(6) B NN (x, k) = Tp 2 M 1

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1 MMAR 26 12th IEEE International Conference on Methods and Models in Autoation and Robotics August 26 Międzyzdroje, Poland Synthesis of Sliding Mode Control of Robot with Neural Networ Model Jaub Możaryn and Jerzy E. Kure Abstract A synthesis of sliding ode control for a robot ar based on a robot odel identified by neural networs is presented. The proposed structural feed-forward neural networs estiate the eleents of the Lagrange-Euler atheatical odel of robot, and they can be directly used for a synthesis of a odel-based control syste. I. INTRODUCTION One of the fundaental tass of a robot anipulator is the fast and accurate tracing of the reference trajectory. A robot has the coplicated structure of a atheatical odel that is nonlinear and consists of coupling between joints. One of probles of a odel-based control syste design is the exact nowledge of a robot atheatical odel, the calculation of which is difficult. It requires the nowledge of exact values of robot ineatical and dynaical paraeters, that are hard to obtain 8. Recently, there has been research done to find a reliable identification ethod of a robot atheatical odel and its paraeters 5, 14. The popular approach is to rewrite a atheatical odel of a robot to be linear in relation to the dynaic paraeters called a base paraeter set (BPS) 12. Thereafter, a BPS can be identified using different estiation techniques 16 eg. least square techniques, Kalan filtering, gradient techniques. However, the drawbac of these identification ethods is the requireent of coplicated atheatical functions that describe a atheatical odel in ters of the unnown paraeters. To overcoe probles of coplicated coputations and a nowledge of atheatical equations, there are suggested ethods of analysis and synthesis of the dynaical nonlinear systes to identification of a robot atheatical odel. Therefore, there is the growing interest in use of neural networs 1, 11, 15, 18. Their advantages are the approxiation of the ultivariable nonlinear functions, and the easy adaptation of odel paraeters. Moreover, the neural networ s design does not require an exact nowledge of functions that describe the odel, but only values of odel variables. Their structure can reseble a plant odel. There are several techniques of designing robot control systes that are robust with respect to odel uncertainties and disturbances. One such ethod is a sliding ode control. Many publications present different approaches to the proble of a sliding ode control of a robot ar eg. 1, 2, 6, 7, 17. This paper presents a synthesis of a robot sliding ode Jaub Mozaryn and Jerzy E. Kure are with Institute of Autoatic Control and Robotics, Warsaw University of Technology, Warszawa, `sw.andrzeja Boboli 8, POLAND, j.ozaryn@chtr.pw.edu.pl control syste with a atheatical odel whose coefficients are estiated using neural networs. The structure of a neural networ odel for identification can be easily designed based on a robot atheatical odel, and used in the control syste. A sliding ode control algorith was chosen, because it is robust with respect to inaccurate odel paraeters. The presented control algorith and the identification ethod are discrete tie, and can be used in digital control systes. This paper is organized as follows. In section 2, a robot atheatical odel in a for of Lagrange-Euler equations is described. Section 3 introduces a sliding ode control of the robot. In sections 4 and 5, a neural networ odel of the robot is presented and its synthesis with a sliding ode control syste. The siulations of a two degree of freedo robot with a proposed controller are presented in section 6. Finally concluding rears are given. II. DISCRETE TIME ROBOT MODEL The discrete tie odel of a robot with n degrees of freedo, based on Lagrange-Euler equations 3 can be presented as follows τ() = Tp 2 M()q( + 1) 2q() + q( 1) +V () + G(), (1) where τ() R n is the