DRAFT 1
A Geometric Approach to Trajectory Design
for an Autonomous Underwater Vehicle:
Surveying the Bulbous Bow of a Ship
R.N. Smith, Member IEEE, D. Cazzaro, L. Invernizzi, G. Marani S.K. Choi and
M. Chyba
Abstract
In this paper, we present a control strategy design technique for an autonomous underwater vehicle
based on solutions to the motion planning problem derived from differential geometric methods. The
motion planning problem is motivated by the practical application of surveying the hull of a ship
for implications of harbor and port security. In recent years, engineers and researchers have been
collaborating on automating ship hull inspections by employing autonomous vehicles. Despite the
progresses made, human intervention is still necessary at this stage. To increase the functionality of
these autonomous systems, we focus on developing model-based control strategies for the survey
missions around challenging regions, such as the bulbous bow region of a ship. Recent advances in
differential geometry have given rise to the field of geometric control theory. This has proven to be an
effective framework for control strategy design for mechanical systems, and has recently been extended
to applications for underwater vehicles. Advantages of geometric control theory include the exploitation
of symmetries and nonlinearities inherent to the system.
Here, we examine the posed inspection problem from a path planning viewpoint, applying recently
developed techniques from the field of differential geometric control theory to design the control
This research is supported in part by National Science Foundation grant DMS-0608583, and in part by Office of Naval
Research grants N00014-03-1-0969, N00014-04-1-0751 and N00014-04-1-0751.
R.N. Smith: Computer Science Department, Robotic Embedded Systems Laboratory, 3710 S. McClintock Avenue, Ronald
Tutor Hall (RTH 512), University of Southern California, Los Angeles, CA 90089
M. Chyba: Mathematics Department, College of Natural Sciences, University of Hawai‘i, Honolulu, HI 96822, USA
D. Cazzaro, L. Invernizzi: College of Engineering, Sant’Anna School of Advanced Studies, Pisa, Pi, 56127, Italy
G. Marani, S.K. Choi: Autonomous Systems Laboratory, College of Engineering, University of Hawai‘i, Honolulu, HI 96822,
USA
February 18, 2010 DRAFT

2 DRAFT
strategies that steer the vehicle along the prescribed path. Three potential scenarios for surveying a
ship’s bulbous bow region are motivated for path planning applications. For each scenario, we compute
the control strategy and implement it onto a test-bed vehicle. Experimental results are analyzed and
compared with theoretical predictions.
Index Terms
Bulbous Bow Survey, Autonomous Underwater Vehicle, Trajectory Design, Geometric Control
Theory
I. INTRODUCTION
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,
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SMITH et al.: BULBOUS BOW SURVEY USING GEOMETRIC CONTROL 3
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 [1].
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 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.
Fig. 1. USS George H.W. Bush (CVN 77), [2].
In an effort to increase the functionality of au-
tonomous systems, such as that described in [1], 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. 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 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
February 18, 2010 DRAFT

4 DRAFT
bow provides an interesting control theory problem for which to consider motion planning and
trajectory design, due to its peculiar shape. To our knowledge, there is no previous research
specifically dedicated to this topic.
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. The foundation
upon which we build our path planning approach is that of differential geometric control theory.
Previous research has shown that geometric control theory is a useful and effective way to
design and compute control strategies for many simple mechanical control systems1, including
the submerged rigid body (e.g., [3], [4]). Additionally, the rigid body submerged in an ideal
fluid is used as a running example throughout [5], [6], [7] and [8]. This differential geometric
architecture, and associated path planning techniques have been extended to include external
forces, such as viscous damping and restoration forces resulting from buoyancy and gravity, in
the series of publications [9], [10] and [11].
Supported by the research presented in the aforementioned references, differential geometry
provides the framework and structure necessary to consider an agile AUV capable of moving
in all six degrees-of-freedom (DOF). This a priori consideration of path planning can outweigh
the computational 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 accommodate under-actuated scenarios, such as thruster
failure for a fully-actuated vehicle, or standard path planning for an under-actuated torpedo-
shaped vehicle. A complete theoretical analysis of our proposed approach with experimental
results has been thoroughly exposed in [11]. In this paper, we present the results of several
experiments conducted on a test-bed AUV; the Omni-Directional Intelligent Navigator (ODIN),
which is owned and maintained by the Autonomous Systems Laboratory, College of Engineering,
University of Hawai‘i.
1A simple mechanical control system is one that has a Lagrangian expressing the energy of the system as potential minus
kinetic.
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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 nonlinearities 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 [12] and [13].
We continue our presentation in Section II by developing the equations of motion for a
rigid body submerged in a viscous fluid in both the traditional manner as well as in the
language of differential geometry. These geometric equations are not a new formulation of the
standard equations, but simply a translation and slight abstraction into the differential geometric
architecture. In Section II-A we provide the technical specifications and physical characteristics of
ODIN, the test-bed vehicle used for our experiments. In Section III, 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 implementable controls for ODIN.
Three scenarios for surveying the bulbous bow are motivated and described in Section IV. 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.
II. EQUATIONS OF MOTION
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 
I
: (O
I
; fs
1
; s
2
; s
3
g), shown in Fig. 2, is a right-
handed, orthogonal coordinate system defined with the s
1
and s
2
axes lying in the horizontal
February 18, 2010 DRAFT

6 DRAFT
plane perpendicular to the direction of gravity, while the s
3
axis is orthogonal to the s
1
s
2
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 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.
O
I
s
1
s
2
s
3
O
B
B
1
B
2
B
3
b
Fig. 2. Earth-fixed and body-fixed coordinate reference
frames.
The body-fixed frame 
B
: (O
B
; fB
1
; B
2
; B
3
g),
shown in Fig. 2, is a right-handed, orthogonal
reference frame defined with the origin O
B
located
at a chosen location, and the body axes B
1
; B
2
and B
3
coinciding with the principle axes of
inertia. The the longitudinal (B
1
) and transverse
(B
2
) axes are taken positive to the fore and star-
board, respectively. The configuration of a rigid
body in six DOF can be described using  =
(x; y; z; ; ; )
t
= (b;2)t, where 2 = (; ; )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 2 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 R
2
such that _
2
= R
2

