerationwhichavoidstheintensevibrationinhighspeedNCmachiningBasedonthediscretecharacteristicofthedatasamplinginterpolationtheacedeccontroldiscretemathematicalmodelisalsosetupandthediscreteexpressionofthetheoreticaldecelerationlengthisobtainedfurthermoreAimingatthequestionofhardlypredeterminingthedecelerationpointinaccdeccontrolbeforeinterpolationtheadaptiveacedeccontrolalgorithmisdeducedfromtheexpressionsofthetheoreticaldecelerationlengthTheexperimentalresultprovesthattheacedeccontr01modelhasthecharacteristicofeasyimplementation,stablemovementandlowimpactThemodelhasbeenappliedinmultiaxeshighspeedmicrofabricationmachiningSUCCESSfUllyKeywords：Hi曲speedNCmachining，Accelerationanddeceleration(acedec)controlmodel，Cubicspeedcurve，Discretemathematicalmodel，Adaptiveaccelerationanddecelerationcontrolalgorithmdni：101631jzusA071351Documentcode：ACLCnumber：U441+4INTRODUCTIONNC(numericalcontr01)machiningisnowdevelopingtowardshighspeedandhighefficiencyInhighspeedmachining。eachmotionaxismustaccelerateintomovingstateandrealizeprecisestopinfewsecondsSo，researchingonanefficientaccelerationanddeceleration(acedec)controlmethodtomeetthedemandofhighspeedmachiningisoneofthecriticalproblemsinmodemhighperformanceNCsystemThecommonlyusedmethodsinmostdomesticeconomicalCNC(computernumericalcontr01)sys-ternarelinearaccdecmodeandexponentialacedecmodeButvibrationiseasilycausedbydiscontinuityofacceleration，whichaffectsmachiningqualityandequipmentlife(HuetaL，1999；Zhang，2002)TolCorrespondingauthorProjectsupportedbytheHi1echResearchandDevelopmentPro-gram(863)ofChina(No2006AA042233)，theNationalNaturalScienceFoundationofChina(No50575205)andtheNaturaIScienceFoundationofZhejiangProvmceosY104243andY105686)，Chinadecreasethevibration，thes-curveacedecmotionplanningmethodisadoptedinadvancedCNCsystemTheacceleration(ordeceleration)stageinthes-curveaccdecmotionplanningmethodiscomposedofincreasingaccelerationphase，constantaccelerationphaseanddecreasingaccelerationphase(orincreasingdecelerationphase，constantdecelerationphaseanddecreasingdecelerationphase)ThroughgradedcontrolofaccelerationineachstagemachiningfeedratecanbechangedsmoothlyHoweveLthea1gorithmistoocomplex(KaanandYusuf,200l：NamandYang2004)Thetrigonometricfunctionacedecmethodismoreflexible，butthealgorithmisalsocomputationextensiveandmorecomplex，whichisrelativelydi幣culttosatisfyrealtimerequirement(GunandLi，2003)Themethodselectingpolynomialfunctionscangeneratesomanykindsofacedeccharacteristicsand，furthermore，canmakethechar-acteristicsofdecelerationbeindependentfromthoseofaccelerationInordertoachievehighperformancemotioncontrol，themotionprofilesmustbematched堡萬方數(shù)據(jù)Lengeta1JZhejiangUnivSciA200893)：358-365tothesystemlimitssuchasthemaximumaccelera-tionandthemaximumvelocityIfpositiontrajecto-riesofwhichthevelocityprofilesaresmootharegeneratedbythemethodselectingpolynomialfunc-tionsitrequiresalotofcomputations(InabaandSakakibara1985；Park1996)ThedigitalconvolutionmethodismuchmoreemcienlthanthemethodselectingpolynomialfunctionsandiseasilyimplementedbyhardwareBut，inthevelocityprofilesgeneratedbythemethodtheaccelerationintervalisalwaysthesameasthedecelerationintervalandthecharacteristicsofthedecelerationaredependentonthoseoftheacceleration(KhalsaandMahoney,1990；ChenandLee1998)AsimpletoefficientsstoredmethodforgeneratingvelocityprofilesisproposedAccordingtothedesiredcharacteristicsofaccdec，eachsetofcoefficientsiscalculatedandstoredGivenamovingdistanceandacedecintervals，avelocityprofilehavingthedesiredcharacteristicsofaccdeccanbee幣cientlygeneratedbyusingthesetoemcientsButforlongandshortdistancesthesameacedecintervalsareselected；thee幣cientvelocityprofilecannotbecalculatedforthevarieddistancemovements(JeonandHa，2000)ThispaperisorganizedasfollowsSection2proposesacubicpolynomialacedeccontrolmodelforhighspeedNCmachiningBasedonthediscretecharacteristicofthedatasamplinginterpolation，thecubicpolynomialaccdeccontroldiscretemathe-maticalmodelissetupinSection3TheadaptiveacedeccontrolalgorithmforpredeterminingthedecelerationpointofarbitraryroutesegmentisdeducedinSection4ExperimentalresultsarepresentedinSection5andconclusionsaresummarizedinSection6struttedbythecubicpolynomial，359y(甜)=(q+2a2u+3a3u2+4a4u3)fm(1)Itisassumedthattmistheacceleratingordeceleratingdurationtimewhichistakenassynchronizedmotionaxestoaccelerateordeceleratefromthebeginningspeedtotheendspeed，妒ttm，fO，tmTheotherkineticcharacteristiccurvesofaccel-erationandierkcanbeobtainedbydifferentiatingthefeedratecurve，A(u)撕=(2a2+6asU+12a46as24a4u)t：鏟。