南京工程學院自動化學院本科畢業設計（論文）1附錄A：英文資料AGeneralizedApproachfortheAccelerationandDecelerationofCNCMachineToolsJaeWookJeonDepartmentofControlandInstrumentationEngineeringSungkyunkwanUniversity300Chunchun-Dong,Jangan-Gu,SuwonCity,KoreaAbstractManytechniquesfortheaccelerationanddecelerationofCNCmachinetoolshavebeenproposedinordertomakeCNCmachinetoolsperformgivenmachiningtasksefficiently.Sincetheyshouldbecalculatedinalimitedtime,mostofthemarenotcomputationallyintensive.However,theseprevioustechniquescannotgeneratevelocityprofileshavingsomekindsofaccelerationanddecelerationcharacteristicsthoughtheycangeneratevelocityprofileshavingvariousaccelerationanddecelerationcharacteristics.Thispaperproposesageneralizedapproachforgeneratingvelocityprofileswhichcannotbegeneratedbyprevioustechniquesaswellasvelocityprofilesgeneratedbythem.Accordingtothedesiredcharacteristicsofaccelerationanddeceleration,eachsetofcoefficientsiscalculatedandisstored.Givenamovingdistance,anaccelerationinterval,andadecelerationintervalforCNCmachinetools,avelocityprofilehavingthedesiredcharacteristicsofaccelerationanddecelerationcanbeefficientlygeneratedbyusingthesecoefficients.Itwillbeshownhowtogeneratetypicalvelocityprofilesbytheproposedtechnique.I.INTRODUCTIONThedemandforbetteraccuracyinthemanufacturingofcomplicatedpartsandthedesiretoincreaseproductivityhavedevelopedCNCsystemssothatCNCmachinetoolsmovemoreaccuratelyandmorequickly.SincethecombinedcharacteristicsofthecontrolandthemachinetooldeterminethefinalaccuracyandproductivityoftheCNCsystem,therearemanyfactorstoconsiderforimprovingthesequantities.Oneoftheimportantfactorsisefficientlygeneratingvelocityprofileswhichhavethedesiredaccelerationanddecelerationcharacteristicswhichinturnaredeterminedaccordingtogivenmachiningtasks.ManyresearchershaveproposedtechniquesforgeneratingvelocityprofilesofCNC南京工程學院自動化學院本科畢業設計（論文）2machinetools.Oneofthemisgeneratingvelocityprofilesbytheselectionofpolynomialfunctions1.Thistechniquecangeneratesomanykindsofvelocityprofilesandfurthermorecanmakethecharacteristicsofdecelerationbeindependentfromthatofacceleration.Themajorproblemofthistechniqueiscomputationalloadtoincreasealmostexponentiallywiththeorderofpolynomialintheaccelerationorthatinthedeceleration.Duetotimeconstraints,itisverydifficulttoapplythesetechniquestocontrollingCNCsystems.Otherprevioustechniquesforgeneratingvelocityprofilesisbasedonadigitalconvolution2-6.Thesetechniquesaremuchmoreefficientthanthetechniquesselectingpolynomialfunctionsandareeasilyimplementedbyahardware.But,invelocityprofilesgeneratedbythesetechniques,theaccelerationintervalisalwayssameasthedecelerationintervalandthecharacteristicsofdecelerationaredependentonthatoftheacceleration.Thus,somevelocityprofilescannotbegeneratedbythesetechniques.Inthispaper,ageneralizedapproachfortheaccelerationanddecelerationofCNCmachinetoolsisproposed.Theproposedtechniqueisassimpleandefficientasthetechniquesbasedonadigitalconvolutionandcangeneratevelocityprofileswhichhavemorevariouscharacteristicsofaccelerationanddecelerationthanthetechniquesbasedonadigitalconvolutioncan.Thatis,theproposedtechniquecangeneratevelocityprofilesofwhichthedecelerationcharacteristicsareindependentfromtheaccelerationcharacteristics.First,somecoefficientsarecalculatedandarestoredaccordingtotheaccelerationcharacteristic,thedecelerationcharacteristic,andtheaccelerationinterval,andthedecelerationinterval.Thengivenadesiredmovingdistance,avelocityprofilewhichhasthedesiredcharacteristicsisgeneratedbycalculatingthepositionincrementduringeachsamplingtime.Thepositionincrementduringeachsamplingiscalculatedbymultiplyingthestoredcoefficientswiththeproductofonesamplingtimeandthevelocityafteracceleration.InsectionII,existingtechniquesforgeneratingvelocityprofilesofCNCmachinetoolswillbeexplained.InsectionIII,itwillbeexplainedhowtogenerateadesiredvelocityprofilebytheproposedtechnique.Theproposedtechniqueandotherexistingtechniqueswillbecompared.InsectionIV,itwillbeshownthattheproposedtechniquecangeneratesomeusefulvelocityprofilesforCNCmachinetools.II.PREVIOUSTECHNIQUES南京工程學院自動化學院本科畢業設計（論文）3Fortheillustrationoftheprevioustechniques,letusconsideronesingle-axiscontrolsystemofwhichthemaximumvelocity,themaximumacceleration,andthesamplingtimeareVmax，Amax，andTsrespectively.IfthissystemmovesthegivendistanceSatthemaximumvelocityVmax,thenthemovementtimeT1intherectangularvelocityprofilewillbeT1=S/Vmax=pTs,(1)Selectinganintegernwhichisthesmallestintegeramongintegerswhichareequaltoorgreaterthanp,theresultingrectangularvelocityprofileisconstructedasinFig.1.ThevelocityandpositionequationsforthisprofileareThepositionincrementduringeachsamplingtimeis000()()(1)/,1(4)sssPkTPkTPkTSnknwhere0()sPkTand0(1)sPkTarethepositioncommandsatthekthandthe(1)kthsamplingtimesrespectively.Thatis,thepositionincrementduringeverysamplingtimeisthesame.However,sincenophysicalsystemcanachievetheaboverectangularvelocityprofileduetoimpulseacceleration,theaccelerationintervaltoincreasevelocityfromtheresttoaspecifiedvalueandthedecelerationintervaltodecreasevelocityfromthespecifiedvaluetotherestareneeded.II-A.SelectionofPolynomialFunctionsGivenanaccelerationintervalTs=naTs,adecelerationintervalTd=ndTswherena=nd,andadistanceS,atrapezoidalvelocityprofilewhichhasthelinearaccelerationanddecelerationcharacteristicscanbeconstructedasinFig.2.Ifthisprofilehastheconstantvelocityinterval,thenn=S/(VmaxTs)islargerthanna.ThevelocityandpositionequationsforthisprofileareAnd1()asaassssasaasSttTnTTSVtTtnTnTStnSnTtnTTnTTnnT(5)00()/()0,(2)()/()0,(3)ssVtSnTtnTPtSnTtnT南京工程學院自動化學院本科畢業設計（論文）4Fig.1.ArectangularvelocityprofileformovingadistanceSFig.2.AtrapezoidalvelocityprofileformovingadistanceSIntheabovederivation,itisassumedthattheaccelerationintervalisthesameasthedecelerationintervalandtheconstantvelocityintervalispresent.Inthecasethattheaccelerationintervalisdifferentfromthedecelerationintervalortheconstantvelocityintervalisnotpresent,similarequationsarepossibletoderive.Thistrapezoidalvelocityprofilewhichhast