SICOMAT : a system for SImulation and COntrol analysis of MAchine Tools

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1 SICOMAT : a ytem for SImulation and COntrol analyi of MAchine Tool Maxime Gautier, Minh Tu Pham, Wiama Khalil, Philippe Lemoine, Philippe Poignet To cite thi verion: Maxime Gautier, Minh Tu Pham, Wiama Khalil, Philippe Lemoine, Philippe Poignet. SICO- MAT : a ytem for SImulation and COntrol analyi of MAchine Tool. IEEE/ASME International Conference on Advanced Intelligent Mechatronic, AIM 2, Jul 2, Como, Italy., pp , 2, <.9/AIM >. <hal > HAL Id: hal Submitted on 9 Feb 29 HAL i a multi-diciplinary open acce archive for the depoit and diemination of cientific reearch document, whether they are publihed or not. The document may come from teaching and reearch intitution in France or abroad, or from public or private reearch center. L archive ouverte pluridiciplinaire HAL, et detinée au dépôt et à la diffuion de document cientifique de niveau recherche, publié ou non, émanant de établiement d eneignement et de recherche françai ou étranger, de laboratoire public ou privé.

2 SICOMAT: A ytem for SImulation and COntrol analyi of MAchine Tool M. Gautier*, M.T. Pham*, W. Khalil*, Ph. Lemoine *, Ph. Poignet ** Λ Intitut de Recherche en Communication et en Cyberntique de Nante (I.R.C.Cy.N.) rue de la Noë, BP 92, 4432 Nante Cedex 3, France. ΛΛ Laboratoire d'informatique, de Robotique et de Microlectronique de Montpellier (L.I.R.M.M.) 6 rue Ada, Montpellier Cedex 5, France Maxime.Gautier@irccyn.ec-nante.fr Abtract Thi paper preent a oftware package for the imulation and the control analyi of machine tool axe. Thi package which i called SICOMAT (SImulation and COntrol analyi of MAchine Tool), provide a large variety of toolboxe to analyze the behavior and the control of the machine. The oftware take into account everal element uch a the flexibility of bodie, the interactioetween everal axe, the effect of numerical control and the availability to reduce model. I. Introduction The need for the manufacturer to reduce the cot for deigning and evaluating a prototype goe through the definition of a good imulation tool. Therefore we develop a imulation environment able to guide the deigner in the choice of the component interfering in the complete mechatronic ytem i.e. flexible mechanical tructure, actuator, enor and controller. The ever increaing joint peed and acceleration of machine tool lead to develop accurate model taking into account elaticity of tructure and joint. A good compromie between the complexity of a ditributed elaticity approach a finite element model and the implicity of neglecting the elaticity with a rigid body model i to conider a multi-body model with lumped elaticitie and rigid bodie baed on a technology analyi of the ytem' component []. A block diagram and a linear tate pace decription of one axi or everal decoupled axe are implemented. A decription of multi-body ytem baed on the robotic formalim i alo implemented [2]. The originality of thi approachitoprovide a ytematic method to automatically calculate the geometric, kinematic and dynamic model of coupled or decoupled multi-axe machine tool, whatever the number of rigid and elatic degree of freedom (dof) of the ytem. Moreover the oftware contain everal tool to completely tudy a given tructure from the modeling (modal analyi toolbox and model reduction toolbox) to the control imulation. An other ignificant point of thi package i the poibility to tune the controller gain conidering the computer numerical controller (CNC) component (PID with feedforward controller tructure, delay, quantization), the mechanical tructure and the drivechain component included in the imulator. Two method are available in SICOMAT. The firt one i a time approach baed on interactively calibrating tep repone. The econd one i baed on a frequency automatic approach, which guarantee tability margin for a given cloed loop bandwidth uing optimization technique [3][4]. The paper i organized a follow: ection 2 decribe the global environment of the imulator and the tructure modeling, ection 3 preent the different toolboxe available in the imulator: model reduction, modal analyi, control deign technique (time and frequency approache). Finally, ection 4 give an example of a high-peed machine with a linear direct drive axi. II. Decription of SICOMAT A. Main menu of the imulator Fig.. Main window of the imulator SICOMAT i deigned to eaily et up the imulation environment (mechanical model, CNC, actuator and enor model) a well a the parameter value, uch a the number of mae and pring for the mechanic, the geometrical and inertial parameter, the repreentation of the mechanical tructure with linear tate-pace model or with imulink block or with robotic formalim [2]. The uer may elect the deired component in a data bae that contain commonly ued motor, gearboxe, tranmiion and o on, or define hi own component. The electric actuator

