Effective Sampling for Robot Motion Planning Through the Lens of Lattices

TL;DR

This paper introduces a deterministic sampling method based on lattice theory for robot motion planning, significantly improving finite-time guarantees and computational efficiency over existing methods.

Contribution

It presents a novel deterministic sampling approach using the $A_d^*$ lattice, enhancing finite-time guarantees and speed in motion planning.

Findings

Achieves at least an order-of-magnitude speedup over existing methods.

Provides strong finite-time guarantees for motion planners.

Offers deep mathematical insights into sampling strategies.

Abstract

Sampling-based methods for motion planning, which capture the structure of the robot's free space via (typically random) sampling, have gained popularity due to their scalability, simplicity, and for offering global guarantees, such as probabilistic completeness and asymptotic optimality. Unfortunately, the practicality of those guarantees remains limited as they do not provide insights into the behavior of motion planners for a finite number of samples (i.e., a finite running time). In this work, we harness lattice theory and the concept of $(\delta,\epsilon)$-completeness by Tsao et al. (2020) to construct deterministic sample sets that endow their planners with strong finite-time guarantees while minimizing running time. In particular, we introduce a highly-efficient deterministic sampling approach based on the $A_d^*$ lattice, which is the best-known geometric covering in dimensions $\leq 21$. Using our new sampling approach, we obtain at least an order-of-magnitude speedup over existing deterministic and uniform random sampling methods for complex motion-planning problems. Overall, our work provides deep mathematical insights while advancing the practical applicability of sampling-based motion planning.