Fast Ergodic Search With Kernel Functions

Ergodic search enables optimal exploration of an information distribution with guaranteed asymptotic coverage of the search space. However, current methods typically have exponential computational complexity and are limited to Euclidean space. We introduce a computationally efficient ergodic search method. Our contributions are two-fold as follows: First, we develop a kernel-based ergodic metric, generalizing it from Euclidean space to Lie groups. We prove this metric is consistent with the exact ergodic metric and ensures linear complexity. Second, we derive an iterative optimal control algorithm for trajectory optimization with the kernel metric. Numerical benchmarks show our method is two orders of magnitude faster than the state-of-the-art method. 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-s-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.