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

Derek Phung, Yuri Shimane

arXiv:2610.10803v1eess.SYcs.SY

TL;DR

Earth–Moon vertical-orbit CR3BP designs do not by themselves establish long-duration behavior in a high-fidelity ephemeris model. This work develops geometry-constrained reference generation and receding-horizon station keeping, then tests the approach on L2 1036 and broader orbit sets. L2 1036 was maintained in all 200 five-year Monte Carlo runs at the tested navigation-error levels.

  • Problem

    A CR3BP design alone does not establish whether a vertical-orbit trajectory will retain its intended geometry after transition to a high-fidelity ephemeris model.

    Source: INTRODUCTION · p. 1

  • Method

    The workflow generates high-fidelity references under rotating-frame geometry constraints and uses receding-horizon MPC, executing only the first planned impulse before replanning from a new state estimate.

    Source: INTRODUCTION · p. 2, Model Predictive Control · pp. 6–7

  • Results

    All 200 five-year L2 1036 Monte Carlo runs maintained the orbit successfully under low and high navigation error.

    Source: CONCLUSION · p. 19

  • Takeaways & Limitations

    Deterministic reference-generation effort and closed-loop maintenance effort can differ substantially, so they should be evaluated separately and then combined at the mission level.

    Source: CONCLUSION · p. 19

    Baseline generation uses deterministic gravity with lunar harmonics through degree and order 4 and no SRP, while station-keeping models continuous SRP without Earth or Moon shadowing.

    Source: High-Fidelity Ephemeris Model · p. 4

Abstract

from arXiv · show

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.

INTRODUCTION

The paper addresses the gap between periodic CR3BP designs and long-duration high-fidelity references, developing a workflow that combines geometry-constrained reference generation, receding-horizon station keeping, and dynamical interpretation. It applies that workflow to a representative orbit, a 90-epoch sweep, and 200 catalog cases.

INTRODUCTION

  • Vertical-orbit out-of-plane motion supports high-latitude lunar observation and has been considered for cislunar navigation, while a CR3BP design alone does not establish long-duration behavior in a high-fidelity model.The work extends geometry-constrained high-fidelity references to multi-year vertical orbits, tests start-epoch robustness and catalog cases, and analyzes long-duration station keeping and maneuver directions.Source: INTRODUCTION · pp. 1–2

CR3BP Initial Conditions and Epoch Mapping

  • Catalog periodic CR3BP states are repeated in phase, mapped at a selected epoch into Moon-centered rotating and J2000 frames, and used to initialize high-fidelity trajectory nodes.The final high-fidelity reference is not simply the repeated CR3BP solution.Source: CR3BP Initial Conditions and Epoch… · p. 4

High-Fidelity Baseline Optimization

Model Predictive Control

  • Station keeping recursively solves a finite-horizon problem from successive state estimates, implements only the first planned impulse, and rebuilds the problem at the next opportunity.For L2 1036, opportunities occur at the upper and lower out-of-plane amplitude extrema.Source: Model Predictive Control · pp. 6–7

Uncertainty and Disturbance Models

  • The station-keeping simulations model independent zero-mean Gaussian initial and navigation errors, maneuver-execution errors, and spacecraft disturbances; runs count as successful if the horizon completes without loss of a feasible solution.The disturbance cases vary SRP-parameter uncertainty and attitude-deadband and attitude-slew errors.Source: Uncertainty and Disturbance Models · pp. 7–8

Finite-Time Lyapunov Exponent

  • The FTLE measures local amplification over one CR3BP period along the high-fidelity reference using the largest singular value of a scaled state-transition matrix.The state is scaled with position and velocity scales of 10,000 km and 0.700200 km/s, respectively.Source: Finite-Time Lyapunov Exponent · p. 8

CR3BP Stability Directions and Maneuver Angles

  • Stable and unstable directions are defined by the smallest- and largest-magnitude monodromy eigenvalues, transported to matched orbit phases, and compared with planned impulsive maneuver directions.The comparison uses velocity components of the transported six-dimensional directions and includes only planned first maneuvers meeting the 0.15 cm/s execution threshold.Source: CR3BP Stability Directions and… · pp. 8–9

NUMERICAL RESULTS

  • The numerical analyses examine L2 1036 in detail, extend baseline-generation tests to start-epoch robustness and the L1/L2 family, and compare station keeping across selected long-duration references.The introduction reports 90 consecutive daily start epochs and 200 catalog orbits in the broader baseline-generation analyses.Source: INTRODUCTION · p. 2, NUMERICAL RESULTS · p. 9

Detailed L2 1036 Analysis

The L2 1036 workflow produces a 5.5-year high-fidelity reference with negligible deterministic correction and supports five-year MPC station keeping under both tested navigation-error levels. Geometry sensitivity, start-epoch testing, dynamical analyses, and catalog-wide runs characterize the scope and behavior of the approach.

