Family of controllers for attitude synchronization on the sphere

In this paper we study a family of controllers that guarantees attitude synchronization for a network of agents in the unit sphere domain, i.e., S2. 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 constructed as functions of distance functions in S2, and we provide conditions on those distance functions that guarantee that i) a synchronized network of agents is locally asymptotically stable for an arbitrary connected network graph; ii) a synchronized network is asymptotically achieved for almost all initial conditions in a tree network graph. When performing synchronization along a principal axis, we propose controllers that do not require full torque, but rather torque orthogonal to that principal axis; while for synchronization along other axes, the proposed controllers require full torque. 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 under actuated rigid bodies, such as satellites.