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JournalofZhejiangUnivem時(shí)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(wàn)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(wàn)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(wàn)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
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