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International Journal of Machine Tools&Manufacture44(2004)1381–1389Kinematic and dynamic synthesis of a parallel kinematic high speeddrilling machineReuven Katz,Zhe LiÃEngineering Research Center,Department of Mechanical Engineering,University of Michigan,Ann Arbor,MI48109-2125,USAReceived2March2003;received in revised form20February2004;accepted22April2004AbstractTypically,the term‘‘high speed drilling’’is related to spindle capability of high cutting speeds.The suggested high speed drill-ing machine(HSDM)extends this term to include very fast and accurate point-to-point motions.The new HSDM is composed of a planar parallel mechanism with two linear motors as the inputs.The paper is focused on the kinematic and dynamic synthesis of this parallel kinematic machine(PKM).The kinematic synthesis introduces a new methodology of input motion planning for ideal drilling operation and accurate point-to-point positioning.The dynamic synthesis aims at reducing the input power of the PKM using a spring element.Keywords:Parallel kinematic machine;High speed drilling;Kinematic and dynamic synthesis1.IntroductionDuring the recent years,a large variety of PKMs were introduced by research institutes and by indus-tries.Most,but not all,of these machines were based on the well-known Stewart platform[1]configuration. The advantages of these parallel structures are high nominal load to weight ratio,good positional accuracy and a rigid structure[2].The main disadvantages of Stewart type PKMs are the small workspace relative to the overall size of the machine and relatively slow oper-ation speed[3,4].Workspace of a machine tool is defined as the volume where the tip of the tool can move and cut material.The design of a planar Stewart platformwas m entioned in[5]as an affordable way of retrofitting non-CNC machines required for plastic moulds machining.The design of the PKM[5]allowed adjustable geometry that could have been optimally reconfigured for any prescribed path.Typically,chan-ging the length of one or more links in a controlled sequence does the adjustment of PKM geometry.The application of the PKMs with‘‘constant-length links’’for the design of machine tools is less common than the type with‘‘varying-length links’’.An excellent example of a‘‘constant-length links’’type of machine is shown in[6].Renault-Automation Comau has built the machine named‘‘Urane SX’’.The HSDM described herein utilizes a parallel mechanism with con-stant-length links.Drilling operations are well introduced in the litera-ture[7].An extensive experimental study of high-speed drilling operations for the automotive industry is reported in[8].Data was collected fromhundreds con-trolled drilling experiments in order to specify the para-meters required for quality drilling.Ideal drilling motions and guidelines for performing high quality drilling were presented in[9]through theoretical and experimental studies.In the synthesis of the suggested PKM,we follow the suggestions in[9].The detailed mechanical structures of the proposed new PKM were introduced in[10,11].One possible configuration of the machine is shown in Fig.1;it has large workspace,high-speed point-to-point motion and very high drilling speed.The parallel mechanism pro-vides Y,and Z axes motions.The X axis motion is pro-vided by the table.For achieving high-speedÃCorresponding author.Tel.:+1-734-647-7325;fax:+1-734-615-0312.E-mail address:lizhe@(Z.Li).performance,two linear motors are used for driving the mechanism and a high-speed spindle is used for drilling.The purpose of this paper is to describe new kinematic and dynamic synthesis methods that are developed for improving the performance of the machine.Through input motion planning for drilling and point-to-point positioning,the machining error will be reduced and the quality of the finished holes can be greatly improved.By adding a well-tuned spring element to the PKM,the input power can be mini-mized so that the size the machine and the energy con-sumption can be reduced.Numerical simulations verify the correctness and effectiveness of the methods pre-sented in this paper.2.Kinematic and dynamic equations of motion of the PKM moduleThe schematic diagram of the PKM module is shown in