Compensation of friction and backlash effects in an electrical actuator

Transcription

1 SPECIAL ISSUE PAPER 1 Compenation of friction and backlah effect in an electrical actuator R Merzouki*, J C Cadiou and N M Sirdi Laboratoire de Robotique de Veraille, Vélizy, France Abtract: In thi paper, non-linear oberver, baed on an etimation of the friction force and the diturbed torque tranmitted due to the dead zone, are developed for ytem preenting mechanical imperfection uch a friction and backlah. Then an adaptive controller uing thee non-linear oberver i preented, to compenate for mechanical diturbance on-line. Simulation and experimental reult applied on an electrical actuator are given to upport the theoretical demontration. eyword: friction force, non-linear oberver, backlah effect, tranmitted torque, electrical actuator, dead zone, adaptive control NOTATION 1 INTRODUCTION a area contact width (m) The preence of non-linearitie in mechanical ytem F global friction torque (N m) make them difficult to control with high accuracy. F friction torque error (N m) Among thee imperfection are friction, which depend j dead zone magnitude (rad) on the relative velocity of the motion, and the backlah J motor inertia (N m) phenomenon iued from the dead zone between two m J load inertia (N m) involved part. elaticity contant (N m/rad) Among reearch dealing with friction effect, Friedland N reducer contant [1] ha developed an algorithm in order to etimate the w non-linear tranmitted torque (N m) Coulomb friction force. Thi algorithm i a reduced-order w non-linear tranmitted torque error (N m) oberver containing two non-linear function, where one correpond to the Jacobean of the other. A good a Coulomb friction torque (N m) choice of non-linear function in the oberver allow a tiction torque (N m) an aymptotic tability of the error. Amin et al. [] 1 a vicou friction coefficient (N m /rad) have developed two type of oberver: the firt conider e input poition error (rad) the friction force a a contant and the econd i ued e ė input velocity error (rad/) to etimate the relative velocity of the motion during e e output poition error (rad) the contact. Canuda de Witt et al. [3] propoed a ė output velocity error (rad/) model of friction that include different effect, uch h input reducer poition (rad) a hyterei behaviour and the tiction effect. They e ḣ input reducer velocity (rad/) developed an adaptive control in order to etimate and e ḧ input reducer acceleration (rad/) then compenate for the friction effect. e h output reducer poition (rad) The preence of the dead zone in mechanical ytem ḣ output reducer velocity (rad/) introduce an hyterei phenomenon between the input ḧ output reducer acceleration (rad/) and the output poition. Thi decribe the backlah ḣ Stibeck velocity (rad/) phenomenon, which caue non-table behaviour in the tri m friction parameter controlled ytem. Backlah i inherent in mechanical ytem, epecially when tarting the motion, but if it increae due to wear it will diturb the performance of The MS wa received on 7 April 3 and wa accepted after reviion the ytem. In uch a cae, compenation for thee effect for publication on 8 September 3. * Correponding author: Laboratoire de Robotique de Veraille, 1 1 i due to mechanical or control method. For a long avenue de l Europe, 7814 Vélizy, France. time, mechanical olution exited to eliminate thee Proc. Intn Mech. Engr Vol. 18 Part I: J. Sytem and Control Engineering The Charleworth Group, Hudderfield

