KIM AND LEWIS: OPTIMAL DESIGN OF NEURAL-NETWORK CONTROLLER which is guaranteed negative as long as either(54)or(55) hold F+J(=02+1((58 ≥(∈M+1H成/{m(q)+hm(分}≡B2 (54) where sqn(a) is a signum function The weighting matrices are as follows: WP≥V∈M+n+是WM≡B (55) Q1 where B: and Bw are convergence regions. According to a Q1 standard Lyapunov theory extension[11], this demonstrates uni- formly ultimate boundedness ofet),r(t), and w(t) Q21=Q12 Remarks Q21=Q 1)The OCTC is globally asymptotically stable if h(ar)is fully known, whereas the neural-adaptive controller is Q2=1_「300 UUB. In both cases, there is a convergence of trackinger- rors. UUB is a notion of stability in the practical sense that Solving the matrices K and A using MatLab [15] yields is usually sufficient for the performance of closed-loop systems, provided that the bound on system states is small K=-3 A 6 1897919404(60 2)Robotic manipulators are subjected to structured and/or The motion problem considered is for the robot end-effector unstructured uncertainties in all applications. Structured to track a point on a circle centered at x= y=0.05 m and incertainty is defined as the case of a correct dynamical radius 0.05 m, which turns 1/2 times per second in slow motion model but with parameter uncertainty due to tolerance and two times per second in fast motion. It was pointed out that variations in the manipulator-link properties, unknown control-system performance may be quite different in low-speed loads, and so on Unstructured uncertainty describes the and high-speed motion. Therefore, we carry out our simulation case of unmodeled dynamics that result from the presence for two circular trajectories of high-frequency modes in the manipulator, nonlinear The desired positions in low speed are friction. The adaptive optimizing feature of the proposed neural controller is suitable even without full knowledge x(t)=1+0.05sin(t) of the system dynamics (t)=10+0.05co(mt) (61) 3)From Barron results [2], there exist lower bounds of order (1/NA)2/n on the approximation error EM if only the and the high-speed positions profiles are parameters of a linear combination of basis functions are adjusted. Our stability proof shows that the effect of the xd(t)=1+0.05sin(47 bounds on the approximation error can be alleviated by (t)=1+0.050(4mt) the judicious choice of weighting matrices Q and R 4)It is emphasized that the neural-weight values may be By solving the inverse kinematics, we obtain the desired joint- initialized at zero, and stability will be maintained by the angle trajectory in fast motion optimal controller(t) in the performance-measurement The responses of the OCTC, where all nonlinearities are loop until the neural network learns. This means that there exactly known, are shown in Fig.3 without disturbances and is no off-line learning or trial and error phase, which often friction. The simulation was performed in low speed and high requires a long time in other works speed. After a transient due to error in initial conditions, the 5)The advantage of the CMAc control scheme over other position errors tend asymptotically toward zero existing neural-network architectures is that the number To show the effect of unstructured uncertainties, we dropped of adjustable parameters(i.e, weight values)is signifi- a term(m1 +m2)glici in gravity forces. The simulation re antly less, since only weights in the output layer are to sults are shown in Fig. 4(a) in low speed. Note that there is be adjusted. It is very suitable for closed-loop control. a steady-state error with OCT b)shows the effect of external disturbances and friction forces which is difficult to V. SIMULATION RESULTS model and compensate. This is corrected by adding a CMAC neural network as follows The dynamic equations for an 7-link manipulator can be The CMAC can be characterized by found in [9]. The cost functional to be minimized is number of input spaces:c=[g i J()=2/(2 Qz+u Ru)dt number of partitions for each space: Ni= 3, i · number of association points:NA=3×3×3×3 An external disturbance and frictions are receptive field-basis functions: pi, i (i)= exp[(, T()=[8sin(2)8cx(切) (57) mn,2=0, 1,t=1,KIM AND LEWIS: OPTIMAL DESIGN OF NEURAL-NETWORK CONTROLLER 27 which is guaranteed negative as long as either (54) or (55) holds (54) (55) where and are convergence regions. According to a standard Lyapunov theory extension [11], this demonstrates uni￾formly ultimate boundedness of , , and . Remarks: 1) The OCTC is globally asymptotically stable if is fully known, whereas the neural-adaptive controller is UUB. In both cases, there is a convergence of tracking er￾rors. UUB is a notion of stability in the practical sense that is usually sufficient for the performance of closed-loop systems, provided that the bound on system states is small enough. 2) Robotic manipulators are subjected to structured and/or unstructured uncertainties in all applications. Structured uncertainty is defined as the case of a correct dynamical model but with parameter uncertainty due to tolerance variations in the manipulator-link properties, unknown loads, and so on. Unstructured uncertainty describes the case of unmodeled dynamics that result from the presence of high-frequency modes in the manipulator, nonlinear friction. The adaptive optimizing feature of the proposed neural controller is suitable even without full knowledge of the system dynamics. 3) From Barron results [2], there exist lower bounds of order on the approximation error if only the parameters of a linear combination of basis functions are adjusted. Our stability proof shows that the effect of the bounds on the approximation error can be alleviated by the judicious choice of weighting matrices and . 4) It is emphasized that the neural-weight values may be initialized at zero, and stability will be maintained by the optimal controller in the performance-measurement loop until the neural network learns. This means that there is no off-line learning or trial and error phase, which often requires a long time in other works. 5) The advantage of the CMAC control scheme over other existing neural-network architectures is that the number of adjustable parameters (i.e., weight values) is signifi￾cantly less, since only weights in the output layer are to be adjusted. It is very suitable for closed-loop control. V. SIMULATION RESULTS The dynamic equations for an -link manipulator can be found in [9]. The cost functional to be minimized is (56) An external disturbance and frictions are (57) (58) where sqn is a signum function. The weighting matrices are as follows: (59) Solving the matrices and using MatLab [15] yields (60) The motion problem considered is for the robot end-effector to track a point on a circle centered at 0.05 m and radius 0.05 m, which turns 1/2 times per second in slow motion and two times per second in fast motion. It was pointed out that control-system performance may be quite different in low-speed and high-speed motion. Therefore, we carry out our simulation for two circular trajectories. The desired positions in low speed are (61) and the high-speed positions profiles are (62) By solving the inverse kinematics, we obtain the desired joint￾angle trajectory in fast motion. The responses of the OCTC, where all nonlinearities are exactly known, are shown in Fig. 3 without disturbances and friction. The simulation was performed in low speed and high speed. After a transient due to error in initial conditions, the position errors tend asymptotically toward zero. To show the effect of unstructured uncertainties, we dropped a term in gravity forces. The simulation re￾sults are shown in Fig. 4(a) in low speed. Note that there is a steady-state error with OCTC. Fig. 4(b) shows the effect of external disturbances and friction forces, which is difficult to model and compensate. This is corrected by adding a CMAC neural network as follows. The CMAC can be characterized by: • number of input spaces: ; • number of partitions for each space: ; • number of association points: ; • receptive field-basis functions: with , and ;