3 dynamic and aturation, the effect of delay and quantization due to CNC are alo conidered [3]. The oftware i entirely boxe-driven and enable the uer to deign and modify a virtual prototype and to eaily check it performance in a CAD approach without the need of the real prototype. The main window of the imulator i hown on Figure. B. Trajectory generator Several mechanical model are implemented in the library of SICOMAT. They are choen through 3 boxe, Figure. The box 'Number of axi' allow to chooe machine with one or two decoupled linear axi, or with two coupled axe. The boxe 'Model on X axi' and 'Model on Y axi' define the number of mae (M) and pring (S) of the lumped model for each axi repectively. The boxe 'Decription on X axi' and 'Decription on Y axi' allow tochooe between a block diagram or a teady tatepace or a robotic formalim decription of the mechanical model. In that lat cae, ome parameter of the decription are defined in 2 file whoe name are given in the box 'Aociated function to the model'. Clicking on the 'OK' button run the automatic calculation of the Simulink block which are linked together and aociated with Matlab routine to imulate the ytem while tracking different trajectorie (traight line, circular interpolation, ), to perform frequency and time analyi and to diplay reult. An example of a ingle axi i repreented on Figure 2. time u()>time_f*.2 STOP poition et point Trajectory generator poition_reference Scope poition_reference Error poition_ref vel_ref_motx poition_meaured Poition controller Speed controller vel_ref vel_me vel_me2 Ui_ref Ui_ref load_poition poition_meaured load_velocity Power converter Motor Mechanical tructure poition Scope poition Fig. 3. An example of a machine axi and the radiu of the driving wheel upported by the motor haft repectively, with: x pm = R pm pm () J pc, pc, x pc, R pc are repectively the inertia, the angular and linear poition and the radiu of the driving wheel upported by the rack and gearwheel, with : x pc = R pc pc (2) = Rpc R pm i the ratio of the belt tranmiion, m c i the ma of the belt. J pig, pig, R pig are repectively the inertia, the radiu and the angular poition of the gearwheel. k eq i the global torion tiffne between m and pig. x cre, m cre are repectively the linear poition and the ma of the rack. x M tand for the load poition. M load i the load ma. Internal material damping, vicou and Coulomb friction force are alo conidered: μ eq i the global material damping coefficient between m and pig, μ 2eq i the global material damping coefficient between x pc and x M, f eq i the global vicou friction coefficient between the load and the motor. C.2 Linear tate pace model Quit Modify Parameter Modify Scheme Save Scheme Analyi The tate pace model cae obtained from the Lagrangian equation a recalled here: Fig. 2. Example of a ingle axi MẌ + μ _X + KX = F (3) C. Modeling of the mechanic C. Decription of the ytem The example of a machine compoed of a motor, a tranmiion gearing with two driving wheel and a belt i given to illutrate the modeling method, Figure 3. The motion tranformation from rotation to tranlation i achieved with a rack and a gearwheel. In thi example, the model i implified to 2 Mae and one Stiffne (2MS), but more accurate model are available in the imulator. The following notation are ued in Figure 3: mot i the driving torque. J m, m are the inertia and the angular poition of the rotor of the motor repectively. J pm, pm, x pm, R pm are the inertia, the angular and linear poition X i the (nx) vector of degree of freedom of the ytem, n i the number of degree of freedom F i the vector of driving force, μ i the damping matrix, M and K are the ma and tiffne matrice repectively. In the cae of the example, (3) i written with: x = m x M ;F = mot M = Jeq J 2eq M 2eq = M cre + M load + J pig + J apig + J pc R 2 pig (4) (5) (6) J eq = J m + J am + J pm + m c R 2 pm (7)

