  This paper presents a generalization of conventional sliding mode control
designs for systems in Euclidean spaces to fully actuated simple mechanical
systems whose configuration space is a Lie group for the trajectory-tracking
problem. A generic kinematic control is first devised in the underlying Lie
algebra, which enables the construction of a Lie group on the tangent bundle
where the system state evolves. A sliding subgroup is then proposed on the
tangent bundle with the desired sliding properties, and a control law is
designed for the error dynamics trajectories to reach the sliding subgroup
globally exponentially. Tracking control is then composed of the reaching law
and sliding mode, and is applied for attitude tracking on the special
orthogonal group SO(3) and the unit sphere S3. Numerical simulations show the
performance of the proposed geometric sliding-mode controller (GSMC) in
contrast with two control schemes of the literature.
