Finite-time control for body-fixed hovering of rigid spacecraft over an asteroid

A finite-time control scheme for autonomous body-fixed hovering of a rigid spacecraft over a tumbling asteroid is presented in the framework of geometric mechanics. This control scheme guarantees convergence of configuration (pose) and velocity tracking errors in finitetime with (Hölder) continuous feedback. The configuration space for the spacecraft is the Lie group SE(3), which is the set of positions and orientations of the rigid spacecraft in three-dimensional Euclidean space. The asteroid's trajectory is assumed to be available through the spacecraft's on-board navigation. The relative configuration between the spacecraft and the asteroid is described in terms of exponential coordinates on the Lie group SE(3). With this feedback control, the spacecraft achieves a desired relative configuration with respect to the asteroid autonomously and in finite time, without requiring explicit reference states. Finite-time convergence of the proposed control scheme for the closed-loop system is theoretically proved using Lyapunov stability analysis. A numerical simulation demonstrates successful application of this finite-time control scheme for coupled translational and rotational maneuvers about a selected tumbling asteroid, leading to asteroid body-fixed hovering. The asymptotic tracking control was also implemented to compare with the performance of the finite-time control.