dynamic modeling and control of an autonomous underwater vehicle_library submission
DESCRIPTION
Masters Thesis on guidance and control of underwater vehiclesTRANSCRIPT
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Dynamic Modeling and Control of an Autonomous
Underwater Vehicle (AUV)
Submitted in partial fulfillment of degree of
Bachelor and Master of Technology in
Aerospace Engineering
By
Chintan S. Raikar
Roll no: 08d01007
Under the Guidance of
Prof. Leena Vachhani
Prof. Hemendra Arya
Department of Aerospace Engineering
Indian Institute of Technology Bombay
June, 2013
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Abstract
In recent years, there have been intensive efforts toward the development ofunderwater vehicles. Autonomous underwater vehicle (AUV) has potential application in
marine exploration, defense and reconnaissance and oil industry. The model of underwater
vehicles strongly affects the dynamic performance as well as accurate control, navigation
and guidance of underwater vehicles. Accurate modeling of underwater vehicle is therefore
of prime importance for precision control and execution of path planning missions. The
model of underwater vehicles strongly affects the dynamic performance as well as accurate
control, navigation and guidance of underwater vehicles. This work deals with dynamic
modeling and control of Matsya AUV in which the hydrodynamic derivatives are determined
both theoretically and experimentally, based on the assumption that the motions in
different directions are decoupled. The dynamic model generated has been verified
experimentally and dynamic model is linearized using Jacobian method. Various operating
points are chosen for linearization and a PID controller is developed to control the heave and
heading motions of vehicle on the linearized model. Comparison between linear and non
linear models has been reflected in the simulations.
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Acknowledgement
I would like to express my sincere gratitude to Prof. Leena Vachhani and Prof.
Hemendra Arya for their constant guidance during the course of this project. I would like this
opportunity to deluge my deepest gratitude to them for giving me such an innovative and
challenging project. They have been always there to discuss about our ideas and their moral
support always encouraged me carrying out our project work.
I would also like to thank AUV-IITB team of IIT Bombay for their hard work and
dedication shown in development of Matsya underwater vehicle. I would thank Anay Joshi,
Sneh Vaswani for their constant help while conducting experiments. All the members of the
team have shown immense support for this project without which completion would not
have been possible. I would extend my gratitude towards my lab mates Satyaswaroop,
Shripad Gade, G Sai Jaideep from Controls and Dynamics Laboratory for very interesting
discussion regarding this topic.
At last I would like to acknowledge my parents for their constant moral support during
testing times. This project has added new dimension to my approach while working on
problems and I would take this experience to further goals and objectives in my career.
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Table of Contents
1. Introduction....................................................................................................................12. Development of Dynamic Model ...................................................................................5
2.1 Rigid Body Dynamics ................................................................................................. 52.1.1 Translational Motion .................................................................................. 6
2.1.2 Rotational Motion ...................................................................................... 7
3. Derivation of Dynamic Matrices ..................................................................................103.1 Mass and Inertia Matrix .......................................................................................... 10
3.2 Coriolis and Centripetal Matrix ............................................................................... 10
4. Hydrodynamic Forces and Moments............................................................................124.1 Radiation Induced Forces ........................................................................................ 12
4.1.1 Added Mass and Inertia ............................................................................ 12
4.1.2 Added Coriolis and Centripetal Matrix ...................................................... 13
4.1.3 Hydrodynamic Damping ........................................................................... 14
4.1.3.1 Potential Damping ..................................................................... 15
4.1.3.2 Skin Friction ............................................................................... 15
4.1.3.3 Wave Drift Damping ................................................................... 15
4.1.3.4 Damping Due to Vortex Shedding .............................................. 15
4.1.4 Restoring Forces and Moments ................................................................. 16
5. Calculation of Hydrodynamic Derivatives.....................................................................175.1 Strip Theory for Estimating Hydrodynamic Derivatives .......................................... 17
6. Dynamic Model of Matsya 1.0 and Matsya 2.0............................................................. 207. Parameter Calculations for Matya................................................................................ 22
7.1 Assumptions on AUV Dynamics ............................................................................. 22
7.2 Determination of Dynamic Model of Matsya ......................................................... 23
7.3 Propulsive Forces and Moments ........................................................................... 24
7.4 Estimation of damping Coefficients ....................................................................... 25
8. System Identification for calculation of damping parameters......................................278.1 Evaluation of damping parameters for surge, heave and sway .............................. 28
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8.2 Evaluation of damping parameters for roll, pitch and yaw axes ............................... 31
8.2.1 Pitch damping calculation ......................................................................... 32
8.2.2 Roll damping calculation ........................................................................... 33
8.2.3 Yaw damping calculation .......................................................................... 34
9. Validation of Dynamic Model.......................................................................................359.1 Heave Experiments .................................................................................................. 35
9.2 Surge Experiments ................................................................................................... 35
9.3 Sway Experiments .................................................................................................... 36
9.4 Open loop Roll experiments ..................................................................................... 36
9.5 Open loop Pitch Experiments ................................................................................... 37
9.6 Open loop surge and depth control ......................................................................... 38
9.7 Simultaneous Roll and pitch excitaion ...................................................................... 39
10.Linearization of Dynamic Model...................................................................................4110.1 Formulation of Jacobian Matrix ............................................................................. 41
10.2 Jacobian for 6 DOF systems.................................................................................... 43
10.3 Linearization of dynamic model of Matsya 2.0 ....................................................... 44
10.4 Open loop simulation of linearized model .............................................................. 46
11.Controllability Analysis of Linear model.......................................................................5411.1 Controllability ........................................................................................................ 54
11.2 Design of PID Controller for Depth control ............................................................. 55
11.3 Design of PID Controller for Yaw control ................................................................ 56
12.Conclusions...................................................................................................................5813.Future Work..................................................................................................................6014.Bibliography..................................................................................................................6115.Appendix1.....................................................................................................................64
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LIST OF FIGURES
Figure 1 Inertial earth fixed frame XYZ and body fixed frame X0Y0Z0for a rigid body ..................6
Figure 2 Matsya 1.0...................................................................................................................20
Figure 3 Matsya 2.0...................................................................................................................21
Figure 4 Curve fit for surge drag ................................................................................................30
Figure 5 Curve fit for sway drag .................................................................................................30
Figure 6 Curve fit for heave drag ...............................................................................................31
Figure 7 Open Loop Pitch response from wet tests ....................................................................32
Figure 8 Open loop roll stability from wet tests .........................................................................33
Figure 9 Open loop yaw identification .......................................................................................34
Figure 10 Damping Parameters .................................................................................................34
Figure 11 Open loop roll experiments ........................................................................................36
Figure 12 open loop pitch experiments ......................................................................................37
Figure 13 Simultaneous Surge and depth control .......................................................................38
Figure 14 Observed response for simultaneous heave and depth control ...................................39
Figure 15 Open loop simulation of simultaneous pitch and roll ..................................................39
Figure 16 Observed response for Simultaneous pitch and roll in open loop ................................40
Figure 17 Open loop roll performance by linearized model 2 .....................................................49
