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

arXiv:2610.10803 · eess.SY, cs.SY · Submitted 2026-10-07 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Today's paper: "High-Fidelity Baseline Design and Station Keeping Analyses for Earth-Moon Vertical Orbits".

Dev: 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.

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

Title and authors: Rosa: Let's talk about the title and the authors for a minute. We have "High-Fidelity Baseline Design and Station Keeping Analyses for Earth-Moon Vertical Orbits" by Derek Phung and Yuri Shimane.

Dev: The title tells you exactly what we’re dealing with: high fidelity, baseline design, station keeping analysis, and specifically focusing on vertical orbits around the Moon. It sets the scope right away.

Taro: And looking at the authors, they seem to be tackling a problem that spans both orbital mechanics and control systems. You have Phung and Shimane working together on this kind of complex dynamic system.

Rosa: It means this isn't just a simple trajectory calculation; it’s about creating something robust enough for real mission use, considering all the complexities of the Earth-Moon system dynamics.

Dev: The implication is that they aren't just giving you a theoretical orbit; they are providing a tested, validated reference path that accounts for the physical realities of the environment.

The paper's summary: Rosa: So, what does the actual summary of this paper tell us about how they achieved this high-fidelity design and station keeping?

Dev: They take those CR3BP initial conditions—like those from the JPL Three-Body Periodic Orbit Catalog—and they use them to define their catalog initial conditions for the Earth-Moon system, specifically using a representative case like L2 one thousand thirty-six <ref:2610.10803#pg3,from the JPL Three-Body Periodic Orbit Catalog>.

Taro: They model the spacecraft as an infinitesimal third body moving under Earth and Moon gravity, using specific normalization parameters like mass parameter µ = m2/(m1 + m2), distance unit DU, and time unit TU for their calculations.

Rosa: Then they move into the optimization part where they use a nonlinear optimal control problem formulation, dividing each revolution into four coast arcs to minimize some cost function related to the maneuver magnitudes.

Dev: They solve this nonlinear program using a trust-region sequential convex programming method called SCvx* through SCPLib with Clarabel as the subproblem solver to find these locally optimal solutions.

Taro: And they also use Monte Carlo analyses, running two hundred catalog cases, which means they’re testing how this reference holds up under various uncertainties like navigation error and disturbances over time <ref:2610.10803#pg1>.

The paper's improvements: Rosa: Now, the authors point out some key things they improved or highlighted in their approach to make this work more effective.

Dev: They emphasize that the optimization under rotating-frame geometry constraints is a specific approach they’re using to construct those dynamically consistent references, which is a big step because it deals with how you define consistency in that frame.

Taro: The paper also mentions that they use dynamical information, specifically the finite-time Lyapunov exponent along the reference and stable or unstable CR3BP directions, to characterize how the station-keeping controller selects particular maneuver directions.

Rosa: That’s important because it moves beyond just finding *a* correction; it tells you *why* the controller picks a certain direction based on local stability and amplification along that path.

Dev: And they also show that navigation error ends up having the largest effect on maneuver cost and tracking dispersion, which is a very practical piece of information for mission planning.

Conclusion: Rosa: So, wrapping this up, what’s the main implication of this whole "High-Fidelity Baseline Design and Station Keeping Analyses for Earth-Moon Vertical Orbits" work?

Dev: The main thing is that they successfully demonstrate a workflow that works for generating these high-fidelity SPK references over multi-year periods, and crucially, they show that the system remains maintainable even under realistic uncertainties in five-year Monte Carlo simulations.

Taro: I think what stands out is how they connect the planned first maneuvers to those stable and unstable CR3BP velocity directions, suggesting an AI could prioritize maneuvers aligned with the stable axis based on that statistical preference.

Rosa: And I also want to emphasize that they show a big difference between the effort needed for generating a deterministic reference path versus the effort required for closed-loop maintenance, which means you can't just lump those two costs together when planning a mission.

Dev: To close out, this paper gives us a solid foundation for designing and testing these references under uncertainty, showing that with the right configuration of MPC and careful baseline design, you get a predictable result for these vertical orbits.

eess.SY, cs.SY

Submitted: 2026-10-07

Updated: 2026-10-07

Comments: 20 pages, 11 figures, 7 tables. Submitted to the 2027 AAS/AIAA Space Flight Mechanics Meeting

Code: https://github.com/eXplorationLogisticsControl/spk-validations

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 90/100

The gist: 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

Key concepts

High-Fidelity Baseline Generation: This process creates a precise starting point for the orbit by mapping initial conditions to physical time, optimizing them using a detailed gravitational model that accounts for lunar harmonics, and exporting validated reference data called Spacecraft and Planet Kernel (SPK) references.
Receding-Horizon MPC: This control method is used for station-keeping. It solves an optimal control problem repeatedly over a finite time horizon, allowing the spacecraft to make decisions at each step based on the latest state estimates, helping it reject navigation errors and disturbances.
Finite-Time Lyapunov Exponent (FTLE): The FTLE measures how quickly small deviations from the planned reference orbit grow or shrink over a short period. It is used to characterize the local amplification of errors along the high-fidelity reference trajectory, indicating areas where tracking might be unstable.

Terminology

Fidelity — 50× in this paper · explained in 2 episodes
Robustness — 8× in this paper · explained in 5 episodes
Alignment — 4× in this paper
Near-Rectilinear Halo Orbits — 3× in this paper
Spacecraft and Planet Kernel — 3× in this paper
Consistency — 2× in this paper · explained in 3 episodes
Jet Propulsion Laboratory — 2× in this paper
Monte Carlo Analysis — 2× in this paper · explained in 2 episodes
Model Predictive Control (MPC) — 1× in this paper · explained in 7 episodes

Summary

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.

Improvements for AI systems

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.
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.
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.
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.
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.

Abstract

Earth-Moon vertical orbits are periodic solutions of the circular restricted three-body problem (CR3BP) with substantial out-of-plane motion, considered for high-latitude lunar observation and cislunar navigation. This work develops an end-to-end framework for high-fidelity reference generation under rotating-frame geometry constraints and receding-horizon model predictive control. A representative L 2 vertical orbit yields a 5.5-year reference with negligible deterministic correction and remains maintainable in five-year Monte Carlo simulations under realistic uncertainties. The workflow produces locally optimal solutions across 90 consecutive daily start epochs and 200 catalog cases. Finite-Time Lyapunov Exponent along the reference and stable/unstable CR3BP directions interpret station-keeping behavior.