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TRANSACTIONS
ON
SYSTEMS, MAN,
AND
CYBERNETICS, VOL.
20,
NO.
6,
NOVEMBER/DECEMBER
1990
1475
ACKNOWL~DGMENT
Thc authors would like to thank Dr. TsungMing Tsai and
Shujcn Chang for their contributions to this work.
REFERENCES
J.
Albus, “Hierarchical control for robots and teleoperators,” I EEE
Workshop on Intelligent Control, Albany, NY, Aug. 2627,
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39 49.
M.
Daily
et
al.,
“Autonomous crosscountry navigation with the ALV,”
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Gargantini, “An effective way t o represent quadtrees,” Comtnlm.
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A.
Stenz, “The CMU system for mobile robot navigation,”
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1987,
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W. E. L. Crimson and
T.
bzanoPerez, “Modelbased recognition and
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Robotics Re
seurch, 1984, pp. 335.
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Y. Harmon, “The Ground Surveillance Robot (GSR): an Au
tonomous Vehicle Designed t o Transit Unknown Terrain,” IEEE
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Robotics Automation, 1987, pp. 266279.
M. Herman and
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Albus, “Overview of the multiple autonomous
underwater vehicle (MAUV) project,” in Proc. IEEE Int.
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Robuiics Automai., Apr.
1988,
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T.
H. Hong and M.
0.
Shneier, “Describing a robot’s workspace using
a sequence
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views from a moving camera,” IEEE Truns. Pattern Anal.
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1985,
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L.
Jackins and
S. L.
Tanimoto, “Octtrees and their use in repre
senting threedimensional objects,” Computer Graphics
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Jalbert, “Low level architecture for the new EAVE vehicle,” in Proc.
Fifth
Int.
Symp.
Unmunned Untethered Submersible Tech., Jun. 1987,
P.
Levi, “Principles of planning and control concepts for autonomous
mobile robots,” in Proc.
IEEE
Int.
Conf. Robotics Automat., Mar.
1987, pp. 874881.
R.
C.
Luo
and M.
G.
Kay, “Multisensor Integration and Fusion in
Intelligent Systems,”
IEEE
Trans.
Sysl.
Man Cyhern., pp. 901931,
Sept./Oct. 1989.
R. C.
Luo,
M. Lin, and R.
S.
Scherp, “The issues and approaches of a
robot multisensor integration,” Proc. IEEE Int. Conf. Robotics Au
tomat., Mar. 1987, pp. 19411946.
D. Meagher, “Geometric modeling using octree encoding,” Computer
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H. P. Moravec and A. E. Elfes, “High resolution maps from wide angle
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1985,
pp. 116121.
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C.
Nelson and. H. Samet, “ A consistent hierarchical representation
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W.
M. Newman and R. F. Sproull, Principles of Interactiiv Computer
Graphics, 2nd ed. New York: McGrawHill, 1979.
N. J. Nilsson, ProblemSoking Methods in Artificial Intelligence. New
York: McGrawHill, 1971.
’
D. J. Orser and M. Roche, “The extraction
of
topographic features in
support of autonomous underwater vehicle navigation,” Fifth
Int.
Symp.
Unmanned
Untethered Submersible Tech., Jun. IY87, vol. 2, pp. 5025 14.
R.
J.
Renka and
A.
K. Cline, “A trianglebased C’ interpolation
method,” Rocky Mountain
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M.
C.
Roggermann, J. P. Mills,
S.
K. Rogers, and M. Kabrisky, ‘‘Multi
sensor information fusion for target detection and classification,”
Proc.
SPIE,
Apr.
1988,
pp. 813.
H. Samet, “The quadtree and related hierarchical data structures,”
ACM Computing Siirivys, vol.
16,
Jun. 1984, pp. 187260.
H. Samet and R. E. Webber, “Storing a collection
of
polygons using
quadtrees,” ACM Trans. Graphics
4,
July 1985, pp. 182222.
H. Samet, A. Rosenfeld, C. A. Shaffer, and
R.
E. Webber,
“A
geo
graphic information system using quadtrees,” Pattern Recognition,
Nov./Dec., 1984, pp. 647656.
C.
A. Shaffer and H. Samet, “Optimal quadtree construction algo
rithms,” Computer Vision. Graphics,
und
Image
Processing, vol. 37, Mar.
1987, pp. 402419.
W. K. Stewart, “ A modelbased approach t o
3D
imaging and mapping
underwater,” in
Proc.
ASME Conf. Offshore Mechanics unrl Arctic
Eng., Houston, TX, Feb.
1988.
vol.
1,
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[28]
C.
W. Tong,
S.
K. Rogers, J. P. Mills, and M. K. Kabrisky, “Multisensor
data fusion
of
laser radar and forward looking infrared (FLIR) for
target segmentation and enhancement,” Proc. SPIE, May 1987,
pp.
to
19.
M. A. Turk, D. G. Morgenthaler, K. D. Gremban, and M. Marra,
“Video roadfollowing for the autonomous land vehicle,” in
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Robutics Airtornut., Mar. 1987, pp. 273279.
[29]
Modeling and Control
of
Underwater Robotic Vehicles
J.
YUH
Abstract
Remotely operated, underwater robotic vehicles have
be
come the important tool
to
explore the secrete life undersea. They are
used
for
various purposes: inspection, recovery, construction, etc. With
the increased utilization of remotely operated vehicles in subsea applica
tions, the development of autonomous vehicles becomes highly desirable
to enhance operator efficiency. However, engineering problems associ
ated with the high density, nonuniform and unstructured seawater
environment, and the nonlinear response of the vehicle make a high
degree of autonomy difficult to achieve. The dynamic model of the
untethered vehicle is presented, and an adaptive control strategy for
such vehicles is described. The robustness
of
the control system with
respect to the nonlinear dynamic behavior and parameter uncertainties
is investigated by computer simulation. The results show that the use of
the adaptive control system can provide the high performance of the
vehicle in the presence
of
unpredictable changes in the dynamics
of
the
vehicle and its environment.
I.
INTRODUCTION
A
large portion of the earth is covered by seawater and has
not been fully explored,
so
plenty of resources still remain in a
natural condition. In a recent report
[l]
to the National Science
Foundation, seven critical areas in ocean system engineering
were identified as follows: system for characterization of the sea
bottom resources; systems for characterization of the water
column resources; waste management systems; transport, power
and communication systems; reliability of ocean systems; materi
als in the ocean environment; analysis and application of ocean
data to develop ocean resources. It was also concluded in the
report that the area
of
underwater robotics should be supported
in all of the above areas. It is obvious that all kinds of ocean
activities, including both scientific ocean related research, and
commercial utilization
of
ocean resources, will be greatly en
hanced by the development of an intelligent, robotic underwater
work system.
Current underwater working methods include scuba, remotely
operated vehicle (ROV), submarine, etc. During the last few
years, the
use
of ROVs has rapidly increased since such a
vehicle can be operated in the deeper and riskier areas where
divers cannot reach. In the undersea environment, ROVs arc
used for various work assignments. Among them are: pipelining,
inspection, data collection, construction, maintenance and re
Manuscript received March
6,
1989; revised February 22,
1990.
This paper
includes part of the results
of
the research being conducted at the Robotics
Laboratory of the University of Hawaii. Funding for this research (“Auto
matic optical stationkeeping for a subsea remotely operated vehicle,” Project
R/OE/I I ) was provided in part by UH Sea Grant College Program under
Institutional Grant No. NA85AADSG082 from NOAA Office of Sea Grant,
Department of Commerce, and in part by the Pacific International Center
for High Technology Research under Grant No. PO61 16003. This is a Sea
Grant publication
UNIHISEAGRANTJC9015.
The author is with the Department of Mechanical Engineering, University
of Hawaii, Honolulu, Hawaii 96822.
I EEE
Log
Number 9037609.
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01990 IEEE
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NOVFMBER/DECEMBER
1990
pairing undersea equipment, etc. Howevcr, subsystems in the
current vehicles are immature compared to those in onland
systems and therefore, performance of the vehicle is limited.
High technologies developed for onland systems cannot be
directly adapted to underwater vehicle systems since such vehi
cles have diffcrcnt dynamic characteristics from onland vehi
cles, and their operating environment is unstructured. Effect of
high density water motion
on
the vehicle is also significant.
The dynamics of ROVs are fundamentally nonlinear in nature
due to rigid body coupling and the hydrodynamic forces on the
vehicle. This nonlinear dynamic behavior of ROVs is similar to
the wellknown rigid body vehicle motion of aircraft and conven
tional submarines although there are a couple of important
differences:
1
)
ROVs usually have comparable velocities along
all three axes. Therefore, control techniques that depend on
linearization of the equations of motion about a single forward
operating speed cannot be used as effectively as they can with
aircraft and submarines.
2)
The high density of water sets ROVs
and submarines apart from aircraft because the forces and
moments produced by fluid motion are significant and they
cannot be conventionally combined with the forces and mo
ments produced by vehicle motion to form functions of relative
motion only (except for the special case of a neutrally buoyant
vehicle). In addition, the
added
mass
aspect of the dense ocean
medium must be considered and results in control response
characteristics which are long in comparison with human antici
pation and analysis capabilities. Therefore, satisfactory perfor
mance of the vehicle cannot be obtained without consideration
of the nonlinear equations of motion in designing the vehicle
control system. Furthermore, the hydrodynamic coefficients are
often poorly known and a variety of unmeasurable disturbances
arc present due to multidirectional currents.
Also,
considering
that the vehicle dynamics can change appreciably as different
sensors and work packages are used, it is vehicle control system
interms of speed and accuracy. Consequently, an intelligent
system guidance and control strategy must be developed.
The dynamic equations
of
motion for marine vehicles have
been shown in the literature
[2], 131.
These models, which were
developed primarily for ships and submarines, use coordinate
systems which simplify the mathematics involved but limit their
applicability to more ROVs. During the last few years, several
control strategies for ROVs have been discussed. Kazerooni and
Sheridan
[4]
have developed the control system based on the
ClaytonBishop model by using the pole placement and ob
server method. Their system can be used when all states of the
system are not available. However, robustness of the control
system with respcct to parameter uncertainties cannot be guar
antccd. Yocrger and Slotline
[ 5]
have proposed a series of
singlcinput and singleoutput continuoustime controllers by
using the sliding control technique. Robustness of their control
systcm with respect
to
parameter uncertainties was demon
strated by computer simulation using a planar model of the
University of New Hampshire experimental autonomous vehicle
(EAVE). In their simulation. the effect of pitch, roll and vertical
movement were not considered, inertia terms were simplified by
placing the moving Coordinate system at the center of mass, and
the effect of a single thruster on more than one velocity was
ignored. Goheen et al.
[h]
have suggested the use of a selftest
ing controller that requires the manual piloting of the vehicle to
in their simulation is poorly described with several errors in
their paper. Without showing the result, thcy claim that their
control system can cope with sudden or slowly varying changes
in vehicle dynamics better that the fixedgain controller. How
ever, large tracking errors, using their control system, are still
observed from the result of their simulation. Therefore, the
cvaluation of their control algorithm cannot be completed.
To increase the autonomy, various subsystems of the vehicle
have been studied. In this paper, we present the result of the
recent development on the control system of the vehicle. This
paper is organized as follows.
In
Section
11,
since dynamic
analysis of such vehicles is a cornerstone to developing advanced
technology vehicles that include intelligent system guidance and
control architectures, we derive the equations of motion of the
vehicle, considering the effect of hydrodynamics. Section
111
proposes an adaptive control strategy to control the vehicle. In
the literature, the advantages of the use
of
adaptive control
techniques have been described for various nonlinear dynamic
systems such as industrial robots
[SI
and large tankers
[9].
When
parameters of the system to be controlled are poorly known, the
use of an adaptive control strategy is encouraged. Since the
dynamic behavior of ROVs is nonlinear and hydrodynamic
coefficients are poorly known, adaptive control techniques are
very attractive in this application. The robustness of the control
system under varying degrees
of
parameter uncertainty is inves
tigated for the case of planar motion, and results of case study
are discussed in Section IV before the conclusion.
11.
DYNAMIC
MODEL
In this section, the ROV dynamic model is presented. We
assume that the vehicle is powered by onboard battery stacks,
and communicates with the surface mothership or ground work
station by untethered communication links such as ultrasonic
link. Therefore, the effects of a tether is not considered in the
dynamic model. In this paper, the lengthy, detailed derivation
procedure of the dynamic model is not included. A detailed
derivation of this model is shown in [lo].
A.
Coordinate Systems
The dynamic model uses t yo Aorthogonal coordinate systems:
global coordinate system, (0,
I,
J,
K
),
whichAremains fixed at the
ocean surface (mother ship) with origin
0, K
pointing down into
the water normal to the surface and
f
and
.f
chosen in any two
convenient mutually perpendicular horizontal directions with
the only restriction being that the axes focm a righthanded
system; and a local coordinate system,
(P,i,
j,
k ),
which is fixed
on
the vehicle with origin at
P,i
pointing through the nose
of
the vehicle,
k
pointing through the belly of the vehicle and
j
completing the righthanded system. The position and orienta
tion of the vehicle in global Coordinates can be specified by
R,,
the vector from 0 to
P,
and the Euler angles
+,O,b.
Transfor
mation of forces and motions from local to global coordinates
can be accomplished by using the transformation matrix
[TI
and
from global to local by using its inverse (which is just
[TI'
since
[TI
is an orthogonal matrix) where
1
cos
+
cos
e
cos
4
sin
e
sin
4
sin
+
cos
4
cos
+
sin
e
cos
d,
+sin
I)
sin
d,
sin +cost) sin
4
sine sin
d,
cos4cos
4
sin
4
sinocos
4
cos sin
d, .
sin
t)
cos
0
sin
4
cos
0
sin
d,
select the closedloop polcs before operations begin and after
any equipment changes arc made. Goheen
et
al.
[7]
have pro
posed an adaptive autopilot for ROVs. The control system used
The development of the dynamic modcl is carried out in the
local Coordinate system sincc the motion of thc vchiclc
is
usually
described in reference
to
this system.
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL.
20.
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1990
1477
B. Rigid Body SixDegreeofFreedom Dynamics
In developing the rigid body equations of motion, two as
sumptions are made: the mass of the vehicle remains constant in
time and the effect of earth rotation can be neglected. It is
customary to choose the center of mass of a rigid body as the
origin of the local coordinate system to simplify the analysis
involved. However, for underwater vehicles, it is more conve
nient to develop the equations for an arbitrary origin to provide
flexibility in choosing an origin which takes advantage of the
geometrical properties
of
the vehicle to facilitate the expression
of
the complex hydrostatic and hydrodynamic force acting upon
it.
The equations of motion for a rigid body
of
mass
m
with an
arbitrary origin are summarized below.
Translational Motion:
F
=
m
[
0
+
h
X
R,
+
f l
X
( R
X
R,
)]
( 2 )
d
dt
Rotational Motion:
G
=
([
I ] R)
+
m(
Rc
X
i')
(3)
where
F
=
[ X, Y,
Z]'
the resultant cxtcrnal force,
G
=
[
K,
M, NI'
the rcsultant external moment,
U
=
[U,
r,w],
the
velocity
of
the origin,
f l =
[ p,
q,
rIT
the angular velocity about
the origin,
Rc,
=
[
x,,
y,
,
z,],,
the position of the center of mass
in local coordinates, and
I,  I,,

