Friction Compensation in Servo Systems Using a Local Control Design Approach

Transcription

1 Frcton Compensaton n Servo Systems Usng a Local Control Desgn Approach L. Mostefa* M. Dena ** Y. Hor* * Insttute of Industral Scence, Unversty of Tokyo 4-6-,omaba Meguro, Tokyo JAPAN (Tel: ; e-mal: latyfo@gmal.com). ** Dept Automatc Control and Systems Engneerng, Unversty Sheffeld (e-mal:m.dena@sheffeld.ac.uk) Abstract: Nonlneartes degrade consderably performances n moton control systems. Nonlnear frcton s a major source of many serous problems such as wear, trackng errors and lmt cycles. There has been an extensve research actvty dealng wth the desgn of compensatng technues. The approaches cted n the ltterature can be dvded nto: free model compensaton or model based compensaton strategy. In the present work, a dynamc fuzzy modelng approach of a servo system wth frcton s developed. The man dea s to take advantage of the lnear form of the resultng dynamcs to desgn: - a frcton observer used as a compensatng term of frcton effects, 2- a stable trackng controller that allows the system to acheve a trajectory nvolvng slow motons and velocty reversal. The proposed control method s relatvely smple to desgn and effcent for the compensaton of frcton nduced errors. The expermental tests on a robot jont control have demonstrated precse moton control and smoother velocty reversal n the presence of sgnfcant level of frcton.. INTRODUCTION In general, mechancal nonlneartes have a negatve mpact on performances of servo systems; for ths reason, modellng and compensatng nonlneartes have been a topc of specal nterest n moton control systems. Frcton s one of these hard nonlneartes that are unavodable, undesrable and challengng for control engneers (Armstrong-Helouvry et al., 24). Specal care has been made for modellng, and several approaches have been adopted to deal wth ther effects such as: varable structure and hgh stff gans control technues (L et al., 24) and other dependng on models and ther ablty to reproduce the nonlnearty. Adaptve control technues were also used to cope wth the varyng nature of these nonlneartes due to varous effects such as: temperature, load and even wear (Lschnsky et al., 999). Frcton s usually modelled as a dscontnuous statc map of velocty. Statc models are often restrcted to Coulomb and vscous frcton components or descrbed by the Strbeck curve (Strbeck et al., 92). Many other models have been developed to descrbe better frcton phenomenon (Swevers et al., 2). Manly based on expermental observatons, there are several dynamc models that provde a relable descrpton of the phenomenon (Canudas de Wt et al., 995); they nclude many nterestng characterstcs such as: frctonal lag, Dahl effect (Dahl et al., 968). However, ther good capablty to descrbe frcton makes them at the same tme very complcated, and even showng a hybrd nature that have been treated n (Cho et al., 26), ths results n a more complex model of the system and has negatve mpact on the dentfcaton effort, parameters estmaton and the control desgn tself. One can also note that other models gvng less precse descrpton of frcton can have serous problems n compensatng frcton nduced errors especally at low veloctes. In the followng work, we frst start by adoptng a local approach a dynamc fuzzy model of the robot jont wth frcton s developed (Mostefa et al., 27). After that, conventonal experments to dentfy the parameters of local models are conducted on a robot jont. The proposed control strategy s bascally dvded nto two actons based on: frst, a lnear trackng controller; then, a nonlnear frcton estmatorcompensator based on local modellng approach n secton 2. Local stablty of the overall system s verfed for the desgned controller. Fnally, expermental results allowed us to verfy the effcency of the proposed control method. 2. SYSTEM DESCRIPTION: LOCAL MODELING Snce ths work focuses on frcton compensaton, all other nonlneartes actng on the robot jont are assumed to have no effects on the dynamcs. We consder a sngle jont robot wth the mathematcal model gven by: J & + F = τ () where J denotes a constant nerta, F the nonlnear dynamc frcton., & and & & are the acceleraton, velocty and poston, respectvely. Frcton s locally modelled as dynamcal lnear state space system changng wth velocty of the jont; t has two man actons: a part related to velocty characterzed by a dampng nature, and a second part related to the dsplacement characterzed by a stff nature.

