28 EEE TRANSACTIONS ON SYSTEMS. MAN. AND CYBERNETICS-PART C: APPLICATIONS AND REVIEWS VOL 30. NO. 1. FEBUARY 2000 Fig 3. Performance of OCTC (34): (a) tracking error for slow motion and(b)tracking error for fast motion(solid: joint 1, dotted: joint 2) 051 Time(second Time(second) Fig 4. Performance of OCTC(34): (a) tracking error with modeling error for slow motion and(b) tracking error with disturbance and friction for slow motion Time(second) Fig. 5. Performance of CMAC neural network controller(40)(a)tracking error for slow motion and ( b)tracking error for fast motion(solid: joint 1, dotted Joint 2) learning rate in the weight-tuning law: VI. CONCLUSION We have developed a hierarchical, intelligent control scheme for a robotic manipulator using the HJB optimization process 08000 and the CMac neural network. It has been shown that the entire closed-loop system behavior depends on the user-specified per and=0.0001 formance index Q and R, through the critic-gain matrix A. The simulation time. 20s Lyapunov function for the stability of the overall system is au The results in Figs. 5 and 6 clearly show the ability of the CMac tomatically generated by the weighting matrices. In the deriva neural-network controller to me uncertainties, both struc- tion of the optimal-computed torque controller, it has been as- tured and unstructured. Note that the problem noted umed that nonlinearities in the robotic manipulator are com- with octc does not arise here. as all the nonlinearities pletely known. However, even with the knowledge of nonlinear- sumed unknown to the cmac neural controller ities, it is difficult to achieve the control objective in the pres-28 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 30, NO. 1, FEBUARY 2000 (a) (b) Fig. 3. Performance of OCTC (34): (a) tracking error for slow motion and (b) tracking error for fast motion (solid: joint 1, dotted: joint 2). (a) (b) Fig. 4. Performance of OCTC (34): (a) tracking error with modeling error for slow motion and (b) tracking error with disturbance and friction for slow motion (solid: joint 1, dotted: joint 2). (a) (b) Fig. 5. Performance of CMAC neural network controller (40): (a) tracking error for slow motion and (b) tracking error for fast motion (solid: joint 1, dotted: joint 2). • learning rate in the weight-tuning law: and ; • simulation time: 20 s. The results in Figs. 5 and 6 clearly show the ability of the CMAC neural-network controller to overcome uncertainties, both struc￾tured and unstructured. Note that the problem noted in Fig. 4 with OCTC does not arise here, as all the nonlinearities are as￾sumed unknown to the CMAC neural controller. VI. CONCLUSION We have developed a hierarchical, intelligent control scheme for a robotic manipulator using the HJB optimization process and the CMAC neural network. It has been shown that the entire closed-loop system behavior depends on the user-specified per￾formance index and , through the critic-gain matrix . The Lyapunov function for the stability of the overall system is au￾tomatically generated by the weighting matrices. In the deriva￾tion of the optimal-computed torque controller, it has been as￾sumed that nonlinearities in the robotic manipulator are com￾pletely known. However, even with the knowledge of nonlinear￾ities, it is difficult to achieve the control objective in the pres-