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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

This work presents a localization scheme for use in wireless sensor networks (WSNs) that is based on a proposed connectivity-based RF localization strategy called the distributed Fermat-point location estimation algorithm (DFPLE). DFPLE applies triangle area of location estimation formed by intersections of three neighboring beacon nodes. The Fermat point is determined as the shortest path from three vertices of the triangle. The area of estimated location then refined using Fermat point to achieve minimum error in estimating sensor nodes location. DFPLE solves problems of large errors and poor performance encountered by localization schemes that are based on a bounding box algorithm. Performance analysis of a 200-node development environment reveals that, when the number of sensor nodes is below 150, the mean error decreases rapidly as the node density increases, and when the number of sensor nodes exceeds 170, the mean error remains below 1% as the node density increases. Second, when the number of beacon nodes is less than 60, normal nodes lack sufficient beacon nodes to enable their locations to be estimated. However, the mean error changes slightly as the number of beacon nodes increases above 60. Simulation results revealed that the proposed algorithm for estimating sensor positions is more accurate than existing algorithms, and improves upon conventional bounding box strategies.

A wireless sensor network is a large-scale

Numerous sensor network applications require location awareness, whereas in sensor networks, nodes are deployed into an unplanned infrastructure in which no

Several localization strategies have been proposed, ranging from solutions dependent on hardware support by GPS and the presence of an established infrastructure, to range-free solutions that utilize signal strength, hop count to known landmarks or a priori knowledge about density of nodes in a network. Most of these strategies share a common feature: they use beacon nodes that know their own locations. Other sensor nodes identify their locations based on information provided by these beacon nodes. Furthermore, in localization techniques, centralized localization approaches depend on sensor nodes transmitting data to a central location, where computation is performed to determine the location of each node. Consequently they generate high communication costs and inherent delay. However, distributed localization schemes do not require centralized computation and each node determines its location using limited communication with nearby nodes. For instance, beacon-based distributed algorithms such as diffusion, bounding box, gradient multi-literation and APIT, typically start with a group of beacons. Nodes in the network obtain a distance measurement to a few beacons, and then use these measurements to estimate their locations.

One of the most well-known localization methods for WSNs is Convex Position Estimation (CPE) proposed by Dohetry

To overcome the disadvantages of CPE, in this paper DFPLE is proposed to minimize the mean error and computational price in estimating WSNs location. Like CPE, DFPLE is based on a bounding box algorithm to estimate the candidate of location. DFPLE expands the bound for location estimation using three cases of beacon node positioning. Therefore unlike the CPE has four bound points, DFPLE has dynamic number of bound points. Then, instead of using linear programming to find smaller area of estimation, DFPLE exploits the capability of Fermat Point calculation to refine the area of feasible set of solution and finally achieves minimal computational price.

The rest of this paper is organized as follows. Section 2 discusses existing location discovery algorithms, including range-based and range-free schemes. Section 3 then describes the proposed DFPLE algorithm. Next, Section 4 summarizes the performance analysis and simulation. Conclusions are finally drawn in Section 5, along with recommendations for future research.

Several effective location discovery protocols for wireless sensor networks have been proposed in recent years [

Range-based schemes use absolute point-to-point distance or angle information to calculate locations between neighboring sensors. Such schemes estimate the absolute distance between a sender and receiver according to received signal strength or by time-of-flight of a communication signal. Common approaches for distance/angle estimation include Time of Arrival (ToA), Time Difference of Arrival (TDoA), Angle of Arrival (AoA) and Received Signal Strength (RSS). The accuracy of such estimation methods, however, depends on the transmission medium and surrounding environment, and they usually require complex hardware.

