 Research
 Open Access
Hopdistance relationship analysis with quasiUDG model for node localization in wireless sensor networks
 Deyun Gao^{1},
 Ping Chen^{2},
 Chuan Heng Foh^{3}Email author and
 Yanchao Niu^{1}
https://doi.org/10.1186/16871499201199
© Gao et al; licensee Springer. 2011
 Received: 31 December 2010
 Accepted: 17 September 2011
 Published: 17 September 2011
Abstract
In wireless sensor networks (WSNs), location information plays an important role in many fundamental services which includes geographic routing, target tracking, locationbased coverage, topology control, and others. One promising approach in sensor network localization is the determination of location based on hop counts. A critical priori of this approach that directly influences the accuracy of location estimation is the hopdistance relationship. However, most of the related works on the hopdistance relationship assume the unitdisk graph (UDG) model that is unrealistic in a practical scenario. In this paper, we formulate the hopdistance relationship for quasiUDG model in WSNs where sensor nodes are randomly and independently deployed in a circular region based on a Poisson point process. Different from the UDG model, quasiUDG model has the nonuniformity property for connectivity. We derive an approximated recursive expression for the probability of the hop count with a given geographic distance. The border effect and dependence problem are also taken into consideration. Furthermore, we give the expressions describing the distribution of distance with known hop counts for inner nodes and those suffered from the border effect where we discover the insignificance of the border effect. The analytical results are validated by simulations showing the accuracy of the employed approximation. Besides, we demonstrate the localization application of the formulated relationship and show the accuracy improvement in the WSN localization.
Keywords
 Sensor Node
 Wireless Sensor Network
 Localization Error
 Localization Algorithm
 Communication Range
1 Introduction
In recent years, wireless sensor networks (WSNs) which generally consist of a large number of small, inexpensive and energy efficient sensor nodes have become one of the most important and basic technologies for information access [1]. WSNs have been widely used in military, environment monitoring, medicine care, and transportation control. Spatial information is crucial for sensor data to be interpreted meaningfully in many domains such as environmental monitoring, smart building failure detection, and military target tracking. The location information of sensors also helps facilitate WSN operation such as routing to a geographic field of interests, measuring quality of coverage, and achieving traffic load balance. In many monitoring applications, the sensor nodes must be aware its location to explain 'what happens and where'.
While specialized localization devices exist such as GPS, given the large number of sensor nodes involved in building a single WSN, it is cost ineffective to equip every sensor node with such a sophisticated device. Therefore, seeking for an alternative localization technology in WSNs has become one major research in WSNs [2]. Over the past few years, many localization algorithms have been proposed to provide sensor localization [3]. These localization protocols can be divided into two categories: rangebased and rangefree. The former is defined by methods that use absolute pointtopoint distance estimates (range) or angle estimates for computing locations. The latter makes no assumption about the availability or validity of such information. Recently, rangefree localization methods have attracted much attention because no extra sophisticated device for distance measurement is needed for each sensor node. Despite the challenge in obtaining virtual coordinates purely based on radio connectivity information [4, 5], attempts have been made in developing a practical solution to achieve localization. A few representative protocols of this rangefree scheme include DVHop [6], APIT [7], DRLS [8], MDSMAP [9], and LSSOM [10]. Most of the rangefree localization schemes, such as DVHop, need to compute the average distance per hop to estimate a node's location. In other words, the performance of these localization schemes relies on the accuracy of the employed hopdistance relationship. Since the determination of an accurate hopdistance relationship depends on various complex factors such as node deployment, node density, and wireless communication technology that cannot be easily quantified, the deduction process is tedious and unlikely to produce an exact close form relationship using, say the geometric methods [11].
Due to lack of any predetermined infrastructure and selforganized nature, in most cases, the sensor nodes are randomly and independently deployed in a bounded area. For simplicity, the vast majority of studies based on the idealized unitdisk graph (UDG) network model, where any two sensors can directly communicate with each other if and only if their geographic distance is smaller than a predetermined radio range. Examples of these research include georouting protocols [12, 13], localization algorithms [8, 14], and topology control techniques [15, 16]. Similarly, most of the works related to the hopdistance relationship have been investigated assuming the UDG model [11, 17–23]. The probability that two randomly selected stations with a known distance can communicate in K or less hops with omnidirectional antennas has been analyzed by Chandler [17]. Bettestetter and Eberspacher, derived the