Adaptive Control for Free-floating Space Flexible Robot Based on Fuzzy CMAC

Journal of Information & Computational Science 11:1 (14) 141 149 January 1, 14 Available at http://www.joics.com Adaptive Control for Free-floating Space Flexible Robot Based on Fuzzy CMAC Wenhui Zhang, Xiaoping Hu, Yamin Fang Institute of Technology, Lishui University, Lishui 33, China Abstract The tracking control problems of free-floating flexible space robot are studied by the paper, an adaptive Fuzzy CMAC PID control method is proposed by this paper. Dynamics equation of free-floating flexible space robot is established base on Lagrange principle, Controller based on Fuzzy CMAC is designed to used to adaptive learn and compensate inverse-model of flexible space robot, learning rules of network parameters are designed to adaptive adjust on line. Error function is provided by PID controller. The controller can improved the control accuracy and the asymptotic convergence of tracking error. The simulation results show that the control method had important engineering value. Keywords: Fuzzy CMAC; Inverse-model Control; PID Control; Space Flexible Robot 1 Introduction With the development of space transportation technology, space manipulator highlights the great advantage in terms of load carrying research, its research has also been widespread concerned in the by experts [1-5]. but because space manipulator system is nonlinear strongly coupled systems, there are the more intense dynamic coupling effects and uncertainty between robotic arm movement and the position of the carrier, To eliminate the impact of these nonlinear factors, advanced control strategies by has been applied in robot tracking control [6-9], such as adaptive control, fuzzy control, neural network control and so on [1-13]. Flexible space robot has a stronger coupling, it is difficult to obtain dynamic model in freefloating mode, although adaptive control method can get better control effect, the determination of the parameter linearization and regression matrix requires a lot of calculation [18]-[19]. Because fuzzy control and neural network control method are not required to know the exact model of the control object, and has the ability to learn, and has made some achievements in the field of space control application. A fuzzy control strategy was proposed by [14]-[15], by adaptive law were designed to adjust fuzzy rules; but due to too many fuzzy rules were obtained, the Project supported by Zhejiang Provincial Natural Science Foundation of China (No. LZ1F1 and No. LY13F). Corresponding author. Email address: hit zwh@16.com (Xiaoping Hu). 1548 7741 / Copyright 14 Binary Information Press DOI: 1.1733/jics1743

14 W. Zhang et al. / Journal of Information & Computational Science 11:1 (14) 141 149 amount of computation would be increased geometrically, on the contrary, the rules too little can not guarantee control accuracy. Back-propagation error algorithm was proposed to adjust the network weights by [7], because of offline training, it difficult to guarantee real-time and can not be applied. For the lack of the above methods, a fuzzy the CMAC network adaptive control scheme is proposed for flexible space manipulators. The fuzzy CMAC(Cerebellum Model Articulation Controller) neural network is used to learn quickly the inverse dynamics model of the non-linear flexible arm. Learning algorithm is designed to improved Hebb learning rule online. Simulation results show that the proposed scheme is effective. Dynamic Model of Flexible Space Robot Without loss of generality, freely floating space flexible manipulator system of plane motion is established. the system structure is shown in Fig. 1. O XY is the translational inertial coordinate. y x 1 θ x O g1 B 1 O B P u(x,t) x y 1 θ 1 r O 1 r 1 Y α C O O g O X Fig. 1: Model of space flexible robot Because the model is more complex, the model needs to be reasonable assumed to analysis. The flexibility of link is only considered by the paper, the moment of inertia of the motor is not considered, so when system dynamics analysis is conducted, the following assumptions are made: 1) The density of each connecting rod and the system body is equally distributed; ) The size of the flexible rod ends of actuators is not considered, it is regarded as a concentrated mass; 3) The length of the link is much larger than its cross-sectional dimension; 4) The lateral bending deformation of the link is only considered, axial deformation and shear deformation is ignored. The elastic displacement mode function expansion of flexible beam is described as: n u(x, t) = ϕ i (x )q i (t) i = 1,,, n (1) i=1