(2)【，(“)=(+Again，integratingEq(1)withrespecttotimeyieldsthedisplacementcurvefunctionas，S(zf)=a0+aI+a2u2+a3u3+a4u4(3)Theboundaryconditionsare，s(0)=0，v(o)=K，礦(1)=圪，A(O)=A(1)=0，whereKand圪standforthebeginningspeedinacestage(ordecstage)andtheendspeedThenthecurvilinearfunctionsarederivedas，(“)=6(圪一K)(12-)4，彳(u)=：6K(V+,3-(KV,)一(uK-)甜u：2)+t2m(,V(uK一圪)材，(4)=K+3(KK咖2+2(K一圪弦，s(“)=tmKU+(圪一V。)tmU3+05(K-Vo)tmU4CUBICPOLYNOMIALACCDECCONTROLequalFtoromtheE孟q&imumuac=0ce5lerathteioanc彳ceml瓤eraTtihoennfm：ains(4)，let，彳(“)MODELbededucedas，Tomeettheneedofhighspeedmachining，thefeedratemustbechangedsmoothlyandtheaccelerationmustbecontinuousTheboundaryconditionsare：(1)thedisplacementatthebeginningtimeis0；(2)boththebeginningspeedandtheendspeedarethesameasrequired；(3)theaccelerationsbothatthebeginningtimeandtheendtimeare0Theaccdecfeedratecurve如nctioniscontm=3KKI“24。)=nT(5)玎isarealnumber,denotingthetheoreticaltimesofthetheoreticalrunningtimetmtointerpolationperiodTwhendecelerating(oraccelerating)fromKtoWhent=tm，thetheoreticalacceleration(orde-celeration)lengthSlisobtainedfromEq(4)as，萬方數(shù)據(jù)Lengeta1JZhejiangUnivSciA200893)：358-365墨=(圪+V。)tm2=3lK2一曙II(4A。)Therelationofspeed(叻，acceleration0)andjerkisexpressedasshowninFig1FromFig1，inthecubicpolynomialacedeccontrolmodeltheaccelerationiSconsecutive。whichavoidstheOccur-renceofintensevibrationinhighspeedmachiningThecalculationsofjerk，acceleration，speedanddisplacementinvaryingspeedprocessaresimpleandeasytorealizebecauseoffeWfourfundamentaloperations礦匕二上二金一p#毒三7l一一Fig1Speed(，acceleration似)andjerk(J)ofthecubicpolynomialaccdeccontrolmodelCUBICPOLYNOMIALACCDECCONTROLDISCRETEMODELCharacteristicsofdatasamplinginterpolationThecontrollingwithdatasamplinginterpolationisatypeofdiscretecontrollingmodeThedatasampiinginterpolationisbasedonapproximatingacurvewithstraightlinesegments，whoselengthsarepro-portionaltothelocalaxialplannedvelocitiesTheacedeccon仃OlalgorithmiSperformedonthedesiredfeedratecommand礦firstlyThenthefeedratecornmandV(afteracedeccontrolalgorithm)issenttotheinterpolatortocomputethetravelingdistaneecomponentforeachaxisintheCartesiancoordinates(KYand刁ThetravelingdistancecomponentsZ厶Yand覘alongtheXYandZaxesaretransmittedtothemotioncontrolroutine邪positioncommandstothepositioncontrolloopforthedesiredmachiningfinallyWhendatasamplinginterpolationisusedinpo-sitioncontr01itiStheconditiontOmakesurethateverysynchronizedmotionaxisreachesthedestina-tionsimultaneouslyandtheirmovementiScontinuousSOthateachaxisrunningtimetisjustintegraltimesthatoftheinterpolationperiod互ietimepartitioningruleThisconditioncanberealizedbyadjustingthecommandfeedrateofeachsynchronizedmotionaxis(ieacceleratingordecelerating)(Guoeta1，2003)ThroughtheaboveanalysesnshouldbeanintegerSetNastheminimumintegernotsmallerthan玎N=ceil(n)NnReplacingtherealnumber，1withtheintegerNinEq(5)yields，=3IKKI(24。)=NT(6)ConstructionofdiscretemodelGivenanycurvegivenbythefunctiony=flt)，theinterval【to，糾isdividedintoNdivisionsPickingtheleftendpointstodeterminetheheightsisshowninFig2a；theapproximateareaunder少鋇Dontheintervalto，明is，r兒燦嘻小娟棚孚)孚Ototlt2r卜l，l肛ltMtOtotlt2缸lnt#-ltutFig2ImplementcurveofJft)(a)Leftendpointsusedfortheheightsofrectangles；(b)Rightendpointsusedfortheheightsofrectangles萬方數(shù)據(jù)Lengeta1JZhejiangUnivSciA20089f3J：358-365ThiswayofapproximatingtheareaunderthecurveiscalledtheleftsummethodChoosingthefightendpointstodeterminetheheightsisshowninFig2b；theapproximateareaunder尸砜f)ontheintervalto，明is，Ij刪，喜小+z可tN-to警ThiswayofapproximatingtheareaunderthecurveiscalledtherightsummethodSupposethattheslopeofierkiSKsandinitialvaluesofjerk,acceleration，speedanddisplacementareJo，Ao，goandSo，respectivelyThediscretemodelofjerkcurve，implementedbyIeftsummethod，iS。J(iT)=厶+0(f一1)T，1iN(7)Thediscretemodelsofacceleration，speedanddisplacementcurve，realizedbyrightsummethod，are，iA(iT)=40+，(7)r，IiN+I，=1I礦(f丁)=Vo+A(jT)T，j=l，s(fr)=so+V(jT)T，j=l1iN1f361州恥器一嘉N(N齋：、(+1)(+2)r2+1)(+2)r2lfNThediscretemodelsofaccelera