Halo Orbit Maintenance around L1 Point of the Sun-Earth System Using Optimal Control and Lyapunov Stability Theory

Abstract

This paper addresses the maintenance of a halo orbit around the L1 point of the Sun-Earth system in a circular restricted three-body problem. To this effect, a trajectory tracking problem is formulated and solved by designing various controllers using the linear quadratic method and Lyapunov stability theory. The linear quadratic formulations are performed using two approaches: the first one linearizes the equations of motion at several operating points, while the second approach uses a state-dependent coefficient system matrix that requires solving the state-dependent Riccati equation (SDRE). To handle the nonlinearity and to reduce the computational complexity as compared to the linear quadratic method, the controller is also derived using Lyapunov stability theory. The proposed controllers are tested for their effectiveness in reducing the orbit insertion errors as well as for disturbance rejection. The disturbances being considered are primarily due to the eccentricity of Earth’s orbit around the Sun, solar radiation pressure, and the gravitational pull of the Moon. The simulation results are presented to delineate the performances of the proposed controllers. The superiority of a Lyapunov theory-based controller over the LQR-based controllers is demonstrated.