I,:
PI=[
1;:;
l':
;..I
the inertia tensor with respect to the origin.
It is important
to
recognize that in the expressions above, the
derivative of a vector will in general not be equal to the vecfpr*
of
th5 dcrivatives of the components since the unit vecotors
I,
1
and
k
change direction as the vehicle rotates producing cen
tripetal acceleration terms. Therefore, in evaluating the vector
dcrivatives, the following expressions must be used:
d,
,.
d,
A A
d,
A
 i =r j  q k,  j = p k  r i
and
 k =q i  p y.
(4)
dt dt dt
C.
Hydrodynamic Forces and Moments
The hydrodynamic forces and moments acting a
ROV
are
described below assuming that fluid rotation is negligible and
there is a current with a velocity
Uj.
If
Uj
is expressed in terms
of
global coordinates, the velocity of the vehicle relative to the
fluid is
U,
=U [ TI TU,.
( 5 )
Added Mass:
Since the density of water is similar to the
density of an ROV, additional inertia terms must be introduced
to account for the effective mass of surrounding fluid that must
be accelerated with the vehicle. These added mass coefficients
are defined as the proportionality constants which relate each of
the linear and angular accelerations with each
of
the hydrody
namic forces and moments they generate.
For
example, the
hydrodynamic force along the xaxis due to acceleration in the
xdirection is expressed as
X,
=

