Approximate continuous-time optimal control in obstacle avoidance by time/space discretization of non-convex state constraints

This paper addresses an approximate version of the optimal control problem with non-convex state constraints via discretization of time and space, where the specific application pursued is the obstacle avoidance problem. First, it is pointed out that the standard continuous-time cost function with the final state fixed is not suitable to the optimal control problem under the non-convex state constraints, and then a new cost function including intermediate target states is proposed. Next, for the optimal control problem with this cost function, where the non-convex state constraints are discretized with respect to time axis and state space, an optimal continuous-time control is given in an explicit form, including the intermediate target states obtained by solving a discrete optimization problem. Efficient algorithms such as the breadth first search algorithm can be applied to this discrete problem. Thus the continuous-time trajectory as well as the discretized state at each discrete time is simultaneously optimized. Finally, we illustrate the effectiveness of the proposed approach with numerical simulations