Stabilisation of infinitesimally rigid formations of multi-robot networks
TL;DR: It is shown that infinitesimal rigidity is a sufficient condition for local asymptotical stability of the equilibrium manifold of the multivehicle system.
Abstract: This article considers the design of a formation control for multivehicle systems that uses only local information. The control is derived from a potential function based on an undirected infinitesimally rigid graph that specifies the target formation. A potential function is obtained from the graph, from which a gradient control is derived. Under this controller the target formation becomes a manifold of equilibria for the multivehicle system. It is shown that infinitesimal rigidity is a sufficient condition for local asymptotical stability of the equilibrium manifold. A complete study of the stability of the regular polygon formation is presented and results for directed graphs are presented as well. Finally, the controller is validated experimentally.
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TL;DR: This paper proposes gradient descent algorithms for a class of utility functions which encode optimal coverage and sensing policies which are adaptive, distributed, asynchronous, and verifiably correct.
Abstract: This paper presents control and coordination algorithms for groups of vehicles. The focus is on autonomous vehicle networks performing distributed sensing tasks where each vehicle plays the role of a mobile tunable sensor. The paper proposes gradient descent algorithms for a class of utility functions which encode optimal coverage and sensing policies. The resulting closed-loop behavior is adaptive, distributed, asynchronous, and verifiably correct.
2,198 citations
TL;DR: A survey of formation control of multi-agent systems focuses on the sensing capability and the interaction topology of agents, and categorizes the existing results into position-, displacement-, and distance-based control.
1,751 citations
TL;DR: A new technique based on complex Laplacian is introduced to address the problems of which formation shapes specified by inter-agent relative positions can be formed and how they can be achieved with distributed control ensuring global stability.
Abstract: The paper concentrates on the fundamental coordination problem that requires a network of agents to achieve a specific but arbitrary formation shape. A new technique based on complex Laplacian is introduced to address the problems of which formation shapes specified by inter-agent relative positions can be formed and how they can be achieved with distributed control ensuring global stability. Concerning the first question, we show that all similar formations subject to only shape constraints are those that lie in the null space of a complex Laplacian satisfying certain rank condition and that a formation shape can be realized almost surely if and only if the graph modeling the inter-agent specification of the formation shape is 2-rooted. Concerning the second question, a distributed and linear control law is developed based on the complex Laplacian specifying the target formation shape, and provable existence conditions of stabilizing gains to assign the eigenvalues of the closed-loop system at desired locations are given. Moreover, we show how the formation shape control law is extended to achieve a rigid formation if a subset of knowledgable agents knowing the desired formation size scales the formation while the rest agents do not need to re-design and change their control laws.
360 citations
TL;DR: In this article, it is shown that a framework in an arbitrary dimension can be uniquely determined up to a translation and a scaling factor by the bearings if and only if the framework is infinitesimally bearing rigid.
Abstract: A fundamental problem that the bearing rigidity theory studies is to determine when a framework can be uniquely determined up to a translation and a scaling factor by its inter-neighbor bearings While many previous works focused on the bearing rigidity of two-dimensional frameworks, a first contribution of this paper is to extend these results to arbitrary dimensions It is shown that a framework in an arbitrary dimension can be uniquely determined up to a translation and a scaling factor by the bearings if and only if the framework is infinitesimally bearing rigid In this paper, the proposed bearing rigidity theory is further applied to the bearing-only formation stabilization problem where the target formation is defined by inter-neighbor bearings and the feedback control uses only bearing measurements Nonlinear distributed bearing-only formation control laws are proposed for the cases with and without a global orientation It is proved that the control laws can almost globally stabilize infinitesimally bearing rigid formations Numerical simulations are provided to support the analysis
309 citations
TL;DR: Within this framework, the development on this topic is systematically reviewed and the representative outcomes can be sorted out from four aspects: 1) agent dynamics; 2) network topologies; 3) feedback and communication mechanisms; 4) collective behaviors.