Figure 18 Surge and depth performance for linearized model ....................................................50
Figure 19 Open loop positive roll and negative pitch .................................................................51
Figure 20 Open loop roll with surge and pitch............................................................................52
Figure 21 Surge and Depth control with an initial positive pitch ................................................53
Figure 22 depth control comparison for linear and non linear model .........................................56
Figure 23 Yaw command of 90 degrees .....................................................................................57
Figure 24 Definition of Reference frames ...................................................................................65
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LIST OF TABLES
Table 1 Strip theory estimates for 2D surface ............................................................................18
Table 2 Dimensions and vehicle specifications for Matsya 1.0 and 2.0 .......................................21
Table 3 Hydrodynamic parameters for Matsya ..........................................................................24
Table 4 Damping Coefficients for Matsya ..................................................................................26
Table 5 Surge Drag force form wet tests ....................................................................................28
Table 6 Sway Drag force from wet tests ....................................................................................29
Table 7 Heave Drag force from wet tests ...................................................................................29
Table 8 Curve fitted parameters ................................................................................................31
Table 9 Open loop Heave tests ..................................................................................................35
Table 10 Open loop surge tests..................................................................................................35
Table 11 Open loop sway tests ..................................................................................................36
Table 12 Eigenvalues for linearized system for variable surge speeds ........................................45 Table 13 Eigenvalues for linearized system for variable surge speeds ........................................45
Table 14 Eigenvalues for linearized system for variable surge speeds ........................................46
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Nomenclature
AUV Autonomous Underwater Vehicle
Body frame x co-ordinate
Body frame y co-ordinate Body frame z co-ordinate Angle of rotation about the xBaxis Angle of rotation about the yBaxis Angle of rotation about the xBaxis Body frame state vector
Inertial frame state vector
velocity state vector corresponding to the vehicle1 Vector defining the linear velocities2 Vector defining the angular velocities Position along the x axis Position along the y axis Position along the z axis
1 Vector defining the position of vehicle
2 Vector defining the attitude in Euler angles1 Position vector transformation matrix2 Velocity vector transformation matrix mass and inertia matrix() Coriolis and centripetal matrix() hydrodynamic damping matrix
gravitational and buoyancy vector
External force and torque vector External force0 Absolute angular momentum0 Velocity of particle
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Time derivative in inertial frame Time derivative in body frame Angular velocity
centre of gravity vector
Mass density of a rigid body0 External Moment Force applied to vehicle along the x axis Force applied to vehicle along the y axis Force applied to vehicle along the z axis Torque applied to vehicle along the x axis
Torque applied to vehicle along the y axis Torque applied to vehicle along the z axis The vehicles linear velocity along the x axis The vehicles linear velocity along the yaxis The vehicles linear velocity along the zaxis vehicle roll rate vehicle pitch rate vehicle yaw rate x co-ordinate of centre of gravity vector with respect to origin y co-ordinate of centre of gravity vector with respect to origin z co-ordinate of centre of gravity vector with respect to origin x co-ordinate of centre of buoyancy vector with respect to origin y co-ordinate of centre of buoyancy vector with respect to origin z co-ordinate of centre of buoyancy vector with respect to origin rigid body mass matrix added mass matrix rigid body Coriolis and centripetal matrix added Coriolis and centripetal matrix External force and torque vector of rigid body
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Kinetic energy of vehicle Kinetic energy of added mass Force along the x axis due to added mass
Force along the y axis due to added mass
Force along the z axis due to added mass Torque along the x axis due to added mass Torque along the y axis due to added mass Torque along the z axis due to added mass Potential damping term Damping due to skin friction Damping due to wave drift Damping due to vortex shedding Linear damping matrix Quadratic damping matrix Speed of the vehicle Reynolds Number Characteristic length Volume of fluid displaced1 1 Position of thruster with respect to origin
PWM Pulse width modulation
Equilibrium state vector , , , , , Equilibrium values of , ,, , ,
(t) Input vector() Nominal input vector System matrix Control Matrix Output Matrix Feed forward matrix Jacobian matrix
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Linear damping parameter in x-direction Quadratic damping parameter in x- direction Added inertia/mass in x-direction due to surge speed
,,,,, Non linear functions describing dynamic of AUV in
, , ,,,
Controllability matrix
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1.INTRODUCTION
In recent years, autonomous underwater vehicles (AUVs) have an increasingly
pervasive role in underwater research and exploration [1]. These vehicles generally have a
streamlined, torpedo-shaped body, and are intended for long-distance missions where their
low drag enables high speeds and coverage of a large distance. Hydrodynamic fins are used
to direct the vehicle and rely on forward motion to generate the forces required to change
orientation. Some AUVs use a combination of fins and through-body thrusters for control of
the vehicle. Through-body thrusters enable orientation control at low speeds, w hile the fins
provide control at higher speeds. Examples of these AUVs include the NPS Aries [2], Otter[3],
and C-SCOUT [4]. Inspired from the above examples, Matsya is an autonomous underwater
vehicle (AUV) developed by a team of students at the Indian Institute of Technology Bombay
(IITB). Developed over a design cycle of seven months, Matsya is capable of localizing itself in
an underwater environment and complete some predefined real life tasks for the Robosub
2012 competition. The thesis investigates the Matsya prototype as a basic test bench for
design and validation of dynamic model and thereby conducts some experiments on real life
situations.
Underwater vehicles have immense applications such as underwater surveillance,
marine life exploration, pipe line repairs et al. Today, Indias interest in oil and gas
exploration and fisheries is well known [5]. Development of unmanned underwater vehicle is
crucial for future of oil and gas exploration. Whitecomb [6] says that low cost AUV and ROV
systems are about to replace manned hydrographic survey launches in deep sea exploration.
Besides these, the military applications of underwater vehicles are numerous especially in
underwater reconnaissance and intelligence gathering operations [6]. The Maya AUV of India
is a recent advancement by Defense and Research Organization (DRDO) in oceanographic
studies and environmental monitoring of coastal waters and estuaries [7].
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Underwater vehicles are designed to work over large number of operating points.
Aircrafts and submarines are usually linearized about different constant forward speeds.
Linear control theory and gain scheduling techniques are applied to each of these operating
points. However, such models do not consider nonlinearities caused by quadratic drag and
lift forces. A linear approximation of non linearity will have both structural and parametric
non linearity which in case of mechanical systems is directly included in the model. This work
considers non linear modeling and control of autonomous underwater vehicles [7,8].
Most open-frame underwater vehicles have the following characteristics: two or three
symmetry planes, low operation velocities (< 1m/s), passively stable in roll and pitch angular
motions, and creeping and uncoupled motions. For this type of underwater vehicle, the 6-
DOF motion dynamic equations might be simplified [9]. As a result, an approximate
uncoupled scalar dynamic model is obtained which is sufficiently precise for control system
design.
The design and development of an autonomous undersea vehicle (AUV) is a
complex and expensive task. If the designer relies exclusively on prototype testing to
develop the vehicles geometry and controllers, the process can be lengthy and poses the
additional risk of prototype loss [10]. Every design iteration involves changes to the
prototype vehicle which may take days, followed by further testing. As a result, designers of
AUVs rely increasingly on computer modeling as a design tool[12], particularly for the initial
phases of vehicle development. An AUV simulation environment may include a number of
elements such as a collision detection module, a mission planner, a controller and a
dynamics model.
The function of the dynamics model is to represent the vehicles interact ion with the
fluid in which it moves [11]. Use of such model allows the designer a means for determining
the inherent motion characteristics of a proposed vehicle before prototyping. Also, a
controller can be devised to improve the vehicles natural behavior. However, the usefulness
of the results is predicated on the ability to model the vehicle accurately when little or no
experimental data is available. This, in turn, requires a thorough understanding of the
vehicles dynamics which can be broken down into three sub-tasks:
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i) Derivation of mathematical equation governing the motion of vehicleii) Determination of hydrodynamic characteristics of a vehicleiii) The computational solution of the system of equations, for a known set of
control inputs, to obtain the ensuing motion
The hydrodynamic characteristics of AUVs have been quantified through the use of
hydrodynamic derivatives, which are determined using analytical, empirical or
experimental methods [12,13]. The hydrodynamic derivatives are coefficients in the
mathematical model which quantify the forces acting on the vehicle as a function of its
attitude and motion. A number of methods have been proposed for the determination of
hydrodynamic coefficients [12, 13]. They can be broadly classified into test-based and
predictive methods. The former include direct experimental determination based on wind-
tunnel or tow-tank model tests [22]; as well as testing of full-size captive vehicles [10].
