Understanding the Prediction Gap in Multi-hop Localization

by

Cameron Dean Whitehouse

B.A. (Rutgers University) 1999B.S. (Rutgers University) 1999

M.S. (University of California, Berkeley) 2003

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Computer Science

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:Professor David Culler, Chair

Professor Eric BrewerProfessor Todd Dawson

Fall 2006

The dissertation of Cameron Dean Whitehouse is approved:

Chair Date

Date

Date

University of California, Berkeley

Fall 2006

Understanding the Prediction Gap in Multi-hop Localization

Copyright 2006

by

Cameron Dean Whitehouse

1

Abstract

Understanding the Prediction Gap in Multi-hop Localization

by

Cameron Dean Whitehouse

Doctor of Philosophy in Computer Science

University of California, Berkeley

Professor David Culler, Chair

Wireless sensor networks consist of many tiny, wireless, battery-powered sensor nodes that enable

the collection of sensor data from the physical world. A key requirement to interpreting this data

is that we identify the locations of the nodes in space. To this end, many techniques are being

developed forranging-based sensor localization, in which the positions of nodes can be estimated

based on range estimates between neighboring nodes.

Most work in this area is based on simulation, and only recentapplications of ranging-

based localization in the physical world have revealed whatwe call thePrediction Gap: localization

error observed in real deployments can be many times worse than the error predicted by simulation.

The Prediction Gap is a real barrier to sensor localization because simulation is an essential tool

for designing, developing, and evaluating sensor technology and algorithms before they are actually

used in costly, large-scale deployments.

The goals of this dissertation are 1) to close the PredictionGap and 2) to identify its causes

in sensor localization. We first establish the existence andmagnitude of the Prediction Gap by

building and deploying a sensor localization system and comparing observed localization error with

predictions from the traditional model of ranging. We then develop new non-parametric modeling

techniques that can use empirical ranging data to predict localization error in a deployment. We

show that our non-parametric models do not cost significantly more than traditional parametric

models in terms of data collection or simulation, and solve many of the prediction issues present in

existing simulations.

In order to identify the causes of the Prediction Gap in sensor localization, we create

hybrid models that combine components of our non-parametric models with traditional parametric

2

models. By comparing localization error from a hybrid modelwith a purely parametric model,

we isolate the effects of that component of our data. We use this technique to identify the causes

of the Prediction Gap for six different localization algorithms from the literature, and conclude by

developing a new parametric model that captures the true characteristics of our empirical ranging

data.

Professor David CullerDissertation Committee Chair

i

Dedicated to my family

ii

Acknowledgments

Thanks to all of my collaborators who provided technical, intellectual, practical, and advisorial

help with various aspects of this work. These include but arenot limited to Alec Woo, Chris Karlof,

Fred Jiang, Cory Sharp, Rob Szewczyk, Jason Hill, Scott Klemmer, Sarah Waterson, Gilman Tolle,

Jonathan Hui, Phil Buonadonna, Phoebus Chen, Mike Manzo, Matt Welsh, Sam Madden, Pra-

bal Dutta, Joe Polastre, Naveen Sastry, Tye Rattenbury, Kris Pister, Deborah Estrin, Joe Heller-

stein, Bhaskar Krishnamachari, Feng Zhao, Carlos Guestrin, David Wagner, Shankar Sastry and, of

course, my dissertation committee.

Special thanks to Arianna for her constant support.

This work was funded in part by the National Defense Science and Engineering Graduate Fel-

lowship, The UC Berkeley Graduate Opportunity Fellowship,the Siebel Fellowship, the DARPA

NEST contract F33615-01-C-1895, and Intel Research at Berkeley.

iii

Contents

List of Figures v

1 Introduction 11.1 Sensor Field Localization . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 21.2 The Prediction Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 31.3 Outline of the Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 5

2 Background 72.1 Sensor Field Localization . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 7

2.1.1 Single-hop Localization . . . . . . . . . . . . . . . . . . . . . . . .. . . 82.1.2 Multi-hop Localization . . . . . . . . . . . . . . . . . . . . . . . . .. . . 11

2.2 Ranging Theory: The Noisy Disk Model . . . . . . . . . . . . . . . . .. . . . . . 122.3 Physical Range Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 13

2.3.1 Radio Signal Strength . . . . . . . . . . . . . . . . . . . . . . . . . . .. 132.3.2 Acoustic Time of Flight . . . . . . . . . . . . . . . . . . . . . . . . . .. 152.3.3 Interferometric Ranging . . . . . . . . . . . . . . . . . . . . . . . .. . . 162.3.4 RF Time of Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Localization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 182.5 Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 21

2.5.1 Simulation-based Studies . . . . . . . . . . . . . . . . . . . . . . .. . . . 222.5.2 Ranging Characterization Studies . . . . . . . . . . . . . . . .. . . . . . 242.5.3 Localization Deployment Studies . . . . . . . . . . . . . . . . .. . . . . 26

3 Establishing the Prediction Gap 283.1 The Ranging Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 293.2 Noise Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 313.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 343.4 Dealing with Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 383.5 The Localization Algorithm . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 423.6 Distributed Programming . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 433.7 Implementation and Debugging . . . . . . . . . . . . . . . . . . . . . .. . . . . 463.8 Deployment Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 483.9 Comparing Theoretical and Observed Localization Error. . . . . . . . . . . . . . 51

iv

3.10 The Prediction Gap Established . . . . . . . . . . . . . . . . . . . .. . . . . . . . 54

4 Closing the Prediction Gap 554.1 Modeling the sensors and environment . . . . . . . . . . . . . . . .. . . . . . . . 56

4.1.1 Parametric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .564.1.2 Non-parametric Models . . . . . . . . . . . . . . . . . . . . . . . . . .. 57

4.2 Empirically Profiling the Physical World . . . . . . . . . . . . .. . . . . . . . . . 594.2.1 Traditional Data Collection . . . . . . . . . . . . . . . . . . . . .. . . . . 594.2.2 High-fidelity Data Collection . . . . . . . . . . . . . . . . . . . .. . . . . 614.2.3 Generality of an Empirical Profile . . . . . . . . . . . . . . . . .. . . . . 63

4.3 Comparing Non-parametric Predictions and Observed Localization Error . . . . . 65

5 Explaining the Prediction Gap 705.1 The Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 71

5.1.1 Identifying Ranging Irregularities . . . . . . . . . . . . . .. . . . . . . . 725.1.2 Creating Hybrid Models . . . . . . . . . . . . . . . . . . . . . . . . . .. 725.1.3 Parameters and Topology . . . . . . . . . . . . . . . . . . . . . . . . .. . 74

5.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 765.3 Analyzing Each Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 80

5.3.1 Bounding Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3.2 DV-Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.3.3 MDS-Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.3.4 GPS-Free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.3.5 Robust Quads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.3.6 MDS-Map(P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6 Removing the Prediction Gap 926.1 Existing Models of Irregularity . . . . . . . . . . . . . . . . . . . .. . . . . . . . 93

6.1.1 Non-uniformity of Nodes . . . . . . . . . . . . . . . . . . . . . . . . .. . 936.1.2 Radio Irregularity Model . . . . . . . . . . . . . . . . . . . . . . . .. . . 966.1.3 Gaussian Packet Reception Rate Model . . . . . . . . . . . . . .. . . . . 996.1.4 Shadowing and Multi-path . . . . . . . . . . . . . . . . . . . . . . . .. . 103

6.2 Towards a New Parametric Model . . . . . . . . . . . . . . . . . . . . . .. . . . 1046.2.1 A Geometric Noise Distribution . . . . . . . . . . . . . . . . . . .. . . . 1066.2.2 An Exponential Model of Connectivity . . . . . . . . . . . . . .. . . . . 1076.2.3 Verifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113

7 Conclusions 1157.1 Advantages of Our Modeling Techniques . . . . . . . . . . . . . . .. . . . . . . 1167.2 Parametric vs. Non-parametric Models . . . . . . . . . . . . . . .. . . . . . . . . 1177.3 Extending Analysis to Other Areas . . . . . . . . . . . . . . . . . . .. . . . . . . 118

Bibliography 124

v

List of Figures

2.1 Single- and Multi-hop Localization . . . . . . . . . . . . . . . . .. . . . . . . . . 92.2 Error Increase over Distance . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 142.3 Localization Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 20

3.1 The Ultrasound Ranging Hardware . . . . . . . . . . . . . . . . . . . .. . . . . . 303.2 Raw Time of Flight Readings . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 313.3 Averaging Ranging Data . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 323.4 The MedianTube Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 333.5 The Effect of Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 353.6 The Effect of Calibration . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 373.7 Capture Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 393.8 The Prevalence of Capture . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 413.9 The Hood Programming Abstraction . . . . . . . . . . . . . . . . . . .. . . . . . 453.10 Stages of Development and Debugging . . . . . . . . . . . . . . . .. . . . . . . . 473.11 The Final Deployment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 483.12 Localization Error Vectors . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 493.13 The Localization Error Gap . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 503.14 The Shortest Path Error Gap . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 523.15 The Node Degree Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53

4.1 The Non-parametric Model . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 584.2 Traditional Data Collection . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 604.3 The Data Collection Topology . . . . . . . . . . . . . . . . . . . . . . .. . . . . 624.4 Profiling Multiple Environments . . . . . . . . . . . . . . . . . . . .. . . . . . . 644.5 Closing the Localization Error Gap . . . . . . . . . . . . . . . . . .. . . . . . . . 664.6 Closing the Shortest Path Error Gap . . . . . . . . . . . . . . . . . .. . . . . . . 674.7 Closing the Node Degree Gap . . . . . . . . . . . . . . . . . . . . . . . . .. . . 68

5.1 Ranging Irregularities . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 735.2 Ordering of Localization Algorithms . . . . . . . . . . . . . . . .. . . . . . . . . 765.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 775.4 Causes of the Prediction Gap . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 795.5 The Bounding Box Algorithm . . . . . . . . . . . . . . . . . . . . . . . . .. . . 805.6 Shortest Path Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 81

vi

5.7 The Effect of Density on Shortest Paths . . . . . . . . . . . . . . .. . . . . . . . 835.8 Anchor Corrections and Range Irregularities . . . . . . . . .. . . . . . . . . . . . 855.9 Robust Quads and Stitching Failure . . . . . . . . . . . . . . . . . .. . . . . . . 885.10 Robust Quads Phase Transition . . . . . . . . . . . . . . . . . . . . .. . . . . . . 895.11 Robust Quads Overly Restrictive . . . . . . . . . . . . . . . . . . .. . . . . . . . 90

6.1 Radio Transitional Region . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 946.2 Non-Uniformity and the Transitional Region . . . . . . . . . .. . . . . . . . . . . 956.3 Uniformity of Ultrasound Nodes . . . . . . . . . . . . . . . . . . . . .. . . . . . 976.4 Radio Irregularity Model . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 986.5 Sources of Non-isotropic Ultrasound . . . . . . . . . . . . . . . .. . . . . . . . . 996.6 Ultrasound Connectivity Contours . . . . . . . . . . . . . . . . . .. . . . . . . . 1006.7 Gaussian Packet Reception Rate Model . . . . . . . . . . . . . . . .. . . . . . . 1016.8 Ultrasound Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 1026.9 Shadowing, Multi-path and Bit Errors . . . . . . . . . . . . . . . .. . . . . . . . 1036.10 Ultrasonic Emanation Pattern . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 1056.11 Geometric Noise Distribution . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 1086.12 Connectivity Characteristics of the Geometric Noise Model . . . . . . . . . . . . . 1106.13 Power Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 1116.14 Complete Parametric Model . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 1126.15 Top-down Parametric Model Evaluation . . . . . . . . . . . . . .. . . . . . . . . 114

7.1 A Tracking Deployment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 1207.2 Profiling the PIR Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 122

1

Chapter 1

Introduction

Sensor arrays have long been used to measure phenomena that are distributed through

space. Perhaps the most common sensor array today is the digital camera: a CCD can consist of

hundreds of thousands of light sensors, called pixels, eachof which measures the light emanating

from a different point in space. The digital camera, however, is an exceptional case; the straight-line

propagation of light allows a lens to focus light from multiple different points in space onto a sensor

array that is spatially concentrated, often with all the pixels located within one square centimeter.

Other stimuli which cannot be focused with a lens, such as temperature, humidity, and pressure,

must be measuredin place; each sensor must actually be located at the point in space where a

measurement is to be taken. This fundamentally changes the computer system needed to operate

an array of in-place sensors: when they must be distributed over meters or even kilometers, it is no

longer feasible to run a wire to each sensor.

Wireless sensor networksenable one to collect data from a spatially distributed array of

sensors. Each node in the network consists of a sensor, a microprocessor, a battery and a radio. The

microprocessor can sample from the sensor and process the data as needed, as well as communicate

with neighboring nodes. In simplest form, all wireless sensor nodes periodically sample from their

own sensors and the data is routed through the wireless network to a gateway node, at which point

it is permanently archived in a database. This system allowssensors to be distributed through space

by replacing power lines with batteries, replacing data lines with a wireless network, and coupling

each sensor with a microprocessor for decentralized control.

This system architecture enables a plethora of new sensing applications. For example,

humidity gradients can be measured in a forest, helping scientists identify the effects of trees on

the water cycle [94]. Vibrations of large structures such asbuildings, boats, and bridges, can be

2

monitored to understand the effects of earthquakes, and to identify points of structural damage

[44, 46]. Networks of heat-sensing nodes can be dropped froman airplane around a wildfire to

identify areas of rapid movement [20]. Weather sensing nodes can be used for precision agriculture

to monitor the light, temperature, and moisture levels at different points in a vineyard [11].

A complete sensor network application requires a number of different system components

to operate correctly, including multi-hop routing, time synchronization, localization of the nodes,

power management, etc. Furthermore, each specific application carries a different set of constraints

and demands on these components. For example, a network usedfor precision agriculture may

require a long network lifetime, but can tolerate high network loss and latency and the nodes can

be localized manually at deployment time. On the other hand,a network dropped over a wildfire

may only need to function for a few hours but the locations of the nodes must be determined by

the nodes themselves. Lastly, a network for structural monitoring may need extremely precise time

synchronization to correlate vibrations in space and high sampling rates to capture high-frequency

vibrations, straining the network bandwidth. Thus, while the primary goal of a wireless sensor

network is simply to replace a wired network, its spatially-distributed and resource-constrained

nature makes it difficult to find a one-size-fits-all solution; as one moves along the different axes

of the application space, one must manage anew the complex interactions and trade-offs among

hardware capabilities, resource constraints, and application requirements.

1.1 Sensor Field Localization

In this study, we focus mainly on the task ofsensor field localization, which is the pro-

cess of identifying the locations of the nodes in a wireless sensor network. Location information is

critical for essentially all sensor network applications;without knowing the location of a sensor, the

data being produced cannot be interpreted. As such, localization is one of the critical system com-

ponents required by the applications mentioned above. However, it is also an application of its own.

In fact, a large fraction of the applications for wireless embedded systems falls under the category of

asset or object localization, identification, and tracking. As an application, localization in turn also

requires a broad array of other system components and services, sometimes including specialized

hardware and drivers, collision detection, multi-hop routing, neighborhood management, modeling

and simulation techniques, etc.

There are several ways to localize sensor nodes. For example, nodes can be localized

at deployment time using a GPS or DGPS receiver that is attached to the person deploying the

3

nodes [18]. A survey-grade device can be used to localize thenodes after deployment [28]. A

closed-loop system including a pan/tilt laser that is detectable by the nodes can provide similar

accuracy without human intervention [32, 75]. Coarse locations can be obtained by simply placing

beacon nodes with known positions throughout the deployment area and allowing nodes to estimate

their positions based on the beacons within radio range.

Each of these localization techniques achieves a differentbalance of human effort before

deployment, node effort after deployment, and localization accuracy. The rest of this study focuses

on a particular kind of solution calledranging-basedsensor localization.Rangingis the process of

estimating the distance between two nodes. Each range estimate is used to constrain the location of

one node with respect to the location of a neighboring node. When enough constraints exist within a

network of nodes, the locations of all nodes are over-constrained and their relative positions can be

solved for analytically. If a small number ofanchor nodeswith known positions are in the network,

the locations of all nodes can be determined within the global coordinate system defined by the

anchor nodes.

Ranging-based localization systems require no special infrastructure such as a laser sys-

tem or a dense blanket of beacon nodes and no human effort is required during deployment. In-

stead, these systems require the design of new range sensors, localization algorithms, and signal-

processing techniques. The rational behind focusing on this particular kind solution is that the cost

of solving the ranging-based solution once can be amortizedover time, providing both high accuracy

and simple operation for many deployments.

1.2 The Prediction Gap

Accurate models of the sensors and the environment are extremely important for any

sensor network application because they are necessary for designing, simulating, and analyzing the

data processing algorithms. Although over 100 algorithms for ranging-based localization have been

proposed, however, none of them have addressed the issue of accurate ranging models. Nearly

all localization studies are based on a very simple model of ranging called theNoisy Disk, which

predicts that a node will obtain a range estimate to all nodescloser than a maximum rangedmax

and that all range estimates will exhibit zero-mean Gaussian noiseN (0, σ). However, this is often

not the case. As a result, the localization error of an algorithm can be several times worse in a real

deployment than predicted by simulation; one empirical deployment from the literature documents

errors that were up to 8 times worse [92]. The difference between real-world localization error and

4

that predicted by simulation is what we call thePrediction Gap.

The Prediction Gap is an important, long-standing problem in the localization literature

because real deployments are unpredictable and may not meetapplication requirements, even if

predicted to do so in simulation. Worse, since everything known about the range sensors and envi-

ronment is built into the model used for simulation, any additional error observed in the real world

is unexplained, and therefore difficult to improve upon; since simulation does not necessarily pre-

dict reality, any improvements made on the algorithm in simulation will not necessarily translate

into improvements in the real world. Similarly, a simulation-based comparison of two algorithms

in simulation will not necessarily predict which algorithmwill perform better in a real deployment.

Without simulation, the user is reduced to development through trial-and-error in the field, which

can be especially problematic for mission critical deployments which can only be deployed once,

such as forest fire tracking, or for large deployments with 1000’s of nodes where the cost of re-

deployment is prohibitive. Ultimately, the Prediction Gapis a quantification of the inadequacy of

simulation for ranging-based localization today; a large Prediction Gap is an ominous sign in a field

where real-world deployments are costly, difficult, and scarce, and most studies are solidly grounded

in simulation.

The Prediction Gap, however, is not a problem unique to ranging-based localization,

which is only one instance of a class of sensor network applications that perform real-timedata

processing. Unlike the more common class ofdata collectionapplications, which collect an opaque

data stream for later human processing, data processing applications must actually process the sen-

sor data and infer some physical property about the world, with no human intervention. This means

that they must be prepared for noise, failures, and other anomalies in the data because they cannot

utilize post-hocapplication development, in which the user can repeatedly clean the data and tune

the algorithms until the application works. Any application that must operate autonomously with

no human intervention and no closed-loop feed-back must be grounded in an accurate model of the

sensors and environment. This is acutely evident, for example, in multi-hop routing, for which the

community is still improving on models of the radio link layer that took years to develop. It is also

evident in applications like tracking, which use passive infrared or magnetometer sensors to infer

the locations of moving objects in the world. Finally, it is particularly true for applications where

algorithmic properties can change dramatically at scale and/or with different network topologies,

and thus where simulation, not deployment, must be the primary mode of development.

5

1.3 Outline of the Solution

One solution to this problem for ranging-based localization is to create a more accurate

model of the environment and range sensor. However, doing sowould require that wealreadyun-

derstand the causes of the Prediction Gap; in order to build anew model, we must understand which

aspects of the environment and range sensor are different from the idealized Noisy Disk model and

are causing additional errors in our localization algorithm. It is possible to collect ranging data,

analyze it, and postulate a new model, a process we callbottom-upmodeling. However, actually

validating any new model is a difficult process that requiresa real-world localization deployment to

be compared with a simulation using the new model, a process we call top-downmodeling. This

validation process must be repeated whenever the environment, range sensor, or algorithm changes.

Instead of trying to create an improved parametric model of ranging, we use non-parametric

models which take ranging data collected in the real world and use it directly in simulation. This

process avoids the need to reduce the empirical data set to a simple set of parameters and also main-

tains the integrity of the data set more than a parametric model might, preserving any anomalies and

subtle structural artifacts that may be overlooked. In Chapter 4, we validate this modeling process

by comparing localization error from a real-world deployment to that predicted by simulation, and

find that our non-parametric model predicts the real deployment much more accurately than the

Noisy Disk model. Because the non-parametric model does notassume any structure of the data,

this validation process does not need to be repeated whenever the deployment scenario changes.

Instead, we only need to collect a new set of empirical ranging data with the new environment or

range sensor.

The main advantage of non-parametric models is that they canbe created without requir-

ing the user to model the structure of the data. The corresponding disadvantage is that they do not

provide any insight into this structure. Thus, non-parametric models are very useful for creating

realistic simulations, but are not very useful for inspiring new algorithmic designs. To remedy this

shortcoming, Chapter 5 demonstrates a technique for identifying which components of the data

are affecting the localization algorithm. The technique makes use ofhybrid models, which use

parametric models for some components of the data and non-parametric models for others. If we

hypothesize that the Prediction Gap is caused by a particular aspect of the Noisy Disk model, such

as the assumption of Gaussian noise, we can create a controlled experiment in which one simulation

uses the Noisy Disk model and another simulation uses a hybrid model with the Noisy Disk model

of connectivity and a non-parametric model of empirical noise. If the two simulations produce the

6

same localization error, we conclude that the Gaussian noise assumption is asufficientmodel of

empirical noise. On the other hand, if the two produce different localization error, we can conclude

that empirical noise is not Gaussian. Thus, hybrid parametric/non-parametric models enable a sci-

entific process through which we can identify the structuralcomponents of the data that most affect

a particular localization algorithm.

After achieving a sufficient understanding of the data set, anew parametric model can be

derived. This algebraic form would useful, for example, when deriving properties or performing

mathematical proofs about an algorithm. In Chapter 6, we derive a model based on a geometric

distribution of noise and a log-normal model of attenuationthat includes shadowing and multi-path

effects. We validate the model by comparing its predicted localization error to that of an empirical

deployment, and find that it predicts better than the Noisy Disk model and as well as the non-

parametric model.

The Prediction Gap is not a problem specific to ranging-basedlocalization and our solu-

tion can be applied to any data processing application. In Chapter 7, we show how these techniques

can be extended to model a passive infrared sensor, for example, to simulate and analyze a tracking

application.

7

Chapter 2

Background

Localization is a very broad and active area of research thatspans the areas of hardware

design, signal processing, and algorithms and has been applied in a diverse set of application do-

mains including sensor networks, ubiquitous computing, military applications, and inventory man-

agement. In this chapter, we place our work within the context of this larger body of literature,

outlining key concepts and pointing out important previousstudies. We first define the problem of

localization in Section 2.1 and an ontology of different versions of this problem. We then describe

the task of ranging, including a common theoretical model ofranging in Section 2.2 and several

existing implementations of range sensors in Section 2.3. In Section 2.4 we provide an overview

of six localization algorithms from the literature, and in Section 2.5 we explore the literature for

existing evidence or explanations of the Prediction Gap.