, we can state the formulation of the kinematic equations
of motion for a rigid body moving in six DOF as
_ =
2
6
4
R 0
33
0
33 R2
3
7
5
2
6
4



3
7
5 : (1)
Equivalently, formulating this system on the differentiable manifold Q, Eqs. (1) are written
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SMITH et al.: BULBOUS BOW SURVEY USING GEOMETRIC CONTROL 7
as the forward kinematic map  : Q! SE(3), with
_b = R ; (2)
_
R = R
^
: (3)
In Eq. (3), the operator ^ : R3 ! so(3) is defined by ^y z = y  z. The space so(3) is the Lie
algebra associated to the Lie group SO(3), and is the space of skew-symmetric 3 3 matrices
(i.e., so(3) = fR 2 R33jRt = Rg).
From standard references on rigid body dynamics (e.g., [14], [15], [16] and [17]), we have
that the equations of motion for a submerged rigid body in an ideal fluid are given by
M
B
_v Cor
B
(v)v = (t); (4)
where M
B
and Cor
B
respectively represent the rigid body inertia and the Coriolis and centripetal
force matrices, and (t) represents the external control forces. The external controls are the forces
and moments applied by the actuators of the vehicle and can be written as  = ('

; 


)
t, where
'

= (X; Y; Z)
t and 


= (K;M;N)
t and we adopt the standard SNAME notation for the forces
(X; Y; Z) and moments (K;M;N), [18].
To equivalently state Eqs. (4) using the geometric representation, we begin by expressing the
translational (T
trans
(t)) and rotational (T
rot
(t)) portions of the body kinetic energy as
T
trans
(t) =
1
2
mk _b(t)k2R3 ; Trot(t) =
1
2
J
b
k
(t)k2R3 ; (5)
where m is the mass of the vehicle and J
b
is its inertia. Now, let
: R+ ! Q be a differentiable
curve at q
0
2 Q. By use of the forward kinematic map  : Q! SE(3) we induce a differentiable
curve
1
=  
: R+ ! SE(3) at (b
0
; R
0
) , (q
0
). If we assign a nonnegative number
KE(v
0
) to the tangent vector v
0
=
0
q0
2 T
q0Q (Tq0Q is the tangent space to the manifold Q at
q
0
), we define the kinetic energy of the rigid body at time zero along the curve
1
. Repeating
this process for every tangent vector v and every point q of the manifold Q, we generate the
function KE : TQ ! R, which defines the kinetic energy. Here TQ is the union of all the
tangent spaces, and is referred to as the tangent bundle. It is shown in [8] that there exists a
C
1, positive-semidefinite tensor field G such that KE(v
q
) =
1
2
G(v
q
;v
q
), which is analogous
to the definition of the kinetic energy in Q = SE(3). This G is the inner product that we will
February 18, 2010 DRAFT

8 DRAFT
use in our equations.
Thus, the kinetic energy of a rigid body in an interconnected-mechanical system is represented
by a tensor field on the configuration manifold Q. We refer to this object as the kinetic energy
metric for the system. In a similar fashion, we can construct the kinetic energy for the fluid as
another tensor field. The sum of the later and G defines the total kinetic energy for the submerged
rigid body.
To simplify both the standard and the geometric representations of the equations of motion,
we make two non-limiting assumptions. First, we choose O
B
to coincide with the center of mass
of the AUV, and secondly, the axes of the body-fixed frame to correspond with the principle
axes of inertia of the vehicle. These assumptions lead to M
B
being a diagonal matrix.
Since the AUV is submerged in a viscous fluid we must introduce terms to account for the
added mass, viscous damping and restoring forces. The added mass is a pressure-induced force
due to the inertia of the surrounding fluid and is proportional to the acceleration of the rigid
body. At low speed and assuming three planes of symmetry, as is common for most AUVs, the
added mass matrix can be assumed to be diagonal, M
a
= diag(M
f
; J
f
).
Now, we have that the kinetic energy metric for the submerged rigid body is the unique
Riemannian metric on Q = SE(3) given by:
G =
0
B
@
M 0
0 J
1
C
A ; (6)
where M = mI
3
+ M
f
and J = J
b
+ J
f
. In the sequel, we will use m
i
= m + M
i
f
and
j
i
= J

i
b
+ J

i
f
, for i = 1; 2; 3. Thus, M = diag(m
1
;m
2
;m
3
) and J = diag(j
1
; j
2
; j
3
).
As with any Riemannian metric, associated to G is its Levi-Civita affine connection: the unique
affine connection2 that is both symmetric and metric compatible. The Levi-Civita connection (see
e.g., [19]) provides the appropriate notion of acceleration for a curve in the configuration space
by guaranteeing that the acceleration is in fact a tangent vector field along a curve
. The
connection also accounts for the Coriolis and centripetal forces acting on the system. Explicitly,
if
(t) = (b(t); R(t)) is a curve in SE(3), and
0(t) = ((t);
(t)) is its pseudo-velocity as
2An affine connection transports tangent vectors to a manifold from one point to another along a curve.
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SMITH et al.: BULBOUS BOW SURVEY USING GEOMETRIC CONTROL 9
given in Eqs. (2) and (3), the accelerations are given by
r

0

0
=
0
B
@
_ +M1


M

_
+ J1


 J
+  M

1
C
A ; (7)
where r denotes the Levi-Civita connection and r

0

0 is the covariant derivative of
0 with
respect to itself3. We refer to r

0

0 as the geometric acceleration with respect to r and note
that Eq. (7) is Newton’s Second Law expressed geometrically as a =
P
i
F
i
=m.
To conclude this overview of the rigid body dynamics, note that Eq. (4) is equivalent to
r