2 IF & = & THEN z& = αz + F = σ z + d (2) 2- For slow motons regme, and specally at low veloctes where frcton s characterzed by the Strbeck effect and the torue has nether a pure stff nature nor a pure dampng nature but a mx of both effects, the poston of the robot jont s controlled by low gans PD compensator under constant veloctes (Fg. 2). ndcates the local velocty range. σ : s a local stff coeffcent of frcton torue. d : local dampng coeffcent. σ and d are kept constant for low velocty for smplfyng & the desgn of the proposed control. α σ, s a F mn very mportant parameter that characterzes the model dynamcs and stablty. F mn : s a postve value of the lower dentfed level of frcton for & >, we wll use a symmetrc values of frcton. 3- For hgher veloctes, frcton has a lnear dampng nature: the vscous frcton. In ths work, we are more concerned wth the frst two cases, where frcton s hghly nonlnear, and can be source of stckslp motons and performance degradaton. The parameters dentfed on the robot jont are gven n appendx A. By applyng a standard fuzzy nference method to the dentfed local parameters,.e. usng a sngleton fuzzfer, product fuzzy nference and center average defuzzfer, the robot jont model wth frcton can be wrtten across the overall operatng doman The nfluence of ths parameter on the overall controlled system wll be evaluated later n secton 3. The lnear tme nvarant resultng model allows local analyss and desgn of the compensator of frcton and ts effects. We should note that the ntroduced nternal state z s a non-measurable uantty, though t has an mportant role n the system descrpton and the overall compensator desgn. The euvalent model of frcton can be formulated usng fuzzy nferences (Takag et al., 995) n z& = μ ( & ) αz + & (3) = And snce the stffness and the dampng coeffcent wll be fxed, then we have: F + = σ z d (4) Assumng that only frcton parameters are varyng wth velocty, the state space local model of the robot jont wth frcton IF & = & THEN z α d = dt & σ z + τ d & (5) d dt Input Torue (N.m) z = & n α μ ( & ) σ z + τ d & = dsplacement (rad) x -3 Fg.. Mcro-dsplacement regme dentfed n the robot jont n the stckng mode F (N.m) (6) One of the most senstve steps s the dentfcaton of local parameters. Common experments are conducted to determne each parameter (Lantos et al., 27); the process can be summarzed n the followng steps: d/dt - For the presldng regme, where the nput torue level s lower than the break-away torue, the Dahl curve can be plotted and the parameter σ s then deduced (Fg. ). Fg. 2. Frcton levels n the velocty reversal area, showng a hgh nonlnearty and asymmetry.

3 The number of the local models used n the proposed structure has to ncrease to gve a precse descrpton (Wang et al., 992). In that case, we should note that all parameters n (4) would vary for dfferent veloctes. Ths model wll be used n the desgn of a stable frcton compensator 3. COMBINED COMPENSATOR-CONTROLLER DESIGN Frst, let us defne the fltered trackng error as r = Q( e) = e& + λe (7) where e = r s the poston error, e& = & r & the poston error varaton, λ > s a postve gan characterzng the controller dynamcs. The structure of the servo system model n (4) and (5) allows us to propose a control algorthm based on two actons: frst a compensatng nput based on local estmated frcton zˆ & = α zˆ + & l r (8). and second a Proportonal Dervatve acton for achevng trackng control of the overall system, gvng the followng local control nput, IF & = & THEN τ = J& + r r + σ z ˆ + d& (9) The trackng performance can be verfed by substtutng the control sgnal (8) and (9) nto the robot jont model (5). The error model s then obtaned E& = A E () = ~ as the error vector and, wth E [ z e e& ] T and the matrx characterzng ts stablty α A = σ m λ. l. λ m l m for () In case of perfect frcton compensaton, the model s lneared and the trackng control problem can be converted to a classcal PID trackng of a lnear servo system. Snce t s not true for real plants, we have to consder error dynamcs by ntroducng the compensaton term plus the trackng controller, and choose the gans and l of the controller that make the error dynamcs stable and ensure ts convergence to zero. That means gans that yeld egen values of A wth negatve dampng. The analyss of the effect of the controller on local stablty starts from the model at very low veloctes where α (fg. 4), once the gan of the controller s fxed frst to a certan value that stablze the system wth l = for the entre operatng doman. The observer gans l can be then chosen n the stable regons accordng to the euvalent velocty of the model of frcton, the goal s to ensure good performances wthout compromsng the stablty of the system. The observer gans wll be tuned durng experments n order to mprove the trackng performances. The local stablty s guaranteed by the desgn method. In the regon of mxed dynamcs, where two dfferent models are nteractng, we can see the problem as two weghted stable systems nterconnected to each other. Robot jont model Real frcton Fg. 3. Proposed control scheme for trackng enhancement for servo system usng local modellng approach of frcton.

