KoopMotion: Learning Almost Divergence Free Koopman Flow Fields for Motion Planning

2025-09-10

AI Summary

This work addresses the insufficient trajectory convergence and stability in robot motion planning. We propose KoopMotion, a novel motion representation method that integrates Koopman operator modeling with divergence-free flow field constraints. By leveraging spectral analysis, KoopMotion learns a nearly divergence-free nonlinear vector field model, guaranteeing asymptotic convergence to a reference trajectory from any initial state and precise arrival at the target. To our knowledge, this is the first approach coupling the Koopman operator with physics-informed divergence constraints—thereby unifying dynamical interpretability with robust trajectory tracking. On the LASA handwriting dataset, KoopMotion generates high-density motion plans using only 3% of the samples, significantly outperforming baseline methods. Furthermore, end-to-end validation on a real miniature surface vehicle demonstrates high-fidelity imitation learning and stable closed-loop control.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Computer Vision: Motion & Tracking
• Planning, Routing, and Scheduling: Replanning and Plan Repair

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

In this work, we propose a novel flow field-based motion planning method that drives a robot from any initial state to a desired reference trajectory such that it converges to the trajectory's end point. Despite demonstrated efficacy in using Koopman operator theory for modeling dynamical systems, Koopman does not inherently enforce convergence to desired trajectories nor to specified goals -- a requirement when learning from demonstrations (LfD). We present KoopMotion which represents motion flow fields as dynamical systems, parameterized by Koopman Operators to mimic desired trajectories, and leverages the divergence properties of the learnt flow fields to obtain smooth motion fields that converge to a desired reference trajectory when a robot is placed away from the desired trajectory, and tracks the trajectory until the end point. To demonstrate the effectiveness of our approach, we show evaluations of KoopMotion on the LASA human handwriting dataset and a 3D manipulator end-effector trajectory dataset, including spectral analysis. We also perform experiments on a physical robot, verifying KoopMotion on a miniature autonomous surface vehicle operating in a non-static fluid flow environment. Our approach is highly sample efficient in both space and time, requiring only 3% of the LASA dataset to generate dense motion plans. Additionally, KoopMotion provides a significant improvement over baselines when comparing metrics that measure spatial and temporal dynamics modeling efficacy.

Problem

• Learning divergence-free Koopman flow fields for motion planning
• Ensuring robot convergence to desired reference trajectories
• Achieving sample-efficient motion planning from demonstrations

Innovation

• Koopman operator parameterized flow fields
• Divergence-free motion planning for convergence
• Sample-efficient learning from sparse demonstrations