⚡ electrical engineering

Spectral Koopman Approach for Reconstructing State-space Geometry of Cislunar Restricted 3-Body Problem

This paper proposes a novel spectral Koopman approach using path integrals to compute principal eigenfunctions, whose zero level curves globally reconstruct the phase-space geometry of the planar Cislunar Restricted 3-Body Problem, offering a global linear representation that contrasts with traditional local trajectory-based analyses.

Original authors: Subhrajit Sinha, Sriram S. K. S. Narayanan, Raktim Bhattacharya, Umesh Vaidya

Published 2026-06-10

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CC BY 4.0

Original authors: Subhrajit Sinha, Sriram S. K. S. Narayanan, Raktim Bhattacharya, Umesh Vaidya

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to navigate a complex, swirling river system where the water flows in unpredictable, chaotic patterns. This is what space travel looks like in the "cislunar" region—the space between Earth and the Moon. Unlike the smooth, predictable orbits we use for satellites near Earth, this area is a chaotic dance between the gravity of Earth, the Moon, and the Sun.

Traditionally, scientists have tried to map this chaos by looking at individual paths (trajectories) one by one. It's like trying to understand the flow of a river by dropping a single leaf in the water and watching where it goes. If you want to know the whole picture, you have to drop millions of leaves, which is slow and computationally expensive.

The New Approach: The "Koopman" Lens
This paper proposes a different way to look at the problem. Instead of tracking individual leaves, the authors use a mathematical tool called the Koopman operator.

Think of the Koopman operator as a special pair of glasses or a "lens" that transforms the chaotic, non-linear river into a simple, straight-line highway. Even though the water (the spacecraft) is swirling wildly, this lens allows us to describe the entire system using simple, linear rules. It's like taking a tangled ball of yarn and magically straightening it out so you can see the whole pattern at once.

The Secret Map: "Path Integrals"
To put on these special glasses, the authors use a method called the path integral formulation.

The Analogy: Imagine you want to know the "personality" of a specific spot in the river. Instead of just looking at the water right there, you trace every possible path a drop of water could take to get there and every path it could take away from there.
The Result: By doing this math, they calculate special "eigenfunctions." Think of these as invisible magnetic fields or terrain maps that reveal the hidden structure of the space.

What Did They Find?
The authors applied this method to the Earth-Moon system and discovered that the "zero-level curves" (the lines where these invisible maps read zero) perfectly trace out the highways of space.

The Gateways (L1 and L2):
- Near the points between Earth and Moon (called Lagrange points), the maps show "funnels."
- Unstable Funnels: These are like slides that push spacecraft away from Earth toward the Moon.
- Stable Funnels: These are like vacuum cleaners that pull spacecraft in from the Moon back toward Earth.
- The paper shows that these funnels connect to form a "transport tube." If you launch a spacecraft into this invisible tube, it will naturally flow from Earth to the Moon without needing much fuel.
The Safe Zones (L4 and L5):
- At the triangular points (60 degrees ahead and behind the Moon), the maps show closed loops.
- These are like swirling eddies in the river where things get trapped in a stable circle. Spacecraft here can orbit in a "tadpole" shape without falling out. The math confirms these areas are safe and stable, unlike the chaotic gateways.
The Dead Ends (L3):
- On the far side of Earth (opposite the Moon), the maps show that the paths are fragile and tend to leak out into deep space rather than forming a useful highway.

Why This Matters
The paper claims that by using this "Spectral Koopman" approach, they can reconstruct the entire geometry of the space between Earth and the Moon just by looking at the data.

Old Way: "Let's simulate a million different paths to see where they go." (Slow, local, piece-by-piece).
New Way: "Let's calculate the invisible map that defines the flow, and the highways appear instantly." (Global, fast, structural).

The Bottom Line
The authors successfully demonstrated that this mathematical lens can reveal the "invisible highways" of the solar system. It shows exactly where the natural currents of gravity will take a spacecraft, allowing for much more efficient mission planning. They didn't just find a new path; they found a way to see the shape of the entire river system at once.

Technical Summary: Spectral Koopman Approach for Reconstructing State-space Geometry of Cislunar Restricted 3-Body Problem

Problem Statement
The cislunar environment, governed by the gravitational interplay of the Earth, Moon, and Sun, presents a highly nonlinear dynamical system characterized by libration points, invariant manifolds, and chaotic transport. Traditional mission design and trajectory analysis for this region rely on the Circular Restricted Three-Body Problem (CR3BP) and numerical optimization techniques (e.g., differential correction, multiple shooting). However, these existing methods face three persistent limitations:

Fidelity Mismatches: Discrepancies between simplified models and high-fidelity reality.
Computational Cost: The need for costly recomputation when navigation updates occur.
Scalability Issues: Poor scalability of Monte Carlo methods for long-horizon uncertainty propagation.

