Linear stability of the relative equilibria of a spacecraft around an asteroid

The gravitational orbit-rotation coupling dynamics of spacecraft around an asteroid is of great value and interest in the future asteroid missions. The linear stability of the relative equilibria of the full dynamics of a spacecraft around an asteroid with orbit-rotation coupling is studied with the geometric mechanics. Based on the linear system matrix, the linear stability of the relative equilibria is investigated and the stability conditions are given explicitly. It is found that the linearized system decouples into two entirely independent subsystems, which correspond to the motions within and outside the equatorial plane of the asteroid respectively. The linear stability of the relative equilibria is investigated numerically with respect to the system parameters through comparisons with the traditional spacecraft dynamics without orbit-rotation coupling. The relations between the full dynamics and the traditional spacecraft dynamics, as well as the effect of the orbit-rotation coupling, are assessed. We find that when the orbitrotation coupling is strong, the mass distribution of the spacecraft dominates the stability. Whereas when the orbitrotation coupling is weak, both the mass distribution and the traditional stationary orbit stability have significant effects on the stability.