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

Periodic trajectory tracking for control-affine driftless systems on compact Lie groups