Informed Sampling for Asymptotically Optimal Path Planning (Consolidated Version)

Anytime almost-surely asymptotically optimal planners, such as RRT*, incrementally find paths to every state in the search domain. This is inefficient once an initial solution is found, as then only states that can provide a <italic> better</italic> solution need to be considered. Exact knowledge of these states requires solving the problem but can be approximated with heuristics. This paper formally defines these sets of states and demonstrates how they can be used to analyze arbitrary planning problems. It uses the well-known <inline-formula><tex-math notation="LaTeX">$L^2$ </tex-math></inline-formula> norm (i.e., Euclidean distance) to analyze minimum-path-length problems and shows that existing approaches decrease in effectiveness <italic>factorially</italic> (i.e., faster than exponentially) with state dimension. It presents a method to address this curse of dimensionality by <italic>directly</italic> sampling the prolate hyperspheroids (i.e., symmetric <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -dimensional ellipses) that define the <inline-formula><tex-math notation="LaTeX">$L^2$</tex-math></inline-formula> <italic>informed</italic> set. The importance of this direct informed sampling technique is demonstrated with Informed RRT*. This extension of RRT* has less theoretical dependence on state dimension and problem size than existing techniques and allows for <italic>linear</italic> convergence on some problems. It is shown experimentally to find better solutions faster than existing techniques on both abstract planning problems and HERB, a two-arm manipulation robot.