An approach for time-free optimal two-impulse trajectories using primer vector theory and Mathieu transformation

In this paper, a semi-analytical approach combining primer vector theory and Mathieu transformation for a time-free optimal two-impulse transfer problem is proposed. The Mathieu transformation provides analytical expressions of the primer vector and its derivative for each coast arc. Using necessary optimality conditions of primer vector theory, continuous property of positions, and Pontryagin's necessary conditions for optimality, twenty seven algebraical equations with twenty seven unknown constants are derived for the time-free two-impulse transfer problem. By solving such twenty seven unknown parameters using a nonlinear least square method and a genetic algorithm, the two-impulse transfer trajectory is determined. Two numerical examples are given to demonstrate the effectiveness of the proposed approach. The approach can be also extended to time-free N-impulse (N > 2) transfer problems by adding thirteen algebraical equations with thirteen unknown constants for each additional mid-coast arc.