Stabilization of collective motion in synchronized, balanced and splay phase arrangements on a desired circle
01 Jul 2015-pp 731-736
TL;DR: The feedback controls are derived and the asymptotic stability of the desired collective circular motion is proved by using Lyapunov theory and the LaSalle's Invariance principle.
Abstract: This paper proposes a design methodology to stabilize collective circular motion of a group of N-identical agents moving at unit speed around individual circles of different radii and different centers. The collective circular motion studied in this paper is characterized by the clockwise rotation of all agents around a common circle of desired radius as well as center, which is fixed. Our interest is to achieve those collective circular motions in which the phases of the agents are arranged either in synchronized, in balanced or in splay formation. In synchronized formation, the agents and their centroid move in a common direction while in balanced formation, the movement of the agents ensures a fixed location of the centroid. The splay state is a special case of balanced formation, in which the phases are separated by multiples of 2π/N. We derive the feedback controls and prove the asymptotic stability of the desired collective circular motion by using Lyapunov theory and the LaSalle's Invariance principle.
Citations
More filters
TL;DR: This paper presents suitable feedback control laws for each of these motion coordination tasks by considering a second-order rotational dynamics of the agent.
34 citations
TL;DR: A methodology to stabilize synchronized, balanced, or symmetric phase patterns of unicycle-type agents on a desired common circle, while restricting their trajectories to a certain region of interest, is proposed in this note.
Abstract: A methodology to stabilize synchronized , balanced , or symmetric phase patterns of unicycle-type agents on a desired common circle, while restricting their trajectories to a certain region of interest, is proposed in this note. These phase patterns are characterized by the motion of the collective centroid of the group of agents and derived by optimizing the average linear momentum of the group. Under a mild assumption on initial states of the agents, we design control laws by exploiting the concept of barrier Lyapunov function in conjunction with bounded phase potential functions. We show that the agents asymptotically stabilize to a desired phase arrangement and their trajectories remain bounded during stabilization. We obtain bounds on the different quantities of interest in the postdesign analysis and show that these bounds depend on the initial conditions and can be altered by adjusting the controller gains. We also prove convergence when the control input is saturated to a prespecified value. Finally, we provide a discussion on the application and limitation of the proposed approach and characterize the feasible initial conditions.
12 citations
Cites background or result from "Stabilization of collective motion ..."
...Moreover, the controllers in [22] and [23] demand higher control effort....
[...]
...These results are more general than [22] and [23] that had focused on stabilization of synchronized and balanced phase arrangements using QLF-based design approach and do not address the issue of bounded trajectories....
[...]
01 Dec 2017
TL;DR: This work synthesizes a feedback for a fully connected network of identical Liénard-type oscillators such that the phase-balanced equilibrium — the mode where the centroid of the coupled oscillators in polar coordinates is at the origin — is asymptotically stable, and thephase-synchronized equilibrium is unstable.
Abstract: We synthesize a feedback for a fully connected network of identical Lienard-type oscillators such that the phase-balanced equilibrium — the mode where the centroid of the coupled oscillators in polar coordinates is at the origin — is asymptotically stable, and the phase-synchronized equilibrium is unstable. Our approach hinges on a coordinate transformation of the oscillator dynamics to polar coordinates, and periodic averaging theory to simplify the examination of multiple time-scale behavior. Using Lyapunov- and linearization-based arguments, we demonstrate that the oscillator dynamics have the same radii and balanced phases in steady state for a large set of initial conditions. Numerical simulation results are presented to validate the analyses.
6 citations
Cites result from "Stabilization of collective motion ..."
...This is similar to the balanced set defined in [4], [7]....
[...]
Posted Content•
TL;DR: This paper considers the collective circular motion of multi-agent systems in which all the agents are required to traverse different circles or a common circle at a prescribed angular velocity, and presents suitable feedback control laws by considering a second-order rotational dynamics of the agent.
Abstract: This paper considers the collective circular motion of multi-agent systems in which all the agents are required to traverse different circles or a common circle at a prescribed angular velocity. It is required to achieve these collective motions with the heading angles of the agents synchronized or balanced. In synchronization, the agents and their centroid have a common velocity direction, while in balancing, the movement of agents causes the location of the centroid to become stationary. The agents considered are initially moving at unit speed around individual circles at different angular velocities. It is assumed that the agents are subjected to limited communication constraints, and exchange relative information according to a time-invariant undirected graph. We present suitable feedback control laws for each of these motion coordination tasks by considering a second-order rotational dynamics of the agent. Simulations are given to illustrate the theoretical findings.
