Towards Practical Finite Sample Bounds for Motion Planning in TAMP

TL;DR

This work tackles the problem of determining finite, practically usable sample counts for sampling-based motion planners within Task and Motion Planning (TAMP). It connects radius PRM theory to sample-complexity notions, leveraging α-net covering arguments and Blumer's bounds, and introduces a numerical bound algorithm to produce tighter, less conservative sample counts than conventional worst-case estimates. The authors extend the analysis toward KNN PRMs by studying the decay of the effective connection radius with increasing samples, and they present an adaptive, sampling-bound strategy (aPRM) to guide PRM usage inside a TAMP solver. Empirical results show the numerical bound is tight in planar settings (within a few orders of magnitude) but loosens with higher dimensionality; as a heuristic for KNN PRMs in planar problems, it improves planning time and guides sampling in a principled way, marking a meaningful step toward practical finite-sample bounds in TAMP. The work lays a foundation for tighter, more realistic bounds and suggests directions for extending the theory to more complex planners and problem classes.

Abstract

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.

Figures (5)

• Figure 1: Left: An $\alpha$-net, where the points are interpreted as centers of balls, with radius $\alpha$, that collectively cover the rectangle. Middle: An example of a $\delta$-clear path. Right: A depiction of how a covering as dictated by Lemma \ref{['lem:coverings_find_passages']} yields a roadmap that finds paths in narrow passages.
• Figure 2: Left: A comparison of the sufficient number of samples required to find an $0.5$-net in a unit cube with probability $1 - \gamma$ as computed by Lemma \ref{['lem:blumer_e_net_bound']} and numerical computation done by Algorithm \ref{['alg:numerical_bound']}. The right-hand side in the maximization in Lemma \ref{['lem:blumer_e_net_bound']} dominates for nearly all $\gamma$ plotted, since the dotted lines are nearly horizontal. Right: A depiction of the search conducted by the numerical algorithm.
• Figure 3: Left: A depiction of the effective connection radius for a PRM graph with $K=1$. Note that we have a conservative approximation that fails to capture 'long distance edges' between isolated vertices. Right: A plot of the ratio of the connection radius to the net radius as a function of the number of sampled vertices with a fixed failure probability of $\gamma = 0.1$ and $K=256$, as determined by their numerical procedures. The conditions for Theorem \ref{['thm:rad_prm_bound']} are satisfied for all sample counts with ratios above 4.0 (dotted red line). All curves are below the line, so the conditions of Theorem \ref{['thm:knn_prm_bound']} are not satisfied for any realistic $n$.
• Figure 4: The largest connection radius computed by the numerical bound by setting the probability of failure $\gamma = 0.01$ is represented by corresponding dotted vertical bars. Their right-to-left ordering is the same as the ordering of the histograms.
• Figure :

Theorems & Definitions (22)

• definition thmcounterdefinition: Path Clearance
• definition thmcounterdefinition: $\alpha$-net
• lemma thmcounterlemma: $\alpha$-nets find paths in $2\alpha$-clear passages
• lemma thmcounterlemma: Number of random samples to form an $\alpha$-net
• theorem 1.1: Radius PRM Guarantee
• definition thmcounterdefinition: Connection Radius
• lemma thmcounterlemma
• proposition thmcounterproposition
• corollary thmcountercorollary
• theorem 1.2: Vacuous KNN PRM Guarantee