LQR Design For Formation Flying Near Halo Orbits Exploiting Quasi-Periodic Symmetry In Toroidal Coordinates

Joaquin G. Lopez-Cepero; Julio C. Sanchez; Rafael Vazquez

REVIEW 3 major objections 3 minor 13 references

LQR Design For Formation Flying Near Halo Orbits Exploiting Quasi-Periodic Symmetry In Toroidal Coordinates

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Quasi-periodic symmetry reduces infinite-horizon LQR for halo-orbit formation flying to a single-period computation.

desk verdict A genuinely new quasi-periodic LQR reduction with a clean proof, but the load-bearing stabilizability hypothesis is asserted rather than verified and the validation is too thin to fully support the accuracy claims. read the letter →

arxiv 2608.10726 v1 pith:66PW3OMF submitted 2026-08-11 physics.space-ph cs.SYeess.SYmath.OC

Joaquin G. Lopez-Cepero , Rafael Vazquez , Julio C. Sanchez This is my paper · ORCID

classification physics.space-phcs.SYeess.SYmath.OC

keywords linearquadraticregulatortoroidalcoordinatesquasi-periodicsymmetryhaloorbitscircularrestrictedthree-bodyproblemdifferenceRiccatiequationformationflyingcislunarspace

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to show that an infinite-horizon linear-quadratic regulator for relative motion near halo orbits can be designed with only one orbital period of offline computation. The trick is to write the relative dynamics in non-singular toroidal coordinates, where the time-varying coordinate frame rotates by a fixed angle every period. The paper proves, via a uniqueness argument on the stabilizing solution of the difference Riccati equation, that the Riccati matrices and optimal gains at later periods are just rotated copies of the first-period ones. If correct, this removes the need to store an infinite, non-repeating gain sequence and makes the optimal controller practical for onboard cislunar formation flying. Simulations on the nonlinear CR3BP and on an ephemeris model with solar perturbation support the claim, with a 40% reduction in total control effort compared with an impulsive baseline.

What carries the argument

The central object is the orthogonal matrix $\Gamma = \mathrm{blockdiag}(R_{\theta_\lambda}, R_{\theta_\lambda})$, describing how the non-singular toroidal frame returns after one reference-orbit period: $R(t+T) = R(t) R_{\theta_\lambda}$, where $R_{\theta_\lambda}$ is a rotation by the center-mode angle $\theta_\lambda$. This quasi-periodic symmetry of the coordinate transformation turns the otherwise non-periodic linear time-varying dynamics into a quasi-periodic system with $A_{k+N_p} = \Gamma^T A_k \Gamma$ and $B_{k+N_p} = \Gamma^T B_k$. The argument then invokes the standard uniqueness of the bounded stabilizing solution of the difference Riccati equation under uniform stabilizability and detectability, and shows the rotated sequence $\Gamma^T P_k \Gamma$ satisfies the same recursion, forcing $P_{k+N_p} = \Gamma^T P_k \Gamma$ and $K_{k+N_p} = K_k \Gamma$.

What would settle it

Compute sliding-window controllability and observability Gramians of $(A_k, B_k)$ along the L1 halo orbit over many periods: if their singular values are not uniformly bounded away from zero and infinity, or if $R(t)$ becomes singular, the uniqueness argument for the Riccati solution fails and the relation $P_{k+N_p} = \Gamma^T P_k \Gamma$ is not guaranteed.

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Extended reading notes

Core claim

On the discrete-time linear time-varying system obtained in toroidal coordinates, the state transition and input matrices satisfy $A_{k+N_p} = \Gamma^T A_k \Gamma$ and $B_{k+N_p} = \Gamma^T B_k$, where $\Gamma$ is an orthogonal matrix that rotates the toroidal frame by the center-mode angle over one orbital period. When the state weight $Q$ is invariant under this rotation, $\Gamma^T Q \Gamma = Q$, the paper proves that the unique stabilizing solution of the infinite-horizon difference Riccati equation obeys $P_{k+N_p} = \Gamma^T P_k \Gamma$, and consequently the optimal feedback gains satisfy $K_{k+N_p} = K_k \Gamma$. Therefore all future gains are obtained from the gains of a single period by a fixed rotation. The paper validates this controller on an L1 Northern halo orbit in the Earth-Moon system, first against the full nonlinear CR3BP and then against a high-fidelity ephemeris model, showing successful reconfiguration on the invariant torus with about 40% lower total delta-v than an impulsive targeting-plus-station-keeping baseline at comparable tracking accuracy.

