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arxiv: 2606.22362 · v1 · pith:F2T4O2THnew · submitted 2026-06-21 · 🌌 astro-ph.EP · astro-ph.IM

Autonomous Orbit Determination Analysis of a Conceptual Cislunar Navigation Constellation based on Inter-Satellite Range Measurement

Haohan Li , Yuxuan Miao , Xiyun Hou , Bosheng Li , Jinjun Zheng , Hao Yu , Kanglian Zhao , Huan Yan This is my paper

Pith reviewed 2026-06-26 10:08 UTC · model grok-4.3

classification 🌌 astro-ph.EP astro-ph.IM
keywords cislunar constellationautonomous orbit determinationinter-satellite rangerelative planarity factorout-of-plane parametersLagrange point orbitsnavigation accuracyshort-arc determination
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The pith

Out-of-plane parameters control autonomous orbit determination accuracy in a four-satellite cislunar constellation using inter-satellite ranges.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines autonomous orbit determination for a proposed constellation consisting of one L3 orbit, one L4 orbit, one L5 orbit, and one near-Moon orbit. It models determination solely from inter-satellite range data and finds that out-of-plane amplitudes and phases are the dominant design factors for short-arc accuracy. A new relative planarity factor Pr is defined that shows negative correlation with accuracy, demonstrating that greater coplanarity degrades performance. Long-arc solutions prove largely insensitive to the same parameters. A sympathetic reader would care because reliable self-navigation without external references is required for sustained cislunar operations.

Core claim

With a model of autonomous orbit determination based on inter-satellite range measurements alone, the out-of-plane design parameters are identified as the main parameters influencing the AOD accuracy. For short arcs, increasing out-of-plane amplitude improves accuracy while initial phases exert more complex effects. The relative planarity factor Pr, which correlates negatively with AOD accuracy, is introduced to quantify how coplanarity of the constellation reduces determination performance. For long-arc AOD the influence of design parameters becomes insignificant.

What carries the argument

The relative planarity factor Pr, a metric proposed to evaluate how constellation coplanarity affects range-only autonomous orbit determination accuracy.

If this is right

  • Increasing out-of-plane amplitude improves short-arc AOD accuracy.
  • Coplanarity of the four orbits, as quantified by Pr, significantly reduces AOD accuracy.
  • Initial out-of-plane phases exert a complex influence on short-arc accuracy.
  • Long-arc AOD accuracy shows little dependence on the examined design parameters.
  • Out-of-plane parameters are the primary drivers of overall AOD performance in this model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Constellation designers would need to enforce sufficient out-of-plane separation to preserve navigation observability.
  • The Pr metric could be tested against real perturbed trajectories to check whether its correlation holds beyond the ideal case examined.
  • Similar planarity analysis might apply to range-only determination in other restricted three-body regimes.

Load-bearing premise

Inter-satellite range measurements are the only data source and contain no unmodeled biases or external perturbations that would invalidate the solutions.

What would settle it

A set of range-only simulations in which AOD error covariances remain unchanged when out-of-plane amplitudes are varied would falsify the claim that those parameters dominate accuracy.

Figures

Figures reproduced from arXiv: 2606.22362 by Bosheng Li, Haohan Li, Hao Yu, Huan Yan, Jinjun Zheng, Kanglian Zhao, Xiyun Hou, Yuxuan Miao.

