Ergodic Exploration over Meshable Surfaces

TL;DR

The paper introduces a mesh-based ergodic trajectory optimization framework that extends ergodic search to any surface approximable by a triangle mesh by using Laplace-Beltrami eigenfunctions as the basis functions. It proves convergence of the discrete ergodic objective to the continuous one as the mesh refines and demonstrates comparable performance to analytic bases on planes and spheres while enabling effective exploration on more complex geometries such as torus, bunny, and wind turbines; it also shows superior exploration quality compared to HEDAC under equivalent conditions. The approach relies on finite element-style discretization on meshes, computation of eigenpairs of the mesh Laplacian, and a direct-collocation trajectory optimization with a Gaussian information model. The results indicate strong generality across surface geometries and topology, offering a practical tool for information-driven exploration in search, rescue, and inspection tasks with complex surfaces. The work lays groundwork for future physical deployments and multi-robot extensions, addressing computational trade-offs and sensor-model considerations in dynamic environments.

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.

Figures (6)

• Figure 1: Ergodic Search over a Complex Meshable Surface The proposed approach plans dynamically feasible trajectories that search with respect to an information map over any surface that a triangle mesh can approximate. Shown is an inspection trajectory for the Stanford bunny with a uniform information map.
• Figure 2: Simple $\mathbb{R}^2$ Surface Search Comparison Searching over 2D Euclidean space and comparing the (a) normal ergodic search with analytical solutions to the basis functions (ergodicity of $0.00254$) to (b) our mesh approximation and numerical methods to compute the basis functions (ergodicity of $0.00304$ with respect to the analytical basis functions)
• Figure 3: Sphere Surface Search Comparison Classical ergodic objective vs. our mesh-based ergodic objective for a sphere. The trajectory is darker/thicker to indicate lower velocity and lighter/thinner to indicate higher velocity. The information gathered is visualized as a heat map with red indicating more information. The collected information is comparable with (a) the classical ETO ergodicity of $0.0718$ and (b) our proposed mesh-based ETO ergodicity of $0.0762$ with respect to the analytical basis functions.
• Figure 4: HEDAC Comparison Comparing our mesh-based ergodic search (planned for only 100 timesteps, dotted) to the HEDAC approach ivic2023multi (with 0 to 1000 timesteps, solid) for the wind turbine mesh. Blue is ergodicity achieved; Red is the walltime used to compute the trajectories.
• Figure 5: Wind Turbine Ergodic Search Trajectory (a) our mesh-based ergodic search and (b) the HEDAC approach ivic2023multi on a wind turbine mesh for 100 timesteps. Our approach achieved an ergodicity of $1.17 \times 10^{-6}$ and HEDAC achieved an ergodicity of $5.43 \times 10^{-5}$. HEDAC leaves two of the blades unexamined.

Theorems & Definitions (8)

• Lemma 1
• proof
• Corollary 1
• proof
• Corollary 2
• proof
• Theorem 1
• proof