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Ph.D. DISSERTATION

Cooperative Localization Schemes forWireless Sensor Networks in anAnchor-deficient Environment

앵커가부족한무선센서네트워크환경에서의협력위치추정기법연구

BY

YU WON-TAE

AUGUST 2018

DEPARTMENT OF ELECTRICAL ENGINEERING ANDCOMPUTER SCIENCE

COLLEGE OF ENGINEERINGSEOUL NATIONAL UNIVERSITY

Ph.D. DISSERTATION

Cooperative Localization Schemes forWireless Sensor Networks in anAnchor-deficient Environment

앵커가부족한무선센서네트워크환경에서의협력위치추정기법연구

BY

YU WON-TAE

AUGUST 2018

DEPARTMENT OF ELECTRICAL ENGINEERING ANDCOMPUTER SCIENCE

COLLEGE OF ENGINEERINGSEOUL NATIONAL UNIVERSITY

Cooperative Localization Schemes forWireless Sensor Networks in anAnchor-deficient Environment

앵커가부족한무선센서네트워크환경에서의협력위치추정기법연구

지도교수김성철

이논문을공학박사학위논문으로제출함

2018년 8월

서울대학교대학원

전기컴퓨터공학부

유원태

유원태의공학박사학위논문을인준함

2018년 8월

위 원 장: 박 세 웅 (인)부위원장: 김 성 철 (인)위 원: 김 남 수 (인)위 원: 신 요 안 (인)위 원: 김 용 화 (인)

Abstract

Many applications of wireless sensor network (WSN) require accurate location

information of the sensor node. The sensor node can identify its location by the posi-

tioning device such as the global positioning system (GPS). However, it is impractical

to obtain location information via GPS in certain situations such as a battleground or

indoor environment. Alternatively, the sensor node can estimate its position by utiliz-

ing signals from nearby anchor nodes that have knowledge of their own locations. If

the number of anchor nodes is insufficient, it is difficult to estimate each node’s lo-

cation using the anchor nodes’ information by employing the existing algorithms, the

localization problem becoming complicated. Thus, in a wireless environment where

location information of few anchor nodes is available, each node is required to coop-

erate with other nodes, and self-organizing localization is the crucial feature of WSNs.

In this dissertation, I investigate several schemes for accurate localization in anchor-

deficient environments. First, I propose a recursive self-organizing localization scheme,

solely based on the neighbors’ connectivity information. This scheme utilizes a mass

spring-relaxation (MSR) algorithm in which each node finds its location by iteratively

balancing the geometric relationships with neighboring nodes until the system reaches

an equilibrium state. I propose a simple distance correction factor to consider the ac-

curacy of distance measurements, and adopt the adaptive step size control based on the

gradient method to improve the system stability. The proposed scheme improves the

system performance in terms of convergence speed, system stability, and estimation

accuracy.

Additionally, I consider mobile anchor assisted localization in the situation of bad

condition such as lack of anchor. This method assumes that the mobile anchors do not

have energy restrictions and can move on the ground or fly. They periodically broadcast

their location to support localization of nearby sensor nodes, and localization perfor-

i

mance is highly dependent on the mobile anchor trajectory. Therefore, I study a dy-

namic path planning method of the mobile anchor in outdoor wireless sensor networks.

The objective of the path planning is to steer the mobile anchor to the positions which

minimize the estimation uncertainty of the sensors. The method is based on a sin-

gle mobile anchor and does not require prior knowledge of the network environment.

The mobile anchor determines waypoints using the Cramer-Rao lower bound (CRLB),

which gives the minimum achievable variance of the estimated location of the sensors.

To reduce the complexity of CRLB calculations, I consider several objective functions

based on the Fisher information matrix. In addition, I focus on the minimum spanning

tree over the wireless sensor network to determine energy-efficient paths and guaran-

tee localization of every node. Simulation results confirm that the proposed method

improves the localization accuracy when compared to static path planning algorithm

and guarantees the localization of all the node in the network.

keywords: Localization, wireless sensor networks, mass-spring relaxation, UAV,

path-planning, CRLB, minimum spanning tree

student number: 2014-30308

ii

Contents

Abstract i

Contents iii

List of Tables iv

List of Figures v

1 INTRODUCTION 1

2 SELF-ORGANIZING LOCALIZATION WITH ADAPTIVE WEIGHTS

FOR WIRELESS SENSOR NETWORKS 6

2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Conventional Mass-spring Relaxation Method . . . . . . . . . . . . . 9

2.4 An Improved Algorithm for Mass-spring Relaxation . . . . . . . . . . 12

2.4.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.2 Distance Correction Factor . . . . . . . . . . . . . . . . . . . 12

2.4.3 Adaptive Step Size Control . . . . . . . . . . . . . . . . . . . 17

2.4.4 Location Estimation Process . . . . . . . . . . . . . . . . . . 18

2.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.1 Simulation Settings . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 21

iii

2.5.3 Experiment Settings . . . . . . . . . . . . . . . . . . . . . . 25

2.5.4 Experiment Results . . . . . . . . . . . . . . . . . . . . . . . 27

3 DYNAMIC PATH PLANNING FOR MOBILE ANCHOR ASSISTED LO-

CALIZATION 30

3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Position Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.1 The Mobile Anchor Movement Model . . . . . . . . . . . . . 32

3.3.2 Cramer-Rao Lower Bound . . . . . . . . . . . . . . . . . . . 33

3.4 Optimal Mobile Anchor Trajectory for Localization . . . . . . . . . . 36

3.4.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.2 Optimization Problem . . . . . . . . . . . . . . . . . . . . . 36

3.4.3 Trajectory Optimization . . . . . . . . . . . . . . . . . . . . 37

3.4.4 MST-based Path Planning . . . . . . . . . . . . . . . . . . . 39

3.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5.1 Simulation Settings . . . . . . . . . . . . . . . . . . . . . . . 42

3.5.2 Path Planning for Single Sensor Node . . . . . . . . . . . . . 43

3.5.3 Path Planning for Multiple Sensor Nodes . . . . . . . . . . . 45

3.5.4 Experiment Settings . . . . . . . . . . . . . . . . . . . . . . 50

3.5.5 Experiment Results . . . . . . . . . . . . . . . . . . . . . . . 51

4 CONCLUSION 53

Abstract (In Korean) 62

iv

List of Tables

2.1 Comparison of the system stability performance between the classical

MSR and the proposed MSR (λ = 0.015/m2, σ = 4dB) . . . . . . . 21

2.2 Comparison of the system stability performance between the classical

MSR and the proposed MSR (λ = 0.075/m2) . . . . . . . . . . . . . 27

3.1 Comparison of communication cost for position estimation of each

algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 Comparison of the performance between the MST-based trajectory and

the proposed algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 51

v

List of Figures

1.1 Change in variance of position estimation according to mobile anchor

position: (a) low and (b) high variance . . . . . . . . . . . . . . . . . 3

2.1 Example of node location estimation using MSR algorithm; true and

estimated positions are marked by ‘�’ and ‘◦’, respectively. . . . . . . 11

2.2 Relationship between distance and CRLB. . . . . . . . . . . . . . . . 13

2.3 Data generation for feedforward neural network. . . . . . . . . . . . . 15

2.4 Compared the results of the proposed method with those of the feed-

forward neural network. . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 RMSE versus node density. (σ = 4 dB) . . . . . . . . . . . . . . . . 22

2.6 RMSE versus shadowing factor. (λ = 0.015/m2) . . . . . . . . . . . 24

2.7 Snapshot of Experiment area. . . . . . . . . . . . . . . . . . . . . . . 25

2.8 RSS of measured data. (the reference distance d0 = 2 m) . . . . . . . 26

2.9 System performance of different algorithms, as a function of the num-

ber of sensor nodes: (a) RMSE versus node density, (b) Average vari-

ance of the position error . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1 The CRBs, assuming np = 2 and σdB = 4 dB . . . . . . . . . . . . 35

vi

3.2 The process of MST-based path planning: (a) find the MST of WSN

graph G, (b) set root nodes and measure the number of nodes con-

nected through each branch to root, (c) determine the optimal trajec-

tory by connecting partial solutions . . . . . . . . . . . . . . . . . . . 40

3.3 Trajectory optimization in single-node scenario: (a) position estima-

tion over time steps, (b) trajectory determined using D-optimality . . . 44

3.4 RMSE of path planning approaches according to the number of sensor

nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5 Path length of the approaches according to the number of sensor nodes 47

3.6 Trajectory of mobile anchor and position estimation of sensor nodes . 49

3.7 The sensor placement . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.8 The performance for each algorithm: (a) The MST-based trajectory,

(b) The trajectory of the proposed algorithm . . . . . . . . . . . . . . 52

vii

Chapter 1

INTRODUCTION

Applications based on wireless sensor networks (WSNs) are widely used in various

fields such as environmental monitoring, military operations, industrial automation,

and traffic management. These networks are typically intended to retrieve information

from sensors distributed over a desired area. Hence, the position of each sensor node

is important for data analyses. Each sensor in a WSN can provide its location using

positioning devices such as global positioning system (GPS). However, for certain ap-

plications such as field military operations or environmental monitoring, the use of

GPS is impractical due to several aspects including inherent expenses, power require-

ments, and susceptibility to jamming [1]. Alternatively, the location of a sensor node

can be estimated by using signals from nearby reference nodes (or anchor nodes) that

have knowledge of their own locations. If sensors are within the communication range

of anchor nodes, their location can be estimated using their relative position and the

measured distance. Range measurements that rely on Time of Arrival (ToA), Time

Difference of Arrival (TDoA), or Angle of Arrival (AOA) require additional hard-

ware for the arrival time or angle of the received signals, whereas the received signal

strength (RSS) of signals can be measured by any receivers during data communica-

tion without presenting additional bandwidth or energy requirements [2]. Therefore,

considering the hardware limitations of low-cost sensors constituting WSNs, I assume

1

that the range measurement depends on the RSS.

