An Optimal Control Approach to the Minimum-Time Trajectory Planning of Robotic Manipulators

Abstract

Trajectory planning is a classic problem in robotics, with different approaches and optimisation objectives documented in the literature. Most of the time, the path is assumed, i.e., pre-defined, and optimisation consists of finding the timing of motion under a number of constraints. The focus of this work is on the minimum-time manoeuvring of robotic manipulators. A nonlinear optimal control approach is proposed that does not require the provision of either a pre-defined path or a pre-defined control structure and allows the inclusion of dynamic constraints. The solution (path and timing of motion) is obtained by transforming the optimal control problem into a nonlinear programming problem. The proposed approach is applied to a two-link manipulator for illustration purposes. The optimisation is carried out both without and with obstacles. The minimum-distance and minimum-time solutions are compared, and some classic results are obtained, including the trapezoidal pattern of the joint velocity and the bang–bang structure of the control torques. The effects of limitations on the jerks of actuators and the rate of change in torque inputs are discussed. The application to a four-link manipulator is also included to show the ‘scalability’ of the approach, together with a comparison with a classic path-and-motion-planning method, to highlight the characteristics and performance of the proposed approach. Finally, the possibility of enforcing a number of via-points along the path is demonstrated. The proposed method allows the computation of the path and motion simultaneously with the computation time, which is 1–30 times the manoeuvre time, on a standard PC with the current implementation.