Distributed efficient localization in swarm robotics using Min–Max and Particle Swarm Optimization

Accepted Manuscript

Distributed Efficient Localization in Swarm Robotics Using Min-Max and Particle Swarm Optimization Alan Oliveira de Sa, ´ Nadia Nedjah, Luiza de Macedo Mourelle PII: DOI: Reference:

S0957-4174(15)00809-X 10.1016/j.eswa.2015.12.007 ESWA 10415

To appear in:

Expert Systems With Applications

Received date: Revised date: Accepted date:

10 December 2014 9 December 2015 10 December 2015

Please cite this article as: Alan Oliveira de Sa, ´ Nadia Nedjah, Luiza de Macedo Mourelle, Distributed Efficient Localization in Swarm Robotics Using Min-Max and Particle Swarm Optimization, Expert Systems With Applications (2015), doi: 10.1016/j.eswa.2015.12.007

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Highlights • We propose an efficient localization method for swarm robotic systems.

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• The method, based on Min-Max and PSO algorithms, presents low positioning error.

• We demonstrate that the method overcomes the drawback of the Min-Max method.

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• It is shown that the processing time grows with the swarm connectivity.

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Distributed Efficient Localization in Swarm Robotics Using Min-Max and Particle Swarm Optimization

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Alan Oliveira de S´ a1 Center of Electronics, Communications and Information Technology,

Admiral Wandenkolk Instruction Center, Brazilian Navy, Rio de Janeiro, RJ, Brazil

Nadia Nedjah2

Department of Electronics Engineering and Telecommunication, Engineering Faculty, State University of Rio de Janeiro, RJ, Brazil

Luiza de Macedo Mourelle3

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Department of System Engineering and Computation, Engineering Faculty, State University of Rio de Janeiro, RJ, Brazil

Abstract

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In a wireless sensors network in general, and a swarm of robots in particular, solving the localization problem consists of discovering the sensor’s or robot’s positions without the use of external references, such as the Global Positioning

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System – GPS. In this problem, the solution is performed based on distance measurements to existing reference nodes also known as anchors. These nodes have knowledge about their respective positions in the environment. Aiming

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at efficient yet accurate method to approach the localization problem, some bio-inspired algorithms have been explored. In this sense, targeting the accu-

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racy of the final result rather than the efficiency of the computational process, we propose a new localization method based on Min-Max and Particle Swarm Optimization. Generally, the performance results prove the effectiveness of the

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proposed method for any swarm configuration. Furthermore, its efficiency is demonstrated for high connectivity swarms. Specifically, the proposed method 1 [email protected] 2 Corresponding

author, E-mail: [email protected]

3 [email protected]

Preprint submitted to Journal of LATEX Templates

December 29, 2015

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was able to reduce the localization average error by 84%, in the worst case, considering a configuration of 10 anchors and 100 unknown nodes and by almost 100%, in the best case, considering 30 anchors and 200 unknown nodes. This

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proves that for high connectivity networks or swarms, the proposed method provides almost exact solution to the localization problem, which is a big shift forward in the state-of-the-art methods.

Keywords: Localization, Wireless Sensor Network, Swarm Robotics, PSO

1. Introduction

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Many applications of Swarm Robotic Systems (SRSs) require that a robot is

able to find out its position. This position may be either absolute, i.e. based on an universal reference system, or relative to other robots, i.e. based on a local coordinate system. The robot position information is necessary, for example, to establish behaviors of self-assembly, where each robot of the swarm must be

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positioned within a pre-defined organization, or self-healing, where the robots reorganize themselves to reconstitute a formation that has been undone (Rubenstein, 2009). Similarly, Wireless Sensor Networks (WSNs), whose prospect of

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application is broader than SRSs, have attracted great attention from the industry point of view. However, in most cases, WSNs have little use when it is

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not possible to estimate the sensors positions (Sun and Su, 2011). In both cases of SRSs and WSNs, the basic devices, i.e. robots or sensors, respectively, have common characteristics: they are of a reduced size, have access

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to limited energy source and must be of low cost. Thus, the straightforward solution that consists of endowing each basic device with a Global Positioning

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System (GPS) is often not viable. The localization problem consists of inferring the position of a set of robots

or sensors when no external reference, such as GPS, is available. Many of the localization methods depend on the ability of a node to measure its distance to some reference nodes, also known as anchors, whose positions are known. Com-

mon techniques for distance measurement are based on either the received signal

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strength (RSS), the one-way/roundtrip propagation time of a signal or the comparison of the propagation time of two signals that are known to have different propagation speeds (Mao et al., 2007; Lymberopoulos et al., 2006). Typically,

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localization algorithms based on propagation time measurements achieve better accuracy than the RSS based algorithms (Mao et al., 2007).

As the measurement techniques presented rely on signal propagation characteristics, a threshold distance for such measurements has to be considered. In

the simple case, wherein all reference nodes are within the distance measurement

threshold, the measurements are direct and made in a single hop. However, in

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the cases where one or more reference nodes are outside this threshold, distance measurements are obtained indirectly using a multi-hop strategy. For this purpose, algorithms such as Sum-dist and DV-hop are used to estimate the length of the multi-hop path between the nodes (Langendoen and Reijers, 2003). De-

pending on the network topology, and swarm connectivity, the use of one and multi-hop can be combined. In (Langendoen and Reijers, 2003), a three-step

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approach is proposed to address the multi-hop localization:

1. Estimate the distances of each unknown node to the reference nodes.

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2. Compute a rough estimation of the position of each node using the measurements obtained in step 1.

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3. Refine the position estimate of each node using the position known so far and distance information informed by the neighboring nodes.