Figure 1
Figure 1. Figure 1: 3D configuration of Constellations A to D in 30 days [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An example of an L3 Lissajous orbit lasting 180 days in the high-fidelity model of [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Examples of L4 (a) and an example of L5 (b) TLPOs lasting 180 days in the [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: DRO family in the CRTBP (a) and an example of a DRO in the high-fidelity model [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Four families of halo orbits in CRTBP model (a) and an example of a L1-north [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of the FLCE of the example DRO and NRHO orbits [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Relative geometry of the L3+L4+L5+DRO constellation and the upper limits of the [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Variation of Pr and DAGDOP with respect to A3 in simulation 1 r(Pr, DAGDOP) = −0.9806 (a) Variation of Pr with respect to A4 and A5 (b) Variation of DAGDOP with respect to A4 and A5 [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Contour maps of Pr and DAGDOP with respect to A4 and A5 in simulation 2 r(Pr, DAGDOP) = −0.7520 5.1.2 Influence of the out-of-plane initial relative phase angles: ∆ϕ4 and ∆ϕ5 Next, we study the effects on the DAGDOP factor of the initial relative phase angles ∆ϕ4 and ∆ϕ5. In this subsection, we fix ϕ3 and change ∆ϕ4 and ∆ϕ5 to analyze their influence. The difference between simulation 3 and simulation 4 li… view at source ↗
Figure 10
Figure 10. Figure 10: Contour maps of Pr and DAGDOP in simulation 3 (ϕ3 = 0) r(Pr, DAGDOP) = −0.4436 Similar conclusions can still be drawn from [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Contour maps of Pr and DAGDOP in simulation 4 (ϕ3 = π/2) r(Pr, DAGDOP) = −0.4442 [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Variation of Pr and DAGDOP with respect to ϕ3 in simulation 5 ∆ϕ5 affect significantly only at the local maxima and minima. By comparing the variation curves of DAGDOP and Pr across the tests, a strong negative correlation between the two can be observed: where DAGDOP reaches a local minimum, Pr attains a local maximum, and vice versa. The values of correlation coefficients between them also prove this re… view at source ↗
Figure 13
Figure 13. Figure 13: Variation of DAGDOP with respect to ∆ϕ4 and ∆ϕ5 in a 7-day arc 6 Discussion 6.1 DAGDOP and AOD accuracy In the above analysis, we employ the DAGDOP factor to reflect the AOD accuracy. In previous studies [28], the agreement between the OD accuracy and this factor has already been demonstrated. Here we also demonstrate its validity in our study. The AOD accuracy of the navigation constellation can also be … view at source ↗
Figure 14
Figure 14. Figure 14: Variation of the DAGDOP and the OD RMS with respect to A3 RMSm is calculated by: RMSm = vuutX N k=1 [PITH_FULL_IMAGE:figures/full_fig_p031_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Variation of DAGDOP with respect to the in-plane parameters of the L4 and L5 satellites • The in-plane parameters of L3 satellite As shown in [PITH_FULL_IMAGE:figures/full_fig_p032_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Variation of DAGDOP with respect to the in-plane parameters of L3 and DRO satellites As shown in [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Variation of DAGDOP with the support of three GNSS satellites 6.4 The Choice of Satellite Maneuver Timing Navigation satellites, including the L4 and L5 TLPO satellites, require periodic station￾keeping maneuvers throughout their operational lifetime [31]. After each maneuver, the orbits must be re-determined as quickly as possible, which corresponds to the short-arc AOD scenario in our study. Consequentl… view at source ↗
read the original abstract

With the community's increasing interest in the cislunar space, building a navigation constellation servicing the whole cislunar space has become a pressing need. Previous studies mainly focus on constellations using orbits close to the Moon, which limits the servicing volume of the constellation. In this work, a four-satellite constellation using one L3 orbit, one L4 orbit, one L5 orbit and an orbit close to the Moon is proposed. The orbit determination accuracy is an important factor to be considered when designing parameters of the constellation. In this study, the mode of autonomous orbit determination (AOD) based on inter-satellite range data is considered. With such a model, the out-of-plane design parameters are identified as the main parameters influencing the AOD accuracy. For the AOD based on short arcs, we find that the increase of the out-of-plane amplitude can improve the AOD accuracy, and the out-of-plane initial phases have a more complex influence. A novel relative planarity factor (RPF) $P_\text{r}$, which has negative correlation with the AOD accuracy, is proposed as the metric to evaluate the variation of AOD performance. Using $P_\text{r}$, we demonstrate that the coplanarity of the constellation can significantly reduce the AOD accuracy. For the long arc AOD, the influence of different parameters is insignificant.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes a four-satellite cislunar navigation constellation using one L3 orbit, one L4 orbit, one L5 orbit, and one orbit close to the Moon. It analyzes autonomous orbit determination (AOD) accuracy based solely on inter-satellite range measurements, identifying out-of-plane design parameters (amplitude and initial phase) as the dominant influences. For short-arc AOD, larger out-of-plane amplitudes improve accuracy while phases have complex effects; a novel relative planarity factor Pr is introduced with negative correlation to AOD accuracy, showing that coplanarity degrades performance. Long-arc AOD exhibits insignificant parameter sensitivity.

Significance. If the simulation results hold under realistic conditions, the work provides a timely geometric design metric (Pr) for cislunar constellations that could simplify optimization of AOD performance, addressing a pressing need for navigation infrastructure beyond lunar orbits. The identification of out-of-plane parameters as primary drivers offers a concrete, falsifiable guideline for constellation geometry.

