A simple iterative positioning algorithm for client node localization in WLANs
- Luis M Trevisan^{1},
- Marcelo E Pellenz^{1}Email author,
- Manoel C Penna^{1},
- Richard D Souza^{2} and
- Mauro SP Fonseca^{1}
https://doi.org/10.1186/1687-1499-2013-276
© Trevisan et al.; licensee Springer. 2013
Received: 5 June 2013
Accepted: 7 November 2013
Published: 5 December 2013
Abstract
The ability to determine in real-time the geographic location of client nodes is an important tool in wireless networks, allowing instantaneous mobile tracking, implementation of location-aware services and also efficient channel and power allocation planning. Among existing classical cooperative localization techniques for wireless networks, the maximum likelihood estimator (MLE) is theoretically the best. However, the gradient-based algorithms that are commonly used for maximum likelihood estimation are quite sensitive to the initial values and cannot achieve the theoretical optimal performance. In this paper, we propose a new iterative positioning algorithm based on received signal strength information that employs a location ordering strategy and a numerical nonlinear optimization method. The algorithm performance is evaluated through simulation for different network scenarios. A real wireless network scenario is also implemented in order to demonstrate the algorithm effectiveness. The proposed algorithm, while presenting a simplified implementation, can achieve better positioning estimates than the classical MLE approach based on the conjugated gradient.
Keywords
1 Introduction
Wireless networks have found widespread application in many scenarios including entertainment, medicine, security, automation, emergency services, among other uses. These networks may operate under different architectures, including structured mode using access points (APs) and unstructured modes using ad hoc and mesh topologies. The knowledge of node positions in a wireless network has many applications. In wireless mesh networks (WMNs) [1], the positioning information allows the creation of efficient scheduling algorithms to reduce collisions and interferences. The positioning information can be used to estimate the interference a node causes in other nodes of the network and thus improve the multiple access strategies, scheduling mechanisms, channel allocation algorithms, and routing protocols.
In wireless local area networks (WLANs), the positioning information can be used for location-based services [2], mobile device tracking [3–5], and physical layer authentication [6]. Even though there exist more precise location techniques based on the angle-of-arrival (AoA), time-of-arrival (ToA), and time-difference-of-arrival (TDoA) [7, 8], algorithms based on the received signal strength (RSS) measurements are still very attractive from a practical point of view because this metric is available in every radio interface and does not require any additional hardware features nor explicit cooperation from the localized node. Among existing classical localization techniques for wireless networks, the maximum likelihood estimator (MLE) is theoretically the best. However, the gradient-based algorithms that are commonly used for MLE are quite sensitive to the initial position estimation of the unknown nodes and cannot achieve the theoretical optimal performance. The development of a positioning algorithm less susceptible to the initial coordinate estimations motivates our investigation.
In this paper, we propose an iterative positioning algorithm for localization of client nodes in WLANs. The pairwise range estimation is based on RSS. We consider a centralized processing unit (central node), as usually implemented in many networks for operation management. It is assumed that each network node collects received power information and MAC addresses from neighbor nodes within its transmission range. Such information is sent to the central node that executes the localization algorithm. The main contributions of the proposed algorithm are the use of a selection and ordering strategy for the reference (or anchor) nodes and also the application of the numerical nonlinear optimization method of Nelder-Mead [9], that does not require the derivative of the cost function. The algorithm performance is evaluated through computer simulations for different network scenarios. A real wireless network scenario is implemented in order to demonstrate the algorithm effectiveness. The proposed algorithm, while presenting a simplified implementation, can achieve better positioning estimates than the classical MLE approach based on the conjugated gradient [10].
The rest of this paper is organized as follows: in Section 2, we present a brief description of the localization technique principles and describe the path loss channel model with shadowing which is employed in the algorithms using RSS. In Section 3, we describe the classical localization techniques for wireless networks. The proposed algorithm is presented in Section 4. The performance results are presented and discussed in Section 5, while Section 6 concludes the paper.