Fig.2.In consistent with the machine tool conventions,the z -axis is along the direction of tool movement.The PKM module has two inputs (two lin-ear motors)indicated as part 1and part 6,and one output motion of the tool.The positioning and drilling motion of the PKM module in this application is char-acterized by _y1¼_y 6(y axis motion for point-to-point positioning)and _y1¼À_y 6(z axis motion for drilling).Motion equations for both rigid body and elastic body PKM module are developed.The rigid body equations are used for the synthesis of input motion planning of drilling and input power reduction.The elastic body equations are used for residual vibration control after point-to-point positioning of the tool.2.1.Equations of motion of the PKM module with rigid linksUsing complex-number representation of mechan-isms [12],the kinematic equations of the tool unit (indi-cated as part 3which includes the platform,the spindleand the tool)are developed as follows.The displace-ment of the tool is y 3¼ðy 1þy 6Þ=2z 3¼r sin bð1Þandb ¼arccos ðy 6Ày 1Àb Þ=ð2r Þð2Þwhere b is the distance between point B and point C,r is the length of link AB (the lengths of link AB,CD and CE are equal).The velocity of the tool is _y3¼ð_y 1þ_y 6Þ=2_z 3¼r _b cos bð3Þwhere_b¼ð_y 1À_y 6Þ=ð2r sin b Þð4ÞThe acceleration of the tool is€y 3¼ð€y 1þ€y 6Þ=2€z 3¼r €b cos b Àr _b 2sin b ð5Þwhere€b ¼_b 2cos b þð€y 6À€y 1Þ=ð2r ÞÀsin bð6ÞThe dynamic equations of the PKM module are developed using Lagrange’s equation of the second kind [13]as shown in Eq.(7).d d t @T @q j À@T@q j ¼Q j ðj ¼1;2;...k Þð7Þwhere T is the total kinetic energy of the system;q j and _qj are the generalized coordinates and velocities;Q j is the generalized force corresponding to q j .k is the num-ber of the independent generalized coordinates of the system.Here,k ¼2,q 1¼y 1and q 2¼y 6.After deri-vation,Eq.(7)can be expressed as X n i ¼1m i €y gi @_y gi @_q j þ€z gi @_z gi @_q j þI gi €h i @_h i@_q j ()¼Q jðj ¼1;2;...;k Þð8ÞFig.2.Schematic diagram of the PKMmodule.Fig.1.Schematic diagram of the HSDM.1382R.Katz,Z.Li /International Journal of Machine Tools &Manufacture 44(2004)1381–1389where n is the number of the moving links;(m i ,I gi )are mass and mass moment of inertia of link i ;(y gi ,z gi )are the coordinates of the center of mass of link i ;h i is the rotation angle of link i in the PKM module.The gen-eralized force Q j can be determined byQ j ¼À@V @q j þX ni ¼1F 0i@r i@q jð9Þwhere V is the potential energy and F 0i are the non-potential forces.For the drilling operation of the PKM module,we haveQ 1Q 2 ¼Àg P 5i ¼2m i@_z gi @_y 1ÀF cut @_zg 3@_y 1þF 1Àg P 5i ¼2m i @_z gi @_y 6ÀF cut @_z g 3@_y 6þF 68>>><>>>:9>>>=>>>;ð10Þwhere F cut is the cutting force,F 1and F 6are the inputforces exerted on the PKM by the linear motors.Eqs.(1)to (10)form the kinematic and dynamic equa-tions of the PKM module with rigid links.2.2.Equations of motion of the PKM module with elastic linksThe dynamic differential equations of a compliant mechanism can be derived using the finite element method and take the form of½M n Ân f €D g n Â1þ½C n Ân f _D g n Â1þ½K n Ân f D g n Â1¼f R g n Â1ð11Þwhere [M ],[C ]and [K ]are system mass,damping and stiffness matrix,respectively;{D }is the set of general-ized coordinates representing the translation and rotation deformations at each element node in global coordinate system;{R }is the set of generalized exter-nal forces corresponding to {D };n is the number of the generalized coordinates (elastic degrees of freedomof the mechanism).In our FEA model,we use frame element shown in Fig.3in which EI e is the bending stiffness (E is the modulus of elasticity of the material,I e is the moment of inertia),q is the material density,l eis the original length of the element.d i (i ¼1,2,...,6)are nodal displacements expressed in local coordinate system(x ,y ).The mass matrix and stiffness matrix for the frame element will be 6Â6symmetric matrices which can be derived fromthe kinetic energy and strain energy expressions as Eqs.