Development of General-Purpose Root-Finding Module for General Mission Analysis Tool

Abstract

A general-purpose root-finding module for designing spacecraft trajectories is developed to have similar accuracy to that of other well-known root-finding modules, and greater speed. Three quasi-Newton root-finding algorithms are implemented: the Newton–Raphson method, the Broyden’s method and the Generalized Broyden’s method. Based on the proposed root-finding module, General Mission Analysis Tool (GMAT) of National Aeronautics and Space Administration’s Goddard Space Flight Center (NASA/GSFC) is functionally extended by integrating the Broyden’s method and the Generalized Broyden’s method into its differential corrector module. Non-trivial spacecraft trajectory design problems, such as Lambert’s problem for trans-lunar trajectory and minimum-time transfer problem to the Mars are solved to analyze the performance of the proposed module. The numerical performances of each root-finding algorithm are quantitatively analyzed by the total number of function evaluations, the total number of iterations, convergence error, mean convergence rate, and running time. The overall comparative analysis shows that the Broyden’s method and the Generalized Broyden’s method are about 20–40% faster than the Newton–Raphson method and solutions from each algorithm have similar numerical accuracy. We also show that for selected test problems, the Generalized Broyden’s method converges in less running time than others with similar numerical accuracy in GMAT. The updated differential corrector module was released in R2014a version of GMAT.