4 J am i the inertia of the motor haft. ψ μeq + feq n 2 Rpig 2 b μ = K = ψ feq R pig feq R pig μ 2eq + f eq keq n 2 Rpig 2 b keq R pig keq R pig k eq!! (8) (9) Multiplying (3) by the invere of the ma matrix lead to: Ẍ = M F M μ _X M KX () And the tate equation follow: Where z = A = _ X X _z = Az + BF () i the tate vector and: n I n M K M μ ; B = n M (2) I n i the (nxn) identity matrix. n i the (nxn) matrix of zero. C.3 Simulink block diagram Thi decription ue the elementary block diagram of the oftware Simulink. The uer draw the model jut a with pencil and paper by uing click-and-drag moue operation. Thi method avoid to expre differential equation in a language or program. Simulink include a comprehenive library of ink, ource, connector, linear and nonlinear component. Thi approach provide a graphical inight which can facilitate the undertanding of the model, but it i limited, for ake of implicity, to model from the rigid one to the 8 Mae and 7 Spring model. The block diagram of the example (3), (4), (5), (6), (7), (8), (9), equivalent to the tate equation (), i given on Figure 4. Motor Torque Fig. 4. /Jeq Inertia Coulomb & Vicou Friction Rpig/nb Rpig/nb Rpig/nb keq Stiffne mueq /M2eq Inertia2 Coulomb & Vicou Friction2 2 motor velocity motor poition 4 load poition 3 load velocity Example 2m implemented a a Simulink block diagram C.4 Robotic formalim Another apect of SICOMAT i to propoe an original method of decriptioaed on robotic formalim. [2] propoe to adapt method, which are derived from robotic in order to provide ytematic and automatic model for ytem with lumped elaticitie. One main application of the method concern high-peed machine tool and robot with elatic joint. The method can provide the kinematic and dynamic model of uch ytem. To achieve thi goal the method adapt ome well-known tool and notation, which are widely ued for rigid robot. The technique conit on uing a decriptioaed on the modified Denavit-Hartenberg (MDH) [5] to define the kinematic of the ytem. Given the robot geometric parameter and it dynamic parameter, the geometric, the kinematic and the dynamic model are calculated automatically by the SYMORO (SYmbolic MOdelling of RObot) oftware package [6]. Thi oftware provide the different model under C or MATLAB code ueable directly in the MAT- LAB/Simulink environment. It hould be noticed that for identification purpoe, the tandard dynamic model can be rewritten in term of a minimal et of parameter called the bae inertial parameter which are the only parameter that cae identified [7]. Thee parameter cae obtained from the claical inertial parameter by eliminating thoe, which have no effect on the dynamic model and by regrouping ome other. Thi procedure reduce the complexity of the dynamic model and will be ued ytematically hereafter in order to decreae the imulation time. Uing the bae parameter SYMORO calculate the minimal dynamic model ued for imulation or control. D. Electrical drive modeling Mot of actual machine tool actuator are ynchronou motor with PWM converter and include current loop with field oriented control in the d-q frame. Thu it i aumed that the current loop cae modeled on q axi by a firt order tranfer function, between the current etpoint and the driving torque. The aturation of the current reference and the upplied voltage are imulatedtaking into account the effect of the electromotive force (back emf E) [3]. E. Computer Numerical Control (CNC) The controller implemented in indutrial CNC are uually baed on a cacade tructure with a fat inner loop for torque control and outer loop for peed and poition control. Thi tructure improve the robutne of the controller with repect to the diturbance and modeling error. The peed control loop i claically implemented with proportional and integral (PI) controller (PI or IP tructure) with anti-windup trategy. The poition loop i deigned with a P controller. In order to decreae the tracking error, velocity and acceleration feedforward ignal might be added to the control loop. Several apect of the control are conidered in the imulator uch acontinuou or dicrete time control law, pole placement method, time or frequency approache. III. Analyi and control tuning of a machine with SICOMAT Four boxe are all uer-electable on Figure 2. Ξ Modify Parameter