probability of the distance of two randomly chosen nodes deployed in a rectangular region within one or two hops [18]. However, when the hop counts are larger than two, only simulation results are available. The distribution parameters are computed by the iterative formula which extends from [19] with a linear formation. Ekici et al. [20] studied the probability of the khop distance in two dimensional network based on the approximated Gaussian distribution. Dulman et al. [11] derived the relationship between the number of hops separating two nodes and the physical distance between them in one and twodimensional topologies considering the UDG model. In the study, the approximated approach based on a Markov Chain in twodimensional case is rather complicated to compute. Zhao and Liang [21] collected the hopdistance joint distribution from Monte Carlo simulations in a circular region and proposed an attenuated Gaussian approximation for the conditional probability distribution function (pdf) of the Euclidean distance given a known hop count. Ta et al. [22] provided a recursive equation for the two randomly located sensor nodes that are khop neighbors given a known distance in homogeneous wireless sensor networks. Ma et al. [23] proposed a method to compute the conditional probability that a destination node has hopcount h with respect to a source node given that the distance between the source and the destination is d.
Despite the current efforts, no fixed communication range exists in actual network environment for the reasons such as multipath fading and antenna issues. Therefore, a certain level of deviation occurs between the intended operation and actual operation in wireless sensor networks when the UDG model is assumed in a protocol design. To deal with this problem, a practical model called the quasi Unitdisk Graph (quasiUDG) model is proposed recently [24]. The quasiUDG model can be characterized by two parameters, the radio range R and the quasiUDG factor α. For any two nodes in the quasiUDG model, if their distance is longer than R, no direct communication link exists between the two. Otherwise, if their distance is between αR and R, a communication link exists with a probability of p_{ l } , and p_{ l } = 1 when their distance is shorter than αR. Given this newly proposed practical property of connectivity, it warrants an investigation of the hopdistance relationship with the quasiUDG model for the rangefree localization schemes to capture practical connectivity characteristics.
In this paper, we focus on exploiting the connectivity property of the quasiUDG model and analyze the relationship between the hop counts separating two nodes and their geographic distance with a specific node density in a WSN. We seek approximation technique to provide a scalable solution for the twodimensional case. We further demonstrate the application of the developed hopdistance relationship to a rangefree localization scheme.
In our WSN setup, we consider that sensor nodes are deployed into a circular region S_{ b } with the radius R_{ b } , where the deployment position follows a Poisson point process with a certain density λ. We set ${p}_{l}=\frac{\alpha}{1\alpha}\left(\frac{R}{d}1\right)$ such that a longer distance between two nodes has a lower probability to form a direct communication link. With this setup, we formulate the probability that a pair of nodes with a known distance resulting a particular hop count. Additionally, we also develop the probability that a pair of nodes with a known distance gives a particular hop count. Finally, in our analysis, we present a quantitative evaluation for the border effect of geographic distance distribution with a given hop count.
The rest of this paper is organized as follows. In Section 2, we present our analytical model deriving an approximate recursive formula for the hopdistance relationship considering the quasiUDG model. Section 3 extends our analytical model by taking the border effect and dependence problem into consideration. Section 4 formulates the probability distribution of distance with known hop counts. In Section 5, we demonstrate the use of our developed hopdistance relationship by applying the relationship to a least squares (LS) based localization algorithm. Finally, we report results in Section 6 and draw important conclusions in Section 7.
2 The probability of the hop count given a known distance
In general, the hopdistance relationship is influenced by the density of sensor nodes and their deployment strategy, as well as the radio communication characteristics. Considering the more practical quasiUDG model, it is recognized that the formulation for the hopdistance relationship with the consideration of quasiUDG model is tedious and unlikely to produce an exact close form. We seek approximation using a recursive approach to derive an approximated hopdistance relationship. In this section, we focus on analyzing the probability that a particular pair of sensor nodes forms a certain hop count with a known distance.
Suppose that N sensor nodes are deployed randomly in circular region S_{ b } with a radius R_{ b } . The number of nodes in any region is a Poisson random variable with an average node density of $\lambda =\frac{N}{{S}_{b}}=\frac{N}{\left(\pi {R}_{b}^{2}\right)}$. Assume that the communication range of a node is R, the communication model between any pair of nodes follows the quasiUDG model with a factor of α where 0 < α < 1.
With the quasiUDG model, the communication area between two nodes with the distance d can be further divided into three cases shown as follows.

