Time-optimal control with direct collocation and variable discretization

Abstract: This paper deals with time-optimal control of nonlinear continuous-time systems based on direct collocation. The underlying discretization grid is variable in time, as the time intervals are subject to optimization. This technique differs from approaches that are usually based on a time transformation. Hermite-Simpson collocation is selected as common representative in the field of optimal control and trajectory optimization. Hereby, quadratic splines approximate the system dynamics. Several splines of different order are suitable for the control parameterization. A comparative analysis reveals that increasing the degrees of freedom in control, e.g. quadratic splines, is not suitable for time-optimal control problems due to constraint violation and inherent oscillations. However, choosing constant or linear control splines points out to be very effective. A major advantage is that the implicit solution of the system dynamics is suited for stiff systems and often requires smaller grid sizes in practice.