GLIDING ROBOTIC FISH: DESIGN, COLLABORATIVE ESTIMATION, AND

APPLICATION TO UNDERWATER SENSING

By

Osama Nasr Ennasr

A DISSERTATION

Michigan State University

in partial fulfillment of the requirements

Submitted to

for the degree of

Electrical Engineering – Doctor of Philosophy

2020

ABSTRACT

GLIDING ROBOTIC FISH: DESIGN, COLLABORATIVE ESTIMATION, AND

APPLICATION TO UNDERWATER SENSING

By

Osama Nasr Ennasr

Autonomous underwater robots have received significant attention over the past two decades due to
the increasing demand for environmental sustainability. Recently, gliding robotic fish has emerged
as a promising mobile sensing platform in versatile aquatic environments. Such robots, inspired by
underwater gliders and robotic fish, combine buoyancy-driven gliding and fin-actuated swimming
to realize both energy-efficient locomotion and high maneuverability. In this dissertation, we first
discuss the design improvements for the second-generation gliding robotic fish “Grace 2”. These
improvements have transformed the robots to underwater sensing platforms that can be modified to
fit the requirements of a specific application with relative ease.

We focus on the application of detecting and tracking live fish underwater, which is an important
part of fishery research, as it helps scientists understands habitat use, migratory patterns, and
spawning behavior of fishes. The gliding robotic fish has demonstrated its ability to detect special
acoustic signals emulating tagged fish through a series of trials in Higgins Lake, Michigan. These
tests have also validated a gliding-based strategy for navigating to a GPS waypoint, and offered
insight into the limitations of the current design. Additional improvements are proposed to allow
these robots to glide at larger depths and perform more interesting working patterns underwater.

Motivated by the problem of tracking real fish, we consider the case where multiple robots
localize and track a moving target without the need for a centralized node. We present theoretical
treatment on how a network of robots can infer the location of an emitter (or target), and then track
it, through a time-difference-of-arrival (TDOA) localization scheme in a fully distributed manner.
In particular, we utilize a networked extended Kalman filter to estimate the target’s location in a
distributed manner, and establish that successful localization can be achieved under fixed and time-
varying undirected communication topologies if every agent is part of a network with a minimum

of 4 connected, non-coplanar agents. We further propose a movement control strategy based on
the norm of the estimation covariance matrices, with a tuning parameter to balance the trade-off
between estimation performance and the total distance traveled by the robots.

Finally, motivated by the distributed localization problem, we investigate a more general problem
of distributed estimation by a network of sensors. Specifically, we consider the class of consensus-
based distributed linear filters (CBDLF) where each sensor updates its estimate in two steps: a
consensus step dictated by a weighted and directed communication graph, followed by a local
Luenberger filtering step. We show that the sub-optimal filtering gains that minimize an upper
bound of a quadratic filtering cost are related to the convergence of a set of coupled Riccati
equations. Then we show that the convergence of these coupled Riccati equations depends on
the notion of squared detectability for the networked system, and proceed to provide necessary
conditions that link the convergence of the coupled Riccati equations to the network topology and
consensus weights of the communication graph.

Dedicated to my parents

Nasr & Hana

for all the effort they put in to make this happen.

iv

ACKNOWLEDGEMENTS

First, I would like to express my warmest gratitude to my advisor, the director of Smart Microsystems
Lab at Michigan State University, Prof. Xiaobo Tan. During my junior year at Michigan State
University, Prof. Tan gave me the opportunity of exploring academic research as an undergraduate
research assistant, and provided me with continuous motivation and support throughout my Ph.D.
journey in the field of underwater robotics. Now, upon the completion of my Ph.D. training, with
the experience and knowledge I gained over the past years, I am excitedly looking forward to starting
a new chapter of my research career with the Department of Defense and the Geospatial Research
Lab of the Engineering Research and Development Center. I owe much of my development to Prof.
Tan, and genuinely thank him for his invaluable mentoring and guidance throughout my career and
personal life.

I want to thank my academic committee members, Prof. Khalil, Prof. Mukherjee, and Prof.
Bopardikar at Michigan State University for their insightful comments in the area of control and
robotics.
I also thank our collaborators at the U.S. Geological Survey for their kindness and
generosity in facilitating the experiments for this project. Specifically, I would like to thank Dr.
Chris Holbrook for all his effort and insight in advancing our robots for acoustic telemetry.

In Smart Microsystems Lab, I owe many people a debt of gratitude. The research in gliding
robotic fish needs diverse expertise and extensive collaboration. Specifically, I would like to give
special thanks to John Thon, without whom this work would have been impossible. The countless
hours we spent together improving and debugging our robots have allowed me to navigate all the
personal and professional challenges that I had to deal with over the years. I would also like to
thank Demetris Coleman and Ryan Johnson for their help in advancing the electrical design of our
robots, as well as my fellow researchers in our lab, Mohammed Al-Rubaiai, Maria Castano, Jason
Greenberg, Pratap Solanki, and Thassyo Pinto for their help and support throughout my Ph.D.
journey.

I would also like to thank the other students in Smart Microsystems Lab and the faculty, students

v

and post-docs in my department, whose comments and discussions have helped my research. Thanks
also to all of the administrators and staff members in the ECE department who have helped me over
the years.

I also want to acknowledge the support from the funding agencies that made this work possible:
the National Science Foundation (IIS 1319602, CCF 1331852, ECCS 1446793, IIS 1715714, IIS
1848945), the Great Lakes Fishery Commission (2014_Tan_44058), the United States Geological
Survey Innovation Center for Earth Sciences, and the Department of Defense Science Mathematics
and Research for Transformation Scholarship.

Most of all, I want to thank my mother, Hana Khreishi, my father, Nasr Ennasr, and my siblings,
Haitham, Nadia, and Areej, for their unconditional love in both good times and bad times. I could
not have finished my Ph.D. program without their understanding, support, and encouragement.

vi

TABLE OF CONTENTS

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INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

LIST OF TABLES .
LIST OF FIGURES .
CHAPTER 1

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
1.1 Acoustic Telemetry in Aquatic Environments . . . . . . . . . . . . . . . . . . .
1.2 Beyond Presence-Absence Information . . . . . . . . . . . . . . . . . . . . . . .
1.3 Overview of Contributions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1
Improvements to Gliding Robotic Fish . . . . . . . . . . . . . . . . . . .
1.3.2 Application to Acoustic Telemetry . . . . . . . . . . . . . . . . . . . . .
1.3.3 Distributed Localization and Tracking of a Moving Target
. . . . . . .
1.3.4 Distributed Estimation of Linear Time-Invariant Systems . . . . . . . .
1.4 Dissertation Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix
x
1
1
4
6
6
7
7
8
9
CHAPTER 2 DESIGN IMPROVEMENTS FOR GLIDING ROBOTIC FISH . . . . . 11
2.1 Overview of Gliding Robotic Fish . . . . . . . . . . . . . . . . . . . . . . . . . . 12
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Design Improvements .
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2.2.1 Mechanical Improvements
2.2.2 Electrical and System Improvements . . . . . . . . . . . . . . . . . . . . 16
. 18
2.3.1
Swimming-based Navigation . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.2 Gliding-based Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Summary & Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3 Motion and Navigation .

CHAPTER 3 CHARACTERIZATION OF ACOUSTIC DETECTION USING A

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3.1 Methods .

GLIDING ROBOTIC FISH AS A MOBILE RECEIVER PLATFORM 26
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3.1.1 Mobile Receiver .
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3.1.2 Field Tests
. 29
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3.1.3 Mobile Detection Range Tests . . . . . . . . . . . . . . . . . . . . . . . . 30
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.4 Data Analysis .
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3.2 Results
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3.3 Discussion .
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3.4 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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4.1 Problem Setup .
4.2 Distributed Estimation .

CHAPTER 4 TIME-DIFFERENCE-OF-ARRIVAL (TDOA)-BASED DISTRIBUTED
TARGET LOCALIZATION BY A ROBOTIC NETWORK . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 46
. 48
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4.2.1 Decentralized Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.2 Distributed Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
. 60

4.3 Target Tracking with Coordinated Robotic Network Movement . . . . . . . .

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vii

4.3.1 Optimal Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 Simulation Results .
. 63
4.5 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

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5.1 Problem Setup .
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5.2 Necessary Conditions
5.3 Numerical Examples .
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CHAPTER 5 NECESSARY CONDITIONS FOR THE CONVERGENCE OF THE
DISTRIBUTED KALMAN FILTER AND ITS COUPLED RICCATI
EQUATIONS .
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. 72
. 74
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
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. 81
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. 84
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5.4 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
CHAPTER 6 SUMMARY AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . 87
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3.1 Example 1 .
5.3.2 Example 2 .

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6.1 Summary .
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6.2 Future Work .
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APPENDICES .

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APPENDIX A STOCHASTIC STABILITY FOR DISTRIBUTED

TDOA-BASED LOCALIZATION OF A MOVING TARGET . . . 91

APPENDIX B SUPPLEMENTAL RESULTS FOR TDOA-BASED

DISTRIBUTED LOCALIZATION AND TRACKING OF
A MOVING TARGET . . . . . . . . . . . . . . . . . . . . . . . .

APPENDIX C SUPPLEMENTAL RESULTS FOR THE DISTRIBUTED

KALMAN FILTER . . . . . . . . . . . . . . . . . . . . . . . . .

. 102

. 109

BIBLIOGRAPHY .

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. 113

viii

LIST OF TABLES

Table 2.1: Reported parameters of first-generation gliding robotic fish “Grace” in [1]. . 13

Table 2.2: Mechanical specifications for Grace 2.

. . . . . . . . . . . . . . . . . . . . . 16

Table 2.3: Selected components for Grace 2.5.

. . . . . . . . . . . . . . . . . . . . . . . 19

Table 3.1: Summary of mobile range detection trials with GRACE at Higgins Lake
in 2016, 2017, and 2018. Mean heading reflects the bearing from the first
GPS measurement toward the last GPS measurement in each run. Wind
speed and direction (heading; note that this is the direction the wind
vectors are following, not “blowing from” as is customary) are coarse-
scale regional estimates based on the NCEP’s Global Forecast System, as
described in methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Table 3.2: Summary of parametric coefficients (linear terms) and smoothing terms
from GAM used to determine if detection range was related to distance
from transmitter (Tag Distance, in meters), direction of robot travel rel-
ative to transmitter (toward or away), robot depth (Depth, in meters
from water surface), or robot pitch (Pitch, in degrees from horizontal).
Included for each parametric coefficient is the estimate, standard error
(SE), test statistic (Z), and p-value for the null hypothesis that the cor-
responding parameter is zero. Included for each smoothing term is the
estimated degrees of freedom (EDF), test statistic ( 
2), and approximate
p-value for the null hypothesis that the smoothing term is zero. Bold faced
p-values are significant at significance level of 0.05.

. . . . . . . . . . . . . . 38

Table 3.3: Speed of the robot relative to ground from the field trials.

. . . . . . . . . . 38

ix

LIST OF FIGURES

Figure 1.1: Acoustic tag surgically implanted into fish (Source: Great Lakes Fishery

Commission). .

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[18].

[17].

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Figure 1.2: Example of OceanServer Iver2 propelled AUV used in acoustic telemetry

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Figure 1.3: Exampe of a Liquid Robotics wave glider AUV used in acoustic telemetry

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

3

4

5

Figure 1.4: Acoustic receivers deployed across the Great Lakes (Source: Great Lakes

Acoustic Telemetry Observation System).

. . . . . . . . . . . . . . . . . . .

Figure 2.1: The SLOCUM AUV [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Figure 2.2: Robotic fish from Smart Microsystems Lab at Michigan State University.

. 12

Figure 2.3: First-generation prototype of Gliding Robotic Fish ”Grace“ from [1].

. . . 13

Figure 2.4: Second-generation (Grace 2) prototypes of gliding robotic fish.

. . . . . . . 14

Figure 2.5: Side view of the Grace 2.5 design with the interface open depicting (a)
3D printed interface, (b) electronics board, (c) water-proof switching
connector, (d) water inlets, (e) gliding mechanism, (f) quick-disconnect
system, (g) adjustable balance weights, and (h) batteries.

. . . . . . . . . . 15

Figure 2.6: A gliding robotic fish modified to carry (a) a propeller, through the

introduction of (b) an auxiliary port. . . . . . . . . . . . . . . . . . . . . . . 17

Figure 2.7: Schematics of working patterns: (a) gliding in sagittal plane, and (b)

spiraling in 3D space.

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Figure 2.8: A gliding robotic fish prototype performing a spiral during experiments

in MSU swimming pools.

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Figure 2.9: Yaw and bearing angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Figure 2.10: Tail bias vs. angle difference 
. . . . . . . . . . . . . . . . . . . . . . . . . . 22

Figure 2.11: Flapping amplitude vs. angle difference 
.

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Figure 2.12: Tail deflection 
 vs. angle difference 
.

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x

Figure 2.13: Front view for components for next-generation design depicting individ-
ually sealed for (a) electronics, (b) battery packs, (c) movable mass, and
(d) buoyancy control tank.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Figure 3.1: Grace 2 carrying a forward-facing acoustic receiver.

. . . . . . . . . . . . . 27

Figure 3.2: Grace 2 at Higgins Lake, Michigan, 3 November 2017. . . . . . . . . . . . . 28

Figure 3.3: A gliding robotic fish prototype during field experiments in Higgins Lake. . 28

Figure 3.4: A gliding robotic fish prototype during field experiments in Higgins Lake. . 29

Figure 3.5: Lake tests (Panel A) Map of Higgins Lake, Michigan showing regions
(red rectangles) where gliding robotic fish GRACE was tested 11 Nov
2016 (panel B), 03 Nov 2017 (panel C), and 14-15 June 2018 (panels D-F)
with stationary transmitter locations (green circles), stationary receiver
locations (yellow circles), and GPS locations (pink/red circles) recorded
by the gliding robotic fish during each run. GPS points are shaded along
a gradient from start (pink) to end (red) of each run. . . . . . . . . . . . . . 31

Figure 3.6: Depth, pitch, and temperature profiles of gliding robotic fish Grace 2
during a 10-minute segment of run 2018-1 in Higgins Lake, with depths
recorded by the on-board sensors (open symbols) and estimated depths
at the time of test tag signal transmissions (red symbols). Horizontal bro-
ken lines show depth (panel A) and temperature (panel C) of stationary
transmitter at site T6.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Figure 3.7: Depth profiles of gliding robotic fish Grace 2 during a selected 100– to
300–m segment of each run (panels A-D) in Higgins Lake, with locations
of test tag transmissions that were detected (closed symbols) or not
detected (open symbols) by the on-board receiver.

. . . . . . . . . . . . . . 37

Figure 3.8: Estimated efficiencies (proportion of coded signals detected) of detecting
an acoustic transmitter on acoustic receivers affixed to stationary moor-
ings (red symbols; red shaded regions are GAM-based 95% confidence
regions) and gliding robotic fish Grace 2 (black line; grey shaded regions
are GAM-based 95% confidence regions) in Higgins Lake, Michigan dur-
ing field tests in 2016 (panel A), 2017 (panel B), and 2018 (panels C-E).
Vertical grey bars show distances between robot and the transmitter
when each coded signal was detected (1) or not detected (0). . . . . . . . . . 40

Figure 3.9: Partial effects of robot depth and pitch on detection probability. Esti-
mated partial effects of robot depth and pitch on the log odds of detection
probability when the robot is moving away from (panels A, B) or toward
(panels C, D) a stationary transmitter in Higgins Lake, Michigan.

. . . . . 41

xi

Figure 3.10: Estimated efficiencies (proportion of coded bursts detected) of detecting
an acoustic transmitter on acoustic receivers affixed to gliding robotic fish
Grace 2 in Higgins Lake, Michigan, during field tests in which evidence
(GAM model results) suggested that detection range was influenced most
by depth when the robot was moving away from the transmitter (panel
C) and pitch when the robot was moving toward the transmitter (panel
A, B, D).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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Figure 4.1: Network of 3 agents monitoring the target with agent 1 as the reference

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node.

node.

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Figure 4.2: Network of 4 agents monitoring the target with agent 1 as the reference

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Figure 4.3: The 4 possible configurations for node 1 to be connected to the network

using the minimum number of edges.

. . . . . . . . . . . . . . . . . . . . . 59

Figure 4.4: Illustration of the simplified model of a propelled underwater robot with
steering control, with a view on the sagittal plane. The robot has similar
yaw control in the horizontal plane.

. . . . . . . . . . . . . . . . . . . . . . 64

Figure 4.5: The fixed communication topology among a network of 8 agents. It is
clear that no agent in the network has a sufficient number of neighbors
(and therefore TDOA measurements) to estimate the process on its own. . . 65

Figure 4.6: Initial setup for simulation environment with a fixed communication
topology. The big circle represents the moving target, while the small
circles (overlapping one another at the origin) represent the initial es-
timates of the target’s location for each robot. The ellipsoids represent
the mobile robots, while the thin lines connecting them represent the
communication links.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 66

Figure 4.7: Simulation results of the robots for the two extreme values of 
 under
a fixed communication topology where (a) the robots are constantly
moving with target (
 = 0), and (b) the robots are staying put (
 = ∞). . . . 66

Figure 4.8: Comparison of the Mean Squared Estimation Error (MSEE) of for each
agent in the network for different values of 
 under a fixed communica-
tion topology. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

.

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Figure 4.9: Comparison of average covariance norms for different 
 values under a

fixed communication topology.

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Figure 4.10: Comparison of the collective distance traveled by the entire network for

different 
 values under a fixed communication topology.

. . . . . . . . . . 68

xii

Figure 4.11: Initial setup for distance-based communication links.

. . . . . . . . . . . . 69

Figure 4.12: Final setup for distance-based communication links. . . . . . . . . . . . . . 69

Figure 4.13: Mean Squared Estimation Error (MSEE) under a time-varying commu-

nication graph with continuous tracking (
 = 0) . . . . . . . . . . . . . . . . 70

Figure 5.1: The minimal communication links required to satisfy the detectability

requirement of all (
,O
) pairs for Example 1. . . . . . . . . . . . . . . . . 82

Figure 5.2: Comparison of the norms of the covariance matrices under the consensus-
then-filter distributed Kalman filter for Example 1. 

 represents the
network that satisfies (1/

(



. . . . . . . . . . . . . . . . . . . . . . 83

))2.




Figure 5.3: Comparison of the norms of the covariance matrices under the consensus-
then-filter distributed Kalman filter for Example 2. 

 represents the
network that satisfies (1/

(



. . . . . . . . . . . . . . . . . . . . . . 85

))2.




xiii

CHAPTER 1

INTRODUCTION

There is growing interest in monitoring aquatic environments, due to the emerging problems
of environmental pollution and expanding demand for sustainable development. Such pollution
involves various types of contaminants, including industrial waste, chemicals, and bacteria, to name
a few. The pollution could happen in different aquatic environments, such as ponds, rivers, lakes,
and even the ocean. For that reason, underwater robots are increasingly drawing attention in aquatic
sensing. The past decade or so has seen great progress in the use of robotic technology in aquatic
environmental sensing.

In this dissertation, we focus on the application of these robots to localization and tracking
of acoustically tagged aquatic animals. For the remainder of this introduction, we discuss the
existing methods for acoustic telemetry in aquatic environments, and conduct a brief survey of the
technologies used in that field. We then present the contributions of this research to advancing
acoustic telemetry. Specifically, we discuss the design improvements to gliding robotic fish that
facilitated carrying out acoustic telemetry experiments, and theoretical work on how networks of
robots can collectively localize and track a moving target or estimate the state of a dynamical
system.

1.1 Acoustic Telemetry in Aquatic Environments

Tracking of aquatic animals presents an important problem in fishery research and is paramount
to improving our understanding of aquatic ecosystems. Acoustic telemetry is often used by re-
searchers to collect information about aquatic animals, and study their migration, habitat use, and
survival. Acoustic telemetry systems consist of two main components: transmitters which broad-
cast a series of sound pulses (or pings) into the surrounding water, and receivers that listen to such
pings and record the time and corresponding ID of any detection. Animals can be tagged with
acoustic transmitters (see Fig. 1.1) and their movements can be tracked by stationary networks

1

Figure 1.1: Acoustic tag surgically implanted into fish (Source: Great Lakes Fishery Com-
mission).

of hydrophones or acoustic receivers that can identify acoustic-tagged animals in their vicinity
[4, 5, 6, 7].

Recent improvements in acoustic telemetry technology have advanced studies of long-term
spatiotemporal ecology and behavior of aquatic organisms [8, 9].
In particular, fish movement
tracking information obtained via acoustic telemetry has provided insight far superior than what can
be obtained by direct observations or traditional survey methods (i.e., trawls, gill nets). Observations
from such networks have already been used to improve control and assessment of invasive species,
gain new insights into spawning behavior and habitat requirements of fish, and describe movements
of high-valued stocks [8, 7]. In the future, telemetry promises to address many critical uncertainties
in fishery research and aquatic ecosystems. However, logistical and economic constraints may
preclude use of receiver networks to fill gaps in understanding of animal movements, especially in
large systems and extreme environments.

Due to the limited detection range of these receivers, such acoustic telemetry systems are limited
to fixed areas. Autonomous underwater vehicles (AUVs) are becoming more common telemetry
assets as receivers can be self-contained and attached as a payload, or they can be integrated for real-
time detection while simultaneously measuring physical and biological properties of the aquatic
environment [10, 11]. For example, buoyancy-driven gliders with integrated or externally mounted

2

Figure 1.2: Example of OceanServer Iver2 propelled AUV used in acoustic telemetry [18].

receivers have been used to study sturgeon and shark habitats [12, 13, 14], stereo-hydrophone
acoustic receiver systems have been integrated into propelled AUVs to track and follow leopard
sharks [15], and receivers and hydrophones have also been externally mounted and integrated into
a propeller-driven AUV [16] and a wave glider for real-time detections of tagged marine life [17].
Figs. 1.2–1.3 show examples of different AUVs equipped with acoustic receivers.

