Tham khảo tài liệu 'robot manipulators trends and development 2010 part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 112 Robot Manipulators Trends and Development p -rWT q q q lM-1 q BTPe 56 which implies that V -eTQe . Equation 56 gives the adaptive update rule for the parameter estimate vector p because p is equal to zero. Substituting p p - Ộ into 56 gives the adaptive update rule p rwT q q q M-1 q BTPe 57 for the parameter estimate vector Ộ. Detailed stability analysis Craig 1988 shows that the tracking error vector e approaches to zero asymptotically. The adaptive computed-torque controller has some restrictions required for the implementation. That is the controller needs to measure accurate acceleration q and to ensure that M-1 q exists. To avoid these restrictions other adaptive control schemes are introduced in following sections. Adaptive inertia-related control Slotine Li 1987 proposed an adaptive inertia-related control scheme that does not need to measure joint acceleration and ensure inversion of the estimated inertia matrix. Consider the control input T IM q qd Ao C q q qd Ae g q Kvr 58 where the auxiliary signal r is defined as r Ae e with A being an n X n positive-definite diagonal matrix. Using q q d Ae - r q qd Ae - r and property 4 the robot dynamic equation 1 can be rewritten as T Y - p- M q r - C q q r 59 where Y0p M q qd Ae C q q qd Ae g q 60 and Y - is an n X r matrix of known time functions. Equation 60 is the same type of parameter separation that was used in the formulation of the adaptive computed-torque controller. However here Y - is independent of the joint acceleration q . Similar to the formulation 60 we also have M q qd Ae C q ĩ qd Ae g q Y - p. 61 To form the error system substituting the control input 58 into the equation of motion 1 yields Motion Control 113 M q q C q q q g q M q qd Ae C q q qd Ae g q Kvr . 62 Substituting q qd - e and lị lịd - e into 62 and using 60 and 61 the equation 62 can be rewritten as M q r C q q r Kvr Y . p 63 where p p-ip is the parameter error. To show the convergence of the tracking error to zero Slotine Li .
