Limited benefit of cooperation in distributed relative localization

Important applications in robotic and sensor networks require distributed algorithms to solve the so-called relative localization problem: a node-indexed vector has to be reconstructed from measurements of differences between neighbor nodes. In a recent note, we have studied the estimation error of a popular gradient descent algorithm showing that the mean square error has a minimum at a finite time, after which the performance worsens. This paper proposes a suitable modification of this algorithm incorporating more realistic "a priori" information on the position. The new algorithm presents a performance monotonically decreasing to the optimal one. Furthermore, we show that the optimal performance is approximated, up to a 1 + \eps factor, within a time which is independent of the graph and of the number of nodes. This convergence time is very much related to the minimum exhibited by the previous algorithm and both lead to the following conclusion: in the presence of noisy data, cooperation is only useful till a certain limit.Comment: 11 pages, 2 figures, submitted to conferenc

Average resistance of toroidal graphs

The average effective resistance of a graph is a relevant performance index in many applications, including distributed estimation and control of network systems. In this paper, we study how the average resistance depends on the graph topology and specifically on the dimension of the graph. We concentrate on $d$-dimensional toroidal grids and we exploit the connection between resistance and Laplacian eigenvalues. Our analysis provides tight estimates of the average resistance, which are key to study its asymptotic behavior when the number of nodes grows to infinity. In dimension two, the average resistance diverges: in this case, we are able to capture its rate of growth when the sides of the grid grow at different rates. In higher dimensions, the average resistance is bounded uniformly in the number of nodes: in this case, we conjecture that its value is of order $1/d$ for large $d$. We prove this fact for hypercubes and when the side lengths go to infinity.Comment: 24 pages, 6 figures, to appear in SIAM Journal on Control and Optimization (SICON

Distributed estimation from relative measurements of heterogeneous and uncertain quality

This paper studies the problem of estimation from relative measurements in a graph, in which a vector indexed over the nodes has to be reconstructed from pairwise measurements of differences between its components associated to nodes connected by an edge. In order to model heterogeneity and uncertainty of the measurements, we assume them to be affected by additive noise distributed according to a Gaussian mixture. In this original setup, we formulate the problem of computing the Maximum-Likelihood (ML) estimates and we design two novel algorithms, based on Least Squares regression and Expectation-Maximization (EM). The first algorithm (LS- EM) is centralized and performs the estimation from relative measurements, the soft classification of the measurements, and the estimation of the noise parameters. The second algorithm (Distributed LS-EM) is distributed and performs estimation and soft classification of the measurements, but requires the knowledge of the noise parameters. We provide rigorous proofs of convergence of both algorithms and we present numerical experiments to evaluate and compare their performance with classical solutions. The experiments show the robustness of the proposed methods against different kinds of noise and, for the Distributed LS-EM, against errors in the knowledge of noise parameters.Comment: Submitted to IEEE transaction

Distributed Estimation from Relative and Absolute Measurements

International audienceThis note defines the problem of least-squares distributed estimation from relative and absolute measurements, by encoding the set of measurements in a weighted undirected graph. The role of its topology is studied by an electrical interpretation, which easily allows distinguishing between topologies that lead to "small" or "large" estimation errors. The least-squares problem is solved by a distributed gradient algorithm: the computed solution is approximately optimal after a number of steps that does not depend on the size of the problem or on the graph-theoretic properties of its encoding. This fact indicates that only a limited cooperation between the sensors is necessary

Distributed relative localization using the multi-dimensional weighted centroid

A key problem in multi-agent systems is the distributed estimation of the localization of agents in a common reference from relative measurements. Estimations can be referred to an anchor node or, as we do here, referred to the weighted centroid of the multi-agent system. We propose a Jacobi Over—Relaxation method for distributed estimation of the weighted centroid of the multi-agent system from noisy relative measurements. Contrary to previous approaches, we consider relative multidimensional measurements with general covariance matrices not necessarily diagonal. We prove our weighted centroid method converges faster than anchor-based solutions. We also analyze the method convergence and provide mathematical constraints that ensure avoiding ringing phenomena