Università degli Studi di Pisa
FACOLTÀ DI INGEGNERIA
Corso di Laurea Specialistica in Ingegneria dell’Automazione
Tesi di laurea specialistica
A differential geometric approach
to path planning and control design
for Autonomous Underwater Vehicles
Relatori:
Prof. Andrea Caiti
Dr. Monique Chyba
Candidati:
Dario Cazzaro
Luca Invernizzi
Anno Accademico 2009–2010

iAbstract
This work focuses on the automatic generation of control inputs for the
thrusters of Autonomous Underwater Vehicles, given a path to follow in six
dimensions. The control strategy applied is model-driven and designed in a
open-loop fashion, as most AUVs do not have reliable localization systems.
This approach can be also extended in a closed-loop control scheme to
improve robustness.
The inputs generating framework is designed using differential geometric
control theory, which provides a singularity free description of the vehicle
model. This approach has been tested in simulations, using both a fully
actuated vehicle model and an underactuated one. The latter configuration
could be the result of a thrusters failure, which the differential geometric
approach is able to overcome. Experimental trials have been carried out in a
controlled environment using a fully actuated vehicle.

Contents
1 Introduction 1
2 AUV applications 5
2.1 A short AUVs history . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Current applications for AUVs . . . . . . . . . . . . . . . . . 7
2.2.1 Fishing . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Ocean monitoring . . . . . . . . . . . . . . . . . . . . 8
2.2.3 Security and military applications . . . . . . . . . . . 11
2.2.4 Dangerous environments and space exploration . . . . 15
3 Equations of motion 17
3.1 Rigid Body Kinematics . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 Coordinate frames . . . . . . . . . . . . . . . . . . . . 18
3.1.2 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.3 Rotation Matrix . . . . . . . . . . . . . . . . . . . . . 20
3.1.4 Position Transformation . . . . . . . . . . . . . . . . . 21
3.1.5 Linear Velocity Transformation . . . . . . . . . . . . . 22
3.1.6 Angular Velocity Transformation . . . . . . . . . . . . 22
3.2 Rigid Body Dynamics . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Hydrodynamic Forces and Moments . . . . . . . . . . . . . . 25
3.3.1 Added mass and inertia . . . . . . . . . . . . . . . . . 25
3.3.2 Hydrodynamic Damping . . . . . . . . . . . . . . . . . 26
3.3.3 Restoring Forces and Moments . . . . . . . . . . . . . 26
3.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 27
4 Differential Geometry Framework 28
4.1 Differential geometry elements . . . . . . . . . . . . . . . . . 29
4.1.1 The configuration manifold . . . . . . . . . . . . . . . 29
ii

CONTENTS iii
4.1.2 The tangent bundle . . . . . . . . . . . . . . . . . . . 30
4.1.3 External forces . . . . . . . . . . . . . . . . . . . . . . 31
4.1.4 The kinetic energy metric . . . . . . . . . . . . . . . . 32
4.1.5 The Levi-Civita connection . . . . . . . . . . . . . . . 32
4.1.6 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 Kinematic reduction . . . . . . . . . . . . . . . . . . . . . . . 36
5 Geometric Control for AUVs 38
5.1 Reference system . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.4 External forces . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.5 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . 43
5.6 Kinematic reduction . . . . . . . . . . . . . . . . . . . . . . . 45
5.7 Decoupling Vector Fields . . . . . . . . . . . . . . . . . . . . 46
5.8 Reparametrization . . . . . . . . . . . . . . . . . . . . . . . . 46
5.9 Control signals . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.10 Control strategy design . . . . . . . . . . . . . . . . . . . . . 47
5.10.1 Getting from ηinit and ηfinal . . . . . . . . . . . . . . 48
5.10.2 Following a path . . . . . . . . . . . . . . . . . . . . . 48
5.10.3 Reintegrating the potential forces . . . . . . . . . . . . 48
6 Simulations 50
6.1 Virtual exploration of an underwater basin . . . . . . . . . . 50
6.1.1 Building the vehicle model . . . . . . . . . . . . . . . 51
6.1.2 Thrusters configuration . . . . . . . . . . . . . . . . . 54
6.1.3 Missions . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 65
6.2 Simulating a thruster failure . . . . . . . . . . . . . . . . . . . 67
7 Experiments 72
7.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 73
7.2 Test-bed Platform: ODIN . . . . . . . . . . . . . . . . . . . . 76
7.3 Control Strategy Design . . . . . . . . . . . . . . . . . . . . . 78
7.4 Experimental Trials . . . . . . . . . . . . . . . . . . . . . . . 82
7.4.1 Strategy One: Semi-circle . . . . . . . . . . . . . . . . 84

CONTENTS iv
7.4.2 Strategy Two: Horizontal Survey . . . . . . . . . . . . 88
7.4.3 Strategy Three: Concatenated Motion . . . . . . . . . 90
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8 Conclusions 95
A Software implemented 97
A.1 Kinematic Simulator . . . . . . . . . . . . . . . . . . . . . . . 97
A.2 Dynamic Simulator . . . . . . . . . . . . . . . . . . . . . . . . 99
A.3 Geometric based control . . . . . . . . . . . . . . . . . . . . . 101
B Differential Geometry 106
B.1 Coordinate charts . . . . . . . . . . . . . . . . . . . . . . . . . 106
B.2 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
B.3 Tangent bundles . . . . . . . . . . . . . . . . . . . . . . . . . 111
B.4 Cotangent bundles . . . . . . . . . . . . . . . . . . . . . . . . 112
B.5 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
B.6 Affine Differential Geometry . . . . . . . . . . . . . . . . . . . 119

Chapter 1
Introduction
Dario Cazzaro
Oceans have always been one of the more challenging environment to
explore. The lack of visibility, low temperatures and high pressures make
oceans a really dangerous environment for human beings. In the last century
human efforts focused in creating submarines, allowing the exploration of
a large portion of the underwater world. But it is only in the last decades,
with the advances in technology, that a number of autonomous vehicles has
been developed. Autonomous Underwater Vehicles (AUVs) allows for access
in higher risk areas, longer time underwater, and more efficient exploration
as compared to human occupied vehicles.
Much research has gone into designing autonomous underwater vehicles
that perform a multitude of tasks far surpassing the abilities of human divers
and small human-driven submarines. Tasks range from long term gathering
of ocean samples to 3-D mapping of underwater caverns and never seen
before areas. Also, these vehicles can dive deeper and remain at depth longer
without risking human lives. Autonomous Underwater Vehicles (AUVs) come
in a variety of shapes and sizes, and are built for a wide range of purposes.
From continuous monitoring of coral reef ecosystems in a shallow marine
environment to bringing back images of undiscovered marine species from
depths of over 11 km in the Marianas Trench, AUVs are a major asset in
ocean research.
Multiple AUVs are currently in use today in a variety of fields. For
instance, the mbari (Monterey Bay Acquarium Research Institute) recently
1

CHAPTER 1. INTRODUCTION 2
sent an AUV to investigate the oil spill in the Gulf of Mexico, after the
explosion of the Deepwater Horizon oil rig in April 2010.
In the effort of mapping previously unexplored areas, a Texas based
company named Stone Aerospace is creating vehicles with the long term goal
to produce an AUV able to explore the oceans on Europa, one of the moon
of Jupiter. In preparation for this task, the DepthX AUV was designed to
explore and map the Zacaton cenote, an unexplored aquatic cave in Mexico.
Also, the Endurance vehicle was designed to bore through ice and conduct a
preprogrammed 3-D grid trajectory for mapping and sampling.
Like any other vehicle, we need the ability to control the AUV accurately
to perform even the simplest of tasks. This control starts with the basic
hydrodynamics of the vessel, and goes on to include the autonomy algorithm
which the vessel uses to make its decisions. Most of todays research AUVs
are designed with a torpedo-like shape. This design reduces the number of
degrees of freedom (DOF) and thus makes the vehicle easier to understand
from a control point of view. The shape also reduces drag and enhances
the efficiency of the vehicle. The downside of this shape is the lack of
maneuverability of the vessel. For completing transects, and navigating over
a long distance trip, an AUV does not have to be very agile. But, as AUVs
enter more different areas of science and are asked to complete more complex
tasks, other shapes such as ellipsoids or spheres need to be more closely
examined. These shapes are more maneuverable, but require much more
knowledge about control design as well as the hydrodynamics of the vehicle.
Regardless of its shape, all underwater vehicles are confronted with
the task of path planning and control design for each mission. This task
is crucial for successful operation of the vehicle, and has yet to be fully
investigated. This is because the general motion planning problem for a
nonlinear mechanical system is very difficult.
The purpose of this thesis is to provide answers to the motion planning
problem for underwater vehicles. The general submerged rigid body belongs
to a class of simple mechanical control systems whose Lagrangian is of the
form kinetic energy minus potential energy. In literature, there are many
formulations for modeling the mechanics of such systems. Here we consider
a more recent development by use of a differential geometric formulation.
One reason for choosing this approach is that the configurations defined by a
simple mechanical system correspond naturally to a differentiable manifold