3 citations
TL;DR: This paper investigates the circular motion of a group of n(n ≥ 2) nonholonomic robots over time-varying communication networks to achieve a balanced circular motion centered at a location determined by each robot under the assumption that the global ranking of any robot is unknown.
Abstract: This paper investigates the circular motion of a group of n(n ≥ 2) nonholonomic robots over time-varying communication networks. We aim to achieve a balanced circular motion centered at a location determined by each robot under the assumption that the global ranking of any robot is unknown. A back-stepping based controller is firstly designed to make all the robots rotate around a common center, the position of which is obtained through executing a consensus algorithm by each robot. Then, the maximum and minimum consensus algorithm and a distributed modified ordinal ranking algorithm are applied to set the rotation radius, angular velocity, and orientation parameters in a distributed manner such that all the robots can uniformly space on the common circle. At last, the effectiveness of the proposed algorithms is illustrated through a simulation example.
3 citations
References
More filters
TL;DR: In this article, the authors review 25 years of research on the Kuramoto model, highlighting the false turns as well as the successes, but mainly following the trail leading from Kuramoto's work to Crawford's recent contributions.
2,795 citations
"Stabilization of collective motion ..." refers background in this paper
...which is also referred to as the phase order parameter [20]....
[...]
TL;DR: This paper shows that the system's equilibrium formations are generalized regular polygons and it is exposed how the multivehicle system's global behavior can be shaped through appropriate controller gain assignments.
Abstract: Inspired by the so-called "bugs" problem from mathematics, we study the geometric formations of multivehicle systems under cyclic pursuit. First, we introduce the notion of cyclic pursuit by examining a system of identical linear agents in the plane. This idea is then extended to a system of wheeled vehicles, each subject to a single nonholonomic constraint (i.e., unicycles), which is the principal focus of this paper. The pursuit framework is particularly simple in that the n identical vehicles are ordered such that vehicle i pursues vehicle i+1 modulo n. In this paper, we assume each vehicle has the same constant forward speed. We show that the system's equilibrium formations are generalized regular polygons and it is exposed how the multivehicle system's global behavior can be shaped through appropriate controller gain assignments. We then study the local stability of these equilibrium polygons, revealing which formations are stable and which are not.
669 citations
"Stabilization of collective motion ..." refers background in this paper
...collective circular motion of multivehicle system under cyclic pursuit is given in [15]....
[...]
TL;DR: 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.
Abstract: 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
528 citations
"Stabilization of collective motion ..." refers background or methods in this paper
...On the other hand, in [2], the synchronized and balanced collective motions of the agents, initially rotating around individual circles of same radius, are stabilized on a common circle, which is centered at the prescribed location and have a radius similar to that of initial individual circles....
[...]
...holds [1], [2], where bN/2c is the largest integer less than or equal to N/2....
[...]
...All other critical points of U are isolated in the shape manifold TN/S1 and are saddle points of U [2]....
[...]
...Previous work in this direction [2] has focused on achieving the common circular motion about a desired center when the angular frequencies of the initial circular motions of the agents, are same....
[...]
...The m-th harmonic of the phase order parameter pθ , which plays an important role in stabilizing the splay phase arrangement, is given by [1], [2]...
[...]
Book•
01 May 2000
TL;DR: In this paper, the authors present an overview of the history of linear algebra and its applications in computer graphics and computer networks, including the following: 1.1 Introduction to Systems of Linear Equations and Matrices. 2.1 Determinants by Cofactor Expansion. 3.2 Evaluating determinants by Row Reduction. 4.3 Properties of the Determinant Function.
