EEE TRANSACTIONS ON SYSTEMS. MAN. AND CYBERNETICS-PART C: APPLICATIONS AND REVIEWS VOL 30. NO. 1. FEBUARY 2000 Optimal Design of CMAc Neural-Network Controller for robot manipulators Young h. Kim and Frank L. Lewis, Fellow, IEEE Abstract-This paper is concerned with the application neural-network based, closed-loop control can be found [12 of quadratic optimization for motion control to feedback For indirect or identification-based, robotic-system control, sev control of robotic systems using cerebellar model arithmetic eral neural network and learning schemes c can be found in the lit computer (CMAC) neural networks. Explicit solutions to the I control erature. Most of these approaches consider neural networks as of robotic systems are found by solving an algebraic Riccati equa- very general computational models. Although a pure neural-net- tion. It is shown how the CMAC's can cope with nonlinearities work approach without a knowledge of robot dynamics may be through optimization with no preliminary of -line learning phase promising, it is important to note that this approach will not be punov stability analysis, so that both system-tracking stability and very practical due to high dimensionality of input-output space error convergence can be guaranteed in the closed-loop system. In this way, the training or off-line learning process by pure con- The filtered-tracking error or critic gain and the Lyapunov nectionist models would require a neural network of impractical function for the nonlinear analysis are derived from the user input size and unreasonable number of repetition cycles. The pure in terms of a specified quadratic-performance index Simulation connectionist approach has poor generalization properties results from a two-link robot manipulator show the satisfactory In this paper, we propose a ne performance of the proposed control schemes even in the presence that integrates linear optimal-control techniques and CMAC neural-network learning methods. The linear optimal control Index Terms--CMAC neural network, optimal control, robotic has an inherent robustness against a certain range of model uncertainties [9]. However, nonlinear dynamics cannot be taken nto consideration in linear optimal-control design. We use L. INTRODUCTION the Cmac neural networks to adaptively estimate nonlinear I hERE has been some work related to applying optimal ncertainties, yielding a controller that can tolerate a wider control techniques to the nonlinear robotic manipulator. range of uncertainties. The salient feature of this H-J-B control These approaches often combine feedback linearization and op- design is that we can use a priori knowledge of the plant timal-control techniques Johansson [6] showed explicit solu- dynamics as the system equation in the corresponding linear tions to the Hamilton-Jacobi-Bellman(H-J-B)equation for optimal-control design. The neural network is used to improve optimal control of robot motion and how optimal control and performance in the face of unknown nonlinearities by adding adaptive control may act in concert in the case of unknown nonlinear effects to the linear optimal controller. or uncertain system parameters. Dawson et al. [5] used a gen- The paper is organized as follows In Section II, we will re- ral-control law known as modified computed-torque control view some fundamentals of the CMAC neural networks.In Sec MCTC)and quadratic optimal-control theory to derive a pa- tion I, we give a new control design for rigid robot systems rameterized proportional-derivative(PD)form for an auxiliary using the H-J-B equation In Section IV, a CMAC controller oput to the controller. However, in actual situations, the robot combined with the optimal-control signal is proposed In Sec- dynamics is rarely known completely, and thus, it is difficult to tion v, a two-link robot controller is designed and simulated in express real robot dynamics in exact mathematical equations or to linearize the dynamics with respect to the operating point Neural networks have been used for approximation of non linear systems, for classification of signals, and for associative memory For control engineers, the approximation capability of Let R denote the real numbers, n the real n-vectors, and tification-based control. More work is now appearing on the a∈as础l-=√+…+ and the norm of a matrix use of neural networks in direct, closed-loop controllers that AE mxn as (A!l=vAma[AT A] where Ama[ 1 and Amin[l yield guaranteed performance [13]. The robotic application of are the largest and smallest eigenvalues of amatrix. The absolute ue is denoted as· aii and B∈我m×n, the Frobenius norm is supported by NSF Grant ECS-952167. defined by非1=tr(44)=∑吗 with tr() as the trace 种如需分+mmmm ykim50@hotmail.com;flewis@arri.uta.edu) Publisher Item Identifier S 1094-6977(00)00364-3 Ar2≤‖Arl22 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 30, NO. 1, FEBUARY 2000 Optimal Design of CMAC Neural-Network Controller for Robot Manipulators Young H. Kim and Frank L. Lewis, Fellow, IEEE Abstract—This paper is concerned with the application of quadratic optimization for motion control to feedback control of robotic