2 Localization techniques overview
Consider a wireless network scenario where a specific node with unknown position, called unknown node, should be localized. The other nodes in its vicinity are assumed to have known positions and are called reference nodes. It is possible to estimate the unknown node coordinates by applying a localization technique. This procedure starts with the reference nodes transmitting their own coordinates to the unknown node. Then, the unknown node must estimate its relative distance to each of the references based on any technique of range estimation. Finally, the unknown node applies a combining technique over the distances estimates and received coordinates in order to estimate its own position. Alternately, if the unknown node does not cooperate in the localization process, the procedure is executed by the set of reference nodes based on opportunistic measurements collected when the unknown node transmits. We do not include in our analysis the class of Bayesian localization algorithms [11–14] as those assume the cooperation among all nodes. In our scenario, we consider that the unknown node may be non-cooperative.
Basically, the location discovery approaches consist of two phases [15]. In the first phase, the relative distance between two nodes can be estimated using methods as RSS, ToA, TDoA, and AoA [7]. Distance estimation techniques based on RSS measures the signal power at the receiver. Assuming a known transmit power, the propagation loss is computed using a theoretical or empirical model and this loss is translated into a distance estimate. This technique is mainly used for radio frequency (RF) signals and is subject to different types of errors: additive noise, multipath fading, and shadowing. A time-based method (ToA/TDoA) estimates the relative distance based on the time or time difference the RF signal takes to travel from the transmitter to the receiver node. This technique assumes that the signal propagation speed is known and can be applied to different signal types including RF, acoustic, infrared, and ultrasound. Finally, the AoA technique is applied to estimate the angle at which signals are received and is generally used in combination with other techniques. An interesting overview of these algorithms for wireless position estimation is presented in [16].
In the second phase, the estimated relative distances should be combined to obtain the node position. The first combining approach is to use a method called hyperbolic trilateration, where the node position is estimated calculating the intersection point of three circumferences. Each of them with a center at one reference, and with radius equal to the estimated distance between its center reference and the unknown node. A second strategy denoted triangulation can be used if the signal angle-of-arrival is available instead of the distance. The node position is computed using simple trigonometry laws. A third combining method called multilateration calculates the node position by minimizing the differences between the noisy measured distances and estimated distances, using MLE. The multilateration technique is a generalization of the trilateration by using more than three references [15].
In real-world scenarios, the RSS measurements are highly dependent of multipath fading and shadowing, so that RSS is considered a poor range estimator compared with ToA and AoA [7]. However, the RSS measurement is an inexpensive technique because the signal power, generally available at any RF transceiver, can be measured during normal data communication without additional transmissions or bandwidth requirements. Therefore, the technique does not require any additional hardware setup nor extra energy consumption. Technically, the random shadowing behavior of the wireless channel becomes the most relevant source of error for a RSS-based location system. Measurement errors affects the TOA distance measurement in an additive way while RSS is affected in a multiplicative way [7]. Finally, even though it is less precise, due to its low cost and ease of practical deployment, the RSS technique is used for range estimation in the proposed localization framework.
3 Problem formulation
where ${P}_{t}^{\mathit{\text{dBm}}}$ is the transmit power, λ = c / f is the signal wavelength, f is the carrier frequency and c = 3 × 10^{8} m/s.
Consider a wireless ad hoc network with m reference nodes and n unknown nodes. The positioning algorithm aims to locate the n unknown nodes as close as possible to their real position based on peer-to-peer range estimation and on the known coordinates of the references. The core algorithm mechanism is the atomic multilateration. Because of the shadowing effect, the trilateration in a real scenario will not have a unique intersection point.
in which becomes clear that the estimated distance $\widehat{{d}_{\mathit{\text{ij}}}}$ is affected by the shadowing in a multiplicative way, or equivalently, that the estimation error $\left(\widehat{{d}_{\mathit{\text{ij}}}}-{d}_{\mathit{\text{ij}}}\right)$ is proportional to the range.
The candidate position that minimizes $f\left({\stackrel{~}{\mathbf{\text{u}}}}_{i}\right)$ is the estimated position of the unknown node.
4 Proposed algorithm
In this section, we present the details of the proposed localization algorithm, but first we discuss the influence of ordering in the localization accuracy.