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The optimization using bio-inspired techniques are often applied to the problem of localization, in the case of a single hop (de S´ a et al., 2014) as well as in that of multi-hop (Ekberg, 2009). In this paper, we solve the localization

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problem based on a multi-hop strategy. The method exploits an optimization process using the Particle Swarm Optimization (PSO) (Engelbrecht, 2005). The methods acts in three stages: During the first stage, the Sum-Dist (Langendoen and Reijers, 2003) is used to estimate the multi-hop distances to the reference nodes; Then, during the second stage, an initial position estimate is made using the Min-Max method (Langendoen and Reijers, 2003; Savvides et al., 2002); 4

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After that, during the third stage, the refinement of the positions is performed by means of an optimization process using PSO, wherein the objective function has been precisely tailored as to guide the optimization process towards a robot

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localization with an acceptably low error ratio. When compared to existing localization methods that use optimization algorithms (Naraghi-Pour and Rojas, 2014; Carli et al., 2014; Ekberg and Ngai,

2011; Ekberg, 2009), the contribution of this work consists of the use of the information provided by the Min-Max method. This is done not only to assess

the initial positions of the unknown nodes, but also to further refine these posi-

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tions. In this novel approach, the bounds found by the Min-Max to establish the

possible area for each unknown node are considered in the objective function, thus enhancing the convergence of the swarm solution to the actual position of the unknown nodes. Moreover, in contrast with some techniques presented in Section 2, it is important to emphasize that the method proposed herein solves the localization problem in a completely distributed manner. The solution is

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collectively found by the swarm, as opposed to a dictated solution found by a host as it is the case in most of the existing methods. The proposed distributed

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strategy makes the method more resilient, and requiring a relatively low number of reference nodes. It does not need any infrastructure with sensors or extra references in the environment. Furthermore, it does not require a previous survey

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of the environment to collect ambient’s data. Generally speaking, the performance results, presented and discussed later in

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this paper, prove beyond doubt that the proposed method is absolutely effective in finding an acceptable localization for any sensor network and robotic swarm configuration. Furthermore, its efficiency is demonstrated for high connectivity

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networks and swarms. Specifically, the proposed localization method was able to reduce the node positioning average error by 84%, considering a low connectivity configuration, and by almost 100%, considering a high connectivity case. This means that for high connectivity configurations, the proposed method was able an almost exact solution to the localization problem, which is a big shift forward when compared to existing methods. 5

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The rest of this paper is organized as follows: First, in Section 2, some related works are presented. Later, in Section 3, we briefly describe the main steps of PSO. In Section 4, we show the proposed distributed localization method that

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exploits the Min-Max technique and PSO. Then, in Section 5, we report on the performance results obtained. Finally, in Section 6, we present some concluding remarks along with some possible future work.

2. Related Work

The importance of the localization information to swarm robotics and wire-

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less sensor networks systems, conjugated with the limitations in terms of hardware, cost and energy requirements that are typical of these devices, have motivated the search for more efficient yet accurate methods to solve the localization problem.

The localization methods can be organized in two classes: range-free and

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range-based. The former considers methods that do not need distance measurements to perform the localization process. On the other hand, the latter exploits methods that use distance measurements to estimate the node posi-

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tions. Some related works are presented in Sections 2.1 and 2.2, associated with the range-free and range-based strategies, respectively.

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2.1. Range-free Localization

A simple range-free method, presented in (Tesoriero et al., 2010), uses the

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RFID (Radio Frequency Identification) technology to locate robots in an indoor environment. In this solution, the area is divided into a grid of small cells, where passive RFID tags are installed. The robot is endowed with a passive RFID

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reader. As the robot moves through the mapped area, the passive RFID reader acquires the closer RFID tag, and the robot position is estimated correlating the obtained ID with a database that contains the coordinates of each RFID tag. The highlight of this solution is the simplicity and the low computational cost required to establish the robot position. On the other hand, the localization

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process is centralized, the robot does not have autonomy to estimate its own position, and it requires an array of RFID tag installed over all its operational environment. Another RFID based solution is presented in (Calderoni et al.,

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2015), for indoor localization in a hospital environment. In this solution, a set of fixed passive RFID devices measures the power transmitted by an active RFID

tag, and sends these measurements to a server that estimates the location of the active RFID using Random Forest classifiers. This system, intended to assess the room where the active RFID is located rather than its precise position,

requires less RFID devices installed over the operational area than in (Tesoriero

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et al., 2010).

In (Yun et al., 2009), the authors present a solution, based on the method introduced by (Kim and Kwon, 2005; Bulusu et al., 2000), where the position of an unknown node is computed as the weighted average of the positions of all onehop reachable reference nodes. In this case, the weight of a given reference node is determined by a Fuzzy Logic System (FLS). The input of the modeled FLS

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is the RSS coming from a reference node. The output of the FLS is the weight associated with the respective reference node. Simulation results showed that

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the FLS-based method obtained an average positioning error better then those obtained with the methods proposed by (Bulusu et al., 2000; Kim and Kwon, 2005). It is worth mentioning that the number of reference nodes required by

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this method is relatively large, which is approximately 67% of all nodes. In (Nallanthighal and Chinta, 2014), the authors propose the Improved Grid-

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Scan Localization Algorithm (IGSL). Firstly, based on the connectivity with a set of one-hop anchors, the unknown node establishes an estimative region (ER) where it lies in. After that, the ER is divided into a grid, and the size

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of the ER is reduced, based the contribution of a set of two-hop and farther anchors, eliminating some of the ER‘s grid points. Finally, the node’s position is estimated as the average of all the remaining ER’s grid points. The IGSL does not require any additional hardware to measure the RSS from anchors, but requires a relatively large number of anchors (45%) to achieve 85% of accuracy. A range-free localization method that does not require a large number of 7

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reference nodes is presented in (Stella et al., 2014). The authors introduce a technique where the localization process is performed based on the analysis of the RSS fingerprint of signals emitted by a set of heterogeneous reference nodes.

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The reference nodes are wireless devices of different technologies such as WLAN (Wireless Local Area Network), GSM (Global System for Mobile Communications) and UMTS (Universal Mobile Telecommunication System). To estimate

its own position, an unknown node measures the local RSS fingerprint and com-

pares it with a RSS fingerprint map in order to determine the area which best

matches the measured RSS fingerprint. The advantage of this method is that

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the references can be obtained from an infrastructure that commonly already exists. A drawback is that it requires a previous survey of the RSS fingerprints of the entire area of interest.

