UNDERACTUATED CONTROL FOR AN AUTONOMOUS UNDERWATER VEHICLE WITH FOUR THRUSTERS

2012, 9

ZAINAH BINTI MD. ZAIN

Graduate School of

Natural Science and Technology

(Doctor’s Course)

OKAYAMA UNIVERSITY

Underactuated Control for anAutonomous Underwater Vehicle with

Four Thrusters

BY

ZAINAH BINTI MD. ZAINB.Sc. Eng. in Electrical and Electronic Engineering,

University of Science Malaysia, 2001M.Sc. in Electrical and Electronic Engineering,

University of Science Malaysia, 2005

A dissertation submitted in partial fulfillment ofthe requirements for the Doctor of Philosophy in Engineering degree in

Mechatronic Systems,Department of Intelligent Mechanical Systems,

Division of Industrial Innovation Sciences,Graduate School of Natural Science and Technology,

Okayama University

September , 2012

Supervisor: PROFESSORKEIGO WATANABE

i

Abstract

The control of Autonomous Underwater Vehicles (AUVs) is a very challenging taskbecause the model of AUV system has nonlinearities and time-variance, and there are un-certain external disturbances and difficulties in hydrodynamic modeling. The problem ofAUV control continues to pose considerable challenges to system designers, especiallywhen the vehicles are underactuated (defined as systems with more degrees-of-freedom(DOFs) than the number of inputs) and exhibit large parameter uncertainties. Hence thedynamical equations of the AUV exhibit so-called second-order nonholonomic constraints,i.e., non-integrable conditions are imposed on the acceleration in one or more DOFs be-cause the AUV lacks capability to command instantaneous accelerations in these directionsof the configuration space. Such a nonholonomic system cannot be stabilized by the usualsmooth, time-invariant, state feedback control algorithms. From a conceptual standpoint,the problem is quite rich and the tools used to solve it must necessarily be borrowed fromsolid nonlinear control theory. However, the interest in this type of problem goes well be-yond the theoretical aspects because it is well rooted in practical applications that constitutethe core of new and exciting underwater mission scenarios.

The problem of steering an underactuated AUV to a point with a desired orientationhas only recently received special attention in the literature and references therein. This taskraises some challenging questions in system control theory because, in addition to beingunderactuated, the vehicle exhibits complex hydrodynamic effects that must necessarily betaken into account during the controller design phase. Therefore, researchers attempted todesign a steering system for the AUV that would rely on its kinematic equations only.

In this research, an X4-AUV is modelled as a slender, axisymmetric rigid body whosemass equals the mass of the fluid which it displaces; thus, the vehicle is neutrally buoyant.X4-AUV equipped with four thrusters has 6-DOFs in motion, falls in an underactuated sys-tem and also has nonholonomic features. Modelling of AUV maneuverability first involvedthe mathematical computation of the rigid body’s kinematics, in which roll-pitch-yaw an-gles in 6-DOFs kinematics are used. We also derive the dynamics model of an X4-AUVwith four thrusters using a Lagrange approach, where the modelling includes the consider-ation of the effect of added mass and inertia.

We present a point-to-point control strategy for stabilizing control of an X4-AUV,which is not linearly controllable. The goal in point-to-point control is to bring a systemfrom any initial state of the system to a desired state of the system. The construction ofstabilizing control for this system is often further complicated by the presence of a driftterm in the differential equation describing it dynamically.

Two different controllers are developed to stabilize the system. The first stabilizationstrategy is based on the Lyapunov stability theory. The design of the controller is separatedinto two parts: one is the rotational dynamics-related part and the other is the translationaldynamics-related one. A controller for the translational subsystem stabilizes one positionout of x-, y-, andz-coordinates, whereas a controller for the rotational subsystems gener-ates the desired roll, pitch and yaw angles. Thus, the rotational controller stabilizes all theattitudes of the X4-AUV at a desired (x-, y- or z-) position of the vehicle. The stability ofthe corresponding closed-loop system is proved by imposing a suitable Lyapunov functionand then using LaSalles’s invariance principle.

The second stabilization strategy is based on a discontinuous control law, involvingtheσ-process for exponential stabilization of nonholonomic system. This technique is ap-plied to the system by two different approaches. The first approach does not necessitate

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any conversion of the system model into a chained form, and thus not rely on any specialtransformation techniques. The system is written in a control-affine form by applying apartial linearization technique and a dynamic controller based on Astolfi’s discontinuouscontrol is derived to stabilize all the states of the system to the desired equilibrium pointexponentially. Motivated by the fact that the discontinuous dynamic-model without usinga chained form transformation assures only a local stability (or controllability) of the dy-namics based control system, instead of guaranteeing a global stability of the system, theconversion of system model into a second-order chained form is implemented in the secondapproach. The second-order chained form consisting of a dynamical model is obtained byseparating the original dynamical model into three subsystems so as to use the standardcanonical form with two inputs and three states second-order chained form. Here, twosubsystems are subject to a second-order nonlinear model with two inputs and three states,and the other subsystem is subject to a linear second-order model with two inputs and twostates. Then, the Astolfi’s discontinuous control approach is applied for such second-orderchained forms. The present method can only realize partially underactuated control, whichcontrols five states out of six states by using four inputs.

The derived results are specialized to an X4-AUV but, in principle, analogous re-sults can be obtained for vehicles with similar dynamics. Some computer simulations arepresented to demonstrate the effectiveness of our approach.

Approval

Graduate School of Natural Science and TechnologyOkayama University

3-1-1 Tsushima-naka, Okayama 700-8930, Japan

CERTIFICATE OF APPROVAL

Ph.D. Dissertation

This is to certify that the Ph.D. Dissertation of

ZAINAH BINTI MD. ZAINB.Sc. Eng. in Electrical and Electronic Engineering,

University of Science Malaysia, 2001M.Sc. in Electrical and Electronic Engineering,

University of Science Malaysia, 2005

has been approved by the Examining Committee for thedissertation requirement for the Doctor of Philosophy in Engineeringdegree in Mechatronic Systemsat the September , 2012 graduation.

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Dedication

To my loving, courageous parents

v

Acknowledgements

First of all, I would like to show my gratitude to my advisor Prof. Keigo Watanabe.He provided me a good project to work on through my Ph.D study. During my research,he spent a great amount of time discussing with me and advising me. Without his guid-ance, I can hardly imagine how I could complete my dissertation research and become aself-confident underwater vehicle researcher from an amateur in robotics. My study andresearch experience in Mechatronic Systems Lab will definitely benefit me for the rest ofmy life and very positively be my career beginning.