Abstract: Collective control of a multiagent system is concerned with designing strategies for a group of autonomous agents operating in a networked environment. The aim is to achieve a global control objective through distributed sensing, communication, computing, and control. It has attracted many researchers from a wide range of disciplines, including the literature of automatic control. This paper aims to give a general framework that is able to accommodate many of these outcomes. Within this framework, the development on this topic is systematically reviewed and the representative outcomes can be sorted out from four aspects: 1) agent dynamics; 2) network topologies; 3) feedback and communication mechanisms; and 4) collective behaviors. Thus, the state-of-the-art approach and technology is described. Moreover, within this framework, further interesting and promising directions on this research topic are envisioned.
243 citations
References
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Book•
01 Oct 1989
TL;DR: The Poincare-Bendixson Theorem as mentioned in this paper describes the existence, uniqueness, differentiability, and flow properties of vector fields, and is used to prove that a dynamical system is Chaotic.
Abstract: Equilibrium Solutions, Stability, and Linearized Stability * Liapunov Functions * Invariant Manifolds: Linear and Nonlinear Systems * Periodic Orbits * Vector Fields Possessing an Integral * Index Theory * Some General Properties of Vector Fields: Existence, Uniqueness, Differentiability, and Flows * Asymptotic Behavior * The Poincare-Bendixson Theorem * Poincare Maps * Conjugacies of Maps, and Varying the Cross-Section * Structural Stability, Genericity, and Transversality * Lagrange's Equations * Hamiltonian Vector Fields * Gradient Vector Fields * Reversible Dynamical Systems * Asymptotically Autonomous Vector Fields * Center Manifolds * Normal Forms * Bifurcation of Fixed Points of Vector Fields * Bifurcations of Fixed Points of Maps * On the Interpretation and Application of Bifurcation Diagrams: A Word of Caution * The Smale Horseshoe * Symbolic Dynamics * The Conley-Moser Conditions or 'How to Prove That a Dynamical System is Chaotic' * Dynamics Near Homoclinic Points of Two-Dimensional Maps * Orbits Homoclinic to Hyperbolic Fixed Points in Three-Dimensional Autonomous Vector Fields * Melnikov's Method for Homoclinic Orbits in Two-Dimensional, Time-Periodic Vector Fields * Liapunov Exponents * Chaos and Strange Attractors * Hyperbolic Invariant Sets: A Chaotic Saddle * Long Period Sinks in Dissipative Systems and Elliptic Islands in Conservative Systems * Global Bifurcations Arising from Local Codimension-Two Bifurcations * Glossary of Frequently Used Terms
5,220 citations
"Stabilisation of infinitesimally ri..." refers background in this paper
...Theorem 3 (Wiggins 1990, p. 195): If the origin is stable under (18), then the origin of (16)–(17) is also stable....
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TL;DR: A theoretical framework for design and analysis of distributed flocking algorithms, and shows that migration of flocks can be performed using a peer-to-peer network of agents, i.e., "flocks need no leaders."
Abstract: In this paper, we present a theoretical framework for design and analysis of distributed flocking algorithms. Two cases of flocking in free-space and presence of multiple obstacles are considered. We present three flocking algorithms: two for free-flocking and one for constrained flocking. A comprehensive analysis of the first two algorithms is provided. We demonstrate the first algorithm embodies all three rules of Reynolds. This is a formal approach to extraction of interaction rules that lead to the emergence of collective behavior. We show that the first algorithm generically leads to regular fragmentation, whereas the second and third algorithms both lead to flocking. A systematic method is provided for construction of cost functions (or collective potentials) for flocking. These collective potentials penalize deviation from a class of lattice-shape objects called /spl alpha/-lattices. We use a multi-species framework for construction of collective potentials that consist of flock-members, or /spl alpha/-agents, and virtual agents associated with /spl alpha/-agents called /spl beta/- and /spl gamma/-agents. We show that migration of flocks can be performed using a peer-to-peer network of agents, i.e., "flocks need no leaders." A "universal" definition of flocking for particle systems with similarities to Lyapunov stability is given. Several simulation results are provided that demonstrate performing 2-D and 3-D flocking, split/rejoin maneuver, and squeezing maneuver for hundreds of agents using the proposed algorithms.
