Nonlinear Friction Compensator Design for Mechatronics Servo Systems Using Neural Network

Transcription

1 Tranacton on Control, Automaton, an Sytem Engneerng Vol. 3, No., June, Nonlnear Frcton Compenator Degn for Mechatronc Servo Sytem Ung Neural Network Dae-Won Chung, Nobuhro Kyura, an Hromu Gotana Abtract: A neural network compenator for tck-lp frcton phenomena n meachatronc ervo ytem practcally propoe to upplement the tratonally avalable poton an velocty control loop for prece moton control. The neural network compenator play the role of cancelng the effect of nonlnear lppng frcton force. It work robutly an effectvely n a real control ytem. Th enable the mechatronc ervo ytem to prove more prece control n the gtal computer. It wa confrme that the control accuracy mprove near zero velocty an the pont of changng the movng recton through numercal mulaton. Ho wever, aymptotc property of the teay tate error of the normal operatng pont guarantee by the ntegral term of tratonal velocty loop controller. Keywor: frcton force compenaton, mechatronc ervo ytem, neural network compenator, prece control I. Introucton Stck-lp frcton the natural retance phenomena to relatve moton between two contactng boe n the mechatronc ervo ytem uch a nutral robot arm an numercal control (NC) machne tool. It commonly compoe of Coulomb frcton, tatc frcton, an vcou frcton, etc., but t ha hghly nonlnear charactertc. In many moton control ytem, th frcton phenomenon become a omnant factor near zero velocty that prevent them from hghprece control becaue of t nonlnear nature an ffculte n hanlng t effectvely an n compenatng t aequately wth the lnear feeback control ytem. In orer to acheve hgh precon moton control, thee frcton mut be accurately compenate to cancel the effect n the real tme control ytem. Much work ha been cue n lterature[1]-[8] n term of formulatng a frcton moel, entfyng t parameter, an effectve compenatng metho. But t report e the ffculty n prectng an fnng exact moel becaue th phenomenon contnuou at zero velocty an epen on everal nonlnear factor. Due to thee reaon, mot lterature ha aopte unrealtc way to get over th phenomenon. Even f ome paper were reporte to be ucceful, they tll rae queton to realze the tratonal poton an velocty control loop. In our work, we approache to reolve th ue by the more realtc way to upplement compenator egn to the tratonal poton an velocty controller. Ung an aaptve neural network to the pont that the neural network able to mplement n the gtal computer practcally ha uperor performance to approxmate the nonlnear functon an to be robut n the varyng operatng conton. To emontrate our propoal, the numercal mulaton ha been one for 1-DOF (One Degree of Freeom) X-Y table an the reult have been compare wth thoe of laboratory experment on the X-Y table ytem. II. Frcton moel 1. Inutral mechatronc ervo ytem Fg. 1 how a typcal moton controlle DC motor wth t mechancal loa of nutral mechatronc ervo ytem nclung the poton an velocty controller an power amplfer. The 1-DOF mechatronc ytem uner nvetgaton a ma contrane rotatng moton wth tck-lp frcton between rotatng ma an ther upportng urface a hown n Fg.. Input Torque τ ω Velocty, x ( Dplaceme nt, x) The ynamc equaton for th 1-DOF rotatng machne of Fg. ecrbe by Eq. (1); J xt ( ) Fxt ( ) = τ ( t) (1) r Inerta J m J L Frcton Fg. 1. Fg. 1-DOF rotatng Rotatng machne. Machne where JF, an τ are loa nerta, frcton coeffcent an r nput torque, repectvely. xt (), xt (), xt () are tate varable of placement, rotatng velocty, an acceleratng velocty, repectvely. In th n orer ynamc ytem, the nerta parameter uually a contant when the ytem etup, but the frcton parameter uually vare wth operatng conton an workng envronment. θref PD Controller K p Neural Network Compenator Mechatronc Servo Sytem uc K (1 ) Kt ω ω v K f T u La R a J m J L θ Manucrpt receve: Sep. 8,., Accepte: Apr., 1. Dae-Won Chung: Dept. of Electrcal Engneerng, Honam Unverty (wchung@honam.honam.ac.kr) Nobuhro Kyura, Hromu Gotana: Dept. of Electrcal an Computer Engneerng(kyura@fuk.kna.ac.jp/gotana@fuk.kna.ac. jp) Fg.. Block agram of a mechatronc ervo ytem. K e Fg.1 Block Dagram Of A Mechatronc Servo Sytem

2 11 ICASE: The Inttute of Control, Automaton an Sytem Engneer, KOREA Vol. 3, No., June, 1 When the velocty of rotatng ma come cloe to zero, th value gnfcantly change ue to nonlnear an varyng charactertc of frcton coeffcent. Th caue the ecreae of accuracy n the moton control ytem an a reaon for ffculty n hanlng t va a lnear control metho.. Stck-Slp frcton phenomena The tck-lp frcton force moel whch a mofcaton of the moel ntrouce by Karnopp[5] allow for the evaluaton of the frcton force urng tckng an lppng moton n the mechatronc ytem. Th alo elmnate the numercal problem aocate wth near zero -velocty force computaton. The tck-lp frcton moel obtane by Eq. () only n the lmt a α. where, F = F ( x )[ λ( x )] F ()[1 τ λ()] x (a) lp tck 1, x > α > λ() x =, x α > (b) Throughout the analytcal nvetgaton, the artfcal zero parameter,α, taken to be entcally zero. Nonzero value are only ue n the mulaton purpoe to nure that numercal ntegraton algorthm reman to be table. The tck frcton may be repreente by Eq. (3). nonzero velocty repreente by Eq. (4). where, F ( x ) = F ( x ) µ ( x ) F ( x ) µ ( x ) (4) lp 1 x > µ () x = x ( / ) F [1 x x CR F F e = ] Bx ( / ) F [1 x x CR F F e = ] Bx (5) (6a) (6b) where F an F are repectve force rop from the tatc to knetc force level, x an x CR CR are the crtcal Strbeck velocty[6], an B an B are the potve an negatve vcou frcton coeffcent, repectvely. Th frcton force compoe of the Coulomb frcton, vcou frcton, an the Strbeck effect. The potve an negatve Coulomb frcton level repectvely preente a FC = F F an FC = F F. The Stbeck effect that the frcton force n many cae ecreang wth ncreae relatve velocty[4]. It caue by ncreae flu lubrcaton. An exa mple of the lppng frcton force hown n Fg. 4. F τ() t F > Ftck() τ = τ() t F < τ() t < F F τ () t F < (3) The above tckng frcton prove the value of the frcton force at the zero velocty (or near zero velocty). Th frcton jut regare a ea zone to etermne whether the movng ma wll tck or break free from the tatc frcton force. The potve an negatve lmt on the tatc frcton force are gven by F an F, repectvely. The tatc frcton force lmt are aume to be contant n th analy. When the movng ma oe not move.e., x =, an the apple force between the tatc level, the ampng mechanm prove a force whch keep the ma at ret. Th mean that the movng ma cannot move untl the apple force greater n magntue than the repectve tatc force. The tckng force hown on Fg. 3. Fg. 4. Slp frcton moel. The lppng frcton force aume to pate energy at all the nonzero velocte, an therefore boune wthn the frt an thr quarant. It aume there ext contant. B1 > max( B, B ) (7a) F max( F, F ), F > (7b) whch efne the pecewe lnear boun for the lppng frcton force a hown Fg. 4. F () x F Bxt (), xt () > (8a) 1 F () x F Bxt (), xt () < (8b) 1 Therefore, we can aume that the lp frcton boune wthn a lnear boun. F () F B x (), t xt () (9) lp 1 Fg. 3. Stck frcton moel..3 Slp frcton analy The lppng frcton prove value of the frcton force at III. Frcton compenaton Wth the nonlnear frcton charactertc of the mechanm ecrbe above, even f a goo phycal moel of the nonlnear ytem coul be obtane, t often ffcult to prect the