4.1 The Influence of ordering
Taget localization sequence and the localization sequences given by the F _{ proximity } and R _{ proximity } factors
Target location sequence | F _{ proximity } (i) | Estimated location sequence | R _{ proximity } (i) | Estimated location sequence |
---|---|---|---|---|
15 | 47.6830 | 8 | 81.2738 | 15 |
14 | 46.6297 | 15 | 63.4042 | 14 |
13 | 46.1839 | 7 | 51.9312 | 13 |
12 | 45.5853 | 14 | 43.9526 | 12 |
8 | 44.9661 | 6 | 39.1365 | 8 |
7 | 44.6653 | 13 | 38.2657 | 7 |
6 | 44.0120 | 5 | 38.0920 | 11 |
5 | 43.8848 | 12 | 36.8669 | 6 |
11 | 43.2998 | 4 | 35.1000 | 5 |
4 | 43.2586 | 11 | 33.6129 | 10 |
10 | 42.8085 | 3 | 33.1326 | 4 |
9 | 42.8002 | 10 | 31.1058 | 3 |
3 | 42.5210 | 2 | 30.0848 | 9 |
2 | 42.5205 | 9 | 29.1206 | 2 |
1 | 42.4264 | 1 | 27.2395 | 1 |
Based on simulation experiments, we identified that the distance estimates between two unknown nodes not yet localized can be used to improve the sequence ordering used by the localization algorithm. This is done by adding to F_{proximity} the constant parameter $\gamma =(n-|\mathcal{L}\left|\right)\xb7max\left(\hat{D}\right)$, where n is the number of unknown nodes, $\left|\mathcal{L}\right|$ is the cardinality of , and $\hat{D}=\phantom{\rule{2.77626pt}{0ex}}\left[\phantom{\rule{0.3em}{0ex}}\widehat{{d}_{\mathit{\text{ij}}}}\right]$ is the matrix of estimated distances. The scaling factor α has the objective to locate unknown nodes that are closer to a great number of reference nodes. Therefore, if there are two unknown nodes with the same value of R_{proximity}(i), the one that has a larger number of closer reference nodes will be localized first. This procedure is a strategy to give a localization reliability metric for each of m reference nodes. The performance was evaluated for values 1 ≤ α ≤ 3 and the best result obtained through simulations was with α = 1.15. The R_{proximity} values for a node at the different positions indicated in Figure 2 are also presented in Table 1. Note that the estimated location sequence obtained based on R_{proximity} is very close to the target location sequence.
4.2 Algorithm
The proposed algorithm locates nodes individually, following the sequence ordering obtained using the R_{proximity} factor, where the node with the highest R_{proximity} factor is located first. The pseudocode is presented in Algorithm 1. The set of reference nodes, , have known coordinates, r_{ j }. These nodes collect the power measurements from unknown neighbor nodes to form matrix $\widehat{\mathbf{P}}$. The reference power ${P}_{0}^{\mathit{\text{dBm}}}$ and the pathloss exponent η are predefined parameters. Based on the power measurements, the R_{proximity} factor is computed for all unknown nodes. Then, the iterative localization procedure is started, where the unknown nodes are localized in the order based on the R_{proximity} factor. The coordinates are estimated by minimizing the cost function $f\left({\stackrel{~}{\mathbf{\text{u}}}}_{k}\right)$. The localized node becomes part of the reference set for the next iteration of the algorithm.
Moreover, in our implementation, we use of the Nelder-Mead nonlinear optimization numerical method [9] for minimizing the weighted cost function, therefore without requiring the derivative of the cost-function, as is the case of MLE methods based on the conjugate gradient [10].
4.3 Initial candidate position problem
The intersection of these three straight lines defines the initial candidate position of unknown node k for the localization algorithm, as marked by the triangle in Figure 7. In a specific scenario, where the three references are aligned, the lines are parallel. In such case, the initial candidate position is defined as the mean computed among the coordinates defined by the intersection of the line defined by the references with the parallel lines [19].
