Dynamic Coverage Verification in Mobile Sensor Networks Via Switched Higher Order Laplacians

Abstract

Abstract-In this paper, we study the problem of verifying dynamic coverage in mobile sensor networks using certain switched linear systems. These switched systems describe the flow of discrete differential forms on time-evolving simplicial complexes. The simplicial complexes model the connectivity of agents in the network, and the homology groups of the simplicial complexes provides information about the coverage properties of the network. Our main result states that the asymptotic stability the switched linear system implies that every point of the domain covered by the mobile sensor nodes is visited infinitely often, hence verifying dynamic coverage. The enabling mathematical technique for this result is the theory of higher order Laplacian operators, which is a generalization of the graph Laplacian used in spectral graph theory and continuous-time consensus problems.

Towards simplicial coverage repair for mobile robot teams

“…Among these is [6] who consider a topological variation of the evader-pursuer problem where the objective is to ensure that the evader, initially occupying a coverage hole, cannot go undetected indefinitely within some fenced region. Additionally, in [7], the authors consider a switched dynamical system using higher-order Laplacians and estab lish that for a team of agents constrained to some domain that each point in that domain will be visited infinitely often.…”

“…In generating the proposed algorithm, we leverage the recent results of [4] and [7]. In [4] the authors employ a decentralized subgradient method for metric-free hole localization by solving the following LP (10) where x E ker Ck.…”

“…Additionally, we leverage the results of [7]. In this work, the authors consider the combinatorial Laplacian flow…”

Run to Potential: Sweep Coverage in Wireless Sensor Networks

“…Although mobile nodes do not exchange information with each other, they need to communication with the static sensors to avoid duplicate coverage. Muhammad et al [17] provide a verification method for sweep coverage.…”

Testing contractibility in planar rips complexes

“…A recent example of this approach is the analysis by Carlsson and others [7,9,29] of a large set of nine-dimensional feature points extracted from digital photographs (the "Mumford data set"). Rips complexes of points in the plane have also been used to model coverage problems in sensor networks [30,46,45]; this particular use of Rips complexes was the original motivation for our work.…”