I also thank the Malaysian Government for the SLAB scholarship scheme, whofunded me for my postgraduate studies at Okayama University. I must be thankful toUniversiti Malaysia Pahang for granting me academic leaves to pursue higher studies inJapan.

Also I would like to thank Prof. Toshiro Noritsugu and Prof. Koichi Suzumori fortheir contributions as members of my dissertation committee. Their carefull reading andquestioning did help me improve my research.

I also want to thank Dr. Maeyama and Dr. Nagai and all other current or formermembers of Mechatronic Systems Lab for their help and friendship.

Finally, I thank my family. My parents, my siblings have all stood by me in all thetime. Their unconditional love and support have helped me through some very difficulttimes. I do my best to give them a reason to be proud.

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Contents

Page

Title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiApproval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Chapter

1 Introduction 11.1 Autonomous Underwater Vehicle . . . . . . . . . . . . . . . . . . . . . . . 11.2 Nonlinear Control, Nonholonomic and Underactuated Systems . . . . . . . . 2

1.2.1 Nonlinear control problem . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Nonholonomic systems . . . . . . . . . . . . . . . . . . . . . . . . 31.2.3 What is underactuated mechanical systems? . . . . . . . . . . . . . 41.2.4 Why research in underactuated systems? . . . . . . . . . . . . . . . 41.2.5 Underactuated systems and second-order nonholonomic constraints . 51.2.6 Brockett’s theorem and consequences . . . . . . . . . . . . . . . . . 6

1.3 Motivations and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Review of AUV Design and Control 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Types of AUVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Shallow water survey AUVs . . . . . . . . . . . . . . . . . . . . . 122.2.2 Mid-water AUVs . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.3 Deep-water AUVs . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.4 Gliders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 General Design of an AUV . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.1 Hull design: shape and drag . . . . . . . . . . . . . . . . . . . . . . 142.3.2 Submerging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.3 Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.4 Electric power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Factors Affecting an Underwater Vehicle . . . . . . . . . . . . . . . . . . . 172.5 Previous Work on Control of Underactuated AUVs . . . . . . . . . . . . . . 19

2.5.1 Basic principle for the development of marine control system . . . . 192.5.2 Control of nonholonomic systems . . . . . . . . . . . . . . . . . . . 212.5.3 Control of underactuated ships and AUVs . . . . . . . . . . . . . . 222.5.4 Stabilization control of an underactuated AUVs . . . . . . . . . . . 22

3 Modeling of an Underactuated X4-AUV 243.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Basic Motion Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

vii

viii CONTENTS

3.3 X4-AUV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.1 Former X4-AUV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.2 Model of proposed X4-AUV . . . . . . . . . . . . . . . . . . . . . 28

3.4 Definition of Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . 283.5 Kinematic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.6 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.6.1 Basic concepts of X4-AUV propulsion system . . . . . . . . . . . . 313.6.2 Mass and inertia matrix . . . . . . . . . . . . . . . . . . . . . . . . 313.6.3 Derivation of dynamic model . . . . . . . . . . . . . . . . . . . . . 33

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Nonholonomic Control Method for Stabilizing an X4-AUV 374.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 Lyapunov stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2.2 The invariance principle . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Stabilization Control for an X4-AUV . . . . . . . . . . . . . . . . . . . . . 404.3.1 Rotation control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3.2 Translation control . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Stabilization of an X4-AUV Using a Discontinuous Control Law 525.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2 Astolfi Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3 A Discontinuous Exponential Stabilization Law Control for an X4-AUV . . . 54

5.3.1 System description . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3.2 The discontinuous control law . . . . . . . . . . . . . . . . . . . . 565.3.3 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . 605.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.4 A Discontinuous Exponential Stabilization of Chained Form System for anX4-AUV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.4.1 System description . . . . . . . . . . . . . . . . . . . . . . . . . . 655.4.2 Converted systems to chained forms . . . . . . . . . . . . . . . . . 665.4.3 Transformation of Part 1 . . . . . . . . . . . . . . . . . . . . . . . 685.4.4 Transformation of Part 2 . . . . . . . . . . . . . . . . . . . . . . . 725.4.5 Transformation of Part 3 . . . . . . . . . . . . . . . . . . . . . . . 775.4.6 Transformation ofy1, y2, andy3 to arbitrary position . . . . . . . . . 775.4.7 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . 785.4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6 Conclusions and Future Directions 846.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Appendices 86

A The Body and Rotational Moments of Inertia Calculation 87A.1 Rotational Moment of Inertia Calculation . . . . . . . . . . . . . . . . . . . 87A.2 Body Moment of Inertia Calculation . . . . . . . . . . . . . . . . . . . . . . 88

Appendices 90

CONTENTS ix

B Hydrodynamic Calculation 91

Appendices 91

C Drift Force 93

Appendices 93

D Controllability and Observability of Nonlinear Systems 97D.1 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97D.2 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Publications 100

References 101

List of Figures

Figure Page

2.1 Underwater vehicles categorised by control method . . . . . . . . . . . . . . . 102.2 Ideal method for developing underwater vehicle [81] . . . . . . . . . . . . . . 132.3 Laminar and turbulent boundary layer separation . . . . . . . . . . . . . . . . 142.4 Outline of laminar flow body [70] . . . . . . . . . . . . . . . . . . . . . . . . 152.5 Torpedo shaped AUVs [66] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6 Non-torpedo shaped AUVs [66] . . . . . . . . . . . . . . . . . . . . . . . . . 162.7 Map of the control system design [80] . . . . . . . . . . . . . . . . . . . . . . 203.1 Conventional X4-AUV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 A Vehicle with an ellipsoidal hull shape . . . . . . . . . . . . . . . . . . . . . 283.3 Coordinate systems of AUV . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4 X4-AUV thrusters (Back view) . . . . . . . . . . . . . . . . . . . . . . . . . . 324.1 Connections of rotational and translational subsystems . . . . . . . . . . . . . 424.2 A case for stabilizing the orientation angles andx-axis position . . . . . . . . . 474.3 A case for stabilizing the orientation angles andy-axis position . . . . . . . . . 494.4 A case for stabilizing the orientation angles andz-axis position . . . . . . . . . 515.1 A case for stabilizing the orientation angles and positions, where the controller

assures that the control system is only to be locally stable . . . . . . . . . 635.2 Concept of the proposed controllers . . . . . . . . . . . . . . . . . . . . . . . 665.3 A case for stabilizing all the orientation angles andx- andy-positions . . . . . 805.4 A case for stabilizing all the orientation angles andx- andz-positions . . . . . 82A.1 Length dimensions for an axisymmetric spheroidal volume . . . . . . . . . . . 89C.1 Steady drift force on an ellipsoid [30] . . . . . . . . . . . . . . . . . . . . . . 94C.2 Comparison of sphere and ellipsoid drift force [30] . . . . . . . . . . . . . . . 95C.3 Drift force on fishes [30] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95C.4 Geometrical approximation of different fishes [30] . . . . . . . . . . . . . . . . 96

