Direct control of the endpoint of the manipulator under non-smooth uncertainty and reference trajectories

Abstract

The problem of tracking the reference trajectory for the manipulator endpoint is considered under the following assumptions: the matrices of the mechanical system have uncertain parameters; external non-smooth disturbances act on the system; and only the vector of generalized coordinates of the manipulator is measured. This vector is uniquely recalculated into the vector of positions of the endpoint. A direct method for the synthesis of a tracking system is proposed, which does not require solving the inverse position problem. The manipulator model is represented in the canonical input-output form considering the vector of the endpoint. The input is a vector of generalized moments developed by the actuators. The complexity of regulation consists of the uncertainty of the input matrix. To stabilize tracking errors under these conditions, a method of linearization by dynamic feedback has been developed. For estimating mixed variables and disturbances, an observer of the minimum possible dynamic order is proposed. A cascade procedure for tuning S-shaped smooth and nonlinear (sigmoid) corrective actions of the observer is presented. This procedure provides an estimation of non-smooth disturbances in a given time with a given accuracy. To smooth primitive trajectories that define, as a first approximation, the desired movement of the endpoint in the workspace, tracking differentiators with sigmoid local feedbacks are used. To tune a tracking differentiator of arbitrary dimension, a decomposition procedure is developed. This procedure considers physical constraints on the velocity and higher derivatives of the manipulator endpoint. The variables of such a differentiator generate smoothed trajectories and their derivatives of any desired order. These trajectories become achievable for the robot. They are used in the tracking system of the control plant as new reference trajectories. The simulation results for a 3-link manipulator demonstrate the excellent performance of the proposed algorithms.