autonomous, wireless sensor network-assisted target search and mapping

8/10/2019 Autonomous, wireless sensor network-assisted target search and mapping

1/135

Autonomous, Wireless Sensor Network-Assisted

Target Search and Mapping

by

Steffen Beyme

Dipl.-Ing. Electrical Engineering, Humboldt-Universitt zu Berlin, 1991

A THESIS SUBMITTED IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF

Doctor of Philosophy

in

THE FACULTY OF GRADUATE AND POSTDOCTORAL

STUDIES

(Electrical and Computer Engineering)

The University Of British Columbia

(Vancouver)

October 2014

c Steffen Beyme, 2014


2/135

Abstract

The requirements of wireless sensor networks for localization applications are

largely dictated by the need to estimate node positions and to establish routes to

dedicated gateways for user communication and control. These requirements add

significantly to the cost and complexity of such networks.

In some applications, such as autonomous exploration or search and rescue,

which may benefit greatly from the capabilities of wireless sensor networks, it

is necessary to guide an autonomous sensor and actuator platform to a target, for

example to acquire a large data payload from a sensor node, or to retrieve the target

outright.

We consider the scenario of a mobile platform capable of directly interrogating

individual, nearby sensor nodes. Assuming that a broadcast message originates

from a source node and propagates through the network by flooding, we studyapplications of autonomous target search and mapping, using observations of the

message hop count alone. Complex computational and communication tasks are

offloaded from the sensor nodes, leading to significant simplifications of the node

hardware and software.

This introduces the need to model the hop count observations made by the mo-

bile platform to infer node locations. Using results from first-passage percolation

theory and a maximum entropy argument, we formulate a stochastic jump process

which approximates the message hop count at distance rfrom the source. We show

that the marginal distribution of this process has a simple analytic form whose pa-rameters can be learned by maximum likelihood estimation.

Target search involving an autonomous mobile platform is modeled as a stochas-

tic planning problem, solved approximately through policy rollout. The cost-to-go

ii


3/135

at the rollout horizon is approximated by an open-loop search plan in which path

constraints and assumptions about future information gains are relaxed. It is shownthat the performance is improved over typical information-driven approaches.

Finally, the hop count observation model is applied to an autonomous mapping

problem. The platform is guided under a myopic utility function which quantifies

the expected information gain of the inferred map. Utility function parameters are

adapted heuristically such that map inference improves, without the cost penalty of

true non-myopic planning.

iii


4/135

Preface

Chapters 2 to 4 are based on manuscripts that to date have either been published, or

accepted or submitted for publication, in peer-reviewed journals and conferences.

All manuscripts were co-authored by the candidate as the first author, with revi-

sions and comments by Dr. Cyril Leung. In all these works, the candidate had the

primary responsibility for conducting the research, the design and performance of

simulations, results analysis and preparation of the manuscripts, under the supervi-

sion of Dr. Cyril Leung. The following list summarizes the publications resulting

from the candidates PhD work:

S. Beyme and C. Leung, Modeling the hop count distribution in wirelesssensor networks,Proc. of the 26th IEEE Canadian Conference on Electri-

cal and Computer Engineering (CCECE), pages 16, May 2013.

S. Beyme and C. Leung, A stochastic process model of the hop count dis-tribution in wireless sensor networks, Elsevier Ad Hoc Networks, vol. 17,

pages 6070, June 2014.

S. Beyme and C. Leung, Rollout algorithm for target search in a wirelesssensor network,Proc. of the IEEE 80th Vehicular Technology Conference ,

Sept. 2014. Accepted.

S. Beyme and C. Leung, Wireless sensor network-assisted, autonomousmapping with information-theoretic utility, 6th IEEE International Sym-posium on Wireless Vehicular Communications, Sept. 2014, Accepted.

iv


5/135

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

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

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis Organization and Contributions . . . . . . . . . . . . . . . 4

2 Stochastic Process Model of the Hop Count in a WSN . . . . . . . . 6

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Motivation and Related Work . . . . . . . . . . . . . . . 7

2.1.2 Chapter Contribution . . . . . . . . . . . . . . . . . . . . 8

2.1.3 Chapter Organization . . . . . . . . . . . . . . . . . . . . 92.2 Wireless Sensor Network Model . . . . . . . . . . . . . . . . . . 9

2.2.1 Stochastic Geometry Background . . . . . . . . . . . . . 9

2.2.2 First-Passage Percolation . . . . . . . . . . . . . . . . . . 11

v


6/135

2.3 Stochastic Process Model for the Hop Count . . . . . . . . . . . . 14

2.3.1 Jump-type Lvy Processes . . . . . . . . . . . . . . . . . 142.3.2 Maximum Entropy Model . . . . . . . . . . . . . . . . . 17

2.3.3 Maximum Likelihood Fit of the Hop Count Process . . . . 19

2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . 21

2.4.2 Hop Count Distribution . . . . . . . . . . . . . . . . . . . 22

2.4.3 Localization of Source Node . . . . . . . . . . . . . . . . 23

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Rollout Algorithms for WSN-assisted Target Search . . . . . . . . . 33

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.1 Background and Motivation . . . . . . . . . . . . . . . . 34


3.1.3 Chapter Organization . . . . . . . . . . . . . . . . . . . . 37

3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 Wireless Sensor Network Model . . . . . . . . . . . . . . 37

3.2.2 Autonomous Mobile Searcher . . . . . . . . . . . . . . . 39

3.2.3 Formulation of Target Search Problem . . . . . . . . . . . 40

3.3 Approximate Online Solution of POMDP by Rollout . . . . . . . 44

3.3.1 Rollout Algorithm . . . . . . . . . . . . . . . . . . . . . 443.3.2 Parallel Rollout . . . . . . . . . . . . . . . . . . . . . . . 44

3.4 Heuristics for the Expected Search Time . . . . . . . . . . . . . . 45

3.4.1 Constrained Search Path . . . . . . . . . . . . . . . . . . 46

3.4.2 Relaxation of Search Path Constraint . . . . . . . . . . . 46

3.5 Information-Driven Target Search . . . . . . . . . . . . . . . . . 50

3.5.1 Mutual Information Utility . . . . . . . . . . . . . . . . . 50

3.5.2 Infotaxis and Mutual Information . . . . . . . . . . . . . 51

3.6 A Lower Bound on Search Time for Multiple Searchers . . . . . . 52

3.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 533.7.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . 53

3.7.2 Idealized Observations . . . . . . . . . . . . . . . . . . . 55

3.7.3 Empirical Observations . . . . . . . . . . . . . . . . . . . 56

vi


7/135

3.8 Statistical Dependence of Observations . . . . . . . . . . . . . . 57

3.8.1 Mitigation of Observation Dependence . . . . . . . . . . 593.8.2 Explicit Model of Observation Dependence . . . . . . . . 61

3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4 WSN-assisted Autonomous Mapping . . . . . . . . . . . . . . . . . . 67

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1.1 Background and Motivation . . . . . . . . . . . . . . . . 68


4.1.3 Chapter Organization . . . . . . . . . . . . . . . . . . . . 70

4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2.1 Wireless Sensor Network Model . . . . . . . . . . . . . . 70

4.2.2 Autonomous Mapper . . . . . . . . . . . . . . . . . . . . 72

4.3 Mapping Path Planning . . . . . . . . . . . . . . . . . . . . . . . 74

4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.4.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . 76

4.4.2 Simulation of Map Inference . . . . . . . . . . . . . . . . 77

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . 81

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2.1 Parametric Models of the Hop Count Distribution . . . . . 84

