{"id":"1682031f-0902-4a9d-9352-4ef1a0e8ad44","arxiv_id":"2502.05140","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Mass-optimal forced periodic trajectories in the Earth-Moon CR3BP can be generated by warm-starting from energy-optimal solutions, and the thrust-limited reachable set appears to contain the linearized energy-limited set in the xy-plane.","lead":"This paper computes fuel-efficient, repeating spacecraft paths in the Earth-Moon system that use low thrust, and compares two ways of defining which paths are reachable. It finds that the thrust-limited family of paths covers a larger set of starting positions in the orbital plane than the linearized energy-limited family, which matters for planning cislunar missions.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed superset is an artifact of comparing a nonlinear thrust-limited set against a linearized energy set; exactly, any thrust-limited trajectory is energy-feasible, so the true energy-limited set contains it.","rationale":"The reader's weakest assumption identified the linearized energy hyperellipsoid as one of two unreliable proxies and called for a nonlinear energy baseline. My stress-test sharpens this into a definitive mathematical objection: for the exact definitions in Eqs. (20)-(21), every thrust-limited trajectory is automatically energy-feasible, so the exact thrust-limited reachable set is contained in the exact energy-limited reachable set. Consequently, the central claim as worded in the abstract is false regardless of sampling quality; the only way to obtain the reported superset is to compare against the linearized ellipsoid, which is not the true energy-limited set. The paper does acknowledge this in one sentence in the Results, but the abstract and conclusions do not carry the qualification. The methodological contribution (computing mass-optimal forced periodic trajectories with ASSET) may still be useful, but the headline reachable-set result does not establish a physical superset relation and would need to be reframed as 'the sampled nonlinear thrust-limited set extends beyond the linearized energy ellipsoid in the tested directions.' Because the central claim is not merely unproven but contradicted by the paper's own definitions, I recommend rejecting the paper in its current form, with the path to revision being a like-for-like nonlinear comparison and a corrected abstract.","tokens_in":9421,"tokens_out":5195,"duration_ms":51657,"concrete_test":"Compute the nonlinear energy-optimal reachable set boundary with the same ASSET/PSIOPT solver used for the thrust-limited set: for each of the 12 PSO direction vectors psi, maximize psi^T delta x0 subject to the periodic boundary constraint and J_E <= 1/2 * u_max^2 (tf - t0). Plot these points against Fig. 4. If any thrust-limited PSO point lies outside the nonlinear energy set, the superset claim survives; if, as the energy inequality predicts, the nonlinear energy boundary contains all thrust-limited points, the claimed superset is a linearization artifact.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's central claim, stated in the abstract and conclusions, is that the thrust-limited mass-optimal reachable set is a superset of the energy-limited energy-optimal reachable set in the xy-plane. This cannot be true for the exact sets defined in Eqs. (20)-(21). If a mass-optimal control satisfies ||u*_M(t)|| <= u_max on [t0, tf], then J_E(u*_M) = 1/2 * integral ||u*_M||^2 dt <= 1/2 * u_max^2 (tf - t0). Thus that control is feasible for the energy-limited problem, so the minimal energy cost for the same delta x0 is no larger than the energy limit. The state delta x0 therefore belongs to the energy-limited reachable set of Eq. (20). The exact thrust-limited reachable set is a subset, not a superset, of the exact energy-limited reachable set. The observed 'superset' in Fig. 4 can only mean that the linearized hyperellipsoid of Eq. (34) is smaller than the true energy-limited set in some directions. The paper's own Results section concedes this: 'if the same solution method is used to find an energy-optimal reachable set and then a mass-optimal reachable set, the energy-optimal reachable set would be a superset of the mass-optimal reachable set.' The abstract omits the qualifier 'linearized,' making the headline claim false as stated. The unsampled directions are a secondary concern; the primary issue is the asymmetric comparison between a nonlinear optimization and a first-order linearization.