understanding the prediction gap in multi-hop localization
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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
es in
Cen
timet
ers
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28
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48
15
34
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24
27
8
12 6
25
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43
13
37
42
17
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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1
1.5
2
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Sho
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Model 1Model 3Model 5
(a) Shortest Path Errors
0 5 10 15 20 250
10
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30
40
50
60
70
80
90
100
Average Degree
Per
cent
Loc
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%)
Model 1Model 3Model 5
(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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0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3
3.5
4
Nor
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Cor
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ath
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or
Average Degree
Model 1Model 5
High Variance
Low Variance
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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50
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80
90
100
Average Degree
Per
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Model 1Model 3Model 5
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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1 2 3 4 5 6 7 8 9 10 11 120
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0.4
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(a) 12 Nodes as Transmitters
1 2 3 4 5 6 7 8 9 10 11 120
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Node ID
Pro
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ecep
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0−100cm101−200cm201−300cm301−400cm401−500cm
(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.
98
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.
103
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 ).
106
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).
109
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
200300
400500
600
0.4
0.6
0.8
10
0.2
0.4
0.6
0.8
1
x 10−3
Distance (cm)Ranging Yield
Pro
babi
lity
Den
sity
(ke
rnel
sm
ooth
ing)
(a) Yield of Geometric Noise Model
0100
200300
400500
600
0
0.2
0.4
0.6
0.8
10
0.5
1
1.5
2
x 10−3
Distance (cm)Ranging Yield
Pro
babi
lity
Den
sity
(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.
111
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
50 100 150 200 250 300 350 400 450 5000
10
20
30
40
50
60
True Distance (cm)
Tim
e of
Flig
ht (
mse
c)
(a) Data Generated from Complete Model
0100
200300
400500
600
0
0.2
0.4
0.6
0.8
10
0.2
0.4
0.6
0.8
1
1.2
1.4
x 10−3
Distance (cm)Ranging Yield
Pro
babi
lity
Den
sity
(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.
113
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.
114
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.
115
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
117
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
118
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
122
(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.
123
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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