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高速钻床的运动
的运动学和动力学
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并联运动高速钻床的运动学和动力学合成,高速钻床的运动,的运动学和动力学
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International Journal of Machine Tools received in revised form 20 February 2004; accepted 22 April 2004 Abstract Typically, 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. # 2004 Elsevier Ltd. All rights reserved. Keywords: Parallel kinematic machine; High speed drilling; Kinematic and dynamic synthesis 1. Introduction During 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 confi guration. 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 defi ned as the volume where the tip of the tool can move and cut material. The design of a planar Stewart platform was mentioned in 5 as an aff ordable way of retrofi tting non-CNC machines required for plastic moulds machining. The design of the PKM 5 allowed adjustable geometry that could have been optimally reconfi gured 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 themachinenamedUraneSX.TheHSDM 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 from hundreds con- trolled drilling experiments in order to specify the para- metersrequiredforquality drilling.Idealdrilling 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 confi guration 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- videdbythetable.Forachievinghigh-speed ? Corresponding author. Tel.: +1-734-647-7325; fax: +1-734-615- 0312. E-mail address: lizhe (Z. Li). 0890-6955/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2004.04.007 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 developedforimprovingtheperformanceofthe machine. Through input motion planning for drilling and point-to-point positioning, the machining error will be reduced and the quality of the fi nished 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 eff ectiveness of the methods pre- sented in this paper. 2. Kinematic and dynamic equations of motion of the PKM module The 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 1 and 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 _ y y1 _ y y6(y axis motion for point-to-point positioning) and _ y y1 ?_ y y6(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 links Using complex-number representation of mechan- isms 12, the kinematic equations of the tool unit (indi- cated as part 3 which includes the platform, the spindle and the tool) are developed as follows. The displace- ment of the tool is y3 y1 y6=2 z3 rsinb ? 1 and b arccosy6? y1? b=2r2 where 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 _ y y 3 _ y y1 _ y y6=2 _ z z3 r_b bcosb ? 3 where _ b b _ y y1? _ y y6=2rsinb4 The acceleration of the tool is y y 3 y y1 y y6=2 z z3 rb bcosb ? r_b b2sinb ? 5 where b b _ b b2cosb y y6? y y1=2r ?sinb 6 The dynamic equations of the PKM module are developed using Lagranges equation of the second kind 13 as shown in Eq. (7). d dt T _ q qj ? ? T qj Qjj 1;2;.k7 where T is the total kinetic energy of the system; qjand _ q q jare the generalized coordinates and velocities; Qjis the generalized force corresponding to qj. k is the num- ber of the independent generalized coordinates of the system. Here, k 2, q1 y1and q2 y6. After deri- vation, Eq. (7) can be expressed as X n i1 mi y y gi _ y ygi _ q qj z zgi _ z zgi _ q qj ? Igih hi _h hi _ q qj () Qj j 1;2;.;k8 Fig. 2.Schematic diagram of the PKM module. Fig. 1.Schematic diagram of the HSDM. 1382R. Katz, Z. Li / International Journal of Machine Tools (mi,Igi) are mass and mass moment of inertia of link i; (ygi,zgi) are the coordinates of the center of mass of link i; hiis the rotation angle of link i in the PKM module. The gen- eralized force Qjcan be determined by Qj ? V qj X n i1 F0i ri qj 9 where V is the potential energy and F0iare the non- potential forces. For the drilling operation of the PKM module, we have Q1 Q2 ? ?gP 5 i2 mi _ z zgi _ y y1 ? Fcut _ z zg3 _ y y1 F1 ?gP 5 i2 mi _ z zgi _ y y6 ? Fcut _ z zg3 _ y y6 F6 8 : 9 = ; 10 where Fcutis the cutting force, F1and F6are the input forces 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 links The dynamic diff erential equations of a compliant mechanism can be derived using the fi nite element method and take the form of M?n?nfD Dgn?1 C?n?nf_D Dgn?1 K?n?nfDgn?1 fRgn?111 where M, C and K are system mass, damping and stiff ness matrix, respectively; D is the set of general- izedcoordinatesrepresentingthetranslationand 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 freedom of the mechanism). In our FEA model, we use frame element shown in Fig. 3 in which EIeis the bending stiff ness (E is the modulus of elasticity of the material, Ieis the moment of inertia), q is the material density, le is the original length of the element. di(i 1, 2,.,6) are nodal displacements expressed in local coordinate system (x, y). The mass matrix and stiff ness matrix for the frame element will be 6 ? 6 symmetric matrices which can be derived from the kinetic energy and strain energy expressions as Eqs. (12) and (13) d dt T _d d ? ? T d ? m?efd dg12 U d ? k?efdg13 where T is the kinetic energy and U is the strain energy of the element; fdg d1d2d3d4d5d6?