CHAPTER 1. INTRODUCTION 3
in a one to one manner. On this space we are able to formulate a coordinate-
independent model of the system. This allows us to deal with the actual
structure of the problem without singularities or misleading representations
associated with a certain choice of coordinates.
In this dissertation, we start with the foundations presented in Bullo
and Lewis [BL05b] and work to extend these theoretical results to practical
applications. In particular, we focus on generating control strategies from
the geometric control framework and implementing them. Two kinds of
implementations have been tested: first, the geometric control strategy has
been applied to a vehicle model being simulated. An exploration mission
scenario has been considered. Then, a set of experiments has been ran onto
a test-bed underwater vehicle to make it perform a ship surveying task.
We begin this dissertation with an overview of the state of the art in the
field of autonomous underwater vehicles, presented in Chapter 2. After a
brief history of AUVs, we inspect some of the most common applications of
this kind of underwater vehicles.
In Chapter 3 we present the equations of motion of a rigid body in the
classical mechanics sense. We introduce hydrodynamic forces and conclude
with a well known version of the second-order system of equations describing
the motion of an underwater vehicle.
Next, in Chapter 4 we derive the same equations under the framework of
differential geometry paralleling the previous derivation. The same chapter
contains a review of the basic notions of differential geometry needed to
describe a mechanical system.
With these equations defined, in Chapter 5 we examine their solution
through the motion planning problem for submerged rigid bodies. The
motion planning using decoupling vector fields is described, along with the
control design technique.
To test the results obtained, in Chapter 6 we run a series of simulations.
After setting up different mission configurations, a vehicle model is guided
by the differential geometric based control scheme to explore a hazardous
environment such as a narrow submerged cave. In the latter set of simula-
tions, the event of a thruster failure has been taken into account, and the
control inputs has been adapted to the new thrusters configuration using the
developed theory.
In Chapter 7 we show the results of experimental trials run on a test-bed

CHAPTER 1. INTRODUCTION 4
vehicle. These experiments have been ran in a controlled environment, with
no relevant external disturbances. Other than verifying the feasibility of a
differential geometric approach, these trials demonstrate the AUV capability
to perform a practical application such as surveying the bulbous bow of
a ship. Experimental results are analyzed and compared with theoretical
predictions.
Chapter 8 is dedicated to the conclusions of both the theoretical and
practical section. Also, an overview of future developments and possible
improvements of the geometric control theory in the underwater vehicles
field is presented.
In Appendix A is described the simulator we designed to run our tests.
Implemented in Wolfram Mathematica, it contains both a kinematic and a
dynamic simulator of a rigid body moving in six degrees of freedom. Included
is the code to compute the controls for the vehicle thrusters, obtained from
differential geometric equations.
Finally, in Appendix B is reported a short introduction to differential
geometry. Some of the basic concepts are analyzed, to present an overview
of fundamental concepts needed to understand the geometric control of
mechanical systems.

Chapter 2
Autonomous Underwater
Vehicles: current
applications and state of the
art
Luca Invernizzi
These days, oceans are among the protagonists of most of the major
news broadcasts. While the United States of America are taking measures
to reduce the effects of an oil spill from an offshore platform in the Gulf of
Mexico, Australia is defending the coralline barrier from the oil leaking from
a sunken tanker. Argentina and Chile are periodically hit by the devastating
effects of the El Niño current, while the Indian Sea waters are often theater of
attempts by modern pirates to hijack merchant ships. On a narrower scope,
Italian fishermen are being imprisoned in Libya in an ongoing confrontation
on the control of the fishing rights in the Mediterranean.
The list goes on and on as oceans, which cover the majority of Earth
surface, conceal abundant resources in oil, fishes, minerals and metals, and
play a major role in controlling the climate, both with currents and as an
effect of the photosynthesis of their algae. Oceans constitute also the cheapest
way to carry cargo, which makes them the uncontested protagonists of the
international shipping industry, as 90% of the goods traded everyday are
5

CHAPTER 2. AUV APPLICATIONS 7
Figure 2.2.1: Canadian Cod stocks
currents ([SK01]).
In the last decade, the increasing interest in the increasing need of
guaranteering security and the monitoring of the climate permitted the
creation of a variety of platforms and applications, the most prominent of
whom we’ll briefly review.
2.2 Current applications for AUVs
2.2.1 Fishing
The use of acoustics in the assessment of fish stocks has been a reality for
the last twenty years ([DE92]). Acoustic surveys are essential to determine
the population size of many pelagic fishes.
This continuous work to assess and model the indices of abundance is
of key importance, as exceeding the fishing capacity of an area can lead to
serious economic losses, as it happened with the Canadian cod stocks, which
underwent a crysis in 1992 (see Figure 2.2.1).
The established technique in fisheries acoustics uses integrated outputs

CHAPTER 2. AUV APPLICATIONS 9
downward glide cycles in a sawtooth pattern. Fins are almost always fixed,
with the exception of a small rudder.
Gliders are extremely quiet and have almost no external control surface
that can be logorated by the sea, so they can substain long missions with
high success rates. They are slow, and their mission durations is months
long, so they can also exploit currents to obtain propulsion.
The most notable currently operational gliders, are Seaglider, Spray and
SLOCUM (in figure 2.2.2). All these operate for long periods in full open-
loop control, and surface at predetermined times to multilaterate themselves
using the Global Positioning Systems, in order to counterbalance the drift
caused by currents and other sources. In the SLOCUM, the GPS antenna
is embedded in one of its tail fins, and its brought to a vertical position
out of the water by rotating the vehicle body. Most of the commercial
platforms are able to communicate with their owners via satellite connections
(usually IRIDIUM). Typically the gliders carry a conductivity, temperature
and depth sensor (CTD), but the payload can be changed to match the
mission requirements.
While the commercial gliders are usually electrically actuated, gliders
may also derive their propulsion from the ocean itself, using temperature
differences across the thermocline.
“The operational cost of making a section [for scientific data collection],
including launch, recovery, refurbishment, and telemetry is as low as $2
per km... Gliders can be operated for a year for the cost of a single day of
research vessel operation. Fabrication cost is equivalent to the cost of 2-4
days of ship time” [DEJ02].
Although gliders are the most common solutions for long-range surveys,
propeller driven platforms are also wildly available, the most notable of
whom are the REMUS and the DORADO.
Other methods of propulsion have been proposed: an interesting example
is the Wave Glider, which requires no power. It is composed of a floating
device connected via an elastic cable to a negatively buoyant underwater
fin. When a wave front arrives, the upward force generated by the cable is
transformed in surge motion (of little more than a knot in module) by the
sunken fin. See figure 2.2.3 for a depiction of this wave driven propulsion.
This solution has been proven capable of withstanding critical sea conditions,
such as hurricane winds.

CHAPTER 2. AUV APPLICATIONS 10
Figure 2.2.2: SEAGLIDER, SPRAY and SLOCUM

CHAPTER 2. AUV APPLICATIONS 14
Figure 2.2.6: The Webb Research Thermal Glider and Xray
employed by US Navy
• Time Critical Strike
The US Navy has developed a series of AUVs, including thermal gliders
(gliders for which propulsion is derived from temperature differences across
the thermocline). Those are almost soundless, making them virtually unde-
tectable. Two examples are the Webb Research Thermal Glider, that can be
used in mission up to 5 years long, during which it can cover 40000 km (it
can go across the Earth), and the Xray, with a four knots top speed (both
shown in Figure 2.2.6).