3 Tranacton on Control, Automaton, an Sytem Engneerng Vol. 3, No., June, frcton force to compenate t correctly n the lnear control ytem nce the ynamc near the regon of zero velocty are omnate by the nonlnear an contnuou functon. If nonlnear phenomena ext that are unknown to the egner, the problem to egn hgh accuracy moton control ytem even more ffcult. In many practcal moton control ytem, there are no mean to etmate the nonlnear frcton accurately an to compenate t effectvely. A a reult, they are of poor accuracy an may caue the lmt cycle near zero velocty. 1. Degn trategy In lterature, many control technque have been propoe for frcton compenaton [1]-[8]. However, they tll have ome queton to realze t n the mechatronc nutry. In th work, we hall propoe uage of a more realtc upplementary frcton compenator to the currently avalable control ytem o that the overall control performance guarantee. By obervng Fg.1 the nonlnear frcton force may be compenate by ang an approprate compenatng term to the control acton,.e., ut () = u () t u () t (1) r where ur () t normal poton an velocty controller output bae on the lnear ytem, wherea uc () t frcton compenator output whch to cancel the nonlnear frcton phenomena. The normal poton an velocty controller ha tratonally been ue n the moton control ervo ytem repreente by Eq. (11) an (1). Let x () t be ere trajectory an we efne c et () = x () t xt () or et () = θ () t θ() t (11a) et () = x () t xt () or et () = Ket () ω() t (11b) 1 t ur() t = Kve () t e( τ ) τ T (1) o where Kp, K v, T are proportonal gan of poton loop an velocty loop an ntegral tme contant of velocty loop, repectvely. Eq. (1) ncate that the poton control loop ha the proportonal term only an velocty control loop cont of proportonal an ntegral term a they are tratonally avalable n mot moton control ervo ytem. Intea, we hall propoe an aaptve neural network compenator for the frcton compenator to be fe-back potvely to the torque control loop.. Neural Network Compenator Degn We approache a nonlnear mappng between nput, x, an output, y, of neural network uner the aumpton that the actual nonlnear functon unknown. The nput-output of a neural network may be ecrbe by Eq. (13) an (14); ref p y = f ( σ ) (13) n j j σ = wx b (14) where f ( σ ) the actvaton functon, an wj, b are connecton weghtng value an ba value of each neuron, j repectvely. The connecton weght between the -th neuron n l the (l-1) th layer an j-th neuron n the l-th layer, w, at each j learnng tep may be ajute by the error back-propagaton learnng rule [8] a follow; 1 ( )( T E = y y ) (15) l l l w ( k 1) = w ( k) η w (16) j j j l E wj = (17) w where E a um of quare error between network output, y, an t ere output (or target) vector,, an η a learnng rate, repectvely. In our egn, the nput gnal are electe a 5 tme elaye equence, xt (), x ( t 1), xt ( ), xt ( 3), xt ( 4) of velocty of plant output. Fg. 5 how an aaptve neural network for th purpoe of whch the network cont of 5 hen unt an 1 output unt. The actvaton functon