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Modeling of an Autonomous Underwater Robot with Rotating Thrusters

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Mathematical modeling, simulation and control of an underwater robot are a very complex task due to its nonlinear dynamic structure. In this paper, the authors present kinematic and dynamic modeling of an underwater robot with two rotating thrusters. Through a virtual environment implemented in MATLAB and LabVIEW, the performance of the proposed robot under real operating conditions was demonstrated.
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Volume 6 • Issue 1 • 1000162
Open Access
Research Article
Jebelli et al., Adv Robot Autom 2017, 6:1
DOI: 10.4172/2168-9695.1000162
Advances in Robotics
& Automation
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ISSN: 2168-9695
Keywords: AUV; Hydrodynamics coecients; Added mass;
Graphical user interface (GUI); Virtual reality (VR)
Introduction
ese days, underwater vehicles have wide range of applications
in dierent marine industries. Controllability and maneuverability of
these vehicles which are strongly aected by various forces including
hydrodynamic forces applied on them. At the design stage, knowing
and calculating these forces are of very high importance which a
mistake in calculations and analysis of forces can have a lot of damage
to be followed including time, the cost of re-design and total change in
the body and following to that some changes in internal and external,
mechanical and electronic components of that. erefore, simulating
a robot in a real system and implementation and real monitoring of
a robot performance can signicantly help the designer in observing
the robot performance in dierent condition. But it needs a good
understanding of the robot environment modeling body coordinate
and accurate calculation of the external forces applied on the body.
In this project, we implement a virtual simulation of the robot in
real conditions aer the accurate analysis of the applied forces on
the designed robot body and coding achieved equations in MATLAB
and the environment of LabVIEW which is able to give us the whole
performance of the robot for the moment which this display control
in the actual operation of the robot in the water is a very accurate
indicator of the robot performance in tests that enables us to manually
apply necessary orders to robot through a wireless transceiver of the
robot by changing the parameters of this display.
Design the Body
To successfully design the body, one should establish a primary plan
of the body structure with the location of each sub-part, knowing that
the device will be made of one piece with a camera on the top and engines
attached on the sides. In this work (Figure 1); in this case, all thrusters
should be put “on” at the same time to create a simultaneous vertical-
horizontal movement; it will then lead to high power consumption.
Our rst contribution was to use a pair of mobile thrusters instead of
xed vertical/horizontal ones, thus saving energy. In practice, the two
thrusters shown in Figure 2 could be oriented within a specic angle
based on the vertical and horizontal forces needed to move the device
in a predened direction. Any change of angle should be made possible
by the instant movement of a servo motor which consumes much less
energy than the constant movement of a thruster.
Mass Shier
As shown in Figure 3, a mass shier was included, rst because of
the thruster’s movement and second, to make the maneuver possible in
the direction of the pitch. e whole set can have two movement modes.
In the rst mode, the body will have, by the help of the mass shier, a
constant horizontal movement and its movement towards vertical and
horizontal directions should be performed through a change in the
*Corresponding author: Jebelli A, Faculty of Engineering, University of Ottawa,
Canada, Tel: +16135625700; E-mail: ajebelli@uottawa.ca
Received March 01, 2017; Accepted March 10, 2017 ; Published
March 17, 2017
Citation: Jebelli A, Yagoub MCE, Dhillon BS (2017) Modeling of an Autonomous
Underwater Robot with Rotating Thrusters. Adv Robot Autom 6: 162. doi:
10.4172/2168-9695.1000162
Copyright: © 2017 Jebelli A, et al. This is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.
Abstract
Mathematical modeling, simulation and control of an underwater robot are a very complex task due to its non-
linear dynamic structure. In this paper, the authors present kinematic and dynamic modeling of an underwater robot
with two rotating thrusters. Through a virtual environment implemented in MATLAB and LabVIEW, the performance
of the proposed robot under real operating conditions was demonstrated.
Modeling of an Autonomous Underwater Robot with Rotating Thrusters
Jebelli A*, Yagoub MCE and Dhillon BS
Faculty of Engineering, University of Ottawa, Canada
Figure 1: Isometric scheme of the designed robot.
Figure 2: The designed robot in the sea.
Adv Robot Autom, an open access journal
ISSN: 2168-9695
Citation: Jebelli A, Yagoub MCE, Dhillon BS (2017) Modeling of an Autonomous Underwater Robot with Rotating Thrusters. Adv Robot Autom 6: 162.
doi: 10.4172/2168-9695.1000162
Page 2 of 10
Volume 6 • Issue 1 • 1000162
thruster’s angle. In the second mode, the thrusters should be kept xed
in a horizontal direction and the vertical movement should be made
possible through a change in the body angle in the pitch direction. e
rst mode is used to maneuver or pass obstacles while the second mode
is used to preserve and store energy as well as to change direction in
relatively large depths.
Physical Device
According to Figure 4, the center of the physical device is matched
with the reference point of the vehicle. Its axis comes out from the front
end of the vehicle and its y axis extends to the right; the z axis completes
the right hand rule. Let this system be called B and its axes b1, b2 and b3.
Gravity Model
Assuming the Earth as a perfect sphere, the gravity in terms of
height is determined by the following equation:
2
gR
µ
=
(1)
where μ is the earth constant (3.986005 × 1014 m3/s2). R, the distance
between the two masses, depends on the earth's radius R [1,2].
R = Re + h (2)
Modeling the Forces
In this work, it has been assumed that the uid in which the vehicle
is moving has a steadily and xed density ρ = 1000 kg/m3. e external
forces applied on the vehicle can be then evaluated [3].
Buoyancy force
According to Archimedes' principle, the buoyancy force, applied
vertically to the body and in the upward direction, equals the weight of
the displaced uid by the body:
FBuoyancy = ρVg (3)
where ρ is the density of the uid and V it’s volume.
Weight force
e weight force is calculated as
FW=mBg (4)
where mB is the vehicle mass.
rust force
As shown in Figure 5, two engines on either side of the vehicle are
used to provide the thrust force. e forces produced by the le and
right engines are noted Fr and FL, respectively.
Added mass force
As the device moves within the uid, a certain amount of liquid
will move with it [4]. As inertial and Coriolis matrices relate the
accelerations and angular/linear velocities of the body to the forces
applied to the device, the added mass matrices and added Coriolis relate
the accelerations and angular/linear velocities to the hydrodynamic
force arising from the liquid displacement and applied to the device.
When a vehicle moves inside the uid, a dynamic pressure
distribution is created around it. Bernoulli's law states that the pressure
applied on a dierential surface dS depends on the uid particle
velocity on the dierential surface and also to the water column height
above it and this pressure applies a dierential force dF and dierential
moment dM on the dierential surface dS. e force and dierential
moment are called the force and added mass moment.
When a force is applied to the uid particles, its adjacent particles
are accelerated due to a viscosity. at is why when a device moves
in the uid, the uid particles which are located exactly on the device
move at the same rate of the vehicle and the particles that are far from
the vehicle move with dierent velocities. In fact, there is a velocity
distribution on the dierential surface dS and the farthest particles
Figure 3: Mass shifter: location of the cube dumbbells, ball Screw, and stepper
motor.
Figure 4: Physical coordinates system. Figure 5: Propulsion.
e
Adv Robot Autom, an open access journal
ISSN: 2168-9695
Citation: Jebelli A, Yagoub MCE, Dhillon BS (2017) Modeling of an Autonomous Underwater Robot with Rotating Thrusters. Adv Robot Autom 6: 162.
doi: 10.4172/2168-9695.1000162
Page 3 of 10
Volume 6 • Issue 1 • 1000162
have a zero velocity [5]. According to Regan et al., [1] the force due
to the added mass is
( )
AAA
F Mv C vv= +
(5)
where FAis the added mass, MAthe added mass matrix, v the velocity
and CA the added Coriolis matrix.
Hydrodynamic force
As stated refs. the main hydrodynamic forces that are applied on a
subsurface vessel include Quadratic drag forces and lineal skin friction
forces. Although the equations of a vehicle with six degrees of freedom
are highly non-linear, a series of simplications are usually used.
e most common simplication is to assume that the linear
and quadratic forces in the direction i depend on the velocity vector
components in that direction. us, the hydrodynamic forces are
calculated as follows :
uu
u
vv
v
ww
w
hyd
ppp
qqq
r
rr
Xuu
Xu
Y vv
Yv
Z ww
Zw
FKp K pp
Mq M qq
Nr N rr









