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arxiv: 2603.11538 · v2 · submitted 2026-03-12 · 🧮 math.OC

Families of Two-Impulse Optimal Rendezvous Transfers Between Elliptic Orbits

Beom Park , Kathleen C. Howell , Jaewoo Kim , Jaemyung Ahn This is my paper

Pith reviewed 2026-05-15 12:22 UTC · model grok-4.3

classification 🧮 math.OC
keywords two-impulse rendezvousoptimal transferselliptic orbitsnumerical continuationprimer vector theoryfuel optimizationsolution familiesKeplerian orbits
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The pith

Optimal two-impulse rendezvous solutions between elliptic orbits connect into continuous families when the problem is re-parameterized.

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

The paper establishes that isolated fuel-optimal two-impulse transfers are actually linked members of larger continuous families once the search is re-parameterized and continued numerically. Enforcing a subset of first-order optimality conditions produces one-parameter families that can be traced as orbital geometry changes. These families are classified with Hessian criteria and Primer Vector Theory and then projected onto porkchop plots that link angular and temporal variables. A reader cares because the approach supplies a global map of the solution landscape instead of scattered points, showing how branches appear, merge, or vanish and revealing nearby near-optimal alternatives.

Core claim

The classical fuel-optimal two-impulse rendezvous problem between Keplerian orbits is revisited by enforcing a subset of first-order necessary optimality conditions and tracing the resulting one-parameter families via numerical continuation. The families are classified using Hessian-based criteria and Primer Vector Theory, and projected onto porkchop plots to connect the angular and temporal domains. Representative case studies reveal the emergence, merging, and disappearance of locally optimal branches under variations in orbital geometry, supplying a global map of the solution landscape and clarifying the robustness of optimal solutions.

What carries the argument

Numerical continuation of one-parameter families obtained by enforcing a subset of first-order necessary optimality conditions, classified via Hessian criteria and Primer Vector Theory.

If this is right

  • The global map shows how locally optimal branches emerge, merge, or disappear under orbital geometry changes.
  • Near-optimal transfers can be identified in the vicinity of any nominal trajectory.
  • Projection onto porkchop plots links angular and temporal domains for each family.
  • Classification by Hessian criteria and Primer Vector Theory distinguishes the character of each branch.
  • The structure clarifies robustness of any given optimal solution across small geometry perturbations.

Where Pith is reading between the lines

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

  • The same continuation approach could map families for three-impulse or continuous-thrust rendezvous once the optimality conditions are adjusted.
  • Mission designers could select trajectories from the interior of a family rather than from its endpoint to gain margin against model errors.
  • Connecting families to porkchop plots suggests a practical way to visualize fuel-time trade-offs across entire solution sets instead of single points.
  • Detecting where branches merge or terminate may flag bifurcation points where small geometry changes produce qualitatively new optimal transfers.

Load-bearing premise

Enforcing only a subset of first-order necessary optimality conditions permits reliable numerical continuation of the families without missing branches or encountering singularities as orbital geometry varies.

What would settle it

A numerical continuation run that produces an isolated optimal solution with no connecting branch when eccentricity or argument of perigee is varied smoothly would falsify the claim that all optima belong to traceable continuous families.

read the original abstract

The classical fuel-optimal two-impulse rendezvous problem between Keplerian orbits is revisited from a family-based perspective. Conventional approaches often yield isolated optimal solutions whose mutual relationships remain unclear; yet, when re-parameterized appropriately, seemingly unrelated optima are revealed to be connected members of continuous solution families. To expose this structure, the proposed framework enforces a subset of first-order necessary optimality conditions and traces the resulting one-parameter families via numerical continuation. The families are classified using Hessian-based criteria and Primer Vector Theory, and are projected onto porkchop plots to connect the angular and temporal domains. Representative case studies reveal the emergence, merging, and disappearance of locally optimal branches under variations in orbital geometry, supplying a global map of the solution landscape. This complementary perspective clarifies the robustness of optimal solutions and identifies alternative near-optimal transfers in the vicinity of a nominal trajectory.

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

1 major / 0 minor

Summary. The paper revisits the fuel-optimal two-impulse rendezvous problem between Keplerian elliptic orbits by re-parameterizing the problem to reveal that isolated optimal solutions belong to continuous one-parameter families. The framework enforces a subset of first-order necessary optimality conditions, traces the families via numerical continuation, classifies them using Hessian-based criteria and Primer Vector Theory, and projects the results onto porkchop plots to link angular and temporal domains. Representative case studies illustrate the emergence, merging, and disappearance of locally optimal branches under changes in orbital geometry, providing a global map of the solution landscape.

Significance. If the numerical continuation reliably captures all branches without missing solutions due to singularities, the work supplies a complementary global perspective on the two-impulse rendezvous solution space that clarifies relationships among optima and identifies nearby near-optimal transfers. This could strengthen robustness analysis in mission design, though the absence of explicit validation data or error metrics in the provided description limits immediate assessment of practical impact.

major comments (1)
  1. [Abstract] Abstract: The central claim that 'all locally optimal solutions are connected members of continuous families' rests on numerical continuation of a subset of first-order necessary conditions. No analysis is provided of the regularity of the continuation map when orbital parameters (eccentricity, relative inclination) drive the primer vector or Hessian toward degeneracy, leaving open the possibility that branches are silently dropped or require ad-hoc re-initialization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback and for identifying an important aspect of our continuation framework. We address the single major comment below and have revised the manuscript to incorporate additional discussion on regularity conditions.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that 'all locally optimal solutions are connected members of continuous families' rests on numerical continuation of a subset of first-order necessary conditions. No analysis is provided of the regularity of the continuation map when orbital parameters (eccentricity, relative inclination) drive the primer vector or Hessian toward degeneracy, leaving open the possibility that branches are silently dropped or require ad-hoc re-initialization.

    Authors: We agree that a formal regularity analysis of the continuation map strengthens the presentation. In the revised manuscript we have added a dedicated paragraph in Section 3.2 that derives the non-degeneracy conditions from Primer Vector Theory (primer vector magnitude bounded away from zero) and the second-variation test (Hessian positive definite on the tangent space). We also describe the practical safeguards implemented in the continuation algorithm: step-size adaptation and explicit monitoring of the smallest singular value of the Jacobian; when these indicators approach critical thresholds the continuation is halted and the point is flagged rather than continued. Across all reported case studies no such thresholds were crossed, so no ad-hoc re-initializations were required and no branches were dropped. These additions directly address the concern while preserving the original numerical results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard optimality conditions and numerical continuation

full rationale

The paper enforces a subset of first-order necessary optimality conditions (standard in optimal control) and applies numerical continuation to trace one-parameter families of two-impulse transfers. This process does not reduce any claimed result to its inputs by construction, nor does it involve fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations. Primer Vector Theory and Hessian-based classification are invoked from established external literature. The central claim of connected solution families emerges from the continuation procedure itself rather than being presupposed, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard assumptions from optimal control and orbital mechanics; no new free parameters or invented entities are introduced in the abstract description.

axioms (2)
  • domain assumption Spacecraft motion follows Keplerian two-body dynamics
    Implicit in the statement of elliptic orbits and rendezvous transfers.
  • standard math First-order necessary conditions from optimal control theory are sufficient to define the families
    Used to enforce the subset for numerical continuation.

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Reference graph

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    work page 1970