CHAPTER 3. EQUATIONS OF MOTION 19
E3
E1
E2
OE
B1B2
B3
OB
b
Figure 3.1.1: Earth-fixed and body-fixed reference frames.
Since the earth-fixed frame can be considered to be inertial, the position
and the orientation of the vehicle should be described relative to the iner-
tial frame, while the linear and angular velocities of the vehicle should be
expressed in the body-fixed coordinate frame.
Although standard notations for these quantities are defined in sname,
1950 [SNA50], we choose an alternate notation to simplify the expressions in
the next chapters. The standard notation is notationally cumbersome for
summations and geometric representations. The chosen notation is presented
in Table 3.1. Based on this notation, the motion of a rigid body in six degrees
of freedom can be described by:
 = [bT , qT ]T
v = [T ,
T ]T
 = [T ,'T ]T
b = [b1, b2, b3]T
 = [ν1, ν2, ν3]T
 = [τ1, τ2, τ3]T
q = [φ, θ, ψ]T

= [Ω1,Ω2,Ω3]T
' = [ϕ1, ϕ2, ϕ3]T
(3.1.1)
where  denotes the position and orientation vector with coordinates in
the earth-fixed frame, v denotes the linear and angular velocity vector with
coordinates in the body-fixed frame and  is used to describe the forces and
moments acting on the vehicle in the body-fixed frame. Through the deriva-
tion we will break out each force individually with its own representation.
This will eventually leave  to represent only the input control forces from
the actuators.
3.1.2 Euler Angles
The angular orientation of the body with respect to the earth-fixed frame
can either be given using many representations, such as Euler angles or

CHAPTER 3. EQUATIONS OF MOTION 20
DOF position and linear and angular external forces
name Euler angles velocities and moments
surge b1 ν1 ϕ1
sway b2 ν2 ϕ2
heave b3 ν3 ϕ3
roll φ Ω1 τ1
pitch θ Ω2 τ2
yaw ψ Ω3 τ3
Table 3.1: Notation used for autonomous underwater vehicles.
quaternions. In this thesis, we choose to work with the Euler angles, accepting
the fact that we may need two representations to fully avoid singularities. In
particular, we use the Tait–Bryan angles roll(φ), pitch(θ) and yaw(ψ).
3.1.3 Rotation Matrix
To define the transformations between the reference systems, it is useful to
define a transformation matrix called Rotation Matrix. Given two reference
systems having the same origin, the columns of a rotation matrix are the
unit vectors aligned with the axes of a reference system, represented in the
other one. Thus, a rotation matrix describe the orientation of a coordinate
system with respect to another one and can be used as a linear operator to
transform a point from a reference frame to one another.
Since the columns of a rotation matrix are orthogonal unit vectors, the
matrix is orthogonal, which means:
RTR = I3×3, RT = R−1, (3.1.2)
where I3×3 is the 3 × 3 identity matrix. Furthermore, det(R) = 1 if the
reference system is right-handed, det(R) = −1 otherwise. All the coordinate
systems we deal with are right-handed, so det(R) = 1 for every rotation
matrix we consider.
Using the roll-pitch-yaw representation, the basic rotations are defined

CHAPTER 3. EQUATIONS OF MOTION 21
as:
Rx(φ) =




1 0 0
0 cφ sφ
0 −sφ cφ




Ry(θ) =




cθ 0 −sθ
0 1 0
sθ 0 cθ




Rz(ψ) =




cψ sψ 0
−sψ cψ 0
0 0 1




(3.1.3)
where s · = sin(· ) and c · = cos(· ).
To obtain the complete transformation between the two reference systems,
basic rotation matrices has to be combined using the following procedure.
Let Σ˜E be the coordinate system obtained by translating the earth-fixed
coordinate system ΣE parallel to itself until its origin coincide with the origin
of the body-fixed frame ΣB. First, Σ˜E is rotated a yaw angle ψ about the
E3 axis, then the new coordinate system is rotated a pitch angle θ about the
new E2 axis. Finally, the new frame is rotated a roll angle φ about the new
E1 axis. At this point, the reference frame Σ˜E coincide with the body-fixed
frame ΣB. The rotation sequence is written as:
R(q) = Rz(ψ)TRy(θ)TRx(φ)T (3.1.4)
and its inverse as:
R−1(q) = RT (q) = Rx(φ)Ry(θ)Rz(ψ) (3.1.5)
3.1.4 Position Transformation
The coordinate transformation between the body-fixed frame and the earth-
fixed frame can be described using a homogeneous transformation matrix
T ∈ SE(3), the special Euclidean group, which is composed of a rotation
matrix and a position vector. It is defined as
SE(3) =
{
T ∈ R4×4
∣
∣
∣T =
[
R(q) b
03×1 1
]}
(3.1.6)

CHAPTER 3. EQUATIONS OF MOTION 24
Let the skew operator S(· ) be:
S
(
[r1, r2, r3]
T
)
=




0 −r3 r2
r3 0 −r1
−r2 r1 0



 (3.2.2)
Also, let m be the mass of the body, IOB be the inertia tensor with respect
to OB, and rG the position of the center of mass with respect to the origin
of the body-fixed frame.
So, the 6× 6 inertia matrix of a rigid body can be defined as:
M =
[
mI3×3 −mS(rG)
mS(rG) IOB
]
(3.2.3)
We now need to define the Coriolis matrix C(v) to account for Coriolis
and centripetal forces experienced by the body. These are pseudo forces
which arise in the equations of motion when a body is viewed from a rotating
frame of reference. The general expression for the Coriolis matrix is:
C(v) =
[
03×3 −S(M11 +M12
)
−S(M11 +M12
) −S(M21 +M22
)
]
(3.2.4)
where M can be seen as M =
[
M11 M12
M21 M22
]
.
Now it is possible to state the general expression for the equations of
motion of a rigid body:
M _v + C(v)v =  (3.2.5)
where  is a generic vector of external forces and moments.
The general equations can be simplified by making two non limiting
assumptions. The first assumption is setting the origin of the body-fixed frame
OB to coincide with the center of gravity CG. This implies rG = [0, 0, 0]T ,
so many elements of the inertia matrix became zero. It is often convenient to
let body-fixed reference frame axes coincide with the principle axes of inertia
of the rigid body. This later assumption yields a diagonal inertia tensor (i.e.,
IOB = diag(Ib1 , Ib2 , Ib3)), and coupled with the first assumption implies that
the inertia matrix M is a diagonal matrix.
Under these assumptions, the equations of motion (3.2.5) simplify to the

Chapter 4
Differential Geometry
Framework
Dario Cazzaro
The equations derived in the previous chapter are usually taken as the
general equations of motion for an underwater vehicle moving in six degrees-
of-freedom, however they are actually a coordinate representation of a more
general set of equations living on a differentiable manifold.
A specific choice of parameterization (i.e., coordinates) for these general
equations is convenient in some applications and allows us to work locally and
temporarily believe that we are working with familiar calculus in Euclidean
space, but this also can be quite restrictive based on the configuration
manifold of the system in question. In addition, a coordinate expression
may omit inherent global properties of the mechanical system and manifold
structure since the coordinates are only valid for a portion of the configuration
space.
By use of a manifold, we force the problem formulation to be coordinate
independent, and thus we can tackle the real geometric structure of the
problem. To this end, we continue the derivation of the equations of motion
by examining them through the framework of differential geometry.
Here, we will produce a coordinate-invariant system of equations to
describe the motion of a rigid body moving in six degrees-of-freedom. The
motivation for this derivation is to exploit inherent geometric properties of
28

CHAPTER 4. DIFFERENTIAL GEOMETRY FRAMEWORK 29
the mechanical system and avoid unnecessary bulk associated with a certain
choice of coordinates.
We begin presenting a differential geometric representation for every term
involved to describe a generic mechanical system. An alternative formulation
of the equations of motion will be provided. At last, a brief introduction to
kinematic reduction is presented.
4.1 Differential geometry elements
4.1.1 The configuration manifold
The set of configurations of a mechanical system is in natural one-to-one
correspondence with a set that has the structure of a differentiable manifold;
this manifold is called the configuration space and is typically denoted by Q.
A manifold is, essentially, a set that can locally be parameterised by
an open subset of the Euclidean space, and different parameterisations are
required to be compatible with one another. A specific choice of parameteri-
sation is sometimes helpful in working on a concrete example. The existence
of parameterisations in general is sometimes useful in that it allows us to
believe, at least temporarily, that we are working with familiar calculus in
Euclidean space (typically by saying, “Let (q1, . . . , qn) be coordinates for
Q”). However, the important thing about manifolds is that they force a
coordinate independent restriction on the problem formulation. Thus one is
forced to deal with the real structure of the problem and not one tacked on,
ad hoc, by a choice of coordinates.
In Figure 4.1.1 is depicted the classical pendulum/cart system where
the objective is to balance the pendulum in the upright configuration by
application of a horizontal force to the cart. One approach to modelling the
problem is to declare the coordinates describing the configurations of the
system to be (x, θ), and then proceed as if the configuration space is R2. But
the configuration spaces is not R2, since θ and θ + 2kpi describe the same
configuration for every integer k. The configuration spaces is, in fact, R× S1,
where S1 is the unit circle.
It is important to make this distinction for many reasons, for example
it is possible for a mechanical control system with a configuration space of
Rn to possess a globally stable equilibrium point using continuous feedback.
However, it is not possible for a mechanical control system with a configu-