5 The drive chain of a machine tool uually contain a motor, a drive pulley or a gearbox and a tranmiion, which cae a ball-crew gear, a drive pulley or a rack and gearwheel, and a load. Manufacturer' data are given in file regrouped in a databae. The name of the file needed for a model are given and cae modified with thi function. Ξ Modify Scheme The main imulator, Figure 2, i automatically built from four elementary Simulink cheme given in a library: the trajectory generator, the power converter, the controller and the mechanical tructure. Thee block cae changed to modify the imulator, taking new one from the library. Ξ Save Modelbox Any new or modified imulator cae aved a a new imulator. Ξ Analyi box Thi box i dedicated to analyze the flexible mode of the mechanical tructure, to reduce the model order, to compute the time and frequency repone and to optimize the controller tuning. Clicking on thi box open the following menu: tep. Firtly gain value are calculated uing a implified rigid model. Secondly thee value initialize an optimization procedure which ue the complete model implemented in the imulator to perform an accurate tuning. There are two baic approache of deigning the digital controller: the direct dicrete-time deign or the continuou controller redeign [9]. We chooe the econd one becaue it allow continuou time technique to be ued, pecially the deign in the frequency domain which enure tability margin. Firtly, acontinuou time control law i defined. Secondly it i dicretized and then it i approximated by continuou tranfer function, uch that the gain cae calculated with a continuou controller which take into account the effect of digital control in the bandwidth of the cloed loop. Detail cae found in [3], [4]. At firt, the gain K p, T i of the peed loop,are tuned in order to get a phae margin ' mv at a frequency! vit which depend on the frequency f meca of the firt flexible mode of the proce (uually! vit =2ßf meca =(4 to )). Then the gain K v of the poition controller i calculated in order to get a phae margin ' mp. The tuning mut lead to a behavior for the peed loop imilar to the Figure 6. T bov () i the open loop velocity tranfer function. Fig. 5. Function of the Analyi box Fig. 6. Phae margin of the peed loop with the rigid model and the IP controller A. Modal analyi of the Sytem The "undamped ytem" function calculate the natural frequencie a the eigen value of (M K) which define the undamped ytem (μ = in (3)). The "damped ytem" function calculate the damping coefficient and the natural frequencie of the full implemented mechanical model calculated from the 'linmod' Matlab function [8]. The component of the eigen vector are alo calculated to be interpreted a the contribution of each degree of freedom on the ytem. Thi i a help to ee the dominant mode of the model and to chooe the proper bandwidth for the controller. B. Controller tuning B. Initial tuning on a implified rigid model The uer problem i then to tune the controller gain conidering both the CNC component and the mechanical tructure. A ignificant point of SICOMAT i to propoe an automatic tuning procedure compoed of two B.2 Accurate tuning of the controller taking into account the complete mode Thee value initialize next an optimization procedure uing the full model to perform an accurate tuning. At thi tage, the complete model implemented in the imulator i taken into account: lumped model of the flexible mechanic, digital control and location of meaurement. The 'dlinmod' Matlab procedure automatically calculate high order dicrete tranfer function which make impoible to get analytical expreion for the gain. They are calculated with optimization technique, uing the procedure 'fmin', baed on the Nelder-Mead implex method [8] in order to atify the phae margin and bandwidth. In cae of the complete model, the delay and the ocillatory pole and zero introduced by thenumerical control and the flexible mechanic makethe gain tuning very delicate. Too mall gain margin may be obtained in pite of a good phae margin(typically 6dB gain margin i needed). The tuning procedure conit then in computing again the gain by optimization with gain margin criterion, tarting from the value obtained with the phae margin criterion.

