tailieunhanh - Control of Robot Manipulators in Joint Space - R. Kelly, V. Santibanez and A. Loria Part 8

Tham khảo tài liệu 'control of robot manipulators in joint space - r. kelly, v. santibanez and a. loria part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 196 8 PD Control with Desired Gravity Compensation Kelly R. 1997 PD control with desired gravity compensation of robotic manipulators A review The International Journal of Robotics Research Vol. 16 No. 5 pp. 660-672. Topics on bifurcation of equilibria may be consulted in Parker T. s. ChuaL. o. 1989 Practical numerical algorithms for chaotic systems Springer-Verlag. Guckenheimer J. Holmes p. 1990 Nonlinear oscillations dynamical systems and bifurcation of vector fields Springer-Verlag. Wiggins s. 1990 Introduction to applied nonlinear dynamical systems and chaos Springer-Verlag. Hale . Kocak w. Kbs Dynamicsand bifurcations Springer-Verlag. Jackson E. A. 1991 Perspectives of nonlinear dynamics Vol. 1 Cambridge Uni versit yPress. studyof the Lyapunovfunction for global asymptotic stability presented in Section is taken from Kelly R. 1993 Comments on Adaptive PD controller for robot manipulators CEEẼ T Ễ ns tiynsr nl obotScs i ndr9 is ma iiii Vol. 9 No. 1 February pp. lir-isr. p robiems 1. CansiCerthe model oirhy idrelounboluin sludied Jq o PfTh the followiap numeriial hiluhs I. mgl qd Tr I. Consider theu ol PD cinhnl with Cesireh grahitychmpshsatíonand suppose ihefoiiowingmitial conditions ạ 0 0 and ậ 0 0. From this we have ặ 0 7t 2. Assume that kp 4 tf. a Obtain an upper-bound on qft 2 Hint Use . 2. Consider the PD control law with desired gravity compensation for the Chilihrm lesthhrd leytReequaUon Jq mgl sm ợ T . Problems 197 The equilibria of the closed-loop equation are q q T s 0 T where s is the solution of kpS mgl sinf J - sin ợd - s 0 . a Show that s satisfies 2mgl b Simulate the system in closed loop with the following numerical values J 1 m I 1 g 10 kp 1 4 kv 1 and with initial conditions 0 tf 8 and j 0 0. For the desired angular position qd 71 2 verify by simulation that IÍIII -x q t Ạ qd- c Obtain by simulation the approximate value of limt .TO i . Verify that Hindoo qd - q t 3. Consider tbemodelof theideal pendulum .