High-Fidelity Baseline Design and Station Keeping Analyses for Earth-Moon Vertical Orbits

The gist: This work develops an end-toend framework for high-fidelity reference generation under rotating-frame geometry constraints and receding-horizon model predictive control for Earth–Moon vertical orbits.

High-Fidelity Baseline Generation

The baseline generation process involves mapping CR3BP initial conditions to a physical epoch, optimizing them in the high-fidelity ephemeris model under rotating-frame geometry constraints, and exporting them as validated Spacecraft and Planet Kernel (SPK) references. The Earth–Moon CR3BP is used to define these catalog initial conditions. The spacecraft is modeled as an infinitesimal third body moving under the gravity of the Earth and Moon, whose masses are m1 and m2, respectively. The normalized frame angular velocity is ω = [0 0 1]T. The high-fidelity ephemeris model (HFEM) propagates the spacecraft in Moon-centered J2000 using JPL ephemerides, physical parameters, and frame definitions. The natural dynamics are x˙ = f(x, t) = [v − µM∥r∥3/2r + Xb∈Bab + aSH + aSRP]. Baseline generation uses the deterministic gravitational model with lunar harmonics through degree and order 4 and no SRP.

High-Fidelity Baseline Optimization

The high-fidelity baseline optimization follows the formulation of Shimane and Ho. Each CR3BP revolution is divided into four coast arcs, such that for Nrev revolutions, Nnode = 4Nrev + 1 with time interval between each node ∆t = TC/4. The nonlinear optimal control problem (OCP) is min xk,∆vk,σk Nnode X−1 k=0 σk (5a). Constraint (5b) enforces the dynamics via multiple shooting. Constraint (5c) bounds the impulse magnitude from above by umax and assigns a slack σk to each maneuver magnitude, which is minimized through (5a). The nonlinear program is solved using SCvx∗, a trust-region sequential convex programming method23 through SCPLib, with Clarabel24 as the convex subproblem solver.

Station-Keeping and Dynamical Analysis

Station keeping then solves a finite-horizon OCP recursively from successive state estimates to reject navigation, execution, and disturbance errors. The MPC formulation is full-state targeting MPC from,14 originally conceived for Near-Rectilinear Halo Orbits (NRHO), where a finite-horizon impulsive problem is solved at each control opportunity i. Constraints (7e) and (7f) bound the terminal position and post-impulse velocity to lie within an ellipsoid with radii ϵr and ϵv. The principal closed-loop tracking quantities are the position and velocity deviation magnitudes ∆r(t) =∥r true(t) − r ref(t)∥2, ∆v(t) =∥v true(t) − v ref(t)∥2. The finite-time Lyapunov exponent (FTLE) characterizes local amplification over a finite interval along the high-fidelity reference.

Key Findings and Interpretations

The work makes three main contributions: First, it extends high-fidelity baseline optimization under rotating-frame geometry constraints to multi-year Earth–Moon vertical orbit references, validating the resulting Spacecraft and Planet Kernel (SPK) products. Second, it demonstrates long-duration receding-horizon station keeping along the generated vertical orbits under navigation, maneuver-execution, solar-radiation-pressure, and attitude-disturbance uncertainty, including a five-year Monte Carlo analysis. Third, it interprets the MPC maneuver directions by combining the finite-time Lyapunov exponent along the actual high-fidelity reference with a comparison between planned first MPC maneuvers and stable and unstable CR3BP velocity directions evaluated at matched orbit phases. The analysis shows that navigation error has the largest effect on maneuver cost and tracking dispersion. Furthermore, the planned first MPC maneuvers also show a persistent ensemble-level preference for the corresponding stable CR3BP velocity axis over the unstable axis, even though no such alignment is imposed in the OCP. The comparison between reference generation and station-keeping reveals that deterministic reference generation effort and closed-loop maintenance effort can differ substantially.

