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Journal of Zhejiang Univem时SCIENCE AISSN 1673-565X(Print)；ISSN 1862-1775(Online)wwwzjuedu cnIzus；wwwspringerlinkcornE-mail：jzuszjueducnLeng et a1J Zhejiang Univ SciA 2008 913)：358-365Research on cubic polynomial acceleration and decelerationcontrol model for high speed NC machiningHongbin LENGt,Yi-jie WU竹，Xiaohong PAN(Institute ofModern Manufacture Engineering,Zhefiang Univers魄Hangzhou 31002L China)E-mail：lenghb2002zjueducn；w埘11 1 6zjueducnReceived July 2，2007；revision accepted Nov3，2007；published onlme Dec30，2007Abstract：To satisfy the need of high speed NC(numerical contr01)machining，an acceleration and deceleration(acedec)control model is proposed。and the speed curve is also constructed by the cubic polynomialThe proposed con仃ol model providescontinuity ofaccelerationwhich avoids the intense vibration in high speed NC machiningBased on the discrete characteristic ofthe data sampling interpolationthe acedec control discrete mathematical model is also set up and the discrete expression of thetheoretical deceleration length is obtained furthermoreAiming at the question of hardly predetermining the deceleration point inaccdec control before interpolationthe adaptive acedec control algorithm is deduced from the expressions of the theoreticaldeceleration lengthThe experimental result proves that the acedec contr01 model has the characteristic of easy implementation,stable movement and low impactThe model has been applied in multiaxes high speed micro fabrication machining SUCCESSfUllyKey words：Hi曲speed NC machining，Acceleration and deceleration(acedec)control model，Cubic speed curve，Discretemathematical model，Adaptive acceleration and deceleration control algorithmdni：101631jzusA071351 Document code：A CLC number：U441+4INTRODUCTIONNC(numerical contr01)machining is now developing towards high speed and high efficiencyInhigh speed machining。each motion axis must accelerate into moving state and realize precise stop in fewsecondsSo，researching on an efficient accelerationand deceleration(acedec)control method to meet thedemand of high speed machining is one of the criticalproblems in modem high performance NC systemThe commonly used methods in most domesticeconomical CNC(computer numerical contr01)sys-tern are linear accdec mode and exponential acedecmodeBut vibration is easily caused by discontinuityof acceleration，which affects machining quality andequipment life(Hu et aL，1 999；Zhang，2002)Tol Corresponding authorProject supported by the Hi1ech Research and Development Pro-gram(863)of China(No2006AA042233)，the National NaturalScience Foundation ofChina(No50575205)and the NaturaI ScienceFoundation of Zhejiang ProvmceosY104243 and Y105686)，Chinadecrease the vibration，the s-curve acedec motionplanning method is adopted in advanced CNC systemThe acceleration(or deceleration)stage in the s-curveaccdec motion planning method is composed of increasing acceleration phase，constant accelerationphase and decreasing acceleration phase(or increasing deceleration phase，constant deceleration phaseand decreasing deceleration phase)Through gradedcontrol of acceleration in each stagemachiningfeedrate can be changed smoothlyHoweveL the a1gorithm is too complex(Kaan and Yusuf,200 l：Namand Yang2004)The trigonometric function acedecmethod is more flexible，but the algorithm is alsocomputation extensive and more complex，which isrelatively di币cult to satisfy realtime requirement(Gun and Li，2003)The method selecting polynomialfunctions can generate so many kinds of acedeccharacteristics and，furthermore，can make the char-acteristics of deceleration be independent from thoseof accelerationIn order to achieve highperformancemotion control，the motion profiles must be matched堡万方数据Leng et a1JZhejiang Univ SciA 200893)：358-365to the system limits such as the maximum accelera-tion and the maximum velocityIf position trajecto-ries of which the velocity profiles are smooth aregenerated by the method selecting polynomial