Minimum-Time and Minimum-Fuel Low-Thrust Trajectory Design for Satellite Formation in Low-Earth Orbits

Indirect formalism of optimal control theory is used to generate minimum-time and minimum-fuel trajectories for formation of two spacecraft (deputies) relative to a chief satellite. For minimum-fuel problems, a hyperbolic tangent smoothing method is used to facilitate numerical solution of the resulting boundary-value problems by constructing a one-parameter family of smooth control profiles that asymptotically approach the theoretically optimal, but non-smooth bang-bang thrust profile. Impact of the continuation parameter on the solution of minimum-fuel trajectories is analyzed. The fidelity of the dynamical model is improved beyond the two-body dynamics by including the perturbation due to the Earth’s second zonal harmonic, J
2
. In addition, a particular formation is investigated, where the deputies are constrained to lie diametrically opposite on a three-dimensional sphere centered at the chief.