Each of these vehicles has its unique advantages and disadvantages. Propelled AUVs (e.g.,
REMUS-100 [16], and OceanServer Iver2 [18]) can overcome large currents and surface waves,
and are capable of traveling at higher speeds than other unmanned vehicles. The main drawback
for these vehicles is the power requirement needed for the propellers, limiting deployment duration
in the field. Another popular class of AUVs is wave gliders, where a surface float that utilizes wave
energy to move forward is attached to an underwater sub via a tether [11]. The energy-efficient nature
of wave gliders, such as the Liquid Robotics Wave Glider used in [17], allows them to be deployed
for months at a time. Moreover, accurate positioning of the surface float is readily available through
GPS, along with reliable communication for the duration of the mission. The main limitation
of wave gliders is that any underwater instrumentation needs to be tethered to the surface float,
limiting the depth at which such instrumentation can be deployed, and introducing tether dynamics

3

Figure 1.3: Exampe of a Liquid Robotics wave glider AUV used in acoustic telemetry [17].

that should be considered for accurate representation of the underwater environment. Underwater
gliders are another class of energy-efficient AUVs that travel by changing their buoyancy and center
of gravity. Underwater gliders, such as the Slocum glider [3], have been widely used in a variety
of underwater applications [12, 10]. Like wave gliders, the energy efficient nature of these robots
makes them ideal for long-duration missions. However, unlike wave gliders, there is no tether
involved with these robots, allowing for sampling of water environment at different depths up to the
depth rating of the vehicle. The biggest challenges with these robots involve underwater localization
and communication since RF signals do not penetrate water effectively. These vehicles often record
data while underwater, and surface occasionally to download or offload this data.

1.2 Beyond Presence-Absence Information

In addition to carrying acoustic receiver into areas that are not covered by stationary receivers,
AUVs can also carry a variety of environmental sensors. Measurements from these sensors can
further help characterize the interaction between aquatic animals and their environment. These
sensors allow researchers to monitor the external environment of fishes, allowing them to address

4

Figure 1.4: Acoustic receivers deployed across the Great Lakes (Source: Great Lakes Acoustic
Telemetry Observation System).

biological questions that were previously difficult (or impossible) to answer.

Acoustic receivers provide information about the location of tagged fish. When a signal
is identified, the tag’s unique ID code is saved with the date and time. The data from any
single receiver provide a record of each visit to that location by a tagged fish. Researchers often
deploy many receivers over large regions to understand the movement patterns of tagged fish
(see Fig. 1.4). Consequently, the fish positioning resolution is defined by the detection range of
individual receivers, which is usually on the order of 100 – 1000 m.

Fine-scale fish positioning (at the resolution of a few meters to tens of meters) can be achieved
with a network of receivers using time-difference-of-arrival (TDOA) methods. When the same
signal is detected by multiple receivers, it is possible to infer the location of the tag from the
differences in detection times. Such methods can provide a more detailed understanding of fish
behavior. Networks of AUVs equipped with acoustic receivers hold great potential in understanding
aquatic ecosystems, and advancing the field of acoustic telemetry. To effectively utilize such
networks, algorithms should rely on distributed processing of information and cooperation between

5

the robots to achieve specific tasks.

1.3 Overview of Contributions

The contributions of this research reside mainly in improving the design of the gliding robotic
fish, investigating its applicability to acoustic telemetry, and theoretical treatment of distributed
localization and estimation using robotic networks. The details are as follows.

1.3.1

Improvements to Gliding Robotic Fish

Given the ever-changing and unpredictable nature of the environments in which they operate,
robots employed for underwater missions need to be both highly energy-efficient and maneuverable
[19]. Gliding robotic fish, a new type of underwater robots, has emerged as a promising mobile
sensing platform for monitoring aquatic environment. Such a robot combines the advantages of
both underwater gliders and robotic fish and features long operation duration along with high
maneuverability. This makes the gliding robotic fish well suited for a variety of underwater sensing
applications.

While the first-generation design of these robots was successfully used in sampling harmful oil
and algae bloom [20, 21], some aspects of the design needed to be re-addressed. The design of
the gliding robotic fish was improved to increase the serviceability of the robots by incorporating
a resealable design. This facilitated the rapid testing of different electrical designs, and allowed
regular inspection and maintenance to be performed on these robots. The computational power,
data storage capacity, and battery capacity of these robots were also dramatically improved to
facilitate carrying out longer missions. The new design also allows for a larger and replaceable
sensor payload that can be customized to fit different applications.

Additional improvements to the design are proposed, which would allow for further modularity

of the system and reduction of the manufacturing and fabrication costs.

6

1.3.2 Application to Acoustic Telemetry

Gliding robotic fish can be modified to carry an acoustic receivers to detect, and potentially track,
tagged fish in areas where it would be difficult to install stationary receivers. The successful
development of gliding robotic fish has allowed for the investigation of the robot’s capability in
detecting and tracking tagged fish.

A series of field trials have been conducted in Higgins Lake, Michigan, to evaluate the use of
these robots as mobile receiver platforms for detecting acoustic signals in a challenging environment.
A self-contained acoustic receiver was attached to the a gliding robotic fish prototype, and the
detection efficiency of the mobile receiver was compared to that of stationary receivers. Further
trials were conducted to study the factors that affect the detection efficiency of the mobile receiver,
such as depth underwater, robot heading relative to transmitter, and pitch angle of the robot.

While the detection ranges for this mobile receiver varied between trials, the results obtained
from these trials showed that the detection efficiency of the mobile receiver is comparable to that
of stationary receivers at distances up to 600 m. Furthermore, the results show that the detection
efficiency of the mobile receiver degrades significantly if the mobile receiver is facing away from
the tag. The results also indicate that the depth and pitch of the robot also affect the detection
efficiency of the mobile receiver. However, the impact of these variables is less paramount than
that of the distance between the receiver and transmitter, or that of their alignment.

These trials suggest that such mobile receivers can be utilized for active tracking of tagged
animals. However, care must be taken when designing tracking controller in order to maximize
the alignment between the receiver and the transmitter. The trials have also offered insight into
the limitations of the current design, and additional work is proposed to further characterize the
performance of such mobile receivers.

1.3.3 Distributed Localization and Tracking of a Moving Target

Motivated by the problem of tracking live fish through acoustic signals by a robotic network,
analytical work is completed on distributed target localization and tracking using a network of

7

mobile agents. Time-difference-of-arrival (TDOA)-based algorithms are widely used for precise
localization of a target, examples of which extend beyond acoustic telemetry in fishery research
[22], to wireless ranging radar systems and cellular positioning systems [23]. However, much of
the work on TDOA-based localization in the literature adopts a centralized approach, in which a
reference node is chosen and the times of arrival (TOA) of the emitted signal for all other nodes
in the network are subtracted from the reference node’s TOA, generating TDOA measurements at
fusion hub.

Due to power and bandwidth constraints in robotic networks, centralized information processing
is often infeasible, particularly for large-scale and unreliable networks. Moreover, some sensors
cannot transmit their measurements to the reference node due to their limited communication ranges.
This has motivated us to investigate a completely distributed approach for localization of a moving
target. It is shown that source localization is not possible (centrally, and therefore distributively)
when the total number of TDOA measurement is insufficient. Then, we demonstrate that the target’s
position can be successfully estimated in a distributed manner if and only if every agent is part of a
connected network that collectively has a sufficient number of TDOA measurements, even if each
agent has an insufficient number of measurements.

Estimation of the target’s location is achieved without relying on a centralized node, and without
requiring all agents in the network to be communicating with one another. Necessary and sufficient
conditions on the underlying bidirectional communication graph are derived to ensure successful
localization of the target. Moreover, tracking of the moving target is examined through a coordinated
moving strategy that can be tuned to emphasize estimation performance or energy conservation,
depending on the application.

1.3.4 Distributed Estimation of Linear Time-Invariant Systems

The problem of distributed localization of a moving target has motivated the analysis of a more
general application of networked systems:
the distributed estimation problem. Specifically, we
examine the problem of estimating the state of a linear time-invariant system that is monitored by

8

a network of sensors. It is assumed that the interactions between the sensing sites are dictated by
a weighted directed graph. Under this distributed approach, each sensor site only has access to its
information and to that of the sensors in its neighborhood. While the analytical results obtained
here do not directly apply to the distributed localization problem, these results serve as a building
block for later investigation of time-varying dynamical systems monitored by sensor networks.

This distributed filtering problem has received significant attention over the past years, and
different ideas have been proposed to effectively deal with this problem[24, 25, 26, 27, 28, 29].
We focus on single-time-scale algorithms in which data fusion occurs once per time step of
plant dynamics due to bandwidth and power constraints presented in networked applications. In
particular, we consider the class of consensus-based distributed linear filters (CBDLF) where each
sensor updates its estimate in two steps: a consensus step dictated by a weighted and directed
communication graph, followed by a local Luenberger filtering step.

We then present sub-optimal filtering gains that minimize an upper bound of a quadratic filtering
cost, and show that these filtering gains are related to the convergence of a set of coupled Riccati
equations. While it has been shown that the convergence of these coupled Riccati equations is
related to the feasibility of a set of linear matrix inequalities (LMIs) [30], these conditions do
not, in general, shed light on the network conditions required to satisfy such LMIs. The work
presented in this dissertation utilizes some of the results developed in the Markovian jump linear
systems (MJLS) literature to connect the convergence properties of these coupled Riccati equations
to certain conditions on the network topology and consensus weights.

1.4 Dissertation Organization

The remainder of this dissertation is organized as follows. In Chapter 2, the design concept
for the gliding robotic fish is revisited, and the design improvements that the new generation of
gliding robotic fish has seen are highlighted. Then, the capability of this robot in advancing acoustic
telemetry in fishery research is presented in Chapter 3 through a series of field trials that were carried
out in Higgins Lake, Michigan. In Chapter 4, a detailed discussion on localization of a moving

9

target using time-difference-of-arrival (TDOA) measurements by a network of agents is presented.
This is motivated by the problem of tracking real fish, where multiple robots need to localize and
track a moving target, without the need for a centralized node to collect all measurements and
propagate an estimate of the target’s position back to the network. This problem is expanded on in
Chapter 5, where we consider the more general problem of distributed estimation of a dynamical
system monitored by a network of sensors. Finally, concluding remarks are provided in Chapter 6.

10

CHAPTER 2

DESIGN IMPROVEMENTS FOR GLIDING ROBOTIC FISH

The design concept of a gliding robotic fish comes from energy-efficient underwater gliders and
highly maneuverable robotic fish. On the one hand, underwater gliders are known for their great
energy-efficiency and long-duration operation in oceanographic applications [31]. An underwater
glider utilizes its buoyancy to enable motion without any additional propulsion and adjusts its center
of gravity to achieve certain attitude, which results in glide and thus horizontal travel. Since energy
is needed only for buoyancy and center-of-gravity adjustment when switching the glide profile,
underwater gliders are very energy-efficient, as proven by the great success of the Seaglider [2] and
SLOCUM [3] (Figure 2.1). The maneuverability of underwater gliders, however, is quite limited.
The large size (1– 2 m long), heavy weight (50 kg and above), and high cost [31] of these vehicles
also impede their adoption in the application of networked sensing and in versatile environments
like ponds and inland lakes.

On the other hand, over the past two decades, there has been significant interest in developing
robots that propel and maneuver themselves like real fish do. Often called robotic fish (Figure 2.2),
they accomplish swimming by deforming their body and using fin-like attachments [32]. In many
designs, a fish-like flapping tail is used to provide propulsion force and a biased tail angle is applied

Figure 2.1: The SLOCUM AUV [3].

11

Figure 2.2: Robotic fish from Smart Microsystems Lab at Michigan State University.

to realize turning. Due to the similarity to real fish and the fact that the rotation of the tail fin
is usually enabled by a motor that is easy to control and fast in response, robotic fish typically
have high maneuverability (i.e., small turning radius). However, as the forward propelling force is
generated from the flapping motion of the fins, such robots require constant actuation for swimming
and cannot work for extended periods of time without battery recharge.

2.1 Overview of Gliding Robotic Fish

Inspired by the design and merits of both underwater gliders and robotic fish, a new type
of underwater robots, gliding robotic fish, has emerged as an underwater sensing platform [1].
A gliding robotic fish combines both mechanisms of gliding and swimming, featuring energy
efficiency and high maneuverability at the same time. Such a robot would realize most of its
locomotion through gliding like underwater gliders, by utilizing its buoyancy to enable motion
without any additional propulsion, while it can adjust its center of gravity to achieve a certain
attitude.
It would use actively controlled fins to achieve high maneuverability, during turning
and orientation maintenance. Of course, fins can also provide additional propulsive power during
locomotion if needed.

Table 2.1 lists some of the aspects reported in [1] for an experimental prototype of the gliding
robotic fish (Fig. 2.3). The smaller dimension of Grace compared to traditional underwater gliders
proved to be valuable in relatively shallow water environment, such as lakes, rivers, and ponds,

12

Figure 2.3: First-generation prototype of Gliding Robotic Fish ”Grace“ from [1].

Table 2.1: Reported parameters of first-generation gliding robotic fish “Grace” in [1].

Parameter
Hull dimensions (L x W x H)
Tail to nose length
Wingspan
Weight

Payload

Description
65 x 15 x 18 cm
90 cm
75 cm
9 kg
Turner Design Cyclops
7F sensor (crude oil or
blue green algae sensor)
Temperature sensor

where larger gliders are not quite suitable due to their large size and high cost. This prototype
was successfully deployed for oil spill detection in Kalamazoo river, Michigan, and for monitoring
algae bloom in Wintergreen lake, MI [20, 21].

However, there were several aspects of the first-generation design that needed improvement.
First, the hull design of Grace in [1] was completely sealed, which made servicing the robot pro-
hibitive, as accessing the internal electronics would require destroying the robot’s shell. Second,
the payload carrying capabilities of the first-generation design were limited to one or two small
components, and would need external attachments (e.g., external weight, foam blocks) to accom-

13

Figure 2.4: Second-generation (Grace 2) prototypes of gliding robotic fish.

modate the buoyancy of the payload. Finally, the battery capacity of the original design limited its
application to the order of several hours to one day.

2.2 Design Improvements

The second-generation gliding robotic fish, which we refer to as “Grace 2”, has seen several
electrical and mechanical enhancements to improve its applicability and prolong its operational
time. In this section, we discuss these design improvements that were implemented on two new
prototypes (see Fig. 2.4). Some of these improvements were done incrementally over the period
of this study, which we subversion as 2.0, 2.1, and 2.5. We will use the term “Grace 2” when
we discuss fundamental changes to the design when compared to first-generation prototype in [1],
while we use “Grace 2.x” when discussing improvements that were accommodated in that specific
subversion.

2.2.1 Mechanical Improvements

The new mechanical design for Grace 2 addressed the issues of serviceability and payload carrying
capacity1. To address the issue of serviceability, a 3D printed interface was introduced between the
1While the development of the Grace 2 robots involved several people, the majority of the
fabrication and mechanical work on these robots was completed by John Thon from the Smart
Microsystems Lab at Michigan State University. The majority of the electrical improvements was

14

Figure 2.5: Side view of the Grace 2.5 design with the interface open depicting (a) 3D
printed interface, (b) electronics board, (c) water-proof switching connector, (d) water inlets,
(e) gliding mechanism, (f) quick-disconnect system, (g) adjustable balance weights, and (h)
batteries.

nose of the robot and the rest of its body. This interface, when bolted together, formed a water-tight
seal by compressing a custom O-ring at the interface. To maximize the use of space, batteries were
stored in the robot’s nose, and a water-proof connector was introduced to operate as a switch for
the system.

Two measures were taken to allow for larger, and possibly variable, payload carrying capacity.
First, a larger buoyancy-control tank, with a volume of approximately 190 mL, was utilized for the
second-generation robots. To deal with the serviceable nature of the robot, a 3D printed piece was
designed to fit the buoyancy-control tank and held water-tight quick disconnects that mated with the
water inlets from the nose. The movable mass and the buoyancy tank were fixed together to form
what we refer to as the “gliding mechanism”, which could slide in and out of the robot’s main body
for maintenance and inspection. The second measure allowed for variable weights to be attached
to the bottom of the gliding mechanism. These weights are typically adjusted and moved around
before deployment in order to accommodate the buoyancy and weight distribution of the payloads.
completed by the author.

15

Table 2.2: Mechanical specifications for Grace 2.

Component
Hull dimensions (L x W x H)
Tail to nose length
Wingspan
Weight
Robot hull material
Buoyancy Control
Pitch Control

Payload

Description
103 x 20 x 30 cm
140 cm
60 cm
20 kg
Carbon fiber with 3D printed interface
190 mL
+ 40◦, -25◦
Dissolved Oxygen and Temperature sensor
Photosynthetically Active Radiation sensor
Freshwater Blue Green Algae sensor
Chlorophyll sensor
Acoustic receiver

Fig. 2.5 depicts a side view of a Grace 2 prototype and its different components.

The need for larger buoyancy control, payload carrying capacity, and battery capacity required
a larger body hull that can accommodate all these components. Table 2.2 lists the mechanical
specifications of the Grace 2.

2.2.2 Electrical and System Improvements

The serviceable nature of the new design allowed us to rapidly prototype and test out different
electrical systems and components to improve the robot’s functionality. The Grace 2 design
implemented a changeable sensor harness that could be replaced to accommodate a different set of
sensors, and a connector was introduced between each of the shell components and the on-board
electronics.

The initial electrical design for Grace 2.0 utilized an architecture similar to the first-generation
Grace, with the exception of introducing a second microcontroller (MCU) dedicated to reading and
communicating with the sensor bundle. The power system for the Grace 2.0 design used linear
regulators to regulate battery voltages to 3.3V and 5V, which provided power to the electronics
and the sensors. However, due to the relatively high power demand of the sensors along with the
big difference between the battery voltage and the regulated voltages, the linear regulators became

16

Figure 2.6: A gliding robotic fish modified to carry (a) a propeller, through the introduction
of (b) an auxiliary port.

very hot in a very short period of time, causing the electronics to behave erratically and the entire
system to shut off. The electrical design for Grace 2.1 used a new power system architecture,
in which the linear regulators were replaced with switching regulators. The introduction of these
regulators drastically improved the stability of the electronics and allowed us to perform meaningful
experiments (which we present later in Chapter 3). Additionally, the wiring harness inside the robot
was simplified to only two connectors, one for the shell components (namely, the GPS, tail servo, and
pressure sensor), and one for the sensor bundle. The Grace 2.1 design relied on the same dual-MCU
architecture used in Grace 2.0, in which one MCU was dedicated to reading and communicating
with the set of environmental sensors, while another MCU was responsible for controlling the
actuators, reading essential sensors (i.e., depth, GPS, and IMU), and communicating with a base
station.

The design in Grace 2.0 and 2.1 did not implement on-board storage of the data. Instead, the
robot periodically sent essential data (i.e., UTC time, GPS position, and IMU info) to a laptop over
the wireless XBee channel. When the robot performed a dive, all these parameters along with the
robot depth were temporarily stored in memory until the next time the robot surfaces, at which time
the data was sent over the wireless channel.

17

The dual MCU architecture was somewhat limiting since all tasks needed to be implemented on
this embedded structure. Furthermore, the lack of on-board storage also limited the applicability
of these robot. Therefore, a new revision of the electronics was introduced in Grace 2.5 in which
a Raspberry Pi (RPi) computer was included to handle high-level tasks (e.g. communicating with
the base station, mission control, data storage). The RPi computer freed the MCUs to perform
low-level tasks only, such as reading sensor value and controlling actuators. Internal storage onto
an 8 GB mirco-SD card was also integrated with the latest design, and the robot logged all data
at 5 s intervals. The introduction of the RPi computer also allowed for rapid testing of different
ideas without needing to open the robot and adjusting the low-level microcontroller code.
It
also provided the capability of implementing an sophisticated algorithms and systems, such as the
Robot Operating System (ROS). Finally, the Grace 2.5 design was modified to support an additional
“auxiliary module” through an additional water-proof connector that can be used to connect another
sensor or actuator, such as a propeller (see Fig. 2.6). The components used in the final design of
Grace 2.5 are listed in Table 2.3.

2.3 Motion and Navigation

For gliding robotic fish, there could be various interesting motions generated by integrating the
gliding and swimming mechanisms. Two main steady-state motion profiles are typically used as
the regular working patterns for navigation and water sampling. One is gliding in the sagittal plane,
and the other is spiraling in the 3D space.

Gliding is also the common operating mode for traditional underwater gliders (Fig. 2.7(a)). In
the zig-zag trajectory, the gliding robotic fish only consumes energy during the transition between
descending and ascending, which makes the gliding robotic fish highly energy-efficient. The
other motion, spiral in the 3D space, allows for sampling water columns using gliding robotic fish
(Fig. 2.7(b)). A water column is a conceptual narrow volume (like a narrow cylinder) of water
stretching vertically from the surface to the bottom. Water column sampling is a routine surveying
method in environmental studies. This motion is achieved by incorporating the steady gliding with

18

Table 2.3: Selected components for Grace 2.5.

Component name

Processors

Battery

Data Storage
Mass actuator

Buoyancy-control actuator
Tail servo motor
Pressure sensor
GPS Unit
Wireless module
Wireless antenna
IMU
Dissolved Oxygen and Temperature
sensor
Photosynthetically Active Radiation
sensor
Chlorophyll sensor
Algea sensor
Sensor bundle connector
Aux port connector
Switch/charge connector

Component model
Raspberry Pi Zero-W
(2x) Microchip dsPIC30F6014A
(3x) Batteryspace High Power Polymer Li-Ion,
18.5v @ 185 Wh (Approximately 5 days of contin-
uous actuation with 1400 dive cycles at 5 minute
intervals)
8GB micro SD
Actuonix L16-P Miniature Linear Actuator with
Feedback
ServoCity 6" Stroke 180 lb Thrust Heavy Duty
Linear Actuator
ServoCity HS-7980TH
Honeywell 40PC100G2A
GPS 18x LVC
XBee-Pro 900HP RP-SMA
900MHz Duck Antenna RP-SMA
VectorNav VN100S
In-Situ RDO PRO-X

LI-COR LI-192SA
Turner Designs Cyclops-7 Submersible sensors-C
Turner Designs Cyclops-7 Submersible sensors-P
SubConn MCIL16-F/M
SubConn MCBH6-F
SubConn MCBH4-F

(a)

(b)

Figure 2.7: Schematics of working patterns: (a) gliding in sagittal plane, and (b) spiraling in
3D space.