major comments (2)
  1. [AOD model description] Section describing the AOD model and measurement assumptions: the central claims on out-of-plane parameter dominance and the negative correlation of Pr with AOD accuracy rest on the assumption that inter-satellite ranges are the sole observable and are free of unmodeled biases, clock errors, or dynamical perturbations (SRP, third-body gravity, lunar gravity errors). No evidence is provided that these effects were injected into the simulations or that the observability analysis accounts for them; if present at realistic levels, the reported sensitivities and utility of Pr as a design metric could change substantially.
  2. [Short-arc AOD results] Short-arc AOD simulation results (results section): the reported improvement with out-of-plane amplitude, complex phase effects, and the demonstration that coplanarity reduces accuracy via Pr are derived from idealized bias-free measurements. Without additional runs that include perturbations, these findings are load-bearing for the design recommendations but lack the validation needed to confirm they are not artifacts of the noise-free model.
minor comments (3)
  1. [Abstract and introduction] The abstract and introduction should explicitly state the orbit propagator fidelity, measurement noise model (e.g., white noise standard deviation), and arc lengths used in the simulations to allow reproducibility.
  2. [Definition of Pr] Clarify the exact mathematical definition of the relative planarity factor Pr (including how it is computed from the four orbital planes) in the main text rather than relying on the abstract description.
  3. [Figures] Figure captions for any constellation geometry or error plots should include the specific parameter values (amplitudes, phases) corresponding to each curve to aid interpretation of the Pr correlation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and for recognizing the potential utility of the relative planarity factor Pr. We address the two major comments point by point below, proposing targeted revisions to clarify assumptions while preserving the scope of this conceptual study.

read point-by-point responses

  1. Referee: [AOD model description] Section describing the AOD model and measurement assumptions: the central claims on out-of-plane parameter dominance and the negative correlation of Pr with AOD accuracy rest on the assumption that inter-satellite ranges are the sole observable and are free of unmodeled biases, clock errors, or dynamical perturbations (SRP, third-body gravity, lunar gravity errors). No evidence is provided that these effects were injected into the simulations or that the observability analysis accounts for them; if present at realistic levels, the reported sensitivities and utility of Pr as a design metric could change substantially.

    Authors: We agree that the analysis is performed under idealized assumptions with bias-free inter-satellite ranges and a simplified dynamical model. The manuscript is a conceptual numerical study focused on geometric observability effects rather than end-to-end realism. We will revise the AOD model section to explicitly list the measurement assumptions, add a dedicated limitations paragraph, and state that the reported sensitivities and Pr correlation apply to this noise-free case. The identification of out-of-plane dominance remains a valid geometric insight within the modeled conditions, but we will not claim direct applicability to perturbed scenarios. revision: partial

  2. Referee: [Short-arc AOD results] Short-arc AOD simulation results (results section): the reported improvement with out-of-plane amplitude, complex phase effects, and the demonstration that coplanarity reduces accuracy via Pr are derived from idealized bias-free measurements. Without additional runs that include perturbations, these findings are load-bearing for the design recommendations but lack the validation needed to confirm they are not artifacts of the noise-free model.

    Authors: The short-arc results are generated from the idealized model described in the methods. We will expand the results and conclusions sections to stress that the amplitude, phase, and Pr findings are specific to bias-free measurements and to recommend that the Pr metric be validated with realistic perturbations in subsequent studies. No new perturbed simulations are available for this revision; the current work is positioned as providing baseline geometric guidance rather than validated operational recommendations. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent simulation and geometric definition

full rationale

The paper defines a four-satellite cislunar constellation, performs AOD simulations using inter-satellite range measurements as the sole observable, identifies out-of-plane parameters as dominant via those simulations, and introduces a geometrically defined RPF Pr that is then shown to correlate negatively with the simulated AOD accuracy. No quoted step reduces a claimed prediction or result to an input by construction (e.g., no fitted parameter renamed as prediction, no self-definitional loop, no load-bearing self-citation chain). The central claims rest on external simulation outputs rather than tautological re-expression of the model inputs, making the derivation self-contained against the paper's own benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

The analysis rests on standard cislunar dynamical models and numerical AOD simulations whose free parameters are the out-of-plane amplitudes and phases; the RPF is introduced as a derived geometric quantity without external validation.

free parameters (2)
  • out-of-plane amplitude
    Varied to demonstrate improvement in short-arc AOD accuracy
  • out-of-plane initial phase
    Varied to show complex influence on AOD accuracy
axioms (1)
  • domain assumption Cislunar orbital motion can be propagated accurately enough for AOD studies using standard restricted three-body or ephemeris models.
    Implicit in the AOD simulation framework described in the abstract.
invented entities (1)
  • Relative planarity factor Pr no independent evidence
    purpose: Scalar metric that quantifies constellation coplanarity and correlates negatively with AOD accuracy
    Newly defined in the paper; no independent evidence supplied in abstract.

pith-pipeline@v0.9.1-grok · 5796 in / 1457 out tokens · 36376 ms · 2026-06-26T10:08:13.428175+00:00 · methodology

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