# Local Synchronization of Sampled-Data Systems on Lie Groups

## TL;DR

This paper introduces a smooth distributed control law for local synchronization of identical driftless kinematic agents on Lie groups, achieving exponential convergence when initialized close to each other, with analysis extending from commutative to general matrix Lie groups.

## Contribution

It proposes a novel control law for synchronization on Lie groups and provides a comprehensive analysis including special cases and generalizations using the Baker-Campbell-Hausdorff theorem.

## Key findings

- Synchronization achieved exponentially fast near initial conditions.
- Closed-loop dynamics are linear in exponential coordinates for commutative Lie groups.
- Characterization of equilibria and settling times for complete graphs.

## Abstract

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.