research article an algorithmic approach to wireless...

Research ArticleAn Algorithmic Approach to Wireless Sensor NetworksLocalization Using Rigid Graphs

Shamantha Rai B and Shirshu Varma

IIIT-Allahabad Allahabad 211012 India

Correspondence should be addressed to Shamantha Rai B raishamanthgmailcom

Received 12 January 2016 Revised 26 April 2016 Accepted 11 May 2016

Academic Editor Jian-Nong Cao

Copyright copy 2016 S Rai B and S Varma This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In this work estimating the position coordinates of Wireless Sensor Network nodes using the concept of rigid graphs is carried outin detail The range based localization approaches use the distance information measured by the RSSI which is prone to noise dueto effects of path loss shadowing and so forth In this work both the distance and the bearing information are used for localizationusing the trilateration technique Rigid graph theory is employed to analyze the localizability that is whether the nodes of theWSNare uniquely localized The WSN graph is divided into rigid patches by varying appropriately the communication power range ofthe WSN nodes and then localizing the patches by trilateration The main advantage of localizing the network using rigid graphapproach is that it overcomes the effect of noisy perturbed distance Our approach gives a better performance compared to robustquads in terms of percentage of localizable nodes and computational complexity

1 Introduction

The manual configuration of WSN nodes is not alwaysfeasible because it might be deployed in human unattendedgeographic regions [1] while even GPS cannot be installed ineveryWSN node because it is costly hardware and it does notsupport indoor localization [2] These limitations lead to thedevelopment of localization algorithms where certain nodesknown as anchor nodes know their positions in prior timeand the remaining nodes estimate positions based on theseanchor nodes These localization techniques are termed ascooperative localization or in-network localization or self-localization [3] Usually geographical localization methodsuse various information from the deployed WSN such asranging information like intersensor distances intersensorangle (bearings) and neighbor informationThemain reasonwhy ranging based information is used for localization is dueto the fact that it does not need extra hardware which makesit cost-effective

The WSN localization problem is more or less simi-lar to the Euclidean graph realization problem where theedge length of the corresponding two vertices and distancecalculated from the coordinates of the two vertices must

be equal Thus if the graph 119866 of the WSN is uniquelyrealizable then the WSN can be localizable to the globalhomogeneous coordinate transformations of rotation reflec-tion and translation [4] Rigid graphs have certain propertiesof connectivity which makes them more suitable in usingthem for WSN localization Rigid graphs are more efficientthan the prior graph theory related distance measurementlocalization techniques because their distances remain intactwith homogeneous coordinate transformations In somecases the whole WSN cannot be localized due to con-straints in hardware and deployment issues Therefore densedeployment is needed to have a fully localizable network[5] The process of localization is a costly affair becauseit consumes a lot of resources like energy computationaltime and so forth Therefore in the recent years researchis carried on analyzing whether the given WSN graph isable to be localized based on certain graph theoreticalproperties and formal analysis This process is called thelocalizability analysis From the WSN graph determined isit possible to localize the network This query is called thenetwork localizability The localization algorithms developedusing rigid graphs can be either centralized or distributedCentralized algorithms developed in [6] provide accurate

Hindawi Publishing CorporationJournal of SensorsVolume 2016 Article ID 3986321 11 pageshttpdxdoiorg10115520163986321

2 Journal of Sensors

location estimates but are having issues related to scalabilitylarge amount of computational complexity and low reliabilityas compared to distributed algorithms like [7ndash9] This workis an extension of our previous work [10] In this work wedeal with both distance and bearing information for accurateand efficient localization we also deal with error analysissystematically and also compared our results with the robustquads technique To localize networks in human unattendedareas only distributed algorithms can be employedThereforein this work we have developed a distributed localizationtechnique which involves the following aspects

(1) The main problem in keeping the communicationpower range to a minimal value is that sufficient rigidgraphs cannot be generated thereby the algorithmwill not be able to localize all the nodes uniquely Itis possible to vary the power level ofWSN nodes as in[11] so that the WSN can be made into a localizablenetwork from a nonlocalizable one We suitably varythe power level of the WSN nodes such that itscommunication radius is increased This increase incommunication radius will obviously increase theaverage node degree of the WSN and thereby we canexpect a bigger number of rigid trilateration patchesneeded for iterative trilateration localization

(2) Placing of anchor nodes is not feasible in someoutdoor localization scenarios hence an anchor-freelocalization technique is designed in this work

(3) In order to achieve higher location estimation accu-racy in this approach both distance and angle infor-mation between the neighboring WSN nodes areused

(4) The patches obtained are subjected to global coordi-nate transformations of translation and rotation andtrilateration is used to localize these patches

2 Graph Theoretical Framework andLocalization Using Rigid Graphs

The rigid graphs as the name suggests fall into the categoryof graph theory hence denoted using the graph theorynotations In our work [12] we discussed how rigid graphtheory can be applied to localization and topology controlin WSN In this section the graph theoretical frameworkdepicting WSN nodes in the form of graph is shown

A 119889-dimension point formation at 119901 = column1199011 1199012

119901119899 usually denoted as 119865(119901) which is in the Euclidean

coordinate system set of 119899 number of points 1199011 1199012 119901

119899

in 119877119889 is considered in [5 13 14] In terms ofWSN the points

correspond to the positions of WSN nodes which forms thenetwork The communication link is the intersensor nodedistance In this coordinate of points communication linksbetween points 119901

119894and 119901

119895are deduced where 119894 and 119895 are

integers of set 119894 = (1 2 119899) Length of the communicationlink is the Euclidean distance between the points 119901

119894and 119901

119895

Some of the standard definitions are given below

Definition 1 (sensor graph) A WSN graph 119873(119878119863) where 119878

represents the WSN nodes and119863 the distance betweenWSNnodes can be represented by a graph 119866 = (119881 119864) with avertex set 119881 and an edge set 119864 where each vertex 119894 isin 119881 isrelated with a sensor node 119878

119894in the network graph and each

edge (119894 119895) isin 119864 relates to a sensor pair 119878119894 119878119895for which the

intersensor distances 119889119894119895are known One calls 119866 = (119881 119864) the

underlying graph of the sensor network119873(119878119863)

Definition 2 (framework and realization) A 119889-dimensional(119889 isin 2 3) representation of graph is mapping of a graph 119866 =

(119881 119864) to the point formations 119901 119881 rarr 119877119889 Given graph 119866 =

(119881 119864) and a119889-dimensional representation of it the pair (119866 119901)

is termed as a 119889-dimensional framework A distance set119863 for119866 is set of distances 119889

119894119895gt 0 defined for all edges (119894 119895) isin 119864

Given the distance set 119863 for the graph 119866 a 119889-dimensional(119889 isin 2 3) representation 119901 of 119866 is a realization if it results in119901(119904119894) minus119901(119904

119895) = 119889

119894119895for all pairs of (119894 119895) isin 119881 where (119894 119895) isin 119881

Definition 3 (equivalent and congruent frameworks) Twoframeworks 119866(119881 119901) and 119866(119881 119902) are equivalent if 119901(119904

119894) minus

119901(119904119895) = 119902(119904

119894) minus 119902(119904

119895) holds for every pair 119894 119895 isin 119881

connected by an edge Two frameworks 119866(119881 119901) and 119866(119881 119902)

are congruent if 119901(119904119894) minus 119901(119904

119895) = 119902(119904

119894) minus 119902(119904

119895) holds for

every pair 119894 119895 isin 119881 nomatter whether there is an edge betweenthem

Definition 4 (flexible graph and rigid graph) A framework119866(119881 119901) is called generic if the set containing the coordi-nates of all its points is algebraically independent over therationales A framework 119866(119881 119901) is called rigid if there existsa sufficiently small positive constant 120598 such that if 119866(119881 119901)

is equivalent to 119866(119881 119902) and 119901(119904119894) minus 119901(119904

119895) lt 120598 for all

119894 isin 119881 then 119866(119881 119902) is congruent with 119866(119881 119901) A graph119866(119881 119864) is called rigid if there is an associated framework119866(119881 119901) that is generic and rigid A framework is calledflexible if one has a continuous deformation starting fromthe known configuration to another such that edge lengthsare preserved as shown in Figures 1(a) and 1(b) If no suchdeformation exists then it is called rigid A framework119866(119881 119901) is globally rigid if every framework equivalent to119866(119881 119901) is also congruent with 119866(119881 119901) A graph 119866(119881 119864)

is called globally rigid if there is an associated framework119866(119881 119901) that is generic and globally rigid

Definition 5 (minimal rigid graph) A rigid framework isminimally rigid if it becomes flexible after an edge is removedas shown in Figure 1(c)

Definition 6 (redundantly rigid framework) A rigid frame-work is redundantly rigid if it remains rigid upon the removalof any edge as shown in Figure 1(d)