2.1 Sensor Field Localization

In the problem of sensor field localization, a sensor field is usually defined asn nodes in a

two-dimensional plane, although most of the definitions andsolutions in this area can be straightfor-

wardly extended to three dimensions. The firstm nodes are termedanchor nodesand have known

locations in a global coordinate system. Anchor nodes may belocalized using Differential GPS

(DGPS) [18], surveyor-quality laser range finders [28], tape measures [97], or other means, but due

to the challenges of manual localization, the number of manually localized nodes is generally much

smaller than the number of nodes in the network:m << n.

In ranging-based localization, each nodei can obtain a distance estimatedij to another

8

neighborj that is some function of the true distancedij between them.

dij = f(dij) (2.1)

Some range estimates, however, arefailures, which means that a pair of nodes does not obtain a

valid distance estimate at all. Failures can occur for a variety of reasons, including hardware failure,

noise filtering, or a low signal to noise ratio. These failures can be denoted by a null valuedij = ø.

The nodes and the distance estimates between them form a graph G = (V,E), where|V | = n and

eij ∈ E =

ø failure betweeni andj

f(dij) otherwise(2.2)

The task ofsensor field localizationis to derive the positions of then − m unlocalized nodes from

the ranging graphG.

The general problem of sensor field localization can be divided into four sub-classes.Cen-

tralized localization assumes that the entire graphG can be collected to a single location and can be

used to localize all nodes.Distributed localization refers to the process in which each node derives

its own location using only locally available information,which is typically only a subsection of the

graphG. Absolutelocalization requires all nodes to be localized in a single global coordinate system

defined by the anchor nodes.Relativelocalization requires each node to be localized relative toits

neighbors in a locally-defined coordinate system. The benefit of relative localization techniques is

that they can be used even whenm = 0. If m ≥ 3, a local-global coordinate transform can be

derived to convert any relative localization algorithm to an absolute localization algorithm.

2.1.1 Single-hop Localization

In Single-hoplocalization, each nodei has a direct range estimate to at least three anchor

nodes. For each such anchor nodej, this estimatedij can be used to relate the unknown coordinates

(xi, yi) of the unlocalized node to the known coordinates(xj , yj) of the anchor node using the

standard distance formula:

d2ij = (xi − xj)

2 + (yi − yj)2

d2ij = x2

i − 2xixj + x2j + y2

i − 2yiyj + y2j (2.3)

A system of three or more such equations can be linearized by subtracting one of the

equations from the rest to remove the quadratic terms, leaving two variables in two or more linear

equations. For example, the following three distance equations

9

(a) Single-hop Localization

(b) Multi-hop Localization

Figure 2.1: Single- and Multi-hop Localization differ in that in a) single-hop localization, allnodes have range estimates to at least three anchor nodes while in b) multi-hop localization, nodesmust use range estimates to other unlocalized nodes. Here, black nodes are anchor nodes whilewhite notes need to be localized.

10

d2ij = x2

i − 2xixj + x2j + y2

i − 2yiyj + y2j

d2ik = x2

i − 2xixk + x2k + y2

i − 2yiyk + y2k

d2ih = x2

i − 2xixh + x2h + y2

i − 2yiyh + y2h (2.4)

can be reduced to the following two equations, which are linear inxi andyi.

d2ij − d2

ih = xi ∗ (2xh − 2xj) − yi ∗ (2yh − 2yj) + y2j − y2

h

d2ik − d2

ih = xi ∗ (2xh − 2xk) − yi ∗ (2yh − 2yk) + y2k − y2

h (2.5)

These two equations can be solved for the two unknown variablesxi andyi directly. This process

is known astri-lateration. If more than two equations remain, the linear system can be solved

approximately using least squares in a process often calledmulti-lateration.

All of the values needed to solve this set of equations, i.e. the range estimatesdij and the

anchor coordinates(xj , yj), are immediately available to the node through local radio communi-

cation. Therefore, single-hop localization can be trivially executed in a distributed fashion, where

each node localizes itself using only locally available information. It can be difficult to apply to

sensor field localization, however, because of the high proportion of nodes that must be manually

localized in order to fully cover a sparsely connected sensor network with anchors. In a grid-like

network where each node is connected to all eight of its immediate grid neighbors, more than one in

every four nodes would need to be manually localized for every node in the network to be connected

to at least three anchor nodes.

An alternative to tri-lateration is calledRF profiling, developed for the RADAR local-

ization system at Microsoft [5]. RF profiling requires a pre-deployment stage in which the radio

signal strength (RSS) of each beacon is recorded at each position in the two dimensional region to

be localized. The readings taken at a particular position can be called the RF profile of that position.

At a later time, a node with unknown location matches the RF profile of its current position to the

profiles of the positions already recorded. RF profiling is a single-hop technique because it still

requires each mobile node to have direct radio connectivitywith several anchor nodes. One disad-

vantage over tri-lateration is that the user must profile theentire two-dimensional region in which

localization is to take place. However, this tedious process also allows the technique to deal with

environmental sources of systematic error such as walls andfurniture that can disturb RF signal

propagation. Indeed, this technique was motivated by an initial implementation of RADAR which

11

found that RSS was inadequate for distance estimation indoors, even with an attenuation model that

accounted for walls and other objects [5]. RF profiling allowed RADAR to achieve approximately

4m localization error indoors using nothing but RSS from 802.11 base stations.

Single-hop localization is well understood and several commercial systems and academic

prototypes have been built. GPS is a well known system that uses an expensive infrastructure of

highly synchronized satellites and multi-lateration to find the position of mobile nodes on the earth’s

surface using RF time of flight [36]. Cricket performs multi-lateration indoors using ultrasonic

time of flight [70]. Besides RADAR, several systems have employed RF profiling for RSS-based

localization using several different types of radios, including 802.11 [19,29,49], VHF [15], cellular

radios [84], and most recently low-power wireless sensor networks [57]. A recent study has shown

that Bayesian inference can achieve similar results without pre-collecting a complete RF profile

[58], although this technique does not remove the requirement of dense anchor nodes.

2.1.2 Multi-hop Localization

In Multi-hop localization, nodes are not directly adjacent to multiple anchor nodes and

must usenon-adjacentanchors and range estimates for localization, as illustrated in Figure 2.1. This

makes the problem fundamentally more difficult than single-hop localization for two reasons. First,

Equations 2.4 can no longer be linearized because the remaining quadratic terms in Equations 2.5

such asy2j can be eliminated only ifj is an anchor node andyj is a constant value . Second, the range

estimates and anchor node coordinates required to make Equations 2.5 a fully constrained system

of equations are not necessarily available through single-hop communication and must be obtained

through distributed routing or dissemination algorithms.Multi-hop localization is therefore neither

a simple linear optimization nor is it computable with only local information.

One main difference between single- and multi-hop localization is that multi-hop algo-

rithms must be evaluated at scale. In single-hop localization, the accuracy observed in a single-cell

deployment can be generalized to larger multi-cell deployments because each cell is roughly inde-

pendent. In other words, a system shown to work in one room canreasonably be expected to work

across an entire building if each room has enough anchors. The performance of a multi-hop algo-

rithm on a sensor network of 10 nodes, however, can be completely different from its performance

on 1000 nodes. Furthermore, each algorithm cannot be evaluated on only one topology, but must

be evaluated on a broad range of network topologies. For these two reasons, multi-hop localiza-

tion research is mainly focused on theoretical analysis andsimulation with relatively few empirical

12

systems and prototypes.

2.2 Ranging Theory: The Noisy Disk Model

A rangingmodelis the functionf(dij) from Equation 2.1. For theoretical analysis and

simulation of multi-hop localization, range estimation isalmost universally modeled with theNoisy

Disk, which has two components: noise and connectivity. The connectivity component is parame-

terized by a valuedmax and states that a node will obtain a range estimate to all nodes withindmax

and to no nodes beyonddmax. The noise component models the differences between the range es-

timates and the true distances using a Normal distribution with standard deviation parameterσ. In

some instances, a variant of this model has used a uniform noise distribution instead of the normal

distribution. When using Gaussian noise, the Noisy Disk defines the distance estimatedij between

nodesi andj in terms of the true distancedij as

dij =

N (dij , σ) dij ≤ dmax

ø otherwise.(2.6)

The connectivity component of the Noisy Disk model is also known as the Unit Disk model of

connectivity.

Researchers generally acknowledge that noise is not perfectly Gaussian or uniform and

that connectivity is not perfectly disk-like. Regardless,the Noisy Disk model is universally con-

sidered to be good enough for the simulation and evaluation of multi-hop localization algorithms

and is ubiquitous in the sensor localization literature. Theoretical analyses have successfully used

the Noisy Disk model to derive the maximum likelihood solution to localization [103], lower

bounds on localization error [13, 81], and specific properties about localization algorithms [60].

The Noisy Disk Model is most commonly used to evaluate and compare algorithms in simula-

tion [1,2,4,21,42,67,84,87]. Several projects have collected empirical ultrasound data [82] or RSS

data [66,90] to derive realistic values for the parametersdmax andσ, which are then used to simulate

the behavior of various localization algorithms. Other studies use these parameters for sensitivity

analysis by, for example, measuring localization accuracywhile varyingdmax from 1.1 to 2.2 times

the average node spacing andσ from 0 to 50% ofdmax or similar values [50,63,79,82].

13

2.3 Physical Range Sensors

Any signal that changes reliably over distance can be used asa range sensor. For example,

magnetic fields can be used to localize objects in three dimensions with millimeter accuracy [71],

although the limited range of a few feet makes it difficult to apply to sensor field localization. The

physics of the sensor determines how closely the resulting range estimates match the Noisy Disk

model of ranging. This section describes the physical principles underlying several types of range

sensors that are particularly well suited for sensor field localization: they all have relatively long

range in roughly all directions and use small, cheap and low-power hardware that require simple

signal processing that can be performed on a sensor node. We do not discuss ranging techniques

that assume a subset of more powerful nodes, such as laser range finding techniques [32,76,100] or

mobile nodes [38,93].

2.3.1 Radio Signal Strength

Radio signal strength (RSS) is the power with which a radio signal is received. If the

transmission power is known, RSS can be used to estimate distance based on a model of signal

attenuation over distance. One such model can be derived from simple principles of physics: as-

suming an isotropic transmitting antenna and a near-ideal environment, the radio signal should

emanate from the transmitter in a sphere. Therefore, signalstrength at a receiver with distancer

from a transmitter is proportional toAr

AswhereAr is the aperture (surface area) of the receiver and

As is the surface area of a sphere with radiusr. More precisely,

RSS =PtAr

(4πr)2(2.7)

wherePt is the transmission power. SinceAr is constant and the area of a sphere is proportional to1r2 , RSS will decrease by a factor of1

4 every timer doubles, i.e. the received power will decrease

by 10 log10(4) = 6dB asr doubles.

The model above assumes acoefficient of attenuationα = 2, based on the rate of growth

of the area on a sphere. However, in reality antennas are not isotropic and RF power does not radiate

in a sphere. Furthermore, a real deployment environment is never ideal, and so the coefficient of

attenuation can be significantly higher than 2. Other more realistic models have been proposed that

combine theory with empirical observation, including Nakagami and Rayleigh [72] fading models.

Perhaps the most commonly used path loss model is the the log-normal model [72], which postulates

14

−5 0 5 10 15 20 25 30 35 40 451.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75Average Signal Strength over Distance

Distance (ft)

Sig

nal S

tren

gth

(V)

SmallError

LargeError

Figure 2.2: Error Increase over Distance depends on bothnoiseand attenuation rate. As theattenuation rate flattens out, differences in signal strength become small relative to noise levels.

logarithmic attenuation over distance and Gaussian noise,as given by

RSS(d) = RSS(d0) + 10α log10(d

d0) + Xσ (2.8)

whered0 is a reference distance andXσ is a Gaussian random variable. In contrast to the theoretical

model, the coefficient of attenuationα in these models is a parameter derived from empirical data.

RSSnoiseis the amount by which RSS can vary at a single distance in a particular en-

vironment and, together with the coefficient of attenuation, determines the overall ranging error of

RSS. Because RSS attenuates at an ever decreasing rate, the difference in signal strength between

1m and 2m will be equal to or larger than the difference between 10m and 20m. Thus, as distance

increases, changes in signal strength due to distance become small relative to noise, even if the level

of noise remains the same over distance. A constant level ofnoisetherefore results in ever increas-

ing error when signal strength is used to estimate distance; if RSS noise is sufficient that we cannot

tell the difference between 1 and 2m, we also cannot tell the difference between 10m and 20m. This

effect is illustrated in Figure 2.2, which shows how noise at5–10 foot distances translates into small

error, while similar noise levels at 20–40 foot distances translates into large error.

In many ways, RSS ranging is the ideal range technology for wireless sensor networks:

15

it requires no additional hardware and almost no computational costs. However, even with a sin-

gle pair of stationary nodes in a stationary environment, RSS is subject to high levels of noise.

Furthermore, individual radios can vary significantly in both transmission strength and receptivity,

especially in low-power radios [34, 95], and the effect of reflectors and attenuators can dominate

the effect of distance on RSS, giving RSS a reputation for being extremely “noisy” and unsuitable

for multi-hop localization [10, 31, 87]. Although projectssuch as RADAR have used RSS with

RF profiling, most empirical studies that use RSS directly for range estimation have yielded in-

conclusive or negative results, even outdoors. One study explored RSS ranging outdoors in both

an open and a heavily wooded environment using two 802.11 nodes, but only promised a stan-

dard deviation of error near 50% of the range at best [90]. RSSranging was shown to be effective

for indoor localization to within 1.8m in another study, butonly when the nodes had a 2-3 meter

spacing and RSSI was measured using the Berkeley VaritronixFox receiver, a high-fidelity Wi-Fi

propagation analyzer [66]. The low-power radios that are common in sensor networks are even

more difficult to use for ranging. Several studies that characterized RSS data using low power ra-

dios decided not to use these radios for localization [34, 40] or later rejected RSS in favor of other

ranging technologies [83, 98]. Today, for real deploymentsthat require sensor field localization,

more costly alternatives to RSS such as acoustic, RF time of flight, or laser are being developed to

localize nodes outdoors in open spaces with only 10 or even 2 meter spacing [32, 48, 68, 82, 88].

This reflects a general lack of confidence in RSS ranging in thecommunity, although no conclusive

results have shown RSS ranging to be impossible. We demonstrate in a separate study that RSS

can indeed be used for multi-hop, sensor field localization and can even achieve results comparable

to GPS, achieving near 4m average accuracy on a 49 node 50x50mnetwork [97]. However, this

was only possible by using the techniques described in subsequent chapters to create a predictable

deployment environment.

2.3.2 Acoustic Time of Flight

The time of flight (TOF) of an acoustic signal is the difference between transmission time

tt and receive timetr. TOF can be multiplied by the speed of sound to infer the distance between

the transmitter and receiver according to the following equation

dij = (tr − tt) ∗ c (2.9)

wherev is approximately331.5(0.6θ)m/sec andθ is the temperature in degrees Celsius. TOF can

be measured in two common ways. If the transmitter and receiver are operating in the same time

16

base, the transmitter can send an acoustic pulse at a known time and the receiver can simply observe

the time at which it is received [27]. In a sensor field, this technique requires time synchronization

between nodes, which has been shown to be accurate to within microseconds, although often at a

significant cost in bandwidth and/or energy [59]. Alternatively, the transmitter could send an acous-

tic pulse and a radio message simultaneously [70]. The RF pulse arrives within 10s of nanoseconds

and its reception time is a reasonable estimate oftt for short distances, and the receiver can measure

the time difference of arrival (TDOA) of the two signals to estimate the true TOF.

As a ranging technique, acoustic TOF is generally more robust to environmental influ-

ences than RSS because attenuation and reflection of the signal does not affect the TOF of the line

of sight signal; it only affects the volume with which it is received when it arrives. The beginning

of a weaker signal may be more difficult to detect, however, and weaker signals may therefore have

higher error on average. One way to avoid this problem is to modulate the outgoing signal and mea-

sure thephaseof the received signal, allowing one to infer the arrival time of beginning of the signal

even if it is not directly observable. Girod demonstrated that if the signal is not self-correlating, this

modulation technique not only provides increased precision upon detection, but also robustness to

multi-path reflections and interference from other transmitters, achieving 5 centimeter accuracy in

environments as adverse as a forest, at distances of 10s of meters [28]. As with almost any ranging

technique, however, acoustic TOF can always yield very higherrors when the line of sight signal is

blocked but a reflection of the signal is not.

Because of its increased robustness, acoustic TOF has been used with more success in both

single- and multi-hop localization than RSS. Many early single-hop localization systems such as

AT&T’s Active Bats [30] and MIT’s Cricket [70] used ultrasonic TOF, as do many robotic systems

including CMU’s Millibots [62] and the popular Pioneer robot series [22]. More recently, UCLA’s

AHLoS [82] localization system and a similar system by UIUC [48] are using acoustic TOF towards

multi-hop, sensor field localization. UCLA’s Acoustic ENSBox [28] uses wideband acoustics in the

audible frequencies to localize nodes even in the presence of obstacles such as trees.

2.3.3 Interferometric Ranging

Although radio signal strength and ultrasonic ranging are most commonly used and cited,

a new ranging technique has recently been proposed and showspromise for wireless sensor net-

works. In Radio Interferometric Ranging, two non-colocated nodesA andB transmit radio signals

at different frequenciesf1 andf2, such that the difference between them is small|f1−f2| << f1, f2.

17

These signals interact to create a low beat frequency which can be detected at receiversB andC.

The phase will be different at each receiver, however, and the relative phase offset will be

2πdAD − dBD + dBC − dAC

c/f(mod2π) (2.10)

wherec is the speed of light andf = (f1 + f2)/2. In other words, the phase difference between

the signals received atC andD is a function purely of the four distances among the four nodes.

Repeating this process with other sets of four nodes createsan over-constrained system from which

the distances between all nodes can be derived.

Because the phase at two different nodes must be compared, this system requires precise

time synchronization in the network. Also, Equation 2.10 assumes a line of sight signal from both

transmitters to both receivers. Any interference from other transmitters or from reflected signals can

change the phase and even the beat frequency observed at the receiver and can cause large errors.

Even distant objects in the environment can therefore be an obstacle for this technique, since the

range of interferometric ranging has been shown to be much larger than even the effective radio

range [47]. In theory, errors due to multi-path are detectable because each distance is being mea-

sured multiple times; since the multi-path effects will be different for each pair of transmitters and

receivers, erroneous range estimates will be inconsistentwith other estimates and can be eliminated.

This technique for dealing with multi-path errors, however, has not yet been demonstrated.

Without multi-path problems, this technique has been shownto produce range errors with

a standard deviation of error near 3 centimeters over rangesof up to 160 meters with the Chipcon

CC1000 radio, which is used with the mica2 platform. This technique therefore combines the long

range obtained with radio signal strength and the high accuracy obtained with ultrasonic ranging

without adding significant hardware or computational coststo the sensor nodes.

2.3.4 RF Time of Flight

Another promising technique for sensor networks is RF time of flight. Historically, RF

time of flight has been reserved for systems like GPS that can achieve accurate time synchronization

between multiple nodes in an open, outdoor environment. Ultra-wideband (UWB) radios have

recently been demonstrated to remove both of these requirements to some degree [53]. By using

round-trip time, participating nodes do not need to be synchronized in time, and the short duration

of a UWB signal allows the line-of-sight signal to be identified from the midst of reflected signals

because it is the first signal to arrive.

18

More recently, Lanzisera [51] demonstrated TOF on a 2.4GHz radio with requirements

compatible with IEEE 802.15.4 radios, which are often used in sensor networks. NodeA repeat-

edly modulates a code that is not self-correlated, similar to the acoustic TOF system by Girod [28].

Another nodeB receives this signal, buffers it, and retransmits what was received back toA. Be-

causeA andB are not time synchronized,B will not likely begin receiving at the very beginning of

the transmission, but at some arbitrary point during the first cycle of the code. Nonetheless,A can

measure the phase offset between the original transmissionand the signal received fromB, which

should be exactly the time of flight of the radio signal. Repeating this process at multiple frequen-

cies and averaging the resulting range estimates can help reduce the impact of systematic errors due

to multi-path reflections. A prototype system was able to achieve RMS error between 1 and 3m in

environments including hallways and a coal mine.

2.4 Localization Algorithms

There are currently a large number of ranging-based localization algorithms in the litera-

ture, each of which uses a different heuristic to infer node locations based on range estimates. Some

algorithms assume that a network can be decomposed and localized as several sub-networks; other

algorithms assume that range estimates can be added together to create longer range estimates; other

algorithms assume that multi-dimensional coordinates canbe projected onto a two-dimensional

space. Each of these approximations greatly simplifies the sensor localization problem. In this

section, we provide an overview of six representative algorithms.

Most multi-hop localization algorithms, including the sixthat we discuss, fall withing two

main classes of approximations: theshortest-pathand thepatch and stitchapproximations.Shortest

path algorithms approximate the distance between two non-adjacent nodes to be the shortest path

distance through the ranging graphG. For example, if nodei does not have a direct range estimate

dij to nodej, it may use asumof the range estimates through nodesk andm: dik + dkm+ dmj . This

sum constitutes a multi-hop range estimate that is a weak approximation ofdij . The shortest-path

approximation is that the shortest of all such multi-hop range estimates can be considered equal to

a true range estimate.

Shortest paths distances can be efficiently calculated in the network using a distance vector

algorithm similar to those used in routing. All shortest path distances are initialized to infinity:

spij = ∞, ∀i, j and one nodei initiates the algorithm by transmitting a shortest path update message

spii = 0. Every nodej that hears an update message sets its own shortest path distance toi to be

19

the minimum sum over all neighborsk of its ranging estimate tok and the current shortest path

betweenk andi.

spji = mink

djk + spki (2.11)

Whenever its shortest path toi improves, nodej notifies its neighbors with another update message

containing the new valuespji and the algorithm repeats. Each run of the algorithm only calculates

shortest path distances between all nodes andi, the node that initiated the algorithm. Thus, if an

algorithm requires shortest paths to all anchors, the algorithm must be initiated by each anchor

individually.

Once the necessary shortest paths have been created, each algorithm uses them in a dif-

ferent way. TheBounding Boxalgorithm constrains the location of nodei to be withindij of node

j’s x or y coordinates. This constraint can be represented as a box with j in the center and edge

length2dij . If such boxes are formed around multiple anchor nodes, the position of nodei is con-

strained to be within the intersection of these boxes. TheDV-Distancealgorithm is very similar

except that it defines a circle around nodej with radiusdij . Instead of constraining the location

of nodei to bewithin this circle, DV-Distance constrains the position to beon the circle. The in-

tersection of multiple such circles defines the location node i. TheMDS-Mapalgorithm uses the

shortest path distances between all nodes in the network to form a similarity matrix, which indicates

how close each node is to every other node. This matrix is thenused to compute the positions us-

ing Multi-dimensional Scaling (MDS), a statistical technique that embeds a set of data points in a

multi-dimensional space.