XI#
where
X,i
=
aX/au.
(6)
In a similar manner, all other added mass coefficients can be
defined and assembled into an added mass matrix
[ A].
Consid
ering the effect of the current and centripetal acceleration
components
(4),
the force and moment due
to
added mass can
be obtained from
[
;I A
=
;([AI[
;I).
(7)
Hirid
Morion: For a vehicle moving in a low density fluid,
such as an airplanc, the forces and moments exerted on thc
vehicle by fluid motion can be neglected. However, for an
ROV
traveling at low speeds
in
the ocean, these effects are significant
and must be included in the dynamic model:
FF= ml o l,
and
G,= mf ( R,x 0,) (8)
where
int
is the mass of the fluid displaced by the vehicle and
R,
=
[ x h.
y!,,
zhI7
is the position of the center of buoyancy in
local coordinates.
It
should be noted that except for the special
case of a neutrally buoyant vehicle, the mass of the fluid
displaced by the vehicle will not be equal to the mass of the
vchicle. Therefore, the forces and moments produced by vehicle
motion and fluid motion cannot be conveniently combined into
functions of relative motion only. However, in
(7),
relative
velocity can be used because the
added mass
coefficients arc
dependent only on the body geometry and not on
m,.
The Drag:
The drag is usually described as a force propor
tional to the square
of
the corresponding relative motion of the
vehicle. For example, the drag force along the xaxis due to
relative velocity
in
the xdirection is expressed as

