---
title: 'KoopMotion: Flow Planning with Koopman Operators'
url: https://www.emergentmind.com/topics/koopmotion
type: topic
---

# KoopMotion: Flow Planning with Koopman Operators

Searching arXiv for the specified paper to ground the article in the cited source.
KoopMotion is a flow field-based motion planning framework that learns smooth, stable, and convergent dynamical system policies from demonstrations by parameterizing motion flow fields with Koopman operators. It is designed to drive a robot from any initial state to a desired reference trajectory such that it converges to the trajectory’s end point, addressing a limitation of Koopman operator theory noted in learning from demonstrations: Koopman does not inherently enforce convergence to desired trajectories nor to specified goals [2509.09074]. The framework represents motion flow fields as dynamical systems, uses a learned lifting function with Fourier features, and regularizes the learned field to be “almost divergence-free,” with the stated aim of producing smooth motion fields that converge to a desired reference trajectory when initialized away from that trajectory and track it until the end point [2509.09074].

## 1. Conceptual definition and problem setting

KoopMotion formulates motion planning as the learning of a nonlinear discrete-time dynamical system
\[
\mathbf{x}_{k+1} = f(\mathbf{x}_k),
\]
where the system state at time \(k\) is \(\mathbf{x}_k\) and the learned map \(f : \mathbb{R}^d \to \mathbb{R}^d\) propagates the state along demonstrated trajectories [2509.09074]. In the formulation described for the method, the learned \(f\) provides velocity commands directly for control, thereby defining a motion planning flow field [2509.09074].

The central motivation is the mismatch between standard Koopman-based modeling and motion-planning requirements. The source explicitly states that, although Koopman operator theory has demonstrated efficacy for modeling dynamical systems, it does not inherently enforce convergence to desired trajectories or goals, which is required in learning from demonstrations [2509.09074]. KoopMotion addresses this by combining trajectory imitation, a fixed-point constraint at the goal, and a divergence-based regularizer.

Within the broader Koopman literature, the method belongs to a class of approaches that exploit linear evolution in a lifted observable space for analysis and control. Other works have used Koopman formulations for stochastic control [2410.09452], visual control-conditioned representations [2409.03107], and relative motion optimization in space [2207.07079]. KoopMotion is distinguished in the provided material by its explicit emphasis on convergent motion flow fields, trajectory tracking, and endpoint attraction in robotic motion planning [2509.09074].

## 2. Mathematical formulation

KoopMotion uses Koopman operator theory to linearize nonlinear dynamics in a higher-dimensional space via a lifting function \(\Psi\). In the lifted space, the dynamics are written as
\[
\Psi(\mathbf{x}_{k+1}) = \mathcal{K} \Psi(\mathbf{x}_k),
\]
where \(\Psi: \mathbb{R}^d \to \mathbb{R}^{d+\nu}\) is a learned lifting function composed of the state and Fourier features, and \(\mathcal{K} \in \mathbb{R}^{(d+\nu) \times (d+\nu)}\) is the learned linear Koopman operator [2509.09074].

The lifting function is implemented as
\[
\hat{\Psi}(\mathbf{x}_k) = [\mathbf{x}_k, \cos(\mathbf{w}_0^T\mathbf{x}_k + b_0), \ldots, \cos(\mathbf{w}_\nu^T\mathbf{x}_k + b_\nu)],
\]
where \(\mathbf{w}_i\) and \(b_i\) are learnable parameters [2509.09074]. Prediction in the original state space is obtained through a reconstruction or projection step. The source describes the vector field \(\mathbf{\hat{F}}\) as
\[
\mathbf{\hat{F}}(\mathbf{x}_k) := \text{Projection of } \mathcal{K}\Psi(\mathbf{x}_k) \to \mathbb{R}^d.
\]
This provides the motion flow field used for planning and control [2509.09074].

Training minimizes a composite objective
\[
\min_{\mathcal{K}, w, b} \quad \beta_k \mathcal{L}_\text{Koopman} + \beta_d \mathcal{L}_\text{FlowDivergence} + \beta_g \mathcal{L}_\text{Goal},
\]
with default loss weights \(\beta_k = 1\), \(\beta_d = 0.01\), and \(\beta_g = 0.01\) [2509.09074]. The three loss components are:

1. Koopman loss:
\[
\mathcal{L}_\text{Koopman} = \left\| \hat{\Psi}(\mathbf{x}_{k+1}) - \mathcal{\hat{K}}\hat{\Psi}(\mathbf{x}_k) \right\|_2
\]
for temporal consistency in the lifted space [2509.09074].

2. Divergence loss:
\[
\mathcal{L}_\text{FlowDivergence} = \left\| \nabla \cdot \mathbf{\hat{F}} \right\|_2,
\qquad
\nabla \cdot \mathbf{\hat{F}} = \sum_i \frac{\partial \hat{F}_i}{\partial x_i},
\]
which enforces an almost divergence-free flow and is primarily computed at locations along the demonstration trajectories [2509.09074].

