Probabilistic Methods for Initial Orbit Determination and Orbit Determination in Cislunar Space

This paper presents a novel probabilistic framework for cislunar orbit determination that overcomes the limitations of Gauss's method in three-body dynamics by combining kinematic fitting of noisy observations for initial state estimation with a Particle Gaussian Mixture Filter for long-term uncertainty reduction.

The Gist

Imagine you are trying to track a lost hiker in a massive, foggy forest that sits between two giant mountains (the Earth and the Moon). This forest is the Cislunar Space. It's huge—about 1,000 times bigger than the area of space we usually monitor around Earth.

In this forest, the rules of physics are tricky. Because the two giant mountains are pulling on the hiker at the same time, the hiker doesn't move in simple, predictable circles like a planet around the sun. Instead, they might zigzag, loop, or get stuck in chaotic loops.

The problem? We have a telescope on the ground, but it's very far away. We can see the hiker's direction (left/right, up/down), but we can't easily tell exactly how far away they are. It's like trying to guess how far a bird is just by looking at it; it could be 10 feet away or 10 miles away, and it looks the same size in your eyes.

Here is how the authors of this paper solved the problem of finding and tracking these "lost hikers" (satellites or space debris) using a new, clever method.

1. The Old Way vs. The New Way

The Old Way (Gauss's Method):
For a long time, scientists used a famous math trick called "Gauss's Method" to find where things are in space. It's like solving a puzzle with three pieces of information. But this trick only works if the object is moving in a perfect, simple circle (like a car on a flat highway).

• The Problem: In the Cislunar forest, the "highway" is bumpy and curved by the gravity of the Earth and Moon. The old math trick breaks down because the hiker isn't following a simple circle.

The New Way (The "Kinematic Fitting" Trick):
The authors came up with a new approach that doesn't assume the hiker is following a simple path. Instead, they treat the hiker's movement like a movie.

• The Analogy: Imagine you are taking photos of a runner every few minutes for 10 hours. You don't know exactly how fast they are running or where they started, but you have a long list of photos showing where they were.
• The Trick: The authors take all those photos (measurements) and draw a smooth, wiggly line (a polynomial curve) through them. Once they have that smooth line, they can mathematically "speed it up" to figure out how fast the runner is moving at any specific moment.
• The Catch: Since they don't know the exact distance, they have to guess. They guess every possible distance the hiker could be at (from very close to very far). This creates a massive cloud of thousands of "ghost hikers," each representing a different guess.

2. The "Ghost Cloud" (The Initial Estimate)

At the start, the authors have a huge, messy cloud of thousands of possible locations for the hiker. Some ghosts are close, some are far, some are moving fast, some are slow. It's a big, fuzzy mess.

• The Goal: They need to shrink this fuzzy cloud down to find the real hiker.

3. The "Smart Filter" (The PGM Filter)

This is where the magic happens. They use a special tool called the Particle Gaussian Mixture (PGM) Filter.

• The Analogy: Imagine you have a giant bag of marbles (the ghost hikers). You shake the bag, and then you look at a new photo of the hiker.
  • Any marble that is in a place where the hiker couldn't possibly be (based on the new photo) gets thrown out.
  • Any marble that is in a place where the hiker might be gets kept.
  • The "Smart Filter" is special because it knows how to handle the "bumpy highway" of the Cislunar space. Other tools (like standard filters) get confused by the chaos and throw away the wrong marbles, or they get too confident and lose the hiker entirely.
• The Result: With every new photo (measurement), the filter throws out the wrong guesses and keeps the right ones. The "fuzzy cloud" shrinks rapidly until it becomes a tiny, precise dot right on top of the real hiker.

4. Why This Matters: The "Blackout" Test

The authors tested their method with a scary scenario: What if we lose the hiker for a long time?
Imagine the telescope breaks, or the hiker goes behind the Moon for 10 days or even 150 days. During this time, the "ghost cloud" spreads out wildly because the hiker could be anywhere in the chaotic forest.