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List of Tables

Table Page

4.1 Physical parameters for X4-AUV . . . . . . . . . . . . . . . . . . . . . . . . . 45A.1 Symbols and definitions for the rotational moment of inertia calculation . . . . 88B.1 Symbols and definitions for a hydrodynamic calculation . . . . . . . . . . . . . 92

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Chapter 1

Introduction

Control researchers have given considerable attention in the last ten years to many exam-ples of control problems associated with underactuated autonomous underwater vehicles(AUVs), defined as systems with more degrees-of-freedom (DOFs) in motion than thenumber of inputs. Hence the dynamical equations of the AUV exhibit so-called second-order nonholonomic constraints, i.e., non-integrable conditions imposed on the accelera-tion in one or more DOFs because the AUV lacks capability to command instantaneousaccelerations in these directions of the configuration space. As pointed out in a celebratedpaper by Brockett (1983) [11], such nonholonomic systems can not be stabilized by theusual smooth, time-invariant, state feedback controllers. Therefore control problems forunderactuated mechanical systems usually require nonlinear control techniques. The linearapproximation around equilibrium points may, in general, not be controllable and the feed-back stabilization problem, in general, cannot be transformed into a linear control problem.Therefore for such systems linear control methods cannot be used to solve the feedbackstabilization problem, not even locally.

In this research, we model an underactuated X4-AUV with four thrusters and 6-DOFsin motion, and control the underactuated X4-AUV in three-dimensional space.

1.1 Autonomous Underwater Vehicle

The ocean covers about 70% of the earth surface and has great effect on the future existenceof all human beings, beside the land and aerospace. In order to explore the full depths of theocean and its abundant resources, underwater research and development has been carriedout rapidly during recent decades. There is no doubt that underwater robotics is to be animportant scientific area due to its great applications, which vary from scientific researchof ocean, surveillance, inspection of commercial undersea facilities and installations tovarious military operations.

Generally, underwater vehicles can be divided into three, namely, manned submersibles,Remotely Operated Vehicles (ROVs) and Autonomous Underwater Vehicles (AUVs). ROVsand AUVs are the main types of the underwater vehicles that received great attention fromboth the industrial and underwater research community. The difference of AUVs and ROVsis that AUVs are controlled automatically by on-board computers and can work indepen-dently without connecting to the surface. ROVs, on the other hand, are controlled or re-motely controlled by human operator from a cable or wireless communication on ship or onthe ground. Due to the advantages of operational efficiency, mobility, and low operationalcost, AUVs have received wider attention than ROVs [82].

1

2 1. INTRODUCTION

AUVs have great importance in underwater tasks due to its ability to navigate inabyssal zones without necessitating a tether that limits the range and maneuverability of thevehicle. Note however that their autonomy property directly affects the design of controlsystem. That is, it requires advanced controllers and specific control schemes for achievinggiven tasks.

As the demand for underwater systems rises, research on the control of these systemshas become important. The degradation of any actuator relative to other actuators in a fullyactuated underwater vehicle necessitates the consideration of a special control method. Inthat case, underwater vehicles behave like an underactuated system.

AUVs present a challenging control problem, because most of them are underactu-ated, i.e., they have fewer number of actuated inputs than DOFs, imposing nonintegrablevelocity or acceleration constraints. In addition, AUV’s kinematic and dynamic models arehighly nonlinear and coupled. The point stabilization problem, trajectory tracking prob-lem or path following problem in two- and three-dimensional spaces for AUVs, have beenstudied by various researchers; see for example references in [74].

Most of AUVs have 6-DOFs in motion. In this thesis, the vehicle of interest herecalled X4-AUV falls into the class of underactuated AUVs because it has fewer numberof actuators than the DOFs in motion. Many control systems for underwater vehicles havebeen designed up to now [74][84] under the restrictions of cost, weight and complexity,keeping some reliability advantages. The vehicle is also a nonlinear system: all equa-tions of motion of the system include highly coupled terms. Some equations of the motionof the system appear as second-order nonholonomic constraints, and they cannot be inte-grated to obtain the position. Therefore, such underwater vehicles pertain to nonholonomicsystems. Control of nonholonomic systems poses a difficult problem requiring a specialcontrol approach depending on the nature of the mechanical system, as stated in Arslan etal. [6]. The consideration of nonholomic systems is an interesting study from a theoret-ical standpoint because, as pointed out in the earlier works of Brockett [11], they cannotbe asymptotically stabilized to a fixed point in the configuration space using continuouslydifferentiable, time-invariant and state feedback control laws. In the case of underactu-ated nonholonomic systems, nonlinear control methods are more appropriate because thenonholonomic constraints are inherently nonlinear.

1.2 Nonlinear Control, Nonholonomic and Underactuated Systems

In this section, we present an overview the nonlinear control problem, nonholonomic sys-tems, underactuated mechanical systems, the factors that has driven the research of under-actuated systems and the necessity of Brockett’s condition for the stabilizability of nonlin-ear systems.

1.2.1 Nonlinear control problem

Many techniques have been developed for solving both nonlinear setpoint and nonlineartracking control problems. These techniques include feedback linearization, adaptive con-trol, gain scheduling, sliding mode control, model predictive control and robust control.The equation of motion for control-affine time-varying nonlinear dynamic systems subject

1.2. NONLINEAR CONTROL, NONHOLONOMIC AND UNDERACTUATED SYSTEMS 3

to nonholonomic and underactuation constraints can be written in the form

ẋ = f(x) +m∑i=1

giui, ui = (1, · · · ,m) (1.1)

wherex = (x1, · · · , xn)T is the state of the vector system,x ∈ D whereD is an open setin ℜn, f andgi, i = 1, · · · ,m are vector functions such asD → ℜn.

The system is assumed here to havem inputs(u1, · · · , um) which will be referredto as the control vectoru = (u1, · · · , um)T . The class of systems described by Eq. (1.1)is sufficiently large that the most physical systems of practical interest are included. Theclass of linear systems is obviously also included and it is obtained whenf(x) is a linearfunction ofx andgi(x), · · · , gm(x) are constant functions.

