Flow Matching Ergodic Coverage

2025-04-24

AI Summary

Existing ergodic coverage methods are constrained by conventional ergodicity metrics, struggling to simultaneously ensure robustness and coverage accuracy. This paper introduces the first flow-matching-based ergodic control framework, adapting efficient sampling techniques from generative modeling to embodied agent exploration and coverage tasks. Methodologically, we derive a closed-form control policy equivalent to the Linear Quadratic Regulator (LQR). Furthermore, we propose two novel ergodicity metrics: Stein Variational Gradient Flow—enhancing robustness to unnormalized target distributions—and Sinkhorn Divergence Flow—improving coverage accuracy for nonsmooth distributions. We validate our approach across multiple nonlinear dynamical systems and on real-world Franka robot drawing/erasing tasks. Results demonstrate significant improvements in both coverage performance and robustness, without increasing computational overhead.

Technology Category

• Intelligent Robots: State Estimation
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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.

Problem

• Expands ergodic metrics for better control synthesis
• Uses flow matching to enhance coverage performance
• Improves robustness with unnormalized target distributions

Innovation

• Flow matching for ergodic coverage control
• Linear quadratic regulator with closed-form solution
• Stein and Sinkhorn divergence flow metrics