We present hierarchical rejection sampling (HRS) to improve the efficiency of asymptotically optimal sampling-based planners for high-dimensional problems with differential constraints. Pruning nodes and rejecting samples that cannot improve the currently best solution have been shown to improve performance for certain problems. We show that in high-dimensional domains this improvement can be so large that rejecting samples becomes the bottleneck of the algorithm because almost all samples are rejected. This contradicts general wisdom that collision checking is always the bottleneck of sampling-based planners. Only samples in the informed subset of the state space can potentially improve the current solution. For systems without differential constraints the informed subset forms an ellipsoid, which can be parameterized and sampled directly. For systems with differential constraints the informed subset is more complicated and no such direct sampling methods exist. HRS improves the efficiency of finding samples within the informed subset without parameterizing it explicitly. Thus, it can also be applied to systems with differential constraints for which a steering method is available. In our experiments we demonstrate efficiency improvements of an RRT* planner of up to two orders of magnitude.
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