Linear Quadratic Formulation based Station-Keeping in Halo Orbit

A three body problem is an interesting problem in the classical mechanics and dynamical systems theory. When formulated and simplified, for certain cases, interesting trajectories can be obtained around the equilibrium points of the system. This paper deals with such trajectory tracking problem for halo orbit around the L
₁
point of the SunEarth system. The proposed tracking controllers are obtained using the linear quadratic formulation for a given reference trajectory. Two different approaches for such formulation are considered. The first one is based on linearized equations of motion about its operating points. The linearization of dynamics may lead to loss of the system characteristics, when it is far from the operating point, and consequently the controller may not perform satisfactory. To avoid such difficulties, a suboptimal controller using state dependent Riccati equation is also proposed. The efficacy of proposed controllers in the stations keeping is vindicated through numerical simulations, for the orbit insertion errors.