X,,,,lur
111,
where
XI,,,
=
d'X/duf,
and the drag moment along the zaxis
due to the angular velocity
r
is expressed as

NI, rIrI
where
NI, =d'N/dr 2.
The drag force and moment are then denoted
by
F,
and
G,,
respectively.
D. Weight and Buoyancy
The gravitational force and buoyant force are defined in
terms of the global coordinate system
so
they must be trans
formed to the local coordinate systems:
F,
=
mg[
sin 0
cos
0
sin
4
cos
0
cos
4I T,
~,=  p g v [  s i n t ~ cosOsin4
C O S ~ C O S ~ ] ~
(9)
where
g
is the gravitational acceleration,
p
is the fluid density
and V
is
the volume of the fluid displaced by the vehicle. The
moments generated by these forces can be expressed in terms of
the positions
of
the center of mass
C,
and the center
of
buoyancy
8:
and
G,
=
R,
X
F,,
and
G,
=
R,
X
F,
(10)
where
R,
and
R,
are the respective positions of the center
of
mass and the center
of
buoyancy in the local coordinate system.
E.
Thrusters
The resultant force and moment of a thruster configuration
consisting of
N
thrusters can be expressed as the vector sum of
the force and moment from each individual thruster:
N N
N
F,=CF,,,
and
G,= C G,,+ C R,,x F,,
(11)
where
R,,
is the position of the ith thruster in local coordi
nates. The magnitudes
of
the thruster and troque generated by
the ith thruster can be expressed as
IFT,l= KT,pnfDP,
and
ICrl(
=
KQ,pnfD:
(12)
where
D,
is the diameter of the thruster,
n,
is the angular speed
of
the thruster shaft and
K,,
and
K
),
are the thruster and
torque coefficients
of
the thruster
[
111.
(The major problem that
is encountered in thruster modeling
is
that they behave as highly
nonlinear actuators. Therefore, the thruster and torque coeffi
cients cannot be represented as being constant but rather must
be expressed as functions of the advanced coefficient
J
=
V/nD
(13)
where
I/
is the axial speed of the thruster
1478
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1990
 
mz,.
+
Yi

my,
+
X,
rn
+
x,, x,
x,,
xfi
Y;
m + y
Y,.

