An Adaptive CMAC Neural Network Back-Stepping Controller for Robot Manipulators with Unknown Disturbances

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1 An Adaptive CMAC Neural Network Back-Steppin Controller for Robot Manipulators with Unknown Disturbances Jun i, ibiao Wan, eixian Jin, Zhenchu Wan Departent of Mechanical Enineerin, aizhou University, aizhou, Zhejian, 38, China. * Correspondin author, E-ail: lijun@tzc.edu.cn Abstract - he perforance of robot anipulators iht derade sinificantly due to the presence of unknown disturbances. An adaptive Cerebellar Model Articulation Controller (CMAC) neural network back-steppin controller is proposed to iprove the syste perforance. he controller is coposed of a CMAC neural network identification odel and an adaptive back-steppin control. he identification odel is used to approxiate the unknown nonlinear function of the robot anipulators. hen the back-steppin control is able to track the desired trajectories based on the approxiation function. Siulation results show that the proposed controller has stron robustness. Keywords - Adaptive control; Backsteppin control; CMAC; Robot anipulators; Robust I. INRODUCION Robot anipulators are widely used in processin industries, such as surery, anufacturin, aerospace, etc. owever, robot anipulators are nonlinear dynaic syste because of the iprecision anufacturin of the anipulator links, unknown loads, nonlinear friction and external disturbances. herefore, desin of a robust adaptive controller to iprove the perforance of robot anipulators is one of the ost challenin tasks for enineers, especially when robot anipulators need to operate very quickly in response to external disturbances. Recently, the slidin ode control [, ], Model Predictive Control [3] and intellient control [4, 5] are used in robot anipulators so that the robot is able to track different types of reference trajectory. Althouh the robot control field has achieved sinificant proress, there are still soe probles to be solved for practical ipleentations, particularly in the case of sufferin perturbations and lack of accurate syste dynaics. In this paper, we adopt an Adaptive Cerebellar Model Articulation Controller neural network back-steppin Controller (ACMACNNBC) for a -DO robot. he ACMACNNBC is coposed of a CMAC neural network identification odel and an adaptive back-steppin controller. In the ost practice, it is unable to obtain the exact paraeters of the syste and external disturbances. o overcoe this proble, the CMAC neural network identification is applied to approxiate the syste dynaic paraeters. An adaptation law of the controller is synthesized usin the yapunov stability theores in order to uarantee lobal stability of the syste. II. DYNAMICS O ROBO MANIPUAOR As is well known, nelectin friction, the nonlinear dynaic odel of an n riid-link robot anipulator is described by the followin arane for [6]: x M( q) q C( q, q ) qg ( q) d () n where qqq,, represent the joint position, velocity and nn acceleration vectors, respectively. M ( q) is the nn inertia atrix, C( q, q ) is Coriolis and centripetal atrix, G ( q) is the vector of ravity ters, d is a bounded unknown disturbances and is the control input torque. Property. he inertia atrix M ( q ) is syetric, b- ounded and positive definite. Property. he atrix M ( q) C( q, q ) is skew syetric and Cqq (, ) is uniforly bounded. Property 3. G (q) and d are uniforly ultiately boun-ded (UUB). Definin G (q) <G b and d <d b, G b and d b are known positive constants. o siplify the proble, we choose a two-deree of freedo (-DO) robot anipulator as the control plant, which ay serve as representative of any industrial processes. As illustrated in i., et xq [, ], q, the dynaic syste is expressed in Eq.() can be written as [7]: x x x M ( x) M ( x) C( x, x) x M ( x)( G () And the atrices are iven: DOI.53/IJSSS.a ISSN: x online, print