Patch and stitchalgorithms divide the network into small patches that are localized indi-

vidually with respect to a local coordinate system. Typically, the algorithms form a patch around

each nodei consisting of all neighborsNi ⊂ V wherej ∈ Ni ⇐⇒ eij 6= ø. The nodes in

the overlapNij = Ni ∩ Nj between patches for nodesi andj have two coordinates, one in the

coordinate system ofi and one in that ofj. These coordinates can be used to derive a coordinate

transform between the coordinate systems of the two patches, thereby providing the relative loca-

tions ofi andj. The relative locations of non-neighbor nodesi andj can be calculated by cascading

transforms from multiple overlapping patches betweeni andj, or a global stitching order can be

used to localize all nodes within the same coordinate system.

Once the patches are defined, each algorithm uses a differenttechnique to localize them.

The GPS-freealgorithm uses a process callediterative localizationas its patch localization algo-

rithm. In this process, three nodes are assigned initial coordinates in an arbitrary coordinate system.

20

Figure 2.3:Localization Ontology The six multi-hop algorithms that we implemented and analyzedare shown in terms of the ontology of localization problems provided in Section 2.1. We do notanalyze the single-hop algorithms.

These three nodes are used to localize a fourth node. The fournodes can be used to localize a

fifth, and so on. TheRobust Quadsalgorithm is very similar except that it limits each step in the

iterative process to localizing only those nodes with a low probability of localization error. The

MDS-Map(P)algorithm finds the shortest paths between all nodes in a patch and uses MDS to lo-

calize the entire patch at once. As such, MDS-Map(P) uses both the shortest path and the patch and

stitch approximations.

MDS-Map(P) specifies a global stitching order with which thecoordinate systems of all

patches can be transformed into a global coordinate system.The set of stitched patchesS is ini-

tialized to the largest patchS = argmaxi|Ni|. The set of un-stitched patches is set to be all other

patchesS = V − S. At each step, the next patch to be stitched is determined to be the patch inS

with the largest overlap with any patch inS

argmaxi |Ni ∩ Nj | i ∈ S, j ∈ S (2.12)

All six algorithms are shown in Figure 2.3 in terms of the traditional taxonomy of local-

ization algorithms from Section 2.1. All of the algorithms are multi-hop localization algorithms,

meaning that nodes are not assumed to have a range estimate tothree or more anchor nodes. Only

MDS-Map is a centralized algorithm because it needs all range estimates at once. All other algo-

21

rithms are distributed algorithms. Bounding Box and DV-distance are absolute localization algo-

rithms that require at least three anchor nodes in the network and localize all nodes with respect to

the coordinate system that they define. All other algorithmslocalize nodes relative to each other in

a unique but arbitrarily defined coordinate system.

2.5 Model Verification

In the previous two sections, we described both the standardtheoretical model of ranging

and the physics of several common ranging techniques, but have not yet established any relationship

between them. In this section, we explore previous studies in both ranging and localization for

evidence of a verified relationship between the Noisy Disk model and a common ranging technique.

We look specifically for two different types of verification:bottom-up and top-down.

In bottom-upmodel verification, a researcher collects empirical ranging data using a

range sensor and verifies through inspection that its structure is similar to the hypothesized model.

Bottom-up verification can be performed through formal statistical tests. For example, an assump-

tion of Gaussian noise can be tested with the Jarque-Bera test of Normality [6]. Data that would

be used for these tests is typically collected from a single transmitter and receiver pair which are

placed at multiple different distances.

Top-downmodel verification compares the effects of empirical ranging data on localiza-

tion to the effects of a theoretical model, i.e. it defines equivalence to be in terms of the particular

usage of the data. If the empirical data yields the same localization results as the theoretical model,

the model is assumed to be sufficient. Like bottom-up verification, top-down verification can also

be performed through formal statistical tests, such as thet-test. This type of verification has the

benefit of testing not only whether all of the properties of the Noisy Disk model are exhibited by

the empirical ranging data, but also the reverse: whether all of the properties of the empirical data

that affect localization error are captured by the Noisy Disk model. As such, top-down model veri-

fication is more convincing than bottom-up model verification. However, In multi-hop localization,

bottom-up verification is much more common than top-down model verification because researchers

are much more likely to characterize a ranging technology than to use it in a large-scale localization

deployment.

Table 2.1 summarizes selected existing studies along with both their usage of and their

verification of the Noisy Disk model. Columns 1-3 indicate whether the study performed a localiza-

tion simulation, and whether that simulation relied on the Gaussian noise and Unit Disk connectivity

22

models. The 4th column indicates whether the study actuallycollected empirical ranging data. The

6th and 8th columns indicate whether the study used the data to estimate parameters for Gaussian

noise or Unit Disk connectivity models, and the 5th and 7th columns indicate whether the study first

performed any formal tests to verify the assumptions of these models before estimating their param-

eters. Finally, columns 9-11 indicate whether the study performed a localization study using real

hardware, and whether it compared the results of this study to a simulation as a means of top-down

verification of the Gaussian or Unit Disk model.

This table shows that almost no studies performed any verification of the Gaussian or

Unit Disk models (columns 5, 7, 10, and 11), even though everysingle study assumed these models

to some extent (columns 2, 3, 6, 8). The only study that does not appear to assume the Noisy

Disk model is by Simic [91] because it does not evaluate the algorithm that it proposes either in

simulation or on real hardware, nor does it collect empirical ranging data. The derivation of the

algorithm, however, does assume Unit Disk connectivity. Similarly, only one study by Stoleru [92]

performed top-down verification of the Unit Disk model of connectivity, producing a negative result:

connectivity was not sufficiently disk-like to produce results in the real world similar to predictions

in simulation.

2.5.1 Simulation-based Studies

Even though the Noisy Disk model has not been verified, most localization studies use the

Noisy Disk in simulation to evaluate or compare localization algorithms. This is evident from the

fact that nearly all localization studies that use simulation (column 1) also use both Gaussian noise

(column 2) and Unit Disk connectivity (column 3) in that simulation.

Reliance on the Noisy Disk is perhaps most evident in algorithms that explicitly depend

on its particular assumptions and parameterization. For example, in 2001 Doherty proposed an

analytical solution to localization using semi-definite programming by assuming an upper bound

on the distance between two connected nodes, relying on the strict assumption that no node would

underestimate the distance to another node [17]. In 2004, Biswas modified this algorithm by also

assuming that range estimates do not overestimate the distance between two nodes [9]. Both of

these algorithms were evaluated using the connectivity models that they assume.

The only four algorithms that do not use Gaussian Noise in their simulations [10,17,31,79]

are evaluating algorithms that are based on hop count, whichdoes not have a noise component.

Another three algorithms indicated with footnotes in Table2.1 do not use Gaussian noise but also

23

Study Name Simula

tes Lo

caliz

ation

Uses Gau

ssian

Noise

Uses Unit

DiskCon

necti

vity

Collec

tsEm

pirica

l Ran

ging

Data

Botto

m-u

pGau

ssan

Verifi

catio

n

Estim

ates

Gauss

anPar

amet

ers

Botto

m-u

pNois

y DiskVe

rifica

tion

Estim

ates

Noisy Disk

Param

eter

s

Collec

tsLo

caliz

ation

Data

Top-

down

Gauss

ianVe

rifica

tion

Top-

down

UnitDisk

Verifi

catio

n

Bounding Box [91] ✖ – – ✖ – – – – ✖ – –Convex [17] ✔ – ✔ ✖ – – – – ✖ – –Hop-terrain [79] ✔ – ✔ ✖ – – – – ✖ – –MDS-Map [87] ✔ ✔ ✔ ✖ – – – – ✖ – –MDS-Map(P) [86] ✔ ✔ ✔ ✖ – – – – ✖ – –GPS-free [12] ✔ ✔ ✔ ✖ – – – – ✖ – –TPS [1] ✔ ✔ ✔ ✖ – – – – ✖ – –Fading [7] ✔ ✔ ✔ ✖ – – – – ✖ – –Bits [82] ✔ ✔ ✔ ✖ – – – – ✖ – –Semidefinite [9] ✔ ✔ ✔ ✖ – – – – ✖ – –Anisotropic [56] ✔ ✔ ✔ ✖ – – – – ✖ – –Comparison [50] ✔ ✔ ✔ ✖ – – – – ✖ – –APS [63] ✔ ✔1 ✔ ✖ – – – – ✖ – –Anchor-free [69] ✔ ✔1 ✔ ✖ – – – – ✖ – –Scaling [41] ✔ ✔1 ✔ ✖ – – – – ✖ – –APIT [31] ✔ – ✖ ✖ – – – – ✖ – –SpotON [34] ✖ – – ✔ ✖ ✔ ✖ ✖ ✖ – –Robust [26] ✖ – – ✔ ✖ ✔ ✖ ✖ ✖ – –RF-tof [52] ✖ – – ✔ ✖ ✔ ✖ ✖ ✖ – –Acoustic [78] ✖ – – ✔ ✖ ✖ ✖ ✔ ✖ – –Quantized [67] ✔ ✔ ✔ ✔ ? ✔ ✖ ✖ ✖ – –Geolocation [8] ✔ ✔ ✔ ✔ ✖ ✔ ✖ ✔ ✖ – –Millibots [61] ✔ ✔ ✔ ✔ ✖ ✔ ✖ ✔ ✖ – –Sichitiu [90] ✔ ✔ ✔ ✔ ✖ ✔ ✖ ✔ ✖ – –Dynamic [83] ✔ ✔ ✔ ✔ ✖ ✔ ✖ ✔ ✔2 ✖ ✖

Context-aware [45] ✖ – – ✔ ✖ ✔ ✖ ✔ ✔2 ✖ ✖

Relative [66] ✔ ✔ ✔ ✔ ✖ ✔ ✖ ✔ ✔2 ✖ ✖

Robust Quads [60] ✔ ✔ ✔ ✖ – – – – ✔2 ✖ ✖

GPS-less [10] ✔ – ✔ ✔ – – ? ✔ ✔2 ✖ ?Time & Space [25] ✖ – – ✖ – – – – ✔2 ✖ ✖

Resilient [48] ✔ ✔ ✔ ✔ ✖ ✔ ✖ ✔ ✔ ✖ ✖

Aensbox [28] ✖ – – ✔ ✖ ✔ ✖ ✖ ✔ ✖ ✖

Prob Grid [92] ✔ – ✔ ✖ – – – – ✔ – ✔

Table 2.1: that perform localization simulations, collect ranging data, or collect localizationdata. This table indicates whether each study a) assumes andb) validates the Noisy Diskmodel. For each column,✔ indicates “true” and ✖ indicates “false”, – indicates that thecolumn is not applicable, and ? indicates an inconclusive result.

24

do not propose a more realistic, validated noise model. Instead, they use uniformly distributed noise.

APIT is the only study that does not use the Unit Disk model of connectivity. The algo-

rithm is explicitly designed to handle non-disk like connectivity, and uses an irregular radio connec-

tivity model which later became the basis for the Radio Irregularity Model (RIM) [102]. RIM is a

model of radio characteristics that has been derived through a bottom-up verification process. By

using the model to evaluate a localization algorithm, APIT is assuming that, because of this bottom-

up validation, RIM also satisfies the more demanding top-down validation requirements. Because

the simulation results were not compared to a real deployment, however, this has not been verified.

Several studies that evaluate a localization algorithm with the Noisy Disk model actually

derive the model parameters from empirical ranging data. For example, Savvides evaluated the col-

laborative multilateration algorithm in simulation usingparameters derived from ultrasound ranging

data [82], Patwari evaluated a maximum likelihood algorithm using parameters derived from RSS

data [66], and Sichitiu evaluates an algorithm much like iterative multi-lateration using parameters

derived from 802.11 wireless nodes [90]. However, as we willsee in the next section, the data sets

from which the parameters are derived are not verified to conform to the Noisy Disk model.

2.5.2 Ranging Characterization Studies

Almost all studies that characterize a new range sensor, shown in column 4, do so in terms

of both Gaussian noise and Unit Disk connectivity (columns 6and 8). Most such studies, however,

do not use formal statistical tests to validate that the empirical data actually conforms to the Noisy

Disk model (columns 5 and 7).

Only two ranging characterization studies do not estimate standard deviation of noise. The

first does not characterize noise at all, although it does state that the data appears to be Gaussian

distributed [78]. The second is only characterizing connectivity [10]. Similarly, five studies do

not explicitly assume the Unit Disk model of connectivity byestimating maximum range, but also

do not propose a better model of connectivity. Instead, theydo not characterize connectivity at

all. Most of these studies do implicitly assume disk-like radio connectivity, however, by fitting the

empirical data to a RF attenuation model of the form

RSS = a − 11.4688 · b · 10 log10(r) (2.13)

1These simulations used a variant of the Noisy Disk that assumes uniform, not Gaussian, noise.2All nodes in these deployments were within a single hop, and so key aspects of the algorithms may not have come

into effect.

25

Patwari is the only study that attempts to verify the Gaussian noise assumption by plotting

the data in anormality plot, which visually identifies deviations from the Normal distribution [67].

This shows that the radio signal strength data collected in this study appears to be Normally dis-

tributed. However, all of the data was collected with a single transmitter, a HP 8644A signal gener-

ator, and a single Berkeley Varitronics Fox high-fidelity WiFi receiver. The effect of other transmit-

ters and receivers on the noise distribution is therefore unknown, and the results of this verification

are therefore indicated in Table 2.1 as inconclusive.

Bulusu is the only localization study that attempts to verify the Unit Disk model of con-

nectivity by placing a 418MHz Radiometrix radio transmitter in the corner of a grid and measuring

packet reception rates at the other grid positions [10]; 68 of the 78 grid positions measured exhibited

packet reception rates that matched the predictions of the Unit Disk model. All 67 grid positions

within rangedmax were measured and only one of these exhibited lower than expected packet re-

ception. However, only 11 of the 33 grid positions beyonddmax were observed, and 9 of these

11 produced higher than expected packet reception. The other 22 grid positions, if measured, may

or may not have verified the Unit Disk model of connectivity. Furthermore, this experiment was

performed with a single transmitter/receiver pair, reusing the same receiver node at every grid posi-

tion. Therefore, this result is listed as inconclusive in Table 2.1. Later experiments in this study do

appear to have enough data to either confirm or deny the Unit Disk model of connectivity, although

an analysis of the data is not provided.

Other studies from the wireless networking community that characterized similar radios

indicate that the assumptions of Unit Disk model of connectivity do not hold [23, 101]. Unlike

earlier studies that use a single transmitter/receiver pair, Ganesan uses 147 different nodes in a

grid formation and each node acts as both a transmitter and receiver and Zhao uses a single trans-

mitter and up to 60 different nodes in a line as simultaneous receivers. Both of these studies verify

nearly complete connectivity at short distances and nearlyno connectivity at large distances, but also

demonstrate the existence of a largetransitional regionin between, in which levels of connectivity

can be highly variable [104]. These studies show that the transitional region can occupy over 50%

of the radio range, directly contradicting the assumption of Unit Disk model that the transitional

region is negligible. The reason the transitional region isso large was not conclusively identified in

these studies, although independent studies have documented significant differences between sepa-

rate transmitters and receivers in radios [34] and ultrasound [95]. This factor is commonly believed

to be at least part of the cause of non-disk like connectivitywhen multiple transmitters and receivers

are being used in the same experiment.

26

2.5.3 Localization Deployment Studies

Studies that actually perform a localization deployment can compare the observed lo-

calization error with that predicted by simulation for the purpose of top-down model validation.

However, very few localization deployments have been performed and, of those, even fewer are

compared with predictions from simulation.

In most cases, the deployments are small enough that key aspects of the multi-hop local-

ization algorithms are not playing a key role. These cases are noted in Table 2.1 with a footnote.

For example, some deployments used four anchor nodes and only a single mobile node [10,45,83].

Other deployments used multiple nodes all within ranging distance of each other, forming a fully

connected graph for localization [25, 66]. Moore [60] placed all nodes in a cell approximately the

diameter of the maximum range, and the range of the ultrasound device was artificially restricted

in software to limit the number of ranging neighbors that each node could obtain. This artificial

restriction hid the effects of any natural deviation of the range sensor from the Unit Disk model of

connectivity. Furthermore, in these experiments the rangesensors were operating over only very

short distances, which is the most consistent region of operation in terms of noise.

Two of the three deployments that did use multi-hop topologies did not compare the re-

sults with predictions from simulation [28,48]. However, at least one of these still constitute strong

evidence of the Prediction Gap. Kwon et al. estimated the maximum range of their acoustic sensors

to be 22 meters, and accordingly placed 45 nodes in a grid with9.14 meter spacing, a relatively close

distance that would be expected to produce many range estimates and high node degree [48]. After

ranging between all nodes, however, only 35% of the expectedranging estimates were obtained.

This is far less than predicted by the Noisy Disk model and casts doubt on the Unit Disk model of

connectivity. Because of this, the empirical deployment produced poor localization results and the

authors needed to augment the observed range estimates withsimulated estimates in order for the

localization algorithm to work.

To our knowledge, Stoleru et al. have produced the only studybesides our own that

evaluates a localization algorithm in both simulation using the Noisy Disk model and on a physical

sensor network [92]. This study did not evaluate ranging-based localization algorithms; it evaluated

both the Probability Grid and APS DV-Hop connectivity-based algorithms [63]. Even though the

authors hand-calibrated the radios to make the empirical connectivity characteristics as ideal and

disk-like as possible, a comparison revealed that the empirical localization error was up to eight

times worse than predicted by simulation. The authors do notexplain this discrepancy in the study,

27

and indicated that it requires further research. Nonetheless, this result is a strong indication that

the Noisy Disk model does not withstand the tests of top-downvalidation, and provides concrete

evidence for what we call the Prediction Gap.

28

Chapter 3

Establishing the Prediction Gap

On the way to identifying why there is such a large Predictiongap in sensor field local-

ization, we must first establish the gap ourselves. In this chapter, we establish the Prediction Gap by

implementing and deploying a distributed, multi-hop localization system and comparing observed

localization error with predictions by the Noisy Disk model.

Our implementation builds upon and improves some of the besthardware designs and al-

gorithms from existing systems to create a unified system that is designed specifically for the sensor

field localization problem. The goal of this system is not to innovate in the area of localization, but

rather to incarnate the canonical system that underlies thetheory and assumptions in the multi-hop,

ranging-based localization literature.

We describe each stage of the system design and implementation, including hidden chal-

lenges and necessary innovations. In Section 3.1 we describe the ultrasonic ranging hardware,

which combines ideas from several existing ultrasound implementations. We then describe a non-

linear noise filter in Section 3.2 that is designed to reduce the asymmetric noise profile of ultrasonic

ranging. In Section 3.3, we describe our calibration techniques. In Section 3.4, we innovate a new

collision detection scheme to eliminate error due to ultrasound collisions. We describe the localiza-

tion algorithm in Section 3.5, and techniques for a distributed implementation of it in Section 3.6. In

Section 3.7, we describe the incremental process of developing and debugging this system through

simulation, small wired testbeds, and finally real world test environments.

In Section 3.8, we deploy this system using a 49 node network.In Section 3.9, we com-

pare the observed localization error with that predicted bythe Noisy Disk model of ranging. To

ensure a fair comparison, our deployment takes place in an ideal, open outdoor environment, we

use the same topology for the deployment and simulation, andwe derive the parameters for our

29

deployment and simulations from the same empirical data set.

3.1 The Ranging Hardware

The first step to localizing our entire network is to design a simple range sensor that can

estimate the distance between two nodes. Our ultrasonic ranging hardware combines and improves

ideas from several previous ultrasound implementations. Our ultrasonic transducer circuitry is de-

rived from that of the Medusa node [82], which uses 8 ultrasound transducers oriented at different

angles, 4 for transmission and 4 for reception. Our circuitry is similar to the Medusa except that

we add a switchable circuit so that a single transducer can beused to both transmit and receive.

Our nodes measure ultrasonic time of flight by transmitting the acoustic pulse simultaneously with

a radio message so that receivers can measure the time difference on arrival (TDOA) as described in

Cricket [70]. When the transducers are face to face, our implementation can achieve up to 12m range

with less than 5cm standard error. Comparable implementations were able to achieve proportionally

similar results of 3–5m range with 1–2cm accuracy [61, 82, 83]. The differences in magnitude are

due in part to our design decision to reduce the center frequency of the transducer from the standard

40kHz to just above audible range at 25kHz, which increases both maximum range and error.

Ultrasound transducers are highly directional, and small variations from a direct face-to-

face orientation can have large effects on error and connectivity. Two solutions have been proposed

to use ultrasound in multi-hop networks: aligning multipletransducers outward in a radial fashion

[83] or using a metal cone to spread and collect the acoustic energy uniformly in the plane of the

other sensor nodes [61]. We implemented the latter solutionas shown in Figure 3.1 in order to avoid

possible variations in range at different angles from the transducers. In this configuration, our nodes

achieve about 5m range and 90% of the distance estimates are within 6.5cm of the true distances.

The ultrasound transducer is connected to an Atmel Atmega8 1MHz micro-controller

which is used for both transmitting and receiving ultrasound signals. The output of the transducer

is wired to the analog comparator on the micro-controller for detecting incoming signals through

simple threshold detection, and the value of the threshold can be controlled in software through a

digital potentiometer. The input of the transducer is wiredto a pulse width modulator (PWM) on

the Atmega8, which directly keys the 25KHz signal. Both the transducer and the micro-controller

are mounted as a daughter board which is attached to the Mica2Dot mote. Because the radio and

ultrasonic transducer are controlled by different micro-controllers, a single interrupt line is used

for precise time synchronization between them and the two micro-controllers exchange timing and

30

Figure 3.1:The Ultrasound Ranging Hardware is shown here. The white enclosure contains amica2dot and battery and supports a reflective cone above theultrasonic transducer, which pro-trudes from the top.

31

50 100 150 200 250 300 350 400 450 500

10

20

30

40

50

60

True Distance (cm)

Tim

e of

Flig

ht (

mse

c)

Figure 3.2:Raw Time of Flight Readingsshown here were collected using our ultrasound hard-ware and the data collection process described in Section 4.2.2.

ranging information through an I2C communication bus.

We characterized this hardware by collecting time of flight readings at multiple different

distances using multiple different pairs of nodes in a data collection technique described in more

detail in Section 4.2. The raw data is shown in Figure 3.2.

3.2 Noise Filtering

Any technique for range estimation is susceptible to noise,which can often be reduced

or eliminated by filtering a series of successive ranging estimates that are taken consecutively. For

example, Figure 3.3 shows our raw ToF data after averaging each series of 10 consecutive readings.

Averaging the data significantly reduces noise. However, the mean is known to be highly suscep-

tible to outliers; data points extremely far from the mean can skew the mean until it is no longer

representative of the series. Such outliers are common in ultrasound ranging.

The type of noise found in time of flight range estimates is very structured, as illustrated

by the time series of range estimates shown in Figure 3.4 thatwere taken between two nodes that

32

50 100 150 200 250 300 350 400 450 500

10

20

30

40

50

60

True Distance (cm)

Tim

e of

Flig

ht (

mse

c)

Figure 3.3: Averaging Ranging Data over a time series that is collected at the same time cansignificantly reduce noise with respect to the raw ToF data. However, non-Gaussian outliers makethe mean less effective than the non-linear filters demonstrated in Figure 3.5.