=−−












(6)
where
u, v and w are the components of the velocity vector in the
respective directions x, y and z.
p, q and r are physical components of the angular velocity
vector on the respective directions x, y and z.).
Xu and Xu|u| are the linear viscosity and quadratic
coecients along the x direction.
Yv and Yv|v| are linear viscosity and quadratic coecients
along the y direction.
Zw and Zw|w| are linear viscosity and quadratic coecients
along the z direction.
Kp and Kp|p| are linear viscosity and quadratic coecients
along the p direction.
Mq|q| and Mq|q| are linear viscosity and quadratic coecients
along the q direction.
Nr and Nr|r| are linear viscosity and quadratic coecients
along the r direction.
Kinematics
To derive a vector V vs. time in any coordinates system (Table 1),
the following operator D=d/dt is used:
AB
D V D V+ V
ΒΑ
= ω× (7)
en, the transfer matrix form a point B to a given vehicle W can
be stated as in ref:
W
B
cos cos sin sin cos cos sin cos sin cos sin sin
R cos sin sin sin sin cos cos cos sin sin sin cos
sin sin cos cos cos
θψ θψ ψ θψ ψ
θψ θψ ψ θψ ψ
θθ θ
φφ φ+φ


=φ+φ φ−φ

−φ φ

(8)
To calculate the angular speeds, the following equation are used [2]:
φ= + φ + φ
p qsin tan rcos tan
θθ
(9)
= φ− φ
qcos rsin
θ
(10)
φ+ φ
=
qsin rcos
cos
ψθ
(11)
Physical speeds p, q and r are calculated from the following
equation [6]:
=φ−
p sin
ψθ
(12)
= φ+ φ
q cos sin cos
θψ θ
(13)
=− φ+ φ
r sin cos cos
θψ θ
(14)
To model the rotation in three dimensions, the concept in classical
mechanics is to use rotational kinematics. One of the ways to describe
rotation is the quaternion method. e four elements of the quaternion,
q0, q1, q2 and q3, are related to the physical speed components p , q and r
through the following relation:
00
11
22
33
0
0
1
0
2
0
qq
pqr
qq
p rq
qq
qr p
qq
rq p
−−−
 

 

 

=
 

  
  