19

Figure 2.8: A gliding robotic fish prototype performing a spiral during experiments in MSU
swimming pools.

a deflected tail. The non-zero angled tail fin will introduce a turning moment to the steady gliding
robot and lead to a 3D spiraling motion. With the actively-controlled tail, the gliding robotic fish is
capable of spiral motions with tight turning radius. If needed, a gradually changing tail angle will
form spiral-in or spiral-out 3D trajectories that can be used to orient the robot at a certain angle
while gliding.

When performing field tests with these robots (such as the tests we describe in Chapter 3), a
fundamental task is to navigate to a given GPS waypoint. We discuss two navigation schemes that
were implemented on these robots.

2.3.1 Swimming-based Navigation

The first navigation mode relies on continuously flapping the tail to propel the fish forward. The
tail bias and amplitude are updated to drive the robot to the desired location. The task is simplified
by assuming that the robot’s yaw angle, 
, is a good measure of the robot’s heading, which can be
measured directly from the on-board IMU. The bearing angle, 
, to the GPS target location is then
calculated. Figure 2.9 depicts the yaw and bearing angles.

A proportional controller with saturation is then used to compute the bias and amplitude based

20

Figure 2.9: Yaw and bearing angles.

on the difference between the robot’s heading 
 and the bearing angle to target location 



 = 
 − 


(2.1)

Figs. 2.10 and 2.11 show the bias and amplitude of the tail flapping as a function of the difference

. The outputted tail angles from this computed bias and amplitude will always fall within the
limits of the servo motor, which are set to ±45◦.

2.3.2 Gliding-based Navigation

Swimming based navigation offers the benefit of a continuous GPS signal while swimming on the
surface. This simplifies the navigation task and makes swimming a viable strategy for accurate
navigation to target. However, due to the continuous flapping of the tail, swimming is generally
less efficient than gliding and should, therefore, be only used for navigation over a short distance.

21

Figure 2.10: Tail bias vs. angle difference 
.

Figure 2.11: Flapping amplitude vs. angle difference 
.

22

Figure 2.12: Tail deflection 
 vs. angle difference 
.

For that reason, it is desirable to use the gliding mode to navigate from one point to another.
Gliding-based navigation begins by acquiring a valid GPS fix and calculating the bearing angle to
the target location. Once this process is completed, the robot begins diving by typical adjustment
of the mass position and net buoyancy. While diving, the tail fin is deflected at an angle 
 that is
proportional to the difference 
. To prevent damage to the servo motor, the tail deflection angle
is saturated to ±45◦. Fig. 2.12 shows the relationship between the tail deflection angle and the
heading difference 
 during a dive.

When the robot completes a dive, it waits on the surface for another GPS fix before repeating
the process. During that time, the robot applies the swimming-based navigation using the latest
computed heading angle 
 to move towards the target GPS location.

2.4 Summary & Future Work

We discussed the design improvements implemented onto the gliding robotic fish originally
presented in [1]. The modified version significantly improved the serviceability, computational
power, payload-carrying capacity, and battery capacity of the robot. A new power design was

23

Figure 2.13: Front view for components for next-generation design depicting individually
sealed for (a) electronics, (b) battery packs, (c) movable mass, and (d) buoyancy control tank.

introduced to efficiently power the various electronics and environmental sensors. Furthermore, a
new system architecture was implemented that included a Raspberry Pi computer which worked
alongside the embedded MCUs. The new RPi design helped separate high-level tasks (such as
communication, data storage, and task-specific algorithms) from low-level tasks such as control of
the robot’s actuators and communication with the various sensors. This, in turn, facilitated testing
out new ideas at a quicker rate, while also offering the convenience of a Linux-like environment
and all its available tools (e.g., ROS implementation).

The current robot design did, however, suffer from some issues that future work should address.
The main drawback that this design faced was the time and resources needed for fabrication and
manufacturing. Since this was a novel design, almost all mechanical components of the robot
needed to be custom-built (and at times hand-crafted) for this project. While the effect of such an
issue is expected to diminish if these designs are mass-produced, they still presented a bottleneck
for prototyping and rapid testing. As an alternative, future work should focus on reducing the cost
and time needed for fabrication. Currently, a design that utilizes individually sealed components
is being investigated. These components are to be built from readily available material, and
are expected to greatly speed up fabrication (see Fig. 2.13). Moreover, these individually-sealed
components are expected to improve the modularity of the robot, as future designs can use different

24

combinations of these pre-built components depending on the tasks at hand.

The electrical system architecture of the next-generation robots will still use much of the
design elements developed here (e.g., power design and RPi architecture). However, there are still
additional features that need to be implemented in future designs. Specifically, next-generation
robots should focus on implementing underwater acoustic communication, satellite communication,
and solar charging for these robots. Moreover, new electrical systems should also mimic the
modularity of the mechanical components, such that these components offer “plug-and-play”
functionality.

The proposed advancements to the system design will allow for the implementation of more
interesting behaviors by next-generation robots. In particular, strategies for trajectory tracking and
depth maintenance are being investigated [33]. Moreover, the new set of electronics is expected
to facilitate fault detection, such as detecting hitting the bottom of a water body, which are useful
strategies for practical implementation of these robots.

25

CHAPTER 3

CHARACTERIZATION OF ACOUSTIC DETECTION USING A GLIDING ROBOTIC

FISH AS A MOBILE RECEIVER PLATFORM

In this chapter, we present the applicability of Grace 2 to acoustic telemetry. Generally speaking,
TDOA algorithms rely on a target emitting a signal periodically, which is detected by special
receivers deployed either at fixed locations or on mobile robots. If multiple receivers detect the
same signal, it is possible to infer the target’s location using the differences between the detection
times at these receivers, which is explored in detail later in Chapter 4.

As the demand for AUVs to study ecology of fish and other aquatic animals increases, there
remains a growing need to understand how design elements, operational characteristics, and en-
vironmental conditions influence detection performance of telemetry-equipped AUVs.
In this
chapter, we describe detection performance of an acoustic telemetry receiver mounted on Grace 2
during a series of field trials in a freshwater lake (Figs. 3.2– 3.4). Although the primary purpose
of field trials was to field-test hardware and software changes during development, detection per-
formance (efficiency) of the glider-mounted acoustic receiver was also evaluated during each field
test. Specific objectives were to (1) compare the detection performance (efficiency vs range) of
the AUV-mounted receiver to that of stationary receivers and (2) determine if the detection range
performance was related to the direction of travel (i.e., toward vs. away from a remote transmitter),
the robot depth, and the pitch during gliding. Environmental variables (wind, water temperature,
and acoustic noise) were also explored but data limitations prevented incorporating those data into
formal analyses. Results and lessons learned are expected to inform further development of gliding
robotic fish and improve the design of AUV-based telemetry performance assessments.

26

Figure 3.1: Grace 2 carrying a forward-facing acoustic receiver.

3.1 Methods

3.1.1 Mobile Receiver

The ability of Grace 2 in carrying larger payloads has allowed us to mount a forward-facing acoustic
receiver on the robot (Fig. 3.1), and compare its performance to that of stationary receivers.

The GPS sensor (Garmin GPS 18x LVC) was used for obtaining UTC time and establish the
robot’s position when it is on the surface, while the pressure sensor was used to measure robot
depth underwater. The XBee wireless serial interface was used for communication with a laptop
on a boat close to the robot. This channel was used to send commands to the robot or query data
from the robot when it was on the surface.

As was discussed in Chapter 2, the hardware and software capabilities of the robot evolved over
the duration of this study. In the 2016 trial, all robot operations (i.e., control, communication, and
data processing) were implemented on the embedded MCU, and the robot sent its GPS position
along with a UTC time-stamp over the XBee channel every 5 seconds whenever it was on the
surface. When the robot performed a dive, it temporarily stored depth data that was time-stamped
In 2017, the robot also temporarily stored
and sent through the XBee channel upon surfacing.

27

Figure 3.2: Grace 2 at Higgins Lake, Michigan, 3 November 2017.

Figure 3.3: A gliding robotic fish prototype during field experiments in Higgins Lake.

28

Figure 3.4: A gliding robotic fish prototype during field experiments in Higgins Lake.

orientation (yaw, pitch, and roll angles) information whenever it was underwater.
In 2018, we
incorporated a Raspberry Pi Zero W that performed high-level tasks such as communication and
data storage. This allowed us to complement the original broadcasted messages by storing all
available data onto an on-board SD card every 5 seconds. This data consisted of GPS coordinates
and UTC time, environmental sensor readings, orientation of the robot (yaw, pitch, and roll angles),
positions of each actuator, and battery level.

3.1.2 Field Tests

A self-contained acoustic receiver (VEMCO model VR2Tx; 69 kHz) was attached to the robot as
a payload, and field trials were conducted in Higgins lake, Michigan, USA, during 2016 - 2018.
The receiver was mounted at the bottom of the robot, facing forward (Fig. 3.1). The receiver
measured temperature (internal), tilt (degrees from vertical), and environmental acoustic noise

29

every 10 minutes. Detection performance was investigated as a function of distance between a test
transmitter or ‘tag’ and the receiver, robot direction (toward or away from tag), robot depth, and
robot pitch. Wind (from a regional model; see below) and environmental acoustic noise (from the
receiver) were also considered as possible explanatory variables but were not included in analyses.
Tests (‘runs’) were conducted on 11 November 2016, 3 November 2017, and 14-15 June 2018 in
Higgins Lake, Michigan, where water depths ranged between 10-20 m (Fig. 3.5; Table 3.1). These
sites were selected to facilitate field testing of navigation and actuation systems on a given day
and not necessarily with detection range tests in mind. During each trial, an acoustic transmitter
(VEMCO model V8-4H; source power level 147 dB re 1 
Pa @ 1 m) was deployed on a stationary
mooring (depth ranged 1.5–7.0 m among runs) and emitted a uniquely encoded signal every 24
seconds. The specific model tag was used because it provided a detection range amenable to the
field evaluation of the robot in the study system. Specific tag programming (24 s transmit interval)
was selected to ensure that testing would provide sufficient sample size for regression models and
to capture relatively fine-scale changes in variables of interest (e.g., depth, pitch).

For every test, a target GPS waypoint was sent to the robot from a laptop on a boat over the
wireless communication channel, and the robot was tasked with navigating to that location through
a series of dives. A depth of 4.5 m was deemed as the “safe” (maximum) diving depth for the
robot as the depth rating had not been established for the robot hull. Each dive was completed in
approximately 3 minutes on average. Robot heading was controlled by changes in tail position such
that the tail acted as a rudder during gliding to maintain course toward the target location. Between
dives, the robot remained on the surface for 20 seconds to ensure that a GPS lock was established,
and a new GPS position was obtained to calculate the desired heading angle.

3.1.3 Mobile Detection Range Tests

The main objectives for the 2016 trials were to (1) field-test hardware and software and identify
any necessary changes, and (2) establish the range of detection for the mobile receiver and compare
its detection performance with that of stationary receivers deployed at various distances from the

30

Figure 3.5: Lake tests (Panel A) Map of Higgins Lake, Michigan showing regions (red
rectangles) where gliding robotic fish GRACE was tested 11 Nov 2016 (panel B), 03 Nov 2017
(panel C), and 14-15 June 2018 (panels D-F) with stationary transmitter locations (green
circles), stationary receiver locations (yellow circles), and GPS locations (pink/red circles)
recorded by the gliding robotic fish during each run. GPS points are shaded along a gradient
from start (pink) to end (red) of each run.

31

tag. Five acoustic telemetry receivers (VEMCO model VR2Tx 69 kHz) were deployed vertically
(hydrophone up) on stationary moorings, suspended 1.5 m below the surface in 7.3–10.0 m water
depth. Stationary receivers were arranged in a line with the transmitter such that the receivers were
spaced 200, 400, 600, 800, and 1000 m from the transmitter, which was suspended 1.5 m below the
surface in 11.0 m water depth. During the first trial (run 2016-1), the robot began navigating 996 m
from the transmitter, in a direction roughly parallel to the line of stationary receivers and toward the
transmitter. As the trial progressed, winds (approximately 4.9-5.8 m/s, based on a regional model)
moved the robot off course while it attempted to obtain a GPS fix. Magnetic disturbance, resulting
from the internal electronics and actuators, which affected the stability of the measured heading
angle using the on-board Inertial Measurement Unit (IMU), also contributed to the robot veering
off-course. Consequently, the robot was removed from water about 563 m from the transmitter, and
then returned to the water for a second trial (run 2016-2). During the second trial, the robot started
411 m away from the tag, and was removed from the water 288 m from the transmitter.

Prior to the next field test, robot navigation problems observed in 2016 were addressed. First,
the IMU was calibrated to reject larger magnetic disturbances, which improved the stability of
the measured heading angle. Additionally, to combat high winds, the robot was tasked with
“swimming” (by continuously flapping its tail) on the surface while waiting for a GPS fix.

After these corrections, a second field trial was carried out in 2017. During this trial (run
2017-1), the gliding robotic fish navigated from 389 m to less than 1 m from the tag, in a direction
roughly northwest. The tag was suspended 3.0 m below the surface in 19.8 m water depth.
Stationary receivers were not deployed during the test in 2017 due to logistical constraints. While
the windspeeds during this trial were close to those in 2016 (approximately 4.2 m/s, based on a
regional model), the provisions we took with surface swimming allowed the robot to complete
its task. During navigation, the tag transmitted 323 times and the robot repeatedly ascended and
descended between the surface and 4.5 m depth.

A third set of field trials were conducted in 2018 to investigate the effect of the robot’s direction
of travel, depth, and pitch on detection efficiency. In addition, four stationary receivers (R1–R4;

32

Fig. 3.5) were deployed in roughly a square-shaped pattern to explore the utility of obtaining fine-
scale robot tracks using time-difference-of-arrival-based positioning (not evaluated in this work).
Stationary receivers were suspended vertically (hydrophone up) 4.7–19.8 m below the surface in
9.3–24.0 m water depth. A fifth stationary receiver was collocated with the stationary tag; both
were suspended vertically (hydrophone up) 4.6 m below the water surface in 9.1 m water depth.
The robot was tasked with navigating toward the tag during two runs (runs 2018-1, 2018-3) and
away from the tag during one run (run 2018-2). Similar to the trials in 2017, the robot was tasked
with swimming while awaiting a GPS fix on the surface to counteract surface waves that could push
the robot away from its desired course.

Water current and wind data (e.g., velocity and direction) were not measured at the site during
testing, but regional wind data were obtained from NCEP’s Global Forecast System [34] using the
R package RWind. Model-based estimates of mean wind speed and direction during 3-h intervals
(e.g., 00:00–02:59 UTC, 03:00–05:59 UTC, etc.) at 10 m above land surface were obtained for two
locations about 24 km west (85.0◦W, 44.5◦N) and 17 km east (84.5◦W, 44.5◦N) of the study site.
Mean speed and direction between the two locations were used to represent conditions at the study
site during the 3-h interval that contained each run.

3.1.4 Data Analysis

Transmitter detection efficiency curves were estimated for stationary and mobile receivers using
generalized additive models (GAMs) with binomial error structure1. GAMs were used because
they relax distributional assumptions and allow greater flexibility in modelling non-linear responses
than generalized linear models [35]. For example, a GAM allowed comparisons of the shape of
detection range curves among runs and between stationary and mobile receivers. First, the time
of each missed detection (i.e., transmitted but not detected) was estimated by sequencing every
24-s through each detection data set to identify time stamps that were not in the detection file
1Analysis of the acoustic detection data and the GAM models presented in this section were

prepared by Dr. Chris Holbrook from the US Geological Survey.

33

Table 3.1: Summary of mobile range detection trials with GRACE at Higgins Lake in 2016,
2017, and 2018. Mean heading reflects the bearing from the first GPS measurement toward
the last GPS measurement in each run. Wind speed and direction (heading; note that this
is the direction the wind vectors are following, not “blowing from” as is customary) are
coarse-scale regional estimates based on the NCEP’s Global Forecast System, as described in
methods.

Date Time
(UTC - 4)
End

Tag Lo-
cation

Distance to

tag (m)

Start

End

Run
ID

2016-1

2016-2

2017-1

2018-1

2018-2

2018-3

Start
11/11/16
10:47
11/11/16
12:52
11/3/17
12:45
6/14/18
18:06
6/14/18
19:20
6/15/18
10:06

11/11/16
12:38
11/11/16
13:30
11/3/17
14:54
6/14/18
19:00
6/14/18
21:13
6/15/18
11:09

T1

T1

T1

T6

T6

T7

996

411

389

360

44

408

563

288

8

11

361

10

Mean
Heading
(◦)
219 (SSW)

214 (SSW)

303 (WNW)

191 (S)

3 (N)

8 (N)

Wind
Speed
(m/s)

5.8

4.9

4.2

4.1

1.7

1.9

Wind
Heading
(◦)
187 (S)

182 (S)

148 (SE)

163 (SSE)

157 (SE)

324 (NW)

for the receiver. A binary indicator variable was used to represent detection (1) or non-detection
(0). All data types recorded by the robot (i.e., latitude, longitude, depth, pitch, water temperature)
were estimated at time of each detection or non-detection using time-based linear interpolation over
measurements recorded by the robot (Fig. 3.6–3.7). Detection efficiency curves were first described
separately for each run and for stationary and mobile receivers during each run, by fitting a GAM
to data from each run separately, except that runs 2016-1 and 2016-2 were combined. These GAMs
simply estimated the probability of detection (binomial response) as a function of distance between
tag and receiver (Fig. 3.8).

A GAM was fit to the data from runs 2017-1, 2018-1, 2018-2, and 2018-3 to estimate the
detection efficiency as a function of distance between receiver and transmitter, direction of robot
relative to transmitter (toward, away), robot depth (meters below water surface), and robot pitch
(degrees from horizontal). Runs 2016-1 and 2016-2 were not included in the model because (1)
there was little overlap in distances covered between those runs and others (Fig. 3.8); (2) they lacked

34

pitch data; and (3) depth data were incomplete during those runs.Specifically, the model estimated
the log odds (logit) of the probability of detecting each transmitted tag signal as a function of the
predictors






( 
) = 
0 + 


 + 


_



:


 + 


_



:


 + 


 

Ò:


 + 




Ò:


 + 


where 
0 is the intercept; 


 is the fixed effect of direction; 


_



:


 and 


_



:


 are
smoothed functions of distance for each direction and run; 


 

Ò:


 and 




Ò:


 are smoothed
functions of depth and pitch for each direction; and 
 is error assumed from an independent
draw from a normal distribution with zero mean and variance 
2. Direction-specific smoothers
were included to estimate the partial effects of each level of each continuous predictor on the
response (detection efficiency). The run-specific smoother for distance was included to account
for run-specific variability in detection efficiency not explained by other predictors. To eliminate
confounding between run and direction, effects of direction were limited to contrast between runs
2018-1 and 2018-2, which occurred on the same day.This was accomplished by treating runs 18-1
and 18-2 as a single run (“18-1 & 18-2”) in the model. Thus, we assumed that differences between
runs 2018-1 and 2018-2 were attributed to change in direction relative to tag and not other variables
or conditions. This assumption is supported by similar detection ranges of stationary receivers
during those runs (Fig. 3.8C, 3.8D)

An added advantage of a run-specific smoother is that it allows exploration of the unique shape
of each curve that might inform future hypotheses about factors influencing the shape of the curve
during each run. While it is possible that some across-run variation could be explained by wind,
ambient noise, or water temperatures, our observational data set did not have replicate runs (wind,
water temperature) or sufficient within-predictor contrast (ambient noise) and thus we did not
include those variables in the model. Rather, we use those observations to generate hypotheses
from our descriptive analyses.

The model was fit to the data using the gam function in the mgcv library (v. 1.8.31) [36] in R
(v. 3.6.2)[37]. The basis for all smoothing functions was a cubic regression spline with shrinkage
(bs = “cs” in mgcv) and 10 knots. Shrinkage smoothers are useful for identifying non-significant

35

Figure 3.6: Depth, pitch, and temperature profiles of gliding robotic fish Grace 2 during a
10-minute segment of run 2018-1 in Higgins Lake, with depths recorded by the on-board
sensors (open symbols) and estimated depths at the time of test tag signal transmissions
(red symbols). Horizontal broken lines show depth (panel A) and temperature (panel C) of
stationary transmitter at site T6.

smoothers by reducing the effective degrees of freedom to a value as small as zero [35]. All
smoothers were estimated using restricted maximum likelihood. Prior to model fitting, data were
checked for evidence of collinearity (correlations among predictors). Model fit was evaluated
by checking for concurvity (non-independence) among smoothers and evidence of non-normality
among residuals. Significance of each smoothing term was determined based on estimated degrees
of freedom and approximate p-value for the null hypothesis that the smoothing term was zero (see
Table 3.2). The significance level for all tests was 0.05. Partial effects plots (see Fig. 3.9) were
used to assess the influence of each smoother on the log odds of detection efficiency at each level of
the predictor. Plots of fitted values were used to assess the influence of each predictor on detection
efficiency during each run on the probability scale.

36

Figure 3.7: Depth profiles of gliding robotic fish Grace 2 during a selected 100– to 300–m
segment of each run (panels A-D) in Higgins Lake, with locations of test tag transmissions
that were detected (closed symbols) or not detected (open symbols) by the on-board receiver.

37

Table 3.2: Summary of parametric coefficients (linear terms) and smoothing terms from
GAM used to determine if detection range was related to distance from transmitter (Tag
Distance, in meters), direction of robot travel relative to transmitter (toward or away), robot
depth (Depth, in meters from water surface), or robot pitch (Pitch, in degrees from horizontal).
Included for each parametric coefficient is the estimate, standard error (SE), test statistic (Z),
and p-value for the null hypothesis that the corresponding parameter is zero.
Included
for each smoothing term is the estimated degrees of freedom (EDF), test statistic ( 
2), and
approximate p-value for the null hypothesis that the smoothing term is zero. Bold faced
p-values are significant at significance level of 0.05.

Linear Terms
(Intercept)



 = 




 

Smoothing terms


(

_







) : 


 = 





(

_







) : 


 = 







(

_







) : 


 = 2017 − 1

(

_







) : 


 = 2018 − 3


(

_







) : 


 = 2018 − 1, 2018 − 2


( 



Ò) : 


 = 





( 



Ò) : 


 = 




 


(

 

Ò) : 


 = 





(

 

Ò) : 


 = 







Estimate
-4.194
5.19
EDF

6.33E-04

1.432
2.085
2.480
1.223
0.105
1.103
1.256
0.649

SE
0.562
0.573

Z

-7.466
9.071

2

4.03E-04

3.313
6.316
46.726
3.417
0.112
5.804
11.658
1.326

p-value
8.24E-14
1.18E-19
p-value
0.318

1.55E-03
3.62E-04
1.25E-13
2.09E-03

0.120

0.293

8.15E-03
3.06E-04

Table 3.3: Speed of the robot relative to ground from the field trials.

Component
Swimming speed
Glide speed

Description
25 cm/s on average (20 to 30 cm/s)
13 cm/s on average (as low as 5 cm/s against current
and a high of 35 cm/s with current)

3.2 Results

A rough estimate of the robot’s glide speed and swimming speed relative to ground was obtained

from the logged data and is shown in Table 3.3.

Estimated detection efficiency of the robot-mounted receiver was lower than concurrently-
operated stationary receivers during all three runs where stationary receivers covered the full range
of mobile runs(Fig. 3.8A, 3.8C, 3.8D). Scale and shape of detection curves from both stationary
and mobile receivers varied among runs. During 2016, when detection range for both stationary

38

and mobile receivers was larger than during any other run, detection efficiencies for the mobile
receiver were only slightly lower than the stationary receivers up to 600 m from the transmitter, but
substantially lower at 800 m and 1000 m. The shape of the stationary range curve was unexpected
(e.g., higher efficiency at 800 m than 600 m), suggesting that stationary receivers were affected by
processes that were not observed. During 2018-1, estimated range was similar between stationary
and mobile receivers at distances up to 200 m but differed at larger distances due to mobile
detection range declining much faster than stationary detection range. The greatest difference
between stationary and mobile detection range was observed during 2018-2 when the robot was
moving away from the transmitter. During that run, mobile detection range was markedly lower,
even at 50 m—the closest distance between receiver and transmitter during that run.

The GAM model explained 50% of the null deviance. Detection efficiency differed by direction
of travel (Table 3.2) with more than a 5-fold increase in the log-odds of detection when moving
toward the transmitter than away from the transmitter. Significance of distance-based smoothers
suggested that variation in the shape of range curves was attributed to direction of movement and
other run-specific variables not included in the model. Pitch was significant when the robot was
moving toward the transmitter, but not when the robot was moving away from the transmitter
(Fig. 3.9A, 3.9C). Depth was significant when the robot was moving away from the transmitter but
not when the robot was moving toward the transmitter (Fig. 3.9B, 3.9D). Over the ranges of depth
and pitch observed, the effect of depth on detection efficiency moving away from the transmitter
was greater than the affect of pitch moving toward the transmitter (Fig. 3.10).

3.3 Discussion

Understanding detection performance of telemetry receivers can be critical for designing and
conducting a successful animal tracking project [5]. Although it is sometimes possible to derive an
animal location from detection time differences among multiple hydrophones [22], the location of a
receiver at the time of detection is often used to represent the location of a tagged animal at time of
detection for “presence-absence”-type data. Detection range data is useful for interpreting spatial

39

Figure 3.8: Estimated efficiencies (proportion of coded signals detected) of detecting an
acoustic transmitter on acoustic receivers affixed to stationary moorings (red symbols; red
shaded regions are GAM-based 95% confidence regions) and gliding robotic fish Grace 2
(black line; grey shaded regions are GAM-based 95% confidence regions) in Higgins Lake,
Michigan during field tests in 2016 (panel A), 2017 (panel B), and 2018 (panels C-E). Vertical
grey bars show distances between robot and the transmitter when each coded signal was
detected (1) or not detected (0).

40

Figure 3.9: Partial effects of robot depth and pitch on detection probability. Estimated partial
effects of robot depth and pitch on the log odds of detection probability when the robot is
moving away from (panels A, B) or toward (panels C, D) a stationary transmitter in Higgins
Lake, Michigan.

41

Figure 3.10: Estimated efficiencies (proportion of coded bursts detected) of detecting an
acoustic transmitter on acoustic receivers affixed to gliding robotic fish Grace 2 in Higgins
Lake, Michigan, during field tests in which evidence (GAM model results) suggested that
detection range was influenced most by depth when the robot was moving away from the
transmitter (panel C) and pitch when the robot was moving toward the transmitter (panel A,
B, D).

42

ambiguity or uncertainty around detection locations but is not commonly collected [38]. Moreover,
receivers are often assumed omnidirectional, though several processes contributing to directional or
non-uniform detection areas have been described [39]. In our study, decreased detection efficiency
when the robot was moving away from the transmitter was not unexpected due to potential shielding
of the signal based on position and orientation of the receiver on the robot (attached to the bottom
with the hydrophone facing forward) and the importance of line-of-site to acoustic detection. The
magnitude of the effect, however, has important implications for the ability of an AUV in this
configuration to detect tagged fish and for inferences of fish locations based on detection data.

Depth and pitch are critical control parameters of AUVs that are carefully programmed to ensure
mission success in the face of environmental and energetic constraints. When the objective is to
detect acoustic-tagged fish, operational adjustments may be needed to achieve favorable balance
among detection efficiency and other operational processes (e.g., navigation, communication). Our
results suggest that both depth and pitch can influence detection efficiency, even under a narrow
scope of environmental conditions. Vertical gradients of environmental variables known to affect
acoustic signals in water (e.g., salinity, temperature, suspended particulates, entrained air) exist in
most aquatic systems but we were not aware of any such gradients during this study. Although lack
of environmental heterogeneity may limit application of these results to other systems, it may have
improved our ability to detect relatively small effects by minimizing background variability during
our study. Moreover, the potential silver lining of observed variability that is not environmentally-
driven is that it may be related to variables that can be controlled or modified.

We hypothesize that decreasing detection efficiency from the surface to 4 m depth when moving
away from the transmitter was driven by shielding of the signal by the body of the robot at depth.
As described above, this seems reasonable based on position and orientation of the receiver on the
robot. If true, then the depth variable used in our analysis may be a proxy for the difference in
depth between the hydrophone and the transmitter. Although the range in depth difference between
receiver and transmitter in our tests (receiver ranged 0 to 4 m above the tag) may be representative
of some shallow environments, potential clearly exists for much greater vertical separation in many

43

aquatic systems. For an AUV with bottom-mounted receiver, shielding effects are expected to be
largest when the vehicle is deeper than a tagged fish and smallest when the vehicle is shallower
than the tagged fish. Thus, effects of shielding may be minimized by operational parameters (e.g.,
depth range) based on knowledge of the ecology of the target organisms (i.e., remaining near the
surface for pelagic fish) or structural changes to the vehicle (i.e., positioning the receiver on top of
the AUV for surface-oriented fish).

Although the mobile receiver on the gliding robotic fish did not perform as well as stationary
receivers throughout the entire range tested, detection performance over shorter distances (300 m
or less) are likely still adequate for many active tracking needs. In practice, acoustic transmitter
detection ranges vary considerably by hardware and software differences and environmental con-
ditions. Future work may seek to determine if differences between the robot-mounted receiver
and stationary receivers are due to characteristics of the robot (e.g., electrical or mechanical noise)
or interaction of the robot with the environment (e.g., turbulence of flow over the hydrophone).
Regardless of future improvements, however, knowledge of the robot-mounted detection ranges
will be useful for planning future missions, including active tracking with a network of AUVs.

Although these results add to our knowledge of AUV performance, much variation in detection
efficiency remained unexplained in our analysis. Unfortunately, we were not able to account for
the influence of the environmental variables (e.g. wind, water temperature, ambient noise) on
the results because availability of those data were limited. However, we are still able to use
those observations to generate hypotheses from our descriptive analyses. Our observational data
set did not have sufficient replication (multiple runs) over environmental variables (e.g., wind,
ambient noise, thermal stratification). Future work should seek to obtain a balanced study design
with replicate runs over a range of environment variables, so that one can attribute variation in
the detection range to those variables. Obtaining environmental measurements at appropriate
resolution and scale is also important, and these measurements likely need to be recorded locally
and frequently to be relevant to tag transmissions that occur every minute or less.

44

3.4 Summary and Future Work

AUVs are becoming important tools for understanding the relationships between aquatic organ-
isms and their environment. We present results on the applicability of Grace 2 to acoustic telemetry
of tagged fish in fresh and shallow water.

A forward-facing acoustic receiver (VR2Tx) was attached to Grace 2 in order to compare its
detection efficiency of acoustic tags (VEMCO model V8-4H) to stationary receivers. Initial tests
indicate that the detection efficiency for the mobile and fixed receivers are close up to 600 m, with
good detection efficiency at a distance of 300 m (78% for stationary receivers compared to 76%
for the mobile receiver). However, the detection efficiency for the mobile receiver drops faster
than that of the stationary receivers at greater distances. Lowest detection efficiency is observed
when the mobile receiver is facing away from the tag, indicating that alignment between the tag
and the receiver’s detection cone is important. This can help guide the development of tag tracking
controllers and future robot prototypes to minimize shielding of the mobile receiver. As an example,
tag-tracking controllers should consider the directionality of the receiver, as well as the relative
position of the tag with respect to the receiver. Other options to mitigate this effect include the
use of vertically mounted receivers, pointing upwards or downwards [14, 10], using two receivers
mounted back to back, or mounting the receiver on an active bracket that can rotate freely to better
align the receiver towards the transmitters.

On the other hand, there are some questions that need to be addressed to further characterize
the robot’s applicability to active tracking of acoustic signals, and acoustic telemetry in general,
such as capability of overcoming currents, battery life, and maximum depth rating. Understanding
these parameters is important to comparing such prototypes to commercially available tools, and
future work will compare using these robots to that of other vehicles. Future work should also
focus on conducting more replicate runs over a range of environmental variables (e.g., wind, noise,
stratification), and testing real-time control algorithm for active tracking of acoustic tags.

45

CHAPTER 4

TIME-DIFFERENCE-OF-ARRIVAL (TDOA)-BASED DISTRIBUTED TARGET

LOCALIZATION BY A ROBOTIC NETWORK

Motivated by the application of tracking tagged fish by a network of gliding robotic fish, we present
theoretical treatment for the localization and tracking of a moving target. Much of the work on
TDOA-based localization in the literature adopts a centralized approach, in which a reference node
is chosen and the times of arrival (TOA) of the emitted signal for all other nodes in the network
are subtracted from the reference node’s TOA, generating TDOA measurements at fusion hub. If
the propagation speed of the signal is known, the TDOA measurements can be converted to range-
difference measurements, which are then used to estimate the location of the target [40, 41]. This
centralized approach has a long history and is widely used in aerospace systems [42]. Geometric
treatment of the problem for a stationary target was considered in [43] and [44], where the target
location is inferred from the geometric relations imposed by the TDOA measurements. When
the target’s location changes with time, dynamic approaches are generally used for localization, in
which a filter is used to estimate the target’s location. Examples of these methods include utilizing
an Extended Kalman Filter (EKF) in [45], or an Unscented Kalman Filter and Particle Filters in
[46].

Due to power and bandwidth constraints in robotic networks, centralized information processing
is often infeasible, particularly for a large-scale and unreliable networks. Moreover, some sensors
cannot transmit their measurements to the reference node due to their limited communication ranges.
These drawbacks motivated the investigation of distributed strategies for TDOA-based localization.
In [47], decentralized source localization in multihop networks was considered, where a connected
dominating set of nodes work as the network backbone to collect the measurements, and a leader
node of that set is selected to estimate the target’s location, essentially acting as a centralized
estimator of the target’s position. The need for a common reference node is alleviated in [48],
where a network of paired sensors is utilized while requiring all such pairs to be able communicate

46

with one another. As we discuss later in Section 4.2, this decentralized approach can be improved
upon by exchanging estimate information between nodes, allowing for successful estimation without
requiring all agents in the network to be connected.

In Appendix A, we show the boundedness of the estimation error using the stochastic stability
lemma [49, 50]. However, the conditions developed there are difficult to verify. Namely, finding
positive definite matrices which satisfy Lemma A.2.1 are difficult to compute. Moreover, it is not
clear how such matrices relate to the underlying communication graph among agents for exchanging
information. For those reasons, structural observability is used in this chapter to investigate the
network topology conditions for distributed localization of a moving target. It is shown that source
localization is not possible (centrally, and therefore distributively) when the total number of TDOA
measurement is insufficient. Then, we demonstrate that the target’s position can be successfully
estimated in a distributed manner if and only if every agent is part of a connected network that
collectively has a sufficient number of TDOA measurements, even if each agent has an insufficient
number of measurements.

Structural analysis deals with system properties that do not depend on the numerical values of
the parameters, but only on the underlying structure (zeros and non-zeros) of the system matrices
[51, 52, 53]. It turns out that if a structural property holds for one possible choice of non-zero
elements as free parameters, it is true for almost all choices of non-zero elements and, therefore,
is called a generic property of the system. Furthermore, it can be shown that those particular
(non-admissible) choices for which the generic property does not hold, lie on some algebraic
variety with zero Lebesgue measure [54]. While this work is similar to [52] and [55] in that it
employs structural analysis on the system matrices, there is a significant difference. In particular,
the results reported in [52] and [55] treat all non-zero elements as free parameters, which in turn
disguises the importance of the number of TDOA measurements used in this localization scheme.
In Section 4.2, we explicitly consider the role played by using more TDOA measurements, and
prove that the system can be rendered generically observable when a sufficient number of TDOA
measurements are used.

47

The work presented here introduces a fully distributed solution to target localization under fixed
and time-varying undirected communication topologies. This approach does not require a common
reference node or data fusion center, nor does it require each agent to be heavily connected to
estimate the target’s location on its own. Instead, we show that every agent in the network can
successfully localize the target if it is part of a network that has a minimum of 4 connected, non-
coplanar agents. As the target moves away from the convex hull formed by the agents, estimation
performance begins to degrade and it becomes paramount to track the target as it moves through
space. Rather than continuously tracking the target, we further propose an adaptive movement
strategy, where the robotic network moves only when the norm of the estimation covariance matrix
exceeds a certain limit, to balance the trade-off between estimation performance and distance
traveled by the entire mobile network.

4.1 Problem Setup

We consider a moving target in the 3D space with 
(
) =

coordinates at time 
. The target moves randomly in space according to



(
) 

(
) 

(
)(cid:105)(cid:48)
(cid:104)
 
(
),


(
)

(cid:164)
(
)

 +

 0


3

denoting its

(4.1)

 (cid:164)
(
)

(cid:165)
(
)

 =

0 
3

0

0



where 
3 is the 3 × 3 identity matrix and 
(
) ∈ R3 is the process noise, which is assumed to be
zero-mean, white Gaussian noise with covariance matrix 
.

The target emits a signal periodically that gets detected by a group of 
 robotic agents, or nodes,
at different times depending on each agent’s relative distance to the target. At each detection, agent

 records the signal’s TOA and acquires the TOAs of all other agents that can communicate their
information to agent 
. These agents form the set of neighbors of agent 
, which is denoted as
N
. Each agent then subtracts the TOAs of its neighbors from its own TOA, generating a list of
time-difference-of-arrival (TDOA) measurements. Assuming that the propagation speed of the
signal is known, the measurements available for each agent are given by



(

) = Ò
(

) + 

(

),

(4.2)

48

where

with



Ò
,1(
)

...

Ò
,|N
|(
)

 ,

Ò
(
) =

(4.3)

Ò
, 
(
) = (cid:107) 
(
) − 

(
)(cid:107) − (cid:107) 
(
) − 

, 
(
)(cid:107).

(4.4)
Here, 
 is the period at which the signal is emitted, 

(
) ∈ R|N
| is the measurement noise, assumed
to be zero-mean, white Gaussian noise with covariance matrix 

, 

(
) is the position of agent 
,
and 

, 
(
) is the position of the 
-th neighbor of agent 
.

Denoting the target’s state as 
(
) =

, and discretizing the model in (4.1) with
sampling time 
, with slight abuse of notation, we can write the discrete-time model of the target
as


(cid:48)(
)

(cid:164)
(cid:48)(
)(cid:105)(cid:48)

(cid:104)


(
 + 1) = 

(
) + 

(
),

where


 =

(4.5)

.

(4.6)

1 0 0 
 0 0
0 1 0 0 
 0
0 0 1 0 0 

0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1



, 
 =




2
2
0
0




0
0

0

2
2
0
0




0

0
0

2
2
0
0









The time-varying measurement matrix 

(
) can be obtained from (4.2), where



(
) =

(4.7)

 ,

0 0 0
...
...

...

0 0 0


Ò
,1(
)

 

(
)

...


Ò
,1(
)

 

(
)

...


Ò
,1(
)

 

(
)

...


Ò
,|N
|(
)

 

(
)


Ò
,|N
|(
)

 

(
)


Ò
,|N
|(
)

 

(
)

49

and


Ò
, 
(
)

 

(
) =

Ò
, 
(
)

 

(
) =

Ò
, 
(
)

 

(
) =


 (
)

(cid:107) 
(
) − 

(
)(cid:107) − 

(
) − 




(
) − 


(cid:107) 
(
) − 

(
)(cid:107) − 

(
) − 




(
) − 


(cid:107) 
(
) − 

(
)(cid:107) − 

(
) − 




(
) − 




, 
(
)
(cid:107) 
(
) − 

, 
(
)(cid:107) ,

, 
(
)
(cid:107) 
(
) − 

, 
(
)(cid:107) ,

, 
(
)
(cid:107) 
(
) − 

, 
(
)(cid:107) .


 (
)


 (
)

(4.8)

(4.9)

(4.10)

4.2 Distributed Estimation

In this section, we look into the problem of distributed localization of a moving target using
TDOA measurements. The goal is for every agent to estimate the target’s position without requiring
a central node to collect all measurements and propagate an estimate to all agents in the network. To
that end, we first discuss a decentralized estimation scheme, where structural observability analysis
is conducted to derive the minimum number of TDOA measurements required for an agent to
estimate the target state on its own. We will then discuss the distributed estimation scheme where
agents exchange estimate information, and present the necessary and sufficient condition in terms
of network topology for achieving stable estimates.

4.2.1 Decentralized Estimation

In this approach, each agent runs its own filter using its own TDOA measurements. Here, agents
exchange only their locations and TOA values to generate TDOA measurements without exchanging
any other pieces of information. Each node implements an extended Kalman filter (EKF) to estimate
the target’s state

ˆ

(
|
 − 1) = 
 ˆ

(
 − 1|
 − 1),

ˆ

(
|
) = ˆ

(
|
 − 1)

+

(
)(cid:2)

(
) − Ò
( ˆ

(
|
 − 1))(cid:3),

(4.11)

(4.12)

50

where ˆ

(
| 
) is the 
-th node’s estimate of the state at time 
 after the 
-th measurement has been
processed, and 

(
) is its filtering gain, which is computed according to



(
) = 

(
|
 − 1)

(
)(cid:48)

×(cid:2)

(
)

(
|
 − 1)

(
)(cid:48) + 

(cid:3)−1

,

(4.13)

(4.14)

and



(
|
 − 1) = 


(
 − 1|
 − 1) 
(cid:48) + 


(cid:48),



(
|
) = (cid:2)
 − 

(
)

(
)(cid:3) 

(
|
 − 1)(cid:2)
 − 

(
)

(
)(cid:3)(cid:48)

+

(
)



(
)(cid:48),

(4.15)
where 

(
| 
) is the 
-th agent’s error covariance matrix at time 
 after the 
-th measurement has
been processed.

It is well-known (see [56] and [57]) that the estimation error for agent 
 under this scheme,

which propagates as follows,

˜

(
 + 1) = 
(
 − 

(
)

(
)) ˜

(
) + 

(
),

(4.16)
is stable if and only if the pair ( 
, 

(
)) is observable, where ˜

(
) (cid:44) 
(
) − ˆ

(
|
) is the
estimation error for agent 
 and the vector 

(
) collects the terms independent of ˜

(
). In the
following, we will show that the pair ( 
, 

(
)) is unobservable when agent 
 has less than 3
TDOA measurements. To avoid clutter, we will consider only agent 1 of the network, and drop the

 subscript from the following analysis.

As discussed earlier, if a structural property is true for one admissible choice of non-zero
elements, it is true for almost all choices of non-zero elements. Additionally, it can be shown
that the choices of parameters for which the generic property does not hold, lie on a hypersurface
(see Definition 4.2.1) in the free parameter space with zero Lebesgue measure [54]. Due to the
fixed structure of our system matrix 
 in (4.6) and the time-varying measurement matrix 
(
)
in (4.7), it is beneficial to utilize structural analysis when examining the observability of our
system. In the following, we employ a structural approach to establish the minimum number of
TDOA measurements needed to render the process generically observable.

51

Figure 4.1: Network of 3 agents monitoring the target with agent 1 as the reference node.

Figure 4.2: Network of 4 agents monitoring the target with agent 1 as the reference node.

Definition 4.2.1 Let 
 = 
 (
1, . . . , 

) be a polynomial in the 
 variables 
1, . . . , 

 with coeffi-
cients in R. Then the point ¯
 = ( ¯
1, . . . , ¯

) in R
 is called a zero of 
 if 
 ( ¯
1, . . . , ¯

) = 0. The
set of zeros of 
 is called the locus of 
 . A subset 
 of R
 is called a hypersurface in R
 if it is the
locus of a nonconstant polynomial.

First, we consider the case where agent 1 only has two neighbors and, therefore, only two TDOA
measurements as shown in Figure 4.1. The measurement matrix 
(
), in this case, admits the
following structure

(4.17)



 =


4 
5 
6 0 0 0


1 
2 
3 0 0 0
 .
(

 
5)(cid:48)(cid:105)(cid:48)

· · ·

(

 
)(cid:48)

(

 
2)(cid:48)

(cid:104)

O =


(cid:48)




The system is said to be generically observable if the pair ( 
, 

) is observable for almost
all values of 
, 
1, . . . , 
6. In other words, the system is generically observable if and only if the
observability matrix O is full rank for almost all values of 
, 
1, . . . , 
6, where

.

(4.18)

It is well known that rank(O) < 6 if and only if all 6 × 6 minors of O are zero [54]. We can
easily verify that all 6 × 6 minors of O in (4.18) are zero, regardless of the values of 
, 
1, . . . , 
6.
This implies that the process in (4.5) with measurement matrix (4.7) is unobservable when the node
has two or less TDOA measurements1.

1If the node has only one TDOA measurement, and therefore only one neighbor, then there is

only one 6 × 6 minor of O and it is det(O).

