a new range-free localization method using quadratic programming

Computer Communications 34 (2011) 998–1010

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Computer Communications

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A new range-free localization method using quadratic programming

Jaehun Lee, Wooyong Chung, Euntai Kim ⇑School of Electrical and Electronic Engineering, Yonsei University, 134 Sinchon-dong, Seodaemun-gu, Seoul 120-749, Republic of Korea

a r t i c l e i n f o

Article history:Received 3 December 2009Received in revised form 21 October 2010Accepted 22 October 2010Available online 31 October 2010

Keywords:Wireless sensor networksLocalizationProximityQuadratic programming

0140-3664/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.comcom.2010.10.013

⇑ Corresponding author. Tel.: +82 2 2123 2863.E-mail address: [email protected] (E. Kim).

a b s t r a c t

In this paper, we propose a new range-free localization algorithm called optimal proximity distance mapusing quadratic programming (OPDMQP). First, the relationship between geographical distances andproximity among sensor nodes in the given wireless sensor network is mathematically built. Then, thecharacteristics of the given network is represented as a set of constraints on the given network topologyand the localization problem is formulated into a quadratic programming problem. Finally, the proposedmethod is applied to two anisotropic networks the topologies of which are very similar to those of thereal-world applications. Unlike the most of previous localization methods which work well in the isotro-pic networks but not in the anisotropic networks, it is shown that the proposed method exhibits excellentand robust performances not only in the isotropic networks but also in the anisotropic networks.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Recently, wireless sensor networks (WSNs) have been used inmany applications with advances of the micro-electro-mechanicalsystem (MEMS) technology. Interesting applications of WSNs in-clude target tracking [1], habitat monitoring [2], disaster manage-ment [3], and smart home applications [4], etc. In all of these WSNapplications, sensor network is composed of a large number of sen-sors, and the localization of the sensor nodes is one of the key is-sues. The simplest possible localization solution would be toattach a global positioning system (GPS) [5] to all the sensor nodes.However, in many applications, hundreds or thousands of sensornodes might be involved and it is not practical to use a GPS forall the sensor nodes because of cost concerns and some technicalproblems related to line-of-sight (LOS).

Therefore, research has been conducted on localizing WSNs [6–14] and has yielded two paradigms: range-based algorithms andrange-free algorithms. The range-based localization algorithmsuse measurements such as distance or angle between sensornodes. The distance can be measured by a received signal strengthindicator (RSSI), time of arrival (TOA), or time difference of arrival(TDOA) and the angle can be measured by the angle of arrival(AOA) [12]. Thus, range-based localization algorithms are relativelyprecise but require additional hardware and their cost is relativelyhigh [9]. On the contrary, range-free localization algorithms do notuse any measurements such as angle or distance. Instead, theyutilize the connectivity information among sensors, i.e., ‘‘who is

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within the communications range of whom’’ [13]. The range-freeapproaches do not require any additional hardware for measure-ments and thus are cost-effective, but less precise [14].

In this paper, a new range-free localization algorithm calledoptimal proximity distance map using quadratic programming(OPDMQP) is proposed. This work is motivated by [15] in whicha proximity-distance map (PDM) is developed to tackle the aniso-tropic network. The PDM models the relationship between theproximity and geographic distance and shows good localizationperformance regardless of the network topologies. The PDM, how-ever, does not take into consideration the physical constraints thatWSNs have and thus the related information is not fully exploited.For example, if a sensor node is directly connected to an anchor,the sensor should lie within radio range of the anchor. But this con-straint is not exploited in PDM and its solution might locate thesensor out of the radio range of the anchor, which leads to estima-tion error.

The proposed OPDMQP resolves the problem of PDM by embed-ding the constraints of WSNs into the localization problem. First,the relationship between geographical distances and proximityamong sensor nodes in the given wireless sensor network is math-ematically built. Then, the characteristics of the given network isrepresented as a set of constraints on the given network topologyand the localization problem is formulated into a quadratic pro-gramming problem. Later, the proposed method is applied to twoanisotropic networks the topologies of which are very similar tothose of the real-world applications. Unlike the most of previouslocalization methods which work well in the isotropic networksbut not in the anisotropic networks, it is shown that the proposedmethod exhibits excellent and robust performances not only in theisotropic networks but also in the anisotropic networks.

