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

arXiv: https://arxiv.org/abs/2509.09074 · HTML: https://mlsanity.com/paper/2509.09074/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 38 references that are in the map.
- First posted on arXiv 11 Sep 2025.

## Builds on (most cited first, 21 of 21)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [RoboCasa: Large-Scale Simulation of Everyday Tasks for Generalist Robots](https://mlsanity.com/paper/2406.02523/) (2024)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Neural Conservation Laws: A Divergence-Free Perspective](https://mlsanity.com/paper/2210.01741/) (2022)
- [Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability](https://mlsanity.com/paper/1906.03663/) (2019)
- [ImitationFlow: Learning Deep Stable Stochastic Dynamic Systems by Normalizing Flows](https://mlsanity.com/paper/2010.13129/) (2020)
- [Active Learning of Dynamics for Data-Driven Control Using Koopman Operators](https://mlsanity.com/paper/1906.05194/) (2019)
- [Euclideanizing Flows: Diffeomorphic Reduction for Learning Stable Dynamical Systems](https://mlsanity.com/paper/2005.13143/) (2020)
- [Modeling and Control of Soft Robots Using the Koopman Operator and Model Predictive Control](https://mlsanity.com/paper/1902.02827/) (2019)
- [Model-Based Control Using Koopman Operators](https://mlsanity.com/paper/1709.01568/) (2017)
- [Diffeomorphically Learning Stable Koopman Operators](https://mlsanity.com/paper/2112.04085/) (2021)
- [Task Generalization with Stability Guarantees via Elastic Dynamical System Motion Policies](https://mlsanity.com/paper/2309.01884/) (2023)
- [Learning Complex Motion Plans using Neural ODEs with Safety and Stability Guarantees](https://mlsanity.com/paper/2308.00186/) (2023)
- [On the Utility of Koopman Operator Theory in Learning Dexterous Manipulation Skills](https://mlsanity.com/paper/2303.13446/) (2023)
- [Dynamic Modeling of Bucket-Soil Interactions Using Koopman-DFL Lifting Linearization for Model Predictive Contouring Control of Autonomous Excavators](https://mlsanity.com/paper/2107.04314/) (2021)
- [Online Estimation of the Koopman Operator Using Fourier Features](https://mlsanity.com/paper/2212.01503/) (2022)
- [SE(3) Linear Parameter Varying Dynamical Systems for Globally Asymptotically Stable End-Effector Control](https://mlsanity.com/paper/2403.16366/) (2024)
- [Learning of Causal Observable Functions for Koopman-DFL Lifting Linearization of Nonlinear Controlled Systems and Its Application to Excavation Automation](https://mlsanity.com/paper/2104.02004/) (2021)
- [Spectrally Informed Learning of Fluid Flows](https://mlsanity.com/paper/2408.14407/) (2024)
- [EnKode: Active Learning of Unknown Flows with Koopman Operators](https://mlsanity.com/paper/2410.16605/) (2024)

## Built on by (most cited first, 1 of 1)

- [Neural Navigation Functions for Zero-Shot Generalizable Motion Planning](https://mlsanity.com/paper/2606.03756/) (2026)

## Abstract (arXiv)

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. Code at: hrefhttps://alicekl.github.io/koop-motion/colorbluehttps://alicekl.github.io/koop-motion.