Optimal single impulse maneuver for spacecraft cluster flight

The optimal single-impulse maneuver for spacecraft cluster flight is studied in this paper. Based on Gauss' variational equations, the optimal conditions and solutions for time-fixed and time-unfixed case are provided via the periodic condition of nonlinear relative motion. For the time-fixed case, the problem is also treated from the perspective of energy matching condition. In this case, the problem is transformed into seeking for the extremum of a quadratic equation or the solution of a single-root algebraic equation, and the optimal semimajor axis is derived, as well as the optimal velocity impulse. For the time-unfixed case, the problem can be solved by numerical algorithms or transformed into pursuing for the solution of a multi-root algebraic equation with the aid of Fourier-Bessel functions. In the end, the proposed method is validated by several carried out illustrating examples.