 
(15)
Dynamics Analyze
e Linear momentum of a mB mass relative to an arbitrary vehicle,
denoted I, is given by
B
m,
II
BG
pv=
(16)
e system under discussion contains two objects namely, the
oating vehicle, considered as m1 and the weights, considered as m2
that move relatively to m1. According to the Newton’s second law,
the resultant forces acting on a system mB are equal with the linear
momentum changes over time in the inertia system (Figure 6).
Assuming that both objects have constant mass, we get:
( )
12
12
mm
II I
B GG
fD v v
= +
(17)
with
1
I
G
v
the velocity of the center of the 1st object and
2
I
G
v
the velocity
of the center of the 2nd object mass. us,
11
II I I
G GI
Dv DDs=
Degrees of freedom Movement direction Position and Euler angle Linear and angular speed Force and moment
1 Movement along X X u X
2Movement along Y Y v Y
3 Movement along Z Z w Z
4 Rotation along X φp K
5Rotation along Y θq M
6 Rotation along Z ϕr N
Table 1: Variables for subsurface oating vessels with: (1) Surge, (2) Sway, (3) Heave, (4) Roll, (5) Pitch, and (6) Yaw.
Adv Robot Autom, an open access journal
ISSN: 2168-9695
Citation: Jebelli A, Yagoub MCE, Dhillon BS (2017) Modeling of an Autonomous Underwater Robot with Rotating Thrusters. Adv Robot Autom 6: 162.
doi: 10.4172/2168-9695.1000162
Page 4 of 10
Volume 6 • Issue 1 • 1000162
( )
11
I B BI I I
GB GB BI
D Ds s DDs
ω
= +× +
( )
( )
( )
11 1
1
2= + ×+ × +
××+
B B B BI BI B
GB GB GB
BI BI I
GB B
DDs D s Ds
sa
ωω
ωω
(18)
22
II I I
G GI
Dv DDs
=
( )
( ) ( )
22 2 2 2
2
I I B B B BI BI B BI BI I
G GB GB GB GB B
Dv DDs D s Ds s a
ω ω ωω
= + × +× × ++ ×
(19)
Since m2 is moving compared to the reference system B connected
to the mass m1, some of the terms removed in equation (18) cannot not
be removed in this last equation, thus leading to:
( )
( ) ( )
22
11
12 2 2
mm
m m m 2m
= ×+ × ×
+ + + ×+ ×
+ ×× +
B BE BI BE
B GB GB
I B B BI BI
B G GB
B BI BI I
G GB B
fDs s
aa Dùs
v sa
ω ωω
ω
ωω
(20)
Using the relation mB=m1+m2, and reorganizing the equations,
gives
( )
( )
( )
( )
12
12 2
2
12
12 2
2B
mm
mm m
2m m
= ×+ +
×× + +
+ ×+
B BE
B GB G B
BE BI B
GB G B G
BI B I
GB
fD s s
ss a
va
ω
ωω
ω
(21)
( ) ( )
( ) ( )
2
= +× +×
= + × + ×+ × ×
I E E EI EI E EI
B B BE B BE
E E E EI EI E EI EI
B BE B BE
aDvs vs
Dv D s v s
ω ωω
ω ω ωω
(22)
en,
( )
2= + ×+ ×+ × ×
I B E BE E EI E EI EI
B B B B BE
a Dv v v s
ω ω ωω
(23)
Aer substitution,
( )
( )
( )
( )
( )
12
12 2
2
12
12 2 2
BB B
B
mm
m m m 2m
m m 2m
m
= ×+
×× + + + ×
+ + ×+ ×+
××
B BE
B GB G B
BI BI B BI
GB G B G
B B E BE E EI E
GB B B
EI EI
BE
fD s s
ss a
v Dv v v
s
ω
ωω ω
ωω
ωω
(24)
According to the denition of the center of mass:
12
12
12
12
mm mm
+
= →=+
GB G B
GB B GB G B G B
B
ss
s ms s s
m
(25)
with
ωBIBEEI (26)
and
( ) ( )
( ) ( )
( ) ( )
( )
2 22
22 2
BB B
B
m 2m 2m
m m 2m
m
= ×+ ×
××+××+
××+××+
+ ×+ ×
+
×
+
+
×
B BE B EI
B B GB B GB
BE BE BE EI
BGB B GB
EI BE EI EI
BGB B GB
B BE B EI B
G GG
B E BE E EI E
BB B
EI EI
BE
f D ms D ms
ms ms
ms ms
a vv
Dv v v
s
ωω
ωω ωω
ωω ωω
ωω
ωω
ωω
(27)
So fB can be expressed as:
( )
( )
( ) ( )
( )
( )
( )
2
22
BB
B
2
22
B
mm
2m
2m
m 2m
m
= + ×+ ×
+
+
×+ × × +
××+ × ×+
×+ ×+ ×
×+ + ×
××
B E B BE BE
B B B GB
E EI E BE BE
B B B GB
BE EI EI BE
B GB B GB
BE B B EI EI
G B GB B
EI B EI B
GB G G
EI EI
BE
f Dv m D s
v vm s
msms
v mD s m
sa v
s
ωω
ω ωω
ωω ωω
ω ωω
ωω
ωω
(28)
Knowing that
DB ωEIEI×ωBE (29)
( )
( )
( ) ( )
( )
( )
( )
2
2
BB
2
2
B
m 2m
2m
2m
m
= + ×+ ×
×
×+ × ×+
×+ × ×
××+ ×
××
+
+
B E B BE EI
B B B GB
E BE BE BE
B B GB B
EI EI BE
GB B GB
BE B EI BE
G B GB
EI EI EI B
B GB G
EI EI
BE
f Dv m D s
vm s m
sm s
vm s
ms v
s
ωω
ωω ω
ω ωω
ω ωω
ωω ω
ωω
(30
which leads to
( )
( )
( )
( )
( )
2
22
BB
B
2
22 B
mm
2m
2 2m
m 2m m
= + ×+ ×
+ ×+ × × −
× × + ×+ ×
× − + ×+ ×
×
B E B BE BE
B B B GB
E EI E BE BE
B B B GB
EI BE EI B EI
B GB G B
EI B EI B EI
GB G G
EI
BE
f Dv m D s
v vm s
m s vm
sa v
s
ωω
ω ωω
ωω ω ω
ω ωω
ω
(31)
Rotational Dynamics Equations
Angular momentum
e (absolute) angular momentum of mB can be dened as follows:
GP
s
BI B GI I
PG G
l I mv
ω
= +×
(32)
On the other hand, the relative angular momentum of mB is dened
according to:
( )
GP
s
BI B PI P
PG
P rel
l I mv
ω
= +×
(33)
According to White [3], the angular momentum of mB around its
center of mass is given by,
=
BI B GI
GG
lI
ω
(34)
and around the point P, we have
Figure 6: An example of the System B.
Adv Robot Autom, an open access journal
ISSN: 2168-9695
Citation: Jebelli A, Yagoub MCE, Dhillon BS (2017) Modeling of an Autonomous Underwater Robot with Rotating Thrusters. Adv Robot Autom 6: 162.
doi: 10.4172/2168-9695.1000162
Page 5 of 10
Volume 6 • Issue 1 • 1000162
GP
s= +×
BI B GI I
PG G
l I mv
ω
(35)
leading to the following equality:
( )
GP
s= +×
BI B GI I
G GP
P rel
l I mD s
ω
(36)
( )
1 22
GB 1 G B 2
ss= +× + ×
BI B BI I I
B BG
l I mv m v
ω
(37)
A. Euler law:
e Euler law says that the resultant momentum exerted on mB is
equal to the time derivative of angular momentum in the inertia system
[7]
P I BI
BP
M Dl=
(38)
( )
( )
1 22
GB 1 G B 2
ss
= +× +×
B I B BI I I
B BG
M D I mv m v
ω
(39)
which leads to:
( ) ( ) ( )
( ) ( ) ( ) ( )
=+ × +×
+× + × +×
I B BI B B BE B EI BE BE
B BE EI B BE BE B EI EI B EI
DI I D I
I I II
ωω ωω ω
ωω ωω ωω ω
(40)
Each term can be calculated separately,
( ) ( ) ( )
( ) ( ) ( ) ( )
= + × +×
+× +×
IB BI B B BE B EI BE BE
B BE EI B BE BE B EI EI B EI
DI I D I
I I II
ω ω ωω ω
ωω ωω ωω ω
(41)
( ) ( )
( )
11
GB 1 GB 1
ss× = × ×
I I BI E EI
B BE BE
D mv m D s s
ωω
( ) ( )
( )
( )
( )
( )
( ) ( )
( )
( )
( )
( )
11
1
11
11
GB 1 GB 1
GB 1
1 GB 1 GB
1 GB 1 GB
ss
s
ss
ss
× = × ×
= + × ×
= ×× ++ ××
×××+ ×××
I I BI E EI
B BE BE
BE EI E EI
B BE
BE E EI E
BB
BE EI EI EI
BE BE
Dmv m D s s
mv s
m vm v
m sm s
ωω
ωω ω
ωω
ωω ωω
(42)
( )
( )
( ) ( )
( )
1
11
11
GB 1
1 GB 1 GB
1 GB 1 GB
s
ss
2s s
×=
=× + × ×+
××+ ×××
II
B
BE BE E
BB
EI E EI EI
B BE
mDv
m Dv m v
m vm s
ω
ω ωω
(43)
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
22
22
22
22
22
GB 2
2 GB GB
2 G 2 GB
2 GB 2 G
2 GB 2 GB
s
ss
s
s
ss
=×
+× ×
= ×+ × ×
× ×+ ×
+
×+
× ××+ × ××
II
G
B BI I
BE
BE BE E
BB
EI E B EI
B BE
BE EI EI EI
BE BE
Dmv
m D Ds
mv v m v
m v mv s
m sm s
ω
ω
ωω
ωω ωω
(44)
( ) ( )
( )
( ) ( )
( ) ( )
( )
( )
22
22 2
22
2
2
++× +× + +×