Load-bearing premise

The proof assumes the discrete linear time-varying pair $(A_k, B_k)$ in toroidal coordinates is uniformly stabilizable and detectable, and that the toroidal frame $R(t)$ stays invertible along the orbit; the paper does not verify either condition for the specific halo orbit.

Editorial extensions

If this is right

Only one orbital period of Riccati recursion needs to be computed offline; every later gain is a rotation of that period's gains.
Onboard storage shrinks from an infinite gain sequence to one period of gains plus the scalar rotation angle $\theta_\lambda$.
The quasi-periodic LTV system admits a well-defined infinite-horizon LQR with finite memory, which is not possible for a generic LTV system without such symmetry.
On the ephemeris model, the controller keeps RMS position error below 400 m after the first revolution and reduces total delta-v from 2.99 m/s to 1.79 m/s compared with the impulsive baseline.
Tuning the weight matrices trades control effort against perilune error spikes, so the same controller can be made more or less aggressive.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

By the same logic, any quasi-periodic system whose dynamics are invariant up to an orthogonal congruence should admit a one-period Riccati computation; the proof does not use specific CR3BP structure beyond the $\Gamma$ relation.
A numerical verification of uniform stabilizability and detectability on the sampled halo orbit would close the gap between the Proposition's hypotheses and the tested case.
The compact gain representation could be combined with model predictive control terminal ingredients to obtain receding-horizon stability guarantees, as the paper itself suggests as future work.
The 40% delta-v comparison is for one reconfiguration scenario; testing across phasings near perilune, as done for the CR3BP case, would show whether the saving persists on the ephemeris model.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a real result. The single-period gain storage theorem is clean, the algebra checks out, and the 40% fuel saving appears genuine. The main gap is that uniform stabilizability is asserted, not verified, and the validation is thinner than the abstract suggests. With a controllability check and a more honest accuracy statement, this becomes a solid paper.

The genuinely new thing is the Proposition after Eq. (25). The toroidal LTV system has the quasi-periodic structure A_{k+Np} = Gamma^T A_k Gamma and B_{k+Np} = Gamma^T B_k, where Gamma is an orthogonal rotation. The paper shows by direct substitution that if Q is invariant under Gamma, the difference Riccati solution satisfies P_{k+Np} = Gamma^T P_k Gamma, and hence the optimal gains are K_{k+Np} = K_k Gamma. I traced the algebra; the shifted-sequence uniqueness argument is legitimate. This is a real extension of periodic LQR (Lyapunov-Floquet) to the toroidal quasi-periodic case, and it is not present in the cited literature. No fitted parameters hide in the result. Credit where it is due.

The soft spot is not the algebra but the hypothesis. The Proposition relies on uniform stabilizability and detectability to identify the unique bounded stabilizing solution. Detectability is automatic because Q > 0. Uniform stabilizability is never checked for the specific L1 halo orbit. Because the system is quasi-periodic up to orthogonal similarity, uniform stabilizability is equivalent to controllability of the lifted single-period system; a rank check of the one-period controllability Gramian would settle it. The closed-loop simulations suggest stabilizability holds for the tested case, but that is an empirical hint, not proof. If an uncontrollable mode were neutrally stable or unstable, the uniqueness argument would break, and with it the single-period storage claim. The invertibility of R(t) is also stated but not verified; it is unlikely to fail for a non-rectilinear mode, but it should be checked in the same pass.

The numerical validation is single-orbit and single-baseline. The headline 'comparable tracking accuracy' is generous. Table 1 gives LQR RMS 0.394 km versus baseline 0.174 km, and max errors 2.06 km versus 1.69 km. That is roughly a factor of two worse accuracy. The honest statement is that the LQR trades accuracy for a 40% fuel saving. The ephemeris test also uses spline-interpolated gains, which is a reasonable engineering workaround but adds another layer of approximation. No code or data is shipped, which makes reproduction harder.

Who is this for: people working on formation flying near halo orbits, and anyone interested in quasi-periodic LQR reductions. It deserves a serious referee. My recommendation: conditional acceptance, with a required one-period controllability/Gramian check, a comment on R(t) invertibility, a softened accuracy comparison, and ideally a code/data release.

