Generation of Energy-Optimal Low-Thrust Forced Periodic Trajectories in the CR3BP

Colby C. Merrill; Dmitry Savransky; Jackson Kulik

REVIEW 3 major objections 3 minor 12 references

Generation of Energy-Optimal Low-Thrust Forced Periodic Trajectories in the CR3BP

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

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Energy-optimal forced periodic trajectories near the L2 halo orbit form a hyperellipsoid in initial-state space, and lowering perilune distance is among the most expensive deviations.

desk verdict A clean specialization of reachable-set theory to forced periodic orbits, with a useful cost table—but validation covers only one direction and small amplitudes, so the headline numbers are extrapolations. read the letter →

arxiv 2411.11615 v1 pith:GUYN6322 submitted 2024-11-18 math.DS

Colby C. Merrill , Jackson Kulik , Dmitry Savransky This is my paper · ORCID

classification math.DS MSC 70F0770F1570Q05

keywords circularrestrictedthree-bodyproblemlow-thrustpropulsionforcedperiodictrajectoriesenergy-optimalcontrolreachablesethaloorbitEarth-MoonL2linearanalysis

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

This paper aims to establish that, for low-thrust spacecraft near a periodic orbit in the circular restricted three-body problem, the set of initial states that can be forced back to the same state after one period with bounded energy cost is exactly a six-dimensional ellipsoid computed from linearized optimal control. Working near a natural L2 halo orbit in the Earth-Moon system, the authors show that the linear energy cost of such a forced periodic trajectory is a quadratic form in the initial deviation, and that the reachable set is therefore the hyperellipsoid described by that quadratic form. The eigenvectors of the defining matrix rank the directions of deviation by cost. Under a representative 50 mN, 1000 kg spacecraft thrust constraint, the analysis finds that decreasing perilune distance is relatively expensive, which matters for lunar-observation mission design.

What carries the argument

The central object is $E^* = \begin{bmatrix} I_6 & I_6 \end{bmatrix} E \begin{bmatrix} I_6 & I_6 \end{bmatrix}^T$, where $E$ is built from the augmented state transition matrix $\Phi(t_f,t_0)$ and the cost kernel $\int (\Phi^{\lambda_v}_y)^T \Phi^{\lambda_v}_y \, dt$ after eliminating the initial costates through the linearized boundary-value constraint. The eigenvalue decomposition of this symmetric positive semidefinite matrix converts the constrained optimal-control problem into ellipsoidal geometry: each eigenvector names a direction of initial-state deviation and each eigenvalue sets that direction's cost. The largest semiaxis, corresponding to the smallest eigenvalue, gives an almost-free along-track direction, while the fifth eigenvector, which lowers perilune distance, has the second-largest eigenvalue and therefore the second-highest cost.

What would settle it

Take the same L2 halo reference and solve the full nonlinear energy-optimal two-point boundary value problem for initial deviations along the fifth eigenvector at magnitudes spanning the semiaxis extent; if the true cost deviates from $\frac{1}{2}\delta x_0^T E^* \delta x_0$ by more than a few percent in that direction, the ellipsoid's boundary and the perilune-cost ranking would need to be re-drawn.

Watch

Extended reading notes

Core claim

Under the linearized energy-optimal control model, the paper's central claim is that forced periodic trajectories near a natural periodic reference orbit are characterized by the symmetric matrix $E^*$, so that the cost to begin and end at the same initial deviation $\delta x_0$ is $J = \frac{1}{2}\delta x_0^T E^* \delta x_0$. The set of states with $J \le J^*$ is therefore the hyperellipsoid $\{\delta x_0 : \frac{1}{2}\delta x_0^T E^* \delta x_0 \le J^*\}$ with semiaxes $a_i = \sqrt{2J^*/\gamma_i}\, w_i$ from the eigenpairs $(\gamma_i, w_i)$ of $E^*$. For the chosen L2 halo orbit in the Earth-Moon system and a representative low-thrust spacecraft, the paper computes the semiaxis extents and shows numerically, for deviations along the fourth eigenvector, that the linear cost estimate tracks the full nonlinear energy-optimal cost to better than $0.1\%$ in the displayed range. From the eigenvector geometry it concludes that reducing perilune distance is the second-most expensive direction, hence relatively expensive.