=+ ×+ + ×
+ +× +× + +×
+ ×+ × ×
B BE EI BE EI B E BE
G GB B
I I B B BE EI BE BE EI
G G GB
B BE EI BE EI B E BE
G GB B
E EI E EI EI
B B BE
v s Dv
Dv a D s
v s Dv
vv s
ωω ωω ω
ωωω ωω
ωω ωω ω
ω ωω
( )
( ) ( )
( ) ( )
( )
2 2 22
22 2
22
2
2
2
=
+ ×+ × ×+ ×
× × +× × +
××+××+ +×
+ ×+ ×
+
×
B B BE EI BE BE B
G GB GB G
EI B BE BE BE EI
G GB GB
EI BE EI EI B E BE
GB GB B
E EI E EI EI
B B BE
aD s s v
vs s
s s Dv
vv s
ωωω ω
ω ωω ωω
ωω ωω ω
ω ωω
(45)
( ) ( )
( )
( )
( )
( ) ( )
( )
( )
( )
( )
( )
( )
( )
2 2 22 2 2
2 22 2
2 22 2
2 22
22 2
22
GB 2 2 GB 2 GB
2 GB 2GB
2 GB 2 GB
2 GB 2GB
2 GB
2 GB 2 GB
sss
s 2s
2s s
ss
s
ss
×= ×+ × × +
× ××+ ××
+ ××+ ××× +
××× + ×
×× + ×××
+ ×+ ×
I I B B BE
G G GB
EI BE BE B
GB G
EI B BE BE
G GB
BE EI
GB
EI BE EI EI
GB GB
BE B
B
mDv m a m D s
m sm v
m vm s
m sm
sm s
m Dv m
ω
ωω ω
ω ωω
ωω
ωω ωω
ω
( )
( ) ( )
( )
22
2 GB 2 GB
2s s
×
+××+ ×××
EE
B
EI E EI EI
B BE
v
mvm s
ω ωω
(46)
( )
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )
( )
( )
2
×=
=+× ×+× + ×
= ×+ × × + × ×
+×× ×+ ××
+ ×× ×+ × +
× ×+ × ×
+ ×××
II
BGB B
B EI E BI I
B G GB B BE B GB B
B E B EI BE
BG B BG BE B GB
E EI EI EI E
B B GB BE B GB B
EI EI B E
B GB BE B GB B B GB
BE E EI E
B B GB B
EI EI
B GB BE
Dms v
m v s v s ms a
mv v mv s m s
vms smsv
m s s ms Dv ms
v ms v
ms s
ωω
ωω
ωω ω
ωω
ωω
ωω
(47)
By combining the above terms, we obtain:
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( )
( )
( )
= + × +× +
×+×+×+
×+ × × + × ×
+ ×× ×+ ×
×+ × × × +
××
×+ × ×
B B B BE B EI BE BE B BE
B
EI B BE BE B EI EI B EI
B E B EI BE
BG B BG BE B GB
E BE EI EI
B B GB BE B
E EI EI
GB B B GB BE
B E BE E
B GB B B GB B B GB
EI E EI
B B GB
M ID I I
I II
mv v mv s m s
vm s s m
sv m s s
ms Dv ms v ms
v ms
ω ωω ω ω
ωωωωωω
ωω
ωω ω
ωω
ω
ωω
( )
( )
( )
( )
( ) ( )
( )
( )
( )
( )
( )
( )
22 2 2
2 22 2
2 22
22 2
22
22
22
2
22
2
×+
×+ × × +
× ×+ × ×+
××× + ×
×× + ×××
EI
BE
B B BE
GB G GB GB
BE B EI B
GB G GB G
BE BE
GB GB GB
EI BE EI EI
GB GB GB
s
ms a ms D s
ms v ms v
ms s ms
s ms s
ω
ω
ωω
ωω
ωω ωω
(48)
us,
( )
12
1 22
12
12 2
BB B
B G B GB G B G B
B BB
GB G B G
m v m D s D ms m s
mDs mDs mv
= = +=
+=
(49)
Adv Robot Autom, an open access journal
ISSN: 2168-9695
Citation: Jebelli A, Yagoub MCE, Dhillon BS (2017) Modeling of an Autonomous Underwater Robot with Rotating Thrusters. Adv Robot Autom 6: 162.
doi: 10.4172/2168-9695.1000162
Page 6 of 10
Volume 6 • Issue 1 • 1000162
Making the main equation as:
( )
( ) ( )
( ) ( )
( ) ( ) ( )
( )
( ) ( )
( )
( )
( )
22
22 2 2
2
22
22
2
2
× +×
× + × ×+
×× ×− ×
×+ × × × +
× + × ×− ×
×+ × × × +
×+ × × +
××
EI B EI B E B
GB G
EI BE E
BE B GB B
BE EI EI
B GB BE B
E BE BE
GB B B GB BE
B E BE E
B GB B B GB B B GB
EI E EI BE
B B GB BE
B B BE
GB G GB GB
EI
GB
I mv v mv
sm sv
m s sm
sv m s s
ms Dv ms v ms
v ms s
ms a ms D s
ms
ωω
ωω
ωω ω
ωω
ω
ω ωω
ω
ω
( ) ( )
( )
( )
( )
( )
( )
( )
22 2
2 22
22 2
2
22
2
2
2
× ×−
××× + ×
×× + ×××
B EI B
G GB G
BE EI
GB GB GB
EI BE EI EI
GB GB GB
v ms v
ms s ms
s ms s
ω
ωω
ωω ωω
(50)
which can be expressed as a set of dynamic equations
[ ] [ ]
[ ]
{ }
[ ]
[ ]
()
2
22
B
B
B
2
22
m
m
2m 2
2m
m 2m
×
××
×
×
××
×
  
