Exponential Convergence of Infeasibility Proofs for Kinematic Motion Planning

Abstract

Proving motion planning infeasibility is an important part of a complete motion planner. Common approaches for high-dimensional motion planning are only probabilistically complete. Previously, we presented an algorithm to construct infeasibility proofs by applying machine learning to sampled configurations from a bidirectional sampling-based planner. In this work, we prove that the learned manifold converges to an infeasibility proof exponentially. Combining prior approaches for sampling-based planning and our converging infeasibility proofs, we propose the term asymptotic completeness to describe the property of returning a plan or infeasibility proof in the limit. We compare the empirical convergence of different sampling strategies to validate our analysis.

This work is supported in part by the National Science Foundation under Grant No. IIS-1849348.

A Appendix: Comparing PRM and RRT-Connect

Below is an example comparing PRM and RRT-connect in a 2D configuration space with disconnected $\mathcal {C}_{\textrm{rest}}$ regions. PRM’s samples form two classes, the samples connectable to the goal configuration, and the samples not connectable to the goal configuration. RRT-connect’s samples also form two classes, the start tree samples and the goal tree samples. The example configuration space’s $\mathcal {C}_{\textrm{rest}}$ in Fig. 6 has two disconnected components, the region outside of obstacle region 1, and the region in between obstacle region 1 and 2. PRM samples all $\mathcal {C}_{\textrm{free}}$, so with disconnected $\mathcal {C}_{\textrm{rest}}$, the trained manifold is in $\mathcal {C}_{\textrm{obs}}$ (Fig. 6a). RRT-connect samples are connected to either the start configuration or the goal configuration, which is nor guaranteed to cover the entire $\mathcal {C}_{\textrm{free}}$ if there are disconnected regions, thus the learned manifold may not converge into $\mathcal {C}_{\textrm{obs}}$ (Fig. 6b). In our proof, if $\mathcal {C}_{\textrm{rest}}$ has disconnected regions, then we need to use PRM planner. If $\mathcal {C}_{\textrm{rest}}$ is connected, then PRM and RRT-connect are the same.

Fig. 6.

Comparing PRM and RRT-connect in disconnected $\mathcal {C}_{\textrm{rest}}$ configuration space. Manifold trained from PRM samples are guaranteed to converge, while manifold trained from RRT-connect samples may not converge. For this reason, we switch to use PRM for the proofs in this paper to cover the general case.