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The rest of this paper is organized as follows: in Section 2, re-lated studies and the preliminary materials for this paper are given.In Section 3, the proposed method is presented. In Section 4, somesimulations are conducted and the results of the proposed methodare compared with those of previous methods. Finally, a conclusionis drawn in Section 5.

2. Background

2.1. Related studies

Several groups have investigated the localization of WSNs, par-ticularly range-free localization. As a pioneering work, Niculescuand Nath developed the DV-hop approach in which the averagehop distance was computed and applied to the sensor nodes forlocalization [6]. DV-hop method works well in case of isotropicnetwork since the sensor and anchor nodes are placed in the entirearea uniformly. However, it results in large errors in case of aniso-tropic networks since the nodes are not uniformly distributed andthe relationship between hop counts and geographic distances isvery weak. Bulusu et al. proposed a centroid localization algorithm[7]. In the method, the anchor nodes broadcast their positions andeach sensor node computes its position as a center of the con-nected anchor nodes. The centroid algorithm is simple and eco-nomic but requires a lot of anchor nodes since all sensor nodesshould be connected to the anchor node for good localization re-sults. Further, He et al. proposed the APIT localization algorithmin [8]. In the method, location estimation is conducted by isolatingthe environment into triangular regions between anchor nodes andnarrowing down the likely area based on each sensor node’s pres-ence inside or outside of those triangles. The limitation of APITlocalization algorithm is that sensor nodes should be connectedto a lot of anchor nodes. Interestingly, Lim and Hou developed aproximity-distance map (PDM) algorithm to tackle the anisotropicnetwork [15]. In this work, the authors considered localization asan embedding problem which maps the geographic distance intothe proximity measurement, and solved this problem by definingthe linear mapping matrix. The localization was formulated asthe regression problem and recast into the least square problem.

Fig. 1. C-shaped anisotropic sensor netw

The solution of the problem is a proximity-distance map (PDM)and it is sent to all sensor nodes. Each sensor node estimates thedistances from all the anchor nodes based on the PDM and finallylocalizes using the multilateration technique in [6].

In the next section, we formulate the physical constraints ofconnectivity into mathematical expressions and propose a newrange-free localization algorithm by combining those constraintswith PDM.

2.2. Problem formulation

Let us consider a sensor network SN = {node1,node2, . . .,nodeM+N}with M anchor nodes and N sensor nodes and denote the positionof each node as

ni ¼ ðxi; yiÞT for i ¼ 1; . . . ;M þ N: ð1Þ

Here, it is assumed that the positions of M anchor nodes{n1,n2, . . .,nM} are known but the positions of the other N sensornodes {nM+1,nM+2, . . .,nM+N} are unknown. In this localization, theonly available measurement is the proximity information denotingthe number of hops between all the nodes. Let us define the geo-graphic distance between two nodes ni and nj as

di;j ¼ kni � njk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxi � xjÞ2 þ ðyi � yjÞ

2q

; ð2Þ

and denote the proximity information between the ith node and thejth node as

pi;j 2 Z ¼ f0;1;2;3; . . .g; ð3Þ

which indicates the number of hop counts between the two nodes.Then the localization problem can be defined as follows:

Estimate nr ð4Þsubject to ni; di;j; and pk;l;

where i, j 2 {1, . . .,M}, k, l 2 {1, . . .,M + N} and r 2 {M + 1, . . .,M + N}.

ork topology used in the simulation.

Fig. 2. X-shaped anisotropic sensor network topology used in the simulation.

1000 J. Lee et al. / Computer Communications 34 (2011) 998–1010

3. Optimal PDM using quadratic programming

In this section we propose optimal PDM using quadratic pro-gramming (OPDMQP).

3.1. Constraints for range-free localization algorithms

In this subsection, we take into consideration the physical con-straints that proximity has in WSNs. We assume that the commu-nication ranges of all the nodes are equal to R and each node cancommunicate with the others within R. The following theoremsare the constraints that we can use in the range-free localizationalgorithms.

Theorem 1. Let si and sj be the neighbor nodes in a sensor network S,that is, the proximity pi,j equals one (pi,j = 1). Then, the geographicaldistance di,j is smaller than the communication range R (di,j < R).

