Towards General Infeasibility Proofs in Motion Planning

We present a general approach for constructing proofs of motion planning infeasibility. Effective high-dimensional motion planners, such as sampling-based methods, are limited to probabilistic completeness, so when no plan exists, these planners either do not terminate or can only run until a timeout. We address this completeness challenge by augmenting a sampling-based planner with a method to create an infeasibility proof in conjunction with building the search tree. An infeasibility proof is a closed polytope that separates the start and goal into disconnected components of the free configuration space. We identify possible facets of the polytope via a nonlinear optimization procedure using sampled points in the non-free configuration space. We identify the set of facets forming the separating polytope via a linear constraint satisfaction problem. This proof construction is valid for general (i.e., non-Cartesian) configuration spaces. We demonstrate this approach on the low-dimensional Jaco manipulator and discuss engineering approaches to scale to higher dimensional spaces.