Definition 7 (localizability) As already defined above for aWSN graph 119866 = (119881 119864) every node is realized as a pointformation If there is a unique location 119901 119881 rarr 119877

119889 for everynode belonging to the node set119881 such that 119889

119894119895= 119901(119894)minus119901(119895)

for all the edges then one says that such a graph is localizable

Journal of Sensors 3

1 2

34(a)

1

34

(b)

12

34

(c)

1 2

34(d)

Figure 1 In (a) a flexible framework is depicted In (b) a flexible framework whose edge length is varied is shown In (c) a rigid frameworkis shown In (d) a redundantly rigid framework is shown

Thus localizability deals with unique realization of a WSNgraph

The localization problem is summarized by Eren et al [3]as follows

Theorem 8 Let 119873(119878119863) be a sensor network graph in 119889

dimension Let (119866 119901) be the realization framework for theunderlying graph 119873(119878119863) Then the sensor network localiza-tion problem is solvable only if (119866 119901) is generically globallyrigid (GGR)

In 1198772 the rigidity test of a graph can be done combina-

torially by using the combinatorial necessary and sufficientcondition given by [15]

Theorem 9 Let 119866 = (119881 119864) be a graph in 1198772 where |119881| gt 1

then 119866 is generically rigid if and only if there exists a subset1198641015840sube 119864 such that |1198641015840| = 2|119881|minus3 and for subset11986410158401015840 sube 119864

1015840 |11986410158401015840| le

2|119881(11986410158401015840) minus 3|

Based on 119896-connected graph property and redundantlyrigid graph the following necessary and sufficient conditionwas given by [16]

Theorem 10 A graph 119866 = (119881 119864) in 1198772 with |119881| ge 4 is

generically globally rigid if and only if it is 3-connected andredundantly rigid

21 Trilaterations

(i) Generically globally rigid (GGR) graphs are labeledby using the notions of trilateration graphs in 119877

2(ii) Consider a graph119866(119881 119864) in 119877

2 applying trilaterationon the GGR graph 119866 is nothing but addition of newvertex V to119866Then there should be at least three edgesjoining V to the vertices in 119881

Definition 11 (trilateration graphs) A trilateration graph119866(119881 119864) in 119877

2 is set of ordered pairs of vertices (V1 V2 V

119899)

such that the edges (V1 V2) (V1 V3) (V2 V3) are all present in

edge set 119864 and each vertex V119894for 119894 varying from 119894 = 4 5 119899

remains connected to three of the vertices V1 V2 V

(119894minus1)

The trilateration graph can be obtained by using consecu-tive 119899 minus 3-trilateration procedure for the graph of verticesV1 V2 V

3and the edges between them and adding 3

vertices to V119894at 119894 = 1 2 119899 minus 3

(i) The graphs and subgraphs which have this trilatera-tion ordering will reduce the computational complex-ity of localization by a greater margin (polynomialwith sensor nodes) [17]

(ii) GGR graphs of trilateration and quadrilateration canbe obtained by addition of extra edges [18]

(iii) Adding extra edges in sensor network localizationmeans increase of the communication radius byvarying the antenna transmission power

The following theorems are given by [18] to acquire globallyrigid trilateration graphs

Theorem 12 Consider a 2-connected graph in 119866(119881 119864) in 2-119889Then 119866

2 obtained by increasing the communication radius isgenerically globally rigid

Theorem 12 can be extended and be generalized for acycle This is given as follows consider a 2-connected cycle119862 in 2-119889 then 119862

2 obtained by increasing the communicationradius is generically globally rigid

3 Related Work

An extensive theoretical analysis of WSN localization usingrigid graphs which analyzes the conditions when the WSNgraph and the associated WSN nodes become localizable isdealt with in literature [4 17 19] Here WSN localizationis modeled by using the concept of grounded graphs andthe computational time required to localize the network isderived Graphical properties of WSN graphs for localizationusing distance information are studied extensively in [18]where the authors have proved how required localizablegraph can be obtained from a mere connected graph byadjusting the communication power radius of the WSNnodes These adjustments in the network graph will makelocalization possible and that too in linear time Graphicalproperties of unique localizable networks using both distanceand bearing information are studied in [20] Localizabilityanalysis that is whether the given network is localizableor not was studied extensively in [2 21 22] A distributedlinear time algorithm to localize the WSN nodes withoutbeing affected by distance noise was developed in [7] In thisapproach in order to prevent incorrect location coordinatesdue to the flip ambiguity issue concept of robust quads wasused The network nodes are divided into localizable and


WSN deployment Localizability analysis

Adaptively increase thecommunication power radius

of WSN nodes

Divide the network into rigidpatches

Assign the local coordinatesystem to the patches

Do the global translation androtation of the patches

Iterative trilateration on rigidpatches

Refined WSNdeployment Location estimate

Figure 2 Process of localizability and localization

nonlocalizable ones and then a framework for localizationof the partially localizable networks was designed in [9]Sweeps algorithmusing bilateration techniquewas developedfor localizing the sparse networks efficiently in [8] Outliermeasurements of the distance instance used for localizationare explored by the edge verifiability procedure making useof redundant rigid graphs in [22 23]

4 Localizability Analysis and Localization

The generic process of WSN localization using rigid graphconcepts is depicted in Figure 2

(1) Phase 1 In this phase the WSN nodes are deployedby setting their communication power radius to thelowest level

(2) Phase 2 In this phase localizability analysis of thenetwork is done In this process the WSN nodespower level is increased adaptively until sufficientrigid patches are obtained as explained in Section 41The local coordinate system of the obtained patchesis set up as explained in Section 42 The patch isalso subjected to global rotation and translation asexplained in Section 43 Trilateration of the rigidpatch will generate the location estimates of theunknown node as explained in Section 44

(3) Phase 3 In this phase the initial deployment of theWSN nodes will be changed to new refined WSNdeployment due to the adaptive temporary change inthe power levels of the nodes

(4) Phase 4 In this phase the location estimates of thedeployedWSN nodes are found out based on the iter-ative trilateration algorithm explained in Section 44

41 Rigid Patch Formation To localize the given WSN graphwe will have to decompose the given WSN graph intopatches of rigid graphs The minimal rigid graph here will beconstituted only of triangles and quadrilaterals which satisfiesTheorems 8 and 10 A distributed technique is described inAlgorithm 1 to find the rigid patches using which the WSNgraph can be efficiently localized Each WSN node has thefollowing capabilities

(i) It knows its own ID(ii) It knows the distance to each of its neighbors(iii) It can send a message to any selected subset of its

neighbors(iv) It can receive a message from any of its neighbors(v) It performs local computation(vi) It keeps state information

Initially an adjacency matrix for the WSN graph mustbe constructed This adjacency matrix will maintain theintercommunication distance between nodes which are itsneighbors The intercommunication distance will be withinthe maximal communication radius of the WSN node Westart with node ID = 1 and check for its neighbor if it has gota neighbor then that neighbor ID is put into a matrix namedCYCLE From the latter node ID again we find its neighborand put its ID in cycle Continuing this way when a cycle oflength 3 is found we get the smallest rigid component thetriangle The obtained rigid component information is thenmaintained in a matrix called RIGID Continuing with thelast established ID we find the neighbor of that ID and put itin thematrix cycleWhen the neighbor of the new found ID isthe starting node with ID = 1 then we obtain cycle of length 4which is a quadrilateral If the quadrilateral satisfiesTheorems8 and 10 add the quadrilateral to the RIGID matrix if not


Input TheWSN graph of the form 119866 = (119881 119864)Output Rigid Quadrilateral Patches(1) Build adjacency matrix adj(119894) of the WSN graph which will possess distance of WSN nodes

that are within 119877 Consider cycle(119894) and node(119894) Initialize 119894 = 1 119896 = 0119898 = 0 119899 = 0(2) for 119894 = 1 to no of nodes do(3) consider the first node(4) while adj(119894) = 0 do(5) Add the first node to cycle(1)(6) make node(119894 cycle(1)) = 1(7) end while(8) Consider the 1-hop neighbor of first node(9) while adj(cycle(1) 119896) = 0 do(10) add this node to cycle(2)(11) make node(119894 cycle(2)) = 1(12) end while(13) Consider the 1-hop neighbor of the previous node(14) while adj(cycle(2)119898) = 0 do(15) add the this node to cycle(3)(16) if cycle(3) == 119894 then(17) Triangle is obtained and add it to the rigid matrix(18) end if(19) end while(20) Consider the 1-hop neighbor of the previous node(21) while adj(cycle(3) 119899) = 0 do(22) add the this node to cycle(4)(23) if cycle(4) = 119894 then(24) Quad is obtained(25) Check whether obtained quad satisfies Theorem 8 andTheorem 12(26) If it satisfies then add NODE-IDrsquos forming quadrilateral to the rigid matrix(27) end if(28) end while(29) Repeat the process for all the WSN nodes(30) Put all the quads obtained with its NODE-ID in rigid matrix(31) end for

Algorithm 1 Finding the rigid quadrilateral patches

increase the communication power radius until Theorem 12is satisfied thereby obtaining required quadrilateral patchesContinuing with the next node ID which is having commoncommunication links to the previously obtained patch findall the quadrilaterals which constitute the rigid componentsof the WSN graph

42 Local Coordinate System The local coordinate systemassignment for the WSN nodes is inspired from [24] Anyneighboring node of the WSN node whose position has tobe computed has to align at the origin (0 0) of the coordi-nate system and based on this WSN node and remainingneighboring nodes location estimation of the unknownWSNnode is done In Figure 3(a) twoWSN nodes positioned at 119875

1

and 1198752with different coordinate systems (119909

1 1199101) and (119909

2 1199102)

are shown Let us consider WSN nodes 1198751 1198752 1198753 and 119875

4

with their coordinates points (1199091 1199101) (1199092 1199102) (1199093 1199103) and

(1199094 1199104) respectively as shown in Figure 3(b) 119875

2is called

the 1-hop neighbor of 1198751if and only if there is a direct

communication link between 1198752and 119875

1 1198751finds out its

neighboring WSN nodes by sending some beacon packets

The beacon packet will have the fields sender-ID sequence-number and the neighboring nodes information Thus forevery node 119875

119894forall119894 = 1 2 3 119899 there is a neighbor set

119873119895forall119895 = 1 2 3 119899 For the WSN nodes 119875

119894and their

neighboring pair 119873119895distance information 119889

119894is calculated by

RSSI based ranging technique as explained in Section 4 andthe bearing (angle) information 120579

119894is estimated using angle-

of-arrival (AOA) technique as explained below For a WSNnode 119875

1to construct its local coordinate system it needs

minimum of two one-hop neighbors and they must not becollinear It also needs to know the distance information ofits neighbors and the distance 119889

119894gt 0

Consider the WSN nodes 1198751 1198752 and 119875

3as shown in

Figure 3(a) Their positions in coordinate system are

1198751119883

= 0

1198751119884

= 0

1198752119883

= 11988912

1198752119884

= 0


y2

12057921

P2

x2

x1

y1

P1

12057912

(a) Two WSN nodes positioned at 1198751 and 1198752 having differentcoordinate system

P3(x3 y3)

y d13

12057931 12057932

d23

P1(x1 y1)12057913 d12

d34 12057923

12057914x 12057924

P2(x2 y2)

d14 12057941 12057942

d24

P4(x4 y4)

(b) Local coordinate system of WSN nodes positioned at 1198751 1198752 1198753 and1198754 with the availability of distance and angle information

Figure 3 Local coordinate system of WSN nodes

1198753119883

= 11988913cos 12057913

1198753119884

= 11988913sin 12057913

(1)

12057913

can be obtained by applying the triangulation to(1198751 1198752 1198753)

12057913

= cosminus1(1198892

13+ 1198892

12minus 1198892

23)

(21198891311988912)

(2)

The position of node 1198754is calculated using

1198754119883

= 11988914cos 12057914

if 1205791198751

=100381610038161003816100381612057914 minus 120579

13

1003816100381610038161003816

then 1198754119884

= 11988914sin (12057914)

else 1198754119884

= minus11988914sin 12057914

(3)

Using cosine rule 12057914and 1205791198751can be calculated

12057914

= cosminus1(1198892

14+ 1198892

12minus 1198892

24)

(21198891411988912)

1205791198751

= cosminus1(1198892

14+ 1198892

13minus 1198892

34)

21198891411988913

(4)

43 Bearing Heading and Global Transformations Considertwo WSN nodes 119875

1and 119875

2as shown in Figure 4(a) The

bearing information is nothing but the angle between the119909-axis in the local coordinate system of WSN node 119875

1

and the edge line joining 1198751and 119875

2which represents the

communication link between the two WSN nodes [20] Thebearing information is measured in anticlockwise directionfrom the 119909-axis of the WSN nodes local coordinate systemIn the above local coordinate system analysis we discussedeach WSN node having its own local coordinate system InFigure 4(a) the bearing information of 119875

1and 119875

2is 12057912

and12057921 that is the angle between the WSN nodes 119909-axis and the

communication link between 1198751and 1198752 The local coordinate

system of two WSN nodes may not always have the samemapping therefore direction analysis of local coordinatesystem of different WSN nodes is necessary The mapping isdone by obtaining the global coordinates of the nodes Forexample if we want to align the local coordinate system of 119875

1

and 1198752then global coordinate transformation of these nodes

is necessary For global coordinate transformation headinginformation needs to be derived This heading informationis the angle measured in anticlockwise direction between the119909-axis of WSN nodes local coordinate system and the 119910-axisof the global coordinate system [20] Let us suppose that 120601

1

is the heading for the WSN node 1198751 Now the node 119875

1has to

pass its heading information 1206011and bearing information 120579

12

to its neighboring node 1198752so that it can compute its heading

information Upon receiving the heading information andbearing information from the WSN node 119875

1 the WSN node

1198752computes its heading information

1206012= 120587 minus (120579

12minus 1206011) + 12057921 (5)


y2

y1

y2G

12057921 P2 x2G

x2

y1G

P1

12057912

x1

x1G

1206011

1206012

(a) The bearing information between the WSN nodes

y2G

P2

x2G

Θ21

y1G

P1 x1GΘ12

(b) The bearing information between the WSN nodes in the globalcoordinate system

Figure 4 Global coordinate system

Once the WSN nodes map to the global coordinate systemthen the bearing information of their local coordinate sys-tem is transformed into the bearing information in globalcoordinate system Let Θ

12and Θ

21be the global coordinate

bearing information of 1198751and 119875

2 respectively as shown in

Figure 4(b)

Θ21

= 120587 + Θ12

(mod (2120587)) (6)

Global transformation is used to estimate the actual physicallocation of the individual WSN nodes using any anchornode whose reference position will be fixed This anchornode will be the reference for fixed coordinate system ofthe remaining WSN nodes To localize the ordinary nodethe global transformation matrix 119879

119866at every WSN node

of the network translates from local coordinates to globalcoordinate system The global transformation matrix 119879

119866

constitutes translation and rotation matrix of the ordinaryWSN node The global transformation matrix119872

119866is

[119872119866] = [119879

119866] sdot [119877119866] (7)

Substituting in the above equation the components oftranslation and rotation matrix the final global transforma-tion matrix is of the form

[119872119866] =

[[

[

1 0 120572119909

0 1 120572119910

0 0 1

]]

]

sdot[[

[

cosΘ sinΘ 0

minus sinΘ cosΘ 0

0 0 1

]]

]

=[[

[

cosΘ sinΘ 120572119909

minus sinΘ cosΘ 120572119910

0 0 1

]]

]

(8)

44 Localization Procedure The localization is done usingiterative trilateration of the obtained rigid patches followedwith the coordinate transformations as explained in Sections42 and 43 The localization procedure here does not useanchor nodes but starts with three initial nodes based on theconcept of assumption based coordinate system [25]

(1) Consider a quad patch obtained Let it have the threeassumption nodes with coordinates say (119909

1 1199101)

(1199092 1199102) and (119909

3 1199103) Translate the origin to (119909

1 1199101)

by using values 120572119909and 120572

119910which is got by subtracting

all the three coordinate points by (1199091 1199101)

(2) Now rotate the point say (1199092 1199102) from the origin

(1199091 1199101) with an angle Θ as explained in Section 43

Thus obtain the global transformation matrix equa-tion 119872

119866 This will translate the patch from local

coordinate system to the global coordinate system

(3) Trilateration let 1198891 1198892 and 119889

3be the distance

between the ordinary node and the anchor nodesas considered for trilateration as shown in Figure 5Consider

119894 = 1199093minus 1199091

119895 = 1199103minus 1199101

119889 = 1199092

(9)


(0 0 0)

d1

(d 0 0)

d2

d3

(i j 0)

Figure 5 Three-circle intersection based trilateration The big darkcircled point is the unknown coordinate (119909

4 1199104)whose position has

to be estimated based on the known coordinates (small dark circled)1198891 1198892 and 119889

3are the communication radius between the known

coordinates to the unknown

The equation for the center of the circle and thedistance for the anchor nodes as per Figure 5 are givenby

1198892

1= 1199092

4+ 1199102

4+ 1199112

4 (10)

1198892

2= (1199094minus 119889)2

+ 1199102

4 (11)

1198892

3= (1199094minus 119894)2

+ (1199104minus 119895)2

(12)

Subtracting (10) from (11) and (10) from (12) andrearranging them the point (119909

4 1199104) is obtainedwhich

will be the location coordinates of the ordinary node

1199094=

1198892

1minus 1198892

2+ 1198892

2119889

1199104=

1198892

1minus 1198892

3+ 1198942+ 1198952

2119895minus

119894

1198951199094

(13)

(4) Repeat the above steps for the remaining quad patchesobtained

5 Results and Discussions

The simulation parameters are shown in Table 1 119873 numberof nodes are deployed in a square region of 450 times 450m usingrandomuniformdistribution where there is an edge betweennodes if and only if their distance is within communicationrange 119877

51 Node Deployment Node Density and Varying of PowerLevels For simulation nodes are deployed in a square regionof 450 times 450m using random uniform distribution wherethe number of nodes 119873 can be scaled from 100 to 120 in astep size of 10 The power level of every node in the WSN issuitably varied such that the communication power radius 119877

Table 1 Simulation parameters

Parameter Denoted by ValuesDeployment area 119863 450 times 450metersNumber of nodes 119873 100Communication power range 119877 Varied from 70 to 100

0 10

010203040506070809

01 02 03 04 05 06 07 08 09

1

Node density

N = 100

N = 110

N = 120

Prob

abili

ty (n

ge3

)

Figure 6 Node density versus probability of number of neighborsmore than 3

is increased Here the119877 values are increased from 70 to 100 ina step size of 5 This increase in communication radius 119877 willobviously increase the average node degree of the WSN andthereby we can expect a bigger number of rigid trilaterablepatches needed for iterative trilateration localization For thedifferent values of 119877 of 70 75 80 85 90 95 and 100 theWSN instance graph depicting the communication links isshown in Figure 7 It is observed that with increase in 119877 thenode degree of WSN nodes increases thereby increasing theconnectivity of the graph More about the role of node degreeis explained in the next subsection Node density is defined asthe number of nodes per unit square area When119873 = 120 forthe given network topology when the node density is 04 perunit area more than 90 of nodes will have neighbors morethan 3 and similarly for119873 = 100when node density is 05 perunit area as shown in Figure 6

52 Node Degree The connectivity of the WSN graph ismeasured in terms of node degreeThe node degree is definedas the ratio of number of communication links between theWSN nodes and the number of WSN nodes (119873) deployedin the specific area The communication power range isincreased by a value of 5 meters and from 70 meters to100 meters and for each of these 119877 values ten instances ofsimulation run are done as shown in Figure 7 It is observedin the figure that the node degree varies at each differentinstance of simulation run This is mainly due to the randomdeployment of the WSN nodes Due to this variation of nodedegree it is not always possible to obtain higher percentageof rigid patches at lower node degree Therefore to achievehigher percentage of rigid patches the node degree must behighThe average node degree is the mean of the node degreecalculated over the several instances of simulation run doneFor the above simulation setup the average node degree iscalculated for the different communication power range (119877)as shown in Figure 7 In the figure average node degree is


1 2 3 4 5 6 7 8 9 108

1012141618202224

Simulation runs

Nod

e deg

ree

R = 100

R = 95

R = 90

R = 85

R = 80

R = 75

R = 70

Figure 7 Average node degree at different instances of simulationrun

4 6 8 10 12 14 162030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es

N = 130

N = 120

N = 100

Figure 8 Amount of localizable nodes at different node degree fordifferent values of119873

also calculated with 119873 values of 100 110 and 120 to showhow it varies when the WSN nodes in the network scale Itis observed as from the figure that with increasing values 119877 atparticular 119873 value range the average node degree increasesFor 119873 value of 100 the average node degree increases from762 to 1331 and when 119873 = 110 it increases from 797to 1417 and when 119873 = 120 it varies from 8602 to 1621Therefore good network connectivity comes with high nodedegree which can be achieved by higher communicationpower range and also by scaling the network appropriately

53 Node Degree versus Percentage of Localizable NodesThe percentage of localizable nodes depends directly on thenumber of rigid patches that can be generated for particularWSN deployment at a specific communication power rangewhich in turn depends on the average node degree InFigure 8 the percentage of localizable nodes at differentaverage node degree is shown for 119873 values of 100 110 and120 respectively For all the 119873 values when the average nodedegree is less than 7 the percentage of localizable nodes isless than 50 This is due to the fact that with lower averagenode degree the number of rigid patches obtained is lessWhen the average node degree is between 8 and 9 in all

0 5 10 15 20 25 300

102030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es

Proposed approachRobust quads

Figure 9 Percentage of localizable nodes at different node degree

the cases of 119873 the percentage of localizable nodes is above70 percent and up to 80 When the node degree is 9 and119873 = 120 about 90 percent of nodes are localizable andbetween 80 and 90 percent of nodes are localizable for 119873 =

100 and 120 For the node degree of 10 and above up to 16more than 90 of nodes are localizable all the time Thisis due to the fact that with higher average node degree ahigher number of rigid patches can be obtained which canbe easily localized Therefore to localize a network almostcompletely the average node degree must be maintained ashigh between 10 and 16 Each WSN nodes power level can besuitably varied so that its communication power radius 119877 isincreased Here the 119877 values are increased from 70 to 100 ina step size of 5 This increase in communication radius 119877 willobviously increase the average node degree of the WSN andthereby we can expect a bigger number of rigid trilaterablepatches needed for iterative trilateration localization For thedifferent values of 119877 of 70 75 80 85 90 95 and 100 theWSNinstance graph depicting the communication links is shownin Figure 7 The percentage of localizable nodes dependsdirectly on the number of rigid patches that can be generatedfor particular WSN deployment at a specific communicationpower range which in turn depends on the average nodedegree We compare our proposed algorithm with the robustquads of [7] Our method achieves above 90 localizablenetwork when the node degree is between 10 and 14 whilerobust quads require node degree above 22 to achieve morethan 90 localizable network as shown in Figure 9

54 Noise Model and Localization Error

(i) The distance noise model used in this work is givenby

119894119895= 119889119894119895(1 + 120588 times 119899

119891) (14)

where 120588 is the Gaussian noise with zero mean and 119899119891

is the noise factor which is considered here to be 15(high noise)

(ii) The localization error occurs mainly due to the rang-ing errorThe localization error may be defined as theamount of deviation of the position coordinates of the


70 75 80 85 90 95 1000

5

10

15

20

25

30

Communication range (meters)

Nor

mal

ized

loca

tion

erro

r (

)

Proposed Robust quads

Figure 10 Normalized error in percentage at different communica-tion ranges

WSN nodes from the actual position where the nodeis deployed Here we will use the localization errorequation normalized by the communication rangegiven by

120590LE =1

119873times

119873

sum

119894=1

radic(119909119894minus 119909119894)2

+ (119910119894minus 119910119894)2

119873times 100 (15)

where (119909119894 119910119894) is the estimated position coordinate of

the ithWSN node and (119909119894 119910119894) is the actual position of

the ith WSN node

120590LE of our proposed method is compared with robustquads As observed in Figure 10 in the proposed scheme120590LE decreases with increase in communication range due tothe fact that more trilaterable patches are obtained 120590LE isabout 5 of maximum communication range (100) in theproposed approach where as in robust quads it is about 10ofmaximumcommunication rangeThis is because of the factthat location error propagates from the different robust quadsin case of a high noise communication range

55 Computational Complexity Analysis The running timeto estimate robust quads and rigid patches of our approachis O(120575

3) where 120575 is the average node degree The value 120575

in our approach is lower compared to robust quads whichmakes our approach computationally efficient Again in ouralgorithm rigid patches are obtained with a cycle of 119862

4

therefore the worst case to achieve rigid patches is ( 1205753) For

a specific average node degree this worst case efficiency willturn up to be O(1) that is a linear time algorithm Theiterative trilateration procedurersquos complexity isO() where is the number of rigid patches which will be ( 120575

3) in the worst

case

6 Conclusion and Future Scope

In this work using the analytical approaches of rigid graphtheory a localization algorithm using both the distance and

bearing measurement is developed Rigid graph theory isused to analyze the localizability of the WSN nodes in thenetwork In this paper rigid patches of the WSN graph areobtained by varying the communication power range of theWSN nodes which is then applied with trilateration proce-dure to obtain the position coordinates of the WSN nodesTrilateration graphs have the advantage that localization willbe carried out in polynomial time The proposed approachhas lesser average node degree which makes it computa-tionally efficient as the computational complexity dependson the node degree The localization error is normalized tothe communication range and a comparison shows that 5of localization error is obtained in the proposed approachwhereas 10 of the error is obtained in robust quads at a highnoise of 15 As part of future work rigid graph localizationcan be applied to real time context aware applications liketarget tracking autonomous agent and robot tracking andso forth and also implement the localization algorithm in atest bed

Competing Interests

The authors declare that they have no competing interests

References

[1] I F Akyildiz W Su Y Sankarasubramaniam and E Cayirci ldquoAsurvey on sensor networksrdquo IEEE Communications Magazinevol 40 no 8 pp 102ndash105 2002

[2] Y Liu Z Yang X Wang and L Jian ldquoLocation localizationand localizabilityrdquo Journal of Computer Science and Technologyvol 25 no 2 pp 274ndash297 2010

[3] T Eren D K GoldenbergWWhiteley et al ldquoRigidity compu-tation and randomization in network localizationrdquo in Proceed-ings of the 23rd Annualjoint Conference of the IEEEComputerand Communications Societies (Infocom rsquo04) vol 4 pp 2673ndash2684 Hong Kong March 2004

[4] J Aspnes T Eren D K Goldenberg et al ldquoA theory of networklocalizationrdquo IEEETransactions onMobile Computing vol 5 no12 pp 1663ndash1678 2006

[5] Y Zhang Y Chen and Y Liu ldquoTowards unique and anchor-free localization for wireless sensor networksrdquoWireless PersonalCommunications vol 63 no 1 pp 261ndash278 2012

[6] Y Zhu S J Gortler and DThurston ldquoSensor network localiza-tion using sensor perturbationrdquo ACM Transactions on SensorNetworks vol 7 no 4 article 36 2011

[7] D Moore J Leonard D Rus and S Teller ldquoRobust dis-tributed network localization with noisy range measurementsrdquoin Proceedings of the 2nd International Conference on EmbeddedNetworked Sensor Systems (SenSys rsquo04) pp 50ndash61 BaltimoreMd USA November 2004

[8] D K Goldenberg P Bihler M Cao et al ldquoLocalization insparse networks using sweepsrdquo inProceedings of the 12th AnnualInternational Conference on Mobile Computing and Networking(MOBICOM rsquo06) pp 110ndash121 ACM Los Angeles Calif USASeptember 2006

[9] J Fang and A S Morse ldquoMerging globally rigid graphs andsensor network localizationrdquo in Proceedings of the 48th IEEEConference on Decision and Control (CDCCCC rsquo09) pp 1074ndash1079 IEEE Shanghai China 2009


[10] S Varma and B S Rai ldquoGraph rigidity application for localiza-tion in WSNrdquo International Journal of Computer Applicationsvol 49 no 9 pp 15ndash21 2012

[11] P Costa M Cesana S Brambilla and L Casartelli ldquoA coopera-tive approach for topology control in wireless sensor networksrdquoPervasive andMobile Computing vol 5 no 5 pp 526ndash541 2009

[12] B S Rai N Ananad and S Varma ldquoScrutinizing localizedtopology control in wsn using rigid graphsrdquo in Proceedings ofthe 2nd International Conference on Computing for SustainableGlobal Development (INDIACom rsquo15) pp 1712ndash1715 2015

[13] B Hendrickson ldquoConditions for unique graph realizationsrdquoSIAM Journal on Computing vol 21 no 1 pp 65ndash84 1992

[14] S J Gortler A D Healy and D P T Hurston ldquoCharacterizinggeneric global rigidityrdquo American Journal of Mathematics vol132 no 4 pp 897ndash939 2010

[15] G Laman ldquoOn graphs and rigidity of plane skeletal structuresrdquoJournal of Engineering Mathematics vol 4 pp 331ndash340 1970

[16] D K Goldenberg A Krishnamurthy W C Maness et alldquoNetwork localization in partially localizable networksrdquo inProceedings of the IEEE 24th Annual Joint Conference of the IEEEComputer and Communications Societies (IEEE INFOCOM rsquo05)vol 1 pp 313ndash326 Miami Fla USA March 2005

[17] J Aspnes D Goldenberg and Y R Yang ldquoOn the computa-tional complexity of sensor network localizationrdquo in Algorith-mic Aspects of Wireless Sensor Networks S E Nikoletseas and JD P Rolim Eds vol 3121 of Lecture Notes in Computer Sciencepp 32ndash44 Springer New York NY USA 2004

[18] B D O Anderson P N Belhumeur T Eren et al ldquoGraphicalproperties of easily localizable sensor networksrdquo Wireless Net-works vol 15 no 2 pp 177ndash191 2009

[19] B D O Anderson I Shames G Mao and B Fidan ldquoFormaltheory of noisy sensor network localizationrdquo SIAM Journal onDiscrete Mathematics vol 24 no 2 pp 684ndash698 2010

[20] T Eren ldquoCooperative localization in wireless ad hoc and sensornetworks using hybrid distance and bearing (angle of arrival)measurementsrdquo EURASIP Journal on Wireless Communicationsand Networking vol 2011 no 1 article 72 pp 1ndash18 2011

[21] Y Zhang S Liu X Zhao and Z Jia ldquoTheoretic analysis ofunique localization for wireless sensor networksrdquo Ad HocNetworks vol 10 no 3 pp 623ndash634 2012

[22] Z Yang Y Liu and X-Y Li ldquoBeyond trilateration on the local-izability of wireless ad hoc networksrdquo IEEEACM Transactionson Networking vol 18 no 6 pp 1806ndash1814 2010

[23] Z Yang CWu T Chen Y ZhaoW Gong and Y Liu ldquoDetect-ing outlier measurements based on graph rigidity for wirelesssensor network localizationrdquo IEEE Transactions on VehicularTechnology vol 62 no 1 pp 374ndash383 2013

[24] S Capkun M Hamdi and J-P Hubaux ldquoGPS-free positioningin mobile ad hoc networksrdquo Cluster Computing vol 5 no 2 pp157ndash167 2002

[25] C Savarese J M Rabaey and J Beutel ldquoLocation in distributedad-hoc wireless sensor networksrdquo in Proceedings of the IEEEInternational Conference on Acoustics Speech and Signal Pro-cessing (ICASSP rsquo01) vol 4 pp 2037ndash2040 Salt Lake City UtahUSA May 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



location estimates but are having issues related to scalabilitylarge amount of computational complexity and low reliabilityas compared to distributed algorithms like [7ndash9] This workis an extension of our previous work [10] In this work wedeal with both distance and bearing information for accurateand efficient localization we also deal with error analysissystematically and also compared our results with the robustquads technique To localize networks in human unattendedareas only distributed algorithms can be employedThereforein this work we have developed a distributed localizationtechnique which involves the following aspects

(1) The main problem in keeping the communicationpower range to a minimal value is that sufficient rigidgraphs cannot be generated thereby the algorithmwill not be able to localize all the nodes uniquely Itis possible to vary the power level ofWSN nodes as in[11] so that the WSN can be made into a localizablenetwork from a nonlocalizable one We suitably varythe power level of the WSN nodes such that itscommunication radius is increased This increase incommunication radius will obviously increase theaverage node degree of the WSN and thereby we canexpect a bigger number of rigid trilateration patchesneeded for iterative trilateration localization

(2) Placing of anchor nodes is not feasible in someoutdoor localization scenarios hence an anchor-freelocalization technique is designed in this work

(3) In order to achieve higher location estimation accu-racy in this approach both distance and angle infor-mation between the neighboring WSN nodes areused

(4) The patches obtained are subjected to global coordi-nate transformations of translation and rotation andtrilateration is used to localize these patches

2 Graph Theoretical Framework andLocalization Using Rigid Graphs

The rigid graphs as the name suggests fall into the categoryof graph theory hence denoted using the graph theorynotations In our work [12] we discussed how rigid graphtheory can be applied to localization and topology controlin WSN In this section the graph theoretical frameworkdepicting WSN nodes in the form of graph is shown

A 119889-dimension point formation at 119901 = column1199011 1199012

119901119899 usually denoted as 119865(119901) which is in the Euclidean

coordinate system set of 119899 number of points 1199011 1199012 119901

119899

in 119877119889 is considered in [5 13 14] In terms ofWSN the points

correspond to the positions of WSN nodes which forms thenetwork The communication link is the intersensor nodedistance In this coordinate of points communication linksbetween points 119901

119894and 119901

119895are deduced where 119894 and 119895 are

integers of set 119894 = (1 2 119899) Length of the communicationlink is the Euclidean distance between the points 119901

119894and 119901

119895

Some of the standard definitions are given below

Definition 1 (sensor graph) A WSN graph 119873(119878119863) where 119878

represents the WSN nodes and119863 the distance betweenWSNnodes can be represented by a graph 119866 = (119881 119864) with avertex set 119881 and an edge set 119864 where each vertex 119894 isin 119881 isrelated with a sensor node 119878

119894in the network graph and each

edge (119894 119895) isin 119864 relates to a sensor pair 119878119894 119878119895for which the

intersensor distances 119889119894119895are known One calls 119866 = (119881 119864) the

underlying graph of the sensor network119873(119878119863)

Definition 2 (framework and realization) A 119889-dimensional(119889 isin 2 3) representation of graph is mapping of a graph 119866 =

(119881 119864) to the point formations 119901 119881 rarr 119877119889 Given graph 119866 =

(119881 119864) and a119889-dimensional representation of it the pair (119866 119901)

is termed as a 119889-dimensional framework A distance set119863 for119866 is set of distances 119889

119894119895gt 0 defined for all edges (119894 119895) isin 119864

Given the distance set 119863 for the graph 119866 a 119889-dimensional(119889 isin 2 3) representation 119901 of 119866 is a realization if it results in119901(119904119894) minus119901(119904

119895) = 119889

119894119895for all pairs of (119894 119895) isin 119881 where (119894 119895) isin 119881

Definition 3 (equivalent and congruent frameworks) Twoframeworks 119866(119881 119901) and 119866(119881 119902) are equivalent if 119901(119904

119894) minus

119901(119904119895) = 119902(119904

119894) minus 119902(119904

119895) holds for every pair 119894 119895 isin 119881

connected by an edge Two frameworks 119866(119881 119901) and 119866(119881 119902)

are congruent if 119901(119904119894) minus 119901(119904

119895) = 119902(119904

119894) minus 119902(119904

119895) holds for

every pair 119894 119895 isin 119881 nomatter whether there is an edge betweenthem

Definition 4 (flexible graph and rigid graph) A framework119866(119881 119901) is called generic if the set containing the coordi-nates of all its points is algebraically independent over therationales A framework 119866(119881 119901) is called rigid if there existsa sufficiently small positive constant 120598 such that if 119866(119881 119901)

is equivalent to 119866(119881 119902) and 119901(119904119894) minus 119901(119904

119895) lt 120598 for all

119894 isin 119881 then 119866(119881 119902) is congruent with 119866(119881 119901) A graph119866(119881 119864) is called rigid if there is an associated framework119866(119881 119901) that is generic and rigid A framework is calledflexible if one has a continuous deformation starting fromthe known configuration to another such that edge lengthsare preserved as shown in Figures 1(a) and 1(b) If no suchdeformation exists then it is called rigid A framework119866(119881 119901) is globally rigid if every framework equivalent to119866(119881 119901) is also congruent with 119866(119881 119901) A graph 119866(119881 119864)

is called globally rigid if there is an associated framework119866(119881 119901) that is generic and globally rigid

Definition 5 (minimal rigid graph) A rigid framework isminimally rigid if it becomes flexible after an edge is removedas shown in Figure 1(c)

Definition 6 (redundantly rigid framework) A rigid frame-work is redundantly rigid if it remains rigid upon the removalof any edge as shown in Figure 1(d)

Definition 7 (localizability) As already defined above for aWSN graph 119866 = (119881 119864) every node is realized as a pointformation If there is a unique location 119901 119881 rarr 119877

119889 for everynode belonging to the node set119881 such that 119889

119894119895= 119901(119894)minus119901(119895)

for all the edges then one says that such a graph is localizable


1 2

34(a)

1

34

(b)

12

34

(c)

1 2

34(d)










1015840 |11986410158401015840| le

2|119881(11986410158401015840) minus 3|




21 Trilaterations






119899)




(119894minus1)












3 Related Work





of WSN nodes
























1


1 1199101) and (119909

2 1199102)


4



2is called








119894and their








119894gt 0


3as shown in


1198751119883

= 0

1198751119884

= 0

1198752119883

= 11988912

1198752119884

= 0


y2

12057921

P2

x2

x1

y1

P1

12057912


P3(x3 y3)

y d13

12057931 12057932

d23

P1(x1 y1)12057913 d12

d34 12057923

12057914x 12057924

P2(x2 y2)

d14 12057941 12057942

d24

P4(x4 y4)



1198753119883

= 11988913cos 12057913

1198753119884

= 11988913sin 12057913

(1)

12057913


12057913

= cosminus1(1198892

13+ 1198892

12minus 1198892

23)

(21198891311988912)

(2)


1198754119883

= 11988914cos 12057914

if 1205791198751

=100381610038161003816100381612057914 minus 120579

13

1003816100381610038161003816

then 1198754119884

= 11988914sin (12057914)

else 1198754119884

= minus11988914sin 12057914

(3)


12057914

= cosminus1(1198892

14+ 1198892

12minus 1198892

24)

(21198891411988912)

1205791198751

= cosminus1(1198892

14+ 1198892

13minus 1198892

34)

21198891411988913

(4)


1and 119875



1




1and 119875

2is 12057912




1



1


1has to


12



1 the WSN node


1206012= 120587 minus (120579

12minus 1206011) + 12057921 (5)


y2

y1

y2G

12057921 P2 x2G

x2

y1G

P1

12057912

x1

x1G

1206011

1206012


y2G

P2

x2G

Θ21

y1G

P1 x1GΘ12




12and Θ




Figure 4(b)

Θ21

= 120587 + Θ12

(mod (2120587)) (6)




119866


119866is

[119872119866] = [119879

119866] sdot [119877119866] (7)


[119872119866] =

[[

[

1 0 120572119909

0 1 120572119910

0 0 1

]]

]

sdot[[

[

cosΘ sinΘ 0

minus sinΘ cosΘ 0

0 0 1

]]

]

=[[

[

cosΘ sinΘ 120572119909


0 0 1

]]

]

(8)



1 1199101)

(1199092 1199102) and (119909


1 1199101)










3be the distance


119894 = 1199093minus 1199091

119895 = 1199103minus 1199101

119889 = 1199092

(9)


(0 0 0)

d1

(d 0 0)

d2

d3

(i j 0)







1198892

1= 1199092

4+ 1199102

4+ 1199112

4 (10)

1198892

2= (1199094minus 119889)2

+ 1199102

4 (11)

1198892

3= (1199094minus 119894)2

+ (1199104minus 119895)2

(12)




1199094=

1198892

1minus 1198892

2+ 1198892

2119889

1199104=

1198892

1minus 1198892

3+ 1198942+ 1198952

2119895minus

119894

1198951199094

(13)







0 10

010203040506070809

01 02 03 04 05 06 07 08 09

1

Node density

N = 100

N = 110

N = 120

Prob

abili

ty (n

ge3

)





1 2 3 4 5 6 7 8 9 108

1012141618202224

Simulation runs

Nod

e deg

ree

R = 100

R = 95

R = 90

R = 85

R = 80

R = 75

R = 70


4 6 8 10 12 14 162030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es

N = 130

N = 120

N = 100




0 5 10 15 20 25 300

102030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es







119894119895= 119889119894119895(1 + 120588 times 119899

119891) (14)





70 75 80 85 90 95 1000

5

10

15

20

25

30


Nor

mal

ized

loca

tion

erro

r (

)




120590LE =1

119873times

119873

sum

119894=1

radic(119909119894minus 119909119894)2

+ (119910119894minus 119910119894)2

119873times 100 (15)



the ith WSN node





4



3) in the worst

case




Competing Interests


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2

34(a)

1

34

(b)

12

34

(c)

1 2

34(d)










1015840 |11986410158401015840| le

2|119881(11986410158401015840) minus 3|




21 Trilaterations






119899)




(119894minus1)












3 Related Work





of WSN nodes
























1


1 1199101) and (119909

2 1199102)


4



2is called








119894and their








119894gt 0


3as shown in


1198751119883

= 0

1198751119884

= 0

1198752119883

= 11988912

1198752119884

= 0


y2

12057921

P2

x2

x1

y1

P1

12057912


P3(x3 y3)

y d13

12057931 12057932

d23

P1(x1 y1)12057913 d12

d34 12057923

12057914x 12057924

P2(x2 y2)

d14 12057941 12057942

d24

P4(x4 y4)



1198753119883

= 11988913cos 12057913

1198753119884

= 11988913sin 12057913

(1)

12057913


12057913

= cosminus1(1198892

13+ 1198892

12minus 1198892

23)

(21198891311988912)

(2)


1198754119883

= 11988914cos 12057914

if 1205791198751

=100381610038161003816100381612057914 minus 120579

13

1003816100381610038161003816

then 1198754119884

= 11988914sin (12057914)

else 1198754119884

= minus11988914sin 12057914

(3)


12057914

= cosminus1(1198892

14+ 1198892

12minus 1198892

24)

(21198891411988912)

1205791198751

= cosminus1(1198892

14+ 1198892

13minus 1198892

34)

21198891411988913

(4)


1and 119875



1




1and 119875

2is 12057912




1



1


1has to


12



1 the WSN node


1206012= 120587 minus (120579

12minus 1206011) + 12057921 (5)


y2

y1

y2G

12057921 P2 x2G

x2

y1G

P1

12057912

x1

x1G

1206011

1206012


y2G

P2

x2G

Θ21

y1G

P1 x1GΘ12




12and Θ




Figure 4(b)

Θ21

= 120587 + Θ12

(mod (2120587)) (6)




119866


119866is

[119872119866] = [119879

119866] sdot [119877119866] (7)


[119872119866] =

[[

[

1 0 120572119909

0 1 120572119910

0 0 1

]]

]

sdot[[

[

cosΘ sinΘ 0

minus sinΘ cosΘ 0

0 0 1

]]

]

=[[

[

cosΘ sinΘ 120572119909


0 0 1

]]

]