= −
  
  
  
  
+−
  

 
+ ××
 

 
+− ×


BB
BB
E BE
B B B GB
BB B B
B
E BE BE BE
B B GB
BB B B
B
EI E EI BE
B B GB
BB BB
B
B BE EI EI
G B GB
BB
B EI B
GG
f v ms
v ms
vm s
vm s
av
ω
ωω ω
ω ωω
ω ωω
ω
[ ]
()
B
m
 



×

BB
EI
BB
EI
BE
s
ω
ω
(51)
and,
[ ]
[ ]
{ }
{ }
2
2
×
×××
××
×
×
×
   
= +
   
 
 
    
+
    

   
   


+
+
+

B B BB
B
B E B BE
B B GB B
BB B B B
BBE E B BE BE
B GB B
BBB BBB
B EI BE EI B BE
B
BB B B
B EI BE B E
GB
B
BE
B
Mms v I
ms v I
II
I mv v
m
ω
ω ωω
ωω ω ω
ωω
ω
[ ]
{ }
[ ]
{ }
[ ] [ ]
[ ]
22
22 2 2
2
22
2
2
×
××
××
×
××××
×
×× × ×
 
+
  
  
−+
  

 
+
 

  
  
−+
  
  
BB
BB
E EI
GB B B BE
BBB
BB
BE EI E
GB B GB B
BB
BB B
BEI E BE
B GB B GB GB
BB B B
B BB
B BE BE EI
GB G GB GB
svm s
s m sv
ms v ms s
ms v ms s
m
ω
ωω
ωω
ωωω
()
[ ]
()
[ ]
()
[ ]
()
[ ] [ ]
()
( )
22
2
22 2 2
2
2
22
2
×
××
 
 
 
 

 
+ ×
  
 
×− ××
 
  
×+ ×××
  
 

+ ×+ × ×

 
BB
BB
BI BE
GB GB
B
BBB
EI B EI B
G
BB
BB
EI EI
BE B GB
B BB
BB B
EI EI EI
BE B GB BE
BB B B
B EI B
GB G GB G
ss
Imv
sm s
s ms s
ms a ms v
ωω
ωω
ωω
ω ωω
ω
()
()
( )
22
2
 