4,693 citations
"Stabilisation of infinitesimally ri..." refers background or methods in this paper
...The starting point for our article is Olfati-Saber and Murray (2002). Following that article, we use graphs to define formations, but instead of global rigidity we use infinitesimal rigidity and instead of the double integrator model we use the simpler single integrator (kinematic point). More substantially, our stability analysis is complete whereas, being a conference paper, Olfati-Saber and Murray (2002) provides only a sketch. In particular, Olfati-Saber and Murray (2002) have no topological analysis of the equilibrium set and does not note that the equilibrium set is not compact. Moreover, Olfati-Saber and Murray (2002) use a LaSalle argument to prove stability, but since the equilibrium set is not compact, this is open to question....
[...]
...The starting point for our article is Olfati-Saber and Murray (2002). Following that article, we use graphs to define formations, but instead of global rigidity we use infinitesimal rigidity and instead of the double integrator model we use the simpler single integrator (kinematic point). More substantially, our stability analysis is complete whereas, being a conference paper, Olfati-Saber and Murray (2002) provides only a sketch. In particular, Olfati-Saber and Murray (2002) have no topological analysis of the equilibrium set and does not note that the equilibrium set is not compact....
[...]
...An interesting approach to formation control is that of Olfati-Saber (2006)....
[...]
...The starting point for our article is Olfati-Saber and Murray (2002). Following that article, we use graphs to define formations, but instead of global rigidity we use infinitesimal rigidity and instead of the double integrator model we use the simpler single integrator (kinematic point). More substantially, our stability analysis is complete whereas, being a conference paper, Olfati-Saber and Murray (2002) provides only a sketch. In particular, Olfati-Saber and Murray (2002) have no topological analysis of the equilibrium set and does not note that the equilibrium set is not compact. Moreover, Olfati-Saber and Murray (2002) use a LaSalle argument to prove stability, but since the equilibrium set is not compact, this is open to question. Furthermore, Olfati-Saber and Murray (2002) do not address if the trajectories have a limit on the equilibrium set....
[...]
...Remark 2: The proof approach of Olfati-Saber and Murray (2002) is to quotient out the dynamics on the equilibrium manifold so the equilibrium is topologically equivalent to a point....
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16 May 1974
TL;DR: In this article, the authors introduce algebraic graph theory and show that the spectrum of a graph can be modelled as a graph graph, and the spectrum can be represented as a set of connected spanning trees.
Abstract: 1. Introduction to algebraic graph theory Part I. Linear Algebra in Graphic Thoery: 2. The spectrum of a graph 3. Regular graphs and line graphs 4. Cycles and cuts 5. Spanning trees and associated structures 6. The tree-number 7. Determinant expansions 8. Vertex-partitions and the spectrum Part II. Colouring Problems: 9. The chromatic polynomial 10. Subgraph expansions 11. The multiplicative expansion 12. The induced subgraph expansion 13. The Tutte polynomial 14. Chromatic polynomials and spanning trees Part III. Symmetry and Regularity: 15. Automorphisms of graphs 16. Vertex-transitive graphs 17. Symmetric graphs 18. Symmetric graphs of degree three 19. The covering graph construction 20. Distance-transitive graphs 21. Feasibility of intersection arrays 22. Imprimitivity 23. Minimal regular graphs with given girth References Index.
2,924 citations
07 Aug 2002
TL;DR: In this paper, the authors describe decentralized control laws for the coordination of multiple vehicles performing spatially distributed tasks, which are based on a gradient descent scheme applied to a class of decentralized utility functions that encode optimal coverage and sensing policies.
Abstract: This paper describes decentralized control laws for the coordination of multiple vehicles performing spatially distributed tasks. The control laws are based on a gradient descent scheme applied to a class of decentralized utility functions that encode optimal coverage and sensing policies. These utility functions are studied in geographical optimization problems and they arise naturally in vector quantization and in sensor allocation tasks. The approach exploits the computational geometry of spatial structures such as Voronoi diagrams.
2,445 citations
Posted Content•
TL;DR: This paper proposes gradient descent algorithms for a class of utility functions which encode optimal coverage and sensing policies which are adaptive, distributed, asynchronous, and verifiably correct.
Abstract: This paper presents control and coordination algorithms for groups of vehicles. The focus is on autonomous vehicle networks performing distributed sensing tasks where each vehicle plays the role of a mobile tunable sensor. The paper proposes gradient descent algorithms for a class of utility functions which encode optimal coverage and sensing policies. The resulting closed-loop behavior is adaptive, distributed, asynchronous, and verifiably correct.
2,198 citations