0

0
=
6X
i=1
I1
i
(
(t))
i
(t); (8)
where the input control vector fields are the i-th column of the matrix
I1 =
0
B
@
M
1
0
0 J
1
1
C
A : (9)
With no external forces r

0

0
= 0, which represent the geodesics for the affine connection r.
Remark 1. It is important to note that the geometric acceleration is an effective way to consider
acceleration in a general sense, as it is invariant under change of coordinates.
Viscous damping and dissipation encountered by marine vehicles are caused by many factors,
including radiation-induced potential damping from forced body oscillations in the presence of
a free surface, linear and quadratic skin friction, wave drift damping, and vortex shedding. For
a small, slow-moving, fully-submerged AUV which is far from the free surface, pressure drag
(form drag) is dominant; this assumption is further validated based on the speed and shape
of the test-bed vehicle considered here. Since we also assume that the AUV has three planes
of symmetry, the hydrodynamic drag matrix is assumed diagonal and is given by D(v) =
diag(D
1

1
; D
2

2
; D
3

3
; D
4


1
; D
5


2
; D
6


3
). We assume that each of the viscous drag terms
represented by D(v)v to be quadratic with respect to the vehicle’s velocity. Restoring forces and
moments result from the effects of buoyancy and gravity upon the vehicle, and are represented
by g().
3Here we remind the reader that r is not the submerged volume of fluid displaced by a rigid body, but denotes an affine
connection on Q.
February 18, 2010 DRAFT

10 DRAFT
Incorporating the added mass terms and the viscous drag and restoring forces, we can extend
Eq. (4) to
0
B
@
mI
3
+M
f
0
33
0
33 Jb + Jf
1
C
A
0
B
@
_
_

1
C
AD(v)v Cor
B
(v)v + g() = (t): (10)
If we rewrite these equations in the standard Newton-Euler notation (F = ma) and separate the
translational and rotational motion components, we can express Eqs. (10) as
M
_ =
M +D() g(b) +'

; (11)
J
_
=
 J
 M +D


(
)
g(
2
) + 


; (12)
where M 
and J

account for the Coriolis and centripetal forces.
Following this Newton-Euler formulation of the equations of motion, we can extend Eq. (8)
and define the equations of motion for an underwater vehicle submerged in a viscous fluid in
the framework of differential geometry using the Levi-Civita affine connection.
Lemma 1. Let Q = SE(3),r be the Levi-Civita connection on Q associated with the Riemannian
metric G and let the set of input control vector fields be given by I = fI1
1
; :::; I1
6
g. Let
G#(F
drag
(
0
(t))) represent the dissipative forces resulting from hydrodynamic drag (G# is
the inverse of G, and is a tangent bundle isomorphism physically meaning divide by mass).
Let G#(P (
(t))) represent the restoring forces arising from gravity and buoyancy. Then the
equations of motion of a rigid body submerged in a viscous fluid and subjected to dissipative
and restoring forces are given by the forced affine connection control system:
r

0

0
= G#(P (
(t))) + G#(F
drag
(
0
(t))) +
6X
i=1
I1
i
(
(t))
i
(t); (13)
where 
i
(t) represents the controls.
This concludes the general overview on the geometric control framework that will be used to
generate the trajectories and control strategies for our test-bed vehicle. This overview, given with
no intention of being comprehensive, has been presented to give the reader a notion regarding
the concepts and tools that support the geometric control theory for AUVs. The focus of this
paper remains upon the experimental results obtained through the practical implementation of
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Mass 123:8 kg B = gV 1215:8 N C
B
(0; 0;7) mm
Diameter 0:64 m W = mg 1214:5 N C
G
(0; 0; 0) mm
M
u
f
70 kg M v
f
70 kg Mw
f
70 kg
I
xx
5:46 kg m2 I
yy
5:29 kg m2 I
zz
5:72 kg m2
J
p
f
0 kg m2 Jq
f
0 kg m2 Jr
f
0 kg m2
TABLE I
MAIN DIMENSIONS AND HYDRODYNAMIC PARAMETERS FOR ODIN.
control strategies obtained by use of this geometric architecture. For a proof of Lemma 1 and
an thourough analysis of geometric control theory applied to underwater vehicles, we refer the
interested reader to [11].
A. 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, see Fig. 3. Complete
details and technical specifications for this vehicle can be found in [20] or [21], with specifics
related to implementation of geometric control strategies contained in [11].
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 I. These
values were derived from estimations and full-scale model tests performed on ODIN.
Fig. 3. ODIN operating in the pool.
The added mass terms (Mu
f
;M
v
f
;M
w
f
; J
p
f
; J
q
f
; J
r
f
) were
estimated from formulas found in [22] and [23]. Moments
of inertia (I
xx
; I
yy
; I
zz
) were calculated using experiments
outlined in [24]. We used inclining experiments to locate
C
G
, which we take as the center of our body-fixed reference
frame (i.e., C
G
= O
B
). Based on the symmetry of the
vehicle, the center of buoyancy C
B
, is assumed to be the
center of the spherical body of ODIN. The location of C
B
is measured from C
G
= O
B
, and is given in Table I.
Eight Tecnadyne brushless thrusters are attached to the sphere via four fabricated mounts,
each holding two thrusters. These thrusters are evenly distributed around the sphere with four
February 18, 2010 DRAFT

12 DRAFT
oriented vertically and four oriented horizontally. This design provides instantaneous and un-
biased motion in all six DOF, contrary to the more common torpedo-shaped vehicles. Unique
to ODIN’s construction is the control from an eight-dimensional thrust to move in six DOF.
Hence, ODIN operates is an over-actuated condition; redundancy was incorporated in the design
to account for thruster failure or other operational errors. To calculate the six-dimensional thrust
 resulting from the eight-dimensional thrust  (from the thrusters), or vice-versa, we apply a
linear transformation to . We omit the details of this transformation here, but refer the interested
reader to [20]. Along with the tests to determine the values in Table I, we also tested the thrusters.
Each thruster has a unique voltage input to power output relationship. This relationship is highly
nonlinear and is approximated using a piecewise linear function which we refer to as our thruster
model. More information regarding the thruster modeling can be found in [11].
Major internal components include a pressure sensor, inertial measurement 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, yaw, pitch, roll, and depth, 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 (xy) 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.
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III. CONTROL STRATEGY DESIGN
As previously mentioned, the aim of this paper 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, i.e., the bulbous bow. To this end, we are interested in calculating paths that
the AUV can execute given it’s 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 [8] in a
very general manner and in [10] for application to AUVs. The detailed process of adapting the
computed controls for implementation onto the considered test-bed vehicle is described in [11].
To summarize this procedure, we begin by first applying a geometric reduction procedure to the
dynamic system (acceleration control inputs) described by Eqns. (13) 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 [8], 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 differen-
tial 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
February 18, 2010 DRAFT

14 DRAFT
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 [8] and the extension of this result presented in [11].
We continue by briefly outlining the procedure of motion planning via decoupled kinematic
motions. First, we define the initial (
init
) and final (
final
) 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.
This trajectory design process is based on what is commonly known in control literature
as motion planning by use of primitives. This involves the concatenation of several calculated
primitives to create a realizable path connecting 
init
and 
final
. The time-parameterized, con-
catenated path then defines the trajectory from 
init
to 
final
, Determining whether or not the
final configuration is reachable by use of only the kinematic motions defined by the decoupling
vector fields for the given system is non-trivial. We refer the reader to [10] for a complete
characterization of decoupled vector fields, and the corresponding controllability of the system.
After solving the motion planning problem by determining the sequence of integral curves, or
primitives, to follow to get from 
init
to 
final
, we parameterize each segment to start and end
at zero velocity. This parameterization ensures that each concatenated segment begins with the
same initial conditions, and thus guarantees that the entire motion is executable by the vehicle.
From this reparameterized, concatenated, kinematic motion trajectory, we calculate the dynamic
controls that steer the vehicle from 
init
to 
final
. With this heuristic blueprint in mind, we now
present the details of the construction.
Suppose that we have a C1 affine connection control system 
dyn
= (Q;r; I1;Rm), where
m is defined by the number of available DOF. Let the system 
dyn
be kinematically controllable,
in other words, the system can reach any arbitrary final configuration (with zero velocity) from
any initial configuration (with zero velocity) by use of kinematic motions. First, calculate the
decoupling vector fields for the drift-less kinematic system 
kin
= (Q;V = fV
1
; :::; V
m
g;Rm).
This is a straightforward calculation for a given system by following the general outline in [8].
Based on various actuation scenarios for ODIN, a complete characterization of its decoupling
vector fields is presented in [10].
Let FVi
t
represent the flow along the integral curve of the vector field V
i
for a duration [0; t].
Set the initial configuration as 
init
and the final configuration as 
final
. Next, solve the kinematic
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SMITH et al.: BULBOUS BOW SURVEY USING GEOMETRIC CONTROL 15
motion planning problem by concatenating the flows of the decoupling vector fields from 
init
to 
final
by finding k 2 N, a
1
; :::; a
k
2 f1; :::;mg, t
1
; :::; t
k
2 R+ and 
1
; :::; 
k
2 f1;1g such
that
F
kVak
tk



 F
1Va1
t1
(
init
) = 
final
: (14)
For each j 2 f1; :::; kg, choose a C2-reparameterization  : [0; t
j
] ! [0; t
j
] such that  0(0) =

0
(

t
j
) = 0 so that the kinematic motion begins and ends at rest (zero velocity). Define
:
[0; t
j
] ! Q as the integral curve t 7! F
jVaj
t
. Then the dynamic control  : [0; t
j
] ! Rm is
defined by
mX
=1


(t)I1

(
 (t)) = ( 0(t))2r
Vaj
V
aj(
 (t)) + 
00
(t)V
aj(
 (t)): (15)
If we let u : [0; T ]! Rm be the control formed by the concatenation of 
1
; :::; 
k
, then (
; u)
is a controlled trajectory for 
dyn
. This general method of calculating the controls for a dynamic
system following decoupling vector fields was adapted from [8] for our specific application.
Note that for the experiments presented in Section IV, we considered ODIN to be fully-
actuated. Therefore every kinematic motion can be generated as dynamic motion using Eq. (15).
Remark 2. Inherent to this trajectory design technique, the speed along each concatenated
integral curve can arbitrarily be chosen, thus the parameterization depends upon the physical
limits of the thrusters or actuators of the given vehicle. Additionally, with different choices of
(t), to some extent, we can control the time and energy efficiency of the vehicle over the
duration of a given path. However, if the calculated trajectory requires the concatenation of
two or more integral curves, we can never achieve time optimality between the initial and final
configurations. This is a direct result of the trajectory segments being concatenated through
states of zero velocity; obviously such a strategy can never be time optimal.
Unfortunately, in their current form, the calculated controls cannot be directly implemented
onto ODIN. This however, does not imply that these strategies cannot be implemented directly
onto an alternative test-bed vehicle. The geometric control theory technique generates continuous
controls as a function of time, whereas ODIN’s input requires a piece-wise constant control
structure over discretized time intervals. The reason for this type of input is based on the
combination of the refresh rate of the controller and the voltage to thrust relation used for
February 18, 2010 DRAFT

