A dap tive rob ust it e rative lea r ni ng cont rol f or u nce rt ai n robotic syst e ms

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1 Control heory & Applications Vol. 2 No. 5 Oct. 23 Article ID : (23) A dap tive rob ust it e rative lea r ni ng cont rol f or u nce rt ai n robotic syst e ms YANG Sheng- yue, LUO An, FAN Xiao-ping ( Research Center for Automation Engineering, College of Information Science & Engineering, Central South University, Hunan Changsha 4175, China) A bst ract : he uncertain model of the robotic system was decomposed into repetitive and non- repetitive parts, and the normal model of the system was taken into account. By using Lyapunov method, an adaptive robust iterative learning control scheme was presented for the robotic system with both structured and unstructured uncertainties, and the overall stability of the system in the iteration domain was established. In the scheme the bound parameter estimates and the iterative learning control in2 put were adjusted in the iteration domain. he validity of the scheme is illustrated through a simulation example. Key wor ds : iterative learning control ; robust control ; adaptive control ; robot systems CL C num be r : P242; P Docume nt code : A,, (, 4175) : Lyapunov,,..,,.,,.. : ; ; ; 1 I nt rod uction he iterative learning control ( ILC ) methods deal with systems that perform the same tasks repetitively over a finite time interval. he basic idea of the ILC is that the information obtained from the previous trial is used to improve the control input for the next trial. he control input in each trial is adjusted by using the track2 ing error signals obtained from previous trial. As the it2 eration continues the control system eventually learns the task and follows the desired trajectory with little or no error. Depending on whether the system parameters are estimated or not, the ILC schemes may be classified into two categories : nonadaptive and adaptive. In the non2 adaptive case, the current input profile is computed sim2 ply by adding the scaled system error and/ or its time derivative to the previous input profile [13 ], In the adaptive case, however, uncertain parameters are esti 2 mated and used to identify the system dynamics, which is in turn used to generate the control input profile [48 ]. here have been substantial research effort in the adaptive IL C area. Among the reported results, Park et al [4 ] proposed an adaptive iterative learning controller (AIL C) for uncertain robotic systems by using the fact that they are linearly parameterizable. Seo et al [5 ] ex2 tended the result to a class of general nonlinear systems and proposed an intelligent learning control scheme. In the above control schemes [4,5 ], the parameter estimate and the learning control input were updated in the itera2 tion domain. On the other hand, French & Roger [6 ] and J. Y. Chol &J. S. Lee [7 ] gave another AILC in which the parameter estimate is updated in the time domain. However, in these controllers [36 ], the normal mod2 Received date : ; Revised date : Foundation item :supported by the National Natural Science Foundation of China (699753). 转载

2 78 Control heory & Applications Vol. 2 els are not taken into account, and the uncertain parame2 ters are not treated differently. Xu & Viswanathan [8 ] de2 composed a class of MIMO nonlinear dynamical systems into periodic and nonperiodic parts, without taking the normal model into account. In this paper, by using Lyapunov method an adaptive robust iterative learning control scheme is presented for the robotic system with both structured and unstructured uncertainties. In this control scheme, the whole control variable consists of three parts, i. e., the iterative learn2 ing control input, the normal model based computed torque control input, and the robust control input. he bound parameter estimates for the robust input and the learning control input were updated in the iteration do2 main. In the subsequent discussion, the following definition is used. For any matrix A, the induced matrix norm A is defined as A = [ max ( A A) ] 1 2, where max is the largest eigenvalue. 