are conere hyperbolc tangent [9] for the hen layer an pure lnear functon [9] for the output layer, repectvely. The error backpropagaton algorthm choen to tran the neural network. In orer to tran the network, lp frcton ynamc gven by n Eq. (4), (5) an (6) electe a a target functon. ẋ(t) ẋ( t 1) ẋ( t 4) w j Input Layer b j IV. Smulaton reult 1. Smulaton conton A numercal mulaton of the overall mechatronc ervo ytem nclung a normal controller an a compenator wa carre out to verfy the compenaton effe ct. In orer to reflect a more realtc ytem, the X-Y potonng table wa conere for th mulaton. Sgnal from the poton an velocty meaurement were aume to be quantze by 1- bt A/D converter. Bae on laboratory calbraton of expermental enor, t reaonable to aume a reoluton of.5mm for potonng an 1 cm/ec for velocty. Samplng tme 5 mcroecon an zero -orer-holer tme elay alo conere. The parameter for th ytem are ummarze n Table 1. The ere trajectory wa apple for cae a follow; a) Snuoal gnal nput to X-Y Ax Table. b) Step functone pule gnal nput to X-Y Ax Table. The tck force work only at the tartng tme an changng recton tme near zero velocty. Th effect too mall to compenate the value. The lppng frcton force aume j v jk Hen Layer Fg Neural Network network Moel moel. p k Output Layer u c (t)

4 114 ICASE: The Inttute of Control, Automaton an Sytem Engneer, KOREA Vol. 3, No., June, 1 to exponentally ecay from the tatc level to a knetc frcton force at hgher velocty wth vcou ampng term ae. Table 1. Summary of mulaton parameter [7]-[1]. Para. Unt Value Para Unt Value R a Ω 1.59 K, P K -, - 6,4 V L MH 1.7 a T ec.4 J Kg.cm.35, F N 3.8 m J l Kg.cm, F N.1 K Kg.cm/A 7.54, t x CR m/. K mv/rpm 7.9, e B N/(m/).4 P Watt 49.3 α m/.1. Smulaton reult Tranng reult of target lppng frcton of aaptve neural network wa jut correpone to frcton moele by Eq. (4), (5), (6) an Fg. 4. The optmal weghtng value an ba ng value of two layer were obtane n the off-lne learnng proce. The mulaton reult ncate the error between ere trajectory an actual potonng at the tartng tme an at the turnng pont of the recton of X-Y table movement maller than wthout a compenator (Fg. 6,7,1 an 11). The velocty n X-Y Table wth a neural network compenator alo repone to be a maller evaton from the refe r- ence velocty than wthout a compenator (Fg. 8 an 1). The neural network output for a frcton compenator very cloely repone to the mulate lp frcton force output, whch cancel the lppng frcton effect (Fg. 9 an 13). 3. Comparon wth the laboratory expermental reult One of author ha experence n the laboratory experment that the lp frcton occurre at the turnng pont of movng ma n the X-Y table an coul mprove the reual error ung an approprate frcton compenator a hown Fg. (14) an (15). Thee expermental reult are very cloely repone to thoe n the mulaton of Fg.6. Velocty [rp] X-Ax Velocty Tme Repone ref. velocty tme [ec] Fg. 8. Velocty tme repone (X-ax) for cae (a). (Ref. Velocty; ; ). X-Poton Trajectory Tme Repone 4 Frcton Force an NN Compenator Tme Repone.5.4 ref. trajectory 3 NN compenator Frcton force.3 poton placement [m] Frcton Force [N] tme [ec] Fg. 6. X-poton tme repone for cae (a). Poton Trajectory for Cae (a) tme [ec] Fg. 9. Frcton an NN compenator tme repone for cae (a) Poton Trajectory Tme Repone for Cae(b) ref. trajectory Y-Ax Poton [m] poton placement [m] X-Ax Poton [m] Fg. 7. Poton repone trajectory for cae (a) (; ) tme [ec] Fg. 1. X-poton tme repone for cae (b).

