Quaternions in Collective Dynamics

We introduce a model of multiagent dynamics for self-organized motion; individuals travel at a constant speed while trying to adopt the averaged body attitude of their neighbors. The body attitudes are represented through unitary quaternions. We prove the correspondence with the model presented in [P. Degond, A. Frouvelle, and S. Merino-Aceituno, Math. Models Methods Appl. Sci., 27 (2017), pp. 1005--1049], where the body attitudes are represented by rotation matrices. Differently from this previous work, the individual-based model introduced here is based on nematic (rather than polar) alignment. From the individual-based model, the kinetic and macroscopic equations are derived. The benefit of this approach, in contrast to that of the previous one, is twofold: first, it allows for a better understanding of the macroscopic equations obtained and, second, these equations are prone to numerical studies, which is key for applications