(8)



1 1199101)

(1199092 1199102) and (119909


1 1199101)










3be the distance


119894 = 1199093minus 1199091

119895 = 1199103minus 1199101

119889 = 1199092

(9)


(0 0 0)

d1

(d 0 0)

d2

d3

(i j 0)







1198892

1= 1199092

4+ 1199102

4+ 1199112

4 (10)

1198892

2= (1199094minus 119889)2

+ 1199102

4 (11)

1198892

3= (1199094minus 119894)2

+ (1199104minus 119895)2

(12)




1199094=

1198892

1minus 1198892

2+ 1198892

2119889

1199104=

1198892

1minus 1198892

3+ 1198942+ 1198952

2119895minus

119894

1198951199094

(13)







0 10

010203040506070809

01 02 03 04 05 06 07 08 09

1

Node density

N = 100

N = 110

N = 120

Prob

abili

ty (n

ge3

)





1 2 3 4 5 6 7 8 9 108

1012141618202224

Simulation runs

Nod

e deg

ree

R = 100

R = 95

R = 90

R = 85

R = 80

R = 75

R = 70


4 6 8 10 12 14 162030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es

N = 130

N = 120

N = 100




0 5 10 15 20 25 300

102030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es







119894119895= 119889119894119895(1 + 120588 times 119899

119891) (14)





70 75 80 85 90 95 1000

5

10

15

20

25

30


Nor

mal

ized

loca

tion

erro

r (

)




120590LE =1

119873times

119873

sum

119894=1

radic(119909119894minus 119909119894)2

+ (119910119894minus 119910119894)2

119873times 100 (15)



the ith WSN node





4



3) in the worst

case




Competing Interests


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












of WSN nodes
























1


1 1199101) and (119909

2 1199102)


4



2is called








119894and their








119894gt 0


3as shown in


1198751119883

= 0

1198751119884

= 0

1198752119883

= 11988912

1198752119884

= 0


y2

12057921

P2

x2

x1

y1

P1

12057912


P3(x3 y3)

y d13

12057931 12057932

d23

P1(x1 y1)12057913 d12

d34 12057923

12057914x 12057924

P2(x2 y2)

d14 12057941 12057942

d24

P4(x4 y4)



1198753119883

= 11988913cos 12057913

1198753119884

= 11988913sin 12057913

(1)

12057913


12057913

= cosminus1(1198892

13+ 1198892

12minus 1198892

23)

(21198891311988912)

(2)


1198754119883

= 11988914cos 12057914

if 1205791198751

=100381610038161003816100381612057914 minus 120579

13

1003816100381610038161003816

then 1198754119884

= 11988914sin (12057914)

else 1198754119884

= minus11988914sin 12057914

(3)


12057914

= cosminus1(1198892

14+ 1198892

12minus 1198892

24)

(21198891411988912)

1205791198751

= cosminus1(1198892

14+ 1198892

13minus 1198892

34)

21198891411988913

(4)


1and 119875



1




1and 119875

2is 12057912




1



1


1has to


12



1 the WSN node


1206012= 120587 minus (120579

12minus 1206011) + 12057921 (5)


y2

y1

y2G

12057921 P2 x2G

x2

y1G

P1

12057912

x1

x1G

1206011

1206012


y2G

P2

x2G

Θ21

y1G

P1 x1GΘ12




12and Θ




Figure 4(b)

Θ21

= 120587 + Θ12

(mod (2120587)) (6)




119866


119866is

[119872119866] = [119879

119866] sdot [119877119866] (7)


[119872119866] =

[[

[

1 0 120572119909

0 1 120572119910

0 0 1

]]

]

sdot[[

[

cosΘ sinΘ 0

minus sinΘ cosΘ 0

0 0 1

]]

]

=[[

[

cosΘ sinΘ 120572119909


0 0 1

]]

]

(8)



1 1199101)

(1199092 1199102) and (119909


1 1199101)










3be the distance


119894 = 1199093minus 1199091

119895 = 1199103minus 1199101

119889 = 1199092

(9)


(0 0 0)

d1

(d 0 0)

d2

d3

(i j 0)







1198892

1= 1199092

4+ 1199102

4+ 1199112

4 (10)

1198892

2= (1199094minus 119889)2

+ 1199102

4 (11)

1198892

3= (1199094minus 119894)2

+ (1199104minus 119895)2

(12)




1199094=

1198892

1minus 1198892

2+ 1198892

2119889

1199104=

1198892

1minus 1198892

3+ 1198942+ 1198952

2119895minus

119894

1198951199094

(13)







0 10

010203040506070809

01 02 03 04 05 06 07 08 09

1

Node density

N = 100

N = 110

N = 120

Prob

abili

ty (n

ge3

)





1 2 3 4 5 6 7 8 9 108

1012141618202224

Simulation runs

Nod

e deg

ree

R = 100

R = 95

R = 90

R = 85

R = 80

R = 75

R = 70


4 6 8 10 12 14 162030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es

N = 130

N = 120

N = 100




0 5 10 15 20 25 300

102030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es







119894119895= 119889119894119895(1 + 120588 times 119899

119891) (14)





70 75 80 85 90 95 1000

5

10

15

20

25

30


Nor

mal

ized

loca

tion

erro

r (

)




120590LE =1

119873times

119873

sum

119894=1

radic(119909119894minus 119909119894)2

+ (119910119894minus 119910119894)2

119873times 100 (15)



the ith WSN node





4



3) in the worst

case




Competing Interests


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















1


1 1199101) and (119909

2 1199102)


4



2is called








119894and their








119894gt 0


3as shown in


1198751119883

= 0

1198751119884

= 0

1198752119883

= 11988912

1198752119884

= 0


y2

12057921

P2

x2

x1

y1

P1

12057912


P3(x3 y3)

y d13

12057931 12057932

d23

P1(x1 y1)12057913 d12

d34 12057923

12057914x 12057924

P2(x2 y2)

d14 12057941 12057942

d24

P4(x4 y4)



1198753119883

= 11988913cos 12057913

1198753119884

= 11988913sin 12057913

(1)

12057913


12057913

= cosminus1(1198892

13+ 1198892

12minus 1198892

23)

(21198891311988912)

(2)


1198754119883

= 11988914cos 12057914

if 1205791198751

=100381610038161003816100381612057914 minus 120579

13

1003816100381610038161003816

then 1198754119884

= 11988914sin (12057914)

else 1198754119884

= minus11988914sin 12057914

(3)


12057914

= cosminus1(1198892

14+ 1198892

12minus 1198892

24)

(21198891411988912)

1205791198751

= cosminus1(1198892

14+ 1198892

13minus 1198892

34)

21198891411988913

(4)


1and 119875



1




1and 119875

2is 12057912




1



1


1has to


12



1 the WSN node


1206012= 120587 minus (120579

12minus 1206011) + 12057921 (5)


y2

y1

y2G

12057921 P2 x2G

x2

y1G

P1

12057912

x1

x1G

1206011

1206012


y2G

P2

x2G

Θ21

y1G

P1 x1GΘ12




12and Θ




Figure 4(b)

Θ21

= 120587 + Θ12

(mod (2120587)) (6)




119866


119866is

[119872119866] = [119879

119866] sdot [119877119866] (7)


[119872119866] =

[[

[

1 0 120572119909

0 1 120572119910

0 0 1

]]

]

sdot[[

[

cosΘ sinΘ 0

minus sinΘ cosΘ 0

0 0 1

]]

]

=[[

[

cosΘ sinΘ 120572119909


0 0 1

]]

]

(8)



1 1199101)

(1199092 1199102) and (119909


1 1199101)










3be the distance


119894 = 1199093minus 1199091

119895 = 1199103minus 1199101

119889 = 1199092

(9)


(0 0 0)

d1

(d 0 0)

d2

d3

(i j 0)







1198892

1= 1199092

4+ 1199102

4+ 1199112

4 (10)

1198892

2= (1199094minus 119889)2

+ 1199102

4 (11)

1198892

3= (1199094minus 119894)2

+ (1199104minus 119895)2

(12)




1199094=

1198892

1minus 1198892

2+ 1198892

2119889

1199104=

1198892

1minus 1198892

3+ 1198942+ 1198952

2119895minus

119894

1198951199094

(13)







0 10

010203040506070809

01 02 03 04 05 06 07 08 09

1

Node density

N = 100

N = 110

N = 120

Prob

abili

ty (n

ge3

)





1 2 3 4 5 6 7 8 9 108

1012141618202224

Simulation runs

Nod

e deg

ree

R = 100

R = 95

R = 90

R = 85

R = 80

R = 75

R = 70


4 6 8 10 12 14 162030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es

N = 130

N = 120

N = 100




0 5 10 15 20 25 300

102030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es







119894119895= 119889119894119895(1 + 120588 times 119899

119891) (14)





70 75 80 85 90 95 1000

5

10

15

20

25

30


Nor

mal

ized

loca

tion

erro

r (

)