System identification techniques [17, 18] are a less direct, but perhaps more efficient
test-based method and can be applied to free-swimming model or full-size vehicle
tests. An overriding disadvantage of the above methods is the need for a vehicle, as well
as laboratory or in-field testing facilities. These are often not available, either for
reasons of cost or, simply, because the vehicle has not yet been constructed. Predictive
methods offer an attractive alternative to test based methods when the vehicle is still
in the design stages, or when costs prohibit a full-scale testing program. Predictive
methods are most likely to yield reasonable results when applied to streamlined
vehicles since the behavior of these is more easily predicted [13, 16].
1.1Outline of the reportChapter 2 focuses on the theory behind development of dynamic using first principles.
Chapters 3 and 4 describe the definition of various parameters of the dynamic model and
discuss in detail about the hydrodynamic forces and moments exerted on the vehicle. The
evaluation of added inertia parameters of dynamic model and the assumptions involved
are outlined in Chapter 5. Chapter 6 reflects upon the Matysa vehicle of IIT Bombay and
its two variants. Chapter 7 describes the methods used to evaluate parameters for Matsya
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1.0. Evaluation of damping parameters using basic system identification techniques have
been elaborately portrayed in Chapter 8. Chapter 9 includes the experimental results for
validation of dynamic model along with comparison with simulations. The dynamic model
is further linearized and stability analysis is shown in chapter 10. A basic PID controller
design and its results are discussed in Chapter 11. Chapter 12 and 13 describe the overall
brief conclusions from the project along with work that can be taken up in future.
The stage 1 of the project dealt mainly with understanding the development of
dynamic model and evaluation of the dynamic parameters. Methods to evaluate the added
inertia and damping parameters were surveyed. A crude program for simulation of AUV
dynamics as developed. Taking Matysa 1.0 as the test bench, dynamic model was
developed. However, much of damping parameters were taken from vehicles of similar
shape. The dynamic model was simulated for various open loop and closed loop
conditions.
For the second stage, focus was on accurate determination of parameters of dynamic
model. Hence, underwater tests have been conducted for evaluation of damping
parameters. Since, the development of new version of Matysa was in pipeline, all the
parameters have been reevaluated for Matsya 2.0. Also, underwater tests have been
conducted for validating the dynamic model of the vehicle. The dynamic model has been
linearized about various equilibrium points and its performance against non linear model
has been evaluated. A basic PID controller has been designed over the linear model for
control of depth and heading of the vehicle.
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2.Development of Dynamic Model
Dynamic modeling of an underwater vehicle consists of writing and solving the
equations which govern the vehicles motion in 3-D space. This is done by describing the
translational and rotational position and velocity of a vehicle-fixed coordinate frame relative
to an inertial coordinate frame (Earth). The dynamic model is derived from the Newton-Euler
motion equation and is given by,
+ + + = (1)where is a mass and inertia matrix, ()is a Coriolis and centripetal terms matrix,()is a hydrodynamic damping matrix, ()is the gravitational and buoyancy vector, is
the external force and torque input vector, and is the velocity state vector.Newton Euler formulation based on Newtons second law relates mass, acceleration
and force as:- = (2)
Eulers formulation is based on two axioms in terms of expressing Newtons second lawfor law of conservation of linear momentum and angular momentum . Accordingly wehave the following:
= = (3) = = (4)
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2.1Rigid Body DynamicsFor marine vehicles it is desirable to derive the equations of motion for an arbitrary
origin in locally body fixed frame of reference. Since, hydrodynamic and kinematic forces and
moments are given in body fixed frame B, the entire formulation is done in body frame B.
Figure 1 Inertial earth fixed frame XYZ and body fixed frame X0Y0Z0for a rigid bodyCourtesy: Fossen, Thor I. "Guidance and control of ocean vehicles." New York (1994). Pg 22
2.1.1 Translational motionFigure 1 represents a rigid body with its origin at O. The earth fixed frame is defined by
XYZ while body frame 000 is centered at origin O. 0 represents the position vector ofbody frame witch respect to earth frame and represents the position of centre of gravityof rigid body with respect to body frame. Each particle on body has a velocity and positionvector with respect to origin 0.
From the figure 1, it is evident that,
= 0 + 5
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Velocity of centre of mass is given by,
= = 0 + 6Relation between time derivatives of inertial and non inertial frames is given by,
= + (7)Where is time derivative in Earth fixed frame of reference XYZ and is time derivativein body frame of reference 000, and is the angular velocity of body frame .Thus from (6) and (7) and considering that 0 = 0 = 0for a rigid body = 0 + 8
Similarly acceleration vector can be found as:
= 0 + (9) = 0 + 0 + + (10)
Substituting in equation (2), we get
0 + 0 + + = 0 (11)If origin of body frame B, 000is chosen to coincide with vehicles centre of gravity,we have = 0 0 0. Hence, 0 = and 0 = , equation (11) yields, + = (12)
2.1.2 Rotational MotionThe absolute angular momentum 0about origin O is defined in terms of
0 = (13)where is the mass density of the rigid body. = + (14)But Total moment M, is defined as
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0 = (15)0 = 0 0 (16)
The centre of gravity of vehicle is defined as,
= 1 (17)Time derivative of is given by, = (18)From (16),(17) and (18) we have,
0 = 0 0 (19)Absolute angular momentum can be written as,
0 = ( ) = ( 0) + (20)But ( 0) = 0 = 0 (21)From definition of moment of inertia
0,
= 0 (22)0 = 0 + 0 (23)
Time derivative of 0from (19) and using property described in (23)0 = 0 + 0 +( ) 0 + 0 + 0 (24)
Eliminating
0 from (19) and (24),
0 + 0+ 0 + 0 = 0 (25)If origin of body frame B, 000is chosen to coincide with vehicles centre of gravity,
we have = 0 0 0. Hence,0 = and 0 = , equation (25) yields, + = (26)
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Finally, we can consolidate the above derivation by writing (12) and (26) in component
form where,
0 = ,, = 1 0 = ,, = 2 0 = ,, = 1 000 = ,, = 2 000 = , ,
[
+
2 +
2
+
+
+
=
(27)
[ + 2 + 2+ + + = (28)[ + 2 + 2 + + + = (29) + + + (2 2) +
+ + + = (30) + + + (2 2) +
+
+
+
=
(31)
+ + + (2 2) + + + + = (32)
These equations are expressed in more compact form as:
+ = (33)Where = , ,,,, is linear and angular velocity vector in body frame B and = ,, ,,, is generalized vector of external forces and moments. The following
section will discuss in detail about each of these matrices. The following chapters describe
the derivation of these matrices and the additional forces on the vehicle due to motion in
fluid.
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3.Derivation of Dynamic Matrices
The dynamic model is derived from the Newton-Euler motion equation and is given by,
+ + + = where is a mass and inertia matrix, ()is a Coriolis and centripetal terms matrix,
()is a hydrodynamic damping matrix, ()is the gravitational and buoyancy vector, isthe external force and torque input vector, and is the velocity state vector.
