  We treat the periodic trajectory tracking problem: given a periodic
trajectory of a control-affine, left-invariant driftless system in a compact
and connected Lie group $G$ and an initial condition in $G$, find another
trajectory of the system satisfying the initial condition given and that
asymptotically tracks the periodic trajectory. We solve this problem locally
(for initial conditions in a neighborhood of some point of the periodic
trajectory) when $G$ is semisimple and the system is Lie-determined (i.e.
controllable), and only for a class of periodic trajectories (which we call
regular). Finally we present a set of sufficient conditions to ensure the
existence of such trajectories.
