Estimation from Relative Measurements: Electrical Analogy and Large Graphs

Abstract

Abstract-We examine the problem of estimating vectorvalued variables from noisy measurements of the difference between certain pairs of them. This problem, which is naturally posed in terms of a measurement graph, arises in applications such as sensor network localization, time synchronization, and motion consensus.We obtain a characterization on the minimum possible covariance of the estimation error when an arbitrarily large number of measurements are available. This covariance is shown to be equal to a matrix-valued effective resistance in an infinite electrical network. Covariance in large finite graphs converges to this effective resistance as the size of the graphs increases. This convergence result provides the formal justification for regarding large finite graphs as infinite graphs, which can be exploited to determine scaling laws for the estimation error in large finite graphs. Furthermore, these results indicate that in large networks, estimation algorithms that use small subsets of all the available measurements can still obtain accurate estimates.

Angle Fixability and Angle-Based Sensor Network Localization

“…We make the following assumption for problem (4). Assumption 1: Problem (4) is feasible and all sensors do not collide with each other.…”

“…Proof: Sufficiency. Let y = (y 1 , ..., y n ) ∈ R 2n be a solution to (4). Since (Ĝ, x) is angle fixable, we have y = c(I n ⊗ R)x + 1 n ⊗ ξ for some c ∈ R \ {0}, R ∈ O(2), ξ ∈ R 2 .…”

“…In the literature, depending on sensing capabilities of sensors, the sensor network localization (SNL) problems have been studied via relative position-based [4], range-based [1], [5], [6], [7], [8], [9], [10] and bearing (angle of arrival)-based [11], [12], [13], [14], [15], [16], [17] approaches. Among them, bearing-based SNL is a popular topic in recent years due to the advantage that bearing measurements can be easily obtained by vision sensors [15].…”

Transient and limit performance of distributed relative localization

“…A few remarks are due about the formula for J t (A) found in Proposition (2). First of all, J t (A) depends on the eigenvalues of the Laplacian L via the relation…”

“…We refer to this problem as the problem of relative localization. Our formulation of the problem essentially adopts that of Barooah and Hespanha [1], [2], [3], although these authors assume to have an anchor node, in order to avoid the uncertainty about the additive constant, and they resort to infinite graphs as an analysis and modeling tool.…”