(12)and (13)d d t @T @_d À@T@d¼½m e f €d g ð12Þ@U@d¼½k e f d gð13Þwhere T is the kinetic energy and U is the strain energy of the element;f d g ¼½d 1d 2d 3d 4d 5d 6 T ,are the linear and angular deformations of the node at the element local coordinate system.Detailed derivations can be found in [14].Typically,a compliant mechanism is dis-cretized into many elements as in finite element analy-sis.Each element is associated with a mass and a stiffness matrix.Each element has its own local coordi-nate system.We combine the element mass and stiff-ness matrices of all elements and perform coordinate transformations necessary to transform the element local coordinate systemto global coordinate system .This gives the systemm ass [M ]and stiffness [K ]matri-ces.Capturing the damping characteristics in a com-pliant systemis not so straightforward.Even though,in many applications,damping may be small but its effect on the systemstability and dynam ic response,especially in the resonance region,can be significant.The damping matrix [C ]can be written as a linear com-bination of the mass and stiffness matrices [15]to form the proportional damping [C ]which is expressed as ½C ¼a ½M þb ½Kð14Þwhere a and b are two positive coefficients which are usually determined by experiment.An alternate method [16]of representing the damping matrix is expressing [C ]as ½C ¼½M ½C 0ð15ÞThe element of [C 0]is defined as C 0ij¼2f ðsign K ij ÞðK ij =M ij Þ12,where sign K ij ¼ðK ij =K ij Þ,K ij and M ij arethe elements of [K ]and [M ],f is the damping ratio of the material.The generalized force in a frame element is defined asR ei ¼X m j ¼1F xj @x j @d i þF yj @y j @d i þM h j @h j @d i ði ¼1;2;...;6Þð16Þwhere F j and M j are the j th external force and momentincluding the inertia force and moment on the element acting at (x j ,y j ),and m is the number of theexternalFig.3.A planar frame element.R.Katz,Z.Li /International Journal of Machine Tools &Manufacture 44(2004)1381–13891383forces acting on the element.The element generalizedforces f R g e ¼½R e 1R e 2R e 3R e 4R e 5R e 6 T,are then com-bined to formthe systemgeneralized force {R }.The second order ordinary differential equations of motion of the system,Eq.(11),can be directly integrated with a numerical method such as Runge-Kutta method.For the PKM we studied,each link was discreted as 15frame elements.Both Matlab and ADAMS software are used for programming and solving these equations.3.Input motion planning for drillingSuppose we know the ideal motion function of the drilling tool.How to determine the input motor motion so that the ideal tool motion can be realized is critical for high quality drillings.The created explicit input motion function also provides the necessary information for machine controls.According to the study done in [9],the drilling process can be divided into three phases:entrance phase,middle phase,and exit phase.In order to increase the productivity and quality of the drilling,many operation constraints such as minimum tool life constraint,hole location error constraint,exit burr constraint,drill torsion breakage constraint,etc.must be considered and satisfied.Under these conditions,the feed velocity of the tool should be slow at the entrance phase to reduce the hole location errors.The tool velocity should also be slow at the exit phase to reduce the exit burr.At the middle phase,the tool drilling velocity should be fast and kept constant.The retraction of the tool after finishing the drilling should be done as quickly as possible to increase the productivity.Based on these considerations,we assume that the ideal drilling and retracting velocities of the tool are given by Eq.(17).v T ¼v T 121Àcos p t T 1 ð0 t T 1Þv T 1ðT 1 t T 1þT 2Þv T 121þcos p ðt ÀT 1ÀT 2ÞT 3 ðT 1þT 2 t T 1þT 2þT 3Þv T 221Àcos p ðt ÀT 1ÀT 2ÀT 3ÞT 4P3i ¼1T i t P 4i ¼1T i v T 2P4i ¼1T i t P 5i ¼1T i v T 221þcos p ðt ÀT c þT 6ÞT 6P5i ¼1T i t T c8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:ð17Þwhere v T 1is the maximum drilling velocity,T 1,T 2,and T 3are the times corresponding to the entrance phase,the middle phase and the exit phase.v T 2is the maximum retracting velocity.T 4,T 5,and T 6are corre-sponding to accelerating,constant velocity,and decel-erating times for retracting operation.T c ¼P 6i ¼1T i is the cycle time for a single drilling.As a numerical example,suppose we drill a 25.4mm (1in)deep hole with T c ¼0:4s,0.3s for drilling,0.1s for retracting.Set T 1¼T 3¼0:06s,T 4¼T 6¼0:03s.Under these con-ditions,v T 1¼106ðmm =s Þ,v T 2¼À363ðmm =s Þ.The graphical expression of the ideal tool motion is shown in Fig.4.If the link length in PKM r ¼500mm,theangle b ¼53vat the starting point of drilling,the cor-responding input motor velocity relative to the ideal tool motion is shown in Fig.5.Generally,the curve fit-ting method can be used to create the input motion function.But according to the shape of the curve shown in Fig.5,we create the linear motor velocity function manually section by section as shown in Eq.