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a methodology for computing mass-optimal low-thrust forced periodic trajectories in the circular restricted three-body problem (CR3BP) using the ASSET/PSIOPT direct transcription framework, and uses a particle swarm optimization (PSO) scheme to sample the thrust-limited reachable set around a reference orbit. The authors compare this nonlinearly computed mass-optimal reachable set with the linearized energy-limited reachable set obtained from their prior work, and the abstract claims that the thrust-limited mass-optimal reachable set is a superset of the energy-limited energy-optimal reachable set in the xy-plane. The paper reports that the two sets are similar in projection and notes in the Results that the comparison is limited by the different assumptions and constraints of the two methods.","tokens_in":9784,"tokens_out":5240,"duration_ms":49261,"significance":"If the claim is suitably rephrased, the paper makes a useful methodological contribution: it provides a reproducible pipeline for generating mass-optimal forced periodic trajectories in the CR3BP, with explicit PSO hyperparameters, mesh refinement, and reintegration verification. The linearized energy-limited ellipsoid from prior work is a parameter-free construction, and the observation that nonlinear mass-optimal solutions can extend outside this linearized ellipsoid is a concrete, falsifiable finding. However, the headline claim as written is not supported by the paper's own definitions and is in fact contradicted by a simple inequality: every thrust-limited control is also energy-feasible, so the exact energy-limited reachable set must contain the exact thrust-limited reachable set. The paper's significance therefore hinges on reframing the comparison as one between the nonlinear mass-optimal set and the linearized energy ellipsoid, not the exact energy-limited set.","major_comments":[{"comment":"The stated superset claim is false for the exact sets defined in Eqs. (20) and (21). If a mass-optimal control satisfies ||u_M(t)|| ≤ u_max for all t, then J_E(u_M) = (1/2)∫||u_M||² dt ≤ (1/2)u_max²(t_f − t_0), so the same δx0 lies in the exact energy-limited set of Eq. (20). Hence the exact thrust-limited set is a subset, not a superset, of the exact energy-limited set. The result plotted in Fig. 4 can only mean that the linearized hyperellipsoid of Eq. (34) is smaller than the exact energy-limited set in the sampled directions. The abstract omits the qualifier 'linearized' and therefore makes a claim that is mathematically impossible; the Conclusions include the qualifier, but the abstract and the 'superset' wording throughout the paper need to be revised to state that the nonlinear mass-optimal set can extend beyond the linearized energy-limited ellipsoid.","section":"Reachable Set Definitions and Abstract"},{"comment":"The thrust-limited reachable set boundary is approximated by the heuristic JM/(umax(tf−t0)) > 0.95 and is sampled in only 12 azimuth directions, with ψ increased in increments of π/6. The paper itself states that the full set is incomplete due to imperfect sampling. A 95% thrust fraction does not guarantee proximity to the true boundary of the thrust-limited reachable set, and unsampled azimuth directions may contain states that would alter the comparison. As presented, the 'superset' statement is only established for the sampled directions and the chosen proxy threshold, not for the entire xy-plane.","section":"Particle Swarm Optimization and Fig. 4"},{"comment":"The comparison in Fig. 4 is asymmetric: the energy-limited reachable set is the first-order STM hyperellipsoid of Eq. (34), while the mass-limited set is obtained from full nonlinear optimization. The paper concedes in the Results that if the same solution method were used for both problems, the energy-optimal reachable set would be a superset of the mass-optimal reachable set. This concession indicates that the reported 'superset' is likely an artifact of comparing a linearized object with a nonlinear one. To support the paper's central claim, the authors should compute an energy-optimal reachable set with the same nonlinear tool (for example, the same ASSET/PSO pipeline) or provide a quantitative bound on the linearization error in Eq. (34); without this, the observed set difference cannot be attributed to the physical difference between mass and energy optimality.","section":"Energy-Limited Reachable Set and Results"}],"minor_comments":[{"comment":"The symbol ψ is used both for the weight vector in Eq. (36) and for the scalar azimuth angle in Eq. (40), which is confusing; one of these should be renamed.","section":"Equation (40)"},{"comment":"The Introduction contains a typo: 'equilibirum' should be 'equilibrium'; reference [16] contains 'Reahcable' instead of 'Reachable'.","section":"Introduction and References"},{"comment":"The caption should state that the 'Energy Optimal Set' is the projection of the linearized ellipsoid from Eq. (34), not a nonlinear reachable set, to avoid misleading readers.","section":"Figure 4 caption"},{"comment":"The notation N is used both for the normal distribution and for the random vector in Eq. (37); clarifying the distinction between the