 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor A. Ruano under the direction of Editor T. Basar. Research supported by NSF grant ECS-9521673. * Corresponding author. Tel.: 817-272-5957; fax: 817-272-5989. E-mail addresses: ykim50@hotmail.com (Y.H. Kim), #ewis@arri. uta.edu (F.L. Lewis), ddawson@eng.clemson.edu (D.M. Dawson). Automatica 36 (2000) 1355 } 1364 Brief Paper Intelligent optimal control of robotic manipulators using neural networks  Young H. Kim*, Frank L. Lewis, Darren M. Dawson Automation and Robotics Research Institute, The University of Texas at Arlington, 7300 Jack Newell Blvd. South, Fort Worth, TX 76118-7115, USA Department of Electrical and Computer Engineering, Center for Advanced Manufacturing, Clemson University, Clemson, SC 29634-0915, USA Received 26 May 1998; revised 18 May 1999; received in "nal form 12 January 2000 Abstract The paper is concerned with the application of quadratic optimization for motion control to feedback control of robotic systems using 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 equation. It is shown how neural networks can cope with nonlinearities through optimization with no preliminary o!-line learning phase required. The adaptive learning algorithm is derived from Lyapunov stability analysis, so that both system tracking stability and error convergence can be guaranteed in the closed-loop system. The "ltered tracking error or critic gain and the Lyapunov function for the nonlinear analysis are derived from the user input in terms of a speci"ed quadratic performance index. Simulation results on 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.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Robotic manipulators; Optimal control; Closed-loop control; Neural networks 1. Introduction There has been some work related to applying optimal control techniques to the nonlinear robotic manipulator. These approaches often combine feedback linearization and optimal control techniques. Johansson (1990) showed explicit solutions to the H}J}B equation for optimal control of robot motion and how optimal con- trol and adaptive control may act in concert in the case of unknown or uncertain system parameters. Dawson, Grabbe and Lewis (1991) used a general control law known as modi"ed computed torque control (MCTC) and quadratic optimal control theory to derive a parameterized proportional-derivative (PD) form for an auxiliary input to the controller. However, in actual situations, the robot dynamics is rarely completely known, and it is thus di$cult to express real robot dynamics in exact mathematical equations or to linearize the dynamics with respect to the operating point. In this paper, we propose a nonlinear optimal design method that integrates linear optimal control techniques and neural network learning methods. The linear optimal control has an inherent robustness against a certain range of model uncertainties (Lewis & Syrmos, 1995). However, nonlinear dynamics cannot be taken into consideration in linear optimal control design. We use the feed-forward neural networks to adaptively estimate nonlinear uncertainties, yielding a controller that can tolerate a wider range of uncertainties. The salient feature of this H}J}B control design is that we can use a priori knowledge of the plant dynamics as the system equation in the corresponding linear optimal control design. The neural network is used to improve performance in the face of unknown nonlinear character- istics by adding nonlinear e!ects to the linear optimal controller. 0005-1098/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 0 4 5 - 5