Load-bearing premise

The linearized cost estimate is checked against the true nonlinear optimal cost in only one direction in state space, but the paper's ellipsoid sizes and its conclusion that lowering perilune is expensive assume that the same accuracy holds in every direction and at all deviations up to the energy limit.

Editorial extensions

If this is right

For any thrust-limited spacecraft, the semiaxes of the ellipsoid give an immediate feasibility check for a candidate forced periodic trajectory without solving a two-point boundary value problem.
The eigenvector cost ranking supplies a design rule: cheap deviations lie along the large-semiaxis directions, while small-semiaxis directions, including the perilune-lowering fifth eigenvector, should be reserved for high-priority mission objectives.
Because the costates are not periodic when the position and velocity are periodic, the optimal thrust profile repeats only in the state space, not in the control space, so the control must be re-planned each period.
The same $E^*$ matrix gives a full six-dimensional reachable set, so mission designers can map position and velocity bounds at apolune, perilune, and intermediate phases.
The linear analysis makes the energy-constrained reachable set explicit, turning a family of nonlinear optimal-control problems into a single eigenvalue computation.

Reading between the lines

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

The validation is confined to one eigenvector direction; extending the nonlinear Newton-Raphson check to the fifth eigenvector and to off-axis combinations would test whether the ellipsoid's extremal extents are as large as predicted.
The theoretically infinite semiaxis along the first eigenvector suggests a phase-shift direction that linear theory treats as cost-free; in practice any real phase change will carry a small nonzero cost that this model misses.
The same $E^*$ construction should transfer to other reference orbits, such as L1 halos, near-rectilinear halo orbits, or L4/L5 families, where the cost ranking of directions may differ; the perilune conclusion is specific to this L2 halo reference.
The cost ordering across eigenvectors could change if the quadratic control cost were replaced with a mass-optimal or minimum-time objective, so the ranking is tied to the energy-optimal formulation.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Quick take: the paper does one clean thing—it specializes the energy-limited reachable-set formalism from Kulik et al. to forced periodic orbits by imposing δxf = δx0, giving E* = [I I]E[I I]^T and a per-eigenvector cost ranking. That is a real, modest contribution. The derivation is standard and internally consistent, and the Newton-Raphson validation along the 4th eigenvector shows the linear estimate is excellent in that direction.

New and good: the forced-periodic constraint turns a 12-dimensional boundary-condition reachable set into a 6-dimensional hyperellipsoid in initial-state space; Table 1 is a useful cost-direction catalog for a concrete L2 halo reference; the observation that decreasing perilune distance is expensive is physically interesting and worth checking. The paper is honest about relying on prior work and does not oversell novelty.

Soft spots: the validation is narrower than the conclusions. Figure 1 only tests deviations along the 4th eigenvector, up to |dx| = 1e-3 DU, where the largest computed cost is about 1e-4 DU²/TU³. But Table 1 uses J* = 3.51e-4 DU²/TU³, with semiaxes of 4.6e-3 and 1.98e-2 DU—roughly 5 to 20 times the validated range. So the sentence saying J* 'falls in the range of costs with <1% error' is not supported by the data shown. The cost ordering across eigenvectors is also an extrapolation; nonlinear sensitivities are direction-dependent, and the perilune conclusion rests on the linearized 5th-eigenvector ranking. That may well be right, but it is not validated here. Minor: no code or data included, and conditioning of the Φxλ inversion is not discussed.

Bottom line: this is a competent conference-level contribution that deserves referee time in astrodynamics, but the claims about accuracy and the cost ranking should be softened or backed by validation in more directions and at the J* amplitudes. I would not use Table 1 for design yet.

Referee Report

3 major / 3 minor

Summary. This paper studies forced periodic trajectories in the circular restricted three-body problem (CR3BP) under energy-optimal low-thrust control. Using a linearized optimal-control formulation with a quadratic control cost, the authors derive that the set of initial-state deviations δx0 that can be returned to after one reference period with cost at most J* is a hyperellipsoid (1/2)δx0ᵀE*δx0 ≤ J*, with semi-axes given by the eigenvectors and eigenvalues of E* (Eqs. 20–22). They validate the linear estimate against a Newton-Raphson solver for deviations along one eigenvector (Figure 1), then apply the method to an L2 halo orbit in the Earth-Moon system, computing the semi-axis extents for a representative spacecraft (Table 1) and concluding that decreasing perilune distance is relatively expensive because it corresponds to the second-most-expensive eigenvector direction.

