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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



ii

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.



iii

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.



iv

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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x

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].



2

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:



3

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



4

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.



5

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)



6

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



7

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



8

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)



9

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.



11

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.



12

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)



13

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)



14

=

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.



15

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)



16

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)



17

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



18

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,



19

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)



20

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.



21

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



22

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



23

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



24

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)



25

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



26

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.



27

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].



28

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



29

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.



30

Figure 4 Curve fit for surge drag

Figure 5 Curve fit for sway drag



31

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.



32

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



33

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



34

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



35

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



36

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



37

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.



38

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.



39

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



40

Figure 16 Observed response for Simultaneous pitch and roll in open loop



41

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.



42

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)



43

= ,, = ,,

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



44

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.



45

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




46

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


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.



47

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




48

iii) Model 3:- linearization about non zero heave speed


2) Open loop roll performance:-i) Model 1:- linearization about non zero surge speed

Figure 17 Open loop roll performance by linearized model 1



49

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



50

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.



51

4) Miscellaneous simulationsi) Open loop positive roll and negative pitch

Figure 19 Open loop positive roll and negative pitch



52

ii) Open loop surge with roll and pitch

Figure 20 Open loop roll with surge and pitch



53

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.



54

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



55

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.



56

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

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