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Control of Robot Manipulators in Joint Space - R. Kelly, V. Santibanez and A. Loria Part 9

Tham khảo tài liệu 'control of robot manipulators in joint space - r. kelly, v. santibanez and a. loria part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 228 10 Computed-torque Control and Computed-torque-p Control and moreover with a trivial selection of its design parameters. It receives the name computed-torque control. The computed-torque control law is given by T M q qd Kvq Kpq C q q q g q 10.1 where Kv and Kp are symmetric positive definite design matrices and q qd q denotes as usual the position error. Notice that the control law 10.1 contains the terms Kpq Kvq which are of the PD type. However these terms are actually premultiplied by the inertia matrix M qd q . Therefore this is not a linear controller as the PD ince the position and velocitygainsarenot eonodantbut lhey depend explicitly on the position error q. This may be clearly seen when expressing the computed-torque control law given by 10.1 as V.U V .i uv q M qd-nKvq Mgq qd C q q q g q . Computed-torquecontrol was dne ofthqfirst model-based motion control approaches created for manipulators that is in which one makes explicit use of the knowledge of the matrices M q . s q.q cm of the vector g q . Furthermore observe that the desired trajectory of motion qd t and its neoivot iars odt a dll d nd s aswedl delin qoritionandvelocitdmrasurom onte oft and-tut-a-e oool tooomuutetho vontrol action 10.1 . TUt biock-diao-am tVat correspondrtodompoteU-Varqua cont lai of robot monipulators is pdtqeneenin Figure I0VI. Q e pi iv dqr q q Figure 10.1. Block-diagram computed-torque control The equation la obtained by aubetitntmg the eontrol action T from 10.1 in the equation of the robot model III.l to obtain M q q M q qd Kvq Rộ . M.2 10.1 Computed-torque Control 229 Since M g is a positive definite matrix Property 4.1 and therefore it is also invertible Equation 10.2 reduces to q Kvq Kpq 0 which in turn may be expressed in terms of the state vector qT zT T q as 10.3 where I is the kleniuy matrix of dimension n. It is importantto remanothoe the closed-loop Equation 10.3 is represented by a linear autonomous fferentifo equation whose unique equilibrium r T- T o point i