Proof. The proof is clear from the above explanation. h

Theorem 2. Let si and sk be the neighbor nodes of sj but they are notneighbors to each other. Thus, pi,k equals two (pi,k = 2). Then, the geo-graphical distance di,k is larger than the communication range R andsmaller than twice the range (R < di,k < 2R).

Table 1Characteristics of the C-shaped and X-shaped topology.

C-shapedanisotropicnetwork

X-shapedanisotropicnetwork

Avg. connectivitybetween nodes

Mean 11.5531 14.6911Variance 11.5767 21.7005

Avg. distancebetween nodes

Mean 50.9006 48.5236Variance 587.7876 638.9781

Avg. hop countbetween nodes

Mean 4.1037 3.4946Variance 5.1742 3.0493

Proof. It is clear that di,k is larger than R from Theorem 1. Next,assume that di,k is larger than or equal to 2R. Consider two circleswith the radius R, centered at si and sk, respectively. Then, thetwo circles would have no intersections. Then, by Theorem 1, sj

cannot be the neighbor of both of si and sk. It is a contradictionand thus di,k is smaller than 2R. h

In the next subsection, we propose OPDMQP by combining theabove constraints with the PDM.

3.2. Our proposed method

The OPDMQP method borrows the mapping between the prox-imity and geographic distance from [15]. Let pi be the proximityvector between the ith anchor node and the other anchor nodes,

pi ¼ ½pi;1 � � � pi;M�T 2 ZM�1; ð5Þ

where pi,j is the proximity measured from the ith node to the jthnode and pi,i = 0. Then, the overall proximity information can berepresented by an M �M proximity matrix P, which is defined asfollows:

P ¼ ½p1 � � � pM � 2 ZM�M: ð6Þ

Similarly, let di and D be the distance vector and distance ma-trix of anchor nodes, respectively:

di ¼ ½di;1 � � � di;M�T ;D ¼ ½d1 � � � dM �:

ð7Þ

Then, let us think about a linear transformation matrix T calledthe optimal proximity-distance map (PDM) which optimally mapsthe proximity P to the geographical distance D,

Table 2Comparison of the localization error for the C-shaped topology.

DV-hop method [6] PDM method [15] OPDMQP

Error (R) Average 1.536663 0.574742 0.473961Minimum 1.06951 0.36847 0.314505Maximum 2.198505 0.90745 0.672305

Table 3Comparison of the localization error for the X-shaped topology.


Error (R) Average 0.68493 0.454652 0.369022Minimum 0.40158 0.26313 0.25258Maximum 0.826945 0.867525 0.506475

Table 4Comparison of the computation time for two topology.


Time (s) C-shaped 0.0313 0.0781 1.0912X-shaped 0.0313 0.0625 1.0615


T ¼

t1

..

.

tM

266664

377775 ¼

t1;1 � � � t1;M

..

. . .. ..

.

tM;1 � � � tM;M

266664

377775; ð8Þ

where ti,j represents the effect of proximity to the jth anchor nodeon the geographic distance to the ith anchor node. We derive themap by minimizing the following square error for i = 1,2, . . .,M

ei ¼XM

k¼1

ðdi;k � tipkÞ2 ¼ kdT

i � tiPk2: ð9Þ

Now, we fine-tune the map by adding the constraints obtainedin Section 3.1 to the minimization problem (9). Let sj be the prox-imity vector between the jth sensor node nodeM+j (j = 1,2, . . .,N) andthe anchor nodes {node1,node2, . . .,nodeM},

sj ¼ ½pMþj;1 � � � pMþj;M�T 2 ZM�1;

S ¼ ½s1 � � � sN� 2 ZM�N ;ð10Þ

where pM+j,i is the proximity measured from the jth sensor node tothe ith anchor node mentioned above.

Then, the estimated geographical distances between the jthsensor node nodeM+j and the anchor nodes {node1,node2, . . .,nodeM}are given by

lj ¼ ½lj;1 � � � lj;M �T ¼ Tsj; ð11-1Þ

Tsj ¼

t1

..

.

tM

266664

377775½pMþj;1 � � � pMþj;M�

T ¼

t1;1 � � � t1;M

..

. . .. ..

.

tM;1 � � � tM;M

266664

377775

pMþj;1

..

.

pMþj;M

266664

377775

¼t1;1pMþj;1 þ � � � þ t1;MpMþj;M

..

.

tM;1pMþj;1 þ � � � þ tM;MpMþj;M

264

375; ð11-2Þ

lj ¼

t1sj

..

.

tMsj

266664

377775; ð11-3Þ

where lj,i denotes the estimated geographical distance between thejth sensor node nodeM+j (j = 1,2, . . .,N) and the ith anchor nodes nodei

(i = 1,2, . . .,M).Further, let us define the following sets of proximity for the an-

chor node i,

Ii;1 ¼ fj ¼ 1; . . . ;NjpMþj;i ¼ 1g;Ii;2 ¼ fk ¼ 1; . . . ;NjpMþk;i ¼ 2g;

I1 ¼[Mi¼1

Ii;1 and I2 ¼[Mi¼1

Ii;2:

ð12Þ

Then, for r 2 Ii,1, we obtain

tisr ¼ ti;1pMþr;1 þ ti;2pMþr;2 þ � � � þ ti;MpMþr;M < R; ð13Þ

by Theorem 1. Similarly, for s 2 Ii,2, we obtain

R < tiss ¼ ti;1pMþs;1 þ ti;2pMþs;2 þ � � � þ ti;MpMþs;M < 2R; ð14Þ

by Theorem 2. Finally, our goal is to minimize (9) subject to theinequalities (13) and (14). Therefore, we can obtain the OPDM byrecasting the localization problem into the following quadratic pro-gramming (QP) formulation:

minimize ei ¼ kdTi � tiPk2 for i ¼ 1; . . . ;M

subject to Ati < b;

where A ¼ ¼

sTr1

sTr2

..

.

sTrp

sTs1

sTs2

..

.

sTsq

�sTs1

�sTs2

..

.

�sTsq

26666666666666666666666666666664

37777777777777777777777777777775

;

b ¼

b1

b2

..

.

bp

bpþ1

bpþ2

..

.

bpþq

bpþqþ1

bpþqþ2

..

.

bpþ2q

26666666666666666666666666664

37777777777777777777777777775

¼

R

R

..

.

R

2R

2R

..

.

2R

�R�R

..

.

�R

2666666666666666666666666664

3777777777777777777777777775

;

Ii;1 ¼ fr1;r2; . . . ;rpg;Ii;2 ¼ fs1; s2; . . . ; sqg: ð15Þ

Once the QP is solved and the optimal PDM T is obtained, eachsensor node nodeM+j computes the estimates of its geographic dis-tances lj to anchor nodes by Eq. (11) and localizes its positionnj ¼ xj; yj

� �T by multilateration [17]. Several multilateration algo-rithms have been reported for localization in the WSN [6,16,17]and a non-iterative multilateration algorithm reported in [17] isused in this paper. Since lj denotes the distances between nj andthe anchor nodes {node1,node2, . . .,nodeM} given by

Fig. 3. Comparison of the average position error for the C-shaped topology under various anchor populations: (a) 10% anchor nodes, (b) 15% anchor nodes, (c) 20% anchornodes, (d) 25% anchor nodes, (e) 30% anchor nodes and (f) overall summarization.


Fig. 4. Comparison of the average position error for the X-shaped topology under various anchor populations: (a) 10% anchor nodes, (b) 15% anchor nodes, (c) 20% anchornodes, (d) 25% anchor nodes, (e) 30% anchor nodes and (f) overall summarization.



Tsj ¼ lj ¼

lj;1

lj;2

..

.

lj;M

266664

377775 ¼

n1 � nj

�� n2 � nj

�� ...

nM � nj

��

2666664

3777775; ð16Þ

the estimated position nj satisfies

kn1 � njk2 ¼ ðx1 � xjÞ2 þ ðy1 � yjÞ2 ¼ l2j;1;

kn2 � njk2 ¼ ðx2 � xjÞ2 þ ðy2 � yjÞ2 ¼ l2j;2;

..

.

knM � njk2 ¼ ðxM � xjÞ2 þ ðyM � yjÞ2 ¼ l2j;M :

ð17Þ

Subtracting the last equation from the first M � 1 equationsyields

x21 � x2

M � 2ðx1 � xMÞxj þ y21 � y2

M � 2ðy1 � yMÞyj ¼ l2j;1 � l2

j;M ;

x22 � x2

M � 2ðx2 � xMÞxj þ y22 � y2

M � 2ðy2 � yMÞyj ¼ l2j;2 � l2

j;M ;

..

.

x2M�1 � x2

M � 2ðxM�1 � xMÞxj þ y2M�1 � y2

M � 2ðyM�1 � yMÞyj ¼ l2j;M�1 � l2

j;M :

ð18Þ

This can be rewritten into the matrix form

2ðx1�xMÞ 2ðy1�yMÞ2ðx2�xMÞ 2ðy2�yMÞ

..

. ...

2ðxM�1�xMÞ 2ðyM�1�yMÞ

266664

377775

xj

yj

� �¼

x21�x2

Mþy21�y2

Mþ l2j;M� l2

j;1

x22�x2

Mþy22�y2

Mþ l2j;M� l2

j;2

..

.

x2M�1�x2

Mþy2M�1�y2

Mþ l2j;M� l2

j;M�1

26666664

37777775;

ð19Þ

and it can be solved by the standard least square method and thesolution is

nj ¼xj

yj

� �¼ ðFT FÞ�1Fg;

where F ¼

2ðx1 � xMÞ 2ðy1 � yMÞ2ðx2 � xMÞ 2ðy2 � yMÞ

..

. ...

2ðxM�1 � xMÞ 2ðyM�1 � yMÞ

266664

377775

and g ¼

x21 � x2

M þ y21 � y2

M þ l2j;M � l2j;1

x22 � x2

M þ y22 � y2

M þ l2j;M � l2j;2

..

.

x2M�1 � x2

M þ y2M�1 � y2

M þ l2j;M � l2j;M�1

26666664

37777775:

ð20Þ

Table 5Comparison of the computation time under different anchor ratios.

Anchor ratio (%) DV-hop method [6] PDM method [15] OPDMQP

C-shaped10 0.01875 0.05314 0.9437615 0.02031 0.05314 1.0187820 0.01874 0.0547 1.1265825 0.01717 0.05158 1.2209630 0.02031 0.05158 1.40158

X-shaped10 0.02032 0.06094 0.9343715 0.01717 0.05158 1.0093820 0.01404 0.05626 1.104725 0.01874 0.0547 1.2140730 0.01874 0.06094 1.27972

3.3. Protocol

We assume that an anchor node is selected as the sink node.Here, each sensor node finds its position by the followingprocedures:

STEP 1: Each anchor node nodei broadcasts a HELLO message tothe other anchor nodes so that each anchor node nodei knowsits proximity vector pi. In this process, every sensor nodenodeM+j knows its own proximity vector sj.STEP 2: Each anchor node nodei sends an INFO message to thesink node such that its proximity vector pi and location infor-mation are stored in the sink node. The sink node builds theproximity matrix P and distance matrix D at the end of thisstep.STEP 3: Each sensor node nodeM+j which belongs to the setI1 [ I2 sends an INFO message to the sink node such that its

proximity vector sj is stored in the sink node. The sink nodebuilds all constraints, i.e. (13) and (14), at the end of this step.STEP 4: The sink node computes the OPDM matrix T by solvingthe problem (15) with QP.STEP 5: The sink node broadcasts the OPDM matrix T to all thesensor nodes and each sensor node estimates its position by(11) and (20).

The communication costs are M(M + N � 1) HELLO messages instep 1, M � 1 INFO messages in step 2, kN INFO messages in step 3(0 < k 6 1), and N INFO messages in step 5. Note that the HELLOmessages in step 1 and the INFO messages in step 5 are configuredto send under the broadcast transmission mode and the otherINFO messages in step 2 and 3 are configured to send under theunicast transmission mode. The additional communication over-head of the proposed method compared to PDM method is kNINFO messages under unicast transmission mode which dependson the density or average connectivity of the given WSN and isat most N.

4. Experimental results

In this section, some simulations are conducted and the perfor-mance of proposed method is compared with those of the previousmethods: DV-hop [6] and PDM [15]. Four kinds of simulations areperformed.

(a) In the first simulation, twenty-one anchor nodes and sev-enty sensor nodes are distributed in a 100 � 100 squareregion and the localization performances are compared.

(b) In the second simulation, the network size is fixed to onehundred and the anchor node ratio is changed from 10% to30%, observing the variation of the performance.

(c) In the third simulation, the number of anchor nodes is fixedand the number of sensor nodes is changed to observe theimpact on the localization performance.

(d) In the last simulation, the more realistic environments areconsidered. The irregular radio propagation model is usedand the anchor node ratio is increased from 10% to 30%,observing the variation of the performance.

In all simulations, C-shaped and X-shaped anisotropic networksshown in Figs. 1 and 2, respectively, are used. The C-shaped topol-ogy given in Fig. 1 is a typical example of the anisotropic networkand widely used in other papers such as [6,15]. The X-shapedtopology given in Fig. 2 is the more restricted anisotropic networkand has higher average connectivity than C-shaped one. Whenthere are twenty-one anchor nodes and seventy sensor nodes in

Fig. 5. Comparison of the average position error for the C-shaped topology under various sensor populations: (a) 20 anchor nodes and 50 sensor nodes, (b) 20 anchor nodesand 70 sensor nodes, (c) 20 anchor nodes and 90 sensor nodes, (d) 20 anchor nodes and 110 sensor nodes and (e) overall summarization.


each network, the two networks are compared in terms of theaverage connectivity, distance and hop counts between nodes, asgiven in Table 1.

As shown in the table, the X-shaped topology exhibits higherconnectivity and lower hop count and distance between nodesthan the C-shaped topology. Thus, the X-shaped topology is more

Fig. 6. Comparison of the average position error for the X-shaped topology under various sensor populations: (a) 20 anchor nodes and 50 sensor nodes, (b) 20 anchor nodesand 70 sensor nodes, (c) 20 anchor nodes and 90 sensor nodes, (d) 20 anchor nodes and 110 sensor nodes and (e) overall summarization.


densely populated on a small region than the C-shaped one and itis closer to an extreme anisotropic case.

In the first simulation (a), 21 anchor nodes and 70 sensor nodesare distributed, as shown in Fig. 2. The communication range R for

Table 6Comparison of the computation time under different numbers of sensors.

Number of sensors DV-hop method [6] PDM method [15] OPDMQP

C-shaped50 0.01874 0.05002 1.084470 0.01717 0.05002 1.142290 0.02031 0.05782 1.17501110 0.01875 0.05782 1.19375

X-shaped50 0.01717 0.05314 1.067270 0.01876 0.06094 1.0703390 0.02031 0.0547 1.13907110 0.02031 0.0547 1.1547

Fig. 7. Radio propagation model.


all nodes is set to 20 with respect to the 100 � 100 square regionand 10 independent simulation runs are made for each topology.The performance of the proposed method is compared with thoseof the previous methods in Tables 2 and 3 for C-shaped and X-shaped topologies, respectively. From the tables, it can be seen thatthe proposed OPDMQP outperforms the previous methods in termsof the localization error on average as well as in the best and worstcases.

Further, the proposed method is compared with the previousmethod in terms of complexity (computation time), given in Table4. It is observed that the proposed method takes about 30% longerthan PDM.

In the simulation (b), the total size of the network is fixed to 100and the anchor ratio is changed from 10% to 30%. The other param-eters for simulation are same as in the first simulation. Localizationperformances are compared for C-shaped and X-shaped networksin Figs. 3 and 4, respectively, in box plots. The box plots demon-strate the minimum and maximum of localization error, the med-ian, 25% and 75% percentiles of 10 independent simulations.

As in the first simulation, the proposed OPDMQP outperformsthe previous methods not only on average but also in best andworst cases. The proposed one also demonstrates more consistentperformance than the others, as indicated by the box plots. In thesimulation, it is interesting to see that the lower anchor ratiosometimes exhibits the better localization performance than thehigher ratio. However, the improvement of the proposed OPDMQPover the previous methods is maintained regardless of the anchorratios. In particular, when the anchor ratio is 10%, the OPDMQPshows the similar performance as the PDM. But as the anchor ratioincreases, the difference between PDM and OPDMQP also increasesand the reason for that might be that the number of inequality con-straints in the QP formulation (15) increases with the number ofanchor nodes, narrowing down the solution space.

Further, the proposed method is compared with the previousmethod in terms of complexity (computation time). The result isgiven in Table 5. It can be seen that the PDM takes the consistenttime regardless of the change of anchor ratio while the computa-tion time of the OPDMQP increases with the anchor ratio. The rea-son for the increase is that the number of inequality constraints inthe QP formulation (15) increases with the number of anchornodes.

In the simulation (c), the number of anchor nodes is fixed andthe number of sensor nodes is changed to observe the impact ofthe proposed method. The number of anchor nodes is fixed to 20and the number of sensor nodes is varied from 50 to 110. The otherparameters for simulation are same as in the previous two simula-tions. Localization performances are compared in Figs. 5 and 6 forC-shaped and X-shaped networks, respectively, in box plots.

As in the above simulations, the proposed OPDMQP outper-forms the previous methods on average and demonstrates moreconsistent performance than the others, as indicated by the boxplots. It can be seen that the localization error of the proposedOPDMQP decreases as the number of sensor nodes increases. Fur-ther, as the number of sensor nodes increases, the difference be-tween PDM and OPDMQP also increases. The reason might bethat the more sensor nodes exist in the given WSN, the more thereare sensor nodes connected to anchor nodes within two hops.Therefore, the number of inequality constraints in the QP formula-tion (15) increases as the number of sensor nodes increases in caseof the fixed number of anchor nodes.

Also, the proposed method is compared with the previousmethod in terms of complexity (computation time) and the resultis summarized in Table 6. As in the simulation (b), the PDM takesalmost the same time while the computation time of the OPDMQPincreases with the number of sensor nodes. Compared with the re-sult in Table 5, however, the increase of the computation time inTable 6 is less than that in Table 5 and it can be observed thatthe number of anchor nodes has larger effect on computation timethan that of sensor nodes.

Finally, the fourth simulation (d) is concerned about more real-istic environment. An irregular radio propagation model takenfrom [18] is employed to model the fluctuation of the communica-tion range R due to the multipath channel under the real environ-ment. The irregularity is represented by DOI (degree of irregularity)and the communication range is represented by

ð1� DOIÞR <¼ range <¼ R ð21Þ

Fig. 7 shows the example of the radio propagation model.In this simulation, the lower bound in (14) is changed into

ð1� DOIÞR < tiss ¼ ti;1pMþs;1 þ ti;2pMþs;2 þ � � � þ ti;MpMþs;M < 2R;

ð22Þ

to reflect the irregular model. The total network size is fixed to 100and the anchor ratio is varied from 10% to 30%. The DOI value is asassumed to be 0.2. The other parameters for this simulation aresame as in the previous simulations. The localization performancesfor C-shaped and X-shaped networks are compared in box plots ofFigs. 8 and 9, respectively. The box plots demonstrate the statisticsof the 10 independent runs.

As in the previous simulations, the proposed OPDMQP outper-forms the previous methods on average and demonstrates moreconsistent performance than the others, as indicated by the boxplots. It is interesting to see that when anchor ratio is small, theproposed OPDMQP shows similar performances as PDM. However,the proposed OPDMQP exhibits better and more robust localizationperformances than the previous methods in all the otherconditions.

Fig. 8. Comparison of the average position error for the C-shaped topology under noisy radio propagation: (a) 10% anchor nodes, (b) 15% anchor nodes, (c) 20% anchor nodes,(d) 25% anchor nodes, (e) 30% anchor nodes and (f) overall summarization.


Fig. 9. Comparison of the average position error for the X-shaped topology under noisy radio propagation: (a) 10% anchor nodes, (b) 15% anchor nodes, (c) 20% anchor nodes,(d) 25% anchor nodes, (e) 30% anchor nodes and (f) overall summarization.


Fig. 10. Comparison of the average position error under various noisy radio propagation models: (a) C-shaped topology and (b) X-shaped topology.


To observe the influence of irregular propagation model morespecifically, we compared the localization error with differentDOI values. Fig. 10 shows the results of localization with 20 anchornodes, 80 sensor nodes, and various DOI values.

From the Fig. 10, it can be seen that the proposed OPDMQP notonly outperforms the previous methods in ideal situation (DOI = 0),but also shows robust and reliable performances under more real-istic situation. The more the irregularity of propagation increases,the less accurate localization result is. Nevertheless, the proposedOPDMQP results in reliable localization estimates.

5. Conclusions

In this paper, we have proposed a new range-free approachcalled OPDMQP for localization in wireless sensor networks. Unlikemost previous localization methods which focus on the isotropicnetwork, our method is suitable for an anisotropic network. Inthe proposed method, proximity information is transformed intogeographic distances which are used to estimate the positions ofunknown sensor nodes. We modeled the relationship betweenthe geographical distances and the proximity among sensor nodesin the given anisotropic network and formulated the localizationproblem into a quadratic programming problem by imposing twoconstraints concerning the given network topology. The proposedmethod was then applied to two types of anisotropic networkswhich are very like many real-world applications. The simulationresults demonstrated superior performance of the proposed meth-od compared with previous methods.

Acknowledgment

This research was supported by Basic Science Research Programthrough the National Research Foundation of Korea (NRF) fundedby the Ministry of Education, Science and Technology (No. 2010-0012631).

References

[1] S. Lee, H. Cha, R. Ha, Energy-aware location error handling for object trackingapplications in wireless sensor networks, Computer Communications 30 (7)(2007) 1443–1450.

[2] A. Mainwaring, J. Polastre, R. Szewczyk, D. Culler, Wireless sensor networks forhabitat monitoring, in: Proceedings of the ACM Workshop on Sensor Networksand Applications, 2002, pp. 88–97.

[3] N. Pogkas, G. Karastergios, C. Antonopoulos, S. Koubias, G. Papadopoulos,An ad-hoc sensor network for disaster relief operations, in: IEEEConference on Emerging Technologies and Factory Automation, vol. 2,2005, pp. 131–139.

[4] J. Lee, W. Chung, E. Kim, Structure learning of Bayesian networks using dualgenetic algorithm, IEICE Transactions on Information and System E91-D (1)(2007) 32–43.

[5] B. Hofmann-Wellenhof, H. Lichtenegger, J. Collins, Global Positioning System:Theory and Practice, Springer-Verlag, 1993.

[6] D. Niculescu, B. Nath, DV based positioning in ad hoc networks, Journal ofTelecommunication Systems 22 (2003) 267–280.

[7] N. Bulusu, J. Heidemann, D. Estrin, Adaptive beacon placement, in: Proceedingsof the IEEE International Conference on Distributed Computing Systems, 2001,pp. 489–498.

[8] T. He, C. Huang, B.M. Blum, J.A. Stankovic, T. Abdelzaher, Range-freelocalization schemes for large scale sensor networks, in: Proceedings of theInternational Conference on Mobile Computing and Networking, 2003, pp. 81–95.

[9] A. Savvides, C. Han, M.B. Strivastava, Dynamic fine grained localization in ad-hoc networks of sensors, in: Proceedings of the International Conference onMobile Computing and Networking, 2001, pp. 166–179.

[10] S. Li, D. Zhang, A novel manifold learning algorithm for localization estimationin wireless sensor networks, IEICE Transactions on Communications E90-B(12) (2007) 3496–3500.

[11] K. Yedavalli, B. Krishnamachari, Sequence-based localization in wirelesssensor networks, IEEE Transactions on Mobile Computing 7 (1) (2008) 81–94.

[12] G. Mao, B. Fidan, B.D.O. Anderson, Wireless sensor network localizationtechniques, Computer Networks 51 (10) (2007) 2529–2553.

[13] Y. Shang, W. Ruml, Y. Zhang, M. Fromherz, Localization from connectivity insensor networks, IEEE Transactions on Parallel and Distributed Systems 15(11) (2004) 961–974.

[14] S. Yun, J. Lee, W. Chung, E. Kim, S. Kim, A soft computing approach tolocalization in wireless sensor networks, Expert Systems with Applications 36(4) (2009) 7552–7561.

[15] H. Lim, J. Hou, Localization for anisotropic sensor networks, in: Proceedings ofthe IEEE INFOCOM ’05, vol. 1, 2005, pp. 138–149.

[16] R. Nagpal. H. Shrobe, J. Bachrach, Organizing a global coordinate system fromlocal information on an ad hoc sensor network, in: Proceedings of theInformation Processing in Sensor Networks, 2003, pp. 333–348.

[17] K. Langendoen, N. Reijers, Distributed localization in wireless sensornetworks: a quantitative comparison, Computer Networks 43 (4) (2003)499–518.

[18] J.P. Sheu, P.C. Chen, C.S. Hsu, A distributed localization scheme for wirelesssensor networks with improved grid-scan and vector-based refinement, IEEETransactions on Mobile Computing 7 (9) (2008) 1110–1123.

a new range-free localization method using quadratic programming

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