6 B.3 A time approach deign A an alternative to the automatic and optimized gain tuning, a time approach propoed by NC manufacturer i implemented in SICOMAT. The method conit on imulating tep repone tuning the parameter of the controller ucceively. At each iteration, the imulator perform a imulation with the new value of the parameter of the controller. The firt tep i the tuning of the peed loop. The integral gain i fixed to a large value (.5) in order to move the integral effect to low frequencie, outide off the velocity bandwidth. The proportional gain i adjuted running imulation of the velocity cloed loop in order to get a tep repone overhoot cloe to %. Then the integral gain i adjuted to get a tep repone overhoot between 2% and 4%.The econd tep i to adjut the proportional gain of the poition controller in order to get a deired overhoot. At each iteration SICOMAT provide numerical information on the time riing and the overhoot. C. Model reduction The mathematical model obtained with SICOMAT are often highly order becaue they take account the dynamic of rigid bodie, the elaticity of bodie and mechanical tranmiion, the dynamic of electric actuator, the effect of numerical control a delay. One difficultie in deigning controller for complex ytem lie in the problem to repreent a complex model by a equivalent model which ha a maller number of degree of freedom (dof). Three method are implemented in SICOMAT:. The firt one i baed on the calculation of the eigenvalue and eigenvector of A matrix 2, which are orted in order to cancel the m fatet mode of A with matching tatic gain. The tate of the reduced model ha lot it phyical meaning with repect to the dof defined by the initial tate vector. 2. The econd one ue the partial Gramian operator in order to take into account ome controllability and obervability apect of the ytem [][][2][3]. Thi method keep the drawback of the interpretation of the dof of the reduced model. 3. The aim of the third one i to keep the mot ignificant dof of the reduced model among thoe of the initial tate vector X of (3), a a reult of a technological analyi of the mechanical ytem. The reduced model i then calculated by regrouping erial or parallel mae and tiffnee of the full model with very imple rule. Thi method i quite le accurate than the firt two one from a ytem analyi point of view, but keep an intereting phyical meaning. IV. Analyi and imulation of a 5M4S model A. Decription of the ytem SICOMAT ha been devoted to a large number of indutrial cae. A an example, a model compoed of 5 mae and 4 pring (noted 5M4S) will be conidered in thi paper (Figure 7). Thi model repreent a ingle axi compoed of a bed, a double linear ynchronou motor, a linear cale and a load, which may be moved on a perpendicular axi. The two motor' primary are mounted like a gantry type, but very cloe ide to ide on a very rigid tructure in order for the gantry to be conidered a only one equivalent motor. The primary part i a hort Fig. 7. 5M4S Model for a high peed machine tool axi moving coil aembly and the econdary part i a long permanent magnet aembly. The notation ued on Figure 7 are: i the driving force, R o (O o ;x o ;y o ;z o ) i the reference frame, M tructure and k tructure are repectively the ma of the tructure and it tiffne w.r.t. the bed, m econdary and k econdary are repectively the ma of the econdary part of the linear motor and it tiffne w.r.t. the tructure, m prinary i the ma of the primary part of the linear motor, m enor and k enor are repectively the ma of the linear cale enor head and it tiffne w.r.t. the primary part, M load and k load are repectively the ma of the load and it tiffne w.r.t. the primary part. Shortly, the lumped model i compoed of: 5 mae M tructure, m econdary, m primary, M enor, M load and 4 tiffnee k tructure, k econdary, k enor, k load. We will alo conider internal material damping, vicou friction coefficient and Coulomb friction force: μ tructure i the material damping coefficient of the tructure, μ econdary i the material damping coefficient ofthe econdary part of the linear drive, μ enor i the material damping coefficient of the load, μ load i the material damping coefficient of the enor head upport. F vmotor i the vicou friction coefficient between primary and econdary part of the linear drive, F Coulombmotor i the Coulomb friction force between primary and econdary part of the linear drive. Thee imulation are carried out with the following parameter for the 5M4S model: M tructure = 3667kg, m econdary = 232kg, m primary = 92:35kg, M enor = 27:2kg, M load = :8kg, k tructure = e9n=m, k econdary = e2n=m, k enor = 3:8e8N=m, k load =e7n=m, μ tructure =9:5N=(m=), μ econdary = 7:7N=(m=), μ enor = 4:6548N=(m=), μ load = :89N=(m=), F vmotor =88:N=(m=), F Coulombmotor = 42N F vmotor and F Coulombmotor are iued from identification reult. μ tructure, μ econdary, μ enor and μ load are given by technological dataheet.

