Rigid graph control architectures for autonomous formations
01 Dec 2008-IEEE Control Systems Magazine (Institute of Electrical and Electronics Engineers (IEEE Inc))-Vol. 28, Iss: 6, pp 48-63
TL;DR: In this paper, a theory for analyzing and creating architectures appropriate to the control of formations of autonomous vehicles is presented. The theory is based on ideas of rigid graph theory, some but not all of which are old.
Abstract: This article sets out the rudiments of a theory for analyzing and creating architectures appropriate to the control of formations of autonomous vehicles. The theory rests on ideas of rigid graph theory, some but not all of which are old. The theory, however, has some gaps in it, and their elimination would help in applications. Some of the gaps in the relevant graph theory are as follows. First, there is as yet no analogue for three-dimensional graphs of Laman's theorem, which provides a combinatorial criterion for rigidity in two-dimensional graphs. Second, for three-dimensional graphs there is no analogue of the two-dimensional Henneberg construction for growing or deconstructing minimally rigid graphs although there are conjectures. Third, global rigidity can easily be characterized for two-dimensional graphs, but not for three-dimensional graphs.
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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
01 Dec 2010
TL;DR: A robust consensus tracking problem for a class of second-order multi-agent systems has been addressed in the presence of disturbances and unmodeled dynamics using algebraic graph theory, Lyapunov-based analysis, and an invariance-like theorem.
Abstract: In this paper, a robust consensus tracking problem for a class of second-order multi-agent systems has been addressed in the presence of disturbances and unmodeled dynamics. The desired trajectory to be tracked is only provided to a small group of team members. An identifier is developed to estimate the unknown disturbances and unmodeled dynamics. A continuous consensus tracking controller is developed based on this identifier to achieve asymptotic consensus tracking using the local information obtained from neighboring agents. A sufficient condition is derived to ensure consensus tracking and asymptotic stability of the closed-loop system using algebraic graph theory, Lyapunov-based analysis, and an invariance-like theorem. Numerical simulations are provided to demonstrate the effectiveness of the developed robust consensus controller.
298 citations
TL;DR: This paper shows how different mechanisms may lead to clustering behavior in connected networks consisting of diffusively coupled agents, and presents two other mechanisms under which cluster synchronization might be achieved.
238 citations
Cites background from "Rigid graph control architectures f..."
...…March 2011 Accepted 6 May 2011 Available online 23 September 2011 Keywords: Synchronization Clustering Diffusive coupling Self-dynamics Time delay Negative coupling weights a b s t r a c t This paper shows how different mechanisms may lead to clustering behavior in connected networks consisting…...
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References
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TL;DR: A theoretical explanation for the observed behavior of the Vicsek model, which proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.
Abstract: In a recent Physical Review Letters article, Vicsek et al. propose a simple but compelling discrete-time model of n autonomous agents (i.e., points or particles) all moving in the plane with the same speed but with different headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its "neighbors." In their paper, Vicsek et al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.
8,233 citations
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
01 Dec 2001
TL;DR: This paper addresses the control of a team of nonholonomic mobile robots navigating in a terrain with obstacles while maintaining a desired formation and changing formations when required, using graph theory.
Abstract: This paper addresses the control of a team of nonholonomic mobile robots navigating in a terrain with obstacles while maintaining a desired formation and changing formations when required, using graph theory. We model the team as a triple, (g, r, H), consisting of a group element g that describes the gross position of the lead robot, a set of shape variables r that describe the relative positions of robots, and a control graph H that describes the behaviors of the robots in the formation. Our framework enables the representation and enumeration of possible control graphs and the coordination of transitions between any two formations.
1,175 citations
"Rigid graph control architectures f..." refers background in this paper
...An early examination of formation reshaping is provided in [14], and splitting and merging are addressed in [31]....
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...Also, it turns out that the graph-theoretic tools we describe can often be applied to formations with controlled (as opposed to sensed) variables that include both angles and distances [12]–[14]....
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...Nevertheless, if polygonal angle constraints are not present because of the way the graph associated with the formation is structured, then angle and distance constraints can be mixed [14]....
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TL;DR: In this paper, the combinatorial properties of rigid plane skeletal structures are investigated, and the properties are found to be adequately described by a class of graph-structured graphs.
Abstract: In this paper the combinatorial properties of rigid plane skeletal structures are investigated. Those properties are found to be adequately described by a class of graphs.
1,117 citations
"Rigid graph control architectures f..." refers background in this paper
...A key result is Laman’s theorem [20], for which no three-dimensional analog exists....
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...What Does Laman’s Theorem Say? Laman’s theorem [20] requires the idea of an induced subgraph of a graph G = (V, E )....
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10 Dec 2002
TL;DR: In this article, the authors describe a framework for cooperative control of a group of nonholonomic mobile robots that allows them to build complex systems from simple controllers and estimators, and guarantee stability and convergence in a wide range of tasks.
Abstract: We describe a framework for cooperative control of a group of nonholonomic mobile robots that allows us to build complex systems from simple controllers and estimators. The resultant modular approach is attractive because of the potential for reusability. Our approach to composition also guarantees stability and convergence in a wide range of tasks. There are two key features in our approach: 1) a paradigm for switching between simple decentralized controllers that allows for changes in formation; 2) the use of information from a single type of sensor, an omnidirectional camera, for all our controllers. We describe estimators that abstract the sensory information at different levels, enabling both decentralized and centralized cooperative control. Our results include numerical simulations and experiments using a testbed consisting of three nonholonomic robots.
1,068 citations