  This paper presents a novel Lagrangian approach to attitude tracking for
rigid spacecraft using unit quaternions, where the motion equations of a
spacecraft are described by a four degrees of freedom Lagrangian dynamics
subject to a holonomic constraint imposed by the norm of a unit quaternion. The
basic energy-conservation property as well as some additional useful properties
of the Lagrangian dynamics are explored, enabling to develop quaternion-based
attitude tracking controllers by taking full advantage of a broad class of
tracking control designs for mechanical systems based on energy-shaping
methodology. Global tracking of a desired attitude on the unit sphere is
achieved by designing control laws that render the tracking error on the
four-dimensional Euclidean space to converge to the origin. The topological
constraints for globally exponentially tracking by a quaternion-based
continuous controller and singularities in controller designs based on any
three-parameter representation of the attitude are then avoided. Using this
approach, a full-state feedback controller is first developed, and then several
important issues, such as robustness to noise in quaternion measurements,
unknown on-orbit torque disturbances, uncertainty in the inertial matrix, and
lack of angular-velocity measurements are addressed progressively, by designing
a hybrid state-feedback controller, an adaptive hybrid state-feedback
controller, and an adaptive hybrid attitude-feedback controller. Global
asymptotic stability is established for each controller. Simulations are
included to illustrate the theoretical results.
