Optimal Thrust Level for Orbit Insertion

TL;DR

This paper presents an analytical method to optimize the thrust level and direction for minimum-fuel orbital transfers using a constant thrust engine, simplifying the solution process for launcher upper stages.

Contribution

It introduces a closed-loop solution for thrust direction and reduces the optimal control problem to two unknowns, enabling easier trajectory optimization without iterative costate guessing.

Findings

Closed-form solution for thrust direction derived from extremal analysis

Optimal trajectory found with minimal computational effort

Method successfully applied to geostationary transfer orbit case

Abstract

The minimum-fuel orbital transfer is analyzed in the case of a launcher upper stage using a constantly thrusting engine. The thrust level is assumed to be constant and its value is optimized together with the thrust direction. A closed-loop solution for the thrust direction is derived from the extremal analysis for a planar orbital transfer. The optimal control problem reduces to two unknowns, namely the thrust level and the final time. Guessing and propagating the costates is no longer necessary and the optimal trajectory is easily found from a rough initialization. On the other hand the initial costates are assessed analytically from the initial conditions and they can be used as initial guess for transfers at different thrust levels. The method is exemplified on a launcher upper stage targeting a geostationary transfer orbit.