By considering the spacecraft as an extended, rigid body with a prior known attitude instead of a point mass, the attitude-restricted orbital dynamics can improve the precision of the classical point-mass orbital dynamics in close proximity to an asteroid, because it includes the perturbation caused by the gravitational orbit-attitude coupling of the spacecraft (GOACP). The GOACP is defined as the difference between the gravity acting on a non-spherical, extended body (the real case of a spacecraft) and the gravity acting on a point mass (the approximation of a spacecraft in classical orbital dynamics). In-plane equilibrium points that are within the principal planes of the asteroid have been investigated for the attitude-restricted orbital dynamics in previous studies, including equatorial and in-plane non-equatorial equilibrium points. In this study, out-of-plane equilibrium points outside the principal planes of the asteroid were examined. Out-of-plane equilibrium points cannot exist in the classical point-mass orbital dynamics but do exist in the attitude-restricted orbital dynamics owing to the effects of the GOACP. The previously investigated in-plane equilibrium points and the out-of-plane ones examined in this study provide a complete map of the equilibrium points in close proximity to an asteroid with the GOACP. Equatorial and in-plane non-equatorial equilibrium points have extended the longitude and latitude ranges of the classical equilibrium points without the GOACP, respectively, while the out-of-plane ones examined in the present study extend both the longitude and latitude ranges. Additionally, the invariant manifolds of out-of-plane equilibrium points were calculated, and the results indicated that the attitude of spacecraft significantly affects the invariant manifolds. In practice, these equilibrium points can provide natural hovering positions for operations in proximity to asteroids, and their invariant manifolds can be used for transfers to or from the equilibrium points.

A recently proposed orbital dynamics model in the close proximity of an asteroid, which is called "attitude-restricted orbital dynamics" , includes the perturbation caused by the spacecraft’s gravitational orbit-attitude coupling. This orbital model improves the precisionof classical point-mass orbital model with only the non-spherical gravity. Equatorialequilibrium points have been investigated in the previous paper. In this paper, the in-plane non-equatorial equilibrium points, which are outside the asteroid’s equatorial plane but within its longitudinal principal plane, are further studied for a uniformly-rotating asteroid. These non-equatorial equilibrium points are more diverse than those in the classical point-mass orbital dynamics without gravitational orbit-attitude coupling perturbation (GOACP). Two families of them have been found. The equatorial equilibrium points studied before and the non-equatorial ones studied here give a complete map of equilibrium points in the asteroid’s principal planes. Compared with the classical point-mass orbital dynamics without GOACP, the equatorial equilibrium points have extended the longitude range of equilibrium points around an asteroid, while the non-equatorial ones studied here will extend the latitude range. These equatorial and non-equatorial equilibrium points provide natural hovering positions for the asteroid close-proximity operations.

The gravitationally coupled orbit-attitude dynamics, also called the full dynamics, in which the spacecraft is modeled as a rigid body, is a high-precision model for the motion in the close proximity of an asteroid. A feedback control law is proposed to stabilize relative equilibria of the coupled orbit-attitude motion in a uniformly rotating second degree and order gravity field by utilizing the Hamiltonian structure. The feedback control law is consisted of potential shaping and energy dissipation. The potential shaping makes the relative equilibrium a minimum of the modified Hamiltonian by modifying the potential artificially. With the energy-Casimir method, it is theoretically proved that an 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 to the relative equilibrium. The proposed stabilization control law has a simple form and is easy to implement autonomously, which can be attributed to the utilization of natural dynamical behaviors in the controller design.