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Optimal Low-Thrust Orbit Transfers Made Easy: A Direct Approach

Mirko Leomanni, Gianni Bianchini, Andrea Garulli, Renato Quartullo

Abstract

The optimization of low-thrust, multi-revolution orbit transfer trajectories is often regarded as a difficult problem in modern astrodynamics. In this paper, a flexible and computationally efficient approach is presented for the optimization of low-thrust orbit transfers under eclipse constraints. The proposed approach leverages a new dynamic model of the orbital motion and a Lyapunov-based initial guess generation scheme that is very easy to tune. A multi-objective, single-phase formulation of the optimal control problem is devised, which provides a convenient way to trade off fuel consumption and time of flight. A distinctive feature of such a formulation is that it requires no prior information about the structure of the optimal solution. Simulation results for two benchmark orbit transfer scenarios indicate that minimum-time, minimum-fuel and mixed time/fuel-optimal instances of the control problem can be readily solved via direct collocation, while incurring a significantly lower computational demand with respect to existing techniques.

Optimal Low-Thrust Orbit Transfers Made Easy: A Direct Approach

Abstract

The optimization of low-thrust, multi-revolution orbit transfer trajectories is often regarded as a difficult problem in modern astrodynamics. In this paper, a flexible and computationally efficient approach is presented for the optimization of low-thrust orbit transfers under eclipse constraints. The proposed approach leverages a new dynamic model of the orbital motion and a Lyapunov-based initial guess generation scheme that is very easy to tune. A multi-objective, single-phase formulation of the optimal control problem is devised, which provides a convenient way to trade off fuel consumption and time of flight. A distinctive feature of such a formulation is that it requires no prior information about the structure of the optimal solution. Simulation results for two benchmark orbit transfer scenarios indicate that minimum-time, minimum-fuel and mixed time/fuel-optimal instances of the control problem can be readily solved via direct collocation, while incurring a significantly lower computational demand with respect to existing techniques.

Paper Structure

This paper contains 14 sections, 40 equations, 9 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Parametrization of the Orbital Motion
  3. Perturbed Kepler Problem
  4. Ideal Elements
  5. System Model
  6. Eclipse Conditions
  7. Optimal Control Problem
  8. Initial Guess Generation
  9. Lyapunov-based Guidance Scheme
  10. Definition of the Initial Mesh
  11. Numerical Simulations
  12. GTO-GEO Transfer
  13. LEO-GEO Transfer
  14. Conclusions

Figures (9)

  • Figure 1: Illustration of the initial mesh generation strategy, showing the profile of the shadow function \ref{['shadowf']} together with the location the mesh points.
  • Figure 2: Planar projection of the time-optimal GTO-GEO transfer trajectory: thrust phases are colored red, eclipse phases are colored gray.
  • Figure 3: Thrust direction vector profile (interpolated from all non-eclipsed trajectory samples) for the time-optimal GTO-GEO transfer.
  • Figure 4: Planar projection of the fuel-optimal GTO-GEO transfer trajectory: thrust phases are colored red, eclipse phases are colored gray, and optimal coast phases are colored blue.
  • Figure 5: Detail of the throttle control input profile for the fuel-optimal GTO-GEO transfer.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Remark 1
  • Remark 2