Baseline Generation and SPK Validation

  • The 5.5-year L2 1036 reference requires only 1.16 × 10−10 m/s of baseline correction, and SPK validation reports position residuals of order 10−10 km and peak velocity residuals of order 10−12 km/s.The SPK-to-high-fidelity node-to-node position differences remain below about 10−7 km, with velocity differences of order 10−12 km/s.Source: Baseline Generation and SPK Validation · pp. 9–10, Table 3 · p. 10

Geometry Constraints and Start-Epoch Robustness

  • At sbox = 0.15, the L2 1036 case has no active geometry-box nodes and a correction of 6.17 × 10−10 m/s, following 8.318 m/s at sbox = 0.14.This is the smallest tested inactive box for this case; at the production scale sbox = 1, median apolune and perilune radius offsets are about 13,068 and 1,433 km, respectively.Source: Geometry Constraints and Start-Epoch… · pp. 10–11
  • Locally optimal baseline solutions were obtained at all 90 consecutive daily start epochs; 37 needed no refinement, 44 needed one warm start, and 9 needed two or more.Because refinement histories varied, final correction levels were not used to infer a physical dependence of baseline cost on epoch.Source: Geometry Constraints and Start-Epoch… · p. 11, Figure 3 · p. 11
  • Across 1,200 FTLE samples, λFTLE ranged from 0.3273 to 0.3703 day−1; the largest-FTLE location shifted between upper and lower large-|z| orbit portions across four epochs.The lowest FTLE remained near the central crossing.Source: Geometry Constraints and Start-Epoch… · pp. 11–12, Figure 4 · p. 12

Controller Selection and Disturbance Effects

  • Under high navigation error, the balanced Ntarget = 3 controller costs 28.21 m/s/yr versus 27.96 m/s/yr for cost-optimal Ntarget = 5, while reducing 95th-percentile position and velocity deviations from 1475 to 1321 km and 5.44 to 4.72 m/s.Both configurations use terminal tolerances of (25 km, 125 m/s); Ntarget = 3 was carried forward for the subsequent studies.Source: Controller Selection and Disturbance… · p. 12, Table 4 · p. 12
  • All ten navigation/disturbance combinations completed successfully, and at the tested levels navigation uncertainty changed median annualized cost more than any isolated SRP or attitude disturbance.The nonlinear closed-loop response was not necessarily the sum of the individual disturbance responses.Source: Controller Selection and Disturbance… · pp. 12–13, Table 5 · p. 13

Five-Year Station Keeping

  • Across 200 five-year runs, all completed; high navigation error raised median annualized cost by a factor of 8.21 and median maximum position deviation by a factor of 3.57 relative to low error.For high versus low navigation error, median total executed costs were 144.86 and 17.64 m/s, respectively.Source: Five-Year Station Keeping · pp. 13–14, Table 6 · p. 14, Figure 6 · p. 14

Maneuver Alignment with CR3BP Stability Directions

  • Median planned maneuvers favored the stable over the unstable CR3BP velocity axis in both navigation cases; for high error, the median angles were 12.12° and 37.58°, respectively.The MPC formulation has no term favoring either stability direction, and individual maneuvers do not always follow the aggregate preference.Source: Maneuver Alignment with CR3BP… · pp. 15–16, Figure 8 · p. 16

Selected Long-Duration L1 and L2 Vertical Orbits

The selected long-duration cases show that reference-generation effort and closed-loop station-keeping performance can differ, while the conclusion reports robust baseline generation and five-year MPC maintenance within the tested settings.

Selected Long-Duration L1 and L2 Vertical Orbits

  • Under common MPC settings, L1 1072 requires 7.888 m/s of baseline correction and L1 1011 requires 328.451 m/s over their approximately 5.5-year references.L1 1072 accumulates 7.572 m/s during the five-year station-keeping interval; L1 1011 uses four control opportunities per revolution rather than two.Source: Selected Long-Duration L1 and L2… · pp. 16–18
  • Six negligible-correction references cost 26.386–35.527 m/s/yr; L1 1072 costs 31.185 m/s/yr but has 5801.0 km median maximum position deviation, while L1 1011 reaches 375.657 m/s/yr and 22,359.8 km.All eight references complete 100/100 high-navigation-error realizations.Source: Table 7 · p. 18, Selected Long-Duration L1 and L2… · p. 18
  • L1 1011’s comparatively small in-plane motion makes it harder to control, while epoch-varying maximum |z| contributes to its larger tracking deviations.The selected references differ in vertical amplitude and in-plane geometry.Source: Selected Long-Duration L1 and L2… · pp. 16, 18

CONCLUSION

  • For L2 1036, the 5.5-year high-fidelity reference needs 1.16 × 10^-10 m/s of deterministic correction; the tightest tested geometry boxes instead require 12.50 km/s.Sufficiently wide boxes become inactive and recover an effectively ballistic reference.Source: CONCLUSION · p. 19
  • Local optima were found for all 90 consecutive January–March 2026 start dates, though differing refinement amounts prevent interpreting the correction curve as a controlled epoch-cost relationship.Thirty-seven cases needed no refinement, 44 needed one warm start, and 9 needed two or more.Source: CONCLUSION · p. 19
  • A properly tuned MPC configuration maintained L2 1036 in all 200 five-year Monte Carlo runs under low and high navigation error.Navigation error had the largest effect on maneuver cost and tracking dispersion among the tested uncertainty levels; planned first maneuvers favored the corresponding stable velocity axis without that alignment being imposed.Source: CONCLUSION · p. 19
  • The baseline workflow generated approximately six-month references for all 200 catalog cases, and the selected comparisons support assessing reference design and station keeping separately before combining them at mission level.Six selected long-duration references had negligible deterministic correction and a compact station-keeping cost range, while L1 1072 and especially L1 1011 remained operationally expensive or difficult to track.Source: CONCLUSION · p. 19
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