  We present a smooth distributed nonlinear control law for local
synchronization of identical driftless kinematic agents on a Cartesian product
of matrix Lie groups with a connected communication graph. If the agents are
initialized sufficiently close to one another, then synchronization is achieved
exponentially fast. We first analyze the special case of commutative Lie groups
and show that in exponential coordinates, the closed-loop dynamics are linear.
We characterize all equilibria of the network and, in the case of an
unweighted, complete graph, characterize the settling time and conditions for
deadbeat performance. Using the Baker-Campbell-Hausdorff theorem, we show that,
in a neighbourhood of the identity element, all results generalize to arbitrary
matrix Lie groups.