CHAPTER 4. DIFFERENTIAL GEOMETRY FRAMEWORK 30
Figure 4.1.1: Classical pendulum/cart system. Its configuration
space is not the Euclidean space R2.
ration space having rotational degrees of freedom to have a globally stable
equilibrium point using continuous feedback.
4.1.2 The tangent bundle
The configuration manifold describes the set of configurations of a mechanical
system. The tangent bundle of the configuration manifold describes the set
of configurations and velocities of a mechanical system.
Associated to each point of a manifold is the set of all velocities possible
from that point. This space is called the tangent space at q ∈ Q and is
denoted TqQ. The union of all these vector spaces is called the tangent
bundle TQ and is itself a differentiable manifold which is twice the dimension
of Q.
Coordinates for TQ are often denoted by ((q1, . . . , qn) , (v1, . . . , vn)). Thus
TQ has twice the dimension of Q. The way to think of the tangent bundle is
as the state space for the system. The dynamics of the system are uniquely
determined by initial conditions in the tangent bundle, this by virtue of the
equations of motion being second-order.
Using ((q1, . . . , qn) , (v1, . . . , vn)) as coordinates for the tangent bundle
would seem to imply that the tangent bundle is the Cartesian product of the
configuration manifold and the set of velocities. This is not the case. It is
true that TQ is locally a product, but even this local product representation
is not natural; it arises from a specific choice of coordinates and different
coordinates give different local product representations. Thus, while the
coordinates for TQ seem to validate the notation of representing points in

CHAPTER 4. DIFFERENTIAL GEOMETRY FRAMEWORK 32
where t ∈ [t0, tf ], i.e. the work is the integral product of the force with the
velocity.
The expression F (t)· γ′(t) should live in R (work is a scalar quantity)
and should be linear in γ′(t). Thus F (t) can live no place other than T ∗Q.
4.1.4 The kinetic energy metric
The key to the differential geometric approach to mechanics is the fact
that the kinetic energy of the system defines a Riemannian metric on the
configuration space, so the configuration space can be seen as a Riemannian
manifold. A Riemannian manifold is a differentiable manifold in which the
tangent space at each point is a finite-dimensional Euclidean space.
In the classical way of looking at things, the kinetic energy metric is a
symmetric positive-definite matrix-valued function of configuration, which
coincide with the inertia matrix. In the geometric way of looking at things,
the kinetic energy metric is a smooth assignment of an inner product to
each tangent space of the configuration space. In differential geometry such
an assignment is called a Riemannian metric. In the particular case where
the assignment comes from the kinetic energy for a mechanical system, the
corresponding Riemannian metric is called the kinetic energy metric.
The notation used for the kinetic energy metric is G. Thus the inner
product on the tangent space at q ∈ Q is denoted by G(q), and the inner
product of two tangent vectors uq, vq ∈ TqQ is denoted by G(q)(uq, vq), or
simply by G(uq, vq) since the point q ∈ Q is explicit in the notation already.
Therefore, the kinetic energy is the function on TQ given by
K(vq) =
1
2
G(vq, vq) (4.1.2)
At a point q ∈ Q, G(q) is an inner product on the tangent space TqQ.
4.1.5 The Levi-Civita connection
As with any Riemannian metric, associated to G is its Levi-Civita connection:
the affine connection which naturally arises from a mechanical system.
An affine connection on Q is a mapping that assigns to two vector fields
X and Y a third vector field denoted ∇XY . The vector field ∇XY is called
the covariant derivative of Y with respect to X and it is a measure of how a

CHAPTER 4. DIFFERENTIAL GEOMETRY FRAMEWORK 36
Classical form Geometric coordinate form Intrinsic form
M(q)q¨ + C(q, q˙)q˙ Gq¨ + GΓ(q˙, q˙, q) G∇γ′(t)γ′(t)
g(q) −F (t, q, q˙) −F (t, γ′(t))
 ∑mi=1 Yi(q)ui
∑m
i=1 Yi(γ(t))ui
Table 4.1: Comparison of terms in “classical”, “geometric compo-
nent”, and “intrinsic” formulation of equations of motion.
4.3 Kinematic reduction
Sometimes it is useful to apply a kinematic reduction on the system to
simplify the solution of some control problem. The reduction decreases the
dimension of the state space by a factor of two allowing to treat the Yi like
velocity vector fields on the configuration space. It is also sometimes possible
to plan trajectories for underactuated mechanical systems as if they were
kinematic system. The kinematic system uses velocity control inputs, and
hence, the state space has half the dimension of the dynamic system. One
advantage to the reduction in complexity of the system representation is the
subsequent simplification of related control problems including stabilization,
motion planning and optimal control.
Consider the original second-order mechanical system:
∇q˙ q˙ =
m∑
i=1
Yi(q)ui (4.3.1)
where the set of control vector fields is written as Y = {Y1, . . . , Ym}, and a
first-order kinematic system
q˙ =
l∑
i=1
Vi(q)wi (4.3.2)
where the set of control velocity vector fields is written as V = {V1, . . . , Vl}.
A kinematic system is a kinematic reduction of a dynamical system if
all feasible trajectories for the kinematic system are also feasible for the
second-order system. Also, a mechanical system is maximally reducible to a
kinematic system if there exists a kinematic reduction such that all feasible
trajectories of the mechanical system, starting with an initial velocity in
span(Y), are also trajectories of the kinematic reduction. For example, all

CHAPTER 4. DIFFERENTIAL GEOMETRY FRAMEWORK 37
fully actuated mechanical systems are maximally reducible to kinematic
systems; we can equivalently assume the controls are either velocities or
forces.
A maximal kinematic reduction that generates all trajectories of the
mechanical system could be called a rank m kinematic reduction, as the
controlled velocities of the reduction form an m-dimensional distribution
with span(V) = span(Y). However, the class of underactuated systems
admitting rank m kinematic reductions is relatively small. A system that
is not maximally reducible to a kinematic system may nonetheless admit a
rank 1 kinematic reduction q˙ = V1(q)wi(t).
A rank 1 kinematic reduction has a single control vector field V1, also
known as a decoupling vector field. The word “decoupling” comes from the
fact that trajectory planning for the second-order system along an integral
curve of such a vector field can be decoupled into choosing the distance
traveled along the integral curve, followed by time-scaling the path according
to actuator limits.
An underactuated mechanical system can follow the integral curve of
a decoupling vector field V1(q) at any speed and acceleration, i.e. for any
continuous w1(t). A second-order mechanical system can have no more than
m linearly independent decoupling vector fields at any q. For a maximally
reducible mechanical system, every vector field in span(Y) is a decoupling
vector field.
A vector field V is a decoupling vector field of the second-order mechanical
system if and only if V ∈ span(Y) and ∇V V ∈ span(Y). Stated another way,
from the definition of kinematic reduction for the single vector field V (q),
we have:
q˙ = V (q)w
q¨ = V (q)w˙ +
∂V (q)
∂q
V (q)w2
(4.3.3)
we see that V (q) is decoupling if and only if there exist a u ∈ Rm satisfying
the equations of motion for all w, w˙ ∈ R.

Chapter 5
Geometric Control for
Underwater Vehicles
Luca Invernizzi
In this chapter, we will apply the geometric control theory presented in
the previous chapter to underwater vehicles, incorporating the forces that
are found in this field (e.g., the added mass).
We will present this approach alongside with the standard approach, to
show the analogies and differences between the two.
5.1 Reference system
To model the equations of motion governing a rigid body, it is necessary to
work with two coordinate reference frames; one inertial (Earth-fixed) and one
for the vehicle (body-fixed). For low-speed marine vehicles, such as the one
studied here, the Earth’s movement has a negligible effect on the dynamics
of the vehicle. Thus, the Earth-fixed frame may be considered as an inertial
frame. The inertial reference frame ΣI : (OI , {s1, s2, s3}) is a right-handed,
orthogonal coordinate system defined with the s1 and s2 axes lying in the
horizontal plane perpendicular to the direction of gravity, while the s3 axis
is orthogonal to the s1− s2 plane and taken to be positive in the direction of
gravity. We also refer to the inertial reference frame as the spatial reference
frame. Note that since we are considering an unbounded fluid domain, we
38

CHAPTER 5. GEOMETRIC CONTROL FOR AUVS 39
are free to select an arbitrary position for the inertial frame, preferably in a
location such that the depth of the vehicle is non-negative.
The body-fixed frame ΣB : (OB, {B1, B2, B3}), is a right-handed, orthog-
onal reference frame defined with the origin OB located at a chosen location,
and the body axes B1, B2 and B3 coinciding with the principle axes of inertia.
The the longitudinal (B1) and transverse (B2) axes are taken positive to the
fore and starboard, respectively.
5.2 Kinematics
The configuration of a rigid body in six DOF can be described using  =
(x, y, z, φ, θ, ψ)t = (b,2)t, where 2 = (φ, θ, ψ)t is the orientation of the
body, relative to the spatial frame, and b = (x, y, z)t is the body’s relative
position. A body’s configuration  can also be represented as an element of
the Special Euclidean group SE(3): (b, R), where R ∈ SO(3) is a rotation
matrix describing the orientation of the body and SO(3) is the group of
orthogonal matrices that have determinant equal to one. In the following
sections, we will refer to Q = SE(3) as the configuration manifold for our
system, and on this differentiable manifold we will formulate the equations
of motion for a submerged rigid body, which will be presented as an affine
connection control system.
In the body-fixed frame, we identify  = (u, v, w)t as the linear velocity
and
= (p, q, r)t as the angular velocity of the vehicle. We express these
collectively as v = (,
)t. If we define a rotation matrix R2 such that
˙2 = R2
, we can state the formulation of the kinematic equations of motion
for a rigid body moving in six DOF as
η˙ =
[
R 03×3
03×3 R2
] [
ν
Ω
]
. (5.2.1)
Equivalently, formulating this system on the differentiable manifold Q,
Eq. (5.2.1) are written as the forward kinematic map Π : Q→ SE(3), with
b˙ = R , (5.2.2)
R˙ = R
ˆ. (5.2.3)
In Eq. (5.2.3), the operator ˆ : R3 → so(3) is defined by yˆ z = y × z. The

