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In this paper, we develop a \textit{distributed} algorithm to localize anarbitrary number of agents moving in a bounded region of interest. We assumethat the network contains \textit{at least one} agent with known location(hereinafter referred to as an anchor), and each agent measures a noisy versionof its motion and the distances to the nearby agents. We providea~\emph{geometric approach}, which allows each agent to: (i) continually updatethe distances to the locations where it has exchanged information with theother nodes in the past; and (ii) measure the distance between a neighbor andany such locations. Based on this approach, we provide a \emph{linear update}to find the locations of an arbitrary number of mobile agents when they followsome convexity in their deployment and motion.  Since the agents are mobile, they may not be able to find nearby nodes(agents and/or anchors) to implement a distributed algorithm. To address thisissue, we introduce the notion of a \emph{virtual convex hull} with the help ofthe aforementioned geometric approach. In particular, each agent keeps track ofa virtual convex hull of other nodes, which may not physically exist, andupdates its location with respect to its neighbors in the virtual hull. We showthat the corresponding localization algorithm, in the absence of noise, can beabstracted as a Linear Time-Varying (LTV) system, with non-deterministic systemmatrices, which asymptotically tracks the true locations of the agents. Weprovide simulations to verify the analytical results and evaluate theperformance of the algorithm in the presence of noise on the motion as well ason the distance measurements.

Localization in mobile networks via virtual convex hulls