7 B. Mechanical flexible mode analyi The frequency analyi toolbox give the mechanical frequencie, the pulation and the damping coefficient of the ytem. We oberve in thi example (Figure 8) a firt mode near from f meca = 8Hz. The bandwidth of velocity and poition loop cae then repectably chooe equal to! vit = 25:6rad= (2ßf meca =4) and! po = 3:4rad= (2ßf meca =6). the peed loop of ' mv =45 o with a bandwidth of! vit = 26 rad=. For the poition loop, we obtained ' mp =8 o with a bandwidth of! po =33:5 rad=. Parameter K p ( ) T i () K v ( ) Initial tuning Optimized tuning TABLE I Deign of the controller Fig. 8. C. Controller tuning Frequency analyi of the ytem The velocity phae margin ' mv of each axi i choen equal to 45 o with a PI tructure. The frequency and time repone of the velocity loop are given in Figure 9. Open Loop Gain (db) Speed Nichol Chart 2.3 db 6 db 6 db 3 db Open Loop Phae (deg) Fig. 9. db Speed (m/) Step repone of peed loop Time () Frequency and time repone of the peed loop The different loop on Nichol diagram of the Figure 9 are due to the poorly damping mode of the proce. The phae margin ' mp of the poition loop i equal to 8 o. The frequency and time repone of the poition loop are given in Figure. Open Loop Gain (db) Poition Nichol Chart 2.3 db 6 db Open Loop Phae (deg) Fig.. db 3 db 6 db Poition (m) Step repone of poition loop Time () Frequency and time repone of the poition loop The controller tuning i performed in order to achieve the deired phae margin. By electing in the deign controller toolbox, the frequency optimization procedure decribed in the Section 3.2.2, the following parameter are computed in Table I. The performance obtained with thi et of parameter, in term of bandwidth are a phae margin for V. Concluion Thi paper preent the oftware package SICOMAT, which i devoted for the imulation and the prediction of the cloed loop performance of high-peed machine tool axi. The imulation are carried out in the Matlab/Simulink environment. Several toolboxe give the uer the opportunity to completely model and analyze it proce. The mot important contribution for the modeling i the mechanical modeling, which i baed on a multibody tructure model with lumped mae and elaticitie. The analyi i performed through the calculation of the mechanical flexible mode, the controller tuning and the computation of the dynamic behavior. Claical machine tool axe are implemented in a databae or cae eaily created by the uer. Future work will concern the extenion of thi work for any n axe. Reference [] P. J. Barre, Control Strategie For A Numerical Machine Tool Axi In Cae Of Very High Speed Machining, Ph D. Thei, E.N.S.A.M. Pari, 995. [2] W. Khalil, M. Gautier, Modeling of mechanical ytem with lumped elaticity," Proc. IEEE Conf. On Robotic and Automation, SanFrancico, USA, 2. [3] P. Poignet, M. Gautier,W. Khalil, Modeling, Control and Simulation of High Speed Machine Tool Axe," IEEE /ASME International Conference on Advanced Intelligent Mechatronic, Atlanta, USA, pp , 999. [4] M.T. Pham, P. Poignet, M. Gautier, Automatic Tuning of Cacade Structure CNC Controller," 6th International Workhop on Advanced Motion Control, Nagoya, Japan, pp , 2. [5] W. Khalil, J.-F. Kleinfinger, A New Geometric Notation for Open and Cloed-Loop Robot," Proc. IEEE Conf. On Robotic and Automation, San Francico, USA, pp. 74-8, 986. [6] W. Khalil, D. Creuot, A Sytem for the Symbolic Modeling of Robot," Robotica, vol. 5, pp. 53-6, 997. [7] M. Gautier, Numerical calculation of the bae inertial parameter," Journal of Robotic Sytem, vol. 8, No 4, pp , 99. [8] MATHWORKS, Matlab and Simulink Reference Guide, The Mathwork, Inc. Natick, Ma., USA, 998. [9] F.L.L. Lewi, Applied Optimal Control and Etimation, Prentice Hall, 992. [] T. Camboi, Simplification de modele de yteme mécatronique en vue de la commande. Application aux upenion active pour l'automobile, Ph. D. Thei, Univerité de NANTES-Ecole Centrale de NANTES, 998. [] M. G. Safonov, R. Y. Chiang A Schur method for balancedtruncation model reduction," IEEE Tran. Automat. Contr., vol. 34, No 7, p , 989. [2] K. Glover, All optimal Hankel norm approximation of linear multivariable ytem and their L error bound," Int. J. Contr., vol. 39, No. 6, pp. 5-93, 984. [3] B. C. Moore, Principal component analyi in linear ytem: Controllability, obervability, and model reduction," IEEE Tran. Automat. Contr., vol.26,pp.7-3,98.