Emergent homeomorphic curves in swarms

We propose a new decentralized approach for producing emergent curve trajectories in swarms of agents. The central innovation lies in use of a quaternion-based stabilizing embedding, which permits the application of linear control policies to produce globally-stable behaviour from local observations. Agents are modelled as particles in free space using double integrator dynamics. We stabilize an arbitrary number of agents indirectly, through virtual representations bijected onto a circle, and then produce elaborate curve trajectories through a family of topological homeomorphisms. We formulate these virtual agents as embeddings, through a combination of state feedback, rotations, and variable changes. The result is evenly-spaced swarms along the desired trajectories, with each agent only requiring information about the state of its neighbours. Simulations demonstrate the emergent qualities of the swarm as it converges to the desired geometry. The agents maintain separation and automatically compensate for loss or addition of agents in the swarm. We provide mathematical proofs of stability for the embedding, trajectories, and controller.