(18).v M¼v B 1Àcos p t 1ð0 t T 1Þv B þv C Àv B T 2ðt ÀT 1ÞðT 1 t T 1þT 2Þv C 21þcos p ðt ÀT 1ÀT 2ÞT 3 ðT 1þT 2 t T Þv E 21Àcos p ðt ÀT 1ÀT 2ÀT 3ÞT 4 P 3i ¼1T i t P4i ¼1T i v E þv F Àv E T 5t ÀX 4i ¼1T i !P 4i ¼1T i t P5i ¼1T i v F 21þcos p ðt ÀT c þT 6ÞT 6 P5i ¼1T i t T c8>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>:ð18Þwhere v B ¼143:48mm =s,v C ¼165:77mm =s,v E ¼À557:36mm =s,v F ¼À499:44mm =s.When plotting the velocity curve with Eq.(18),no visual difference can be found with the curve shown in Fig.5.Eq.(18)is composed of six parts with four cycloidal functions and two linear functions.If we control the two linear motors to have the same motion as described in Eq.(18),the drilling and retracting velocity of the tool will be almost the same as shown in Fig.4.The absol-ute errors between the ideal and real tool velocity are shown in Fig.6,in which the maximum error is less than 8mm/s,the relative error is less than 1.5%.At the start and the end positions of the drilling,theFig.4.Assumed ideal drilling velocity of the tool.1384R.Katz,Z.Li /International Journal of Machine Tools &Manufacture 44(2004)1381–1389errors are zero.These small absolute and relative errors illustrate the created input motion and are quite acceptable.The derived function is simple enough to be integrated into the control algorithmof the PKM.4.Input motion planning for point-to-point positioningIn order to achieve fast and accurate positioning operation in the whole drilling process,the input motion should be appropriately planned so that the residual vibration of the tool tip can be minimized. Conventionally the constant acceleration motion func-tion is commonly used for driving the axes motions in machine tools.Although this kind of motion function is simple to be controlled,it may excite the elastic vibration of the systemdue to the sudden changes in acceleration.Take the same PKM module used in pre-vious for example.A FEA model is built using ADMAS with frame elements.The positioning motion is the y-axis motion,which is realized by the two linear motors moving in the same direction.Suppose the positioning distance between the two holes is75mm, the constant acceleration is3g(approximated as30m/s2 here).The input motion of the linear motors with constant acceleration and deceleration is shown in Fig.7,in which the maximum velocity is1500mm/s, the positioning time is0.1s.Assuming the material damping ratio as0.01,the residual vibration of the tool tip is shown in Fig.8.In order to reduce the residual vibration and make the positioning motion smoother,a six order poly-nomial input motion function is built as Eq.(19)s in¼c0þc1tþc2t2þc3t3þc4t4þc5t5þc6t6ð19Þwhere the coefficients c i are the design variables which have to be determined by minimizing the residual vibration of the tool tip.Selecting the boundary conditions as that when t¼0,s in¼0,v in¼0,a in¼0; Fig.5.Input velocity corresponding to the ideal toolmotion. Fig.6.Absolute errors between the real and ideal toolvelocities.Fig.7.Original constant acceleration input motion function for positioning.R.Katz,Z.Li/International Journal of Machine Tools&Manufacture44(2004)1381–13891385and when t ¼T p ,s in ¼h ,v in ¼0,a in ¼0,where T p is the point-to-point positioning time,the first six coefficients are resulted:c 0¼0;c 1¼0;c 2¼0c 3¼À6c 4T p þ10c 5T 2p þ15c 6T 3p.3c 4¼À5c 5T p þ9c 6T 2p .2c 5¼6h À3c 6T 6p .T 5p 8>>>>><>>>>>:ð20ÞLogically,set the optimization objective as min !f ðc 6Þ¼D y tool ðt >T p Þs :t :Eq :ð18Þc 6min c 6 c 6maxð21Þwhere c 6is the independent design variable;D y tool ¼y toolmax Ày toolmin ðt >T p Þis the maximum fluctuation of residual vibrations of the tool tip after the point-to-point positioning.Set ½c 6min ;c 6max ¼½À108;108 and start the calculation from c 6¼0.The optimization