Jan 12, 2012 - Accurate localization is essential for a wide range of appli- cations such as .... tification of an anchor placement over a traversal area. As a byproduct ...... Accuracy limitation of hyperbolic multilateration systems. IEEE Trans. on ...
arXiv:1201.2474v1 [cs.NI] 12 Jan 2012
On Quantification of Anchor Placement Yibei Ling
Scott Alexander
Richard Lau
Telcordia Technologies
Telcordia Technologies
Telcordia Technologies
Abstract—This paper attempts to answer a question: for a given traversal area, how to quantify the geometric impact of anchor placement on localization performance. We present a theoretical framework for quantifying the anchor placement impact. An experimental study, as well as the field test using a UWB ranging technology, is presented. These experimental results validate the theoretical analysis. As a byproduct, we propose a two-phase localization method (TPLM) and show that TPLM outperforms the least-square method in localization accuracy by a huge margin. TPLM performs much faster than the gradient descent method and slightly better than the gradient descent method in localization accuracy. Our field test suggests that TPLM is more robust against noise than the least-square and gradient descent methods.
1. I NTRODUCTION Accurate localization is essential for a wide range of applications such as mobile ad hoc networking, cognitive radio and robotics. The localization problem has many variants that reflect the diversity of operational environments. In open areas, GPS has been considered to be the localization choice. Despite its ubiquity and popularity, GPS has its Achilles’ heel that limits its application scope under certain circumstances: GPS typically does not work in indoor environments, and the power consumption of GPS receiver is a major hindrance that precludes GPS applications in resource-constrained sensor networks. To circumvent the limitations of GPS, acoustic/radio-strength and Ultra-wideband (UWB) ranging technologies have been proposed [10], [22], [11]. Measurement technologies include (a) RSS-based (received-signal-strength), (b) TOA-based (timeof-arrival), and (c) AoA-based (angle-of-arrival). RSS-based (received-signal-strength) technologies are most popular mainly due to the ubiquity of WiFi and cellular networks. The basic idea is to translate RSS into distance estimates. The performance of RSS-based measurements are environmentally dependent, due to shadowing and multipath effects [22], [25], [3]. TOA-based technologies such as UWB and GPS measures the propagation-induced time delay between a transmitter and a receiver. The hallmark of TOA-based technologies is the receiver’s ability to accurately deduce the arrival time of the line-of-sight (LOS) signals. TOA-based approaches excel RSS-based ones in both the ranging accuracy and reliability. In particular, UWB-based ranging technologies appear to be promising for indoor positioning as UWB signal can penetrate most building materials [10], [22], [11]. AoAbased technologies measure angles of the target node perceived by anchors by means of an antenna array using either RRS or TOA measurement [22], [11], [2]. Thus AoA-based approaches are viewed as a variant of RRS or TOA technologies.
Rapid advances in IC fabrication and RF technologies make possible the deployment of large scale power-efficient sensor networks. The network localization problem arises from such needs [26], [6]. The goal is to locate all nodes in the network in which only a small number of anchors know their precise positions initially. A sensor node (with initially unknown position) measures its distances to three anchors, and then determines its position. Once the position of a node is determined, then the node becomes a new anchor. The network localization has two major challenges: 1) cascading error accumulation and 2) insufficient number of initial anchors and initially skewed anchor distribution. The first challenge is addressed by utilizing optimization techniques to smooth out error distribution. The second challenge is addressed by using multihop ranging estimation techniques [28], [27]. Centralized optimization techniques for solving network localization include semidefinite programming by So and Ye [24] and second order cone programming (SOCP) relaxation by Tseng [26]. Local optimization techniques are realistic in practical settings. However, they could induce flip ambiguity that deviates significantly from the ground truth [6]. To deal with flip ambiguity, Eren et. al. [9] ingeniously applied graph rigidity theory to establish the unique localizability condition. They showed that a network can be uniquely localizable iff its grounded graph is globally rigid. The robust quadrilaterals algorithm by Moore et al. [21] achieves the network localizability by gluing locally obtained