2 Proble m f or m ulation Consider an uncertain robot system with n rigid bod2 ies : [ M ( q) +M ( q) ] q +[ C ( q, gq) +C ( q, gq) ] gq + [ G ( q) +G( q) ] + d + f = u, (1) where M ( q) R nn, C ( q, gq) R nn and G ( q) R n are the known normal inertia matrix, centripetal plus Coriollis force matrix and gravitational vector, and M ( q),c ( q, gq),g( q) are the corresponding un2 certain parts, respectively ; u is the control input, d and f are the unknown disturbances which are assumed to be bounded, and q, gq, q are the generalized joint position, velocity, and acceleration, respectively. Furthermore, the disturbance d is assumed to be repetitive and f nonrepetitive. For notational convenience, in Eq. (1) and other expressions all over the paper, the time argument t is omitted. Ass ump tion A1 he uncertain models M ( q), C ( q, gq),g( q) are Lipschitz continuous. words, the following inequalities hold : In other M ( q) -M ( q d ) l M q- q d = l M e, C( q, gq) - C( q d, gq d ) l C [q- q d + gq - gq d ] = l C [ e + ge ], G( q) - G( q d ) l G q - q d = l G e, where e = q - q d and l M, l C, l G are the unknown finite Lipschitz constants. q d, gq d, q d represent the desired joint position, velocity, and acceleration, respectively. Ass ump tion A2 Like most other iterative learning control schemes, the initial error is assumed to satisfy the equalities : e () = and ge () =. 3 A dap tive rob ust it e rative lea r ni ng con2 t rol For system (1), the presented adaptive robust itera2 tive learning control scheme is as follows : u j = u c j + u l j + u r j. (2) Where u j denotes the control input at the j-th iteration, u c j is the computed torque control input, u r j the robust control input, and u l j the learning control input. he computed torque control input u c j is generated by u c j = M ( q j ) [ q d - ( k + I) ge j - ke j ] + C ( q j, gq j ) gq j + G ( q j ), (3) where e j = q j - q d, q j, gq j, q j denote the actual joint posi2 tion, velocity, and acceleration at the j- th iteration, re2 spectively; positive parameter. k is a positive feedback matrix and is a Define j = ge j + ke j. Substituting Eqs. (2) and (3) into Eq. (1) yields M ( q j ) [ g j + j ] = u l j + uj r - {M ( q j ) q j +C ( q j, gq j ) gq j +G( q j ) + d +f }. (4) Let M,C,G denote M ( q j ),C ( q j, gq j ), G( q j ), and M d,c d,g d denote M ( q d ), C ( q d, gq d ),G( q d, gq d ) respectively. We can get Where M ( q j ) q j +C ( q j, gq j ) gq j +G( q j ) + d +f = Mg j + R + gh. (5) R = M d q d +C d gq d +G d + d, gh = (M - M d ) q d - Mkge + (C - C d ) gq d +Cge + (G - G d ) + f. Substituting Eq. (5) into Eq. (4) yields ( M +M) g j +M j = u l j + u r j - R - gh. (6) Obviously, the uncertain part R is repetitive and gh is non- repetitive. From Assumption A 1, the following in2 equality holds : gh [l M q d +l C gq d + l G ] e +[k M +

3 No. 5 YANG Sheng- yue et al :Adaptive robust iterative learning control for uncertain robotic systems 79 l C gq d + C ] e + f = E. Where = [ l 1 l 2 l 3 ], l 1 l 2 l 3 = l M q d + l C gq d + l G, = k M + l C q d + C, = f, E = [ e, ge,1 ]. (7) (8) hen, the main aim is to find out the iterative learning algorithm and to determine the robust control input and its bound parameter estimates. in the next section. 