3. Goal convergence loss:
\[
\mathcal{L}_\text{Goal} = \left\| \hat{\Psi}(\mathbf{X}_{T_\text{final}}) - \mathcal{\hat{K}} \hat{\Psi}(\mathbf{X}_{T_\text{final}}) \right\|_2,
\]
which encourages the final state to be a fixed point under the Koopman operator [2509.09074].

The formulation separates three distinct objectives: temporal fidelity to demonstrations, regularization of the geometry of the vector field, and asymptotic behavior at the goal. This suggests that KoopMotion is intended not merely as a predictor of demonstrated trajectories, but as a learned closed-loop dynamical system whose global flow structure matters.

## 3. Divergence structure, tracking, and convergence

The method’s defining property is its use of an “almost divergence-free” regularization. In the source description, this is interpreted as promoting volume-preserving behavior that leads to smoother and more robust flows [2509.09074]. The same material states that, in regions around the demonstrations, an almost divergence-free field means the vector field neither expands nor contracts volumes significantly, which is said to maintain the density of trajectories and minimize spurious attractors or sources [2509.09074].

The method attributes distinct roles to the three training losses. The Koopman loss preserves the temporal evolution of demonstrated trajectories in lifted space; the goal convergence loss makes the endpoint a fixed point, hence an attractor; and the divergence loss regularizes the flow to avoid spurious attractors or repellors along the trajectory [2509.09074]. The source further characterizes the resulting geometry as robust “attraction tubes” around the demonstrated motions that guide the system back to the trajectory if perturbed [2509.09074].

This description is important because a frequent ambiguity in demonstration-based dynamical systems learning is whether a model only reproduces a nominal path or whether it also defines a stable off-trajectory recovery behavior. KoopMotion is presented as addressing both. The demonstrated trajectory is not only replayed; points near, but not on, the demonstration are described as being attracted smoothly toward it, after which the flow follows the demonstrated progression to the end point [2509.09074].

A plausible implication is that KoopMotion should be interpreted as a policy-learning method in which stability is encoded structurally through the learned flow field rather than imposed through a separate tracking controller. The provided text supports this interpretation by emphasizing direct velocity commands, single-goal convergence, and robustness to new initializations [2509.09074].

## 4. Empirical evaluation

The reported empirical study spans a 2D handwriting benchmark, a 3D manipulator end-effector dataset, and hardware experiments on a physical robot. The evaluations are described as including spectral analysis [2509.09074].

### LASA human handwriting dataset

The LASA dataset contains 30 human-drawn trajectories with 7 demonstrations each, representing complex nonlinear 2D motions [2509.09074]. KoopMotion was trained using only 3% of the original data, sub-sampled to 25 points per demonstration, and was reported to reliably reconstruct and generalize the motion flows [2509.09074]. The source states that initial conditions throughout the domain, including states not seen during training, are still reliably attracted to the trajectory and the goal [2509.09074].

The quantitative evaluation used Dynamic Time Warping Distance (DTWD), described as measuring temporal similarity, and Swept Error Area (SEA), described as measuring spatiotemporal alignment [2509.09074]. KoopMotion’s DTWD is reported as comparable to state-of-the-art baselines including GMM-based methods, CLF-DM, CDSP, and DMP, while significantly outperforming them in SEA [2509.09074]. The same source interprets this as indicating better spatial and, critically, temporal correspondence.

The generalization claim is especially specific: every one of 500 random initializations around each trajectory converged robustly to the intended goal, which is presented as evidence for a lack of spurious attractors [2509.09074].

### 3D manipulator end-effector trajectories

The 3D dataset consists of endpoint motions from a teleoperated 7-DOF manipulator in the Robocasa simulator and is described as multi-modal [2509.09074]. KoopMotion is reported to learn a partition of space into distinct flow regions corresponding to different initial segments [2509.09074]. The eigenfunction analysis of the Koopman operator is said to reveal clear segmentation of these regions, while eigenvalue analysis verified system stability with dominant eigenvalues strictly inside the unit circle [2509.09074].

### Physical robot experiments

Hardware validation was performed on a miniature autonomous surface vehicle operating in a non-static fluid flow environment, specifically in a 4.5m x 3m x 1.2m water tank [2509.09074]. KoopMotion’s learned vector field, rescaled to the vehicle’s speed, provided real-time velocity commands through a basic velocity controller [2509.09074]. Reported outcomes include robust path following from positions not seen during training, successful tracking of multi-modal flows, and rendezvous at intended goals from distinct initial locations [2509.09074]. The source also states that no parameter or controller fine-tuning was required for motion execution and that additional robot experiments in the appendix confirmed overall flow field robustness and smoothness [2509.09074].

## 5. Spectral properties and system-theoretic interpretation

Because KoopMotion parameterizes the dynamics through a linear operator in lifted space, the method admits direct spectral analysis [2509.09074]. The stated stability condition is standard for the discrete-time lifted system: it is asymptotically stable if and only if all eigenvalues \(\lambda_i\) of \(\mathcal{\hat{K}}\) satisfy \(|\lambda_i| < 1\) [2509.09074]. The reported experiments found all eigenvalues within the unit circle, which the source interprets as confirming learned-flow stability [2509.09074].