If d ≤ αR, then the two nodes can communicate directly.

If αR < d ≤ R, then the two nodes can communicate with a probability p_{ l }, which is set to (R/d  1)α/(1  α). It means that a longer distance between two nodes has a lower probability to form a direct communication link.

If d > R, then the two nodes cannot communicate directly.
2.1 The case of h= 1
2.2 The case of h≥ 2
We first note that two nodes, named O_{1} and O_{2}, have no direct link but may communicate through h  1 relay nodes. This gives rise to two possibilities, where

O_{2} is not the mhop neighbor of O_{1} if m < h.

Within the communication range of O_{2}, there is a least one (h  1)hop neighbor of O_{1} that has a direct link with O_{2}.
We shall now consider the second possibility in the following. Considering two circles which one centered at O_{1} having a radius of r and the other centered at O_{2} having a radius of R. We denote the distance between the two centers as d and refer the common region of the two circles as S. The quantity P_{ r } (S) is defined as the probability that in the area S, there is no (h  1)hop neighbor of O_{1} that can communicate with O_{2} directly. A differential increment of dr on r can obtain a differential incremental region of dS. Assume that the probability Φ_{ h }(d) of any pair of nodes is independent and statistically identical, we have P_{ r } (S + dS) = P_{ r } (S)P_{ r } (dS). In the following subsections, we calculate P_{ r } (dS) based on three conditions, which are d > R, $\frac{1+\alpha}{2}R<d<R$, and $\alpha R<d\frac{1+\alpha}{2}R$.
2.2.1 O_{1}falls outside the communication range of O_{2}where d> R
We term the circular region centered in O_{2} with the radius αR as $\mathcal{C}\left({O}_{2}\right)$, and the annulus region centered in O_{2} with the larger radius R and the smaller one αR as $\mathcal{A}\left({O}_{2}\right)$. There are two cases needed to be taken into consideration, which are

When dS falls into $\mathcal{A}\left({O}_{2}\right)$ as shown in Figure 2(a), r satisfies d  R ≤ r ≤ d  αR or d + αR ≤ r ≤ d  R. With the definition of the quasiUDG model, every differential region rdrdθ of dS has a corresponding probability p_{ l }to communicate with O_{2}. Therefore, P_{ r }(dS) is given by (3) where${P}_{r}\left(dS\right)=12{\Phi}_{h1}\left(r\right)\lambda rdr\underset{0}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta .$(3)

When dS covers both $\mathcal{C}\left({O}_{2}\right)$ and $\mathcal{A}\left({O}_{2}\right)$, r will be bounded by d  αR ≤ r < d + αR. The part rdrdθ that falls within $\mathcal{C}\left({O}_{2}\right)$ is surely a onehop neighbor of O_{2}. When that part falls within $\mathcal{A}\left({O}_{2}\right)$, it has a corresponding probability p_{ l }that it has a direct link with O_{2}. Then P_{ r }(dS) can be determined by${P}_{r}\left(dS\right)=12{\Phi}_{h1}\left(r\right)\lambda rdr\left[{\phi}_{1}+\underset{\phi 1}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta \right]$(6)
2.2.2 O_{1}falls within the communication range of O_{1}and d satisfies $\frac{1+\alpha}{2}R<d<R$
We use the foregoing strategy for this derivation. We notice that there are three cases needed to be treated individually which are given as follows.

