Shooting and numerical continuation methods for computing time-minimal and energy-minimal trajectories in the Earth-Moon system using low propulsion

Gautier Picot

doi:10.3934/dcdsb.2012.17.245

Abstract

In this article we describe the principle of computations of optimal transfers between quasi-Keplerian orbits in the Earth-Moon system using low-propulsion. The spacecraft's motion is modelled by the equations of the control restricted 3-body problem and we base our work on previous studies concerning the orbit transfer in the two-body problem where geometric and numeric methods were developed to compute optimal solutions. Using numerical simple shooting and continuation methods connected with fundamental results from control theory, such as the Pontryagin Maxium Principle and the second order optimality conditions related to the concept of conjugate points, we compute time-minimal and energy-minimal trajectories between the geostationary initial orbit and a final circular orbit around the Moon, passing through the neighborhood of the libration point $L_1$. Our computations give simple trajectories, obtained by referring to numerical values of the SMART-1 mission.

Highlights

The aim of this article is to study the optimal orbit transfer between quasi-Keplerian orbits in the Earth-Moon system when low propulsion is applied
We provide a numerical checking of the comparison theorem between the first conjugate times in normal and abnormal cases related to the time-minimal problem
Our computations result from an original application of indirect numeric methods, inspired by the Pontryagin maximum principle, to the particular planar restricted 3-body problem and the optimality of these transfers is checked using second order conditions

Summary

Introduction

The aim of this article is to study the optimal orbit transfer between quasi-Keplerian orbits in the Earth-Moon system when low propulsion is applied. Our model is the circular restricted 3-body problem [24] that has provided the framework to numerous dynamical systems studies about space mission design in the solar system, see notably [19, 21]. The physical issue is to maximize the final mass of the spacecraft but, as a first step, we restrict our analysis to the time-minimal control problem and the so-called energy minimization problem, see [8, 16, 17] for complementary results, for two reasons. The energy minimization problem is a L2-regularization of the mass maximization problem which is a non smooth L1problem. This regularization is known to be an important step in the analysis, see [18]. Our work is based on previous works concerning the orbit transfer between

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Conclusion