Orbit Identification

Source: Qiao et al. (2025) "Orbital parameter characterization and objects cataloging for Earth-Moon collinear libration points"
Website: https://cislunarspace.cn

Definition

Orbit Identification is a core problem in cislunar space situational awareness: given a sequence of observed spacecraft states over a period of time, identifying the reference orbit — the periodic or quasi-periodic orbit in the CR3BP model — that the spacecraft is executing.

The essence of this problem is: in the standard orbit catalog established by CR3BP, finding the reference orbit that best matches the observed actual motion, thereby obtaining the physical characteristics (period, amplitude, etc.) of the spacecraft for space object cataloging, collision warning, and space traffic management.

Inverse Relationship with Orbit Design

Orbit identification and orbit design are inverse processes:

Process	Input	Output
Orbit Design	Physical parameters of reference orbit (period, amplitude, etc.)	Actual orbit under ephemeris model (numerical integration)
Orbit Identification	Observed actual orbital state sequence	Corresponding CR3BP reference orbit and its physical parameters

In orbit design, the reference orbit is first obtained in CR3BP, then refined in the ephemeris model using multiple shooting and differential correction methods to obtain the true orbit satisfying the actual gravitational environment.

In orbit identification, the direction is reversed: starting from the actual orbital state sequence, extract physically interpretable parameters (period, amplitude, etc.) to find the corresponding reference orbit.

Limitations of Traditional Methods

The most direct orbit identification method is numerical integration and comparison: select specific state vectors, integrate to obtain a complete orbit, and compare with observations. However, in cislunar space, this method faces fundamental difficulties:

1. Observation Errors

For non-cooperative targets, orbital states come from radar/optical tracking and contain noise errors.

2. Dynamical Model Uncertainty

Unmodeled factors during integration: perturbations (solar radiation pressure, lunar non-spherical gravity, other celestial bodies) and unknown orbital maneuvers of non-cooperative spacecraft.

3. Chaos Sensitivity

CR3BP itself is a non-integrable chaotic system; initial errors cause numerical integration to diverge rapidly. Qiao et al. (2025)'s numerical experiments show: when position error exceeds 10 km and velocity error exceeds 0.1 m/s, integrated trajectories diverge rapidly, making initial Halo orbit identification impossible.

Characteristic Parameter-Based Identification Method

Qiao et al. (2025) propose a six-dimensional characteristic parameter-based orbit identification method that effectively avoids numerical integration divergence:

Core Idea

Convert 6D state $(X, Y, Z, \dot{X}, \dot{Y}, \dot{Z})$ to 6D characteristic parameters $[q_1, p_1, I_2, \theta_2, I_3, \theta_3]$
Characteristic parameters have clear physical meaning, directly related to orbit period and amplitude
On the Poincaré section diagram, reference orbits have a one-to-one correspondence with section coordinates $[I_2^{(0)}, I_3^{(0)}]$
Use optimization to search on the section diagram to find the reference orbit minimizing the mean square error (MSE) between actual and reference orbits

Optimization Model

Given observation sequence $[t_1, t_2, ..., t_n]$ with states $[X_1, X_2, ..., X_n]$ , define MSE:

\text{MSE} = \frac{1}{n}\sum_{i=1}^{n}\left[(I_2^{(i)} - \varphi_{I_2}(\sigma_0; t_0, t_i))^2 + (I_3^{(i)} - \varphi_{I_3}(\sigma_0; t_0, t_i))^2\right]

where $\varphi$ is the integral flow function of the central manifold canonical equations.

Optimization problem:

\min_{\text{MSE}} \quad x = [I_2^{(0)}, I_3^{(0)}, t_0]

\text{s.t.} \quad |I_j^{(0)} - I_j^*| \leq I_{\max}, \quad j=2,3

\quad t_0 \in [0, T_{\max}]

Bayesian Optimization

Since the MSE function is a black-box optimization problem (high computational cost, no explicit derivatives), Qiao et al. (2025) use Bayesian optimization, finding the global optimum within 30 function evaluations, highly efficient.

Sensitivity Analysis

Qiao et al. (2025) systematically analyze two factors affecting orbit identification:

1. Observation Arc Length

Arc Length	Identification Result Characteristics
Short arc (1 hour)	Results dispersed, mainly along equal-energy contours; represents "osculating orbit"
Long arc (1 month)	Results converge to reference orbit; represents "mean orbit"

This parallels the concepts of osculating elements and mean elements in the two-body problem.

2. Observation Errors (State Deviations)

Robustness to state errors:

Position error < 100 km and velocity error < 1 m/s: identification result fluctuations are small
100 km position error and 1 m/s velocity error have equivalent impact on robustness
This points to a direction for future cislunar orbit determination technology: greater emphasis should be placed on improving velocity measurement accuracy

Significance

This method provides a non-iteration-through-long-numerical-integration approach to orbit identification in cislunar space, effective even for non-cooperative targets under low-precision observation conditions. Combined with the Poincaré section distribution map, it can quickly determine which orbit family (Northern Halo, Southern Halo, Lissajous, etc.) a target belongs to and its physical parameters.

Central Manifold
Poincaré Section
Action-Angle Variables
Birkhoff-Gustavson Normal Form
Circular Restricted Three-Body Problem (CR3BP)
Cislunar Space Situational Awareness
Reference Orbit
Non-cooperative Target

References

Qiao C, Long X, Yang L, et al. Orbital parameter characterization and objects cataloging for Earth-Moon collinear libration points[J]. Chinese Journal of Aeronautics, 2025. doi: 10.1016/j.cja.2025.103869.
Wang X, Jin Y C, Schmitt S, et al. Recent advances in Bayesian optimization[J]. ACM Comput Surv, 2023, 55(13s): 1-36.