Significance. If the linearized ellipsoidal bound remains accurate at the amplitudes and directions used in Table 1, the paper offers a computationally cheap, systematic way to characterize low-thrust forced periodic trajectories and to rank the cost of state deviations in the CR3BP. The derivation is self-contained and standard: it follows from optimal-control theory, the state transition matrix, and an eigenvalue decomposition, with no fitted parameters other than the energy budget J*. The main strength is the transparent analytical framework, which could be useful for mission design. However, the quantitative conclusions depend on validation that currently covers only a small portion of the relevant state space, so the significance is conditional on the validation being extended.

major comments (3)

[Validation (Figure 1)] The validation is performed only for deviations along the 4th eigenvector and for |dx| ≤ 1e-3 DU, corresponding to costs up to approximately 1e-4 DU²/TU³. Table 1, however, lists semi-axis extents for J* = 3.51e-4 DU²/TU³, with the 4th-eigenvector extent at 4.6e-3 DU and the 2nd-eigenvector extent at 1.98e-2 DU. The statement that J* "falls in the range of costs with <1% error" is therefore not supported by the displayed data. Please extend the validation to deviations at the semi-axis extents along each eigenvector, or at least along the eigenvectors used for the conclusions, and report the maximum error in J.
[Investigation and Analysis (Table 1, Figure 3)] The conclusion that reducing perilune distance is relatively expensive relies on both the time history of the 5th eigenvector and on the eigenvalue ordering that places it as the second-most-expensive direction. Nonlinear corrections to the cost are generically direction-dependent, and the error at the semi-axis amplitudes could differ between the 4th and 5th eigenvectors. Please verify the cost ranking at J* by computing nonlinear optimal costs for initial deviations along each of the five finite eigenvectors at their semi-axis extents.
[Methods, Eq. (15)] The computation of E* requires inverting the 6x6 block Φx_λ. For a periodic reference orbit this block is related to the monodromy matrix and may be ill-conditioned due to the unit eigenvalue associated with the along-track direction. The paper does not report a condition number or any regularization. Because the near-zero eigenvalue of E* (first row of Table 1) indicates near-singularity, please include a brief conditioning analysis and discuss how the numerical eigenvalues and eigenvectors of E* are affected.

minor comments (3)

[References] Reference [7] contains a typo: "Reahcable" should be "Reachable"; several references also have extra spacing in "V ol" and "T able" that should be corrected.
[Figure 1 caption] The caption states that deviations are made in the direction of the "4th eigenvector in Table 1", but Table 1 lists eigenvectors by row order; this is ambiguous because rows are ordered by extent, not by eigenvector index. Please specify the eigenvector by its components or by a consistent label.
[After Eq. (24)] The phrase "maximum specific power" is imprecise: J* has units of DU²/TU³, which are units of specific energy, not power. Consider using "specific energy per orbit" or "control effort" for clarity.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the reachable-set ellipsoid is derived from the linearized optimal-control STM, and the validation uses an independent Newton-Raphson solve; the main weaknesses are validation-range extrapolation, not circularity.

full rationale

The central object E* in Eq. (19) is built by substituting the linearized boundary-value solution (Eq. (16)) into the costate integral (Eq. (14)); no parameter is fitted to the claimed ellipsoid or to the eigenvector cost ordering. The hyperellipsoid (Eq. (21)) and semi-axes (Eq. (22)) follow directly from the symmetric positive semi-definite quadratic form, so the derivation is self-contained once the STM is computed. Validation against a Newton-Raphson solve of the (nonlinear) energy-optimal boundary-value problem is external to the fitted values, and the paper reports sub-0.1% error over the shown deviation range. The paper does cite the authors' own prior work (Kulik et al. 2024, Ref. [7]) as the basis of the reachable-set formalism, and uses Ref. [11] for STMs, but the relevant formulas are re-derived in this manuscript and are also supported by external references [9,10] and Bryson [8]; thus the self-citation is not load-bearing. The significant limitation is not circularity: the validation in Figure 1 is along only the 4th eigenvector and over |dx| <= 1e-3 DU, while Table 1 reports semiaxes up to about 2e-2 DU for J* = 3.51e-4, so the linear accuracy at the full J* extent, and in other eigenvector directions, is asserted rather than demonstrated. That is an extrapolation and validation gap, not a reduction of the claim to its inputs.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The method rests on standard optimal control and linearization; the key assumptions are the global validity of the linear approximation (validated in one direction) and the invertibility of Φ_xλ. J* is an input scenario, not a fitted quantity. No new physical entities are introduced.