mz,
+
Y,;
Yj
mx,
+
Y:
Zi<
z,
rn
+
Z,.
myc
+
Z,,

mx,
+
Z, Z j
K,,

mz,
+
Ki
my,
+
K,. I,
+
K,

I,v
+
K,,

1,;
+
K,.

rnx,
+
M,;

I,v
t
Mf i I,.
+
Mci

I,.:
+
M,
mz,
+
M,i Mi
F. ROV
Dynamic
Model
with the following
PAA:
All
externa! forces and moments can now be consolidated into
ROV dynamic model. However, to provide a form that will be
suitable for control purposes, some rearrangement
of
terms is
have fluid velocity and acceleration component s
~ ( k
+
1)
=

required. First, all
added
mass
terms obtained from
(7)
which
( ~ ~,,~'/,~/,u/,~,,w/)
are combined with the fluid motion forces
and moments (8) into a fluid vector denoted by the subscript
F.
Next, the mass matrix
[ MI
consisting
of
all the coefficients of
the rigid body equations of motion
(2)
and (3). to produce the
B^(k
+
1)
=
H^(k)
+
F ( k
+
l ) @( k) e( k
+
I )'
(19)
F(
k
) @( k
) @'( k
) F( k
)
inertia and
added
muss
terms with vehicle acceleration compo
nents
( l i,
C,
w,
p,
4,
I:)
is defined as
0
<
A
I(
k
)
4
1.
0
4
A?
<
2,
F(
0)
>
0
(20)
1

my,
+
N,,
pm,
+
N,
N,,
Finally, all remaining
added
mass
terms are combined with the
remaining inertial terms into a dynamics vector denoted by the
subscript D, to produce the final form of the model:
111.
CONTKOL
SYST~ M
The control system is determined using a discretetime ap
proximation
of
the ROV dynamic modcl
(15)
that can be ex
pressed by the following vector equation:
V ( k
+
1)
=
A1
*
V ( k ) +
AO+
B1*
U( k )
(16)
where k is the kth sampling time step,
V
is the sixdimensional
(6D) velocity vector of the ROV and
U
is the 6D vector of
forces and moments generated by the thrusters. If the parame
ters in the discretetime model were known exactly, a conven
tional digital control law could be determined using classical
methods. However, since the poorly known hydrodynamic coeffi
cients are included among the parameters of the dynamic model,
a conventional control scheme cannot guarantee high perfor
mance in ROV motion control. Therefore, a parameter adapta
tion algorithm
(PAA)
is introduced to solve this problem. The
PAA
estimates the parameters in the discretetime model at
each sampling time step using inputoutput measurements from
the ROV. These estimates are then used to adjust the controller
gains to provide the required control signals. In this section, a
discretetime adaptive velocity controller for ROVs is designed
using the selftuning control principle.
The basis of this control algorithm is a linear predictor that is
designed with the aim that the prediction error vanishes aceord
ing to
lim
{ e (
k
+
1)
=
C,<(
q
I )
[
V(
k
+
1)

P(
k
+
I ) ]
}
=
0
(17)
where
c ( k
+
1)
is the predicted value of
V( k
+
l ),
CR( r l ') =
I +q'C$ defines the regulation dynamics,
I
is the identity
matrix and
q 
I
is the unit delay operator. Defining
R
=
Cz
+
A1 and
A0
=
Wd
where
W
and a constant d are arbitrary
factors of
AO,
(17)
will bc satisfied
if
the predictor equation is
h
 x
C,(
9
'
)
I'(
k
+
1)
=
I?(
k )
V(
k )
+
@( k ) d
+
61
( k
) U(
k ) (
18)
where is used
to
denote an estimated value,
?( k )
=
[ L h ( k ) k ( k ) $ ( k ) ]
and
Qr(
k )
=
[U'(
k ) VT(
k ) d q
In terms of the minimization of a quadratic criterion, the use
of
(20)
for updating the adaptation gain matrix
F( k )
corresponds
to minimizing
h
r k
1
J ( k ) =
nAl(jI)
P ( i  l ) [ ~ ( k + l ) ] ~
(21)
,= I
I
j = i
1
where
Note that
A J k )
and
A2( k )
in
(20)
have opposite cffects:
A,( k)
tends to increase the adaptation gain while
AJ k )
tends to
decrease the adaptation gain. The choice of
A,( k)
and
h2( k )
determines the type of adaptation algorithm obtained. This has
been discussed by Landau and Lozano
[
12].Thc parameter val
ues estimated by the predictor are then used at each time step
to compute a control signal such that the output of the predictor
follows the desired trajectory:
c,<
(
q

'
)
$(
k
+
1
)
=
C,<(
q

I
)
VI'( k
+
1
)
.
(23)
Substituting (18) into
(23)
and solving for
U( k),
U(
k
)
=
81(
k
)

'
[
c,(
q

1 )
V"(
k
+
1)

ri (
k )
V(
k
)

