Optimal Design and Tracking Control of Low-Thrust Orbit Transfers Between Planar and Vertical Lyapunov Orbits

This paper investigates the fuel-optimal transfer problem between planar and vertical Lyapunov orbits in cislunar space utilizing a low-thrust engine with variable specific impulse. The equations governing the motion of spacecraft with bounded specific impulse in the Earth-Moon system are derived employing the Pontryagin Minimum Principle. A modified logarithmic homotopy function is utilized to address the bang-bang discontinuity arising from the fuel optimization problem. Given the challenge of predicting transfer time in fuel-optimal problem without prior information, we initiate the optimization process by simplifying the calculation into a constant specific impulse time-optimal transfer scenario. The trajectory is determined through the dynamical evolution configuration. It then serves as an initial guess and is gradually transitioned into the variable specific impulse model, thereby yielding the final transfer trajectory. Finally, for practical applications, a Model Predictive Controller (MPC) combined with an extended Kalman filter is developed for real-time tracking and control of the reference transfer trajectory, while accounting for potential measurement errors in spacecraft state information. The simulation results demonstrate a high success rate and rapid convergence for the trajectory optimization and tracking method, providing valuable insights for missions with similar trajectory designs.