func-tionsit requires a lot of computations(Inaba andSakakibara1 985；Park1 996)The digital convolution method is much more emcienl than the methodselecting polynomial functions and is easily implemented by hardwareBut，in the velocity profilesgenerated by the methodthe acceleration interval isalways the same as the deceleration interval and thecharacteristics of the deceleration are dependent onthose ofthe acceleration(Khalsa and Mahoney,1990；Chen and Lee1 998)A simple toefficients storedmethod for generating velocity profiles is proposedAccording to the desired characteristics of accdec，each set ofcoefficients is calculated and storedGivena moving distance and acedec intervals，a velocityprofile having the desired characteristics of accdeccan be e币ciently generated by using these toemcientsBut for long and short distancesthe sameacedec intervals are selected；the e币cient velocityprofile cannot be calculated for the varied distancemovements(Jeon and Ha，2000)This paper is organized as followsSection 2proposes a cubic polynomial acedec control modelfor high speed NC machiningBased on the discretecharacteristic of the data sampling interpolation，thecubic polynomial accdec control discrete mathe-matical model is set up in Section 3The adaptiveacedec control algorithm for predetermining thedeceleration point of arbitrary route segment is deduced in Section 4Experimental results are presentedin Section 5and conclusions are summarized inSection 6strutted by the cubic polynomial，359y(甜)=(q+2a2u+3a3u2+4a4u3)fm (1)It is assumed that tm is the accelerating or deceleratingduration timewhich is taken as synchronized motionaxes to accelerate or decelerate from the beginningspeed to the end speed，妒ttm，fO，tmThe other kinetic characteristic curves of accel-eration and ierk can be obtained by differentiating thefeedrate curve，A(u)撕=(2a2+6asU+1 2a46as24a4u)t：铲。(2)【，(“)=( + Again，integrating Eq(1)with respect to time yieldsthe displacement curve function as，S(zf)=a0+aI+a2u2+a3u3+a4u4 (3)The boundary conditions are，s(0)=0，v(o)=K，矿(1)=圪，A(O)=A(1)=0，where K and圪stand for the beginning speed in acestage(or dec stage)and the end speedThen the curvilinear functions are derived as，(“)=6(圪一K)(12-)4，彳(u)=：6K(V+,3-(KV,)一(uK-)甜u：2)+t2m(,V(u K一圪)材，(4)=K+3(KK咖2+2(K一圪弦，s(“)=tmKU+(圪一V。)tmU3+05(K-Vo)tmU4CUBIC POLYNOMIAL ACCDEC CONTROL equalFtoromtheE孟q&imumuac=0ce5lerathteioanc彳ceml瓤eraTtihoenn fm：ains(4)，let ， 彳(“)MODEL be deducedas，To meet the need of high speed machining，thefeedrate must be changed smoothly and the acceleration must be continuousThe boundary conditions are：(1)the displacement at the beginning time is 0；(2)both the beginning speed and the end speed are thesame as required；(3)the accelerations both at thebeginning time and the end time are 0The accdec feedrate curve如nction is contm=3 KK I“24。)=nT (5)玎is a real number,denoting the theoretical times ofthe theoretical running time tm to interpolation periodT when decelerating(or accelerating)from K toWhen t=tm，the theoretical acceleration(or de-celeration)length Sl is obtained from Eq(4)as，万方数据Leng et a1J Zhejiang Univ SciA 2008 93)：358-365墨=(圪+V。)tm2=3l K2一曙II(4A。)The relation of speed(叻，acceleration 0)andjerkis expressed as shown in Fig1From Fig1，inthe cubic polynomial acedec control modelthe acceleration iS consecutive。which avoids the Occur-rence of intense vibration in high speed machiningThe calculations of jerk，acceleration，speed and displacement in varying speed process are simple andeasy to realizebecause of feW four fundamental operations 矿匕二上二金一p#毒三 7l 一 一Fig1 Speed(，acceleration似)and jerk(J)ofthe cubicpolynomial accdec control modelCUBIC POLYNOMIAL ACCDEC CONTROLDISCRETE MODELCharacteristics of data sampling interpolationThe controlling with data sampling interpolationis a type of discrete controlling modeThe data sampiing interpolation is based on approximating a curvewith straightline segments，whose lengths are pro-portional to the local axial planned velocitiesTheacedec con仃Ol algorithm iS performed