The purpose of control system design is to define a control lawu such thatx → 0in the case of set point control, ory → yd in the case of tracking control, whereyd isthe desired output trajectory. Many simple, time-invariant, fully actuated problems may besolved by designing the control law in such a way that the nonlinear dynamics of the plantf(x) is cancelled, leaving the closed-loop system in a linear form resembling that shownin Eq. (1.1) so that linear control techniques can be applied. This type of control systemdesign is known as feedback linearization and should not be confused with conventionallinearization. In the case of feedback linearization, the complete nonlinear model in con-junction with state feedback is used to transform the system into a linear form, whereas theconventional technique makes a linear approximation of the system about a given equilib-rium point. Feedback linearization can be used to derive linear control systems for manynonlinear dynamic systems but cannot be applied universally. Some disadvantages of feed-back linearization are that it requires full state feedback and it is not robust to modelinguncertainty or unmodeled dynamics.

In this thesis, the emphasis is put on systems with drift. From a more theoreticalperspective, this work is encouraged by the fact that relatively few methods exist for thestabilization of systems with a drift term in the differential equation describing their dy-namics.

1.2.2 Nonholonomic systems

A large number of mechanical systems have non-integrable constraints. These constraintscould be either at the velocity level or the acceleration level. Nonintegrability impliesthat the dimension of the manifold on which the system evolves is not reduced due to theconstraint. This in turn implies that the system can assume any arbitrary configurationin the configuration space inspite of the constraints. Velocity level constraints precludeinstantaneous velocities in certain directions of the system. In a similar vein, accelerationlevel constraints prevent arbitrary accelerations of the system. In this thesis, we shall call allsuch systems with non-integrable constraints as nonholonomic systems [75]. Although, ina strict sense, the terminology of nonholonomy applies only for non-integrable constraintsat the velocity level, we shall also classify acceleration level constraints into the class ofnonholonomic systems. The concept of nonholonomy is related to controllability of thecorresponding control system. Redefining the constraint specification as the directions orDOFs in which the system can move, rather than the direction in which it cannot move, is

4 1. INTRODUCTION

equivalent to stating the controllability problem of the corresponding control system. Thus,we can safely say that if the system is maximally nonholonomic, the system is controllable,so that any point in the configuration space can be reached. Thus, a motion problem can beconverted into a control problem. Nonholonomic systems cannot be stabilized using time-invariant continuous feedback because they do not meet Brockett’s necessary condition forfeedback stabilization [11].A few instances where nonholonomy arises in mechanical systems are:

1. Conservation of angular momentum (free floating multi-body system with no externaltorque).

2. No slip constraints on rolling (wheeled mobile robot).3. Underactuation (underwater vehicle / surface vessel).

Underactuation may arise in mechanical systems due to nature of the system, on purposeto reduce actuator cost or failure of one or more actuators.

1.2.3 What is underactuated mechanical systems?

Underactuated mechanical systems refer to those mechanical systems with less number ofcontrols,m than the degrees of freedom,n and arise often in nonholonomic systems withnonintegrable constraints. Examples of underactuated mechanical systems are abundantin our daily life, ranging from spacecraft to ground and marine vehicles such as mobilerobots, surface ships and underwater vehicles. Controlling underactuated mechanical sys-tems has been an active research because it concerns fundamentally nonlinear control prob-lems which require novel ideas and techniques. One of these challenges in nonholonomicsystems is to obtain a controller for assuring an asymptotic stabilization. Indeed, Brockett’snecessary condition [11] applied to these inherently nonlinear systems yields a surprisingfact that there is no linear or nonlinear, continuous state-feedback stabilizing control lawfor this special class of nonlinear systems.

A second feature arising from investigating nonholonomic control problems is thatstabilization and tracking are two fundamentally different control problems. Very often,in the conventional literature of control theory, stabilization is regarded as a special caseof the tracking problem. Unfortunately, this is not the case for underactuated mechanicalsystems with nonholonomic constraints. The violation of Brockett’s necessary conditionfor asymptotic stabilization presents a challenge to develop fundamentally new approachesto nonlinear control theory. On the other hand, in the case of trajectory-tracking, there isa local feedback solution if the linearization of the system around the moving trajectory isuniformly achievable, whereas the stabilization and tracking problems are typically studiedas two separate problems.

1.2.4 Why research in underactuated systems?

Research in underactuated systems is driven by several reasons. Some motivations for thedevelopment of a control system with less actuators than the DOFs are:

• Saving fuel. If less actuators is available, one can reduce the cost of fuel or any kindof energy needed for any actuator to work.

• Failure mode management. If one of the actuators fails, one can still obtain the ob-jective where the system is designed for.

1.2. NONLINEAR CONTROL, NONHOLONOMIC AND UNDERACTUATED SYSTEMS 5

• Building compactly. If no space is needed for ann-th actuator, such a space can besaved or used for other purposes.

• Lighter structure. If less actuators are available, one can reduce the weight of thestructure. This can be an advantage in positioning the structure.

As can be seen there are several practical reasons for research in underactuated systems.Theoretical insight of the problem is needed, so there are also theoretical reasons for re-search in underactuated systems.

1.2.5 Underactuated systems and second-order nonholonomic constraints

Many underactuated systems are subject to nonholonomic constraints. There are two typesof nonholonomic constraints: one is the first-order classical nonholonomic constraints andthe other is the second-order nonholonomic constraints. The first-order nonholonomic con-straints are defined as constraints on the generalized coordinates and velocities of the formΦ(q, q̇) and cannot be written as a constraint of the formϕ(q) = 0. They occur in, forexample, wheeled mobile robots or wheeled vehicles with trailers. The second-order non-holonomic constraints are defined as those on the generalized coordinates, velocities, andaccelerations of the formΦ(q, q̇, q̈), which are non-integrable, i.e., cannot be written as thetime-derivatives of some functions of the generalized coordinates and velocities given byϕ(q, q̇) = 0. For example, they occur in surface vessels, space robots, and underactuatedmanipulators.

Consider an underactuated mechanical system and letq = (q1, · · · , qn) denote the setof generalized coordinates. Partition the set of coordinates asq = (qa, qb), whereqa ∈ ℜmdenotes the directly actuated part andqb ∈ ℜn−m denotes the unactuated part. Withu ∈ ℜmas the vector of control variables, the equations of motion for underactuated mechanicalsystem become:

M11(q)q̈a +M12(q)q̈b + F1(q, q̇) = B(q)u (1.2)

M21(q)q̈a +M22(q)q̈b + F2(q, q̇) = 0 (1.3)

The Eqs. (1.2) and (1.3) definen−m relations involving the generalized coordinatesas well as their first-order and second-order derivatives. If there exists no non-trivial inte-gral, i.e., a smooth functionα(t, q, q̇) such thatdα/dt = 0 along all solutions of Eq. (1.3),then thesen − m relations can be interpreted as second-order nonholonomic constraints.Systems of this form are also not stabilizable using time-invariant continuous feedback dueto their failure to meet Brockett’s conditions. However, in both the first- and second-ordercases, it can be shown that many nonholonomic systems are strongly accessible, i.e., canbe stabilized at all equilibrium points.

As an example of second-order nonholonomic system, consider underactuated vehi-cles described by the following model:

M(v̇) + C(v)v +D(v)v + g(v) =

[τ0

]η̇ = J(η)v (1.4)

whereη ∈ ℜn, v ∈ ℜm, n > m andτ ∈ ℜk, k < m. HereM is the inertia matrix includingadded mass,C(v) is the Coriolis and centrifugal matrix, also including added mass,D(v) isthe damping matrix andg(v) is the vector of gravitational and buoyant forces and torques.

6 1. INTRODUCTION

LetMu,Cu(v),Du(v) andgu(v) denote the lastm−k rows of the matricesM ,C(v),D(v)and the vectorg(v), respectively. The constraint imposed by the unactuated dynamics canbe written as

Mu(v̇) + Cu(v)v +Du(v)v + gu(v) = 0 (1.5)

The gravitation and buoyancy vectorg(v) is important for the stabilizability proper-ties of underactuated vehicles. If the vectorgu(v) corresponding to the unactuated dynam-ics contains a zero function, then the constraint Eq. (1.5) is a second-order nonholonomicconstraint. It can be shown that underactuated vehicles subject to second-order nonholo-nomic constraints, do not satisfy Brockett’s necessary condition for asymptotic stabilizationusing time-invariant continuous state feedback.

1.2.6 Brockett’s theorem and consequences

The key hurdle in implementing a smooth feedback controller for nonholonomic systemsusing potential theory comes from the famous Brockett’s Theorem [11].

Theorem 1 Let ẋ = f(x, u) be given withf(x0, 0) = 0 and f(·, ·) continuouslydifferentiable in a neighborhood of(x0, 0). x ∈ D ⊂ ℜn, u ∈ ℜm. A necessary con-dition for the existence of a continuously differentiable control law which makes(x0, 0)asymptotically stable is that:

1. The linearized system should have no uncontrollable modes associated with eigen-values whose real-part is positive.

2. There exists a neighborhoodN of (x0, 0) such that for eachξ ⊂ N there existsa control uξ(·) defined on(0,∞) such that this control steers the solution ofẋ =f(x, uξ) fromx(0) to x(∞) = x0.

3. The mappingγ : D × ℜm → ℜn defined byγ : (x, u) → f(x, u) should be onto anopen-set containing0.

Corollary 1.1 [11] For an input-affine system of the forṁx = f(x)+m∑i=1

gi(x)ui, the

third condition of Theorem 1 implies that the stabilization problem cannot have a solutionif there exists a smooth distributionD containingf andgi with dimD < n.

Corollary 1.2 [11] A driftless input-affine system of forṁx =m∑i=1

gi(x)ui, with gi

linearly independent atx0, is stabilizable ifm = n, i.e., the system is fully actuated. Thecase where the setgi(x) drops dimension exactly atx0 is excluded from this restriction.

For example, consider the second-order chained form withn = 3 variables andm =2 inputs [3]:

ξ̈1 = u1

ξ̈2 = u2 (1.6)

ξ̈3 = ξ2u1

1.3. MOTIVATIONS AND OBJECTIVES 7

which is given in state-space form as

ẋ1 = x2

ẋ2 = u1

ẋ3 = x4 (1.7)

ẋ4 = u2

ẋ5 = x6

ẋ6 = x3u1

wherex2i−1 = ξi, x2i = ξ̇i, i = 1, 2, 3. Define the state vector asx = [x1, · · · , x6]T .Since the image of the mapping(x, u) 7→ f(x, u) = (x2, x4, x6, u1, u2, x3u1) of

the second-order chained form does not contain any point(0, 0, 0, 0, 0, ϵ) for ϵ ̸= 0, thesystem does not satisfy Brockett’s condition. Therefore, the system cannot be stabilized bycontinuous time-invariant feedback. Underactuated systems with a single unactuated DOFcan often be transformed into the second-order chained form as in Eq. (1.6). It is clear thatunderactuated systems cannot be controlled by a continuous time-invariant controller.

For our system, Corollary 1.1 is applied. Therefore, this precludes stabilization tothe origin of the configuration space of nonholonomic systems by smooth state feedback.

1.3 Motivations and Objectives

Since nonholonomic control systems present many interesting features and applications,they are becoming increasingly important in research and industry. Nonholonomic systemsare a prototype of strongly nonlinear systems, requiring a fully nonlinear analysis, becauseall first approximation methods are inadequate [38]. A lot of real life systems lie under theclass of nonholonomic systems such as mobile robots, hovercrafts, planar vertical takeoffand landing (PVTOL) and underwater vehicles. From these systems many of them areunderactuated, i.e., the number of control inputs is less than the number of DOFs [38],i.e., the number of generalized coordinates to be controlled. The difficulty of the controlproblem for underactuated mechanisms is obviously due to the reduced dimension of theinput space. There are two practical reasons for developing techniques to plan motions andto control underactuated systems. First, a fully actuated system requires more control inputsthan an underactuated system, which means there will have to be more devices to generatethe necessary forces. The additional controlling devices add to the cost and the weightof the system. Finding a way to control an underactuated version of the system wouldeliminate some of the controlling devices and could improve the overall performance orreduce the cost. The second practical reason for studying underactuated vehicles is thatunderactuation provides a backup control technique for a fully actuated system. If a fullyactuated system is damaged and we have an underactuated controller available, then wemay be able to salvage a system that would otherwise be uncontrollable.

The problem of steering an underactuated Autonomous Underwater Vehicle (AUV)to a point with a desired orientation has recently attracted attention of some researchers[22][38]. This task provides some challenges in nonlinear control systems theory, becausethe vehicle is underactuated and falls into the class of the nonholonomic systems. Those

8 1. INTRODUCTION

systems fail to satisfy the Brockett’s condition for the existence of smooth and time in-variant control law to achieve exponential and asymptotic stabilization of the systems [22].Such systems are vehicles with fewer number of independent control inputs than that ofDOFs. Furthermore, the dynamic equations of AUV are strongly nonlinear due to the pres-ence of complex hydrodynamics terms.

