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.
Using KoopMotion, we learn motion policies (black) from demonstrations (red). Using a Koopman operator theoretic approach, we are able to learn a smooth underlying dynamical system that captures the spatial and temporal structure of the demonstrations.
We introduce additional novel loss terms to our optimize over, so that the surrounding vector field (gray) guides the system (a robot) toward the desired demonstration when away from the (green) initial conditions along the (blue) trajectories, and to stop at the demonstration end-point.
SharpC
Leaf2
Snake
S Shape
BendedLine
J2
We can also learn flow fields where demonstrations start from different groups of initial conditions, and exhibit different speeds.
G Shape convergence from original domain
G Shape cnvergence from increased domain
N Shape convergence from original domain
N Shape convergence from increased domain
N Shape convergence from an even larger domain
@article{li2025koopmotion,
author = {Alice Kate Li, Thales C. Silva, Victoria Edwards, Vijay Kumar, and M. Ani Hsieh},
title = {KoopMotion: Learning Almost Divergence Free Koopman Flow Fields for Motion Planning},
journal = {Conference on Robot Learning (CoRL)},
year = {2025},
url={https://arxiv.org/abs/2509.09074},
}