Fast Ergodic Search with Kernel Functions

TL;DR

This paper presents a novel kernel-based ergodic search method that generalizes to Lie groups, offering linear complexity and significantly faster performance for exploration tasks in high-dimensional spaces, demonstrated on a peg-in-hole task.

Contribution

We introduce a kernel-based ergodic metric applicable to Lie groups, with proven consistency and linear complexity, and derive optimality conditions for nonlinear systems enabling efficient trajectory optimization.

Findings

At least 100x faster than existing algorithms.

Successfully applied to a peg-in-hole insertion task.

Achieved 100% success rate in the demonstration.

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.