3.1Mass and Inertia matrixThe mass and inertia matrix consists of a rigid body mass and an added mass,
respectively MRBand MA
= + = 11 1221 22 (34)The rigid body mass term can be written as,
=
+
+
(35)
=
0
0
0 0 00
0 0 00
0 0
(36)
3.2Coriolis and Centripetal MatrixCoriolis and Centripetal Matrix have contribution due to rigid body mass and added
mass and inertia.
= + (37)These matrices are obtained through use of Kirchhoffs flow equation and property of
kinetic energy of a rigid mass.
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Kinetic energy in quadratic form is given by, = 12 38
= 12 1 = , , 2 = , ,
=1
2 1111 + 1122 + 2211 + 2222 (39)Kirchhoffs equation in flow in vector form are given by,
1+ 2 1 = 1 (40) 2 + 2
2 + 1 1 = 2 (41)
1=
11
1 +
12
2 (42)
2 = 211 +222 (43)
=
2 11 1 + 2
2=
033 1
1 2
12 (44)
Substituting (42 & 43) in (44) we get,
= 2 22 1 02 45 =
0
0
0+ +
0
0
0 + +
0
0
0 + +
+ + 0 +
+
+ + + 0
+
+ + + + 0
The next chapter describes the various external hydrodynamic forces and moments
due to motion in fluid.
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4.Hydrodynamic Forces and Moments
An underwater vehicle may experience two classes of hydrodynamic forces:
Radiation Induced Forces:- Forces on the body when the body is forced to oscillate with
wave excitation frequency and there are no incident waves
Diffraction forces:- Forces on body when body is restrained from oscillating and there
are incident regular waves
4.1Radiation Induced ForcesThe radiation Induced forces can be identified as sum of the following parameters
a) Added mass due to inertia of surrounding fluidb) Radiation induced dampingc) Restoring forces due to Weight and Buoyancy
4.1.1 Added Mass and InertiaThe concept of added mass and inertia is commonly misunderstood as finite amount
mass and inertia of fluid particles attached to the body of underwater vehicle which amount
to overall new mass and inertia of vehicle [19]. However, it should be understood as pressure
induced forces and moments due to forced harmonic motion of the body which is in
proportion to acceleration of body.
For completely submerged vehicles, added mass is constant. For any vehicle to pass
through water, it should induce motion in otherwise stationary fluid. This implies that in
order for the vehicle to move, the fluid particle should deviate and as a consequence the
fluid surrounding vehicle must possess some kinetic energy given as:
= 12 (47)
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The added inertia matrix is defines as,
=
=
11
12
21 22 (48)
= = The contribution of added mass to dynamics of AUV is further confirmed on
substituting MA in equation (46). Further using the Kirchhoffs fluid dynamic equations in
component form,
= (49)The added inertia force is given by,
= + + + + + 2++ + 2
+ + () (50)Each of the terms in equation (50) is a contribution of added inertia and added mass to
be reflected in dynamics of vehicle as given in equations (27 - 32).
4.1.2 Added Mass Coriolis and Centripetal MatrixSimilar to rigid body Coriolis matrix, the hydrodynamic added mass Coriolis matrix
satisfies the skew symmetric condition.
= 033 (111 + 122)(111 + 122) (211 + 222) (51)
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=
0
00
0
3
2
0
0030
1
0
002
1
0
0320
3
2
30130
1
2102
1
0
(52)
Where,
1 = + + + + + 2 = + + + + + 3 = + + + + + 1 = + + ++ + (53)
1 =
+
+
+
+
+
1 = + + + + +
4.1.3 Hydrodynamic DampingHydrodynamic Damping for underwater vehicles is mainly caused by the following
phenomena:
= Radiation induced potential damping due to forced body oscillations.
= Linear Skin friction due to laminar boundary layers and quadratic skin frictiondue to quadratic boundary layers
= Wave drift damping = Damping due to vortex shredding = + + + 54
However, it is difficult to give a general expression of hydrodynamic damping matrix
()and hence it is commonly written as, = + (55)
where is a linear damping matrix and is non linear damping matrix account forhigher order terms.
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4.1.4 Potential DampingThe radiation induced damping term is usually referred to as potential damping.
However, the contribution from potential damping terms is very small as compared to
dissipative terms like viscous damping for underwater vehicles. Potential damping is
prominent for surface vehicles such as ships. Hence, this work does not take into account for
potential damping. Also, it is very difficult to evaluate the contribution of potential damping
due to lack of proper theory and expensive experimental setups.
4.1.4.1 Skin FrictionContribution due to skin friction is consideration with both laminar and turbulent
boundary layer contributing to drag on the vehicle. The laminar skin friction drag is the sole
contributor to the linear damping matrix.
=
(56)
4.1.4.2
Wave drift Damping
Like potential damping, wave drift mainly affects surface and shallow water vehicles.
Wave drift damping can be interpreted as added resistance for surface vehicles advancing in
waves. Wave drift damping force is proportional to square of significant wave height. Wave
drift mainly affects the surge motion of vehicle rather than sway and yaw motion.
4.1.4.3 Damping due to vortex shreddingIn a viscous fluid frictional forces are present such that the total energy of system is not
conserved accounting for the frictional losses. The viscous force due to vortex shedding and
turbulent boundary layer is together modeled as non linear damping forces.
= 12 (57)
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Where U is velocity of vehicle, A is the projected area and is the dragcoefficient. The drag coefficient depends on Reynolds number which is a function ofvelocity, characteristic length D and viscosity of fluid .
= (58)
=
|||||||||||| (59)
4.1.5 Restoring Forces and MomentsIn hydrodynamic terminology, gravitational and buoyant forces are called restoring
forces. Gravitational forces act through center of gravity of the vehicle = , . while the buoyancy forces act through center of buoyancy of vehicle.
Restoring force vector in matrix form is given by,
=
+
+
(60)
The forces mentioned above are in body frame of reference and are defined as follows,
= 112 00 = 112 00 (61)
Where = = and is the volume of fluid displaced which is same asthe volume of the vehicle for an underwater vehicle.
= + + (62)
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5.Calculation of Hydrodynamic Derivatives
There are several methods that will produce results for hydrodynamic parameters
based on a given geometry. The methods include analytical, experimental, computational,
and semi-empirical approaches. The distinctions between the modeling methods are further
described below.
i) Analytical: - Analytical methods for determining model parameter values includeimplementing strip theory or solving Laplace's equation [19].
ii) Experimental: - These studies include sea trials and tow-tank tests. These methodsare costly due to the expense of constructing scale vehicle models and operating
the experimental facility. Further, the added mass and inertia terms are difficult to
obtain from sea trials [10, 22].
iii) Computational: - Computational fluid dynamics (CFD) involve solving the Navier-Stokes flow equations numerically using a computer. CFD programs are less
expensive than tow-tank and sea trial testing and more broadly applicable than
analytical methods, however they require an expert to grid the model and validate
the results [15].
iv) Semi-empirical: - These methods use experimentally derived guidelines forestimating model parameter values for vehicles with generic shapes [13].
This study will use analytical strip theory for calculation of hydrodynamic parameters
for its simplicity and scalability to underwater vehicles.