2 PPMrl cos PMrl cos Mq () P Mrl cos P Mrl sin Mrl ( )sin Cqq (,) Mrl sin ( Mr Ml ) cos Mr cos( ) G () q Mr cos( ) where P =.33M r +M l, P =.33M r, M n, θ n, l n represent ass, joint position and lenth of link n, for n =,. In addition, r is the distance between the center of ravity of the link and the axis of the joint and r is the distance between the center of ravity of the link and the axis of the joint. is ravitation constant. Y r M È Y É r M É È i. (). a wo-do of Robot Manipulator ç III. DESIGN O E CONROER ç A. Description of CMAC he structure of CMAC includes input space, association eory space, receptive-field space, weiht eory space and output space. Accordin to Ref. [8, 9], the sinal propaation and the basic function are introduced in each space of the CMAC as follows: () Input space: [,,..., ] is the input. Each input state variable can be quantized into discrete reions (called an eleent) accordin to a iven control space. () Association eory space: Several eleents are accuulated as a block, each block perfors a receptivefield basis function. he Gaussian function is usually chosen as the receptive-field basis function jcji ji( j) exp,fori,,, b ji (3) (4) (5) (6) where ji ( j ) denote the ith block receptive-field basis function for the jth input with the center of C ji and the width b ji, is the nuber of the blocks. (3) Receptive-field space: In this space, the th ultidiensional receptive-field function is defined as ji j j ( ) ( ),for,,, N (7) where N is the nuber of receptive-field. he function can be written in a vector notation as (, Cb, ) [,..., ] N (8) where C [ C,..., C,,...,,...,,... ] C C C C, and b [ b,..., b,,...,,...,,... ] b b b b. (4) Weiht eory space: Each location of the receptive-field space contains adjustable weihts values in the weiht eory space with N coponents can be expressed in a vector as W [ W, W,..., ] W (9) N where W denotes the connectin weihts value of the th receptive-field. (5) Output space: the output of CMAC NN is the alebraic su of the activated weihts in weiht eory space, which can be written in a vector for as y W ( ) () B. Desin of ACMACNNBC Assuin that y d and α are desired value of state variables x and x, respectively, the error states between the state variables and the desired values can be defined as follows: z x yd () and z x () By substitutin () into the tie derivative of (), we can acquire results in the followin trackin error dynaics as z x y x y z y (3) d d d Select the virtual controller as z y, where λ d >. Consider the followin yapunov function candidate V z z. By differentiatin yapunov function and usin (3), the first derivative of the yapunov function can be written as Vz z z ( xy ) z ( xy ) z ( x y ) d d d z ( z y z z z z (4) ro () and (), it can be obtained: z x M Cx M ( G M (5) DOI.53/IJSSS.a ISSN: x online, print

3 o ake z as sall as possible, the followin control input torque τ is selected as z z (6) where λ is a positive constant, is the nonlinear robot function. In this case, consider the second yapunov function as follows V V z Mz (7) In view of (5) and usin Property, differentiatin (7) yields V V z Mz z Mz z Mz z zz zzm( x ) zcz z zz zz ( CxCz (8) M ( G ) z z z z z ( f ) z ( G where f C M. Substitutin (6) into (8) yields V z zz zz ( f ) z ( G (9) he CMAC is adopted to approxiate the unknown function f so that we can conclude that asyptotic stability of the syste is uaranteed. he next step is to select an appropriate CMAC adaptive weiht law, and then stability synthesis can be perfored for closed-loop syste by yapunov ethod. Assue that the nonlinear function is appr-oxiated by CMAC NN for soe constant known ideal weiht W fro () and expressed as W ( ) () Define the estiate of the value of () as ˆ ˆ W ( ) () with W ˆ the current values of the NN estiated weihts. It s provided by the adaptive weiht law. So estiation error of the weiht is W W Wˆ () he ideal weiht is bounded by known positive values W ax as follows: W W (3) ax Since W is a constant, take a derivative with (), it can be obtain W Wˆ. o speed up the converence of the controller, the adaptive weihts law is defined as ˆ ˆ W kg z W z G ( ) (4) where G is a positive definite atrix and k is a positive constant. Select the lobal yapunov function candidate as V V trw G W. By differentiatin the yields of V, we obtain V V tr W G W z z z z z ( f ) z ( G tr W G W z ( G tr W G W z z z z z ( f ˆ ˆ ) (5) here exists optial error between the estiation function ˆ and the unknown function f in [] such that f ˆ (6) where ε is a sall positive constant. Obtained fro ()- (), the forula (5) can be written as ˆ ˆ V z zz zz ( f W ( ) W ( )) z ( G tr W G W z z z z z ( f ˆ ) z W ( ) ( ) z G d tr W G W z z z z z ( f ˆ ) z ( G tr W ( G W z ( )) hen, usin (4), (6), and Property 3 yields V z zz z z z( Gb db) k z trw ( W W) Since Cauchy Schwarz inequality trw ( WW) ( W, W) W. W W WW ( W W ) ax (7) he forula (7) can be expressed as V z z z( z (8) kw ( W Wax ) ( ( Gb db))) his is a neative as lon as the ter in braces positive, and copletin the square in braces ter yields z kw ( W Wax) ( ( Gbdb)) (9) kw ( W ) kw 4 z ( ( G) ax ax which is uaranteed to be positive as lon as kw ax 4 ( ( Gb db)) z (3) Or b b DOI.53/IJSSS.a ISSN: x online, print

