In this paper we study a family of controllers that guarantees attitude synchronization for a network of elements in the unit sphere domain, i.e. $\mathcal{S}^2$. We propose distributed continuous controllers for elements whose dynamics are controllable (i.e. control with torque as command), and which can be implemented by each individual agent without the need of a common global orientation frame among the network, i.e. it requires only local information that can be measured by each individual agent from its own orientation frame. The controllers are specified according to arbitrary distance functions in $\mathcal{S}^2$, and we provide conditions on those distance functions that guarantee that i) a synchronized network of agents is locally asymptotically stable for an arbitrary network graph; ii) a synchronized network can be achieved for almost all initial conditions in a tree graph network. We also study the equilibria configurations that come with specific types of network graphs. The proposed strategies can be used in attitude synchronization of swarms of fully actuated rigid bodies, such as satellites.

A preliminary version of this work was submitted to the 2015 IEEE Conference on Decision and Control, "Family of Controllers for Attitude Synchronization in S2"