Referee Report

3 major / 3 minor

Summary. The paper designs an infinite-horizon LQR for relative motion control near periodic halo orbits in the CR3BP, formulated in non-singular toroidal coordinates. The main theoretical contribution is a Proposition showing that, for a quasi-periodic LTV system whose state transition and input matrices satisfy A_{k+Np}=Γ^T A_k Γ and B_{k+Np}=Γ^T B_k, and for a state weight Q satisfying Γ^T Q Γ=Q, the unique stabilizing solution of the difference Riccati equation satisfies P_{k+Np}=Γ^T P_k Γ. Consequently the optimal gains are determined by one orbital period of offline computation and can be extended online by a fixed rotation. The controller is tested on the full nonlinear CR3BP dynamics for an L1 Northern halo orbit in the Earth-Moon system and on a high-fidelity ephemeris model with solar perturbation, and is compared with an impulsive targeting plus station-keeping baseline, reporting a roughly 40% reduction in total ΔV at comparable tracking accuracy.

Significance. If the Proposition holds for the specific orbit used, the paper removes the apparent need to store an infinite non-repeating gain sequence for quasi-periodic toroidal LTV dynamics. The proof itself is a clean direct substitution into the Riccati recursion and is correct conditional on uniform stabilizability and detectability; no fitted or data-driven quantity enters the periodicity argument. The paper also demonstrates a working implementation on a realistic ephemeris model, and the comparison against an impulsive baseline gives a concrete, falsifiable performance claim. The main significance is therefore the compact gain representation and the accompanying numerical demonstration, both of which are valuable for practical cislunar formation flying.

major comments (3)

[§4.3, Proposition after Eq. (25)] The Proposition is load-bearing for the single-period gain storage claim, but its key hypothesis, uniform stabilizability and detectability of the discrete LTV pair (A_k,B_k), is asserted and never verified for the L1 halo orbit used in §5. Since Q is positive definite, detectability is automatic; the real condition to check is uniform stabilizability. For the quasi-periodic structure (A_{k+Np},B_{k+Np})=(Γ^T A_k Γ,Γ^T B_k), this is equivalent to a rank condition on the single-period controllability Gramian of the lifted system, and a short numerical check would settle it. As written, the uniqueness argument in Eqs. (28)-(30) guarantees P_{k+Np}=Γ^T P_k Γ only if that condition holds. A successful closed-loop simulation is consistent with stabilizability but does not prove it, and if the condition failed, the central claim that one period of offline computation suffices would not be guaranteed.
[§6, Table 1 and Fig. 8] The abstract and conclusions state that the proposed controller attains a tracking accuracy 'comparable' to the baseline, but the numbers in Table 1 do not directly support that wording: after the first revolution, the LQR RMS error is 0.394 km versus 0.174 km for the baseline, and the maximum error is 2.060 km versus 1.687 km. The 40% ΔV reduction is meaningful, but 'comparable' needs an operational definition, for example the same RMS or maximum-error threshold, or a Pareto comparison from the sensitivity analysis in Fig. 9. As reported, the baseline is roughly twice as accurate on RMS, so the comparison should be framed as a trade-off rather than an equivalence.
[§6.2, LQR controller performance on ephemeris model] The ephemeris validation consists of a single reconfiguration trajectory with gains obtained by cubic-spline interpolation of CR3BP-derived eigenvectors and gains. This introduces an approximation that is not quantified: the ephemeris trajectory is not exactly periodic, and the interpolation is performed over one CR3BP revolution while the actual revolution durations vary. The claim that the single-period gain representation remains valid on the ephemeris model would be substantially stronger if the authors reported a second epoch or reference phasing, or directly checked that the interpolated gains stabilize the linearized ephemeris model over several revolutions. As it stands, the validation demonstrates feasibility but not robustness of the one-period storage claim outside the CR3BP model.

minor comments (3)