4 W Wax W ax 4 ( ( Gb db)) k (3) Owin to all paraeters in above (3) and (3) such as k, ε, W ax, λ, G b and d b are all positive constants. herefore, z and W are deonstrated to be UUB. As a result, the fact that error z is bounded, which uarantees z and z bounded. With x and x bounded, which results in the desired trajectory y d bounded. Moreover, Wˆ and ˆ are also bounded because of W bounded. hus, V is neative outside a copact set so that the closed-loop syste is asyptotically stable. IV. SIMUAION RESUS In order to verify the better effect of the proposed controller than the conventional PID controller, the siulation tests have been carried out on the -DO robot anipulator. he paraeters of the robot anipulator are chosen as: M =.5k, M =.765k, l = l =.5, r = r =.5 and =9.8/s. he values of external disturbances and the desired trajectories are set as d= [.5sinπt.5sinπt],D= [sin.πt sin.πt], respectively. he paraeters of PID controller proportional, interal, differential are selected as k p =.5, k I =.5 and k D =.3, respectively. he paraeters of the ACMACNNB controller are selected as below: λ =3, λ =5, k=.5 and G is a second-order unit positive atrix. he CAMCNN contains 3-nodes for each robot link to approxiate unknown dynaics with centers of receptive-field function C ji evenly space [-, ]. he widths are all selected as b ji=.5. he NN weihts are initialized as.5 in siulation test. Siulation results are shown in i. -i.8 As you can see in i. -3, the actual trajectories are trackin the desired trajectories by PID. he responses of the ACMACNNBC are perfored in i he CMAC NN adaptive weihts are depicted in i. 6-7 by usin the adaptive law (4). o study the advantaes of the proposed controller over the conventional PID controller, we copare the position trackin error of the two systes in the i he results show that the proposed controller can respond converence rates faster and ake saller error than the PID controller with external disturbances. lin k : p ositio n(rad ) tie(s) i. (). ink with PID Controller link : position(ra link : position(ra tie(s) i. (3). ink with PID Controller tie(s) link : position(ra link : NN weihts link : NN weihts i. (4). ink with ACMACNNB Controller tie(s) i. (5). ink with ACMACNNB Controller tie(s) i. (6). the NN Adaptive Weihts for ink tie(s) i. (7). the NN Adaptive Weihts for ink DOI.53/IJSSS.a ISSN: x online, print

5 link : position error(ra ACMACNNBC PID tie(s) i. (8). Coparison of Both Controllers for ink CONIC O INERES he authors confir that this article content has no conflicts of interest. ACKNOWEDGMEN his work was supported by the Science-echnoloy oundation of aizhou, Zhejian province, China (No. 3KY). link : position error(ra ACMACNNBC PID tie(s) i. (9). Coparison of Both Controllers for ink V. CONCUSION o achieve stability for the robot anipulator with unknown external disturbances in the course of the coplex environent work, an adaptive controller has been developed based on back-steppin control theory. he architecture of the proposed controller consists of a backsteppin control and a CMAC NN identification odel which copes with the unknown paraeters and external disturbances. he overall control syste is very easy to ipleent because of no a priori paraeters of the robot anipulator dynaics. Another advantae of this approach is the fact that it tracks well with the desired trajectories and the converence rates can be iproved even if the existence of unknown external disturbances. Moreover, the stability of the closed-loop syste is uaranteed by yapunov stability theores so that the syste has stron robustness. REERENCES [] R. J. ian, Desin of an enhanced adaptive self-oranizin fuzzy slidin-ode controller for robotic systes, Expert Systes with Applications, vol. 39, pp ,. []. Sun,. Pei, and Y. Pan, Neural network-based slidin ode adaptive control for robot anipulators, Neurcoputin, vol. 74, pp ,. [3]. Bossi, C. Rottenbacher, and G. Mii, Multivariable predictive control for vibratin structures: An application, Control Enineerin Practice, vol. 9, pp ,. [4] A. orouzantabar,. A. alebi, and A. K. Sedih, Adaptive neural network control of bilateral teleportation with constant tie delay, Nonlinear Dynaics, vol. 67, pp. 3-34,. [5]. Chen, Z. G. ou, M. an, Adaptive neural network trackin control for anipulators with uncertain kineatics, dynaics and actuator odel, Autoatica, vol. 45, pp. 3-38, 9. [6] Y.. Ki,.. ewis, Neural network output feedback control of robotic anipulators, IEEE ransactions on robotic and autoation, vol. 5, pp. 3-39, 999. [7] J. K. iu, X.. Wan, Advanced slidin ode control for echanical systes: Desin, Analysis and MAAB Siulation. Spriner: Berlin,. [8] Y.. Pen, Robust intellient slidin odel control usin recurrent cerebellar odel articulation controller for uncertain nonlinear chaotic systes, Chaos, Solitons & ractals, vol. 39, pp. 5-67, 9. [9] C.. su, C. M. Chun, and C. M. in, C. Y. su, Adaptive CMAC neural control of chaotic systes with a PI-type learnin alorith, Expert Systes with Applications, vol. 36, pp , 9. [] C. M. in, Y.. Pen, Missile uidance law desin usin adaptive cerebellar odel articulation controller, IEEE ransactions on Neural Networks, vol. 6, pp , 5. DOI.53/IJSSS.a ISSN: x online, print