Changbin Yu

Rigid graph control architectures for autonomous formations

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 Lamans 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.

Cluster consensus control of generic linear multi-agent systems under directed topology with acyclic partition

This paper investigates the cluster consensus control for generic linear multi-agent systems (MASs) under directed interaction topology via distributed feedback controller. Focus of this paper is particularly on addressing the following problem which is of both theoretical and practical interests but have not been considered in the existing literature: under what kind of interaction among the clusters can the cluster consensus control be achieved regardless of the magnitudes of the coupling strengths among the agents? Directed acyclic interaction topology among the clusters is proved to have this property. As opposed to the algebraic conditions provided in the existing literature, conditions for guaranteeing the cluster consensus control in this paper are presented in terms of purely the graphic topology conditions and thus are very easy to be verified.

Control of Minimally Persistent Formations in the Plane

This paper studies the problem of controlling the shape of a formation of point agents in the plane. A model is considered where the distance between certain agent pairs is maintained by one of the agents making up the pair; if enough appropriately chosen distances are maintained, with the number growing linearly with the number of agents, then the shape of the formation will be maintained. The detailed question examined in the paper is how one may construct decentralized nonlinear control laws to be operated at each agent that will restore the shape of the formation in the presence of small distortions from the nominal shape. Using the theory of rigid and persistent graphs, the question is answered. As it turns out, a certain submatrix of a matrix known as the rigidity matrix can be proved to have nonzero leading principal minors, which allows the determination of a stabilizing control law.

Three and higher dimensional autonomous formations: Rigidity, persistence and structural persistence

In this paper, we generalize the notion of persistence, which has been originally introduced for two-dimensional formations, to R^d for d>=3, seeking to provide a theoretical framework for real world applications, which often are in three-dimensional space as opposed to the plane. Persistence captures the desirable property that a formation moves as a cohesive whole when certain agents maintain their distances from certain other agents. We verify that many of the properties of rigid and/or persistent formations established in R^2 are also valid for higher dimensions. Analysing the closed subgraphs and directed paths in persistent graphs, we derive some further properties of persistent formations. We also provide an easily checkable necessary condition for persistence. We then turn our attention to consider some practical issues raised in multi-agent formation control in three-dimensional space. We display a new phenomenon, not present in R^2, whereby subsets of agents can behave in a problematic way. When this behaviour is precluded, we say that the graph depicting the multi-agent formation has structural persistence. In real deployment of controlled multi-agent systems, formations with underlying structurally persistent graphs are of interest. We analyse the characteristics of structurally persistent graphs and provide a streamlined test for structural persistence. We study the connections between the allocation of degrees of freedom (DOFs) across agents and the characteristics of persistence and/or structural persistence of a directed graph. We also show how to transfer DOFs among agents, when the formation changes with new agent(s) added, to preserve persistence and/or structural persistence.

Control of a three-coleader formation in the plane

This paper considers a formation of three point agents moving in the plane, where the agents have a cyclic ordering with each one required to maintain a nominated distance from its neighbour; further, each agent is allowed to determine its movement strategy using local knowledge only of the direction of its neighbour, and the current and desired distance from its neighbour. The motion of the formation is studied when distances are initially incorrect. A convergence result is established, to the effect that provided agents never become collinear, the correct distances will be approached exponentially fast, and the formation as a whole will rotate by a finite angle and translate by a finite distance.

Brief paper: Formation control using range-only measurements

This paper proposes algorithms to coordinate a formation of mobile agents when the agents are not able to measure the relative positions of their neighbors, but only the distances to their respective neighbors. In this sense, less information is available to agents than is normally assumed in formation stabilization or station keeping problems. To control the shape of the formation, the solution advanced in the paper involves subsets of non-neighbor agents cyclically localizing the relative positions of their respective neighbor agents while these are held stationary, and then moving to reduce the value of a cost function which is nonnegative and assumes the zero value precisely when the formation has correct distances. The movement schedule is obtained by a novel vertex-coloring algorithm whose computation time is linear in the number of agents when implemented on the graphs of minimally rigid formations. Since in some formations, it may be that an agent is never allowed to be stationary (e.g. it is a fixed-wing aircraft), or because formations may be required to move as a whole in a certain direction, the results are extended to allow for cyclic localization of agents in this case. The tool used is the Cayley-Menger determinant.

