Emergent Hypotrochoidal and Epitrochoidal Formations in a Swarm of Agents

This paper proposes a decentralised controller for a swarm of agents that generates hypotrochoidal and epitrochoidal curves in 3D space. The dynamics of the agents is modelled as a double integrator where the control input, i.e. the acceleration profile, contains two terms that account for the interactions between the agents in terms of position and velocity, respectively. The term related to the interaction between velocities is weighted by a skew-symmetric matrix responsible for coupling the velocities displacements. Such a matrix defines an axis of rotation centred on the centroid of the swarm, around which the swarm organises itself to generate the desired trajectories, which can be selected by properly setting two free parameters of the model. In particular, the motion of each agent is analysed by separating its dynamics along the rotation axis from that in the perpendicular plane. It is shown that the agents achieve consensus or periodic motion along the axis of rotation while performing the desired evolutions in the perpendicular plane. Numerical simulations are presented to illustrate the proposed framework.