In (Li et al., 2014), the Sequence Based Localization (SBL) technique is used. First, the operational area is divided into regions, based on a set of bisector lines, perpendicularly drawn between each pair of reference nodes. Then, the

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region wherein the unknown node resides, is identified by a sequence number that is obtained ranking the RSS from each reference node. Thus, the unknown

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node position is considered to be the centroid of the identified region. The advantage of this method is the low computational cost to perform localization. On the other hand, its resolution is low but enough to identify the room inside

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a building, where the unknown node resides. Some localization techniques based on image processing have also been de-

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veloped. In (Son et al., 2015) a vehicle endowed with three cameras locate itself based on images captured from the environment. The technique requires a previous survey of the operational area, in order to populate a database of images

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that will be used as reference to compute the vehicle’s position. The high computational cost, due to the memory space required to store the image database, and the size and cost of the sensors, i.e. a set of cameras, makes this solution

not attractive for swarm robotic systems with the characteristics mentioned in Section 1. Another image based technique is presented in (Rampinelli et al., 2014), where 11 static cameras and 11 image servers, are used to assess the 8

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position of a robot while moving inside three rooms. In (Niculescu and Nath, 2001), the authors introduce the DV-Hop method where the distance to anchors are estimated in a multi-hop approach. To esti-

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mate the distance to anchors, this method counts the number of hops between the unknown node and an anchor, and multiplies it by an average hop length. After discovering the coordinates and distance to anchors, the unknown nodes estimate their positions using the Lateration method, solved by the Linear Least

Square (LLS) technique. In (Safa, 2014), an extension of the DV-Hop method,

referred as Hybrid DV-Hop (HDV-Hop) is proposed. In HDV-Hop, the Latera-

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tion method is solved by the Non-Linear Least Square (NLLS) technique, pro-

viding a higher accuracy than the LLS, used by (Niculescu and Nath, 2001). On the other hand, due to the high computational cost of the NLLS, the positions of all nodes are computed in a central, and more powerful, station. Comparing with (Niculescu and Nath, 2001; Safa, 2014), an advantage of the solution herein proposed is that it includes a refinement phase, that uses an optimization algo-

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rithm to improve the quality of the estimated positions through a collaborative process between neighboring nodes.

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Considering the techniques presented in this section, the advantage of the solution proposed in this paper is that it does not require any infrastructure, such as the wireless devices used by (Stella et al., 2014), the RFID devices

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of (Tesoriero et al., 2010; Calderoni et al., 2015) or either the cameras used by (Rampinelli et al., 2014). Also, the localization process is performed in a

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completely distributed fashion, what makes it more resilient than in (Tesoriero et al., 2010; Calderoni et al., 2015; Safa, 2014), where the localization process is centralized. It uses less reference nodes than in (Yun et al., 2009), and does

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not need a previous survey of the operational area such as (Stella et al., 2014; Son et al., 2015). It is noteworthy mentioning that, in general, the range-free solutions as (Calderoni et al., 2015; Nallanthighal and Chinta, 2014; Li et al., 2014; Stella et al., 2014; Tesoriero et al., 2010; Yun et al., 2009; Kim and Kwon, 2005; Bulusu et al., 2000), are simpler and, in some cases, require less resources, but their results are not as accurate as those yield by a range-based solution 9

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(Yun et al., 2009), such as the algorithm proposed in this paper. 2.2. Range-based Localization

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In (Niculescu and Nath, 2001), the authors introduce the ad-hoc positioning algorithm. This is a multi-hop localization algorithm developed for networks where the unknown nodes, in most cases, are not able to directly measure their

distance to anchors. To deal with this constraint, the authors present three

alternative multi-hop methods to estimate the distance to anchors: DV-Hop, DV-Distance and Euclidean distance. The DV-Hop is briefly described in Sec-

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tion 2.1, as a range-free algorithm. In the DV-Distance method, also referred

as Sum-Dist (Savvides et al., 2002; Langendoen and Reijers, 2003), the distance to anchor is computed as the length of the shortest path between the unknown node and the given anchor. In the Euclidean method, the distance is obtained by computing the true Euclidean distance between the unknown node and an anchor, based on the geometry of the nodes along the path. For all of the three

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methods, the anchors coordinates are disseminated in the network. Once knowing the coordinates and distance to anchors, the unknown nodes estimate their

technique.

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positions through the Lateration method, using the Linear Least Square (LLS)

In (Rabaey and Langendoen, 2002), the authors propose the Robust Posi-

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tioning algorithm, which is quite similar to the ad-hoc positioning algorithm, but has an additional stage to refine the estimated position of an unknown nodes. First, the DV-Hop method is used to obtain the distance to anchors. Then, the

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Lateration method is used to estimate the initial position of the unknown nodes, considering the distance to anchors and their respective coordinates. After that,

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the positions are refined using the Lateration method, via a distributed process where, at each iteration, every unknown node re-computes its position based on the coordinates and distance of its one-hop neighbors. Another range-based algorithm, called N-hop Multilateration, is proposed in

(Savvides et al., 2002). In this algorithm, the distances to anchors are evaluated using the Sum-Dist (Savvides et al., 2002; Langendoen and Reijers, 2003) 10

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method, which computes the shortest-path length to the anchors, as performed by the DV-Distance method. To compute the initial position of the unknown nodes, the authors introduce the Min-max (Langendoen and Reijers, 2003),

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(Savvides et al., 2002) method, which estimates the initial position based on the distance to anchors and their respective coordinates. It requires less com-

putational resources than the Lateration method. The position refinement is performed using the Lateration method, based on the coordinates and distances to the one-hop neighbors.

The Lateration method, performed by Robust Positioning and N-hop Multi-

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lateration algorithms to refine the node‘s positions, is also solved using a standard LLS approach, which, in some cases, fails due to the eventual arising of a non-invertible matrix within the computation (Langendoen and Reijers, 2003). This constraint is solved by the algorithm proposed in this paper, due to the fact that the refinement is achieved by an optimization process, which always obtains a valid solution even if it is, eventually, sub-optimal.