33

0 20 40 60 80 100 120 140 160 180 2000

15

30

45

60

75

90

105Filtered Time of Flight Estimates

Time (sec)

Dis

tanc

e E

stim

ate

(cm

)

Raw Distance EstimatesFiltered Distance Estimates

False positives

Outliers

Normal Noise

Figure 3.4:The MedianTube Filter chooses the minimum value within a small range of the median.This eliminates outliers and false positives and exploits the fact that the signal is often detected justafter, and rarely just before, it arrives.

are one foot apart. First, the data has outliers on both the positive and negative end of the noise

distribution. The negative outliers are false positives; they represent detections of ultrasound before

the ultrasound actually arrived, possibly due to ambient noise. The positive outliers are detections

that do not occur until well after the signal has arrived, possibly because the incoming signal has a

low amplitude.

Besides the outliers, the rest of the points are fairly well concentrated. Due to the nature

of time of flight, the lowest of these values is most likely to be the correct distance estimate; an

ultrasound pulse is very likely to be detected shortly afterit actually arrives, but is very unlikely

to be detected very shortly before it arrives. Because of this asymmetry with ultrasound, we use

a filter calledmedianTube, which reduces a time series of successive ranging estimates to be the

smallest value within some pre-defined range of the median value of the series. In other words, the

filter first removes outliers which are too far from the medianvalue, and then chooses the smallest

of the remaining values. The result of the medianTube filter on the data in Figure 3.4 with a sliding

window of 20 samples is illustrated with a dashed line.

Even though the median is known for being very robust to a small number of outliers, we

34

found through testing that medianTube performs significantly better than a simple median. This is

because the median cannot detect when a series of readings isdominated by outliers due to a noisy

signal between two nodes. On the other hand, when the medianTube filter identifies most readings

from a series to be outliers, the entire series can be eliminated. Figure 3.5(a) shows the data from

Figure 3.2 after the series of range estimates from each transmitter/receiver pair has been filtered

using a simple median, and Figure 3.5(b) shows the same data after using the medianTube filter.

3.3 Calibration

Calibration is the process of forcing a system to conform to agiven input/output mapping.

This is often done by adjusting the physical devices internally but can equivalently be done by

passing the system’s output through acalibration functionthat maps the actual device response to

the desired response. For our localization system, the actual response is the ultrasonic TOFtij

between the transmitteri and receiverj and the desired response is the distancedij . The calibration

function must therefore be of the form

dij = f(tij, β) (3.1)

Whereβ ∈ ℜp are the parameters that describe the system.

As described in Section 2.3.2, sound travels at a constant rate and multiplying ToF by

the constant value of approximately 340 should convert the time of flight in seconds to the distance

traveled in meters. Our calibration function must be slightly more complex, however, due to several

other factors that affect TOF:

1. A non-zero delayδ between the transmission time of the radio message and the ultrasound

pulse changes the measured ToF by a constant factor.

2. The timesτT andτR required for the diaphragms of the transducers to begin oscillating during

transmission and reception is non-zero, and add two constant factors to the measured ToF.

3. The volume of a transmitterVT and the sensitivity of the receiverSR affects the speed with

which the signal can be detected. This latency can be incorporated as a multiplying coefficient

of ToF because volume decreases with distance.

4. Received volume is also affected by signal attenuators inthe environment such as grass or

carpet and signal reflectors such as walls.

35

50 100 150 200 250 300 350 400 450 500

10

20

30

40

50

60

True Distance (cm)

Tim

e of

Flig

ht (

mse

c)

(a) Median Filter

50 100 150 200 250 300 350 400 450 500

10

20

30

40

50

60

True Distance (cm)

Tim

e of

Flig

ht (

mse

c)

(b) MedianTube Filter

Figure 3.5:The Effect of Filtering on our raw ToF data is shown here using a) a simple medianfilter and b) the medianTube filter. The medianTube filter is much more effective because it canidentify a series of data that demonstrates little self-consistency.

36

5. A difference in transmission frequency and the receiver’s center frequency,|fT − fR|, has a

near-linear affect on the effective received volume.

6. The relative orientations of the sounder and microphone,ΦT andΦR, will affect the volume

with which the acoustic tone is received according to some non-linear functionfO(·).

We therefore arrive at the following complete model of the system response for a trans-

mitter/receiver pair:

dij = δ + τT + τR + VT · tij + SR · tij+|fT − fR| · tij + fO(ΦT ,ΦR) · tij+Attenuationenv · tij (3.2)

To simplify this equation, we collapse all additive terms such asτT andτR into a single

parameterβ1 and all multiplicative terms into a parameterβ2. Thus, all physical aspects of our

system can be modeled by a linear calibration function

dij = β1 + β2 · tij (3.3)

The exact coefficients can be estimated from empirical data to capture average node orientation,

environmental influence, etc. To calibrate the ToF readingsshown in Figure 3.5(b), each measure-

ment is combined with the true distance at which it was measured using the equation above. We can

combine all such equations to form a fully constrained linear system

dij = β1 + β2 · tij (3.4)

dik = β1 + β2 · tik (3.5)

djk = β1 + β2 · tjk (3.6)

...

which can be trivially solved forβ1 andβ2 using least squares. The distance estimates produced

from the ToF readings after calibration are shown in Figure 3.6(a).

Because a single set of parameters are being used for all nodes, this process assumes

that all nodes are the same. Any variations in transmitter strength, receiver sensitivity, or relative

orientations are not captured individually by the parameters, which only represent the average value

of all such variations. Therefore, we call this processuniformcalibration.

37

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

550

True Distance (cm)

Est

imat

ed D

ista

nce

(cm

)

(a) Uniform Calibration

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

550

True Distance (cm)

Est

imat

ed D

ista

nce

(cm

)

(b) Joint Calibration

Figure 3.6:The Effect of Calibration on medianTube-filtered data is shown here using a) uniformcalibration and b) joint calibration. Joint calibration does not do significantly better than uniformcalibration, indicating that node variability is low.

38

In other work, we show that a different parameterization canbe used with the same data

set to estimate linear coefficients for each transmitter andreceiver individually [96]; we must use a

different set of parameters for each nodei andj to create equations of the form

dij = βti1

+ βrj1

+ βti2· tij + βr

j2· tij (3.7)

instead of Equation 3.3. Because these coefficients represent the volume and sensitivity of each

transducer individually, the resulting range errors have been shown to be significantly lower for

some data sets. We call this processjoint calibration. However, we found that joint calibration does

not significantly affect our ultrasound ranging data set, indicating that there is only a small degree

of variation between individual nodes with our ultrasound hardware. The ToF data from Figure 3.5

is shown after joint calibration in Figure 3.6(b).

3.4 Dealing with Collisions

In the previous sections, we describe our techniques for estimating the distance between

two nodes. In this section, we address a problem that occurs when there are more than two nodes:

the ultrasound signals from two transmitters can collide. Pure ultrasonic tones cannot be differenti-

ated, and so an interfering tone from one transmitter will change the estimated TOF from the other

transmitter, causing large ranging errors.

One technique to avoid ranging errors due to collisions is tohave each node encode a

unique signature in the ultrasound pulse through frequencyor amplitude modulation. The receiver

can be assured that the signal has been received without collision if the signature can be accurately

decoded. This even allows the reception of multiple ultrasound signals simultaneously if the signa-

ture can be accurately decoded in the presence of interference from other transmissions [26].

Another solution is to simultaneously send both the ultrasound pulse and a radio message,

such that a radio message is always being sent if an ultrasound pulse is being sent. If the radio signal

can be accurately decoded, the receiver can be assured that no other radio message, and therefore

no other ultrasound pulse, was being sent simultaneously [70]. One advantage of this technique is

that it reuses the modulator/demodulator on the radio instead of building redundant functionality on

the acoustic transducer. It relies on the fact that radio collisions always result in corruption of all

messages involved.

Our ultrasound implementation initially employed the latter of these two techniques, the

RF envelope, because of its simplicity; the ultrasound pulse does not need to be modulated or

39

Figure 3.7:Capture Experiment Setupconsisted of three nodes in an equilateral triangle. NodesA and B both transmit packets that overlap in time while the third node attempts to receive them.

decoded, reducing both memory requirements and power consumption. However, collisions were

still a problem because radio collisions donotalways result in corruption; when two nodes transmit

simultaneously with the FSK radio used on the mica2dot, a third node will sometimes receive one

message completely uncorrupted, even if both are received with almost the same signal strength.

This phenomenon is known ascaptureand can be the cause of very large ranging errors: if two

nodesA andB transmit both RF and ultrasound signals that overlap in timeand the RF signal from

A is received clearly, but the ultrasound pulse fromB arrives first, the TOF estimate will be an

arbitrary value related to the distance of neitherA norB from the receiver.

We performed an experiment of controlled collisions to measure how often this phe-

nomenon might cause ranging errors by generating two radio packets with a precise time difference

∆t. As illustrated in Figure 3.7, two transmittersA andB and a receiver are placed in an isosceles

triangle and the two transmitters are synchronized to transmit at timestA andtB such that the time

between them was∆t = tB − tA. Thus, when∆t is positiveA transmits first and when it is neg-

ativeB transmits first.∆t is varied from−23ms to 23ms at 1ms intervals and 10 collisions were

generated at each value of∆t. The packets are17.9ms long, so0.5ms intervals are used around

40

∆ = 0ms and∆ = ±18ms for higher resolution data. Time synchronization was only accurate to

1ms. While a 0dBm transmit power was used on both nodes, slightlymoving one of the senders

or adjusting the antenna orientation changed the power relationship between the two senders at the

receiver. It was fairly difficult to find a “null” point at which neither transmitter was received due

to the difference in received energy from the transmitters being below the SNR threshold of the

receiver. In fact, we confirmed that the power difference required to cause one transmitter to be

received over the other one was not observable using the 10bit ADC to sample the RSSI pin on

the radio. In our experiments, we deliberately move nodeB such that its signal was stronger than

nodeA at the receiver. Figure 3.8 summarizes the findings from thisexperiment. TheY axis is the

percentage of packets received while theX axis is the∆t between packet start times. In the left half

of the graphsB sends first while on the right half of the graphsA sends first. At the two edges of

the graphs where|∆t| > 17.9ms the messages do not overlap in time and both are received without

corruption. At0 < ∆t < 17ms, A is sent first and, becauseB is stronger, it corrupts the tail end

of A’s message when it arrives. However, when−17ms < ∆t < 0, B is sent first and, due to the

capture effect, is not corrupted byA at all onceA is sent. Therefore, the message fromB is received

without corruption even though it overlaps in time with the message fromA.

This experiment shows that, with the CC1000 radios, the capture effect is quite common

and can cause ranging errors up to 50% of the time. To remedy this situation, we implemented an

application-level collision detection protocol in which each node sends ranging messages in batches

of ten with a small random delay between each message. The maximum random delay is about

10 times the length of the packets, making the probabilitypm of two individual messages colliding

about 110 . Therefore, the probabilitypb thateverymessage in a batch from one node collides with

every message in a batch from another node decreases exponentially as the lengthn of the batch

grows, as

pb = pnm

Using batches allows receiving nodes to detect and discard data from ranging collisions with high

probability: any node that hears messages from two overlapping batches can discard all ranging

messages from both batches.

Using these techniques, all nodes in a network can obtain range estimates to their neigh-

bors in a singleranging phase, in which each node randomly sends ranging messages to all neigh-

bors, which collect the range estimates, filter them, apply acalibration function, and observe the

occurrence of collisions. In the next two sections, we describe how to use these range estimates to

41

Figure 3.8: The Prevalence of Captureis indicated in this experiment. NodesA and B bothtransmit with a time difference of∆t. When−17ms < ∆t < 0, the messages overlap but a thirdnode can still hearB’s message with no corruption.

42

derive the locations of the nodes.

3.5 The Localization Algorithm

After the ranging phase is complete, the network contains the ranging graphG described

in Section 2.1, with distance estimates between each node and its neighbors. This graph must be

used in thelocalizationphase to derive the positions of the non-anchor nodes.

To localize the nodes in our system, we implemented a decentralized version of the Ad-

hoc Positioning System’s (APS)DV-distancealgorithm [63] using TinyOS [55] and nesC [24]. DV-

distance is only one of many ranging-based algorithms that have been proposed for localization, but

we use it in this study because it represents a large class of algorithms that use shortest-path distance

to estimate true distance [80, 82, 87, 91, 92]. DV-distance has also been shown to yield comparable

localization error to the other distributed localization algorithms [50].

DV-distance estimates the distance between a node and an anchor to be the sum of the

distances on the shortest path through the network between them. These shortest path distances

are then used by each node to solve the linear system in Equations 2.4, effectively reducing the

multi-hop localization problem to single-hop localization.

To find the shortest path distances, DV-distance uses a distributed distance vector algo-

rithm. Before the algorithm begins, each node must have range estimates to all of its neighbors and

must initialize its shortest path distance to each anchor tobe∞. Each anchor node initiates a run of

the algorithm by broadcasting a shortest path (SP) message with the following information:

• the source node IDi

• the anchor node IDj

• the anchor node location

• the shortest path distance estimatespij from i to j

When the anchor node initiates the algorithm,i andj are set to the anchor’s ID and the shortest path

estimate is0. Whenever a nodek hears a SP message, it compares its own shortest path estimate

spkj to the sum of the shortest path from the message sender and therange estimate to the sender

spij + dki. If the latter is shorter than the former, the node updates its shortest path estimate and

sends a new SP message with the updated information. In this way, a single message from an anchor

43

triggers an iterative algorithm through which all nodes acquire the anchor node’s position as well as

a shortest path estimate to that anchor.

Shortest path distances, of course, are only estimates of the true distance between a node

and an anchor node; they must route around non-convex network topologies and their ”zig-zag”

nature should always make them longer than the true distance. APS uses acorrection factorto

correct for such systematic biases, exploiting the fact that anchors know both the true distance to

other anchors and the shortest path distance estimate, by sending theratio of these to all nodes

surrounding each anchor. A nodei near anchorj that receives a correction factor for anchork can

multiple its own shortest path estimate tok by the correction factor. This is intended to remove

systematic bias from shortest path estimates, assuming that the same factors that affect the shortest

path fromj to k also affect the shortest path fromi to k, although there is no guarantee.

3.6 Distributed Programming

Sensor field localization can be seen as a distributed programming problem; each node

must perform local operations using data that is resident onother nodes in the network. However,

distributed programming on a sensor network can be difficult, and we require new programming

abstractions to create a DV-distance implementation that can run on sensor nodes.

Traditional abstractions like distributed shared memory or tuple spaces are difficult to

apply to sensor networks because of the unreliable, bandwidth-limited, geographically constrained

communication model. Furthermore, these traditional abstractions are not necessary for most sensor

algorithms, which are typically based on local communication among neighboring nodes; each node

selects a subset of the nodes within radio range, maintains state about them, and shares data with

them. However, this concept of a node and its neighborhood isstill not a programming primitive

in the sensor network community. Neighborhood-based algorithms are typically implemented as

compositions of other more primitive parts such as neighbordiscovery, data caching, and messag-

ing protocols. This can make programming a distributed application like localization challenging.

To facilitate this process, we define a concrete relationship between these concepts in a single uni-

fied programming abstraction calledHood, which allows developers to think about and implement

algorithms directly in terms of neighborhoods and data sharing instead of decomposing them into

lower-level programming abstractions.

A neighborhood is defined with Hood by a set of criteria for choosing neighbors and a

set of variables to be shared. For example, Hood can define a one-hop neighborhood over which

44

light readings are shared and a two-hop neighborhood over which both locations and temperatures

are shared. Once the neighborhoods are defined, Hood provides an interface to read the names

and shared values of each neighbor. Beneath this interface,Hood is managing discovery and data

sharing, hiding the complexity of the membership lists, data caches, and messaging.

Attributesare the elements of a node’s state that are shared with its neighbors, such as

sensor readings or geographic location. When a node updatesits own attribute, the value is said to

be reflectedto its co-neighbors, much like traditional reflective memory (RM) [89]. Exactly how

data is reflected is determined by thepush policy. Typically, this is simply to broadcast the value

once each time it is set, but could also be to broadcast periodically or reliably. When an attribute is

received at a co-neighbor, it is passed through thefiltersof each neighborhood defined on that node.

Filters examine each shared attribute to determine which nodes are valuable enough to place in the

neighbor listand which attributes of those nodes need to be cached. For each node in the neighbor

list, a mirror is allocated, which represents the local view of that neighbor’s state. It contains both

reflections, which are cached versions of that neighbor’s attributes, and scribbles, which are local

annotations about that neighbor. Scribbles are often used to represent locally derived values of a

neighbor such as a distance estimate or link-quality estimate.

A node can define multiple neighborhoods with different variables shared over each of

them, although the members of each neighborhood may overlap. Figure 3.9 shows an example of

a node that is sharing itsMag andLocationattributes. It defines two neighborhoods: theTracking

Neighborhoodconsists of three nodes that haveMag values that exceed some threshold, and with

which it shares both theMag andLocationattributes. TheRouting Neighborhoodconsists of the

eight nodes that are closest geographically, and with whichit sharesthe Locationattribute. The

Receive Link QualityandTransmit Link Qualityare scribbles that are maintained locally about each

node in the Routing Neighborhood.

All data sharing and data caching in our implementation of the DV-distance algorithm is

taken care of by the Hood abstraction. We define aRanging Neighborhoodcontaining all neighbors

to which a node can obtain a valid range estimate. The mirror for each neighbor contains its series

of range estimates, calibration coefficients, and the result of the medianTube filter for that neighbor.

TheAnchor Neighborhoodcontains the four nearest anchor nodes, and the mirror for each anchor

contains its location, a shortest path estimate to that anchor, and up to three anchor corrections

from that anchor node. When a shortest path update message arrives from a neighboring node, the

neighborhood manager checks the Ranging and Anchor neighborhoods to see if a local shortest path

estimate needs to be updated. If so, the new shortest path information is added to the neighborhood

45

Mag Nbr 1 Mag Nbr 2 Mag Nbr 3 Rte Nbr 1 Rte Nbr 2 ... Rte Nbr 8

Mirrors

Tracking Neighborhood

Mirrors

Routing NeighborhoodAttributes

Mirrors

Mirrors

MirrorsMirrors

MagMagMagMag

Loca-

tion

Loca-

tion

Loca-

tion

Loca-

tion

Loca-

tion

Loca-

tion

Loca-

tion

Rx Link

Quality

Quality

Rx Link

Quality

Quality

Rx Link

Quality

Tx

Quality

...

...

...

- Attr -- Attr -

- Attr -- Attr -- Refl- - Refl- ReflRefl -Refl -

Refl -Refl - - Refl -- Refl -- Refl- Refl - Refl- Refl

- Scrib -- Scrib - - Sc- Sc

- Sc- Sc

Figure 3.9: The Hood Programming Abstraction provides a high-level interface forneighborhood-based data sharing. In this figure, theTracking Neighborhoodon the left containsthree nodes with which this node sharesMag values andLocationvalues. TheRouting Neighbor-hoodon the right contains eight nodes.

46

and automatically shared with all other neighbors.

3.7 Implementation and Debugging

Our implementation of the DV-distance algorithm runs in four fully decentralized phases.

To initiate each experiment, the network is flooded with parameters such as transmission power and

calibration coefficients. The four nodes in the corners of the network are designated as anchor nodes

and are given their true positions, at which point they initiate aranging phasein which all nodes

estimate the distance to each of their direct ranging neighbors. The anchors then initiate ashortest

path phase, in which anchors initiate the distributed shortest path algorithm described above. Then,

the anchor nodes initiate aanchor correction phase, in which anchor correction factors are broadcast

in a regional flood. When all flooding is complete, each node estimates its own position in the

localization phase. The phase transitions are initiated by the anchor nodes, which listen to network

traffic to determine when each preceding phase is ending. From the time that the anchor nodes

are given their positions, the entire process is automated with no human intervention or central

computer and completes in less than five minutes for each deployment. During each experiment,

a laptop eavesdrops on the network to reveal current progress and, afterward, an automated script

retrieves all ranging estimates, shortest paths, and estimated locations that were stored in RAM on

the nodes.

Development of this system took place in several phases. First, it was debugged on a large

scale using the TOSSIM simulator for TinyOS and the TinyViz visualization component [54]. The

visualization in Figure 3.10(a) shows the system on a 10-node network, where the blue arrows indi-

cate localization error vectors, which is the difference between the estimated position and the true

position. The grey circles indicate estimated error. Thesesimulations were scaled up to 150 nodes.

Subsequently, the algorithm was programmed onto Berkeley’s Mica2Dot mote [35], which consists

of a ChipCon CC1000 FSK 433Mhz radio and an Atmel Atmega128 4MHz micro-controller and

was equipped with the ultrasound hardware described in Section 3.1. Because development was

not taking place in a simulated environment, we made most of the functions and variables in our

code remotely accessible from the PC through the Active Message interface. We could access these

functions and variables as well as reprogram the nodes through a wired testbed that we built us-

ing the Crossbow EPRB programming device and an Ethernet switch, as shown in Figure 3.10(b).

However, because the testbed was not mobile and was in a space-confined location, these tests were

limited to localization with a maximum of 12 nodes. Once the system worked on this wired, in-

47

(a) Simulation The localization implementation was first tested and debugged in

a simulated environment, scaling to 150 nodes, using TOSSIMsimulator and the

TinyViz visualization tool, shown here.

(b) Indoor Testbed The localization system was first tested in real hardware on

the wired testbed shown here, which allowed debugging commands and repro-

gramming of all nodes on the network while simultaneously testing the code with

real hardware.

Figure 3.10:Stages of Development and Debugginginclude simulation and small, wired testbeds.

48

Figure 3.11:The Final Deployment involved 49 nodes over a 13x13m area on a paved surface.

door testbed, we tested the system in an outdoor environmentshown in Figure 3.11, first in a 16-

node topology and scaling upwards to 25-, 36-, and 49-node topologies. In this environment, we

used the wireless communication channel for both application data and debugging commands, and

reprogramming of the network was performed manually.

3.8 Deployment Details

In our final deployment, 49 nodes were deployed over a 13x13m area in a 7x7 grid, in

which each of the grid positions was perturbed by Gaussian noise with σ = 0.5m. We used a

randomly perturbed grid to avoid artifacts of the rigidity of a strict grid or the network partitions

common in completely random topologies. To avoid performing this one deployment with a topol-

ogy on which our system would by chance perform unusually well or badly, we generated 100

random topologies, simulated the algorithm on each of them,and chose to use the random topology

which yielded the median average error in simulation. The longest shortest path in the selected

topology was eight hops long, and even longer paths appearedin the real deployment.

The main deployment took place outdoors in the parking lot shown in Figure 3.11. We

49

0 50 100 150 200 2500

0.02

0.04Empirical Error Distribution (kernel smoothing)

Error (cm)P

roba

bilit

y

200 400 600 800 1000 1200 1400 1600200

400

600

800

1000

1200

1400

1600

Dis

tanc

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49

Figure 3.12:Localization Error Vectors are shown in this graph by arrows; the true node positionsare the beginning of each arrow. The anchor nodes are indicated by “X”’s and the gray linesindicate ranging connectivity. Nodes 33, 16, and 43 were dead nodes. The median error for this runwas 47.8cm, and the top graph is a kernel smoothing of the error distribution.

used a system of tape measures to deploy the nodes with an estimated accuracy of about 2.5cm.

After we measured the topology and placed the nodes, we executed the localization system on the

network eight times. The median localization error for thisdeployment was 78.0cm, while the upper

and lower quartiles of error were 131.2cm and 40.5cm, respectively. The actual topology used in

this deployment can be seen in Figure 3.12, along with the localization errors resulting from one of

the several runs of the localization algorithm.

50

Noisy Disk Deployment0

50

100

150

200

250Upper Quartile, Median, and Lower Quartile

Loca

lizat

ion

Err

or (

cm)

Figure 3.13:The Localization Error Gap is illustrated by the dramatic difference in localizationerror predicted by the Noisy Disk model in simulation and that observed in the real deployment. Thebox indicates the median localization error and the error bars indicate the 10th and 90th percentiles.

51

3.9 Comparing Theoretical and Observed Localization Error

We reused the topology from our empirical deployment in simulation to see how the

observed localization error compares to the localization error predicted by the Noisy Disk model

of ranging. To derive the Noisy Disk parameters for simulation, we used the ranging data that was

collected in the same location as the deployment, which is also the data set that was used to derive

the calibration coefficients for the empirical deployment,as described in Section 3.3. We used the

valuedmax = 450cm based on the maximum distance that our ranging technology could robustly

reach, as demonstrated by the data in Figure 3.6(a). We used the valueσ = 4.9cm based on a

maximum likelihood fit of the nominal ranging errors. We excluded the outliers from this fit by

fitting the ranging error to a Gaussian mixture model with 2 means, and choosing the valueσ with

the highest posterior probability. If the dataset contained no outliers, this procedure would yield the

same valueσ as the standard maximum likelihood estimator.

Even though the Noisy Disk parameters were derived from the data in a way similar to

derivations from previous work described in Section 2.5.2,the localization errors in simulation

were significantly different than those observed in the empirical deployment. The median localiza-

tion error in simulation was 15.2cm, with upper and lower deciles of error of 34.7cm and 6.0cm,

respectively. This is significantly lower than the error distribution actually observed in the empirical

deployments, with a median of 78.0 and upper and lower deciles of 208.3cm and 23.2cm. Both

error distributions are shown in Figure 3.13.

A deeper analysis reveals that the simulation differs from the deployments not only in

terms of localization error, but in terms of intermediate values that are generated, as well. For

example, we can characterize shortest path error as a ratio of the shortest path distance to the true

distance between two nodes. If the shortest paths are all exactly correct, this ratio should be exactly

1. The shortest path error for both simulation and deploymentare similar, but the variance of the

shortest path error is much higher in the deployment than predicted by simulation. While the center

80% of the shortest path errors vary by less than 3% of the truedistance in simulation, they vary

by up to 19% in the empirical deployment. The shortest path error distributions are illustrated in

Figure 3.14.

In localization, a node’sdegreeis the number of neighboring nodes with which it can ob-

tain a range estimate. Figure 3.15 shows that the median nodedegree in simulated localization runs

is 12 while in the empirical deployment it is only 6. This difference can be extremely significant;

studies have shown that localization algorithms can behavevery differently in high density networks

52

Noisy Disk Deployment0.95

1

1.05

1.1

1.15

1.2Upper Quartile, Median, and Lower Quartile

Sho

rtes

t Pat

h E

rror

(cm

)

Figure 3.14:The Shortest Path Error Gap is primarily exhibited by a difference in variance. Insimulation, shortest paths vary by 3% while in the real deployment they vary by 19%.

53

Noisy Disk Deployment2

4

6

8

10

12

14

16

18Upper Quartile, Median, and Lower Quartile

Nod

e D

egre

e

Figure 3.15:The Node Degree Gapindicates that nodes in simulation have on average 12 rangingneighbors while in the real deployment they had only 6.

54

than in low density, where the threshold between the two is approximately a node degree of about

9 [50].

3.10 The Prediction Gap Established

In this section, we provided concrete evidence of the Prediction Gap. The localization sys-

tem we designed was representative of the canonical system used in most localization simulations

and theoretical analysis in the literature. It was carefully designed to provide range estimates that

are as good or better than most existing implementations, and we provided analysis and solutions for

problems that were previously not addressed, including collision avoidance and non-linear noise fil-

tering for the asymmetric noise profile of time of flight ranging. The localization algorithm we used

is representative of a large class of existing localizationalgorithms, and has been experimentally

shown to produce comparable localization results and to have similar failure modes. We extended

the algorithm by carefully building a distributed implementation, which required new programming

abstractions and an incremental development process through simulation, emulation, small wired

testbeds, and ultimately real deployments. By combining range sensors and a distributed localiza-

tion algorithm, this system is a precise and complete representation of many canonical ideas from

the localization literature, and the unexpected performance observed in Section 3.9 cannot be at-

tributed to implementation or design issues, but rather to alack of understanding in the literature of

how such a system behaves.

Similarly, our comparison with the Noisy Disk model was preceded by a very careful

analysis and thorough characterization of our range sensor. The characterization captured aspects

of many different transmitter/receiver pairs at random orientations and in multiple different paths

through the actual deployment environment. We captured an abundance of data at a high resolution:

at least one point every 2.5cm over the entire range of the sensor. This data set was used to set the

calibration and filtering coefficients in our deployment, and the same data set was used to estimate

the Gaussian noise and Unit Disk parameters that we used in simulation. Thus, differences between

our simulation results and observed deployment results cannot be attributed to using an unrealistic

simulation scenario. Rather it should be attributed to a failure of the Noisy Disk model to capture

the structure of our empirical ranging data. This argument is supported by the fact that our com-

parison corroborates previous studies which found that simulation does not accurately predict true

deployments. In the next chapter, we will take this analysisone step further by identifying the cause

of the Prediction Gap that we observed in this chapter.

55

Chapter 4

Closing the Prediction Gap

The previous section shows that traditional simulation of localization using the Noisy

Disk of ranging model does not accurately predict the localization errors observed in the empirical

deployment. This difference is what we can been calling thePrediction Gap, and is a long-standing

problem in the localization literature for three reasons:

1. Real deployments are unpredictable. If an application such as tracking specifies a maximum

allowable localization error, a real deployment may not meet that requirement even if it is

predicted to do so in simulation. This can be a problem for mission critical deployments

which can only be deployed once, such as forest fire tracking,or for large deployments with

1000’s of nodes where the cost of redeployment is prohibitive.

2. Comparison of algorithms is inconclusive. Besides predicting the localization error of a par-

ticular real deployment, simulation is also used to comparealgorithms and to analyze the sen-

sitivity of an algorithm to different noise levels or topologies. Because there is no concrete

relationship between simulation and empirical deployment, the conclusions from simulation-

based analysis may not hold in the real world.

3. Empirical error is difficult to explain. If everything known about the environment and range

sensor is incorporated into a theoretical model which produces low errors in simulation, then

the cause of any additional error observed in the real deployment is not known. Furthermore,

if the cause of the additional error is not know, it is difficult to reduce.

The first step to addressing these issues is to reevaluate ourmodel of the sensors and

the environment and to create one that accurately predicts empirical localization error. There are

56

several challenges to improving the traditional parametric Noisy Disk model. Instead, we choose

to use non-parametric models, which take data collected in the real world and use it directly in

simulation, avoiding the need to reduce complex empirical data to a simple set of parameters. This

can produce accurate simulations without committing in advance to a particular parametric form of

the empirical data.

4.1 Modeling the sensors and environment

4.1.1 Parametric Models

Parametric models like the Noisy Disk specify astructurethat can only change in a certain

number of ways, as enumerated by the model’s parameters. Many techniques have been developed

for choosing the best parameters to fit a model to a data set. For example, least squares fitting

chooses the parameters that minimize the squared difference between the observed data points and

predictions from the model [65]. Robust estimation techniques are similar, but they place lower

weight on points that are not well predicted by the model [37]. Maximum likelihood techniques

maximize the probability of the data points given the parameters [73]. Indeed, most machine learn-

ing techniques, including neural networks [77], the expectation-maximization (EM) algorithm [73],

and support vector machines [16] are all parameter estimation techniques that assume the user has

already determined the general structure of the data in someparametric form.

In contrast, very few formal techniques exists in the way of choosing the model itself. This

is a natural dichotomy because a model defines a clear parameter space, but the space of all models is

typically not well defined. Defining the space of all models would require aneighborhood function

that defines a transformation from one model to other similarmodels and creates a well-behaved

space of models over which an algorithm may search. This is possible in some cases such as neural

networks and Bayesian networks with structures that facilitate search using genetic algorithms [3]

or other techniques [33], even though the search over such structures has been shown to be NP-

Complete at least in some cases [14]. Defining a good neighborhood function over algebraic models

like Equation 2.6 to create a searchable space of algebraic formulae is much more difficult. Instead,

when creating an algebraic model of a process such as ranging, scientists typically resort tofirst

principlesof physics; each aspect of the hardware, signal propagation, and environment are modeled

according to algebraic formulae from traditional physics.In our case, the ranging process is captured

by our model in Equation 3.2.

57

The problem with the technique of first principles is that it leaves us with no further

recourse when we find our model to be insufficient. In general,the model is known to be a simpli-

fication of the real physical process; the calibration function we used in Equation 3.3, for example,

does not explicitly account for the orientation or frequency variations of our original model in Equa-

tion 3.2 and instead incorporates these into the noise parameter, along with many other aspects of

the physical world that are too complex to model. Improving our model requires us to identify

which of these physical processes produce salient effects that cannot be treated as noise, or that

change our assumed noise distribution. This task is made more difficult by the fact that a physical

property of the range sensor and environment may affect one localization algorithm but not another

and that this effect may be exhibited in one network topologybut not another. Therefore, to create

an improved model from first principles, the scientist must not only understand the physical world

but also its complex interaction with network topology and the implicit assumptions of a particular

algorithm. In our case, it is not immediately clear which aspects of our environment and range

sensor are causing unexpectedly higher error in deploymentthan with the Noisy Disk.

4.1.2 Non-parametric Models

Non-parametric models differ from parametric models in that the structureof the data

is not assumed in advance, but is instead determined by the data being modeled. Non-parametric

models are also calleddistribution freemodels because they do not assume the data conforms to

some predetermined distribution. Several forms of non-parametric models exist, the most common

of which include histograms [85], kernel regression [39], and wavelet analysis [43]. In this section,

we show how to usestatistical sampling, in which we generate data for simulation by randomly

drawing measurements from an empirical data set.

We define the distributionM(d, ǫ) to be the set of all observed ranging estimates for

distances in the interval[d − ǫ, d + ǫ]. This set is our non-parametric model and represents an

empirical distribution of range estimates at distanced. For example, the setM(350cm, 5cm) is

represented by all the range estimates between the verticalbars in Figure 4.1.

We can generate a ranging estimatedij for simulation from this model by simply drawing

a random sampled from the setM(dij , ǫ). Using the value ofd directly, however, would not be

accurate; the value ofǫ increases the variance ofdij becauseM(dij , ǫ) includes range estimates

from both longer and shorter distances thandij . Instead, we use theerror of the sample, which is

the differenced − da whereda is the actual distance at whichd was measured. Thus, a simulated

58

Figure 4.1:The Non-parametric Model is essentially a binning of empirical data. The blue dotsindicate observed data points. All dots between the two red lines are binned into a set calledM(350cm, 5cm), which is randomly sampled to simulateddij = 350cm in simulation.

59

error measurement can be generated fromM(dij , ǫ) as

dij = dij + (d − da) (4.1)

Besides range estimates, the setM(d, ǫ) also includesranging failures, denoted byø,

which are ranging instances when a pair of nodes fail to obtain a distance estimate. This is necessary

to model the probability of connectivity at distanced; if 50% of all ranging estimates taken at

distanced are ranging failures then randomly sampling fromM(d, ǫ) should yield a 50% chance of

drawingø.

4.2 Empirically Profiling the Physical World

4.2.1 Traditional Data Collection

Becaused ∼ M(dij , ǫ), the simulation is using the empirical distribution of ranging

estimates at distancedij if and only if M(dij , ǫ) accurately represents the noise and connectivity

characteristics at that distance. The challenge in using this sampling technique is to collect ranging

error and connectivity data with a high enough resolution sothat small values ofǫ can be used. For

example, if we want to useǫ = 2.5cm and ultrasound ranging has a maximum range of 10m, we

must take empirical ultrasound measurements at 400 different distances. The typical data collection

process, however, makes it difficult to collect data with such high spatial resolution: one usually

places a transmitter and receiver a known distance apart, collects range estimates, and repeats at

a small number of increasing distances. The data set collected by Sichitiu [90] in Figure 4.2, for

example, collects ranging data with up to 10m spacing. The low spatial resolution of this data would

make it difficult to use values ofǫ smaller than 5 meters. Furthermore, to use this data set withour

non-parametric model to simulate ranging at a distance ofd = 25m, we would need to assume that

data collected atd = 20m andd = 30m has roughly the same characteristics asd = 25m.

The problem with the traditional data collection process isthat it requires a linearly in-

creasing amount of time as the number of distances are measured. Even if all readings can be taken

in 60 seconds at each distance, measuring 400 different distances would require almost 7 hours.

Not only does this make it difficult to collect samples from multiple different combinations of range

sensors and environments, but it makes it impossible to collect a complete sample of an outdoor

environment, for example, before the temperature, humidity, and wind conditions change. Of the

authors mentioned in Section 2.5.2 that collected empirical ranging data, most collected data at no

more than 15 different distances.

60

Figure 4.2:Traditional Data Collection results are illustrated here, in which RSS data was col-lected at multiple different distances by taking a single pair of nodes and placing them at progres-sively larger distances [90]. Because of the low resolution, this data would be difficult to use withour non-parametric model.

61

Another problem with the traditional data collection approach is that it only measures a

small number of points in the total space of noise factors: ranging data is measured with a single

transmitter and a single receiver, usually in the same orientation, and in a single line through space.

This means that any idiosynchracies of the particular transmitter and receiver are present in all data

collected at all data points and any physical aspects of the testing environment, such as a wall several

meters away or the orientation of the nodes, may produce systematic errors in the entire data set.

Of the empirical data collection studies mentioned in Section 2.5.2, all authors collected data with

a single transmitter and receiver.

4.2.2 High-fidelity Data Collection

Instead of measuring each distance with a single pair of nodes, we designed a data collec-

tion process that could measure several hundred distances as well as different transmitter/receiver

pairs, node orientations, and paths through the environment. All measurements are taken at once

with√

400 = 20 nodes in a special topology where each pair of nodes measuresa different dis-

tance. By adding a few additional nodes, we can get multiple pairs of nodes at each distance. We

generated such topologies usingrejection sampling[74], i.e., we generated thousands of topologies

until one of them measured a uniform distribution of distances. For example, we used the topology

in Figure 4.3(a), which required 25 nodes to obtain 2.5cm resolution over 5m, to characterize our

ultrasound range sensor. Figure 4.3(a) shows a histogram ofthe distances that are measured by this

topology.

All nodes are placed at random orientations in this topologyand each node transmitsN

times in turn while all other nodes receive. To remove the bias of each distance being measured

by only two pairs of nodes (the reciprocal pairs A/B and B/A),this procedure is repeated five times

with different mappings of nodes to the topology locations.These mappings are also generated using

rejection sampling to ensure that the same distances are notalways measured by the same pairs. The

procedure generates10·N total measurements at each distance with 10 different transmitter/receiver

pairs. Therefore, with the topology in Figure 4.3(a) and valuesN = 10 andǫ = 0.05m (two inches),

the setM(δ, ǫ) is likely to include 400 empirical measurements.

Unlike the conventional pairwise data collection technique described above, the empirical

measurements inM(δ, ǫ) are taken with dozens of transmitter/receiver pairs, capturing a broad

spectrum of node, antenna, and orientation variability. Furthermore, the measurements are taken

over several different paths through the environment, capturing variability due to dips, bumps, rocks

62

0 1 2 3 4 50

1

2

3

4

5

Distances are in meters

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P Q

R

S

T

U

V

WX

Y

(a) TopologyThis specially generated topology with 25 nodes measures

300 different distances with at least 1 distance every .025mbetween

0.4m and 5.2m.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.50

1

2

3

4

5

6

7

8

Distribution of Measured Distances

Distance (m)

#Mea

sure

men

ts

(b) Histogram This histogram shows that the distances measured by the topology

are uniformly distributed over the ultrasonic range.

Figure 4.3:The Data Collection Topology

63

Indoors Grass Elevated EveningMax Rangedmax (cm) 344.5 231.1 351.6 364.5Noiseσ (cm) 6.2 6.3 6.44 4.7

Table 4.1:Generalizing Noisy Disk Parameterscan be difficult, because the Noisy Disk parame-ters are very different for each of the different environments in Figure 4.4

or other environmental factors. Finally, this technique captures connectivity characteristics by fixing

the number of transmissions and measuring the number of readings at each distance. In contrast,

the conventional pairwise technique described above requires the experimenter to take readings at

every possible distance, hiding the degradation of rangingconnectivity with distance.

The rejection sampling algorithms required on average twelve hours to compute the topol-

ogy and node mappings. We measured the topology positions using tape measures by first measuring

out two right triangles to create a square and then placing two tape measures along the vertical edges

of the square and one which ran horizontally between them. Tolocate each position, the horizontal

tape measure was slid up or down to find the correct Y coordinate, and the X coordinate was found

on the horizontal tape measure itself. This process could becompleted in about 1 hour and required

two people, as opposed to the traditional process which requires only a single experimenter.

For each mapping of nodes to topology locations, ranging between all nodes took place

over a period of about 5–10 minutes. Another 20 minutes was required to collect the data to a

central base station over the wireless network. The nodes were then collected and redistributed in

a new mapping of nodes to topology positions. The data collection process extended over a period

of about 3.5 hours, but could be reduced to about 1 hour if we stored the data to external flash on

the nodes and retrieved it after the experiment, or if we usedthe faster 802.15.4 radios that are now

common in sensor networks for faster data collection.

4.2.3 Generality of an Empirical Profile

Once data is collected in a particular environment, it can only be used to simulate a de-

ployment in that same environment. We used the data collection process described above to collect

data in several different environments, including indoors, outdoors in a grassy field, on pavement

during the day and at night, and in a network raised above grass to approximate free space. Some

of these environments are shown in Figure 4.4.

Each of these environments yields data with very different characteristics, as becomes

64

(a) Indoors (b) Grass

(c) Elevated (d) Evening

Figure 4.4: Profiling Multiple Environments using the data collection techniques described inSection 4.2.2 reveals that data collected in one environment may be very different than other envi-ronments.

65

evident when we fit the data with the Noisy Disk model using theapproach described in Section 3.9.

Note, for example, that the grass environment has a maximum range only 63% the length of the

range in a parking lot, while the elevated nodes yield 37% more noise than the parking lot. Table 4.1

shows the maximum rangedmax and noise parameterσ derived from the data collected in each of

the environments in Figure 4.4, showing how significantly different environments can affect the

time of flight measurements.

Although an empirical profile from one environment cannot begeneralized to other en-

vironments, this is not a limitation only of non-parametricmodels. As described in Section 4.1.1,

the parameters of parametric models must also be derived from empirical data and, as with non-

parametric models, these parameters cannot be generalizedto environments other than the one in

which that data was collected.

4.3 Comparing Non-parametric Predictions and Observed Localiza-

tion Error

The empirical profile of our deployment environment was earlier used to derive calibra-

tion coefficients in Section 3.3 and to derive Noisy Disk parameters for simulation in Section 3.9. In

this section, we use the same empirical profile as a non-parametric model of our deployment envi-

ronment. Similar to our comparison between Noisy Disk simulation and the deployment, we reuse

the topology from our empirical deployment in simulation tosee how the observed localization er-

ror compares to the localization error predicted by the non-parametric model of our range sensor

and environment. Because the empirical profile was collected with a resolution of approximately

2.5cm, we choose the parameterǫ = 5cm for our model.

The non-parametric model of our environment predicts the localization error from our true

deployment much more accurately than the Noisy Disk model. This is demonstrated by Figure 4.5,

which shows the error distributions for the deployments andboth simulations. The Noisy Disk

simulation predicts a median error of about 15, with upper and lower deciles of error of 34cm and

6cm, respectively. This is much smaller than the observed median error of 78cm, 90th percentile of

208cm, and 10th percentile of 23cm. Indeed, the predicted median error is lower than the observed

10th percentile. The non-parametric simulation produces amuch more accurate prediction, with

median error of 67cm, 90th percentile of 174cm, and 10th percentile of 22cm.

The localization error distribution from the deployment isstill significantly different than

66

Noisy Disk Non−parametric Deployment0

50

100

150

200

250Upper Quartile, Median, and Lower Quartile

Loca

lizat

ion

Err

or (

cm)

Figure 4.5:Closing the Localization Error Gap can be performed with non-parametric modeling,although all three distributions are still statistically different. Non-parametric simulation does notexplain the shortcomings of the Noisy Disk model, but this will be performed in Chapter 5.

67

Noisy Disk Non−parametric Deployment0.95

1

1.05

1.1

1.15

1.2

1.25Upper Quartile, Median, and Lower Quartile

Sho

rtes

t Pat

h E

rror

(cm

)

Figure 4.6:Closing the Shortest Path Gapwith non-parametric simulation yields similar medianshortest path errors and similar variance, although all three distributions are statistically different.

the distribution from the non-parametric simulation, as determined by a two-sidedt-test withα =

0.05. However, almost no simulation technique can be expected toproduce exactly the same error

distribution as the real world, and the predicted results are qualitatively very close or at least, in

contrast to the Noisy Disk simulation, represent the correct order of magnitude.

The non-parametric model is not only a better predictor of the overall algorithmic behav-

ior, but is also a better predictor of the internal structureof the algorithm. As shown by Figures 4.6

and 4.7, the non-parametric model more accurately predictsthe distribution of shortest path distance

errors and node degrees, respectively, than does the Noisy Disk model. The distribution of node de-

grees produced by the non-parametric simulation and the deployment are statistically equivalent,

according to a two-sidedt-test withα = 0.05 andp = 0.45.

The reason why the overall localization error and the internal algorithmic behavior is

more accurately predicted by the non-parametric model is not immediately clear from these results;

they only show that the non-parametric model is an improvement over the parametric model. This

improvement will be more completely explained in Chapter 5,when we combine the two techniques

68

Noisy Disk Non−parametric Deployment2

4

6

8

10

12

14

16

18Upper Quartile, Median, and Lower Quartile

Nod

e D

egre

e

Figure 4.7:Closing the Node Degree Gapwith non-parametric simulation yields similar node de-gree and variation. The distributions for non-parametric simulation and the empirical deploymentare statistically equivalent.

69

through hybrid parametric/non-parametric models.

70

Chapter 5

Explaining the Prediction Gap

In the previous chapter we saw that careful data collection in combination with non-

parametric models can close the prediction gap left by certain parametric models like the Noisy

Disk. This addresses the first two concerns listed in the introduction of Chapter 4: 1) deployments

will be more predictable and 2) conclusions drawn from simulation-based analysis and comparison

will be more meaningful. However, the third concern is stillnot addressed: the Prediction Gap

is still difficult to explain. The techniques in Chapter 4 didnot identify the aspects of our real-

world environment and range sensor that caused the error notpredicted by the Noisy Disk. Without

knowing the cause of this error, it is still difficult to reduce it.

Any aspect of our ranging data that does not conform to the traditional Noisy Disk model

can be called a rangingirregularity. Typically, a real-world range sensor may have dozens of ranging

irregularities due to manufacturing flaws, changing environments, or unforeseen physical dynamics.

In this section, we develop a scientific approach to identifythe irregularities in our empirical ranging

data that contribute to increased localization error.

This analysis is complicated by the fact that all localization algorithms may react dif-

ferently to ranging irregularities; an irregularity may cause significant error for one algorithm, not

affect another algorithm at all, and even improve the error results for a third algorithm. To com-

pletely explore the ranging irregularities in our ranging data set, we formulate an experiment and

repeat it with six different localization algorithms from the literature. The experimental setup allows

us to consider a particular irregularityX in isolation, and to answer the question:

Question Is ranging irregularityX a significant cause of error leading to the Prediction Gap?

The experimental setup consists of two steps. First, we evaluate the algorithm with the

Noisy Disk ranging model. Then, we add ranging irregularityX to our model and evaluate the

71

algorithm again. This experimental method isolates the effect of irregularityX and a comparison of

the results from the two steps verifies one of two possible hypotheses:

H0 Irregularity X is not a significant cause of error, and localization error inboth trials will be the

same.

H1 Irregularity X is a significant cause of error, and localization error in both trials will be different.

A key aspect of this methodology is that the experimenter does not need to know how or

why irregularityX may affect the localization algorithm, and may not even havea clear idea of what

irregularityX is. For example, the experimenter may want to test if the empirical noise distribution

is different from the model, without knowing exactly how to characterize the difference between the

two. We do not require the experimenter to modify the parametric form of the Noisy Disk model

to try to capture irregularityX. Instead, we combine the Noisy Disk model with anon-parametric

model of irregularityX. The techniques we use to add only a single irregularity at a time will be

described in more detail in the next section.

5.1 The Experimental Setup

The experiment we use to identify the cause of the PredictionGap for a particular algo-

rithm has four steps:

1. We compare the empirical ranging data to the Noisy Disk model and hypothesize which

ranging irregularity is causing the Prediction Gap.

2. We develop a hybrid model that incorporates the ranging irregularity into the Noisy Disk

model.

3. We derive Noisy Disk parameters from the empirical ranging data to ensure a fair comparison

between the ideal and empirical components.

4. We evaluate the localization algorithm using both the hybrid model and the Noisy Disk model.

By comparing the resulting localization errors, we can isolate the effects of the ranging irreg-

ularity.

In Section 5.1.1, we compare our empirical ultrasound data to the Noisy Disk model

and identify four ranging irregularities that may be causing the Prediction Gap. In Section 5.1.2,

72

we incorporate these irregularities into hybrid models. Wederive Noisy Disk parameters from our

empirical data in Section 5.1.3. The results of performing this experiment on six different algorithms

are presented in Section 5.2.

5.1.1 Identifying Ranging Irregularities

We can hypothesize ranging irregularities through inspection of our empirical ultrasound

data, illustrated in Figure 5.1. Figure 5.1.a shows that theprobability of successfully obtaining a

range estimate at each distance does not match the Unit Disk model of connectivity: many pairs

that are closer thandmax do not in fact obtain a ranging estimate with some probability while others

farther thandmax do. Figure 5.1.b contains a histogram of ranging error that illustrates non-Gaussian

ranging error, including a larger number of extreme underestimates and overestimates than would

be predicted by the Normal distribution. Based on these observations, we hypothesize four types of

ranging irregularities that might be causing the prediction gap:

Extreme overestimates:an excess of range estimates that are longer than the true distance

by more than two standard deviations

Extreme underestimates:an excess of range estimates that are shorter than the true distance

by more than two standard deviations

Long-range proficiency: the existence of range estimates between nodes farther thannomi-

nal rangedmax

Short-range deficiency: the existence of range failures between nodes closer than nominal

rangedmax

The causes of these irregularities are unknown, but may include irregular environmental

attenuation, variance in node orientation, or irregular amplifying pathways.

5.1.2 Creating Hybrid Models

We can isolate each of the four irregularities described above by creating a series of five

different ranging models, each incorporating one ranging irregularity more than the previous one:

Model 1 (Noisy Disk): No irregularities

Model 2: Model 1 + Extreme overestimates

73

−500 0 500 1000 1500 20000

50

100

150

200

250

300

350

1600

Error (cm)

Fre

quen

cy o

f Occ

urre

nce

0 200 400 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance (cm)

Pro

babi

lity

of C

onne

ctiv

ityTrueDmax

Overestimates

Underestimates

Short Ranges

Long Ranges

Figure 5.1:Ranging Irregularities are evident from the empirical data, including (a) short-rangedeficiency and long-range proficiency and (b) extreme underestimates and overestimates.

Model 3: Model 2 + Extreme underestimates

Model 4: Model 3 + Long-range proficiency

Model 5: Model 4 + Short-range deficiency

After evaluating a localization algorithm with this seriesof models, the localization error

produced by Models 1 and 2 can be compared to evaluate the effect of extreme overestimates.

Similarly, the error produced by Models 4 and 5 can be compared to evaluate the effect of short-

range deficiency.

Because we are adding each irregularity to a model with all previous irregularities, Model

5 will incorporate all four irregularities simultaneously. Thus, Model 1 is the pure Noisy Disk model

and Model 5 is the pure empirical data. Models 2-4 are on the spectrum between these two extremes.

By structuring our experiments in this way, we are able to observe the effects of each irregularity as

well as their cumulative effects, and can compare the ideal model directly to the empirical model.

However, we are also assuming that the effects of the different ranging irregularities are indepen-

dent. A more complete study would present a comparison between all24 combinations of ranging

irregularities, although we have found through experimentation that the independence assumption

is reasonable.

74

To create Model 1, we estimatedij according to the Noisy Disk formula listed in Equa-

tion 2.6

dij =

N (dij , σ) dij ≤ dmax

ø otherwise.(5.1)

Model 3 contains both extreme overestimates and underestimates, i.e. Model 3 is empiri-

cal noise coupled with Unit Disk connectivity. To create this model, we sample an empirical valued

from a new set of empirical observationsM(dij , ǫ), which includes only those that werenot ranging

failuresM(dij , ǫ) − ø

dij =

d ∈ M(dij , ǫ) dij ≤ dmax

ø otherwise.(5.2)

Model 2 is similar to Model 3 except that the range estimated should be normally dis-

tributed if it is less than zero and empirically distributedif it is greater than zero. We can achieve

this distribution by replacingd with normally distributed noise whenever it is an underestimate

dij =

d ∈ M(dij , ǫ) d ≥ 0, dij ≤ dmax

N (dij , σ) d < 0, dij ≤ dmax

ø otherwise.

(5.3)

The noise in Model 4 is always distributed according to the empirical distribution, and

range estimates are always obtained at distances less thandmax. However, range estimates are also

obtained at distances greater thandmax with the same probability as empirical range estimates. To

achieve this distribution, we use

dij =

d ∈ M(dij , ǫ) dij ≤ dmax

d ∈ M(dij , ǫ) dij > dmax

(5.4)

Model 5 uses both the empirical noise and connectivity distributions, achieved with pure

random sampling from the empirical data set

dij = d ∈ M(dij , ǫ) (5.5)

5.1.3 Parameters and Topology

Using these five models, we evaluate all six algorithms on a simulated 18m x 18m square

topology with 100 nodes and four anchors, one in each corner.Nodes are placed in a grid topology

75

with Gaussian noise added to the grid positions to avoid exhibiting artifacts of the network partitions

that are likely in a completely random topology or of the strict rigidity of a true grid, neither of which

are representative of the canonical deployment. Regardless of the ranging model used, all networks

have an average degree of 9, meaning that all nodes have an average of 9 neighbors. This number of

neighbors has been shown to be a transition point between high and low density networks [50]. The

only exception to this rule are trials with Model 4 from Section 5.1.2, for which it was impossible

to hold bothdmax and average network degree constant. These trials have a higher network degree

of 12 because the model incorporates long-range proficiencyand all nodes closer thandmax are

connected.

This experiment only measures a single point along the dimensions of network size, an-

chor density, topology, etc. The chosen topology is sufficiently representative of the canonical

sensor field deployment, however, because smaller topologies exhibit predominantly edge effects

(nearly all nodes are near an edge) and larger deployments can be subdivided into a network of this

type by placing anchor nodes appropriately throughout the network. The purpose of this study is

not to explore the effects of network size and anchor density, which has been done in other stud-

ies [50,63,81], but to explore the effects of ranging irregularities.

In all of our experiments, the non-parametric models and Noisy Disk parametersσ and

dmax are produced from the ultrasound profile that was collected in our deployment environment, as

described in Section 3. In order to explore the effects of outliers on these localization algorithms, we

use mean filtering instead of the special medianTube filter that we designed to remove these outliers.

To derive the parameterσ, we fit a mixture of Gaussians to the ultrasound data and choose the

parameterσ with the highest likelihood, similar to Section 3.9. Unlikethat section, which followed

standard convention in the literature by settingdmax to the maximum obtainable range of the sensor,

we now setdmax to a value that would achieve the same average degree as the empirical data. We

call this value theeffective rangeof the empirical data, which can be calculated asreff =√

PΠ

where

P =

∫ r=dmax

r=0Π ∗ r2 ∗ p(r) (5.6)

andp(r) is the empirical probability of successfully obtaining a range estimate at distancer. Nodes

using the Unit Disk model of connectivity with parameterdmax = reff should have the same

number of neighbors as nodes using the empirical ultrasounddata.

76

Rank Ideal Empirical1 DV-Distance MDS-Map

MDS-Map2 MDS-Map(P) Bounding Box

DV-Distance3 Bounding Box MDS-Map(P)

Robust Quads4 GPS-Free GPS-Free

Robust Quads

Figure 5.2:The Ordering of Localization Algorithms is not the same for the ideal ranging modeland the empirical model.

5.2 Experimental Results

We perform the experiment described above on six localization algorithms from the lit-

erature: Bounding Box, DV-distance, MDS-Map, GPS-Free, Robust Quads, and MDS-Map(P).

These algorithms were chosen because they represent the twomain classes of approximations used

by multi-hop localization algorithms: theshortest-pathand thepatch and stitchapproximations.

The algorithms themselves were introduced in Section 2.4 and are more completely described in

Section 5.3. The results of our experiment on these six algorithms reveal several broad findings:

• The Prediction Gap is evident with all six algorithms; no algorithm using empirical ranging

data produced localization error within a factor of two of the Noisy Disk prediction.

• The cause of the Prediction Gap is different for each of the six algorithms; irregularities do

not have the same effects on all algorithms.

• The ranking of the algorithms is different with the Noisy Disk model and empirical ranging

data; an algorithm that appears to be better with the Noisy Disk model may actually be worse

with empirical ranging data, or vice versa.

The median errors and the median percentage of nodes localized in 30 trials of each al-

gorithm with all five ranging models are shown in Figure 5.3. We used a one-tailedt-Test with

α = 0.05 to compare adjacent models, and the statistically significant changes are indicated in the

figure with* ’s. For example, the Bounding Box and DV-Distance algorithms are both significantly

affected by Models 3 and 4, but not by Models 2 and 5. Therefore, with these algorithms the differ-

ence between Models 2 and 3 and Models 3 and 4 are marked with a* while the difference between

77

BBox DV−D MDS GPS−F RQds MDS(P)0

200

400

600

800

1000

1200

Loca

lizat

ion

Err

or (

cm)

Model 1Model 2Model 3Model 4Model 5

* *

*

*

*

*

*

*

*

(a) Localization Error

BBox DV−D MDS GPS−F RQds MDS(P)0

10

20

30

40

50

60

70

80

90

100

110

Per

cent

Nod

es L

ocal

ized

(%

)

* * *

(b) Percent Localized

Figure 5.3:Experimental Resultsfor each algorithm along thex-axis, with each of five differentranging models, showing (a) the median localization error and (b) the median percentage of nodeslocalized. Statistically significant changes are indicated with * ’s.

78

Models 1 and 2 and Models 4 and 5 are not marked. The ranging irregularities that cause changes

in each algorithm are summarized in Table 5.4.

All algorithms perform relatively well when evaluated using the Noisy Disk model. The

fact that localization error for all algorithms gets significantly worse as ranging irregularities are

introduced, and that no algorithm improves, demonstrates that the Prediction Gap is a problem with

all localization algorithms, not just the DV-distance algorithm as demonstrated in Chapter 3.

Perhaps most surprising is the extremely high sensitivity of all six algorithms to small

changes in the ranging model. Simulation with a theoreticalmodel is never an exact replica of

reality and is never expected to produce exactly the same algorithmic response as empirical noise.

However, we do typically expect simulation to be

1. indicative: results should be within a constantfudge factorof empirical results

2. decisive:an algorithm that performs better in simulation should alsoperform better in reality

The results of our experiment show that simulation with the Noisy Disk does not meet either of these

expectations. Localization error increases by less than 70% for some algorithms and over 800% for

others, indicating that localization error in simulation is notindicativeof error in a real deployment.

To test for decisiveness, we used a one-tailedt-Test withα = 0.05 to derive a statistically signif-

icant ranking of all algorithms. An equivalent ranking between two algorithms indicates that they

are both statistically better and worse than the same set of other algorithms, and that there is no

difference between their own localization errors. The resultant orderings are not the same when us-

ing purely theoretical and purely empirical models, as shown in Table 5.2. This conclusively shows

that simulation using the Noisy Disk ranging model is neither indicative nor decisive, meaning that

is has almost no value when trying to design, tune, and deploya localization algorithm in the real

world. This result motivates the more detailed analysis presented in the next section that identifies

what aspects of empirical noise are not captured by the NoisyDisk model, yet are having significant

impact on localization simulations.

Besides the trends mentioned above, the true value of these results is that we can explain

the cause of the Prediction Gap for each algorithm. The numeric results and the significance testing

illustrated in Figure 5.3 allows us to accept one of the two hypotheses above for each combination

of ranging irregularity and localization algorithm. For example, we can conclude that the Bounding

Box and DV-Distance algorithms are sensitive (H1) to extreme underestimates and long-range profi-

ciency (Models 3 and 4) because those models caused statistically significant changes in localization

79

Extrem

eOve

resti

mat

es

Extrem

eUnd

eres

timat

es

Long

-rang

ePro

ficien

cy

Short-

rang

eDefi

cienc

y

BBox Error ✖ ✔ ✔ ✖

BBox % Localized ✔ ✔ ✔ ✔

DV-D Error ✖ ✔ ✔ ✖

DV-D % Localized ✖ ✖ ✖ ✖

MDS Error ✔ ✖ ✔ ✖

MDS % Localized ✖ ✖ ✖ ✖

GPS-F Error ✔ ? ? ?GPS-F % Localized ? ? ? ?RQds Error ✔ ✖ ✔ ✖

RQds % Localized ✖ ✖ ✔ ✖

MDS(P) Error ✔ ✖ ✔ ✖

MDS(P) % Localized ✖ ✖ ✖ ✖

Figure 5.4:Causes of the Prediction Gapfor each algorithm are summarized here. For each col-umn,✔ and✖ indicate that the ranging irregularity effects or does not effect the final localizationerror and the percentage of nodes localized by a particular algorithm. The? indicates that theexperiment produced an inconclusive result.

80

(a) Consistent Bounds (b) Inconsistent

Figure 5.5: The Bounding Box Algorithm (a) constrains each node to be within its multi-hopdistance to each anchor (b) sometimes resulting in inconsistent constraints.

error for those algorithms. On the other hand, these algorithms and not sensitive (H0) to extreme

overestimates and short-range deficiency (Models 2 and 5). Table 5.4 summarizes which ranging ir-

regularities caused statistically significant changes in localization error and the percentage of nodes

localized for each algorithm. Only two algorithms, MDS-Mapand MDS-Map(P), show the same

pattern of statistically significant changes in both localization error and the percentage of nodes

localized. This indicates that, due to the different approximation algorithms and their correspond-

ingly different assumptions, each algorithm is affected differently by each ranging irregularity. In

the next section, we use our findings to analyze each approximation algorithm and to identify which

assumptions they make that might not hold true with empirical ranging data.

5.3 Analyzing Each Algorithm

5.3.1 Bounding Box

The Bounding Box algorithm [82, 91] uses the shortest path distancespij to constrain

the unknown coordinates of nodei in terms of the known coordinates of anchor nodej. These

81

(a) Normal (b) Noise (c) Range

Figure 5.6:Shortest Paths(a) zig-zag and should always be longer than the true distance. However,(b) extreme underestimates and (c) long-range proficiency combine to significantly shorten them.

constraints take the following form:

xj − eij < xi < xj + eij

yj − eij < yi < yj + eij

These are very loose constraints which only require that a node bewithin a certain distance from

an anchor node, notat a certain distance. Furthermore, the constraints are placed on thex and

y coordinates independently, so the union of constraints from all anchor nodes defines a box, as

depicted in Figure 5.5(a). The location of each node is then approximated to be the center of the

box defined by the union of all constraints.

The simulation results in Figure 5.3(a) show that localization error for the Bounding Box

algorithm significantly increases when subject to extreme underestimates and long-range profi-

ciency. Simultaneously, the percentage of nodes that the algorithm is able to localize drops sig-

nificantly. However, the algorithm is largely impervious toextreme overestimates and short-range

deficiency. We can explain these trends through a deeper analysis of how the shortest path approxi-

mation is affected by noise and connectivity irregularities.

Intuitively, shortest path distances are always longer than the true distance because of their

zig-zagnature, as illustrated in Figure 5.6(a). Shortest paths straighten out as the network density

82

increases and should asymptotically approach the true distance as density goes to infinity. However,

any shortest path algorithm will preferentially choose underestimated ranging estimates and avoid

overestimated range estimates, i.e. given a choice betweentwo similar paths, the algorithm will

choose the one with a higher proportion of underestimates because it will be shorter. This effect

is illustrated in Figure 5.6(b). Therefore, the positive and negative errors in range estimates do

not necessarily cancel out; shortest path algorithms are highly sensitive to underestimates and are

largely impervious to overestimates.

For this reason, the extreme underestimates in our empirical ranging data can cause dis-

proportionate errors in shortest path estimates; they effectively create shortcuts through the network

and divert many shortest paths, causing widespread errors through awormholeeffect. This is illus-

trated in Figure 5.7(a), which shows average shortest path distance errors for Models 1, 3, and 5.

Shortest paths from Model 3, which contains noise irregularities, can be as much as 50% shorter on

average than those from Model 1, even though the nominal empirical ranging error is only around

5–10%; the few range estimates that are extremely underestimated cause errors in a large number of

shortest paths. The same graph shows that the shortest pathsget even shorter with increased network

density. This is because the distance vector algorithm has more reasonable alternatives to a shortest

path in a very dense network than in a very sparse one and will therefore have more opportunities

to use underestimates or avoid overestimates.

The effects of underestimates are exacerbated by the effects of long-range proficiency.

The shortest path algorithm prefers to use long ranges and largely ignores short ranges because

long ranges tend to decrease the shortest path distance, as illustrated in Figure 5.6(c). Therefore,

the fact that our empirical ultrasound data has more long ranges and less short ranges, i.e. that it

has both long-range proficiency and short-range deficiency,means that shortest path distances will

be decreased further, even though network density remains the same. This effect is illustrated in

Figure 5.7(a) where, for each density, the shortest path distances are shorter for Model 5 than for

Models 1 or 3.

Thus, extreme underestimates and long-range proficiency combine to yield shortest path

distances which are up to 50% shorter than the true distances. This is extremely detrimental to

the Bounding Box algorithm because it can result ininconsistentbounds; when the shortest path

distances become too short, the upper and lower bounds defined by two or more anchor nodes

do not overlap. In this scenario, nodes cannot be localized,as seen in Figure 5.5(b). As noise

and connectivity irregularities are introduced, the percentage of nodes localized by Bounding Box

quickly drops to the extent that Bounding Box cannot localize most nodes when subject to Model

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(b) Bounding Box: percentage localized

Figure 5.7:The Effect of Density(a) is to make shortest paths get shorter as density increases andas ranging irregularities from Models 3 and 5 are introduced. In (b), this leads to many nodes notbeing localized by the Bounding Box algorithm.

84

5, even at low densities. The shortest path errors at each density are strongly correlated with the

percentage of nodes localized by bounding box, as shown by the two graphs in Figure 5.7.

5.3.2 DV-Distance

Like the previous algorithm, DV-Distance [63] approximates the distance between a node

i and an anchor nodej to be the shortest path distancespij. DV-Distance then uses this value to

constrain the position of nodei in terms of the position of each anchor nodej with an equation of

the following form:

(xi − xj)2 + (yi − yj)

2 = sp2ij

In contrast to Bounding Box, this strict equality relates both x andy coordinates in the same equa-

tion, forming a circular constraint. A system of at least three such equations can be linearized and

solved with least squares for the coordinatesxi andyi, as explained in Section 3.5. In this way,

DV-Distance directly solves for node position by using the shortest path distances to reduce the

multi-hop localization problem to single-hop localization.

Because DV-Distance makes the same shortest-path approximation as Bounding Box, it is

susceptible to the same ranging irregularities: extreme underestimates and long-range proficiency.

DV-distance solves directly for a point estimate of each node’s position, however, and does not

suffer from the problem of inconsistent bounds as Bounding Box does. Therefore, DV-Distance is

always able to localize all non-anchor nodes in the network,as shown in Figure 5.3(b).

To deal with noise in the shortest paths, DV-distance exploits the fact that each anchor

node can calculate the ratio of the shortest path distance and the true distance to all other anchors.

This ratio is broadcast to the network as acorrection factor, which can be used by neighboring

nodes to adjust shortest path estimates before localization. The correction factor at anchork for

anchorj would be of the following form:

Ckj =

√

(xj − xk)2 + (yj − yk)2

spjk

(5.7)

A nodei near the anchor nodek can improve its own shortest path estimate toj by multiplying it

by the correction factor

spij = Ckj · spij (5.8)

Anchor correction factors are intended to fix exactly the kind of systematic errors in short-

est path distances that trouble Bounding Box. Indeed, Figure 5.8 shows that corrections factors

cause the median shortest path distance error with Model 5 tobe very similar to that achieved with

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Figure 5.8: Anchor Corrections reduce systematic bias, making the shortest path distancesofModel 5 approximate those of Model 1. However, they do not remove the high variance that rangingirregularities cause in the shortest path algorithm.

Model 1. However, the results in Figure 5.3(a) show that DV-Distance’s localization error is affected

by empirical ranging data almost as much as Bounding Box.

Correction factors in fact are not very effective in the faceof ranging irregularities because

they do not remove the highvariance in shortest path distances caused by Model 5. Figure 5.8

shows that corrected shortest path distances from Model 5 can be off by a factor of two both before

and after correction factors. The reason for this high variance is that, as stated in Section 5.3.1, a

small number of noise or connectivity irregularities causeerrors in a large number of shortest path

distances. However, they do not cause errors in all shortestpaths, nor are all irregularities equally

damaging. Therefore, some shortest path distances will be greatly affected by ranging irregularities

while others will be unaffected. Anchor corrections apply to all shortest paths regardless, correcting

any general bias in shortest path errors but not correcting the variance.

5.3.3 MDS-Map

MDS is an analytical technique to deriven-dimensional positions ofn objects given a

complete similarity matrixD with the metric distances between them. MDS-Map [87] is a ranging-

based sensor localization algorithm that uses MDS by makingtwo approximations: 1) all range

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failures eij = ø can be approximated by shortest path distancesspij to convert the incomplete

matrix E into a complete similarity matrixD 2) the locations of all nodes can be approximated by

a 2-dimensional projection of then-dimensional positions derived through MDS. This procedure

requires the entire graphG, so MDS-Map is a centralized algorithm.

Even though MDS-Map also uses the shortest path approximation, results in Figure 5.3(a)

show that it is much more robust to underestimated ranges than Bounding Box and DV-Distance.

One shortcoming of the previous two algorithms is that they only use shortest pathsspij between

nodes and anchor nodes; DV-Distance uses one shortest path per anchor node, while Bounding

Box only uses at most four shortest paths in total: those thatdefine the highest and lowest bounds

on its x andy coordinates. In contrast, MDS-Map uses edges between all nodes simultaneously,

dramatically increasing the number of constraints used to determine a node’s location and reducing

the ability of a few underestimates to have significant influence.

MDS-Map also shows a marked increase in sensitivity to extreme overestimates with

respect the Bounding Box and DV-distance, which were both impervious to them. This is because

Bounding Box and DV-Distance use shortest paths to estimateprimarily long distances between

nodes and anchor nodes while MDS-Map estimates the shortestpath differences betweenall pairs

of nodes, most of which are relatively close together. Shortest paths of one or two hops are more

likely to be affected by extreme overestimates than shortest paths of many hops; the shortest path

algorithm can usually choose between many alternative routes for long paths but not for short ones.

The first two algorithms therefore showed a general bias towards underestimated shortest paths

while MDS-Map shows instead a much higher total variance, with many underestimatedandmany

overestimated edges.

5.3.4 GPS-Free

While the previous three algorithms are what we callshortest pathapproximations, the

next three are what we callpatch and stitchapproximations. The patch localization algorithm used

by GPS-Free uses the center nodei of the patch and two connected neighbor nodes{ j, k | eij , eik, ejk ∈E} to bootstrapa coordinate system by assigning the following coordinates:

(xi, yi) = (0, 0)

(xj , yj) = (dij , 0)

(xk, yk) = (dik cos(γ), dik sin(γ))

87

where

γ = arccos((dij)

2 + (dik)2 − (djk)2

2(dij ∗ dik)) (5.9)

These three nodes become anchor nodes that define a local coordinate system. The other nodes in

the patch are localized usingiterative multi-lateration[82]: any node connected to at least three

anchors is first localized and then its new position estimateis used to localize other nodes. This

process repeats until all possible nodes are localized. GPS-Free chooses the two bootstrapped an-

chorsj andk to maximize the total number of neighbors in the patch that can be localized through

iterative localization. This criteria does not uniquely specify the pair, and in our implementation we

randomly chose any pair that met this criteria.

The only significant change to the GPS-Free localization error is caused by extreme over-

estimates. However, this algorithm is most likely sensitive to several ranging irregularities, but the

effect is not statistically significant because the magnitude of the error has already approached that

of random node placement. Therefore, we indicated most results of this study asinconclusivein

Table 5.4. The extreme sensitivity of this algorithm to noise can be attributed to Iterative Multi-

Lateration, GPS-Free’s patch localization algorithm, which exhibits the same trend when subject to

ranging irregularities. Iterative Multi-Lateration is highly sensitive to noise irregularities because

each step in the process uses only a few range estimates, and subsequent steps build upon the results

of earlier steps. There are very few extremely underestimated or overestimated ranges in the data

set, but if one of them is used in an early stage of localization, the errors it causes will propagate to

all other nodes in the cluster.

The more surprising fact about GPS-Free localization is that the percentage of localized

nodes is actuallyimprovedwhen subjected to long-range proficiency and short-range deficiency.

These irregularities do not change the average degree of thenetwork or the average number of

nodes localized in each cluster. Instead, it makes the neighbors in each cluster slightly farther away

from each other on average. This decreases the number of times that the intersection of two clusters

Nij is a co-linear set, causing a slight increase in the number ofpatches that can be stitched.

5.3.5 Robust Quads

Robust Quads is a patch and stitch approximation that attempts to improve on GPS-Free

by preventing some of the biggest errors in iterative multi-lateration. The robust quads algorithm

defines a parameter of robustnessθ which is usually set to3σ, whereσ is the nominal standard

deviation of ranging noise. A triangle of neighborsi,j, andk is defined to be robust ifd cos2(φ) > θ,

88

Figure 5.9:Robust Quadswith this cluster cannot localize the top six and the bottom three nodes inthis cluster to a common coordinate system. At only 66% localized, this patch has a high probabilityof stopping the stitching process.

whered is the shortest edge andφ is the smallest angle in△ijk. A clique of four nodesi,j,k,l is

defined to be arobust quadif all triangles between these nodes form robust triangles.To localize a

patch, this algorithm first finds all robust quads from the(|Ni|

4

)

quadrilaterals in the patch. It forms an

overlap graphGo = (Vo, Eo) where each vertexq ∈ Vo is a robust quad and vertices are connected

iff the two quads have at least three nodes in common, i.e.eqp ∈ Eo ⇐⇒ |q ∩ p| = 3, q, p ∈ Vo.

Three nodes from one robust quad are used to bootstrap a coordinate system, as in GPS-Free, and to

localize the fourth node in that quad. Then, all other nodes in the patch are localized using iterative

multi-lateration, with the order in which nodes are localized defined by a breadth-first search through

the overlap graph. Because quads are fully connected, a quadthat overlaps with a localized quad is

guaranteed to have at most one unlocalized node that is connected to three localized nodes. Robust

Quads does not specify how to choose the root of the breadth-first search. In our implementation,

we choose any quad that contains the center nodei of the patch. As with GPS-Free, we use the

stitching order defined by MDS-Map(P).

Because Robust Quads tends to localize nodes using long ranges and avoids localizing

nodes using short ranges, it is very sensitive to extreme overestimates and relatively robust to ex-

treme underestimates. The dominant characteristic of Robust Quads, however, is that with Models

1–3 less than 25% of the nodes can be localized to a global coordinate system in our topology. This

is not true for the Robust Quads patch localization algorithm, which localizes on average 65% and

55%, respectively. Therefore, we can assume that the reasonmost nodes cannot be localized is

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Figure 5.10:Robust Quadsgoes very quickly from localizing almost no nodes to localizing almostall nodes. Long-range proficiency triggers this phase transition at lower densities than Models 1 or3.

because of a failure in the stitching algorithm.

Through inspection of several networks in which no more patches could be stitched to-

gether, we found that many of the patches on the fringe of the localized section were similar to the

patch illustrated in Figure 5.9. This patch contains five robust quads, the top three of which overlap

and the bottom two of which do not. Therefore, the top six nodes can be localized to a common

coordinate system while the bottom three can not, i.e. 66% ofthe nodes could be localized. This

causes a problem during the stitching process when only halfof the cluster overlaps with a cluster

that is localized to the global coordinate system; the cluster cannot propagate the coordinate system

to the other side of the cluster, and stitching stops. In a network where most patches are localized to

only 50–60%, the probability of each patch stopping the stitching process is high enough that only

a small number of patches can be stitched.

Subjecting Robust Quads to long-range proficiency improvesthe percentage of nodes that

can be localized to a global coordinate system. In the patch localization process, the percentage

of nodes localized increases to about 80%. This is because a patch with more long ranges is more

likely to have robust triangles. This is similar to the reason why GPS-Free localized more nodes

90

(a) Not a Robust Quad, but robust (b) Not a quad, but robust

Figure 5.11:Robust Quads is Overly-restrictivein the sense that it will not allow the three anchornodes in either of these topologies to localize the fourth node, even though it could be performedrobustly.

when subjected to long-range proficiency. While a change from 50–60% to 80% is not very high, it

is enough to trigger aphase transitionin which the probability of each patch stopping the stitching

process becomes low enough that most patches can be stitched. This phase transition is evident in

Figure 5.10. With both Models 1 and 3, the transition occurs between average degrees of 10 and

12. When long-range proficiency is introduced, however, thetransition occurs between degrees of 6

and 7.5. Long-range proficiency also causes the average localization error to increase, approaching

that of GPS-Free. This is presumably because the new nodes that are being localized are those with

higher localization error.

It is worth mentioning that the entire patch in Figure 5.9 canbe localized with respect to

a single coordinate system without any danger offlip or discontinuous flexambiguities described

by Moore [60]. The patch is not completely localized, however, because the Robust Quads patch

localization algorithm is too restrictive in that it does not allow multi-lateration even in cases where

it can be performed robustly, as illustrated in Figure 5.11.

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5.3.6 MDS-Map(P)

MDS-Map(P) uses MDS-Map as a patch localization algorithm.The shortest paths be-

tween all nodes in patchi are calculated and combined with the range estimates to forma complete

similarity matrix Di, which is used to localize the nodes in the patch relative to each other. No

anchor bootstrapping is required. The local coordinate systems are then stitched together to form a

global coordinate system using the algorithm described earlier. The original MDS-Map(P) proposal

suggests using patches of nodes that are more than one hop from the center of each patch. In our

implementation, we use one-hop patches to make the algorithm comparable to the GPS-Free and

Robust Quads algorithms.

The characteristics of MDS-Map(P) follow the same trends asthose of MDS-Map: the

algorithm is more sensitive to connectivity irregularities than to noise irregularities, although sig-

nificantly affected by both. The main difference between thetwo algorithms is that MDS-Map(P)

has consistently higher error, regardless of the ranging model used, because the stitching algorithm

is amplifying the errors introduced during the patch localization process. By comparing the local-

ization of MDS-Map and MDS-Map(P) on a network that requiredchains of up to 25 stitches, we

can infer that the stitching process amplifies errors by a factor of two or less.

One problem with GPS-Free and Robust Quads is that they use a chain of calculations

(i.e. iterative multi-lateration), each of which depend ona small number of ranging estimates.

A single noisy ranging estimate that causes an error early oncan be very damaging due to error

propagation. MDS-Map(P) makes headway on this problem by replacing the fragile chain with a

relatively robust single computation for patch localization. However, it does not do the same for

the stitching algorithm. All algorithms including MDS-Map(P) use a greedy stitching algorithm

that stitches the un-stitched patch that has the largest overlap with a stitched patch. This stitching

algorithm is not robust to noise: a single badly localized patch can (and does) cause severe stitching

error, which propagates through all subsequent stitches.

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Chapter 6

Removing the Prediction Gap

In this chapter, we close the Prediction Gap by building a parametric model of our raw

ultrasound data. In the previous sections, we have seen thatthe traditional parametric model is

insufficient and that non-parametric models provide much more accurate simulation. However,

parametric models are preferable to non-parametric modelsfor several tasks, including analytical

proofs and the mathematical derivation of error bounds. Thevalue of a parametric model in this

regard, however, depends on its balance of accuracy and simplicity. Capturing all irregular features

of the data may require a sophisticated model with a large number of parameters, which would be

difficult to use for analysis. Simplifying the model and reducing the number of parameters, on the

other hand, can reduce the accuracy of the model and make any such analysis meaningless. The goal

of this chapter is to explore the tradeoff between accuracy and simplicity in the context of ultrasonic

range sensors by creating a simple model that captures the most salient features of our empirical

ultrasound data.

The aspects of our data that we would like to capture are motivated by the hybrid model

analysis in Chapter 5, which identified four irregularitiesin ranging connectivity and noise char-

acteristics that can effect localization error. Some of these irregularities are only slight deviations

from existing models. For example, even though ranging noise roughly conforms to the Normal dis-

tribution, a relatively small number of outliers can be devastating to some localization algorithms.

This indicates that our model must capture not only the general structure of the data, but must also

exhibit small deviations from that structure.

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6.1 Existing Models of Irregularity

The Unit Disk model of connectivity is known to be a simplification of true link-layer

characteristics. Instead of only two regions of connectivity, where all nodes are completely con-

nected at close distances and all nodes are completely disconnected at far distances, classical models

would also predict a third region of unreliable connectivity where nodes are connected with some

probability 0 < p < 1. This so calledtransitional regionoccurs when the signal to noise ratio

(SNR) is high enough that packets can still be received but low enough that the probability of bit

errors becomes substantial. This region is expected to be small, and classical models would predict

that in conditions where a low-power radio has about 20m range, it might have a transitional region

of about 2m [104]. However, several recent empirical studies have demonstrated that the transi-

tional region can actually occupy over half of the maximum usable range [23,99,101], as illustrated

in Figure 6.1. Effects of the transitional region can therefore dominate certain aspects of wireless

networking, changing the expected pattern of network floods, the structure of routing trees, and the

techniques used for reliable data collection. Because of its large impact, several new models have

been created to explain the unexpectedly large size of the transitional region.

In this section, we attempt to use several of these models to explain the ultrasound con-

nectivity irregularities identified in previous sections.Short-range deficiency and long-range pro-

ficiency in ultrasound have many properties in common with the transitional region in radio links.

One key difference is that this region occupies the entire useful range of ultrasound while it only

occupies a portion of radio range, so any explanation to be borrowed from the wireless networking

literature would need to be especially prominent in ultrasound. It would also need to be consistent

with the many differences between the physical models of radios and range sensors.

6.1.1 Non-uniformity of Nodes

One popular belief is that the transitional region in radio connectivity is caused by a non-

uniformity of nodes; variations in radio circuitry, antenna orientation, etc. may cause some nodes to

transmit with more power than others or to be more sensitive receivers, causing each node to have a

different maximum range. Even if the individual nodes have anarrow transitional region, therefore,

the transitional region for all nodes in aggregate would be wider than expected, as illustrated in

Figure 6.2.

A similar phenomenon could occur with ultrasonic range sensors: variance in the time

constants of the oscillators or in the orientation of the reflective metal cone could cause some nodes

94

Figure 6.1:The Transitional Region for radio connectivity violates the Unit Disk model of connec-tivity and can impact several wireless networking protocols. With low-power radios, it can occupyover half of the usable range, as shown in this figure. Figure reproduced with permission [99].

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(a) Narrow Transitional Region for Individual Nodes (b) Wide Aggregate Transitional Region

Figure 6.2:Non-Uniformity of Nodes is believed to cause a wider than expected transitional re-gion; if a) each node has a narrow transitional region individually, but different range then b) thetransitional region in the aggregate would appear wide.

to transmit louder or receive with a lower SNR than others, causing each node to have a different

range. To test this theory, we analyzed the probability thateach node would obtain a range estimate

in each of five different regions:

• 1-100cm

• 101-200cm

• 201-300cm

• 301-400cm

• 401-500cm

The data used for this analysis is the same data collected in Section 4.2.2 that is shown in Figure 3.2.

The data collection process required that each node range toall other 24 nodes at up to five different

distances. A high variance among nodes would be observed if some nodes could range only at

short distances, other nodes could range at both short and mid-distances, and the rest could range

at all distances. Figure 6.3, however, shows that our ultrasound range sensors do not exhibit this

trend. There is some variation among nodes, but the transitional region for each individual node is

comparable to the aggregate transitional region for all nodes. All nodes are almost equally likely to

96

obtain a range estimates in the five different regions. For clarity, this figure only shows data for 12

random nodes of the 25 used in the data collection process.

Our analysis of ranging errors in Section 3.3 also supports the conclusion that there is low

variance among the individual ultrasound devices. In that section, we calibrated the nodes using

a single set of calibration parameters for all nodes, and then calibrated them again using different

calibration parameters for each node to account for any variance among them. If there was a high

variance among devices, the second calibration process would produce much lower range errors.

However, we saw that the two calibration methods produced very similar range errors, indicating

that there is little variation among nodes.

6.1.2 Radio Irregularity Model

Another explanation in the wireless networking literaturefor the unexpectedly large tran-

sitional region in low-power radio links is called the RadioIrregularity Model (RIM) [102], which

argues that the radio range is non-isotropic; instead of having a single range in all directions, the

range boundary varies based on the angle of transmission, asshown in Figure 6.4. According to this

model, a node would only be connected to a portion of the neighbors between the outer and inner

limits of the range boundary, which would create the illusion of a transitional region.

Non-isotropic propagation could be caused by shadowing andmulti-path effects, as well

as attributes of the radio itself. Especially with platforms like the mica mote, the layout of the hard-

ware components with respect to the radio and antenna could effect the propagation of low-power

radio signals. A similar phenomenon could feasibly be observed with our ultrasound transducers; if

the reflective cone were off-center, as shown in Figure 6.5, the acoustic signal would be stronger in

one direction than another.

To test the RIM model against our ultrasound data, we analyzed the pattern of connectivity

for each of our nodes in the two-dimensional topology shown in Figure 4.3(a). The RIM model

would predict that, once a node fails to range to one node, it will fail to all other nodes at the same

angle of transmission. However, the contour map in Figure 6.6, which illustrates the probability with

which a node can obtain a range estimate to a neighbor, shows that the probability of ranging does

not decrease monotonically: some nodes yield a higher probability of obtaining a range estimate

than closer nodes that are at the same angle of transmission.A weak reflection of acoustic energy

in a particular direction would affect all nodes in that direction, and so isolated spatial holes in

connectivity violate the predictions of the RIM model.

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(b) 12 Nodes as Receivers

Figure 6.3:Uniformity of Ultrasound Nodes This graph shows the probability that a node canobtain a range estimate at each of 5 different distances, when acting as a) the transmitter and b) thereceiver. All nodes are equally likely to obtain range estimates in each of the five regions, showingthat non-uniformity is the not the cause of the wide transitional region in ultrasound.

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Figure 6.4:The Radio Irregularity Model postulates that transmitters have non-isotropic rangewhich varies with the angle of transmission. This could cause the illusion of a wide transition regionbecause nodes at the same distance could be either connectedor disconnected. Figure reproducedfrom [102] with permission.

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Figure 6.5:Sources of Non-isotropic Ultrasoundcould include, for example, an off-center reflec-tive cone, as shown in this figure. This would cause a non-isotropic transmission range, as predictedby the RIM model.

6.1.3 Gaussian Packet Reception Rate Model

Because the cause of the transitional region is not obvious,early work in reliable multi-

hop routing with low-power radios attempted to build a statistical model of the transitional region,

without explaining the phenomenon [99]. At each distance, the model used the mean and standard

deviation of the packet reception rates for all pairs of nodes observed at that distance. This model

applied to the data in Figure 6.1 can be illustrated by the mean and error bars in Figure 6.7.

This model does not explain the transitional region and doesnot necessarily provide an

accurate description. In fact, it yields probabilities outside the range[0, 1]. However, the model de-

scribed the transitional region in radio links well enough to be useful for reliable multi-hop routing,

and may similarly be useful for multi-hop localization withultrasound. Applying the model to our

ultrasound data, however, reveals that it does not capture the bi-modality of the connectivity data;

most nodes have either a very high or very low probability of obtaining a range estimate, but the

Gaussian assumption of this model can only capture one or theother, as shown in Figure 6.8. At far

and near distances, the model places most probability density of unconnected and well connected

pairs, respectively. At mid distances, the model places most probability density in between, where

only a small percentage of nodes can be found.

100

Figure 6.6:Ultrasound Connectivity Contours show that connectivity is non-monotonically de-creasing: close nodes will often be disconnected while far nodes are connected, even when angle oftransmission is held constant. This is not explained by the RIM model.

101

Figure 6.7:The Gaussian Packet Reception Rate Modeluses aggregate statistics like the meanand variance of packet reception rates at each distance to model the transition region. Figuremodified from [99] with permission.

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0 100 200 300 400 500 600

0

0.2

0.4

0.6

0.8

1

Distance (cm)

Pro

babi

lity

of R

angi

ng

Figure 6.8:Ultrasound Connectivity cannot be modeled with an average and standard deviationat each distance because the model does not capture the bi-modal distribution observed at mostdistances.

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Figure 6.9:Shadowing and Multi-path can combine to create a wide transitional region. Each pairof nodes will observe different signal attenuation due to shadowing and multi-path, causing them toapproach the noise floor at very different distances. Figurereproduced with permission [104].

6.1.4 Shadowing and Multi-path

The last model we examine explains the unexpectedly wide transitional region in terms of

natural variation in signal attenuation [104]. The radio signal is assumed to attenuate according to a

log-normal path loss model given by

PL(d) = PL(d0) + 10n log10(d

d0) + Xσ (6.1)

whered0 is a reference distance,n is the coefficient of attenuation, andXσ is zero-mean Gaussian

noise. Zuniga et al. assume thatXσ is a constant value for a particular transmitter/receiver pair

due to effects like shadowing and multi-path,1 and that the Normal distribution only hold in the

aggregate over all pairs [104]. Thus, a particular transmitter/receiver pair will consistently receive a

stronger or weaker signal than the path loss model would predict, bounded with high probability by

PL(d) ± 2σ

1Shadowingis signal attenuation due to an obstacle between the transmitter and receiver andmulti-path is signalenergy received from reflective surfaces.

104

.

The probability of a bit error while receiving a radio message is given by

Pe =1

2exp(−SNR

2) (6.2)

This value is approximately0 when the SNR is high and exponentially approaches12 as SNR goes

to zero. If the transitional region is defined to be the distance where the packet reception rate is

0.1 < p < 0.9, the beginningds and endde of the transitional region can be derived as

ds = 10Pn+10 log10(−1.28 ln(2(1−0.9

18f )))−Pt+PL(do)+Xσ

−10n (6.3)

de = 10Pn+10 log10(−1.28 ln(2(1−0.1

18f )))−Pt+PL(do)+Xσ

−10n (6.4)

wheref is the number of bytes in a packet. The value ofXσ for a particular transmitter/receiver

pair can greatly change the position of the transitional region, as shown in Figure 6.9. In this

scenario, whenXσ = 0 the transitional region occurs between 18–20m. WhenXσ = −2σ, it

occurs between 11–12m, and whenXσ = +2σ it occurs between 29–32m. Thus, according to

this model, the cause of the transitional region is not inherent in the nodes themselves or in the

angle of transmission, but in the two-dimensional space between the transmitter and receiver that

causes a particular combination of shadowing and multi-path effects. Non-isotropic transmission

and reception, as described by the RIM model above, may contribute to this effect.