2 R MERZOUI, J C CADIOU AND N M SIRDI diturber effect by changing all imperfect part on the The econd imperfection i backlah and i repreented ytem. Some control olution have alo been propoed, by a variable dead zone from to 4. Finally, the tran- e.g. by Brandenburg and Schafter [4], who tudied mitted motion to the output axi i via a tring ytem the influence and the partial compenation of imultaneouly with changeable tiffne. acting backlah and Coulomb friction in a On thi bench tet, the meaure of input and output peed and poition control of an elatic two-ma ytem. poition i taken by two incremental coder, a hown Recker et al. [5] and Tao and okotovic [6] worked on in Fig. 1a. the adaptive control of ytem with backlah. Different mathematical model were propoed, uch a that of Tao and okotivic [6], who modelled an invere backlah model baed on an hyterei cycle. Cadiou and M Sirdi. Etimation of the friction force [7] have developed a differentiable model baed on the In order to explain the origin of the friction force dead zone characteritic. oberver, two different mode are tudied. The firt one In thi paper, two non-linear oberver have been i the nearly tatic mode, which correpond to the lowdeveloped, to etimate the friction force and the di- velocity variation. The econd one correpond to the turber tranmitted torque. After that, an adaptive con- dynamic mode, where the velocity variation i important. troller i preented, to compenate for the diturbance effect. In mot application, the friction and the backlah non-linearitie could not be accurately known,..1 Nearly tatic mode o only an etimation of thee effect could be poible. Conider that during mechanical motion between two A mathematical model of imperfection i given, rep- urface in contact, the preure ditribution P(x) [8] i reenting an invere igmoid to repreent the diturber given by Fig. b. In thi cae, the ditribution i choen torque oberver [8] and Tutin model for the cae of a a half-ellipe in the plane and i limited by a and friction [9]. Simulation and experimental reult are pre- +a, where a i a poitive value that define the maximum ented in thi paper, which are applicable to a bench tet of the deflection [3]. The maximum preure contact contructed in the author laboratory, and an important act at the centre and decreae progreively with the number of mechanical imperfection are given. width contact. The eaiet repreentation of thi lat ditribution could be formulated a follow: DEFINITIONS AND MODELLING x P(x)=P S1 (1) a.1 Decription of the bench tet For the nearly tatic mode, the friction force F depend The experimental bench tet of Fig. 1a correpond to 1 on the normal force N, o that an electrical actuator, divided into two part. The firt part repreent the motor part and i driven by a d.c. F =mn () motor. The econd part decribe the reducer part which 1 regroup three important mechanical imperfection. where m i the friction parameter at low velocitie. The The firt imperfection i tatic and vicou friction, normal force N could be deduced by integration of the where coefficient can be changed for different applicould then be expreed a aumed preure ditribution P(x). The friction force cation by uing many brake part made of aluminium, metal, etc. Thee coefficient could alo be identified approximately by uing claical identification algorithm F = P +a am P(x) dx (3) [7], e.g. the recurive leat mean quare method. 1 a Fig. 1 (a) Bench tet; (b) bloc cheme of the bench tet Proc. Intn Mech. Engr Vol. 18 Part I: J. Sytem and Control Engineering The Charleworth Group, Hudderfield

3 COMPENSATION OF FRICTION AND BACLASH EFFECTS IN AN ELECTRICAL ACTUATOR 3 Fig. (a) Surface in contact; (b) preure ditribution Replacing equation (1) by equation (3) give poitive contant. The rule of thi filter i to cut the high frequencie of e, iued a a reult of the preence of mechanical imperfection. F = P +a x am P 1 a S1 a dx (4) The formulation of the m oberver and the friction force F etimator can now be rewritten from equation Putting x=a in a give dx=a co a da, with aµ (9) and (1), which take the ame mathematical [ p/, p/]; therefore formulation a their correponding model, adding the term k e, a hown below: F =P m P +p/ a co a da (5) 1 p/ mˆ = l ḣ mˆ + pa l ap ḣ ign (ḣ ) k e tri tri F =P am P +p/ A1+co a 1 B da (6) p/ Fˆ = pp am +a ḣ F = pp am (7) 1 (11) where P repreent the maximum preure over the contact urface and i uppoed known in thi cae... Dynamic mode When P(x ±a) =, evolution of the friction force will depend principally on the relative velocity ḣ between the urface in contact: F =a ḣ (8) where a i the vicou friction coefficient. The global friction force F=F +F will therefore be expreed for 1 both mode a follow: F= pp am +a ḣ (9) From equation (11) and (9), the etimated friction error and the oberved parameter error m are given a m = l m +re ḣ tri F = pp a m Putting G= pp a thu give the etimated friction error a (1) (13) F =F Fˆ =Gm (14)..3 Friction oberver.3 Decription of the tranmitted torque Conider a low-pa filter formulated a Figure 3 how an approximation of a tranmitted 1 torque via a dead zone of magnitude j, with flexible mˆ = ( ḣ tri /l)p+1a pa ḣ tri ap l k e (1) B link. The difference between the input poition h and e the output poition N h of the reducer part i put a where a i the Coulomb friction, uppoed known, p i Dh, with N a the reducer contant. The continuou and the Laplace contant, ḣ i the Stribeck velocity given derivable function, a igmoid, i ued to decribe the tri in reference [7], ḣ /l i the cut-off frequency, mˆ correpond to the oberver of parameter m, e =h hd, tri where hd repreent the deired output, and k, l are tranmitted torque characteritic inide and outide the dead zone [8], and i eay to implement in the control cheme. Proc. Intn Mech. Engr Vol. 18 Part I: J. Sytem and Control Engineering The Charleworth Group, Hudderfield