4 M ai n res ults his would be discussed he mains results of this paper are described as fol2 lowing theorem and corollaries. If the system satisfied Assumptions A 1 and A2 de2 scribed in Section 1, then the learning algorithm, adap2 tive update law and the system s convergence are given as heorem 1. heore m 1 If the iterative learning algorithm and robust control input are given as u l j = uj l j, u1 l =, (9) uj r = - ( P+ C +^ j ) j - ^ j Esgn ( j ), (1) where is a positive constant, ^and ^are the j-th esti2 mates of parameters 3 ( 3 = C ) and, respec2 tively, and if the adaptive updating laws of ^ j and ^ j are given as ^ j = ^ j j 2, ^ 1 =, (11) ^ j = ^ j j, ^ 1 = [,, ], (12) where the parameter = E = [ e, ge,1 ], then system ( 1) is asymptotically stabilized under the control scheme (2). Proof Choose a Lyapunov function V j = (R - uj l 2 + j P j + ^ j ^ j - 2 ) d t, (13) where P is a symmetry positive matrix. hen = ( uj l - uj l - 1) ( uj l + uj l R) d t + ( ( ^ ( ^ j - j P j - j - 1 P j - 1 ) d t + j - ^ j - 1 ) ( ^ j + ^ j ) d t + ^ j - 1 ) ( ^ j + ^ j ) d t. From (9), (11) and (12) we have = ( ( - j ) ( j + 2 ( uj l - R) ) d t + j P j - j - 1 P j - 1 ) d t + ( j 2 ) ( - j ( ^ j - 3 ) ) d t + ( j ) ( - j + 2 ( ^ j - ) ) d t. he equation (6) can be converted into u l j - R = ( M +M) g j +M j - ( u r j - gh) = (14) Mg j +M j - ( u r j - gh). (15) Substituting (15) into ( 14), the following inequality is derived easily. ( - 2 jmg j - 2 jm j ) d t + ( j P j d t + 2 j ( uj r - gh) d t + 2 j 2 ( ^ j - 3 ) d t + 2 j j d t - j - 1 P j - 1 d t. Since we have 2 ( jm j 2 jmg j d t = jm 2 j ( ^ j - )d t - j j gm j - 2 jm j ) d t - 2 j j + j P j d t ( ^ j - 3 ) d t + j d t - j - 1 P j - 1 d t = - j gm j d t, j ( u r j - gh) d t + 2 j ( ^ j - )d t -

4 71 Control heory & Applications Vol. 2 2 j ( P +( gm - 2 C) - 2 M ) j d t + j ( C +C) j d t + ( j () M () j - j ( ) M ( ) j ( ) ) + 2 j ( uj r - gh) d t + 2 j 2 ( ^ j - 3 ) d t + 2 j ( ^ j - )d t - 2 j j d t - j - 1 P j - 1 d t, where M () = M ( q () ), M ( ) = M ( q ( ) ). It is well known that matrix and the inertia M is positive. A2, we get j () V j - V j - i gm - 2 C is a skew-symmetric =. hus, we have P j d t j ( uj r - gh) d t + j ( C +C) j d t + From Assumption 2 j 2 ( ^ j - 3 ) d t + 2 j ( ^ j - )d t - 2 j From (1) we get 2 2 j ( u r j - gh) d t = j d t j ( P + C + ^ j ) j d t - j^ j Esgn ( ) j d t - 2 j ghd t. j - 1 P j - 1 d t. Substituting this equation into the above inequality yields 2 j (C) j d t - 2^ j j j d t + 2 j 2 ( ^ j - 3 ) d t jghd t + 2 j ( ^ 2 j j d t - j - 1 P j - 1 d t. j ^ j Esgn ( j ) d t - j - )d t - Since all the desired and estimative bound parameters are positive, 3 = C, and j sgn ( j ) j and j j = j 2 for j is a column vector, such that j (C) j d t ^ j j j d t = ^ j j 2 3 d t, j 2 d t, j^ j Esgn ( ) j d t j ^ j Ed t. So the following inequality can be obtained easily - 2 j ^ j Ed t - 2 j ghd t + 2 j ( ^ j - )d t - 2 j j d t - For the parameter = j - 1 P j - 1 d t. E = [ e, ge,1 ], be2 sides expression (7), the following inequality holds - 2 j j d t - j - 1 P j - 1 d t. When and only when j - 1 ], the equality theorem is completed. =, j = for all t [, = holds. he proof of the Since Assumption 2 is too strict and difficult to achieve in practice. this constrain. Following corollaries can remove Corolla ry 1 If Assumption 2 is not found, and the condition lim j () = j is satisfied, then the system ( 1) is also asymptotically stabilized under the control scheme ( 2), in which the robust control and iterative learning input and the adap 2 tive update law, as in the theorem, are given as (9) (12). Proof For lim j () j lim j =, we have j () M () j =. So Corollary 1 can be provided easily, and the proof of the corollary is the same as the proof of the theorem. In practice, the condition given above is usually unre2 alizable. And when the norm of the initial error j (), j = 1,2,, whereis a small enough positive constant. can be achieved as in Corollary 2. hen the convergence of the system Corolla ry 2 If the following condition about initial error j (), j = 1,2, is satisfied, then the system is stable and following in2 equality is found.