16 DRAFT
ODIN’s thrusters. Also, it is important to keep the thrusters operating in a steady state to reduce
their transient output response from constant changes in input voltage. In addition to this 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.
Example 1. Consider the simple motion of a five meter pure surge relative to the body-fixed
reference frame. If we assume that 
init
= (0; 0; 0; 0; 0; 0), then 
final
= (5; 0; 0; 0; 0; 0). Since
ODIN is assumed to be fully-actuated, we have control over all six DOF. This implies that we
have six input control vector fields I1
6
= fI1
1
; I1
2
; I1
3
; I1
4
; I1
5
; I1
6
g, and every vector field is a
decoupling vector field. Hence, the decoupling vector fields for this system are the constant mul-
tiples and linear combinations of the set D = fX
1
= (1; 0; 0; 0; 0; 0); X
2
= (0; 1; 0; 0; 0; 0); X
3
=
(0; 0; 1; 0; 0; 0); X
4
= (0; 0; 0; 1; 0; 0); X
5
= (0; 0; 0; 0; 1; 0); X
6
= (0; 0; 0; 0; 0; 1)g.
To achieve the desired motion, we need only consider integral curves of the decoupling vector
field X
1
. Here, there is only one integral curve to follow, and FX1
5
(
init
) = 
final
. Thus, the
trajectory
(t) : [0; 5] ! Q is simply the straight line from (0; 0; 0; 0; 0; 0) to (5; 0; 0; 0; 0; 0).
For this example, we choose to reparameterize
(t) using (t) : [0; 30] ! [0; 5] so that the
motion lasts for 30 seconds. This reparameterization gives an average velocity for the trajectory
of 0:167 m/s. From the data and analysis presented in [11], the drag coefficient applicable for
this motion is D
1
= 231 kg/m. Hence, we can calculate the dynamic control structure by use of
Eq. (15) as

1
(t)I1
1
(
 (t)) = ( 0(t))2fr
X1X1(
 (t)) + 
00
(t)X
1
(
 (t)); (16)
where (t) = t
2
(45t)
2700
, which ensures that the vehicle will begin and end the motion with zero
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SMITH et al.: BULBOUS BOW SURVEY USING GEOMETRIC CONTROL 17
velocity. Computing the covariant derivative, Eq. (16) becomes

1
(t) = D
1
(
0
(t))
2
+m
1

00
(t)
= 231

t(30 t)
900
!
2
+ 196

15 t
450

=
7(11t
4 660t3 + 9900t2 16800t+ 252000)
270000
:
(17)
Thus, the control strategy calculated by use of our geometric methods for realizing a pure
surge displacement of five meters for ODIN is given by 1(t) = 7(11t
4660t3+9900t216800t+252000)
270000
.
This control has a positive portion to accelerate the vehicle from rest and realize the motion
and a negative section to slow the vehicle to a complete stop in the prescribed location. The
value of 1(t) is zero at t = 0; 25:05 and 30 seconds. For implementation onto ODIN, we need
to convert 1(t) into a PWC control. Consider,
s
1
=
1
25:05
Z
25:05
0

1
(t)dt; s
2
=
1
4:95
Z
30
25:05

1
(t)dt: (18)
Calculating, we get s
1
= 9:98 N and s
2
= 3:84 N. Thus, we can write the PWC control as
s
1
(t) =
8
>><
>>:
9:98 N t 2 [0; 25:05]
3:84 N t 2 [25:05; 30]
: (19)
The final step to implement this onto ODIN is to simply connect these two constant controls via
a linear junction lasting 0:9 seconds to avoid instantaneous switching of the thrust command
sent to the thrusters. The duration of this junction is based upon the refresh rate of ODIN’s CPU
and the avoidance of large voltage changes which may damage ODIN’s thrusters (see [20] for
detailed information).
The resulting discretized control structure which is implemented onto ODIN is listed in Table
II, where the six-dimensional control structure is given by  = (1; 2; :::; 6).
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.
February 18, 2010 DRAFT

18 DRAFT
Time (s) Applied Thrust (6-dim.) (N)
0 (0,0,0,0,0,0)
0.9 (9.98,0,0,0,0,0)
25.95 (9.98,0,0,0,0,0)
26.85 (-3.84,0,0,0,0,0)
31.8 (-3.84,0,0,0,0,0)
32.7 (0,0,0,0,0,0)
TABLE II
DISCRETIZED CONTROL STRUCTURE FOR 5M PURE SURGE.
Algorithm 1 Implementation of Control Strategies Designed using Differential Geometry
1: Apply a kinematic reduction to Eqs. (13).
2: Solve the motion planning problem from 
init
to 
final
via inverse kinematics.
3: Calculate the continuous control strategy (t) for the dynamic system by use of the geometric
theory, i.e., Eq. (15).
4: Discretize (t) to obtain an implementable piece-wise constant control structure 
0
(t).
5: Upload 
0
(t) to ODIN’s CPU.
6: while ODIN autonomously implements 
0
(t) in full open-loop. do
7: On-board, ODIN converts 
0
(t) from a 6-dimensional control to an 8-dimensional control.
8: The 8-dimensional control is converted from force (N) to voltage (V).
9: The voltage controls are send to the eight independent thrusters.
10: Position and orientation data are collected.
11: end while
12: Collected data are post-processed and analyzed
IV. EXPERIMENTS
In the sections to follow, we present experimental results for three geometrically 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 4 displays a
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SMITH et al.: BULBOUS BOW SURVEY USING GEOMETRIC CONTROL 19
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 imple-
mented 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.
To survey the uniquely-shaped bulbous bow, we propose two separate missions. The first is
a semi-circular trajectory as depicted in Fig. 5. 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. 8, or to perform repeated transects up and down the longitudinal axis of the
bulb, as shown in Fig. 6. 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. The third proposed strategy is the concatenation of the
previous two, as seen Fig. 8. We now continue by presenting the computed controls and the
implementation results for these missions.
13 m
10 m
10 m 2.5 m
LWL
Fig. 4. Side view of a bulbous bow on a ship and its
dimension.
February 18, 2010 DRAFT