52

Now we consider the case where agent 1 has three neighbors and, therefore, three TDOA
measurements as shown in Figure 4.2. The measurement matrix 
(
), in this case, admits the
following structure





 =


1 
2 
3 0 0 0

4 
5 
6 0 0 0

7 
8 
9 0 0 0

 .

(4.19)

Checking all 6 × 6 minors of O, we observe that some minors of O are not identically zero and

are all of the form



3(
3 (
5
7 − 
4
8) + 
2 (
4
9 − 
6
7)
+
1 (
6
8 − 
5
9))2,

(4.20)

for some 
 ∈ R. Therefore, we conclude that rank(O) = 6 for almost all values of 
 and 
1, . . . , 
9,
and that the pair ( 
, 
(
)) is generically observable if the agent has a minimum of 3 TDOA
measurements2. Furthermore, the set of values that render the pair unobservable is a hypersurface
in the free parameter space where the expression in (4.20) is zero. Interestingly, this means that the
process is generically observable except when:

• The sampling time used for discretization of the system in (4.5) is 0, or

• All of the points that satisfy

det



(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)


1 
2 
3

4 
5 
6

7 
8 
9



(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = 0,

i.e., when the 3 TDOA measurements are linearly dependent, and the 4 agents are coplanar.

For all agents in the network to be able to estimate the target’s position under this decentralized
scheme, we would require that all such pairs ( 
, 
1(
)),( 
, 
2(
)),. . . ,( 
, 

(
)) to be observ-
able, i.e., we would require the pair (

 ⊗ 
, 

) to be observable, where ⊗ denotes the Kronecker
2If the node has more than 3 TDOA measurements, it can be verified that rank(O) = 6 if

rank(

) ≥ 3.

53

product, and

 .

0

. . .



(
)


1(
)


ˆ

(
|
)(cid:48)(cid:105)(cid:48)

0



(
) (cid:44)

. . . 
(
)(cid:48)(cid:105)(cid:48)

. . .

(cid:104)
(cid:104)

(4.21)

Let ˆ
(
|
) =
wide state 
(
) =
entries are all 1. The dynamics of this network-wide state can be derived as follows

denote the network-wide estimate of the network-
= 1
 ⊗ 
(
), where 1
 ∈ R
 is the column vector whose

ˆ
1(
|
)(cid:48)

(
)(cid:48)


(
 + 1) =1
 ⊗ ( 

(
) + 

(
))
= (

 ⊗ 
) (1
 ⊗ 
(
))
+ (

 ⊗ 
)(1
 ⊗ 
(
))
= (

 ⊗ 
) 
(
) + (

 ⊗ 
)
(
),

(4.22)
with 
(
) (cid:44) 1
 ⊗ 
(
) representing the network-wide process noise. Denoting the 
-th agent’s
estimation error by ˜

(
) (cid:44) 
(
) − ˆ

(
|
), and the network-wide estimation error ˜
(
) (cid:44)
˜
1(
)(cid:48)

˜

(
)(cid:48)(cid:105)(cid:48)

(cid:104)

. . .

, the dynamics of ˜
(
) are given by
˜
(
 + 1) = (

 ⊗ 
) (
6
 − 
(
)

(
)) ˜
(
)

(4.23)

+ 
(
),

where 
(
) is a block-diagonal matrix of the filter gains 
1(
) . . . 

(
), and the vector 
(
)
collects the terms independent of ˜
(
). This network-wide estimation error can be stabilized if
the pair (

 ⊗ 
, 

(
)) is generically observable, where each agent needs to have a sufficient
number of neighbors to estimate the process using only its own TDOA measurements.

Under this formulation, each agent can estimate the target’s location when it has a minimum
of 3 TDOA measurements, corresponding to each agent having a minimum of 3 neighbors. This
decentralized approach requires each agent to be heavily connected such that the target system is
observable using each agent’s own measurements. Next, we discuss how the number of required
communication links can be reduced, and argue that it is possible to estimate the target’s location
without the need for heavily connecting the agents.

54

4.2.2 Distributed Estimation
Consider the dynamical system in (4.22), and noting that for a stochastic matrix 
 ∈ R
×
,

1
 = 1
, we can rewrite (4.22) as


(
 + 1) =1
 ⊗ ( 

(
) + 

(
))

=
1
 ⊗ 

(
) + 1
 ⊗ 

(
)
= (
 ⊗ 
) 
(
) + (

 ⊗ 
)
(
).

(4.24)

For this modified dynamical system in (4.24), a centralized filter can be designed with estimation

error dynamics that can be expressed as

˜
(
 + 1) = (
 ⊗ 
) (
6
 − 

(
)

(
)) ˜
(
)

+ 
(
),

(4.25)

where 

(
) is the filter gain, which can stabilize the error dynamics if the pair (
 ⊗ 
, 

(
)) is
generically observable. In the following, we will show that it is possible to obtain a network-wide
estimation error with dynamics similar to (4.25) by averaging the estimates among neighboring
agents.

Let 
 ∈ R
×
 be a stochastic matrix with entries 

 
 > 0 if 
 = 
 or if agents 
 and 
 can
exchange information; otherwise 

 
 = 0. We assume here that every agent has access to its own
information (i.e., 


 > 0), and that the communication links are bidirectional, namely, if agent 

can send its information to agent 
, then the reverse is also true, 

 
 , 
 

 > 0. Every agent in the
network implements a filtering scheme similar to the decentralized case, but followed by updating
its estimate by averaging the estimates from neighbors and itself. The filter implemented by each
agent in the network is then given by

55


=1





(cid:20)
(cid:18)

one-step formulation of agent 
’s estimate can as

The 
-th agent’s estimation error is then given by



 



 ˆ
 
(
)+

ˆ

(
 + 1) =


=1


 
(
)


 
(
) − Ò 
( ˆ
 
(
))

(cid:19)(cid:21)


 
(cid:2)
(
 − 
 
(
)
 
(
)) ˜
 
(
)
+ 
 
(
)(cid:3) .
Denoting the network-wide estimation error by ˜
(
) (cid:44)(cid:104)

˜

(
 + 1) =






=1

˜
1(
)(cid:48)

. . .

.

(4.29)

(4.30)

(4.31)

˜

(
)(cid:48)(cid:105)(cid:48)

, then

ˆ

(
|
 − 1) = 
 ¯

(
 − 1),
ˆ

(
|
) = ˆ

(
|
 − 1)

+

(
)(cid:2)

(
) − Ò
( ˆ

(
|
 − 1))(cid:3),






 
 ˆ
 
(
|
).

¯

(
) =

(4.26)

(4.27)

(4.28)

Denoting ˆ

(
|
 − 1) by ˆ

(
), and substituting (4.27) and (4.28) into (4.26), we can express a

˜
(
 + 1) = (
 ⊗ 
) (
6
 − 
(
)

(
)) ˜
(
)

+ 
(
),

which is similar to (4.25), except that here the gain matrix 
(
) is restricted to be block-
diagonal. As explained in [52], computing such a constrained gain is possible via an iterative
cone-complementary optimization algorithm; see [58] and [59] for details. In [60], the authors
derived a suboptimal filtering gain inspired by the Markovian jump linear system filtering problem,
where



(
) =

(
|
 − 1)

(
)(cid:48)

×(cid:2)

(
)

(
|
 − 1)

(
)(cid:48) + 

(cid:3)−1,

(4.32)

56

and



(
|
 − 1) = 
 ¯

(
 − 1) 
(cid:48) + 


(cid:48),



(
|
) = (cid:2)
6 − 

(
)

(
)(cid:3) 

(
|
 − 1)

×(cid:2)
6 − 

(
)

(
)(cid:3)(cid:48)




+

(
)



(
)(cid:48),


 
 
 
(
|
).

¯

(
) =

(4.33)

(4.34)

(4.35)


=1

Therefore, to ensure the convergence of the networked filter, one has to ensure that the networked
system is observable [52]. To that end, we investigate the conditions on the matrix 
 and, therefore,
the topology of the undirected communication graph among agents, that would render the pair
(
 ⊗ 
, 

) observable, and therefore ensure the convergence of the networked filter. We first
note the following property regarding the powers of the matrix 
 from [61].

Lemma 4.2.2 Let [
 
]
 
 denote the (
, 
) element of the matrix 
 
, where 
 is the stochastic
matrix representing the communication topology with 


 > 0. Then, [
 
]
 
 > 0 if there is a path
between agents 
 and 
 of length less than or equal to 
; otherwise [
 
]
 
 = 0.

The pair (
 ⊗ 
, 

) is observable if and only if rank(O) = 6
. Here,

(4.36)

(cid:104) O1

(cid:105)

where 
 = 6
 −1. Equivalently, denoting O
 as the block column representing agent 
’s subsystem,



. From the structure of 

, it is easy to see that rank(O) =
we can write O =

=1 rank(O
).
We are now ready to present the main results in this chapter.

. . . O




O =



(
)



(
)(
 ⊗ 
)


(
)(
 ⊗ 
)2

...



(
)(
 ⊗ 
) 




(
)



(
)(
 ⊗ 
)


(
)(
2 ⊗ 
2)

...



(
)(
 
 ⊗ 

),





=



57

Theorem 4.2.3 For a network of agents with time-invariant and undirected topology, the system
under the proposed distributed TDOA-based localization is observable if and only if every agent is
part of a (sub)network that has at least 4 connected, non-coplanar agents.

Proof: Sufficiency: We consider the case where agent 
 is part of a network that has only 4
connected agents, and note that the following results can be easily extended to the cases where the
network has more agents. Since it is always possible to renumber the agents, we will only consider
the subsystem corresponding to agent 1, and write



O1 =

(cid:2)
23(cid:3)


1(
)

...

41 
4(
) 
23

 .

(4.37)

This agent can be connected to the network in 4 possible ways that are shown in Fig. 4.33.
For the ease of presentation, in the following, we examine Case (d), where the agent has only one
neighbor, and show that rank(O1) = 6. The other three cases are discussed in Appendix B.1.

Case (d)
The structure of 

(
) for 
 = 1, . . . , 4 is as

,


1 
2 
3 0 0 0

(cid:104)
(cid:105)
−
1 −
2 −
3 0 0 0
 ,
−
4 −
5 −
6 0 0 0
 ,
(cid:105)
(cid:104)−
7 −
8 −
9 0 0 0

0 0 0

0 0 0


7


8


9


4


5


6

.


1(
) =


2(
) =


3(
) =


4(
) =

3The graphs shown in Fig. 4.3 represent the minimum number of links required for each graph
to be connected. It is possible to add more edges among agents in these graphs and adding more
links will only help in terms of observability.

58

Figure 4.3: The 4 possible configurations for node 1 to be connected to the network using the
minimum number of edges.

Recalling that

it can be shown that the pair

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

rank





...


 

−1



(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) ,


 


...


 

+
−1

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)



(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)


1(
)

3(
)

4(
)


(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = rank

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)
,

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)


1(
) 
6

3(
) 
6

4(
) 
6


1(
)

3(
)

4(
)

...

59

= 6.

is generically observable, implying that

rank

for almost all values of 
 and 
. Additionally, rank(O1) = 6 for almost all values of 
 and 
.
Specifically, this generic property holds for all values of 
 and 
, except for when the 4 agents are
coplanar. Since it is always possible to renumber the agents, then if every agent is connected to the
network that has a minimum of 4 connected, non-coplanar agents, then the pair (
 ⊗ 
, 

(
)) is
generically observable. The proof for the other cases follows a similar approach, and is presented
in Appendix B.1.

Necessity: If an agent is disconnected, or is part of a network that has less than 4 agents,
then there are not enough pieces of information to estimate the target’s position centrally, let alone
(cid:4)
distributively, and the error dynamics in (4.25) – and therefore (4.31) – cannot be stabilized.

For a time-varying, undirected communication graph, Theorem 4.2.3 can be extended to offer

a scalable approach that is somewhat robust to communication link dropout.

Corollary 4.2.4 For a time-varying undirected network, if every agent remains part of a network
that has a minimum of 4 connected, non-coplanar agents then the networked system under the
proposed distributed estimation scheme is observable.

Proof: As the network connectivity changes, if every agent remains part of a network that has
a minimum of 4 connected, non-coplanar agents, then it can be shown that rank(O
) = 6 for
every agent in the network. This, in turn, ensures that the networked system is observable by
(cid:4)
Theorem 4.2.3.

4.3 Target Tracking with Coordinated Robotic Network Movement

In this section we look into the second part of the problem. We begin the discussion by

investigating the agent formation needed for optimum localization of the target.

4.3.1 Optimal Formation
For a network of 
 agents, there is a total number of 
(
 − 1)/2 possible agent pairs. Let
I0 = {(
, 
)|1 ≤ 
 < 
 ≤ 
} denote the set of all agent pairs and I = {(
, 
)| 
 < 
, 

 
 > 0}, a

60

subset of I0, represent the set of agent pairs used for estimation. From the definition of 

 
, it is
clear that I depends on the communication topology among agents.

and is given by the inverse of the Fisher information matrix [62]. With 
 (cid:44) (cid:104)

The Cramer-Rao bound (CRB) is a lower bound for the covariance matrix of unbiased estimators
, the

(cid:105)


(cid:48)

. . .


(cid:48)
1




Fisher information matrix is given by

(cid:20)(cid:18) 


 
 ln 
 (
| 
)

(cid:19)(cid:18) 


 
 ln 
 (
| 
)

(cid:19)(cid:48)(cid:21)


 = 


,

(4.38)

where 
 (
| 
) is the probability density function (PDF) of 
 given 
 and 
[·] denotes the expectation
on 
. Note that the measurement noise 

 is assumed to be Gaussian with zero mean. Assuming
that 

 = 
2


 
|N
|, one arrives at a 3 × 3 Fisher information matrix by Chan and Ho4 [63]

where


 =



(cid:48),

1

2


(cid:104)

(cid:105)

(
, 
)∈I ,



 



 =
. . .


 
 = 

 − 
 
 ,

 − 


(cid:107) 
 − 

(cid:107) .



 =

(4.39)

(4.40)

(4.41)

(4.42)

Clearly, 

 is a unit-length vector pointing from agent 
 to the target, and the matrix 
 depends on
the target position, agents’ positions, and the set I of agent pairs used for localization.

Since the CRB is a square matrix, we seek a formation of agents that minimizes of the trace of

the CRB,

tr(
−1) = 
2

 tr

(cid:16)[

(cid:48)]−1(cid:17)

,

(4.43)

which is a lower bound for the sum of variances of unbiased estimators for all elements of the
target’s position 
. In order to obtain the lowest possible CRB, and therefore a better performance
by the estimator, all agent pairs in the network need to be considered, and I = I0, requiring a
fully-connected network. The necessary and sufficient conditions to achieve a minimum CRB are

4Here, the propapagation speed of the transmitted signal is normalized to one.

61

presented in [62]. It is known that the optimum 2D formation is that of a uniform angular array
where all agents are equally distributed around a circle of radius 
, for some arbitrary 
 > 0,
centered around the target [62]. Similarly, in the 3D case, the optimal configurations for a complete
network are the 3D equivalent of uniform angular arrays, known as the Platonic solids.

Remark 4.3.1 The biggest drawback in achieving optimal estimation is that it requires the complete
connectivity of the corresponding agent network. This requirement can be relaxed if one is interested
in sub-optimal performance. We note that since the performance depends on the shape of the
formation, one can search the shape space of the agent formation space to arrive at topology-
specific shapes that minimize (at least locally), the associated CRB. In this work, we specify the
formation to be that of a Platonic solid, even when the network is not completely connected.

The goal of tracking the moving target by the network is to ensure adequate localization using
the distributed estimator. However, it is not necessary to continuously move the robots with the
target. In order to balance the trade-off between the cumulative distance traveled and the estimation
performance, each agent can utilize the norm of the error covariance matrix, (cid:107)

(cid:107), and only apply
the tracking control if the (cid:107)

(cid:107) > 
 for some constant 
 > 0. Note that 
 is a design parameter
a user can set depending on the specific problem.
In particular, this parameter allows the user
to mediate between two extreme cases, 1) minimizing energy and remaining stationary (
 = ∞),
and (2) maximizing estimation performance by continuously tracking the moving target (
 = 0).
While this switching control strategy is not guaranteed to drive each agent to its corresponding
desired location, it can greatly reduce the total distance traveled by all agents, as is illustrated in the
following section.

In order to guarantee that all agents move together, or keep still at the same time, we exploit
the average-based consensus scheme in (4.28) and (4.35) between TDOA measurements. Namely,
every 
 << 
 seconds, each robotic agent 
 in the network communicates and averages its estimate
and covariance matrix with its neighbors N
. This ensures that all agents agree on their covariance

62

matrices and coordinate their movements5.

4.4 Simulation Results

In this work, we use each robot’s estimate of the target’s state to arrive at the desired position of
each robot. To facilitate the simulation, we consider a simplified model of a propelled underwater
vehicle with steering control in yaw and pitch. The model used for each agent in the network is
given by:

(4.44)

(cid:104)

(cid:105)



















where 

 =
denotes the position of agent 
 in the inertial frame, 

 represents the
agent’s linear velocity along its body-fixed 
 axis, which stretches from the robot center to the
front of the robot. The angle 

 is the angle between the inertial 
 axis and agent 
’s body-fixed 

axis, while 

 is the robot’s yaw angle measured as the angle between the inertial 
 axis and the
projection of the body-fixed 
-axis onto the inertial 
 − 
 plane. The terms 

,
 and 

,
 are the
rates of change of the angles 

 and 

, while 

, 

,2, and 

,3 represent the control inputs for the
robot. Figure 4.4 illustrates the the robot position and relevant angle in the sagittal (
 − 
) plane.
The estimated position of the moving target is fed into a feedback linearization control strategy
to drive each agent to its desired position (see Appendix B.2 for details). The desired state for agent
5Exchanging information on a much quicker time scale is only needed to ensure that the agents

coordinate their movement, and is not a requirement for the distributed estimator.












(cid:164)


(cid:164)


(cid:164)


(cid:164)


(cid:164)


(cid:164)


(cid:164)

,

(cid:164)

,






=



 cos (

) cos (

)


 sin (

) cos (

)



 sin (

)







,




,




,2


,3



63

Figure 4.4: Illustration of the simplified model of a propelled underwater robot with steering
control, with a view on the sagittal plane. The robot has similar yaw control in the horizontal
plane.


 can be obtained from its estimate of target’s state:


ਭ

 =

ˆ

 + 


(cid:107) ˆ(cid:164)

(cid:107)

ਭ

 =

 = tan−1

ਭ

 = sin−1

ਭ

(cid:18) ˆ(cid:164)

(cid:18) ˆ(cid:164)



(cid:19)
(cid:19)





ˆ(cid:164)





(cid:107) ˆ(cid:164)

(cid:107)

(4.45)

(4.46)

(4.47)

(4.48)

(4.49)

where 

 is a constant vector that defines the position of agent 
 relative to the target.

64

It is clear
Figure 4.5: The fixed communication topology among a network of 8 agents.
that no agent in the network has a sufficient number of neighbors (and therefore TDOA
measurements) to estimate the process on its own.

The proposed approach was simulated for a network of 
 = 8 agents with initial positions:


1 =


2 =


3 =


4 =


5 =


6 =


7 =


8 =

1 −2 −5
5 −1 4
7 4 −2

(cid:105)(cid:48)
(cid:104)−7 3 5
(cid:105)(cid:48)
(cid:104)
(cid:105)(cid:48)
(cid:104)
(cid:105)(cid:48)
(cid:104)
(cid:104)−5 2 −2
(cid:105)(cid:48)
(cid:105)(cid:48)
(cid:104)
(cid:105)(cid:48)
(cid:104)
(cid:104)−1 2 −1
(cid:105)(cid:48)
(cid:104)

0 −6 7
7 −5 1

(cid:104)

(cid:105)(cid:48)

(cid:105)(cid:48)

as shown in Figure 4.6. The measurement noise covariance matrix for each agent was set to


 = 
|N
|. Estimators of each agent were initialized with ˆ

 =
and 

 = 100
6.
, with a process noise covariance
The initial position of the moving target was set to
matrix 
 = 0.01
3. The emitted signal from the source was assumed to have a period of 
 = 1 s,
while the average-based consensus was carried every 
 = 0.1 s. The desired formation for all agents
was set to that of a cube, i.e. a Platonic solid with 8 vertices, with a the radius of the underlying
sphere set to 10.

17 −5 −15

0 0 0 0 0 0

First we utilized the fixed communication topology shown in Fig. 4.5. Figs. 4.7a and 4.7b
show the final network configuration for the two extreme cases 
 = 0 and 
 = ∞, respectively.

65

Figure 4.6: Initial setup for simulation environment with a fixed communication topology.
The big circle represents the moving target, while the small circles (overlapping one another
at the origin) represent the initial estimates of the target’s location for each robot. The
ellipsoids represent the mobile robots, while the thin lines connecting them represent the
communication links.

(a)

(b)

Figure 4.7: Simulation results of the robots for the two extreme values of 
 under a fixed
communication topology where (a) the robots are constantly moving with target (
 = 0), and
(b) the robots are staying put (
 = ∞).

66

(a) 
 = 0

(b) 
 = 30

(c) 
 = 100

(d) 
 = ∞

Figure 4.8: Comparison of the Mean Squared Estimation Error (MSEE) of for each agent in
the network for different values of 
 under a fixed communication topology.

When 
 = 0, the network prioritizes minimizing the covariance norms over the total traveled
distance, causing the network to continuously track the moving target with its desired formation.
On the other hand, when 
 = ∞, the robots attempt to hold their positions to minimize the overall
traveled distance without any consideration to the covariance norms. Figs. 4.8 and 4.9 show the
estimation performance (estimation error and covariance in (4.31) and (4.35), respectively) for
different values of 
, while Fig. 4.10 depicts the corresponding cumulative distance traveled by all
agents in the network. From these figures, it can be seen that as 
 increases, the overall traveled
distance decreases while the covariance norms increase, causing the network to remain stationary
more often in exchange for less reliable estimation, as is also captured in Fig. 4.9.

67

Figure 4.9: Comparison of average covariance norms for different 
 values under a fixed
communication topology.

Figure 4.10: Comparison of the collective distance traveled by the entire network for different

 values under a fixed communication topology.

68

0100200300400500600Time (s)050100150200250300Figure 4.11: Initial setup for distance-based communication links.

Figure 4.12: Final setup for distance-based communication links.

69

-20-6-10-4z0-2y1002x-10-5405101520Figure 4.13: Mean Squared Estimation Error (MSEE) under a time-varying communication
graph with continuous tracking (
 = 0) .

For the case of a time-varying communication network, we implemented a distance rule where a
communication link exists between any two agents if the distance between them is strictly less than
21. To ensure that no agent has a sufficient number of measurements to estimate the process on its
own, we deformed the desired tracking formation so that the distance between some agents would be
large enough. Figs. 4.11–4.12 show the initial and final setup, respectively, as the agents constantly
track the target position (i.e., 
 = 0). Initially the communication graph is fully connected as all
agents are close enough. As the distance between the robots increases, some of the communication
links are lost. Finally, it is clear from Fig. 4.13 that successful estimation is achieved even as the
communication graph changes and some of the communication links are dropped.

4.5 Summary and Future Work

We investigated the problem of distributed localization of a moving target by a network of
agents using TDOA measurements from first observability principles. Structural observability
principles were utilized to highlight the importance of having enough measurements to accurately
localize the moving target. We showed that a decentralized approach without exchanging estimates

70

requires every agent in the network to be heavily connected to localize the target on its own. On the
other hand, under the distributed approach, we showed that when the agents exchange information
and fuse their estimates, then it is indeed possible for every agent in the network to estimate the
target’s position without needing to be heavily connected. Specifically, we showed that if every
agent is connected to a network with a minimum of 4 connected, non-coplanar agents, then the
process is rendered generically observable and each agent can accurately localize the target. This
work can be extended to time-varying communication topologies that are commonly encountered
when the communication links depend on the distances between the robots as they move. The
distributed filtering approach offers a degree of robustness to communication link dropouts, and
has been shown to perform well if the minimum connectivity condition holds. Furthermore,
we proposed a movement control strategy that aims to balance the trade-off between estimation
performance and total distance traveled by the network. A single parameter, 
, is specified to tune
the network performance between the two extreme cases of continuously tracking the target (best
estimation performance and high energy consumption), or remaining stationary (maximum energy
conservation at the expense of accurate localization). Through simulation, we notice a graceful
degradation in estimation performance for larger values of 
 in order to reduce energy expenditure.
This work can be expanded in several ways, including the analysis of directed communication
graphs. Additionally, different distributed control strategies can be introduced for formation control
and tracking of the target with obstacle and collision avoidance. Other interesting problems include
online-tuning of the parameter 
 to change the network objective and react to changes in the
environment (e.g., to reduce energy expenditure further when the robots are low on battery).
Future work should ultimately seek experimental validation of the proposed approach using mobile
acoustic receivers using Grace 2, similar to the setup presented in Chapter 3. Finally, future work
can consider stochastic models for the moving target, such as the correlated random walk model
[67], along with different filtering methodologies, such as the particle filtering [68].

71

CHAPTER 5

NECESSARY CONDITIONS FOR THE CONVERGENCE OF THE DISTRIBUTED

KALMAN FILTER AND ITS COUPLED RICCATI EQUATIONS

The problem of distributed localization of a moving target in Chapter 4 motivated the study of a more
general application of sensor networks: the distributed estimation problem. In many applications
involving large-scale complex systems, the state of the system is monitored by a group of sensors
spatially distributed over large networks where the communication between sensors is limited. To
model such a scenario, consider the discrete-time linear time-invariant (LTI) dynamical system


(
 + 1) = 

(
) + 
(
)

(5.1)

where 
 is the discrete-time index, 
 ∈ B(R
) 1 is the system matrix, 
(
) ∈ R
 is the state vector,
and 
(
) ∈ R
 is a driving noise assumed to be Gaussian with zero mean and positive definite
covariance 
. The state of the system is monitored by a network of 
 agents indexed by 
, each of
which is equipped with a sensor that receives a partial measurement of the state that is modeled by



(
) = 


(
) + 

(
),


 = 1, . . . , 
,

(5.2)

where 

(
) ∈ R

 is the measurement made by sensor 
 with 

 ∈ B(R
, R

), and 

 ∈ R

 is the
measurement noise assumed to be Gaussian with zero mean and positive definite covariance 

.
An important problem in such networks is to develop distributed algorithms for state estimation
of the process in (5.1), where the goal is for each agent to estimate the entire system state using
its respective local measurements and the information obtained from neighbors. While this model
does not directly capture the time-varying nature of the distributed localization problem, it allows
us to consider a wider range of system dynamics (stable, unstable, and marginally stable systems),
1For Banach spaces X and Y, we set B(X, Y) as the Banach spaces of all bounded linear
operators of X into Y, and use B(X) (cid:44) B(X, X). We denote by R
 the 
-dimensional real
Euclidean spaces and B(R
, R
) the normed bounded linear space of 
 × 
 real matrices, with the
uniform induced-norm represented by (cid:107).(cid:107), and use B(R
) (cid:44) B(R
, R
).

72

as well as a more general view into the relationship between system dynamics, agent measurement
models, and network requirements.

This distributed filtering problem has received significant attention over the past years, with
early studies considering two-time-scale algorithms with multiple data fusion iterations between
two consecutive time steps of the plant dynamics [24, 25]. Bandwidth and energy limitations
motivated the study of single-time-scale algorithms where the data fusion occurs once per time
step of plant dynamics. In [26, 27], the authors used state augmentation to cast the distributed
estimation problem as a problem for designing a decentralized stabilizing controller for an LTI
plant and provided conditions for successful estimation under their proposed approaches. In [28],
the authors alleviated the need for state augmentation and proposed a distributed filter where each
sensor performs consensus only on the portion of the state it cannot estimate on its own. In the
most general case, this would require the partitioning of the system dynamics into observable and
unobservable components and enforcing consensus on the unobservable modes at each sensor site.
In [29], the authors introduced the idea of consensus-based distributed linear filters (CBDLF), where
each sensor updates its estimate in two steps: a local update step using its own observations followed
by a consensus step with its neighbors. The authors went on to provide sub-optimal filtering gains
to minimize an upper bound on a quadratic filtering cost, and provided some sufficient conditions
for the convergence of their scheme in terms of the feasibility of a set of linear matrix inequalities
(LMIs).

The gains obtained in [29] are similar, in some sense, to the ones obtained from applying a
centralized Kalman filter, and the convergence results are tied to the convergence properties of a
set of coupled Riccati equations. However, the work in [29] does not, in general, shed light on
the network conditions required to satisfy such LMIs. In this work, we utilize some of the results
developed in the Markovian jump linear systems (MJLS) literature to connect the convergence
properties of these coupled Riccati equations to certain conditions on the network topology and
consensus weights.

73






=1

5.1 Problem Setup

For ease of exposition, in this section we consider a CBDLF where consensus is performed
before local filtering, and discuss in Appendix C.3 how these results hold when the order is reversed
to consider the CBDLF in [29].

Here, in the first step, each agent produces an intermediate estimate of the state by performing
a convex combination of its own estimate, ˆ

(
), and the state estimates by all other agents within
its communication range, i.e.,



(
) =



 
 ˆ
 
(
),


 = 1, . . . , 
,

where 

 
 are nonnegative scalars summing up to one (




=1 

 
 = 1) with 

 
 > 0 if 
 = 
 or
if agent 
 can obtain information from agent 
; otherwise, 

 
 = 0. After all agents complete the
consensus step, each agent then updates its estimate by performing a local filtering step based on
the intermediate estimate 
(
), i.e.,

(5.3)

(cid:18)

(cid:19)



(
) − 



(
)

.

(5.4)

ˆ

(
 + 1) = 


(
) + 



where 

 is the filter gain.

Let 

(
) = 
(
) − ˆ

(
) denote the estimation error for agent 
 at time 
. Combining (5.3) and

(5.4), we have2



(
 + 1) = ( 
 − 



)



 
 
 
(
) + 
(
) − 



(
).

(5.5)






=1










=0


=1

74

In Appendix C.2, we show that it is possible to derive filtering gains that minimize an upper

bound of the following finite horizon quadratic filtering cost function


(
) =


{(cid:107)

(
)(cid:107)2},

(5.6)

with 
{.} denoting the expected value. These sub-optimal filtering gains can be computed in two
steps under the same communication topology and consensus weights used in (5.3)–(5.4). The
2We assume that the initial state 
(0), and the noises 

(
) and 
(
) are independent for all


 ≥ 0.

filtering gains require each agent to store a matrix 

(
) that represents, as we discuss in Section 5.2,
an upper bound of the error covariance for agent 
.

During the consensus step, the agents compute an intermediate covariance by performing a

convex combination of their own 

(
) and that of their neighbors, i.e.,










=1

75


(
) =



 
 
 
(
).

(5.7)

Then, during the filtering step, each agent computes the local filtering gain to be used in (5.4)


=1

using the intermediate covariance



(
) = 

(
)
(cid:48)


 (

 + 


(
)
(cid:48)


)−1,

where (cid:48) represents the matrix transpose, and updates its covariance upper bound according to



(
 + 1) =

(
) 
(cid:48) + 
 − 

(
)
(cid:48)

×

 + 

)−1


(
) 
(cid:48).

(


(
)
(cid:48)

(5.8)

(5.9)

Substituting 
(
) from (5.7) into (5.9) yields a set of coupled Riccati equations. The CBDLF in
(5.3)–(5.4) and (5.7)–(5.9) is referred to as the distributed Kalman filter – due to its similarity to the
centralized Kalman filter – and provides an attractive solution to the distributed filtering problem.
However, the numerical tools used to analyze the convergence of this filter and its associated
coupled Riccati equations do not offer much insight into the required network connectivity and
consensus weights.

5.2 Necessary Conditions

For a centralized estimation problem, the detectability of the pair ( 
, 
) plays an important role
in the convergence of the Riccati equation that governs the centralized Kalman filter. Therefore, it
is natural for us to seek a similar notion for the distributed problem. In what follows, we consider
the noise-free dynamics of the estimation errors in (5.5), namely



(
 + 1) = ( 
 − 



)



 
 
 
(
).

(5.10)

Denoting the network-wide estimation error by 
(
) = [
1(
)(cid:48) · · · 

(
)(cid:48)](cid:48), one can show that

the noise-free dynamics for this network-wide error are given by


(
 + 1) = B
(
),

(cid:16)

R

(cid:17)

(5.11)

where B (cid:44) diag[ 
 − 



](
 ⊗ 

) ∈ B
3, with ⊗ denoting the Kronecker product and

 ∈ B(R
) is the consensus weights matrix whose (
, 
) entry is 

 
. Clearly, the system in (5.11)
is asymptotically stable if 

(B) < 1, where 

(.) denotes the spectral radius of a linear operator.

Definition 5.2.1 (Detectability) Let 
 = {
1, . . . , 

} be the set of measurement matrices of all
sensors. For the system in (5.1)–(5.2) and the consensus weights 
, we say ( 
, 
, 
) is detectable
if there exists a set of gain matrices 

, 
 = 1, ..., 
, such that 

(B) < 1.

As we discuss later in this section, detectability in the sense of Definition 5.2.1 is not sufficient
to ensure the convergence of the coupled Riccati equations in (5.7) and (5.9), even though it would
ensure exponential convergence of 
(
) to zero in (5.11).

To facilitate the analysis and discussion of these coupled Riccati equations, we make use of
the following notation. Let H
,
 denote the linear space made up of all 
-sequences of matrices

 = (
1, . . . , 

) with 

 ∈ B(R
, R
). For 
 ∈ H
,
 we write 
(cid:48) = (
(cid:48)

) ∈ H
,
 and
say that 
 ∈ H
 (cid:44) H
,
 is symmetric if 
 = 
(cid:48). For a symmetric 
 ∈ H
, we use the notation

 ∈ H
0 (respectively, 
 ∈ H
+) when all of the matrices 

 are positive semidefinite (respectively,
positive definite)4. For any 
 ∈ H
, define the operators E(
) = (E1(
), . . . , E
(
)) ∈ B(H
),
and L(
) = (L1(
), . . . , L
(
)) ∈ B(H
), where

1, . . . , 
(cid:48)

E
(
) (cid:44)



 


 ∈ B(R
),

(5.12)






=1

L
(
) (cid:44) Ɖ
E
(
)Ɖ(cid:48)


 ∈ B(R
),

(5.13)
3For a set of 
 matrices, 

 ∈ B(R
), we denote by diag[

] the 

 × 

 block-diagonal
4For 
, 
 ∈ H
, we write 
 ≥ 
 if 
 − 
 = (
1 − 
1, . . . , 

 − 

) ∈ H
0, and that 
 > 
 if

matrix with 

 in the diagonal.

 − 
 ∈ H
+.

76

for some Ɖ
 ∈ B(R
), for 
 = 1, ..., 
. It can be verified that the spaces H
 and R

2 are uniformly
homeomorphic, and that any operator Z in B(H
) can be represented in B(R

2) [30]. The
following result is important in the analysis of the coupled Riccati equations.
Lemma 5.2.2 (Proof in [30]) Consider the operator L(.) defined above. We have that



(L) = 

(A),

(cid:16)

R

2(cid:17)

where A = diag[Ɖ
 ⊗ Ɖ
](
 ⊗ 


 ∈ H
+ such that 
 − L(
) > 0.


2) ∈ B

. Furthermore, 

(A) < 1 iff there exists some

With the above notation, we present the following lemma regarding the upper bounds for the

error covariances along with a stronger definition for detectability.

Lemma 5.2.3 Define '(
) ∈ H
0, where '
(
) denotes the noise-free covariance of the estimation
error in (5.10) for agent 
 at time 
. Let 
(
) ∈ H
0 be defined recursively as


(
 + 1) = L(
(
)),


(0) = '(0),

(5.14)

with Ɖ
 = ( 
 − 



). Then,

'(
) ≤ 
(
),

∀
 ≥ 0.

Furthermore, define (
) ∈ H
+ and 
(
) ∈ H
+, where 
(
) denotes the covariance of agent 
’s
estimation error with noise in (5.5), and 

(
) is defined recursively as



(
 + 1) = L
(
(
)) + 
 + 

 

 
(cid:48)


, 
(0) = (0),

(5.15)

with Ɖ
 = ( 
 − 



). Then,

(
) ≤ 
(
),

∀
 ≥ 0.

Proof: See Appendix C.1.

(cid:4)

Definition 5.2.4 (S-detectability) For the system in (5.1)–(5.2) and the consensus weights 
, we
say ( 
, 
, 
) is square detectable (S-detectable) if there exists a set of gain matrices 

, 
 = 1, ..., 
,
such that 

(L) < 1 with Ɖ
 = 
 − 



.

77

Note that S-detectability implies that there exists a set of gain matrices such that upper bounds
for the noise-free covariances in (5.14) are asymptotically stable. Using Lemma 5.2.3, it is easy to
check that if ( 
, 
, 
) is S-detectable, then it is detectable in the sense of Definition 5.2.1, since













(cid:107)
(
)(cid:107)2 =

(cid:107)

(
)(cid:107)2 =

tr ('
(
)) ≤

tr(

(
)),


=1


=1


=1

and thus, if 

(L) < 1 then 

(B) < 1. However, detectability in the sense of Definition 5.2.1
does not imply S-detectability. Indeed, consider the case with 
 = 1, 
 = 2, 
 = 
1 = 
2 = 1,

1 = 0.3, and 
2 = −0.25, then 

(B) = 0.975 while 

(L) = 1.026.

This notion of S-detectability is similar in some way to the notion of mean square detectability
in MJLS. This allows us to leverage some of the results there in analyzing the conditions on
the consensus weights and network connectivity needed for the stability of the coupled Riccati
equations. Specifically, it has been shown in [30] that these coupled Riccati equations converge
to a stabilizing solution5 only if there exists a set of gains 

 such that 

(L) < 1 with Ɖ
 =

− 



. That is, the coupled Riccati equations converge to a stabilizing solution only if their noise-
free counterparts are stable. This is precisely the definition of S-detectability in this distributed
estimation context.

The remainder of this section is aimed at providing the necessary conditions for S-detectability
of ( 
, 
, 
) in terms of network topology and communication weights.. Before presenting the
main result, however, we review some of the relevant results from MJLS. Specifically, we make use
of the notion of weak detectability (W-detectability), which has been shown to play an important
role in the LQR control problem of MJLS [64, 65].

For some 
 ∈ H
0, consider the system


(
 + 1) = L(
(
)),

5
 ∈ H
+ is called a stabilizing solution if 

 = 
E
(
) 
(cid:48) + 
 − 
E
(
)
(cid:48)

(5.16)

 (

E
(
)
(cid:48)

 +


)−1

E
(
) 
(cid:48) for all 
 = 1, . . . , 
, and the gains 

 defined in (5.8) with 
 = E
(
) stabilize
(5.14).

78

with Ɖ
 = 

, and introduce the functional


(
) =







(

(0)(cid:48)

(
))

where the matrices 
(
) ∈ H
0 are defined recursively by

E
(
(
)) 

,



(
 + 1) = 
(cid:48)


 

 + 
(cid:48)


=1

(5.17)

(5.18)



(0) = 0.

Definition 5.2.5 (W-detectability) Consider the system (5.16). We say that ( 
, 
, 
) is W-
detectable if there exist integers 

, 
 
 ≥ 0 and scalars 0 ≤ 
 < 1, 
 > 0 such that 
(

) ≥


 (cid:107) 
(0)(cid:107) whenever (cid:107) 
(
 
)(cid:107) ≥ 
 (cid:107) 
(0)(cid:107), where the norm is defined as (cid:107) 
(cid:107)2 =

 

(
(cid:48)


 

).

Associated with the matrices 
(
) in (5.18), are the set of observability matrices O =

{O1, · · · , O
}, where

(cid:104)O
(0)
for each 
 ∈ {1, . . . , 
}, where 

(
) =

−1

O
 =

· · · O
(
2
 − 1)(cid:105)(cid:48)

(5.19)

=0 O
(
), and the matrices O(
) are defined recursively

as

O
(
 + 1) = 
(cid:48)


E
(O(
)) 

,

O
(0) = 
(cid:48)


 

.

(5.20)

With the above definition of W-detectability, the following results (Lemmas 5.2.6 and 5.2.7)

are adapted from [64] and [65].
Lemma 5.2.6 If ( 
, 
, 
) is S-detectable, then ( 
, 
, 
) is W-detectable.

Lemma 5.2.7 ( 
, 
, 
) is W-detectable iff for some 
(0) ∈ 



(O
), for any 
 ∈ {1, . . . , 
}, we
have that lim
→∞ (cid:107)
(
)(cid:107)2 = 0.

In our distributed estimation problem, we have 

 = 
 for all 
, and the condition in Lemma 5.2.7
is equivalent to the detectability of the pairs ( 
, O
) for all 
, allowing us to employ these results in
deriving necessary conditions on the network for the distributed estimation problem.

For a weight matrix 
, define recursively,

(
−1)


(
)

 
 =










=1


(0)

 
 = 

 
 ,

(5.21)



 
 ,

79

where 

 
 is the Kronecker delta, and recall the definition of a source component [28] in the network.
Definition 5.2.8 (Source component) A set of agents S = {
1, . . . , 

}, S ⊆ {1, . . . , 
}, is a
source component of the network if these exists 
 > 0, such that 
(
)

 
 > 0 for any 
, 
 ∈ S, and for
any 
 ∈ S and 
 ࢡ S, 
(
)
Definition 5.2.9 (Detectable source component) A source component S = {
1, . . . , 

} is de-
tectable if the pair ( 
, 
S) is detectable, where 
S =


 
 = 0 for all 
 > 0.

(cid:105)(cid:48)

(cid:104)

.


(cid:48)

1

. . . 
(cid:48)




We can present our main result.

Theorem 5.2.10 ( 
, 
, 
) is S-detectable only if:

1. Every source component in the network is detectable.

 ))2 for each 
 = 1, . . . , 
, where 


2. 


 < (1/

( 




 is the unobservable partition of 
 using



.

Proof: First, we show that the first statement holds. To that end, if ( 
, 
, 
) is S-detectable, then
it is necessarily W-detectable, and from Lemma 5.2.7 we have that ( 
, O
) is detectable for each 
.
It is easy to check that O
(
) in (5.20) can be written as

(
)

 
 
(cid:48)

(cid:48)

O
(
) =






 
 
 

 .


=1

Let S be a source component, and denote by OS the observability matrix corresponding to the
pair ( 
, C
). It is known that each agent 
 ∈ {1, . . . , 
} is either part of a source component, or
there exists a directed path to 
 from some agent 
 in a source component.

On the one hand, for some agent 
 ∈ S, it follows from Definition 5.2.8 that there exists an 
 > 0

 
 > 0 for every 
 ∈ S. It can then be checked that for any 
 ∈ S, 



(O
) = 



(OS),

such that 
(
)
and S is a detectable source component since for any 
(0) ∈ 



(OS), lim
→∞ (cid:107)
(
)(cid:107)2 = 0.

On the other hand, if 
 ࢡ S, then there is no directed path from 
 to 
 ∈ S, and 



(O
) ⊆




(OS). However, since ( 
, O
) and ( 
, O 
) are detectable, with 



(O 
) = 



(OS), then S
is a detectable source component.

80

Now we show that the second statement holds. Indeed, if ( 
, 
, 
) is S-detectable, then there
exist gain matrices 

, 
 = 1, . . . , 
, such that 

(L) < 1. From Lemma 5.2.2, this implies that

 − L(
) > 0 for some positive definite matrices 
 ∈ H
+. Following the same approach as in
[66], we have that



−( 
 − 



)(cid:48) 





 
 
 
( 
 − 



) > 0,

⇒ 

−(√


=1




)( 
 − 



)(cid:48)

(√




)( 
 − 



) > 0,

for 
 = 1, . . . , 
, and therefore (√


 
, 

) is detectable. We can then find a similarity transforma-
tion to transform the pair (√


 
, 

) into
¯

 = 

(cid:0)√






√




 
(cid:1) 
(cid:48)
(cid:104)








 =

(cid:105)

0

,

¯

 = 


(cid:48)

 =





0






 





 ,

represents the unobservable modes. Clearly, if (√


 
, 

) is detectable then

√


 



 must be stable, and 


 < (1/

( 






where
√


 




 ))2.

(cid:4)

These necessary conditions allow us to consider the weakest communication topologies that
will ensure S-detectability, as well as the maximum values for the self-weights used in consensus.

