Cislunar State and Uncertainty Propagation via the Modified Generalized Equinoctial Orbital Elements

arXiv:2603.20110 · math.DS, astro-ph.EP, math.PR · Submitted 2026-03-20 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: Today's paper: "Cislunar State and Uncertainty Propagation via the Modified Generalized Equinoctial Orbital Elements".

Jocelyn: The complex cislunar dynamical environment poses challenges for spacecraft navigation and Space Domain Awareness (SDA) operations, where accurate modeling of state and uncertainty evolution is essential.

Vera: First, who's behind it and why it matters.

Paper summary: Vera: So, wrapping up the discussion on "Cislunar State and Uncertainty Propagation via the Modified Generalized Equinoctial Orbital Elements," the authors of this work are Maaninee Gupta and Kyle J. DeMars, and they put forward a framework based on M-GEqOEs.

Jocelyn: They developed this set specifically to handle the nonlinear dynamics of Earth, Moon, and Sun interactions in cislunar space while offering better uncertainty characterization than conventional Gaussian methods when propagating states over time.

Subrahmanyan: The main implication here is that this methodology provides a robust way to model trajectories in the cislunar domain by directly embedding perturbations into generalized orbital elements, which helps preserve the Gaussian nature of uncertainty.

Vera: It suggests that for future space missions and SDA operations, adopting this M-GEqOE approach could lead to more reliable state and uncertainty predictions over longer propagation intervals in the cislunar environment.

Jocelyn: I think what this paper really points toward is the necessity of moving beyond Gaussian assumptions when dealing with these highly nonlinear regions, which is a key challenge for tracking objects near the Moon.

Subrahmanyan: The work demonstrates that by using the Henze–Zirkler test, researchers can directly evaluate non-Gaussianity from simulation statistics, providing a more honest assessment of the uncertainty than older divergence measures.

Vera: It’s about having a modeling tool that respects the actual dynamics of the environment rather than forcing data into an idealized Gaussian box.

Jocelyn: And for those listening who are involved in pulsar surveys or spacecraft tracking, this means better tools for catalog maintenance and conjunction assessment when dealing with objects in complex orbital regimes.

Subrahmanyan: The future work suggested by the authors would likely involve extending these M-GEqOEs to even more complex dynamical situations or integrating them into broader Space Domain Awareness systems.

Vera: It seems like a solid foundation for applying this method where high fidelity and uncertainty accuracy are paramount, especially in environments like cislunar space.

Jocelyn: This paper is definitely worth paying attention to if you're interested in the operational side of orbital mechanics and how we can improve our tracking capabilities there.

Conclusion: Vera: So, we've been diving deep into the technical details of how these M-GEqOEs work to track spacecraft in cislunar space, and now it's time to look at what this paper is actually about and why it matters.

Jocelyn: I mean, the title itself says "Cislunar State and Uncertainty Propagation," which sounds pretty dense, but the core idea seems to be using these modified orbital elements to handle that complexity.

Subrahmanyan: The paper focuses on improving how we model the state and its uncertainty over time when dealing with Earth, Moon, and Sun perturbations in that environment. It's about making sure our predictions are as accurate as possible.

Vera: Exactly, Subrahmanyan; it’s not just about tracking a ship; it's about understanding the chaotic nature of those close gravitational interactions and how that translates into errors we have to account for.

Jocelyn: And the authors, Gupta and DeMars, they tackle this by using a lower-complexity approach—the M-GEqOEs—to incorporate high-fidelity dynamics without getting bogged down in overly complicated math right away.

Subrahmanyan: That’s interesting because it suggests a way to get better fidelity without needing a massive computational overhead, which is crucial for long-term mission planning and understanding the solar system's evolution.

Vera: It gives us a practical tool that seems to handle both the physical forces and the associated uncertainty in a way that keeps things behaving nicely, specifically by preserving Gaussian behavior in those uncertainty estimates.

Jocelyn: That preservation of Gaussian behavior is what really stands out; it means our confidence levels stay more reliable as we propagate the state forward through these nonlinear regions where things get tricky.

Subrahmanyan: From a theoretical standpoint, this methodology offers a cleaner path to understanding how small initial uncertainties can grow or shrink in the cislunar regime, which has huge implications for long-term stability studies of any object in that vicinity.