If 0 < r < R  d, dS will be the annulus region and the entire section of dS will fall within $\mathcal{A}\left({O}_{2}\right)$, which gives${P}_{r}\left(dS\right)=12{\Phi}_{h1}\left(r\right)\lambda rdr\underset{0}{\overset{\pi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta $(8)

If Rd ≤ r < dαR or d+αR ≤ r < R+d, dS will not be the annulus region but the entire section of dS will still fall within $\mathcal{A}\left({O}_{2}\right)$. Then we can obtain P_{ r }(dS) by (3).

If dαR ≤ r < d+αR, dS will cover both $\mathcal{C}\left({O}_{2}\right)$ and $\mathcal{A}\left({O}_{2}\right)$. In this case, we can determine P_{ r }(dS) by (6).
2.2.3 O_{1}falls within the communication range of O_{2}and d satisfies $\alpha R<d\frac{1+\alpha}{2}R$
There are four cases needed to be considered when O_{1} falls within the communication range of O_{2} and d satisfying the condition $\alpha R<d\frac{1+\alpha}{2}R$, which are

If 0 < r < dαR, dS will be the annulus region and the entire section of dS will fall within $\mathcal{C}\left({O}_{2}\right)$. Then we can determine P_{ r }(dS) by (8).

If dαR ≤ r < Rd, dS will still be the annulus region but it covers both $\mathcal{C}\left({O}_{2}\right)$ and $\mathcal{A}\left({O}_{2}\right)$. Therefore, we have${P}_{r}\left(dS\right)=12{\Phi}_{h1}\left(r\right)\lambda rdr\left[{\phi}_{1}+\underset{{\phi}_{1}}{\overset{\pi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta \right]$(9)

If Rd ≤ r < d+αR, dS will not be will the annulus region and it covers both $\mathcal{C}\left({O}_{2}\right)$ and $\mathcal{A}\left({O}_{2}\right)$. The probability P_{ r }(dS) can be obtained by (6).

If d+αR ≤ r < R+d, dS will fall within the region $\mathcal{A}\left({O}_{2}\right)$, and hence we can compute P_{ r }(dS) by (3).
2.3 Determination of Φ _{ h }(d) for h≥ 2
where knowing d, Ω(d) can be determined by one of the following expressions, which are

For d > hR or d < αR :$\Omega \left(d\right)=0;$(12)

For R < d ≤ hR :$\begin{array}{lll}\hfill \Omega \left(d\right)& ={\int}_{dR}^{d\alpha R}{\Phi}_{h1}\left(r\right)r\underset{0}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta dr\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ +{\int}_{d\alpha R}^{d+\alpha R}{\Phi}_{h1}\left(r\right)r\left({\phi}_{1}+\underset{{\phi}_{1}}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta \right)dr\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ +{\int}_{d+\alpha R}^{d+R}{\Phi}_{h1}\left(r\right)r\underset{0}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta dr\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$(13)

For $\frac{1+\alpha}{2}R<d\le R$:$\begin{array}{lll}\hfill \Omega \left(d\right)& ={\int}_{0}^{Rd}{\Phi}_{h1}\left(r\right)r\underset{0}{\overset{\pi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)\phantom{\rule{2.77695pt}{0ex}}d\theta dr\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ +{\int}_{Rd}^{d\alpha R}{\Phi}_{h1}\left(r\right)r\underset{0}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta dr\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ +{\int}_{d\alpha R}^{d+\alpha R}{\Phi}_{h1}\left(r\right)r\left({\phi}_{1}+\underset{{\phi}_{1}}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta \right)dr\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ +{\int}_{d+\alpha R}^{d+R}{\Phi}_{h1}\left(r\right)r\underset{0}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta dr\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$(14)