5.3 Numerical Examples

5.3.1 Example 1

We consider an unstable system with 
 = 2, monitored by 
 = 3 sensors, where

(cid:105)
(cid:105)

(cid:104)
(cid:104)

81

2 0

0 2

 ,


 =


1 =


2 =

1 0

0 1

,

,


3 =

1 0

0 1

 .

Clearly, we have that ( 
, 
1) and ( 
, 
2) are not detectable, implying that agents 1 and 2 cannot
estimate the process on their own. On the other hand, it is easy to check that the pairs


1


3



(cid:170)(cid:174)(cid:174)(cid:172) ,(cid:169)(cid:173)(cid:173)(cid:171)
,


2


3



(cid:170)(cid:174)(cid:174)(cid:172)

( 
, 
3) ,(cid:169)(cid:173)(cid:173)(cid:171)
,


1


2



(cid:170)(cid:174)(cid:174)(cid:172) ,(cid:169)(cid:173)(cid:173)(cid:171)
,

are all detectable. Therefore, to satisfy the condition that ( 
, O1) is detectable, the minimal
connectivity requirement we need is 
12 > 0 or 
13 > 0. Similarly for ( 
, O2) to be detectable,
we need 
21 > 0 or 
23 > 0. Finally, ( 
, O3) is detectable without needing a path from any
other agent. Figure 5.1 represents the communication networks with minimum links that satisfy
these conditions. It is easy to check in Figure 5.1 that every source component in those graphs is
detectable, and adding more communication links between the nodes will not affect the detectability
of ( 
, O
).

Note that the spectral radius of the unobservable partitions for agents 1 and 2 is 

( 


1) =


( 


2) = 2, and therefore we require 
11 < 0.25 and 
22 < 0.25. To show the importance of the
values of self weights 


, we simulate the response of a distributed Kalman filter using (5.7)–(5.9)
with the filtering gains shown in (5.8) for the case when agent 3 is disconnected from agents 1 and
2. Figure 5.2 shows the norms of the covariance matrices under different values for 
11 and 
22,

Figure 5.1: The minimal communication links required to satisfy the detectability require-
ment of all (
,O
) pairs for Example 1.

82

Figure 5.2: Comparison of the norms of the covariance matrices under the consensus-then-
filter distributed Kalman filter for Example 1. 

 represents the network that satisfies
(1/

(



))2.




where we used the following weights





 =

0.24 0.76 0
0.76 0.24 0
1
0

0



0.25 0.75 0
0.75 0.25 0
1
0

0

 ,



 =



 =

 ,




0.26 0.74 0
0.74 0.26 0
1
0

0

for the case that satisfies the condition 
11, 
22 < 0.25, and

(cid:104)−15 15

(cid:105)(cid:48)

,

(cid:104)

(cid:105)(cid:48)

for the cases that violate this condition, and the simulations were initialized with 
(0) =
ˆ

(0) =

, 

(0) = 10
, 
 = 0.1
, and 

 = 0.1
.

0 0

Figure 5.2 shows that 
3(
) converges very quickly to its steady state value without agent 3
needing additional information from its neighbors, since it can estimate the process on its own. An
interesting behavior is observed for the coupled dynamics for 
1(
) and 
2(
) as 


 approaches
(1/

( 



 ))2. We see that covariance matrices for agents 1 and 2 converge to steady values when

83




 < 0.25 and grow unbounded for 


 > 0.25, highlighting the importance of the consensus
weights when performing distributed estimation.

5.3.2 Example 2

We consider an unstable system with 
 = 3, monitored by 
 = 3 sensors, where




 =

0.8147 0.8147 0.2785
0.9058 0.9058 0.5469
0.9575

0

0

(cid:104)
(cid:104)
(cid:104)


1 =


2 =


3 =

1 0 0

0 1 0

0 0 1

(cid:105)
(cid:105)
(cid:105)

,

,

.

 ,

It can be checked that ( 
, 
1) and ( 
, 
2) are observable, implying that agents 1 and 2 can
estimate the process on their own. On the other hand, ( 
, 
3) is not detectable, and agent 3
requires information from agents 1 or 2 to successfully estimate the process. Therefore, to satisfy
the condition that ( 
, O3) is detectable, the minimal connectivity requirement we need is 
31 > 0
or 
32 > 0.

Note that the spectral radius of the unobservable partitions for agent 3 is 

( 



3) = 1.7205, and
therefore we require 
33 < 0.3378. We simulate the response of a distributed Kalman filter using
(5.7)–(5.9) with the filtering gains shown in (5.8) for the case when agent 3 is obtains information
from both agents 1 and 2. Figure 5.3 shows the norms of the covariance matrices under different
values for 
33 where we used the following weights

for the case that satisfies the condition 
33 < 0.3378, and






 =

0
1

1
0
0
0
0.34 0.34 0.32



 =

0
1

1
0
0
0
0.33 0.33 0.34

84

 ,
 ,

Figure 5.3: Comparison of the norms of the covariance matrices under the consensus-then-
filter distributed Kalman filter for Example 2. 

 represents the network that satisfies
(1/

(



))2.




(cid:104)−2.2691 −2.0143 1.6173

(cid:105)(cid:48)

(cid:104)

(cid:105)(cid:48)

for the case which violates that condition, and the simulations were initialized with 
(0) =

, ˆ

(0) =

0 0 0

, 

(0) = 100
, 
 = 
, and 

 = 
.

Figure 5.3 shows that 
1(
) and 
2(
) converge to their respective steady state values without
agent needing additional information from its neighbors, since they can estimate the process on their
own. An interesting behavior is observed for the coupled dynamics for 
3(
) as 
33 approaches
(1/

( 



3))2, further highlighting the importance of the consensus weights.

5.4 Summary and Future Work

We consider the problem of distributed estimation of an LTI system by a network of sensors using
consensus-based distributed linear filters, where the agents update their estimates by performing
consensus followed by local filtering. We present a distributed version of the centralized Kalman
filter, in which the filtering gains minimize an upper bound of a quadratic filtering cost. These
filtering gains are computed using a set of coupled Riccati equations that, in turn, can be updated
in a distributed manner under the same network. We then provide the notion of S-detectability that
is necessary for the convergence of these coupled Riccati equations, and build on that to provide
necessary conditions in terms of the minimum network connectivity and a limit on the self-weights

85

used during the consensus step. Future work will first focus on the other half of this problem, in
which we consider the sufficient conditions for S-detectability and their implication on the network
in order to ensure successful estimation and asymptotic convergence of the distributed Kalman
filters. Future work will also seek to extend these results to distributed estimation of time-varying
systems, and investigate the conditions for convergence of the coupled Riccati equations with
time-varying matrices.

86

CHAPTER 6

SUMMARY AND FUTURE WORK

6.1 Summary

In this dissertation, we discussed the role gliding robotic fish play in environmental moni-
toring, and focus on their application to acoustic telemetry of fish movement. We presented the
improvements of mechanical and electrical designs of the gliding robotic fish to enhance their
serviceability, operational time, and computational power.. These improvements were facilitated
by the introduction of a water-tight interface that allowed for opening and closing the robot for
regular maintenance and inspection. The new serviceable nature of these robots allowed us to
rapidly test new electrical systems. The final design of the robot featured a Raspberry Pi computer,
which operates on a Linux-like environment that facilitated the development of high-level tasks,
such as navigation and data storage.

The new design allowed for a larger sensor payload to be integrated into the robot, as well as for
carrying an acoustic receiver for detecting acoustic signals. Gliding robotic fish prototypes were
tested in Higgins Lake to investigate the applicability of these robots to tracking tagged animals in
fresh and shallow water. The field tests were also helpful in evaluating gliding- and swimming-
based navigation strategies that drive the robot to a specified GPS location. The lake experiments
showed that the robot can indeed carry an acoustic receiver and detect acoustic signals reliably at
a distance of approximately 300 m. Furthermore, these tests shed the light onto the importance of
aligning the acoustic receiver towards the tag to maximize detection, as the detection performance
of the mobile receiver degraded when the receiver was facing away from the target. The depth and
pitch of the robot also affected the detection performance of the mobile receiver. However, the
effect of these variables was less prominent than facing the receiver away from the transmitter.

Motivated by the problem of tracking fish movement, we then investigated the problem of
localization and tracking of a moving target by a network of robots. A distributed EKF was

87

presented to estimate the target’s position based on the TDOA measurements model, without the
need for centralized processing of these measurements. Under the proposed scheme, we showed
that any given agent in the network can accurately estimate the target’s position when it is part
of a connected network with 4 non-coplanar agents. We then presented a movement strategy for
the network to actively track the target, ensuring adequate localization of the target as it moves in
space. A tuning parameter was introduced to achieve a balance between estimation performance
and energy consumption.

The distributed localization problem motivated the study of the more general distributed esti-
mation problem, in which a network of sensors are used to monitor and estimate the state of an LTI
system. Under this general setup, it is assumed that interactions between agents are dictated by a
directed and weighted communication graph. We then presented a suboptimal distributed filtering
strategy in which each agent updates its estimates in two steps: a consensus step followed by a
filtering step. We showed that the suboptimal filtering gains are related to the convergence of a set
of coupled Riccati equations. We then went on to discuss the necessary conditions in terms of graph
topology and consensus weights needed for the convergence of these coupled Riccati equations.

6.2 Future Work

The work presented here can be further improved in future studies. The main drawback that the
current robot design faces involves the time and resources needed for fabrication and manufacturing.
Future designs will focus on reducing the cost and time needed for fabrication. As for the electrical
side, future designs will incorporate underwater acoustic communication, satellite communication,
and solar charging for these robots. Furthermore, additional work will be done to fully characterize
the operational specifications of these robots, such as their capability of overcoming currents,
battery life, and maximum depth rating.

Additional field trials will also be carried out to further characterize the performance of gliding
In particular, future work will focus on conducting
robotic fish as a mobile receiver platform.
more replicate runs over a range of environmental variables (e.g., wind, noise, stratification), and

88

examine different methods to mitigate shielding the acoustic signals from the receivers. As an
example, future work can compare the detection performance of vertically mounted receivers, as
well as investigating navigation controllers that take into account the relative positions between
the tag and receiver. Ultimately, future work will seek experimental validation of TDOA-based
localization and tracking of an acoustic tag using a network of gliding robotic fish operating as
mobile receivers. Future work can also consider stochastic models for the moving target, such as
the correlated random walk model [67], along with different filtering methodologies, such as the
particle filtering [68].

Finally, future work will focus on the other half of the distributed estimation problem. Specifi-
cally, sufficient conditions for the convergence of the coupled Riccati equations in terms of network
topology and consensus weights will be investigated. It will also be of interest to extend the results
obtained here to time-varying systems, and consider the necessary and sufficient conditions for the
convergence of the coupled Riccati equations with time-varying matrices.

89

APPENDICES

90

APPENDIX A

STOCHASTIC STABILITY FOR DISTRIBUTED TDOA-BASED LOCALIZATION OF A

MOVING TARGET

Analyzing the stability of the extended Kalman filter has been investigated in [56] and [69]. A similar
framework to the one presented in [56] can be utilized for analyzing the stochastic stability of the
EKF under the weighted averaging scheme in (4.26)–(4.28), and (4.32)–(4.35) when implemented
by each agent in the network.

Throughout this appendix, (cid:107) · (cid:107) denotes the Euclidian norm of real vectors or the spectral
norm of real matrices, 
{
} is the expectation value of 
, and 
{
|
}, the expectation value of 

conditioned on 
. Moreover, R
 denotes the real 
–dimensional vector space, 

 denotes the 
 × 

identity matrix, and 
 ⊗ 
 denotes the Kronecker product of matrices 
 and 
.


 = ˆ

(
), ˜



Let 
 denote the dimension of in which the target is moving in space (
 = 2 for 2D, and

 = 3 for 3D), and let 
 = 2
 denote the dimension of the target’s state. For ease of writing, let


 = 
(
), and similarly let ˆ



 = 

(
), Furthermore, let

 denote agent 
’s measurement at time instant 
 (rather than 

(
) in (4.2), for consistency with



discrete-time literature), and for notational consistency, let Ò
(·) denote the measurement model
Ò
(·) in (4.3). Finally, let 



 denote the measurement noise for agent 
 with covariance matrix 

.
The one step formulation of the each agent’s estimation in (4.29) can be written as


 = 

(
), and 




 = ˜

(
), 



(cid:104)






=1

(cid:16)


)(cid:17)(cid:105)

ˆ



+1 =



 



 ˆ
 



 + 

 






 − Ò 
( ˆ
 


 


(A.1)

Similarly, a one step formulation of the 
–th agent’s error covariance matrix in (4.33)–(4.35) can
be derived by substituting (4.35) and (4.34) into (4.33), and denoting 

(
|
 − 1) by 










+1 =



 



(

 − 
 






)
 


 



(

 − 
 






 




 





 
 
 







(cid:104)






=1

For notation consistency, we redefine 




 as





 = 




 







 


(cid:20)




 + 

(cid:17)−1


=1


)
 

(cid:105) +


(cid:16)





 




 





91



(cid:21)

+ 



 (A.2)

(A.3)

In the averaged EKF formulation, we note the use of the linearized measurement function, 



.

It is possible to expand the measurement function Ò


Ò
(

) − Ò
( ˆ




) = 




)

) + 

(

 , ˆ




 via

(

 − ˆ



The 
-th agent’s estimation error is defined as


 = 

 − ˆ


˜






which can be obtained by subtracting (A.1) from (4.5), yielding

(cid:104)






=1



 



(
 − 
 






) ˜
 


 



 + 
 



 + 
 





(cid:105)

˜



+1 =

where


 

(

 , ˆ



 = −




)




 = 


 − 







 





 + 
 


.
A.1 Error Bounds for the Averaged EKF

which follows the form in (4.30), with 
 
(
) = 
 


(A.4)

(A.5)

(A.6)

(A.7)

(A.8)

The basis of the convergence analysis is the stochastic stability lemma [49, 50], which is given

as follows:
Lemma A.1.1 If there exist real numbers ¯
, 
, 
 > 0 and 0 < 
 ≤ 1, and there is a stochastic
process 

(

) with the following properties:


(cid:107)

(cid:107)2 ≤ 

(

) ≤ ¯
(cid:107)

(cid:107)2


{

+1(

+1)|

} − 

(

) ≤ 
 − 


(

)

(A.9)

(A.10)

then the random process 

 is exponentially bounded in mean square with probability one, as in


{(cid:107)

(cid:107)2} ≤ ¯






{(cid:107)
0(cid:107)2} (1 − 
) 
 + 





(1 − 
)


(A.11)

for every 
 ≥ 0. Moreover, the stochastic process is bounded with probability one.


−1



=1

92

(cid:104)

(cid:104)

(cid:105)


(cid:105)

Proof: The proof of this lemma is provided in [50] and [56].

(cid:4)

Let ˜

 =

˜
1





. . .

˜







denote the network-wide estimation error of all agents in the

network. Using (A.6), we can obtain the dynamics for the network-wide estimation error

˜

 = (
 ⊗ 
)(
2

 − 

 

) ˜

 + (
 ⊗ 
2
) 

 + (
 ⊗ 
2
)



(A.12)

(cid:104)

(cid:105)


1



. . . 
 




and 

 = blkdiag

are block diagonal matrices
Here, 

 = blkdiag
for the filtering gain and linearized measurement matrices, respectively, while 
 is a stochastic ma-
trix representing the averaging weights, [
]
 
 = 

 
. The terms 

 and 

 are the concatenations
of the terms 




, respectively. From (A.7) and (A.8), we can write

. . . 
 





 and 




1





 = −(

 ⊗ 
)

 
(

 , ˆ

)


 = (

 ⊗ 
)

 − (

 ⊗ 
)

 



(A.13)

(A.14)

where

and



 = 1
 ⊗ 

 ,


 = 1
 ⊗ 

 ,

(cid:104)



 =


1




. . . 






(cid:105)


,

(cid:104)

(cid:20)






=1


 )
(cid:105)



(cid:21)

.

(cid:104)


1



. . . 





(cid:105)

. Using

(A.15)


(

 , ˆ

) =


1(

 , ˆ
1

)


. . . 

(

 , ˆ



Similarly, we define the network-wide error covariance, 

 = blkdiag

the matrix inversion lemma, it can be noted that (A.2) is equivalent to





+1 =

we get



 




 






 − 

 






 






 





+ 






+1 = (√


 ⊗ 
)

(√


 ⊗ 
)
 − (√


 ⊗ 
)

 




 (√

 




 ⊗ 
)
 + (√




√




 ⊗ 


)

(A.16)

93

Furthermore, we restrict 

 to have a block diagonal structure in order to capture the distributed
nature of this problem. The network-wide gain matrix 

 can then be expressed as

(cid:104)

where the matrix 
 = blkdiag
matrix.



 = 

 



. . . 

(cid:105)


1


 (

 

 




 + 
)−1

(A.17)

is the network-wide measurement noise covariance

Before stating the main result of this section, we present the following lemmas that will aid in

analyzing the averaged EKF.

N (cid:44)



Lemma A.1.2 Consider the system given by (4.5), (4.2) and the averaged EKF in (A.1)–(A.3). Let

=1 N
, and assume that there exist positive real numbers ¯
, ¯Ò, ¯
, 
, 
, 

, 

 > 0 such that

the following inequalities hold for all 
 ≥ 0:

(cid:107) 
(cid:107) ≤



 ≤
(cid:107)

(cid:107) ≤


 ≤ 

 ≤

¯






¯Ò

¯



 ≥ 
2

 ≥ 
2


 
2


 
|N|

then there exists a real number 0 < 
 < 1 such that the inequality

(
2

 − 

 

)
(
 ⊗ 
)
 
−1
with 

 given by (A.17), holds for 
 ≥ 0.


+1 × (
 ⊗ 
)(
2

 − 

 

) ≤ (1 − 
)
−1




Proof: Rearranging the terms in (A.16)



+1 =(√


 ⊗ 
)(cid:104)(
2

 − 

 

)

(
2

 − 

 

)

+(

 ⊗ 



)(cid:105) (√

+

 

 

(
2

 − 

 

)


 ⊗ 
)


(A.18)

(A.19)

(A.20)

(A.21)

(A.22)

(A.23)

(A.24)

(A.25)

94

Now, we consider the term 

 

 

(
2

 − 

 

)
. With (A.17), it can be verified that



(
2

 − 

 

)
 = 

 − 

 




 (

 

 




 + 
)−1

 



which is symmetric, and applying the matrix inversion lemma, we get

 
−1

)−1 > 0



(
2

 − 

 

)
 = (
−1


 



because 
−1




> 0. Additionally, from (A.17), and since 

 > 0 and 
 > 0,



 

 = 

 




 (

 

 




 + 
)−1

 ≥ 0

From (A.27) and (A.28), it can be seen that



 

 

(
2

 − 

 

)
 ≥ 0

Therefore, from (A.25) and (A.29), the inequality


 ⊗ 
)(cid:104)(
2

 − 

 

)

(
2

 − 

 

)
 + (

 ⊗ 



)(cid:105) (√



+1 ≥ (√


 ⊗ 
)


holds, and can be rewritten as


 ⊗ 
)(
2

 − 

 

)×



 + (
2

 − 

 

)−1(

 ⊗ 



)(
2

 − 

 

)−
(cid:105)
(cid:104)


+1 ≥ (√
× (
2

 − 

 

)
(√


 ⊗ 
)


From (A.17), and the assumed bounds (A.18)–(A.23), it is easy to see that

(A.26)

(A.27)

(A.28)

(A.29)

(A.30)

(A.31)

(A.32)

which leads to



+1 ≥ (√


 ⊗ 
)(
2

 − 

 

) ×

(cid:107)

(cid:107) ≤ ¯
 ¯Ò

2




 +

(cid:19)2

(cid:18)



2


1 + ¯
 ¯Ò

2



2



95

 (
2

 − 

 

)
(√


 ⊗ 
)


Taking the inverse of both sides, and multiplying from the left by (
2

 − 

 

)
(√
from the right by (√


 ⊗ 
)(
2

 − 

 

) we get

(
2

 −

 

)
(√


 ⊗ 
)
 
−1


+1(√


 ⊗ 
)×(
2

 −

 

) ≤

Since 
 is stochastic,

(
2

 − 

 

)
(
 ⊗ 
)
 
−1

+1
× (
 ⊗ 
)(
2

 − 

 

) ≤ (
2

 − 

 

)

× (√

 ⊗ 
)(
2

 − 

 

)
≤ (1 − 
)
−1


 ⊗ 
)
 
−1


+1(√




i.e., inequality (A.24) with

1 − 
 =

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)1 +

(cid:19)2

(cid:18)

¯




2


1 + ¯
 ¯Ò

2


−1

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

1 +

(cid:19)2

(cid:18)

¯




2


1 + ¯
 ¯Ò

2





 ⊗ 
)
 and

−1


−1




(A.33)

(A.34)

(A.35)

(cid:4)

It is worth noting that conditions (A.18)–(A.20) and (A.22)–(A.23) are easy to satisfy, since we
assume knowledge of the system’s matrices. Condition (A.21) is specifically addressed in Section
III-B.

Lemma A.1.3 Let the assumptions in Lemma A.1.2 hold, and assume that there exist positive real
numbers 
 
, 
 
 > 0, such that the nonlinear function 
(
1, 
2) in (A.13) is bounded by

(cid:107) 
(
1, 
2)(cid:107) ≤ 
 
(cid:107)
1 − 
2(cid:107)2

(A.36)

for 
1, 
2 ∈ R2

 with (cid:107)
1 − 
2(cid:107) < 
 
, then there is a positive real number 

 > 0 such that





 (
 ⊗ 

)
 
−1


+1 [2(
 ⊗ 
)(
2

 − 

 

) ˜

 + (
 ⊗ 
2
) 

] ≤ 

(cid:107) ˜

(cid:107)3




(A.37)

holds for (cid:107) ˜

(cid:107) ≤ 
 
.