20 DRAFT
Fig. 5. Side view of a bulbous bow on a ship. Also depicted
is the semi-circle trajectory for the inspection of the front of
a bulbous bow.
Fig. 6. Front view of a bulbous bow. Also depicted is the
trajectory to survey the sides of the bulb.
For all of the experiments, the initial configuration is taken to be 
init
= (0; 0; 1:5; 0; 0; 0), with
the origin of the earth-fixed frame positioned on the free surface. In particular, 
init
is located
at the origin of the earth-fixed fram, 1:5 m below the water’s surface. All control strategies
presented here are designed such that the vehicle begins at 
init
with zero velocity, and ends at

final
with zero velocity. The presented experiments were perofrmed in the diving well a 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. 4. We scale
the height of the bulb from 10 m to 2:5 m, which implies that the 2:5 m radius of the hemisphere
scales to approximately 0:5 m. For the motion parallel to the free surface, we scale the 10 m
length of the bulb to 5 m. As the aim of this paper 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. The calculations are similar to those carried out in Example 1, and the interested reader
is directed to [11] for detailed calculations of multiple control strategies.
A. Strategy One: Semi-circle
The first strategy we wish to construct is the semi-circle trajectory shown in Fig. 5, to inspect
the front and the sides of the bulbous bow. This motion is constructed by simultaneously applying
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SMITH et al.: BULBOUS BOW SURVEY USING GEOMETRIC CONTROL 21
Fig. 7. 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.
Fig. 8. Overall path for complete inspection of a bulbous
bow.
controls in both pure heave and pure surge. The pure heave control is designed so that the vehicle
realizes a net 2:5 m pure heave. The surge control is designed such that the vehicle begins at
rest, realizes a negative pure surge of 0:5 m, then moves 0:5 m in the positive surge direction
to culminate with zero net displacement in pure surge. The final configuration for the vehicle
is 
f
= (0; 0; 4; 0; 0; 0) meters. We parameterize this motion to begin and end at rest, and based
on the operational velocity of ODIN, the duration of the motion is 10:7 seconds. We present the
computed PWC control strategy in Table III as a six-dimensional control, corresponding to the
six DOF in the body-fixed reference frame.
Time (s) Applied Thrust (6-dim.) (N) Time (s) Applied Thrust (6-dim.) (N)
0 (0,0,0,0,0,0) 6.5 (32.7, 0, 30.04, 0, 0, 0)
0.9 (-32.7, 0, 30.04, 0, 0, 0) 7.7 (32.7, 0, -24.56, 0, 0, 0)
2.8 (-32.7, 0, 30.04, 0, 0, 0) 8.6 (-30.99, 0, -24.56, 0, 0, 0)
3.7 (30.99, 0, 30.04, 0, 0, 0) 9.8 (-30.99, 0, -24.56, 0, 0, 0)
4.9 (30.99, 0, 30.04, 0, 0, 0) 10.7 (0,0,0,0,0,0)
5.8 (32.7, 0, 30.04, 0, 0, 0)
TABLE III
PIECE-WISE CONSTANT CONTROL STRATEGY TO PERFORM A SEMI-CIRCLE TRAJECTORY.
Figure 9 displays the implementation results of the control strategy given in Table III. The
February 18, 2010 DRAFT

22 DRAFT
0 2 4 6 8 10−40
−20
0
20
40
X (N)
0 2 4 6 8 10−1
−0.5
0
0.5
1
Y (N)
0 2 4 6 8 10−40
−20
0
20
40
Z (N)
Time (s)
0 2 4 6 8 10−1
−0.5
0
0.5
1
x (m)
0 2 4 6 8 10−1
−0.5
0
0.5
1
y (m)
0 2 4 6 8 101
2
3
4
5
z (m)
Time (s)
0 2 4 6 8 10−40
−20
0
20
40
φ
0 2 4 6 8 10−40
−20
0
20
40
θ
0 2 4 6 8 10−40
−20
0
20
40
ψ
Time (s)
Fig. 9. Strategy One: Semi-circle trajectory. Solid (blue) line represents actual evolution, dash-dot (red) line represents the
theoretical evolution.
first column of plots in Fig. 9 give the control forces (in Newtons) that were applied by ODIN
during the implementation. In the second and third columns, we display the evolution of all six
DOF of the vehicle during the test. The solid (blue) line denotes the actual evolution of ODIN.
The dash-dot (red) line represents the theoretical evolution of ODIN. The theoretical evolution
represents the trajectory that ODIN was prescribed to follow based on the concatenated integral
curves that were chosen to solve the motion planning problem.
First, we examine the controls applied during the experiment. Note that for the surge control
X , the magnitude of the control does not quite match the values given in Table III. This is a
result of implementing a six-dimensional control strategy onto a vehicle that is driven by eight
thrusters. The linear transformation applied to convert from six dimensions to eight dimensions,
and vise-versa, has a nonzero null space. This means that there are infinitely many transformations
which convert the controls. ODIN’s on-board computer choses one of these transformations for its
computations. More information regarding this transformation can be found in [11], with specific
DRAFT February 18, 2010

SMITH et al.: BULBOUS BOW SURVEY USING GEOMETRIC CONTROL 23
details related to ODIN found in [25]. We remark that ODIN performs this transformation twice
during the implementation. First, ODIN computes the 8-dimensional thrust from the prescribed
six-dimensional controls. After implementation, ODIN transforms the actual applied controls
from the eight thrusters into a six-dimensional output control for post-processing.
Next, consider the evolution of ODIN during the experiment. The main intent of this strategy
was to realize both heave and surge motions. For the surge motion, the experimental results match
well with the theoretical trajectory. We see deviation between the actual and theoretical evolutions
begin around t = 4 seconds. This occurs because ODIN did not reach the full 0:5 m displacement.
The actual evolution is seen to be just out of phase of the theoretical prediction. This is probably a
result of a small error in the drag coefficient calculated for ODIN. Overall, the surge motion was
executed very well. Similarly for the heave motion, we see an excellent correspondence between
theoretical predictions and experimental results. The sway motion displays a slight deviation,
which is a result of an initial yaw angle offset at the beginning of the experiment. Such an offset
cannot be corrected since we are operating in open-loop. This type of implementation coupled
with potential transient thruster response results in the discrepancies seen in the plots for the
Euler angles. Such discrepancies are minimal and expected. Research is active in the design
of a robust feedback controller for AUVs to track a given trajectory in the presence of initial
disturbances such as yaw offsets and disturbance inputs. Initial results in this area can be found
in [12] and [13]. It is an area of future work to implement such a controller onto ODIN for
trajectory tracking experiments.
B. Strategy Two: Horizontal Survey
The second strategy we implemented is a common control strategy for a seabed survey for
applications such as coral reef monitoring. It is simply composed of a pitch angle of  = 20
and a five meter surge while maintaining this pitch angle and constant depth.4
This control strategy provides the ability to survey the top of the bulb, while choosing  = 20
and moving in the positive surge direction will allow for survey of the bottom of the bulb. The
basic idea for this control strategy is to pitch the vehicle so that the forward-looking camera (or
sensor) points downward, then apply a pure surge control (relative to the earth-fixed reference
4The pitch angle of 20 is chosen based on the physical limitations of the test-bed vehicle ODIN.
February 18, 2010 DRAFT