120590LE =1

119873times

119873

sum

119894=1

radic(119909119894minus 119909119894)2

+ (119910119894minus 119910119894)2

119873times 100 (15)



the ith WSN node





4



3) in the worst

case




Competing Interests


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










y2

12057921

P2

x2

x1

y1

P1

12057912


P3(x3 y3)

y d13

12057931 12057932

d23

P1(x1 y1)12057913 d12

d34 12057923

12057914x 12057924

P2(x2 y2)

d14 12057941 12057942

d24

P4(x4 y4)



1198753119883

= 11988913cos 12057913

1198753119884

= 11988913sin 12057913

(1)

12057913


12057913

= cosminus1(1198892

13+ 1198892

12minus 1198892

23)

(21198891311988912)

(2)


1198754119883

= 11988914cos 12057914

if 1205791198751

=100381610038161003816100381612057914 minus 120579

13

1003816100381610038161003816

then 1198754119884

= 11988914sin (12057914)

else 1198754119884

= minus11988914sin 12057914

(3)


12057914

= cosminus1(1198892

14+ 1198892

12minus 1198892

24)

(21198891411988912)

1205791198751

= cosminus1(1198892

14+ 1198892

13minus 1198892

34)

21198891411988913

(4)


1and 119875



1




1and 119875

2is 12057912




1



1


1has to


12



1 the WSN node


1206012= 120587 minus (120579

12minus 1206011) + 12057921 (5)


y2

y1

y2G

12057921 P2 x2G

x2

y1G

P1

12057912

x1

x1G

1206011

1206012


y2G

P2

x2G

Θ21

y1G

P1 x1GΘ12




12and Θ




Figure 4(b)

Θ21

= 120587 + Θ12

(mod (2120587)) (6)




119866


119866is

[119872119866] = [119879

119866] sdot [119877119866] (7)


[119872119866] =

[[

[

1 0 120572119909

0 1 120572119910

0 0 1

]]

]

sdot[[

[

cosΘ sinΘ 0

minus sinΘ cosΘ 0

0 0 1

]]

]

=[[

[

cosΘ sinΘ 120572119909


0 0 1

]]

]

(8)



1 1199101)

(1199092 1199102) and (119909


1 1199101)










3be the distance


119894 = 1199093minus 1199091

119895 = 1199103minus 1199101

119889 = 1199092

(9)


(0 0 0)

d1

(d 0 0)

d2

d3

(i j 0)







1198892

1= 1199092

4+ 1199102

4+ 1199112

4 (10)

1198892

2= (1199094minus 119889)2

+ 1199102

4 (11)

1198892

3= (1199094minus 119894)2

+ (1199104minus 119895)2

(12)




1199094=

1198892

1minus 1198892

2+ 1198892

2119889

1199104=

1198892

1minus 1198892

3+ 1198942+ 1198952

2119895minus

119894

1198951199094

(13)







0 10

010203040506070809

01 02 03 04 05 06 07 08 09

1

Node density

N = 100

N = 110

N = 120

Prob

abili

ty (n

ge3

)





1 2 3 4 5 6 7 8 9 108

1012141618202224

Simulation runs

Nod

e deg

ree

R = 100

R = 95

R = 90

R = 85

R = 80

R = 75

R = 70


4 6 8 10 12 14 162030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es

N = 130

N = 120

N = 100




0 5 10 15 20 25 300

102030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es







119894119895= 119889119894119895(1 + 120588 times 119899

119891) (14)





70 75 80 85 90 95 1000

5

10

15

20

25

30


Nor

mal

ized

loca

tion

erro

r (

)




120590LE =1

119873times

119873

sum

119894=1

radic(119909119894minus 119909119894)2

+ (119910119894minus 119910119894)2

119873times 100 (15)



the ith WSN node





4



3) in the worst

case




Competing Interests


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










y2

y1

y2G

12057921 P2 x2G

x2

y1G

P1

12057912

x1

x1G

1206011

1206012


y2G

P2

x2G

Θ21

y1G

P1 x1GΘ12




12and Θ




Figure 4(b)

Θ21

= 120587 + Θ12

(mod (2120587)) (6)




119866


119866is

[119872119866] = [119879

119866] sdot [119877119866] (7)


[119872119866] =

[[

[

1 0 120572119909

0 1 120572119910

0 0 1

]]

]

sdot[[

[

cosΘ sinΘ 0

minus sinΘ cosΘ 0

0 0 1

]]

]

=[[

[

cosΘ sinΘ 120572119909


0 0 1

]]

]

(8)



1 1199101)

(1199092 1199102) and (119909


1 1199101)










3be the distance


119894 = 1199093minus 1199091

119895 = 1199103minus 1199101

119889 = 1199092

(9)


(0 0 0)

d1

(d 0 0)

d2

d3

(i j 0)







1198892

1= 1199092

4+ 1199102

4+ 1199112

4 (10)

1198892

2= (1199094minus 119889)2

+ 1199102

4 (11)

1198892

3= (1199094minus 119894)2

+ (1199104minus 119895)2

(12)




1199094=

1198892

1minus 1198892

2+ 1198892

2119889

1199104=

1198892

1minus 1198892

3+ 1198942+ 1198952

2119895minus

119894

1198951199094

(13)







0 10

010203040506070809

01 02 03 04 05 06 07 08 09

1

Node density

N = 100

N = 110

N = 120

Prob

abili

ty (n

ge3

)





1 2 3 4 5 6 7 8 9 108

1012141618202224

Simulation runs

Nod

e deg

ree

R = 100

R = 95

R = 90

R = 85

R = 80

R = 75

R = 70


4 6 8 10 12 14 162030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es

N = 130

N = 120

N = 100




0 5 10 15 20 25 300

102030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es







119894119895= 119889119894119895(1 + 120588 times 119899

119891) (14)





70 75 80 85 90 95 1000

5

10

15

20

25

30


Nor

mal

ized

loca

tion

erro

r (

)




120590LE =1

119873times

119873

sum

119894=1

radic(119909119894minus 119909119894)2

+ (119910119894minus 119910119894)2

119873times 100 (15)



the ith WSN node





4



3) in the worst

case




Competing Interests


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(0 0 0)

d1

(d 0 0)

d2

d3

(i j 0)







1198892

1= 1199092

4+ 1199102

4+ 1199112

4 (10)

1198892

2= (1199094minus 119889)2

+ 1199102

4 (11)

1198892

3= (1199094minus 119894)2

+ (1199104minus 119895)2

(12)




1199094=

1198892

1minus 1198892

2+ 1198892

2119889

1199104=

1198892

1minus 1198892

3+ 1198942+ 1198952

2119895minus

119894

1198951199094

(13)







0 10

010203040506070809

01 02 03 04 05 06 07 08 09

1

Node density

N = 100

N = 110

N = 120

Prob

abili

ty (n

ge3

)





1 2 3 4 5 6 7 8 9 108

1012141618202224

Simulation runs

Nod

e deg

ree

R = 100

R = 95

R = 90

R = 85

R = 80

R = 75

R = 70


4 6 8 10 12 14 162030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es

N = 130

N = 120

N = 100




0 5 10 15 20 25 300

102030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es







119894119895= 119889119894119895(1 + 120588 times 119899

119891) (14)





70 75 80 85 90 95 1000

5

10

15

20

25

30


Nor

mal

ized

loca

tion

erro

r (

)




120590LE =1

119873times

119873

sum

119894=1

radic(119909119894minus 119909119894)2

+ (119910119894minus 119910119894)2

119873times 100 (15)



the ith WSN node





4



3) in the worst

case




Competing Interests


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2 3 4 5 6 7 8 9 108

1012141618202224

Simulation runs

Nod

e deg

ree

R = 100

R = 95

R = 90

R = 85

R = 80

R = 75

R = 70


4 6 8 10 12 14 162030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es

N = 130

N = 120

N = 100




0 5 10 15 20 25 300

102030405060708090

100

Average node degree

Perc

enta

ge o

f loc

aliz

able

nod

es







119894119895= 119889119894119895(1 + 120588 times 119899

119891) (14)





70 75 80 85 90 95 1000

5

10

15

20

25

30


Nor

mal

ized

loca

tion

erro

r (

)




120590LE =1

119873times

119873

sum

119894=1

radic(119909119894minus 119909119894)2

+ (119910119894minus 119910119894)2

119873times 100 (15)



the ith WSN node





4



3) in the worst

case




Competing Interests


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










70 75 80 85 90 95 1000

5

10

15

20

25

30


Nor

mal

ized

loca

tion

erro

r (

)




120590LE =1

119873times

119873

sum

119894=1

radic(119909119894minus 119909119894)2

+ (119910119894minus 119910119894)2

119873times 100 (15)



the ith WSN node





4



3) in the worst

case




Competing Interests


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article an algorithmic approach to wireless...

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