Coordination for Linear Multiagent Systems With Dynamic Interaction Topology in the Leader-Following Framework

This paper considers mainly the leader-following consensus for multiple agents with general linear system dynamics under switching topologies. Three different settings are systematically considered. We first consider the setting that the underlying interaction topologies switch arbitrarily among the possible weakly connected digraphs and then extend it to a more general setting that the weak connectivity of the interaction topologies is kept for some disconnected time intervals with short length due to the communication constraints among agents. Exponentially, consensus control is proved to be achieved, and the convergence rate can be specified as well for both settings in spite of the relaxed conditions on the system dynamics of each individual agent which even allow that each agent has exponentially unstable mode, while for the last case where the weak connectivity is only maintained on the joint of the interaction topologies, consensus control is proved to be achieved when the system matrix of each individual agent satisfies certain stability conditions.

Leader-follower formation via complex Laplacian

The paper introduces complex-valued Laplacians for graphs whose edges are attributed with complex weights and studies the leader-follower formation problem based on complex Laplacians. The main goal is to control the shape of a planar formation of point agents in the plane using simple and linear interaction rules related to complex Laplacians. We present a characterization of complex Laplacians that preserve a specific planar formation as an equilibrium solution for both single integrator kinematics and double integrator dynamics. Planar formations under study are subject to translation, rotation, and scaling in the plane, but can be determined by two co-leaders in leader-follower networks. Furthermore, when a complex Laplacian does not result in an asymptotically stable behavior of the multi-agent system, we show that a stabilizing matrix, which updates the complex weights, exists to asymptotically stabilize the system while preserving the equilibrium formation. Also, algorithms are provided to find stabilizing matrices. Finally, simulations are presented to illustrate our results.

UAV Formation Control : Theory and Application

Unmanned airborne vehicles (UAVs) are finding use in military operations and starting to find use in civilian operations. UAVs often fly in formation, meaning that the distances between individual pairs of UAVs stay fixed, and the formation of UAVs in a sense moves as a rigid entity. In order to maintain the shape of a formation, it is enough to maintain the distance between a certain number of the agent pairs; this will result in the distance between all pairs being constant. We describe how to characterize the choice of agent pairs to secure this shape-preserving property for a planar formation, and we describe decentralized control laws which will stably restore the shape of a formation when the distances between nominated agent pairs become unequal to their prescribed values. A mixture of graph theory, nonlinear systems theory and linear algebra is relevant. We also consider a particular practical problem of flying a group of three UAVs in an equilateral triangle, with the centre of mass following a nominated trajectory reflecting constraints on turning radius, and with a requirement that the speeds of the UAVs are constant, and nearly (but not necessarily exactly) equal.

Control of Minimally Persistent Leader-Remote-Follower and Coleader Formations in the Plane

This paper solves an n -agent formation shape control problem in the plane. The objective is to design decentralized control laws so that the agents cooperatively restore a prescribed formation shape in the presence of small perturbations from the prescribed shape. We consider two classes of directed, cyclic information architectures associated with so-called minimally persistent formations: leader-remote-follower and coleader. In our framework the formation shape is maintained by controlling certain interagent distances. Only one agent is responsible for maintaining each distance. We propose a decentralized control law where each agent executes its control using only the relative position measurements of agents to which it must maintain its distance. The resulting nonlinear closed-loop system has a manifold of equilibria, which implies that the linearized system is nonhyperbolic. We apply center manifold theory to show local exponential stability of the desired formation shape. The result circumvents the non-compactness of the equilibrium manifold. Choosing stabilizing gains is possible if a certain submatrix of the rigidity matrix has all leading principal minors nonzero, and we show that this condition holds for all minimally persistent leader-remote-follower and coleader formations with generic agent positions. Simulations are provided.