Analytical Solution for Optimal Low-Thrust Limited-Power Transfers Between Non-Coplanar Coaxial Orbits

Sandro Da Fernandes,Francisco Das Chagas Carvalho

doi:10.5028/jatm.v10.904

Abstract

In this paper, an analytical solution for time-fixed optimal low-thrust limited-power transfers (no rendezvous) between elliptic coaxial non-coplanar orbits in an inverse-square force field is presented. Two particular classes of maneuvers are related to such transfers: maneuvers with change in the inclination of the orbital plane and maneuvers with change in the longitude of the ascending node. The optimization problem is formulated as a Mayer problem of optimal control with the state defined by semi-major axis, eccentricity, inclination or longitude of the ascending node, according to the class of maneuver considered, and a variable measuring the fuel consumption. After applying Pontryagin’s maximum principle and determining the maximum Hamiltonian, short periodic terms are eliminated through an infinitesimal canonical transformation. The new maximum Hamiltonian resulting from this canonical transformation describes the extremal trajectories for long duration transfers. Closed-form analytical solution is then obtained through Hamilton-Jacobi theory. For long duration maneuvers, the existence of conjugate points is investigated through the Jacobi condition. Simplified solution is determined for transfers between close orbits. The analytical solution is compared to the numerical solution obtained by integration of the canonical system of differential equations describing the extremal trajectories for some sets of initial conditions. Results show a great agreement between these solutions for the class of maneuvers considered in the analysis. The solution of the two-point boundary value problem of going from an initial orbit to a final orbit, based on the analytical solution, is also discussed.

Highlights

In the last thirty years, important space missions have made use of low-thrust propulsion systems
A comparison between the complete, nonlinear and linear, firstorder analytical solutions, derived in the preceding sections, and a numerical solution obtained by integrating the canonical system of differential equations governed by new Hamiltonian function H*, given by Eqs. 14 and 15, is discussed and the accuracy of the analytical approach is established
An approximated first-order analytical solution for optimal time-fixed low-thrust limited power transfers between elliptic coaxial non-coplanar orbits in an inverse-square force field is determined through canonical transformation theory

Summary

Introduction

In the last thirty years, important space missions have made use of low-thrust propulsion systems. The two pioneer missions were NASA-JPL Deep Space 1 and ESA’s SMART-1. Deep Space 1 was the first interplanetary spacecraft to utilize Solar Electric Propulsion. It was developed by NASA in the New Millennium Program to testing new technologies for future Space and Earth Science Programs. It was launched on October 24, 1998, and its mission terminated on December 18, 2001, when its fuel supply exhausted. It was used to test solar electric propulsion and other deep-space technologies. It was launched on September 27, 2003, and its mission ended on

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