Synchronization in complex networks of phase oscillators: A survey <s> Flocking, Schooling, and Vehicle Coordination <s> Abstract This paper presents a Lie group setting for the problem of control of formations, as a natural outcome of the analysis of a planar two-vehicle formation control law. The vehicle trajectories are described using the planar Frenet–Serret equations of motion, which capture the evolution of both the vehicle position and orientation for unit-speed motion subject to curvature (steering) control. The set of all possible (relative) equilibria for arbitrary G -invariant curvature controls is described (where G = SE (2) is a symmetry group for the control law), and a global convergence result for the two-vehicle control law is proved. An n -vehicle generalization of the two-vehicle control law is also presented, and the corresponding (relative) equilibria for the n -vehicle problem are characterized. Work is on-going to discover stability and convergence results for the n -vehicle problem. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Flocking, Schooling, and Vehicle Coordination <s> A simple model with a novel type of dynamics is introduced in order to investigate the emergence of self-ordered motion in systems of particles with biologically motivated interaction. In our model particles are driven with a constant absolute velocity and at each time step assume the average direction of motion of the particles in their neighborhood with some random perturbation $(\ensuremath{\eta})$ added. We present numerical evidence that this model results in a kinetic phase transition from no transport (zero average velocity, $|{\mathbf{v}}_{a}|\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$) to finite net transport through spontaneous symmetry breaking of the rotational symmetry. The transition is continuous, since $|{\mathbf{v}}_{a}|$ is found to scale as $({\ensuremath{\eta}}_{c}\ensuremath{-}\ensuremath{\eta}{)}^{\ensuremath{\beta}}$ with $\ensuremath{\beta}\ensuremath{\simeq}0.45$. <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Flocking, Schooling, and Vehicle Coordination <s> This paper proposes a design methodology to stabilize isolated relative equilibria in a model of all-to-all coupled identical particles moving in the plane at unit speed. Isolated relative equilibria correspond to either parallel motion of all particles with fixed relative spacing or to circular motion of all particles with fixed relative phases. The stabilizing feedbacks derive from Lyapunov functions that prove exponential stability and suggest almost global convergence properties. The results of the paper provide a low-order parametric family of stabilizable collectives that offer a set of primitives for the design of higher-level tasks at the group level <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Flocking, Schooling, and Vehicle Coordination <s> This paper proposes a design methodology to stabilize relative equilibria in a model of identical, steered particles moving in the plane at unit speed. Relative equilibria either correspond to parallel motion of all particles with fixed relative spacing or to circular motion of all particles around the same circle. Particles exchange relative information according to a communication graph that can be undirected or directed and time-invariant or time-varying. The emphasis of this paper is to show how previous results assuming all-to-all communication can be extended to a general communication framework. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Flocking, Schooling, and Vehicle Coordination <s> Coordinated Control and Estimation for Multi-agent Systems: Theory and Practice Daniel J. Klein Chair of the Supervisory Committee: Professor Kristi A. Morgansen Aeronautics & Astronautics The problems studied in this thesis center on coordinated control and estimation tasks for multi-agent systems. The work begins technically, with a Kuramoto-inspired study of coordinated phasor centroid control for a group ofN homogeneous phase integrators. A main result is the development of a controller which guarantees asymptotic stability of the phasor centroid to a known reference vector. An extensions of the splay state to non-balanced group states is then explored. The work then moves on to a group of N heterogeneous agents wherein some agents act as leaders to the others, which obey a standard sinusoidal protocol. One interesting point here is that unlike in the related controlled linear agreement problem, symmetry of the communication topology does not determine uncontrollability. While these results are derived in continuous time, the next contributions shows that many of the desirable coordination properties can be preserved in discrete time. The work then moves to a more applied focus. A simulation study of coordinated target tracking is conducted, considering questions of control, estimation, and communication for distributed systems. Finally, the design and development of a computer vision system which enables a demonstration of some of the theory-oriented work on the University of Washington’s Underwater Fin-actuated Autonomous Vehicle testbed is discussed. <s> BIB005 </s> Synchronization in complex networks of phase oscillators: A survey <s> Flocking, Schooling, and Vehicle Coordination <s> This paper considers the integration of communication and control with respect to the task of coordinated heading control for a group of N vehicles. The heading control employed on each vehicle is a discretization of the well-known Kuramoto model of nonlinearly coupled oscillators. Stability for both all-to-all and random one-to-all