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In (Ekberg, 2009), the author presents the Swarm-Intelligent Localization algorithm (SIL), which uses the PSO to estimate the positions of nodes in a static

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WSN. Firstly, the distance to anchors and anchors coordinates are obtained using a multi-hop strategy, using Sum-dist. After that, each unknown node uses the PSO to estimate a self coarse position, minimizing a fitness function

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that considers the distance to anchors and their respective coordinates. Finally, each unknown node uses the PSO once again, to refine its position, minimizing

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another fitness function that considers the distances and coordinates of the onehop neighbors, two-hop neighbors and anchors. An extension of the proposed algorithm to mobile sensor networks was later demonstrated in (Ekberg and

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Ngai, 2011). An advantage of the algorithm herein proposed, comparing with SIL, is that it requires less computational cost to estimate the coarse positions of the nodes once, for this task, it uses the Min-Max method, simpler than the

PSO. Another contribution of the proposed algorithm is the enhancement of the fitness function used to refine the nodes position. To this end, some information provided by Min-Max method are added to the fitness function leading the 11

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possible solutions to a more restricted area, wherein the actual position resides. Some other localization methods (Naraghi-Pour and Rojas, 2014; Carli et al., 2014) estimate the unknown node’s position minimizing a function that com-

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putes the euclidean distance errors to a set of one-hop neighbors. In (NaraghiPour and Rojas, 2014), the solution is obtained through the Distributed Ran-

domized Gradient Descent (DRGD), that is proposed based on the Gradient

Descent (GD) algorithm. In (Carli et al., 2014) the solution is obtained through an online Bayesian filter, based on the Extended Kalman Filter (EKF). Comparing to the approach of (Naraghi-Pour and Rojas, 2014; Carli et al., 2014),

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the fitness function minimized in the present method includes the use of a confidence factor and adds two more terms that contributes positively with the localization process: one term uses the information provided by the two-hop neighbors, as in (Ekberg, 2009), and the other term considers the area found by Min-Max method.

In (Xiong et al., 2015), the unknown nodes keep sending massages directly to

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a set of anchor nodes. Then, the time-difference-of-arrival (TDOA) of the messages received by the anchors are processed in a server on the backbone network.

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After that, based on a Gaussian measurement noise model, the unknown node position is obtained by means of the maximum likelihood estimation (MLE). One contribution of the method proposed by (Xiong et al., 2015) is that it is

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suitable for fully asynchronous WSN, thanks to a compensation operation that cancels the relative clock offsets and skews among anchors. A drawback of the

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method is that the unknown nodes need to be directly connected, i.e. in onehop, to the anchor nodes. This fact increases the number of anchors required when the communication range of the nodes is small comparing with the cov-

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ered area dimensions. I this sense, the multi-hop approach of the method, herein proposed, contributes to the reduction of the required number of anchors. Also, a central server is not needed to compute the nodes positions. Another method, that uses a cluster matching technique, is presented in

(Rashid et al., 2015). In this method, an infrared (IR) sensor, fixed in a frame, scans the environment assessing the coordinates of the robots that are in its line 12

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of sight, obtaining a first network topology with absolute positions. Then, each robot measures the distances and the angles to each of its neighbors, bringing up a second network topology, with relative positions. Finally, both network

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topologies are merged so that absolute positions may be obtained for a number of robots higher than those belonging to the first network topology that may

have missed robots that were out of the sensor’s line of sight. Comparing with this solution, an advantage of the algorithm herein proposed is that it requires

simpler sensors, given that it does not need to measure angles between robots, just distances. Also, no additional sensor, outside the swarm, is required.

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It is noteworthy mentioning that the range-based localization methods, in

general, provide better accuracy than the range-free methods (Yun et al., 2009), however the nodes require additional hardware to obtain the distance measures. 3. Particle Swarm Optimization

The PSO has its origin in the simulation of simplified social models, such as

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the movement of bird flocks and fish schools (Kennedy, 1995; Shi, 1998). The particle is the basic element handled by the algorithm. A particle represents

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a possible solution of a given problem. The swarm represents a set of possible solutions. Each iteration, the position of each particle of the swam is updated according to (1), where x` and v` are the position and velocity of particle `,

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respectively.

x` (t + 1) = x` (t) + v` (t + 1)

(1)

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The velocity v` of particle ` is updated based on three components: the

inertia of the particle; cognition of the particle, which is based on the best solution obtained by the particle so far; and social term, which is based on the

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global best solution found by the swarm. The velocity of particle ` is defined in

(2): v` (t + 1) = ωv` (t) + ϕ1 r1 (t)(m` − x` (t)) + ϕ2 r2 (t)(mg − x` (t)),

(2)

wherein ω is a parameter that weighs the particle inertia, ϕ1 and ϕ2 are the parameters that weigh the cognitive and social components, respectively, r1 and 13

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r2 are random numbers in [0, 1], m` is the best position visited by particle ` so far, and mg is the best position inferred by the swarm considering the experience of all the particles.

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In order to better explore the multi-dimensional search space, a velocity maximum limit is imposed for each dimension j, as stipulated in (3): 0 ≤ v`j ≤ γ(maxj − minj ),

(3)

wherein maxj and minj are the maximum and minimum limits of the search space with respect to dimension j and γ ∈ [0, 1] .

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The overall computation performed by the standard PSO, during a mini-

mization process of objective function f (x), where x represents a particle position considering a swarm of N particles, is given in Algorithm 1. Algorithm 1 PSO begin best ← ∞; for each particle ℓ, 1 ≤ ℓ ≤ N do Set randomly position xℓ and velocity vℓ ;

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mℓ ← xℓ ; if f (mℓ ) < best then

repeat

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mg ← mℓ ; best ← f (mℓ ); end end

for each particle ℓ, 1 ≤ ℓ ≤ N do

Update velocity vℓ , as in (2) and (3); Update position xℓ , as in (1);

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f itness ← f (xℓ );

mℓ ← xℓ , whenever f itness < f (mℓ ); mg ← xℓ , whenever f itness < f (mg );

end until Stopping condition; return mg ;

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end

It is worth mentioning that some variants of the classical PSO, not explored

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in this paper, have been developed in order to improve the performance of the algorithm, especially with regard to the avoidance of convergence to a local optimum, such as in (Cui et al., 2008, 2012).

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4. Proposed Localization Method The localization proposed method can be applied to problems with two or three dimensions. However, for the sake of clarity of the analysis presented, but

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without loss of generality, we focus on the two dimensions case. As a premise,

the problem is regarded as multi-hop. This means that the unknown nodes, in most cases, cannot measure their distance to the reference node directly. Also,

all nodes are considered static and no distance measurement errors are taken into account.

