Exploration of Alternative State Vector Choices for Low-Thrust Trajectory Optimization

Abstract

As is well known in celestial mechanics, coordinate choices have significant consequences in the analytical and computational approaches to solve the most fundamental initial value problem. The present study focuses on the impact of various coordinate representations of the dynamics on the solution of the ensuing state/costate two-point boundary-value problems that arise when solving the indirect optimal control necessary conditions. Minimum-fuel trajectory designs are considered for 1) a geocentric spiral from geostationary transfer orbit to geostationary Earth orbit, and 2) a heliocentric transfer from Earth to a highly eccentric and inclined orbit of asteroid Dionysus. This study unifies and extends the available literature by considering the relative merits of eight different coordinate choices to establish the state/costate differential equations. Two different sets of orbit elements are considered: equinoctial elements and a six-element set consisting of the angular momentum vector and the eccentricity vector. In addition to the Cartesian and spherical coordinates, four hybrid coordinate sets associated with an osculating triad defined by the instantaneous position and velocity vectors that consist of a mixture of slow and fast variables are introduced and studied. Reliability and efficiency of convergence to the known optimal solution are studied statistically for all eight sets; the results are interesting and of significant practical utility.