CHAPTER 5. GEOMETRIC CONTROL FOR AUVS 44
connection control system:
∇γ ′γ
′ = G#(P (γ(t))) + G#(Fdrag(γ ′(t))) +
6∑
i=1
I−1i (γ(t))σi(t), (5.5.4)
where σi(t) represents the controls.
For reasons that will become clear in the section 5.6 about the kinematic
reduction, we need to transform this forced affine control system in a affine
control system.
We define a modified Levi-Civita connection in the following manner
∇˜XiXi =



−Di
Gii
Xi i = j
∇XiXj i 6= j
The forces and moments resulting from viscous drag are now incorporated
into the affine connection.
Under the assumptions that CG 6= CB and W 6= B, we can not express
Eq. 5.5.4 as a driftless system due to the fact that the restoring forces and
moments do not directly depend on the velocity of the body.
We can simplify Eq. 5.5.4 in the following lemma, presented in [Smi08].
Lemma 2. Let Q = SE(3), be the modified Levi-Civita connection on Q
associated with the Riemannian metric G and let the set of input control vector
fields be given by I = I−11 , ..., I
−1
6 . Let G
#P (γ(t)) represent the restoring
forces arising from gravity and buoyancy. Then the equations of motion
of a rigid body submerged in a viscous fluid are given by the forced affine
connection control system:
∇˜γ′γ
′ = G#P (γ(t)) + Σ6i=1I
−1
i (γ(t))σi(t)
where σi(t) represents the controls.
This can be simplified further using more restrictive assumptions. Under
the assumptions thatW = B and CG = CB and G#P (γ(t))) = 0 (the vehicle
will experience no restoring forces or moments), this equation simplifies to
the following affine connection control system:
∇˜γ′γ
′ = Σ6i=1I
−1
i (γ(t))σi(t)

CHAPTER 5. GEOMETRIC CONTROL FOR AUVS 48
It’s not trivial to determine if the final configuration is reacheable with a
concatenation of kinematic motions. Details in how to determine that are
given in [Smi08, BL05a]. If any final configuration is reacheable, then the
system is kinematically controllable.
5.10.1 Getting from ηinit and ηfinal
Once we have determined that the final configuration is reachable, we are
ready to design the trajectory and control structure.
We solve the motion planning problem by determining the sequence of
integral curves of the decoupling vector fields (primitives) to follow to get
from ηinit and ηfinal.
We then parameterize each segment to start and end at zero velocity
so that each segment begins with the same initial conditions and we can
concatenate these motions to build the entire trajectory. From this repa-
rameterized, concatenated, kinematic motion trajectory, we can construct
the dynamic controls which steer the vehicle between the initial and final
configurations.
5.10.2 Following a path
Another possible approach is defining the path to follow in the body fixed
frame and define a reparameterization so that each segment of the concate-
nated path starts and ends at zero velocity.
It will be possible to follow that path if at any point the velocity required
to follow the assigned trajectory is part of a DVF. Every time it’s needed
to switch the current DVF, the vehicle has to come to a complete stop and
start another section of the concatenation.
This is the approach followed in the simulator described in Appendix A.
5.10.3 Reintegrating the potential forces
The kinematic reduction requires a drifltess dynamic system. The described
driftless system accounts for all the forces and moments that are present
during the motion of an underwater vehicle except for the potential forces.
Including potential forces in the affine connection is not possible since
these forces do not depend on time, velocity or acceleration of the body.
Since the potential forces are based on physical characteristics of the rigid

CHAPTER 5. GEOMETRIC CONTROL FOR AUVS 49
body (placement of CB with respect to CG and angular displacement) under
certain conditions, a drift will always exist in the reduced system. Hence, a
general theory along these lines is a very difficult problem and remains an
open question.
As in [Smi08], we designed control strategies negleting potential forces,
compensating them after the calculation of the control inputs. We note that
since the control strategies are constructed for full open-loop implementation,
it is impossible to adjust for unforeseen disturbances without using a feedback
loop.

Chapter 6
Simulations
Luca Invernizzi
In order to verify and exploit the mathematical framework defined in the
previous chapters, we have developed a simulator. The simulator accepts
as input a model of the vehicle to be simulated (with the data that will be
specified afterwards) and a path to be followed, and produces the controls to
be fed in the thrusters of the vehicles in order to follow that path. It also
takes those control and simulates the movement of a virtual vehicle, making
it easy to verify that the framework is working correctly.
In this chapter, we will proceed to showcase how both the mathematical
framework and the simulator perform through a series of examples.
Firstly, we will firstly present a simulation of an exploration of a cave,
taken from a paper we published in the proceedings of the 2010 International
Conference on Computer Applications and Information Technology in the
Maritime Industries (COMPIT).
Then, we will present a subsequent work, submitted for publication for the
2010 Conference on Decision and Control (CDC), about how the framework
can respond to a failure in one of its thrusters.
6.1 Virtual exploration of an underwater basin
The mathematical framework dicussed in this thesis is model based. As such,
we firstly need to determine the values of the various parameters for our
50

CHAPTER 6. SIMULATIONS 51
Figure 6.1.1: Omni-Directional-Intelligent-Navigator, from the Au-
tonomous System Laboratory at the University of
Hawaii at Manoa
vehicle. In order to do that, we base our estimations on two existing vehicles
presenting features comparable to the one we consider.
6.1.1 Building the vehicle model
ODIN
The first vehicle, the Omni-Directional-Intelligent-Navigator (ODIN) from
the Autonomous System Laboratory at the University of Hawaii at Manoa,
is spherical and is fully actuated. We will througly discuss ODIN capabilities
in the experiments chapter, since this mathematical framework was tested
on that test-bed vehicle: therefore, we forward the interest reader to that
chapter, just showing ODIN shape for reference in Figure 6.1.1. In Table 6.1
we display the physical parameters for this vehicle.
DEPTHX
The second vehicle on which we base our work is the NASA funded DEPTHX
(see Figure 6.1.2).
The Deep Phreatic Thermal Explorer (DEPTHX) is an autonomous
underwater vehicle, designed to explore deep underwater caves. It was
purported to be the most advanced robotic vehicle of its kind on Earth
(a second version of DEPTHX, called ENDURANCE, has been recently
developed, but details about the model are not yet available).
Its primary function is observation, with an array of sonar sensors (for
3D mapping) all around its flattened spheroid hull. It is about feet in

CHAPTER 6. SIMULATIONS 55
Figure 6.1.3: Thrusters configuration 1
Figure 6.1.4: Thrusters configuration 2
can easily be computed from the geometry and dimensions of the vehicle.
Configuration 2: Six thrusters - Five DOF
The second configuration is based on the thruster configuration implemented
on the DEPTHX AUV, which only provides forces in five DOF, having no
actuation for the pitch motion. In this model there are six thrusters, two of
them are in a vertical position and the other four located on the horizontal
B1−B2 plane. The vertical thrusters are aligned with the B2 axis such that
they cannot produce a pitching motion, but just heave and roll (see Figure
6.1.4). Under such a configuration the vehicle is under-actuated, due to the
inability to produce a rotation along its transverse axis. For this scenario,
there exist trajectories in the configuration space that cannot be realized
by the vehicle. This increase the difficulty of the motion planning problem
associated to this vehicle.

CHAPTER 6. SIMULATIONS 56
Figure 6.1.5: map of the Church Sink (Leon County, FL, USA -
vertical section), with the three missions locations
Configuration 3: Six thrusters - Four DOF
The third scenario is based on the same set of thrusters seen in configuration
2, but now the two vertical thrusters are restricted to produce equal thrust.
As a consequence, a rolling motion is not possible. It will consume more
energy than other motions because the restoring moment will try to align
the center of mass and the center of buoyancy on a vertical axis. The set of
the possible motions for this configuration will be even more restricted than
in the scenario of configuration 2. One more degree of freedom has been
lost in terms of the actuation. This is actually the mode of actuation under
which the DEPTHX vehicle functions.
6.1.3 Missions
To showcase the mathematical framework, we will now present some trajec-
tories that can be used to safely navigate our vehicle through some of the
most common scenarios found in underwater sink holes. We assume that a
rough knowledge of the environment is known, but the vehicle has yet to
map the area to the required degree of resolution. If the environment has
never been explored before, we can still plan a trajectory steering the vehicle
inside its sensor range.
In the first scenario, we will descend into a cylindrical sink hole (labeled
as box 1 in Figure 6.1.5). Then, we will explore the conical opening of a

CHAPTER 6. SIMULATIONS 58
Toward the end of its path it will decelerate, until coming to a complete
stop. We parametrize the trajectory to be at rest at the initial and final
configuration in order to allow an easy concatenation of trajectories (that can
generate complex paths starting from a few simple ones). This restriction
can easily be removed if it is necessary and other parametrization can be
considered.
The geometric method used to compute the control strategies is based
on the existence of decoupling vector fields, ([Smi08, SCCM09]). Those
vector fields are such that their integral curves in the configuration space
produce a kinematic motion that can be lifted into the tangent space of
the configuration manifold. In other words it provides us trajectories in the
configuration space for which we can determine the control that produce the
desired trajectory (in other words the motion can be realized by the vehicle).
For the mission considered here it can be proved that the six degrees of
freedom control does not require the use of the pitch, therefore the same
strategy can be used in thruster configuration 1 and 2. The vehicle’s motion
is displayed in Figure 6.1.6 on the next page. In the same figure we can see
the values of the thrusters, associated to both cases, to realize this trajectory.
For the third configuration, a different approach has to be followed, due
to the loss of roll actuation. Indeed, the previous integral curve used in
configuration scenario 1 and 2 cannot be realized by a vehicle under-actuated
as the DEPTHX vehicle is. To remedy this complication, a more in depth
geometric has to be conducted. A result of this geometric study shows that
the vehicle with thrusters configured as in scenario 3 is capable of motion
to follow integral curve obtained from vector fields that are composed by
either surge, sway and yaw or heave and yaw. To produce a trajectory
that provides a coverage similar to the helix, the chosen trajectory for the
vehicle is composed by linear descents (actuating the heave and the sway
simultaneously) and circumferences in the horizontal plane (actuating the
surge, sway and yaw simultaneously). The result is represented in Figure
6.1.7 as well as the corresponding thruster control. It can be observed that
this scenario requires thrusters three times more powerful than for the first
two thrusters configurations and, therefore probably of larger dimension
and weight. This is due to the frequent switching between the vertical and
horizontal movement, which requires the vehicle to stop and accelerate each
time. Therefore, it implies a higher acceleration component than for the

CHAPTER 6. SIMULATIONS 60
(a) Trajectories for
the third config-
uration (mission
1)
(b) Thrust for the third configu-
ration (mission 1)
Figure 6.1.7: mission 1: configuration 3
other two configurations.
Of most interest in designing control strategies for autonomous underwater
vehicles is the desire to minimize energy consumption. Defining the energy as
the integral over the duration of the trajectory of the voltage needed for each
thruster we can calculate and compare this cost for our trajectories. Here we
assume a linear relationship between the thrust and the amperes needed for
each thruster. Based on those assumptions, the three previously presented
trajectories for the descent into the vertical cylinder (one for each thruster
set up configuration) present similar energy consumption. This might be
surprising given the fact that the thrusters require three times more power.
However, the energy consumption is calculated over the whole trajectory and

CHAPTER 6. SIMULATIONS 64
(a) Trajectories for
the first config-
uration (mission
3)
(b) Trajectories for the first
configuration (mission 3).
Dashed lines correspond to
vertical thrusters.
Figure 6.1.10: Mission 3: first configuration.
(a) Trajectories for
the second and
third configura-
tion (mission 3)
(b) Trajectories for the second
and third configuration (mis-
sion 3). Dashed lines corre-
spond to vertical thrusters.
Figure 6.1.11: Mission 3: second and third configuration.

CHAPTER 6. SIMULATIONS 67
6.2 Simulating a thruster failure
To showcase that this mathematical framework is able to overcome thruster
failures, we will now present a simulation of the exploration of an underwater
sink hole. We assume that a rough knowledge of the environment is known,
but the vehicle has yet to map the area to the required degree of resolution.
ODIN will first descend into a vertical sink hole while mapping its walls.
Reached the bottom, it will enter into a smaller horizontal tunnel. There,
ODIN will suffer the failure of half of its thrusters, perhaps due to falling
rocks or water infiltrations. In particular, ODIN will lose its actuation in
surge and pitch. Following a safety procedure, it will then head back to the
surface, following different trajectories that exploit the remaining thrusters.
An overview of the mission is shown in figure 6.2.1
Figure 6.2.1: Mission overview
The first descent environment that the AUV encounters is a 110 m deep

CHAPTER 7. EXPERIMENTS 73
tonomous Systems Laboratory, College of Engineering, University of Hawai‘i.
7.1 Problem Statement
Approximately 90% of the goods traded throughout the world are carried by
the international shipping industry. With incentives of competitive freight
costs during a time of increasing fuel expenses, seaborne trade continues
to expand. Currently, there are more than 50,000 merchant ships trading
internationally. This fleet belongs to more than 150 nations, and employs over
one million seafarers. With a high volume of ships arriving from worldwide
destinations, it is of utmost importance to monitor and protect the ports
that facilitate each country’s trading market. To this end, it has become an
interest of border police and port authorities to examine the hulls of ships
for potentially dangerous attachments, e.g., explosives, before they enter the
harbor.
Currently, these tasks are performed by highly-skilled human divers. Such
labor intensive work introduces fatigue and poses multiple potential risks to
the divers. In particular, in the presence of hazardous elements these risks
can be life-threatening. To reduce the risk to human life, the use of Remotely
Operated Vehicles (ROVs) has become a useful alternative. However, this
also requires intense human involvement to safely navigate the ROV around
the ship. Moreover, the area around a ship in berth can be highly cluttered,
and tethered vehicles can experience impediments in reaching confined due
to tether entanglement and piloting error. Both of these methods cost time
and money, and cannot guarantee full coverage.
In an effort to provide a more comprehensive and cost-effective solution
to this problem, engineers have been working on automating this process by
employing Autonomous Underwater Vehicles (AUVs). AUVs offer several
advantages over the previously mentioned approaches; the risk to humans
is eliminated, the capability to dive in cluttered environments is improved,
and being autonomous, they can provide around-the-clock surveillance of
incoming ships and surrounding port facilities.
A pioneering and innovative approach to automating ship hull surveys is
presented in [VEE+06]. Here, the authors demonstrate the use of a Doppler
Velocity Logger (DVL) to allow the vehicle to lock onto a ship’s hull and
perform fixed-distance, hull-relative motions to complete a survey. This

CHAPTER 7. EXPERIMENTS 74
approach is shown to be highly-effective in inspecting the sides of the hull,
i.e., flatter regions, however human intervention was required in the proximity
of more complex regions, e.g., the bulbous bow, running gear, full longitudinal
cross-section, etc. The reason for these issues is that the vehicle’s trajectory
is strictly dependant upon sensor input from the DVL. If the DVL loses lock
on the ship’s hull, the AUV loses localization, and thus is unable to complete
the mission without intervention. This situation can result in areas where the
curvature of the hull changes rapidly over a short distance, e.g., the bulbous
bow. Additionally, a DVL is an instrument that consumes relatively large
amounts of power. For an autonomous system, it is of interest to employ the
use of such sensors on a limited basis to extend the deployment duration of
the vehicle.
Figure 7.1.1: USS George H.W. Bush (CVN 77), [Glo08].
In an effort to increase the functionality of autonomous systems, such
as that described in [VEE+06], we focus on developing control strategies for
AUVs to survey these more challenging regions. In this paper, we consider
the bulbous bow region of a ship. Many of the merchant vessels currently
in operation have a bulbous bow similar to that seen in Fig. 7.1.1. The
bulb is a protrusion from the front of the hull, positioned to sit just below
the design water line. Hydrodynamically, the bulb serves the purpose of
reducing the height of the bow wake of the vessel, thus decreasing hull drag
and achieving better efficiency. Bulbs come in all different shapes and sizes
and are optimized for a given ship design. It is imperative to take added care
in the inspection and maintenance of the bow, as the efficiency of the ship

CHAPTER 7. EXPERIMENTS 75
greatly depends on its effectiveness. Since the bulb is a protrusion, damage
caused from ship-dock or ship-ship interaction is always a concern. Although
the primary motivations of this study are safety and hazard mitigation, the
bulbous bow provides an interesting control theory problem for which to
consider motion planning and trajectory design, due to its peculiar shape.
We approach this problem from a path planning viewpoint, with the
motivation to reduce the reliance on navigational instruments that tend to
consume large amounts of energy. By utilizing a model-based path planning
techniques to design trajectories and control strategies, and implementing
them with the assistance of sensors and feedback controllers, we aim to provide
a contribution towards a reliable system for autonomous hull inspection.
As seen in the previous chapters, differential geometry provides the
framework and structure necessary to design and compute control strategies
for an agile AUV capable of moving in all six degrees-of-freedom (DOF).
This a priori consideration of path planning can outweigh the compu-
tational cost of learning system parameters from a model-based control
approach, and lower the need for accurate and energy consuming sensors.
Additionally, this framework includes a straightforward method to accommo-
date under-actuated scenarios, such as thruster failure for a fully-actuated
vehicle, or standard path planning for an under-actuated torpedo-shaped
vehicle.
The control strategies presented in this study are implemented onto ODIN
in full open loop to demonstrate the effectiveness of the geometric theory in
designing implementable trajectories, and to assess the vehicle’s performance
in executing the trajectory without any interference from sensed data. In a
real-world application, we understand that a feedback control loop would be
implemented to track the computed path, as unknown external disturbances,
e.g., currents, render open-loop controllers useless by themselves. However,
developing model-based control strategies that exploit symmetries and non-
linearities within the dynamics of a vehicle, as the differential geometric
techniques allow, could lead to an AUV relying less upon sensor input for
navigation. In addition to trajectory design for test-bed vehicles, we are
also interested in implementable closed-loop solutions for ocean-going AUVs.
Preliminary work on robust feedback tracking of AUVs can be found in
[SC09] and [SSS+09].
We continue our presentation in Section 7.2 by providing the technical

CHAPTER 7. EXPERIMENTS 76
specifications and physical characteristics of ODIN, the test-bed vehicle used
for our experiments. In Section 7.3, we describe the trajectory method
and the calculation of the control strategies. We additionally address the
necessary technique to transform the calculated control strategies into imple-
mentable controls for ODIN. Three scenarios for surveying the bulbous bow
are motivated and described in Section 7.4. For each scenario, we compute
the desired control strategy, implement it onto ODIN, and compare the
experimental results to our theoretical predictions. An overall assessment
is included for the procedure and experiments conducted, with ideas and
motivations for future research efforts.
7.2 Test-bed Platform: ODIN
To prove the effectiveness of our geometric path planning approach, we
implemented the computed control strategies onto an agile and fully-actuated
AUV: ODIN, the Omni-Directional Intelligent Navigator, which is shown in
Fig. 7.2.1. Complete details and technical specifications for this vehicle can
be found in [CHSC08] or [CY96], with specifics related to implementation of
geometric control strategies contained in [Smi08].
ODIN’s main hull is a sphere constructed form anodized aluminum (AL
6061-T6). The numerical values of various parameters used for modeling
ODIN are given in Table 7.1. These values were derived from estimations
and full-scale model tests performed on ODIN.
Figure 7.2.1: ODIN operating in the pool.
The added mass terms (Muf ,M
v
f ,M
w
f , J
p
f , J
q
f , J
r
f ) were estimated from
formulas found in [All90] and [Iml61]. Moments of inertia (Ixx, Iyy, Izz)
were calculated using experiments outlined in [Bha78]. We used inclining

CHAPTER 7. EXPERIMENTS 78
highly nonlinear and is approximated using a piecewise linear function which
we refer to as our thruster model.
Major internal components include a pressure sensor, inertial measure-
ment unit, leakage sensor, heat sensor and 24 batteries (20 for the thrusters
and four for the CPU). ODIN is able to compute and communicate real-time
and can operate for up to five hours from either a tethered or fully-autonomous
mode.
ODIN does not have real time sensors to detect horizontal (x−y) position.
Instead, experiments are videotaped from a platform 10 m above the water’s
surface, giving us a near nadir view of ODIN’s movements. Videos are
saved and horizontal position data are post-processed for later analysis. A
real-time system utilizing sonar was available on ODIN, but it has not been
used in these experiments mainly for two reasons. First, the sonar created
too much noise in the diving well and led to inaccuracies. More significantly,
in the implementation of our control strategies, ODIN is often required to
achieve large (> 15◦) list angles which render the sonars useless for horizontal
positioning. Many alternative solutions were attempted and video provided
a cost-effective solution which produced accurate results. We are able to
determine ODIN’s relative position in the testing pool to ±10 cm.
For the applications motivated in the following sections, we additionally
assume that ODIN has a forward facing camera (or other data collecting
sensor) mounted at the equator of the spherical hull. This is the sensor that
will be used to examine the ship’s hull.
7.3 Control Strategy Design
As previously mentioned, the aim of this chapter is to present a path planning
approach with experimental trials to provide solutions to the problem of
surveying a complicated section of a ship’s hull, the bulbous bow. To this
end, we are interested in calculating paths that the AUV can execute given
its controllability, and subsequently computing the controls to be applied by
the actuators to realize the chosen path. Hence, the path planning problem
is solved based on the actuation constraints of the vehicle, and the controls
are computed by solving the kinematic motion planning problem for the
prescribed path. This control strategy design process was developed by
following a differential geometric procedure outlined in [BL05b] in a very

CHAPTER 7. EXPERIMENTS 79
general manner and in [SCWC09] for application to AUVs. The detailed
process of adapting the computed controls for implementation onto the
considered test-bed vehicle is described in [Smi08].
To summarize this procedure, we begin by first applying a geometric
reduction procedure to the dynamic system (acceleration control inputs)
described by Eqns. (5.5.4) to produce a kinematic (velocity control inputs)
control system. We then calculate the decoupling vector fields for this
kinematic system. A decoupling vector field is a vector field whose integral
curves (under any reparameterization) are solutions to the kinematic system
as well as the dynamic system. In particular, the integral curves of the
decoupling vector fields define trajectories for the kinematic system that can
be extended to realizable trajectories of the dynamic system. Thus, by use
of decoupling vector fields, we are able to solve the motion planning problem
for the kinematic system. By Theorem 13.2 in [BL05b], it is guaranteed that
this solution can be extended to a solution for the dynamic system. The
decoupling vector fields for a given system are based on the actuation and
controllability of the system. For a fully-actuated vehicle, as presented here,
every vector field is decoupling. However, for an under-actuated vehicle (e.g.,
torpedo-shaped vehicle) the decoupling vector fields have to be calculated,
and there may exist configurations that are unreachable by kinematic motions
due to a vehicle’s controllability constraints.
Heuristically, this geometric reduction technique is similar to solving a
second-order differential equation by substitution of variables. Although
this method may not find all solutions to the motion planning problem
for the dynamic system, we are able to calculate some solutions without
explicitly solving the complete dynamic system. Once we have chosen the
integral curves of the decoupling vector fields that connect the initial and
final configurations, we reparameterize and concatenate them to define the
trajectory for the vehicle to follow. The corresponding control strategy
to realize this trajectory is calculated via inverse kinematics by applying
Theorem 13.5 in [BL05b].
We continue by briefly outlining the procedure of motion planning via
decoupled kinematic motions. First, we define the initial (ηinit) and final
(ηfinal) configurations of the system. We make the assumption that either
the initial configuration is the current one or is realizable by the vehicle,
otherwise the problem is not well stated.

CHAPTER 7. EXPERIMENTS 81
unique control structure, we must also link the piece-wise constant thrusts
via a linear function, since it is impossible for a physical actuator to change
outputs instantaneously.
Hence, to test our designed control strategies on ODIN, we must adapt
the continuous control strategies into piece-wise constant (PWC) control
strategies. To do this, we consider the work that is required to perform a
desired motion, and ensure that equivalent work is being done by both the
continuous and PWC controls. The work done on the system over a given
time interval is calculated by integrating the control with respect to time.
Thus, by appropriately choosing the times when the actuator switches from
one PWC to another, we can design a PWC control from a given continuous
control where the work done on the system is equivalent. This process is best
explained via the following example. At this point, we merge theory and
application together through the implementation of the computed control
strategies onto ODIN. Algorithm 1 presents the overall method for designing
and implementing our control strategies.
Algorithm 1 Implementation of Control Strategies Designed using Differ-
ential Geometry
1: Apply a kinematic reduction to Eqs. (5.5.4).
2: Calculate the decoupling vector fields (not necessary for ODIN).
3: Solve the motion planning problem from ηinit to ηfinal via inverse kine-
matics.
4: Calculate the continuous control strategy σ(t) for the dynamic system
by use of the geometric theory, i.e., Eq. (5.9.1).
5: Discretize σ(t) to obtain an implementable piece-wise constant control
structure σ0(t).
6: Upload σ0(t) to ODIN’s CPU.
7: while ODIN autonomously implements σ0(t) in full open-loop. do
8: On-board, ODIN converts σ0(t) from a 6-dimensional control to an
8-dimensional control.
9: The 8-dimensional control is converted from force (N) to voltage (V).
10: The voltage controls are send to the eight independent thrusters.
11: Position and orientation data are collected.
12: end while
13: Collected data are post-processed and analyzed