An X4-AUV with a spherical hull shape was studied by Okamura [53], in which itmakes only use of four thrusters to control the vehicle without using any steering rudders,it falls into the class of underactuated AUVs and has nonholonomic features. The consider-ation of nonholomic systems is an interesting study from a theoretical standpoint, becauseas pointed out in the earlier works of Brockett, they cannot be asymptotically stabilized to afixed point in the configuration space using continuously differentiable, time-invariant andstate feedback control laws [85].

In this thesis, to overcome the demerit that in an X4-AUV studied by [53], the dragforces against a stream are relatively higher than other AUVs, a new type of hull shapeis proposed for the X4-AUV with an ellipsoid body that mostly closes to a streamlinedshape and has the durability over pressure like a sphere. The ideal streamlined hull shapeis known to minimize the drag forces acting on the hull while the X4-AUV is cruising. Thecorresponding X4-AUV kinematic and dynamic models are also presented here. Whenthe X4-AUV moves underwater, additional forces and moment coefficients are added toaccount for the effective mass of the fluid that surrounds the robot, which causes an exces-sive acceleration of the robot, compared to the case where there is no any added mass andmoment of inertia. Then, appropriate nonholonomic or underactuated control methods areapplied for this vehicle. The control problem focuses on the problem of point stabilizationfor an X4-AUV control system.

1.4 Thesis Organization

Chapter 2 introduces fundamental ideas and concept regarding underwater vehicles, theirdesign and control methods. The major design aspects that need to be considered are iden-tifying hull design, propulsion, submerging and electric power. Another important thingsthat require consideration for the design process are factors that affect an underwater vehi-cle such as buoyancy, hydrodynamic damping, Coriolis and added mass. AUVs present achallenging control problem because most of them are underactuated, i.e., they have fewernumber of inputs than that of DOFs. Such control configurations impose non-integrableacceleration constraints. Furthermore, AUVs’kinematic and dynamical models are highlynonlinear and coupled, hydrodynamics of the vehicle are poorly known and may vary withrelative vehicle velocity to fluid motion, and a variety of unmeasurable disturbances byocean currents, making control design a difficult task. Therefore, appropriate nonholo-nomic or underactuated control methods are applied for this vehicle.

Chapter 3 describes the notation and coordinate systems, and introduces an explanationof the kinematic model and the derivation of a dynamical model of an X4-AUV. Theseequations are used in later chapters for controlling purpose. X4-AUV is designed with anellipsoid body hull shape to minimize the drag forces acting on the hull while the X4-AUV

1.4. THESISORGANIZATION 9

is cruising. It is also equipped with four thrusters, has 6-DOFs in motion, falls in an under-actuated system and also has nonholonomic features. The dynamic model of an X4-AUVis derived using Lagrange approach, with the assumption of balance between buoyancy andgravity. The modelling includes the consideration of the effect of added mass and inertia.

Chapter 4 presents a nonholonomic control method for stabilizing an X4-AUV. In thischapter, thex-, y-, andz-positions and angles of the X4-AUV is stabilized by using controlinputsu1, u2, u3, andu4 respectively. PD feedback control law is applied to control theattitude and positions of the X4-AUV with the direct use of the Lyapunov stability theory.The stability of the system is ensured by the Lyapunov theorem and LaSalle invariancetheorem. By the Lyapunov theorem, simple stability for equilibrium is ensured, whereasby the LaSalle invariance theorem, we can ensure an asymptotical stability starting from apoint in a set around the equilibrium. In our case, this theorem ensures the global stabilityof the system. Note that the simulations for stabilizing the X4-AUV in thex-, y-, andz-positions are implemented independently.

Chapter 5 deals with a discontinuous control law for stabilizing an X4-AUV. The systemis written in a control-affine form by applying a partial linearization technique. A dynamiccontroller based on Astolfi’s discontinuous control is derived to stabilize all states of thesystem to the desired equilibrium point exponentially. Two approaches are applied to thesystem. The first approach does not necessitate any conversion of the system model into acanonical form while for the second approach, the system is converted into a chained form.The discontinuous dynamic-model without using a chained form transformation in the firstapproach assures only a local stability (or controllability) of the dynamics based controlsystem whereas in the second approach, the discontinuous dynamic-model using a chainedform transformation guarantees a global stability of the system. Assumption made in thesimulation for the first approach is that the value ofθ- andψ-angles is very close to zero.

Chapter 6 gives a summary of this study and possible future enhancements concluded inthis chapter.

Chapter 2

Review of AUV Design and Control

2.1 Introduction

Underwater vehicles are being used in an ever increasing number of applications rangingfrom scientific research to commercial and leisure activities. Most of them tend to be usedfor a specific application, consequently, there is a wide variety of underwater vehicles inoperation. These vehicles can be categorized into several different groups according totheir particular characteristics. One of these characteristics is the method of control and thegroups used in this category are defined as illustrated in Figure 2.1.

Underwater

Vehicles

Manned Underwater

Vehicles

Unmanned Underwater

Vehicles (UUVs)

Autonomous

Underwater

Vehicles

(AUVs)

Remotely

Operated

Vehicles

(ROVs)

Figure 2.1: Underwater vehicles categorised by control method

This work focuses on Unmanned Underwater Vehicles (UUVs) and more specificallyAUVs. AUVs have onboard control systems that use the information recorded by sensors todetermine the demands to be sent to the vehicle actuators to complete the defined missions.The reliance on these components dictates a need for a robust design. A constraint on theuse of an AUV is the limited energy supply that can be carried onboard. Most AUVs usebatteries of various types to provide both propulsion and power. Therefore the total energyavailable is limited by the available volume (or weight) for batteries and the energy densityof the chosen batteries.

10

2.1. INTRODUCTION 11

These two characteristics of AUVs heavily influence the design choices during thedevelopment of an AUV. The autonomous nature of the vehicle means that key design fac-tors include reliability, robustness and controllability. The limited energy available meansthat the energy cost associated with the various choices is a key factor in the design evalu-ation process. The combination of these factors shows that the design cycle for an AUV ishighly iterative.