The Receiver Signal Strength Indicator (RSSI), initially used for power control in wireless networks, can also serve as a tool for distance estimation. The idea is that given a predefined transmission power, a signal propagation model that maps transmission power and a distance to a received power, one can estimate the distance from a receiver to a sender by identifying the strength of received power. The benefit of using RSSI for localization in sensor networks is obvious: trilateration can be achieved for all nodes using only three beacons, and nodes only perform passive listening. Unfortunately, existing signal propagation models are lacking, thereby significantly limiting localization accuracy; a receiver usually needs to perform sophisticated algorithms to synthesize the RSSI values from multiple senders to achieve adequate accuracy.

Time of Arrival (ToA) and Time Difference of Arrival (TDoA) measure signal arrival time or the difference in arrival times to calculate distance based on transmission time and speed. They can be applied to many different kinds of signals such as RF, acoustic and ultrasound signals. The ToA is less accurate than TDoA as processing delays and non-LOS (Line-of-Sight) propagation can generate errors. The ToA also requires synchronization to accurately measure time-of-flight.

The TDoA utilizes signal propagation speed, which is more robust than the signal attenuation characteristic used by RSSI. Ideally, when a sender and receiver are synchronized, the Time-of-Flight (ToF) measurement is already sufficient to identify the distance between the receiver and sender. However, synchronization, whose precision can match the radio signal speed, is hard to achieve. The TDoA mechanism is frequently utilized in cellular networks for localizing a handset. Since it only requires that the difference between arrival times is observed at several receivers, TDoA eliminates the need for synchronization between a handset and receivers. However, the receivers must be synchronized. Recent literature has defined TDoA as the difference between arrival times of two signals. This definition for TDoA actually applies to ToF, as the propagation time of one signal is measured and another signal is utilized for time synchronization. Employing different signal types for ranging has the limitation that one of them may not work properly in an environment that favors another. Therefore, ranging mechanisms relying on only one signal could be desirable for sensor location surveys.

The Angle of Arrival (AoA) scheme requires measurement of the angle at which a signal arrives at a base station or a sensor. It is used initially in cellular networks that require each receiver is equipped with additional gear (e.g., an antenna array) to detect the bearing of a sender’s signal. _{1}_{2}_{1}_{2}_{1}_{2}_{1}_{2}

In range-free localization schemes, the nodes determine their location without time, angle, or power measurements. Therefore, hardware design is dramatically simplified; however, such schemes are there- fore extremely cost effective. In such schemes, errors may be masked by network fault tolerance, redundancy computation, and aggregation. Bulusu proposed an outdoor localization scheme “Centroid”, an outdoor location scheme in which the nodes determine the location as the centroid of its proximate anchor nodes [

This section describes how the proposed DFPLE strategy uses the Fermat point of the triangle for an irregular wireless sensor network [

There are

Every sensor node has a unique ID.

Sensor nodes are deployed randomly.

There are

Each beacon node is equipped with a GPS and, thus, knows its own location.

The other (

For sake of effective-performance, the transmission power of a beacon node is modulated by the variable radius method. That is, the power level of beacon nodes can be modulated to high power level up to increase the communication range of beacon nodes to

The DFPLE consists of four main phases on its operation: gathering beacon node location phase, estimating location, refining estimated location and error estimation. Each step of phases described as follows:

To gather information about other beacon nodes within communication range, beacon nodes must increase power to extend their communication range to

The beacon nodes gather the ID and location information of neighboring beacon nodes by exchanging beacon frames.

Beacon nodes reduce power to their original level.

Normal nodes record all neighbors (including normal node ID, beacon node ID, and locations) within communication range.

Neighboring beacon nodes provide other beacon node locations, which are collected in Phase I. When the beacon node is beyond the communication range of normal nodes, the neighboring beacon node that is farthest from the normal beacon node is considered the beacon node.