4 4 R MERZOUI, J C CADIOU AND N M SIRDI decribed by the following equation ytem: J ḧ +F=C J m ḧ e +f m ḣ e +C=U Fig. 3 Approximation of tranmitted torque () where J, ḧ are the inertia of the reducer part uppoed known and the output reducer acceleration repectively, J, f, ḧ, ḣ are the inertia of the motor part, the m m e e vicou input friction, which are uppoed known, the input reducer acceleration and velocity repectively, U i the control torque, F i the friction force, uppoed unknown in the actuator and ha to be etimated, and h and h are the input and output meaured poition e of the reducer part repectively. C repreent the tran- mitted torque to the load via a flexible axi and a known dead zone. It may be expreed a Then the mathematical formulation of thi tranmitted torque via a dead zone i defined by C= Dh+w (1) 1 e c C= where i the tiffne of the flexible part and w i the (15) 1+e c DhB diturber torque, decribed before. where C i the approximate tranmitted torque repreented Note that in Fig. 3. After decompoing the expreion of the tranmitted torque C into two part, Dh=h N h e () C=C +w (16) which repreent the difference between the input and the output poition of the reducer part. N i the reducer with C the linear tranmitted torque, given by contant. C = Dh (17) Sytem () can then be expreed a follow: and w the diturber and non-linear tranmitted torque, J ḧ +F=h N h +w formulated a follow: e J ḧ +f ḣ =U h +N h w 1 e c Dh m e m e e w= 4j (18) 1+e c Dh (3) The repreentation of w depend on the value of e =h hd the contant c, which depend on the magnitude j. e =h hd Expreion (18) correpond to the bet approximation e e e of the tranmitted torque via a dead zone, a hown z=h N h e in Fig. 3. e =z z According to the imulation and experimental tet, z d the bet decreaing value of w i obtained inide [ j, j ] w =w ŵ after putting F =F Fˆ c= 1 j (19) (4) For the next ection, the parameter c i calculated where hd i the deired output poition, hd i the deired e on-line by giving an approximative and initial value to input poition, z i the difference between the input and j. Then the torque w i etimated after oberving the the output poition of the reducer part and z i it d evolution of the magnitude j on-line. deired value, ŵ i the etimated diturber torque and w correpond to the etimated error of the diturber torque. Now, ytem (3) may be expreed a 3 CONTROL OF THE BENCH TEST INCLUDING FRICTION AND BACLASH J ḧ +F=z+w The dynamic model of the bench tet given in Fig. 1, J ḧ +f ḣ +J ḧ +F=U m e m e including the friction and backlah imperfection, i (5) Proc. Intn Mech. Engr Vol. 18 Part I: J. Sytem and Control Engineering The Charleworth Group, Hudderfield

