Analytical approach to spacecraft formation-flying with low-thrust relative spiral trajectories

Abstract

This work addresses the growing need for an intuitive, systematic approach to low-thrust spacecraft formation flying by extending shape-based continuous thrust trajectory design methods to the relative motion of two spacecraft. There is widespread interest in distributed space systems for their low costs, broad capabilities, and high redundancy. This trend introduces a new challenge for trajectory design when combined with the increasing prevalence of low-thrust, high specific impulse electric propulsion systems. That challenge is met herein with a geometrically intuitive, semi-analytical solution to the low-thrust problem. Beginning with the equations of relative motion of two spacecraft, an unperturbed chief and a continuously-thrusting deputy, a thrust profile is constructed which transforms the equations into a form that is solved analytically. The resulting relative trajectories are the family of sinusoidal spirals, which provide diversity for design and optimization based upon a single thrust parameter. Closed-form expressions are derived for the trajectory shape and time-of-flight for two prescribed relative velocity behaviors, and used to develop a novel patched-spirals trajectory design and optimization method. The example problem of a servicer spacecraft establishing and reconfiguring a formation around a target in geostationary earth orbit is used to demonstrate the application of the patched spirals technique as well as the advantages of the relative spiral trajectories over impulsive maneuvers. The sensitivity of the trajectory solutions to deviations from the underlying assumptions, uncertainties in the state, and errors in thrust are studied through high-fidelity simulation.