For $\alpha R<d\le \frac{1+\alpha}{2}R$:$\begin{array}{lll}\hfill \Omega \left(d\right)& =\phantom{\rule{2.77695pt}{0ex}}{\int}_{0}^{d\alpha R}{\Phi}_{h1}\left(r\right)r\underset{0}{\overset{\pi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta dr\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ +\phantom{\rule{2.77695pt}{0ex}}{\int}_{d\alpha R}^{Rd}{\Phi}_{h1}\left(r\right)r\left({\phi}_{1}+\underset{{\phi}_{1}}{\overset{\pi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta \right)dr\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ +\phantom{\rule{2.77695pt}{0ex}}{\int}_{d\alpha R}^{d+\alpha R}{\Phi}_{h1}\left(r\right)r\left({\phi}_{1}+\underset{{\phi}_{1}}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta \right)dr\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ +\phantom{\rule{2.77695pt}{0ex}}{\int}_{d+\alpha R}^{d+R}{\Phi}_{h1}\left(r\right)r\underset{0}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta dr.\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$(15)
3 The border effect and dependence problem
In the above analysis, we do not consider borders of a WSN. However, in a realistic scenario, the deployment area of WSNs is finite and hence borders exist. It is known that the probability Φ_{ h }(d) derived assuming that both involved nodes are not near the border of a WSN may give a slightly different result when one or both of them fall near the border. This is known as the border effect. One common handling of the border effect is to consider the toroidal distance metric in the simulation experiment where a node closed to the border can communicate directly with some nodes at the opposite border [25]. While this special setup eliminates the border effect, it creates discrepancy between the study and practical setups which may lead to a certain level of errors.
Clearly, nodes which are closer to the border cover smaller regions than those at least d away from the border, and therefore intuitively the quantity for Ω(d) should be smaller with the consideration of the border effect. Apparently, the border effect gives a different level of impacts in the measure of Φ_{ h }(d) with a different distance between an involved node and the border. However, it is tedious to derive all cases considering the border effect. For simplicity, we take two key cases of the border effect into consideration. Assuming the center of deployment area is O, we consider two annulus near the border in the following.

The first annulus, called ${\mathcal{A}}_{1}\left(o\right)$, is between the circles with radius of R_{ b }R and R_{ b }αR.

The second annulus, called ${\mathcal{A}}_{2}\left(o\right)$, is between the circles with radius of R_{ b }R and R_{ b }αR.
We set an average metric ζ(h) which varies from 0 to 1 for each hop to determine the decrement of Ω(d). For the circle area with the radius R_{ b }  R, which can be called $\mathcal{C}\left(o\right)$, we can set ζ(h) = 1 accordingly.
4 Distance distribution with known hop counts
In this section, assume that sensor nodes are randomly deployed in a circular region, we derive equations to determine the probability density function of distance d with a known hop count ${f}_{\mathcal{H}}\left(d\right)$.
where r_{0} = 0 when h = 1, and r_{0} = αR when h > 1.
5 Localization Applications
With the development of the hopdistance relationship for the quasiUDG model, in this section, we show the application of this new relationship to a particular localization algorithm using LS based localization algorithms [26], and we call this newly designed localization algorithm enhance weighted least squares (EWLS).
In a particular localization scenario in WSNs, we assume that there is a number of nodes whose locations are known, and they shall be called anchor nodes. Other nodes that have no knowledge of their locations are called unknown nodes. Consider that an unknown node j can obtain the location x _{ i }, hop h_{ ji } and average hopdistance c_{ i } of an anchor node i. The distance between nodes j and i can be calculated as d_{ ji } = c_{ i }h_{ ji } . In our test scenario, we place an anchor node o in the center and add several other anchor nodes in the map.
6 Result discussions
In this section, we compare the analytical and statistical results through simulation experiments to illustrate the performance of our proposed hopdistance model. To illustrate the benefit of applying our model to LSbased localization algorithms, we compared our enhanced algorithm of EWLS to two classical LSbased localization algorithms namely LS [26] and PDM [27].
6.1 Impacts of boarder effects and dependence
Comparisons between analytical and simulation results of Φ_{ h }(d)
Hops  2  3  4  5  6  7  8  9  