quadrilaterals, thus effectively reducing the likelihood of flip ambiguities. Kannan et al. [12], [13] formulate flip ambiguity problem to facilitate robust sensor network localization. Lederer et al. [15], [16] and Priyantha et al. [23] revolve around anchor-free localization problem in which none of the nodes know their positions. The goal is to construct a global network layout by using network connectivity. They develop an algorithm for constructing a globally rigid Delaunay complex for localization of a large sensor network with complex shape. Bruck, Gao and Jiang [6] establish the condition of network localization using the local angle information. They show that embedding a unit disk graph is NP-hard even when the angles between adjacent edges are given. Patwari et al. [22] used the Cramer-Rao bound (CRB) to establish performance bounds for localizing stationary nodes in sensor networks under different path-loss exponents. Dulman et al. [8] propose an iteration algorithm for the placement of three anchors for a given set of stationary nodes: in each iteration, the new position of one chosen anchor is computed using the noise-resilience metric. Bishop et al. [2], [4], [5], [3]
study the geometric impact of anchor placement with respect to one stationary node. Bulusu et al. proposed adaptive anchor placement methods [7]. Based on actual localization error at different places in the region, their algorithms can empirically determine good places to deploy additional anchors. Our work primarily focuses on the geometric impact quantification of an anchor placement over a traversal area. As a byproduct, it also forms the basis for optimal anchor selection for mitigating the impact of measurement noise. To our best knowledge, only a few papers [8], [7], [2], [22], [4], [5], [3] mentioned about the geometric effect of anchor placement but under a different context from this paper. The idea of quantifying the geometric impact of anchor placement on localization accuracy over a traversal area has not been considered before. The rest of this paper is structured as follows. Section 2 shows examples to illustrate the anchor placement impact on localization accuracy. Section 3 proposes a method for quantifying the anchor placement impact. Section 4 presents a two-phase localization method Section 5 conducts a simulation study to validate the theoretical results. Section 6 presents the field study using the UWB ranging technology. Section 7 concludes this paper. 2. E FFECT OF A NCHOR P LACEMENT Throughout this paper we use the term anchor to denote a node with known position. The positions of anchors are stationary and available to each mobile node (MN hereafter) in a traversal area [22]. The goal is to establish the position of a MN through ranging measurements to available anchors. The problem is formulated as follows: Let pi = (xi , yi ), (1 ≤ i ≤ m) denote the known position of the ith anchor, and p the actual position of the MN of interest. The p distance between p and pi is thus expressed as di = d(p, pi ) = (x − xi )2 + (y − yi )2 . In practical terms, the obtained distance measurement is affected by measurement noises. Thus the obtained distance is as dbi = di +i where di the true distance, and i is widely assumed to be a Gaussian noise with zero mean and variance σi2 [30], [29], [4], [14], [8], [22], [11]. Define a function f (x, y) =
m X
(2)
where η refers to the iteration step size, i denotes the ith iterations. One starts with an initial value (x(0) , y (0) ), then applies (2) iteratively to reach a local minimum (ˆ x, yˆ), i.e., f (ˆ x, yˆ) ≤ f (x(i) , y (i) ), 1 ≤ i ≤ n. There are three issues associated with GDM: 1) initial value; 2) iteration step size and 3) convergence rate. The convergence rate could be sensitive to the initial value as well as the iteration step size. While in general there is no theoretical guidance for selecting the initial value in practice, the value obtained by a linearized method is often chosen heuristically as the initial value. This nonlinear optimization problem in (1) can be linearized by the least-square method [14], [19] as follows: (0) x (AT A) (0) = AT M, where (3) y b2 b2 d1 − d2 + r22 − r12 (y2 − y1 ) 2 2 b2 b2 (y3 − y1 ) , M = d1 − d3 + r3 − r1 , ··· ··· 2 2 2 2 b b (ym − y1 ) d1 − dm + rm − r1 (4) p AT is the transpose of A, and ri = x2i + yi2 , 1 ≤ i ≤ m. The least-square method (LSM hereafter) uses noisy measurements to estimate the position of MN (x(0) , y (0) )T .
(x2 − x1 ) (x3 − x1 ) A = 2 ··· (xm − x1 )
100
80
60
40
20
0
(x − xi )2 + (y − yi )2 − dbi
2 2
0
,
where m is the number of accessible anchors, (xi , yi ) is the position of the ith anchor, and dbi noisy ranging between the ith anchor and the MN. The localization problem [22], [8], [14] is formulated as min f (x, y), < is the real number set.
20
Fig. 1.
i=1
(x,y)∈
