Sufficient Conditions for the Existence of Resolution Complete Planning Algorithms

This paper addresses theoretical foundations of motion planning with differential constraints in the presence of obstacles. We establish general conditions for the existence of resolution complete planning algorithms by introducing a functional analysis framework and reducing algorithm existence to a simple topological property. First, we establish metric spaces over the control function space and the trajectory space. Second, using these metrics and assuming that the control system is Lipschitz continuous, we show that the mapping between open-loop controls and corresponding trajectories is continuous. Next, we prove that the set of all paths connecting the initial state to the goal set is open. Therefore, the set of open-loop controls, corresponding to solution trajectories, must be open. This leads to a simple algorithm that searches for a solution by sampling a control space directly, without building a reachability graph. A dense sample set is given by a discrete-time model. Convergence of the algorithm is proven in the metric of a trajectory space. The results provide some insights into the design of more effective planning algorithms and motion primitives.