vector of control signals, q R n is the vector of generalized joint coordinates, M() = M(q()) = ij R n n is the robot inertia atrix, V () = V (q(), q( 1)) = v i R n is the vector of Coriolis and centrifugal effects, G() = G(q()) = g i R n is the vector of the gravity loading, is the discrete tie, T p is the sapling tie, t = T p. The presented odel (1) can be rewritten in the state space for, as where x() = A(x, ) = x( + 1) = A(x, ) + B(x, )τ() y() = Cx() + Dτ() q( 1) q(), x 2 2q() q( 1) T 2 p M 1 ()P (q, ) P (q, ) = V () + G(), B(x, ) = Tp 2 M() 1, C = I, D =., (2), IEEE Conference Nuber:

2 III. SLIDING MODE CONTROL OF ROBOT For a robot let us denote x = x x r, (3) where x r R n is the vector of the reference generalized state variables in all joints. The sliding ode control law can be described by the equation τ() = τ eq () + τ nl (), (4) where τ eq () = (LB(x())) 1 L(A(x()) x r ( + 1)) (5) τ nl = (LB(x())) 1 T p Ksign(s( x()))+ +T p Γs( x()) s( x()) The so-called switching function is in the for (6) s( x) = L x = Λ I x, (7) where Λ = diag l i. The design of a sliding ode control syste requires tuning of the paraeters K = i R n, i >, Γ = γ i R n, γ i >, 1 T p γ i >, Λ R n n, l i < 1, i = 1... n, to allow proper functioning. It affects the dynaics and the robustness of the sliding ode control syste. IV. NEURAL NETWORK ROBOT MODEL In the robot odel (1) the unnown nonlinear eleents of M(), V (), G() should be identified. The odel (1) can be rewritten as follows M()γ() + P () = τ(), (8) where γ() = q(+1) 2q()+q( 1) T p. 2 It consists of n independent equations in the for n ij ()γ i () + p i () = τ i (), i = 1... n. (9) j=1 In the proposed approach, the structure of the neural networ should be designed to identify eleents p i () and ij (). Inputs to a neural networ are q( 1), q(), and γ i (). The output of a neural networ is τ i (). For identification of coefficients in (9), feed forward neural networs can be used. The structure of the neural networ for identification is shown in Fig. 1. The robot atheatical odel has specific properties 3, 4, 15. Such properties can be considered during the design of the structure of neural networs. It is ostly iportant for the inertia atrix M(). Property 1: Each eleent of the inertia atrix M() is the function described as ij () = f(q s+1,..., q n, ), s in(i, j). (1) Property 2: The inertia atrix M() is syetric and positive definite. The first property defines the set of generalized coordinates that influence each eleent of the inertia atrix. There are iportant conclusions fro this property. Each eleent of the inertia atrix M() is irrespective of the position in the first joint, and an eleent nn has a constant value. This conclusion can siplify the structure of the neural networ odel. The second property can be used to test if neural networs properly identify eleents of the inertia atrix, and thereupon can be ipleented in a control syste. A presented neural networ robot odel can be used for identification and estiation of eleents of M NN () = NNij (), P NN () = p NNi (), that correspond to the atrices in (8). V. APPLICATION OF NEURAL NETWORK ROBOT MODEL IN SLIDING MODE CONTROL OF ROBOT In the sliding ode control (4) for a robot, the odel described in state space (2) is used. Since calculation of the robot odel is coplicated, we propose to use the odel, which is approxiated by neural networs described in the previous chapter. The structure of the neural networ for an estiation of robot odel coefficients in (9) is shown in Fig. 2. An estiated robot odel can be written in a state space, as x( + 1) = A NN (x, ) + B NN (x, )τ() y() = Cx() + Dτ() where A NN (x, ) = x = 2 2q() q( 1) Tp 2 M 1 NN ()P NN() B NN (x, ) = Tp 2 M 1 NN (), C = I, D =., VI. COMPUTER SIMULATIONS, (11) In order to train and test the proposed neural networ odels data saples were generated fro the siulation of the robot with two degrees of freedo and revolute joints 4. The robot s physical paraeters are given in Table I, the robot is presented in Fig. 3. For the calculation of the training and testing data, the reference trajectory for every joint was set as the su of three different sine functions, according to the following forula q ri () = 3 (a ij sin(ϖ ij T p + ϕ ij )), (12) j=1 where i = 1,..., n is the joint nuber, a ij is the aplitude, ϖ ij is the angular velocity and ϕ ij is the phase. The values of paraeters for training and testing trajectories are given in Table II. The robot was siulated with a given trajectory in tie interval T = 1sec, and T p =.1sec. Thus, there were 632

