Exploiting Scaling Constants to Facilitate the Convergence of Indirect Trajectory Optimization Methods

TL;DR

This paper introduces scaling techniques to improve the convergence and efficiency of indirect low thrust trajectory optimization, reducing reliance on guesses and simplifying calculations for fuel and time-optimal problems.

Contribution

It develops novel scaling factors based on energy optimal solutions to enhance convergence and reduce computational effort in indirect trajectory optimization methods.

Findings

Scaling factors improve initial residuals and convergence speed.

Energy optimal solutions serve as effective initial guesses.

Simplifications do not affect optimality, enhancing efficiency.

Abstract

This note develops easily applicable techniques that improve the convergence and reduce the computational time of indirect low thrust trajectory optimization when solving fuel- and time-optimal problems. For solving fuel optimal (FO) problems, a positive scaling factor -- $\Gamma_{TR}$ -- is introduced based on the energy optimal (EO) solution to establish a convenient profile for the switching function of the FO problem. This negates the need for random guesses to initialize the indirect optimization process. Similarly, another scaling factor-$\beta$-, is introduced when solving the time-optimal (TO) problem to connect the EO problem to the TO. The developed methodology for the TO problem was crucial for the GTOC11 competition. Case studies are conducted to validate the solution process in both TO and FO problems. For geocentric cases, the effect of eclipses and $J_2$ perturbations were also considered. The examples show that EO can provide a good guess for TO and FO problems and that introducing the constants can reduce the initial residuals and improve convergence. It is also shown that the equation for the Lagrangian multiplier of mass and the associated boundary condition can be ignored for both FO and TO cases without affecting optimality. This simplification reduces the problem dimensions and improves efficiency.