$\mathcal{C}\left(o\right)$  CAD  0.34  0.36  0.85  1.49  2.12  2.76  3.36  3.9 
ω(h)  1.0  0.77  0.70  0.65  0.63  0.60  0.58  0.54  
${\mathcal{A}}_{1}\left(o\right)$  CAD  0.42  0.38  0.86  1.52  2.13  2.69  3.31  3.98 
CAD*  0.66  0.59  0.88  1.59  2.21  2.79  3.45  4.03  
ω(h)  0.95  0.77  0.70  0.65  0.62  0.61  0.59  0.57  
${\mathcal{A}}_{2}\left(o\right)$  CAD  0.35  0.49  1.17  1.76  2.33  2.89  3.40  4.05 
CAD*  0.74  0.75  1.19  1.85  2.45  3.06  3.61  4.16  
ω(h)  0.92  0.77  0.69  0.65  0.62  0.61  0.59  0.58 
In Table 1 we use cumulative absolute difference (CAD) to measure the sum of absolute differences between the analytical results and statistical data. We set $CAD={\sum}_{d}{\Phi}_{h}\left(d\right)Si{m}_{h}$, where Φ_{ h }(d) and Sim_{ h } are the probabilities of two nodes giving a hop count of h with a known distance of d obtained from the analysis and simulation, respectively. Moreover, we denote CAD* as the CAD measurement between analytical results without the border effect consideration and statistical data. For ${\mathcal{A}}_{1}\left(o\right)$ and ${\mathcal{A}}_{2}\left(o\right)$, we can see that the CAD* of each hop is larger than that of CAD because of the impact of the border effect.
6.2 The validation of distribution of distance by a known hop count
6.3 Localization accuracy comparisons
In the following, we further compare the localization accuracy among EWLS, LS and PDM under various scenarios. In these simulation experiments, we set N = 400, and sensor nodes are deployed uniformly in the circle area with the radius R_{ b } = 200. The connectivity of nodes follows the quasiUDG model. The localization error is calculated as $\xi ={\sum}_{j}\parallel {x}_{j}{\widehat{x}}_{j}\parallel /\left(Nn\right)$.
7 Conclusions
The hopdistance relationship information can effectively improve the performance of the protocols for wireless sensor networks in many aspects. However, most studies focus on the UDG model which significantly deviates from the real world. In the paper, we presented an analytical modeling to formulate the hopdistance relationship considering the quasiUDG model. Senor nodes are randomly distributed in a circular region according to a Poisson point process. The probability of a particular hop count given a known distance Ω_{ h }(d) was studied, and the border effect and dependence problem was considered in our analysis. Precisely, we derived the probability density function of a random variable describing the distance between two arbitrary nodes with a given hop count. Simulation results confirmed that our analytical results gave excellent accuracy. From the results, we further illustrated impact of the border effect.
Furthermore, we demonstrated the application of our developed hopdistance relationship considering the quasiUDG model in WSN localizations. We designed a LSbased localization algorithm using our developed relationship and compared its performance with other popular LSbased localization algorithms. We again confirmed that the explicit use of our developed relationship in the computation of localization algorithms improved the localization accuracy.
A Appendix
where 0 < d < 2R_{ b } .
Declarations
Acknowledgements
The authors gratefully acknowledge the support of the Program of Introducing Talents of Discipline to Universities ("111 Project") under grant No. B08002, and the support of the National Natural Science Foundation of China (NSFC) under Grants No. 60802016, 60833002 and 60972010, the support by "the Fundamental Research Funds for the Central Universities" under grant No. 2009JBM007.
Authors’ Affiliations
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