However, in a wide network environment having only a few reference nodes, ef-

fective signalling may be constrained. In such cases, the localization problem becomes

difficult. Several decentralized localization algorithms that could be implemented in a

network environment exhibiting an insufficient number of anchor nodes have been

researched [3-8]. Among these, I investigated the mass-spring-relaxation (MSR) algo-

rithm [8] as deemed suitable for sensor positioning due to its cost-effectiveness. The

concept of this algorithm is to find the equilibrium position of a node that minimizes

the difference between the estimated distance and the real distance between nodes.

The MSR algorithm estimates the nodes’ location by modelling the complex localiza-

tion problem with the dynamic nature of the physical spring system. Because of the

benefits of the MSR algorithm, it is used to develop several algorithms to improve the

quality of localization [8-14]. However, these researches assume that a sufficient num-

ber of anchor nodes are uniformly deployed or the link connectivity between nodes

is guaranteed to be consistent. If the assumption fails due to a low link quality, these

problems would result in oscillation or false estimation of the nodes’ location. To over-

come these drawbacks, I propose a self-organizing localization scheme, wherein each

node estimates its own position using the location information of its one-hop neigh-

boring nodes. I analyse the inherent problems exhibited by various papers based on the

MSR scheme and propose two algorithm improvements, a distance correction factor

and an adaptive step size based on the gradient method, to solve them. I evaluated the

accuracy of the proposed algorithm through the simulation in various scenarios. Then,

to verify the feasibility of the proposed algorithm, the experiment was conducted us-

ing commercial ZigBee devices in realistic environment. Both simulations and experi-

ments show that, as compared to the conventional MSR algorithm, the enhanced MSR

algorithm delivers better performance of localization accuracy and a reduction in the

number of estimation iterations. Due to the adaptive step size control, it was possible

to maintain a stable system even in a poor localization environment.

2

Recent research has introduced mobile anchor assisted localization [15, 16]. This

method assumes that the mobile anchors do not have energy restrictions and can move

on the ground or fly. In addition, the GPS-equipped mobile anchors cover a monitoring

area and periodically broadcast their location to support localization of nearby sensor

nodes. Therefore, the localization accuracy of nodes in the WSN can be improved by

carefully designing the path of the mobile anchors. Thus, mobile anchor assisted lo-

calization has become one of the mainstream methods for estimating node position.

Various path planning schemes have been proposed to improve the localization per-

formance and include static and dynamic approaches [17]. In static approaches, the

mobile anchor follows a predefined trajectory based on a preliminary analysis of the

monitored area [18-20]. In contrast, dynamic approaches rely on the node distribution,

and aim to minimize both the path length and energy consumption.

(b)(a)

x

Y

x

Y

Figure 1.1: Change in variance of position estimation according to mobile anchor po-

sition: (a) low and (b) high variance

In these respects, I investigate a dynamic path planning approach for a single mo-

3

bile anchor to accurately localize multiple nodes in WSNs, without requiring prior

knowledge of the monitored region and WSN configuration. Figure 1.1 shows the ef-

fects of the mobile anchor trajectory on the sensor node localization error. Thus, I

focus mainly on developing path planning method for optimizing the mobile anchor

trajectory to achieve the best localization performance. The mobile anchor determines

the trajectory based on the analysis of bounds on the localization accuracy of nodes. In

this process, the mobile anchor analyzes the limits on localization accuracy by using

the Cramer-Rao lower bound (CRLB), which defines a lower bound on the variance

achievable by any unbiased location estimator [21, 22]. Then, the mobile anchor dy-

namically determines the trajectory and broadcasts the beacon signal to the node for

accurate localization.

There are available many dynamic path planning methods to reduce implemen-

tation cost by using a single instead of multiple mobile anchors [23-27]. In [23], the

author proposes virtual force-based dynamic path planning of a mobile anchor in three-

dimensional spaces. The mobile anchor trajectory is based on the real-time calculation

of a virtual force exerted upon the anchor by unknown sensor nodes, where the loca-

tion of the nodes is detected by directional antennas mounted on the mobile anchor.

In [24, 25], the path is determined using fuzzy logic, where the area for path plan-

ning is divided into a set of symmetric hexagons that can be visited by the mobile

anchor. In [26, 27], the authors propose cooperative path planning for several mobile

anchors using a predictive model based on the Fisher information matrix (FIM). Still,

this approach should be carefully designed given that multiple mobile anchors should

be controlled simultaneously.

As I mentioned earlier, the performance of localization can be quantified by com-

paring its covariance to the CRLB. However, the CRLB is not a function of the es-

timation method but of the geometry of the problem. To overcome these drawbacks,

I focus on specific criteria relying on a single-valued objective function based on the

FIM [28, 29]. The FIM is referred to as the inverse of the CRLB. Then, to find the

4

optimal node position with minimum variance, I formulate the optimization problem

using the FIM.

After successfully localizing a node, the mobile anchor should select the next node

to be visited. This selection determines the energy consumption for movement and the

localization time, and may not guarantee the localization of every sensor in the WSN.

To determine an optimal trajectory, the Traveling Salesman Problem (TSP) can be

used [30]. However, this is a well-known NP-complete problem. Therefore, I use the

Minimum Spanning Tree (MST), which allows to approximate the TSP and reduce

the complexity [31-33]. This algorithm computes the spanning tree of the weighted

graph for a network, whose final cost is the sum of the weights among the edges in

the tree, and its solution is the path connecting all the nodes at minimum cost. In the

proposed method, I assign weights to each edge based on the distance measurements

among sensor nodes, and the shortest path for the mobile anchor is determined as the

MST in the WSN. Still, a centralized MST can incur in considerable communication

overheads to transfer the measurements among nodes to the mobile anchor. Hence, I

apply the distributed MST algorithm proposed by Gallager, Humblet and Spira [34] for

determining the minimum-weight tree. Based on the distributed MST, I construct the

path planning problem without requiring prior information such as the node distribu-

tion and WSN size. This path planning approach enables an energy-efficient solution

without omitting any node in the WSNs.

The rest of this paper is organized as follows: In Chapter 2, I describe the problem

statement and the basic MSR algorithm for the self-organized localization. The pro-

posed algorithm is introduced to account for the accuracy of the measured distances

and ensure the algorithm’s stability. In Chapter 3, the optimization problem for path

planning and trajectory generation in WSN with only one mobile anchor are intro-

duced. This algorithm predicts the variance of estimation according to mobile anchor

trajectory and control the trajectory to improve the accuracy of the location estimation

of nodes. Finally, the concussion of the dissertation is represented in Chapter 4.

5

Chapter 2

SELF-ORGANIZING LOCALIZATION WITH ADAP-

TIVE WEIGHTS FOR WIRELESS SENSOR NETWORKS

2.1 Motivation

This study proposes a self-organizing localization scheme based on the mass-spring

relaxation scheme. The sensor nodes are modelled as masses and regarded as if they

would be connected with the neighboring nodes by springs that would force the nodes

to move towards the equilibrium positions. Therefore, an unknown position could be

determined using the MSR algorithm by calculating the related forces exerted by the

neighboring nodes. However, in a noisy environment, errors in distance measurements

adversely affect the calculation of the force acting on the node, causing system oscil-

lation and affecting the localization of other nodes.

To solve these problems, I devise a distance correction factor to account for the

accuracy of the measured distances. In the conventional MSR method, the spring con-

stant indicates the uncertainty of the measurements. In many researches, only the un-

certainty of the node location information is taken into account by the spring constant.

However, there is also substantial inherent uncertainty in the measured distance be-

tween the nodes. According to [35, 36], longer distances cause lower levels of con-

6

fidence that lead to stochastic errors of the estimated distance following the Cramer-

Rao Lower Bound (CRLB). If no correction mechanism for the measurement error

exists, this assumption precludes finding the exact equilibrium springs’ length. How-

ever, many studies have not taken this drawback into account, only considering the

node’s state. Hereby, the proposed distance correction factor quantifies the uncertainty

of the measured distance. The correction factor for a pair of nodes is inversely propor-

tional to the distance between the nodes. Thus, the larger distance measurements are

down-weighted in the individual optimization function within the localization process.

Second, to ensure the algorithm’s stability and improve the location estimation

accuracy, I apply an adaptive step size control approach that varies the step in relation-

ship to the converging state of the estimated position. In the location updating process,

the residual springs’ forces are used to refine the estimated location of the nodes. Nu-

merous conventional papers [8-10] have reported a solution to this process simply by

applying a fixed step size. If the estimated location moves in the direction of the re-

sultant force by an infinitesimal amount, the convergence process may take longer to

reach equilibrium, or the estimated position may diverge and affect the other nodes.

To resolve this problem, the gradient descent method is utilized to prevent oscillation

and reduce the convergence time. Among various gradient algorithms, the Barzilai and

Borwein (BB) gradient method [37] is adopted, which requires minimal memory and

computational resources. The step size is derived from a two-point approximation to

the secant equation underlying the quasi-Newton method.

2.2 Problem Statement

Assume that there areN nodes with unknown locations randomly distributed in a two-

dimensional WSN. The M anchor nodes are located at the boundaries of the network

area to simulate a network with an insufficient number of anchor nodes. These anchor

7

nodes can identify their location via GPS or some other mechanism. Furthermore, it

is assumed that once all the nodes are deployed, they will remain in their locations.

Each node has a unique ID and a transmission radius R. The nodes located within the

transmission radius of another node can be denoted as “one-hop neighbors” and it can

be assumed that the communication link between them is guaranteed. Thus, the nodes

with unknown locations must be able to accurately determine their own locations by

exchanging information with their one-hop neighbors.

Let xi be a two-dimensional position vector that denotes the true position of node

i, for which n(i) denotes a set of its one-hop neighbors and dij denotes the actual

distance between the node i and its one-hop neighboring node j. I denote δij as the

measured distance between nodes i and j and assume that the measured distance be-

tween any pair of nodes is symmetrical, i.e., δij = δji. In this paper, I assumed the

usage of the RSS range estimation method, the measured distances being based on the

log-distance path loss model [2], expressed as follows

Pij = P0 − 10nplogdijd0

+Xij (2.1)

where P0 is the power measured at a reference distance d0 from the node and Pij is

the power received by the node j from the node i. Furthermore, np is the path-loss

exponent and Xij ∼ N(0, σ2

)represents the log-normal shadowing effect where σ2

is the variance of the shadowing. Given Pij in Eq. (2.1), the distance between node i

and node j is derived as

δij = 10(P0−Pij)/(10np) (2.2)

The objective of the proposed scheme is to localize the nodes with unknown po-

sitions in a distributed and self-organized manner. The nodes that cannot communi-

cate directly with the anchor nodes must use inaccurate information obtained from the

neighboring nodes whose locations are unknown. Therefore, the estimation process

8

consists in updating the location of the node using an iterative estimation procedure.

Let x(k)i be a two-dimensional position vector that denotes the estimated position of

node i at the k-th iteration. The estimated distance d(k)ij denotes the Euclidean distance

between two nodes and is represented as

d(k)ij = ‖x(k)

i − x(k)j ‖2 =

√(x(k)i − x

(k)j

)T (x(k)i − x

(k)j

)(2.3)

where (·)T is the matrix transpose operation.

2.3 Conventional Mass-spring Relaxation Method

The MSR method is efficient in solving the problem outlined in Section 2.2. It resolves

the complex localization problem by applying a virtual spring connection between a

pair of nodes, which has similar properties to the second law of Newton.

In the MSR method, the localization problem is viewed as the task of finding the

equilibrium position of a mass connected with springs to fixed bodies. In the formula-

tion, the mass corresponds to the node i and the fixed bodies correspond to its one-hop

neighboring nodes. In other words, while estimating x(k)i , it is assumed that the one-

hop neighboring nodes have fixed positions. I construct the spring sij in such a way

that the free length of sij is equal to the actual distance dij and the deformation length

can be expressed as l(k)ij = d(k)ij − dij at the k-th iteration of the proposed localization

scheme. In practice, as the actual distances corresponding to the nodes are unknown,

the actual distances are replaced by the measured distances and subsequently, the de-

formation length is expressed as l(k)ij = d(k)ij − δij .

The springs contain a certain potential energy owing to their deformation, whereas

the stored energy in a spring sij becomes zero if d(k)ij = δij . Hence, the force F(k)ij acts

on the node i towards the direction that minimizes the potential energy stored in the

spring sij . The force F(k)ij can be expressed as

F(k)ij = mj

(d(k)ij − δij

)u(k)ij , j ∈ n(i) (2.4)

9

where u(k)ij is the unit vector from node i to node j. mj is the spring constant that

is determined by the measurement uncertainty. Particularly, mj is denoted by the lo-

cation accuracy of node j: If node j is the anchor node that has the correct location

information, mj is set to 1, whereas if the location is unknown, mj is set to 0.5.

The net force F(k)i acting on node i at the k-th iteration is expressed as

F(k)i =

∑j∈n(i)

F(k)ij (2.5)

The MSR algorithm aims to correct the estimated position by minimizing the energy

of the springs connected with the neighboring nodes. Thus, the estimated location of

the node i at the (k + 1)-th iteration is updated according to the total force F(k)i as

x(k+1)i = ωF

(k)i + x

(k)i (2.6)

Here, ω is the step size governing the convergence speed of the MSR algorithm. Each

node i updates its estimated location by “moving” in the direction of the resultant

force. Most of the conventional researches empirically set the step size to 1/2|n(i)|,

that is inversely proportional to the number of neighboring nodes. When the estimated

position of each node reaches the location where ‖Fi‖ = 0, the estimation process

ends. This location is defined as an equilibrium location. Figure 2.1 shows the example

of a process of node location estimation.

In a real environment, it is difficult to find the location exhibiting an exact ‖Fi‖

zero value due to various errors. Hence, the node decides to end the estimation process

through the pre-set threshold ∆th and determines its final estimated position as

xi = x(k+1)i , if ‖x(k+1)

i − x(k)i ‖2 ≤ ∆th (2.7)

After completing the process of individual update, the node broadcasts its final

estimated location information xi to the neighboring nodes.

10

j

kq

p

i

Figure 2.1: Example of node location estimation using MSR algorithm; true and esti-

mated positions are marked by ‘�’ and ‘◦’, respectively.

11

2.4 An Improved Algorithm for Mass-spring Relaxation

2.4.1 Outline

The localization process will reach a static equilibrium by minimizing the total force

within the springs. Accurate measurement of the force is very important for both the

estimation accuracy and system stability. However, existing MSR studies lacked accu-

rate estimations of the virtual forces acting on the nodes. Moreover, the forces must be

considered as variable throughout the iterative process. In this dissertation, I provide

several improvements to the classical MSR algorithm to reduce the estimation errors

and guarantee the system stability.

2.4.2 Distance Correction Factor

As mentioned in Section 2.3, the constantmj represents the spring constant modelling

the measurement uncertainty. Numerous researches assumed that the characteristics

of the springs are the same for all nodes, thus the spring constant could be ignored

or treated for simplicity as a single system-level constant. Certain researches [9, 11]

consider the constant as modelling the accuracy of the location information depending

on whether the node connected to the corresponding spring is an anchor node or not.

However, these studies simply set constants for each node type, and did not consider

the accuracy of the distance measurement.

In real systems, the accuracy of the location information depends on various pa-

rameters and rarely shows a constant value over the entire WSN. The proposed MSR

algorithm localizes the nodes using the distance measured by the RSS method, using

the log-normal shadowing model. Therefore, the accuracy of the location information

is determined by the accuracy of the distance estimation. The CRLB of the distance

measurements based on the log-normal shadowing signal propagation model in Eq.

(2.1) is calculated as

12

0 5 10 15 20 25

Actual Distance dij (m)

0

20

40

60

80

100

120

140

Measure

d D

ista

nce δ

ij (m

)

analysis of measurements

CRLB

Measurements

Figure 2.2: Relationship between distance and CRLB.

13

√V ar (δij) ≥

σdij10nplog10(e)

(2.8)

where V ar (δij) is the variance of the measured distance δij . Equation (2.8) states that

the uncertainty of the measured distance increases with the actual distance dij . More-

over, Fig. 2.2 shows the relationship between the distances and CRLB values of the

measurements, assuming np = 2.3 and σ = 4 dB. I can observe that the measure-

ments precision decreases as the distance between the nodes increases.

According to this observation, I suggest that an additional factor should be in-

cluded in the MSR algorithm to reflect the accuracy of the distance measurement de-

pending on the actual distance. Inspired by the distance correction factor used in [6],

the proposed formula for the net force exerted on node i at the k-th iteration becomes

F(k)i =

∑j∈n(i)

e−(δij/δi,max)F(k)ij (2.9)

where δi,max is the distance between the node i and its farthest neighbor. This equation

implies that the proposed algorithm down-weights the information from farther nodes

in the localization process of individual nodes.

To evaluate the distance correction factor, I applied the feedforward neural network

(FNN) [38]. I considered various FNN structures and trained neuron networks based

on a large number of known input-output data sets. These data sets are divided into

three categories: training sets, cross-validation sets, and test data sets. The data sets

are generated, assuming a wireless channel environment that has the np set to 2.3, and

σ(dB) set to 4, and composed of M neighboring nodes and one target node, as shown

in Fig. 2.3. The input data is (M × 2)-element vector [|F1|, ...|FM |, θ1, ..., θM ]. The

output data represents a two-dimensional vector [xe, ye], which is the correction vector

between the real position and the estimated position.

14

Figure 2.3: Data generation for feedforward neural network.

15

Figure 2.4 shows the location estimation accuracy of the proposed factor and the

trained output. The label “Inv. MSR” considers the accuracy of the measured distance

by introducing the weight which is inversely proportional to the distance. The error

between the desired output and the trained output decreases with the number of layers

and the number of neurons and is under 0.5 m. The proposed factor exhibit better

performance than the “Inv. MSR”. In addition, The proposed factor achieves system

performance similar to FNN 1 layer case and a performance difference of about 0.3 m

with the 3 layer case. The FNN provides more accurate results, but it is advantageous

to use the distance correction factor in terms of the amount of computation.

3 4 5 6 7 8 9 10 11 12 13

The number of anchor nodes

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Localiz

ation E

rror

(m)

Feedforward 5

Feedforward 10

Feedforward 20

Feedforward 10-10

Feedforward 20-20

Feedforward 10-10-10

Feedforward 20-20-20

Prop. MSR

Inv. MSR

Classic MSR

Figure 2.4: Compared the results of the proposed method with those of the feedforward

neural network.

16

2.4.3 Adaptive Step Size Control

In the location updating process outlined in Eq. (2.6), the node updates its location

according to the resultant force in each iteration, whereas the step size determines the

algorithm convergence speed. In numerous studies, a fixed step size is applied to the

location updating process. The step size has a significant impact on the algorithm per-

formance. Too small values of the step size would cause the algorithm to converge very

slowly, whereas a large step size could cause a diverging process, affecting the system

stability. To resolve this problem, a gradient-based adaptive step size control is applied

for implementing a reliable localization system. I adopt the Barzilai and Borwein (BB)

gradient method that requires minimal memory and computational resources.

The objective of the proposed algorithm is to estimate the node’s location by min-

imizing the potential energy of the virtual springs acting on the node. The potential

energy W (x(k)i ) of the node i can be expressed by

W(x(k)i

)=

∫F(k)i dl (2.10)

Thus, I are interested in solving the following optimization problem

minimize∑

W(x(k)i

)(2.11)

The principle of the BB method is based on using the information in the previous

iteration to determine the step size in the next iteration. The step size in this method is

derived from the two-point approach to the secant equation based on the quasi-Newton

method, specifically:

x(k+1)i = x

(k)i −B−1k F

(k)i (2.12)

where B−1k = −ω(k)i I and I is the identity matrix. With the Taylor series expansion

for the quadratic approach, optimal Bk can be determined by

Bk = arg minB−1∈R

‖sk−1 −B−1k yk−1‖2 (2.13)

17

where sk−1 = x(k)i − x

(k−1)i , yk−1 = F

(k)i − F

(k−1)i , and ‖ · ‖ denotes the Euclidean

norm. This minimum value is obtained by

ω(k)i =

sTk−1yk−1

yTk−1yk−1(2.14)

As the proposed algorithm operates with the gradient descent, there is a non-zero

probability that the system will converge to a local, rather than a global, minimum.

This tendency is affected by the initial estimate, and it is very important to determine

the initial position. This is a common problem with algorithms with similar mech-

anisms. In addition, when the gradient is shallow during optimization and less than

the threshold ∆th during localization, the convergence in gradient descent could show

the wrong position before reaching the optimal position. The BB method can mitigate

this tendency by appropriately controlling the step size, considering the update state

of the estimation and the variation in the total force. While this is not addressed in this

dissertation, the issue requires further investigation.

2.4.4 Location Estimation Process

A problem, common to most decentralized schemes and thus to the MSR method, is

the need to assign an initial position x(0)i to every node. This requirement could be

fulfilled using certain existing algorithms [39-41]. In this paper, for simplicity, I as-

sume that the initialization of the algorithm is carried out using the DV-hop method

[39]. Initially, each node measures the distances from its neighbors and determines its

minimum hop count information through hello packets from the anchor nodes. Subse-

quently, it estimates its initial position x(0)i through the DV-hop scheme.

After initialization, each node calculates the net force solely based on the distance

and location information of the neighboring nodes and estimates its position. Algo-

rithm 2 details the proposed localization method.

18

Algorithm 1 Location Estimation Process

1: Initial position vector x(0)i , i ∈ {1, . . . , N}, derived from DV-hop [39].

2: Set k ← 1, N = {1, . . . , N}

3: while N 6= Φ do

4: for each node i ∈ N do

5: Calculate the force F(k)i using Eq. (2.9)

6: if k = 1 then

7: ω(k)i = 1

2|n(i)|

8: else

9: Calculate ω(k)i based on Eq. (2.14)

10: end if

11: Update the location x(k+1)i = ω

(k)i F

(k)i + x

(k)i

12: if ‖x(k+1)i − x

(k)i ‖2 < ∆th then

13: xi = x(k+1)i

14: N = N − i

15: end if

16: end for

17: Set k ← k + 1

18: end while

19

2.5 Performance Evaluation

2.5.1 Simulation Settings

In this section, I compare the localization performance of the proposed scheme versus

the performance of the dw-MDS which is commonly used as benchmark. A sensor

network with anchor nodes, M = 9 is considered, and the anchor nodes are located

at the center and edges of a 100 m × 100 m square region. The nodes with unknown

locations, N = [130, 135, 140, 145, 150 , 155, 160], are randomly distributed within

the same square area, with node density λ varying from 0.013/m2 to 0.016/m2.

As mentioned above, I assume that the measured distances are based on the log-

distance path loss model. Based on the CC2420 Micaz Radio Module [42], I assume

that P0 = −56 dBm, d0 = 1m and, the sensitivity level for receiving and performing

correct demodulation is −87 dBm. The performance evaluation of the localization

accuracy for both proposed solutions is presented under the conditions of the log-

normal shadow fading environment that has np set to 2.3 and σ(dB) set to 4 and 8.

The threshold ∆th for ending the location update is set as 0.1. I ran 10000 Monte

Carlo simulation trials for each scenario to analyse the system performance in terms

of system stability and computed the root mean square error (RMSE) as follows:

RMSE =

√∑Ni=1 (xi − xi)

T (xi − xi)

N(2.15)

In the comparisons, the label “MSR” refers to the classical MSR algorithm [8],

whereas “MSR+DCF” labelled data considers the accuracy of the measured distance

by introducing the distance correction factor proposed in this paper. “MSR+Adp” in-

dicates adding an adaptive weight to the classical MSR algorithm. The label “Prop.

MSR” refers to the full version of the proposed scheme, including the distance correc-

tion factor and the adaptive weight enhancements. In addition, to compare the perfor-

mance of the proposed algorithm with those of the dw-MDS [6], I configure that the

20

total number of iterations of dw-MDS will be equal to the average number of iterations

of the proposed scheme. This is because the proposed scheme is an individual iterative

process, unlike dw-MDS.

2.5.2 Simulation Results

Table 2.1 shows the performance comparison of the four algorithms in terms of conver-

gence rate and system stability. A failure case is declared when one or more nodes can

not find the location due to oscillation. In terms of the number of required iterations

for determining the location update, although the application of the distance correction

factor (MSR+DCF) is not effective in reducing the average number of iterations, it has

a measurable effect on suppressing the oscillation by correcting errors that adversely

affect the algorithm. The integration of an adaptive step size (MSR+Adp and Prop.

MSR cases) achieves a reduction in the number of estimation iterations by more than

half and a perfectly stable performance.

Figure 2.5 shows the RMSE for each algorithm as a function of the node density,

assuming σ = 4 dB. Two advanced MSRs (MSR+DCF, MSR+Adp) exhibit better

performance than the classical MSR. The proposed scheme (Prop. MSR), based on

combining the correction factor and the adaptive step size, exhibits about 7 % higher

performance than the conventional scheme (MSR). In addition, the proposed scheme

Table 2.1: Comparison of the system stability performance between the classical MSR

and the proposed MSR (λ = 0.015/m2, σ = 4dB)

MSR MSR+DCF MSR+Adp Prop. MSR

Average ] of iterations 14 13 7 7

Min ] of iterations 3 2 2 2

Max ] of iterations 44 41 17 16

Number of failures 1040 450 0 0

21

0.013 0.0135 0.014 0.0145 0.015 0.0155 0.016

Node Density ( )

5.5

6

6.5

7

7.5

8

Localiz

ation E

rror

(m)

RMSE of algorithms

DV-hop

MSR

MSR+DCF

MSR+Adap.

proMSR

dw-MDS

Figure 2.5: RMSE versus node density. (σ = 4 dB)

22

achieves the performance close to dw-MDS by approximately 2 % difference. In the

dw-MDS, all the nodes compute the global cost function value for localization in a

distributed manner and cyclically send it to the next neighbor. On the other hand, the

proposed scheme requires only information of one-hop neighboring nodes, which is

more efficient for signaling overhead and energy consumption than dw-MDS.

Figure 2.6 illustrates the localization performance by considering the shadowing

factor, assuming λ = 0.015/m2. Owing to an increase in the measured distance error

caused by an increased shadowing measured factor, the localization error of all the

schemes increased. The localization error of the MSR+DCF decreases faster than the

classical MSR. As the shadowing factor increases, the distance between the nodes

increases, thus the error of the measured distance becomes relatively large. Therefore,

the application of the distance correction factor is effective. Moreover, the integration

of the adaptive weight (MSR+Adp and Prop. MSR cases) achieves a reduction in the

number of required estimation iterations and a significantly stabilized performance,

as presented in Table 2.1. Finally, the proposed scheme (Prop. MSR), combining the

advantages brought by the distance correction factor and adaptive weight, exhibits

about 7 % performance enhancement compared to the classical MSR algorithm and

system performance similar to dw-MDS.

23

4 4.5 5 5.5 6 6.5 7 7.5 8

Shadowing factor (dB)

6

8

10

12

14

16

18

20

22

Loca

lizat

ion

Err

or (

m)

RMSE of algorithms

DV-hopMSRMSR+DCFMSR+Adap.proMSRdw-MDS

Figure 2.6: RMSE versus shadowing factor. (λ = 0.015/m2)

24

2.5.3 Experiment Settings

Figure 2.7: Snapshot of Experiment area.

To verify the proposed approach in a realistic WSN scenario, I confirmed the appli-

cability of the proposed algorithm to reality through experiments. The verification was

conducted using Zigbee-based sensor devices considered in previous simulations. The

64 sensors are located within the 20 m × 20 m square area in a gym (Fig. 2.7). Four

points at the edge of experiment region were selected as anchor nodes and N points

were randomly selected as sensor nodes, with node density λ varying from 0.075/m2

to 0.12/m2. I measured an RSS value between any pair of sensors and calculated a

measured RSS value to a corresponding distance by applying indoor path loss model.

Figure 2.8 depicts the measured RSSs over the log distance, showing np = 2.33, and

σ(dB) = 4.756. The remaining parameters were the same as in the previous scenario.

25

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

log10(di/d

0)

-80

-70

-60

-50

-40

-30

-20

-10

0

Receiv

ed s

ignal str

ength

Measurement results

Figure 2.8: RSS of measured data. (the reference distance d0 = 2 m)

26

2.5.4 Experiment Results

I analyzed the feasibility of the proposed algorithm in the case of various node den-

sity. Figure 2.9 illustrates the system performance for each algorithm as a function of

the number of nodes. Figure 2.9a shows the RMSE of the position estimation for the

sensor nodes. The proposed scheme (Prop. MSR) exhibits about 12 ∼ 16% higher

performance than conventional MSR. This performance approaches the performance

of dw-MDS. Moreover, the application of the adaptive weight reduces the number of

iterations required for estimation and improves the stability of the localization system,

as presented in Table 2.2. Figure 2.9b shows the variance of the position error for each

algorithm as a function of the number of nodes. All algorithms have reduced variance

as the number of nodes increases, while the variance is less than 1 m for cases with

more than 40 nodes. When considered together with the RMSE results in Fig. 2.9a,

the proposed algorithm achieves stable and accurate performance with the number of

iterations reduced by more than half.

Table 2.2: Comparison of the system stability performance between the classical MSR

and the proposed MSR (λ = 0.075/m2)

MSR MSR+DCF MSR+Adp Prop. MSR

Average ] of iterations 8 7 6 5

Min ] of iterations 2 2 1 1

Max ] of iterations 21 20 12 11

Failure rate 4 % 2 % 0 % 0 %

27

0.075 0.0825 0.09 0.0975 0.105 0.1125 0.12

Node Density ( )

1

1.5

2

2.5

3

3.5

4

4.5

Lo

ca

liza

tio

n E

rro

r (m

)

RMSE of algorithms in the experiment

DV-hop

MSR

MSR+DCF

MSR+Adap.

proMSR

dw-MDS

(a)

0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12

Node Density ( )

0

1

2

3

4

5

6

7

8

9

Va

ria

nce

of

the

estim

atio

n (

m)

Variance of algorithms in the experiment

DV-hop

MSR

MSR+DCF

MSR+Adap.

proMSR

dw-MDS

(b)

Figure 2.9: System performance of different algorithms, as a function of the number

of sensor nodes: (a) RMSE versus node density, (b) Average variance of the position

error

28

To summarize the simulation and experimental results, I analyzed the performance

of the proposed algorithm in terms of the convergence rate, system stability, and accu-

racy. If only Figures 2.5, 2.6, and 2.9 were considered, the improvement in the perfor-

mance of the proposed algorithm is insignificant. However, as shown in Tables 2.1, and

2.2, the proposed algorithm reduces the number of iteration and prevents oscillation.

The application of the distance correction factor has a measurable effect in suppressing

oscillation by correcting errors that adversely affect the algorithm. When the RMSE

of MSR+Adp and Prop. MSR are compared, it can be confirmed that the accuracy is

improved by the distance correction factor. In addition, adaptive step size control not

only implements a perfectly stable system but also provides more accurate location

estimation with the number of iterations reduced by more than half.

29

Chapter 3

DYNAMIC PATH PLANNING FOR MOBILE ANCHOR

ASSISTED LOCALIZATION

3.1 Motivation

In this chapter, I study a dynamic path planning approach for a single mobile anchor to

accurately localize multiple nodes in WSNs, without requiring prior knowledge of the

monitored region and WSN configuration. The localization accuracy depends not only

on the influence of the environment but also on the geometrical relationship among

the position information transmitted by the anchor to a node. Given that the effect of

various environmental parameters is difficult to overcome, the mobile anchor should

move to an optimal position for maximizing the localization accuracy. Thus, the pro-

posed algorithm aims to determine the optimal trajectory maximizing the localization

accuracy of scattered sensor nodes. The basic idea of this study is based on the ge-

ometric dilution of precision [43], where the mobile anchor calculates the CRLB for

each node from its own trajectory, and the estimated position of the node. However,

because the CRLB, which is the inverse of the FIM, is not a function of the estimation

method, it has limitations on the analysis of the uncertainty that can be obtained for a

particular system. Furthermore, maximizing the FIM to determine the trajectory that

30

minimizes this bound involves maximizing a matrix, thus requiring a highly complex

process. Therefore, I study several matrix norms for the FIM and formulate the trajec-

tory optimization problem.

The mobile anchor determines the sequence of waypoints considering one node at

a time, and when the localization of that node is finished, it considers the next node. If

multiple nodes are simultaneously within the communication range of the mobile an-

chor, it must select which node to visit first. A random selection may increase energy

consumption for movement and lead to omitting some nodes. Therefore, I apply the

MST to find the minimal path linking nodes in the WSN. In the proposed method, the

weight of each edge in the spanning tree represents the distance among sensor nodes.

Consequently, the MST retrieves the shortest trajectory for the mobile anchor to com-

municate with all the nodes in the WSN. As mentioned above, I adopt the distributed

MST to prevent communication overheads.

3.2 Problem Statement

I consider a static wireless sensor network, where N sensor nodes with unknown loca-

tions are randomly distributed. A single mobile anchor is located in the network area

to localize sensors and identifies its location via GPS or some other mechanism. The

movement model of the mobile anchor will be described in the next section.

N sensor nodes with unknown locations are randomly distributed and remain in

their locations. The location of the node i is denoted by xi which is a two-dimensional

position vector. The mobile anchor moves within a monitored area and periodically

broadcasts beacon signals containing information on its position. The received sig-

nal strength indicator (RSSI) is assumed for measuring the distance, considering the

limitations of low-cost sensor. The node measures the received signal strength and cal-

culates the effective propagation loss based on the known transmit power of anchor

nodes and the mobile anchor. Then, by using theoretical and empirical models, the

31

node can translate the loss into a distance estimate. The received signal power Pi at

node i is assumed to be based on the log-distance path loss model and can be expressed

as

Pi = P0 − 10np logdid0

+Xi, (3.1)

where P0 is the power measured at a reference distance (d0 = 1 m) from the mobile

anchor, and di denotes the actual distance between the node i and the mobile anchor.

Furthermore, np is the path-loss exponent and Xi ∼ N(0, σ2

)represents the log-

normal shadowing effect where σ2 is the variance of the shadowing. Given Pi in Eq.

(3.1), the measured distance δi between sensor i and the mobile anchor is derived as

δi = 10(P0−Pi)/(10np). (3.2)

As I mentioned above, the FIM is adopted to estimate the uncertainty of the node’s

position estimate. For this process, one should note that the path-loss exponent can be

obtained by conducting the distance measurement at the additional process [44-46] or

be treated as an additional unknown parameter. In this dissertation, for simplicity, I

assume that the path-loss exponent is known a priori.

3.3 Position Estimation

3.3.1 The Mobile Anchor Movement Model

This section describes the motion of the mobile anchor to then explain the proposed

position estimation. The mobile anchor moves within a monitored area and periodi-

cally broadcasts beacon signals containing information on its position. The position

of the anchor at time step k is denoted by b [k] = [x[k], y[k]]T , its trajectory can be

updated by

b [k + 1] = b [k] + vT [k]∆T

cos θ[k]

sin θ[k]

,vT [k] ∈ [0, VT ], θ[k] ∈ [−ϕ,ϕ], (3.3)

32

where ∆T is the time interval between time step k and k+ 1, vT [k] is the velocity, VT

is the maximum speed, θ[k] is the relative angle with respect to the moving direction,

and ϕ denotes the maximum steering angle of the mobile anchor.

The sensor node receiving the beacon signal determines its distance to the mobile

anchor using Eq. (3.2). Then, the location of the mobile anchor and the distance are

stored in the sensor node. Let x(m)i be a two-dimensional position vector that denotes

the estimated position of node i, with m being a stored information index. When more

than two positions of the mobile anchor have been received, the node estimates its lo-

cation using the stored information. For node i, the trajectory of the mobile anchor and

distance are denoted by B(m)i = [b(1)i , . . . , b(m)

i ], and δ(m)i = [δ

(1)i , . . . , δ

(m)i ] for m

stored registers, respectively. The node updates these location and distance informa-

tion whenever it receives beacon signals from the mobile anchor.

As the localization is independently conducted by each sensor node, I use least-

squares trilateration as the localization algorithm. This estimation algorithm consists

of updating the location of the node considering the latest beacon signal transmitted

from the mobile anchor. Then, the node defines the convergence to its final position

based on a predefined threshold ∆th as

xi = x(m)i , if ‖x(m)

i − x(m−1)i ‖2 ≤ ∆th. (3.4)

After converging to a final estimated position, each node retrieves its position xi

to the mobile anchor, and the mobile anchor moves toward the next node in the WSN.

3.3.2 Cramer-Rao Lower Bound

The CRLB states that, the inverse of the fisher information matrix F(m)i gives the

minimum achievable variance of the estimated location of node i:

cov(x(m)i ) = E

{[x

(m)i − xi][x

(m)i − xi]

}≥(F(m)i

)−1, (3.5)

F(m)i = −E

[ ∂2∂x2

i

log f(P(m)i |xi)

],

33

where f(P(m)i |xi) is the probability density function of the measured received power

matrix, P(m)i = [P

(1)i , . . . , P

(m)i ], at node i. Then, given Eq. (2.1), the FIM for the

RSS is calculated as [47]

F(m)i = F

(B

(m)i

)=( 10npσ log 10

)2 Fxx Fxy

F Txy Fyy

, (3.6)

where Fxx =m∑p=1

(x(m)i − x(p)

)2(d

(p)i )4

,

Fxy =m∑p=1

(x(m)i − x(p)

)(y(m)i − y(p)

)(d

(p)i )4

,

Fyy =

m∑p=1

(y(m)i − y(p)

)2(d

(p)i )4

,

where b(p)i = [x(p), y(p)]T is the p-th stored mobile anchor position in the memory of

node i, and d(p)i is the Euclidean distance between the mobile anchor and node i given

by

d(p)i = ‖x(m)

i − b(p)i ‖2 =

√(x(m)i − b(p)i

)T (x(m)i − b(p)i

)(3.7)

where (·)T is the matrix transpose operation.

Based on these formulation, the trace of the covariance of the node i’s location

estimate satisfies

σ2i , tr{cov

(x(m)i

)}= V ar

(x(m)i

)+ V ar

(y(m)i

)(3.8)

≥([Fxx − FxyF−1yy F

Txy

]−1)+([Fyy − FxyF−1xx F

Txy

]−1)

The CRBs are shown in Fig. 3.1 when there are four reference points located in the

corners of a 5 m by 5 m square, assuming np = 2 and σdB = 4 dB. The minimum

value is 2.52 that is the standard deviation of location estimates in a channel with

34

σdB/np = 2 is limited to this value.

Using the FIM properties, the mobile anchor can predict the accuracy in the esti-

mated position of the node at the next candidate movement. Then, the mobile anchor

adjusts the next waypoint such that it reduces the CRLB, i.e., the proposed algorithm

aims to find the trajectory that minimizes the variance of the position estimation.

2.5

5

3

3.5

4

4

Low

er

bound 4.5

53

y (m)

5

4

5.5

2 3

x (m)

211

0 0

Figure 3.1: The CRBs, assuming np = 2 and σdB = 4 dB

35

3.4 Optimal Mobile Anchor Trajectory for Localization

3.4.1 Outline

As mentioned above, localization performance is highly dependent on the mobile an-

chor trajectory, so the trajectory analysis is important for both the estimation accuracy

and path planning efficiency. The trajectory optimization problem is formulated by us-

ing a real-valued scalar function based on the FIM. Furthermore, as the mobile anchor

considers one sensor node at a time, it is necessary to determine the route for visiting

the nodes to minimize the path and prevent the omission of nodes. In the following, I

formulate the optimization problem to find optimal waypoints for the mobile anchor.

Then, I propose an energy-efficient path planning scheme visiting all the nodes in the

WSN.

3.4.2 Optimization Problem

An information-theoretic framework based on the FIM can exploit its geometrical

characteristics to find the sequence of mobile anchor positions that optimize the es-

timation accuracy of node positions. As the covariance of the estimated position is a

matrix, the procedure that maximize the FIM involves computationally expensive pro-

cess. To reduce this cost, the FIM can be compressed into a real-valued scalar function

using summary statistics. For the proposed method, I consider various optimality cri-

teria that use the FIM for the calculation of a single-valued objective function. The

optimization problem for path planning at each time step can be formulated as

minimize f(F(B

(m)i , ρ (vT [k], θ[k])

))subejct to vT [k] ∈ [0, VT ],

θ[k] ∈ [−ϕ,ϕ]. (3.9)

In this optimization problem, f(·) is a objective function based on the FIM. F (·, ·)

is a estimation of the FIM according to the mobile anchor’s trajectory B(m)i and the

36

next candidate movement ρ (vT [k], θ[k]). The two constraints impose the speed and

steering limits for the mobile anchor.

Given that the FIM is symmetric and positive semidefinite, its quadratic form de-

fines an ellipsoid representing the estimation uncertainty. In fact, the FIM eigenvalues

are related to the error ellipsoid of position estimation, and the lengths of the ellipse

axes are determined by these eigenvalues.

Consider the following single-valued objective functions based on the FIM. The

D-optimality criterion maximizes the FIM determinant and consequently minimizes

the volume of the uncertainty ellipsoid for localization:

f(F(m)

)= − log

(det{F(m)

}). (3.10)

The E-optimality criterion minimizes the length of the largest axis of the ellipsoid

by minimizing the largest eigenvalue of the FIM inverse:

f(F(m)

)= max

a

(eig{

(F(m))−1}). (3.11)

The A-optimality criterion minimizes the trace of the FIM inverse and suppresses

the average variance of the estimates:

f(F(m)

)= trace

{(F(m))−1

}. (3.12)

3.4.3 Trajectory Optimization

Nevertheless, it is not convenient to estimate all values at the positions of the mobile

anchor within a defined time step. Therefore, the path planning for the mobile anchor

can be formulated using the gradient descent method:

b[k + 1] = b[k]− µ(m)i

∂f(F(m)i

)∂x

(m)i

, (3.13)

where µ(m)i is a time-varying step according to the velocity. To simplify the calculation,

the gradient of objective function f(F(m)

)can be determined by the first-order finite

37

difference approximation:

∂f(F(m)i

)∂x

(m)i

≈f(F(m)i

(B

(m)i + ε(θ[k])

))− f

(F(m)i

(B

(m)i

))ε

, (3.14)

where ε(θ[k]) is a two-dimensional vector representing the movement in the direction

of steering angle θ[k] by distance ε, which is a small positive real value. From this

formulation, the optimal steering angle can be estimated as

θ[k] = argminθ[k]≤|ϕ|

∂f(F(m)i

)∂x

(m)i

. (3.15)

Therefore, the optimization objective is to determine the steering angle of the mobile

anchor that retrieves the fastest convergence to the minimum of objective function

f(F(m)i

). After finding the optimal steering angle, the other motion parameter, namely,

the velocity of the mobile anchor should be optimized. Specifically, velocity vT [k] at

time step k can be determined by finding the position that minimizes the objective

function within maximum displacement VT ·∆T in the direction of estimated steering

angle θ[k]:

vT [k] = argmin0<vT [k]≤VT

F(m)i

(B

(m)i + ρ

(vT [k], θ[k]

)), (3.16)

where ρ(vT [k], θ[k]

)is a two-dimensional vector representing the movement in the

direction of steering angle θ[k] at velocity vT [k]:

ρ(vT [k], θ[k]

)= vT [k]∆T

cos θ[k]

sin θ[k]

. (3.17)

Finally, the updated position of the mobile anchor at time step k + 1 can be ex-

pressed as

b[k + 1] = b[k] + ρ(vT [k], θ[k]

). (3.18)

38

3.4.4 MST-based Path Planning

The proposed path planning using the distributed MST does not require intervention of

the mobile anchor or prior information of the WSN environment. In fact, path planning

is performed by communication among the nodes before the mobile anchor starts nav-

igating the monitored area to determine the node positions. First, each node measures

the distance from neighboring nodes, and I assume that the measured distance between

any pair of sensors is symmetrical, i.e., δij = δji. Then, the MST algorithm is applied

on each node, providing a different subtree per node. Likewise, each subtree has a lead-

ing node, with respect to which the minimum-weight trajectory is determined. Then,

an extended tree is generated by associating the partial solutions among neighboring

nodes and combining trees whenever possible. This process is iterated by defining new

leader nodes until the extended tree covers the entire WSN. The optimal mobile an-

chor trajectory is determined by following Algorithm 2, where the nodes, edges and

inter-node distances of the g raph G are represented byN , E andW , respectively.

