  Orbit-attitude hovering of a spacecraft at the natural relative equilibria in
the body-fixed frame of a uniformly rotating asteroid is discussed in the
framework of the full spacecraft dynamics, in which the spacecraft is modeled
as a rigid body with the gravitational orbit-attitude coupling. In this
hovering model, both the position and attitude of the spacecraft are kept to be
stationary in the asteroid body-fixed frame. A Hamiltonian structure-based
feedback control law is proposed to stabilize the relative equilibria of the
full dynamics to achieve the orbit-attitude hovering. The control law is
consisted of two parts: potential shaping and energy dissipation. The potential
shaping is to make the relative equilibrium a minimum of the modified
Hamiltonian on the invariant manifold by modifying the potential artificially.
With the energy-Casimir method, it is shown that the unstable relative
equilibrium can always be stabilized in the Lyapunov sense by the potential
shaping with sufficiently large feedback gains. Then the energy dissipation
leads the motion to converge asymptotically to the minimum of the modified
Hamiltonian on the invariant manifold, i.e., the relative equilibrium. The
feasibility of the proposed stabilization control law is validated through
numerical simulations in the case of a spacecraft orbiting around a small
asteroid. The main advantage of the proposed hovering control law is that it is
very simple and is easy to implement autonomously by the spacecraft with little
computation. This advantage is attributed to the utilization of dynamical
behaviors of the system in the control design.