results in c 6¼À108mm =s 6.Consequently,c 5¼7:5Â107mm =s 5,c 4¼À1:425Â107mm =s 4,c 3¼8:5Â105mm =s 3,c 2¼c 1¼c 0¼0.It can be seen that the opti-mization calculation brought the design variable c 6to the boundary.If further loosing the limit for c 6,the objective will continue reduce in value,but the maximum value of acceleration of the input motion will become too big.The optimal input motions after the optimization are shown in Fig.9.The correspond-ing residual vibration of the tool tip is shown in Fig.10.It is seen fromcom paring Fig.8and Fig.10that the amplitude and tool tip residual vibration was reduced by 30times after optimization.Smaller residual vibration will be very useful for increasing the position-ing accuracy.It should be mentioned that only link elasticity is included in above calculation.The residual vibration after optimization will still be very small if the compliance from other sources such as bearings and drive systems caused it 10times higher than the result shown in Fig.10.5.Input power reduction by adding spring elements Reducing the input power is one of many considera-tions in machine tool design.For the PKM westudied,Fig.8.Residual vibration of the tool tip before theoptimization.Fig.9.Optimal polynomial input motion function forpositioning.Fig.10.Residual vibration of the tool tip after the optimization.1386R.Katz,Z.Li /International Journal of Machine Tools &Manufacture 44(2004)1381–1389two linear motors are the input units which drive the PKM module to perform drilling and positioning operations.One factor to be considered in selecting a linear motor is its maximum required power.The input power of the PKM module is determined by the input forces multiplying the input velocities of the two linear motors.Omitting the friction in the joints,the input forces are determined from balancing the drilling force and inertia forces of the links and the spindle unit.Adding an energy storage element such as a spring to the PKM may be possible to reduce the input power if the stiffness and the initial (free)length of the spring are selected properly.The reduction of the maximum input power results in smaller linear motors to drive the PKM module.This will in turn reduce the energy consumption and the size of the machine structure.A linear spring can be added in the middle of the two links as shown in Fig.11(a).Or two torsional springs can be added at points B and C as shown in Fig.11(b).The synthesis process is the same for the linear or tor-sional springs.We will take the linear spring as an example to illustrate the design process.The general-ized force in Eq.(10)has the formofQ 1Q 2 ¼Àg P 5i ¼2m i @_z gi @_y 1ÀF cut @_z g 3@_y 1Àk ðl Àl 0Þ1þr sin b @_b @_y 1 !þF 1Àg P 5i ¼2m i @_z gi @_y 6ÀF cut @_z g 3@_y 6Àk ðl Àl 0Þ1þr sin b @_b @_y 1 !þF 68>>>><>>>>:9>>>>=>>>>;ð22Þwhere l 0and k are the initial length and the stiffness ofthe linear spring.The input power of the linear motors is determined by P 1P 2 ¼F 1Á_y1F 6Á_y 6ð23ÞIn order to reduce the input power,we set the opti-mization objective as follows:min !f ðv Þ¼P2i ¼1D P is :t :l min l 0 l maxk min k k maxð24Þwhere v is a vector of design variables including the length and the stiffness of the spring,D P i ¼P i max ÀP i min ði ¼1;2Þ.For the PKM module we studied,the mass properties are listed in Table 1.The initial values of the design variables are set as l 0¼451:36mm,k 0¼5N =mm.The domains for design variables are set as ½l min ;l max ¼½400;500 mm,½k min ;k max ¼½1;20 N =mm.The PKM module is dri-ven by the input motion function described as Eq.(18).Through minimizing objective (24),the optimal spring parameters are obtained as l 0¼433:93mm and k ¼14:99N =mm.The input powers of the linear motors with and without the optimized spring are shown in Fig.12,in which the solid lines represents the input power without spring,the dotted lines represents the input power with the optimal spring.It can be seen from the result that the maximum input power of the right linear motor is reduced from 122.37to 70.43W.A 42.45%reduction is achieved.For the left linear motor,the maximum input power is reduced from 114.44to 62.72W.A 45.19%reduction is achieved.The effectiveness of the presented method by adding a spring element to reduce the input power