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.

翻译：在配置空间中使用基于采样的运动规划器（如PRM）时，难以确定需要多少样本才能保证PRM持续找到解。这在任务与运动规划（TAMP）中尤为重要，因为需要连续解决大量运动规划问题。通过借鉴确定性采样与样本复杂度理论的现有成果，我们尝试证明一个样本数量上界，该上界能以高概率保证找到解。我们还提出了一种数值算法，基于推导边界时所应用的样本复杂度定理证明，计算更紧致的样本数量。实验表明，在平面规划问题上，我们的数值边界算法紧致度在两个数量级内，但随着问题维度增加，紧致度会下降。当作为启发式方法部署于TAMP规划器中进行样本调度时，我们在平面问题上也观察到规划时间的改善。尽管实验表明边界紧致化仍需大量工作，本文提出的思想向实用样本边界迈出了重要一步。