on the desiredfeedrate command矿firstlyThen the feedrate cornmand V(after acedec control algorithm)is sent to theinterpolator to compute the traveling distanee component for each axis in the Cartesian coordinates(KY and刁The traveling distance componentsZ厶Y and觇along the XY and Z axes are transmitted tothe motion control routine邪position commands tothe position control loop for the desired machiningfinallyWhen data sampling interpolation is used in po-sition contr01it iS the condition tO make sure thatevery synchronized motion axis reaches the destina-tion simultaneously and their movement iS continuousSO that each axis running time t is just integral timesthat ofthe interpolation period互ietime partitioningruleThis condition can be realized by adjusting thecommand feedrate of each synchronized motion axis(ieaccelerating or decelerating)(Guo et a1，2003)Through the above analysesn should be an integerSet N as the minimum integer not smaller than玎N=ceil(n)NnReplacing the real number，1 with the integer N inEq(5)yields，=3IKK I(24。)=NT (6)Construction of discrete modelGiven any curve given by the function y=flt)，theinterval【to，纠is divided into N divisionsPicking theleft endpoints to determine the heights is shown inFig2a；the approximate area under少钡D on the intervalto，明is，r儿灿嘻小娟棚孚)孚O to tl t2 r卜l，l肛l tM tO to tl t2 缸l n t#-l tu tFig2 Implement curve ofJft)(a)Left endpoints used forthe heights of rectangles；(b)Right endpoints used forthe heights of rectangles万方数据Leng et a1J Zhejiang Univ SciA 2008 9f3J：358-365This way of approximating the area under thecurve is called the left sum methodChoosing the fight endpoints to determine theheights is shown in Fig2b；the approximate areaunder尸砜f)on the intervalto，明is，Ij删，喜小+z可tN-to警This way of approximating the area under thecurve is called the right sum methodSuppose that the slope of ierk iS Ks and initialvalues of jerk,acceleration，speed and displacementare Jo，Ao，go and So，respectivelyThe discrete modelofjerk curve，implemented by Ieft sum method，iS。J(iT)=厶+0(f一1)T，1iN (7)The discrete models of acceleration，speed anddisplacement curve，realized by right sum method，are，iA(iT)=40+，(7)r，I，the route segment is the decelerationstageFrom Eq(1 0)，the discrete expression of themaximum deceleration length S1 can be computedIfS1冼，the end feedratecannot be reached at the endpointwhere L iS the length of the route segmentForthe deceleration stagethe theoretical decelerationpoint should be calculated beforehand，otherwiseovershoot would resultLet St=厶IS=(一1)Kr2+(N+1)VET2=L，【tm=3(,一K)(24。)=NTThen，the value of N can be computed，and thebeginning speed K is modified as，矿：2L-(N+I)VeT1(一1)丁The discrete model of the deceleration curve isexpressed as，咿)=t+丽(Ve-而Vs)(j3丽N+2)f+丽3+(V1e)(-K+)2)f2 一揣f3砂N(+1)(+2)7If圪圪，the route segment iS the accelerationstageFrom Eq(1 0)，the discrete expression of themaximum acceleration length SI can be calculatedIfSl，the end speed圪also cannot be achievedThevalue of M can be calculated and the end speed Ze iSmodified as described above，y：2L-(M-1)VJe (M+1)TThe discrete model of the acceleration curve isexpressed as，矿(f丁)：K+!兰二型!丝坐+型二竺2 f：、7 5M(M+1)(M+2) (M+1)(M+2)一而2(vo丽-v,)M(Mf3，1fM+1)(M+2)Both acedec stages existWhen both acedec stages exist，the minimumlength品is obtained from Eq(11)Suppose thatis the actual maximum speed of the route segmentCombination of Eqs(6)and(1 1)yields!垡二塑：MT24Ilax堑垡二型：胛 2钆吉(M+)k丁+三(M-1)K丁+丢(+1)K丁=工万方数据Leng et a1J Zhejiang Univ SciA 2008 9r3J：358-365Then，values of M，N，Vmx can be computedThe interpolation algorithm is used in this paperdiscrete model of the acceleration curve is expressedaSEXPERIME】NTAL ItESU【TSy(fn：形+!垡二型!