5.1Strip theory for estimating hydrodynamicsStrip theory, also known as slender body approximation, can be applied to slender
bodies in order to estimate the hydrodynamic parameters (such as added mass and inertia)
for a body using the 2D sectional properties. Strip theory can also approximate other
parameters in the equations of motion, such as damping coefficients. Strip theory takes the
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hydrodynamic parameters of the 2D shape and integrates the parameters over the length of
the vessel [19,22]. The expressions for the hydrodynamic coefficients are as follows,
11 = = 112, 2
2
22 = = 222, 2
2
33 = = 332, /2
/2 44 = = 442,
/2
/2 (63)
55 = = 112, /2
/2 66 = = 112,
/2
/2
Proper calculated assumption of a 2D area for body has proven to give satisfactory
results.
Coefficient Circle Ellipse Square
112 2 2 4.752 222 2 2 4.752
332
2
2 4.75
2
Table 1 Strip theory estimates for 2D surface
= , 2 = , ,As per data given in table 1, for applying the same model for a non circular, ellipsoidal
or square vehicle an equivalent square length must be found out before using the direct 2D
approximations. Since, the vehicle under study is MATSYA, it has a roughly rectangular prism
shape a hence, the equivalent square length will be,
2 = (64)According to [13], the strip theory is modified to obtained much more accurate results,
hence a parameter 0is defined,
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0 = 2 (65)Where is the length of the vehicle, and 2is the length of equivalent squareSimilarly parameter 1is defined as1 = 1.51 0.150 1 (66)The added mass is then found by an empirical relation,
= 8 01
= 442, /2
/2= 2 332,
/2
/2+ 2 222,
/2
/2 (67)
= 552, /2
/2= 2 332,
/2
/2+ 2 112,
/2
/2 (68)
= 662, /2
/2= 2 112,
/2
/2+ 2 222,
/2
/2 (69)
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6.Dynamic Model of Matsya 1.0 and Matsya 2.0
For the purpose of this project, Matsya 1.0 and Matsya 2.0 have been taken as baseline
vehicles over which dynamic modeling have to be implemented. This section focuses on
derivation of dynamic model of Matsya 1.0 and Matsya 2.0 and identification of the
hydrodynamic parameters. This section will differentiate the utility of the two vehicles
designed and developed by Autonomous Underwater Vehicle Team of IIT Bombay. Matsya
1.0 was designed to understand basic underwater navigation and control problems and
would only navigate in shallow waters while Matsya 2.0 is an advanced prototype with
objective of executing manipulation tasks.
The basic difference lies in the degrees of freedom of the two vehicles with 1.0 having
5 degrees of freedom and no control in sway direction.
Figure 2 Matsya 1.0
Matsya 2.0 has 6 cross body thrusters with control over 5 degrees of freedom. Since
the roll axis is inherently stable due to mechanical construction of vehicle, this degree of
freedom is not controllable via thrusters.
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Figure 3 Matsya 2.0
Most of the experimental validation was performed on Matsya 2.0 with various open
validation experiments being conducted. The upcoming sections will discuss the calculations
of parameters for both the variants of Matsya and comparison between the two vehicles.
Parameters Matysa 1.0 Matsya 2.0
Mass 20.2 kg 23
Weight 197.96 N 225.4 N
Buoyancy 217.56 N 227.36 N
Centre of Gravity =
0 0 0 0 0 0Centre of Buoyancy =
0 0 0.1 0 0 0.1
Length 1.0 m 0.891 mBreadth 0.53 m 0.70 mHeight 0.42 m 0.46 m
Table 2 Dimensions and vehicle specifications for Matsya 1.0 and 2.0
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7.Parameter Calculations for Matsya
For the purpose of this project, Matsya 1.0 and Matsya 2.0 have been taken as baseline
vehicles over which dynamic modeling have to be implemented. This section focuses on
derivation of dynamic model of Matsya 1.0 and Matsya 2.0 and identification of the
hydrodynamic parameters.
7.1Assumptions on AUV DynamicsObtaining the parameters of the dynamic model is a difficult time consuming process.
Therefore assumptions on the dynamics of the AUV are made to simplify the dynamic model
and to facilitate modeling. The following assumptions are made:
i) Relative less speed -- Lift forces are neglected because vehicle operates at very smallspeed. The maximum speed of vehicle was analytically found to be 1 m/s while 0.6 m/s
for Matsya 2.0 and this was confirmed during underwater experiments (dry tests).
ii) AUV symmetric about three planes -- The AUV is symmetric about the x-z plane andclose to symmetric about the y-z plane. Although the AUV is not symmetric about the
x-y plane it is assumed that the vehicle is symmetric about this plane, therefore it is
assumed that the degrees of freedom are decoupled. The AUV can be assumed to be
symmetric about three planes since the vehicle operates at relatively low speed.
iii) The -frame is positioned at the center of gravity, = 0 0 0iv) No environmental disturbances The AUV is assumed to be working in clean
environments without any disturbances due to wind and gusts. As the vehicle operates
at depth below 5-6 meters, such assumptions may hold.
v) Decoupled degrees of freedom - Decoupling assumes that a motion along one degreeof freedom does not affect another degree of freedom. Decoupling is valid for the
model that does not include ocean currents since the AUV is symmetric about its three
planes, the off- diagonal elements in the dynamic model are much smaller than their
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counterparts and the hydrodynamic damping coupling is negligible at low speeds.
When the degrees of freedom are decoupled the Coriolis and centripetal matrix
becomes negligible, since only diagonal terms matter for the decoupled model.
7.2Determination of Dynamic model for MATSYAUsing the assumptions stated in 7.1 and applying analytical and computational tools
like Solidworks and ANSYS the dynamical model for Matsya is obtained. Following lists out
the obtained hydrodynamic and rigid body matrices:
1) Rigid mass and Inertia matrix
=000
0
0
000
0
0
000
0
0
00
000
00000
0000
0 (70)
2) Coriolis and Centripetal matrix
=0
0
00
0
0
00
0
0
00
0
00
0
00
000 (71)
3) Restoring Forces
=
(
+
)
( + ) (72)
4) Added Inertia and Coriolis MatrixAs per the discussion in section 7.1 the added mass matrix has diagonal
elements with no contribution from off diagonal elements. Since, the speed of
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vehicle is very low and having 3-axes plane of symmetry, such an approximation is
valid.
= ()
=0
00
0 0
0 00 0
0 0 00
0 0
0 0 0 (74)
The hydrodynamic parameters were calculated as per strip theory
approximation discussed in section 6.1.
Parameters Matsya 1.0 Matsya 2.0
-5.26 -12.39 -4.39 -20.39 -8.8 -20.39
-0.1209 -0.019 -0.74 -0.117 -0.43 -0.112
Table 3 Hydrodynamic parameters for Matsya
7.3Propulsive Forces and MomentsThe vector
of propulsion forces and moments depends on specific configuration
of actuators such as propellers and rudders. Considering that this study considers a
vehicle based on thrusters and without any control surfaces, the force and torque vector
is defined by,
= (75)
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The dimension depends on the number of thrusters and is mapping matrixwhich defines the position of the individual thrusters. A mapping matrix for Matsya 1.0 is
given by,
=00
1010
00
1220
00
1330
10
00
04
100
0
05 (76)
For Matysa 2.0
=1
4
2
3
0 0 1 1 0 0
0 0 0 0 1 1
1 1 0 0 0 0
0 0 0 0 0 0
0 0 0 0
0 0 0 0
T T
T T
x x
y y
(77)
No dynamic model of thruster is done since it is very fast response system as compared
to AUV. However, the forward and backward thrust of vehicle is not the same for same
power consumption. Hence, during the simulation of dynamic model a separate
compensation is included for generation of thrust force.