Vera: It means we have a more trustworthy way to make those long-term predictions for things like lunar missions or even deep space probes operating near the Moon.

Jocelyn: So, essentially, these M-GEqOEs are a more robust language for describing orbits in this messy gravitational setting than the traditional methods we've been using.

Subrahmanyan: And that robustness allows us to push the boundaries of what we can predict reliably in our understanding of solar system dynamics.

Vera: It really puts our observational data into a better context, giving us tools to interpret those noisy measurements with greater confidence.

Jocelyn: Next up, we're going to look at how they tested this concept using Monte Carlo simulations and their Henze–Zirkler test for uncertainty characterization.

School of Aeronautics and Astronautics, Purdue University

math.DS, astro-ph.EP, math.PR

Submitted: 2026-03-20

Updated: 2026-03-20

Comments: Submitted to Celestial Mechanics and Dynamical Astronomy

DOI: 10.1007/s10569-026-10333-y

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 79/100

The gist: The complex cislunar dynamical environment poses challenges for spacecraft navigation and Space Domain Awareness (SDA) operations, where accurate modeling of state and uncertainty evolution is

Key concepts

Modified Generalized Equinoctial Orbital Elements (M-GEqOEs): These are a specific set of orbital parameters derived from standard equinoctial elements, modified to handle conservative and non-conservative forces found in cislunar space. They offer a low-complexity framework for tracking spacecraft motion under complex gravitational influences.
Generalized Coordinate Representation: The M-GEqOEs use a generalized coordinate system that is designed to maintain Gaussian behavior in how uncertainty evolves over time. This is particularly useful for accurately characterizing the spread of possible states in sensitive orbital regions.
Henze–Zirkler (HZ) Test: This test is used instead of standard divergence measures to check if the uncertainty distribution remains Gaussian. It analyzes sample statistics like the mean and covariance to directly identify non-Gaussian behavior in the propagated state uncertainties.

Terminology

Fidelity — 23× in this paper · explained in 2 episodes
Accuracy — 7× in this paper · explained in 1 episode
Non-Gaussianity — 7× in this paper · explained in 2 episodes
Space Domain Awareness — 7× in this paper
Consistency — 2× in this paper · explained in 3 episodes
Monte Carlo Analysis — 2× in this paper · explained in 2 episodes
Near Rectilinear Halo Orbit — 2× in this paper
Robustness — 2× in this paper · explained in 5 episodes
Chaotic Dynamics — 1× in this paper · explained in 2 episodes
Computational Tractability — 1× in this paper · explained in 2 episodes
Computational overhead — 1× in this paper · explained in 2 episodes
Core — 1× in this paper · explained in 2 episodes
Elliptic Lunar Frozen Orbit — 1× in this paper
Generalization — 1× in this paper · explained in 13 episodes
NASA Jet Propulsion Lab — 1× in this paper

Summary

The complex cislunar dynamical environment poses challenges for spacecraft navigation and Space Domain Awareness (SDA) operations, where accurate modeling of state and uncertainty evolution is essential. The Modified Generalized Equinoctial Orbital Elements (M-GEqOEs) are explored as a low-complexity methodology to incorporate high-fidelity Earth–Moon–Sun perturbations while demonstrating improved preservation of Gaussian behavior in uncertainty propagation compared to conventional methods.

The gist: The M-GEqOE set allows the direct inclusion of conservative perturbations, offering a low-complexity methodology for modeling dynamics in this complex regime, and the generalized coordinate representation maintains Gaussian behavior over longer propagation intervals and exhibits improved uncertainty characterization in sensitive and highly nonlinear regions of the trajectories.

Modeling Dynamics via M-GEqOEs

The Modified Generalized Equinoctial Orbital Elements (M-GEqOEs) are derived from the Generalized Equinoctial Orbital Elements (GEqOEs) to provide a framework for incorporating both conservative and non-conservative perturbations that manifest in cislunar space. The formulation is built upon generalized orbital motion, where the perturbed two-body motion is represented by Equation (1), which includes perturbing accelerations, denoted as ap(r, r˙, t). The M-GEqOE set itself is defined as a specific collection of elements:

**: **

: The first element is the generalized semi-latus rectum, ˜p = h˜2/µC (Equation 15). This element incorporates the generalized angular momentum, h˜. 2. The sixth element is the classical true longitude, L = ω + omega + θ (Equation 16). 3. The second and third elements parameterize the eccentricity vector: p1 = ˜e sin Ψ and p2 = ˜e cos Ψ (Equations 17, 18). Finally, q1 and q2 orient the equinoctial reference frame relative to the inertial frame: q1 = tan(i)/2 sin(omega) and q2 = tan(i)/2 cos(omega) (Equations 20, 21). These elements define the axes of the equinoctial reference frame, Σeq. 4. The time derivatives for these elements are detailed in Equation (30), which directly incorporate all significant perturbations necessary for modeling the dynamics, independent of whether they are expressed as perturbing potentials or forces. 5. The transformation between Cartesian coordinates and M-GEqOE coordinates is crucial for initialization and propagation, allowing the state vector xC to be mapped into xM = [˜p p1 p2 q1 q2 L] (Equation 31). 6. Conversely, the transformation from M-GEqOEs back to Cartesian coordinates is detailed in Equations (35) through (40), enabling visualization and comparison against direct propagation methods. 7. To ensure validity throughout propagation, a constant offset Uoffset is applied to the third-body gravitational potential to maintain nonnegativity of the effective potential energy, Ueff (Equation 46).

High-Fidelity Cislunar Dynamics Simulation

The work achieves high-fidelity simulation by modeling the gravitational influence of the Earth, Moon, and Sun. In Cartesian coordinates, this is achieved using an N-body ephemeris model based on an inertial frame centered on the primary gravitational body (Equation 42). To mitigate numerical error accumulation during propagation, a nondimensionalization scheme is employed using characteristic quantities associated with the Earth–Moon Circular Restricted Three-Body Problem (CR3BP), such as the characteristic length l∗, mass m∗, and time t∗. For M-GEqOE propagation in Section 3.2, the gravitational influences of the Earth, Moon, and Sun are included; when treating Earth as central, Moon is a point-mass third-body perturbation; conversely, for orbits near the Moon’s vicinity, the Moon is treated as central with Earth providing the perturbing potential. Solar gravity is incorporated as an external perturbing force P in Equation (5), which manifests directly in the equations evolving the M-GEqOEs (Equation 30).

Uncertainty Characterization using Henze–Zirkler Test

To assess uncertainty evolution, Monte Carlo simulations with N = 10,000 samples are used for both M-GEqOE and Cartesian coordinates. Instead of relying on measures like Kullback-Leibler divergence or the Bhattacharyya coefficient, which assume a Gaussian approximating distribution, the paper employs the Henze–Zirkler (HZ) test to identify non-Gaussianity directly from sample statistics. The HZ statistic is calculated using the ensemble mean mx(t) and covariance P xx(t), and squared Mahalanobis distances d2i(t) between samples (Equations 47, 48, 49).

Improvements for AI systems

Here are specific improvements that can be made to AI systems by leveraging the methodologies and findings from this scientific paper, along with what these improved systems could achieve:

Improved State Estimation in Cislunar Space via M-GEqOEs:
Enhanced Uncertainty Characterization using the Henze–Zirkler (HZ) Test:
High-Fidelity Trajectory Prediction Under Complex Perturbations (Earth, Moon, Sun):
Robust Modeling of Nonlinear Dynamical Regimes in SDA:

Specific Improvements and Capabilities:

A system utilizing the M-GEqOE formulation can perform state estimation and tracking for spacecraft in cislunar space with significantly higher fidelity than conventional Cartesian propagation methods.
The improved system can maintain a more realistic representation of uncertainty, specifically by better preserving Gaussian behavior over extended propagation intervals, which is crucial for long-term mission planning and Space Domain Awareness (SDA).
The system can accurately model and propagate spacecraft states under the combined gravitational influences of Earth, Moon, and Sun (N-body dynamics), utilizing Nondimensionalization schemes derived from the Earth–Moon Circular Restricted Three-Body Problem (CR3BP) to mitigate numerical errors.
The AI system will be capable of identifying regions in a trajectory where uncertainty departs significantly from Gaussianity using the HZ test, allowing for more nuanced risk assessment and maneuver detection in complex, nonlinear cislunar environments.
The improved system can provide more reliable predictions for maneuvers and conjunction assessments by using coordinate representations (M-GEqOE) that are inherently better suited to the underlying dynamics than standard Cartesian coordinates when dealing with strong non-linearities near perilune passages of resonant orbits.