of the machine is verified.Torsional springs may be sued to reduce the inertial effect and the size of the springattachment.Fig.11.The PKM module with (a)linear and (b)torsion spring elements.Table 1Mass properties of the PKM module m 1 5.00kg J 1–m 2 1.55kg J 23:489Â104kg mm 2m 314.21kg J 3–m 4 1.55kg J 43:489Â104kg mm 2m 5 1.55kg J 53:489Â104kg mm 2m 65.00kgJ 6–R.Katz,Z.Li /International Journal of Machine Tools &Manufacture 44(2004)1381–138913876.ConclusionsThe paper presents a new type of high speed drilling machine based on a planar PKM module.The study introduces synthesis technology for planning the desir-able motion functions of the PKM.The method allows both the point-to-point positioning motion and the up-and-down motion required for drilling operations.The result has shown that it is possible to reduce substan-tially the residual vibration of the tool tip by optimiz-ing a polynomial motion function.Reducing residual vibration is critical when tool-positioning requirement for the HSDM is in the range of several microns.By adding a ‘‘well-tuned’’optimal spring to the structure,it was possible to reduce the required input power for driving the linear motors.The simulation has demonstrated that more than 40%reduction in the required input power is achieved relative to the struc-ture without the spring.The reduction of requiredinput power may allow choosing smaller motors and as a result reducing costs of hardware and operations.In order to better understand the properties of the HSDM and to complete its design,further study is required.It will include error analysis of the machine as well as the control strategies and control design of the system.7.AcknowledgementsThe authors gratefully acknowledge the financial support of the NSF Engineering Research Center for Reconfigurable Machining Systems (US NSF Grant EEC95-92125)at the University of Michigan and the valuable input fromthe Center’s industrial partners.References[1]D.Stewart,A platformwith six degrees of freedom ,Proceedingsof the Institution of Mechanical Engineers,1965–1966,pp.371–381.[2]J.-P.Merlet,Parallel manipulators:state of the art and perspec-tives,Advanced Robotics 8(6)(1994)589–596.[3]V.Gopalakrishnan,D.Fedewa,M.Mehrabi,S.Kota,N.Orlan-dea,Parallel structures and their applications in reconfigurable machining systems,Proceeding of Year 2000PKM International Conference,Ann Arbor,Michigan,USA,2000,pp.87–97.[4]P.H.Yang,K.J.Waldron,V.Dutt,E.Orin,Design of a threedegree of freedom motion platform for a low cost driving simu-lator,Journal of Applied Mechanics and Robotics 3(4)(1996)26–30.[5]L.J.Plessis,J.A.Snyman,W.J.Smit,Optimization of the adjust-able geometry of planar Stewart platform machining center with respect to placement of workpiece relative to toolpath,Proceed-ing of Year 2000Parallel Kinematic Machines International Conference,Ann Arbor,Michigan,USA,2000,pp.316–329.[6]pany,F.Pierrot,unay,C.Fioroni,Modeling andpreliminary design issues of 3-axis parallel machine-tool,Pro-ceeding of Year 2000PKM International Conference,Ann Arbor,MI,USA,2000,pp.14–23.[7]J.S.Stephenson,J.S.Agapiou,Metal Cutting Theory and Prac-tice,Marcel Dekker Inc,New York,1997.[8]X.Lin,J.Ni,Drill design for high throughput hole making,Proceedings of 1999International Conference on Advanced Manufacturing Technology,Xi’an,China,1999,pp.162–166.[9]R.J.Furness,A.G.Ulsoy,C.L.Wu,Supervisory control of drill-ing.Transactions of the ASME,Journal of Engineering for Industry 118(1996)10–19.[10]R.Katz,Y.Koren,F.Pierrot,Z.Li,Bi-axial coplanar drillingapparatus,US Patent 6,557,235,B1.[11]R.Katz,Z.Li,F.Pierrot,Conceptual design of a high speeddrilling machine (HSDM)based on a PKM module,Technical Report #37,ERC/RMS,The University of Michigan,2001.[12]A.G.Erdman,G.N.Sandor,S.Kota,Fourth ed.,MechanismDesign-Analysis and Synthesis,Vol.1,Prentice-Hall,Upper Saddle River,NJ,2001.[13]D.Greenwood,Principles of Dynamics,Prentice-Hall,Engle-wood Cliffs,NJ,1988.Fig.12.Input powers of (a)the right linear motor and (b)the left linear motor with and without the optimal spring (the solid lines represents the input power without spring,the dotted lines represents the input power with the optimal spring).1388R.Katz,Z.Li /International Journal of Machine Tools &Manufacture 44(2004)1381–1389。