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Families of Two-Impulse Optimal Rendezvous Transfers Between Elliptic Orbits

Beom Park, Kathleen C. Howell, Jaewoo Kim, Jaemyung Ahn

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.

Families of Two-Impulse Optimal Rendezvous Transfers Between Elliptic Orbits

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.
Paper Structure (38 sections, 1 theorem, 52 equations, 32 figures, 8 tables)

This paper contains 38 sections, 1 theorem, 52 equations, 32 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Problem Formulation
  4. Locating : Existing Strategies
  5. Geometry-Based Analysis
  6. Numerical Methods: Grid Search and Optimization
  7. Optimization Formulation and Solution Dimensionality
  8. Motivation and Dimensionality
  9. Families of
  10. Gradient Information
  11. Optimality Domains and Asymptotic Behaviors
  12. Angular Domain ()
  13. Temporal domain ()
  14. Framework for Families of
  15. Seed Initialization
  16. ...and 23 more sections

Key Result

Proposition 1

At an asymptotic case $t \rightarrow \infty$, the optimality constraints in the $T-t$ domain (Eqs. eq:dJ_dt-eq:dJ_dtof) approach the optimality constraints in the $M_{1}-M_{2}$ domain (Eq. eq:angular_optimality).

Figures (32)

  • Figure 1: Cost landscape and selected transfer geometries for the baseline transfer scenario (Table \ref{['table:sample_orbits_first']}).
  • Figure 2: Comparison of cost landscapes in the angular domain ($d_1=\mathrm{s}$) as $t \rightarrow \infty$. Markers indicate stationary points classified by Hessian information (Table \ref{['table:colormap']}).
  • Figure 3: Comparison of cost landscapes in the angular domain ($d_1=\mathrm{l}$) as $t \rightarrow \infty$. Markers indicate stationary points classified by Hessian information (Table \ref{['table:colormap']}).
  • Figure 4: Representative tiot geometries in the near-asymptotic regime ($t = 25\cdot 10^3$ seconds).
  • Figure 5: Conceptual framework for constructing tiot families.
  • ...and 27 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof