An Effective Area-Based Localization Algorithm for Wireless Networks

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI10.1109/TC.2014.2366744, IEEE Transactions on Computers

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An Effective Area-based LocalizationAlgorithm for Wireless Networks

Noureddine Lasla, Mohamed F. Younis Senior Member, IEEE, Abdelraouf Ouadjaout and NadjibBadache

Abstract—Area-based localization algorithms use only the position of some reference nodes, called anchors, to estimate theresidence area of the remaining nodes. Existing algorithms use a triangle, a ring or a circle as the geometric shape thatdefines the node’s residence area. However, existing algorithms suffer from two major problems: (1) in some cases, they mightmake wrong decisions about a node presence inside a given area, or (2) they require high anchor density to achieve a lowlocation estimation error and high ratio of localizable nodes. In this paper, we overcome these shortcomings by introducinga new approach for determining the node’s residence area that is geometrically shaped as a half-symmetric lens. A novelHalf Symmetric Lens based localization algorithm (HSL) is proposed. HSL yields smaller residence areas, and consequently,better location accuracy than contemporary schemes. HSL further employs Voronoi diagram in order to boost the percentageof localizable nodes. The performance of HSL is validated through mathematical analysis, extensive simulations experimentsand prototype implementation. The validation results confirm that HSL achieves better location accuracy and higher ratio oflocalizable nodes compared to competing algorithms.

Index Terms—Localization, Wireless networks, Anchor-based position estimation.

F

1 INTRODUCTION

Localization is one of the technical areas that havereceived increasing attention in recent years due tothe boom in location-based services that mobile userscan benefit from and to the wealth of applications ofad-hoc and sensor networks [1], [2], [3]. For example,applications like asset tracking, search and rescue anddigital battlefield require the knowledge of node’slocation in order to achieve the design goal. On theother hand, wireless sensor networks (WSNs) havegained popularity in recent years due to the growinglist of applications [4]. Most notable among theseapplications are those serving in unattended setupswhere human’s presence is risky and/or impractical.Examples include surveillance of vast borders militaryreconnaissance, and security monitoring of strategicinstallations. In these applications, battery-operatedsensors are randomly deployed in an area to monitortheir surroundings and report their findings to a base-station. In order for the sensor data to be consideredin situation assessment they need to be correlated towhere they are collected from and thus it is importantfor the location of the individual sensors to be known.

Although GPS receivers would provide a globalcoordinate system to which the networks node will

• N. Lasla, A. Ouadjabout and N. Badache are with the Center ofResearch on Scientific & Tech. Info. (CERIST), and the Universityof Sciences and Technology Houari Boumediene (USTHB), Algeria.E-mail: {n.lasla,a.ouadjaout,badache}@cerist.dz

• M. Younis is with Dept. of Computer Science and Elec. Eng., Univ.of Maryland, Baltimore County, Baltimore, MD, USA.E-mail:[email protected]

be related, these receivers require line-of-sight andthus cannot operate in in-door environments and arenegatively affected by cloud, terrain and trees [5]. Inaddition, the small form factor, constrained cost andlimited energy supply of the employed nodes make itimpractical for each node to have an on-board GPS re-ceiver and the network has to rely on relative localiza-tion schemes where a node’s position is topologicallydefined based on proximity to its neighbours and/orreference points. Furthermore, the reliance on WiFiaccess points and cellular telecommunication struc-ture is impractical or infeasible in many applicationsof ad-hoc and sensor networks where the networkoperates using non-standard protocols and withininaccessible deployment area, e.g., in combat field,where telecommunication infrastructure is limited orthe use of its service is prohibited.

Localization methodologies can be generally clas-sified into range-free and range-based. Range-basedschemes [6], [7], [8] work by measuring point-to-pointdistance or angle between pairs of communicatingnodes, or between a node and an anchor (a referencenode with a known position). On the other hand,range-free localization [9], [10], [11], [12], [13] do notrequire point-to-point measurements and generallyuses connectivity to estimate an approximate distanceor uses received signal strength indication (RSSI) in-formation to infer the near-far relationships betweennodes and anchors, which know their exact positionvia GPS or manual placement in the field of interest.Most localization schemes provide the coordinatesof a node relative to the anchors without boundingthe error that these coordinates may be subject to




in the x- and y-directions. Area-based localization isa special class of range-free methodologies in whichthe localization process determines a residence areawithin which the node is located.

The residence area represents a geographical regioncontaining the node. The process of defining thisregion for a node involves drawing a set of geometricshapes using the positions of the reachable anchors.Then, the node tests whether it lies inside or out-side these geometric shapes. The results of such atest determine the boundary of the node’s residencearea, i.e., the intersection of shapes that the nodeis located in and do not overlap with the shapesthat do not contain the node. If the node does notbelong to any of the formed geometric shapes, it iscalled non-localizable. Note here that the presencetest in area-based approaches relay on the proxim-ity or the near-far information among anchors andnodes. Such information is obtained by using theRSSI measurements from radio transceiver withoutthe need of additional hardware. Some recent studies[11], [12], [14], [15] show that RSSI is an effectivemetric that indicates near-far relationship, and workswell for localization problems especially in outdoorenvironment. For instance, the experimental resultsfor MICA and MICAz motes reported in [11], [14],show that RSSI values decrease monotonically withincreasing distance between the sender and receiver.In addition, a very recent study [16] has demonstratedthat it is also possible to indicate near-far relationshipamong nodes based on RSSI, without extra hardwareand even in an environment with obstacles, by usingpower scanning techniques. The final location of thenode can be estimated by computing the centroid ofits residence area. The efficiency of the area-basedalgorithms depends on the size of the calculatedresidence area; the smaller the size of the area is, thebetter the accuracy is likely to be.

Area-based localization is deemed effective in ap-plications in which determining the exact position isnot a must, the cost and complexity of range estima-tion hardware is unwarranted, and/or for which theincreased communication and computation overheadof multi-literation is not bearable. Examples of theseapplications include:

1) Secure detention of criminals [17]: A WSN canbe used to track inmates inside a large prisonby determining the zones within which eachinmate is. Area-based location will suffice indetecting any escape attempts and identifyingthe involved individuals.

2) Livestock Watch [18]: In a cattle grazing field, itis important to track livestock since the grazingfield can be close to highways or railroads. Area-based localization can provide an assessment ofthe safety of the livestock and ensure timelyaction when one is endangered.

Existing area-based localization algorithms employone of three types of primitive geometric shapes todraw the nodes’ residence areas, namely, a triangle[11], a ring [12], and a circle [13]. However, thesealgorithms suffer from two major shortcomings. Thefirst is that in some cases, an incorrect decision ismade about the presence of a node inside a given area,which affects the correctness of the obtained residenceareas. For example, it was shown in [11] that nearly14% of the decisions made about the presence of anode inside or outside a given triangle are wrong.Meanwhile, to draw the residence area, the authorsof [13] assume that the radio propagation model is aperfect circle with known radius. This is an unrealisticassumption [19] and causes major inaccuracy as weillustrate in the next section. The second shortcom-ing is the need for high anchor density in order toachieve a low localization error and high percentageof localizable nodes. For example, the approaches of[11] and [12] require at least three anchors to draw thenode residence area, which means that dense anchordeployment is required to achieve good accuracy andlow ratio of non-localizable nodes.

To address the above issues, in this paper we pro-pose a new area-based localization algorithm, calledHSL (Half Symmetric Lens based localization algo-rithm). HSL is based on the geometric shape of half-symmetric lens which can be simply drawn using thelocation information of only two anchors. Basically,HSL defines the residence area of a node as halfof the overlapped area of two circles. As we con-firm with detailed mathematical analysis, simulationsexperiments and prototype implementation, such aresidence area is smaller than the one defined bycompeting-schemes in the literature and thus HSLwould yield better location accuracy. To resolve theproblem of non-localizable nodes, where a node doeshear only from one anchor, HSL uses the Voronoidiagram to divide the network into a set of cells usingthe communication range of anchors, and locates eachnode within one Voronoi cell by considering the fact itlies outside the range of anchors that it could not hearfrom. This allows a non-localizable node to initiallydefine a Voronoi cell as its residence area and thenexcludes the area covered by unreachable anchors tofurther diminish the uncertainty about its position.

The rest of the paper is organized as follows. Section2 discusses the related work. A detailed description ofHSL is provided in Section 3. Section 4 derives ananalytical estimate of the performance of HSL andcompares it to competing schemes. The simulationresults are reported in Section 5. Section 6 presents aprototype implementation using MicaZ motes. Finally,Section 7 concludes the paper.

2 RELATED WORKSA class of area-based localization schemes relies onreceived signal strength index (RSSI) measurements




(a) (b) (c) (d)

Fig. 1. Basic geometric shape construction in (a) triangle, (b) ring and (c) circle area-based algorithms. (d) Negative information in circularbased algorithms

to serve in indoor environments. This class is referredto as fingerprinting and is often applied using WLANto find the most likelihood area where the node mayreside in [20], [21]. A fingerprint is a vector of RSSImeasurements using beacons from a set of anchors(access points). First, a map of fingerprints in selected(relatively small) regions in the area of interest is tobe prepared through an offline phase where a setof measurements are collected. Interpolation is usedto associate a fingerprint to the other regions withinthe area. To estimate the node residence area, thealgorithms tries to find the closest region whose fin-gerprint closely matches the RSSI values experiencedby the node.

Area-based localization schemes determine a resi-dence area where the node lies within, and providean estimate of the node’s position inside such anarea. Thus, area-based localization can only give anapproximate position with accuracy that depends onthe size of the identified residence area; the smallerthe size of the area is, the better the localizationaccuracy would be. The centroid of, or a random pointwithin the residence area may be chosen as a positionestimate. The study of [22] shows that the averagelocalization error using the random point may belarger than that of using the centroid based positionestimate. Therefore, most of the published area-basedalgorithms use the centroid method.

The simplest centroid-based scheme has been pro-posed in [10]. In this algorithm, each anchor transmitsa beacon to announce its coordinates. Upon hearingthe beacons of neighboring anchors a node calculatesthe centroid based on the coordinates of these anchors.The main advantage of this solution is its simplicityand ease of implementation. However, this algorithmrequires high anchors density in order to ensure thatevery node hears from multiple anchors. In order toimprove the accuracy of the centroid method, moresophisticated algorithms were proposed to shrink thesize of the node residence area. The idea is to definea regular geometric shape that defines the coverageof the transmission of anchors. Based on the locationof the anchors and their coverage regions, the nodechecks whether it is located inside or outside theseregions. The node’s residence area is then defined asthe intersection of the coverage regions of the anchors

that it hears from. According to the geometric shapeof the coverage regions and how they are drawn, wedistinguish three classes of algorithms; triangle, ringand circle based.

2.1 Triangle based AlgorithmsHe et al., [11] have proposed to redraw the noderesidence area as a set of triangles made up of verticesformed by all the possible subset of three neighbour-ing anchors (see Fig. 1(a)). By executing the belongingtest, which checks whether a node is inside each ofthe formed triangles or not, the node residence areacan be redefined as the intersection of the trianglesin which the node resides. A node’s presence insideor outside a given triangle, can be assessed usingperfect Point In Triangle (PIT) test [11]. If there existsa direction such that a point adjacent to point S is fur-ther/closer to points A1, A2, and A3 simultaneously,then S is outside the triangle 4A1A2A3. Otherwise,S is inside 4A1A2A3. However, in a network withstationary nodes, the perfect PIT test is infeasible sinceit requires node movement. To deal with this problem,the authors have proposed an Approximate Point InTriangle (APIT) test that uses a neighboring nodeinformation to emulate the node movement in perfectPIT test. If no neighbour of S is further from/closerto all three anchors A1, A2 and A3 simultaneously,S assumes that it is inside 4A1A2A3. Otherwise, Sassumes it lies outside this triangle. The proximity as-sessment information is derived from RSSI exchangedbetween neighboring nodes. However, by relying onlyon neighboring nodes the number of emulated direc-tion becomes limited by the number of neighbors andAPIT may make an incorrect assessment. It is shownthat when the node density per radio range is 6, thepercentage of such an assessment error in APIT canreach 14%. Therefore, the accuracy of the estimatedposition diminishes when the density of nodes is low.Moreover, the APIT test requires exchanging extraRSSI information messages between each node andall its neighbors, i.e., two rounds of broadcast.

2.2 Ring based AlgorithmsLiu et al., [12] proposed a localization algorithm calledROCRSSI, which redraws the node residence area




as the overlap of multiple rings. Fig. 1(b) shows anillustration. The belonging test checks the presenceof a node S inside a ring formed by the intersectionof two concentric circles centred at the point formedby one neighbor anchor A1. The radius of the innerand the outer circles are the distances between A1

and two other anchors A2 and A3. Based on thecomparison of RSSI values between A1−A2, A1−A3

and A1 − S, the node S can check whether it isinside or outside the formed ring. If RSSI(A1, A2) >RSSI(A1, S) > RSSI(A1, A3) , then S is likely to fallwithin the shadowed ring area. This process is re-peated for each set of three neighboring anchors, andthe final residence area is reduced and formed as theintersection of all the rings that the node falls within.To perform the belonging test, each anchor shouldinform the nodes of the measured RSSI values withits neighbor anchors. Thus, the extra exchanged RSSImessages and the need of anchors with high powerradio transmission, make this scheme unsuitable forresource-constrained networks.

2.3 Circle based Algorithms

In this class of algorithms [13], [23], [24], [25], a nodecalculates its residence area based on the assumptionthat the radio coverage area is modelled as a perfectcircle with a known radius R. So, each node candeduce whether it is within each of the areas coveredby the radio range of its neighbor anchors (see Fig.1(c)). The overlapping region of all the circles definesthe node’s residence area. The obtained residence areamay also be narrowed-down using the coverage areaof farther anchors (anchors within two-hops) which isdiscarded from the node’s residence area since theyare not directly reachable.

However, modelling the radio range as a perfectcircle with known radius is not realistic and doesnot often hold in practice. Furthermore, this approachmight deem a node non-localizable. For instance, letus consider the example shown in Fig. 1(d). In thefigure, the area marked by the dotted lines representsthe anchors’ practical communication region, whilethe circle marks the ideal scenario. A node S can hearfrom anchor A1, and hence its initial residence area isdetermined as a circle C1 centred at A1 with radius R.The node S discovers a two-hops anchor A2, and usesthis negative information to refine its residence areaby discarding the intersection region of the two circlesC1 and C2, where C2 is the circle centred at A2 withradius R. It is clear that S becomes non-localizable asit is outside the refined residence area. Therefore, thenegative information can be misleading and might notbe helpful in refining the nodes’ residence area.

In summary, existing area-based location schemesare prune to increased localization error and requiresrelatively high count of anchors as most of themrequire at least three anchors to perform the local-

(a) (b)

Fig. 2. Overview of the half-symmetric lens construction: (a) node sresides in half plane HP (A1), (b) node s resides in half-symmetriclens HSL(A1, A2)

ization process. The proposed HSL approach opts toovercome these shortcomings as we explain next.

3 HSL-BASED LOCALIZATION

In this section, we present our proposed HSL local-ization algorithm. HSL avoids the shortcomings ofexisting schemes by introducing a new shape thatleads to a smaller residence area. To localize a node,HSL requires only two neighboring anchors and thusincreases the ratio of localizable nodes. To boost thisratio even more, HSL employs a Voronoi diagrambased technique. In the balance of this section, nodesare assumed to be randomly deployed and haveunique IDs. Each anchor is assumed to know itsposition and the position of all other anchors.

3.1 Approach Overview

Let A = {A1, A2, ...Am} be a set of neighboring an-chors of node S in two dimensional Euclidean plane.For any two anchors Ai and Aj , let B(Ai, Aj) bethe perpendicular bisector of segment Ai Aj , whichdivides the plane into two halves HPAi

and HPAj

containing Ai and Aj , respectively. Thus, S is in HPAi

if it is closer to Ai than Aj , or S is in HPAj if it is closerto Aj than Ai (see Fig. 2(a)). Let us now draw twocircles CAi

and CAjcentred at Ai and Aj respectively,

with the same radius that equals to the distance dijbetween Ai and Aj . The intersection of CAi and CAj

creates a geometric shape called symmetric lens orVesica piscis [26]. The bisector B(Ai, Aj) divides thesymmetric lens into two half-symmetric lens, namely,HSL(Ai, Aj) and HSL(Aj , Ai), containing Ai and Aj ,respectively, as depicted in Fig. 2(b). Therefore, (i) ifS is in HP (Ai), (ii) if S is the closer node to Ai thanAj and (iii) if S is the closer node to Aj than Ai,then S must be in HSL(Ai, Aj). The same appliesfor HP (Aj) and Aj . Otherwise, S is outside thesymmetric lens SL(Ai, Aj) defined by the coordinateof Ai and Aj . We note that the circles drawn toconstruct HSL are totally different from those usedin the circular-area based algorithms discussed in theprevious section.

To derive the near-far information between nodes,we use the Received Signal Strength power as Indica-tor (RSSI). Formally, a node S is inside the symmetric




lens of two anchors Ai and Aj , if and only if:

RSSIAiS > RSSIAiAjand RSSIAjS > RSSIAiAj

If indeed node S lies in the symmetric lens forthe two neighboring anchors Ai and Aj , node Shas to find in which sub-area it resides. Basically, ifRSSIAiS > RSSIAjS node S concludes that is inHSL(Ai, Aj) as illustrated in Fig. 2(b). Otherwise, it isin HSL(Aj , Ai). It is worth noting that the probabilityof having RSSIAiS = RSSIAjS is very minute inpractice; nonetheless, if it happens that the node hasto be located on the bisector and the accuracy of theestimated coordinates will be quite high. In addition,if the node is not within the symmetric lens areaSL(Ai, Aj), then it is in one of the following twogeometric area:

1) The node S is inside one of the circles cen-tred at Ai and Aj and not in the symmetriclens SL(Ai, Aj); this area have a crescent-likeshape and we denote it by CR. Such a scenariocan be checked by simply comparing RSSIAiS

and RSSIAjS to RSSIAiAj . If the RSSIAiS >RSSIAiAj it could be concluded that S is in thearea within the circle of radius dij centred atAi minus SL(Ai, Aj) , denoted as CR(Ai, Aj).Similarly, if RSSIAjS > RSSIAjAi

, S would bein CR(Aj , Ai).

2) Otherwise, the node is outside the area definedby the union of the two circles centred at Aiand Aj with the same radius that equals to dij .We refer to this area as excluded region and wedenote it by ER(Ai, Aj) .

3.2 Non-localizable Node ProblemLet us consider the example shown in Fig. 3, where A1

and A2 are neighbors of node S. We can see that thetwo circles and the half-symmetric lens area definedby the two anchors A1 and A2 do not contain node S.In this situation, node S cannot locates itself even withthe presence of two neighboring anchors, and it isconsidered non-localizable. To overcome this problem,we partition the network area into a set of disjointregions in such a way to allow node S to locate itselfwithin one of those regions using the informationabout the anchors which it has heard from. Therefore,a non-localizable node becomes localizable and canuse the half-symmetric lens to refine its residence areawithin the identified regions. The partitioning is basedon Voronoi tessellation [27].

Let A = {A1, A2, ..An} be a set of anchors in thetwo-dimensional bounded network area. The Voronoicell of an anchor Ai with respect to a set of anchorsA, denoted V N(Ai), is the set of points in the planewhich are closer to Ai than any anchor in A \ {Ai}.If the Voronoi cell of each anchor is constructed withrespect to all other anchors in the network, the setof Voronoi cells will be a partition of the node field.

Fig. 3. Illustrating the non-localizable node problem, where the near-far relationship with anchors does not enable node S to determine itsresidence area with acceptable accuracy

From the above definition, we can conclude that eachnode is located in the Voronoi cell of its closest anchor(using only one anchor). Using near-far relationshipinferred from the RSSI information, each node can de-termine its closest anchor, and hence the Voronoi cellwhere it resides in. In this way, each node can locateitself within an initial residence area, i.e., Voronoi cell.After that, the initial residence area will be refined bykeeping only the intersection of all the half symmetriclenses that the node belongs to.

3.3 Algorithm DescriptionAt network setup, the individual anchors use theposition information to determine the Voronoi cells.The position of anchors can be provided throughdeterministic placement. Alternatively, an anchor canbe equipped with GPS receivers and relatively longhaul communication capability so that it can reachother anchors and inform them about its coordinates.The anchor positions are then used to form a Voronoidiagram and determine for each anchor its Voronoicell. This can be done in a centralized manner or usinga distributed algorithm [27], [28], [29].

To perform localization, each anchor starts broad-casting a beacon message containing its coordinatesand its Voronoi cell (the coordinates of polygon ver-tices). Upon receiving the beacon of anchors A, anode S that is within A’s reachable range, adds A toits neighbor anchors list denoted by ALS . Each rowin ALS includes the following information about theneighbor anchor: (1) the anchor’s ID, (2) the anchor’scoordinates, (3) the anchor’s Voronoi cell and (4) theRSSI corresponding to the received beacon messagefrom the anchor.

The Voronoi cell of the nearest anchor, which hasthe strongest RSSI in ALS , represents the initial node’sresidence area. The initial residence area is refined byconsidering the half symmetric lens, defined based onthe coordinates of its neighbor anchors. This processis referred to as the symmetric lens presence test, andinvolves checking if the node S is inside or outsidethe symmetric lens area defined by the coordinates ofevery pair of neighboring anchors. A summary of theHSL presence test is given in Fig. 4.

An example on how the node constructs its resi-dence area is illustrated in Fig. 5. The three anchorsA1, A2 and A3 are neighbors of node S, and accord-ing to the Voronoi diagram, the network is divided




init: Ai and Aj aretwo neighboring

anchors of node S

isr(S,Ai) >r(Ai, Aj)

andr(S,Aj) >r(Ai, Aj)

isr(S,Ai) >

r(S,Aj)

inHSL(Ai, Aj)

inHSL(Aj , Ai)

isr(S,Ai) >r(Ai, Aj)

inCR(Ai, Aj)

isr(S,Aj) >r(Ai, Aj)

inCR(Aj , Ai)

outER(Ai, Aj)

r(Ai, Aj) representsthe RSSI betweennode Ai and Aj

YesNo

Yes

No

Yes

No

Yes

No

Fig. 4. Flowchart that summarize the HSL presence test

(a) (b) (c)

Fig. 5. Residence area construction in HSL: (a) node s is within theV N(A1), (b) node s in half symmetric lens HSL(A1, A2), (c) nodes in HSL(A1, A2) and not in SL(A1, A3)

into three sub-areas V N(A1), V N(A2) and V N(A3).Initially, the node S locates itself within the Voronoicell of its closest anchor V N(A1) (see Fig. 5(a)). Then,the node S considers SL(A1, A2) and RSSIA1S andRSSIA2S to conclude that lies inside HSL(A1, A2),as shown in Fig. 5(b). The residence area is furtherrefined by considering other anchors. Basically, nodeS realizes that it is not in the SL(A3, A1) (see Fig. 5(c)).This is repeated for all anchors in order to shrink theresidence area and increase the localization accuracy.The pseudo code of HSL is provided in Appendix A.

3.4 Estimating CoordinatesIn some applications it may be useful to furtherestimate the coordinates of the node, especially whenthe residence area of the individual nodes is irregularin shape. Recall that the residence area is definedby the intersection of multiple regular shapes, e.g. acircle, and depends on the location and the size ofthese regular shapes. Fig. 5 is an example scenarioof what a residence area may look like for HSL.To do so, a node S, typically estimates its positionas the centroid of the obtained residence area RS .However, this would involve significantly complexgeometry and heavy computation, and may not suitvery resource-constrained nodes. To cope with limita-tion of computation resources, we propose the use ofgrid scan algorithm to provide an approximate area.

First the initial residence area which is defined bythe vertices of the Voronoi cell, is mapped to a gridarray. Basically, the smallest rectangle that contains

0

500

1000

1500

2000

2500

3000

2 3 4 5 6 7

Mea

n re

side

nce

area

(m2 )

Number of neighbour anchors

HSL

PIT

ROCRSSI

DRLS

Fig. 6. Average size of the residence area in HSL, PIT, DRLS andROCRSSI

the Voronoi cell is determined and is divided to smallsquare-shaped cells. The set of grid cells, denoted byG, that the Voronoi cell covers are used to definethe residence area. The initial residence area is thenrefined by excluding cells based on the HSL. Then, thenode’s position is estimated as the centroid of the gridcells that represent the final residence area. Assumethat the grid scanning yields k cells, then the node’scoordinates Se is calculated by:

se = {1

k

k∑i=1

Xi,1

k

k∑i=1

Yi}

where (Xi, Yi) are the coordinates of center of celli of the residence area. Appendix B provides moredetailed description of the grid scan algorithm andcomparison of its performance to analytical geometrybased calculation of the residence area.

4 PERFORMANCE ANALYSIS

To analyze the performance and assess the advan-tage of HSL over competing range-free localizationscheme, we define the location uncertainty metric asthe area that the node’s position could be in, i.e., sizeof the residence area. Obviously, the bigger the resi-dence area is, the higher the level of uncertainty aboutwhere the node is located would be, and consequentlythe lower the localization accuracy would be.

Before analyzing the location uncertainty mathe-matically, we have first performed an extensive sim-ulation and compared the obtained results with thedifferent area-based approaches discussed in Section2. In this simulation, the average sizes of the obtainedresidence area for one node, as a function of thenumber of its neighboring anchors, are plotted inFig. 6. The comparison is made with DRLS, ROCRSSIand the perfect version of APIT, namely PIT, whichyields better performance than APIT. We have runthe simulation 1000 times for each number of neigh-boring anchors, and the results are plotted with 95%confidence interval. The anchors are deployed inside acircle centred at node’s position with radius R = 40musing a uniform random distribution.

From the figure we observe that HSL yields betterperformance as the average size of the obtained resi-dence area is the smallest under the different number




(a) (b) (c)

Fig. 7. (a) Polar coordinates representation for the positions of anchors and the node, (b) Condition for a triangle to contains the origin(node), and (c) Condition for an half symmetric lens to contains the origin (node)

of neighboring anchors. The closest performance toHSL is that of PIT, whereas DRLS and ROCRSSI yieldweaker performance, and this is due mainly to the ba-sic geometric shape used to draw the residence area inthese baseline approaches. To confirm the simulationresults in Fig. 6, we conduct mathematical analysis ofthe size of the residence area in HSL and PIT. We havechosen PIT since it yields better performance thanDRLS and ROCRSSI. In the following three subsec-tions, we first derive the expected location uncertaintyfor PIT in the case where a node has three neighboringanchors. After that, the expected location uncertaintyof HSL, when the node has two neighboring anchors,is given. Finally, in the third subsection we make acomparison between the obtained results in order toshow which algorithm gives the best performance.

4.1 Expected Location Uncertainty for PITIn PIT, each node calculates its residence area asthe intersection of a set of triangles that contain thenode. The size of the intersection area depends onthe number of anchors that made the triangles andtheir position in the plane. In this analysis, we focuson the case of just one triangle when being in therange of three anchors A,B and C, and calculatethe expected size of the node residence area as ameasure of the location uncertainty metric. Assumingthat the position of the three anchors are based on auniform random distribution within a circle of radiusR and centered at the node S, the location uncertaintybecomes the expected area of one random triangleinside a circle that contains the origin (i.e., the positionof node S), as shown in Fig. 7(a).

The area of a triangle made up of three pointsA (x1, y1), B (x2, y2) and C (x3, y3) is given by theabsolute value of the determinant:

(1)T (A,B,C) =1

2

∣∣∣∣∣∣∣∣x1 y1 1

x2 y2 1

x3 y3 1

∣∣∣∣∣∣∣∣

T (A,B,C) =1

2(|(x3y1)− (x2y1) + (x1y2)− (x3y2)

+ (x2y3)− (x1y3)|)

Let (ri, θi), i = 1, 2, 3 be the polar coordinates of pointsA, B and C, respectively. Then,

x1 = r1 cos (θ1), x2 = r2 cos (θ2), x3 = r3 cos (θ3) (2)

y1 = r1 sin (θ1), y2 = r2 sin(θ2), y3 = r3 sin (θ3) (3)

where, 0 ≤ r1, r2, r3 ≤ R and 0 ≤ θ1, θ2, θ3 ≤ 2π.

We look now for the condition that the trianglecontains the origin. Because of the symmetry of thesystem, we can set θ1 = 0 and 0 ≤ θ2 ≤ π. For triangle4(A,B,C) to contain the origin, the following twoconditions must hold:• Anchor C must lie on the opposite side of anchorB of the line connecting anchor A and the origin,

• Anchor C must lie on the opposite side of anchorA of the line connecting anchor B and the origin.

In other words, as shown in Fig. 7(b), anchor C mustlie in the shaded (red) area, where π ≤ θ3 ≤ π + θ2.

Let P be the probability that a triangle containsthe origin. The joint probability distribution function(PDF) of a random point inside a circle (i.e., the jointdistributions of random variable X and random vari-able Y representing the x and y Cartesian coordinateof the random point) is given by:

fX,Y (x, y) =

{1

πR2if x2 + y2 ≤ R2

0 otherwise

In a polar coordinate system r = g1(x, y) =√(x2 + y2) and θ = g2(x, y) = arctan (x/y). Then,

x = h1(r, θ) = r cos (θ) and y = h2(r, θ) = r sin (θ).Thus, the joint PDF of a point using the polarcoordinate system is given by:

fR,Θ(r, θ) = fX,Y ( h1(r, θ), h2(r, θ) ) |J(r, θ)|

where |J(r, θ)| is the Jacobian of the transformationgiven by: ∣∣∣∣∣∂x/∂r ∂x/∂θ

∂y/∂r ∂y/∂θ

∣∣∣∣∣Then,

fR,Θ(r, θ) =

{ r

πR2if 0 ≤ θ ≤ 2π, 0 ≤ r ≤ R

0 otherwise




We can rewrite the non-zero part as:

fΘ(θ) =1

2πand fR(r) =

2r

R2

From this, the probability for the triangle to containthe origin is given by:

P =

∫ π

0

dθ2

π

∫ π+θ2

π

dθ3

π

∫∫∫ R

0

1

R62r22r22r3 dr1dr2dr3

P =8

R62ππ

∫ π

0

dθ2

∫ π

π

+θ2dθ3

∫∫∫ R

0

r1r2r3dr1dr2dr3

P =8

2R6π2

π2

2(R2

2)3 =

1

4

From equations (1), (2) and (3), the area of a triangleusing the polar coordinates is given by :

T =1

2

∣∣∣∣∣∣∣∣r1 cos(θ1) r1 sin (θ1) 1

r2 cos(θ2) r2 sin (θ2) 1


∣∣∣∣∣∣∣∣

The area of a triangle that contains the origin is givenby:

T0 =1

2

∣∣∣∣∣∣∣∣r1 cos(0) r1 sin (0) 1



∣∣∣∣∣∣∣∣

=1

2|r1r2 sin (θ2)− r2r3 cos (θ3) sin (θ2)

− r1r3 sin (θ3) + r2r3 cos (θ2) sin (θ3)|

=1

2|r1r2 sin (θ2)− r1r3 sin (θ3) + r2r3 sin (θ3 − θ2)|

Let G be a random variable that represents the area ofa random triangle, and let C be the event such as thetriangle contains the origin. Then the expected valueE(G | C ) is given by:

E(G | C )

=1

P

∫ π

θ2=0

∫ π+θ2

θ3=π

∫∫∫ R

ri=0

T0 fR(r1) fR(r2) fR(r3)

fΘ(θ2) fΘ(θ3) dr1dr2dr3 dθ2dθ3

=16

R6π2

∫ π

θ2=0

∫ π+θ2

θ3=π

∫∫∫ R

ri=0

T0 r1r2r3dr1dr2dr3dθ2dθ3

=8

R6π2

∫ π

θ2=0

∫ π+θ2

θ3=π

∫∫∫ R

ri=0

|r1r2 sin(θ2)−r1r3 sin(θ3)

+ r2r3 sin(θ3 − θ2)| r1r2r3 dr1dr2dr3 dθ2dθ3

Let us calculate the first part of the integral:∫ π

θ2 =0

∫ π+θ2

θ3 =π

∫∫∫ R

0

r1r2 sin(θ2) r1r2r3 dr1dr2dr3dθ2dθ3

=

∫ π

θ2=0

∫ π+θ2

θ3=π

∫∫∫ R

ri=0

r3 r21r

22sin(θ2)dr1dr2dr3dθ2dθ3

=

∫ π

θ2=0

∫ π+θ2

θ3=π

R2

2

R3

3

R3

3sin(θ2) dθ2dθ3

=R8

18

∫ π

θ2=0

∫ π+θ2

θ3=π

sin(θ2)dθ2dθ3 =R8

18

∫ π

0

θ2sin(θ2)dθ2

=R8

18(−θ2cos(θ2) + sin(θ2) )|π0=

R8

18π

The other two parts of the integral give the samevalue. Thus,

E(G | C ) = |3R8

18π

8

R6π2| = 4

3πR2 = 0.424 R2

It is worth noting that for R = 40 the value matchesthe one obtained through simulation in Fig. 6 (PITunder 3 anchors).

4.2 Expected Location Uncertainty for HSLIn HSL, the residence area is determined based onthe intersection of two circles centered at two anchorsA and B with the same radius that is equal to thedistance between them d. The intersection shouldcontain the node S. Such intersection represents thelocation uncertainty of S. As we mentioned earlier,the two anchors lie inside a circle of radius R andcentered at S and their positions are determined basedon a random uniform distribution. In the followingwe calculate the expected area of that intersectionwhen it contains the node S, i.e., uncertainty areausing just two anchors A and B, and then extend theanalysis by factoring in the third anchor C in order tocompare with PIT. Again we use (ri, θi), i = 1, 2, 3as the polar coordinates and (xi, yi), i = 1, 2, 3as the Cartesian coordinates of anchors A, B and C,respectively.

Consider A at distance d from B, the area ofthe resulting half symmetric lens using A and B,HSL(A,B), is given by: 1

2

(2π3 −

√3

2

)d2 [30]. Thus,

the expected area of the half symmetric lens thatwe seek is proportional to the squared distance d2

between A and B. Let us now derive the conditionfor HSL(A,B) to contain the origin. Given anchor Aat (r1, 0) (see Fig. 7(c)), it is clear that for HSL(A,B) tocontain the origin the following two conditions mustbe fulfilled:

1) Anchor A must be closer to S than to B, meaningthat B must be outside the circle centered at Awith radius r1.

2) Anchor B must be closer to S than to A, whichmeans that B must lie in the side that containsS of the line equidistant from S and A.

Then, the admissible region for B, as shown in Fig.7(c), is the difference between the whole disc C(S,R)centered at S with radius R and the two red (thickline-marked) circular segments Cs1 and Cs2 centeredat A and S, respectively.

Thus, the probability P for the intersection to con-tain S is given by:

P =Area(C(S,R))− [Area(Cs1) +Area(Cs2)]

Area(C(S,R))




The area of minor segment CS defined by an angle θin a circle of radius r is given by:

CS =1

2(θ − sin θ)r2.

For Cs1, the radius is r1, θ = 2π3 and the area would

thus be 12 (

2π3 −

√3

2 )r21 . Meanwhile, for Cs2, the radius

is R and θ = 2 cos−1 r12R , and thus

Area(Cs2) =R2

2(θ − sin θ )

=R2

2

(2 cos−1 r1

2R− sin

(2 cos−1 r1

2R

) )Substituting

P =

∫ R

r1=0

1

πR2

(πR2 −

(1

2

(2π

3−√3

2

)r21

+1

2

(2 cos−1 r1

2R− sin (2 cos−1 r1

2R))R2

) )2r1 dr1

P =2

πR2

∫ R

r1=0

πR2

−

(1

2

(2π

3−√3

2

)r21 +

1

2

(2 cos−1 r1

2R

− sin (2 cos−1 r1

2R)

)R2

)r1dr1

P =3√3 + 2π

6π= 0.609

The area of the half symmetric lens is given by(2π3 −

√3

2

)d2 where d is the Euclidean distance be-

tween the two anchors A and B. The Euclidean dis-tance d is defined as follow:

d2 = (x1 − x2)2 + (y1 − y2)

2

Giving anchor A at (r1, 0)

d2 = (r1 − x2)2+ (y2)

2

Let G be a random variable that represents the areaof HSL(A,B), and let C be the event such that theintersection contains the origin. Then, the expectedvalue E(G | C ) is the integral of the area of HSLover the region satisfying the condition, divided bythe probability that HSL(A,B) contains the origin.Let AHSL(r1, x2, y2) denote the area of HSL(A,B).Then

AHSL(r1, x2, y2) =

(2π

3−√3

2

)d2

=

(2π

3−√3

2

)(r1 − x2)

2+ (y2)

2

Thus,E (G | C )

=1

P

∫admissible region

AHSL (r1, x2, y2) fG (g) dg

=1

P

1

2

(2π

3−√3

2

)∫admissible region

d2fG (g) dg

=1

P

1

2

(2π

3−√3

2

)(∫C(S,R)

d2fG (g) dg

−(∫

CS1

d2fG (g) dg +

∫CS2

d2fG (g) dg

) )

=1

P

1

2

(2π

3−√3

2

)(I1 − (I2 + I3))

(4)

Where fG (g) is the probability density function of Ggiven by:fG (g) = fRfXfY

=

{2rπR2 , x2 + y2 ≤ R2 and 0 ≤ r ≤ R

0, otherwise

From Fig. 7(c) we can derive the bound of the integralas follow:

For C(S,R): 0 ≤ r1 ≤ R , −R ≤ x2 ≤ R and−√R2 − x2 ≤ y2 ≤

√R2 − x2

For CS1(A, r1): 0 ≤ r1 ≤ R , −r1 ≤ x2 ≤ − r12 and−√r21 − x2 ≤ y2 ≤

√r21 − x2

For CS2(S,R): 0 ≤ r1 ≤ R, r12 ≤ x

2≤ R and

−√R2 − x2 ≤ y2 ≤

√R2 − x2

Then,

(5)

I1 =1

πR2

∫ R

r1=0

2r1dr1

∫ R

x2=−Rdx2

−∫ √R2−x2

y2=−√R2−x2

(r1 − x2)2+ (y2)

2dy2

= R2

(6)

I2 =1

πR2

∫ R2

r1=0

2r1dr1

∫ − r12

x2=−r1dx2

−∫ √r21−x2

y2=−√r21−x2

(0− x2)2+ (y2)

2dy2

=1

144

(8− 3

√3

π

)R2

(7)

I3 =1

πR2

∫ R

r1=0

2r1dr1

∫ R

x2=r 12

dx2

−∫ √R2−x2

y2=−√R2−x2

(0− x2)2+ (y2)

2dy2

=7

18− 9√3

16πR2




From equations (4), (5), (6) and (7):

E (G | δ)

=3(

1144

(3√

3π − 8

)+ 11

18 + 9√

316π

)(2π3 −

√3

2

)π

3√3 + 2π

R2

Thus, the expected area of a half symmetric lensthat contains the origin is:

(4π − 3√3)(20π + 21

√3)

36(2π + 3√3)

R2 =1, 769

2R2 = 0.88 R2

We would like to note that when setting R = 40 thevalue matches the one obtained through simulation inFig. 6 (HSL under 2 anchors).

4.3 Comparison between HSL and PITIn the previous two subsections, the expected locationuncertainty of PIT and HSL are derived. Because thederivation was under different number of neighboringanchors (three for PIT and two for HSL), the obtainedresults cannot be directly compared. Therefore, in thissubsection, the analysis is extended to compare theexpected residence areas based on the same numberof anchors.

Let us consider the following assumptions:1) The deployment region have a rectangular shape

(L×M) of surface D2) Three anchors A, B and C are deployed in D3) A node S is deployed in D such as at least two

anchors, say A and B, are within its communi-cation range

4) To eliminate the boundary effect, we assume thatthe node S is deployed in the inner rectangle ofthe deployment region (L−R×M −R)

We summarize the following probabilities that werederived earlier in this section:

1) Pn = πR2/D is the probability that the anchorC is neighbor of S

2) PHSL(A,B) = 0.609 is the probability that node Sis within an HSL(A,B)

3) PPIT = 1/4 is the probability that node S iswithin triangle 4(ABC)

To simplify the presentation we set R = 1.

4.3.1 The Expected Residence Area in PITIn PIT, we distinguish between three possible scenar-ios for residence area for S, according to whether thethird anchor C is a neighbor of S and to whether Sis within the triangle formed by its three neighboringanchors.

1) The anchor C is not a neighbor of S (Fig. 8(a)). Inthis case, PIT cannot decides about the positionof S, and its residence area will thus be thewhole deployment area D.

2) The anchor C is a neighbor of S and S lies withinthe triangle formed by its three neighbor anchors

(a) (b) (c)

Fig. 8. Possible residences areas in PIT, (a) case 1, (b) case 2 and(c) case 3

(Fig. 8(b)). In this case the residence area of S isthe triangle 4(A,B,C) and its expected area is0.42 as shown above.

3) The anchor C is a neighbor of S; however S doesnot lie within the triangle formed by its threeneighbor anchors (Fig. 8(c)). In this case, PIT failsto determine the position of S, and its residencearea will be the whole deployment area D.

The expected node residence area of each of thethree enumerated cases in PIT, is the area for each casemultiplied by its associated probability of having it.• Case 1: D (1− Pn)• Case 2: 0.42 Pn• Case 3: D Pn (1− PPIT )Thus, the expected residence area of PIT in the

previous defined network assumptions is given by:

E(RPIT ) = D (1− Pn) + 0.42 Pn +D Pn (1− PPIT )

4.3.2 Expected Residence Area in HSLIn HSL we distinguish between four possible casesfor the residence areas of S according to whether thethird anchor C is located within the range of S, i.e.,C is less than R units away from S, and whether Sfalls within the different formed HSLs using anchorsA and B.

1) The anchor C is not a neighbor of S and Sis within HSL(B,A) (Fig. 9(a)). For this casethe residence area of S is HSL(B,A) and itsexpected size is 0.88, as shown above in thissection.

2) The anchor C is not within the range of S andS does not lie within HSL(B,A) (Fig. 9(b)). Inthis case the HSL cannot help in determining theposition of S, and the residence area would bethe whole deployment area D.

3) The anchor C is a neighbor of S and S is withinat least one of the three formed HSLs, namely,HSL(B,A), HSL(B,C) or HSL(A,C). This caseis illustrated in Fig. 9(c) and for which the sizeof the residence area is smaller than or equal to0.88.

4) The anchor C is a neighbor of S and S isnot within any of the three formed HSLs,i.e., HSL(B,A), HSL(B,C) and HSL(A,C), asshown in Fig. 9(d). In this case HSL cannotdetermine the position of S, and its residencearea is the whole deployment area D.




(a) (b) (c) (d)

Fig. 9. Possible residences areas in HSL, (a) case 1, (b) case 2, (c)case 3 and (d) case 4

0

10

20

30

40

50

10 20 30 40

Expectedresidence

area

(m2)

Deployement area

PITPIT simHSLHSL sim

Fig. 10. Validating the analytical performance of HSL and PIT to thatobtained through simulation

The expected node residence area of each of the fourenumerated cases in HSL, is the area for each casemultiplied by the probability of having it.• Case 1: 0.88 (1− Pn)• Case 2: D (1− Pn) (1− PHSL(B,A))• Case 3: 0.88 Pn + 0.88 Pn + 0.88 Pn• Case 4: D Pn (1−PHSL(B,A)) (1−PHSL(B,C)) (1−PHSL(C,A))

Then the upper bound of the expected node res-idence area for HSL based on the stated networkassumptions, is given by:

E(RHSL)≤ 0.88 (1−Pn)+D (1−Pn) (1−PHSL(B,A))

+ 0.88 Pn + 0.88 Pn + 0.88 Pn

+D Pn(1−PHSL(B,A)) (1−PHSL(B,C)) (1

− PHSL(C,A))

In Fig. 10, we plot the expected residence area ofboth PIT and HSL as a function of D. We plot alsothe simulation results of both PIT and HSL based onthe previously stated network assumptions. We haverun the simulation 10000 times for each value of D.In each run the position of the node S and the threeanchors are randomly selected and the residence areais calculated based on HSL and PIT. The average overthe 10000 runs is plotted in Fig. 10. The simulationresults confirm the accuracy of our analysis as the gapbetween the simulation and analytical results is verysmall. Fig. 10 also demonstrates the advantage of HSLover PIT where HSL yields localization accuracy thatis up to 280% better than PIT.

5 SIMULATION EXPERIMENTS

In this section, the performance of HSL is comparedthrough simulation to that of APIT, ROCRSSI anda representative circular-area based algorithm called

DRLS [13]. The algorithms are evaluated in terms ofthe following metrics:

1) Ratio of localizable nodes: is defined as thepercentage of nodes successfully located withinthe residence area by the localization algorithm.

2) Estimation error: is defined as the average Eu-clidean distance between the real position of anode and its estimated position.

The performance of HSL is evaluated based on tworadio propagation models. The first is the free spacemodel that considers a perfect reception of signalsover distance. The second model is the log-distancepath loss with shadowing [31], which captures thepath loss of RF signals inside a building or in denselypopulated areas, and formally expressed as:

Preceived[dBm] = Pd0 [dBm]+10β log10 (d/d0)+Xg (8)

where Preceived is the received power in dBm at dis-tance d, Pd0 is the received power at reference distanced0, β is a path loss exponent and Xg is a Gaussianrandom variable with zero mean and σ standarddeviation. According to [31], a particular environmentcan be modelled by appropriate setting of β and σ.For this simulation we have considered two settings,urban area [β = 3, σ = 5] and obstructed factory[β = 4, σ = 6.8].

In the simulation environment, which is developedin Python , n nodes and m anchors are deployed usinga uniform random distribution within an area of sizeD. We have considered two node densities by chang-ing D, high (250m× 250m), and low (400m× 400m).We also study the effect of the ratio of anchors definedas

m

m+ nin order to assess the resource requirements

for the compared approaches. In the simulation re-sults, each plotted point represents the average of100 executions. We plot the 95% confidence intervalon the graphs. The number of nodes and the node’stransmission range are set to 300 and 40m respectively.

We define γ =RαRβ

as a parameter in DRLS, where Rα

is the communication range set in DRLS, and Rβ isthe real communication range. As the performance ofDRLS depends on γ, which captures the possible in-correct estimation of the actual communication range(Rα), we simulate three versions of DRLS, with γ = 1,γ = 1.1 and γ = 1.2 (hereafter called DRLS-1, DRLS-1.1 and DRLS-1.2, respectively).

5.1 Ratio of Localizable NodesFig. 11 shows the ratio of localizable nodes as afunction of the percentage of anchors in the networkwhile assuming a free space model. Figs. 11(a) and11(b) show that HSL and DRLS-1 yield superior per-formance that grows as the ratio of anchors increases.This is intuitive as only one anchor is required inHSL and DRLS for a node to determine in whichVoronoi cell and circle, respectively, it belongs to. For




0

20

40

60

80

100

10 15 20 25 30 35 40

Rat

io o

f Loc

aliz

ed N

odes

(%)

Ratio of Anchors(%)

HSL

DRLS

APIT

ROCRSSI

(a)

0

20

40

60

80

100

10 15 20 25 30 35 40

Rat

io o

f Loc

aliz

ed N

odes

(%)

Ratio of Anchors(%)

HSL

DRLS

APIT

ROCRSSI

(b)

0

20

40

60

80

100

10 15 20 25 30 35 40

Rat

io o

f Loc

aliz

ed N

odes

(%)

Ratio of Anchors(%)

HSL

DRLS γ=1.0

DRLS γ=1.1

DRLS γ=1.2

(c)

Fig. 11. Localizable nodes ratio assuming a free space model for network size (a) (250× 250), (b) (400× 400), and (c) DRLS (250× 250)

0

10

20

30

40

50

60

70

80

90

100

10 15 20 25 30 35 40

Mea

n E

rror

(m)

Ratio of Anchors(%)

HSL

DRLS

APIT

ROCRSSI

(a)

0

20

40

60

80

100

120

140

160

10 15 20 25 30 35 40

Mea

n E

rror

(m)

Ratio of Anchors(%)

HSL

DRLS

APIT

ROCRSSI

(b)

0

2

4

6

8

10

12

14

16

18

10 15 20 25 30 35 40

Mea

n E

rror

(m)

Ratio of Anchors(%)

HSL

DRLS γ=1.0

DRLS γ=1.1

DRLS γ=1.2

(c)

Fig. 12. Average estimation error assuming a free space model for network size (a) (250× 250), (b) (400× 400), and (c) DRLS (250× 250)

10

20

30

40

50

60

70

80

90

100

10 15 20 25 30 35 40

Mea

n E

rror

(m)

Ratio of Anchors(%)

HSL

DRLS

APIT

ROCRSSI

(a)

0

10

20

30

40

50

60

70

80

90

100

10 15 20 25 30 35 40

Mea

n E

rror

(m)

Ratio of Anchors(%)

HSL

DRLS

APIT

ROCRSSI

(b)

8

10

12

14

16

18

20

10 15 20 25 30 35 40

Mea

n E

rror

(m)

Ratio of Anchors(%)

HSL

DRLS γ=1.0

DRLS γ=1.1

DRLS γ=1.2

(c)

Fig. 13. Average estimation error for log-distance path loss model in (a) urban area, (b) obstructed factory and (c) DRLS in obstructed factory

small ratio of anchors, some nodes may not have anyneighboring anchor, and thus become non-localizable.This explains why the ratio of localizable nodes is lessthan 100% for low anchor count. On the other hand,ROCRSSI and APIT require at least two and threeanchors, respectively. In addition, the three anchorsin APIT have to form a triangle within which thesensor lies. The same applies to ROCRSSI, whereasthe nodes must be within the ring formed by the twoanchors. These strong constraints lead to low ratioof localizable nodes. Fig. 11(c) studies the effect ofincreasing γ on the ratio of localizable nodes in DRLS.As seen in the figure, this ratio is not affected byγ. This is due to the fact that by increasing γ, onlythe radius of the circle used to estimate the residencearea grows. Since the number of neighboring anchorsin this case will not be affected, and in DRLS anode needs to hear from at least one anchor to belocalizable, the ratio of localizable nodes stays thesame. The results about the ratio of localizable nodes

in the case of log-distance path loss model are givenin Appendix C, where the performances are degradedand more than 20% of nodes become non-localizable.

5.2 Estimation Error

Fig. 12 shows the estimation error as the ratio ofanchors varies while assuming a free space model.From Figs. 12(a) and 12(b), we observe that as the ratioof anchors increases, the ratio of localizable nodes in-creases, and hence lower estimation error is achieved.Also, because the higher number of localizable nodesin HSL and the used geometric shape to determinethe residence area, HSL outperforms all the other ap-proaches. In Fig. 12(c), we observe that the estimationerror in DRLS grows when γ increases. Higher valuesof γ imply larger residence area, and hence higherestimation errors are obtained. From the previous twofigures, we can note that HSL performs better thanDRLS-1, even if they have the same ratio of localizablenodes. This is due to the basic geometric shape used




by HSL that yields a smaller residence area. Moreover,in real deployment setup, it is not possible to perfectlyestimate the actual communication range and thussetting γ to 1 is not practical. In Figs. 11(c) and 12(c),the superiority of HSL over DRLS is clearly shown.

Fig. 13 shows the average estimation error in urbanarea and obstructed factory based on the log-distancepath loss model. The results clearly show the negativeeffect of noises caused by multipath and shadowingon the performance of area based algorithms. Yet theresults confirm the advantage of HSL over the otheralgorithms even in the presence of noises. Fig. 13(c)shows how noises dramatically affect DRLS in thecase of incorrect estimation of radio communicationrange, and confirm the superiority of HSL. Additionalsimulation results are given in Appendix C.

6 PROTOTYPE-BASED VALIDATION

The efficiency of HSL is further studied through pro-totype implementation using MEMSIC MicaZ motes[32] equipped with CC2420 RF transceiver [33] anda half-wave external monopole antenna. The transmitpower of CC2420 is set to the maximum value 0dBm.The experiment was conducted in an outdoor setupto experience a relatively obstruction-free RF signalpropagation. The deployment area was divided into(4× 8) grid of squares each of side 2.5m. In order toavoid ground reflections that produce energy holes,the study in [34] recommends that the heights of thetransceiver is to be decreased. To do so while get-ting accurate RSSI measurements that increase withthe distance, placing the transceivers at a height of0.08m was shown to be a good choice. Therefore, wetried heights around such a value and observed thevariation of RSSI. Based on multiple trials, the MicaZmotes was finally placed at 0.10m from the ground. Inthe experiment we have deployed 6 MicaZ motes asanchors denoted by A1, A2, A3, A4, A5 and A6 placedat (2.5, 0), (0, 5), (7.5, 7.5), (2.5, 15), (10, 17.5) and(0, 20), respectively. We have also deployed one MicaZmote as a node denoted by S placed at (5, 10). NodeS is to determine its position using the anchors. Theanchors start by beaconing their positions. Node Sand each anchors record the measured RSSI of allreceived beacons.

The recorded RSSIs values have then been usedto calculate the residence area using both HSL andROCRSSI. ROCRSSI is picked as a baseline approachin these experiments since its operation requirementsis close to those of HSL. Other schemes like PITrequires node movement or a high anchor density toperform the belonging test. To compare the perfor-mance of HSL and ROCRSSI under different numberof anchors, we execute them for all the possible subsetof anchors. In other words, we use all combinations of2, 3, 4, 5 and 6 anchors and compare the average sizeof the residence area for both schemes. The obtained

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Fig. 14. Average size of the residence area for HSL and ROCRSSIbased on experimental results

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Fig. 15. The obtained residence area in both (a) HSL and (b)ROCRSSI, when the number of anchors is 4

results for each number of anchors are plotted in Fig.14. From the figure, we can observe that for HSL,the average size of the residence area in the case oftwo neighboring anchors is about 87m2, which is veryclose to the theoretical analysis result (0.88 R2), whereR = 10m and presents here the distance betweenthe node S and the farthest anchors A6. In addition,HSL outperforms ROCRSSI regardless the numberof anchors, and yields better results even with fewanchors.

An examples of the shapes of the obtained residencearea for both HSL and ROCRSSI is shown in Fig. 15. Inthe figure, the gray areas represents the initial regionswhere the node is located in, the red areas representsthe excluded regions and the blue area represents thefinal obtained residence area. The size of the residencearea in HSL is very small compared to that obtainedusing ROCRSSI, which confirm the simulation results.

7 CONCLUSION

In this paper, we have proposed HSL, a new dis-tributed area-based localization algorithm for wirelessnetwork. HSL is designed to achieve high ratio oflocalizable nodes and low error position estimateunder lower anchor density compared to the leadingschemes in the literature such as APIT, ROCRSSI, andDRLS. HSL is based on the geometric shape of half-symmetric lens which can be simply drawn using onlythe location information of two anchors. The presencetest is based on checking whether a node is withinthe symmetric lens of different combinations of twoneighboring anchors. To resolve the problem of non-localizable nodes, HSL uses the Voronoi diagram todivide the network into a set of cells, and locates




each node within one Voronoi cell. This allows a non-localizable node to use the negative information todiminish its location uncertainty.

The mathematical analysis of the size of the resi-dence area has verified the superior performance ofHSL compared to a competing scheme PIT, whereHSL yields localization accuracy that is up to 280%better than PIT. Simulation results have also shownthat HSL outperforms APIT, ROCRSSI, and DRLSin terms of location accuracy and ratio of localiz-able nodes. Finally, a prototype implementation usingMicaZ motes was performed and the results haveconfirmed the previous mathematical analysis andsimulation experiment.

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[30] Area of a symmetric lens (vesica-piscis).[Online]. Available: http://math.ucsd.edu/ wgar-ner/math4c/randprob/areaprob/pdf/twocircles1.pdf

[31] T. Rappaport, Wireless Communications: Principles and Practice.Upper Saddle River, NJ, USA: Prentice Hall PTR, 2001.

[32] Memsic micaz sensor mote specification. [On-line]. Available: http://www.memsic.com/wireless-sensor-networks/MPR2400CB

[33] Cc2420 datasheet. [Online]. Available:http://inst.eecs.berkeley.edu/cs150/Documents/CC2420.pdf

[34] T. Stoyanova, F. Kerasiotis, A. S. Prayati, and G. D. Pa-padopoulos, “Evaluation of impact factors on rss accuracyfor localization and tracking applications in sensor networks,”Telecommunication Systems, no. 3-4, pp. 235–248, 2009.

Noureddine Lasla is a Ph.D. student at the USTHB, Algeria. He isalso a research assistant at the Research Center CERIST, Algeria.He received his M.S. degree in Computing Science from NationalInstitute of Informatics (ESI) in 2008. His current research interest islocalization, deployment and coverage in wireless sensor network.

Mohamed F. Younis is an associate professor in the Departmentof CSEE at the University of Maryland Baltimore County (UMBC).Before joining UMBC, he was at Honeywell International Inc. Whileat Honeywell, he led multiple projects for building integrated fault-tolerant avionics, in which a novel architecture and an operatingsystem were developed. He has five granted and two pendingpatents. He has published more than 200 technical papers in ref-ereed conferences and journals. He is a senior member of the IEEE.

Abdelraouf Ouadjaout is a Ph.D. student at the USTHB, Algeria.He is also a research assistant at the Research Center CERIST,Algeria. He received his M.S. degree in Computing Science fromthe University of Science and Technology Houari Boumediene in2009. His current research interests are security in wireless sensornetworks and software verification of embedded software.

Nadjib Badache Joined USTHB University of Algiers, in 1983, asassistant professor and then professor, where he taught operatingsystems design, distributed systems and networking with researchmainly in distributed algorithms and mobile systems. From 2000 to2008, he was head of LSI laboratory, where he conducted manyprojects on routing protocols, energy efficiency and security in MobileAd-hoc Networks and WSN. Since March, 2008 he is director ofCERIST and professor at USTHB University.




APPENDIX APSEUDO CODE OF THE HSL ALGORITHM

Algorithm 1 Residence area of node S1: let A be the set of neighbor anchors.2: let HSL be the set of the half symmetric-lens areas

where the node S is inside them3: let CR be the set of the crescent areas where the

node S is inside them4: let ER be the set of the excluded regions where

the node S is outside them5: for each Ai, Aj ∈ A where Ai 6= Aj do6: if RSSIAiS > RSSIAiAj

and RSSIAjS >RSSIAiAj

then7: if RSSIAiS > RSSIAjS then8: HSL = HSL ∪ {HSL(Ai, Aj)}9: else if RSSIAjS > RSSIAiS then

10: HSL = HSL ∪ {HSL(Aj , Ai)}11: end if12: else13: if RSSIAiS > RSSIAiAj then14: CR = CR ∪ {CR(Ai, Aj)}15: else16: if RSSIAjS > RSSIAiAj

then17: CR = CR ∪ {CR(Aj , Ai)}18: else19: ER = ER ∪ {{C(Ai)} ∪ {C(Aj)}}20: end if21: end if22: end if23: end for

APPENDIX BGRID SCAN ALGORITHMIn order to provide an approximate residence areafor very resource-constrained nodes, we present inthis section the grid scan algorithm. First the initialresidence area, which is defined by the vertices ofthe Voronoi cell, is mapped to a grid array. Basically,the smallest rectangle that contains the Voronoi cellis determined and is divided to small square-shapedcells. The set of grid cells, denoted by G, that theVoronoi cell covers are used to define the residencearea. When Algorithm 1 defines the basic residencearea for a node with respect to every pair of anchors(half-symmetric-lens, crescent and excluded region),the grid cells is then scanned in order to mark onlythose belonging to the intersection of all the definedareas for the anchor pairs. In other words, the grid-cells are eliminated from the Voronoi cell reflecting theshrinking of the residence area. In order to determinewhether a cell is located inside the residence area inHSL, three distinct algorithms, namely, Algorithm 2,Algorithm 3 and Algorithm 4, are applied for half-symmetric-lens, crescent and excluded region, respec-tively. The final node’s residence area is then defined

as the set of cells kept (valid cells) after executing thegrid scan algorithm. Therefore, the estimated node’sposition is defined as the average over all the validcells.

Algorithm 2 Grid Scan for half symmetric-lens areasfor each HSL(S,Ain, Aout) ∈ HSL do

2: for each Gk ∈ G doif dAoutGk

> dAinAoutthen

4: G = G\{Gk}. {Gk is invalid}else if dAinGk

> dAoutGkthen

6: G = G\{Gk}. {Gk is invalid}end if

8: end forend for

Algorithm 3 Grid Scan for crescent areasfor each CR(Ain, Aout) ∈ CR do

for each Gk ∈ G do3: if dAoutGk

< dAinAoutthen

G = G\{Gk}. {Gk is invalid}else if dAinGk

> dAoutAoutthen

6: G = G\{Gk}. {Gk is invalid}end if

end for9: end for

Algorithm 4 Grid Scan for excluded regionsfor each ER(A1, A2) ∈ ER do

for each Gk ∈ G doif dA1Gk

< dA1A2 then4: G = G\{Gk}. {Gk is invalid}

else if dA2Gk< dA1A2

thenG = G\{Gk}. {Gk is invalid}

end if8: end for

end for

B.1 Grid Scan Algorithm PerformanceIn this section we assess the performance of the gridscan algorithm under different grid sizes. We haveconducted an extensive simulation and calculated theaverage size of the obtained residence areas for onenode and the associate storage overhead when vary-ing the number of anchors. The results are plottedin Figs. 16 and 17, using four different sizes of thesquares that constitute the grid, namely 2, 5, 10, and15 unites. We have run the simulation 100 times foreach number of neighboring anchors, and the resultsare plotted with 95% confidence interval. The anchorsare deployed inside a circle centred at node’s posi-tion with radius R = 40m using a uniform randomdistribution. The obtained results are compared with




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Fig. 16. Average size of the residence area for HSL using grid scanalgorithm for different grid squares sizes

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Fig. 17. Illustrating the the number of saved cells when usingdifferent grid squares sizes

the area sizes calculated using analytical geometry(available through a Python library).

Fig. 16 compares the average size of a residence areafor different grid configurations and using analyticalgeometry (denoted perfect in the figure). The figureshows that the accuracy of the area size estimation ofthe grid scan improves when the cell size diminishes.While this is intuitive, the effect of the cell size is notvery significant for networks with low anchor ratio.

Fig. 17 captures the storage overhead as a functionof the cell size. The y-axis reports the average numberof cells that is used to represent a residence area.Clearly the storage requirement will be proportionalto the number of cells per residence area. The plotindicates that small cell sizes will boost the neededstorage significantly. Considering both Figs 16 and 17,one can conclude that for a network with few anchorsthe cell size will be mostly determined by the storageconstraints that a node has, i.e., using relatively largecells; while a high density of anchors will favour smallcells. Based on the experiments, a cell size of 5 seemsto be a reasonable choice that balances interest in bothaccuracy and storage resources.

APPENDIX CADDITIONAL SIMULATION RESULTS

Fig. 18 shows the ratio of localizable nodes in the caseof log-distance path loss model. In the presence ofnoises, the radio signals quality and the communica-tion range are negatively affected, which reduce the

probability of receiving beacon packets from anchornodes. In the case of obstructed factory (see Fig.18(b)), fewer nodes are localizable compared to thecase of urban area (see Fig. 18(a)). This is because inurban area less noises are introduced compared to theobstructed factory. By comparing this results to thatobtained in the case of free space model (see Fig. 11),the ratio of localizable nodes is decreased and morethan 20% of nodes become non-localizable.

In order to show the distribution of error, Fig.19 plots the cumulative distribution function (CDF)of localization error for the different area-based al-gorithms in free space, urban area and obstructedfactory setups, where the ratio of anchors if fixedto 23%. The plots are consistent with the presentedresults in section 5 and show the negative impact ofnoises on the average localization error of area basedalgorithms. The results also confirm the performanceadvantage of HSL over APIT, ROCRSSI and DRLSeven in the presence of noises, as shown in Figs. 19(b)and 19(c).

In order to show the contribution of applying theVoronoi technique to the performance of HSL, Figs. 20and 21 plot the mean error and the CDF of localizationerror, respectively, when such a technique is appliedto all the other baseline algorithms. In other words,in both figures HSL, APIT, ROCRSSI and DRLS applythe Voronoi technique. The results clearly show thatby applying the Voronoi technique to the other al-gorithms, the localization error decreases and betterperformance is obtained since more nodes becomelocalizable and have an initial residence area (Voronoicell). Yet HSL still sustains its performance advantageover the other algorithms even when they apply theVoronoi technique, due to the basic half symmetriclens shape used by HSL which yields smaller resi-dence area than the other approaches.




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Fig. 18. Localizable nodes ratio assuming a log-distance path loss model in (a) urban area [β = 3, σ = 5], (b) obstructed factory [β = 4, σ =6.8], and (c) DRLS in obstructed factory [β = 4, σ = 6.8]

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Fig. 19. CDF of localization error assuming (a) free space model, (b) log-distance model in urban area [β = 3, σ = 5] and (c) log-distancemodel in obstructed factory [β = 4, σ = 6.8]

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Fig. 20. Mean error when applying Voronoi technique to DRLS, ROCRSSI and APIT assuming (a) free space model, (b) log-distance modelin urban area [β = 3, σ = 5] and (c) log-distance model in obstructed factory [β = 4, σ = 6.8]

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Fig. 21. CDF of localization error when applying Voronoi technique to DRLS, ROCRSSI and APIT in (a) free space, (b) urban area [β =3, σ = 5] and (c) obstructed factory[β = 4, σ = 6.8]