5.2.2 Statistical Dependence of Hop Count Observations . . . . 85

5.2.3 Simulation-based Observation Models . . . . . . . . . . . 85

5.2.4 Multi-Modal Observations . . . . . . . . . . . . . . . . . 85

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

A Necessary Condition for the Hop Count Process . . . . . . . . . . . 97

B Proof of Strong Mixing Property . . . . . . . . . . . . . . . . . . . . 99

vii


8/135

C Constrained-Path Search as Integer Program . . . . . . . . . . . . . 102

C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102C.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

C.2.1 Minimizing the Expected Search Time . . . . . . . . . . . 104

C.2.2 Maximizing the Detection Probability . . . . . . . . . . . 106

C.3 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . 107

C.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

D Infotaxis and Mutual Information . . . . . . . . . . . . . . . . . . . 114

D.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

D.2 Proof of Equivalence . . . . . . . . . . . . . . . . . . . . . . . . 114

D.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

E Pseudocode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

viii


9/135

List of Tables

Table 4.1 Map average entropy and MSE,wi adapted according to (4.16) 78

Table 4.2 Map average entropy and MSE,wi adapted according to (4.17) 78

ix


10/135

List of Figures

Figure 2.1 Realization of a random geometric graph . . . . . . . . . . . 10

Figure 2.2 CDFs of the translated Poisson model and the empirical hopcount in a 2D network, for mean node degree 8 . . . . . . . . 25

Figure 2.3 CDFs of the translated Poisson model and the empirical hop

count in a 2D network, for mean node degree 16 . . . . . . . 26



Figure 2.5 Kullback-Leibler divergence (KLD) between empirical distri-

bution and translated Poisson distribution . . . . . . . . . . . 28


count in a 1D network, for mean node degree 8 . . . . . . . . 29





Figure 2.9 CDFs of the normalized localization error, given 8 hop count

observations, for mean node degrees 8 and 40. . . . . . . . . . 32

Figure 3.1 Dynamic Bayes Network representing a POMDP . . . . . . . 42

Figure 3.2 CDF of the search time for rollout under random action selec-

tion, compared to the search time for myopic mutual informa-tion utility. Rollout horizonH= 4, hop counts generated by

model M3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

x


11/135

Figure 3.3 CDF of the search time for rollout under random action se-

lection, compared to the search time for non-myopic mutualinformation utility. Rollout horizonH= 4, hop counts gener-

ated by model M3 . . . . . . . . . . . . . . . . . . . . . . . 58

Figure 3.4 CDFs of the search time for rollout under 3 different base poli-

cies: random, constant and greedy action selection. Rollout

horizonH=4, hop counts generated by model M3 . . . . . . 59

Figure 3.5 CDF of the search time for rollout under random action, com-

pared to 2 parallel rollout approaches: random and constant

action selection, random and greedy action selection. Rollout

horizonH=

4, hop counts generated by model M3 . . . . . . 60

Figure 3.6 CDF of the search time for rollout under random action, com-

pared to 2 parallel rollout approaches: random and constant

action selection, random and greedy action selection. Rollous

horizonH=2, hop counts generated by model M3 . . . . . . 61

Figure 3.7 CDFs of the search time for rollout under random action, with

horizonH= 4, for the 3 hop count generation models M1, M2

and M3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Figure 3.8 CDFs of the search time for rollout under random action, with

horizonH=4 and for the 3 hop count generation models M1,

M2 and M3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Figure 3.9 Deviation of local average hop count from the model mean . . 65

Figure 3.10 Correlation between hop count observations, as a function of

the distance from the source node and the inter-observation

distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Figure 4.1 True map, inferred map and autonomous mapper path . . . . . 79

Figure 4.2 Average entropy per map element . . . . . . . . . . . . . . . 80

Figure 4.3 Mean square error between true and inferred map . . . . . . . 80

Figure C.1 Search path of minimum expected search time . . . . . . . . . 109

Figure C.2 Search path of maximum probability of detection . . . . . . . 110

xi


12/135

Figure C.3 Cumulative probability of detection for the two search policies

of minimum expected search time and maximum probabilityof detection, and for the optimal search without path constraint 111

Figure C.4 Search path for unpenalized objective function . . . . . . . . 112

Figure C.5 Cumulative probability of detection for unpenalized objective

function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

xii


13/135

List of Acronyms

The following acronyms are frequently used in this thesis:

KLD Kullback-Leibler divergence

MLE Maximum likelihood estimation

MSE Mean square error

POMDP Partially observable Markov decision process

WSN Wireless sensor network

xiii


14/135

Acknowledgments

The experience of graduate school at UBC has been rewarding and fulfilling, and I

owe a debt of gratitude to many people here, whose support made this possible.

I reserve special thanks for my thesis supervisor and mentor, Dr. Cyril Leung,

whom I have had the privilege to work with. His invaluable insight and continued

encouragement have been an inspiration throughout this journey of thesis research.

He let me explore with great freedom and offered the guidance and the support

without which this work would not have progressed to this point.

I would like to thank my supervisory committee members, Dr. Vikram Krish-

namurthy and Dr. Z. Jane Wang, for their advice and much appreciated feedback.

I would also like to thank the faculty, staff and fellow students in the Depart-

ment of Electrical and Computer Engineering, who all contributed to create a stim-

ulating research environment.This work was supported in part by the Natural Sciences and Engineering Re-

search Council (NSERC) of Canada under Grants OGP0001731 and 1731-2013

and by the UBC PMC-Sierra Professorship in Networking and Communications.

Finally, I owe enduring gratitude to my parents, for the love and encouragement

they have provided. My greatest thanks go to my wife, Beatriz, and to our children,

Carl and Alex, whose love and patience have let me see this thesis through.

xiv


15/135

To my Family

xv


16/135

Chapter 1

Introduction

1.1 Background and Motivation

Both target localization and mapping are important applications of wireless sensor

networks (WSNs) [80] as well as autonomous robotics [110]. Much research con-

tinues to be devoted to advance the state-of-the-art in these fields and to expand and

enhance the operational capabilities of these complementary technologies. This

thesis considers the joint application of wireless sensor networks and autonomous

robotics.

Some applications, such as autonomous exploration or search and rescue, would

benefit from both the pervasive sensor coverage which only WSNs can provide, and

the mobility of a single or multiple autonomous sensor and actuator platforms. In

a joint application, a typical mission would involve the ad hoc deployment of a

WSN (due to the circumstances of the mission, often in a random manner) and an

autonomous platform able to locate one or several targets, or inferring a map,

by interacting with the WSN. The objective can range from collecting a large data

payload from a sensor node that has reported an event of interest (but energy con-

straints prevent the actual, recorded observation data from being forwarded through

the wireless sensor network), moving more sophisticated sensing or actuating ca-pabilities closer to the site of interest (as in planetary exploration), to retrieval of

the target outright. A survey of related applications at the intersection of WSNs

and autonomous robotics can be found in [91]. A method for localization and

1


17/135

autonomous navigation using WSNs was recently described in [22].

Many WSNs used for localization are organized around the concept of location-aware sensor nodes which forward individual target location estimates to one or

several sink nodes, using multi-hop communication. There can be varying degrees

of local cooperation between nodes to enhance the sensing and localization perfor-

mance[80]. The purpose of the sinks is to aggregate, or fuse, information from

multiple sensor nodes to improve the location estimates and ultimately, to serve

as gateways through which a user communicates with the WSN. The sinks can

be dedicated nodes, or may be dynamically selected from the population of sen-

sor nodes (for example based on the available amount of energy) to act as fusion

centers. Node location awareness can be addressed by the use of special-purpose

localization hardware (e.g. GPS or time/angle of arrival) and associated protocols.

The need to establish node positions and to maintain multi-hop routes from the

sensor nodes to the statically or dynamically assigned fusion centers thus drives

many of the hardware and communication requirements of WSNs and contributes

significantly to their cost and complexity.

