Flow Matching Ergodic Coverage

TL;DR

This paper addresses the limitation of fixed ergodic metrics in ergodic coverage by introducing flow matching, a flow-based optimization framework that recasts ergodic coverage as an LQR problem with a closed-form solution. By defining reference flows from Fourier, Stein variational gradient, and Sinkhorn divergence, the method enables robust, scalable control synthesis over unnormalized and irregular target distributions without extra computational overhead. The approach yields improved performance on benchmarks across diverse dynamics and is validated through hardware demonstrations on a Franka robot, highlighting practical applicability for distribution-aware exploration tasks. This work bridges flow-based inference with embodied motion planning, offering a flexible toolkit for distribution tracking in robotics and related domains, with open-source code and demonstrations.

Abstract

Ergodic coverage effectively generates exploratory behaviors for embodied agents by aligning the spatial distribution of the agent's trajectory with a target distribution, where the difference between these two distributions is measured by the ergodic metric. However, existing ergodic coverage methods are constrained by the limited set of ergodic metrics available for control synthesis, fundamentally limiting their performance. In this work, we propose an alternative approach to ergodic coverage based on flow matching, a technique widely used in generative inference for efficient and scalable sampling. We formally derive the flow matching problem for ergodic coverage and show that it is equivalent to a linear quadratic regulator problem with a closed-form solution. Our formulation enables alternative ergodic metrics from generative inference that overcome the limitations of existing ones. These metrics were previously infeasible for control synthesis but can now be supported with no computational overhead. Specifically, flow matching with the Stein variational gradient flow enables control synthesis directly over the score function of the target distribution, improving robustness to the unnormalized distributions; on the other hand, flow matching with the Sinkhorn divergence flow enables an optimal transport-based ergodic metric, improving coverage performance on non-smooth distributions with irregular supports. We validate the improved performance and competitive computational efficiency of our method through comprehensive numerical benchmarks and across different nonlinear dynamics. We further demonstrate the practicality of our method through a series of drawing and erasing tasks on a Franka robot.

Figures (15)

• Figure 1: Similarity between flow-based generative inference and flow matching ergodic coverage.
• Figure 2: The bandwidth of the RBF kernel in Stein variational gradient flow affects the resulting ergodic trajectories. Overly small bandwidth could cause the mode collapse issue while overly large bandwidth could also lead to under-exploration.
• Figure 3: The entropy regularization weight $\epsilon$ in Sinkhorn divergence (\ref{['eq:sinkhorn_divergence']}) affects the resulting ergodic trajectories. Smaller values make the divergence closer to the Wasserstein distance, improving coverage performance at the cost of slower computation feydy_interpolating_2019.
• Figure 4: Results for Benchmark Q1. Quantitative (left) and qualitative (right) results for the comparison between our method with existing trajectory optimization methods for the Fourier ergodic metric. Our method consistently reaches the same desired level of ergodicity with less time. The white line in the violin plot is the median of the results, and the black dot in the trajectory plot is the initial position.
• Figure 5: Benchmark Q2.A. Quantitative (left) and qualitative (right) results show that our method is more accurate and consistent under normalization errors compared to SOTA methods. The white line in the violin plot is the median of the results, and the black dot in the trajectory plot is the initial position.

Theorems & Definitions (13)

• Definition 1: Trajectory empirical distribution
• Definition 2: Ergodic system petersen_ergodic_1989walters_introduction_2000
• Lemma 1
• Definition 3: Discrepancy measure
• Definition 4: Ergodic metric
• Lemma 2
• Definition 5: Ergodic coverage
• Definition 6: Flow-based sampling
• Definition 7: Flow matching
• Definition 8: Flow-based ergodic coverage