Affine formation of multi-agent systems over directed graphs

This paper proposes a simple, distributed control protocol for multi-agent systems with relative sensing capability over a directed network to achieve an affine formation. In contrast to the linear consensus protocol, the control protocol in this paper takes both positive and negative weights that partially encode the target formation shape and is thus able to achieve a formation pattern rather than just consensus. Having possibly negative weights in a graph, the associated Laplacian is called the generalized Laplacian. The connection between the generic rank property of the generalized Laplacian and the connectivity of a directed graph is established, based on which a necessary and sufficient graphical condition is developed to ensure the emergence of an affine formation by the proposed distributed control protocol.