7.4 Estimation of Damping CoefficientsAs per the discussion in section 5.1.3, a major contribution to drag of vehicle is due
to skin friction and cortex shedding. The generalized drag coefficient for a body is given as
= 22 (78)Where is a drag coefficient and is the projected surface area. Drag coefficient is
a function of shape of the body, viscosity of fluid and Reynolds number.
= 0 + 22 (79)0 is a function of body shape and independent of velocity, whereas 2 is
dependent on projected surface area and velocity of body and is used as non linear
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damping coefficient as described in section (5.1.3). Following table lists out the damping
parameter for Matsya 1.0.
Damping Parameter Matsya 1.0
0.82 1.05 1.05 0.01 0.013 0.015|| 1.37|| 2.28|| 3.28|| 4.48e-3|| 6.08e-3|| 6.08e-3
Table 4 Damping Coefficients for Matsya
Some of the damping coefficients may not be obtained analytically due to complex analysis
involved. Hence, these were derived for Matysa 1.0 from vehicles [13, 15] of similar mass and
shape. However, as seen in Stage 1 of the project the data was erroneous and led to false
results. The following section used system identification techniques for calculation of damping
parameters for Matsya 2.0.
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8.System identification for calculation of damping parameters
System identification is the art and science of building mathematical models of
dynamic systems from observed input-output data [24]. To validate the dynamic model of
the AUV, the mass and damping parameters used in the dynamic model need to be
estimated. System identification of a dynamical system generally consists of the following
four steps
1. Data acquisition
2. Characterisation
3. Identification/estimation
4. Verification
The first and most important step is to acquire the input/output data of the system to
be identified. Acquiring data is not trivial and could be very much laborious and expensive.
This involves careful planning of the inputs to be applied so that sufficient information about
the system dynamics is obtained. If the inputs are not well designed, then it could lead to
insufficient or even useless data. The second step defines the structure of the system, for
example, type and order of the differential equation relating the input to the output. This
means selection of a suitable model structure. The third step is identification/estimation,
which involves determining the numerical values of the structural parameters, which
minimise the error between the system to be identified, and its model. Common estimation
methods are least squares (LS), instrumental-variable (IV), maximum-likelihood (MLE) and
the prediction-error method (PEM) [25,26].
The final step, verification, consists of relating the system to the identified model
responses in time or frequency domain to instill confidence in the obtained model. Residual
(correlation) analysis, Bode plots and cross-validation tests are generally employed for model
validation [27].
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Decoupling between the degrees of freedom is used to treat every degree of freedom
separately.
+ + + = (80)Where represents mass and inertia associated with considered degree of freedom
and and are linear and quadratic damping parameters, are gravity and buoyancyforces while is term representing external forces.
To determine the behavior of the AUV, all the parameters in the above equation need
to be known. The input force/torque is assumed to be known and can be calculateddirectly from duty cycle/voltage measurements. The gravity and buoyancy matrix is known
from underwater neutral buoyancy tests. Both the rigid body inertia and added inertia are
known as described in previous sections. The remaining parameters and which areunknown can be obtained through static and dynamic experiments.
8.1Evaluation of damping parameters for surge, heave and swayDrag force was computed in both directions for surge and heave degrees of freedom
using wet tests in swimming pool of IIT Bombay. In the equation (80), only the drag
parameters are unknown. The inertia, body forces and external forces have been modeled in
previous sections. Hence, the vehicle is tested at various surge, sway and heave speeds to
record the acceleration and velocities. The data is recorded at a sampling rate of 2 Hz. The
following tables lists the results of the experiments conducted.
Velocity
(m/s)
Drag Force (N)
Surge Forward Surge Backward
0.1 2.12 2.23
0.2 4.78 5.35
0.3 7.65 8.62
0.5 19.6 22.1
Table 5 Surge Drag force form wet tests
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Velocity
(m/s)
Drag Force (N)
Sway Left Sway Right
0.1 4.46 4.97
0.2 8.77 10.52
0.35 16.33 19.6
Table 6 Sway Drag force from wet tests
Velocity
(m/s)
Drag Force (N)
Heave Down Heave Up
0.1 3.15 3.19
0.2 7.1 7.42
0.3 12.7 13.8
0.4 19.6 22.4
Table 7 Heave Drag force from wet tests
It is observed that the maximum speed achievable is along the surge direction while both
heave and sway direction have higher drag forces.
First a quadratic fit of the data is made in Matlab, which estimates the unknown
terms of the drag force equation
=
1
2 +
2
+
3 (81)
Where 1 is the quadratic damping term, 2 is the linear damping term and 3 is theequation offset for basic fitting.
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Figure 4 Curve fit for surge drag
Figure 5 Curve fit for sway drag
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Figure 6 Curve fit for heave drag
Quadratic Fit
parameter
Surge Sway Heave
1 73.91 29.2 73.752 10.53 34.34 18.073 1.648 0.734 0.5875
Table 8 Curve fitted parameters
8.2Evaluation of damping parameters for roll, pitch and yaw axesFor deducing parameters for roll pitch and yaw, the open loop roll and pitch stability
experiments were conducted. The open loop pitch and roll response recorded are illustrated
in Figure 7 and Figure 8. Also the pitch and roll rates were recorded using IMU data and
substituting all the values in equation (80) the damping parameters for roll, pitch and yaw
was calculated.
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8.2.1 Pitch damping calculation
Figure 7 Open Loop Pitch response from wet tests
Following data was seen during open loop pitch response:-
= 1 = 2 = 1.897 2 = 2.45 = 1
= 0.0826
2
Using system Identification technique, we evaluated
= . || = .
-20
-10
0
10
20
30
40
50
0 2 4 6 8 10 12
Pitch
angle(degrees)
time
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8.2.2 Roll damping Calculation
Figure 8 Open loop roll stability from wet tests
Following data was seen during open loop roll response:-
= 0.34 = 1.7381 = 0.0347 2
= 1.34
= 0.2635
= 0.0008
2
Using system identification technique, we evaluated
= . || = .
15, 0
-20
-10
0
10
20
30
40
50
0 2 4 6 8 10 12 14 16
RollAngle(degrees)
time
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8.2.3 Yaw damping calculation
Figure 9 Open loop yaw identification
The yaw damping coefficients are found in a similar manner,
= . || = . Parameters Values
-1.153
|
| 73.91
34.34|| 29.2 18.07|| 73.75 4.5|| 2.23e-3 4.06|| 8.68e-1 1.02|| 1.08e-3
Figure 10 Damping Parameters
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9.Validation of Dynamic model
Through the open loop experiments conducted in Chapter 8, the damping parameters
were successfully identified. Thus, all the parameters of the model described in Chapter 3
with suitable assumptions as explained in Chapter 7 have been identified. We validate the
dynamic model by conducting some open loop experiments and corresponding simulation of
the dynamic model.
9.1Heave ExperimentsThe open loop heave was tested by giving various combinations of 10 bit PWM input to
the heave thrusters. The maximum limit on PWM input is 512. The table lists the observed
and simulated settling speeds for various combinations.
PWM Observed Speed (m/s) Simulated Speed (m/s)
100 0.11 0.135
200 0.24 0.22
300 0.31 0.29
Table 9 Open loop Heave tests
The maximum observed speed in heave direction was 0.36 m/s
9.2Surge ExperimentsPWM Observed Speed (m/s) Simulated Speed (m/s)
100 0.20 0.21
200 0.25 0.31
400 0.4 0.45
500 0.5 0.5
Table 10 Open loop surge tests
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9.3Sway ExperimentsPWM Observed Speed (m/s) Simulated Speed (m/s)
100 0.1 0.21
200 0.19 0.31
400 0.4 0.45
500 0.5 0.5
Table 11 Open loop sway tests
9.4Open loop Roll experimentsRoll control was tested by giving a set initial deflection on 40 degrees and the
corresponding response was recorded from the IMU data. In dynamic model simulation a
similar simulation was performed. The settling time and the peaks recorded were found to
match to a great extent.
Figure 11 Open loop roll experiments
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9.5Open loop Pitch experimentsOpen loop pitch performance was tested by giving a set initial deflection of 40 degrees
and the corresponding response was recorded from the IMU data. In dynamic model
simulation a similar simulation was performed. The settling time and the peaks recorded
were found to match to a great extent.
Figure 12 open loop pitch experiments
The simulation results obtained have matched the performance shown by vehicle in
the experiments. The experiments were carried for variety of pitch and roll angles and the
data obtained matched the simulation results.
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9.6Open loop surge and depth control
Figure 13 Simultaneous Surge and depth control
For open loop surge and heave, it was observed in the simulation that the vehicle
initially pitches before to settling to the required depth. Similar experiments were performed
in underwater tests there was indeed a pitch and surge axis coupling which gave rise to
inherent pitching of vehicle while surge and depth control were simultaneously activated.
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Figure 14 Observed response for simultaneous heave and depth control
9.7Simultaneous roll and pitch excitation
Figure 15 Open loop simulation of simultaneous pitch and roll
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Figure 16 Observed response for Simultaneous pitch and roll in open loop
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10.Linearization of Dynamic Model
Though the dynamics of underwater vehicle system is highly coupled and non-linear in
nature, decoupled linear control system strategy is widely used for practical applications. In
modeling systems, we see that nearly all systems are nonlinear, in that the differential
equations governing the evolution of the system's variables are nonlinear. However, most of
the theory we have developed has centered on linear systems. To design a linear control
system, first it is necessary obtain a linear model of the system to which these techniques will
be applied. The model is linearized over a set of surge speeds ranging from (0.1 m/s 0.5
m/s). We can heave and sway speed as in general they will be very less compared to surge
speed. However, to prevent the loss of generalization, linearization is done on all surge,
heave and sway axis. The state vector for equilibrium is given as:-
= Where , , , , , are the equilibrium values of , ,,,, respectively.
For equilibrium, , , = 0to ensure stability of vehicle. Due to inherent mechanical rolland pitch stability we cannot have a non-zero p, q as an equilibrium point. So having a
equilibrium point will non-zero pitch rate and roll rate is not sustainable and system itself is
not stable at this points and the neighborhood region. Also having a non-zero yaw rate is not
feasible, as AUVs specifically used a tight yaw control for navigation which requires yaw rate
to settle to zero.
1) Equilibrium about surge :- = 0 0 0 0 02) Equilibrium about sway :-
=
0
0 0 0 0
3) Equilibrium about heave :- = 0 0 0 0 010.1 Formulation of Jacobian Matrix
In this section we develop Jacobian linearization of a nonlinear system," about a
specific operating point, called an equilibrium point.
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Consider a non linear equation given by:-
= , , (82)Suppose for a given nominal input
(
), there is a nominal state trajectory denoted by
(), we have, = , , (83)For the case of equilibrium, = , , = 0 (84)When the initial state or inputs deviate from nominal state or inputs, we have
= 0+ (85)
= 0+ (86)
We will obtain a linear perturbation model to approximately describe x(t). Linear
model is simpler, easier to analyze and provides more insights.
We write the perturbed state as,
= + (87) = +
(88)
= , , , . (89)= + , ,+ , , (90)
Now using Taylor series expansion,
( ), , (t) ( ), , (t)
1 1
( ( ), , ( )) ( ( ), , ( )) | ( ) | ( )
H.O.T
n ni i
x t t u j x t t u k
j kj k
f ff x t t u t f x t t u t x t u t
x x
The higher order terms vanish as and vanish. Thus we have, + (91)
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= ,, = ,,
Equation (91) is more famously known as Jacobian linearization of non linear system.
10.2 Jacobian for 6 DOF systemThe Jacobian matrix for 6-DOF system of equation defined in Chapter 3, is given by
= , ,,,, + 1 = , ,,, , + 2 = , ,,,, + 3 (92) = , ,,,, + 4
=
,
,
,
,
,
+
5
= , ,,,, + 6where,,,,,are the functions describing the dynamic model of the vehicle.
=
93
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10.3 Linearization of dynamic model of Matsya 2.01) Case 1:- Linearization about constant surge speedsComputing the Jacobian matrix for the 1
stcase where nominal point is non zero velocity
in surge direction:-
=
| |2 0 0 0 ( ) 0
0 ( ) 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
u u u
v
w
p
q
r
X X u B W
Y B W
Z
K
M
N
(94)
The complete linearized model can be represented as follows:-
1 1
20.53 147.8 0 0 0 0.3 0 0 0 1 1 0 0
0 34.34 0.3 0 0 0 0 0 0 0 1 1
0 0 18.07 0 0 0 1 1 0 0 0 0( ) ( )
0 0 0 4.5 0 0 0 0 0 0 0 0
0 0 0 0 4.06 0 0.35 0.35 0 0 0
0 0 0 0 0 1.02
A A
u u u
v v
w wM M M M
p p
q q
r r
1
2
3
4
5
6
0
0 0 0.23 0.23 0 0
T
T
T
T
T
T
(95)
1
cos cos cos sin sin sin cos sin sin cos cos cos 0 0 0
sin cos cos cos sin sin sin cos sin sin cos sin 0 0 0
sin cos sin cos cos 0 0 0
(M M ) 0 0 0 1 sin tan cos tan
0 0 0 0 cos si
A
x
y
z
n
sin cos0 0 0 0
cos cos
u
v
w
p
q
r
(96)
This is of the state space form,
= + = +
Where A is the Jacobian Matrix, B is control Matrix, C is the output matrix. D is zero
matrix for the dynamic model derived above.
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The eigenvalues for above Jacobian is [75.43,34.34,18.07,4.5,4.06,1.02]and are all negative. Hence the linearization results in a stable system. The table below
represents the eigenvalues for different surge speeds.
Surge Speed Eigenvalues Stable/Unstable
0.1 [16.31,34.34,18.07,4.5,4.06,1.02] Stable0.2 [31.09,34.34,18.07,4.5,4.06,1.02] Stable0.3 [45.87,34.34,18.07,4.5,4.06,1.02] Stable0.5 [75.43,34.34,18.07,4.5,4.06,1.02] Stable
Table 12 Eigenvalues for linearized system for variable surge speeds
Case 2:- Linearization about constant sway speed
=
| |
0 0 0 ( ) 0
0 2 ( ) 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
u
v v v
w
p
q
r
X B W
Y Y v B W
Z
K
M
N
Sway Speed Eigenvalues Stable/Unstable
0.1 [1.53,40.18,18.07,4.5,4.06,1.02] Stable0.2 [1.53,46.02,18.07,4.5,4.06,1.02] Stable0.3 [1.53,51.86,18.07,4.5,4.06,1.02] Stable
Table 13 Eigenvalues for linearized system for variable surge speeds
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Case 3:- Linearization about constant heave speed
=
0 0 0 ( ) 0
0 ( ) 0 0 0
0 0 2 0 0 00 0 0 0 0
0 0 0 0 0
0 0 0 0 0
u
v
w ww
p
q
r
X B W
Y B W
Z Z wK
M
N
Heave Speed Eigenvalues Stable/Unstable
0.1 [16.31,34.34,23.91,4.5,4.06,1.02] Stable0.2 [16.31,34.34,29.75,4.5,4.06,1.02] Stable0.3 [16.31,34.34,35.59,4.5,4.06,1.02] Stable
Table 14 Eigenvalues for linearized system for variable surge speeds
10.4 Open Loop Simulations of Linearized modelThe linearized model developed is simulated in open loop environment for different
combination of inputs and compared with the non linear counterpart.
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1) Open loop pitch performance:-i) Model 1:- linearization about non zero surge speed
Figure 14 Open pitch performance by linearized model 1
ii) Model 2:- linearization about non zero sway speed
Figure 15 Open pitch performance by linearized model 2
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iii) Model 3:- linearization about non zero heave speed
Figure 16 Open pitch performance by linearized model 3
2) Open loop roll performance:-i) Model 1:- linearization about non zero surge speed
Figure 17 Open loop roll performance by linearized model 1
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ii) Model 1:- linearization about non zero sway speed
Figure 17 Open loop roll performance by linearized model 2
iii) Model 1:- linearization about non zero heave speed
Figure 18 Open loop roll performance for linearized model 3
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Open loop roll performance of linear system exactly matches with the open loop roll
stability performance of non linear system. All linearized models (1,2,3) show exactly similar
performance. This is because, in linearized model all the axes are essentially decoupled and
hence there is no effect of a non-zero surge velocity on pitch or roll motions. In the upcoming
results, only the performance of linear model 1 would be shown as other models have
exactly same performance.
3) Simultaneous surge and depth performance
Figure 18 Surge and depth performance for linearized model
The coupling between pitch and surge axis as found in non linear system is not
replicated in linear model. This is due neglecting of various Coriolis and centripetal terms
during the linearization process.
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4) Miscellaneous simulationsi) Open loop positive roll and negative pitch
Figure 19 Open loop positive roll and negative pitch
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ii) Open loop surge with roll and pitch
Figure 20 Open loop roll with surge and pitch
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iii) Surge and depth control with initial pitch condition
Figure 21 Surge and Depth control with an initial positive pitch
Most of performance in surge, heave and sway direction is satisfactory. However,
coupling between pitch, surge and depth is not replicated in the linearized model.
As the model is linear and stable (eigenvalues are negative), we can go about the
design of a linear controller for pitch, yaw and depth axis.
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11.Controllability analysis of linear model
The linear model of underwater vehicle is very stable as seen from the eigenvalue
evaluation in tables 11 to 13. In this section, we proceed to check the controllability and
observability of the states. The system is represented in state space form as defined in
equation (9596).
11.1 ControllabilityThe state of a system, which is a collection of the system's variables values, completely
describes the system at any given time. In particular, no information in the past of a system
will help in predicting the future, if the states at the present time are known.
For a linear system defined by,
= + The controllability matrix is given by,
= 2 . 1 (97)If the determinant of controllability matrix , is non-zero then all the states of the
system are controllable
Controllability for linearized model 1:-
For the system designed, controllability matrix is given by,
0.0014 0 0 0 0 0
0 0.0007 0 0 0 0
0 0 0.0013 0 0 00 0 0 0 0 0
0 0 0 0 0.0122 0
0 0 0 0 0 0.0266
C
(98)
The 4th
row of the matrix is a zero row which implies that 4th
state ie roll rate is not
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controllable. However, due good dynamic stability of roll axis it should not pose a problem.
Underwater vehicle in general do not use a roll maneuver for their missions.
Controllability for linearized model 2:-
0.0185 0 0 0 0.0001 0
0 0.0004 0 0 0 0
0 0 0.0013 0 0 0
0 0 0 0 0 0
0 0 0 0 0.0122 0
0 0 0 0 0 0.0266
C
(99)
The second linearized model also gives the same result. Since, the model is linear and
more or less captures the overall system dynamics, a simple linear PID controller could beused for basic depth and heading control.
11.2 Design of PID controller for depth controlConverting the dynamic model from state space to classical laplace domain we have
the following transfer function:-
= .+ . ( + )Hence a linear PID control can be used in series with the plant transfer function:-
Constraints involved while design were as follows:-
i) Settling time < 30 secondsii) Peak overshoot < 20 %Nichols Ziegler tuning method has been used. In this first a step response of plant
transfer function is evaluated the various parameters in designing of controller.
The tuned PID parameters for linear model are = 345, = 1023 = 20.34The same PID controller was applied on non linear model and responses are recorded
as shown in figure 22.
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Figure 22 depth control comparison for linear and non linear model
The linear model has an overshoot of 18% for a depth setpoint of 1 meter. However,
the non linear model does not exhibit such an overshoot. Moreover, the settling time for non
linear model is 50% more. This is due to neglection of Coriolis terms which induced pitch and
depth coupling. Hence the thrust force is not complete used for reaching to particular depth.
Thus, as described in Chapter 10 the coupling between depth and pitch is not evident in
linearized model and observed in the above simulation.
11.3 Design of PID controller for yaw controlA similar controller is developed for heading direction. The following figure shows
result for yaw command of 90 degrees. The yaw transfer function is given by,
= 0.06036 + 0.2677(3 4)
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The constraints used for designing were
1) Settling time < 20 seconds2) Peak overshoot < 10%
Figure 23 Yaw command of 90 degrees
Both the constraints werent achieved as attaining the peak overshoot would increase
the settling time. Hence, attaining the settling time was used as a hard constraint. The
simulation results show similar responses for linear and non linear model.
Consolidating the observations from the experiments and the simulations, we find that
linearized model was able to capture dynamics of vehicle to a large extent. However, it fails
to capture the pitch and heave motion coupling. This may due to neglection of quadratic
damping and coriolis terms that were neglected during process of linearization.
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12.Conclusion
A comprehensive study of kinematic and dynamic model of generalized underwater
vehicle has been performed. For the purpose of validation of the study Matsya AUV has been
chosen as a test bench platform. The following assumptions are made to the dynamic model:
the AUV moves with a relatively low speed, there is symmetry about the three planes, the
body-fixed frame of the AUV is positioned at the center of gravity. With these assumptions
decoupling the degrees of freedom is possible, where only the surge, heave and yaw degrees
of freedom will be controlled. Since one is dealing with a decoupled model only the
mass/inertia and damping terms need to be estimated.
Estimation of added mass/inertia is done using strip theory and was found to be a
match with AUVs comparable to size and weight of MATSYA. Similarly drag analysis has been
performed for identification of damping parameters. In first stage of the project, the
damping parameters were taken of vehicles of same size as MATYSA. However, that lead to
incorrect modeling of vehicle and the open loop results were erroneous. Hence, a basic
system identification technique is employed during modeling of Matsya 2.0. Wet tests wereconducted to record data for different experiments. The pressure sensor and IMU data is
used as an aid to find the drag forces on vehicle. Thus, the entire model was successfully
identified. Validation of model has been done by performing various open loop stability
experiments. The overall data obtained from experiments has found to be good match with
simulations performed on the dynamic model.
For design of controller, initially the model was linearized using Jacobian linearization
technique. Stability analysis of linearized model has been done and the system is found to be
completely stable. A basic PID controller for depth and heading is designed and comparison
between its application to linear and non linear model is shown. The response for depth
control shows the difference in two models implying the non-linearities due to Coriolis and
damping terms havent been incorporated in the linear model. Also, the coupling effect
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between depth and pitch is not reflected in the linear model. Hence