Localizability and distributed protocols for bearing-based network localization in arbitrary dimensions

Abstract

This paper addresses the problem of bearing-based network localization, which aims to localize all the nodes in a static network given the locations of a subset of nodes termed anchors and inter-node bearings measured in a common reference frame. The contributions of the paper are twofold. Firstly, we propose necessary and sufficient conditions for network localizability with both algebraic and rigidity theoretic interpretations. The analysis of the localizability heavily relies on the recently developed bearing rigidity theory and a special matrix termed the bearing Laplacian. Secondly, we propose a linear distributed protocol for bearing-based network localization. The protocol can globally localize a network if and only if the network is localizable. The sensitivity of the protocol to constant measurement errors is also analyzed. One novelty of this work is that the localizability analysis and localization protocol are applicable to networks in arbitrary dimensional spaces.Distributed localization of sensor networks is a core problem in many multi-agent coordination tasks. Network localizability and distributed protocols are two fundamental problems for any network localization problems. Network localizability characterizes whether or not a network can be possibly localized given the anchor locations and interneighbor relative measurements, whereas distributed protocols are used for localizing the network in a distributed manner if the network is localizable. According to the types of the relative measurements used for localization, the existing works can be divided into three classes: distance-based, bearing-based, and position-based. Distance-based network localization has been studied extensively so far (see [1][2][3][4] and the references therein). The analysis of the localizability in distance-based network localization relies heavily on the distance rigidity theory. It has been shown that a network in an n-dimensional space can be uniquely localized if the network is globally rigid and has at least n + 1 anchors in a general position [1]. More recently, bearing-based network localization has also attracted extensive research attention [5][6][7][8][9][10][11]. The analysis of the localizability in bearing-based network localization relies on the analogous bearing rigidity theory [12][13][14][15]. Finally, position-based network localization, where the inter-neighbor distance and local bearing measurements are used together for network localization, has been studied in [16] by using a complex graph Laplacian.Although bearing-based network localization has been studied by many researchers, the two fundamental problems, network localizability and distributed protocols, have not yet been fully explored. It was shown in [7-10] that a network is localizable when the network is bearing rigid and has at least two anchors. This condition is, however, sufficient but not necessary when the number of anchors is greater than two [10, Cor 10]. A necessary and sufficient condition for network localizability was proposed...

Bearing-Based Formation Control of A Group of Agents with Leader-First Follower Structure

“…All trajectories of (23) converge to the desired positions except for those starting at p 2 (0) = p * 2b . Moreover, the undesired equilibrium p 2 = p * 2b , p 3 = p * 3b is unstable.…”

“…A bearing-only stabilization control law for formations with undirected graphs in R d has been proposed in [21]. Further applications of bearing rigidity theory in formation maneuvering and network localization have also been discussed in [22], [23]. However, in these works, only undirected graphs were considered.…”

Formations on directed cycles with bearing‐only measurements

“…The problem considered in this paper is related to cyclic pursuit [25][26][27][28][29][30] (in which the interaction topology is also described by a cycle graph) and bearing-only localization problems. [31][32][33] However, this paper attempts to study a bearing-only-based formation control problem with directed cycle sensing graphs. To the best of our knowledge, this problem has not been studied in the existing literature.…”

Angle Fixability and Angle-Based Sensor Network Localization

“…Lemma 9 implies that at least 3 anchors are required for ASNL in R 2 . Note that for SNL based on bearings measured in the global coordinate frame [12], [13], [15], [16], the minimum number of anchors required for unique localizability in R 2 is 2. This is because the angle between two global bearings can be determined as clockwise or counterclockwise in the global coordinate frame.…”