Swarm fixed-time reference tracking: a discrete model

Abstract

This paper proposes a discrete-time kinematic model for a swarm/flock of multidimensional agents combining two problems that are usually addressed separately: trajectory tracking and shape control. Relying only on local information, for the first time in the discrete-time domain, the model provides a time control on the evolution of the swarm/flock, that is: in fixed-time, the swarm centroid tracks a given reference trajectory; in finite-time, the connections graph becomes complete and the agents reach an aggregation inside a region of attraction centred on the moving centroid. The shape of such a region can vary among polytopes, oriented ellipsoids/spheres and straight lines by only changing the eigenvalues of a coordinates-coupling matrix. Compared to classic shape control methods where agents inter-distances must be provided, this model is more suitable for large swarms. Finally, numerical simulations and a laboratory experiment are proposed to highlight the described model properties.