 
+ ×××
 
 
B
BB
BB
EI EI
GB GB
ms s
ωω
Simulation Platform
e equations obtained in the previous section and describing the
movement of the vehicle were coded in MATLAB. e Runge-Kutta
method was used to solve them with a time step size of 20 ms. is
value was tuned to simultaneously assure relatively fast convergence
and acceptable accuracy. Aer completing the coding in MATLAB, the
Lab View soware was used to design a Graphical User Interface (GUI)
as well as a virtual reality (VR) to virtually observe the maneuvers of
the vehicle.
Initial conditions
As shown in Figure 7, this part should be set before running the
simulation. Here the user can set the initial conditions of the vehicle
including the following cases:
Initial latitude and longitude and height
Initial velocity
Initial physical angle
Initial angular velocity
Simulation monitoring
As displayed in Figure 8, it is possible to perform some of the
settings related to the platforms including setting the frequency of the
loops or choosing the processor.
Setting the simulation parameters
As seen in Figure 9, this part includes four main components:
Setting the environmental parameters (Figure 10). In this part, the
simulation parameters include:
Determining the Greenwich longitude and the radius of the earth.
Determining the rotational velocity of the earth.
• Determining the earth chamfer: If this value is zero, the Earth
is assumed a perfect sphere.
Determining the uid density in which the vehicle moves.
Setting the volume mass parameters of the vehicle: this part, as
seen in Figure 11, consists in
the vehicle.
Figure 7: Initial conditions.
•
determining the total volume of
Adv Robot Autom, an open access journal
ISSN: 2168-9695
Citation: Jebelli A, Yagoub MCE, Dhillon BS (2017) Modeling of an Autonomous Underwater Robot with Rotating Thrusters. Adv Robot Autom 6: 162.
doi: 10.4172/2168-9695.1000162
Page 7 of 10
Volume 6 • Issue 1 • 1000162
Figure 8: Platform settings.
Figure 9: Settings parameters.
Determining the total volume and the moment of inertia of the
entire vehicle regardless of the moving mass.
Determining the mass and the moment of inertia of the moving
mass.
Setting the hydrodynamic parameters
In this part, the hydrodynamic parameters of the device are set by
the user. is part includes three main components:
Determining linear drag coecients (Figure 12a)
Determining quadratic drag coecients (Figure 12b)
Setting the added mass elements (Figure 13).
Locating the critical points versus the reference point
In this part the user should determine the location
contains the critical points of the vehicle versus the
B. ese important points include
e location of the center of the moving mass.
e buoyancy location,
e pressure location.
e le and right engine force location.
Control and displays
is part is composed of three sets that include (Figure 15):
e physical angular velocity.
•
vector,
reference
which
point
(Figure 14):
Adv Robot Autom, an open access journal
ISSN: 2168-9695
Citation: Jebelli A, Yagoub MCE, Dhillon BS (2017) Modeling of an Autonomous Underwater Robot with Rotating Thrusters. Adv Robot Autom 6: 162.
doi: 10.4172/2168-9695.1000162
Page 8 of 10
Volume 6 • Issue 1 • 1000162
e latitude and longitude.
e physical angle versus the North East Down (NED)
coordinate system.
e vertical velocity that resents the rate of reduced or increased
height.
e height.
VR control
Figure 10: Environmental parameters.
Figure 11: Setting device parameters.
Figure 12: Linear drag and Quadratic drag.
Two controllers have been implemented to control the VR
environment, including the perspective and setting the distance
between the two cameras located on the back of the device (eye) versus
the body.
Vehicle control tools
Using these tools, the user can control the device in the GUI
environment, i.e., determining:
e right engine speed.
Figure 13: Added mass.
Figure 14: The critical points’ location vector.
Figure 15: Control.
Adv Robot Autom, an open access journal
ISSN: 2168-9695
Citation: Jebelli A, Yagoub MCE, Dhillon BS (2017) Modeling of an Autonomous Underwater Robot with Rotating Thrusters. Adv Robot Autom 6: 162.
doi: 10.4172/2168-9695.1000162
Page 9 of 10
Volume 6 • Issue 1 • 1000162
Figure 16: VR Environment.
Figure 17: Displayers status (part 1 presents the main data of the vehicle, and
part 2 presents the vehicle control tool).
Figure 18: The status of displayers
Figure 19: Status of displayers.
e right engine speed relative to the body.
e le engine speed.
e le engine speed relative to the body.
e moving mass location.
Data
In this part, the diagrams of the movement data are visible to the
operator. ese data include:
e velocity.
e rotational velocity.
e body angle to the magnetic north and horizon.
e acceleration.
VR environment
In this part, a three-dimensional model of the body is created
in 1-scale. is graphic model is directly connected to the equation
solution subsystem and displays the movement changes of the vehicle
including the translational and rotational movements (Figure 16).
Results
Figure 17 shows the status of displayers in a position where the robot
depth’s is 5 m, with a constant speed of 0.2 m/s and with a direction
angle of 30° towards the north and with an horizontal movement.
Figure 18 shows the displayers in a status where the robot is at rest,
at a depth of 2 m, and with a 180° direction.
Adv Robot Autom, an open access journal
ISSN: 2168-9695
Citation: Jebelli A, Yagoub MCE, Dhillon BS (2017) Modeling of an Autonomous Underwater Robot with Rotating Thrusters. Adv Robot Autom 6: 162.
doi: 10.4172/2168-9695.1000162
Page 10 of 10
Volume 6 • Issue 1 • 1000162
Figure 20: Status of displayers.
Figure 20 shows the status of the displayers in a position where
the robot is changing its depth from 5 m to 7 m. e image shows the
moment when it has reached the depth of 6.4 m.
Conclusion
is work deals with the modeling of the operation of an
underwater robot with two rotating thrusters. To eciently control
the device movement at dierent depths with the help of a mass
shier, an accurate monitoring system was implemented. is virtual
environment gives the user the ability to observe and adjust, in real
time, the required parameters for controlling the robot in dierent
situations. It also plays an eective role in enhancing the performance
