Bounded trajectories near collinear-like equilibrium points of elongated asteroids using linear control

A new method of obtaining bounded trajectories near collinear-like equilibrium points of elongated asteroids using a simple linear control is proposed. The characteristic equation of the linearized equation of motion near the collinear-like equilibrium points is reduced based on the property of the second derivatives of the effective potential. According to the reduced characteristic equation, the motion is decoupled into an in-plane motion and a out-of-plane motion. The solutions to the decoupled equation of motion are derived and the corresponding Lissajous trajectories are chosen as desired trajectories. A simple linear control, which tracks the Lissajous trajectories, is formulated with the stability conditions of gains. The effectiveness of the proposed method is validated by simulations using high fidelity models of 216 Kleopatra and 951 Gaspra.