5 Practical results
Power measurements in each scenario
Scenario | d _{i 1} | d _{i 2} | d _{i 3} | P _{1} | ${\mathit{\sigma}}_{{\mathit{P}}_{\mathit{1}}}^{\mathit{2}}$ | P _{2} | ${\mathit{\sigma}}_{{\mathit{P}}_{\mathit{2}}}^{\mathit{2}}$ | P _{3} | ${\mathit{\sigma}}_{{\mathit{P}}_{\mathit{3}}}^{\mathit{2}}$ |
---|---|---|---|---|---|---|---|---|---|
(m) | (m) | (m) | (dBm) | (dBm^{2}) | (dBm) | (dBm^{2}) | (dBm) | (dBm^{2}) | |
A | 5 | 5 | 5 | –37.84 | 0.80 | –38.04 | 0.66 | –41.83 | 1.32 |
B | 10 | 10 | 10 | –45.38 | 0.57 | –45.57 | 0.63 | –50.31 | 1.32 |
C | 20 | 20 | 20 | –50.93 | 1.48 | –50.65 | 0.49 | –54.70 | 1.16 |
D | 50 | 50 | 50 | –54.86 | 1.04 | –55.47 | 0.64 | –57.76 | 1.54 |
E | 20 | 10 | 50 | –51.01 | 1.69 | –44.42 | 0.78 | –56.03 | 0.62 |
Wireless interface parameters
${\mathit{P}}_{\mathit{t}}^{\mathit{\text{dBm}}}$ | 15 | Transmission power |
---|---|---|
${G}_{t}^{\mathit{\text{dBi}}}$ | 4 | Transmitter antenna gain |
${G}_{r}^{\mathit{\text{dBi}}}$ | 4 | Receiver antenna gain |
${L}_{c}^{\mathit{\text{dB}}}$ | 2 | Cable loss |
f_{0} (GHz) | 2.417 | Carrier frequency (channel 4) |
Actual and estimated distances
Scenario | Distance | R _{1} | R _{2} | R _{3} |
---|---|---|---|---|
A | d _{ ij } | 5 | 5 | 5 |
${\hat{d}}_{\mathit{\text{ij}}}$ | 4.5521 | 4.6815 | 5.8643 | |
B | d _{ ij } | 10 | 10 | 10 |
${\hat{d}}_{\mathit{\text{ij}}}$ | 9.5012 | 10.2626 | 13.6512 | |
C | d _{ ij } | 20 | 20 | 20 |
${\hat{d}}_{\mathit{\text{ij}}}$ | 15.8156 | 16.1516 | 23.2995 | |
D | d _{ ij } | 50 | 50 | 50 |
${\hat{d}}_{\mathit{\text{ij}}}$ | 24.3975 | 23.7469 | 31.6195 | |
E | d _{ ij } | 20 | 10 | 50 |
${\hat{d}}_{\mathit{\text{ij}}}$ | 17.2376 | 8.6479 | 27.7609 |
6 Conclusions
In this paper, we present a new localization strategy based on RSS measurements. The algorithm is centralized and iterative: centralized because a node must accumulate the RSS measurements and run the location algorithm for all the other nodes; iterative because each node is localized one at a time and once located it becomes a reference to the location of other unknown nodes in the next iterations. The Nelder-Mead nonlinear optimization numerical method is employed to find the candidate coordinates for the unknown node, by minimizing the cost function. It is shown that the proposed algorithm can achieve better practical position estimates than the classical MLE strategy that employs gradient-search-based methods.
Endnotes
^{a} There are two constants that are used in our proposed algorithm and that were numerically obtained (α and ρ). These values were obtained after several tests considering up to n = 30 unknown nodes randomly located within an area of 15×15 m^{2} (the node positions are random and follow a uniform distribution). For each number of unknown nodes, an ensemble of topologies were generated and several values of α and ρ were tested. The impact of ρ in the performance of the proposed algorithm is larger than the impact of α.
^{b} Among existing classical cooperative localization techniques for wireless networks, the MLE is theoretically the best in the sense that it asymptotically achieves the Cramer-Rao bound (CRB). However, the gradient based iterative algorithms that are commonly used to achieve MLE positions are quite sensitive to the initial values and cannot achieve the theoretical optimal performance [14]. In this case the optimum solution is the minimum mean square error (MMSE) algorithm, which requires the prior distribution on the location of the nodes.
Declarations
Acknowledgements
This work was partially supported by CAPES and CNPq (Brazil).
Authors’ Affiliations
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