5 COMPENSATION OF FRICTION AND BACLASH EFFECTS IN AN ELECTRICAL ACTUATOR 5 The control cheme to the bench tet i given by Fig. 4. where and are the PD contant of the C controller and Fˆ the etimated friction force given in Fig P1 D1 1 C and C are proportional derivative (PD) controller bloc with tranient bloc of H and H. According to the control cheme of Fig. 4, 1 In order to linearize the firt equation of ytem (5), the z and w value given in ytem (4) are replaced by hd= ė + e (3) e D P etimated value a follow: where and repreent the C controller contant P D J ḧ +F=e +z +w +ŵ (6) of Fig. 4. z d By replacing the expreion for U in equation (3), For that, z i choen a d the following equation i obtained: z =J ḧd+fˆ ŵ (7) d By replacing expreion (7) in equation (6), the output ytem equation i given by J ë +F =e z +w (8) which i the equation deduced after linearization of the reducer part model. Now, to linearize the econd equation of ytem (5), the following affectation are ued: h e = z +N h J m ë z + Af m + D1 B ė z + A k 1 + P1 B e z +( J m N +J D1 D )ë +( f N + N )ė m P1 D D1 P D1 +( N )e +F = (33) P1 P1 P Thi lat equation decribe the linearization of the motor part of our bench tet. From equation (11) and (9), the etimation error of the friction parameter m and the friction force are given by the following equation ytem: ḣ e = z +N ḣ ḧ e = z +N ḧ m = jm +k e F =Gm (34) The econd equation of ytem (5) will become J m z + f m ż+( J m N +J )ḧ +f m N ḣ +F=U (3) Thu, the control law U for global ytem () i expreed a U= J m z d+ Af m + D1 B żd+ 1 zd (9) where G= pp a and j= l ḣ tri The torque etimator ha the ame formulation a it model (18) and i given by the following expreion: ŵ= 4ĵ 1 e c Dh 1+e c Dh +(J N +J )ḧd+( f N + N )ḣd m m D1 + N hd k e ė e +Fˆ P1 1 z D1 e P1 e (31) w = 4j (35) with c uppoed known. In the cae of Dh&, w will take the following expreion: (36) Fig. 4 Control cheme of the bench tet including friction and backlah Proc. Intn Mech. Engr Vol. 18 Part I: J. Sytem and Control Engineering The Charleworth Group, Hudderfield

6 6 R MERZOUI, J C CADIOU AND N M SIRDI after chooing a backlah magnitude model given by dj = (37) dt with j the magnitude contant. Thu, the oberver magnitude can be expreed by the output poition error and output velocity error, a follow: dĵ dt = k ė k 3 e (38) where k and k are poitive contant. 3 The etimation error of the backlah magnitude i Fig. 5 Equivalent cheme of the global ytem in a cloed loop given a dj dt = dj dt dĵ dt =k ė +k 3 e (39) Thu, the global ytem will be a combination of all the following equation: J ë +F =e z +w The characteritic equation of the global ytem (4) i defined by the following equation: with C 8 p8+c 7 p7+c 6 p6+c 5 p5+c 4 p4+c 3 p3 +C p+c 1 p= (41) J m ë z + Af m + D1 B ė z + A k 1 + P1 B e z C 8 =4J a 3 C 7 =4J (a +ja 3 ) +( J m N +J D1 D )ë C +( f m N P1 D D1 P + D1 N )ė C6 =4J (j+1)a +4b 3 +k a 3 +( P1 N P1 P )e +F = dj dt =k ė +k 3 e 5 =4J a +a 3 (4Gk +k 3 )+4J ja +k a +4(b +jb ) 3 C =4J ja +k ja +k a +4(b +jb )+a k j w = 4j C =4(b +jb )+k a +a (4Gk +k )+a k j m = jm +k e C =4jb +a (4Gk +k )+k jb 3 3 F =Gm C =k ja 1 3 (4) To define the condition on the controller contant and which i repreented in Fig. 5, with on contant k k, k,, ue of the Routh criterium, 1 3 determine the tability limit for the global ytem to b = J m 3 converge to the equilibrium tate (e, e, w, e F ). b =j J m + f m + D1 4 SIMULATION RESULTS b =j 1 Af m + D1 B +k 1 + P1 The imulation tet are performed on a mechanical model repreentative of the bench tet of Fig. 1b. The b =j Ak 1 + P1 B parameter of the model for the imulation tet are a = J N J =15, =.5, =15, =.3 3 D1 D m P1 D1 P D a = + f N N =1 Nm/rad, J =. 97 N m P1 D D1 P m D1 m j( J N +J ) f =. 43 Nm/rad, m J =7.5Nm m D1 D a = N a =8Nm, a =1.5 N m, a =16Nm/rad 1 P1 P P1 1 j( f N + N ) j =.1 rad, k 1 =1, k =.1, k 3 =1 m P1 D D1 P D1 a =Gk j( N ) and N =59 P1 P1 P Proc. Intn Mech. Engr Vol. 18 Part I: J. Sytem and Control Engineering The Charleworth Group, Hudderfield