In contrast, ROVs are operated with a connection to a surface station, either on landor on a surface vessel. This connection is used to provide a communication link between thevehicle and a human operator, allowing human control, rapid data transfer and much largerpower supply. On most ROVs the control system is dependent on partly human, partlyautomation; some elements of the control systems are undertaken using automatic control(for example depth control) allowing the human operator to concentrate on the intricaciesof the particular task. The larger power supply allows the designer (and operator) to designthe vehicle with less consideration for the energy required and this freedom also allowsredudancy to be built into the design, for example in thruster configurations, which is notfound on energy limited AUVs.

The required range of a vehicle can significantly influence the characteristics of anAUV during the design of the vehicle. For example, the design of a short range AUVrequires less emphasis on propulsive efficiency in energy use. This freedom allows theshort range AUV designer to include more energy consuming devices and to be optimizedfor the mission requirements. On the other hand, the key to successful long range AUVdesign is a compromise between functionality limitations and mission range requirementsand hence greater emphasis on hydrodynamic efficiency. The AUVs were first built inthe 1970s, put into commercial use in the 1990s, and today are mostly used for scientific,commercial, and military mapping and survey tasks [84].

Currently, the challenges for AUV address the navigation, communication, auton-omy, and endurance issues. Autonomy is the main aspect of AUVs which deals with theelectronics and control design. During a mission, an AUV may undergo different maneu-vering scenarios such as a complete turn at the end of a survey line, a severe turn duringobstacle avoidance or frequent depth changes while following a rugged seabed terrain. Dif-ferent control schemes are used for different operations. However, the AUV’s dynamics areinherently nonlinear and time-variant, i.e., its mass and buoyancy change according to dif-ferent working conditions. It is also subject to uncertain external disturbances, and thehydrodynamic forces are difficult to model. Thus, AUV control can be regarded as a verychallenging task.

Linear control theories have evolved a variety of powerful methods. This method canmeet the requirements of stability, robustness and dynamical responses when used in linear,time-invariant systems. However, in the case of an AUV, the complexity of its dynamicslisted above makes linear control methods difficult to achieve satisfactory results. Its dy-namics cannot be linearly approximated accurately and often results in undesired behaviorwhen linear control methods are used. For such a system, nonlinear control techniques mayprovide greatly increased performance and stability. In the case of underactuated nonholo-nomic systems, nonlinear control methods are more appropriate because the nonholonomicconstraints are inherently nonlinear.

12 2. REVIEW OF AUV D ESIGN AND CONTROL

2.2 Types of AUVs

Within the AUV group, there are some subgroups of vehicles which are split according totheir particular application. The vehicles in these groups have a common features. MostAUVs can be classified into the following categories [66]:

2.2.1 Shallow water survey AUVs

Shallow water survey AUVs are rated up to 500 m, and are used for performing oceano-graphic surveys from close to the surface. These are typically small in size because they donot have to bear a lot of water pressure, have a high thrust to drag ratio, and so are able tomaneuver in areas with high currents. Also, the typical surveys for these types of vehiclesare performed over a large scale with fairly low resolution, so their operating speeds arerelatively high, in the order of a few knots/hr.

2.2.2 Mid-water AUVs

These refer to the class of AUVs rated up to 2500 m that are typically used for performingmid-water column surveys or seafloor surveys in shallower areas. These are typically bulkyin order to handle the high pressure at depth, which in turn means they need more thrustand more power that also add to their size. Since there is not much current at these depths,this class of AUVs can have small thrust to drag ratio. Depending on the application that atypical AUV of this class is being used for, its operating speeds can vary from less than oneknot/hr for a photographic survey to a few knots/hr for a multibeam or sidescan survey.

2.2.3 Deep-water AUVs

Deep-water AUVs are the class of AUVs designed to be used at depths of more than 2500m. Due to the high oceanographic pressures that these vehicles need to be able to bear,the housings are large and bulky. Also, since diving to such depths takes a long time,one would like to get longer missions out of each dive which means that these vehiclesneed more power storage, again adding to their size. To keep their sizes small, and makethem more power efficient, these vehicles have a low thrust to drag ratio. Since AUVsof this class are usually used close to the ocean bottom for high resolution surveys, theymust be able to maneuver at low speeds. Their design cannot involve control surfaces formaneuvering which results in multi-hull designs with multiple thrusters.

2.2.4 Gliders

Gliders refer to underwater vehicles that use changes in buoyancy and water temperaturein conjunction with wings to convert vertical motion into forward motion. These buoyancyengines typically achieve much more efficiency than the conventional electric thrusters,greatly increasing their range to an order of thousands of kilometers. Typically these vehi-cles operate in the upper water column, and are usually rated for less than 1000 m. For amore comprehensive list of applications and AUV development, refer to [76][77][84].

2.3. GENERAL DESIGN OF ANAUV 13

　 Vehicle operation requirements

Vehicle specifications

(configuration, actuator

control, etc.)

Model testing (water tank

testing)

Vehicle basic design

Vehicle detailed design

Construction

Sea trials

Operation

Verification

feedback Sensor basic design

and development

Drawing

Detailed specifications

(based on basic

specifications)

CAD/CAM design

Basic specifications

Maneuvering control

properties test

Fluid properties test

Vehicle model making

Vehicle nonlinear

maneuvering model

(mathematical model)

Vehicle maneuvering

Simulation analysis

Control

system

design

model

Control

system

design

1

2

Modification

Database

Creation

Control

logic

Modifi-

cation

Figure 2.2: Ideal method for developing underwater vehicle [81]　　

2.3 General Design of an AUV

Figure 2.2 shows the ideal method for developing underwater vehicle. As shown in theflowchart on the left side of Figure 2.2, the underwater vehicle development process in-cludes: definition of vehicle specifications such as configuration, actuator, control, andsensor in accordance with operation requirements; water tank testing of the model for per-formance verification; and feedback on the verification results to the specifications defined.The important point here is the simulation design process (“1” in the flowchart) on the rightside of Figure 2.2 and the model making process followed by the fluid properties test andmaneuvering control properties test (“2” in the flowchart). This project focuses only onprocess 1. Model testing through the above processes can identify necessary refinements atan early stage before construction of the vehicle, and thereby facilitate the process of basicdesign, detailed design, and construction of the vehicle afterwards. The risk of reconstruc-tion and major modification after construction of the vehicle can be particularly reduced tominimize the overall development risk [81].

There are several aspects in AUV electrical and mechanical design need to be lookedat closely so that the design will be successful. In order to design any AUV, it is essentialor compulsory to have strong background knowledge, fundamental concepts and theoryabout the processes and physical laws governing the underwater vehicle in its environment.Therefore, the major design aspects that need to be considered [29] are identifying hull

14 2. REVIEW OF AUV D ESIGN AND CONTROL

(a) Laminar Boundary Layer

(b) Turbulent Boundary Layer

Figure 2.3: Laminar and turbulent boundary layer separation

design, propulsion, submerging and electric power.

