Body-fixed orbit-attitude hovering at equilibria near an asteroid using non-canonical Hamiltonian structure

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.