A new flocking model through body attitude coordination

Abstract

Communicated by N. Bellomo and F. BrezziWe present a new model for multi-agent dynamics where each agent is described by its position and body attitude: agents travel at a constant speed in a given direction and their body can rotate around it adopting different configurations. In this manner, the body attitude is described by three orthonormal axes giving an element in SO(3) (rotation matrix). Agents try to coordinate their body attitudes with the ones of their neighbours. In this paper, we give the individual-based model (particle model) for this dynamics and derive its corresponding kinetic and macroscopic equations.

Quaternions in Collective Dynamics

“…We derive the corresponding macroscopic equations from the associated mean-field equation, which we refer to as the Self-Organized Hydrodynamics based on Quaternions (SOHQ), by reference to the Self-Organized Hydrodynamics (SOH) derived from the Vicsek dynamics (see [21] and the discussion below). Our model is inspired by the one in [17] where the body attitude is represented by elements of the rotation group SO (3). The macroscopic equations obtained in [17] present various drawbacks.…”

“…Our model is inspired by the one in [17] where the body attitude is represented by elements of the rotation group SO (3). The macroscopic equations obtained in [17] present various drawbacks. First, some of the terms in the equations do not have a clear interpretation.…”

“…Analogously, the simulation of the macroscopic equations also benefits from the computational efficiency of the quaternions: each grid point will require less than half the memory to store the information of the system than is required when using matrices; we will avoid orthogonalizing matrices by just normalizing 4-dimensional vectors; and, importantly, all the terms in the quaternion formulation are explicit, while in the matrix formulation they are not and require the inversion of an operator (see (16)). In summary, by considering a quaternionic representation, we render the results in [17], which uses the matrix representation, amenable to a numerical study.…”

“…In contrast with the use of rotation matrices in [17], the use of the quaternion representation makes the modeling more difficult on the individual-based level; first, it is not clear how to define a mean body attitude based on quaternions and, second, we need to consider nematic alignment rather than polar alignment. However, the macroscopic equations obtained are easier to interpret than in [17] and provide the right framework to carry out numerical simulations.…”

“…However, the macroscopic equations obtained are easier to interpret than in [17] and provide the right framework to carry out numerical simulations. The main contributions of the present paper include (i) deriving the macroscopic equations, (ii) finding the right modeling at the individual-based level, and (iii) proving the equivalence of the models and results obtained here with those in [17] for the rotation-matrix representation.…”