[Eq. (17) and Eq. (37)] The sign convention for the rotation matrix R_{θλ} and the resulting Γ is not made explicit in the implementation. Since Eq. (31) gives K_{k+Np}=K_k Γ while Eq. (37) uses K(t)=K(s_curr)Γ^p, a reader cannot immediately tell whether the online rotation is by +mθ or -mθ; please state the convention clearly.
[Fig. 2 and Fig. 6] The bottom plots are labeled 'Acc. ΔV [mm/s]' but the text describes the quantity as accumulated control effort; please clarify whether the plotted value is cumulative and why the unit is mm/s rather than m/s.
[§6, Sensitivity analysis] There is a typo in 'Tuning appropiately the values' and the sensitivity figure would benefit from an explicit statement of whether the plotted curves correspond to the same simulation time window and the same initial/final states as Table 1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the quasi-periodic symmetry reduction is derived algebraically from the stated invariance condition, and the Riccati uniqueness step rests on an external textbook theorem.

full rationale

The central claim is the Proposition following Eq. (25): P_{k+Np} = Γ^T P_k Γ and K_{k+Np} = K_k Γ. This is established by direct substitution into the difference Riccati equation (Eqs. 28-30), using only the algebraic relations A_{k+Np}=Γ^T A_k Γ, B_{k+Np}=Γ^T B_k from Eq. (26), orthogonality of Γ, and the design invariance condition Γ^T Q Γ = Q. None of these quantities is fitted to make the result true: Q is chosen beforehand, Γ is constructed from the monodromy eigenvector rotation, and no simulated outcome enters the proof. The uniqueness of the stabilizing Riccati solution is invoked from a standard external reference (Anderson and Moore), not from the authors' own prior work, so the proof is not a self-citation chain. The toroidal-coordinate formalism is adopted from Elliott and Bosanac (Ref. 5), but the periodicity relation T(t+T)=T(t)Γ and the shifted Riccati argument are derived in the present paper from Eq. (19) and the eigenvector relation Eq. (15). The 40% fuel saving and the simulation errors are post-hoc performance results, not inputs to the derivation. The main genuine gap — uniform stabilizability of the toroidal LTV pair is asserted rather than verified for the specific halo orbit — is a correctness/verification concern, not a circularity: the claim would fail if the assumption were false, but it is not secretly assumed as the conclusion. No step reduces by construction to its own input, so the circularity score is 0.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central symmetry proof rests on three nonstandard premises: the availability of a center mode, the invertibility of the toroidal transformation, and uniform stabilizability and detectability. The first two are properties of the selected orbit; the third is a standard control-theoretic condition that is asserted, not established. The Q invariance condition is a design assumption. No new physical entities are introduced.

free parameters (4)

CR3BP LQR weights Q, W = Q=diag(1e-2,1e-2,1e-2,1e-3,1e-3,1e-3), W=diag(1,1,1)
Chosen by hand to shape reference tracking and control effort; not fitted to data. The CR3BP validation results depend on them.
Ephemeris LQR weights Q, W = Q=diag(100,100,100,0.1,0.1,0.1), W=diag(1,1,1)*1e-2
More aggressive position weighting chosen by hand to counter ephemeris mismatch; not fitted to data.
Sampling time step Delta t = 26.89 minutes
Defines the discrete-time dynamics and the number of steps per period; chosen for the simulation, not optimized.
Initial separation scale epsilon = 1.12 km from reference orbit
Sets the initial state Z0=[0,0.2566e-4,0,0,0,0]^T; chosen as the test scenario, not fitted.

assumptions (6)

domain assumption The reference halo orbit has a non-trivial center mode with complex conjugate eigenvalues e^{+-i omega} on the unit circle.
Required for the toroidal coordinate construction from the oscillatory mode; the selected orbit has stability index close to unity. Invoked in 'Non-singular toroidal coordinates definition', Eq. (15).
domain assumption The toroidal coordinate basis vectors rr and ri remain non-aligned so that R(t) is invertible for all t.
Necessary for T(t) invertible and for the toroidal state transformation to be non-singular. Invoked in 'Non-singular toroidal coordinates definition'.
ad hoc to paper The pair (A_k, B_k) is uniformly stabilizable and detectable for the discrete-time LTV system.
Assumed in the Proposition to invoke uniqueness of the bounded stabilizing solution of the difference Riccati equation; not verified for the toroidal dynamics. Invoked in 'Controller design, LQR problem resolution'.
ad hoc to paper The weight matrix Q satisfies Gamma^T Q Gamma = Q (e.g. equal weights for the alpha and beta coordinates).
Imposed to make the Gamma-symmetry of the Riccati solution hold; it is a sufficient design condition, not a general requirement.
standard math Standard Riccati equation theory for infinite-horizon LTV systems (existence, uniqueness, and stability of the solution).
Background theorem from Anderson and Moore [10], used to identify the candidate as the unique stabilizing solution of the shifted system.
standard math CR3BP equations and monodromy matrix properties, including Eq. (6) for STM over periods.
Standard astrodynamics background used throughout the modeling and the derivation.