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 fi nite element analy- sis. Each element is associated with a mass and a stiff ness 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 system to global coordinate system. This gives the system mass M and stiff ness K matri- ces. Capturing the damping characteristics in a com- pliant system is not so straightforward. Even though, in many applications, damping may be small but its eff ect on the system stability and dynamic response, especially in the resonance region, can be signifi cant. The damping matrix C can be written as a linear com- bination of the mass and stiff ness matrices 15 to form the proportional damping C which is expressed as C? aM? bK?14 where a and b are two positive coeffi cients which are usuallydeterminedbyexperiment.Analternate method 16 of representing the damping matrix is expressing C as C? M?C0?15 The element of C0 is defi ned as C0 ij 2fsignKij Kij=Mij 1 2, where signKij Kij= Kij ? ? ? ?, Kij and Mijare the elements of K and M, f is the damping ratio of the material. The generalized force in a frame element is defi ned as Re i X m j1 Fxj xj di Fyj yj di Mhj hj di ? i 1;2;.;6 16 where Fjand Mjare the jth external force and moment including the inertia force and moment on the element acting at (xj,yj), and m is the number of the externalFig. 3.A planar frame element. R. Katz, Z. Li / International Journal of Machine Tools Fig. 5.Input velocity corresponding to the ideal tool motion. Fig. 6.Absolute errors between the real and ideal tool velocities. Fig. 7.Original constant acceleration input motion function for positioning. R. Katz, Z. Li / International Journal of Machine Tools c1 0; c2 0 c3 ?6c4Tp 10c5T2 p 15c6T3 p ?. 3 c4 ?5c5Tp 9c6T2 p ?. 2 c56h ? 3c6T6 p ?. T5 p 8 : 20 Logically, set the optimization objective as min ! fc6 Dytoolt Tp s:t:Eq: 18 c6min? c6? c6max21 where c6is the independent design variable; Dytool ytoolmax? ytoolmint Tp is the maximum fl uctuation of residual vibrations of the tool tip after the point-to- point positioning. Set c6min; c6max? ? 108; 108? and start the calculation from c6 0. The optimization results in c6 ?108mm=s6. Consequently, c5 7:5? 107mm=s5,c4 ?1:425 ? 107mm=s4,c3 8:5 ? 105 mm=s3, c2 c1 c0 0. It can be seen that the opti- mization calculation brought the design variable c6to the boundary. If further loosing the limit for c6, the objectivewillcontinuereduceinvalue,butthe 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 from comparing Fig. 8 and Fig. 10 that the amplitude and tool tip residual vibration was reduced by30timesafteroptimization.Smallerresidual 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 10 times 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 we studied, Fig. 8.Residual vibration of the tool tip before the optimization. Fig. 9.Optimal polynomial input motion function for positioning. Fig. 10.Residual vibration of the tool tip after the optimization. 1386R. Katz, Z. Li / International Journal of Machine Tools 22 where l0 and k are the initial length and the stiff ness of the linear spring. The input power of the linear motors is determined by P1 P2 ? F1? _ y y1 F6? _ y y6 ? 23 In order to reduce the input power, we set the opti- mization objective as follows: min ! fv P 2 i1 DPi s:t:lmin? l0? lmax kmin? k ? kmax 24 where v is a vector of design variables including the lengthandthe stiff nessofthespring, DPi Pimax? Pimini 1;2. For the PKM module we studied, the mass properties are listed in Table 1. The initialvaluesofthedesignvariablesaresetas l0 0 451:36 mm,k0 5 N=mm.Thedomainsfor design variables are set as lmin;lmax? 400; 500? mm, kmin; kmax? 1; 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 parametersareobtainedasl0 433:93 mmand k 14:99 N=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 122.37 to 70.43 W. A 42.45% reduction is achieved. For the left linear motor, the maximum input power is reduced from 114.44 to 62.72 W. A 45.19% reduction is achieved. The eff ectiveness of the presented method by adding a spring element to reduce the input power of the machine is verifi ed. Torsional springs may be sued to reduce the inertial eff ect and the size of the spring attachment. Fig. 11.The PKM module with (a) linear and (b) torsion spring elements. Table 1 Mass properties of the PKM module m15.00 kgJ1 m21.55 kgJ23:489 ? 104kg mm2 m314.21 kgJ3 m41.55 kgJ43:489 ? 104kg mm2 m51.55 kgJ53:489 ? 104kg mm2 m65.00 kgJ6 R. Katz, Z. Li / International Journal of Machine Tools & Manufacture 44 (2004) 138113891387 6. Conclusions The 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 required input 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. Acknowledgements The authors gratefully acknowledge the fi nancial support of the NSF Engineering Research Center for Reconfi gurable Machining Systems (US NSF Grant EEC95-92125) at the University of Michigan and the valuable input from the Centers industrial partners. References 1 D. Stewart, A platform with six degrees of freedom, Proceedings of the Institution of Mechanical Engineers, 19651966, pp. 371381. 2 J.-P. Merlet, Parallel manipulators: state of the art and perspec- tives, Advanced Robotics 8 (6) (1994) 589596. 3 V. Gopalakrishnan, D. Fedewa, M. Mehrabi, S. Kota, N. Orlan- dea, Parallel structures and their applications in reconfi gurable machining systems, Proceeding of Year 2000 PKM International Conference, Ann Arbor, Michigan, USA, 2000, pp. 8797. 4 P.H. Yang, K.J. Waldron, V. Dutt, E. Orin, Design of a three degree of freedom motion platform for a low cost driving simu- lator, Journal of Applied Mechanics and Robotics 3 (4) (1996) 2630. 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 2000 Parallel Kinematic Machines International Conference, Ann Arbor, Michigan, USA, 2000, pp. 316329. 6 O. Company, F. Pierrot, F. Launay, C. Fioroni, Modeling and preliminary design issues of 3-axis parallel machine-tool, Pro- ceeding of Year 2
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