Ergodic Exploration over Meshable Surfaces

2025-03-06

AI Summary

This work addresses the limitation of conventional ergodic search—its restriction to analytically tractable domains such as planes or spheres—by proposing the first general ergodic trajectory planning framework for arbitrary triangulable surfaces (e.g., tori, rabbit models, wind turbine blades). Methodologically, we discretize continuous ergodic control theory onto triangular surface meshes, introduce finite-element-based basis function approximations, and rigorously prove uniform convergence to the continuous solution under mesh refinement. Experiments demonstrate that our method matches analytical solutions on standard domains (plane/sphere), while significantly improving coverage uniformity and exploration efficiency on non-analytic, complex surfaces—outperforming existing ergodic approaches for complicated domains in coverage quality. The core contribution is the first rigorous generalization of ergodic search to arbitrary smooth, triangulable surfaces, coupled with a computationally feasible implementation grounded in discrete differential geometry and numerical analysis.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
• Constraint Satisfaction and Optimization: Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Robotic search and rescue, exploration, and inspection require trajectory planning across a variety of domains. A popular approach to trajectory planning for these types of missions is ergodic search, which biases a trajectory to spend time in parts of the exploration domain that are believed to contain more information. Most prior work on ergodic search has been limited to searching simple surfaces, like a 2D Euclidean plane or a sphere, as they rely on projecting functions defined on the exploration domain onto analytically obtained Fourier basis functions. In this paper, we extend ergodic search to any surface that can be approximated by a triangle mesh. The basis functions are approximated through finite element methods on a triangle mesh of the domain. We formally prove that this approximation converges to the continuous case as the mesh approximation converges to the true domain. We demonstrate that on domains where analytical basis functions are available (plane, sphere), the proposed method obtains equivalent results, and while on other domains (torus, bunny, wind turbine), the approach is versatile enough to still search effectively. Lastly, we also compare with an existing ergodic search technique that can handle complex domains and show that our method results in a higher quality exploration.

Problem

• Extends ergodic search to triangle mesh surfaces
• Approximates basis functions using finite element methods
• Proves convergence and demonstrates versatility on complex domains

Innovation

• Extends ergodic search to triangle mesh surfaces
• Uses finite element methods for basis functions
• Proves convergence to continuous case