Like the approach described by (Langendoen and Reijers, 2003), as sketched

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in the introduction, the method is divided in three stages. In the remainder

of this section we introduce the underlying details of the three stages of the method; then we describe the technique used to establish the confidence factor, used in the fitness function of the third stage of the algorithm; subsequently, we

4.1. Distance to anchors

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present the complete algorithm of the proposed method.

The method first stage (STAGE-1) estimates the distance to the reference

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nodes, wherein the Sum-Dist algorithm is used. The algorithm starts with the anchor nodes transmitting a message containing the anchor coordinates and the total distance traveled by the message so far (i.e. 0 until this moment). The

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message is passed from node to node, so that the length of each hop is added to the total distance traveled by the message. The message received by a node is

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only stored and then retransmitted if and only if, the distance traveled by the message is the smallest known so far, to a given reference node. Thus, at the end of STAGE-1, each unknown node i will be aware of the position coordinates

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(ˆ xr , yˆr ) of each reference node r and the distance of the shortest route li,r that exists between them. It is noteworthy that li,r , i.e. the shortest distance traveled by a message

from unknown node i till anchor r, is not necessarily the actual distance between them. For instance, in a scenario with randomly distributed nodes, it is common

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that nodes that compose the path between i and r are not aligned, causing that li,r is greater than the actual distance between the two nodes. This inaccuracy may impact in the initial position estimation, performed as described in Section

when the estimated position is refined by the PSO. 4.2. Initial node position

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4.2. However, this error is mitigated in the third stage, described in Section 4.3,

The second stage (STAGE-2) of the method estimates the initial/coarse po-

sition of each unknown node using the Min-Max method, introduced by (Sav-

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vides et al., 2002). It is noteworthy that in (Savvides et al., 2002) the Min-Max

method remained nameless and was first referred as Min-Max in (Langendoen and Reijers, 2003).

In this method, the estimated distances li,r , obtained in STAGE-I, are used to establish the bounds of the region Si where the node i resides. Figure 1 shows how the bounds of this area are computed, considering, for sake of clarity,

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only the x dimension. In the mentioned figure, it is possible to note that the estimated distance between the node i and the anchor r1 is li,1 = d1,2 + d2,i .

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Also, the estimated distance between the node i and the anchor r3 is li,3 = di,3 . Considering x ˆ1 the coordinate of r1 in dimension x, it is possible to state that, with regard to this anchor, the possible region to node i resides between

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x ˆ1 −(d1,2 + d2,i ) and x ˆ1 +(d1,2 + d2,i ). Performing the same analysis with regard to node r3 , it is possible to state that the possible region to node i lies between x ˆ3 − di,3 and x ˆ3 + di,3 . Considering both anchors simultaneously, we conclude

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that the position of node i lies between max[ˆ x1 −(d1,2 + d2,i ) , x ˆ3 −di,3 ] = x ˆ3 −di,3

and min[ˆ x1 + (d1,2 + d2,i ) , x ˆ3 + di,3 ] = x ˆ1 + (d1,2 + d2,i ). So, the estimated

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coordinate xi of node i is such that x ˆ3 − li,3 ≤ xi ≤ x ˆ1 + li,1 . The same procedure is applied in the y dimension.

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di ,3

di ,3

3

2

Anchor Referência

i d1,2 + d2,i

Unknown Desconhecido

d1,2 + d2,i

Possible region for Região possível paranode o nói:i: to [x1 + (d1,2 + d2,i )] from de [x3 – di ,3] até

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1

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Figure 1: Example of the computation of the Si bounds in dimension x

Based on the aforementioned, the first step of this method is to create, for each reference node r a region Bi,r , denominated as bounding box. The boundaries of this regions are computed, in each dimension, by adding and subtracting li,r from the position of each reference node (ˆ xr , yˆr ), according to

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(4).

Bi,r : [ˆ xr − li,r , yˆr − li,r ] × [ˆ xr + li,r , yˆr + li,r ].

(4)

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Let Si be the region corresponding to the intersection of all these bounding boxes, at least three, as defined in (5): \

Bi,r : [max(ˆ xr −li,r ), max(ˆ yr −li,r )]×[min(ˆ xr +li,r ), min(ˆ yr +li,r )]. (5) ∀r

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Si =

∀r

∀r

∀r

∀r

The initial position ui of unknown node i is computed as the center point of Si .

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Figure 2 illustrates the Min-Max method to estimate the position of an

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unknown node i from three anchors r1 , r2 and r3 .

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r2=(x2, y2)

r1=(x1, y1) li,1

li,2

ui=(xi, yi)

li,3 r3=(x3, y3)

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Si

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Figure 2: Graphical illustration of the Min-Max method

Note that all steps of the Min-Max method are composed by low complexity operations, including the process to obtain the intersection area Si , that is done by simple searches for a minimum/maximum value in sets with R elements, wherein R is the total number of anchor nodes.

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Other alternatives to Min-Max consists of using an optimization algorithm, as the PSO used in (Ekberg, 2009), or a deterministic method such as the

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Lateration (Langendoen and Reijers, 2003). However, even with the use of those processes, the initial position computation remains dependent on li,r , which is predominantly inaccurate. Since both methods are impacted by the inaccuracy

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of li,r , we opted by using the Min-Max approximation due to its simplicity. Furthermore, it requires a reduced computational cost when compared to both

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PSO and Lateration method.

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4.3. Node position refinement The third stage (STAGE-3) is an iterative and distributed process that allows

for the accuracy improvement of the unknown node positions. During this stage, each unknown node reevaluates its position based on the positions of its neighbors as well as the inferred distances to those neighbors, which also are updated at each iteration. Since, ideally the measures of distance between nodes remain constant in a static swarm, the estimated positions of the unknown nodes 18

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tend to be adjusted gradually, in order to comply with those distance measures, making the estimated positions converge to the real positions. In this stage, the PSO is used to drive the refinement process. The objective function, optimized

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by the PSO, is a composite of three main terms. First, we must consider, for a given unknown node i, the error introduced in the distance to each of its neighboring node n (Ekberg, 2009). In this case, node

n is considered a neighbor node of node i, which can be an anchor or unknown node, if and only if it is accessible from i via one single hop, i.e. it is within the distance measurement threshold L. This is defined in (6):

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g(i) = (di,n − ||pn − pi ||)2 ,

(6)

wherein di,n is the distance measured between unknown node i and its neighbor node n, pn is the position of n and pi is the estimated position of node i. Thus, the first term of the objective function considers the contribution of all neighboring nodes n, normalizing the error g(i) by a confidence factor that

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establishes how precise is the knowledge passed by each neighbor n, depending

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on how close it is to an anchor. This term is defined as in (7): X g(i) f1 (i) = , ζn

(7)

n∈Vi

wherein Vi is the set of neighboring nodes of i and ζn is the confidence factor assigned to neighboring n. The technique to compute the confidence factor of

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a node is described in 4.4.