96

Proof: From (A.13) and (A.32), a bound can be established on (cid:107) 

(cid:107)1

(cid:107) 

(cid:107) ≤ ¯
 ¯
 ¯Ò

2


(cid:107) 
(

 , ˆ

)(cid:107)

For (cid:107) ˜

(cid:107) = (cid:107)

 − ˆ

(cid:107) ≤ 
 
, using (A.36), one gets

(cid:107) 

(cid:107) ≤ ¯
 ¯
 ¯Ò

2



 
(cid:107) ˜

(cid:107)2 (cid:44) 
(cid:48)(cid:107) ˜

(cid:107)2 ≤ 
(cid:48)
 
(cid:107) ˜



(cid:107)

which in turn leads to

(cid:34)

(cid:32)

(cid:33)





 (
 ⊗ 

)
 
−1


≤ 
(cid:48) 1




+1 [2(
 ⊗ 
)(
2

 − 

 

) ˜

 + (
 ⊗ 
2
) 

]
1 + ¯
 ¯Ò2

2


+ 
(cid:48)
 


(cid:107) ˜

(cid:107)3

2 ¯


(cid:35)
(cid:32)

(cid:34)

(cid:33)

(cid:35)

2 ¯


1 + ¯
 ¯Ò2

2


+ 
(cid:48)
 


i.e., inequality (A.37) with



 = 
(cid:48) 1



(A.38)

(A.39)

(A.40)

(A.41)

(cid:4)

Lemma A.1.4 Let the assumptions in Lemma A.1.2 hold, then there is a positive real number


 > 0 such that


{




 
−1


+1

} ≤ 



(A.42)

for all 
 ≥ 0.

Proof: Using the definition of 

 given by (A.14), and since 

 and 

 are uncorrelated, we can
expand the expectation expression in (A.42) via

 
−1

+1

} = 
{

+ 
{



+1(

 ⊗ 
)

}


 (

 ⊗ 
)
 
−1


 (

 ⊗ 
)
 
−1


+1(

 ⊗ 
)

 

}


{



(A.43)


 








From the bound in (A.18)–(A.23) and (A.32), equation (A.43) can be converted to an inequality
via

(cid:32) ¯
 ¯
 ¯Ò

(cid:33)2


2



{








}

(A.44)


{




 
−1


+1

} ≤ 1





{






 (

 ⊗ 
)
(

 ⊗ 
)

} + 1




97

Because both sides of (A.44) are scalars, we can take the trace of the right-hand side without
changing its value

(cid:16)(

 ⊗ 
)
{

 



}(

 ⊗ 
)
(cid:17) + 1




(cid:32) ¯
 ¯
 ¯Ò

(cid:33)2

(cid:16)





2



}(cid:17)


{

 



tr

(A.45)

(A.46)

(cid:4)


 
−1


+1

} ≤ 1

 tr
2



2



=





{



Defining

leads to 
{




 
−1


+1

} ≤ 

.

+ N ¯
2 ¯
2 ¯Ò2


2




 (cid:44) 2



2







+ N ¯
2 ¯
2 ¯Ò2


2


With the above results, we can now present the following theorem regarding the stochastic

stability of the averaged EKF.

Theorem A.1.5 Consider the movement model for the target given by (4.5) and the nonlinear
measurement model for each agent given by (4.2), along with the averaged EKF in (A.1)–(A.3),
and let the assumptions of Lemmas A.1.2 and A.1.3 hold. then the network-wide estimation error
˜

 given by (A.12) is exponentially bounded in mean square with probability one.

Proof: We choose

From (A.21), we have



( ˜

) = ˜

 
−1




˜



1
¯


≤ 

( ˜

) ≤ 1




(A.47)

(A.48)

which satisfies the first condition of Lemma A.1.1 with 
 = 1
establish an upper bound on 
{

+1( ˜

+1)| ˜

}. Inserting (A.12) into (A.47) yields

¯
 and ¯
 = 1


. What remains is to



+1( ˜

+1) = [(
 ⊗ 
)(
2

 − 

 

) ˜

+

(
 ⊗ 
2
) 

 + (
 ⊗ 
2
)

]
 
−1
× [(
 ⊗ 
)(
2

 − 

 

) ˜


+(
 ⊗ 
2
) 

 + (
 ⊗ 
2
)

]




98

(A.49)

Taking the conditional expectation 
{

+1( ˜

+1)| ˜

} and applying Lemmas A.1.2, A.1.3, and A.1.4
for (cid:107) ˜

(cid:107) ≤ 
 
 we obtain

Defining 
 = min

(cid:16)


 
, 

2 ¯




(cid:17)


{

+1( ˜

+1)| ˜

} ≤ (1 − 
)

( ˜

)
+ 


 
(cid:107) ˜

(cid:107) + 



, then for (cid:107) ˜

(cid:107) ≤ 



(cid:107) ˜

(cid:107)(cid:107) ˜

(cid:107)2 ≤ 

2 ¯


(cid:107) ˜

(cid:107)2 ≤ 

2



( ˜

)

Applying inequality (A.51) to (A.50) yields


{

+1( ˜

+1) − 

( ˜

) ≤ − 

2



( ˜

) + 



(A.50)

(A.51)

(A.52)

which satisfies the second condition of Lemma A.1.1, ensuring that the network-wide estimation
error ˜

 remains bounded in mean square.
(cid:4)

A.2 Boundedness of Averaged Covariance Matrix

The boundedness of the covariance matrix in inequality (A.21) under the proposed method

must be addressed. To that end, consider the operator originally presented in [70].
Lemma A.2.1 Let 
 denote the set of matrices {
1, . . . , 

} and 
 denote the set of matrices
{
1, . . . , 
 
}. Define the following operator


+1( 
, 
) =

 


with



 
 
 
 
 



 ( 
, 
) 
 
 


(A.53)









=1


=1

1( 
, 
) =

 




 
 
 
 
 
 
 
 


If there exist positive-definite matrices 
 = {
1, . . . , 
 
} such that 

 > 
 

following statements hold for all 
 = 1, . . . , 
:

(A.54)
1( 
, 
), then the

1. For all bounded positive semi-definite matrices 

 ≥ 0,


( 
, 
) = 0.

lim
→∞ 
 


2. For all bounded positive semi-definite matrices 



0 ≥ 0, and 

 ≥ 0, the sequences 




+1 =

1( 
, 

) + 

 are bounded.

 


99




2 =

=


=1












=1
+



 
 
 
 
 


1 
 
 
 + 





 
 
 
 
 


1 ( 
, 
0)
 
 




 
 
 

 
 
 
 
 + 




=1
2( 
, 
0) + 
 

= 
 


1( 
, 
) + 








=1

(cid:17)


 ≤(cid:16)






 + 

1 − 


and by induction, it can be shown that





+1 = 
 



+1( 
, 
0) +


 ( 
, 
) + 



 


From the proof of Part 1, there exist constants 
, 
 > 0 and 0 < 
 < 1 such that 
 



 
 

 and 

 ≤ 


, which is used to show that


( 
, 
0) ≤

Proof:

1. Since 

 ≥ 0 and 

 > 0, then there exists a constant 
 > 0 such that 

 ≤ 


,∀
. Then it
1( 
, 
) < 

, then there exists a
1( 
, 
) ≤ 
2

, and


( 
, 
). Additionally, since 
 

1( 
, 
) ≤ 
 

. Then, 
 


2( 
, 
) ≤ 
 
 



( 
, 
) ≤ 
 
 


is clear that 
 

constant 
 ∈ (0, 1) such that 
 

sequentially, 
 



( 
, 
) ≤ 

 

. Therefore, 
 



( 
, 
) ≤ 


 

 and lim
→∞ 
 



( 
, 
) = 0.

2. From the definition of 




+1, we have





(cid:4)

Corollary A.2.2 For given matrices 
, 
, 
, 
 = {
1, . . . , 

}, and time varying matrices 

 =
{
1


 }, define the operator

, . . . , 
 






(
) =






 



(
 − 
 






)
 
(
 − 
 


 






 




 




 





 
 
 







(cid:104)






=1

(cid:20)


)
 

(cid:105) +






=1

100



(cid:21)

+ 





(A.55)

where


(cid:16)


 











 + 

(cid:17)−1


 = 











) and 

 =










(cid:20)

(cid:21)

If there exist positive-definite matrices 
 = {
1, . . . , 
 
} such that 

 > 
 

for all 
 ≥ 0, where 
 

0 ≥ 0, the sequences 






1( 
(
 − 

 

), 
)
1(·, ·) is defined in Lemma A.2.1, then for all positive semi-definite matrices

+1 = 




(

) are bounded.

Proof: This corollary can be proven by directly applying the results in Lemma A.2.1, defining


 = 
(
 − 


(cid:4)

+ 



.









=1 

 




 





 
 
 




Examining (A.2), it can be seen that the error covariance matrix under the averaged scheme

 remains bounded under the


(

). Therefore, by Corollary A.2.2, 



takes the form 


proposed scheme if the conditions of Lemma A.2.1 are satisfied.


+1 = 



101

APPENDIX B

SUPPLEMENTAL RESULTS FOR TDOA-BASED DISTRIBUTED LOCALIZATION

AND TRACKING OF A MOVING TARGET

B.1 Cases (a), (b), and (c) in Theorem 4.2.3

We show that rank(O1) = 6 for the remaining three cases in Fig. 4.3.

Case (a)

This case is the easiest to examine, as agent 1 has a sufficient number of neighbors. The measurement
matrix 
1(
) has the following structure

and from the discussion on decentralized approach, the pair ( 
, 
1(
)) is generically observable.
It immediately follows that rank(O1) = 6, since

(B.1)




1(
) =


1 
2 
3 0 0 0

4 
5 
6 0 0 0

7 
8 
9 0 0 0



(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:2)
2(cid:3)
(cid:2)
7(cid:3)

[
]11 
1(
) 

11 
1(
) 
2

...

11 
1(
) 
7

 ,


(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

= 6,

rank (O1) = rank

for almost all values of 
 and 
.

102

Case (b)
The structure of 

(
) for 
 = 1, . . . , 4 is

(cid:104)


1(
) =


2(
) =


3(
) =

4(
) =

We first note that the pair

(cid:105)

,

(cid:105)
(cid:105)

,

.


5

8


6

9


1 
2 
3 0 0 0
−
1 −
2 −
3 0 0 0
0 0 0

4
0 0 0

7


 ,
(cid:104)−
4 −
5 −
6 0 0 0
(cid:104)−
7 −
8 −
9 0 0 0

1(
)


2(
)

(cid:169)(cid:173)(cid:173)(cid:171)
,

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)


1(
)

2(
)

...


1(
) 
6

2(
) 
6



(cid:170)(cid:174)(cid:174)(cid:172) ,

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

rank

= 6,

is generically observable, which can be shown following the same analysis in the discussion for the
decentralized approach. This in turn, ensures that

for almost all values of 
 and 
, and it immediately follows that rank(O1) = 6.

103

Following similar procedures to those presented in the decentralized approach discussion, it


1(
) =


2(
) =


3(
) =


4(
) =

,

.


7


8


9

0 0 0


4 
5 
6 0 0 0


1 
2 
3 0 0 0
 ,
(cid:104)−
1 −
2 −
3 0 0 0
(cid:105)
−
4 −
5 −
6 0 0 0
 ,
(cid:105)
(cid:104)−
7 −
8 −
9 0 0 0


(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)
,
(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) ,


(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)
(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)


1(
) 
6

2(
) 
6

3(
) 
6


1(
)

3(
)

4(
)


1(
)

2(
)

3(
)

= 6,

Case (c)
The structure of 

(
) for 
 = 1, . . . , 4 follows

can be shown that the pair

is generically observable, and that

rank

...

for almost all values of 
 and 
. Therefore, rank(O1) = 6 for almost all values of 
 and 
.
B.2 Formation Control

In this section, we provide the details for the dynamic feedback linearization controller for the
system shown in (4.44). We first augment the system dynamics with another integrator on the 



104

input channel

(cid:164)

 = 

,1,

(B.2)

with 

,1, 

,2, and 

,3 constituting the new control inputs to the 
th robot.

Then we define the position coordinates of agent 
 as the outputs to perform feedback lineariza-

tion

The augmented robot model in (4.44), (B.2) and output (B.3) can be written in compact form

where



 =

,


 (

) =

 =





,1


,2


,3

 =



















 .


1(

)

2(

)

3(

)

(cid:164)

 = 
 (

) + 
(

)

,


 = 
(

),









































,




,


(B.3)





 cos (

) cos (

)


 sin (

) cos (

)



 sin (

)







,




,


0
0
0

(B.4)

(B.5)



,

105

(cid:104)

and 
(

) =

(cid:105)

, with


1


2


3









,

,


2 =


1 =

0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0

 
(

) = 

3 
 


 
(

) = 

2 
 


 
(

) ࣔ 0, and 

3 
2














0
0
0
0
0
0
0
0
1



Noting that 

1 
 





and that 

1 
2
defined as





 
(

) ࣔ 0, 

2 
2




A(

) =






1 
2


1 
2


1 
2











1(

) 

2 
2

2(

) 

2 
2

3(

) 

2 
2











1(

) 

3 
2

2(

) 

3 
2

3(

) 

3 
2








1(

)

2(

)

3(

)

is nonsingular as long as 

 ࣔ 0, and 

 ࣔ ± 


 ,


3 =

.

(B.6)


 
(

) = 0 for 
 = 0, 1 and for all 
 = 1, 2, 3,

 
(

) ࣔ 0, the decoupling matrix A(

),

(B.7)

(B.8)

2 . Now, we define 

 as 

 = 
(

), where


,1 =
1(

)


,2 =
 
 
1(

)

 
1(

)


,3 =
2


,4 =
2(

)


,5 =
 
 
2(

)

 
2(

)


,6 =
2


,7 =
3(

)


,8 =
 
 
3(

)

 
3(

)


,9 =
2

106

The model can then be written in terms of the transformed state 



(cid:164)

,1 =

,2
(cid:164)

,2 =

,3
(cid:164)

,3 =
3
(cid:164)

,4 =

,5
(cid:164)

,5 =

,6
(cid:164)

,6 =
3
(cid:164)

,7 =

,8
(cid:164)

,8 =

,9
(cid:164)

,9 =
3

(cid:164)

 =


 
1(

) + 3



=1


 
2(

) + 3



=1



 
 
2


 
1(

)



 
 
2


 
2(

)


=1



 
 
2


 
3(

)


 
3(

) + 3

 + A(

)

,

 =


(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) .
 

,


(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

 −
 

 +

0
¯
 0
¯

0

¯
 0
0
0

0
¯
 0
¯

0



¯
 0
0
0

107

Combining the equations for (cid:164)

,3, (cid:164)

,6 and (cid:164)

,9,

1(

)

2(

)

3(

)

(cid:164)

,3
(cid:164)

,6
(cid:164)

,9


3



3



3





allows for the selection of linearizing control


1(

)

2(

)

3(

)
Substituting (B.11) into (B.10), allows us to rewrite (B.9) as



 = A−1(

)


3



3



3



(B.9)

(B.10)

(B.11)

(B.12)

where

¯
 =



 ,

0 1 0
0 0 1
0 0 0



0
0
1

 ,

¯
 =

(B.13)

generating a linear map from the agent’s virtual control 

 to its 3D position.

The estimated position of the moving target is fed into a feedback linearization control strategy
to drive each agent to its desired position. The desired state for agent 
 can be obtained from its
estimate of target’s state:


ਭ

 =

ˆ

 + 


(cid:107) ˆ(cid:164)

(cid:107)

ਭ

 =

 = tan−1

ਭ

 = sin−1

ਭ

(cid:18) ˆ(cid:164)

(cid:18) ˆ(cid:164)







ˆ(cid:164)





(cid:107) ˆ(cid:164)

(cid:107)

(cid:19)
(cid:19)

(B.14)

(B.15)

(B.16)

(B.17)

(B.18)

where 

 is a constant vector that defines the position of agent 
 relative to the target. Additionally,
since the model used for estimation is that of a point particle moving in space, we assume that the
target follows a straight line, and set 
ਭ

,
 = 0. In order to drive the transformed state
of agent 
, 

, to its desired value 
ਭ


,
 = 
ਭ

 ), we can design a gain matrix 
, such that


 = 
ਭ

 = 
(
ਭ



 −



¯
 0
0
0

0
¯
 0
¯

0

¯
 0
0
0

0
¯
 0
¯

0

 


is stable, driving 

 to 
ਭ


 , and therefore driving agent 
 to its desired location.

108

APPENDIX C

SUPPLEMENTAL RESULTS FOR THE DISTRIBUTED KALMAN FILTER

In this appendix, we provide the proofs for Lemma 5.2.3, the derivation for the sub-optimal filtering
gains in (5.8), and show that the results in Chapter 5 hold when the order of the filtering steps is
reversed.

C.1 Proof of Lemma 5.2.3

Remark C.1.1 (From [29]) Given a set of 
 nonnegative scalars 

 summing up to one, a set of

 vectors 

, and a set of 
 matrices 

, the following holds

(cid:32) 



(cid:33)(cid:32) 



(cid:33)(cid:48)







 







 






=1


=1


=1

≤


 
(cid:48)


 




(cid:48)

.

For the noise-free case, the matrix '
(
 + 1) can be explicitly written as









=1


=1



 
 
 
(cid:170)(cid:174)(cid:172)×


 
 
 
(cid:170)(cid:174)(cid:172)(cid:48)

.

'
(
 + 1) =(cid:169)(cid:173)(cid:171)( 
 − 



)
(cid:169)(cid:173)(cid:171)( 
 − 



)




Using Remark C.1.1, it follows that

'
(
 + 1) ≤ ( 
 − 



)



 
 ' 
(
)( 
 − 



)(cid:48).

Defining '(
) ∈ H
0, with '
(
) shown above, we have


=1

'(
 + 1) ≤ L('(
))

with Ɖ
 = ( 
 − 



), and the first statement can then be proved by induction.

109

The second statement can be proved in a similar manner. Since the noises have zero mean and

are independent with respect to themselves and 
(0), using Remark C.1.1 it can be checked that


(
 + 1) = 
(cid:8)

(
 + 1)

(
 + 1)(cid:48)(cid:9)

≤ L
((
)) + 
 + 

 

 
(cid:48)

,

with (
) = (1(
), . . . , 
(
)) ∈ H
+ and Ɖ
 = ( 
 − 



). The second statement can then be
proved by induction.

C.2 Sub-optimal Filtering Gains

From Lemma 5.2.3, it immediately follows that


(
) ≤ ¯
(
)

where 
(
) is defined in (5.6), and ¯
(
) is defined as









¯
(
) =

tr(

(
)),

(C.1)

with 

(
) defined in (5.15). In what follows, we will show that the optimal cost ¯
∗(
) is obtained
using the gains


=0


=1


∗

 (
) = 
E
(
∗(
))
(cid:48)


 (

 + 

E
(
∗(
))
(cid:48)


)−1,

(C.2)

with 
∗(0) = (0), and 
∗(
) ∈ H
+ is computed using (5.15) with the gains 
∗


 (
).

Let ¯
(
) denote the cost when an arbitrary set of gains 

(
) is used, and let 
(
) ∈ H
+

denote the matrices obtained when these arbitrary gains are used in (5.15).

To show the desired result, we need to show that

¯
(
) − ¯
∗(
) =

tr(

(
) − 
∗


 (
)) ≥ 0










=0


=1

110

We do that by induction, and assuming 
(
) ≥ 
∗(
) for 
 ≥ 0. It can be shown through

matrix manipulation (see [30]) that





∗(
 + 1) − 

(
 + 1)
= ( 
 − 
∗
− (

 − 
∗


 

)E
(
∗(
) − 
(
))( 
 − 
∗

 )(

 + 

E
(
(
))
(cid:48)


 

)(cid:48)

)(

 − 
∗

 ).


) > 0, it immediately follows that 

(
+1) ≥ 
∗

Since 
(
) ≥ 
∗(
) and (

+

E
(
(
))
(cid:48)

 (
+1)
and ¯
(
) − ¯
∗(
) ≥ 0. Finally, we show that these sub-optimal gains and the matrices 
∗(
) can
be computed in the distributed manner as in (5.7)–(5.9). Indeed, substituting 
∗

 (
) in (5.15), and
after performing some matrix manipulation, we can write


 (
 + 1) = 
E
(
∗(
)) 
(cid:48) + 
 − 
E
(
∗(
))
(cid:48)

∗

×

 + 

)−1

E
(
∗(
)) 
(cid:48),

(

E
(
∗(
))
(cid:48)

(C.3)

which is the same expression obtained when substituting (5.7) into (5.8) and (5.9).

C.3 Reversing Order of Consensus and Filtering Steps

When the order in which the consensus and filtering steps is reversed as in [29], the noise-free

estimation error dynamics are given by



(
 + 1) =



 
(cid:0)
 − 
 
 
 
(cid:1) 
 
(
).






=1

(C.4)

First we show that the dynamics in (C.4) are asymptotically stable iff the dynamics in (5.5) are
asymptotically stable. From (C.4), the network-wide error dynamics can be expressed as


(
 + 1) = B2
(
),

(cid:16)

R

(cid:17)

(cid:16)

R

(cid:17)

(C.5)

(cid:16)

R

(cid:17)

where B2 (cid:44) (
 ⊗ 

)diag[ 
 − 



] ∈ B




] ∈ B
asymptotically stable.

and diag[ 
 −
, it follows that 

(B) = 

(B2) and (5.11) is asymptotically stable iff (C.5) is

. Since (
 ⊗ 

) ∈ B

111

The notion of S-detectability in Definition 5.2.4 implies that there exist certain gain matrices such
that the dynamics of the upper bounds for the noise-free covariances in (5.14) are asymptotically
stable. To show that notion holds regardless of the order of the consensus and filtering steps, we
need to consider the upper bounds of the noise-free covariances for (C.5). Define the operator
J(.) ∈ B(H
) as






=1

J
(
) (cid:44)



 
 Ɖ 


 Ɖ(cid:48)


 ∈ B(R
).

The upper bound for the noise-free covariances '
(
) can be written in a compact form as


(
 + 1) = J(
(
)),


(0) = '(0),

(C.6)

with Ɖ
 = ( 
 − 



).

It has been shown in [30] that 

(J) = 

(L). Therefore, the dynamics in (5.14) are asymp-
totically stable iff (C.6) is asymptotically stable, and the notion of S-detectability in Definition 5.2.4
does not depend on the order in which the steps are taken. Furthermore, it has been shown in [30]
that the coupled Riccati equations presented in [29], which are used to calculate the sub-optimal
filtering gains for this case, converge to a stabilizing solution iff the equations in (C.3) converge to
a stabilizing solution. Therefore, the necessary conditions we derive in Section 5.2 for the stability
of the distributed Kalman filter do not depend on the order of the consensus and filtering steps.

112

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