Furthermore, standard local linearization techniques fail to capture the global phase-space geometry organized by strong nonlinear structures like resonances and transport channels. While operator-theoretic approaches (specifically the Koopman operator) offer a global linear representation of nonlinear systems, existing data-driven approximations (e.g., Extended Dynamic Mode Decomposition) often require careful basis selection and struggle to recover the intrinsic global spectrum accurately.

Methodology
The authors propose a novel approach based on the path integral formulation of the Koopman spectrum to reconstruct the global phase-space geometry of the planar CR3BP without relying on ad hoc dictionary selection.

Koopman Operator Framework: The system dynamics $\dot{x} = f(x)$ are analyzed through the evolution of observables. The Koopman operator $U_t$ provides a linear description of the nonlinear flow. The principal eigenfunctions $\phi_\lambda$ of this operator satisfy the equation $\frac{\partial \phi_\lambda}{\partial x}f = \lambda \phi_\lambda$ .
Path Integral Formulation: Instead of approximating the operator in a finite-dimensional subspace, the authors compute the principal eigenfunctions directly. By decomposing the vector field into linear ($Ax$) and nonlinear ( $F_n(x)$ ) components, the nonlinear part of the eigenfunction $h_\lambda(x)$ is expressed as a path integral:
$h_\lambda(x) = \int_0^\infty e^{-\lambda \tau} w_\lambda^\top F_n(s_\tau(x)) d\tau$
where $s_\tau(x)$ represents the system trajectory.
Numerical Implementation: The eigenfunctions are approximated by evaluating the integral over a finite time horizon $T$ . Unstable eigenfunctions (associated with $\lambda_u > 0$ ) are computed using forward trajectories, while stable eigenfunctions (associated with $\lambda_s < 0$ ) are computed using backward trajectories.
Geometry Reconstruction: The authors utilize the property that the joint zero-level curves of these principal eigenfunctions correspond to the invariant manifolds (stable, unstable, and center) of the system's equilibrium points (Lagrange points).

Key Contributions

Global Linear Representation: The paper demonstrates that the Koopman framework can generate a global linear representation of the CR3BP, overcoming the limitations of local linearization.
Direct Computation of Eigenfunctions: By leveraging the path integral approach, the method computes principal eigenfunctions directly from time-series data, bypassing the need for manual dictionary design required by methods like EDMD.
Invariant Manifold Identification: The work establishes that the zero-level sets of the computed eigenfunctions encode the phase-space characteristics, specifically identifying stable, unstable, and center manifolds associated with the five Lagrange points ( $L_1$ through $L_5$ ).
Transport Channel Visualization: The approach successfully visualizes the "invisible highways" of the cislunar environment, showing how unstable manifolds of one equilibrium can align with stable manifolds of another to form transport channels.

Results
The proposed method was applied to the planar 4D CR3BP model (Earth-Moon system). The simulation results include:

Reconstruction of Zero-Velocity Curves (ZVCs): The method recovers the energetically allowed regions and forbidden zones defined by the Jacobi constant.
Local Geometry at Equilibria:
- $L_1$ and $L_2$ : The eigenfunctions clearly delineate transport gateways. The unstable manifolds extend outward through the "necks" (allowing escape from Earth or entry to the exterior), while stable manifolds bend inward (capturing trajectories).
- $L_3$ : Exhibits fragile manifolds that leak toward external regions rather than forming strong transport corridors.
- $L_4$ and $L_5$ : The center eigenfunctions dominate, producing closed loops corresponding to stable tadpole and Trojan orbits, consistent with the linear stability of these triangular points.
Global Phase-Space Partitioning: The zero-level curves of the eigenfunctions partition the phase space spectrally. Unlike energy-based Jacobi surfaces which define where motion is possible, the Koopman curves reveal how trajectories are asymptotically organized within those regions, identifying separatrices and invariant boundaries.
Transport Mechanism: The results illustrate a transport channel between $L_1$ and $L_2$ , where the unstable manifold of $L_1$ and the stable manifold of $L_2$ bound an invariant tube, facilitating transfer between the Earth and Moon regions.

Significance and Claims
The paper claims that the spectral Koopman approach offers a powerful tool for cislunar dynamics by shifting the focus from individual trajectories to the evolution of observables. The primary significance lies in the ability to:

Recover Global Geometry: Reconstruct the global phase-space geometry of the CR3BP from time-series data, capturing nonlinear structures that local methods miss.
Enable Downstream Tasks: Provide a foundation for uncertainty propagation, reachability analysis, and control design. Because the Koopman operator is linear, the authors posit that uncertainty analysis and control strategy design for the nonlinear CR3BP can be simplified and made more scalable compared to traditional nonlinear methods.
Data-Driven Discovery: Open a new direction for data-driven discovery in space dynamics, where the linearity of the operator allows for the analysis and design of control strategies for inherently nonlinear systems.

The authors conclude that while the current work focuses on recovering the geometry of the CR3BP, the framework holds promise for future applications in real-time trajectory optimization and robust control in complex multi-body environments.

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