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An algebraic approach to the minimum-cost multi-impulse orbit transfer problem

Martin Avendano, Verónica Martín-Molina, Jorge Martín-Morales, Jorge Ortigas-Galindo

Abstract

We present a purely algebraic formulation (i.e. polynomial equations only) of the minimum-cost multi-impulse orbit transfer problem without time constraints, while keeping all the variables with a precise physical meaning. We apply general algebraic techniques to solve these equations (resultants, Gröbner bases, etc.) in several situations of practical interest of different degrees of generality. For instance, we provide a proof of the optimality of the Hohmann transfer for the minimum fuel 2-impulse circular to circular orbit transfer problem, and we provide a general formula for the optimal 2-impulse in-plane transfer between two rotated elliptical orbits under a mild symmetry assumption on the two points where the impulses are applied (which we conjecture that can be removed).

An algebraic approach to the minimum-cost multi-impulse orbit transfer problem

Abstract

We present a purely algebraic formulation (i.e. polynomial equations only) of the minimum-cost multi-impulse orbit transfer problem without time constraints, while keeping all the variables with a precise physical meaning. We apply general algebraic techniques to solve these equations (resultants, Gröbner bases, etc.) in several situations of practical interest of different degrees of generality. For instance, we provide a proof of the optimality of the Hohmann transfer for the minimum fuel 2-impulse circular to circular orbit transfer problem, and we provide a general formula for the optimal 2-impulse in-plane transfer between two rotated elliptical orbits under a mild symmetry assumption on the two points where the impulses are applied (which we conjecture that can be removed).

Paper Structure

This paper contains 8 sections, 1 theorem, 67 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Keplerian motion
  3. Multi-impulse orbit transfers
  4. Minimum $\Delta v^2$ Lambert problem
  5. Optimality of the Hohmann transfer
  6. Two rotated ellipses
  7. Numerical tests
  8. Conclusions

Key Result

Theorem 1

Let $q,q_1,\ldots,q_m:\mathbb{R}^k\to\mathbb{R}$ in $\mathcal{C}^\infty$ and $p\in\mathbb{R}^k$ a common zero of $q_1,\ldots,q_m$ be such that the vectors $\nabla q_1(p),\ldots,\nabla q_m(p)$ are linearly independent. Then $p$ is a local extremum of $q$ on the manifold defined by $\{ q_1=\cdots=q_m=

Figures (4)

  • Figure 1: Ellipse with focus $F$ and a satellite $S$ on it.
  • Figure 2: $\Delta_0+\Delta_1$ of the best transfer and an example of one of these transfers.
  • Figure 3: (a) Angle of separation between the semi-major axis of the initial orbit and the point where the first impulse is applied. (b) Fuel comparison between the best transfer and the best one from apogee to apogee.
  • Figure 4: Fuel comparison between the best transfer and the solutions provided by Eq.\ref{['sol-caso2a-t2']}

Theorems & Definitions (1)

  • Theorem 1: Lagrange multipliers