Local Synchronization of Sampled-Data Systems on Lie Groups
It provides a theoretical framework for synchronization on Lie groups, which is relevant for robotic systems and multi-agent coordination, but the results are local and incremental over existing synchronization methods.
This paper proposes a smooth distributed nonlinear control law for local synchronization of identical driftless kinematic agents on matrix Lie groups with a connected communication graph. The control law achieves exponential synchronization when agents are initialized sufficiently close, with results generalized to arbitrary matrix Lie groups via the Baker-Campbell-Hausdorff theorem.
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.