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Cite this review

Pith. "Pith review of LQR Design For Formation Flying Near Halo Orbits Exploiting Quasi-Periodic Symmetry In Toroidal Coordinates." pith.science (2026). https://pith.science/paper/66PW3OMF

@misc{pith2026260810726,
  author       = {Pith},
  title        = {Pith review of: LQR Design For Formation Flying Near Halo Orbits Exploiting Quasi-Periodic Symmetry In Toroidal Coordinates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/66PW3OMF}},
  note         = {Machine review of arXiv:2608.10726}
}

read the original abstract

This work presents the design of a Linear Quadratic Regulator for relative motion control near periodic orbits in the Circular Restricted Three-Body Problem (CR3BP), formulated in non-singular toroidal coordinates. The key result exploits the rotational quasi-periodicity of the toroidal coordinate transformation. A uniqueness argument on the stabilizing solution of the associated difference Riccati equation proves the optimal control gains need only be computed for a single orbital period. The controller is validated against the full non-linear CR3BP dynamics on an L1 Northern halo orbit in the Earth-Moon system, and further against a high-fidelity ephemeris model, demonstrating successful reconfiguration maneuvers on the invariant torus with low control effort. Compared with an impulsive targeting and station-keeping baseline, the proposed controller attains a comparable tracking accuracy while reducing the total control effort by approximately 40%.

Figures

Figures reproduced from arXiv: 2608.10726 by the authors.

**Figure 2.** Evolution of the relative error with respect to the [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Total reconfiguration cost as a function of initial [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Invariant curve of the reference orbit in the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Recovered trajectory for the ephemeris model at th [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: Evolution of the relative error with respect to the [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: Trajectory followed on the α −β plane for the ephemeris model simulation, together with the invariant curve at initial time [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: Position error comparison between the LQR and the b [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

**Figure 9.** Figure 9: Performance evolution of the LQR controller on the [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Options for staging orbits i n cislunar space,

R. Whitley and R. Martinez, “Options for staging orbits i n cislunar space,” 2016 IEEE Aerospace Con- ference, IEEE, 2016, pp. 1–9

work page 2016
[2]

Earth-M oon Near Rectilinear Halo and Butterﬂy Orbits for Lunar Surface Exploration,

R. Whitley, D. Davis, L. Burke, and K. C. Howell, “Earth-M oon Near Rectilinear Halo and Butterﬂy Orbits for Lunar Surface Exploration,” AAS/AIAA Astrodynamics Specialist Conference, 2018

work page 2018
[3]

Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies

V . Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies . Academic Press, 1967

work page 1967
[4]

Mo del predictive control for formation recon- ﬁguration exploiting quasi-periodic tori in the cislunar e nvironment,

A. Capannolo, G. Zanotti, M. Lavagna, and G. Cataldo, “Mo del predictive control for formation recon- ﬁguration exploiting quasi-periodic tori in the cislunar e nvironment,” Nonlinear Dynamics , V ol. 111, No. 8, 2023, pp. 6941–6959

work page 2023
[5]

Describing relative motion n ear periodic orbits via local toroidal coordi- nates,

I. Elliott and N. Bosanac, “Describing relative motion n ear periodic orbits via local toroidal coordi- nates,” Celestial Mechanics and Dynamical Astronomy , V ol. 134, No. 2, 2022, p. 19

work page 2022
[6]

Optimal periodic co ntroller for formation ﬂying on libration point orbits,

H. Peng, J. Zhao, Z. Wu, and W . Zhong, “Optimal periodic co ntroller for formation ﬂying on libration point orbits,” Acta Astronautica, V ol. 69, No. 7-8, 2011, pp. 537–550

work page 2011
[7]

C ontrol design for rendezvous operation near Halo Orbits using Lyapunov-Floquet theory and AUTO,

J. G. L´ opez-Cepero, J. Galan-Vioque, and R. V azquez, “C ontrol design for rendezvous operation near Halo Orbits using Lyapunov-Floquet theory and AUTO,” CEAS EuroGNC 2024, 2022

work page 2024
[8]

