Steering laws and continuum models for planar formations

Abstract

Abstract-We consider a Lie group formulation for the problem of control of formations. Vehicle trajectories are described using the planar Frenet-Serret equations of motion, which capture the evolution of both vehicle position and orientation for unit-speed motion subject to curvature (steering) control. The Lie group structure can be exploited to determine the set of all possible (relative) equilibria for arbitrary G-invariant curvature controls, where G = SE(2) is a symmetry group for the control law. The main result is a convergence result for n vehicles (for finite n), using a Lyapunov function which for n = 2, has been previously shown to yield global convergence. A continuum formulation of the basic equations is also presented.

Coordinated patterns of unit speed particles on a closed curve

“…Extension to many particles are made in [19]. Sepulchre, Paley and Leonard [20] noticed that patterns of many constant speed particles can be achieved in the plane by extending methods previously developed for coupled oscillators [21].…”

Group Coordination and Cooperative Control of Steered Particles in the Plane

“…Consequently, the closed-loop vector field is invariant under an action of the symmetry group SE(2) and the closed-loop dynamics evolve on a reduced quotient manifold (shape space). Equilibria of the reduced dynamics are called relative equilibria and can be only of two types [JK03]: parallel motions, characterized by a common orientation for all the particles (with arbitrary relative spacing), and circular motions, characterized by circular orbits of the particles around the same fixed point.…”

“…We consider a continuous-time kinematic model of N > 1 identical particles (of unit mass) moving in the plane at unit speed [JK03]:…”

“…In this paper, we consider a kinematic model of identical (pointwise) particles in the plane [JK03]. The particles move at constant speed and are subject to steering controls that change their orientation.…”

Oscillators as Systems and Synchrony as a Design Principle

“…For example, autonomous underwater vehicles (AUVs) are used to collect oceanographic measurements in network formations that maximize the information intake, see, e.g., [15]. In ongoing work [25,17,26], we study a continuous-time kinematic model of N identical, self-propelled particles subject to planar steering controls, first considered in [9,10]. In complex notation, this model is given bẏ…”