24 DRAFT
0 10 20 30 40−20
−15
−10
−5
0
5
10
X (N)
0 10 20 30 40−1
−0.5
0
0.5
1
Y (N)
0 10 20 30 40−5
0
5
M (N
m)
Time (s)
0 10 20 30 40−6
−5
−4
−3
−2
−1
0
1
x (m)
0 10 20 30 40−4
−2
0
2
4
y (m)
0 10 20 30 400
0.5
1
1.5
2
2.5
3
z (m)
Time (s)
0 10 20 30 40−40
−20
0
20
40
φ
0 10 20 30 40−40
−20
0
20
40
θ
0 10 20 30 40−40
−20
0
20
40
ψ
Time (s)
Fig. 10. Strategy Two: Horizontal survey. Solid (blue) line represents actual evolution, dash-dot (red) line represents the
theoretical evolution.
frame) while maintaining the desired pitch angle to survey along top or bottom of the bulb. In
this case, we prescribe a 5 m pure surge and a final configuration for the vehicle of 
f
=
(5; 0; 1:5; 0;20; 0). The duration of this motion is 38:6 seconds. The six-dimensional PWC
control strategy is given in Table IV.
Time (s) Applied Thrust (6-dim.) (N) Time (s) Applied Thrust (6-dim.) (N)
0 (0,0,0,0,0,0) 31.9 (-10.7, 0, 7.6, 0, -2.91, 0)
0.9 (0.45, 0, 1.2, 0, -2.91, 0) 32.8 (4.6, 0, 2.02, 0, -2.91, 0)
5.9 (0.45, 0, 1.2, 0, -2.91, 0) 37.7 (4.6, 0, 2.02, 0, -2.91, 0)
6.8 (-10.7, 0, 7.6, 0, -2.91, 0) 38.6 (0,0,0,0,0,0)
TABLE IV
PIECE-WISE CONSTANT CONTROL STRATEGY TO SURVEY THE TOP PORTION OF THE BULBOUS BOW.
For this implementation, we had issues with yaw stabilization throughout implementation of
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SMITH et al.: BULBOUS BOW SURVEY USING GEOMETRIC CONTROL 25
the control strategies and frequently had large deviations at the culmination of the mission. To
compensate for this, we employed a feedback controller for only the yaw control. ODIN has
a Proportional-Derivative (PD) controller on-board, which can be activated to provide feedback
in depth, roll, pitch and/or yaw. ODIN does not have an on-board compass, so yaw angles, and
associated errors, are measured relative to the yaw angle at the initialization of the motion, which
is assumed to be zero. This initial angle is what the feedback controller is working to maintain.
In Fig. 10, we see a large deviation in sway (y) due to the feedback controller maintaining
an non-zero yaw angle throughout implementation the experiment. Note that during the first
 8 seconds, the vehicle should be performing a pure pitch, however we notice artifacts in
the roll evolution, also caused by this initial yaw angle offset. The deviation in sway begins
to appear when the vehicle executes the translational portion of the implementation. Since we
supply the controls to realize a given motion in open-loop, and ODIN has no on-board sensors
for horizontal positioning, running a feedback control loop on yaw alone will not eliminate
deviation in sway caused from such an initialization error. Additionally, since the horizontal
positioning is decoupled from the actual vehicle, they may have sightly different frames of
reference based on the assumed initial yaw angle. In particular, a translational motion relative
to the vehicle may differ from a translational motion relative to the camera atop the diving
platform. Additional results for this implementation are presented in Fig. 10, and our discussion
continues with considereing the evolution of other parameters.
The initial seven seconds of this strategy is devoted to stabilize the pitch angle to the prescribed
20. During this time we see that the x; y and z evolutions remain stable with some disturbance
seen in . The pitch angle did not quite stabilize during the initial seven seconds, but it quickly
levels out from about t = 12 seconds onward. Note that this pitch angle is slightly more than the
20 prescribed. This excess pitch attributes to the slight rise to the surface seen in the depth
evolution. Regardless, the actual depth evolution remains around the prescribed 1:5 m for the
duration of the trajectory. For the surge evolution, we see that ODIN approximately realized the
predicted 5 m displacement. Error here is again due to an error in the estimation of the drag
coefficients for the vehicle. Overall, the implementation of this control strategy matched well
with the desired trajectory to be performed.
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26 DRAFT
C. Strategy Three: Concatenated Motion
Here, we combine the control strategies presented in Sections IV-A and IV-B into a single
implementable trajectory. Since both of the previous strategies were designed to begin and end
at rest, we can simply concatenate them together. The final configuration for this motion is