7 COMPENSATION OF FRICTION AND BACLASH EFFECTS IN AN ELECTRICAL ACTUATOR 7 Figure 6a decribe the tracking of the output poition in Fig. 8b before compenation and in Fig. 9b. After for a deired output ignal hd(t)=.5 in (, pt) and a adding oberver, the output tracking error i reduced PD controller applied on the ytem of Fig. 1. Figure 6b and i uniform for each period. However, a tatic error repreent the output poition error before the adaptive i till preent, due to the preence of the flexibility effect compenation of the imperfection effect. It can be een in the mechanical ytem. Figure 1 decribe the that the reel output poition ignal i le deformed at hyterei cycle between the input and the output reducer the peak area due to the preence of the diturbance. poition. After compenating for the dead zone effect, Thi deformation i compenated after introducing the the width of the cycle i reduced. nowing that the adaptive compenation, a hown in Fig. 7a. Then, the flexibility effect i till preent in the modelling, the real output poition ignal approache the deired one, relation between the input and output poition i not with a poition error decribed a in Fig. 7b. Thi error exactly linear after compenation. Figure 11a repreent i epecially due to the flexibility effect, which are the control ignal before and after compenation. The not conidered in the adaptive compenation. Figure 8a ignal before compenation i not a clear a the one decribe the tracking of the input ignal before compenation after compenation. Thi i due to the preence of the for the mechanical imperfection effect. The imperfection, defined by the diturber torque of Fig. 11b. tracking i very good in the two cae but i clearer in After compenation, a linear repreentation of the torque the cae after compenation ( Fig. 9a). The difference i obtained, which i tranmitted via a flexible axi, a between the error input poition tracking i hown hown in Fig. 11b. Fig. 6 (a) Deired and real output ignal before compenation; (b) output poition error before compenation Fig. 7 (a) Deired and real output ignal after compenation; (b) output poition error after compenation Proc. Intn Mech. Engr Vol. 18 Part I: J. Sytem and Control Engineering The Charleworth Group, Hudderfield

8 8 R MERZOUI, J C CADIOU AND N M SIRDI Fig. 8 (a) Deired and real input ignal before compenation; (b) input poition error before compenation Fig. 9 (a) Deired and real input ignal after compenation; (b) input poition error after compenation Fig. 1 Hyterei backlah behaviour before and after compenation Proc. Intn Mech. Engr Vol. 18 Part I: J. Sytem and Control Engineering The Charleworth Group, Hudderfield

9 COMPENSATION OF FRICTION AND BACLASH EFFECTS IN AN ELECTRICAL ACTUATOR 9 Fig. 11 (a) Control torque before and after compenation; (b) linearization of the diturbed torque after compenation 5 EXPERIMENTAL RESULTS and friction force. Figure 13 how the output velocity ignal before and after the adaptive compenation. Therefore, The experimental tet have been applied on the bench before the compenation cae, undeired ocillation tet of Fig. 1a, uing the following control parameter: around ḣ ()= rad/ are preent and repreent the nonlinearity effect. Thee imperfection are compenated =1, =.1, =1, =5 P1 D1 P D after applying the etimated diturber torque and friction =Nm/rad, j =.48 rad, k =1 force. Finally, the control ignal before and after com- 1 penating the backlah and friction effect are hown k =., k =1, a =8Nm 3 in Fig. 14. In the cae after compenation, the control a =1.5 N m, a =16Nm/rad ignal i clearer than it equivalent before the com- 1 penation due to the adaptive compenation of the In thee tet, the motor reducer wa required to diturbance effect of friction and backlah. move from the initial tatic output poition h ()= p/ rad and output velocity ḣ ()= rad/ to the origin h ()= rad, ḣ ()= rad/. Figure 1 repreent the tracking output poition before 6 CONCLUSION (i.e. the regulation i made by only a PD controller) and after applying the adaptive compenation. The tatic The preence of mechanical imperfection uch a poition error i about.3 rad and i compenated after friction and backlah in controlled ytem make them adding the etimator of the undeired dead zone torque difficult to control with high accuracy. The mechanical Fig. 1 Output poition before and after adaptive compenation Fig. 13 Output velocity before and after adaptive compenation Proc. Intn Mech. Engr Vol. 18 Part I: J. Sytem and Control Engineering The Charleworth Group, Hudderfield

