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Advances in Robot Manipulators Part 7

Tham khảo tài liệu 'advances in robot manipulators part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 232 Advances in Robot Manipulators Therefore it is concluded that the designed neurocontroller provides a good tracking of desired trajectories. Fig. 3. Response of the Adaptive Controller with Gradient-Type tuning. Actual and desired joint angles. Fig. 4. Response of the controller with gradient-type parameter tuning. Representation of tracking errors Design of Adaptive Controllers based on Christoffel Symbols of First Kind 233 6. Appendix Lemma 6 For K diag Kii e Rnxn and d d1 d2 . dn T e Rn if u Ksgn x and kii dj then xTM u d 0 Ge et al. 1998 . Lemma 7 Let V x t be a Lyapunov function so that V x t 0 V x t 0 . If V x t is uniformly continuous Lewis et al. 2003 then V x t 0 as t 66 The following theorem is very important in control of non-linear systems and is due to Desoer and Vidyasagar cf. Desoer Vidyasagar 2008 Theorem 2 Let the closed-loop transfer function H s e Rnxn s be exponentially stable and strictly proper and h t the corresponding impulse response obtained by evaluating the inverse Laplace transform of H s . If u e cn then y h u e cn Fl c y e c y is continuous and y t 0 as t where h u denotes the convolution product of h and u . On the basis of this theorem it is possible to state the following lemma Ge et al. 1998 . Lemma 8 Let e t h t r t where h c1 H s and H s is an nxn strictly proper exponentially stable transfer function. Then r e c e e c F C e e c e is continuous and e t 0 as t . If in addition r 0 as t then e 0 . Ge et al. 1998 . Theorem 3 UUB by Lyapunov Analysis If for system x f x t g t 67 there exists a function V x t with continuous partial derivatives such that for x in a compact set S c Rn V x t is positive definite V x t 0 V x t 0 for x R for some R 0 such that the ball of radius R is contained in S then the system is UUB and the norm of the state is bounded to within a neighborhood of R . The following theorem is a modified version of the uniformly ultimately boundedness theorem of Corless and Leitmann cf. Corless Leitmann 1981 . For