The location of the normal node must meet one of the following three cases:

When the normal node is within communication range of a beacon node, the location of the beacon node is considered the most likely solution [see

When the normal node is within communication range of two beacon nodes, the midpoint of the intersection of their communication ranges is considered the most likely solution [see

When the normal node is within communication range of three beacon nodes, the Fermat point of the triangle which is formed by the intersection of the three circles in which the center of the circles are the beacon node locations is considered the most likely solution [see

Two intersecting circles must have two intersection points. Of course, three circles intersecting circles must have six intersection points. Therefore, a rule is needed for selecting the correct three intersection points needed to construct a triangle. The symbols are given by

Step 1: If
_{2}_{1}

Step 2: If
_{2}_{1}

Step 3: If
_{2}_{1}

The calculations of vertices _{n}_{n}_{B12} ≠ d_{B13} ≠ d_{B23}_{1}_{2} intersect at

From _{B1} = r_{B2} = r_{B3} = r

The FERMAT point is point in Δ

Construct a virtual equilateral triangle associated with each

Construct lines

Finally,

Δ

The centroid of the triangle (

Localization accuracy is determined based on the closeness of a best estimate for the actual position of an unknown node. The closeness of a position estimate to the actual estimate is positively correlated with the accuracy of the algorithm. In this study, performance of the DFPLE algorithm is defined as the mean error (

The proposed DFPLE strategy was simulated using MATLAB in a static wireless sensor network. This simulation was conducted in a 2-D square area (5r5

Most proposed location estimation algorithms generate position estimate errors. Even in idealized setups with no obstacles or external factors, relatively small errors from noisy sensor measurements can induce considerably larger errors in node position estimates. Such errors are related to a set of attributes that in this study are network setup attributes. Network setup attributes include the measurement technology used, accuracy of measurement technology used, network density, uncertainties in beacon node locations and beacon node densities. The simulations focus on the impact of three factors: density of sensor nodes, ratio of beacon nodes and response rate.

The impact on sensor node density is evaluated by increasing the number of sensor nodes from 50 to 200 in a fixed square area (10

The impact on the ratio of beacon nodes is evaluated by increasing the number of beacon nodes from 10 to 95 in a 10

In wireless sensor networks, response rate is also a metric for network performance. When a beacon node sends a query packet to its neighbors inside its communication range, if its neighbor’s location estimation is accurate, this neighbor must respond with a message sent back to the beacon node as soon as it receives the query package. Finally the response rate is calculated. The DFPLE algorithm has an acceptable performance when the number of beacon nodes exceeds 55 (

As mentioned in simulation results, the proposed DFPLE algorithm improves mean error for the randomly selected case of the existing bounding box algorithm (CPE strategy). Specifically, the proposed DFPLE algorithm has better performance than CPE algorithm in terms of location estimation.

This paper has presented DFPLE (Distributed Fermat-point Location Estimation) for WSNs. The proposed method of estimating sensor positions applies a Fermat point algorithm to estimate sensor node positions. Unlike the traditional bounding box algorithm, DFPLE is based on the shortest path from intersection between beacon nodes coverage area. The intersection vertices form a triangle which Fermat point is located to refine the estimated location of sensor nodes. The simulation, comparing DFPLE and CPE, revealed the effects of varying the number of sensor nodes and the proportions of beacon nodes. Simulation results demonstrate that the DFPLE algorithm for estimating sensor positions is more accurate than existing algorithms and improves upon conventional bounding box strategies.

Bounding Box Algorithm.

AoA Measurement.

DFPLE Operation for Sensor Location Estimation.

Calculate the Vertices in a Triangle.

Estimation Locations of Three Beacons.

Formation of Three Virtual Triangles associated with Fermat Point.

Cases of Δ

Types of Δ

Location Estimation of DFPLE and CPE.

Simulation Environment.

Node Density

Number of Beacon Nodes

Number of Beacon Nodes

Symbols for intersection points.

| |
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_{1}, B_{2}, B_{3} |
three circles |

_{1}, P_{2} |
two intersection points of _{1}_{2} |

_{1}, Q_{2} |
two intersection points of _{2}_{3} |

_{1}, R_{2} |
two intersection points of _{1}_{3} |

three vertices of the triangle |