Collective Motion and Oscillator Synchronization

Abstract

Summary. This paper studies connections between phase models of coupled oscillators and kinematic models of groups of self-propelled particles. These connections are exploited in the analysis and design of feedback control laws for the individuals that stabilize collective motions for the group.

Heterogeneous animal group models and their group-level alignment dynamics: An equation-free approach

“…(1)) where the variables ψ k characterize the lumped behavior of each of the three subgroups. It is assumed that the alignment (orientational) dynamics are independent of the translational counterpart (Sepulchre et al, 2005); hence, the dynamical state of an individual can be characterized by its orientation. The functional form for mutual interaction is borrowed from the well-known Kuramoto model (Kuramoto, 1984), a prototypical model for coupled nonlinear oscillators.…”

Consensus on nonlinear spaces

“…The latest published work returns to this question by deriving the same control laws from a general coordination theory on Lie Groups (Sarlette et al [2010b]). The basic investigation that sustained the underlying research over the seven years interval is the extension of the consensus problem, in which agents dynamically seek a value of common agreement, from values on the real line to values on the circle (Sepulchre et al [2004]). A main reference on this topic is Sarlette and Sepulchre [2009b], strongly supported by the results of the thesis of Sarlette [2009].…”

Collective behavior of self-propelling particles with kinematic constraints: The relation between the discrete and the continuous description

“…In [10] the spontaneous emergence of ordered motion has been studied in terms of so called control laws using graph theory. Generalizations of the control laws were considered in [11,12]. In [12] it was shown that the organized motion of SPP with the control laws depending on the relative orientations of the velocities and relative spacing, can be of two types only: parallel and circular motion.…”

“…Generalizations of the control laws were considered in [11,12]. In [12] it was shown that the organized motion of SPP with the control laws depending on the relative orientations of the velocities and relative spacing, can be of two types only: parallel and circular motion. The stability properties of these discrete updating rules (including the T.Vicsek's model) and the dynamics they describe were considered using Lyapunov's theory in [10,11,13,14].…”