W. Zhang et al. / Journal of Information & Computational Science 11:1 (14) 141 149 143 where, ϕ i (x ) is defined as the first order i of the modal functions in term of the flexible arm, q i (t) is defined as modal coordinates corresponded with ϕ i (x ), n is defined as truncating the number of entries. α is defined as attitude, θ 1 and θ is defined as the relative rotation of each joint of the manipulator, q 1 and q are defined as the generalized coordinates case, the total kinetic energy of the system is defined as T, Uniform bending stiffness is defined as E 1, the system elastic potential energy can be described as: U = 1 l E u(x, t) 1 dx x () The Lagrange of the system is defined as: L = T U. (3) If Lagrange equations are used, equation (4) can be gained: [ ] d dl dl = Q i i = 1,,. (4) dt d q i dq i Dynamics model of space robot with flexible manipulator can be described as [18]: [ ] θ D + H = τ. (5) q where, θ = [ θ 1 θ ] T, q = [ q 1 q ] T. The definite and symmetric generalized mass matrix is defined as D. The coupling Coriolis force, centrifugal force and the elastic force of the array of 4 1 is defined as H. The control torque is defined as τ = [ τ 1 τ ] T. 3 The Design of PID Controller Based on Fuzzy CMAC The learning ability of fuzzy CMAC (Cerebellum Model Articulation Controller) neural network combined can approach arbitrary nonlinear function. y = f(x), where x R Nx is a continuous input; y N y is output. Double input and single output of fuzzy CMAC is explained in Fig.. O (n) ij = I (n) ij w ij. (6) 1 3 4 5 X 1 Σ y X Fig. : FCMAC neural network structure

144 W. Zhang et al. / Journal of Information & Computational Science 11:1 (14) 141 149 where, I (n) ij ando(n) ij are defined respectively as input and output of each unit.w ijis weight of association unit. y = O (4) ij. (7) Augmented variable input method is adopted, φ d = [ θ T d i,j q T ] T is defined as the desired joint angles, φ = [ θ T q T ] T is the input vector, e = φ d φ = [ e T θ ] is defined as the error vector, e θ = θ d θ, the stability of the system can be guaranteed by the following controller. τ = D( φ d + K p e + K d ė) + H. (8) where, K p and K d are defined as the feedback gain matrix. In practice, however, it is difficult to get accurately the space robot model, so ˆD and Ĥ are defined as nominal model, the control law is designed as: τ = ˆD( φ d + K p e + K d ė) + Ĥ. (9) The control law (9) is substituted, then ë + K d ė + K p e = ˆD 1 [ D φ + H]. (1) where, D = D ˆD, H = H Ĥ.By the above equation, the modeling uncertainty of system will lead to the reduction of the control performance. To solve the impact of the nonlinear dynamics of flexible space robot model, FCMAC (Fuzzy Cerebellum Model Articulation Controller) is considered to use to approximate the inverse dynamics model of flexible space robot. Equation (9) is written as: τ = D [ θ q ] + H = F ( θ, θ, θ, q, q, q). (11) The FCMAC inverse model control system of the flexible space robot is shown in Fig. 3, where, the total control input τ is composed by τ P D and τ NN. τ = τ P D + τ NN. (1) FCMAC controller θ,θ,θ,q,qq,. θ d + _ K d. + θ PID controller θ d K + + p _ θ τ PD τ NN τ Space flexible manipulators Fig. 3: FCMAC Inverse-model control system

W. Zhang et al. / Journal of Information & Computational Science 11:1 (14) 141 149 145 where, PID feedback controller is defined as τ P D, FCMAC Inverse learning controller is defined as τ NN. The PID feedback controller is design as: FCMAC controller is designed as: τ P D = K d ė + K p e. (13) τ NN = M( θ, θ, θ, q, q, q, w) (14) where, the PID feedback controller played major role in the beginning of learning stage of the network FCMAC, because the inverse dynamic model of the system is not completely learned by the FCMAC, then the PID controller plays the primary control. If when the model is changed by the outside in the control process, the network controller of FCMAC is not immediately available to adapt the inverse dynamics model. Because the PID controller has a good characteristics of dynamic response. The PID controller participate in the compensation control, two controller are combined together to ensure the stability of the system. The uncertain inverse dynamics model is learned by FCMAC with the ability of nonlinear approach, FCMAC will be more and more accurate for the approaching the uncertain inverse dynamics model. PID controller role will become increasingly weak. So, FCMAC network learning error is defined as: E = τ P D (15) The five floors fuzzy CMAC neural network is used as controller; the Gaussian function is adopted as its membership function: µ(x i ) = e ( x i σ ij υ ij ) (i = 1, 6; j = 1, 9) (16) Further, improved and supervised Hebb learning rules are designed by paper, then w i (k + 1) = (1 c)w i (k) + ηr i (k) (17) r i (k) = E(k)y(k)x i (k) (18) n y(k) = y(k 1) + k w i (k)x i (19) w i (k) = i=1 w i(k) n w i (k) i=1 where, r i (k) is defined as progressive signal, which gradually decay with the process. E(k) = τ P D is defined as the output error signal; η > is defined as steps or a learning rate, c 1 is defined as the normal number. Equation (17) and () are used, then () w i (k) = c[w i (k) + η c E(k)y(k)x i(k)] (1)