distribution and the sampled vector would improve readability.","section":"Equations (35) and (38)"},{"comment":"The stopping criterion JM/(umax(tf−t0)) > 0.95 should be justified; a trajectory can be near the reachable boundary with a lower thrust fraction if the control is not fully bang-bang, and the threshold is not derived from any error bound.","section":"Particle Swarm Optimization stopping criterion"}],"recommendation":"major_revision","confidential_remarks":"The manuscript appears to be a conference paper (AAS 25-153) whose central claim is overstated in the abstract. The paper relies heavily on the authors' prior work [11] for the linearized reachable set; the new contribution is mainly the PSO-based mass-optimal sampling. The editor may wish to encourage a revision that reframes the claim as a comparison of the nonlinear thrust-limited set with the linearized energy ellipsoid, rather than with the exact energy-limited set, and that adds a nonlinear energy-optimal set computation or a bound on the linearization error."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing to know: the mass-optimal forced periodic trajectory generator is a genuine and useful piece of work, but the abstract's headline claim—that the thrust-limited reachable set is a superset of the energy-limited set—is false as stated for the exact sets. It only holds because the energy-limited baseline is a linearized approximation. The paper's conclusions quietly say \"linearized\"; the abstract does not.\n\nWhat's new: the direct mass-optimal solve, warmed by an energy-optimal solution without homotopy, and the particle-swarm construction of a thrust-limited reachable set in the CR3BP. That is a real methodological step beyond the prior energy-optimal treatment. The two example trajectories show sane bang-bang structure, and the authors are candid about imperfect sampling.\n\nThe soft spot is the central comparison. Any control with ||u|| ≤ u_max has energy cost at most 1/2 u_max^2 (tf-t0), so the same trajectory is feasible for the energy-limited problem. That means the exact thrust-limited reachable set is a subset, not a superset, of the exact energy-limited set. The \"superset\" in Figure 4 is entirely explained by the linearized energy ellipsoid being smaller than the true nonlinear energy set in some directions. The paper acknowledges this in the Results section, but the abstract overstates it. There are also secondary issues: the thrust-limited boundary is proxied by a 95% thrust fraction, sampled in only twelve azimuth directions, and no like-for-like nonlinear energy-optimal baseline is computed. Error bars, code, and data would make the reachable-set comparison much easier to judge.\n\nThe paper deserves a serious referee because the methodology is worth keeping even though the headline claim needs fixing. The fix is straightforward: either rephrase the claim as a comparison against the linearized energy set, or compute a nonlinear energy-optimal reachable set with the same optimizer and compare honestly. With that revision, this is a solid conference or journal contribution for mission-design readers who care about low-thrust forced periodic orbits in cislunar space. I'd send it out, with a request for major revision on the claim.","headline":"The method is real and useful, but the abstract's superset claim only holds against a linearized baseline; as stated, it is false for the exact reachable sets.","tokens_in":10297,"tokens_out":3434,"would_cite":true,"duration_ms":31665,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["70F07","70Q05","49J15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes that the thrust-limited mass-optimal reachable set is a superset of the energy-limited energy-optimal reachable set in the xy-plane for low-thrust forced periodic trajectories in the Earth-Moon system.","keywords":["forced periodic trajectory","low-thrust trajectory optimization","mass-optimal control","reachable set","circular restricted three-body problem","Earth-Moon system","particle swarm optimization","bang-bang control"],"falsifier":"Recompute the energy-limited reachable set with the same nonlinear optimizer used for the mass-optimal set, and check whether every thrust-limited reachable state in the xy-plane is also energy-limited-reachable; if any mass-optimal thrust-limited state falls inside the nonlinear energy-limited set, the claimed superset fails. A cheaper check is to sample trajectories in additional azimuth directions beyond the twelve used here and see whether any sampled thrust-limited reachable state lies inside the energy ellipsoid.","tokens_in":9202,"feed_emoji":"🛰","tokens_out":10523,"duration_ms":87909,"temperature":0.7,"pith_summary":"Natural periodic orbits in the Earth-Moon system occupy only a limited volume of Cislunar space, and this paper asks whether low-thrust propulsion can