  When using sampling-based motion planners, such as PRMs, in configuration
spaces, it is difficult to determine how many samples are required for the PRM
to find a solution consistently. This is relevant in Task and Motion Planning
(TAMP), where many motion planning problems must be solved in sequence. We
attempt to solve this problem by proving an upper bound on the number of
samples that are sufficient, with high probability, to find a solution by
drawing on prior work in deterministic sampling and sample complexity theory.
We also introduce a numerical algorithm to compute a tighter number of samples
based on the proof of the sample complexity theorem we apply to derive our
bound. Our experiments show that our numerical bounding algorithm is tight
within two orders of magnitude on planar planning problems and becomes looser
as the problem's dimensionality increases. When deployed as a heuristic to
schedule samples in a TAMP planner, we also observe planning time improvements
in planar problems. While our experiments show much work remains to tighten our
bounds, the ideas presented in this paper are a step towards a practical sample
bound.