Abstract: Chapter 1. Systems of Linear Equations and Matrices. 1.1 Introduction to Systems of Linear Equations. 1.2 Gaussian Elimination. 1.3 Matrices and Matrix Operations. 1.4 Inverses Rules of Matrix Arithmetic. 1.5 Elementary Matrices and a Method for Finding A-1. 1.6 Further Results on Systems of Equations and Invertibility. 1.7 Diagonal, Triangular, and Symmetric Matrices. Chapter 2. Determinants. 2.1 Determinants by Cofactor Expansion. 2.2 Evaluating Determinants by Row Reduction. 2.3 Properties of the Determinant Function. 2.4 A Combinatorial Approach to Determinants. Chapter 3. Vectors in 2 Space and 3-Space. 3.1 Introduction to Vectors (Geometric). 3.2 Norm of a Vector Vector Arithmetic. 3.3 Dot Product Projections. 3.4 Cross Product. 3.5 Lines and Planes in 3-Space. Chapter 4. Euclidean Vector Spaces. 4.1 Euclidean n-Space. 4.2 Linear Transformations from Rn to Rm. 4.3 Properties of Linear Transformations from Rn to Rm. 4.4 Linear Transformations and Polynomials. Chapter 5. General Vector Spaces. 5.1 Real Vector Spaces. 5.2 Subspaces. 5.3 Linear Independence. 5.4 Basis and Dimension. 5.5 Row Space, Column Space, and Nullspace. 5.6 Rank and Nullity. Chapter 6. Inner Product Spaces. 6.1 Inner Products. 6.2 Angle and Orthogonality in Inner Product Spaces. 6.3 Orthonormal Bases: Gram-Schmidt Prodcess QR-Decomposition. 6.4 Best Approximation Least Squares. 6.5 Change of Basis. 6.6 Orthogonal Matrices. Chapter 7. Eigenvalues, Eigenvectors. 7.1 Eigenvalues and Eigenvectors. 7.2 Diagonalization. 7.3 Orthogonal Diagonalization. Chapter 8. Linear Transformations. 8.1 General Linear Transformations. 8.2 Kernel and range. 8.3 Inverse Linear Transformations. 8.4 Matrices of General Linear Transformations. 8.5 Similarity. 8.6 Isomorphism. Chapter 9. Additional topics. 9.1 Application to Differential Equations. 9.2 Geometry and Linear Operators on R2. 9.3 Least Squares Fitting to Data. 9.4 Approximation Problems Fourier Series. 9.5 Quadratic Forms. 9.6 Diagonalizing Quadratic Forms Conic Sections. 9.7 Quadric Surfaces. 9.8 Comparison of Procedures for Solving Linear Systems. 9.9 LU-Decompositions. Chapter 10. Complex Vector Spaces. 10.1 Complex Numbers. 10.2 Division of Complex Numbers. 10.3 Polar Form of a Complex Number. 10.4 Complex Vector Spaces. 10.5 Complex Inner Product Spaces. 10.6 Unitary Normal, and Hermitian Matrices. Chapter 11. Applications of Linear Algebra. 11.1 Constructing Curves and Surfaces through Specified Points. 11.2 Electrical Networks. 11.3 Geometric Linear Programming. 11.4 The Earliest Applications of Linear Algebra. 11.5 Cubic Spline Interpolation. 11.6 Markov Chains. 11.7 Graph Theory. 11.8 Games of Strategy. 11.9 Leontief Economic Models. 11.10 Forest Management. 11.11 Computer Graphics. 11.12 Equilibrium Temperature Distributions. 11.13 Computed Tomography. 11.14 Fractals. 11.15 Chaos. 11.16 Cryptography. 11.17 Genetics. 11.18 Age-Specific Population Growth. 11.19 Harvesting of Animal Populations. 11.20 A Least Squares Model for Human Hearing. 11.21 Warps and Morphs. Answers to Exercises. Index.
305 citations
"Stabilization of collective motion ..." refers background in this paper
...Some of the important properties of the inner product 〈·, ·〉, which are relevant in the framework of the present paper can be found in [19], and are listed below:...
[...]
TL;DR: PCOD as discussed by the authors is a cooperative control framework for stabilizing relative equilibria in a model of self-propelled, steered particles moving in the plane at unit speed, which is applicable to time-invariant and undirected interaction.
Abstract: This article describes PCOD, a cooperative control framework for stabilizing relative equilibria in a model of self-propelled, steered particles moving in the plane at unit speed. Relative equilibria correspond either to motion of all of the particles in the same direction or to motion of all of the particles around the same circle. Although the framework applies to time-varying and directed interaction between individuals, we focus here on time-invariant and undirected interaction, using the Laplacian matrix of the interaction graph to design a set of decentralized control laws applicable to mobile sensor networks. Since the direction of motion of each particle is represented in the framework by a point on the unit circle, the closed-loop model has coupled-phase oscillator dynamics.
253 citations
"Stabilization of collective motion ..." refers background in this paper
...The orientation, θk of the velocity vector is also referred to as the phase of the k-th agent [16], [17]....
[...]