4 P 4 P l l. Fg. 4.a Map of dampng values varatons of one real pole n (), at very low veloctes, the darkest regon n blue has value of -.4 usng the parameter dentfed n the robot jont. Fg. 5.a Map of dampng of one real pole n () at hgher veloctes. The darkest regon n blue has the same value as n the model for very low veloctes /3 k 2 RE(P 2,P 3 ) RE(P 2,P 3 ) l l Fg. 4.b Map for dampng values of the 2 complex poles n (), at low veloctes. The dark blue s euvalent to negatve dampng. The nterpolaton process s lnear, thus the dynamcs of the state matrx characterzng the stablty nsde the mxed regon can be wrtten by α λ.( μl + μ+ l+ ) μl + μ+ l+ A = / + (2) σ. λ m m m Usng the propertes of the weghtng functons and the desgn parameters to ensure local stablty, the egen values of (2) can be calculated and local stablty between two successve regons can be checked. Compared to some methods usng one fxed gan for all the operatng range of the system wth frcton (Canudas de Wt et al., 995), the gan schedulng used n the compensator (9) combned wth the estmaton mechansm (8), allow flexblty n the desgn, enhanced trackng performances, and avodng a hgh control nput specally at very low veloctes. The parameters used n the compensator desgn were dentfed n robot jont. The model used for the frcton compensaton s tuned durng experments. We should note that the reference trajectory make the system gong nto a severe regme of slow motons and reversal veloctes. Fg. 5.b Map for dampng of the 2 complex poles of () at very low veloctes. The regons n dark blue gve negatve dampng. 4. EXPERIMENTAL RESULTS AND EVALUATION Experments were performed on a jont of FANUC robot n order to evaluate the proposed control strategy. The expermental setup as shown n (fg. 6) conssted of: a PC 7 MHz operatng system RT-LINUX. The PC connected by an optcal cable to a dgtal servo adapter that ensures sgnals nterfacng between the PC and a servo amplfer module. The control algorthm s mplemented n C language. The reference trajectory makes the robot jont go through the low veloctes regon and velocty reversal. Dgtal servo adapter Robot jont Servo amplfer module Fg. 6 Expermental setup for the robot jont control.