丝塑f+堑垡二竺!f： 、 75 M(M+1)(M+2) (A，+1)(M+2)一丽2而(Vm“丽-K丽)fM M(M+1)(M+2)7The discrete model of the deceleration curve isexpressed as，yo丁)=+墅凳云鬻f+i两3(iV_e丽-Vm“)r2一而2(雨Ve-；两Vm)酉3，1f (+1)(+2)7Based on the abovementioned model，the datasampling parametric interpolator is adopted in thesoftware to implement in the CNC experimental systen_l-whose control system is based on the step motorsystem，which is a framework of the PC-NC systemand similar to the synchronization control of 3axismotion system(Fig31Ace,constant-speed and dee stages all existIf心，acceleration，constant-speed and decel-eration stages all existThe actual maximum speed ofthe route segment is the maximum allowed feedrateR1axThe discrete model of the acceleration curve is Fig3 CNC experimental system based Oil step motorexpressed as，矿(f丁)：形+!垒二竺2堂堡+!坠二坠fz 、7 5M(M+1)(M+2) (M+1)(M+2)一j坠善乓f3，liM M(M+1)(M+2)7The discrete model of the deceleration curve isexpressed as，嘲吒+鬻升丽3(Vo-而Fm=)r2一竖鲁乓f3，liN N(N+1)(+2)7The discrete model of the constant-speed curveis expressed as，v(ir)=Thus the adaptive acedec control algorithm isdeducedTo guarantee the interpolation precision andspeed in high speed NC machining，the parametricIn order to evaluate the effectiveness of theproposed accdec control modelan arbitrary 3D toolpath composed of 1l broken line segments is adoptedin Fig4which iS a part ofthe shoe mold surfaceTheparameters used in experiment are as follows：Themaximum ierk iS set at 8000 mrns3，the accelerationlimits are set at 600 mmsrespectively,the maximumallowed feedrate is given as 50 mmsthe interpolation period iS set at 0004 Sthe impulse equivalent isgiven as 05 Itmthe initial velocity and the final ve-locity are 2 mmsThe experimental results are listedin 1able 1It iS shown that the beginning speed andIiyfAn arbitrary 3D tool path composedFig4 Shoe mold surface万方数据Leng et a1J Zhejiang Univ SciA 2008 9(3)：358-365Table l Experimental data of speed， Without using adaptive accdec Using adaptive acedecNo 粤印 control algorithm control algorithm Actual maximumL【I砌J speed。，(rams)K(mms) K(mms) 圪(rams) 圪(mms)l O852 2000 50000 2000 28725 287252 0375 50000 23115 287253 0376 23115 50000 231154 0752 50000 46310 237265 0751 46310 48679 218006 0750 48679 40145 305577 0752 40145 50000 239848 0376 50000 23724 231289 0375 23724 50000 2372410 0752 50000 25107 232862311523726218003055723984231282372423286955428725237262446930557305572398423724237242667911 O100 25107 2000 9554 2000 9554the end speed can be modified adaptively according tothe length of each broken line segmentand then theactual maximum speed is obtained and the decelera-tion point can be predetermined precisely by adoptingthe proposed adaptive acedec control algorithmTheexperiment curves of displacement，speed and acceleration are shown in Fig5It can be seen that thecubic polynomial acedec control model providescontinuity of accelerationwhich avoided the intensevibration in high speed NC machiningThe calcula-tions ofjerk，acceleration，speed and displacement invarying speed process are enlcient and easy to realizebecause of few four arithmetic operations喜；奶2O40芒30l 20lo0l伽10G 600堇一，盅、lOool；l i阜H寸j：i斗仃i1_|_ij0 28 56 84 112 140 168 196 224 252 280Fig5 Results of displacement,speed and accelerationCurveCONCLUSIONIn this Paper,a cubic polynomial acedec controlmodel for high speed NC machining is proposedAiming at the question of hardly predetermining thedeceleration point in acedec control before interpo1ation，the adaptive acedec control algorithm is deduced furthermoreThe experimental results provethat the proposed acedec control model providescontinuity of acceleration，which avoids the occur-rence of intense vibration in high speed machiningand the deceleration point can be predicted preciselyby the deduced algorithmIn the realtime interpola-tion process，the calculations ofjerk,acceleration，speed and displacement are emcient and easy to realize，which can satis矽realtime demandAt present。the control model has been applied in multiaxes highspeed micro fabrication machining successfullyReferencesChen，CS Lee，AC，1998Design of accelera-tiondeceleration profiles in motion control based ondigital FIR filtersInternationalJournal ofMachine ToolsManufacture，38(7)：799-825 【doi：lO1016S08906955(97)000655】Guo，XG，Li，CX 2003A new flexible acceleration anddeceleration 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