Algorithm 2 MST-based Path Planning

1: Find the MST of G = (N ,E,W ) using the distributed MST algorithm [34]

2: for i ∈N do

3: if (the number of connected nodes in the MST) > 2 then

4: Set node i as root;

5: Set weight of each branch connected to the root according to number of con-

nected nodes;

6: Prioritize each branch in descending order by total weight;

7: end if

8: end for

9: return Priority of branches with respect to root.

39

(a)

set the weights to each branch send a signal to count

the number of sensorsin that branchroot

root

1

2

31

1

1

2

(b)

(c)

Figure 3.2: The process of MST-based path planning: (a) find the MST of WSN graph

G, (b) set root nodes and measure the number of nodes connected through each branch

to root, (c) determine the optimal trajectory by connecting partial solutions

40

Figure 3.2 illustrates the process to determine the optimal mobile anchor trajectory

using distributed MST. Figure 3.2a shows the initial stage, where each sensor node

detects its neighboring nodes and measures the distance to them. Then, the distributed

MST algorithm finds the MST of graph G. Next, Fig. 3.2b shows the designation of

root nodes (highlighted by orange circles), which are those containing more than two

nodes connected. When the mobile anchor reaches a root node, it should select the

optimal among multiple branches. For the root node, the weight of each branch is

determined by the number of connected nodes. To this end, the node at the end of each

branch sends a signal to the other nodes in the branch, and this information is retrieved

from the root node. No signal is generated on the branch connected together on another

root, and this branch is in the last sequence. If there are branches with the same weight

with respect to a root node, they are weighted according to the distance among nodes

constituting the branches. This process aims to prioritize the branches and prevent

inefficient looping trajectories. Therefore, the branch with fewer nodes usually has

the highest priority to be part of the optimal trajectory. Finally, Fig. 3.2c shows the

mobile anchor starting the position estimation at the node highlighted in green. Along

the trajectory, each node points to the next one in the MST for the mobile anchor to

navigate when localization at the current node ends.

41

3.5 Performance Evaluation

3.5.1 Simulation Settings

To evaluate the performance of the proposed path planning method, I performed sim-

ulations in scenarios considering a single and multiple sensor nodes. As mentioned

above, I assumed the measured distances to be based on the log-distance path loss

model and used the air-to-ground propagation channel model [48]. I setP0 = −35 dBm,

d0 = 1 m, and the sensitivity level for receiving and performing correct demodulation

is −68 dBm (a communication range is approximately 16 m). The position estima-

tion accuracy was evaluated under the conditions of log-normal shadow fading envi-

ronment with path-loss exponent np set to 2.66 and variance σ(dB) set to 4 dB. In

the first-order finite difference approximation for finding the optimal values, Steering

angle increments of 5◦ and velocity gradually increasing by 0.1 of the maximum ve-

locity are set. I ran 1000 Monte Carlo simulation trials for each scenario to determine

the accuracy in terms of the root-mean-square error (RMSE) as follows:

RMSE =

√∑Ni=1 (xi − xi)

T (xi − xi)

N(3.19)

where xi and xi are the true and estimated position of sensor node i, respectively.

Likewise, I compared the results of the proposed method with those of the static path

planning SCAN algorithm [18] and the MST-based trajectory, which is commonly

used as benchmark. The MST-based trajectory is an efficient path to visit each node

without considering the estimation accuracy of the node.

42

3.5.2 Path Planning for Single Sensor Node

First, the path planning with the different objective functions is evaluated in a single-

node scenario. The mobile anchor was initially located at b[0] = [0, 0]T m and moved

with steering angle between −25◦ ∼ 25◦ toward the sensor node located at [10, 10]T

m. In addition, I set the maximum velocity of the mobile anchor to VT = 4 m/s and

the time interval to ∆T = 2 s. Instead of the threshold (∆th) termination for the posi-

tion estimation, each simulation was restricted to 20 time steps.

Figure 3.3a shows the RMSE of position estimation using the three optimality

criteria and the straight path. Although the RMSE of all the evaluated approaches are

similar until time step 5, the error using the straight path increases afterwards. This

is because the straight path gradually increase the distance between the mobile an-

chor and sensor node, which adversely affects the estimation accuracy. In addition, as

the mobile anchor moves in a straight path regardless of the estimated node position,

it is eventually out of the communication range of the node. In contrast, the perfor-

mance using the three optimality criteria is very similar, with D-optimality showing

the highest accuracy and fastest convergence to the final estimated position. This can

be because the D-optimality criterion aims to minimize the variance of the position

estimates, thus reducing the volume of the uncertainty ellipsoid. The nature of this op-

timality criterion makes the mobile anchor to move around the node, as shown in Fig.

3.3b.

43

2 4 6 8 10 12 14 16 18 20

Time step

0

2

4

6

8

10

12

14

16

RM

SE

(m

)D-optimalityE-optimalityA-optimalitySCAN

(a)

0 2 4 6 8 10 12 14

x-axis position (m)

0

2

4

6

8

10

12

y-ax

is p

ositi

on (

m)

Ture position of the sensorEstimated position of the sensorMobile anchor trajectory

(b)

Figure 3.3: Trajectory optimization in single-node scenario: (a) position estimation

over time steps, (b) trajectory determined using D-optimality

44

3.5.3 Path Planning for Multiple Sensor Nodes

To verify the proposed approach in a more realistic WSN scenario, I evaluated the path

planning and position estimation performance from multiple sensor nodes. N sensor

nodes are randomly distributed in a square region of 70 m × 70 m. The approaches

were evaluated by varying the number of nodes according to N = 20, 25, 30, 35, and

40, with node density λ varying from 0.004/m2 to 0.008/m2. The initial position,

direction, and steering angle limits of the mobile anchor as well as the time interval

were the same as in the previous scenario. The maximum speed of the mobile anchor

was VT = 6 m/s, and the threshold ∆th indicating the estimation convergence was

set to 1 m.

Figure 3.4 shows the RMSE of position estimation of the sensor nodes. The accu-

racy of the proposed method using the different optimality criteria is independent of the

number of nodes, because the mobile anchor considers each sensor individually and

retrieves accurate positions. In fact, the RMSE results using the MST trajectory and

the SCAN algorithm are higher than that using any optimality criterion in the proposed

method and increase with the number of nodes. This is caused by these algorithms in-

dependency on both the distribution and number of sensor nodes. Consequently, each

node may receive insufficient or inaccurate beacon signals. In the MST-based trajec-

tory, a node with a set of location information that are almost collinear may lead to the

possibility of the flip ambiguity problem, thereby causing a large localization error.

Like the results using a single sensor node, the D-optimality retrieves the best perfor-

mance, which is approximately 52 ∼ 61% higher than that using the SCAN algorithm.

Figure 3.5 shows the average distance traveled by the mobile anchor according to

the number of sensor nodes. The trajectory using the SCAN algorithm is the same

regardless of the number of nodes. For the MST-based trajectory and the proposed

method, increasing the number of nodes lengthens the distance traveled. This is ex-

pected as the mobile anchor moves considers each node individually, and hence the

path length is proportional to the number of nodes. Naturally, the MST-based trajec-

45

4 5 6 7 8

Node density ( ) 10-3

5

10

15

20

25

RM

SE

(m

)

D-optimality

E-optimality

A-optimality

MST

SCAN

Figure 3.4: RMSE of path planning approaches according to the number of sensor

nodes

46

4 5 6 7 8

Node density ( ) 10-3

100

200

300

400

500

600

700

800

Path

length

(m

)

D-optimality

E-optimality

A-optimality

MST

SCAN

Figure 3.5: Path length of the approaches according to the number of sensor nodes

47

tory shows the shortest path length because it represents the path with the minimum

sum of the link distances between the nodes regardless of the estimated state of the

node. The D-optimality criterion achieved the shortest path length among the optimal-

ity criteria, and it shortens the distance traveled by finding the position of each sensor

more quickly.

Table 3.1: Comparison of communication cost for position estimation of each algo-

rithm

min ] of signals max ] of signals average ] of signals

D-optimality 3 8 5

E-optimality 3 10 5

A-optimality 3 10 5

MST 6 40 18

Static algorithm 6 57 21

Table 3.1 shows the number of the signals used for localization of each algorithm.

In three optimality criterion cases, a similar number of information exchanges were

performed to complete the position estimation. On the other hand, in other algorithms,

as the mobile anchor moves along a pre-defined path without considering the estima-

tion accuracy changes of the node. For this reason, the estimated position does not

converge well to any one point, and the position update state satisfies the threshold

∆th after a sufficient amount of information has been exchanged. In addtion, the mo-

bile anchor requests only the location information of the node to calculate the FIM,

and each node needs only the location of the mobile anchor to find the its own loca-

tion, so the signaling overheads of the proposed algorithm is small.