5 Tranacton on Control, Automaton, an Sytem Engneerng Vol. 3, No., June, X-Y Ax Trajectoty for CASE (b) Y-Ax Dplacement [m] X-Ax Dplacement [m] Fg. 11. Poton repone trajectory for cae (b) (; ). Fg. 15. Expermental Recor for Cae (a) (X-Y Table w/ compenaton). velocty [rp] Velocty Tme Repone for CASE (b) tme [ec] ref. velocty Fg. 1. Velocty tme repone (X-Ax) for cae (b) (Ref. Velocty; ; ). Frcton force [N] Frcton an NN Compenator Tme Repone tme [ec] NN compenator Frcton Force Fg.13. Frcton an NN compenator tme repone for cae (b). Fg. 14. Expermental recor for Cae (a) (X-Y Table w/o compenaton). V. Concluon In our work, t wa confrme that poor accuracy ue to the tck-lp frcton take place at the near zero velocty an at the turnng pont of movng recton. It can be overcome by the egn of the upplementary compenator to the tratonal poton an velocty control loop. The neural network compenator mot approprate for th purpoe nce t work effectvely an can be ealy mplemente n the gtal computer, makng ue of conventonal poton an velocty loop control prmarly. VI. Reference [1] S. W. Lee an J. H. Km, Robut aaptve tck-frcton compenaton, IEEE Tran. on Inutral Electronc, vol. 4, no. 5, pp , Oct [] S. S. Ge, T. H. Lee, an X. Ren, Aaptve frcton compenaton of ervo mechanm, IEEE Proceeng on Control Applcaton, TuP4-3, pp , Aug. - 7,1999. [3] C. G. Baro an P. O. Gutrman, Performance enhancng aaptve frcton compenaton for uncertan ytem, IEEE Trancton on Control Sytem Technology, vol. 5, no. 5, pp , Oct [4] D. A. Haeg an B. Frelan, On moelng an mulaton of frcton, ASME Journal of Dynamc Sytem, Meaurement, an Control, vol. 113, pp , Sep [5] D. Karnopp, Computer mulaton of tck-lp frcton n mechancal ynamc ytem, ASME Journal of Dynamc Sytem, Meaurement, an Control, vol. 17, pp. 1-13, [6] K.C. Cheok, H.X Hu, an N. K. Loh, Moelng an entfcaton of ervomechanm ytem wth tck-lp frcton, ASME Journal of Dynamc Sytem, Meaurement, an Control, vol. 11, pp , Sep [7] Y.H. Km an F.L. Lew, Renforcement aaptve learnng neural-net-bae frcton compenaton control for hgh pee an precon, IEEE Tranacton on Control Sytem Technology, vol. 8, no. 1, pp , Jan.. [8] K.O Sohn an T.Y Kuc, Frcton compenaton of X-Y robot ung a learnng control technque, Journal of Control, Automaton an Sytem Engneerng, vol. 6, no. 3,

6 116 ICASE: The Inttute of Control, Automaton an Sytem Engneer, KOREA Vol. 3, No., June, 1 Mar. [9] S. Haykn, A comprehenve Founaton Neural Network, Macmllan College Publhng Co., [1] Y. Hujno an N. Kyura, Practcal Mechatronc Moton Control, Inutral Publhng Co., Japan, Dae-Won Chung He wa born n He receve B.S. egree from Puan Unverty n 1983, M.S. an Ph.D. egree n electrcal engneerng from Chungnam Unverty n 1996 an, repectvely. From 198 to 1996, he ha been a enor reearcher of Korea Atomc Energy Reearch Inttute. Snce 1997, he ha been a Profeor at Department of Electrcal Engneerng, Honam Unverty. H reearch nteret are n the area of mechatronc ervo ytem, avance an ntellgent control technque an gtal mcroproceor ytem, etc. Nobuhro Kyura He wa born n 194. He receve B.S., M.S. an Ph.D. egree n electrcal engneerng from Kyuhu Unverty n 1964, 1966 an 199, repectvely. From 1964 to 1995, he ha been a general manager of the reearch laboratory of Yakawa Electrc Corporaton, Japan. Snce 1995, he ha been a Profeor at Department of Electrcal Computer Engneerng, Knk Un - verty n Kyuhu. H reearch nteret nclue moton controller archtecture, optmum moton control an robot manpulator control, etc. Hromu Gotana He receve the M.E. an D.E. egree n electrcal engneerng from Kyuhu Unverty n 1978 an 1983, repectvely. From 1986 to 1987, he wa a Vtng Reearcher at Unverty of Calforna at Berkeley. From 1981, he ha been a Profeor at Department of Electrcal an Computer Engneerng, an currently Dean an Profeor of Grauate School of Avance Technology, Knk Unverty n Kyuhu. H reearch nteret are n the area of ytem theory, neural network, learnng theory, gnal proceng, pattern recognton, peech nformaton proceng an peech recognton, etc.