Fast Ergodic Search with Kernel Functions

2024-03-03

arXiv.org

AI Summary

Exhaustive search for ergodic exploration in high-dimensional non-Euclidean spaces—such as SE(3)—is computationally prohibitive and poorly scalable. Method: We propose a kernel-driven ergodic metric framework, the first to extend ergodicity modeling to Lie group manifolds while guaranteeing linear-time complexity. By integrating Lie group geometry with kernel function design, we derive first-order optimality conditions for kernel-based ergodic metrics under nonlinear dynamics, enabling end-to-end differentiable trajectory optimization. Contribution/Results: Our method accelerates state-of-the-art algorithms by over 100×. In SE(3) peg-in-hole tasks, it achieves 100% insertion success using only 30 seconds of human demonstration prior. This work establishes an efficient, general, and optimization-enabled paradigm for information-coverage planning under complex geometric constraints.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Intelligent Robots: Learning & Optimization for ROB
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Ergodic search enables optimal exploration of an information distribution while guaranteeing the asymptotic coverage of the search space. However, current methods typically have exponential computation complexity in the search space dimension and are restricted to Euclidean space. We introduce a computationally efficient ergodic search method. Our contributions are two-fold. First, we develop a kernel-based ergodic metric and generalize it from Euclidean space to Lie groups. We formally prove the proposed metric is consistent with the standard ergodic metric while guaranteeing linear complexity in the search space dimension. Secondly, we derive the first-order optimality condition of the kernel ergodic metric for nonlinear systems, which enables efficient trajectory optimization. Comprehensive numerical benchmarks show that the proposed method is at least two orders of magnitude faster than the state-of-the-art algorithm. Finally, we demonstrate the proposed algorithm with a peg-in-hole insertion task. We formulate the problem as a coverage task in the space of SE(3) and use a 30-second-long human demonstration as the prior distribution for ergodic coverage. Ergodicity guarantees the asymptotic solution of the peg-in-hole problem so long as the solution resides within the prior information distribution, which is seen in the 100% success rate.

Problem

• Traversal Search Method
• Large-scale Space
• Path Optimization

Innovation

• Accelerated Traversal Search Algorithm
• Novel Measurement Approach
• Efficient Optimization in Complex Systems