free parameters (1)

Energy budget J* = 3.51e-4 DU^2/TU^3
User-specified maximum specific power for a representative 1000 kg, 50 mN spacecraft; chosen partly because it falls in the validated <1% error range, so the reachable-set size in Table 1 depends on it.

assumptions (4)

domain assumption The 6x6 block Φ_xλ(t_f, t_0) is invertible, so initial costates are uniquely determined by the state boundary conditions (Eq 15).
Eq 15 uses (Φ_xλ)^-1; the existence of a zero eigenvalue in E* (infinite along-track extent) indicates a null direction, so this invertibility is not guaranteed and is used without conditioning analysis.
domain assumption The linearized STM and costate propagation accurately represent the energy-optimal control for all deviations with cost up to J*.
Validation in Figure 1 is performed along one eigenvector only; the paper assumes the <0.1% error holds for the full 6D reachable set.
domain assumption The reference halo orbit is periodic and its numerical initial conditions are accurate enough that the inherent cost (~3.5e-14 DU^2/TU^3) is negligible for computed costs.
Stated in the Validation section; the reference orbit comes from prior numerical work and is not machine-verified.
standard math Pontryagin's minimum principle and the costate equations for the energy-optimal control problem are valid (Eqs 6-8).
Standard optimal control theory from Bryson; not proved in the paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Generation of Energy-Optimal Low-Thrust Forced Periodic Trajectories in the CR3BP." pith.science (2026). https://pith.science/paper/GUYN6322

@misc{pith2026241111615,
  author       = {Pith},
  title        = {Pith review of: Generation of Energy-Optimal Low-Thrust Forced Periodic Trajectories in the CR3BP},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GUYN6322}},
  note         = {Machine review of arXiv:2411.11615}
}

read the original abstract

In this work, we investigate trajectories that require thrust to maintain periodic structure in the circular restricted three-body problem (CR3BP). We produce bounds in position and velocity space for the energy-constrained reachable set of initial conditions. Our trajectories are energy-optimal and analyzed via linear analysis. We provide validation for our technique and analyze the cost of deviating in various directions to the reference. For our given reference, we find that it is relatively expensive to decrease perilune distance for orbits in the Earth-Moon system.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 6 canonical work pages

[1]

write newline

" write newline "" before.all 'output.state := FUNCTION blank.sep after.quote 'output.state := FUNCTION fin.entry output.state after.quoted.block = 'skip 'add.period if write newline FUNCTION new.block output.state before.all = 'skip output.state after.quote = after.quoted.block 'output.state := after.block 'output.state := if if FUNCTION new.sentence out...
[2]

A. Cox, K. C. Howell, and D. C. Folta, ``Dynamical structures in a low-thrust, multi-body model with applications to trajectory design,'' Celestial Mechanics and Dynamical Astronomy , 2019, 10.1007/s10569-019-9891-7

work page doi:10.1007/s10569-019-9891-7 2019
[3]

A. Cox, K. C. Howell, and D. C. Folta, ``Trajectory Design Leveraging Low-Thrust, Multi-Body Equilibria and their Manifolds,'' Journal of Astronautical Sciences , 2020, 10.1007/s40295-020-00211-6

doi:10.1007/s40295-020-00211-6 2020
[4]

Morimoto, H

M. Morimoto, H. Yamakawa, and K. Uesugi, ``Periodic Orbits with Low-Thrust Propulsion in the Restricted Three-body Problem,'' Journal of Guidance, Control, and Dynamics , Vol. 29, No. 5, 2006, pp. 1131--1139, 10.2514/1.19079

doi:10.2514/1.19079 2006
[5]

Tsuruta, M

A. Tsuruta, M. Bando, S. Hokamoto, and D. J. Scheeres, ``New Equilibria and Dynamic Structures Under Continuous Optimal Feedback Control,'' Journal of Guidance, Control, and Dynamics , Vol. 0, No. 0, 2024, pp. 1--12, 10.2514/1.G008270

doi:10.2514/1.g008270 2024
[6]