• Old Filters: When the telescope turns back on, old filters get confused. They think they know exactly where the hiker is, but they are wrong. They lose the hiker forever.
• The PGM Filter: Because it kept thousands of different "ghost" possibilities alive during the blackout, when the telescope turns back on, it can instantly match the new photo to the right ghost. It finds the hiker again, even after a long time.

5. The Bottom Line

The paper shows that you don't need to know the exact distance to a satellite in deep space to track it.

1. Take many photos over a long time.
2. Draw a smooth line through them to guess the speed and direction.
3. Create a cloud of guesses covering all possibilities.
4. Use the "Smart Filter" to whittle down the guesses until you find the truth.

This method is like having a super-smart detective who doesn't need a perfect map to find a criminal. They just need to watch the criminal move for a while, make a bunch of wild guesses, and then use logic to eliminate the impossible ones until only the truth remains. This ensures that as we send more missions to the Moon and beyond, we won't lose track of our spacecraft in the vast, chaotic space between Earth and the Moon.

Technical Summary

1. Problem Statement

The primary challenge addressed is the Initial Orbit Determination (IOD) and subsequent Orbit Determination (OD) of Resident Space Objects (RSOs) in cislunar space (the region between Earth and Moon).

• Limitations of Traditional Methods: Standard IOD methods, such as Gauss's method, rely on Keplerian (two-body) dynamics and the assumption that motion is planar. These assumptions fail in cislunar space where three-body dynamics (Earth-Moon-Sun) dominate, causing non-planar motion and rendering orbital elements (like semi-major axis or eccentricity) less useful or undefined for certain trajectories.
• Data Scarcity and Uncertainty: Cislunar targets are often observed for long durations (10–20 hours) but with significant uncertainty. Range measurements (distance) are often unreliable or unavailable due to the vast distances and signal attenuation, leaving observers with primarily angles-only data (Azimuth and Elevation).
• Non-Gaussian Uncertainty: The chaotic nature of cislunar dynamics (especially near Lagrange points) causes probability density functions (PDFs) of state estimates to become highly non-Gaussian and distorted over time. Traditional filters like the Extended Kalman Filter (EKF) or Unscented Kalman Filter (UKF) often fail to capture these complex distributions, leading to filter divergence or loss of custody of the target.

2. Methodology

The authors propose a combined, minimal-assumption IOD/OD framework consisting of two sequential stages:

A. Initial Orbit Determination (IOD): Kinematic Fitting

Instead of relying on dynamic models (like Gauss's method), the authors use a kinematic fitting approach to generate a probabilistic initial state estimate.

• Data Utilization: The method utilizes a time-series of consecutive ground-based observations (Azimuth, Elevation, and inferred/assumed Range) over a single pass (10–20 hours).
• Polynomial Spline Fitting: The position of the target in the topocentric reference frame is approximated by fitting fourth-order polynomials to the observation data using a Least-Squares estimator.
• State Derivation: Velocities are obtained by taking the time derivative of these fitted polynomials.
• Probabilistic Generation (Monte Carlo): To account for measurement noise and range uncertainty, the process is repeated thousands of times.
  • Range Assumptions: The authors test two minimal assumptions:
    1. Noisy Range: Assuming a high standard deviation (5–10% of true range) for range measurements.
    2. Range Bounding: Assuming the target lies within the cislunar domain bounds ($\rho \sim U[84,328, 550,000]$ km) without specific range data.
  • Particle Cloud: This generates a large particle cloud representing the initial state estimate, which is then modeled as a Gaussian Mixture Model (GMM).

B. Orbit Determination (OD): Particle Gaussian Mixture (PGM) Filter

To track the target and reduce uncertainty over time, the authors employ the Particle Gaussian Mixture (PGM) Filter.

• Why PGM? Standard filters (EKF, UKF, EnKF) assume Gaussianity or struggle with particle depletion in high-dimensional, chaotic systems. The PGM filter addresses the "curse of dimensionality" and inflexibility of component weights in GMMs.
• Algorithm Steps:
  1. Propagation: Particles are propagated using the Circular Restricted Three-Body Problem (CR3BP) dynamics model.
  2. Clustering: The propagated particle cloud is clustered (using k-means) into multiple Gaussian components to form the prior GMM.
  3. Measurement Update: An Ensemble Kalman Filter (EnKF) update is applied to each Gaussian component individually to fuse new angle-only measurements.
  4. Resampling: Particles are resampled from the updated GMM to prevent particle depletion and maintain the distribution shape.