The second term of the objective function, is computed based on nodes that

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are two hops away from unknown node i. According to (Ekberg, 2009), the distance between the unknown nodes i and w, that is positioned at two-hop

distance away from i, is always greater than the threshold of measured distance

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L but smaller than 2L. Thus, the contribution of each node w in the second

term of the objective function can be defined as in (8): h(i) = max(0, L − ||pw − pi ||, ||pw − pi || − 2L)2 ,

(8)

where pw is the position of node w. Therefore, if a possible solution is such that L ≤ ||pw − pi || ≤ 2L, then no value should be added to its fitness. However, 19

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if ||pw − pi || < L, then the square of the distance between pi and the circle of radius L, centered at pw should be added to the fitness. Otherwise, i.e. if 2L < ||pw − pi ||, then the square of the distance between pi and the circle of

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radius 2L, centered at pw should be added to the fitness. Based on the aforementioned explanation, the second term of the objective function is defined as in (9): f2 (i) =

X h(i) , ζw

w∈Wi

(9)

wherein Wi is the set of nodes that are two hops away from i and ζw is the

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confidence factor related to the knowledge informed by node w. Its definition is detailed in Section 4.4.

The third term of the objective function, defined as in (10), is intended to lead the possible solutions within the area Si , as defined by the Min-max technique, during the STAGE-2 of the proposed method (see Section 4.2). Thus,

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if a possible solution falls within Si , then no value should be added to its fitness. However, if it falls outside Si , then a value that is proportional to the square of

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the distance of the solution to the center of Si should be added to its fitness.    0 if pi ∈ Si   (10) f3 (i) =     ρ + (ρ ||p − u ||)2 if p ∈ 1 2 i i i / Si ,

wherein ui is the position of the center of Si , computed by the Min-Max tech-

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nique (see Section 4.2), and ρ1 and ρ2 are empirically set positive parameters. They allow to always favor positions inside Si against those outside even though

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they are both very close the border of Si . Finally, the objective function is then given by (11), which must be mini-

mized: f (i) = f1 (i) + f2 (i) + f3 (i)

(11)

Note that, as there are no errors in distance measurements, the global minimum value of f (i) is 0.

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With the described objective function, each node is able to estimate its position with respect to its closest neighbors, i.e. one-hop and two-hop. Thereby, each robot in the swarm updates its locations in a parallel process. Thus, the

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global goal of the swarm, that is to achieve a zero positioning error for all nodes, follows from the achievement of the objective of each node, that is to be correctly positioned with respect to its neighbors. In other words, a node does not

need to know/compute the position of nodes other than its one and two-hop neighbors.

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4.4. Confidence factor

The confidence factor is intended to give more importance to the contribution informed by nodes that tend to have more accurate positions. Recall that this factor is used in (7) and (9).

The used technique (Ekberg, 2009) is based on the idea that the closer the nodes are to the anchors the more accurate their positions should be. The anchor

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nodes r ∈ R are assumed to have a confidence factor of 1. In this method, the

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confidence factor ζi of node i is calculated according to (12):    1 if i ∈ R    ζi =  P 2    λi,r if i ∈ / R. 

(12)

r∈Ri

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wherein Ri is the set composed by the three closest reference nodes to i in terms of total number of hops, and λi,r is the number of hops between node i

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and anchor r.

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4.5. Algorithm of the proposed method The complete algorithm of the proposed method to the multi-hop localization

problem is shown in Algorithm 2. It includes the three main stages, as presented previously. The algorithm must be executed for each unknown node, causing the full solution to emerge of the distributed processing. In the Algorithm 2, ∆ defines the number of cycles performed during the STAGE-3 while δ defines the number of the PSO iterations done in each of these STAGE-3 cycles. 21

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Algorithm 2 Complete algorithm of the proposed method

5. Performance Results

In order to evaluate the performance of the proposed method, we conducted

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simulations in an area of 100 × 100 measurement units. In this area, 100, 150 and then 200 unknown nodes were randomly distributed. Furthermore,

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were randomly allocated in the same area 10, 20 and then 30 reference nodes respectively. The distance measurement limit L was set to 20 measurement units. For each combination of the reference nodes and unknown nodes, 10

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scenarios were generated, resulting in a total of 90 different simulations. The algorithm of the proposed method was implemented in MATLAB (R2012b

version). The simulations were run on a computer, which included an Intel Core i7 central processing unit (CPU) of 3.07 GHz. The CPU has access to 8GB of random access memory (RAM). The Operating System (SO) used is Windows

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7 for a 64-bits bus based architecture. It is noteworthy to point out that the parameters of the PSO were adjusted empirically through a series of simulations. The set of parameters that allowed us to reach the optimization best results are

throughout this section.

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shown in Table 1. These settings were used to obtain all the results presented

The algorithm of the proposed method was implemented in MATLAB. The

parameters of the PSO were adjusted empirically through a series of simulations. The best results are shown in Table 1.

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Table 1: PSO Parameter setting

Parameter

Value

ω

0.7298

ϕ1

2.05

ϕ2

2.05 0.01

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γ

The values of ∆ and δ, of STAGE-3 of the algorithm, were set to 100 and

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10, respectively. Moreover, the objective function parameters ρ1 and ρ2 were set to 103 and 106 , respectively. In order to evaluate the performance of the proposed method, we computed,

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for each iteration, the mean positioning error (MPE) of all unknown nodes as

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described in (13):

M P Eg =

PI

i=1

||preal (i) − pcalc (i)|| , I

(13)

wherein g represents the number of cycles of the third stage, i identifies the

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unknown node, I is the total number of the unknown nodes, preal is the actual position of the node and pcalc it is the position calculated by the optimization process. As to compile the MPE of the 10 generated scenarios, one for each configu-

ration of the numbers of reference and unknown nodes, the mean value of MPE is computed as in (14): 23

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M M P Eg =

PS

s=1

M P Eg S

(14)

wherein s is the number of the scenario and S is the total number of scenarios.

