Optimal low-thrust orbit transfers connecting gateway with Earth and Moon

Gateway will represent a primary logistic infrastructure in cislunar space. The identification of efficient orbit transfers capable of connecting Earth, Moon, and Gateway paves the way for enabling refurbishment, servicing, and utilization of this orbiting platform. This study is devoted to determining two-way minimum-time low-thrust orbit transfers that connect both Earth and Moon to Gateway. Backward time propagation is proposed as a very convenient option for trajectories that approach and rendezvous with Gateway, and a unified formulation is introduced for minimum-time orbit transfers, using either forward or backward propagation. Two-way transfers between Gateway and a specified low-altitude lunar orbit are first determined, using an indirect heuristic method, which employs the necessary conditions for optimality and a heuristic algorithm. Second, two-way orbit transfers that connect Earth and Gateway are addressed. Because these trajectories exist under the influence of two major attracting bodies, the underlying optimal control problem is formulated as a multi-arc trajectory optimization problem, involving three different representations for the spacecraft state. Multi-arc optimal control problems are associated with several corner conditions to be enforced at the junction time that separates distinct arcs. However, these conditions are shown to be solvable sequentially for the problem at hand, leveraging implicit costate transformation. This implies that the multi-arc problem has a set of unknown quantities with the same size as that of a single-arc optimal control problem, with apparent advantages of a computational nature. The indirect heuristic method is also applied in this second mission scenario. Low-thrust orbit dynamics is propagated in a high-fidelity dynamical framework, with the use of planetary ephemeris and the inclusion of the simultaneous gravitational action of Sun, Earth, and Moon, along the entire transfer paths, in all cases. The numerical results unequivocally prove that the methodology developed in this research is effective for determining two-way minimum-time low-thrust orbit transfers connecting Earth, Moon, and Gateway.