Extended multi-agent consensus protocols for the generation of geometric patterns in the plane

We propose a decentralized control law for multi-agent formations in two dimensions that allows the participating vehicles to display intricate periodic and quasi-periodic geometric patterns. Inspired by the "standard" consensus protocol ẋ = -Lx, these controls rely only on the relative position between the networked agents which are neighbors in the underlying communication graph. Several examples are presented, resulting in non-trivial geometric patterns described by trochoidal curves, similar to those generated by kids around the world using a spirograph. These paths can be useful for coordinated, distributed surveillance and monitoring applications, as well as for the sake of their own aesthetical beauty.