f
= (5; 0; 4; 0; 0; 0), which is realized over a duration of 49:3 seconds. The six-dimensional
PWC control strategy is given in Table V.
Time (s) Applied Thrust (6-dim.) (N) Time (s) Applied Thrust (6-dim.) (N)
0 (0,0,0,0,0,0) 42.3 (30.99, 0, 30.04, 0, 0, 0)
0.9 (0.45, 0, 1.2, 0, -2.91, 0) 43.5 (30.99, 0, 30.04, 0, 0, 0)
5.9 (0.45, 0, 1.2, 0, -2.91, 0) 44.4 (32.7, 0, 30.04, 0, 0, 0)
6.8 (-10.7, 0, 7.6, 0, -2.91, 0) 45.1 (32.7, 0, 30.04, 0, 0, 0)
31.9 (-10.7, 0, 7.6, 0, -2.91, 0) 46 (32.7, 0, -24.56, 0, 0, 0)
32.8 (4.56, 0, 5.02, 0, -2.91, 0) 46.3 (32.7, 0, -24.56, 0, 0, 0)
37.7 (4.56, 0, 5.02, 0, -2.91, 0) 47.2 (-30.99, 0, -24.56, 0, 0, 0)
38.6 (-32.7, 0, 30.04, 0, 0, 0) 48.4 (-30.99, 0, -24.56, 0, 0, 0)
41.4 (-32.7, 0, 30.04, 0, 0, 0) 49.3 (0,0,0,0,0,0)
TABLE V
PIECE-WISE CONSTANT CONTROL STRATEGY FOR THE CONCATENATION OF THE TWO STRATEGIES PRESENTED IN
SECTIONS IV-A AND IV-B.
The general idea for this strategy is that the vehicle begins above the bulb and close to the
bow of the ship, it pitches to point the camera downward, then traverses the length of the bulb
parallel to the free surface. Upon reaching the end of the bulb, the AUV performs the semi-circle
motion to examine the front portion of the bulb. For this concatenated strategy, we choose not
to apply controls at the end of the initial segment to undo the prescribed pitch angle. This is
done in an attempt to create the effect seen in Fig. 8, where the camera points nearly normal to
the surface of the bulb for the first half of the semi-circle trajectory.
Note that the concatenation of two control strategies can give different results from those
obtained by implementing each individually. This is directly related to the ability to stabilize
the vehicle at the junction connecting the two strategies. Based upon the design of the control
strategies, each segment of the concatenation should begin and end with zero velocity, thus
making the concatenation feasible. However, this does not always occur in practice. Additionally,
any errors present in the implementation of the initial leg of the concatenated motion are
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SMITH et al.: BULBOUS BOW SURVEY USING GEOMETRIC CONTROL 27
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)
Fig. 11. Strategy Three: Concatenation of the semi-circle trajectory and the horizontal survey. Solid (blue) line represents
actual evolution, dash-dot (red) line represents the theoretical evolution.
exaggerated through execution of subsequent portions since the initialization point of the later
motions is not in the prescribed location. In an effort to perform concatenated strategies well,
we again activated a feedback controller for the yaw control for this experiment.
The initial 40 seconds of the implementation displayed in Fig. 11 is the strategy presented
in Section IV-B, while the remainder of the experiment is the semi-circle strategy presented in
Section IV-A. During the initial segment of the trajectory, we see similar results to those described
in Section IV-B. The depth remains fairly constant at 1:5 m, the AUV realizes approximately 5
m in surge and the pitch angle is just less than the prescribed 20. We also observe a sway
deviation from an initial yaw offset.
Examining the remaining 20 seconds of the implemented strategy, we see behavior similar to
that presented in Section IV-A, with the exception that the error from the first segment of the
trajectory is introduced as the initial condition for the second leg of the concatenated motion. We
see the initial negative surge of 0:5 m followed by a positive surge evolution of approximately
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28 DRAFT
0:5 m, as prescribed. The depth evolution shows an overshoot in depth by about 1 m. 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 = 40 s, we again notice a spike in the yaw, which corresponds to a time when
the vehicle is releasing the pitch angle.
V. CONCLUSIONS
In this paper, we have presented the equations of motion governing the submerged rigid body
in both the standard formulation as well as a formulation utilizing the architecture of differential
geometry. By use of these geometric equations, we are able to provide solutions to the motion
planning problem for AUVs via a geometric reduction. This geometric control theory technique
has been proven to be an effective path planning tool for AUVs, especially those operating in
an under-actuated condition, see e.g., [10]. Here, we considered a practical application of this
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 vehilces. 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.
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SMITH et al.: BULBOUS BOW SURVEY USING GEOMETRIC CONTROL 29
The experimental results presented here extend the work developed in [11], 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 experimental results presented in
this paper 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 and experimental results.
A reasonable approach to begin the migration is to use our control strategies as the desired
theoretical predictions, and implement a robust, feedback trajectory-tracking controller that can
compensate for the external disturbances. Initial steps in this direction have been taken, and
results can be found in [12] and [13]. Once the theory contained in these references becomes
well-developed and proven technology, we plan to implement a hybrid control scheme onto
ODIN in the pool. We will begin with simple disturbances, such as initial deviations in the state
of the vehicle. From the discussion presented in Sections IV-A to IV-C, a known source of error
comes from an initial offset in the vehicle’s configuration, typically in yaw. Implementing a
hybrid controller as previously described will require many upgrades to ODIN, or the use of an
alternate AUV for sea trials.
Such extensions present the natural question of applicability of the presented techniques to
multiple types of underwater vehicles. First, the theoretical aspect, namely the geometric control,
is independent of the choice of the vehicle. The geometric theory is solely based on the fact
that the underwater vehicle is an example of a simple mechanical control system; this is true for
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30 DRAFT
any underwater vehicle. Generalizing our work to alternate vehicle designs requires only slight
modifications. If the vehicle has three planes of symmetry, which is common for AUVs, the
basic foundations and formulations do not change. Obviously, the physical attributes, such as
mass, inertia and added mass, need to be altered. This corresponds to the generation of a new
kinetic energy metric for the kinematic reduction. Viscous drag coefficients need to be estimated
for the specific vehicle, and the locations of the center of buoyancy and center of gravity need
to be calculated to appropriately account for the restoration forces and moments. Aside from
the obvious physical properties, the only major difference is changing the input control vector
fields. These are the basis upon which the decoupling vector fields, and hence the kinematic
motions, are determined. This alteration is simply done by expressing the location and output
of the actuators of the vehicle in the geometric formulation.
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