Estimating the relative position of mobile agents on Jordan curves from ambiguous proximity data

Abstract

In this paper, we aim at estimating the relative positions of an ensemble of mobile agents. Specifically, the agents are assumed to move along a C 0 Jordan closedcurve of the plane with different speeds. When two of them get sufficiently close in the Euclidean norm they can measure, with a certain degree of precision, their relative Euclidean distance. Based on the knowledge of agent dynamics the prediction-correction algorithm described in this paper enables each agent to recursively estimate the distance on the C 0 curve of all other agents from this intermittent flow of ambiguous and uncertain proximity data. On the basis of these position estimates, any form of decentralized agent formation control could be implemented on the closed curve. The theoretical analysis of the estimation algorithm is validated numerically using a representative example. IntroductionIn this paper, we consider a group of agents moving along a closed curve in the plane, at given (possibly time varying) speeds. Each agent can spot other agents that are close to him in the plane (but not necessarily close on the curve) and gather, with a given sampling rate, a measurement of their Euclidean distance. The problem we consider is that of developing an algorithm that enables each agent to reconstruct the relative position of all the others along the curve, based on these intermittent measurements and on the knowledge of the agents' dynamics. Problems like this are often found when considering a surveillance problem in which a group of heterogeneous artificial agents, equipped with proximity sensors, must patrol a given closed route encircling a sensitive area [9,4,8]. The surveillance is effective if they appropriately space along the path and level out their speed to keep the formation. This formation control problem requires, as a first step, the estimation of the agents' relative position on the curve starting from sampled measurements of their Euclidean distance on the plane. Motivated by similar surveillance * Supported by the RETE DI ECCELLENZA MASTRI, P.O.R. Campania FSE 2007 2013, Asse V.† Department of Electrical Engineering and Information Technology, University of Naples Federico II, 80125, Naples, Italy.problems, in the recent literature on multi-agent systems [14,3,6,2,15], the problem of achieving balanced formations along circular paths has been considered. Specifically, the problem has been tackled with given assumptions on the connectivity among the agents: an allto-all connectivity was assumed in [12,13], while fixed and connected graphs were considered in [7,11]. Recently, an algorithm was proposed to estimate the relative phase among oscillators from proximity data, see [1].All cited authors, although considering different agent dynamics, do make reference to circular paths. In this paper, instead, we aim at extending the algorithm proposed in [1] to cope with generic closed curves. As the agents have, in general, different speeds, the proximity-based interaction between them implies alternate activations and de...