As an approach to reduce the cost and complexity of WSNs used by the ap-

plications considered in this thesis, we assume that the sensor nodes are location-

agnosticand that an autonomous platform, assumed to be location-aware, acts as a

mobile sink by directly interrogating nearby sensor nodes. Instead of discovering

and maintaining routes to the sink, a simple message broadcast protocol is respon-

sible for information dissemination in the WSN. As a consequence, the need for

special-purpose hardware to support node localization, as well as complex routing

strategies and dedicated sink nodes for performing in-situ sensor fusion, is elimi-

nated. By offloading expensive computational and communication tasks from the

sensor nodes to the autonomous platform, a significant simplification of the sen-

sor node hardware and software requirements can be achieved. In this thesis, only

observations of the hop countof a message originating from a source node are

assumed to be available to the autonomous platform, to infer the location of the

source node.

Although we do not pursue the concept further in this thesis, an autonomous

sensor platform allows for the seamless integration of WSN hop count observations

with the platforms on-board sensing capabilities, which are often sophisticated and

2


18/135

complementary to the WSN and may include laser or ultrasound ranging, imaging

etc. Similarly, it is conceivable to use hop count based localization in combinationwith other low-cost methods of WSN node localization, to obtain overall improved

node location estimates. This includes for example methods based on received

signal strength (RSS), which add little or no extra cost to the sensor nodes [12].

The joint application of WSNs and autonomous platforms for target search and

mapping raises the need for anobservation modelwhich relates the statistics of the

hop count of a broadcast message to the distance from the source node in an appro-

priately defined WSN. This problem is central to localization methods referred to as

range-free [102]. However, for many reasonable models of WSNs, the character-

ization of the hop count statistics, given the source-to-sink distance, remains a chal-

lenging, open problem for which only approximations are known, which in many

cases can only be evaluated at significant computational cost [24, 72, 82, 107].

Under certain assumptions, which include linear observations and a Gaussian error

model, the Kalman filter is optimal for localization (these assumptions are some-

what relaxed in the extended Kalman filter) [57]. However, the hop count obser-

vations in a WSN are not well characterized by this model, generally requiring

nonparametric Bayesian methods to compute the a posteriori probability density

function of the target. Typically performed by grid or particle filters, these methods

require a large number of numerical evaluations of the observation model [110]. It

is therefore important, to develop models that have low computational complexity

and characterize the hop count reasonably well. This is the subject of Chapter2of

this thesis.

The search for a (generally moving) target by an autonomous platform based

on observations of the hop count of a broadcast message can be described as a

stochastic planning problem. The general framework for this type of problem is

the partially observable Markov decision process (POMDP) [69]. Unfortunately,

for most problems of practical relevance, solving the POMDP exactly is compu-

tationally intractable. This has given rise to the need for approximate, suboptimal

solutions which can achieve acceptable performance, many of which are based on

online Monte Carlo simulation[85,106]. However, even suboptimal planning al-

gorithms can still present formidable computational challenges. In this thesis, we

propose the use of an efficient heuristic to limit the computational cost of a policy

3


19/135

rollout algorithm [8, 17], which is the subject of Chapter 3. The computational

requirements of Monte Carlo methods such as policy rollout further magnify theneed for observation models of low complexity.

Closely related to localization is autonomous mapping [25, 108], which we

consider in Chapter4. Autonomous mapping platforms are typically equipped with

sensors such as range and direction finders or cameras, and are therefore geared

towards the mapping of physical objects (or obstacles) in the environment. In con-

trast, the pervasive sensor coverage of WSNs enables the dense mapping of quan-

tities such as the concentration of chemicals, vibrations and many others, whose

measurement requires close physical contact with the sensor. With WSN-assisted

mapping, the data association problems[110] inherent in many mapping applica-

tions can be sidestepped quite easily. Another difficult problem in autonomous

mapping is the planning of an optimal path along which observations are made,

such that the mean error between the true and the inferred map is minimized, usu-

ally over a finite time horizon. Due to the curse of dimensionality, this problem is

generally intractable and good approximate techniques to find a near-optimal path

are required for any practical application.

1.2 Thesis Organization and Contributions

Thesis Organization

The subject of Chapter2 is the derivation of an observation model, which relates

the hop count distribution of a broadcast message in a suitably defined WSN to the

source-to-sink distance. We evaluate the model by comparison with the empirical

hop count in a simulated WSN. This model is the basis for Chapter3, in which we

study target search by an autonomous platform as a stochastic planning problem,

and use Monte Carlo techniques to solve it approximately. In Chapter4,we study

the problem of path planning for an autonomous platform which relies on hop count

observations from the WSN to infer the map of sensor measurements. As in the

preceding chapter, approximate solution methods are developed as the path plan-

ning problem is generally intractable. Finally, Chapter5 summarizes conclusions

from this work and provides a few suggestions for future research.

4


20/135

Contributions

The thesis makes the following contributions:

In Chapter2,we formulate a stochastic jump process whose marginal dis-tribution has a simple analytical form and models the hop count of the first-

passage path from a source to a sink node.

In Chapter3, we model the target search as an infinite-horizon, undiscountedcost, online POMDP[69] and solve it approximately through policy rollout

[8]. The terminal cost at the rollout horizon is described by a heuristic based

on a relaxed, optimal search problem.

We show that a target search problem described in terms of an explicit trade-off between exploitation and exploration (referred to as infotaxis [112]), is

mathematically equivalent to a target search with a myopic mutual informa-

tion utility.

A lower bound on the expected search time for multiple uniformly dis-tributed searchers is given in terms of the searcher density, based on the

contact distance [4] in Poisson point processes.

We propose an integer autoregressive INAR(1) process [1] for translated

Poisson innovations, as a model for the statistical dependence of hop count

observations in a WSN.

In Chapter 4, we propose a myopic, information-theoretic utility functionfor path planning in an autonomous mapping application. Utility function

parameters are heuristically adapted to offset the myopic nature of the utility

and achieve improved performance of the map inference.

5


21/135

Chapter 2

Stochastic Process Model of the

Hop Count in a WSN

2.1 Introduction

In this chapter, we consider target localization in randomly deployed multi-hop

wireless sensor networks, where messages originating from a sensor node are

broadcast by flooding and the node-to-node message delays are characterized by

independent, exponential random variables. Using asymptotic results from first-

passage percolation theory and a maximum entropy argument, we formulate a

stochastic jump process to approximate the hop count of a message at distance

rfrom the source node. The resulting marginal distribution of the process has the

form of a translated Poisson distribution which characterizes observations reason-

ably well and whose parameters can be learned, for example by maximum like-

lihood estimation. This result is important in Bayesian target localization, where

mobile or stationary sinks of known position may use hop count observations, con-

ditioned on the Euclidean distance, to estimate the position of a sensor node or

an event of interest within the network. For a target localization problem with a

fixed number of hop count observations, taken at randomly selected sensor nodes,

simulation results show that the proposed model provides reasonably good location

error performance, especially for densely connected networks.

6


22/135

2.1.1 Motivation and Related Work

Target localization in wireless sensor networks (WSNs) is an active area of re-

search with wide applicability. Due to power and interference constraints, the vast

majority of WSNs convey messages via multiple hops from a source to one or

several sinks, mobile or stationary. Localization techniques which exploit the in-

formation about the Euclidean distance from a sensor node, contained in the hop

count of a message originating from that node, are referred to asrange-free[102].

Range-free localization is applicable to networks of typically low-cost, low-power

wireless sensor nodes without the hardware resources needed to accurately mea-

sure node positions, neighbor distances or angles (for example using GPS, time or

angle of arrival). It is therefore an attractive approach in situations where a com-promise is sought between localization accuracy on one hand, and cost, size and

power efficiency on the other.

Various hop-count based localization techniques for WSNs have been pro-

posed; for a survey see e.g. [102]. Relating hop count information to the Eu-

clidean distance between sensor nodes, exemplified by the probability distribution

of the hop count conditioned on distance, remains a challenging problem. Except

in special cases, such as one-dimensional networks [2], only approximations can

be obtained; such approximations are often in the form of recursions, which tend

to be difficult to evaluate [24,72,107].

Moreover, the hop count depends on the chosen path from the source to the sink

and is therefore a function of the routing method employed by the network. For ex-

ample, an approximate closed-form hop count distribution is proposed in [82] and

evaluated for nearest, furthest and random neighbor routing, in which a forwarding

node selects the next node from a semi-circular neighborhood oriented towards the

sink, under the assumption that this neighborhood is not empty. Some of the ex-

isting localization algorithms, such as the DV-hop algorithm [77] and its variants,

define the hop count between nodes as that of the shortest path[24,72,107]. Other

localization algorithms use the hop count of a path established throughgreedy for-

ward routing[20,59,76,113,117], that is, a path which makes maximum progress

toward the sink with every hop. In most cases, the overhead incurred by establish-

ing and maintaining routes is not negligible. Simpler alternatives may be needed

7


23/135

when sensor nodes impose more severe complexity constraints.

This chapter is motivated by the range-free target localization problem in net-works of position-agnostic wireless sensor nodes, which broadcast messages using

flooding under the assumption that node-to-node message delays can be character-

ized by independent, exponential random variables. This is a reasonable assump-

tion in situations where sensor nodes enter a dormant state while harvesting energy

from the environment and wake up at random times, or when the communication

channel is unreliable and retransmissions are required. Under these conditions, a

first-passage pathemerges as the path of minimum passage time from a source to

a sink. Networks of this type can be described in terms offirst-passage percolation

[23,33].

Localization of the source node may be performed by mobile or stationary

sinks able to fuse hop count observations ZN to infer the location XR2 ofthe source node, where pX(x) denotes the a priori pdf ofX. By Bayes rule, the

a posterioripdf of the source location is pX(x|z) pZ(z|x)pX(x), conditioned onobserving the hop countzat the sink position. Knowledge of the observation model

pZ(z|x), that is, the conditional pdf of the hop count, given the source location hy-pothesisx, is essential for the success of this approach, which may be complicated

further by the presence of model parameters whose values are not known a priori

and must be learned on- or off-line. Bayesian localization involves a large number

of numerical evaluations of the observation model, due to the typically large space

of location hypotheses. This creates a need for observation models with low com-

putational complexity, which may outweigh the need for high accuracy in some

applications.

2.1.2 Chapter Contribution

Themain contributionof this chapter is the formulation of a stochastic jump pro-

cess whose marginal distribution has a simple analytical form, to model the hop

count of the first-passage path from a source to a sink, which is at distance r. In

contrast to earlier works [20,24, 72,76,82,107, 113,117], which generally use

geometric arguments to derive expressions for the hop count distribution, our ap-

proach utilizes the abstract model of a jump process and describes the hop count in

8


24/135

terms of the marginal distribution of this process. Starting with a stochastic process

of stationary increments satisfying a strong mixing condition, we make a simplify-ing independence assumption which allows the hop count to be modeled as a jump

Lvy process with drift [6, 29]. We show that, consistent with our assumptions

about the hop count, the maximum entropy principle [52] leads to the selection

of a translated Poisson distribution as the marginal distribution of the hop count

model process.

2.1.3 Chapter Organization

This chapter is structured as follows: in Section2.2,we review relevant concepts

from stochastic geometry and first-passage percolation and introduce our wirelesssensor network model. Our main result, the stochastic process {Zr} which modelsthe observed hop count distribution at distancerfrom a source node, is derived in

Section2.3. We describe how the parameters of this process can be learned using

maximum likelihood estimation. In Section2.4, simulation results are presented

which show that in sensor networks of the type considered in this chapter, the

marginal distribution of the model process approximates the empirical hop count

reasonably well. Furthermore, we study the localization error due to the approx-

imation by comparison with a fictitious, idealized network in which observations

are generated as independent draws from our model. Proofs of propositions used

to derive our model are given in Appendix A and B. This chapter is based on

manuscripts published in [9,10].

2.2 Wireless Sensor Network Model

2.2.1 Stochastic Geometry Background

The geometry of randomly deployed WSNs is commonly described by Gilberts

disk model [35], a special case of the more general Boolean model [104]. Given

a spatial Poisson point process P= {Xi: i N}of density on R2

, two sensornodes are said to be linked if they are within communication range Rof each other.

Gilberts model induces a random geometric graph G,R= {P,ER} with nodeset Pand edge set ER (Figure2.1). The node density and the communication

9


25/135

range R are related through the mean node degree =R2, so that the graph

can be defined equivalently in terms of a single parameter as G. Without loss ofgenerality, it is convenient to condition the point process on there being a node at

the origin. By Slivnyaks Theorem [4], if we remove the point at the origin from

consideration, we have a Poisson point process.

Figure 2.1: Realization of a random geometric graph G restricted to[0, 1]2

Of key interest in the study ofGis continuum percolation [13,100], that is,

the conditions under which a cluster of infinitely many connected nodes emerges

in such a graph, and the probability that the node under consideration (without loss

of generality, the origin) belongs to this cluster. Percolation is widely known to

exhibit a phase transition from a subcritical regime characterized by the existence

of only clusters with a finite number of nodes, to the formation of a single infinite

cluster with probability 1, as is increased above a critical threshold c. Thepercolation probability is defined as the probability that the origin belongs to the

infinite cluster. It is zero for values ofbelow the threshold, and an increasing

function ofabove. The phase transition is interesting in its own right and much

10


26/135

research is devoted to the analysis of critical phenomena [100]. Just above the

critical threshold, the incipient infinite connected cluster has characteristics whichare generally undesirable for a communication network, such as large minimum

path lengths, which diverge at the critical threshold. A universal property of all

systems exhibiting a phase transition is that characteristic quantities display power-

law divergence near the threshold [100]. Therefore, the mean node degree (or

equivalently, the node density or the transmission range) is generally chosen by the

designer so that the network operates sufficiently deep in the supercritical regime,

i.e. it is strongly supercritical. As the simulations will show, the model hop count

distribution is a good approximation of the empirical hop count for large values of

the mean node degree, but deviates more noticeably when the mean node degree

approaches the critical thresholdc.

2.2.2 First-Passage Percolation

The dissemination of messages in some broadcast WSNs has been described in

terms offirst-passage percolation[23,33, 34,64], where every node forwards the

broadcast message to all of its neighbors, so that over time, a message cluster forms

(illustrating the relationship between first-passage percolation and random growth

models[30,84]). This type of information dissemination is commonly referred to

asflooding. We assume that the node-to-node message delays can be characterized

by independent, exponentially distributed random variables with a common mean.

This is a reasonable assumption for WSNs in which the nodes enter a dormant state

[23,39], while replenishing their energy storage by harvesting from unpredictable

environmental sources, and thus transmit at random times. The independence of

the node-to-node delays in the standard first-passage percolation model implies

that transmissions do not interfere with each other. Various approaches to mitigate

interference exist: for simplicity, we postulate the allocation of orthogonal trans-

mission resources with a sufficiently large reuse factor [75], so that interference

can be neglected. In addition, we assume that multiple simultaneous broadcasts

initiated by the same or by different source nodes do not interact, i.e. hop count ob-

servations associated with different broadcasts are independent. Under real-world

conditions, this would not be very efficient and one might consider e.g. schemes

11


27/135

of message aggregation. Finally, unless some form of authentication is introduced,

it is conceivable to disrupt such a system intentionally. Messages convey the cu-mulative number of hops, until observed by a sink. The initial message is tagged

with a unique ID by the source node, so that duplicates can be recognized eas-

ily and discarded by the receiving nodes to prevent unnecessary (and undesirable)

retransmissions.