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The results of the MMPE for each iteration ∆ of STAGE-3, obtained by the proposed method, are shown in Figures 3(a), 3(b) and 3(c), for networks with 10, 20 and 30 anchors, respectively. Each figure also shows the results for networks with 100, 150 and 200 unknown nodes. The MMPE data of Figures

3(a), 3(b) and 3(c) is presented in Table 2. Note that in both Figure 3 and Table 2, the iteration 0 represents the results obtained by Min-Max method, at

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STAGE-2, which is the start point of STAGE-3.

It is possible to observe that in the worst case, i.e. in networks that include 10 reference nodes, the final MMPE is of the order of 1 measurement unit. In networks with 20 reference nodes, the final MMPE is of the order of 10−1 . Nonetheless, in the best case, i.e. when there are 30 reference nodes and 200 unknown nodes, the final MMPE achieved the order of 10−7 measurement units.

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It is noteworthy that these results are obtained in an area of 100 × 100 measurement units. In general, this indicates the effectiveness of the proposed method.

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Note that the accuracy of the solution tends to increase with the number of reference nodes present in the network. Likewise, considering the networks with 30 anchors, an increase in connectivity resulting from the increase of the num-

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ber of unknown nodes also contributes positively with the results, improving

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considerably the accuracy of the final solution.

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1

10

100 unknown 150 unknown 200 unknown

MMPE (log)

MMPE (log)

100 unknown 150 unknown 200 unknown

0

10

0

10

20

40

60

Iteration number

80

100

20

MMPE (log)

−2

10

−4

80

(b) MMPE × iteration, for 20 anchors

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10

−6

60

100 unknown 150 unknown 200 unknown

0

10

10

40

Iteration number

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(a) MMPE × iteration, for 10 anchors

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1

10

20

40

60

Iteration number

80

100

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(c) MMPE × iteration, for 30 anchors

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Figure 3: Evolution of the MMPE at each iteration of STAGE-3

The evolution of the localization process is shown in Figures 4 and 5. The former shows a network of 10 anchors and 100 unknown nodes. The later is

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a scenario with 30 anchors and 200 unknown nodes. Observing the results presented in Figures 4(a) and 5(a), i.e. the results computed by Min-Max in

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STAGE-2, it is possible to verify that the unknown nodes estimated position lie within a convex hull created by the anchors (Savvides et al., 2002), even if the actual position of the nodes is outside that convex hull. This drawback of Min-Max method is overcome by the PSO-based refinement, in STAGE-3, extending the localization area beyond the edges of the mentioned convex hull, as presented in Figures 4(b), 4(c), 5(b) and 5(c). This feature represents a

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Table 2: MMPE data of Figures 3(a), 3(b) and 3(c) 10 anchors

20 anchors

30 anchors

unknown nodes

unknown nodes

unknown nodes

100

150

200

100

150

200

100

0(∗)

9.937

11.038

10.767

6.951

6.499

6.627

5.565

4.328

4.446

10

2.445

2.167

2.054

0.448

0.326

0.379

0.307

0.122

0.100

20

1.987

1.493

1.476

0.248

0.179

0.234

0.182

0.062

0.015

30

1.864

1.249

1.354

0.217

0.162

0.217

0.164

0.058

3.660 10−3

40

1.801

1.141

1.312

0.206

0.159

0.213

0.161

0.058

1.022 10−3

50

1.753

1.004

1.271

0.201

0.158

0.212

0.159

0.058

3.001 10−4

60

1.695

0.979

1.262

0.177

0.158

0.212

0.159

0.058

8.884 10−5

70

1.645

0.973

1.258

0.141

0.158

0.212

0.158

0.058

2.646 10−5

80

1.621

0.970

1.257

0.129

0.158

0.212

0.158

0.058

7.914 10−6

90

1.602

0.969

1.256

0.124

0.158

0.212

0.158

0.058

2.356 10−6

1.579

0.968

1.254

0.123

0.158

0.212

0.161

0.058

6.244 10−7

(∗)

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100

150

200

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∆

Iteration 0 represents the results obtained by Min-Max method, at STAGE-2.

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good contribution, once, according to (Langendoen and Reijers, 2003; NaraghiPour and Rojas, 2014), the localization algorithms usually requires that the unknown nodes resides inside the convex hull formed by a set of anchors. It

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is also important to note that the final MMPE of the proposed algorithm is substantially lower than the MMPE obtained only with the Min-Max method. According to the data presented in Table 2, the PSO-based refinement was able

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to reduce the Min-Max’s MMPE at 84%, in the worst case with 10 anchors and 100 unknown nodes, to almost 100%, in the best case with 30 anchors and 200

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unknown nodes.

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Anchor

Unknown (Estimated) 100

90

90

80

80

70

70

60

60

50

50

40

40

30

30

20

20

10

10

0

0

20

40

60

80

0

100

0

20

X

Anchor

40

60

80

100

X

(a) Min-Max result

(b) After 5 iterations Unknown (Estimated)

90 80 70 60 50 40 30

10 0

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20

0

Unknown (Actual)

Anchor

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100

Y

Unknown (Actual)

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Unknown (Actual)

Y

Y

Unknown (Estimated) 100

20

40

60

80

100

X

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(c) After 100 iterations

Figure 4: Evolution of the localization process in a network with 10 anchors and 100

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unknown nodes

An assessment of the processing time of the algorithm is shown in Figure

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6, where it is possible to relate the time increase to the increase in terms of

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network connectivity. Network connectivity is computed according to (15): connectivity =

(I + R − 1)πL2 , A

(15)

wherein R is the number of reference nodes and A is the surface of the coverage

area.

