Quaternions in Collective Dynamics

Abstract

Abstract. 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.

Alignment of Self-propelled Rigid Bodies: From Particle Systems to Macroscopic Equations

“…Since symmetric matrices with multiple maximal eigenvalues are negligible, we can expect this definition to be well-posed almost surely. The first unexpected link found in [13] between this framework of quaternions and the previous framework of average matrices, is that these two averaging procedures are actually equivalent (when the polar decomposition in formula (7) actually returns a rotation matrix). This is due to the following observations [13]:…”

“…The second link found in [13] between the frameworks of quaternions and rotation matrices, is that these alignment mechanisms are also equivalent. Proposition 8.…”

“…To compute the kernel of the collision operator, in the gradual alignment model we use that the collision operator can be recast as [13]). In the jump-based model the computation of the kernel is straightforward.…”

“…Many refinements have been proposed to incorporate additional mechanism, such as, to cite only a few of them, volume exclusion [18], presence of leaders [21] or polarization of the group [6]. We refer the interested reader to [12,13] and the references therein for more on this topic.…”

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The goal of these lecture notes is to present in a unified way various models for the dynamics of aligning self-propelled rigid bodies at different scales and the links between them. The models and methods are inspired from [12,13], but, in addition, we introduce a new model and apply on it the same methods. While the new model has its own interest, our aim is also to emphasize the methods by demonstrating their adaptability and by presenting them in a unified and simplified way.

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