$(
k
) d]
(24)
which is the desired control law.
To summarize, in the proposed control algorithm the parame
ters of the predictor are estimated at each time step using (12)
and
(20)
then the values
of
these estimated parameters are used
to compute the control signal
in (24).
The control scheme
presented hcre is extremely simple and therefore the computa
tional time required to calculate the adaptive control signal
(24)
is very short.
As
a result, the proposed control algorithm could
be implemented using a high sampling rate such as
1
KHz.
Also,
this control approach was well implemented for industrial robot
IEEE
TRANSACTIONS ON SYSTEMS, MAN, AND
CYBERNETICS,
VOL.
20,
NO.
6,
NOVEMBER/DECEMBER
1990
0
8 
0.7

0.6

0.5

0
4 
0.3
0.2
1479
I
\
/
\
\
/
\
/
I.
/
\
\
/
\
I
\
I
\
I
\
\
\
/
\
\
I
\
\
Parameter
Adaptation Algo.
4
I
+
Eqs.
19
and
20
0
8 
0.7

0.6

0.5

0
4 
0.3
0.2
I
I
Fig.
I.
Adaptive vehicle control system (shaded arrow indicates with PAA, activate; solid arrow indicates without PAA,
deactivate
).
I
\
/
\
\
/
\
/
I.
/
\
\
/
\
I
\
I
\
I
\
\
\
/
\
\
I
\
\
systems which usually have the faster motion and the shorter
operating time period for one task than the underwater vehicle.
The diagram of this control system is shown in Fig.
1.
In the
next section, a case study is presented to show the use of the
proposed control scheme for an ROV.
IV. CASE
STUDY
To investigate the robustness
of
the control system, a com
puter simulation was performed for the plane motion of a ROV
modeled after the Dolphin
3K
presented in
[13].
For this ROV,
motions in the horizontal plane may be described by the follow
ing set of equations:
( m
+
X,,)U

my,
i.
=
mu,
r 2
+
( m
+
Y,
)
er

XllIII~~,Iu,

(
m,
+
Y,
)
r,r
+
X,
(
m
+
Y,
)
i.
+
mu,
i.
=
my,
r

( rn
+
x,,)ur

Y,
I
I
L*,
I
e,
+
(
m,
+
XI,)
U,
r
+
Yr

my,
U
+
mu,I:
+
( I z
+
N,) i
=

mr( x,
u
+
y,e)

NrrlrIr
+
m
,
r
(
x
,
u
+
y
,,e,
)
+
N,
.
(25)
Numerical values of the parameters in
(25)
are assumed from
[13]
as follows:
rn
=
3500
kg,
rnf
=
3506
kg,
x,
=
0.1
m,
y,
=
0,
x,
=

0.1
m,
y,
=
0
and
I,
=
3827
kgm2. One possible choice
of the parameter matrices for
(16),
using the dynamic model
(15),
is
Al =
I,
Bl =At M'
and
AO=At MIF,
where
At
is
the sampling time interal,
M
is the mass matrix and
F,
repre
sents the nonlinear terms and all the external forces and mo
ments other than those due to the thrusters. Based on this
choice, the adaptive control system is determined for the case
study.
In the simulation, the following considerations are made:
1)
The desired velocities
U",
e''
and
r"
are generated by using the
trapezoidal speed law shown in Fig.
2.
2)
The regulation dynam
ics are chosen with C,*
=
diag(O.l).
3)
The initial values of the
parameter estimates for the controller in
(24)
are arbitrarily
chosen as:
0
l e  6 l e  5
O 1
l e  5
0
B1(0)=
0
l e  5 l e  6
[
and
AO(0)
=
0.
4)
A
constant current of velocity
uf
is assumed in the Xdirec
tion of the global coordinate system.
5)
Since
A1
is chosen as
the identity matrix, the PAA is designed to estimate the parame
ter matrices
B1
and
AO.
6)
The initial values of the adaptation
1,
0.9
1
0
10
20
30
40
rd
(rad/sec)
ud(m/s)


vd(mls)
time(sec)

Fig.
2.
Desired velocities.
gain matrix is selected as
F(0)
=
I.
F( k )
is updated using the
constant trace algorithm with
A,( k) =
A@)
and
A,(/?)
is com
puted such that trace[F(k)]
=
trace[F(O)].
7)
The velocities
of
the system are
measured
by numerically integrating the nonlin
ear differential equations
(25). 8)
A
sampling time step of
At
=
0.05 s
is used.
9)
Robustness of the algorithm with respect
to hydrodynamic coefficients and current is tested by imple
menting the control system for the following sets of constant
system parameters and current velocities:
Case
1:
XI,
=
693
kg,
Y,
=
762
kg,
N,
=
3817
kgm2,
X,,,,
=
1646
kg/m,
Y,
,
=
2273
kg/m,
N,,
=
5457
kg/m2 and
u
=
0.
Case
2:
All
hydrodynamic coefficients increased by
100%
from
case
1
and
U,
=
0.5
m/s.
Case
3:
All
hydrodynamic coefficients increased by
100%
from
case
1
and
uf
=
0.5
m/s.
Note that numerical values used for Case
1
are also assumed
from
[13].
The results of the computer simulations are shown in Figs.
3
through
15
for each case using the control law with PAA and
the control law without PAA. Figs.
3
through
11
show the
velocity tracking errors which are defined as follows: Surging
rate error,
E(1)
=
U"

