Effective Sampling for Robot Motion Planning Through the Lens of Lattices

2025-02-07

AI Summary

Existing sampling-based motion planning methods, though probabilistically complete and asymptotically optimal, lack deterministic performance guarantees under finite sample budgets—limiting their practical deployment. To address this, we propose a novel deterministic sampling framework grounded in *Ad*-lattices, introducing *Ad*-lattice covering theory to robotics motion planning for the first time. Our approach establishes a strong, finite-time guarantee mechanism formalized via (δ, ε)-completeness, eliminating reliance on asymptotic convergence. By integrating lattice theory, deterministic geometric covering, and configuration-space modeling, it unifies theoretical soundness with computational efficiency. Experiments in complex environments demonstrate that our method achieves over 10× speedup in planning time compared to state-of-the-art deterministic and uniform random sampling baselines, while significantly improving success rate and real-time responsiveness.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Deterministic Planning
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Enhancing finite-time guarantees in motion planning
• Deterministic sampling for improved efficiency
• Applying lattice theory for optimal sample sets

Innovation

• Deterministic sampling with lattice theory
• Finite-time guarantees for motion planning
• Efficient sampling using A_d^* lattice