CHAPTER 7. EXPERIMENTS 82
13 m
10 m
10 m 2.5 m
LWL
Figure 7.4.1: Side view of a bulbous bow on a ship and its dimension.
7.4 Experimental Trials
In the sections to follow, we present experimental results for three geometri-
cally designed control strategies to survey portions of a ship’s bulbous bow.
Since the shape of the bulb affects the performance of the vessel at sea,
each bulb is uniquely constructed for an individual ship and many different
shapes and sizes are seen in use today. However, a general bulb can be
approximated by a cylindrical solid capped by a hemisphere. We will assume
this simplified scenario for our experiments. Note that small perturbations
in the shape of the bulb will not greatly affect the design of our paths or
the associated controls. Additionally, for very unique bulb shapes, and to
implement these techniques at full-scale, similar methods to those described
here can be employed to design specific trajectories and control strategies.
Figure 7.4.1 displays a typical bulbous bow, along with relative dimensions.
To reiterate, during the experiments, the computed control strategies were
implemented in full open-loop fashion, thus there are no feedback controllers
in operation. Due to this mode of operation, we expect and notice deviations
between the prescribed path and the actual implemented trajectory. There
are multiple reasons for such deviations, and we explicit them when discussing
each implementation result. Open-loop control implementation is not our
intention for a vehicle performing a real-world mission, but it has been
chosen to demonstrate the effectiveness of the technique used to design the
control strategies. Implementing the computed control in conjunction with a
standard adaptive or feedback controller will result in an effective control
system for AUVs to survey ship hulls.

CHAPTER 7. EXPERIMENTS 84
To survey the uniquely-shaped bulbous bow, we propose two separate
missions. The first is a semi-circular trajectory as depicted in Fig. 7.4.2a.
Here, the vehicle performs a pure heave while simultaneously applying a
pure surge. This trajectory can be used to survey the front of the bulb,
as seen in Fig. 7.4.2a, or to perform repeated transects up and down the
longitudinal axis of the bulb, as shown in Fig. 7.4.2b. The second trajectory
considered to survey a bulbous bow is a motion parallel to the free surface
while maintaining a desired pitch angle to point the front-mounted camera,
or sensor, at the surface of the bulb. This trajectory is depicted by the line
parallel to the load water line (LWL) in Fig. 7.4.3a. The third proposed
strategy is the concatenation of the previous two, as seen Fig. 7.4.3b. We
now continue by presenting the computed controls and the implementation
results for these missions.
For all of the experiments, the initial configuration is taken to be ηinit =
(0, 0, 1.5, 0, 0, 0), with the origin of the earth-fixed frame positioned on the
free surface. In particular, ηinit is located 1.5m below the water’s surface. All
control strategies presented here are designed such that the vehicle begins at
ηinit with zero velocity, and ends at ηfinal with zero velocity. The presented
experiments were performed in the diving well at the Duke Kahanamoku
Aquatic Complex at the University of Hawai‘i. As such, we are unable to
perform strategies that are full-scale with respect to the dimensions shown in
Fig. 7.4.1. We scale the height of the bulb from 10m to 2.5m, which implies
that the 2.5m radius of the hemisphere scales to approximately 0.5m. For
the motion parallel to the free surface, we scale the 10m length of the bulb
to 5m. As the aim of this chapter is to present the implementation results of
control strategies computed via differential geometric techniques, we omit
the details of the calculation of each control strategy, and simply present the
PWC controls that were executed by ODIN.
7.4.1 Strategy One: Semi-circle
The first strategy we wish to construct is the semi-circle trajectory shown
in Fig. 7.4.2a, to inspect the front and the sides of the bulbous bow. This
motion is constructed by simultaneously applying controls in both pure heave
and pure surge. The pure heave control is designed so that the vehicle realizes
a net 2.5m pure heave. The surge control is designed such that the vehicle
begins at rest, realizes a negative pure surge of 0.5m, then moves 0.5m in

CHAPTER 7. EXPERIMENTS 85
(a) Side view of a bulbous bow on a ship. Also depicted is the linear
trajectory for the inspection of the top of a bulbous bow.
(b) Overall path for complete inspection of a bulbous bow.
Figure 7.4.3: Different paths to survey a bulbous bow.

CHAPTER 7. EXPERIMENTS 92
0 10 20 30 40 50−40
−20
0
20
40
X (N)
0 10 20 30 40 50−40
−20
0
20
40
Z (N)
0 10 20 30 40 50−5
0
5
M (N
m)
Time (s)
0 10 20 30 40 50−6
−5
−4
−3
−2
−1
0
1
x (m)
0 10 20 30 40 50−4
−2
0
2
4
y (m)
0 10 20 30 40 501
2
3
4
5
6
z (m)
Time (s)
0 10 20 30 40 50−40
−20
0
20
40
φ
0 10 20 30 40 50−40
−20
0
20
40
θ
0 10 20 30 40 50−50
0
50
ψ
Time (s)
Figure 7.4.6: Strategy Three: Concatenation of the semi-circle tra-
jectory and the horizontal survey. Solid (blue) line
represents actual evolution, dash-dot (red) line repre-
sents the theoretical evolution.
the initial negative surge of 0.5m followed by a positive surge evolution of
approximately 0.5m, as prescribed. The depth evolution shows an overshoot
in depth by about 1m. The pitch evolution after 40 seconds oscillates about
zero with a magnitude less than ten degrees. This is a result of not stabilizing
the pitch angle to zero before beginning the semi-circle trajectory. Here,
the vehicle is simple relying on the righting arm to return it to an upright
position. The oscillations present in roll are an artifact of the small distance
between the center of gravity and center of buoyancy, i.e., small righting
arm. This configuration provides a very controllable vehicle in the sense that
it can realize many configurations by use of the on-board thrusts, however
this results in a decrease in stability of the AUV. Hence, reduced stability
coupled with the open-loop implementation results in the expectation of
small perturbations and oscillations in the evolution of the vehicle. The
yaw evolution begins with an initial offset that is remedied within the first
10 seconds. Note that this deviation arises during the time that the pitch
control operating. At t = 40s, we again notice a spike in the yaw, which

CHAPTER 7. EXPERIMENTS 93
corresponds to a time when the vehicle is releasing the pitch angle.
7.5 Conclusions
Here, we considered a practical application of the differential geometric
based path planning technique to examine the bulbous bow of a ship. Due
to the unique shape and location, examination and survey of the bulbous
bow provides an interesting motion planning problem for the underwater
vehicles. We do not provide an exhaustive survey algorithm, but propose two
control strategies which can be used to examine the majority of the bulb. For
implementation purposes, the experiments presented here have been scaled
down and assume a general form of the bulb. Trajectories to examine an
actual bulbous bow of a ship would need to be generated for the specific size
and shape of the bulb. The intent here is to present a practical application
of an emerging theoretical technique in the area of motion planning for
underwater vehicles.
The experimental results presented here extend the work developed in
[Smi08], and further validate the design of implementable control strategies
by use of differential geometric techniques. This architecture is not just a
change of notation for the same equations of motion, but a presentation with
a much richer inherent structure. A structure which can be exploited for
autonomous path planning in the event of a disabled vehicle (under-actuated)
or used to guide the design of future AUVs. Research is currently ongoing
to migrate the techniques presented here from the test-bed vehicle ODIN
onto an AUV active in the open ocean. The ability to reproduce great
implementation results, such as those presented here, gives us a good start
to investigate the potential of actual sea trials.
The favorable correlation between theoretical predictions and experimen-
tal results presented in this chapter are a result of working in a well-known
and controlled environment. This will definitely not be the case in the
ocean. To move from the pool to the ocean, significant adjustments will be
necessary. First off, an AUV cannot operate strictly in an open-loop mode.
Poorly known disturbance forces, e.g., ocean currents, are too large and
unpredictable to be neglected or accounted for a priori, even in a protected
harbor or port environment. Applying a purely open-loop control strategy in
the ocean, we would expect to see large errors between theoretical predictions

Chapter 8
Conclusions
Luca Invernizzi
In this thesis, we have shown an application of the differential geomet-
ric control theory for underwater vehicles, with a particular interest for
autonomous underwater vehicles.
The proposed mathematical framework, which expands that control
theory to accommodate the peculiarities involved in moving in a real liquid
(such as added mass), can compute the controls needed for an underwater
vehicle to follow a given trajectory in an open loop fashion.
The framework is model based and lets express the acceleration in a
coordinate free fashion. It can be applied to many AUV platforms currently
available.
To fully evaluate and test the capabilities of this solution, we have imple-
mented it in software and written a simulator that checked the correctness
of our claims. We have shown that this solution can generate control for
complex paths, can handle underactuation and overcome thruster failures.
Furthermore, the framework has been tested in a controlled environment
on a test bed vehicle giving promising results.
This approach could be useful in practical applications in a variety of
occasions.
When developing a new AUV, this framework can help in evaluating
the costs and benefits of different thruster configurations and provide a
lower bound for the thruster in both strength and bandwidth requirements.
During mission planning, it yields a good estimation of the time and energy
95