3. Key Contributions

1. Minimal-Assumption IOD Framework: A novel method to generate initial state estimates in cislunar space without relying on Keplerian dynamics or precise range data, utilizing only kinematic fitting of angles and bounded range statistics.
2. Robust Tracking in Chaotic Regimes: Demonstration that the PGM Filter effectively maintains custody of targets in highly chaotic trajectories (e.g., near Lagrange points) where UKF and EnKF fail due to non-Gaussian warping of the state PDF.
3. Handling Long Sensor Shutoffs: The framework successfully retains target custody even after extended periods (up to 150 days) of no observations, a scenario where traditional filters typically diverge or become inconsistent.
4. Comparative Analysis: A rigorous comparison showing that while UKF/EnKF may perform adequately in stable, short-term scenarios, the PGM Filter is superior for long-term tracking in cislunar space, particularly when initial state uncertainty is high and non-Gaussian.

4. Results

The framework was tested on three distinct cislunar trajectories:

1. 9:2 Resonant Near-Rectilinear Halo Orbit (NRHO):
   • The PGM filter reduced position uncertainty from ~1,000 km to <10 km and velocity uncertainty from ~5 km/s to <0.5 m/s over six passes.
   • Entropy (uncertainty metric) decreased significantly, outperforming UKF and EnKF in long-term stability.
2. Target Passing Through L2 Lagrange Point:
   • Simulated a 10-day sensor shutoff. The state estimate became highly non-Gaussian (chaotic warping).
   • The PGM filter successfully re-converged the estimate upon the return of observations, whereas UKF became overconfident and inconsistent, and EnKF struggled with the PDF size.
3. 3:1 NRHO with 150-Day Shutoff:
   • An extreme test case. After 150 days of no data, the initial estimate had bifurcated chaotically.
   • The PGM filter retained custody with only two subsequent observations, proving that re-running IOD was unnecessary.
   • Filter Comparison: UKF failed immediately after the shutoff due to overconfidence; EnKF failed due to particle depletion and unwieldy PDF sizes. Only PGM succeeded.

Impact of Range Assumptions:

• The study found that while different range assumptions (noisy vs. uniform bounding) affect the initial precision, the PGM filter converges to similar long-term accuracy regardless of the initial assumption, provided the target remains within the cislunar bounds.

5. Significance and Limitations

Significance:

• Space Situational Awareness (SSA): This work directly supports the National Cislunar Science and Technology Strategy by enabling the tracking of objects in the cislunar domain, which is critical for the safety of upcoming lunar missions (e.g., Artemis, Gateway).
• Operational Resilience: The ability to track targets through long sensor outages without re-initializing IOD is a major operational advantage for space traffic management.
• Methodological Shift: It moves away from dynamic-model-dependent IOD (Gauss) toward a kinematic, data-driven approach suitable for complex multi-body environments.

Limitations:

• Near-Earth Orbits: The current "range bounding" assumption (lower bound ~84,000 km) may fail for targets in highly elliptic orbits that dip below this threshold (e.g., planar mirror orbits). In such cases, the clustering step in the PGM filter can suffer from ill-conditioning, leading to filter inconsistency.
• Clustering Sensitivity: The success of the PGM filter heavily relies on the clustering strategy (k-means). Future work is needed to develop more robust clustering algorithms for ill-conditioned scenarios.
• Model Fidelity: The current study uses the Circular Restricted Three-Body Problem (CR3BP). Future extensions should test the framework on higher-fidelity models (e.g., Elliptical R3BP, Four-Body problems) and real-world sensor data.

Conclusion

The paper presents a robust, minimal-assumption framework for cislunar orbit determination. By combining kinematic polynomial fitting for IOD with the Particle Gaussian Mixture Filter for OD, the authors demonstrate a solution capable of handling the nonlinearity, chaos, and data scarcity inherent in the cislunar environment, outperforming traditional Kalman-based filters in critical long-term tracking scenarios.