systems using cerebellar model arithmetic computer (CMAC) neural networks. Explicit solutions to the Hamilton–Jacobi–Bellman (H–J–B) equation for optimal control of robotic systems are found by solving an algebraic Riccati equa￾tion. It is shown how the CMAC’s can cope with nonlinearities through optimization with no preliminary off-line learning phase required. The adaptive-learning algorithm is derived from Lya￾punov stability analysis, so that both system-tracking stability and error convergence can be guaranteed in the closed-loop system. The filtered-tracking error or critic gain and the Lyapunov function for the nonlinear analysis are derived from the user input in terms of a specified quadratic-performance index. Simulation results from a two-link robot manipulator show the satisfactory performance of the proposed control schemes even in the presence of large modeling uncertainties and external disturbances. Index Terms—CMAC neural network, optimal control, robotic control. I. INTRODUCTION T HERE has been some work related to applying optimal￾control techniques to the nonlinear robotic manipulator. These approaches often combine feedback linearization and op￾timal-control techniques. Johansson [6] showed explicit solu￾tions to the Hamilton–Jacobi–Bellman (H–J–B) equation for optimal control of robot motion and how optimal control and adaptive control may act in concert in the case of unknown or uncertain system parameters. Dawson et al. [5] used a gen￾eral-control law known as modified computed-torque control (MCTC) and quadratic optimal-control theory to derive a pa￾rameterized proportional-derivative (PD) form for an auxiliary input to the controller. However, in actual situations, the robot dynamics is rarely known completely, and thus, it is difficult to express real robot dynamics in exact mathematical equations or to linearize the dynamics with respect to the operating point. Neural networks have been used for approximation of non￾linear systems, for classification of signals, and for associative memory. For control engineers, the approximation capability of neural networks is usually used for system identification or iden￾tification-based control. More work is now appearing on the use of neural networks in direct, closed-loop controllers that yield guaranteed performance [13]. The robotic application of Manuscript received June 2, 1997; revised June 23, 1999. This research was supported by NSF Grant ECS-9521673. The authors are with the Automation and Robotics Research Institute, University of Texas at Arlington, Fort Worth, TX 76118-7115 USA (e-mail: ykim50@hotmail.com; flewis@arri.uta.edu). Publisher Item Identifier S 1094-6977(00)00364-3. neural-network based, closed-loop control can be found [12]. For indirect or identification-based, robotic-system control, sev￾eral neural network and learning schemes can be found in the lit￾erature. Most of these approaches consider neural networks as very general computational models. Although a pure neural-net￾work approach without a knowledge of robot dynamics may be promising, it is important to note that this approach will not be very practical due to high dimensionality of input–output space. In this way, the training or off-line learning process by pure con￾nectionist models would require a neural network of impractical size and unreasonable number of repetition cycles. The pure connectionist approach has poor generalization properties. In this paper, we propose a nonlinear optimal-design method that integrates linear optimal-control techniques and CMAC neural-network learning methods. The linear optimal control has an inherent robustness against a certain range of model uncertainties [9]. However, nonlinear dynamics cannot be taken into consideration in linear optimal-control design. We use the CMAC neural networks to adaptively estimate nonlinear uncertainties, yielding a controller that can tolerate a wider range of uncertainties. The salient feature of this H–J–B control design is that we can use a priori knowledge of the plant dynamics as the system equation in the corresponding linear optimal-control design. The neural network is used to improve performance in the face of unknown nonlinearities by adding nonlinear effects to the linear optimal controller. The paper is organized as follows. In Section II, we will re￾view some fundamentals of the CMAC neural networks. In Sec￾tion III, we give a new control design for rigid robot systems using the H–J–B equation. In Section IV, a CMAC controller combined with the optimal-control signal is proposed. In Sec￾tion V, a two-link robot controller is designed and simulated in the face of large uncertainties and external disturbances. II. BACKGROUND Let denote the real numbers, the real -vectors, and the real matrices. We define the norm of a vector as and the norm of a matrix as where and are the largest and smallest eigenvalues of a matrix. The absolute value is denoted as . Given and , the Frobenius norm is defined by with as the trace operator. The associated inner product is . The Frobenius norm is compatible with the two-norm so that with and . 1094–6977/00$10.00 © 2000 IEEE