Optimal interior Earth–Moon Lagrange point transfer trajectories using mixed impulsive and continuous thrust

Optimal interior Earth–Moon Lagrange point transfer trajectories results with fixed flight time using mixed impulsive and continuous thrust propulsion in the framework of the planar Circular Restricted Three-Body Problem are presented. A virtual spacecraft departs geosynchronous or low-Earth orbit and enters a specified Lyapunov orbit around the interior Earth–Moon Lagrange point. For these transfer trajectory designs, a direct transcription and collocation method is employed to reformulate the continuous dynamic optimization problem into a discrete optimization problem, which then is solved using nonlinear programming software. As the design parameters, flight time and relative weighting factor between impulsive and continuous thrust are adjusted in the transfer trajectory design procedure. For practically implementable transfer trajectory designs, the constraints for coast arc and bounded control thrust are applied in the problem formulation as equality and inequality constraints, respectively. In addition, both quadratic performance index and minimum fuel performance index are used to show the relative comparison and analysis in the transfer trajectory design results. According to the relative weighting factor, the transfer trajectories are classified into direct and spiral departure trajectories whose virtual spacecraft directly departs the Earth parking orbit and departs after generating spirals around the Earth, respectively. The convergence of these transfer trajectory design methods are numerically verified by generating varying numbers of nodes and comparing their design results. Especially, the primer vector theory is utilized to analyze the transfer trajectory design results using minimum fuel performance index. Finally, the progressive homotopy continuation method is applied to design low-Earth departure transfer trajectories. These transfer trajectories show that they are numerically and theoretically valid results with the benefit of mixed impulsive and continuous thrust.