Fast Homotopy Method for Asteroid Landing Trajectory Optimization Using Approximate Initial Costates

This Paper presents a fast homotopy method for solving fuel-optimal asteroid landing trajectories. Similar to the traditional homotopy method, the fuel-optimal control problem is connected with the energy-optimal control problem via a homotopy function of the performance index. Originally, the initial costates are solved more systematically here. A homotopy function of the dynamical equation is proposed to connect the energy-optimal control problem with a gravity-free energy-optimal control problem for which the generalized gravity is zero. The gravity-free energy-optimal control problem is further linked to an approximate optimal-control problem that has an analytical solution, through selecting appropriate coefficients of the performance index and neglecting the thrust magnitude constraint. Based on the initial costates analytically obtained, an algorithm is proposed to solve the original fuel-optimal control problem. The effectiveness and efficiency of the proposed method are validated through simulations of spacecraft landing on 433 Eros with medium thrust and landing on 101955 Bennu with low thrust.