146 W. Zhang et al. / Journal of Information & Computational Science 11:1 (14) 141 149 where, w i (k) = w i (k + 1) w i (k). Then Equation (1) can be written as f i w i = w i (k) η c γ i(e(k), y(k), x i (k)) () f i w i (k) = c w i (k) (3) 4 Simulation Analyses In particular, the paper in order to illustrate the effectiveness of the proposed algorithm, the paper will compare the PID controller and the proposed algorithm by paper. Model parameters of flexible space manipulator are defined as: l =.5 m, l 1 = l = 1.5 m, a = 1 m, E 1 = 3 N m, ρ =.5 kg/m, I = 5 kg m, I 1 =.5 kg m, m = 4 kg, m 1 = kg. The desired trajectories of two joints of the space flexible manipulator are defined as: θ 1d =.5(sin t + sin t); θ d =.5(cos 3t + cos 4t). PD controller gains and parameters: K p = diag(, ); K d = diag(3, 3); η =.8. The initial values of base and manipulator are defined as. The simulation results are shown in Fig. 4-Fig. 7. If FCMAC controller is closed, trajectory tracking Fig. 4 is obtained by only using common PID controller. If FCMAC controller is opened, trajectory tracking Fig. 6 is obtained by using FCMAC controller and PID controller; Fig. 5 is control torque of a PID controller, Fig. 7 is control torque of FCMAC controller. As can be seen from Fig. 4-5, traditional PID controller is difficult to obtain desired trajectory and control torque are also larger. But it can be seen from Fig. 6-7, the design controller based 1. Joint 1/rad.5.5 1. 1 Joint /rad 1 Desired Real 4 6 8 1 Desired Real 4 6 8 1 t/s Fig. 4: Trajectory tracking of PID

W. Zhang et al. / Journal of Information & Computational Science 11:1 (14) 141 149 147 6 Joint 1/N m Joint /N m 4 6 4 4 6 8 1 4 6 8 1 t/s Fig. 5: Control torque of PID 1. Joint 1/rad.5.5 1. Joint /rad 1 Desired Real Desired Real 4 6 8 1 1 4 6 8 1 t/s Fig. 6: Trajectory tracking of the paper 1 Joint 1/N m 5 5 1 4 6 8 1 Joint /N m 5 5 4 6 8 1 t/s Fig. 7: Control torque of the paper

148 W. Zhang et al. / Journal of Information & Computational Science 11:1 (14) 141 149 on of FCMAC the Inverse PID controller can quickly track the desired trajectory in a short time (t=1 s). Because the PID feedback controller played major role in the beginning of learning stage of the network FCMAC, because the inverse dynamic model of the system is not completely learned by the FCMAC, then the PID controller plays the primary control. If when the model is changed by the outside in the control process, the network controller of FCMAC is not immediately available to adapt the inverse dynamics model. Because PID controller has a good dynamic response. The PID controller participate in the compensation control, two controller are combined together to ensure the stability of the system. 5 Conclusions The trajectory tracking problems are considered for free-floating space robot with flexible manipulators. An adaptive Fuzzy CMAC inverse-model control algorithm is proposed by the paper. 1) Nonlinearity dynamics equation of space flexible manipulators is established on the basis of the assumed modes method and Lagrange principle; ) Controller based on Fuzzy CMAC neutral network is designed to adaptive learn and compensate inverse-model; 3) Parameters adaptive laws of Fuzzy CMAC Network base on improved supervisory Hebb learning rules are designed to adjust on line. The simulation results show that the presented controller had engineering value. Acknowledgement The paper is supported from Zhejiang Provincial Natural Science Foundation of China under Grant No. LZ1F1 and No. LY13F. References [1] A. Farzaneh, H. A. Talebi, V. P. Rajnikant, A stable neural network-based observer with application to flexible-joint manipulators [J], IEEE Trans. on Neural Networks, 17(1), 6, 118-19 [] W. H. Zhang, N. M. Qi, H. L. Yin, Neural network adaptive compensation control of free-floating space robot [J], Journal of Astronautics, 3(6), 11, 131-1317 [3] D. King, Space servicing: Past, present and future [C], In: Proceedings of the 6th International Symposium on Artificial Intelligence, Robot and Automation in Space, Montreal, Canada, 1 [4] S. Dubowsky, E. G. Papadopoulo, The kinematics, dynamics and control of free-flying space robotic systems [J], IEEE Transactions on Robotics and Automation, 9(5), 1993, 531-543 [5] A. K. Bejczy, S. T. Venkataraman, Introduction to the special issue on space robotics [J], IEEE Transactions on Robotics and Automation, 9(5), 1993, 51-53 [6] W. B. Xu, X. B. Chen, Artificial moment method for swarm robot formation control [J], Sci. China Ser. F - Inf. Sci., 51(1), 8, 151-1531

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