3 TABLE I PHYSICAL PARAMETERS OF A TWO DEGREE OF FREEDOM ROBOT (DENAVIT-HARTENBERG NOTATION) 4. Lin i α i a i θ i d i Mg r x r y r z I xx g 2 I yy, I zz g 2 I xy, I yz, I xz g 2 Moveent range to to 18 TABLE II TRAJECTORIES PARAMETERS. Lin i a i1 a i2 a i3 ϖ i1 s ϖ i2 s ϖ i3 s ϕ i1 ϕ i2 ϕ i3 Lin 1, training Lin 2, training Lin 1, testing Lin 2, testing , data saples for training and 1, data saples for testing of the neural odels. In all nonlinear layers (NL) of neural networs the neurons are described by the sigoidal activation function y = f nl (x) = tansig(x). (13) In linear layers (L) there are neurons described with the linear activation function Finally siulated control syste was tested for tracing of the reference trajectory. Trajectories, reference trajectories and control signals obtained during siulations are presented in Fig. 4. The accuracy of the presented control syste for positioning was checed using as quality indexes the average absolute error e avi and the axiu absolute error e axi for each joint, calculated as y = f l (x) = x. (14) where x = L i=1 w ix i +b, L is the nuber of neuron inputs, w i is the weight of the i-th input to the neuron, x i is the i-th input to the neuron and b is the threshold offset. The perforance function of the neural networ was chosen as the ean squared error J IIMi = 1 N N (τ i () τ NNi ()) 2, (15) =1 Every coefficient was identified by one nonlinear hidden layer, and one linear output layer. Neural networs were trained using the bacpropagation ethod and the Levenberg-Marquardt ethod to update weights in all layers 9. The structural neural networ odel was trained off-line with nown training data to identify the odel coefficients. Trained neural networs were tested for the positivity of the inertia atrix along training and testing trajectories. Thus, it was possible to invert the estiated inertia atrix M NN (q, ), and consequently it was reasonable to attept to apply the trained neural networs in the control syste. The structure of the neural networs was chosen experientally. There were 2 neurons in each nonlinear layer and 1 neuron in each linear layer. There were 4 training iterations. Afterwards, the trained neural networs were applied in the sliding ode control syste to estiate odel coefficients. e i = q i () q ri (), (16) N =1 e avi = e i(), (17) N e axi = ax e i (), = 1,.., N, (18) where N is the nuber of all data saples. The values of quality indexes (17) and (18) for each joint for both training and testing trajectories, are given in Table III. It is apparent that errors between the reference trajectory and the trajectory fro the siulated robot with sliding ode control with the neural networ odel are generally sall. Nevertheless, there are certain points where these values increase. In these points the control signals have high values which in result can lead to instability of the syste. 633

