Low-Thrust Optimal Escape Trajectories from Lagrangian Point L2 of the Earth-Moon System

This thesis focuses on optimizing escape trajectories from the Lagrangian point of the Earth-Moon system in a higher fidelity model using electric propulsion. Lagrangian points represent specific locations in space where objects placed there tend to remain stationary. At Lagrangian points, the gravitational attraction exerted by two massive bodies exactly balances the centripetal force necessary for a smaller object to move with them. These points are useful for spacecraft to minimize the amount of fuel consumption required to maintain their position. Therefore, the exploitation of these points has become of great interest for the future space missions. In fact, among many uses, it also allows a gateway to Deep Space exploration. Among the countless trajectory available for a specific mission, the ones that require minimal propellant while still satisfying all the other conditions have been employed to optimize scientific results and minimize the associated costs. To achieve this objective, this work addresses the optimization of escape trajectories using electric propulsion with the primary aim of reducing the amount of propellant required and, consequently, maximizing the payload mass. The optimization is achieved using an indirect method based on the optimal control theory. This method transforms the challenge of minimizing propellant usage into a multipoint boundary value problem, which is resolved through an iterative single-shooting process based on Newton’s method. The case addressed in this work belongs to a particular subset of optimal control problems characterized by a discontinuous control law, known as “bang-bang”. An automated tool is used to handle numerical complexities and identify suitable preliminary solutions. The problems relating to the discontinuity of the thrust were addressed, along with the delicacies of the indirect method, which strongly depend on the initial conditions, such as the a priori definition of the thrust structure. Pontryagin’s Maximum Principle allows for adjusting suboptimal solutions when the thrust structure violates them in certain trajectories arcs. The dynamical model encompasses the gravitational influences of four celestial bodies, whereby the spacecraft is influenced by the gravitational forces of the Earth, Moon and Sun. This model relies on JPL’s ephemerides to account for the evolution of the positions of these celestial bodies over time. It also includes solar radiation pressure, the lunisolar gravitational effect and a spherical harmonic model for the Earth as perturbative additional effects. The complex gravitational interactions between the Sun, Earth and Moon dictate the trajectory dynamics near the starting point, i.e. the Lagrangian Point L2. In this highly complex framework, the optimal trajectory is sought among the sub-optimal ones, differentiating the various solutions from each other. In particular, this work focuses on the Earth-Moon system, which presents more complexities related to the Moon's motion of revolution. First, a scenario is analyzed in which the departure date is varied along a whole lunar period maintaining fixed escape time and free terminal energy. Successively, specific dates among them are selected, and the escape durations is varied, still with free terminal energy. Lastly, a more complex analysis is carried out by fixing the terminal energy and letting the escape duration be free.