DeLeo and M

L. DeLeo and M. Pontani, ``Low-Thrust Orbit Dynamics and Periodic Trajectories in the Earth–Moon System,'' Aerotecnica Missili and Spazio , 2022, 10.1007/s42496-022-00122-9

doi:10.1007/s42496-022-00122-9 2022
[7]

Sandel and R

C. Sandel and R. Sood, ``Natural and Forced Spacecraft Loitering in a Near Rectilinear Halo Orbit,'' Journal of Astronautical Sciences , Vol. 71, 2024, 10.1007/s40295-024-00446-7

doi:10.1007/s40295-024-00446-7 2024
[8]

Kulik, M

J. Kulik, M. Zweig, and D. Savransky, ``Comparing Relative Reahcable Sets About Nearly Circular Orbits,'' AAS/AIAA Astrodynamics Specialist Conference , 2024

work page 2024

Show all 12 references

[9]

A. E. Bryson, Applied optimal control: optimization, estimation and control . Routledge, 2018

2018
[10]

Lee and I

S. Lee and I. Hwang, ``Reachable set computation for spacecraft relative motion with energy-limited low-thrust,'' Aerospace Science and Technology , Vol. 77, 2018, pp. 180--188

2018
[11]

Sun and J

H. Sun and J. Li, ``Analysis on reachable set for spacecraft relative motion under low-thrust,'' Automatica , Vol. 115, 2020, p. 108864

2020
[12]

Kulik, W

J. Kulik, W. Clark, and D. Savransky, ``State Transition Tensors for Continuous-Thrust Control of Three-Body Relative Motion,'' Journal of Guidance, Control, and Dynamics , Vol. 46, No. 8, 2023, pp. 1610--1619, 10.2514/1.G007311

2023 doi

Pith tools

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

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION blank.sep after.quote 'output.state := FUNCTION fin.entry output.state after.quoted.block = 'skip 'add.period if write newline FUNCTION new.block output.state before.all = 'skip output.state after.quote = after.quoted.block 'output.state := after.block 'output.state := if if FUNCTION new.sentence out...

[2] [2]

A. Cox, K. C. Howell, and D. C. Folta, ``Dynamical structures in a low-thrust, multi-body model with applications to trajectory design,'' Celestial Mechanics and Dynamical Astronomy , 2019, 10.1007/s10569-019-9891-7

work page doi:10.1007/s10569-019-9891-7 2019

[3] [3]

A. Cox, K. C. Howell, and D. C. Folta, ``Trajectory Design Leveraging Low-Thrust, Multi-Body Equilibria and their Manifolds,'' Journal of Astronautical Sciences , 2020, 10.1007/s40295-020-00211-6

doi:10.1007/s40295-020-00211-6 2020

[4] [4]

Morimoto, H

M. Morimoto, H. Yamakawa, and K. Uesugi, ``Periodic Orbits with Low-Thrust Propulsion in the Restricted Three-body Problem,'' Journal of Guidance, Control, and Dynamics , Vol. 29, No. 5, 2006, pp. 1131--1139, 10.2514/1.19079

doi:10.2514/1.19079 2006

[5] [5]

Tsuruta, M

A. Tsuruta, M. Bando, S. Hokamoto, and D. J. Scheeres, ``New Equilibria and Dynamic Structures Under Continuous Optimal Feedback Control,'' Journal of Guidance, Control, and Dynamics , Vol. 0, No. 0, 2024, pp. 1--12, 10.2514/1.G008270

doi:10.2514/1.g008270 2024

[6] [6]

DeLeo and M

L. DeLeo and M. Pontani, ``Low-Thrust Orbit Dynamics and Periodic Trajectories in the Earth–Moon System,'' Aerotecnica Missili and Spazio , 2022, 10.1007/s42496-022-00122-9

doi:10.1007/s42496-022-00122-9 2022

[7] [7]

Sandel and R

C. Sandel and R. Sood, ``Natural and Forced Spacecraft Loitering in a Near Rectilinear Halo Orbit,'' Journal of Astronautical Sciences , Vol. 71, 2024, 10.1007/s40295-024-00446-7

doi:10.1007/s40295-024-00446-7 2024

[8] [8]

Kulik, M

J. Kulik, M. Zweig, and D. Savransky, ``Comparing Relative Reahcable Sets About Nearly Circular Orbits,'' AAS/AIAA Astrodynamics Specialist Conference , 2024

work page 2024