4 TABLE III AVERAGE AND MAXIMUM ERRORS FOR TRAINING AND TESTING TRAJECTORIES. Lin e avi, training e axi, training e avi, testing e axi, testing VII. CONCLUDING REMARKS This paper presented a synthesis of a sliding ode control syste for a robotic anipulator with a structural neural odel of robot. The obtained results fro control of the two degree of freedo robot show that it is possible to design the sliding ode controller based on the atheatical robot odel calculated using neural net techniques. However, there are certain probles that should be solved in future. There are soe points on the trajectories where errors increase. This can be the result of inaccuracies in the odel and in consequence a loss of odel properties. A possible solution is to define the structure of the neural networ that fulfill and assure the strict properties of the odel eg. syetry of the inertia atrix. Another proble is the length of tie needed to train the neural networ, which was done off-line. More efficient training algoriths and siplifications of neural networ structures should be ipleented. Therefore, the presented approach to robot odel identification and control requires further research. 14 Kozlowsi K., Modelling and Identification in Robotics, Springer Verlag, Berlin, Lewis, F. L., Neural Networ Control of Robot Manipulators and Nonlinear Systes, Taylor & Francis, Poignet Ph., Gautier M., Coparison of Weigthed Least Squares and Extended Kalan Filtering Methods for Dynaic Identification of Robots, Proc. IEEE International Conference on Robotics and Autoation, San Francisco, USA, vol. 2, pp , Możaryn, J., Kure, J. E., Coparison of sliding ode control and decoupled sliding ode control of robot Pua 56, Proc. 9th IEEE Int. Conf. on Methods and Models in Autoation and Robotics MMAR 23, vol. 2, pp , Możaryn, J., Kure, J. E., Coparison of Neural Networ Robot Models with Not Inverted and Inverted Inertia Matrix, Proc. International Conference on Artficial Neural Networs ICANN, Warsaw, 25, vol. 2, pp , 25. REFERENCES 1 Young K.K.D., Controller design for a anipulator using theory of variable structure systes, IEEE Trans. Sys. Man. And Cyb., vol. SMC-8, pp , Slotine J. J., Sastry S.S., Tracing control of nonlinear systes using sliding surfaces with aplication to robot anipulators, International Journal of Control, vol. 38, no. 2, pp , Fu K. S., Gonzalez R. C., Lee C. S. G., Robotics: control, sensing, vision, and inteligence, McGraw-Hill Boo Copany, Tang K.M.W., Tourassis V. D., Systeatic siplification of dynaic robot odels, Proc. Midwest Syp. Circuits Syst., Syracuse, NY, pp , Kozlowsi K., Matheatical Dynaic Robot Models and Identification of Their Paraeters (in polish), Technical University of Poznan Press, Gao W., Hung J. C., Variable Structure Control of Nonlinear Systes:A New Approach, IEEE Transactions on Industrial Electronics, vol. 4, pp , Hung J. Y., Gao W., Hung J. C., Variable Structure Control: A Survey, IEEE Transactions on Industrial Electronics, vol. 4, pp. 2-22, Core P. I., Arstrong-Hlouvry B., A search for consensus aong odel paraeters reported for the PUMA 56 robot, Proc. IEEE Int. Conf. Robotics and Autoation, San Diego, vol. 1, pp , Osowsi, S.: Neural Networs For Inforation Processing (in polish), OWPW, Warszawa, Lewis F. L., Liu K., Yesildire A., Neural Net Robot Controller with Guaranteed Tracing Perforance, IEEE Transactions on Neural Networs, vol. 6, pp , Lewis F. L., Neural Networ Control of Robot Manipulators, IEEE Expert special trac on Intelligent Control, vol. 6, pp , Gautier. M., A Coparison of Filtered Models for Dynaic Identification of Robots Proc. 35th Conference on Decision and Control, Kobe, Japan, pp , Edwards C., Spurgeon S. K., Sliding Mode Control: Theory and Applications, Taylor & Francis,

5 q(-1) q() p NNi () NNi1 () NNi () 1 () NNin () n () Fig. 1. A neural networ odel for the identification of robot atheatical odel eleents. q(-1) q() p NNi () NNi1 () NNin () Fig. 2. A neural networ odel for the estiation of robot atheatical odel eleents. x 2 y 2 z 2 2 a 2 y 1 x 1 2 z 1 a 1 1 y 1 z x Fig. 3. A two degree of freedo robot ar. 635

6 a) position q1deg position q2deg control 1deg control 2deg b) position q1deg position q2deg control 1deg control 2deg Fig. 4. Trajectories and control signals fro the siulation of robot with sliding ode control and neural networ odel (solid line), reference trajectories (dashed line) for each lin, a) training trajectories, b) testing trajectories. 636