5 No. 5 YANG Sheng- yue et al :Adaptive robust iterative learning control for uncertain robotic systems 711 j j j lim - 1 2, (16) where is the maximum eigenvalue of matrix M (), andis the learning parameter. Proof Like the proof of the theorem given above, when we can get If j (), j = 1,2,, - 2 j j d t we have j j d t + 2. j - 1 P j - 1 d t + j () M () j j j > - 1 2, - 2 j j d t + 2 <. So the inequality ( 16) is satisfied, and Corollary 2 is provided. 5 Si m ulation o illustrate the effectiveness of the presented adaptive robust iterative learning control scheme, a two- link pla 2 nar robot is designed to follow a continuous path. he robot system is shown as Fig. 1. he vectors q 1, q 2 de2 note the angles of the two links. We assume that the links are homogeneous rigid bodies with the mass m 1, m 2 and the length l 1, l 2, respectively. he model of this system with disturbance is given in Appendix. Fig. 1 wo-link robot configuration In this simulation example, we select l 1 = 1. m, l 2 =. 5 m, m 1 = 1. kg, m 2 = he desired trajectories are required as 4. 5 kg, s t < 2. 5 s, 5. 5 kg, 2. 5 s t 5 s, q d = [ q d 1, q d 2 ] = [sin (3 t),cos (3 t) ], t [,5 ], and the repetitive and non- repetitive disturbances are as 2 sumed as d = [ d 1, d 2 ] = [2 - t/ 1, 2 - t/ 1 ], t [,5 ], f = [ f 1, f 2 ] = [2rand (1),2rand (1) ], where rand (1) ( - 1,1) is a random function. In the simulation, the learning gain and other parameters were chosen as =,5, = 2, P = 2 3, k = 3 5. he simulation results for the first and fifth iteration are shown in Fig. 2 and Fig. 3, respectively. It is proved that the control scheme presented in this paper is very ef 2 fective. In addition, since the normal model of the un2 certain system has been taken into account, the system can achieve good performance at a few times of itera 2 tions. Fig. 2 Responses at the first iteration (solid :desired dashed :actual)

6 712 Control heory & Applications Vol. 2 Fig. 3 Responses at the fifth iteration (solid :desired dashed :actual) 6 Concl usion By using the Lyapunov method, an adaptive robust it2 erative learning control scheme is presented for the robotic system with both structured and unstructured un2 certainties. Its distinct feature against other iterative learning schemes is that the uncertain model of the robotic system is decomposed into repetitive and non - repetitive parts, the normal model is took into account, and the bound parameter estimates and the iterative learning control input were adjusted in the iteration do2 main. he simulation results show the control scheme is very effective. Ref e re nces : [1] ARIMOO S, KAWAMURA S, MI YAS KI F. Bettering operation of robots by learning [J ]. J of Robotic Systems, 1984, 1 (2) : [2 ] JANG, CHOI C H, AHN H S. Iterative learning control in feed2 back systems [J ]. Automatica, 1995,31 (2) : [ 3 ] L EE J H, L EE K S, KIM W C. Model- based iterative learning con2 trol with a quadratic criterion for time- varying linear systems [ J ]. Automatica, 2,36 (4) : [4] PARK K H, KUC Y, L EE J S. Adaptive learning control of un2 certain robotic systems [J ]. Int J Control, 1996,65 (5) : [5 ] SEO W G, PARK K H, L EE J S. Intelligent learning control for a class of nonlinear dynamics system [J ]. IEE Proc Control heo2 ry and Applications, 1999,146 (2) : [6 ] FRENCH M, ROGERS E. Nonlinear iterative learning by an adap2 tive Lyapunov technique [ A ]. Proc of the 37 th IEEE Conference on Decision and Control [ C ]. ampa, FL : Piscataway, 1998 : [7] CHOI J Y, L EE J S. Adaptive iterative learning control of uncertain robotic systems [J ]. IEE Proc Control heory and Applications, 2,147 (2) : [8 ] XU Jianxin, VISWNAHAN B. Adaptive robust iterative learning control with dead zone scheme [J ]. Automatica, 2,36 (1) : App e ndix he dynamic model of the two-links robot system shown as Fig. 1 can be written as M ( q) q + D ( q, gq) gq 1 gq 2 + G( q) + d + f = u. Where d and f are repetitive and non- repetitive disturbances respec2 tively ; and where M ( q) = m 11 m 12 m 21 m 22, m 11 = 1 3 m 1 l m 2 l m 2 l m 2 l 1 l 2 cos ( q 2 ), m 21 = m 12 = 1 3 m 2 l m 2 l 1 l 2 cos ( q 2 ), m 22 = 1 3 m 2 l 2 2, D ( q, gq) = d 11 d 12 d 21 d 22, d 11 = - m 2 l 1 l 2 sin ( q 2 ) gq 2, d 21 = d 12 = - d 22 =, 1 2 m 2 l 1 l 2 sin ( q 2 ) gq 2, G( q) = g 1 g 2, g 1 = 1 2 m 1 gl 1 cos ( q 1 ) m 2 gl 2 cos ( q 1 + q 2 ) + m 2 gl l cos ( q 1 ), g 2 = 1 2 m 2 gl 2 cos ( q 1 + q 2 ). In above equaltiy, g is the acceleration of gravity. : (1969 ),,,,,,, E- mail csu. edu. cn ; (1957 ),,,,,,, E- mail :an - com ; (1961 ),,,,,,,CIMS, E- mail edu. cn.