of the underwater by using the obtained results to plan/prepare the
robot for future tasks.
References
1. Regan FJ, Anandakrishnan SM, Regan NSF, Anandakrishnan S (1993)
Dynamics of Atmospheric. American Institute of Aeronautics and Astronomy.
2. Goodman R, Zavorotniy (2009) Newton's Law of Universal Gravitation, New
Jersey Center for Teaching and Learning, NJ, USA.
3. White FM (2015) Fluid Mechanics, (8th edn). McGraw-Hill Education, USA.
4. Lee SK, Joung TH (2014) Evaluation of the added mass for a spheroid-type
unmanned underwater vehicle by vertical planar motion mechanism test. Int J
Naval Architect Ocean Eng 3: 174-180.
5. Galdi GP (2011) An Introduction to the Mathematical Theory of the Navier-
Stokes Equations. Springer Monographs in Mathematics.
6. Ridao P, Batlle J, Carreras M (2001) Dynamics Model of an Underwater
Robotic Vehicle. University of Girona, Spain.
7. Chuan ST (1999) Modeling and simulation of the autonomous underwater
vehicleAutolycus. Master's Thesis, Massachusetts Institute of Technology.
Figure 19 shows the displayers in a position where the device is
moving at a depth of 4 m, a speed of 0.35 m/s, and at the same time
spinning to the right side to adjust the direction angle to 160°. e
image shows the moment where the current direction angle of the
device is 99.1°.
Adv Robot Autom, an open access journal
ISSN: 2168-9695
... In [10], the authors present the kinematic and dynamic equations of an underwater robot model with two rotating thrusters. The proposed equations are implemented using the virtual environment realized in MATLAB and Lab-VIEW. ...
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The research of nonlinear hydrodynamic characteristics of the propulsion and steering complex (PSC), which influence the accuracy of the plane trajectory motion of an autonomous underwater vehicle (AUV), is carried out. During the underwater vehicle curvilinear motion, its PSC operates in an oblique incident water flow. This leads to a decrease in the PSC thrust force and negatively affects the controlled trajectory motion of the underwater vehicle. The research was conducted for a specific type of AUV for the plane curvilinear motion mode. The mathematical modeling method was chosen as the research method. To this end, the well-known AUV motion mathematical model is supplemented by the control system that simulates (mimics) its trajectory motion. The developed model consists of four main units: an AUV improved model; the vehicle speed setting unit; the nozzle rotation angle control unit; the unit containing the AUV pre-prepared motion trajectories. The research results of the AUV hydrodynamic parameters for several typical trajectories of its motion are presented. The investigated parameters include the following: the required nozzle rotation angle; the vehicle actual motion trajectory; the vehicle velocity; the propeller shaft moment; the propeller thrust force. As a result of the conducted researches, the dependence diagram of the propeller thrust force on the AUV nozzle rotation angle in the speed range from 0.2 m/s to 1 m/s and during the nozzle rotation in the range of up to 35° was constructed. A three-dimensional matrix, which describes the dependence of the propeller thrust force on the incident water flow angle and velocity of the vehicle, was created. The obtained dependence can be used in the synthesis of automatic control systems regulators of AUV plane manoeuvering (shunting) motion of increased accuracy.
... Reducing the size of these AUVs will increase their availability and maneuverability. However, it is very difficult and challenging to embed electrical and mechanical parts inside these AUVs, thus leading to high costs as well as hard challenges for AUV designers [1][2][3][4][5]. ...
... We first designed an Autonomous Underwater Vehicle (AUV) based on the motion of the white sided dolphins [2][3][4]. ...
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This paper shows added mass and inertia can be acquired from the pure heaving motion and pure pitching motion respectively. A Vertical Planar Motion Mechanism (VPMM) test for the spheroid-type Unmanned Underwater Vehicle (UUV) was compared with a theoretical calculation and Computational Fluid Dynamics (CFD) analysis in this paper. The VPMM test has been carried out at a towing tank with specially manufactured equipment. The linear equations of motion on the vertical plane were considered for theoretical calculation, and CFD results were obtained by commercial CFD package. The VPMM test results show good agreement with theoretical calculations and the CFD results, so that the applicability of the VPMM equipment for an underwater vehicle can be verified with a sufficient accuracy.
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This report takes an in-depth look at the physical laws governing the behavior of an underwater robotic vehicle. A complete 3D kinematics and dynamics matrix-based model is also presented. This model uses the dynamics equations which describe the movement of a rigid body and the main hydrodynamic equations affecting this movement through a fluid environment. This model depends on a set of physical parameters which are obtained from experimentation. This report goes on to describe an identification methodology for slow Underwater Vehicles. As an example, it presents the identified model of an underwater robotic vehicle called GARBI. Real experiments are provided demonstrating the feasibility of the presented model as well as the identification methodology. 1
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Under the assumption that the gravitational force is a central force and diminishes with distance, Newton's inverse square law is shown to be the only one for which all simple finite trajectories are closed orbits.
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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Ocean Engineering, 1999. Includes bibliographical references (p. 128-129).
  • F J Regan
  • S M Anandakrishnan
  • Nsf Regan
  • S Anandakrishnan
Regan FJ, Anandakrishnan SM, Regan NSF, Anandakrishnan S (1993) Dynamics of Atmospheric. American Institute of Aeronautics and Astronomy.
  • F M White
White FM (2015) Fluid Mechanics, (8th edn). McGraw-Hill Education, USA.