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Anchor

Unknown (Estimated) 100

90

90

80

80

70

70

60

60

50

50

40

40

30

30

20

20

10

10

0

0

20

40

60

80

0

100

0

20

40

X

Anchor

60

80

X

(a) Min-Max result

(b) After 5 iterations Unknown (Estimated)

90 80 70 60 50 40 30

10 0

M

20

0

Unknown (Actual)

Anchor

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100

Y

Unknown (Actual)

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Unknown (Actual)

Y

Y

Unknown (Estimated) 100

20

40

60

80

100

X

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(c) After 100 iterations

Figure 5: Evolution of the localization process in a network with 30 anchors and 200

100

30

50

20

0

0

1 2 3 4 5 6 7 8 9 30|200 20|150 30|150 20|200 10|100 20|100 30|100 10|150 10|200

Network Characteristics (anchors | unknown)

Figure 6: Comparison of processing time vs. connectivity

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10 10

Connectivity (Number of nodes)

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Processing Times, normalized (%)

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unknown nodes

100

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6. Conclusions Based on the obtained results, it can be concluded that the proposed method is effective in view the final levels reached by the mean positioning error. In the

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worst case, it is of the order of 1 measurement unit while in the best case, it is

of the order of 10−7 measurement units, considering a total area of 100 × 100 measurement units.

Some considerations must be made about the dual use of the Min-Max method in the algorithm herein proposed. Firstly, the aim of using the Min-

Max method in STAGE-II is due to its simplicity and low computational cost,

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as presented in this paper. Yet, the results presented here demonstrate that the quality of the initial positions assessed through this method is good enough to be applied as a starting point to an optimization algorithm, i.e. the PSO at STAGE-3. Also, the information of the Si area, naturally obtained during the execution of the Min-Max method, contributes positively with the local-

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ization process. We consider that this information, when added to the fitness function, amplifies the differences between the global minimum and an eventual local minimum, providing, in general, a better convergence of the algorithm.

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In this algorithm, the drawback of Min-Max method is overcome by the PSO-based refinement, that is capable to extend the localization area beyond

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the edges of the convex hull region created by the anchors. This feature represents a relevant characteristic of the proposed algorithm, because localization algorithms usually require the unknown nodes to be located inside a convex

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hull formed by a set of anchors. It is worth mentioning that the PSO-based refinement, with the proposed fitness function, was able to reduce the MMPE

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obtained with Min-Max from 84% to almost 100%. Also, in contrast with some other existing localization methods, the proposed

algorithm gathers some desirable features, such as: it is completely distributed, it requires a relatively low number of anchors, no infrastructure with sensors or extra references in the environment is needed, nor a previous survey of the environment to collect ambient’s data.

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As future work, and in order to proceed with the improvement of the algorithm, it is desirable to assess the impact of the error introduced by distance measurements, due to node misalignment during the Sum-Dist applica-

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tion. Other methods for distance estimation, such as the DV-Hop, can be used during the first stage. Moreover, it is desirable to evaluate the use of other opti-

mization algorithms, to be used during the STAGE-3 of the proposed method, as well as new objective function models, especially with regards to the evaluation of the confidence factor.

We also plan to further investigate the performance of the proposed local-

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ization method in problems involving node mobility, as well as problems where

there is noise in the distance measurements. In this sense we consider that experiments, in real world robots, would foment further discussion about the performance of the algorithm in face of communication failures and distance measurement errors, comparing with other real world existing methods, as well

7. Acknowledgement

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as an assessment of it’s impact on the battery use.

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We are eternally grateful to the reviewers and editor that allowed for a great deal of improvement of the content and contribution of this paper. We also are grateful to the State of Rio de Janeiro (FAPERJ, http://www.faperj.br) and

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Federal Research Council in Brazil (CNPq, http://www.cnpq.br for funding

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Xiong, H., Chen, Z., An, W., and Yang, B. (2015). Robust tdoa localization algorithm for asynchronous wireless sensor networks. International Journal of Distributed Sensor Networks, 2015.

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Biography

Alan Oliveira de S is currently a Professor at the Center

of Electronics, Communications and Information Technology, CIAW, Brazilian

Navy. He graduated in Electronic Engineering at the Federal Center for Technological Education of Rio de Janeiro, in 2006. He received his M.Sc. in Electronic Engineering from the State University of Rio de Janeiro in 2015. His research

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interests include swarm robotic systems, swarm intelligence, evolutionary algo-

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rithms, distributed systems and wireless sensor networks.

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Nadia Nedjah graduated in

1987 in Systems Engineering and Computation and in 1990 obtained an M.Sc. degree also in Systems Engineering and Computation. Both degrees were ob-

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tained form University of Annaba, Algeria. Since 1997, she holds a Ph.D. degree from University of Manchester Institute of Science and Technology, UK. She joined the Department of Electronics Engineering and Telecommunications of the Engineering Faculty of the State University of Rio de Janeiro as an Associate Professor in 2005. She is currently a member of the Intelligent System research area in the Electronics Engineering Post-graduate Course of the State Univer35

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sity of Rio de Janeiro, Brazil. She is the Editor-in-Chief of the International Journals of High Performance System Architecture and of Innovative Computing Applications, both published by Inderscience, UK. She published three authored

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books and more than 40 organized books on computational intelligence related topics. She authored more than 90 journal papers and more than 150 conference papers. She is Associate Editor of more than 10 international journals,

such as the Francis & Taylors International Journal of Electronics, Elseviers Integration, The VLSI Journal and Microprocessors and Microsystems and IETs Computer & Digital Techniques. She organized two major conferences related

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to computational intelligence. (More details can be found at her homepage:

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http://www.eng.uerj.br/ nadia/english.html.)

Luiza de Macedo Mourelle is an

associate professor in the Department of System Engineering and Computation

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at the Faculty of Engineering, State University of Rio de Janeiro, Brazil. She is also a member of the Intelligent System research area in the Electronics

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Engineering Post-graduate Course of the State University of Rio de Janeiro, Brazil. Her research interests include computer architecture, embedded systems design, hardware/software co-design and reconfigurable hardware. She received

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her Ph.D. in Computation from the University of Manchester - Institute of Science and Technology (UMIST), England, her M.Sc. in System Engineering and Computation from the Federal University of Rio de Janeiro (UFRJ), Brazil and her Engineering degree in Electronics also from UFRJ, Brazil. (More details can be found at her homepage: http://www.eng.uerj.br/ ldmm.)

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