Universal continuous laws for controlling the manipulation robot
This paper defines continuous manipulator's joint trajectory tracking algorithms that use bounded controls. A standard Lagrangian model of the manipulator's dynamics has the form \[ A(q)\ddot q+ B(q,\dot q)= u,\tag{\(*\)} \] where \(q,\dot q,\ddot q,u\in \mathbb{R}^n\), and \(B(q,\dot q)\) may contain a friction term depending smoothly on joint positions and/or velocities. The controls are assumed to be bounded, \(| u_i|\leq K_i\), \(i= 1,2,\dots,n\). Given a desirable joint trajectory \(q^*(t)\), a control \(u\) is designed enabling a trajectory \(q(t)\) of system \((*)\) to track \(q^*(t)\) asymptotically. Tracking algorithms introduced in this paper have the form \[ u_i(t)= u^*_i(t)- \alpha_i L_i(s_i),\tag{\(**\)} \] where \(\alpha_i\) -- a constant, \(i=1,2,\dots, n\), and include a component \(u^*_i(t)\) corresponding to the desirable trajectory, and a saturation function \(L_i(s_i)\) that depends on the composed tracking error \(s_i= \dot\xi_i- \lambda_i\xi_i\), \(\xi_i= q_i- q^*_i\), and is defined as \[ L_i(s_i)= \begin{cases} \gamma_i s_i\quad & \text{for }\|\gamma_i s_i\|\leq 1,\\ \text{sgn}(s_i)\quad & \text{for }\|\gamma_i s_i\|\geq 1.\end{cases} \] The main result of the paper shows that the control algorithm \((**)\) makes the tracking error system exponentially stable, semiglobally with respect to the design parameters \(\lambda_i\), \(\gamma_i\). A proof of stability presented in the paper is based on a vector Lyapunov function. The most interesting result establishing in this paper says that \(L_i(s_i)\) can be replaced by a suitably growing continuous or smooth bounded function \(C_i(s_i)\).
manipulator's joint trajectory tracking algorithms
scientific article; zbMATH DE number 1294877