Chance-const rained model predictive control for near recti- linear halo orbit spacecraft rendezvous,

J. C. Sanchez, F. Gavilan, and R. V azquez, “Chance-const rained model predictive control for near recti- linear halo orbit spacecraft rendezvous,” Aerospace Science and T echnology, V ol. 100, 2020, p. 105827

work page 2020

Show all 13 references

[9]

Impulsive control of formati ons near invariant tori via local toroidal co- ordinates,

I. Elliott and N. Bosanac, “Impulsive control of formati ons near invariant tori via local toroidal co- ordinates,” Proceedings of the AAS/AIAA Astrodynamics Specialist Virt ual Conference, Online , 2021, pp. 9–12

2021
[10]

B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods . Englewood Cliffs, NJ: Prentice-Hall, 1990

1990
[11]

Ancillary data services of NASA ’s navig ation and ancillary information facility,

C. H. Acton Jr, “Ancillary data services of NASA ’s navig ation and ancillary information facility,” Plan- etary and Space Science , V ol. 44, No. 1, 1996, pp. 65–70

1996
[12]

A look t owards the future in the handling of space science mission geometry,

C. Acton, N. Bachman, B. Semenov, and E. Wright, “A look t owards the future in the handling of space science mission geometry,” Planetary and Space Science , V ol. 150, 2018, pp. 9–12

2018
[13]

Survey constrained model predictive control: Stability and optimality,

D. Q. Mayne, J. B. Rawlings, C. V . Rao, and P . O. Scokaert, “Survey constrained model predictive control: Stability and optimality,” Automatica (Journal of IF AC), V ol. 36, No. 6, 2000, pp. 789–814. 17

2000

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Options for staging orbits i n cislunar space,

R. Whitley and R. Martinez, “Options for staging orbits i n cislunar space,” 2016 IEEE Aerospace Con- ference, IEEE, 2016, pp. 1–9

work page 2016

[2] [2]

Earth-M oon Near Rectilinear Halo and Butterﬂy Orbits for Lunar Surface Exploration,

R. Whitley, D. Davis, L. Burke, and K. C. Howell, “Earth-M oon Near Rectilinear Halo and Butterﬂy Orbits for Lunar Surface Exploration,” AAS/AIAA Astrodynamics Specialist Conference, 2018

work page 2018

[3] [3]

Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies

V . Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies . Academic Press, 1967

work page 1967

[4] [4]

Mo del predictive control for formation recon- ﬁguration exploiting quasi-periodic tori in the cislunar e nvironment,

A. Capannolo, G. Zanotti, M. Lavagna, and G. Cataldo, “Mo del predictive control for formation recon- ﬁguration exploiting quasi-periodic tori in the cislunar e nvironment,” Nonlinear Dynamics , V ol. 111, No. 8, 2023, pp. 6941–6959

work page 2023

[5] [5]

Describing relative motion n ear periodic orbits via local toroidal coordi- nates,

I. Elliott and N. Bosanac, “Describing relative motion n ear periodic orbits via local toroidal coordi- nates,” Celestial Mechanics and Dynamical Astronomy , V ol. 134, No. 2, 2022, p. 19

work page 2022

[6] [6]

Optimal periodic co ntroller for formation ﬂying on libration point orbits,

H. Peng, J. Zhao, Z. Wu, and W . Zhong, “Optimal periodic co ntroller for formation ﬂying on libration point orbits,” Acta Astronautica, V ol. 69, No. 7-8, 2011, pp. 537–550

work page 2011

[7] [7]

C ontrol design for rendezvous operation near Halo Orbits using Lyapunov-Floquet theory and AUTO,

J. G. L´ opez-Cepero, J. Galan-Vioque, and R. V azquez, “C ontrol design for rendezvous operation near Halo Orbits using Lyapunov-Floquet theory and AUTO,” CEAS EuroGNC 2024, 2022

work page 2024

[8] [8]

Chance-const rained model predictive control for near recti- linear halo orbit spacecraft rendezvous,

J. C. Sanchez, F. Gavilan, and R. V azquez, “Chance-const rained model predictive control for near recti- linear halo orbit spacecraft rendezvous,” Aerospace Science and T echnology, V ol. 100, 2020, p. 105827

work page 2020