Existence of the Time Optimal Control for Robotic Manipulators

The minimum time control problem with bounded control has long been of interest to control engineers. There has been some attention on studying this problem in the context of the robotic equation of motion. The exact solution involves solving a highly complex two-point boundary value problem. Before any numerical algorithms are devised, the question of existence of the optimal solution needs to be answered first. In linear systems, well known results exist that provide not only the existence condition but the uniqueness condition and the form of the solution as well. In nonlinear systems, more conditions are required for the existence of the optimal solution. In this paper, we will apply these conditions to the robotic equation of motion and establish a single existence condition. It is found that this condition hinges on the size of the coriolis and the centrifugal terms. If these terms are zero, the domain of existence of time optimal solution can be made arbitrarily large by using suitable bounds on the control.