Family-Wide Results

The same baseline-generation workflow generates references for all 200 approximately six-month family cases. The selected long-duration station-keeping comparison shows that deterministic reference generation effort and closed-loop maintenance effort can differ substantially. Among the selected long-duration L1 and L2 vertical-orbit references, six require only negligible deterministic correction and cluster in a relatively compact station-keeping cost range under high navigation error. L1 1072 and especially L1 1011 show that a maintainable reference can still be operationally expensive or difficult to track. Reference design and station-keeping performance should therefore be evaluated separately and then combined at the mission level. The comparison shows why reference generation and station-keeping should not be collapsed into one metric.

Conclusion

A properly tuned MPC configuration maintains L2 1036 in all 200 five-year Monte Carlo runs under low and high navigation error. Within the tested uncertainty levels, navigation error has the largest effect on maneuver cost and tracking dispersion. The planned first MPC maneuvers also show a persistent ensemble-level preference for the corresponding stable CR3BP velocity axis over the unstable axis, even though no such alignment is imposed in the OCP. The same baseline-generation workflow generates references for all 200 approximately six-month family cases. The selected long-duration station-keeping comparison shows that deterministic reference generation effort and closed-loop maintenance effort can differ substantially. Among the selected long-duration L1 and L2 vertical-orbit references, six require only negligible deterministic correction and cluster in a relatively compact station-keeping cost range under high navigation error. L1 1072 and especially L1 1011 show that a maintainable reference can still be operationally expensive or difficult to track. Reference design and station-keeping performance should therefore be evaluated separately and then combined at the mission level.

References

[1] G. Gomez and J. M. Mondelo, “The Dynamics Around the Collinear Equilibrium Points of the RTBP,” ´ Physica D: Nonlinear Phenomena, Vol. 157, 2001, pp. 283–321, 10.1016/S0167-2789(01)00312-8.

[5] P. Pergola and E. M. Alessi, “Libration Point Orbit Characterization in the Earth–Moon System,” Monthly Notices of the Royal Astronomical Society, Vol. 426, No. 2, 2012, pp. 1212–1222, 10.1111/j.actaastro.vol4577-4585.

1. textbfWeighting maneuver directions using CR3BP stability axes for MPC maneuvers in closed-loop control systems. The paper demonstrates that the planned first MPC maneuver is expressed in J2000 and rotated into the Moon-centered Earth–Moon rotating frame with the three-dimensional frame rotation at the maneuver epoch and provides comparison angles to stable/unstable directions, suggesting an AI can be trained to prioritize maneuvers aligned with the persistent statistical preference for the stable axis.

2. textbfRobust baseline generation by incorporating geometry constraint sensitivity into reference quality assessment. The workflow allows evaluation of how geometry deviations affect correction cost; specifically, "At the tightest tested box scale, the geometry constraints are active throughout the reference and require 12.50 km/s of correction, whereas sufficiently wide boxes become inactive and recover an effectively ballistic reference." This enables an AI to predict the deterministic baseline cost based on a specified tolerance for geometric fidelity rather than assuming perfect adherence to CR3BP initial conditions.

3. textbfAdaptive receding-horizon MPC planning under navigation uncertainty and disturbance models. The system can be configured with full-state targeting MPC where the state estimate is xˆi = x true i + η nav i, allowing the AI to solve the OCP recursively using uncertainties defined in Table 2, ensuring that all 200 runs complete successfully, where each run contains 149 receding-horizon control opportunities.

4. textbfDynamical interpretation of trajectory amplification via Finite-Time Lyapunov Exponent (FTLE). An AI can calculate λFTLE(t) = Λ(t) TC, which is then interpreted alongside maneuver directions to characterize local instability, as the paper notes that The FTLE characterizes local amplification over a finite interval along the high-fidelity reference.

5. textbfLong-term trajectory planning by decoupling reference generation and station-keeping effort. An AI system can assess mission viability by comparing metrics such as deterministic referencegeneration effort and closed-loop maintenance effort, determining that for certain orbits, reference design and station-keeping performance should therefore be evaluated separately and then combined at the mission level.