• Space trajectory optimization
• Earth-Moon
• primer vector theory
• optimal guidance

Over the past years, the commercial interest in returning to the Moon and exploring the deep space has caused the governments to invest more money in the space field, while the main companies, on their side, have requested more and more effort by the research community, looking for highly advanced technologies. Spacecraft Trajectory Design, and more precisely the minimization of fuel needed in Space operations, covers a fundamental role, due to the desire of improving the efficiency of future missions and make them more affordable and repeatable.

The proposed work is inserted in this context, since it aims to provide a full optimization algorithm for fixed-time orbital transfers, with the objective of minimizing the total usage of propellent. Circular Restricted Three Body Problem is used for modelling the dynamics, as well as high-thrust impulsive accelerations, since most of the state-of-the-art works considers this combo as a valuable approximation for preliminary studies in mission design analysis. The problem of how to compute a trajectory satisfying specific constraints is assessed. as its effectiveness strongly impacts the performance of the optimization itself. Two main approaches are described, Single Shooting and Multiple Shooting, focusing especially on giving a mathematical characterization of the two and underlying what distinguish one from the other. Primer Vector theory is then employed to assess optimality and provide a criterion to decrease the cost by inserting intermediate maneuvers along the transfer.

First, a 2-impulses transfer is computed via Orbit Chaining, a technique that exploits the knowledge of the intermediate orbits between the departure and arrival one, which by assumption belong to the same family. The Time of Flight is computed in this phase and kept fixed for the rest of the optimization process. Starting from the initial guess, the algorithm automatically generates the optimal sequence of maneuvers, by iteratively performing two steps: a new maneuver is initialized, based on the investigation of the Primer Vector along the current trajectory, and the position and time of all the intermediate impulses are optimized via a non-linear gradient-based minimization method. The whole sequence is repeated until the path satisfies the Primer Vector optimality conditions.

The algorithm is tested on transfers between Earth-Moon Halo orbits, showing the results obtained, in terms of fuel saving and optimality achievness, for multiple case scenarios regarding different orbits and position on the respective orbits.