The concepts behind this model may explain the connectivitycharacteristics in question,

but the physical model of ultrasound ranging is very different from the model of radio communica-

tion. For example, the probability of an ultrasound error does not approach12 as SNR goes to zero,

and the probability of successful transmission does not decrease exponentially as the packet length

grows. Quite the opposite, the longer an ultrasound transmission, the higher is the probability that

it might be received. Furthermore, any model of ultrasound ranging must not only explain connec-

tivity characteristics but also the observed noise distribution. The following section adapts several

of the underlying concepts from this model to the physical dimensions of ultrasound ranging.

6.2 Towards a New Parametric Model

In this section, we attempt to formulate a parametric model of ultrasonic ranging. Unlike

the models of the transitional region in radio connectivitydescribed above, this model must capture

both connectivity and noise characteristics. As with the radio models, we will assume log-normal

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Figure 6.10:The Ultrasonic Emanation Pattern is the result of reflecting a cone with 25 degreespread into a plane. The acoustic energy is distributed on the frontal surface of the resulting radiallydistributed cone, with surface area of4πd2tan(25o

4 ).

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signal attenuation, as given by Equation 6.1. The coefficient of attenuationn can be derived from

a physical model of ultrasound propagation. The ultrasonictransducers used in our implementation

emanate acoustic energy in a cone with a 25 degree spread. The45 degree metal reflective surfaces

placed above the transducer reflect that energy into a 360 degree, radially distributed cone with a

12.5 degree vertical spread, as illustrated in Figure 6.10.At distanced, the outer edge of this region

has heighth = 2d tan(25o

4 ) and widthw = 2πd. The acoustic energy is therefore distributed over a

surface with areaA = h×w = 4πd2tan(25o

4 ). Thus, the density of acoustic energy should decrease

proportional to 1d2 , i.e. the theoretical coefficient of attenuation for our ultrasonic transducers is2.

6.2.1 A Geometric Noise Distribution

Once the acoustic energy arrives at the receiver, it will detect the first full wavelength of

the signal with probabilitypdetection. If we allow the startup time of the oscillator to be accounted

for by the calibration process, we can make the simplifying assumption that every successive full

wavelength of the signal will also be detected with the same probability pdetection. The number of

wavelengthsω needed to first detect an ultrasound signal will then follow the geometric distribution,

given by

P (ω) = (1 − pdetection)ωpdetection (6.5)

We can assume the probabilitypdetection is related to the signal attenuationPL(d). Unlike

the previous model, however, the probability does not approach 12 as the received power approaches

zero. Instead, the probability of detecting an ultrasound signal approaches zero as attenuation in-

creases. This relationship can be given as

pdetection = exp(−PL(d)

α) (6.6)

whereα is a scaling factor that determines the rate of attenuation over distance. The model is

actually slightly more complicated because all received signals above a certain power rail the

amplifiers in our ultrasound circuit and therefore have approximately the same probability of be-

ing detected. IfPt is the transmission power then the power received at a certain distance is

Pr = Pt − PL(d). This value is lower than the maximum receivable powerPrail by the quan-

tity BR(d) = max(0, Prail − Pr), and the probability with which the signal is detected becomes

pdetection = exp(−BR(d)

α) (6.7)

Before the ultrasound signal arrives and after it has ceased, we can assume that a node

will generate false positives with a constant probabilitypf , i.e. the node will incorrectly detect a

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signal when none is present. The probability of detection can be given for each of three different

regions: 1) before the ultrasound arrives 2) while the ultrasound is being received, and 3) after the

ultrasound pulse has ceased. Assumingv is the speed of sound,f is the ultrasound frequency, and

τ is the time duration of the ultrasound pulse, these probabilities are given by

pbefore(ω, d) = (1 − pf )ωpf (6.8)

pduring(ω, d) = (1 − pf )d·fv · (1 − exp(−BR(d)

α)ω−

d·fv · exp(−BR(d)

α) (6.9)

pafter(ω, d) = (1 − pf )ω−τf · (1 − exp(−BR(d)

α)

τf · pf (6.10)

With our ultrasound implementation,f = 25000, τ = 0.008sec, andPt = 75dB. The transducers

have a maximum sensitivity of−60dB.

We used our model with parametersσ = 17dB and α = 10 to generate data at the

same distances used to collect the empirical data in Figure 3.2. The generated data, shown in

Figure 6.11(a), captures several salient features of the empirical data set, which is reproduced in

Figure 6.11(b) for clarity

• at short distances, ultrasound is detected quickly

• the nominal time to detect ultrasound increases with distance

• only a small number of detections require more than4ms at all distances

• false positives are uniformly distributed both before and after ultrasound arrives

• false positives are more dense before ultrasound arrives than after

6.2.2 An Exponential Model of Connectivity

The geometric distribution appears to capture the speed with which an ultrasound signal is

detected, and therefore accurately captures time of flight noise characteristics. However, it does not

accurately capture connectivity characteristics. Because we used 8-millisecond ultrasound pulses

with a signal period was 40-microsecond, the geometric model would predict that a signal remains

completely undetected with probability

preceived = (1 − exp(−BR(d)

α))

800040 (6.11)

Unfortunately, this is an overestimate. Figure 6.12 shows akernel regression of the frequency with

which each pair obtains a range estimate at each distance. Inthe empirical data set, this frequency

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50 100 150 200 250 300 350 400 450 500

10

20

30

40

50

60

True Distance (cm)

Tim

e of

Flig

ht (

mse

c)

(a) Data Generated from Geometric Noise Distribution

50 100 150 200 250 300 350 400 450 500

10

20

30

40

50

60

True Distance (cm)

Tim

e of

Flig

ht (

mse

c)

(b) Empirical Data

Figure 6.11:The Geometric Noise Modelproduces data shown in a). This data captures many ofthe salient features of the empirical ultrasound data, shown in b).

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follows a binary distribution: most pairs at short distances yield 100% while a small number yield

0% of range estimates. At longer distances, an increasing number yield 0%. In the generated data,

however, almost all pairs yield between 50–100%.

Connectivity is certainly related to signal attenuationPL(d), but the random component

causing the connectivity characteristics are orthogonal to that causing the noise characteristics. This

is evident from the empirical data at short distances, for example, where almost 20% of range es-

timates are not received at all, but those that are received are detected very quickly. This pattern

is possible if connectivity can be determined by a binary indicator, such as an off-center reflec-

tive cone for example, but, given that the cone is not off-center, the speed with which a signal is

detected is affected only by environmental factors. For this reason, we use independent random

noise componentsXσ1 andXσ2 in the equations for deriving noise and connectivity components,

respectively.

In our ultrasound ranging hardware, we used an analog comparator to detect ultrasound

signals above a certain power thresholdPthresh. A more complete model of connectivity would

therefore predict that yield decreases exponentially as the received power approachesPthresh. The

amount that a received signal is above threshold isAT (d) = max(0, Pr − Pthresh), providing the

following probability of detecting a signal

preceived = 1 − exp(−AT (d)

α) (6.12)

The valuespreceived andpdetection are similar, but are dependant on inversely proportional

factors; the amount thatPr is below the railBR(d) increases with distance while the amount that

it is above thresholdAT (d) decreases. The probability with which a signal is detectedpdetection

decreases slowly asPr decreases. WhenPr approachesPthresh, however, the signal can no longer

be detected andpreceived drops very quickly. The different power relationships and their significance

with respect topdetection andpreceived are depicted in Figure 6.13.

We can apply this exponentially decreasing model of connectivity to the geometric noise

model in the previous section by first calculatingpreceived and generating a random valuer = [0, 1].

We can then replace Equation 6.7 with the following equation

pdetection =

exp(−BR(d)α

) r < preceived

pf otherwise(6.13)

Thus, if the acoustic signal is detectable, it will be detected with probabilitypdetection. Otherwise,

it will be detected with the same probability as a false positive.

110

0100

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Pro

babi

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Den

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(ke

rnel

sm

ooth

ing)

(b) Yield of Empirical Ultrasound

Figure 6.12:Connectivity Characteristics of the Geometric Noise Modelare very different fromthose of the empirical ranging data.

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Figure 6.13:Power Relationsillustrated in this diagram are used for modeling ultrasound. Pt isthe transmission power,PL(d) is the path loss at distanced, Pr is the received power,andPrail andPthresh are the maximum and minimum receivable powers, respectively. preceived is proportional toBR(d) = Prail − Pr whilepdetection is proportional toAT (d) = Pr − Pthresh.

112

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10

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60

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Tim

e of

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mse

c)

(a) Data Generated from Complete Model

0100

200300

400500

600

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x 10−3

Distance (cm)Ranging Yield

Pro

babi

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Den

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(ke

rnel

sm

ooth

ing)

(b) Connectivity Characteristics of Complete Model

Figure 6.14:The Complete Parametric Modelcombines the Geometric noise distribution with theexponential model of connectivity. This model produces a) noise characteristics and b) connectivitycharacteristics that represent the main features of the empirical ranging data.

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We used the complete model to generate data at the same distances used to collect the

empirical data in Figure 3.2. The generated data, shown in Figure 6.14, captures empirical con-

nectivity characteristics as well as our simple geometric model of noise. However, the connectivity

characteristics of the generated data now also match those of the empirical data set. Instead of most

range estimates being received between 50–100% of the time,as they were in Figure 6.12(a), most

estimates are received with probabilities of either 0 or 100%.

6.2.3 Verifying the Model

The previous two sections provide abottom-upapproach to modeling, which focuses

primarily on inspection and comparison of the salient features in the data set. However, inspection

is not a complete metric of evaluation for a model because it relies on our understanding of the

data set’s salient features. In this section, we provide atop-downmodel verification process in

which we run a localization algorithm on data generated fromthis model and compare the resulting

localization error with that observed in a real-world deployment.

The results of our comparison are shown in Figure 6.15, whichshows the localization

error in our empirical deployment and compares with the error predicted by the Noisy Disk model,

the non-parametric model, and our new parametric model. Based on the fact that our new parametric

model predicts the empirical deployment results as well as the non-parametric model, we conclude

that this is a sufficiently accurate model of ultrasound ranging.

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Noisy Disk Non−parametric Generated Deployment0

50

100

150

200

250Upper Quartile, Median, and Lower Quartile

Loca

lizat

ion

Err

or (

cm)

Figure 6.15:A Top-down Parametric Model Evaluation is possible by using data generated fromthe model in simulation and comparing with an empirical deployment. This figure shows that themodel predicts the empirical deployment as well as the non-parametric model.

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Chapter 7

Conclusions

The Prediction Gap is a crippling problem in localization for three main reason:

1. The localization error of a real deployment is difficult topredict. This can be a problem for

mission critical deployments which can only be deployed once, such as forest fire tracking,

or for large deployments with 1000’s of nodes where the cost of redeployment is prohibitive.

2. Localization algorithms are difficult to evaluate and improve. Improvements made to an algo-

rithm in simulation do not necessarily translate to improvements in a real deployment because

the sources of error may be very different. Similarly, a comparison of two algorithms in sim-

ulation may not produce the same conclusion as a comparison on a real deployment.

3. Empirical error is difficult to explain. If everything known about the environment and range

sensor is incorporated into a theoretical model which produces low errors in simulation, then

the cause of any additional error observed in the real deployment is not known. If the cause

of the additional error is not known, it is difficult to reduce.

In this study, we thoroughly address the problem of the Prediction Gap. We first establish

the existence and magnitude of the Prediction Gap by building and deploying a sensor localization

system and comparing observed localization error with predictions from the traditional model of

ranging. We then develop new non-parametric modeling techniques that can use empirical ranging

data to predict localization error in a real deployment. These non-parametric models require a

special form of data collection that ensures a thorough, high-resolution profile of our range sensor

and its environment. That empirical profile is then used directly in simulation through statistical

116

sampling techniques. Our non-parametric models are key to closing the Prediction Gap, and solve

many of the issues present in existing simulations.

Once we close the Prediction gap, we proceed to identify its causes by creating hybrid

models that combine components of our non-parametric models with traditional parametric models.

By comparing localization error from a hybrid model with a purely parametric model, we isolate

the effects of a single component of our data. We use this technique to identify the causes of

the Prediction Gap for six different localization algorithms from the literature. We then use this

knowledge to develop a new parametric model that captures the true characteristics of our empirical

ranging data.

7.1 Advantages of Our Modeling Techniques

Traditional techniques for the creation and validation of asensor model in order to simu-

late a deployment can be as difficult as performing the deployment itself. One must first collect data

in a sufficiently representative set of sensor contexts and distill this data down to a simple algebraic

form. This new model must then be validated by comparing its predictions in simulation to a real de-

ployment, and if the application behavior changes at scale or with different network topologies, the

model may need to be validated with several different large networks. Finally, if the model fails the

validation process, the scientist must debug the model by manually comparing the algebraic form

with the raw sensor data in order to identify discrepancies.If any of the algorithm, sensor, stimulus,

or environment change, the validation process must be repeated. Failure to validate a model may

result in a discrepancy between simulation predictions andreal-world deployments, which we call

the Prediction Gap.

The techniques we have demonstrated in this study greatly simplify the process of mod-

eling a sensor. First, we demonstrated that a complete empirical profile of a sensor can quickly

and efficiently be collected by using a large number of sensors simultaneously in the correct con-

figuration. Second, if the data will only be used for simulation, the data does not need to be an-

alyzed and distilled into an algebraic form; the data can be used directly in simulation through

non-parametric modeling techniques. Finally, if a parametric model is needed, hybrid models allow

us to refine a simple parametric model by systematically identifying discrepancies between it and

a non-parametric model. Hybrid models allow us to remove onediscrepancy at a time, in contrast

to the standard validation process which requires all discrepancies to be removed at once. This

methodology combines aspects of bottom-up modeling, in which the user analyzes the raw data and

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makes a structural conjecture, and top-down modeling, in which the user validates that conjecture

based on a particular usage.

The advantages of non-parametric models go beyond not needing to postulate an algebraic

form; they also make fewer assumptions that need to be validated. Validation of a parametric model

verifies that 1) the structural features captured by the model accurately represent the data from a

particular range sensor in a particular environment and 2) the structural features that are not captured

by the model are inconsequential for a particular algorithm. Validation of our non-parametric model,

on the other hand, verifies only that all the data in the setM(d, ǫ) is similarly distributed; if data

from some nodes were very different than data from other nodes, for example, the non-parametric

model we used would not accurately reflect the true deployment. Because the non-parametric model

does not make assumptions about the structure of the data, itdoes not need to be revalidated when

the range sensor or environment changes, as long as the data in M(d, ǫ) continues to be similarly

distributed. Because the non-parametric model does not eliminate “inconsequential” features from

the data, it also does not need to be re-validated when used with a different algorithm that may be

sensitive to these features.

7.2 Parametric vs. Non-parametric Models

As described above, our new non-parametric modeling and hybrid modeling techniques

are crucial inexplainingthe Prediction Gap. However, non-parametric models are notthe only way

to removethe Prediction Gap. In this study, we demonstrated two techniques to achieve this: 1)

to replace parametric models with non-parametric models insimulation, and 2) to make a more

realistic parametric model. Each of these approaches has its advantages and disadvantages, which

we explore here.

One main benefit of parametric models like the Noisy Disk is that they provideinsight

to the algorithm designer by identifying only a small set of ranging characteristics; there may be

hundreds of aspects of the physical world that affect each range estimate, but the model indicates

that only a few of those features are actually important to localization algorithms. Furthermore, the

algebraic form of the model can be useful for theoretical analysis. These strengths of parametric

models, however, are also their weakness; parametric models need to be reevaluated and redevel-

oped for every new range sensor and environment. This is a tedious process requiring data collection

and careful analysis followed by a model verification process that, to be complete, would require

real localization deployments. Another problem is that empirical ranging characteristics like those

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shown in Figures 4.1 can be too complex to capture in parametric form without some amount of

simplification.

Non-parametric models solve both of these problems: new models do not need to be

created for new empirical distributions and complex ranging characteristics can easily be captured.

However, non-parametric models do not provide an algebraicform that can be used for theoretical

analysis nor do they provide any insight into the data or the algorithm. For example, although our

simulations in Section 4.3 closed the prediction gap of our empirical deployment, it did not indicate

what caused the error in the real deployment, or why the non-parametric model was a better predictor

than the Noisy Disk model.

In practice, the use of parametric and non-parametric modeling carry similar costs in all

respects. Both require vast data collection that can be performed in a comparable amount of time.

During simulation, both methods require a single random number to be generated for each range

estimate. The Noisy Disk model requires the user to estimateparametersσ anddmax from the data

while our non-parametric model requires the user to choose avalueǫ. Perhaps the biggest cost of

parametric models is that the user must choose an algebraic form. However, our non-parametric

technique requires the user to properly bin the empirical data into subsets that contain similar data.

We chose to bin our data asM(δ, ǫ), for example, although other bins might be more appropriate.

For example, variations between radios and antennas can be modeled by parameterizing each node

with the quality of its transmitter and receiver. These parameters can be estimated from empiri-

cal data using joint calibration techniques [95]. During simulation, each radio could be randomly

assigned transmitter and receiver parametersT andR and data could be pooled and drawn from

subsetsM(δ, ǫ, T,R). As long as the parametersT andR are assigned according to the true dis-

tribution of radios, this should more accurately model non-uniformity of nodes than using subsets

M(δ, ǫ).

7.3 Extending Analysis to Other Areas

The area of ranging-based sensor localization is particularly vulnerable to the Prediction

Gap and was chosen for this study because it has the followingproperties:

• It relies on a physical sensor that can be noisy, irregular, and easily influenced by the environ-

ment.

• In contrast to sensor systems that perform data collection,a localization algorithm must ac-

119

tually process the sensor data. As such, any theoretical model must accurately represent the

physical sensor.

• Thorough evaluation of a localization system is necessarily performed in simulation because

performing hundreds of large-scale deployments in different topologies is impractical. There-

fore, theoretical models are necessary for research in localization.

• Small deployments do not necessarily demonstrate the Prediction Gap, and large deployments

are rare. Unlike the area of wireless networking, where years of research at the physical, link,

and MAC layers helped build realistic models of the radio that could be used for research at

the routing layer, localization has not had sufficient reality checks for the most commonly

used theoretical models of ranging.

These properties, however, are not unique to ranging-basedlocalization. They are also

true for many other distributed sensing algorithms, such astracking. In August 2005, we deployed

557 nodes with passive infrared (PIR) sensors in a field covering approximately 50,000 square

meters [18]. The sensors themselves are shown in Figure 7.1(a), and part of the deployment area is

shown in Figure 7.1(b). The Markov Chain Monte Carlo Data Association (MCMCDA) algorithm

was used on the output of sensors to track multiple objects moving through the field [64]. This

algorithm considers data from multiple parts of the networksimultaneously to associate correlated

sensor readings due to object motion and to filter out false positives from the sensors.

A naive model of the PIR sensor is that it can detect all objects within a certain radius.

However, this model is clearly not accurate for several reasons. First, each PIR sensor may be

slightly different due to manufacturing or assembly process variations. Second, each node has four

PIR sensors, one in each of the cardinal directions, and should be more likely to detect objects

directly in front of one of the sensors than in one of the corners. Third, the analog output of the PIR

sensor depends on the speed, size, distance, and direction of the object. Therefore, the node should

be more likely to detect close objects, large objects, and objects moving quickly in a direction

perpendicular to one of the sensors. Finally, because PIR sensors pick up any motion, they are

very sensitive to nearby grass and wind. The actual responsefunction of this sensor is much more

complex than the naive model of detection, making it very difficult to predict how the MCMCDA

algorithm would respond to an object as it takes a particularpath through the deployment.

To address this problem, we extended the empirical profilingtechniques described in

Chapter 4 to collect a thorough and complete empirical profile of our PIR sensor. The goal was

120

(a) Motion Sensor

(b) Tracking Deployment

Figure 7.1:A Tracking Deployment a) nodes with passive infrared (PIR) motion sensors depictedare deployed (b) in a 557 network in Berkeley, California.

121

to collect empirical data to represent the sensor response to an object moving near it in any direction

and in any environment. We designed the following data collection process:

• We take four nodes and place them in a 10x10 foot square, each node at a slightly differ-

ent orientation. We consider the first node to define a reference coordinate system, and to

be oriented by definition at 0 degrees. The other three nodes are therefore located at 22 de-

grees, 44 degrees, and 66 degrees, respectively. This layout is represented by the squares in

Figure 7.2(a).

• In the location and at the orientation of each of these nodes,we place three nodes at different

elevations. One node is placed below grass level, one at grass level, and one above grass level.

Thus, 12 nodes were deployed in total.

• One object passed through the sensing ranges of these nodes in a simple “lawn mower” pat-

tern, moving up and down in parallel tracks with 5 foot spacing, as shown by the lines in

Figure 7.2(a). At the end of each line, the position of the object was marked so that it could

be correlated in time with the sensor responses. Thirteen 60foot tracks were used to cover a

60x60 foot area.

• This experiment was repeated at three speeds of 3, 5, and 7 meters per second.

• After this was completed, the 12 nodes were re-deployed witha different permutation of node

ID to node position.

Each pass of the object in this experiment measured the response of a sensor to a single

motion vector at every point in the sensor’s two-dimensional coverage area. However, because each

node has four PIR sensors, this pass actually measured four different motion vectors at each point.

Furthermore, because we placed nodes at four different orientations, we were able to capture a total

of 16 motion vectors at every point in space. These motion vectors were measured at three different

elevations and, because the experiment was repeated with different permutations of nodes to node

positions, the experiment also captured variation in node response due to hardware variations. Thus,

by using multiple nodes simultaneously and by exploiting symmetry in the hardware, we were able

to capture almost a complete profile of sensor responses in little more than an hour. This experiment

did not profile different sizes of objects, although this could easily be measured as well.

The actual measured response of the PIR sensors for a single elevation and set of motion

vectors is shown in Figure 7.2(b). Differences and similarities between the empirical data and the

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(a) PIR Profiling

(b) PIR Response

Figure 7.2:Profiling the PIR Sensora) required a special layout of 12 nodes and a single objectmoving up and down in parallel lines. (b) This procedure measured 16 motion vectors at every pointin the two-dimensional coverage area of the sensor.

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naive model become apparent immediately through inspection, although a more thorough analysis is

required to identify which of these similarities and differences will affect the MCMCDA algorithm.

To that end, this data set can be used to build non-parametricmodels and hybrid models in much

the same way they were built for ranging in Chapters 4 and 5.

Thus, in conclusion, we observe that the Prediction Gap is not a problem specific to

ranging-based localization. It is a problem inherent to alldistributed algorithms that may be de-

ployed in large-scale networks and that process sensor data. All such algorithms will likely be

developed in simulation, will require a model of how a sensorresponds to the stimulus and de-

ployment environment, and will experience difficulty if this model does not accurately represent

the sensor and the environment. This is particularly true for applications that will use a sensor in

a new way, such as using ultrasound for localization or magnetometer and PIR sensors with the

MCMCDA algorithm for tracking, because the traditional models for these sensors may either not

exist or be designed to model the sensor in a different context. The techniques and analysis in this

study can be applied to this entire class of problems to predict real deployment characteristics, to

analyze the Prediction Gap, and to derive new, more accuratemodels of previously uncharacterized

sensors.

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