The paper summary also emphasizes eigenfunction visualization. In multi-modal flows, analyzing Koopman eigenfunctions reveals spatial partitioning of dynamic behaviors and distinct basins of attraction [2509.09074]. In the manipulator experiments, this is tied directly to segmentation of the state space into different flow regions corresponding to different initial segments [2509.09074].

This spectral accessibility distinguishes KoopMotion from many demonstration-learning methods whose internal representations are not directly amenable to operator-theoretic stability analysis. Related Koopman-based works in other domains also exploit spectral or generator structure for analysis and control, including quantum systems [2201.12062], stochastic control [2410.09452], and non-stationary time-series forecasting [2305.18803]. In KoopMotion, however, the role of spectral analysis is explicitly tied to motion-flow stability and basin structure rather than to forecasting or quantum spectral computation [2509.09074].

A plausible implication is that KoopMotion occupies an intermediate position between classical stable dynamical-systems imitation learning and modern operator-learning methods: it uses a learned lifted linear model, but evaluates it through motion-planning criteria such as attraction, convergence, and flow geometry.

## 6. Ablations, sample efficiency, and relation to baselines

The source identifies sample efficiency as a major advantage. KoopMotion is described as highly sample efficient in both space and time, requiring only 3% of the LASA dataset to generate dense motion plans [2509.09074]. In the ablation discussion, reliable dense flow fields were again reported with only 3% of the original data [2509.09074].

The number of Fourier features \(\nu\) was tested over the range 500–1500, with minor impact on DTWD and SEA, which the source interprets as robustness to reasonable feature-dimension choices [2509.09074]. The same ablation notes that including both the divergence and goal losses is critical; without them, flows may not converge or may exhibit spurious attractors [2509.09074]. It is also stated that there is flexibility to trade off trajectory diversity versus convergence by adjusting the loss weights [2509.09074].

The framework is further described as not requiring shape-specific tuning, in contrast to GMM-based approaches that need careful component selection [2509.09074]. The key improvements over baseline methods are summarized in the source as follows:

| Dimension | Reported characterization |
|---|---|
| Trajectory tracking | Comparable or superior DTWD and much lower SEA than GMMs, CLF-DM, DMPs, and modern neural methods |
| Generalization | Globally attracting flows with a single attractor, even from initial states far from demonstrated data |
| Sample and hyperparameter efficiency | Dense fields from sparse data without per-shape tuning |
| Spectral insight | Stability and behavior analysis through Koopman eigenanalysis |
| Physical transfer | Direct sim-to-real transfer requiring minimal adjustment |

These are the method’s own reported comparative claims [2509.09074]. The emphasis on SEA is particularly notable because the source explicitly interprets lower SEA as better spatial and temporal fidelity, not merely shape imitation [2509.09074].

One possible misconception is to treat KoopMotion as a generic Koopman predictor. The provided material argues against that reading: the framework augments Koopman temporal consistency with divergence and goal constraints precisely because plain Koopman modeling does not inherently ensure convergence to a desired trajectory or endpoint [2509.09074].

## 7. Position within Koopman-based research

KoopMotion is part of a wider body of research that uses Koopman operators to linearize nonlinear dynamics in lifted spaces, but its stated contribution is specific to motion planning from demonstrations [2509.09074]. The paper summary explicitly identifies prior areas such as GMM-based dynamical systems learning, Control Lyapunov methods, Neural ODEs, and Normalizing Flow methods as relevant background, although it does not detail them individually [2509.09074].

Across the broader literature, Koopman methods have been used for efficient latent planning from images in DeepKoCo [2011.12690], control-conditioned visual representations in RoboKoop [2409.03107], stochastic optimal control with gEDMD [2410.09452], and convex MPC for contact-rich robotics under global Koopman linearization [2511.06515]. KoopMotion differs from these in the supplied material by centering the construction of a motion flow field that both tracks demonstrated trajectories and converges to a designated endpoint [2509.09074].

The framework also contributes an explicitly geometric interpretation of learned flows through divergence control. The source states that the divergence-free property avoids spurious attractors or repellors and yields robust attraction tubes around demonstrations [2509.09074]. This suggests a synthesis of operator learning and stable dynamical-systems imitation learning in which trajectory fidelity, global convergence, and spectral analyzability are handled within a single learned representation.

In summary, KoopMotion is characterized in the cited work as a Koopman-parameterized, almost divergence-free flow-field method for learning from demonstrations that is intended to provide trajectory tracking, endpoint convergence, spectral interpretability, and sample-efficient dense motion generation [2509.09074]. Its empirical validation spans synthetic and physical robotic settings, including 2D handwriting trajectories, 3D manipulator end-effector motions, and a miniature autonomous surface vehicle in non-static fluid flow, with the strongest reported advantages appearing in sample efficiency, SEA performance, and robust convergence from off-trajectory initial states [2509.09074].

Source: https://www.emergentmind.com/topics/koopmotion