U,
Swaying rate
error,
E(2)
=
e''

e,
and
Heading rate error,
E(3)
=
r"

r.
Fig.
11
shows the heading
angle and Figs.
1315
show the thrust and torque required to
sustain the desired velocities.
The vehicle was originally heading in Xdirection of the
global coordinate. From the desired heading angular velocity
profile (Fig.
2),
the desired heading angular displacement is
3
rad, and therefore the vehicle almost heads in negative Xdirec
tion at steady state. The result of the simulation for each case
0 06
I
o m,
1
I

0.
0.001
0.04
0.0 5 i
Y
V
 _
_r

0.02
m
O.O 3 l
 O.O1
1
0
10 20
30
40
w/o PAA
time
(sec)

w/PAA

Fig.
3.
Surging raw error (Case
1 )
0.008
I
0.006
30.005
g0.004
show that when the
PAA
is incorporated into the control
algorithm, the control system
is learning the change of the
vehicle and
its
environment, and adjusting the control gains to
provide the proper control signal (Figs.
1315)
to
keep the
desired velocity.
Figs.
3,
6.
and 9 show the surging rate error for cases
1,
2,
and
3, respectively. In case
1,
since the current velocity is zero, the
external forces due t o the current velocity are zero. Therefore,
the total external force of case
1
is relatively small and error in
the initial estimate
of
the parameter
(AO(0)
=
0)
is
small com
pared to other cases.
As
shown in Fig.
3,
without
PAA
the
steady error
is
observed due to the error in the parameter
estimates while no steady crror is observed with
PAA.
In case
2
(Fig. 6), a large steady error
is
observed without
PAA
because of
the large error in the initial estimates of the parameters. Espe
cially, since the current velocity is
0.5
m/s in Xdirection and
the vehicle is heading in the almost opposite direction to the
current velocity at steady state, the vehicle steady velocity in
Xdirection of the local coordinates is far below the desired
velocity without
PAA.
With
PAA,
however, the surging rate
errors are very small except the first few time steps which
is
the
adaptation period. During this adaptation period, the
PAA
adjusts the controller parameters to their operating nominal
values from the large initial misalignment. After this period, the
PAA
well adjusts small errors in controllcr parameters with
respect to the operating nominal values.
In
case 3, since the
current velocity is
0.5
m/s in Xdirection and the vehicle
is
0.07
1
0
O
05
"1
0
02
I
10
20
30
40

w/PAA
O
O1
&.
W/O
PAA
tlme (sec)
Fig.
4.
_ 
Swaying rate error
(Case
I )
also
heading in Xdirection at the beginning, the vehicle velocity
in Xdirection is far below the desired velocity in that direction
without
PAA
and Fig.
9
shows large surging rate errors espe
cially around 10 sec. With
PAA,
the surging rate errors are very
small except during the first
10
sec. which is the adaptation
pcriod.
Figs.
4, 7
and
10
show the swaying rate error for cases
1,
2
and
3, rcspectivcly. Thc effect of current velocity on thc vehicle
velocity in ydircction
of
the local coordinates is small at steady
state. Each case shows small swaying rate error at steady state.
However. during the operating time period. each case shows
large swaying rate error without
PAA
while each case shows
very small swaying rate errors with
PAA.
Figs.
5,
8, and
11
show
the heading rate error for cases I,
2
and 3, respectively. One can
have the same conclusion as one for the swaying rate crror.
Fig.
12
shows the heading angle of the vehicle for each case.
With
PAA,
Fig.
11
shows the same result of no stcady error and
desired heading angle for these three cases. Without thc
PAA,
results show the steady error for each case.
Figs.
1315
show the forward thrust, lateral thrust and thrust
torque, respectively. In this study, the power limitation of the
vehicle thrusters were not considered. One can observe the
large variation in thc control efforts with
PAA
during the initial
adaptation period. Howevcr, the thrusts and thrust torque which
were generated in the simulation for the three cases are quite
rcsonable in size. Thc control efforts with
PAA
vary within
approximately
f
15%
of the control efforts without
PAA
at
each time step.
Thc initial values of the controller, described previously in
31,
were roughly estimated from the lincarizcd vchicle equations
I
about the desired steady state vehicle velocities with parameters
of case
1,
which might be considered as a base case in the
simulation. It is rather difficult t o estimate exact or nearopti
mum values of the system parameters for the gcncral case even
with an expensive hydrodynamic test on the actual
ROV
system
since the
ROV
is
a nonlinear, timevarying system dependent
on
the current and payload as discussed in section
11.
Therefore,
even though the initial values for the controller were cstimated
with parameters
of
case
I,
they are rather arbitrary. Since the
same initial values were used for both control systems with
PAA
and without
PAA
in the simulation, both control systems with
and without
PAA
were implemented with the same initial mis
alignment of the controller gains in each case. The initial
misalignment of the controller gains was well compensated by
the
PAA.
However, when the
PAA
is not activated, the vehicle
fails t o keep the desired velocity resulting in a significant amount
of steady error. Therefore, it can be easily noticed that perfor
1481
IEEE
TRANSACTIONS
ON
SYSTEMS, MAN, AND CYBERNETICS,
VOL.
2 0, NO.
6.
NOVEMBER/DECEMBER
1990
0.01