Finally, Fig. 3.6 illustrates the mobile anchor trajectory and both the real and es-

timated sensor node position. The environment in the figure contains 10 nodes and

48

the trajectory was generated using the D-optimality criterion. The numbers next to the

blue triangles and arrows indicate the trajectory sequence of the mobile anchor.

-5 0 5 10 15 20 25 30

x-axis(m)

0

5

10

15

20

25

30

y-ax

is(m

)

Path planning of the Mobile anchor

12

34

5

6

7

89

10

11

12

13

14

1516

17

1819

2021

22 23

24

2526

2728

Mobile anchorReal positionEstimated position

Figure 3.6: Trajectory of mobile anchor and position estimation of sensor nodes

49

3.5.4 Experiment Settings

Figure 3.7: The sensor placement

To verify the proposed approach in a realistic WSN scenario, I confirmed the ap-

plicability of the proposed algorithm to reality through experiments. The verification

was conducted using Zigbee-based sensor devices considered in previous chapter. The

10 sensors are located within the 10 m × 10 m square area, as shown in Fig. 3.7. I

measured RSS values according to the distance and calculated a measured RSS value

to a corresponding distance by applying log-normal path loss model that showing

np = 2.89, and σ(dB) = 4.36. The communication range is set to 5 m. The mo-

bile anchor was initially located at b[0] = [0, 0]T m and moved with steering angle

between −45◦ ∼ 45◦. In addition, I set the maximum velocity of the mobile anchor to

VT = 0.3 m/s and the time interval to ∆T = 4 s.

50

3.5.5 Experiment Results

I compared the performance of the proposed algorithm with the MST-based trajectory.

Table 3.2 shows the RMSE of position estimation of the sensor nodes and the path

length of the mobile anchor for each algorithm. Like previous simulation results, the

MST-based trajectory constructs shorter paths than the proposed algorithm. However,

it is confirmed that the proposed algorithm reduces the localization error by more than

half.

Figure 3.8 shows the trajectory of the mobile anchor and the localization results

for each algorithm. In the MST-based trajectory, the preceding four nodes estimate the

position well, but the estimation error of the other nodes is very large. This is because

the location and the estimated state of each node are not considered at all. Whereas,

because the mobile anchor with the proposed algorithm constructs a path considering

each node individually, the estimation accuracy of each node is relatively constant.

Although the experimental scale is small and simple, the applicability of the pro-

posed algorithm in realistic environment can be confirmed through this experiment.

Table 3.2: Comparison of the performance between the MST-based trajectory and the

proposed algorithm

RMSE Travel distance

MST trajectory 2.63 m 28.2 m

Proposed algorithm 1.12 m 40.07 m

51

02

46

810

x-a

xis

(m)

0123456789

10

y-axis(m)P

ath

pla

nn

ing

of

the

Mo

bile

an

ch

or

1

2

3

4

5 6

7

8

9

10

1

2

3

4567

8

91

0

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

(a)

02

46

810

x-a

xis

(m)

0123456789

10

y-axis(m)

Pa

th p

lan

nin

g o

f th

e M

ob

ile a

nch

or

1

2

3

4

5 6

7

8

9

10

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

19

202122

23 2

4

25

26

27 28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45 4

6

47

48

49

50

51

(b)

Figu

re3.

8:T

hepe

rfor

man

cefo

reac

hal

gori

thm

:(a)

The

MST

-bas

edtr

ajec

tory

,(b)

The

traj

ecto

ryof

the

prop

osed

algo

rith

m

52

Chapter 4

CONCLUSION

In this dissertation, several approaches were proposed to enhance the performance of

localization in wireless sensor networks. These algorithms were introduced for en-

abling each node to estimate its own position in an anchor-deficient environment.

First, I proposed a recursive self-organizing localization scheme using the mass-

spring relaxation method. The location estimation procedure is optimized through an

iterative process using geometric relationships with the neighboring nodes. A correc-

tion factor estimated according to the distance is proposed to reduce the influence of

the distance measurement error. Additionally, the adaptive step size control is utilized

to prevent divergence and facilitate fast convergence towards the correctly estimated

position. The proposed scheme is robust with respect to the anchor-deficient WSN en-

vironment, yielding more accurate results compared to the conventional MSR scheme.

The computation time was successfully reduced, simultaneously improving the pro-

cess of convergence speed and system stabilization by using the adaptive step size

based on the gradient method.

Next, I introduce a path planning method using a single mobile anchor, enhanc-

ing the performance of the mobile anchor assisted localization. The mobile anchor

determines the waypoints to transmit beacon signals enabling accurate localization of

the scattered nodes. Based on the trajectory of the mobile anchor, the mobile anchor

53

can estimate the fluctuation in the variance of the position estimation according to

its movements. Then, the mobile anchor moves to the optimal position such that it

reduces the uncertainty of the estimated node position and broadcasts the beacon sig-

nal with new location information. Using criteria such as D-, A-, and E-optimality, I

formulate the optimization problem of minimizing the uncertainty of the estimation.

The gradient descent is used to solve the optimization problem, and the mobile anchor

determines the trajectory that maximizes the localization accuracy. Moreover, when

multiple sensors are distributed over a region, the traveling distance is minimized us-

ing the distributed MST for motion efficiency, thus reducing energy consumption of

the mobile anchor. It is confirmed that the proposed dynamic path planning method

has a suitable performance regardless of the distribution and number of sensor nodes,

yielding to more accurate results than the conventional static path planning. Specif-

ically, using the D-optimality criterion in the proposed method retrieves the highest

accuracy among the evaluated optimality criteria. As the number of sensors increases,

the travel distance also increase, but the path distance remains below that using the

static path planning SCAN algorithm up to a given number of nodes, but the overall

accuracy of the proposed method is notably higher.

54

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초록

무선 센서 네트워크의 많은 어플리케이션들은 센서 노드들의 정확한 위치 정

보를 필요로 한다. 센서 노드는 위성 위치 확인 시스템과 같은 장치에 의해 자신의

위치를식별할수있지만,전장이나실내환경과같은특정상황에서는많은방해요

소들에의하여위치정보를받는것이매우제한적이다.다른위치추정방법으로는

자신의 위치를 알고있는 인근 앵커 노드로부터의 신호를 이용하여 센서 노드의 위

치를파악할수있다.하지만,앵커노드가부족한환경에서는알고리즘을이용하기

어려운문제가발생하게된다.따라서,본논문은앵커가부족한환경에서의정확한

위치추정을위한여러알고리즘들을제안한다.

첫번째로, 이웃 노드와의 연결성 정보만을 기반으로 하는 재귀적 자가-조직화

위치추정기법을제안한다.이기법은각노드가이웃노드들과의기하학적관계가

평형 상태에 도달할 때까지 반복적으로 위치를 추정해 나가는 질량-스프링 모델을

이용한다. 기존 질량- 스프링 모델의 문제점 분석을 통해 노드간 측정되는 거리의

정확도를 정량화하여 이에 대한 정확도를 보정해 주는 거리 보정 계수를 적용함으

로써,거리오차에대한영향을감소시켰다.또한,여러오차요소로인해발생할수

있는시스템의발진을막기위하여,경사하강법을기반으로한적응형스텝크기제

어 알고리즘을 제안하여 시스템의 안정도를 향상시켰다. 뿐만아니라, 추정 위치에

대한 수렴속도 및 정확도 측면에서도 기존 알고리즘에 비해 향상된 성능을 확인하

였다.

다음으로는 이와 같이 앵커 노드의 정보 이용이 어려운 상황을 보완하기 위한

이동 가능한 모바일 앵커의 활용에 대한 연구를 진행하였다. 모바일 앵커는 해당

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지역을 이동하며 자신의 위치 정보가 담긴 비컨 신호를 주기적으로 전송한다. 각

노드들은이위치정보들을통하여자신의위치를계산하므로,위치추정정확도는

모바일앵커의이동경로에많은영향을받게된다.따라서,앵커노드가없는무선

센서네트워크에서노드들의정확한위치추정을위한모바일앵커의경로계획기

법을연구하였다.이기법은하나의모바일앵커만을고려하였으며,네트워크환경

에대한사전정보를요구하지않는장점이있다.이동경로에따른노드의추정위치

정확도의변화를예측하고,이를최소화시킬수있는정보를제공하기위한위치를

예측하여 모바일 앵커의 이동 경로를 결정하는 기법을 제안한다. 기존 CRLB 계산

을통하여노드의추정위치정확도를예측할수있으나, CRLB계산이지닌한계를

극복하기위해기존에연구된피셔정보행렬을기반으로하는목적함수들을이용

하였다. 또한, 무선 센서 네트워크에서의 최소 스패닝 트리 기법을 활용하여, 모든

노드가정확한위치추정과정을실행할수있으며,동시에에너지소모측면에서도

효율적인경로를구성할수있도록하였다.시뮬레이션결과를통하여제안한기법

이정적경로계획알고리즘과비교하여추정위치정확도가더욱향상되었을뿐만

아니라,네트워크상의모든노드가하나도빠짐없이자신의위치를찾을수있도록

함을확인하였다.

주요어: 위치추정기술, 무선센서네트워크, 질량-스프링 모델, 무인항공기, 칼만필

터,경로계획,크라메르-라오하한,최소신장트리

학번: 2014-30308

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