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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. ᭧

No abstract

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.

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The problem of localization and circumnavigation of a slowly moving target with unknown speed has been considered. The agent only knows its own position with respect to its initial frame, and the bearing angle to the target in that frame. We propose an estimator to localize the target and a control law that forces the agent to move on a circular trajectory around the target such that both the estimator and the control system are exponentially stable. We consider two different cases where the agent's speed is constant and variable. The performance of the proposed algorithm is verified through simulations.

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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.