legitimately expand the set of periodic trajectories available to a spacecraft. The paper develops a numerical method for finding mass-optimal forced periodic trajectories in the circular restricted three-body problem, enforcing that the spacecraft returns to its starting state after one period while thrust magnitude is capped. Its central finding is that the reachable set of these nonlinear, thrust-limited, mass-optimal trajectories is a superset of the reachable set predicted by a linearized energy-optimal analysis, at least when both are projected into the xy-plane. A sympathetic reader would care because propellant-minimizing trajectories that respect thrust limits are what operational spacecraft actually fly, so a wider reachable set means more usable periodic starting states for missions in Cislunar space.","feed_headline":"Low-thrust mass-optimal flights reach a wider set of periodic orbits","feed_subtitle":"Thrust-limited bang-bang trajectories cover more cislunar starting states than the energy-limited ellipsoid predicts.","key_machinery":"The object that carries the argument is the reachable set around a fixed reference trajectory, with two constructions. The energy-limited construction is the symmetric matrix $E^*$: combining the augmented state transition matrix $\\Phi$ with the linearized boundary-value map produces a quadratic cost $\\frac{1}{2}\\delta x_0^T E^* \\delta x_0$ for returning to a perturbed initial state $\\delta x_0$, and the reachable starting states are exactly the hyperellipsoid where this cost is at most $\\frac{1}{2}u_{\\max}^2(t_f-t_0)$. The thrust-limited construction is the nonlinear optimal control problem with cost $J_M = \\int_{t_0}^{t_f} \\|u(t)\\|\\,dt$ and constraint $\\|u(t)\\| \\le u_{\\max}$, whose boundary is sampled by an accelerated particle swarm optimizer with fitness $\\psi^T \\delta x$ and a stopping rule driven by $J_M/(u_{\\max}(t_f-t_0)) > 0.95$. The comparison of the two sets in the xy-plane is the load-bearing comparison of the paper.","core_discovery":"The paper's central claim, stated in the abstract and conclusions, is that the reachable set of nonlinear mass-optimal thrust-limited trajectories is a superset of the linearized energy-optimal energy-limited trajectories in the xy-plane. The energy-limited reachable set is a six-dimensional hyperellipsoid built from a linearized optimal-control analysis: states that can be returned to in one period with energy cost at most $\\frac{1}{2}u_{\\max}^2(t_f-t_0)$ form the ellipsoid $\\{\\delta x_0 : \\frac{1}{2}\\delta x_0^T E^* \\delta x_0 \\le \\frac{1}{2}u_{\\max}^2(t_f-t_0)\\}$. The thrust-limited mass-optimal set is obtained by solving the full nonlinear problem with cost $J_M = \\int_{t_0}^{t_f} \\|u(t)\\|\\,dt$ and constraint $\\|u(t)\\| \\le u_{\\max}$, then sampling the boundary with a particle swarm search that seeks states whose mass cost exceeds 95% of the theoretical maximum. Because the two optimization methods use different assumptions and constraints, the paper argues, the energy-optimal set is not automatically the larger one; the comparison shows the mass-optimal set contains it in the xy-plane. The authors attribute the difference to the alterations that linearization makes to the dynamics, and they note the two sets look similar in that projection.","pith_inferences":["Editorial inference: if the superset holds in other projections, the linearized ellipsoid is a conservative bound on reachable volume, and mission designers could search a wider starting-state region for forced periodic operations than the energy analysis alone would suggest.","Editorial inference: the consistent bang-bang pattern of a long initial burn and a short perilune burn suggests that a reduced control model with two thrust arcs might approximate the mass-optimal family closely enough for fast preliminary design; that is a testable simplification, not a claim of the paper.","Editorial inference: the twelve-direction PSO sampling is a coarse boundary estimate; a denser angular sweep would reveal whether the apparent superset saturates or whether unsampled directions contain thrust-limited reachable states inside the energy ellipsoid."],"forward_implications":["Forced periodic trajectories can be generated directly as mass-optimal solutions by feeding an energy-optimal solution into the mass-optimal problem, without smoothing or homotopy.","Mass-optimal forced periodic trajectories have bang-bang thrust profiles, typically an extended burn at the start and a short burn near perilune.","The xy-plane projection of the thrust-limited reachable set is at least as large as the energy-limited reachable set, so linearized energy analysis may understate which periodic starting states are reachable.","Many energy-optimal trajectories already have thrust magnitudes near the mass-optimal maximum even though they do not enforce