On the Probabilistic Foundations of Probabilistic Roadmaps

Probabilistic roadmap (PRM) planners [5, 16] solve apparently difficult motion planning problems where the robot's configuration space C has dimensionality six or more, and the geometry of the robot and the obstacles is described by hundreds of thousands of triangles. While an algebraic planner would be overwhelmed by the high cost of computing an exact representation of the free space F , defined as the collision-free subset of C, a PRM planner builds only an extremely simplified representation of F , called a probabilistic roadmap. This roadmap is a graph, whose nodes are configurations sampled from F with a suitable probability measure and whose edges are simple collision-free paths, e.g., straight-line segments, between the sampled configurations. PRM planners work surprisingly well in practice, but why? Previous work has partially addressed this question by identifying and formalizing properties of F that guarantee good performance for a PRM planner using the uniform sampling measure (e. Several systematic experimental studies have also compared various PRM planners, in terms of their sampling and connection strategies (e.g., [7, 8, 21]). However, the underlying question " Why are PRM planners probabilistic? " has received little attention so far, and consequently the importance of probabilistic sampling measures for PRM planning remains poorly understood. Since no inherent