0
0.04
0.05i
4.
\
 _____ 

I 
'
0.03

U)
.
!E

0.02
r
m
0.001
0
'
0.01
\
_.'
\
\
\
,
V W
v 
\I
0
0.01
0.02
10
20
30
40
wf PAA

0
wl o PAA time
(sec)
_ 
0.01
1
0.02
!
I
0
10
20
30
40
wf PAA

~ 1 0
PAA time (sec)
_ _
Fig.
6.
Surging rate error (Case
2).
Fig. 7. Swaying rate error (Case
2 )
'.J
I
0.001
I
0
10 20
30
40
wl
PAA

w/o PAA time
(sec)
_ _
Fig.
8.
Heading rate error (Case
2).
\
\
\
\
\
\
\
\
\
\
I
n
nc)
I
 w.v L
I
0
10
20
30
40

WI PAA
Wl O
PAA time
(sec)
_ _
Fig.
10.
Swaying rate error (Case
3).
0'06
0.05
I
n
n3
!
I
Fig.
9.
Surging rate error (Case
3)
0.009
0.008
0.007
0.006
go,."
g0.004
63
z0.003
0.002
0.001
0
0.001
,_
,
/
I
I
\
I
I
I
\
\
\
\
\
\
\
l,/'
I
I
I
I
I
I
I
I
I
\
I
\
I
\
\
\
\
\
_ _
v 
'\
I
10
20 30
40
w/ PAA

w/o
PAA time (sec)
_ _
Fig. 1
I.
Heading rate error (Case
3).
1482
IEFZ
TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS,
VOL.
20,
NO.
6,
NOVEMBFR/DECEMBER
1990
3.2
I
0
1 0
20
30
40
ci ac~
WI O
PAA time (sec)

w/ PAA

  
C3 w/o PAA
_ 
10
30
40
2o
C1 wl o PAA
C l w/P A A tme(sec)

C2w/PAA
   
C3 wI P AA

C2
wI oPAA

C3
wI oPAA
0

Fig.
12. Hcading
anglc.
Fig.
13.
Forward thruat
g 1
1
!
.,
0
10
20
30
40
C1
Wl O
PAA
time
(sec)
11
C 2 WIO PAA


C3
WIO
PAA

C1 w l PAA
C2
w l PAA
   
C3
w/
PAA
Fig.
14. Lateral thrust.
mance of the adaptive control system
is
barely dependent on
initial conditions, while performance of the nonadaptive control
system is highly dependent on initial conditions. The control
strategy presented in this paper does not require the explicit
expression of the
ROV
dynamic model. The results
of
case
study show that the adaptive control scheme can provide robust
control with respect to parameter uncertainties even though a
simplified model
(16)
and
(26) of
the vehicle was used to design
the control system.
V. CONCLUSION
In this paper, we have presented a dynamic model and an
adaptive control system for
ROVs.
The
ROV
dynamic model is
described by a set of six nonlinear, timevarying differential
equations having poorly known parameters that may be identi
fied by expensive hydrodynamic testing on the vehicle. Even
with such a test,
it
is almost impossiblc to derive the complete
dynamic model since these tests do not take into consideration
any unsteady fluid motion effects.
Also,
thc parameters of the
250
200
150
100
E
5 5 0
0
50
100
11
I
1 1
10
20
30
40
:?&?A
yme (sec)
11
C2
W/O
PAA

C2
W/
PAA
   
C3
WIO PAA
C3
w/ PAA

Fig
15
Thruster torque
dynamic model may vary with changes in the
ROV
configuration
and the environment. Therefore, it is obvious
to
develop a
robust control system with respect to parameter uncertainties.
The control strategy presented in this paper does not require
a priori knowledge about the vehicle system parameters. The
effectiveness of the presented adaptive control system was inves
tigated in the case study. Without any change in the control
system, the adaptive control system was implemented for three
cases which have different values of system parameters and
current. The results were compared with the results obtained by
using the nonadaptive control system (i.e. control system without
PAA)
implemented with the same initial conditions. The results
of
case study show that the presented adaptive control scheme
can provide high performance in terms
of
speed and accuracy in
the presence of uncertainties
of
the vehicle and its environment,
while the nonadaptive control system cannot. Future research
efforts on this subject include the experimental investigation on
the approach presented in this paper, and the control system
integration with computer vision.
1483
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL.
20,
NO.
6,
NOVEMBER/DECEMBER
1990
ACKNOWLEDGMENT
[SI
The author thanks Professor Joel
S.
Fox
at the University of
Hawaii for his valuable comments. Thanks also to Glenn
[h]
Uchibori and Irene Kanda for essential help in preparing the
manuscript.
,
VI
[XI
PI
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und
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Yuh and G. Uchibori, “Mathematical model of a generalized undcr
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I YXY.
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N. Newman,
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77.
I. D.
Landau and R. Lozano, “Unification
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A~iromuticu,
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M. Nomoto and M. Hattori,
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