First-passage percolation was introduced in[41] as a model of e.g. fluid trans-

port in random media and has been extensively studied since [47]. In first-passage

percolation, the edges E of a graph Gare assigned independent, identically dis-

tributed non-negative random passage times {(e):e E} with the common lawF

. The passage time for a path in G is then defined as the sum of the edge

passage times

T() = e(e). (2.1)

A path (x, y) from node x to y, defined by the ordered set of edges{(x, v1),(v1, v2), . . . , (vn1, y)} has a hop count of||= n. Let (x, y) be the set of allpaths connectingx and y. Then, thefirst-passage timefromx to y is

T(x, y) = inf(x,y)

{T()} . (2.2)

The first-passage time so defined induces a metric on the graph, with the minimiz-

ing paths referred to as first-passage paths, or geodesics. Because the edge passage

times are non-negative, geodesics cannot contain loops, that is, they are necessar-

ily self-avoiding paths connecting the end points. Rather than studying the random

geometric graph G, it is sometimes more convenient to consider theunit distance

graphon the square latticeZ2, for example when appealing to translational invari-

ance. In fact, most classical results in first-passage percolation were established

in the latter setting[41,95, 96,115]. On the square lattice, first-passage time is

often studied between points mex and nex on the x-axis with m


28/135

first-passage time is a subadditive stochastic process, that is

T0,n T0,m+ Tm,n whenever 0


29/135


30/135

to the subadditivity property of the first-passage time (2.3), we have

T0,n1k=0

Tkn,(k+1)n, N. (2.7)

Suppose now that we approximate the hop countN0,n of the first-passage path by

a stochastic process of the form

N0,n=1k=0

Nkn,(k+1)n, N. (2.8)

A necessary, although not sufficient condition for such a process to approximate

the true hop count without systematic bias is therefore, that N0,n is neither sub-

nor superadditive. In AppendixA, we give a proof that the hop count of the first-

passage path is not subadditive. The proof that the hop count is not superadditive

uses an analogous argument and is therefore is omitted. The process ( 2.8) has the

following property, a proof of which is given in AppendixB:

Proposition 1. The hop count increments {N0,n,Nn,2n, . . .} are strongly mixing.

That is to say, the hop count variables N0,n and Nkn,(k+1)n may be considered

independent for k, or asymptotically independent. We use the mixing property

to motivate a simplifying independence assumption for the increments of our hopcount model process. The assumption of i.i.d. edge passage times implies that

the hop count increments are alsostationary. Stochastic processes with stationary

and independent increments form the class of processes known as Lvy processes

[6,50].

Definition 2.1. A stochastic processXris said to be a Lvy process, if it satisfies

the following conditions:

1. stationary increments: the law ofXrXs is the same as the law ofXrs X0for 0

s


31/135

Among these, jump-type Lvy processes [29]restricted to integer jump sizes

can be consideredprototypes for the hop count process. Lvy processes may fur-thermore include a deterministic component, which in our model represents the

minimum hop count required to reach a sink at distance r> 0 due to the finite,

one-hop transmission rangeR. The jump component of a Lvy process with inte-

ger jumps is characterized by a Lvy measure [6] of the form

m(x) =j

j(xnj), njZ\{0}. (2.9)

The Lvy measure describes the distribution of the jumps of the stochastic process,

in terms of the jump sizes nj

and the associated intensities j

. The measure is

bounded, that is

j

j< . (2.10)

Later it will become necessary to specify a Lvy measure for our model. For

now, we are interested in the general form of the marginal distribution of the Lvy

process, which can be determined from its characteristic function, given by the

Lvy-Khintchineformula [6]. For a measure of the form (2.9), the characteristic

function is

Xr(u) =exp

r

iub +j

j(eiunj 1)

, (2.11)

which describes acompound Poissonprocess [50]

Xr=b r+Nr

i=0

Yi, (2.12)

whereb R denotes the rate of the deterministic, linear drift of the process and Nris a Poisson variable with expectationrwhere= j. The random variables Yi

are independent and identically distributed with fY() = 1m(x), also known asthecompounding distribution.

16


32/135

The expectation of the jump Lvy process islinearin the distancer, as

EXr= iXr(0) =r

b +j

jnj

, (2.13)

which implies, that the average number of hops per unit distance is a constant,

EXr

r =b +

j

jnj. (2.14)

This property of the model process is consistent with the asymptotic behavior of

the hop count in first-passage percolation (2.6), i.e. for sufficiently large distances,

we expect the average hop count to increase linearly in r.Equation (2.13) imposes constraints on the support and the mean of the distri-

bution of the jump component of the processXr, respectively. As the deterministic

componentbrin (2.13) represents the minimum hop count to reach a node at dis-

tancer, the support of the jump component of the process must be restricted to the

non-negative integers. The mean of the jump component ofXris given by (2.13)

asr= rjnj.

2.3.2 Maximum Entropy Model

We now turn to the selection of a specific Lvy measure, whose general structure

is given by (2.9), for the jump component of the process Xr. This can be viewed

as a model identification problem. Appealing to the law of parsimony (Occams

razor, [42, 70]), this heuristic can be used to select among all possible models

one that is consistent with our knowledge and incorporates the least number ofa

prioriassumptions or parameters. This criterion suggests the selection of the Lvy

measurem(x) = (x1), giving rise to the simplePoisson process.A more powerful argument can be based on the maximum entropy principle,

due to Jaynes [51, 52], which states that among all distributions which satisfy an

a priorigiven set of constraints representing our knowledge, we should select the

one which is least informative, or more formally, has the maximum entropy subject

to the constraints. Theentropyof a probability mass function f(k) = {fk: k Z+}

17


33/135

is defined as [19]

H(f) =

k=0

fklogfk. (2.15)

Consider for now the process with the characteristic function

Xr(u) =

p

1 eiu(1p)r

(2.16)

corresponding to the negative binomial distribution NB(r,p), with support on the

non-negative integers. This distribution isinfinitely divisibleand therefore, it is the

distribution of some Lvy process [50]. If and only ifr

=1, the marginal distri-bution of this process isgeometricwith parameter p. Among the discrete distribu-

tions supported on Z+and subject to a mean constraint, the geometric distribution

is well-known to possess maximum entropy [54]. Consequently, the Lvy process

(2.16) achieves maximum entropy, howeveronlyfor r=1.

A maximum entropy Lvy process, on the other hand, corresponds to an in-

finitely divisible distribution which has maximum entropy for all r 0, undersuitable constraints. This condition is satisfied by the Poisson distribution, and

[54, Theorem 2.5] asserts its maximum entropy status within the class of ultra

log-concave distributions [66], which includes all Bernoulli sums and thus, the bi-

nomial family. This maximum entropy result reinforces the choice of a Poisson

variable Yras the jump component of the Lvy process with drift, such that

Xr= b r+Yr. (2.17)

Up to this point, we have ignored that the drift term br is real-valued. In our

application, the drift term represents the minimum hop count to reach a node at

distancerdue to the finite transmission radius R and it is appropriate to replace it

by the integer

r=

r

R

. (2.18)

We obtain the main result of this chapter, the hop count model process {Zr: rR+}

18


34/135

with meanr, as

Zr=r+Yr (2.19)

where ris the minimum hop count and Yris a Poisson variable with a mean given

byr= rr. The processZrhas the marginal probability mass function

pZr(z) =TPois(z;r,r) Pois(zr;r), z r (2.20)

which is referred to as thetranslated Poissondistribution [5].

2.3.3 Maximum Likelihood Fit of the Hop Count ProcessThe maximum likelihood principle is used to obtain an estimate of a distribution

parameter, defined as the parameter value which maximizes the likelihood func-

tion. The maximum likelihood estimator (MLE) has the desirable property of being

asymptotically efficient, that is, unbiased and approaching the Cramr-Rao lower

bound on the variance of for increasing sample sizes [58].

The mean (2.13) of the hop count model process Zr is assumed to increase

linearly with the distancerfrom the source nodex, at ana-prioriunknown rate and

intercept(1,2) , where = R+ [1,), which we want to estimate basedon observations of the hop count at different sites, given x. The mean number ofhops as a function of distanceris approximated as

r=

0 ifr=0,1r+2 ifr>0. (2.21)

The intercept2 1 reflects the fact that for any sensor node other than the source,the hop count must be at least one. Furthermore, at any distancerthe hop count

must be greater or equal than the minimum hop count given by (2.18).

An important property of the hop count process is, that 1,2 are invariant

to the choice of the mean node-to-node passage time, which can be shown by a

simple change of units of time. The process parameters depend only on the density

and the transmission range of the sensor nodes. This implies that the MLE can

19


35/135

be performed off-line at the networks design time, while the mean node-to-node

message delay is allowed to vary (slowly) with operating conditions, such as theaverage energy harvested by the sensor nodes.

An observationz with minimum hop countis modeled by a translated Poisson

variableZwith probability mass function (2.20)

pZr(z) = z

(z)!exp(). (2.22)

where=from (2.19). By substituting=1r+2 we obtain the pmfof the observation, parameterized by the rate 1and intercept2,

pZ(z|r;) =(1r+2 )z

(z)! exp (1r+2 ). (2.23)

We now consider a sample of n independent (though in general, not identically

distributed) observationsz1:n with joint conditional probability

p(z1:n|r1:n;) =n

i=1

pZ(zi|ri;). (2.24)

For maximum likelihood estimation problems, it is often convenient to use the log-

likelihood function, defined as the logarithm of (2.24)and interpreted as a function

of the parameter vectorgiven the observations,

(|z1:n, r1:n) =lnp(z1:n|r1:n;) (2.25)

=n

i=1

lnpZ(zi|ri;). (2.26)

With (2.23), the log-likelihood for our problem becomes

(|z1:n, r1:n) =n

i=1

(zi i) ln(1ri+2 i)

ln(zi i)! (1ri+2 i). (2.27)

20


36/135

If a maximum likelihood estimate exists, then it is

ML=argmax

(|z1:n, r1:n). (2.28)

The partial derivatives of the log-likelihood function with respect toare zero at

any local extremum. Hence, the maximum likelihood estimate is a solution of the

system of likelihood equations

1=

n

i=1

zi i

1ri+2 i 1

ri=0

2

=n

i=1

zi i

1r

i+2 i 1 =0. (2.29)

Multiplication with the common denominator transforms (2.29) into a system of

bivariate polynomials in1and 2

f(1,2) =0

g(1,2) =0 (2.30)

which has the same zeros as (2.29). By Bezouts theorem [105], this system has

deg(f) deg(g) zeros(1,2) C2 which can by easily computed by typical nu-merical solvers even for large polynomial degrees (> 1000), subject to the con-

straint that (1,2), where = R+ [1,). For every zero satisfying theconstraint, negative definiteness of the Hessian of must be ascertained for a lo-

cal maximum of the likelihood to exist. The maximum likelihood estimate for our

problem is identified with the solution (1,2) which globally maximizes , subject

to(1,2) .

2.4 Simulation Results

2.4.1 Simulation Setup

The translated Poisson distribution was applied to model the empirical hop count

observed in simulated broadcast WSNs. For this purpose, our simulation generates

103 realizations of random geometric graphs G on the unit square [0, 1]2, where

21


37/135

the number of nodes is a Poisson variable with density =4000 nodes per unit

area. The source node is located at the center, a setup which minimizes bound-ary effects. We consider mean node degrees ranging from =8 to=40, which

are representative of weakly as well as strongly supercritical networks (the criti-

cal mean node degree isc=c(R)R2 4.52, based on a value ofc(1) 1.44

for the empirical critical node density as given in [63] for R= 1). Node-to-node

delays are modeled by independent exponential random variables with unit mean.

The first-passage paths are computed as the minimum-delay paths from the source

to all nodes within a thin annular region defined by r. The simulation resultsare not sensitive to the exact choice of, provided that


38/135

dependency on the dimensionality of the problem. It is therefore reasonable to

evaluate the suitability of the translated Poisson distribution as a model for the hopcount in one-dimensional WSNs. To this end, we generate realizations of WSNs on

[0, r], by placing nodes uniformly at random with density= /(2R). The source

node is located at 0. Communication rangeR and node-to-node delays are defined

as in the two-dimensional setting described earlier. To relax constraints on the first-

passage path between 0 and r, we extend the network from[0, r]to[A, r+A](thesimulation results show no sensitivity to this extension forA 2R)

The cdfs shown in Figures2.6,2.7and 2.8,corresponding to mean node de-

grees=8, 16 and 40, respectively, demonstrate that the translated Poisson dis-

tribution is also a good fit for the empirical hop count in one-dimensional WSNs.

Similar to our observations in two-dimensional networks, the fit improves as the

distance and mean node degree increase.

2.4.3 Localization of Source Node

In this section, we evaluate the localization errorwhich results from the application

of the translated Poisson distribution as a model for the hop count in a WSN. We

use the same network model and simulation setup as described in Section2.4.2.A

node located at the centerx0 of the unit square [0, 1]2 is the source of a broadcast

message. Hop count observations are made at randomly chosen nodes in the WSN,

conceivably by a mobile sink which can interrogate a node at its current positions

to obtain the nodes hop count information. The mobile sink, which is assumed to

know its own position, then estimates the location of the source node, given all the

hop count observations.

A second, fictitious WSN serves as a baseline for comparison. In this network,

the hop count at an interrogated node a distance raway from the source node is

generated by drawing from the translated Poisson distribution Z TPois(zi;,),with= 1r+2 (2.21)and = r/R(2.18). These observations are idealizedin the sense that their statistics are characterized exactlyby the observation model.

At the mobile sink, let a random variable X [0, 1]2 describe the (unknown)location of the source node and z iNbe the hop count observed at position si[0, 1]2, i =1, . . . , n. The mobile sink applies Bayes rule to compute the posterior

23


39/135

pdf of the source node location as

pX(x|z1:n) =pX(x)n

i=1

TPois(zi;i,i) x [0, 1]2, (2.31)

whereis a constant to normalize the posterior pdf, and pX(x)is the prior distri-

bution of the source location. The parameters of the translated Poisson distribu-

tion arei= 1ri+2 (2.21) andi= ri/R(2.18), with the Euclidean distanceri= si x between the source location hypothesisxand the mobile sink positionsi. The parameters1,2 are determined off-line by maximum-likelihood estima-

tion (2.21). In practice, thea posteriori distribution pX(x|z1:n) is often computedrecursively, i.e. one observation at a time, by a Bayes filter

pX(x|z1:k) =kpX(x|z1:k1) TPois(zk;k,k) x, (2.32)

where pX(x|z0) = pX(x) is the prior distribution of the source location, which isuniform in our simulation. In order for the Bayes filter to be computationally

tractable, we discretize the location hypothesis space [0, 1]2 to a grid ofJ=3232cellsxj, j= 1, . . . ,J. This form of the Bayes filter is referred to as a grid filter [110].

Since in a grid filter, the mode of the distribution is necessarily quantized, we use

theposterior meanas our estimate xof the source location,

x=J

j=1

xjpX(xj|z1:n). (2.33)

Thea posterioripdf of the source location tends to become unimodal and symmet-

ric as the number of randomly taken hop count observations increases, so that the

posterior mean is increasingly representative of the maximum a posteriori estimate

(MAP).

In the same fashion, we perform source node localization in the idealized, base-

line WSN and denote the resulting a posteriori pdf of the source location qX(x|z1:n).Figure2.9shows the cdfs of the normalized localization error

e= xx0/R (2.34)

corresponding to pX(x|z1:n)(2.32) andqX(x|z1:n)withn=8 randomly chosen ob-

24


40/135

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1

.0

r = 0.5R

modelempirical

0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1

.0

r = 1R

modelempirical

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

r = 2R

model

empirical

0 5 10 15 20 25 30 35

0.0

0.2

0.4

0.6

0.8

1.0

r = 8R

model

empirical

hop count

CDF

Figure 2.2: Comparison of the translated Poisson model and the empiri-

cal first-passage hop count (with 95% confidence interval) in a two-

dimensional network with mean node degree =8. CDFs shown for

source-to-sink distancesr {0.5R,R,2R,8R}.

servations, for=8 and=40. We observe that the localization error becomes

smaller with increasing mean node degree, . For =8, the probability that the lo-

calization error exceeds 3Ris approximately 0.075. This is likely due to the heavier

upper tail observed in the empirical hop count distribution for weakly supercritical

WSNs, as seen in Figure2.2.It is known that the robustness of Bayesian inference

suffers, if the distribution of the observation noise has heavier tails than the like-

lihood function modeling it [90]. For=40, the probability that the localization

error exceeds 3Ris less than 0.025.

25


41/135

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1

.0

r = 0.5R

modelempirical

0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1

.0

r = 1R

modelempirical

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

r = 2R

model

empirical

0 5 10 15 20 25 30 35

0.0

0.2

0.4

0.6

0.8

1.0

r = 8R

model

empirical

hop count

CDF



dimensional network with mean node degree=16. CDFs shown for


2.5 Conclusion

In this chapter, we have proposed a new approach to model the hop count distri-

bution between a source and a sink node in broadcast WSNs, in which message

propagation is governed by first-passage percolation and node-to-node delays are

characterized by i.i.d. exponential random variables. We utilize the abstract model

of a stochastic jump process to describe the hop count. By making a simplifying

independence assumption and using a maximum entropy argument, the hop count

model process is shown to have a translated Poisson marginal distribution. Sim-

ulation results confirm that the empirical hop count distribution is generally well

approximated by this model. The simulation results also show that the error re-

sulting from the application of the translated Poisson model in a target localization

26


42/135

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1

.0

r = 0.5R

modelempirical

0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1

.0

r = 1R

modelempirical

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

r = 2R

model

empirical

0 5 10 15 20 25 30 35

0.0

0.2

0.4

0.6

0.8

1.0

r = 8R

model

empirical

hop count

CDF



dimensional network with mean node degree=40. CDFs shown for


problem is small with high probability, although for node degrees approaching the

critical percolation threshold, the performance degrades. Due to its low computa-

tional complexity, we expect this model to be a good candidate for the observation

model in Bayesian target localization applications in low-cost WSNs which rely on

hop count information alone.

While the translated Poisson distribution is the most parsimonious result con-

sistent with our assumptions about the hop count process, the general form of the

jump component of the process is a compound Poisson distribution which, givenadditional information or assumptions, may provide an improved fit of the empiri-

cal hop count.

27


43/135

0

.005

0.0

10

0.0

20

0.0

50

0.1

00

0.2

00

0.5

00

0.5 1.0 2.0 4.0 8.0

normalized distance r/R

KLD

=8

=12

=16

=24

=40

Figure 2.5: Kullback-Leibler divergence (KLD) between empirical distribu-

tion and translated Poisson distribution, for {8,12,16,24,40} andat source-to-sink distances r {0.5R,R,2R,4R,8R}. The KLD de-creases with mean node degree.

28


44/135

0 1 2 3 4 5 6 7

0.0

0.2

0.4

0.6

0.8

1.0

r = 0.5R

modelempirical

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

r = 1R

modelempirical

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

r = 2R

model

empirical

0 5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

r = 8R

model

empirical

hop count

CDF


cal first-passage hop count (with 95% confidence interval) in a one-

dimensional network with mean node degree =8. CDFs shown forsource-to-sink distancesr {0.5R,R,2R,8R}.

29


45/135

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

r = 0.5R

modelempirical

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

r = 1R

modelempirical

0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

r = 2R

model

empirical

0 5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

r = 8R

model

empirical

hop count

CDF



dimensional network with mean node degree=16. CDFs shown forsource-to-sink distancesr {0.5R,R,2R,8R}.

30


46/135

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

r = 0.5R

modelempirical

0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

r = 1R

modelempirical

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

r = 2R

model

empirical

0 5 10 15 20 25 30

0.0

0.2

0.4

0.6

0.8

1.0

r = 8R

model

empirical

hop count

CDF



dimensional network with mean node degree=40. CDFs shown forsource-to-sink distancesr {0.5R,R,2R,8R}.

31


47/135

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

= 8

idealized hop countempirical hop count

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

= 24


0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

= 40


normalized localization error e

CDF

Figure 2.9: CDFs of the normalized localization error e=

x

x0

/R, given

hop count observations at 8 randomly selected nodes, for mean nodedegrees {8,24,40}. We compare localization based on empiricalfirst-passage hop counts with idealized hop counts (independent draws

from the observation model).

32


48/135

Chapter 3

Rollout Algorithms for

WSN-assisted Target Search

3.1 Introduction

An autonomous, mobile platform is tasked with finding the source of a broad-

cast message in a randomly deployed network of location-agnostic wireless sensor

nodes. Messages are assumed to propagate by flooding, with random node-to-node

delays. In WSNs of this type, the hop count of the broadcast message, given the

distance from the source node, can be approximated by a simple parametric dis-

tribution. The autonomous platform is able to interrogate a nearby sensor node to

obtain, with a given success probability, the hop count of the broadcast message.

In this chapter, we model the search as an infinite-horizon, undiscounted cost,

online POMDP and solve it approximately through policy rollout. The cost-to-go

at the rollout horizon is approximated by a heuristic based on an optimal search

plan in which path constraints and assumptions about future information gains are

relaxed. This cost can be computed efficiently, which is essential for the application

of Monte Carlo methods, such as rollout, to stochastic planning problems.

We present simulation results for the search performance under different base

policies as well as for parallel rollout, which demonstrate that our rollout approach

outperforms methods of target search based on myopic or non-myopic mutual in-

formation utility. Furthermore, we evaluate the search performance for different

33


49/135

generative models of the hop count to quantify the performance loss due to the use

of an approximate observation model and the rather significant effect of statisti-cal dependence between observations. We discuss how to explicitly account for

dependence by adapting an integer autoregressive model to describe the hop count.

3.1.1 Background and Motivation

Wireless sensor networks (WSNs) have been an object of growing interest in the

area of target localization. The achievable localization accuracy depends on many

factors, among which are the nature of the observed phenomena, sensor modalities,

the degree of uncertainty about sensor node locations as well as processing and

communication capabilities. Typically, sensor observations are processed in situand reduced to position estimates, which must be routed through the WSN via

multiple hops to dedicated sink nodes, in order to be accessible by the networks

user. In some applications, for example autonomous exploration or search and

rescue, an essential feature is that the position estimates reported by the WSN are

used to assist and guide a mobile sensor and actuator platform (or searcher, for

short) to the target. The main goal for the searcher is to make contact with the

target, for example to acquire large amounts of payload information from a sensor

that has observed and reported an event of interest, or to retrieve the target outright.

Alternative to the use of WSNs as described, it is conceivable for a mobile

searcher in the deployment area of the WSN to interrogate nearby sensor nodes

directly to gather information, based on which the position of the target can be

estimated. Thereby, expensive computational and communication tasks can be of-

floaded from the sensor nodes. A further possibility is to integrate information

obtained from the WSN with the searchers on-board, perhaps more sophisticated

sensing capabilities, thus mitigating the need for the WSN to perform very precise

localization. This can result in a significant simplification of the sensor node hard-

ware and software requirements and reduced node energy consumption. Consistent

with this goal, we assume that the sensor nodes are location-agnostic and unaware

of the distances from, or angles between neighbor nodes. The information made

available by the WSN to the searcher is assumed to be statistically related to the

target location, e.g. noisy measurements of the distance.

34


50/135

In this chapter, we consider the use of randomly deployed WSNs in which,

starting from a source node, messages are disseminated by flooding and the node-to-node transmission delays are random. In such a network, provided that the

node density exceeds a critical threshold, a random cluster of all nodes that have

received the broadcast message grows over time, starting with the source node that

initiated the broadcast upon detection of an event of interest, in a process known as

first-passage percolation [15]. For such a network, it was shown in Chapter 2that

the message hop count distribution, parameterized by the distance from the source

node, can be approximated by a simple stochastic process model.

We are motivated by the problem of a mobile, autonomous searcher which is

given the task of locating a (generally moving) target, that is, the source node of

a broadcast message in a WSN of the type studied in Chapter 2, relying on ob-

servations of the message hop count alone. We focus on the question of how the

searcher can home in and make contact with the target in the shortest expected

time, given only the hop count observations. Making contact with the target is de-

fined here as observing a hop count of zero. The most general framework for this

type of optimal, sequential decision problem under uncertain state transitions and

state observations is the partially observable Markov decision process (POMDP)

[56, 69, 79, 110]. POMDPs optimally blend the need for exploration to reduce

uncertainty with making progress toward the goal state. Unfortunately, for all but

small problems (in terms of the sizes of the state, action and observation spaces),

solving POMDPs is computationally intractable. This is due to the fact that the

solution is quite general, and must be computed for every possible belief state. The

belief state(or information state) is the POMDP concept of expressing the uncer-

tainty about the true, unobservable state as a probability distribution over all states.

For search problems such as the one considered here, it can be more productive to

consider only the current belief state and plan the search with respect to the actu-

ally reachable belief space. This approach is referred to as online POMDP [85].

Online POMDPs have the additional advantage over offline solutions of being able

to (quickly) respond to changing model parameter values. However, in many prac-

tical applications, online POMDPs still present a formidable computational chal-

lenge, compounded by the need to operate in real-time. Therefore, online POMDPs

are most often solved approximately and hence, suboptimally [17]. One class of

35


51/135


52/135

We discuss mitigating strategies and propose an integer autoregressive pro-

cess as a model of the observation dependence. This model is derived byadapting the INAR(1) process [1] to translated Poisson innovations.

3.1.3 Chapter Organization

The chapter is organized as follows: In Section 3.2, our system model is intro-

duced. The main contribution, a heuristic for the expected search time at the rollout

horizon, is derived in Section 3.4.2. Because of its use as a reference for perfor-

mance evaluation, myopic and non-myopic mutual information utilities are briefly

reviewed in Section3.5. We present simulation results in Section3.7that show the

performance improvement of the rollout algorithm over existing techniques. We

also discuss the loss of performance due to statistical dependence of the empirical

hop count observations, and present mitigating strategies, including a model for

the observation dependence, in Section3.8. Conclusions for future work are drawn

in Section 3.9. To simplify the notation for probabilities, we omit the names of

random variables when this is unambiguous.

3.2 System Model

3.2.1 Wireless Sensor Network Model

Randomly deployed WSNs are frequently described by Gilberts disk model[35,

40], which we adopt in this chapter. Without loss of generality, the deployment

area is assumed to be the unit square [0, 1]2 R2. Sensor nodes are distributedaccording to a spatial Poisson point process P of density , restricted to the

deployment area. Two sensor nodes are said to belinkedif they are within com-

munication rangeR of each other. The sensor nodesPand their communication

links ER form a random geometric graph G= {P,ER} [81] with mean node de-gree=R2. The mean node degree must exceed a critical threshold for a large

portion of the network to be connected.

In order to simulate a mobile searcher making hop count observations while

operating in the deployment area of such a WSN, we discretize both the searcher

position and the sensor node coordinates. Rather than placing sensor nodes ran-

37


53/135


54/135


55/135

observed. Time is assumed to be discrete. At every time step k, the searcher may

decide to stay in the cell it is currently visiting, or move to one of the four neighborcells. Any searcher action incurs a cost, that is, an increment of search time.

3.2.3 Formulation of Target Search Problem

A POMDP is a general model for the optimal control of systems with uncertain

state transitions and state observations [69, 110]. In target search problems for-

mulated as POMDPs, the target motion is modeled as a Markov chain, searcher

actions may have uncertain effects, and only noisy observations of at least one

state variable are available. Since a focus of this chapter is the study of a hop count

observation model applied to target search, we can restrict attention to a stationarytarget and assume, that the searcher position is completely determined by the ac-

tions (that is, the searcher position is known without ambiguity). It is worth point-

ing out that certain instances of search problems can be modeled as multi-armed

bandits, for which index policies exist under suitable conditions [71,97]. In this

chapter, online methods of policy rollout will be used to compute an approximate,

suboptimal solution for the target search problem.

Any POMDP can be defined in terms of a tupleS,A, T,J,Z, O, where Sis the state space,Athe set of admissible actions, Tthe state transition function,J

the cost function (commonly, the reward functionR is specified instead), Z is the

set of observations and O defines the state observation law.

Definition 3.2. The POMDP for the target search problem is defined by

S=XY is a finite, joint state space, where Xis the range of the par-tially observed target position andY =X the range of the searcher position.

A state is represented by the vectors= (x,y)T.

A = {Ay}yY is a family of action sets indexed by the set of all possiblesearcher positions. An action set Ay {stay, north, east, south, west,D}de-fines the possible moves the searcher can make from its current position y in

the next time step, and is augmented by an action D denoting detection. If

the target has not been detected by timek, the searchers next action is con-

strained byak+1Ayk\{D}. If the target is detected at time k(by observinga message hop count ofzero), thenak+n=D, for alln>0.

40


56/135

T: SAy S [0, 1]is the transition kernel describing the state dynam-ics. In our model, where the target is assumed to be a stationary sensor nodeand the searcher position is completely determined, we have

T(s, a, s) Pr{Sk+1=s|Sk= s, a} (3.3)

=

x,x y,y a=Dx,x y,y+a otherwise (3.4)

J:AyR specifies the cost of executing the action a,

J(a)0, ifa=D1, ifa Ay (3.5)

Z = N0 {} is the set of observations of message hop counts, N0, aug-mented by a possibility of making no observation, denoted.

O :SAy Z [0, 1]is the hop count observation model, defined as

O(s, a,z)

1, a=D

Pr{Z=z|s}, otherwise(3.6)

where

Pr{0|s} =q, if x=y0, otherwise (3.7)

Pr{|s} =1q, if x=y1qp, otherwise (3.8)

Pr{n|s} =

0, if x=y

qp fZ(n; r(s)), otherwise

(3.9)

forn>0. Here, fZ(n; r(s)) is the translated Poisson distribution (3.2), and

r(s) = cx cy2 is the Euclidean distance between target and searcher po-sitions.

41


57/135

Figure 3.1: Dynamic Bayes Network representing a POMDP

Since the state is only partially observable, the mobile searcher maintains a

belief state,

autonomous, wireless sensor network-assisted target search and mapping

Documents