broadcasts is shown to be dependent on the coupling strength, K, and the time discretization, DeltaT. For a desired system performance, DeltaT imposes a tight deadline by which the state information (M bits) must be propagated through the communication network. Routing optimization with respect to minimizing energy consumption is formulated considering the DeltaT deadline. Multi-hopping is optimal if time permits, but one-to-all broadcasting is more energy efficient when M/DeltaT is sufficiently large. <s> BIB006 </s> Synchronization in complex networks of phase oscillators: A survey <s> Flocking, Schooling, and Vehicle Coordination <s> We study a class of coupled phase models with state-dependent coupling topology. In these models the link weights increase (decrease) in a dynamic fashion according to the phase differences of the coupled systems. We provide sufficient conditions for the systems to be gradients of suitable potential functions and we analyze their clustering properties. The work is motivated by neurophysiological models where the strength of the synapse between two neurons is enhanced if they are simultaneously co-active (close to synchronization) and decreased when they are out of phase. <s> BIB007 </s> Synchronization in complex networks of phase oscillators: A survey <s> Flocking, Schooling, and Vehicle Coordination <s> Previously, we showed using a computational agent-based model that a group of animals moving together can make a collective decision on direction of motion, even if there is a conflict between the directional preferences of two small subgroups of “informed” individuals and the remaining “uninformed” individuals have no directional preference. The model requires no explicit signaling or identification of informed individuals; individuals merely adjust their steering in response to socially acquired information on relative motion of neighbors. In this paper, we show how the dynamics of this system can be modeled analytically, and we derive a testable result that adding uninformed individuals improves stability of collective decision making. We first present a continuous-time dynamic model and prove a necessary and sufficient condition for stable convergence to a collective decision in this model. The stability of the decision, which corresponds to most of the group moving in one of two alternative preferred directions, depends explicitly on the magnitude of the difference in preferred directions; for a difference above a threshold the decision is stable and below that same threshold the decision is unstable. Given qualitative agreement with the results of the previous simulation study, we proceed to explore analytically the subtle but important role of the uninformed individuals in the continuous-time model. Significantly, we show that the likelihood of a collective decision increases with increasing numbers of uninformed individuals. <s> BIB008
A recent research field in control is the coordination of autonomous vehicles based on locally available information and inspired by biological flocking phenomena. Consider a set of n particles in the plane R 2 , which we identify with the complex plane C. Each particle i ∈ V = {1, . . . , n} is characterized by its position r i ∈ C, its heading angle θ i ∈ S 1 , and a steering control law u i (r, θ) depending on the position and heading of itself and other vehicles, see Fig. 2 .(a). For simplicity, we assume that all particles have unit speed. The particle kinematics are then given by BIB001  for i ∈ {1, . . . , n} and i = √ −1. If no control is applied, then particle i travels in a straight line with orientation θ i (0), and if u i = ω i ∈ R is a nonzero constant, then particle i traverses a circle with radius 1/|ω i |. The interaction among the particles is modeled by a possibly time-varying interaction graph G(V, E(t), A(t)) determined by communication and sensing patterns. As shown by BIB002 , interesting motion patterns emerge if the controllers use only relative phase information between neighboring particles, that is, The controlled phase dynamics (7) correspond to the coupled oscillator model (1) with a time-varying interaction graph with weights K · a ij (t) and identically timevarying natural frequencies ω i = ω 0 (t). The controlled Fig. 2 . Panel (a) illustrates the particle kinematics (6). Panels (b)-(e) illustrate the controlled dynamics (6)- (7) with n = 6 particles, a complete interaction graph, and identical and constant natural frequencies: ω0(t) = 0 in panels (b) and (c) and ω0(t) = 1 in panels (d) and (e). The values of K are K = 1 in panel (b) and (d) and K = −1 in panel (c) and (e). The arrows depict the orientation, the dashed curves show the long-term position dynamics, and the solid curves show the initial transient position dynamics. As illustrated, the resulting motion displays "synchronized" or "balanced" heading angles for K = ±1, and translational motion for ω0(t) = 0, respectively circular motion for ω0(t) = 1. phase dynamics (7) give rise to elegant and useful coordination patterns that mimic animal flocking behavior BIB008 and fish schools . A few representative trajectories are illustrated in Fig. 2 . Inspired by these biological phenomena, scientists have studied the controlled phase dynamics (7) and their variations in the context of tracking and formation controllers in swarms of autonomous vehicles. We refer to BIB003 BIB004 BIB005 BIB006 BIB007 BIB008 for other control laws, motion patterns, and their analysis. In the following sections, we will present various tools to analyze the motion patterns in Fig. 2 , which we will refer to as phase synchronization ( 