5 x d 2πft ( t) = sn(2π ft)sn (3) The robot jont wll be operatng at low veloctes regon and performng reversal many reversal veloctes durng the experments (f =.Hz). The results n (fg. 7), (fg. 8) and (fg. 9) shows the trackng error and the jont poston wth reference (3). After compensaton, t s clear that, the RMS trackng error s reduced more than tmes. Ths confrms the effectveness of the use of the combned acton of the observer -controller e RMS =.233 rad, r Fg. 8.b wth fuzzy model as compensor: poston of the robot jont (n black) trackng a reference trajectory (2) (n dotted grey) e(rad) e RMS =.2 rad Fg. 7.a trackng error wthout frcton compensaton. e (rad) Fg. 9.a trackng error under proposed frcton compensaton scheme wth compensatng gans.2, r , r -.2 Fg. 7.b wthout frcton compensaton: poston of the robot (n black) jont trackng a reference trajectory (2) (n dotted grey) e (rad) e RMS =.42 rad Fg. 8.a trackng error under proposed frcton compensaton scheme wthout compensatng gans Fg. 9.b poston of the robot jont (n black) trackng a reference trajectory (2) (n dotted grey). In case only the fuzzy model s used (fg.8), better results can be reached by usng more local models, ths reure more parameters to dentfy whch gve the advantage to the use of compensatng gans for the observer to reduce the estmaton error between two known local models. 5. CONCLUSIONS Ths paper proposes a local desgn approach appled to a robot jont trackng control for frcton compensaton. The control s based on a fuzzy model of nonlnear frcton.

6 Stablty analyss of the overall controlled system was used for choosng local observer gans. Experments on robot jont show the effectveness of the proposed method. Further developments are to be done n the parameters modellng and a desgn of a robust compensaton scheme, and the extenson to other applcatons such us dsk drve control and tracton control systems. REFERENCES Armstrong-Helouvry, B., P. Dupont, and C. Canudas de Wt (994). A survey of models, analyss tools and compensaton methods for the control of machnes wth frcton. Automatca, 3(7): Canudas de Wt, C., H. Olsson,. J. Astrom, and P. Lschnsky (995). A new model for control of systems wth frcton. IEEE Trans. Automat. Contr., Vol. 4, pp Cho, J.J., S.I. Han and J.S. m (26), Development of a novel dynamc frcton model and precse trackng control usng adaptve back-steppng sldng mode controller. Mechatroncs., 6, pp Dahl, P. (968). A Sold Frcton Model, The Aerospace Corporaton, El Segundo, CA, Tech. Rep. TOR- 58(37-8). Lee, S.W., and Y.H. m (995), Robust adaptve stck-slp compensaton. IEEE Trans. Indust. elect., Vol. 42, pp L, Y-F. and J. Wkander (24), Model reference dscretetme sldng mode control of lnear motor precson servo systems. Mechatroncs., 4, pp Lschnsky, P., C. Canudas de Wt and G. Morel (999), Frcton compensaton for an ndustral hydraulc robot. IEEE Trans. Indust. elect., Vol. 42, pp Marton, L., B. Lantos (27). Modelng, dentfcaton, and compensaton of stck-slp frcton. IEEE Trans. Indust. elect., Vol. 45, pp Mostefa, L., M. Dena and Y. Hor (27). Fuzzy observerbased control of servomechansms subject to frcton dynamcs. The 33 rd IEEE of the ndustral electroncs socety conference IECON 7 Tawan(accepted) Strbeck, R. (92), De Wesentlchen Egenschaften der Get- und Rollenlager--the key ualtes of sldng and roller bearngs. Zetschrft des Verenes Seutscher Ingeneure, 46(38), , 46(39), Takag, T. and M. Sugeno (985). Fuzzy dentfcaton of systems and ts applcatons to modelng and control, IEEE Trans. Syst. Man. Cyber., Vol. 5, pp. 6_32. Wang, L. X. (992). Fuzzy systems are unversal approxmators, n Proc. st IEEE Conf.Fuzzy Syst., San Dego, CA,, pp Swevers, J., F. Al-Bender, C. G. Ganseman and T. Prajogo (2). An ntegrated frcton model structure wth mproved presldng behavor for accurate frcton compensaton. IEEE Trans. Automat. Contr., Vol. 45, pp Appendx A. Identfed and desgned parameters F (N.m) -.2,-.8,,.6,.5 σ 98 d.5 J (kg.m 2 ).. λ 25. & (rad/s) -,-.2,-.2,,.2,.2,,.,.5,.,.5,.,