Aerospace Science and Technology 38 (2014) 105–115 Contents lists available at ScienceDirect Aerospace Science and Technology www.elsevier.com/locate/aescte Almost global asymptotic tracking control for spacecraft body-fixed hovering over an asteroid Daero Lee a,1 , Amit K. Sanyal a,2 , Eric A. Butcher b,3 , Daniel J. Scheeres c,4 a Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA b Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 85721, USA c Colorado Center for Astrodynamics Research, Department of Aerospace Engineering Sciences, University of Colorado at Boulder, CO 80309-0431, USA a r t i c l e i n f o a b s t r a c t Article history: Received 11 February 2014 Received in revised form 9 June 2014 Accepted 25 July 2014 Available online 1 August 2014 Keywords: Body-fixed hovering Asymptotic tracking control Lie group SE(3) Exponential coordinates Control saturation Geometric mechanics Almost global asymptotic tracking control for autonomous body-fixed hovering of a rigid spacecraft over an asteroid is proposed in the framework of geometric mechanics. 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 relative motion with respect to the spacecraft is assumed to be available through the spacecraft on-board navigation. The spacecraft tracks a desired relative configuration with respect to an asteroid in an autonomous manner. The relative configuration between the spacecraft and the asteroid is described in terms of exponential coordinates on the Lie group of rigid body motions. A continuous-time feedback tracking control using these exponential coordinates and the relative velocities is presented to perform coupled translational and rotational maneuver over an asteroid in the presence of control force saturation. A Lyapunov analysis guarantees that the spacecraft asymptotically converges to the desired trajectory. Numerical simulation results demonstrate the asymptotic tracking control achieve autonomous body-fixed hovering over a selected asteroid.  2014 Elsevier Masson SAS. All rights reserved. 1. Introduction Spacecraft hovering around small bodies such as asteroids and comets is essential for performing scientific explorations. In the hovering maneuver, the spacecraft may need to use control thrust continuously, overly continuously, or intermittently to null out gravitational and rotational accelerations, creating an equilibrium at a desired position in oder to maintain the desired position [25, 28]. This approach is feasible over small bodies because the nom- inal accelerations on a spacecraft are small [5,6,27]. In general, there are two types of approaches in spacecraft hovering around an asteroid: over-inertial hovering and body-fixed hovering [5,27, 28]. In over-inertial hovering the spacecraft fixes its position rel- ative to the asteroid in the rotating asteroid–Sun frame, creating an artificial libration point in this frame. This hovering mode was implemented by the Hayabusa during most of its mission [28]. As the counterpart to inertial or over-inertial hovering, in aster- oid body-fixed hovering the spacecraft fixes its position relative to 1 Postdoctoral Researcher. 2 Assistant Professor, AIAA member. 3 Associate Professor, Dwight and Aubrey Chapman Professor, AIAA member. 4 Professor, AIAA member. the rotating asteroid and rotates with the asteroid in inertial space. In addition, every maneuver is performed relative to the asteroid body-fixed frame. This hovering mode was also implemented by the Hayabusa mission [10,12,13] during its sampling performances over the surface [5,28]. Body-fixed hovering is necessary for the spacecraft to sample a small body surface by controlling its motion in the asteroid body-fixed frame. Hovering trajectories can be im- plemented for many rotation periods (hours) of the asteroid with a modest control thrust, since the gravitational attraction is relatively weak at the asteroid. In general, the asteroid rotation period is on the order of hours to days at most [28]. Additionally, the spacecraft typically must reorient its attitude to maintain the same attitude relative to the asteroid body-fixed frame for the hovering maneu- ver time. Most recent studies on spacecraft hovering about asteroids have focused mainly on achieving the desired relative position of a point-mass spacecraft with respect to the asteroid without per- forming attitude maneuvers to maintain the desired attitude [2,5, 6,9,13,25,27]. However, this paper presents a rigid body spacecraft hovering scheme for a uniformly rotating asteroid whose rota- tion rate is constant in the asteroid body-fixed frame [11,28,29] where the spacecraft is required to implement maneuvers with large ranges of rotational motion in three-dimensional Euclidean http://dx.doi.org/10.1016/j.ast.2014.07.013 1270-9638/ 2014 Elsevier Masson SAS. All rights reserved.