a thrust bound, so the two solution families are close in thrust usage."],"supporting_citations":[{"why":"Provides the linearized energy-optimal reachable set analysis and the reference trajectory whose reachable sets are compared throughout the paper.","marker":"[11]"},{"why":"Supplies the trajectory optimization toolkit used to solve the nonlinear energy- and mass-optimal control problems with collocation and mesh refinement.","marker":"[13]"},{"why":"Establishes the energy-limited reachable set formulation for spacecraft relative motion that the hyperellipsoid construction relies on.","marker":"[17]"},{"why":"Supplies the companion low-thrust reachable set analysis that justifies the energy-limited ellipsoid form.","marker":"[18]"},{"why":"Introduces particle swarm optimization, the method the authors adapt to sample the thrust-limited reachable set boundary.","marker":"[19]"},{"why":"Provides the accelerated particle swarm variant used in the reachable-set search.","marker":"[20]"},{"why":"Prior comparison of thrust-limited and energy-limited reachable sets for near-circular orbits that motivates the present comparison in the CR3BP.","marker":"[16]"}],"fun_headline_variants":["Mass-optimal thrust beats energy-limited reach in cislunar orbits","Thrust-limited periodic paths cover more ground than energy ellipsoid","Mass-optimal cislunar orbits outsize energy-limited reachable set","Cislunar mass-optimal trajectories supersede energy-limited set","Thrust-limited orbits surpass energy ellipsoid in cislunar space"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The superset conclusion depends on the linearized hyperellipsoid being a faithful stand-in for the true energy-limited reachable set and on the particle-swarm collection of thrust-limited trajectories being a faithful stand-in for the full thrust-limited reachable set; the paper itself notes that the full thrust-limited set is incomplete because the sampling is imperfect.","fun_headline_variants_meta":{"raw":{"variants":["Mass-optimal thrust beats energy-limited reach in cislunar orbits","Thrust-limited periodic paths cover more ground than energy ellipsoid","Mass-optimal cislunar orbits outsize energy-limited reachable set","Cislunar mass-optimal trajectories supersede energy-limited set","Thrust-limited orbits surpass energy ellipsoid in cislunar space"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001035,"raw_usage":{"total_tokens":4359,"prompt_tokens":946,"completion_tokens":3413,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":3321}},"tokens_in":562,"tokens_out":3413,"duration_ms":24178,"temperature":1.0,"reasoning_tokens":3321,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T20:06:35.697604+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the energy-limited reachable set with the same nonlinear optimizer used for the mass-optimal set, and check whether every thrust-limited reachable state in the xy-plane is also energy-limited-reachable; if any mass-optimal thrust-limited state falls inside the nonlinear energy-limited set, the claimed superset fails. A cheaper check is to sample trajectories in additional azimuth directions beyond the twelve used here and see whether any sampled thrust-limited reachable state lies inside the energy ellipsoid.","supporting_citations":[{"cited_title":"Generation of energy-optimal low-thrust forced periodic trajectories in the cr3bp,","cited_arxiv_id":null,"evidence_quote":"Provides the linearized energy-optimal reachable set analysis and the reference trajectory whose reachable sets are compared throughout the paper."},{"cited_title":"ASSET: Astrodynamics Software and Science Enabling Toolkit,","cited_arxiv_id":null,"evidence_quote":"Supplies the trajectory optimization toolkit used to solve the nonlinear energy- and mass-optimal control problems with collocation and mesh refinement."},{"cited_title":"Reachable set computation for spacecraft relative motion with energy-limited low-thrust,","cited_arxiv_id":null,"evidence_quote":"Establishes the energy-limited reachable set formulation for spacecraft relative motion that the hyperellipsoid construction relies on."},{"cited_title":"Analysis on reachable set for spacecraft relative motion under low-thrust,","cited_arxiv_id":null,"evidence_quote":"Supplies the companion low-thrust reachable set analysis that justifies the energy-limited ellipsoid form."},{"cited_title":"Chaos-enhanced accelerated particle swarm optimization,","cited_arxiv_id":null,"evidence_quote":"Provides the accelerated particle swarm variant used in the reachable-set search."},{"cited_title":"Comparing Relative Reahcable Sets About Nearly Circular Orbits,","cited_arxiv_id":null,"evidence_quote":"Prior comparison of thrust-limited and energy-limited reachable sets for near-circular orbits that motivates the present comparison in the CR3BP."}],"review_version":1}