Computation of time-optimal controls applied to rigid manipulators with friction

Abstract

Abstract We present a numerical procedure to compute non-singular, time-optimal solutions for non-linear systems that are linear in the control and have fixed initial and final states and bounded control. Part of our procedure is a new numerically computable test that determines whether bang-bang solutions satisfy Pontryagin's minimum principle. This test reveals the new important fact that, for non-linear systems, with linear control and dimension n, the probability that a bang-bang solution with more than n - 1 switches satisfies Pontryagin's Mini mum Principle is almost zero. Using a parameter optimization procedure we search for bang-bang solutions with up to n - 1 switches which transfer the system from the initial to the final state. If no controls with up to n - 1 switches can be found to satisfy Pontryagin's Minimum Principle the problem is very likely singular. We apply our procedure to the time-optimal control problem for rigid manipulators where friction may be included in the dynamics. We will demon strate that some solutions mentioned in the literature to satisfy Pontryagin's minimum principle do not. A class of time-optimal control problems turns out to be singular. To solve these problems we propose and demonstrate the method of control parametrization.