10 1 R MERZOUI, J C CADIOU AND N M SIRDI Amin, J., Friedland, B. and Harnoy, A. Implementation of friction etimation and compenation technique. In 5th IEEE International Conference on Control Application, Dearborn, Illinoi, September, Canuda de Wit, C., Olon, H., Atröm,. J. and Lichinky, P. A new model for control of ytem with friction. IEEE Tran. Autom. Control, March 1995, 4(3), Brandenburg, G. and Schafer, U. Influence and partial compenation of imultaneouly acting backlah and Coulomb friction in a peed and poition controlled elatic two-ma ytem. In Proceeding of International Conference on Electrical Drive (ICED88), Romania, 1988, pp Recker, D. A., okotovic, P. V., Rhode, D. and Winkelman, J. Adaptive nonlinear control of ytem containing a dead zone. In Proceeding of the 3th Conference on Deciion and Control, Vol. 3, Brighton, December 1991, Fig. 14 Control ignal before and after adaptive compenation pp Tao, G. and okotovic, P. V. Adaptive control of ytem with unknown output backlah. Automatica, 1995, 4(), imperfection effect could be reduced by etimating the neceary diturber torque inide the dead zone and the 7 Cadiou, J. C. and M Sirdi, N.. Modelization and analyi of a ytem with torque tranmitted through a backlah. friction force acting during the motion. After adding In 9th World Congre on The Theory of Machine and thee oberver into the control law, the undeired non- Mechanim (IFTPMM), Vol., Milan, Italy, Augut linearitie can be reduced. For the cae of friction eti- September 1995, pp mation, a dynamic model decribing the friction force 8 Merzouki, R., Cadiou, J. C. and M Sirdi, N.. variation a a function of the output ytem velocity i Compenation of backlah effect in an electrical actuator. preented. Then, the friction oberver correpond to a In the IASTED International Conference on Intelligent filter bloc, where it input i the output poition error Sytem and Control (IASTED), Hawaii, Augut and it output i the friction parameter. For the cae of. backlah, a non-linear and derivable mathematical model 9 Merzouki, R., Cadiou, J. C. and M Sirdi, N.. Adaptive for the diturber torque i preented, where the dead control of an electrical actuator. In the IEEE International zone magnitude approache a contant value. Etimation Conference on Intelligtent Sytem and Control (ICAR1), Budapet, Hungary, Augut 1. of the magnitude variation i oberved a a function of 1 Merzouki, R., Cadiou, J. C. and M Sirdi, N.. the output poition error and output velocity error. A Compenation of tick-lip effect in an electrical actuator. good choice of control ytem parameter allow the con- In Proceeding of the IEEE/RSJ International vergence of the global ytem to the original tate, a Conference on Intelligent Robot and Sytem (IROS), hown in the imulation and experimental reult. Lauanne, Switzerland, October, EPFL, pp Tao, G. and okotovic, P. V. Adaptive control of ytem with backlah. Automatica, 1993, 9(), Tao, G. and okotovic, P. V. Adaptive control of ytem REFERENCES with unknown output backlah. Automatica, 1995, 4(), Friedland, B. A non-linear oberver for etimating parameter 13 Spinnler, G. Conception de Machine, Vol 1 and (Pree in dynamic ytem. Automatica, 1997, 33(8), Polytechnique et Univeritaire Romande). Proc. Intn Mech. Engr Vol. 18 Part I: J. Sytem and Control Engineering The Charleworth Group, Hudderfield