Low-Thrust Minimum Fuel Optimization in the Circular Restricted Three-Body Model

In this paper, low-thrust fuel-optimal transfer trajectories are studied in the Earth-Moon restricted three-body model with constant specific impulse engines. In order to cope with high computational burden and convergence problems caused by multi spirals and large number of bang-bang control structures, homotopy method is implemented by starting from the related and easier energy-optimal problem, for which, the analytical Jacobian is derived, and a combination of Newton and bisection methods is proposed to accurately detect the switching points together with a variable step 7th/8th-order Runge-Kutta integrator. The case study is a low thrust transfer from the GTO to a periodic three-dimensional halo orbit around the Earth-Moon L1 point. Initial costates are first achieved by solving the problem with a high thrust level, which is then reduced through discrete homotopy. The final orbit has about 150 revolutions and the control profile is characterized by about 150 bang-bang structures. The techniques presented in the paper are useful in practical cases where very low-thrust accelerations is used in high nonlinear vector fields.