Abstract of thesis entitled

Localization in Wireless Sensor Networkssubmitted by

King-Yip Cheng

for the degree of Master of Philosophyat The University of Hong Kong

December 2006

Localization in wireless sensor networks is the process of determining the geographical positions of

sensors. Only some of the sensors (anchors) in the networks have prior knowledge about their geo-

graphical positions. Localization algorithms use the location information of anchors and estimates

of distances between neighbouring nodes to determine the positions of the rest of the sensors.

In this work, modifications to Ad Hoc Positioning System (APS) [7] [8] are proposed to im-

prove its performance in anisotropic networks. Only selected anchors instead of all anchors are

included in the multilateration process. The nearest three anchors that form a convex hull embed-

ding the sensor are used to localize the sensor. A heuristic-based Convex Hull Detection Method

(CHDM) is used to detect whether the anchors form a convex hull embedding the sensor. Simula-

tion results suggest that the modifications are considerably more accurate in anisotropic networks

than the original APS. The CHDM is also applicable to localization systems based on proximity-

distance map (PDM) [20].

The effects of the number and placement of anchors are also investigated. The performance

of a PDM-based localization system is severely degraded if anchors are clustered together. A

phased approach, MDS+CHDM, is proposed to alleviate the degradation. In the first phase, MDS-

MAP [11] is used to localize some nodes in the network as it shows less dependence on the number

and placement of anchors. The localized nodes then become secondary anchors. The rest of the

sensors are localized by PDM in the second phase. The phased approach is tested by extensive

simulations.

(Total words: 248)

Signed

King-Yip Cheng

Localization in Wireless Sensor Networks

by

King-Yip Cheng

B.Eng. (Information Engineering), The University of Hong Kong

A thesis submitted in partial fulfillment of

the requirements for the degree of

Master of Philosophy

(Department of Electrical and Electronic Engineering)

at

The University of Hong Kong

December 2006

i

Declaration

I declare that this thesis represents my own work, except where due acknowledgement is made,

and that it has not been previously included in a thesis, dissertation or report submitted to this

University or to any other institution for a degree diploma or other qualifications.

SignedKing-Yip Cheng

To my parents

and

my best friend, Vito

iii

Acknowledgments

I would like to take this opportunity to express my sincere gratitude to my supervisors, Dr.

King-Shan Lui (primary) and Dr. Vincent Tam and Dr. Ricky Y.K. Kwok for their time, guidance

and support given in the course of my study.

Thanks are due to Dr. Lawrence Yeung and Dr. Yi Shang for their valuable comments on this

thesis.

I would also like to thank my colleagues in High Performance Computing Research Laboratory

and Mr. Chiu Hon Sun for the fruitful discussions in the past two years. Working with them is an

unforgettable experience in my life.

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iv

Table of Contents

Page

Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vi

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ix

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Overview of Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . .11.2 Localization in WSN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7

2 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82.2 Multilateration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82.3 Multidimensional Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10

2.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102.3.2 MDS-MAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112.3.3 MDS-MAP(P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .122.3.4 Comments on MDS-MAP,MDS-MAP(P) . . . . . . . . . . . . . . . . . .14

2.4 Ad Hoc Positioning System (APS) . . . . . . . . . . . . . . . . . . . . . . . . . .152.4.1 Location Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .152.4.2 ”DV-Hop” and ”DV-distance” Propagation Methods . . . . . . . . . . . .172.4.3 ”Euclidean” propagation method . . . . . . . . . . . . . . . . . . . . . .182.4.4 Comments on APS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

2.5 Proximity-Distance Map (PDM) . . . . . . . . . . . . . . . . . . . . . . . . . . .192.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192.5.2 Derivation of PDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20

v

Page

2.5.3 Operation of PDM-based Localization System . . . . . . . . . . . . . . .222.5.4 Comments on PDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23

2.6 Convex Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .252.6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .252.6.2 Semidefinite program without relaxation . . . . . . . . . . . . . . . . . .262.6.3 Semidefinite program with relaxation . . . . . . . . . . . . . . . . . . . .282.6.4 Comments on SDP formulations . . . . . . . . . . . . . . . . . . . . . .31

3 Improving APS and PDM in Anisotropic Networks with Convex Hull DetectionMethod (CHDM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33

3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .333.2 Improving APS in Anisotropic Networks . . . . . . . . . . . . . . . . . . . . . .35

3.2.1 Selecting the Nearest 3 Anchors . . . . . . . . . . . . . . . . . . . . . . .383.2.2 Convex Hull Detection . . . . . . . . . . . . . . . . . . . . . . . . . . .413.2.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .443.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .48

3.3 Hybrid Approach: PDM+CHDM . . . . . . . . . . . . . . . . . . . . . . . . . .493.3.1 PDM+CHDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .503.3.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .513.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60

4 Localization with Limited Number of Anchors and Clustered Placement . . . . . . 61

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .614.2 Phased Approach, MDS+PDM . . . . . . . . . . . . . . . . . . . . . . . . . . .634.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67

4.3.1 Effects of anchor placement . . . . . . . . . . . . . . . . . . . . . . . . .674.3.2 Effects of the number of primary anchors . . . . . . . . . . . . . . . . . .704.3.3 Effects of measurement noise . . . . . . . . . . . . . . . . . . . . . . . .724.3.4 Effects of the number of secondary anchors . . . . . . . . . . . . . . . . .744.3.5 Effects of the position of primary anchors . . . . . . . . . . . . . . . . . .74

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78

5 Conclusions and Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81

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vi

List of Figures

Figure Page

1.1 Crossbow MICA2DOT [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

2.1 Localization in Ad Hoc Positioning System (APS) . . . . . . . . . . . . . . . . . . .16

2.2 ”Euclidean” propagation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

2.3 C-shaped topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20

2.4 Anisotropic Network with Clustered Anchors . . . . . . . . . . . . . . . . . . . . . .24

2.5 Geometrical interpretation of non-convexity of lower bound constraint in a 2-D network27

2.6 Geometrical interpretation of convex constraints in a 2-D network . . . . . . . . . . .28

2.7 Feasible set of a simplified convex program . . . . . . . . . . . . . . . . . . . . . . .29

3.1 Uniform Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34

3.2 Distribution of percentage error of measured distances of APS in uniform network . .36

3.3 Distribution of percentage error of measured distances of APS in irregular network . .37

3.4 Performance of APS with uniform topology . . . . . . . . . . . . . . . . . . . . . . .38

3.5 Performance of APS with irregular topology . . . . . . . . . . . . . . . . . . . . . .39

3.6 Performance of modified APS with irregular topology . . . . . . . . . . . . . . . . .40

3.7 Flipping of position estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42

3.8 The spatial relationship of involved anchors and nodex . . . . . . . . . . . . . . . . . 43

3.9 Performance of APS and modified versions under different degree of connectivity . . .45

vii

Figure Page

3.10 Performance of APS and modified versions with different numbers of anchors, aver-age connectivity = 11.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46

3.11 Performance of APS and modified versions with different numbers of anchors, aver-age connectivity = 18.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47

3.12 Performance of APS and modified versions with different degrees of measurement errors47

3.13 Performance of APS with irregular topology . . . . . . . . . . . . . . . . . . . . . .49

3.14 Flow chart of PDM+CHDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52

3.15 Accuracy gained from CHDM with different degrees of connectivity . . . . . . . . . .53

3.16 Number of nodes using DV-distance with different degrees of connectivity . . . . . .53

3.17 Accuracy gained from CHDM with different number of anchors . . . . . . . . . . . .54

3.18 Number of nodes using DV-distance with different degrees of measurement errors . .55

3.19 Accuracy gained from CHDM with different noisy factor . . . . . . . . . . . . . . .55

3.20 Number of nodes using DV-distance with different degrees of measurement errors . .56

3.21 Sensor near the boundary of a convex hull . . . . . . . . . . . . . . . . . . . . . . . .57

3.22 Accuracy gained from CHDM with different value ofk . . . . . . . . . . . . . . . . . 58

3.23 Number of nodes using DV-distance with different values ofk . . . . . . . . . . . . . 59

4.1 50-node network with uniform topologies, 10 anchors,α=0.05 . . . . . . . . . . . . . 63

4.2 200-node network with uniform topologies,α=0.05 . . . . . . . . . . . . . . . . . . . 64

4.3 Effects of anchor placement in uniform networks . . . . . . . . . . . . . . . . . . . .68

4.4 Effects of anchor placement in C-shaped networks . . . . . . . . . . . . . . . . . . .69

4.5 Effects of the number of clustered anchors . . . . . . . . . . . . . . . . . . . . . . . .71

4.6 Effects of measurement noise on uniform networks . . . . . . . . . . . . . . . . . . .72

viii

Figure Page

4.7 Effects of measurement noise on C-shaped networks . . . . . . . . . . . . . . . . . .73

4.8 Effects of the number of secondary anchors . . . . . . . . . . . . . . . . . . . . . . .75

4.9 5 anchors clustered at the tip of ’C’ . . . . . . . . . . . . . . . . . . . . . . . . . . .76

4.10 5 anchors,α=0.05, C-shaped Networks . . . . . . . . . . . . . . . . . . . . . . . . .77

ix

Abstract

Localization in wireless sensor networks is the process of determining the geographical posi-

tions of sensors. Only some of the sensors (anchors) in the networks have prior knowledge about

their geographical positions. Localization algorithms use the location information of anchors and

estimates of distances between neighbouring nodes to determine the positions of the rest of the

sensors.

In this work, modifications to Ad Hoc Positioning System (APS) [7] [8] are proposed to im-

prove its performance in anisotropic networks. Only selected anchors instead of all anchors are

included in the multilateration process. The nearest three anchors that form a convex hull embed-

ding the sensor are used to localize the sensor. A heuristic-based Convex Hull Detection Method

(CHDM) is used to detect whether the anchors form a convex hull embedding the sensor. Simula-

tion results suggest that the modifications are considerably more accurate in anisotropic networks

than the original APS. The CHDM is also applicable to localization systems based on proximity-

distance map (PDM) [20].

The effects of the number and placement of anchors are also investigated. The performance

of a PDM-based localization system is severely degraded if anchors are clustered together. A

x

phased approach, MDS+CHDM, is proposed to alleviate the degradation. In the first phase, MDS-

MAP [11] is used to localize some nodes in the network as it shows less dependence on the number

and placement of anchors. The localized nodes then become secondary anchors. The rest of the

sensors are localized by PDM in the second phase. The phased approach is tested by extensive

simulations.

1

Chapter 1

Introduction

1.1 Overview of Wireless Sensor Networks

With the advance of technology in integrated circuitry, computational devices become smaller,

faster and cheaper. Together with the maturing wireless communication technology, a new net-

working paradigm, wireless sensor network, is introduced. It draws much attention from re-

searchers and industries in recent years because it provides an unprecedented application scope

for users and challenges for researchers.

A sensor network is an ad-hoc network composed of hundreds or even thousands of nodes.

Nodes are capable of sensing at least one phenomenon in the environment, for instance, light in-

tensity, temperature, humidity, seismic waves, etc. Besides sensing the environment, sensor nodes

also have computational power and memory to process the data. Information can be exchanged

between sensors through wireless communication links. Sensors are powered by batteries and

are expected to have a long life-time since recharging is difficult and sometimes impossible for

networks deployed in hostile environment. The low cost of sensors makes them disposable and

suitable for large-scale deployment. Figure 1.1 shows a sensor of series MICA2DOT which is now

commercially available [1]. Although the advancement of hardware is mature enough for sensor

network to be deployed in real world, many issues in software and algorithms for sensor networks

have yet to be addressed.

In recent years, much research effort has been spent in developing different services to make

different sensor network applications realizable. These services take in routing, data processing,

2

Figure 1.1 Crossbow MICA2DOT [1]

3

scheduling, localization, key management, cryptography, etc. Even though counterparts had been

studied for a long time for mobile ad-hoc network formed by handheld devices, modification of

existing algorithms or new algorithms have to be devised to handle the unique constraints imposed

by wireless sensor networks. These constraints include scalability, self-configurability of network,

limited energy of sensor node, limited computational power and memory of sensor node and other

application-specific constraints. This research focuses on one of the services that is critical to many

applications of sensor networks, i.e. localization.

1.2 Localization in WSN

In many applications of wireless sensor networks, precise location information of sensor nodes

is critical to the success of the applications. Most data collected from sensors are only meaningful

when they are coupled with the location information of the corresponding sensors. Consider an

application of habitat monitoring. Thousands of sensors are dropped in the targeted region of

a tropical rain-forest by an aeroplane. Nodes are equipped with sensing devices to monitor the

changes of temperature and humidity of the environment. To make every measurement useful to

scientists, the location where measurements are taken has to be known.

Localization in wireless sensor networks is to determine the geographical positions of sensors

in a wireless sensor network.

The most trivial solution is manual configuration. The location of each sensor is predetermined

before deployment. Sensors are installed to the assigned locations by human. Obviously, this

solution is inscalable as much labour is required for the installation. Furthermore, it is sometimes

infeasible to have manual configuration as the location information of sensors is unknown before

actual deployment. Recalled the previous example of habitat monitoring, sensors are dropped from

an aeroplane which exact locations are only known when sensors land on the forest.

Another solution for localization is equipping every sensor with a GPS receiver. Sensors can

locate themselves individually using the GPS signals. However, installing a GPS receiver for every

sensor node greatly increase the total cost of the sensor network. In addition, the introduction of

GPS receiver increases the energy consumption of a node and hence shortens its life time. Lastly,

4

the location obtained from GPS-receiver may not be precise enough for certain applications and

the accuracy of GPS is affected by various environmental factors. Accuracy can be of tenths of

meters for general GPS. The error can be lowered to less than ten meters for GPS augmentation

systems like Differential GPS (DGPS) but with a higher cost.

In view of the inadequacy of manual configuration and employment of GPS-receiver, re-

searchers propose a framework for localization in wireless sensor networks. In a sensor network,

some of the sensor nodes have prior knowledge about their locations, either through GPS or man-

ual configuration. They are called anchors or beacons. Other nodes that do not have location

information infer their positions by making use of the location information of anchors and other

information available in the network, e.g. measured distance between neighbours, connectivity,

etc.

To measure the distance between neighbouring nodes, each sensor has to be equipped with

a ranging device. There are several ways to measure the distance between two sensors. Since

each sensor is equipped with wireless communication capability in a wireless sensor network, the

strength of received signals from neighbours can be used to estimate the corresponding distances.

Localization algorithms can be roughly classified into three categories based on the mathemat-

ical background. The most prevalent method is trilateration or multilateration.

In a Euclidean plane, if the coordinates of three non-collinear points are known, the coor-

dinates of a pointx can be determined when Euclidean distances between the pointx and the

three non-collinear points are known. However, distance measurements are always subjected to

different kinds of errors in real world, exact solution cannot be directly calculated. In practice,

the coordinates of the point are estimated by minimizing the squared errors between the distance

measurements and the distances between the estimated coordinates of pointx and those of the

three non-collinear points. With multilateration, a sensor can estimate its position after measuring

the distance from at least three anchors. Unfortunately, not every sensor can directly measure the

distance between itself and the anchors. Thus, the essence of localization algorithms based on

multilateration is how to obtain precise distance estimates between anchors and sensors.

5

Another approach is formulating the localization problem as an optimization problem. By

taking in different kinds of constraints, the problem can be formulated as convex programming,

semidefinite programming or linear programming. These problems can be readily solved by ex-

isting solvers efficiently. Since solving optimization problems usually requires intensive computa-

tions, these algorithms are mostly centralized in nature.

Another technique used in localization is multidimensional scaling (MDS) [2] [3]. MDS has its

origin in psychometrics and psychophysics. It is a set of data analysis techniques which transform

distance-like data as coordinate-like data, i.e. a point in a multidimensional space. By ignor-

ing less-significant dimensions, data can be represented as a map in two-dimensional or three-

dimensional space which visualise the relations between data. The nice property of displaying

distance-like data into geographical picture makes MDS a very good tool for localization. The

toughest requirement of MDS is how to obtain all the distance estimates between every pair of

sensors.

In Chapter 2, a treatment of the mathematical backgrounds will be given. Some of the most

representative algorithms will also be discussed.

1.3 Contributions

Although numerous localization algorithms have been proposed in these years, there is a lack

of comparison between different algorithms. In this research work, we implement various repre-

sentative algorithms like Ad Hoc Positioning System (APS) [7] [8], MDS-MAP [11], PDM [20],

etc. We compare their performance against each other under different conditions. We study the

effect of anchor ratios, anchor placement, connectivity, measurement errors and irregularity of net-

works on the accuracy of localization algorithms through extensive simulations. The strengths and

weaknesses of each algorithm are identified. Because of the wide range of sensor network applica-

tions, one sensor network can be significantly different from another one in terms of connectivity,

network topologies, network lifetime requirement, computational power etc. This means that a

universal localization algorithm has to be able to cope with different conditions. However, we find

no absolute winner in our simulations. There is no single algorithm that always gives the most

6

accurate location estimates in all scenarios. Furthermore, accuracy should not be the only metric

used to evaluate an localization algorithm. An algorithm that merely gives accurate result may not

be suitable at all. Besides accuracy, complexity and mode of operation are also important.

Based on our study, we propose modifications on existing algorithms aiming at improving their

performance. In addition, we propose a hybrid approach in localization by incorporating two dif-

ferent localization algorithms which complement each other. Nodes choose their own localization

algorithm by checking their local conditions. From our simulation results, we also find that existing

algorithms either do not perform well with a limited number of anchors and clustered placement

or that is too complex to be employed in a wireless sensor network. In view of this, we design a

localization algorithm which includes two phases. The first phase employ multidimensional scal-

ing which is found to be less sensitive to the number and placement of anchors. However the

computation and communication overheads are too large if the whole network is localized by mul-

tidimensional scaling. The major objective of the first phase is localizing a portion of sensors and

turn them into secondary anchors. Secondary anchors differ from the original anchors deployed

in accuracy of their position information. Nonetheless, the introduction of secondary anchors can

improve the performance of localization in second phase which takes in a more scalable localiza-

tion algorithm. Simulation has been run to justify the phased approach. The contributions of this

thesis can be summarized as follows:

• Representative algorithms like APS, MDS-MAP and PDM-based localization system are

implemented.

• Improving APS with selected anchors with Convex Hull Detection Method (CHDM).

• A hybrid approach in localization by incorporating APS and PDM-based localization system

is proposed .

• A phased approach for localization with limited number of clustered anchors is proposed.

7

1.4 Thesis Organization

In Chapter 2, the mathematical background and some representative algorithms will be dis-

cussed. These algorithms includes APS [7] [8], PDM [20] and mathematical programming based

algorithms [15] [18] [19]. The details behind multilateration and multidimensional scaling will

be revealed. The formulations of localization problem into different mathematical programming

problems will also be given. In Chapter 3, we discuss our proposed modifications to an existing

algorithm called Ad Hoc Positioning System (APS) aiming to improve its accuracy in anisotropic

networks. We introduce a heuristic-based method called Convex Hull Detection Method (CHDM)

to help improving the APS. We also discuss an hybrid approach towards localization which incor-

porates our modified APS and a localization algorithm based on proximity-distance map (PDM).

We present the simulation results of the proposed schemes and discuss the findings from the sim-

ulations. In Chapter 4, the issue of number and placement of anchors will be addressed. The

proposed phased localization algorithm will be introduced afterwards. Simulation results are given

to justify the proposal. Conclusions and future research directions are given in Chapter 5.

8

Chapter 2

Backgrounds

2.1 Overview

In recent years, numerous localization algorithms [7]- [22] have been proposed. In this chap-

ter, the mathematics behind localization will be firstly described. Afterwards, some of the most

representative localization algorithms are introduced. These include Ad Hoc Positioning Sys-

tems(APS) [7] [8], proximity-distance map (PDM) [20], MDS-MAP [11], localization using math-

ematical programming [15] [18] [19].

2.2 Multilateration

With error-free distance measurements from four anchors which are not coplanar, a sensor can

determine its position uniquely in 3-D space. For 2-D space, distance measurements from three

non-collinear anchors are sufficient. Let (u, v, w) be the coordinates of the sensor which we want to

localize. Let (xi, yi, zi) be the coordinates of the anchori anddi be the range measurement between

anchori and the sensor. Assume range measurements fromm anchors are available. Fori=1 tom,

the relationships between the ranges and positions of sensor and anchors are given below:

(u− xi)2 + (v − yi)

2 + (w − zi)2 = di

2 (2.1)

By expanding equation 2.1 and subtracting equation ofi=1 from the rest of equations,m-1 linear

equations can be obtained. Fori=2 tom, we have

2[u(xi− x1) + v(yi− y1) + w(zi− z1)] = d12− di

2 + (xi2 + yi

2 + zi2)− (x1

2 + y12 + z1

2) (2.2)

9

The set of linear equations can be written in matrix form ofAs = b where

s =

u

v

w

,

A =

2u(x2 − x1) 2v(y2 − y1) 2w(z2 − z1)

2u(x3 − x1) 2v(y3 − y1) 2w(z3 − z1)...

......

2u(xm − x1) 2v(ym − y1) 2w(zm − z1)

and

b =

d12 − d2

2 + (x22 + y2

2 + z22)− (x1

2 + y12 + z1

2)

d12 − d3

2 + (x32 + y3

2 + z32)− (x1

2 + y12 + z1

2)...

d12 − dm

2 + (xm2 + ym

2 + zm2)− (x1

2 + y12 + z1

2)

.

If m = 4, A is full rank (anchors are not coplanar) and range measurements are error-free, a

unique and exact solution fors can be obtained by determining the inverse ofA.

s = A−1b

However, distance measurements are always corrupted by certain kind of noise andA may be

rank deficient. Thus, the sensor positions is determined by solving the least square minimization

problem of a system of linear equations.

||As− b||

The minimization problem can be solved with analytic solution

s = A+b

whereA+ is theMoore Penrose inverseof A.

10

2.3 Multidimensional Scaling

2.3.1 Overview

Multidimensional Scaling (MDS) [2] [3] is a set of statistical techniques originally for an-

alyzing the structure of (dis)similarity of data. Through MDS, the (dis)similarities of data are

represented as distances between points. The relations can be visualized as a configuration on a

plane or three-dimensional space. The visualization may give data analyst more insights about the

data than merely inspecting the numerical values.

There are different types of MDS. The classification depends on the type of the distance and

the number of the distance matrices. Distance data can be metric or non-metric. For metric data,

every distance between two points can be expressed as a numeric value. For non-metric data,

only greater-than or less-than relation existed between data. In this section, we only focus on

classical MDS which only uses metric distance data and one distance matrix. Below shows how to

determine the coordinates of sensors from the distance matrix.

Assume there aren sensors situated in ap-dimensional space. Let the coordinates of sensor

i be xi wherexi = (xi1, . . . , xip)T . Without loss of generosity, assumeX = 0 whereX is the

coordinates matrix. The distance matrixD is a square matrix whose entrydij is the Euclidean

distance between nodei and nodej as,

d2ij =

p∑a=1

(xia − xja)2 =

p∑a=1

(x2ia + x2

ja − 2xiaxja) (2.3)

Furthermore, letBn×n be a square matrix where

B = XXT (2.4)

and

bij =

p∑a=1

xiaxja = xTi xj (2.5)

Thus,d2ij of Equation 2.3 can be expressed as

d2ij = bii + bjj − 2bij (2.6)

11

Since∑n

i=1 bij = 0, by summing Equation 2.6 overi, j andi andj, we have

1

n

n∑i=1

d2ij =

1

n

n∑i=1

bii + bjj (2.7)

1

n

n∑j=1

d2ij =

1

n

n∑j=1

bjj + bii (2.8)

1

n2

n∑i=1

n∑j=1

d2ij =

2

n

n∑i=1

bii (2.9)

Solvingbij from 2.6 to 2.9 gives

bij = −1

2(d2

ij −1

n

n∑i=1

d2ij −

1

n

n∑j=1

d2ij +

1

n2

n∑i=1

n∑j=1

d2ij) (2.10)

Now, giving a distance matrix, one can determine the matrixB by Equation 2.10. In addition to

this, we can obtain the matrixX by eigen-decomposition ofB.

B = XXT = QΛQT (2.11)

where columns ofQ are the eigenvectors ofB andΛ is a diagonal matrix whose diagonal compo-

nentsλi are the eigenvalues ofB. It can be shown that the eigenvalues ofB must be positive and

the coordinates matrixX is given by

X = QΛ1/2 (2.12)

andΛ1/2 is a diagonal matrix with diagonal elements√

λi.

2.3.2 MDS-MAP

The previous section shows how to recover a coordinate matrix from a distance matrix, but

more have to be done to apply multidimensional scaling in localization for wireless sensor net-

works. Shanget al. [11] proposed a localization algorithm, MDS-MAP, which is based on multidi-

mensional scaling. The essence of MDS-MAP is the estimation of the distance matrixD in a sensor

network. There are two approaches in estimating the matrixD in MDS-MAP. The choice depends

on the precision of the connectivity information. If sensors can measure the distances between

12

their one-hop neighbours, the matrixD becomes the shortest-path distance matrix wheredij is the

shortest-path distance between nodei and nodej. If no distance measurement is available, the en-

try dij represents the shortest-path hop count between nodei and nodej. The shortest-path distance

can be calculated by the well-known Dijkstra algorithm [23] or Bellman-Ford algorithm [24].

If the distance matrix is error-free, one can recover matrixX with the analytic solution in previ-

ous section. In practice, the distance measurements are always perturbed by noise and the solution

obtained will be of higher dimensions than desired one. Therefore, only firstk (k < m) eigen-

vectors correspond to thek-largest eigenvalues are retained. This results in a lower dimensional

representations and it is the best lower-rank approximation in the least-square sense. The choice

of value ofk depends on the applications. If sensors are placed in a plane, the configuration are

recovered from the first two eigenvectors and eigenvalues. If sensors are placed in a 3-dimensional

space, first three eigenvectors and eigenvalues are used.

It should be noted that the output of MDS only gives a configuration reference to the Cartesian

coordinate system whose origin is the centroid of coordinates matrix. Unless the sensor coordi-

nates are also being referenced to the same coordinate system,X only gives us a relative map. This

should not be surprising since only distance information is provided. Thus, the coordinates matrix

X has to undergo a linear transformation to match with the coordinates system we are working

with. Geometrically, the relative map is shifted, reflected or rotated. To determine the transfor-

mation, one has to know the precise positions of some points in the desired coordinate system.

Therefore, anchors are required. In a 2-D network, at least three anchors are required to obtain the

linear transformation. Four anchors are required in any 3-D network.

2.3.3 MDS-MAP(P)

The performance of MDS-MAP solely depends on how accurate the Euclidean distance ma-

trix is approximated by the shortest-path distance (hop count) matrix, but it is obvious that the

shortest-path distance matrix never reflects the true Euclidean distance unless all nodes are mu-

tually connected. If the Euclidean distance matrix could be obtained by some means, the linear

transformation in the final step would not affect the accuracy. Therefore, the sources of error are

13

the discrepancy between the shortest-path distance (hop count) matrix and the Euclidean distance

matrix. Furthermore, the discrepancy depends on the shape of the underlying networks and the

errors in measuring the distances of neighbouring nodes (if the shortest-path distance is used). The

shortest-path distance will be much larger than the Euclidean distance when network is irregular in

shape. This greatly affects the accuracy of MDS-MAP. In view of this, Shanget al. [12] proposed

a modified version called MDS-MAP(P).

Unlike MDS-MAP for which only one global relative map is calculated for the whole network,

network is divided into regions and a local map is calculated for each region in MDS-MAP(P).

This can avoid the use of shortest-path distances of nodes that are far apart. Shanget al. suggested

to include neighbours within two hops into the local map when the average number of neighbours

is above 12. The operation of MDS-MAP(P) can be considered as applying MDS-MAP separately

for each sub-network. The local relative maps obtained are then refined by minimizing the least

squared errors between the measured distances and the distance calculated from the MDS configu-

ration of nearby nodes. Assume there areN nodes in the relative map, the following least squared

errors are minimized:

minx

∑i,j,i<j

wij(||xi − xj|| − pij)2 ∀i, j ∈ {1, . . . , N} (2.13)

wherewij is the weight,xi is the relative coordinates obtained from MDS andpij is the shortest-

path distance between nodei and nodej or the hop count between nodei and nodej multiplied

by the average hop size. Less weights (wij < 1) are given to nodes that are separated by large hop

counts as their shortest-path distancepij is less reliable. The refined local maps are merged together

to obtain the global map. The final step is the same as MDS-MAP. The relative coordinates are

transformed into absolute coordinates with the location information of anchors. Another variant

MDS-MAP(P,R) was also proposed which is identical to MDS-MAP except the refinement for

local maps is also applied after the global map is obtained.

14

2.3.4 Comments on MDS-MAP,MDS-MAP(P)

MDS-MAP and its variants have the advantage of being less dependent on the number of an-

chors [25]. As stated previously, the position information of anchors is only used in determining the

linear transformation for the relative map. It has nothing to do with the ”shape” of the relative map.

Empirical study [11] suggested that only four anchors are enough for networks whose nodes are

randomly placed in a grid. Further increase in the number of anchors does not provide significant

improvement on the accuracy. On the other hand, MDS-MAP and its variants require all-pair-

shortest-path distance between sensors to perform the localization process. For MDS-MAP, the

distance matrix consists of distance estimates between all nodes of the network. It needs global

information. It also requires substantial communications between sensors to obtain the distance

measurements and hence consuming a lot of energy from sensors. Transmitting information in the

air is considered as the most expensive operation in terms of energy consumption [6]. In addition

to this, MDS-MAP is a centralized algorithm.

Though Shanget al. claimed that improved versions of MDS-MAP can be implemented in

a distributed fashion by dividing the global map into small local maps. The local map is small

enough (consisting 2-hop neighbours only) that individual nodes can handle the computation and

communication requirements. Nonetheless, the map merging process introduced in MDS-MAP(P)

increases the computation and communication requirements. It also raises a question in scheduling

the process.

• Which node should be responsible for the map merging process?

If a random node is chosen, it is virtually centralized computation. All position estimates

will eventually be stored in the selected nodes. The selected node and neighbouring nodes

will become hotspots for communication. Lifetime of these nodes will be shortened.

If the map merging process is distributed to different nodes by some means, details of dif-

ferent local maps have to be transferred from one node to another node. The communication

cost will be even higher than that without load-balancing.

15

Furthermore, the determination of the linear transformation is fundamentally a centralized process

fundamentally as it requires the global map. Therefore, we considers MDS-MAP and its variants

as a family of centralized localization algorithms.

2.4 Ad Hoc Positioning System (APS)

Niculescuet al. [7] [8] propose a distributed localization system called Ad Hoc Positioning

System (APS). It is one of the most representative works in localization in wireless sensor network.

The significance of APS lies in its simplicity and decentralized operation. Even nodes are not

capable of measuring distance between themselves, they can localize itself individually.

2.4.1 Location Estimation

APS adopts multilateration that is similar to the Global Positioning System (GPS). Since

the localization process of GPS requires time synchronization between the receiver’s clock and

the satellites’ clocks, the process is much complicated than the mathematical treatment given in

Section 2.2. However in Ad Hoc Positioning System, there is no need for time synchronization,

the localization process can be linearized as follows:

Figure 2.1 shows the essence of localization in APS.ru is the estimated location,ru is the real

location of sensor,ρi is the estimated distance from anchori andρi is the true distance from the

anchori. 1i is the unit vector ofρi.

ρi = |ri − ru|+ εi (2.14)

ρi = |ri − ru|+ εi (2.15)

1i = − ri − ru

|ri − ru| (2.16)

∆r = ru − ru (2.17)

The position estimate of sensor is iteratively refined by solving a linear system (equation 2.19)

with the linearly approximated distance correction∆ρi for each anchori.

∆ρi = ρi − ρi ' −1i ·∆r + ∆ε (2.18)

16

ur

����� �

�

ur

r∆

ir

iρ

iρ

ρ∆

� !"#$

%&�'

(#)

&)*+,-*./0

(#)

ˆ , estimated,real location of sensor

ˆ , estimated, real distance

ˆ ˆ| |, | |

ˆˆ 1

u u

i i

i i u i i u

i i i

r r

r r r r

r

ρ ρ

ρ ρ

ρ ρ ρ

= − = −

∆ = − − ⋅∆!

Figure 2.1 Localization in Ad Hoc Positioning System (APS)

17

∆ρ1

∆ρ2

...

∆ρn

=

11x 11y

11x 11y

......

1nx 1ny

∆x

∆y

(2.19)

For a sensor to localize itself, it has to obtain the range estimates and absolute positions of at

least three (four) anchors in 2-D (3-D) network. Since not all sensors has three or four anchors

as their one-hop neighbours, some propagation methods have to be used to route the distance

estimates and position information of anchors to sensors which cannot directly communicate with

anchors. Niculescuet al. devise three propagation methods, namely, ”DV-Hop”,”DV-distance” and

”Euclidean” propagation method.

2.4.2 ”DV-Hop” and ” DV-distance” Propagation Methods

”DV-Hop” and ”DV-distance” are very similar to each other in the sense that both use classical

distance vector exchange to propagate the distance information. ”DV-Hop” works as follow:

Every node including anchor keeps a table with entries of{Xi,Yi,hi}. Each entry corresponds

to the coordinates of anchor (Xi,Yi) and the hop count away from the anchor (hi). When the

localization process starts, the tables are empty for ordinary sensors and there is one entry in

each anchor which corresponds to the anchor itself. Packets bearing the corresponding location

information and the hop count (initialized as zero) are distributed by each anchor. Upon receiving

the packet, node stores the coordinates of the anchors and compares the hop count field of the

packet (incremented by one) with that of the table. If the incremented hop count field of the

packet is smaller, the value in the table is updated and packets with the updated hop count field are

forwarded to other one-hop neighbours. Otherwise, the packet is simply discarded. Eventually all

nodes will know the positions of all anchors. Furthermore, the hop count fields in the table reflect

the hop count of shortest path from the corresponding anchors. After an anchori has collected

the coordinates of other anchors, it calculates the correction factorci to determine the average hop

size.

ci =

∑ √(Xi −Xj)2 + (Yi − Yj)2

∑hij

(2.20)

18

A

C

B

D

A’

Figure 2.2 ”Euclidean” propagation method

Sensors then obtain the correction factor from the nearest anchor. Since every sensor has the hop

count of the shortest path from anchors, it can calculates the approximated distance from anchors

by multiplying the hop count and the correction factor. The position of sensors can be estimated

by the procedures given in 2.4.1.

The operation of ”DV-distance” is identical to that of ”DV-Hop” except range measurements

are used in former and hop counts are used in latter. Thushi in the table represents the shortest-path

distance from anchori in ”DV-distance”.

2.4.3 ”Euclidean” propagation method

The third propagation method is ”Euclidean” propagation method. As its name implies, the

true Euclidean distance is propagated instead of path distance or hop count. If an nodeA cannot

directly measure the Euclidean distances from an anchorD (as shown in Figure 2.2), it has to have

at least two neighbours,B andC which already have their Euclidean distances from anchorA. The

Euclidean distance betweenA andD can be calculated by simple trigonometry. However there are

two solutions,A andA′ which yield different Euclidean distances fromD. The final decision is

made locally byA by comparing the Euclidean distance of other one-hop neighbours fromB, C

or D.

19

2.4.4 Comments on APS

Ad Hoc Positioning System provides a truly distributed framework of localization for wireless

sensor networks. Each node can determine its location individually after hop counts and anchor

coordinates exchanges. With ”DV-Hop”, even sensors without ranging capability can localize

themselves. However, the use of correction factor (average hop size) implies that the accuracy

greatly depends on the topologies of the sensor networks. If the variation of hop size is large, the

estimates of distances from anchors will be unreliable and hence the position estimates. In addition,

the shortest path between an anchor and a sensor may be diverted because of obstacle or irregularity

of the network shape. The distance estimates are overestimated. ”DV-distance” also suffered

from the overestimation. Although ”Euclidean” propagation method utilizing the true distance can

improve accuracy comparing to ”DV-Hop” and ”DV-distance, it is possible that some nodes cannot

determine the Euclidean distances from at least three anchors because of insufficient immediate

neighbours. Furthermore, there are error propagation as distance measurements are propagated

from the anchors to remote nodes which cannot directly communicate with anchors. In summary,

the Ad Hoc Positioning system gives fairly good accuracy when the network is regular in shape

and there are a high ratio of anchors to reduce the negative effect brought by error propagation.

2.5 Proximity-Distance Map (PDM)

2.5.1 Overview

Lim et al. [20] proposed a localization system for anisotropic networks based on proximity-

distance map (PDM). The system is based on Ad Hoc Positioning System where multilateration

is used. But a refinement process is introduced to make the estimation of distances from anchors

more accurate. For a sensor network withN nodes, one assumes that there exists a mapping

fp : <2d → < which maps two geographical locations into a measured proximitypij whered is the

dimension of the space. The network is isotropic iffpxi, xj is a function of the Euclidean distance

dij, gp : < → <pij = fp(xi, xj) = gp(dij), ∀i, j ∈ {1, 2, . . . , N}

20

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9

position(1,i)

positio

n(2

,i)

Sensor Positions

Figure 2.3 C-shaped topology

According to the definition, one of the anisotropic topologies is the C-shaped topology when the

hop count or the shortest-path distance between two nodes is used as proximity. Figure 2.3 shows

an instance of C-shaped topology. Recalled the operations of ”DV-Hop”, the proximity, hop count

is multiplied by the correction factorci to estimate the Euclidean distance. Thus we are assuming

the network is isotropic in nature. The mappinggp(dij) = dij/ci is used. For ”DV-distance”, the

Euclidean distance is approximated by the corrected shortest-path distance. The mappinggp(dij) =

dij is used. This suggests another explanation to the poor performance when APS is used in

networks with irregular shapes. Because of the irregular network shape, the measured proximities

differs in various directions or it depends on the geographical locations of the sensor pair,xi and

xj.

2.5.2 Derivation of PDM

In view of this, Limet al. suggests using the relationship between proximities and geographical

distances of anchors to analyze that of the overall network. A linear transformationT called

proximity-distance map (PDM) is devised to preprocess the measured proximities. In a sensor

21

network withM anchors, letpi be the proximity vector of anchori.

pi = [pi1, . . . , piM ]T (2.21)

wherepij is the measured proximity between anchori and j andpii = 0. The vectors of the

anchors form a proximity matrixP .

P = [p1, . . . , pM ] (2.22)

P is a M × M square matrix with zero diagonal entries. The matrix of geographical distance

between anchors are defined similarly.

li = [li1, . . . , liM ]T (2.23)

L = [l1, . . . , lM ] (2.24)

L is a symmetric square matrix with zero diagonal entries.

T is the optimal linear transformation which maps the proximities into the geographical dis-

tances. The transformation is optimal in the sense that the squared errore is minimized.

ei =M∑

k=1

(lik − tipk)2

= ||liT − tiP ||2

The least squared error problem has an analytic solution,

ti = liT P T (PP T )−1 (2.25)

and the PDM,T , is

T = LT P T (PP T )−1 (2.26)

T can be obtained by calculatingP+, the Moore-Penrose inverse ofP . First, P is decomposed

into three matrices by singular value decomposition (SVD) [5].

P = U ·

∑0

0 0

· V T (2.27)

22

U andV are orthogonal matrices and∑

is a diagonal matrix whose diagonal entries (σ1 . . . σr) are

called singular values ofP . All singular values are positive and the number of singular values,r,

is equal to the rank of matrixP .

P+ = P T (PP T )−1 = V ·

∑−1 0

0 0

· UT =

r∑i=1

1

σi

viuTi (2.28)

The calculation ofP+ requires the reciprocals of singular values. Since the measured proximity

is usually corrupted by noise, inverting singular values close to zero may excite the noise. Thus

truncated pseudo inversePγ is used which only includes singular values up to a certain threshold

τ .

P+γ =

k∑i=1

1

σi

viuTi (2.29)

∑k−1i=1 σi∑ri=1 σi

< τ <

∑ki=1 σi∑ri=1 σi

(2.30)

2.5.3 Operation of PDM-based Localization System

The operation of localization system based on PDM is similar to ”DV-Hop” and ”DV-distance”

of APS. Anchors first exchange their location information. Through packet exchange like APS, an

anchori also obtains the measured proximities from other anchors. Each anchor now possesses its

own proximity vectorpi. The measured proximities can be the hop count or shortest-path distance

which depends on the sensor network deployed. However, only a proximity vector cannot derive

the mappingT . After determining its proximity vectors, each anchor has to exchange their vectors

to the other(M−1) anchors to form the proximity matrixP . Each anchor then calculates the trans-

formationT locally and distributes it to the nearby sensors. Since sensors have already obtained

the measured proximities and location information for all anchors during the packet exchange at

the earlier stage, upon receivingT , sensors can process their measured proximitiesps:

ls = Tps = LP+γ ps (2.31)

The position estimate can be obtained by performing the procedures given in Section 2.4.1 withls

instead of measured proximityps.

23

2.5.4 Comments on PDM

PDM aims to capture the dynamic relationship between the measured proximities and geo-

graphical distances. The relationship is assumed to be linear and is approximated by a linear

transformation. With the transformation, sensor can have a better estimation of geographical dis-

tances by transforming the measured proximity. From simulation result, the linear transformation

gives a good approximation of the relationship between the measured proximities and geographi-

cal distances in anisotropic networks. With irregular topologies, vast improvement in accuracy is

obtained when it is compared with that of APS. Furthermore, the operation of localization system

based on PDM remains decentralized.

Even though all these nice properties make PDM-based localization system a very good candi-

date in localization in wireless sensor networks, several points have to be noted. Comparing with

APS, the introduction of PDM incurs overheads in computation and communication. The com-

putational overheads are originated from the calculation of the mapping which requires singular

value decomposition. SVD has a computational complexity ofO(M3) whereM is the number of

anchors. For communication overheads, the major source is the exchange of proximity vectorspi

between anchors. It contributes at leastM × (M − 1) packet exchanges. Furthermore, the success

of PDM relies on an assumption:anchors are distributed across the network.

Since anchors are used to capture the characteristics of geographical distances and proximities

of the network, anchors have to be spread throughout the network to gain the global information.

If anchors are clustered within a region, only characteristics within that particular region can be

embedded in the transformation. Figure 2.4 shows one instance of sensor network with clustered

anchors. Since the anchors (¦) are clustered in the bottom left corner, the mapping only retains

characteristics of the bottom left corner. If the mapping is applied in other regions, for example,

the bottom right corner, because of the dynamic relationship between measured proximity and

geographical distance, the mapping no longer provides a reliable approximation. Blindly applying

the mapping across the network will degrade the accuracy of nodes not residing in the bottom left

corner. In Chapter 4, a detailed discussion and a remedy of this problem will be given.

24

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9

position(1,i)

posi

tion(

2,i)

Sensor Positions

Figure 2.4 Anisotropic Network with Clustered Anchors

25

2.6 Convex Programming

2.6.1 Overview

Besides solving localization problem individually for each sensor, researchers also took a

global approach by considering localization problems of all sensors at the same time. The con-

nectivity information of all sensors are collected and formulated as various constraints. The local-

ization problem can be stated as a quadratic position estimation problem [18] [19] to find sensors

positionsxi satisfying following constraints:

||xi − xj||2 = (dij)2, ||xi − ak||2 = (dik)

2,∀(i, j), (i, k) ∈ Ne

||xi − xj||2 ≥ (rij)2, ||xi − ak||2 ≥ (rik)

2, ∀(i, j), (i, k) ∈ Nl

||xi − xj||2 ≤ (rij)2, ||xi − ak||2 ≤ (rik)

2, ∀(i, j), (i, k) ∈ Nu (2.32)

whereak is the known position of anchorsk. Ne is the set of node pairs,(i, j) and(i, k), where

Euclidean distance estimates,dij, between sensorsi and j or dik between sensori and anchor

k, are available. Similarly,Nl is the set of node pairs,(i, j) and (i, k), where lower bounds of

separationrij, rik between sensors and anchors are available.Nu is the set of node pairs,(i, j) and

(i, k), where upper bounds of separationrij, rik between sensors and anchors are available. Since

distance measurements are always affected by noise, findingxi satisfying all these constraints is

virtually impossible. Thus, Equation 2.32 is considered as a minimization problem whose objective

is to minimize the sum of errors:

min∑

(i,j)∈Ne,i<j

|||xi − xj||2 − (dij)2|+

∑

(i,k)∈Ne

|||xi − ak||2 − (dik)2|

+∑

(i,j)∈Nl,i<j

(||xi − xj||2 − (rij)2)− +

∑

(i,k)∈Nl

(||xi − ak||2 − (rik)2)−

+∑

(i,j)∈Nu,i<j

(||xi − xj||2 − (rij)2)+ +

∑

(i,k)∈Nu

(||xi − ak||2 − (rik)2)+ (2.33)

where(u)− and(u)+ are defined as

(u)− = max{0,−u} and(u)+ = max{0, u}

26

However, this is a non-convex problem which currently does not have effective algorithms

to solve it [30]. Though it can be solved by non-linear solvers, the solution greatly depends on

the initial estimates and search directions [18]. Therefore, instead of formulating localization

problem as Equation 2.33, researchers formulated it as a semidefinite program, a subclass of convex

programs with efficient algorithms for convex optimization problem. The convex formulation also

guarantees a global optimal solution if it is feasible.

Section 2.6.2 and Section 2.6.3 give two semidefinite programs for localization problems in

wireless sensor networks.

2.6.2 Semidefinite program without relaxation

Dohertyet al.[15] is the first to address the localization problem as a semidefinite program. The

formulation utilizes the upper bound relations given in Equations 2.32 as a set of radial constraints

which are convex constraints. They are restated as a linear matrix inequalities (LMI) [32].

||xi − ak||2 ≤ (rik)2 →

rikI (xi − ak)

(xi − ak)T rikI

º 0 (2.34)

However, the lower bound relations are ignored and the equality constraints are relaxed as upper

bound constraints.

||xi − ak||2 = (dik)2 → ||xi − ak||2 ≤ (dik)

2 → dikI (xi − ak)

(xi − ak)T dikI

º 0 (2.35)

The drop of lower bound constraints is due to its non-convexity. Figure 2.5 shows the intersection

of one lower bound constraint and one upper bound constraint. The shaded region is the feasible

set which is a ring and obviously non-convex. Same reason holds for the equality constraint which

can be treated as intersection of a lower bound constraint and a upper bound constraint.

Sensors contribute a number of LMIs which are based on connectivity information of sensors.

These LMIs can be stacked to form a diagonal matrix which gives rise to a large LMI for the whole

network. It is a feasibility problem because there is no objective function defined. The problem

can be solved by interior-point algorithm [33] [34] efficiently. In additions to radial constraints,

27

ikrik

r

Figure 2.5 Geometrical interpretation of non-convexity of lower bound constraint in a 2-Dnetwork

28

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

r

2

r

Figure 2.6 Geometrical interpretation of convex constraints in a 2-D network

Dohertyet al. also introduced other convex constraints like angular constraints, trapezoidal con-

straints and quadrant constraints. Angular constraint can be considered as intersection of three

halfspaces (Figure 2.6b), quadrant constraint is composed by two linear constraints and one LMI

(Figure 2.6c), trapezoidal constraint is composed by four linear constraints (Figure 2.6d). The

shaded regions are the feasible regions for the corresponding constraints. However, sensor nodes

have to be able to measure the relative angles between themselves in order to establish constraints

other than radial one. It imposes extra requirements to the resource-constrained sensors.

Figure 2.7 illustrates the idea behind the convex program by looking at the connectivity infor-

mation of a sensor. There are three radial constraints and the intersection (shaded region) is the

feasible set of a particular sensor. The shaded region is still convex. Since it is a feasibility prob-

lem, the final solution is selected randomly from the feasible set. If measurements are reliable, the

solution is finer when more constraints intersect.

2.6.3 Semidefinite program with relaxation

Biswaset al. [18] also formulated the localization problem as a semidefinite program. Unlike

[15], Biswaset al. took in the bounding away constraints||xi − ak||2 ≥ rik and relaxed the non-

convex problem defined in Equation 2.33 into a semidefinite program. LetX = [x1x2 . . . xn] be

the2× n coordinate matrix that we want to determine. Then

||xi − xj||2 = eTijX

T Xeij,

||xi − ak||2 = (ei;−ak)T [I X]T [I X](ei;−ak),

29

1a

2a

3a

1r

2r

3r

Figure 2.7 Feasible set of a simplified convex program

30

whereeij is a vector with1 at theith entry,−1 at thejth entry and zero elsewhere,ei is a vector

with 1 at the ith entry and zero elsewhere. LetY = XT X. By introducing slack variables,

Equation 2.33 can be written as:

min∑

i,j∈Ne,i<j(α+ij + α−ij) +

∑i,k∈Ne

(α+ik + α−ik)

+∑

i,j∈Nl,i<j β−ij +∑

i,k∈Nlβ−ik

+∑

i,j∈Nu,i<j β+ij +

∑i,k∈Nu

β+ik

subject to

eTijY eij − (dij)

2 = α+ij − α−ij,∀i, j ∈ Ne, i < j,

(ej;−ak)T

I X

XT Y

(ei;−ak)− (dik)

2 = α+ik − α−ik,∀i, k ∈ Ne,

eTijY eij − (rij)

2 ≥ −β−ij ,∀i, j ∈ Nl, i < j,

(ej;−ak)T

I X

XT Y

(ei;−ak)− (rik)

2 ≥ −β−ik,∀i, k ∈ Nl,

eTijY eij − (rij)

2 ≤ β+ij ,∀i, j ∈ Nu, i < j,

(ej;−ak)T

I X

XT Y

(ei;−ak)− (rik)

2 ≤ β+ik,∀i, k ∈ Nu,

α+ij, α

−ij, α

+ik, α

−ik, β

+ij , β

−ij , β

+ik, β

−ik ≥ 0,

Y = XT X. (2.36)

By relaxingY = XT X to Y º XT X, following LMI can be obtained:

Z :=

I X

XT Y

º 0 (2.37)

31

SubstitutingZ into Equation 2.37 yields a semidefinite programming problem as follows:

min∑

i,j∈Ne,i<j(α+ij + α−ij) +

∑i,k∈Ne

(α+ik + α−ik)

+∑

i,j∈Nl,i<j β−ij +∑

i,k∈Nlβ−ik

+∑

i,j∈Nu,i<j β+ij +

∑i,k∈Nu

β+ik

subject to

(1; 0;0)T Z(1; 0;0) = 1

(0; 1;0)T Z(0; 1;0) = 1

(1; 1;0)T Z(1; 1;0) = 2

(0; eij)T Z(0; eij)− (dij)

2 = α+ij − α−ij,∀i, j ∈ Ne, i < j,

(ei;−ak)T Z(ei;−ak)− (dik)

2 = α+ik − α−ik,∀i, k ∈ Ne,

(0; eij)T Z(0; eij)− (rij)

2 ≥ −β−ij ,∀i, j ∈ Nl, i < j,

(ei;−ak)T Z(ei;−ak)− (rik)

2 ≥ −β−ik,∀i, k ∈ Nl,

(0; eij)T Z(0; eij)− (rij)

2 ≤ β+ij ,∀i, j ∈ Nu, i < j,

(ei;−ak)T Z(ei;−ak)− (rik)

2 ≤ β+ik,∀i, k ∈ Nu,

α+ij, α

−ij, α

+ik, α

−ik, β

+ij , β

−ij , β

+ik, β

−ik ≥ 0

Z º 0 (2.38)

The problem can now be readily solved by interior-point algorithms.

2.6.4 Comments on SDP formulations

The formulation of Dohertyet al. focused on the radial constraints and ignored the bound-

ing away constraints due to its non-convexity. Intuitively, the position estimates obtained from

their formulation should be more accurate than those of APS since the former has collected global

information when solving the localization problem. Unfortunately, the drop of bounding away con-

straints degrades the performance. Without the bounding away constraints, the estimated positions

can only fall into the convex hull formed by the anchors. This makes the solution quality depend

on the placement of anchors. The formulation yields good solution when anchors are placed on the

32

perimeter of the network and other sensors fall into the convex hull formed by anchors. Otherwise,

nodes being on the edges of network will be collapsed into the interior of the network and the

estimation errors increase tremendously.

The formulation proposed by Biswaset al. avoids the dependence on anchor placement by

taking bounding away constraints into accounts and the problem is relaxed into a semidefinite

program. When the degree of connectivity is high, sensors have more neighbours (>10), tighter

constraints can be established. The position estimates can be very accurate. Nonetheless, the

scalability becomes a problem when the number of nodes and constraints increase. Every pair

of nodes can contribute either a bounding away constraints or radial constraints. This implies the

number of constraints grow atO(n2) wheren is the number of nodes in the network. For a networks

with 200 nodes, there are about40, 000 constraints. Although some constraints are redundant and

can be removed, solving the optimization problem is still very demanding for a consumer PC.

From our simulation, a consumer PC cannot handle a problem with 200 nodes. The consumer PC

is equipped with a Pentium IV 3.2GHz processor and 1Gb memory. The problem is fed to the

open-source solver SeDuMi [35] which is embedded in MATLAB. However the solver failed to

solve it because of insufficient memory to handle the problem. It is expected that sensor networks

may be composed of not merely hundreds but thousands of nodes. The huge computation and

memory requirements have to be addressed before it can be brought into practice. Furthermore,

solving the optimization problem is intrinsically centralized. This also reduces the practicalities of

SDP-based localization methods.

33

Chapter 3

Improving APS and PDM in Anisotropic Net-works with Convex Hull Detection Method(CHDM)

3.1 Overview

In the last chapter, the mechanism of APS is introduced. The Euclidean propagation method

achieves the relatively more accurate result compared to the DV-Hop and DV-distance propagation

methods. Despite being more accurate, it requires a high density of anchors to determine the

correct Euclidean distances from anchors. Simulation by Niculescuet al. [7] suggested that if

more than 90% of nodes are capable to localize themselves, there should be over 40% of nodes

being anchors when they are uniformly distributed in a square grid with 7.8 neighbours in average.

On the other hand, DV-Hop and DV-distance are simple and distributed, however their accuracy

also highly depends on the density of anchors and the topology of the sensor networks.

In this chapter, we investigate the reasons behind the poor performance of DV-Hop and DV-

distance and give two modifications to improve their performance in anisotropic networks. Fur-

thermore, the modifications are also applicable to PDM-based localization system which will be

discussed in later sections.

34

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9

position(1,i)

positio

n(2

,i)

Sensor Positions

Figure 3.1 Uniform Topology

35

3.2 Improving APS in Anisotropic Networks

Figure 3.1 shows a network with regular shape and uniform density of nodes where DV-Hop

and DV-distance have fairly good performance. Since the shape of the network is convex and the

density of node is roughly the same over the whole network, the straight line between any two node

can be approximated by the shortest path linking the nodes. Hence, the hop count or shortest-path

distance gives a fairly good approximation of the Euclidean distance. However, sensor networks in

real world rarely have regular topologies. Networks in real world usually have irregular topologies.

Their shapes are non-convex and the density of nodes is non-uniform. One example is shown in

Figure 2.3 in previous chapter. The ’C’-shape implies the shortest path between two nodes resided

in the two extremes of the ’C’ has to go along the arc. Obviously, the shortest-path distance exceeds

the Euclidean distance by sheer amount and hence the position estimate obtained is erroneous.

Most Networks with irregular shapes can be considered as several C-shaped networks connected

together. We use simulations to investigate the effect of irregularity of network topology. DV-Hop

and DV-distance are implemented in MATLAB. The two propagation methods are used to localize

the networks shown in Figure 3.1 and 2.3.

There are 200 nodes in total and 20 of them are anchors (10%) in each network. The aver-

age connectivity for both networks is8.3. Anchors are distributed randomly. Since DV-distance

requires range measurements between connected sensors, the error of range measurement is mod-

eled by a Gaussian noise model with a noisy factorα,

d = d× (1 + N(0, α))

where d is the measured distance,d is the true Euclidean distance andN(0, α) is a zero-mean

normal random process with standard deviation ofα. Though it is questionable whether the white

Gaussian noise can sufficiently reflect the noise characteristics of a ranging method, it gives us

an idea about the noise resistance to the most general noise of a particular algorithm. Since this

research focuses on the localization algorithms instead of ranging techniques, the Gaussian model

will also be adopted to model the noise behaviour for other localization algorithms throughout this

thesis.

36

Uniform Network

0

5

10

15

20

25

30

35

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Mor

e

Percentage Error of Measured Distances (%)

Nu

mb

er

of

No

des

DV-distance

DV-Hop

Figure 3.2 Distribution of percentage error of measured distances of APS in uniform network

Figures 3.4 and 3.5 demonstrate the performance of DV-Hop and DV-distance where nodes

are uniformly distributed in a square grid and a ’C’-shape respectively. The ’diamonds’(¦) denote

anchors and the ’crosses’(×) denote the estimated positions of sensors. The ’dots’(•) denote the

true positions of sensors.α equals to0.05. Sensors are connected by solid lines if they are within

communication range between each others.

The average position estimation errors obtained from the DV-distance and DV-Hop are0.2972R

and0.4197R respectively. The position estimation errors are normalized by the radio rangeR, i.e.

error =||xi − xi||2

R,

where xi andxi are the estimated position and the true position of nodei respectively. Since

DV-distance uses distance measurements which are finer than hop counts of DV-Hop, it can be

expected that the former will give more accurate results. Figure 3.2 gives the distribution of av-

erage percentage errors of measured distances of the DV-distance and DV-Hop in the uniform

networks. The average discrepancy for nodei is determined by averaging the percentage errors

of the measured distances between nodei and the anchors. For DV-distance, the discrepancy is

37

Irregular Network

0

5

10

15

20

25

30

35

40

45

50

-30

-25

-20

-15

-10 -5 0 5 10 15 20 25 30 35

Mor

e

Percentage Error of Measured Distances (%)

Nu

mb

er

of

No

de

s

DV-distance

DV-Hop

Figure 3.3 Distribution of percentage error of measured distances of APS in irregular network

within ±3% for most nodes while the discrepancy for DV-Hop has a larger variance. Although

DV-distance outperforms DV-Hop in this scenario, it has to be noted that the performance of DV-

distance depends on the accuracy of the distance measurements which DV-Hop does not have this

dependence. However, the average position estimation errors of both DV-Hop and DV-distance

increase when the underlying network is irregular. The average errors of DV-distance and DV-

Hop go up to1.9120R and2.0762R respectively. Nodes at the two extreme tips of the ’C’ have

the poorest position estimates. Figure 3.3 shows the distribution of average percentage errors of

measured distances of DV-distance and DV-Hop in irregular network. We can see that the range

and variance of the discrepancy shoot up. The correction factorsci in irregular network cannot

correct the measured distances as nice as those of uniform network. In DV-Hop, recall that the

correction factorci is the average hop size calculated by anchori (Equation 2.20) which is used by

all nearby nodes to determine their estimated distances from all anchors. Assume that anchors are

randomly distributed across the network as in our setting. This implies that some anchors are far

away from anchori and some are nearby. Qualitatively, we can consider the correction factorci

38

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9

10

position(1,i)

positio

n(2

,i)

Sensor Positions

(a) DV-Hop

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9

position(1,i)

positio

n(2

,i)

Sensor Positions

(b) DV-distance

Figure 3.4 Performance of APS with uniform topology

being composed by two major components, the average hop size of the long paths and the average

hop size of the short paths. The paths to nearby anchors will be less bendy and thus the average

hop size will be relatively larger. On the other hand, the long paths have higher probability of

passing through the ’C’-bend and gives a relatively smaller average hop size. Therefore, when

an arbitrary node multiplies the correction factorci with small hop count (e.g. anchors that are

nearby), it probably overestimates the Euclidean distance as the average hop size of the short paths

is relatively larger thanci. Similarly, multiplying ci with large hop count (e.g. anchors that are

far away) will most likely overestimate the Euclidean distance since the average hop size of the

long paths is relatively smaller thanci. Furthermore, the long path may bend along the ’C’ which

further worsen the estimate. Though DV-distance does not use hop count, the same argument is

applicable. Both over-estimation and under-estimation cause the significant drops in accuracy.

3.2.1 Selecting the Nearest 3 Anchors

In the last section, we show why DV-Hop and DV-distance perform so badly with irregular

topologies. In the original APS, nodes take in every anchor for multilateration, but we have just

shown that including all anchors actually degrades the performance of the APS under irregular

39

−4 −2 0 2 4 6 8 10 12−2

0

2

4

6

8

10

12

position(1,i)

posi

tion(

2,i)

Sensor Positions

(a) DV-Hop

−2 0 2 4 6 8 10 12−2

0

2

4

6

8

10

position(1,i)

posi

tion(

2,i)

Sensor Positions

(b) DV-distance

Figure 3.5 Performance of APS with irregular topology

40

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

position(1,i)

posi

tion(

2,i)

Sensor Positions

(a) DV-Hop

0 2 4 6 8 10

0

1

2

3

4

5

6

7

8

9

position(1,i)

posi

tion(

2,i)

Sensor Positions

(b) DV-distance

Figure 3.6 Performance of modified APS with irregular topology

networks. The nodes at the two extreme tips of the ’C’ suffer most. Most of their paths towards

anchors can be considered as long paths whose distances are greatly overestimated. As we can see

from Figure 3.5 that the position estimates of nodes at the tips of ’C’ have the largest errors. In view

of this, we propose a simple modification to improve the performance of DV-Hop and DV-distance

with irregular topologies. A sensor node does not include every anchors for multilateration, it only

chooses thenearest 3 (4)anchors in a 2-D (3-D) networks. Choosing less anchors can reduce the

chance of including anchors that are far away if the density of anchors are sufficiently high. If

anchors are close enough, the region covering the anchors and the sensor can be considered as

a small uniform network. Figure 3.6 shows the performance of DV-Hop and DV-distance where

nodes only take in the nearest 3 anchors for multilateration under the same C-shaped topology

given in Figure 2.3. From the figures, we can see that the performance of both DV-distance and

DV-Hop improve. The resulting topologies much resemble to the original topology compared with

those obtained from original APS. Numerically, the average position errors of DV-distance and

DV-Hop are0.7737R and0.9329R respectively. The simple modification yields improvements of

more than 50% for both propagation methods.

41

3.2.2 Convex Hull Detection

The investigation in previous section suggests that selecting the ”appropriate” anchors can

improve the quality of the position estimates, especially in anisotropic networks. Besides choosing

the nearest three anchors, we also explore the effects of choosing anchors that form a convex

hull where the sensor falls within. If nodes fall within the convex hull formed by anchors, the

position estimates obtained from multilateration are more robust against the distance measurement

errors [26] [31]. Consider a simplified scenario as shown in Figure 3.7 which only one distance

estimate is corrupted by noise. Since the anchors (yellow dots) are nearly collinear, a small change

of the position estimate may cause a large position error. The estimated position (red dot) is

flipped across the straight line joining the anchors. On the other hand, when the node is within

the convex hull formed by anchors, it is less likely that the node suffers from the flipping error.

When anchors are abundant, it is less likely all anchors lying in a straight line and causing the

flipping error, but we have shown in previous section that a sensor should not include all anchors

in multilateration. This makes searching a convex hull meaningful when combining it with our

previous modification, choosing the nearest three anchors that form a convex hull embedding the

node-to-be-localized. Though it is also better for anchors to be of similar distance from the sensors

to average out the ranging error [26] [31], it is difficult to search anchors that being nearby and

having similar distance from the sensors in anisotropic networks.

3.2.2.1 Convex Hull Detection Method (CHDM)

Without knowing the exact position of the sensor, we cannot tell explicitly whether it falls

within the convex hull formed by three anchors, and therefore propose a heuristic method to detect

whether a node is within the convex hull.

Let the 3 concerned anchor nodes bea1, a2, anda3. The Convex Hull Detection Method

(CHDM) considers the distances ofx to a1, a2, anda3 and the respective positions of the three

anchors. To facilitate our discussion, let the line connectinga1 anda2 bee12, anda2 anda3 be

e23, etc. The line connectingx, which is assumed to be inside the convex hull at the moment, to

anchor nodea1 is l1. Figure 3.8 shows the spatial relationship of the involved anchors and nodex.

42

erroneous distance estimate

position erro

r

distance error

Figure 3.7 Flipping of position estimate

43

a1

e13

a3

a2

l1

l2 l3

e12

e23

x

Figure 3.8 The spatial relationship of involved anchors and nodex

The following properties should hold for most cases:

l1 < e12 and l1 < e13

l2 < e12 and l2 < e23

l3 < e13 and l3 < e23

where< means shorter in length. These properties are used to test whether nodex falls into the

convex hull. Nodex is considered within the convex hull if all the above properties are satisfied.

3.2.2.2 A Evaluation of the CHDM

Since the CHDM is only a heuristic method, the result is not always true. Experiments are

conducted to test the reliability of CHDM. Four nodes are randomly put in a square grid where

three are considered as anchors and the one left is the sensor. CHDM is applied to test whether the

sensor is within the triangle formed by the anchors. A million of test cases are generated. CHDM

correctly determine whether nodex is within the convex hull in about 80% (795157) of test cases.

Out of the failed test cases, most errors are Type II error (203407) and only very few are Type I

44

error (1436). A Type I error is committed if output of CHDM is positive but nodex is not within

the convex hull. A Type II error is committed if output of CHDM is negative but nodex is indeed

within the convex hull. It is more desirable to have a small Type I error rate and a relatively large

Type II error rate than the opposite. It is unrecoverable once nodes are wrongly determined to be

within convex hull, but nodes can look for another convex hull formed by another combination of

anchors if Type I errors occur.

3.2.2.3 Computational Overheads of the CHDM

The overheads brought by the Convex Hull Detection Method is minimal. CHDM only uses

estimates of distances between the sensor and anchors. It does not require extra information other

than those already available in the original APS. As CHDM is consisted of several simple compar-

isons, the computational complexity of it is also minimal. However, the number of trials of CHDM

executed for a node to find a convex hull is a variable. It is possible for a node to find a convex

hull after a trial, but it is also possible that a node cannot find a convex hull after enumerating all

the combinations of anchors. The overheads of the later case grow exponentially with the number

of anchors. Assume there arem anchors, the number of trials of CHDM ismC3 for the worst

case. The extra computations introduced in the worst case can exhaust the batteries of sensors,

and therefore the number of trial of CHDM is upper-bounded. Every sensor can only include the

k(k < m) nearest anchors, giving a worst case ofkC3. The value ofk depends on the battery life

and computation power of sensors. If a node cannot find a convex hull at all, it uses the nearest 3

anchors to perform the multilateration.

3.2.3 Simulation

To evaluate the modifications made to the original APS, extensive simulations are run. Original

APS, APS with nearest 3 anchors (abbreviated as DV-HopNearest3 or DV-distanceNearest3) and

APS with CHDM (abbreviated as DV-HopCHDM or DV-distanceCHDM) are applied to localize

30 C-shaped networks with 200 nodes. The same topologies are used throughout the simulations

with different measurement errors, numbers of anchors and degrees of connectivity.

45

8 10 12 14 16 18 20 22 24 26 280.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Average Connectivity

Ave

rage

Pos

ition

Err

or (

R)

DV−HopDV−Hop_Nearest3DV−Hop_CHDM

(a) DV-Hop

8 10 12 14 16 18 20 22 24 26 280.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Average Connectivity

Ave

rage

Pos

ition

Err

or (

R)

DV−distanceDV−distance_Nearest3DV−distance_CHDM

(b) DV-distance,α = 0.05

Figure 3.9 Performance of APS and modified versions under different degree of connectivity

3.2.3.1 Effects of Connectivity

Figure 3.9 shows the performance of DV-Hop, DV-distance and their modified versions under

different degrees of connectivity. 20 nodes are randomly picked as anchors. The average connec-

tivity ranges from8 to 26. It is controlled by adjusting the communication range of nodes. The

noisy factorα is set to0.05 for DV-distance. The value ofk presented in Section 3.2.2.3 is set to

10. The number of execution of CHDM is at most10C3 = 120 for each sensor. It can be seen

that the modified versions give significant improvement on accuracy. The average position esti-

mation errors drop to about 50% compared to the original APS. Moreover, the CHDM can further

improve the accuracy. When average connectivity is around 11.7, the average position estimation

errors for DV-Hop, DV-HopNearest3 and DV-HopCHDM are1.2185R, 0.7139R and0.6599R

respectively.

3.2.3.2 Effects of Number of Anchor

Figures 3.10 and 3.11 give the average position errors obtained with different numbers of an-

chors. The number of anchors changes from 10 to 100 (5% to 50%) and the noisy factor remains at

46

10 20 30 40 50 60 70 80 90 1000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Number of Anchors

Ave

rage

Pos

ition

Err

or (

R)

DV−HopDV−Hop_Nearest3DV−Hop_CHDM

(a) DV-Hop

10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

Number of Anchors

Ave

rage

Pos

ition

Err

or (

R)

DV−distanceDV−distance_Nearest3DV−distance_CHDM

(b) DV-distance,α = 0.05

Figure 3.10 Performance of APS and modified versions with different numbers of anchors,average connectivity = 11.7

0.05. The average connectivity is about 11.7 and 18.6. As expected, the position estimation errors

drop when more anchors are available. Our proposed modifications show more improvement as

the number of anchor increases. With increasing anchors, sensors can reach an anchor in a shorter

path and the estimated distance is more reliable. Since variants of DV-Hop use the hop count infor-

mation only, its distance estimates are coarser than those of variants of DV-distance, and therefore

variants of DV-distance benefits more from the increase of anchors.

3.2.3.3 Effects of Measurement Error

The effects of different degree of measurement errors are also investigated. The noisy factor,α,

is changed from 0 to 0.35. Since DV-Hop and its variants do not require range measurement, they

do not affect by the measurement errors. Figure 3.12 shows the relation between the measurement

errors and the location errors. Whenα is less than 0.15, the performances of DV-distance and its

variants are fairly stable. Significant rise of errors is observed when noisy factor grows beyond 0.15

and the errors jump high whenα is beyond 0.2 for both proposed variants and the average error of

47

10 20 30 40 50 60 70 80 90 100

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Number of Anchors

Ave

rage

Pos

ition

Err

or (

R)

DV−HopDV−Hop_Nearest3DV−Hop_CHDM

(a) DV-Hop

10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Anchors

Ave

rage

Pos

ition

Err

or (

R)

DV−distanceDV−distance_Nearest3DV−distance_CHDM

(b) DV-distance,α = 0.05

Figure 3.11 Performance of APS and modified versions with different numbers of anchors,average connectivity = 18.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.5

1

1.5

2

2.5

3

Noisy factor α

Ave

rage

Pos

ition

Err

or (

R)

DV−distanceDV−distance_Nearest3DV−distance_CHDM

(a) DV-distance, connectivity=11.7

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Noisy factor α

Ave

rage

Pos

ition

Err

or (

R)

DV−distanceDV−distance_Nearest3DV−distance_CHDM

(b) DV-distance, connectivity=18.6

Figure 3.12 Performance of APS and modified versions with different degrees of measurementerrors

48

DV-distanceCHDM surpasses that of DV-distancenearest3. The position errors of original APS

starts to climbs up tillα is beyond 0.25.

The ”stable” performance of original APS with smallα does not mean it is less susceptible to

noise. It has the largest average error whenα is small. Furthermore, the position error is probably

dominated by the erroneous path distance estimation rather than the measurement error of ranging.

The poor performance of DV-distanceCHDM with large noisy factor can be accounted to the

Convex Hull Detection Method itself. When measurement errors increase, the accuracy of CHDM

drops, hence the accuracy of the position estimates drops.

3.2.4 Summary

In this section, we have investigated the reason behind the poor performance of APS in irreg-

ular networks. The main reason is the dynamic relation between the path distance (hop count)

and Euclidean distance in irregular networks. The correction factorci cannot correct the path dis-

tance or hop count into a reliable distance estimates. It overestimates when the path is long and

underestimates when the path is short, and thus making the performance a lot worse than that of

uniform networks. In view of this, we propose two simple yet effective modifications to improve

the performance of APS with irregular topologies. By simulations, we show that nodes choosing

the nearest 3 anchors for multilateration can already reduce the error significantly. Including our

heuristic-based Convex Hull Detection Method (CHDM) can further improve the accuracy. The

overheads introduced by CHDM are minimal, but the benefit gained is significant.

49

8 10 12 14 16 18 20 22 24 26 280.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Average Connectivity

Ave

rage

Pos

ition

Err

or (

R)

DV−distance_CHDM, α=0.05DV−distance_CHDM, α=0.1PDM, α=0.05PDM, α=0.1

(a) 20 anchors

8 10 12 14 16 18 20 22 24 26 280.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Average Connectivity

Ave

rage

Pos

ition

Err

or (

R)

DV−distance_CHDM, α=0.05DV−distance_CHDM, α=0.1PDM, α=0.05PDM, α=0.1

(b) 40 anchors

Figure 3.13 Performance of APS with irregular topology

3.3 Hybrid Approach: PDM+CHDM

Lim et al [20] proposed a very powerful localization system based on proximity-distance map

(PDM) for anisotropic networks in 2005. The mechanism is similar to the Ad Hoc Positioning

System (APS). It is distributed and multilateration-based. The detailed description of PDM is given

in Chapter 2. To demonstrate its strength, we compare it against our proposed DV-distanceCHDM.

Figure 3.13 shows the average position errors obtained from PDM and DV-distanceCHDM. The

simulation setting is similar to that of section 3.2.3. Same network topologies are used. There

are 20 or 40 anchors in each network and the noisy factor,α, equals to either 0.05 or 0.1. We

observe that PDM outperforms DV-distanceCHDM consistently. Though PDM has overheads

in calculating the proximity-distance map, its superior performance overshadows the extra costs.

Furthermore, the mechanism of PDM is basically the same as PDM. Actually, we can view APS as

a special case of PDM, but the mapping used by APS is simply a constant scaling of the correction

factorci. All these makes PDM an attractive localization system to replace the APS and its variants,

however, in this section we show that CHDM can further improve the PDM-based localization

system.

50

3.3.1 PDM+CHDM

Recall that PDM tries to capture the characteristics between the proximities and the geograph-

ical distances in different directions. Proximities obtained by sensors are transformed by the map-

ping into more reliable geographical distance estimates. All sensors use the same mapping which

are calculated by anchors. Multilateration is then performed by each sensor to determine its po-

sition estimate. It is questionable whether all nodes should use the global mapping to process its

estimates, especially nodes that are close to the anchors. If nodes are immediate neighbours of an-

chors, their distance estimates are close to the true geographical distance except being corrupted by

the measurement noise. Moreover, proximities obtained from anchors are that close enough should

also be reliable that preprocessing is not necessary. Thus, we propose merging our heuristic-based

CHDM with PDM, a hybrid approach, PDM+CHDM.

In PDM+CHDM, nodes do not blindly transform their proximities with the mapping obtained

from anchors, each node instead performs CHDM to test whether it falls within any convex hull

formed by any three anchors. If a node cannot find any combination of anchors that form a con-

vex hull embedding it, it localizes itself through PDM. Otherwise, it checks whether the furthest

anchors are withink-hop from the sensor. If it does, the sensor will employ DV-distanceCHDM

or DV-Hop CHDM depending on the type of proximity used by the sensor network. The restric-

tion imposed on the hop count from the furthest anchor avoids including proximities obtained

from long path. As previous sections shows that proximities obtained from long path are more

unreliable, especially when the underlying network is anisotropic. Since DV-distanceCHDM and

DV-Hop CHDM are only employed when the furthest anchor is within k-hop, there is no need to

include anchors that are far away in the calculation of the correction factorci. The correction factor

given in equation 2.20 is thus modified as:

ci =

∑ √(Xi −Xj)2 + (Yi − Yj)2

∑hij

,∀hij ≤ k

ci = 1, if∑

hij = 0,∀hij ≤ k (3.1)

51

The value of parameterk depends on the ratio of anchors and the connectivity of networks. If the

furthest anchor is beyondk-hop, the sensor falls back on PDM. Figure 3.14 gives the flow chart of

PDM+CHDM.

3.3.2 Simulation

To justify our proposal, simulations are made to determine whether we can obtain further im-

provements from CHDM. We investigate the effects of anchor ratio, connectivity and measurement

errors. Simulation results from previous sections show that these factors affect DV-distance and

DV-Hop similarly. Since DV-distance is more accurate, we focus on DV-distance and hence the

proximities used will be the shortest-path distance instead of hop count. The parameterk is first

set to 4 and in later section, we discuss how the value ofk affects the performance of the hybrid

approach.

3.3.2.1 Effects of Connectivity

For better illustration, we gives the accuracy gained in percentage in Figure 3.15. The accuracy

gained is calculated as below:

ePDM − ePDM+CHDM

ePDM× 100%,

whereePDM+CHDM is the average position error of sensors which select DV-distance after performing

CHDM. ePDM is the average position error of corresponding nodes but using PDM.

From Figure 3.15, nodes which localizing itself by DV-distance after performing CHDM yield

an average improvement of about 20% when noisy factor is equal to 0.05. Although the hybrid

approach hurts the performance when noisy factor and average connectivity increase, the improve-

ment gained with low connectivity is promising and significant. Figure 3.16 shows the number of

nodes using DV-distance per network. As connectivity increases, more sensors use DV-distance.

It is because the number of k-hop neighbours of an node increases as connectivity, and thus the

chance of finding a convex hull increases.

52

Check whether the sensor falls

within a convex hull and the

furthest anchor is within k-hop

CHDM

Collect proximities from anchors

Positive

The furthest anchor is within k-

hop from the sensor

Yes

Transform the proximities

by the proximity-distance

map and perform

multilateration

Perform multilateration

with anchors that form the

convex hull

PDM+CHDM completes

No

Negative

Figure 3.14 Flow chart of PDM+CHDM

53

8 10 12 14 16 18 20 22 24 26 28−10

−5

0

5

10

15

20

25

30

Average Connectivity

Acc

urac

y ga

ined

(%

)

20 anchors, α=0.05

20 anchors, α=0.1

40 anchors, α=0.05

40 anchors, α=0.1

Figure 3.15 Accuracy gained from CHDM with different degrees of connectivity

8 10 12 14 16 18 20 22 24 26 2850

60

70

80

90

100

110

120

Average Connectivity

Num

ber

of N

odes

Usi

ng D

V−

dist

ance

per

Net

wor

k

20 anchors, α=0.05

20 anchors, α=0.1

40 anchors, α=0.05

40 anchors, α=0.1

Figure 3.16 Number of nodes using DV-distance with different degrees of connectivity

54

10 20 30 40 50 60 70 80 90 100−10

−5

0

5

10

15

20

25

30

35

40

Number of Anchors

Acc

urac

y ga

ined

(%

)

α=0.05, connectivity≈11.7

α=0.1, connectivity≈11.7

α=0.05, connectivity≈18.6

α=0.1, connectivity≈18.6

Figure 3.17 Accuracy gained from CHDM with different number of anchors

3.3.2.2 Effects of Number of Anchors

Figure 3.17 gives the accuracy gained with different number of anchors. Figure 3.18 gives the

average number of nodes using DV-distance per network. In one hand, as the number of anchors

increases, it is easier for a sensor to find anchors that are nearby and forming a convex hull, and

therefore the number of nodes using DV-distance increase. On the other hand, an increasing of

number of anchors means a decreasing of number of normal sensors and hence there is a drop

when the number of anchors is beyond 50. As expected, the accuracy gained increases as anchors

become abundant. Furthermore, the performance agrees with that shown in Figure 3.15 that the

hybrid approach has better performance with low connectivity.

3.3.2.3 Effects of Measurement Errors

Figure 3.19 gives the accuracy gained under different measurement errors. It is interesting

that the graph have a ’U’-shape. When connectivity is low andα is below 0.1 and beyond 0.25,

the hybrid approach can improve the accuracy but it fails when the noisy factor is within 0.1 and

55

10 20 30 40 50 60 70 80 90 10020

30

40

50

60

70

80

90

100

110

120

Number of Anchors

Num

ber

of N

odes

Usi

ng D

V−

dist

ance

per

Net

wor

k

α=0.05, connectivity≈11.7

α=0.1, connectivity≈11.7

α=0.05, connectivity≈18.6

α=0.1, connectivity≈18.6

Figure 3.18 Number of nodes using DV-distance with different degrees of measurement errors

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−30

−20

−10

0

10

20

30

40

50

Noisy factor α

Acc

urac

y ga

ined

(%

)

20 anchors, connectivity≈11.720 anchors, connectivity≈18.640 anchors, connectivity≈11.740 anchors, connectivity≈18.6

Figure 3.19 Accuracy gained from CHDM with different noisy factor

56

0 0.05 0.1 0.15 0.2 0.25 0.3 0.3520

30

40

50

60

70

80

90

100

110

120

Noisy factor α

Num

ber

of N

odes

Usi

ng D

V−

dist

ance

per

Net

wor

k

20 anchors, connectivity≈11.7

40 anchors, connectivity≈11.7

20 anchors, connectivity≈18.6

40 anchors, connectivity≈18.6

Figure 3.20 Number of nodes using DV-distance with different degrees of measurement errors

0.25. Since DV-distanceCHDM is more sensitive to measurement noise, the accuracy gain drops

as noisy factor increases, but as noisy factor further increases, the number of nodes using DV-

distance also declines (as shown in Figure 3.20). The increase of noisy factor indirectly makes

CHDM more ”stringent” in the sense that DV-distanceCHDM is only employed when sensor is

around the centroid of the convex hull formed by anchors. To elaborate on this, we first investigate

why number of nodes using DV-distance declines as noisy factor increases.

Recall that the heuristic-based CHDM consists of six comparisons given in previous section:

l1 < e12 and l1 < e13

l2 < e12 and l2 < e23

l3 < e13 and l3 < e23

A node is said to be within a convex hull if all constraints are satisfied.l1 is the shortest-path

distance from anchor1 (the correction factorci is ignored for brevity). It is the sum ofn distance

measurements wheren is the hop count from the node to anchor1. Since each distance measure-

ment is corrupted by zero-mean Gaussian noise with standard deviation ofα, l1 is also corrupted

57

Figure 3.21 Sensor near the boundary of a convex hull

by zero-mean Gaussian noise but with standard deviation ofα/n.

l1 = l1× (1 + N(0, α/n))

Whenα increases, the probability thatl1 being larger thane12 or e13 also increases. As long as

one constraint is violated, CHDM returns negative, and hence the average number of nodes being

able to find a convex hull decreases. Moreover, sensors obtaining positive results from CHDM

tend to be around the centroid of the convex hull. For instance, Figure 3.21 shows a node which

is within a convex hull but near the boundary. Asl1 andl3 are close toe12 ande23 respectively,

increasing the noisy factor means a higher probability of violating at least one constraints given

above. We conclude that increasing the noisy factor reduces the number of nodes using CHDM and

nodes obtaining positive result from CHDM are more likely to be near the centroid of the convex

hull. Furthermore, the position estimates obtained from DV-distance are more accurate than those

obtained from PDM for these nodes, and therefore the accuracy gained rises.

58

2 3 4 5 6−10

−5

0

5

10

15

20

25

30

k

Acc

urac

y ga

ined

(%

)

connectivity≈18.7, 20 anchors, α=0.05connectivity≈18.7, 20 anchors, α=0.1

Figure 3.22 Accuracy gained from CHDM with different value ofk

3.3.2.4 Effects of parameterk

In previous simulations, the value of parameterk is kept constant at 4, but it is for sure that

the value ofk is critical to the performance of our hybrid approach. In section 3.3.2.1, we observe

that the performance is quite well whenk equals to 4 and connectivity is low. In this section, we

investigate what value should be assigned tok when connectivity is relatively high.

Figure 3.22 and 3.23 show the accuracy gained and the number of nodes using DV-distance

with different value ofk. We find that the performance drops ask increases. By combining the

result obtained in section 3.3.2.1, we observe that the value ofk should decrease as connectivity

increase. As average connectivity increases, an arbitrary node can communicate with more nodes,

and thus the anchors selected for multilateration may be distant from the sensor and the path

distances between the arbitrary node increase. When the path distance gets larger, it becomes less

reliable because of the irregular network topology and hence deteriorating the performance.

59

2 3 4 5 655

60

65

70

75

80

85

90

95

100

k

Num

ber

of N

odes

Usi

ng D

V−

dist

ance

per

Net

wor

k

connectivity≈18.7, 20 anchors, α=0.05

connectivity≈18.7, 20 anchors, α=0.1

Figure 3.23 Number of nodes using DV-distance with different values ofk

60

3.3.3 Summary

In this section, we have demonstrated the strength of the PDM-based localization in anisotropic

networks by comparing it with our DV-distanceCHDM. Though the former consistently outper-

forms DV-distanceCHDM, we propose a hybrid approach which incorporates CHDM into PDM

to further improve its accuracy. We test the hybrid approach through extensive simulations. From

the simulation results, we find that the performance of hybrid approach depends on the value of

parameterk and the connectivity of networks. If the value ofk is chosen properly, the hybrid

approach does improve the PDM-based localization system. If other network parameters like the

number of nodes, noisy factor remain unchanged, the value ofk should decrease as connectivity

increase.

61

Chapter 4

Localization with Limited Number of Anchorsand Clustered Placement

4.1 Overview

In this chapter, a localization algorithm which requires few anchors will be discussed. Most

distributed algorithms based on multilateration require a significant amount of anchors to main-

tain the accuracy of the solution. In the last chapter, we have shown that APS and PDM are no

exceptions. Since deploying more anchors incurs higher costs, trade-off has to be made between

the accuracy and the cost. Furthermore, the placement of anchors also affects the performance of

multilateration based method [27] [28] [25]. Better performance can be achieved if anchors are

distributed uniformly around the perimeter of the sensor network, but it is not always possible to

spread anchors across a sensor network. Consider an application of wireless sensor network in

a battlefield where sensors are used to detect the presence of a hostile force. To ruin the sensor

network covering the battlefield, anchors will become the first target. Even though the outlook

of an anchor may be indistinguishable from other sensors, leaving it in the front line will expose

it to the fire of the enemy. The whole network may be malfunctioned because a few anchors are

destroyed. If anchors are only placed in certain area which is under protection, a sensor network is

still functional when some normal sensors are destroyed in the front line.

62

The proposed algorithm is composed of two localization techniques, multidimensional scaling

(MDS) [2] [3] and proximity-distance map (PDM) [20], in a phased approach inspired by the

work of Ahmedet al. [29]. These two techniques have complementary properties which make the

phased approach superior than simply applying a single approach throughout the network. MDS

can provide relatively accurate result when anchors are very limited in uniform networks. Four

to five anchors are usually enough for localizing a network of 200 nodes in 2-dimension which

many other algorithms fail to give a meaningful result under the same conditions. Furthermore,

the performance of multidimensional scaling also demonstrates less dependence on placement of

anchors. On the other hand, proximity-distance map provides excellent results for anisotropic

topologies. Being distributed, it is a very suitable algorithm for localizing large-scale wireless

sensor networks. However, PDM relies greatly on the characteristics captured by anchors which in

turns require uniform distribution of anchors across the network. The performance of PDM drops

substantially when anchors are squeezed in one region. By exploring these properties of MDS and

PDM, we design a phased approach using MDS to increase the number of anchors and extend its

coverage for PDM.

In the beginning, there are a few nodes that are equipped with GPS receivers and we call these

nodesprimary anchors. In the first phase, a subset of ordinary sensors are selected assecondary

anchors. Nodes which are neither primary nor secondary anchors are called normal sensors. The

definition of primary and secondary anchors will be employed consistently throughout this chapter.

The locations of secondary anchors are determined by MDS. The number of secondary anchors

is controlled such that MDS can be performed on each selected sensors individually. After the

secondary anchors have identified their locations, other ordinary sensors are localized using PDM

based on the location information of both the primary and the secondary anchors in the second

phase.

63

5 6 7 8 9 10 11 12 13 140

0.2

0.4

0.6

0.8

1

1.2

1.4

Connectivity

Err

or (

R)

MDS−MAPSDPPDMDV−Distance

Figure 4.1 50-node network with uniform topologies, 10 anchors,α=0.05

4.2 Phased Approach, MDS+PDM

In this section, the details of the distributed localization algorithm MDS+PDM will be pre-

sented. It combines MDS-MAP and PDM in a phased approach. The choice of integrating MDS-

MAP and PDM is based on our extensive study on existing localization algorithms. We examine

the accuracy obtained by DV-distance(APS), MDS-MAP, SDP and PDM with sufficient anchors

(20% of nodes are anchors). Though Figure 4.1 shows that SDP outperforms the rest of the algo-

rithms, the huge complexity makes it impractical. The next best algorithm is PDM. Unfortunately,

PDM does not perform well when there are only a few anchors as shown in Figure 4.2. On the

other hand, MDS-MAP has less dependence on the number and placement of anchors. We realized

that the accuracy of PDM can be improved by having more anchors. That is, even the primary

anchors are limited, if we can ”add” some secondary anchors to the network, PDM can work com-

parably well with MDS. The secondary anchors have been found based on the primary anchors.

It is in fact a small localization problem since the number of secondary anchors is not large. By

limiting the amount of secondary anchors, it is justifiable to use MDS for finding the locations of

secondary anchors. Our algorithm works as follows. In the first phase, some sensors are selected

as secondary anchors which are localized through multidimensional scaling. In the second phase,

the normal sensors are localized by PDM-based localization method. The mapping is derived from

64

8 10 12 14 16 18 20 22 24 260

0.5

1

1.5

2

2.5

3

3.5

Connectivity

Err

or (

R)

MDS−MAP, 5 clustered anchorsMDS−MAP, 10 clustered anchorsPDM, 5 clustered anchorsPDM, 10 clustered anchors

Figure 4.2 200-node network with uniform topologies,α=0.05

65

the primary and secondary anchors altogether and we assume nodes know the number of primary

anchorskp at deployment. The details of operation are given below:

1. Identification of secondary anchors

Due to the different properties of uniform and anisotropic networks, we adopt different ways

to assign secondary anchors.

• Uniform Networks

In uniform networks, secondary anchors can be more widely spread and can be more

farther away from the primary anchors than in anisotropic networks. Each primary

anchor sends an invitation packet containing its unique ID, a counter initialized to zero

and a valueks controlling the number of secondary anchors, tooneof its neighbours.

Largerks may improve the performance but also increase the complexity at the same

time. It is a tradeoff in determining the value ofkS. Normal sensor receiving this packet

will perform a Bernoulli trial with a success rate ofp. The success ratep roughly

controls the separation between secondary anchors so that they will not be clustered

together. The value ofp can be included in the packet sent by primary anchors or

embedded in sensor nodes before deployment. If the outcome is true, the normal sensor

increments the counter by one and becomes a secondary anchor. The packet will be

forwarded to another neighbour until the counter equals toks. If a secondary anchor

receives a packet originated from other primary anchors, the packet will be ignored and

forwarded to another node. Thus the total number of primary and secondary anchors

will be kp × (ks + 1), kp primary andkp ∗ ks secondary.

• Anisotropic Networks

In a C-shaped network, path distance is an unreliable Euclidean distance estimate when

nodes are far apart. Thus unlike an uniform network, nodes close to the anchors are cho-

sen as secondary anchors. Each primary anchor is responsible to chooseks secondary

anchors. A primary anchor first selects secondary anchors from its direct neighbours.

If the number of one-hop neighbours is less thanks, the residue vacancy will be filled

66

up by the two-hop neighbours or three-hop neighbours untilks secondary anchors are

selected. The total number of primary and secondary anchors is alsokp × (ks + 1).

2. Localization of Secondary Anchors

In this step, secondary anchors have to acquire the proximity information between every

pair of primary and secondary anchors. After sending the invitation packet, each primary

anchor sends packets containing its unique ID and coordinates toall of its neighbours. The

packet also bears a field marking the proximity, i.e. the distance or hop count the packet

has travelled. The value is initialized to be zero. Secondary anchors will simply repeat the

operation of primary anchors, that is sending out packets with its unique ID but leaving the

coordinates field blank.

Every node (including anchors) receiving a proximity packet from an anchor (either primary

or secondary) will store its ID and the proximity value. If a packet from a particular anchor

has been received before, the node examines the proximity and checks whether it is larger

than the stored proximity. If it is larger than the stored value, the packet will be discarded.

Otherwise, the stored value and the proximity field of the packet will be updated and the

packet will be forwarded to other neighbours. Thus the stored proximity always reflects the

shortest-path distance or hop count from a particular anchor.

After an anchorx has discovered its proximities to all anchors, it will send the proximities

it has collected to other anchors and wait for other anchors to repeat the same step. When

all anchors have distributed the proximities to their counterparts, each anchor knows the

proximity information between every pair of anchors. Now, every secondary anchor can

determine its location through classical MDS.

3. Proximity-Distance Map Calculation

After calculating its physical position using MDS, each secondary anchor also knows the

position estimates of other secondary anchors since MDS provides a configuration about the

primary and secondary anchors. Thus the proximity-distance mapT among both primary

and secondary anchors can be calculated immediately as given in Section 2.5.2.

67

4. Localization of Normal Nodes

Each normal sensor nodes uses the mappingT to process the proximity vectorps it has

stored when it aided anchors exchanging proximity information as given in Equation 2.31.

Finally, the node position is calculated by multilateration with the processed proximity vector

and the position information of primary and secondary anchors.

4.3 Simulation

To justify our proposal, extensive simulations are conducted to study the performance of MDS+PDM.

Similar to previous simulations, algorithms are applied to localize 30 200-node-networks. The

connectivity is controlled by varying the communication range of sensors. Nodes are capable of

measuring the distance away from its one-hop neighbours. Measurements are subjected to random

errors as in Equation 3.2. The estimation error is normalized by the communication rangeR. For

MDS+PDM, we randomly pick 20 normal sensors as secondary anchors.

4.3.1 Effects of anchor placement

As we previously pointed out that it is not always feasible to deploy anchors across the network,

thus unlike conventional approaches that assume anchors as spread across the network, we study

the performance when anchors are confined in a small region of the network (see Figure 2.4).

4.3.1.1 Uniform Networks

Figure 4.3 shows the performance of PDM, MDS-MAP and MDS+PDM with anchors dis-

tributed uniformly across the network or anchors clustered together withα=0.05 andα=0.1 respec-

tively. The values presented are the average position error of all nodes obtained from 30 topolo-

gies. The performance of PDM with clustered anchors is much worse than PDM with anchors

distributed across the network. With secondary anchors introduced in MDS+PDM, MDS+PDM

gives better performance for both scenarios. The average error of MDS+PDM with 5 clustered an-

chors andα=0.1 is 0.68R while the corresponding error of PDM is 1.52R, more than two times of

68

8 10 12 14 16 18 20 22 24 260

0.5

1

1.5

2

2.5

3

3.5

4

Connectivity

Err

or (

R)

PDM, Uniform AnchorMDS−MAP, Uniform AnchorMDS+PDM, Uniform AnchorPDM, Clustered AnchorMDS−MAP, Clustered AnchorMDS+PDM, Clustered Anchor

(a)α=0.05, 5 anchors

8 10 12 14 16 18 20 22 24 260

0.5

1

1.5

2

2.5

3

3.5

Connectivity

Err

or (

R)

PDM, Uniform AnchorMDS−MAP, Uniform AnchorMDS+PDM, Uniform AnchorPDM, Clustered AnchorMDS−MAP, Clustered AnchorMDS+PDM, Clustered Anchor

(b) α=0.10, 5 anchors

Figure 4.3 Effects of anchor placement in uniform networks

69

8 10 12 14 16 18 20 22 24 26 280.5

1

1.5

2

2.5

3

3.5

Connectivity

Err

or (

R)

PDM, Uniform AnchorMDS−MAP, Uniform AnchorMDS+PDM, Uniform AnchorPDM, Clustered AnchorMDS−MAP, Clustered AnchorMDS+PDM, Clustered Anchor

(a)α=0.05, 5 anchors

8 10 12 14 16 18 20 22 24 26 280.5

1

1.5

2

2.5

3

3.5

Connectivity

Err

or (

R)

PDM, Uniform AnchorMDS−MAP, Uniform AnchorMDS+PDM, Uniform AnchorPDM, Clustered AnchorMDS−MAP, Clustered AnchorMDS+PDM, Clustered Anchor

(b) α=0.10, 5 anchors

Figure 4.4 Effects of anchor placement in C-shaped networks

that of MDS+PDM. The secondary anchors provide a better capture of the characteristic between

the proximity and geographical distance.

4.3.1.2 C-shaped Networks

Figure 4.4 shows the corresponding performance in C-shaped networks. All anchors are clus-

tered in the top left region as shown in Figure 2.4. Different from the uniform networks, MDS+PDM

does not always outperform the other algorithms in C-shaped networks. PDM overwhelms MDS-

MAP and MDS+PDM when anchors are spread across the network. The average error of PDM

with connectivity about 14.95 andα equals to 0.1 is 0.87R while the corresponding error of

MDS+PDM is about 1.55R, almost doubling the error of PDM. The excellent result from PDM is

due to the accurate characterisation of proximity and geographical distance. Furthermore, the spar-

sity of primary anchors also affects the accuracy of MDS and the position estimates of secondary

anchors. However, MDS+PDM gives better performance when anchors are clustered in C-shaped

network. It is because the position estimates of secondary anchors is much more reliable when

primary anchors are clustered and secondary anchors are distributed around the primary anchors.

With reliable secondary anchors, better solution can be obtained in the second phase. If the position

70

estimates of secondary anchors is not reliable enough, it only degrades the overall performance of

MDS+PDM. In summary, MDS+PDM should be employed when anchors are clustered together

or the underlying network is uniform.

4.3.2 Effects of the number of primary anchors

4.3.2.1 Uniform Networks

Figure 4.5(a) shows the performance of PDM, MDS-MAP and MDS+PDM with different num-

ber of clustered anchors under different degrees of connectivity. MDS+PDM performs consistently

better than PDM with 5 or 10 anchors. Though the gap between MDS+PDM and PDM becomes

smaller when connectivity increases to 25 and 10 anchors are available, MDS+PDM gives more

stable performance. Clearly, adding extra anchors can improve the performance of MDS+PDM.

However, anchors are only used to determine the linear transformation. Thus MDS+PDM gains

less benefits from further increase of anchors. After all, it is less dependent on the number of

anchors. For example, the errors given by MDS+PDM with connectivity = 10.74 andα=0.1 are

0.73R and 0.66R for 5 and 10 anchors respectively. The corresponding errors produced by PDM

are 1.86R and 1.3R.

4.3.2.2 C-shaped Networks

Figure 4.5(b) shows the corresponding performance for C-shaped networks. Though MDS+PDM

still gives stable result with different number of anchors, PDM provides more accurate solution

when sufficient anchors and large connectivity are available. Similar to spreading anchors across

the network, increasing the number of anchors and connectivity imply that more nodes will get

involved in determining the transformation between proximity and geographical distance which

gives a better solution. On the other hand, by increasing connectivity, primary anchors can reach

nodes that are further away. However in C-shaped network, using path distance as an estimate of

Euclidean distance between nodes being far apart is very unreliable. It also makes the position

estimates in the first phase of MDS+PDM become unreliable and affect the overall performance of

MDS+PDM with high connectivity.

71

8 10 12 14 16 18 20 22 24 260

0.5

1

1.5

2

2.5

3

3.5

Connectivity

Err

or (

R)

PDM, 5 anchorsMDS−MAP, 5 anchorsMDS+PDM, 5 anchorsPDM, 10 anchorsMDS−MAP, 10 anchorsMDS+PDM, 10 anchors

(a)α = 0.1, Uniform networks

8 10 12 14 16 18 20 22 24 26 280.5

1

1.5

2

2.5

3

3.5

Connectivity

Err

or (

R)

PDM, 5 anchorsMDS−MAP, 5 anchorsMDS+PDM, 5 anchorsPDM, 10 anchorsMDS−MAP, 10 anchorsMDS+PDM, 10 anchors

(b) α = 0.1, C-shaped Networks

Figure 4.5 Effects of the number of clustered anchors

72

0 0.05 0.1 0.15 0.20.5

1

1.5

2

2.5

3

α

Err

or (

R)

PDMMDS−MAPMDS+PDM

(a) 5 anchors, Average Connectivity=10.74

0 0.05 0.1 0.15 0.20.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

α

Err

or (

R)

PDMMDS−MAPMDS+PDM

(b) 5 anchors, Average Connectivity=17.16

Figure 4.6 Effects of measurement noise on uniform networks

4.3.3 Effects of measurement noise

4.3.3.1 Uniform Networks

Figure 4.6 presents the performance of networks with low and high connectivity under different

degrees of measurement errors. The error of MDS+PDM grows significantly whenα is larger than

0.1. MDS+PDM gives an accuracy of 1.14R whenα equals to 0.15 but strikes to 2.4R whenα

equals to 0.2. Though the error of MDS+PDM increases sharply whenα goes beyond 0.1 for

uniform networks with low connectivity, the error of MDS+PDM is smaller than PDM for all

measurement errors considered. For high connectivity, the error rate changes less abruptly than

that of low connectivity. The error of MDS+PDM grows from 1.01R to 1.59R whenα increases

from 0.15 to 0.2. Furthermore, the performance of MDS+PDM is still better than that of PDM

under all measurement errors considered.

4.3.3.2 C-shaped Networks

Figure 4.7 gives the corresponding statistics for performance of MDS+PDM and PDM in C-

shaped networks, except for the case of high connectivity with exact measurement, MDS+PDM

73

0 0.05 0.1 0.15 0.21.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

α

Err

or (

R)

PDMMDS−MAPMDS+PDM

(a) 5 anchors, Average Connectivity=11.68

0 0.05 0.1 0.15 0.21

1.5

2

2.5

α

Err

or (

R)

PDMMDS−MAPMDS+PDM

(b) 5 anchors, Average Connectivity=18.61

Figure 4.7 Effects of measurement noise on C-shaped networks

74

gives more accurate result than PDM. Unlike the scenario with uniform networks, the gap between

MDS+PDM and PDM grows asα increases. For high connectivity networks, the average errors

of MDS+PDM and PDM withα equals to 0.1 are 1.23R and 1.29R respectively. Asα increases

to 0.2, the difference between MDS+PDM and PDM is more significant, the average error of

MDS+PDM is 2.06R while the average error of PDM is 2.42R.

4.3.4 Effects of the number of secondary anchors

By considering the computation and communication costs of MDS, the number of secondary

anchors should be minimal to give satisfactory performance. To study the effect of the number

of secondary anchors, we vary the number of secondary anchors from 0 (i.e. pure PDM) to 40.

Figure 4.8 shows the corresponding performance for uniform networks and C-shaped networks

respectively. In general, the average estimation error decreases as secondary anchors are introduced

but further increase of secondary anchors beyond 10 does not give any significant improvement.

Under uniform networks of low connectivity withα=0.05, MDS+PDM gives average accuracy of

0.74R and 0.65R when there are 10 and 40 secondary anchors respectively. The corresponding

errors are 1.69R and 1.65R respectively for C-shaped networks. For high connectivity, the error

with α=0.05 under uniform networks is 0.25R for 10 secondary anchors while the error with 40

secondary anchors is 0.28R. The corresponding errors in C-shaped network are 0.92R and 0.97R

for 10 and 40 secondary anchors respectively.

In view of the complexity incurred from the first phase of MDS+PDM and the marginal perfor-

mance gain from increasing the number of secondary anchors, the number of secondary anchors

should be chosen from 10-20 (5% to 10% of network size).

4.3.5 Effects of the position of primary anchors

4.3.5.1 Uniform Networks

For uniform networks, the position of the clustered anchors does not affect the general perfor-

mance of PDM, MDS-MAP nor MDS+PDM. For PDM, the accuracy is mainly affected by the

characterisation of the transformation which is affected by the size of the region covered by the

75

0 5 10 15 20 25 30 35 400.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Number of Secondary Anchors

Err

or (

R)

5 anchors, α=0.05, connectivity=10.745 anchors, α=0.1, connectivity=10.745 anchors, α=0.05, connectivity=20.865 anchors, α=0.1, connectivity=20.86

(a) 5 anchors, Uniform Networks

0 5 10 15 20 25 30 35 400.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Number of Secondary Anchors

Err

or (

R)

5 anchors, α=0.05, connectivity=11.685 anchors, α=0.1, connectivity=11.685 anchors, α=0.05, connectivity=22.395 anchors, α=0.1, connectivity=22.39

(b) 5 anchors, C-shaped Networks

Figure 4.8 Effects of the number of secondary anchors

76

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9

position(1,i)

posi

tion(

2,i)

Sensor Positions

Figure 4.9 5 anchors clustered at the tip of ’C’

anchors instead of the positions. For MDS-MAP, the performance is also independent on the posi-

tion of the anchors as the major purpose of anchors is determining the absolute coordinates for the

relative map. However, we can anticipate that the performance will be affected in C-shaped net-

works as path distance becomes an unreliable estimate of Euclidean distance between two nodes

that are far apart.

4.3.5.2 C-shaped Networks

Instead of putting the cluster of anchors at the top left corner, we placed the anchors randomly

at the tip of the ’C’. Figure 4.9 shows one instance. Figure 4.10 shows the performance of PDM,

MDS-MAP and MDS+PDM withα=0.05. By comparing with Figure 4.4(a), we can see that

the error surges for more than a double when anchors are clustered at the tip of the ’C’. When

anchors are placed at the top left corner, the average errors of PDM, MDS+PDM and MDS-MAP

are 1.62R, 1.22R and 1.87R respectively. The corresponding errors rise up to 4.23R, 4.34R and

4.15R for PDM, MDS+PDM and MDS-MAP when anchors are placed at the tip of the ’C’. The

accuracy drops drastically as the paths connecting most of the nodes and anchors detour around

the ’C’, thus the path distance will greatly over-estimate the true Euclidean distance.

77

8 10 12 14 16 18 20 22 24 26 282.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

Connectivity

Err

or (

R)

PDM, α=0.05MDS−MAP, α=0.05MDS+PDM, α=0.05

Figure 4.10 5 anchors,α=0.05, C-shaped Networks

78

4.4 Summary

In this chapter, we present a phased localization algorithm which performs well with very

few anchors. Reducing the number of anchors deployed can reduce the cost of sensor network

substantially. We first observe that MDS-MAP’s performance is less dependent on the number

and placement of anchors. Despite that, deploying MDS-MAP throughout the sensor network

requires much communications between sensors to estimate the pair-wise distances between them.

To reduce the communication costs, we propose a two-phase approach that only localizes a few

nodes by MDS-MAP with the rest being localized by the distributed algorithm, PDM.

Through extensive simulations, we demonstrate that the proposed algorithm produces results as

accurate as that of MDS-MAP under uniform networks even the number of anchors is minimal. At

the same time, MDS+PDM exhibits less complexity by employing secondary anchors and PDM in

the second phase of localization. We also show that MDS+PDM can enhance the performance of

PDM in C-shaped networks with clustered anchors where PDM is well recognized in localization

in anisotropic networks. Although MDS+PDM gives better results than PDM when anchors are

clustered together, there are rooms for improvement on accuracy. Since MDS-MAP does not

perform very well in anisotropic networks, it makes the position estimates of secondary anchors

less accurate than that of isotropic network. Hence the proximity-distance map derived becomes

less reliable. The major difficulty encountered in localization in anisotropic networks is to obtain

reliable distance measurement between two nodes that are far away from each other.

Simulation results also show that our proposal is less susceptible to anchor placement. The

proposed algorithm can be implemented in a distributed fashion efficiently when the number of

secondary anchors is chosen appropriately. From simulation, we find that choosing secondary

anchors from 2.5% to 5% of network size gives good performance. However, the mechanism

of MDS+PDM requires some prior knowledge about the network. If network is anisotropic, the

secondary anchors should not be far apart. If network is isotropic, the secondary anchors can be

spread across the network to obtain better performance. The prior knowledge may not be available

all the time and is a drawback of the phased approach.

79

Chapter 5

Conclusions and Future Works

In this thesis, existing localization algorithms like APS [7] [8], MDS-MAP [11] and PDM [20]

are evaluated by simulations. Performance of APS is highly dependent on the topologies of the

sensor networks. The accuracy drops when the network is anisotropic. The degradation of accu-

racy is caused by the inaccurate estimates of distances between normal sensors and anchors. The

irregular topologies make the shortest paths between anchors and normal sensors bendy and hence

sensors overestimate the Euclidean distances from anchors. Moreover, sensors also underestimate

the Euclidean distances when the shortest paths between anchors and the sensors are relatively

short and less bendy. To improve the performance of the APS under anisotropic topologies, we

propose not using all anchors in the multilateration process. Only anchors that are near to the sen-

sors are considered. In addition, accuracy can be further improved by selected anchors that form

a convex hull embedding the sensors. To search for a convex hull, a heuristic-based Convex Hull

Detection Method (CHDM) is presented. Through simulations, we showed that the modifications

give significant improvement in anisotropic networks.

The CHDM is not only applicable to APS, we have also demonstrated that CHDM is applicable

to PDM-based localization system. PDM gives much better performance in anisotropic networks

than that of APS. For PDM-based localization system, a mapping is devised to capture the relations

between the measured proximities and geographical distances in various directions. The mapping

is used to transform the measured proximities obtained by sensors into geographical distances, but

we found that the measured proximities obtained from nearby anchors are accurate enough that

80

transforming these proximities may not yield better result. Therefore, we propose that sensors

should execute the CHDM before transforming its proximities by the proximity-distance map. If

the sensor can find a convex hull formed by nearby anchors, it should localize itself through DV-

distance or DV-Hop. Simulation results suggest that including CHDM can further enhance the

performance of PDM-based localization system.

We also investigate the effect of placement of anchors in the thesis. We show that the per-

formance of PDM is susceptible to the placement of anchors. If anchors are clustered together,

the proximity-distance map derived cannot capture the topological information of the whole sen-

sor network, and thus the performance is severely degraded. In view of this, a phased approach,

MDS+PDM is proposed. By observing MDS-MAP [11] is less dependent on the number and

placement of anchors, we use it to localize some of the normal sensors in the networks. The

localized sensors become secondary anchors which are included in the second phase to derive

the proximity-distance map. We have demonstrated that the phased approach yields better result

when the sensor network has limited number of anchors which are clustered together, especially

in isotropic networks. However, there are still rooms for improvement for anisotropic networks as

MDS-MAP does not perform very well with irregular topologies. A possible direction would be

sequential localization that secondary anchors are introduced gradually outward from the cluster

of anchor. First, nodes near the cluster of anchors, say withink hops, may be localized through

PDM. Secondary anchors are introduced afterwards by selecting some sensors from the localized

nodes. Eventually, the value ofk can be grown and more nodes can be localized. The process

continues iteratively until all nodes are localized.

Although numerous localization algorithms for wireless sensor networks have been proposed in

recent years, most existing localization algorithms do not consider the application-specific require-

ments. The wide range of applications of wireless sensor networks makes designing a universal

localization algorithm a difficult task. Thus more effort should be spent in designing tailor-made

localization algorithm for a particular type of application like target tracking, environmental moni-

toring etc. Furthermore, most algorithms assumes sensors are static in most time, future researches

should also focus on localizing sensors with mobility.

81

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84

Publications

King-Yip Cheng, Vincent Tam and King-Shan Lui, “Improving APS with Anchor Selection in

Anisotropic Sensor Networks”.Proceedings of International Conference on Networking and Ser-

vices (ICAS/ICNS), pp.49, Tahiti, French Polynesia, October, 2005.

King-Yip Cheng, King-Shan Lui and Vincent Tam, “Hybrid Approach for Localization in Anisotropic

Sensor Networks”.Proceedings of the 63rd IEEE Vehicular Technology Conference (VTC), pp.344-

348, Melbourne, Australia, May, 2006.

King-Yip Cheng, King-Shan Lui and Vincent Tam, “Localization in Sensor Networks with Limited

Number of Anchors and Clustered Placement”.To appear in proceedings of the IEEE Wireless

Communications and Networking Conference (WCNC), Hong Kong, China, March, 2007.