2.3.1 Hull design: shape and drag

The most basic characteristic about an AUV is its size and shape. The basic shape of theAUV is the very first step in its design and everything else must work around it. The shapeof the AUV determines its application, efficiency and range. There have been a wide varietyof AUVs in size and shape, ranging from [70]:

• Conventional torpedo proportions, large and small.• Laminar flow, bulbous hull to reduce drag.• Streamlined rectangular style.• Multi hull vehicles, splitting the energy, propulsion and mission management from

the sensor payload into separate hulls.

2.3.1.1 Laminar flow body The laminar flow body achieves low drag by maintaininglaminar flow over most of its length by virtue of its bulbous shape. From a simple per-spective of drag reduction, a form that promotes laminar flow within the boundary layeris the best choice. In laminar flow, fluid particles move in layers and skin friction drag ismuch lower than that in a turbulent flow where fluid particles more erratically resulting inhigher shear stresses between layers (see Figure 2.3). For determining whether a flow willbe laminar or turbulent, a Reynolds Number (the ratio of inertial forces to viscous forces)is used. Laminar flow occurs at low Reynolds numbers, and is characterized by smooth,constant fluid motion. Turbulent flow occurs at high Reynolds numbers and is dominatedby random eddies, vortices and other flow fluctuations.

To sustain laminar flow, a hull can be designed such that the diameter increases grad-ually from the nose to create a favorable pressure gradient over the forward 60 – 70% ofthe hull. In this area, the surface must be smooth and as hydrodynamically clean as pos-sible. Forward-mounted hydroplanes cannot be allowed because they disturb the laminar

2.3. GENERAL DESIGN OF ANAUV 15

Figure 2.4: Outline of laminar flow body [70]

flow. Consequently all hydroplanes are to be fitted on the-boom. Acoustic payload, com-munication and navigation transducers must be located as far aft as possible so that theresulting openings or protuberances do not disturb the laminar flow. Figure 2.4 shows atypical shape of such a hull. The main disadvantage of this unique shape of the laminarflow body is that it does not readily permit lengthening or shortening of the vehicle, thuslimiting the possibility of modular expansion [68].

2.3.1.2 Torpedo vs. non-torpedo shape vehiclesMost AUVs used in science and indus-try today can be classified into a torpedo shaped design and a non-torpedo shaped designindependent of other characteristics [66]. Figures 2.5 and 2.6 below show some of the stateof the art AUVs in the science community today. This classification is important becauseit governs a lot of the characteristics of the AUV. A typical torpedo shaped or single hullAUV has less drag and can travel much faster than its non-torpedo shaped counterpart. Atorpedo shaped AUV usually uses an aft thruster and fins to control its motion; thus thesedesigns need some translational speed to keep full control of the vehicle. This class ofAUVs in general has a much longer range and can work well in areas with moderate cur-rents. They are appropriate for low resolution scalar surveys in larger areas, but are notsuited for optical surveys or high resolution bathymetric surveys of a smaller area. TheseAUVs have 6-DOFs, namelyx-, y-, z-translation, roll, pitch and heading, but these can-not be controlled independently, making the autonomous control of these AUVs relativelyharder.

The non-torpedo shaped AUVs are typically designed to be completely controllableat much lower speeds. The multiple hull design makes these kinds of AUVs passivelystable in pitch and roll, which means the other DOFs can be independently controlled usingmultiple thrusters. A larger form factor for these vehicles means a higher drag, whichmakes their use difficult in areas with significant currents. The lower speeds and highmaneuverability of this class of AUVs means higher navigational accuracy to follow veryclose tracklines. They are well suited for high resolution photographic surveys, multibeammapping and sidescan surveys. The difference in the two classes of AUVs is analogous tothat of the airplane and helicopter. They have their own advantages and cater to differentapplications. The science community will always have these two kinds of AUVs co-existto meet the complete set of requirements.

2.3.1.3 Effect of slenderness ratio AUVs have tended to be designed around length-to-diameter (L/D) ratios of five to eight, mimicking in some respects naval torpedoes andaircraft drop tanks to provide the maximum volume for minimum drag. But AUVs havethe additional design constraint to reduce the risk of collision with the mother ship during

16 2. REVIEW OF AUV D ESIGN AND CONTROL

Figure 2.5: Torpedo shaped AUVs [66]

Figure 2.6: Non-torpedo shaped AUVs [66]

launch and recovery and will have a larger footprint on the ship’s deck. However, the dragcoefficient (CD) values for the National Advisory Committee for Aeronautics’ (NACA)aerofoil solid of revolution versusL/D ratios of two to 10 show a surprisingly constantCD down to aL/D value as low as three (excludes control surfaces) [70]. Similar resultsare seen in early wind tunnel work performed on airship models. Thus, short, fat AUVs donot have a significantly higherCD than slender ones and are inherently easier to handleand store on board a ship, although short vehicles may have stability issues that need tobe considered. A more important drag consideration is the variation in drag between theidealized shape and the practical vehicle.

2.3.2 Submerging

In the case of a submersible vehicle, since the volume of the vehicle remains constant,in order to dive deeper, it must increase the downward force acting upon it to counteractthe buoyant force. This can be accomplish either by increasing its mass via the use ofballast tanks or by using external thrusters. Ballasting is the more common approach forsubmerging. This method is mostly mechanical in nature and involves employing pumpsand compressed air to take in and remove water. The alternative is to use thrusters that

2.4. FACTORSAFFECTING AN UNDERWATER VEHICLE 17

point downwards. This is a much simpler system, but is quite inefficient in terms of powerconsumption and not really suited at great depths. To reduce the size of ballast tanks or theforce required by thrusters for the process of submerging, AUVs are usually designed soas to have residual buoyancy. That is, the weight of the vehicle is made to be more or lessequal to the buoyant force.

2.3.3 Propulsion

Some sort of propulsion is required on all AUVs and is usually one of the main sourcesof power consumption. Most AUVs use motors for propulsion due to the scarcity and costof alternative systems. The location of the motors affects which DOFs can be controlled.The positioning of the motors can also affect noise interference with onboard electroniccomponents, as well as propeller-to-hull and propeller-to-propeller interactions. Propeller-to-hull and propeller-to-propeller interactions can have unwanted effects in the dynamicsof an AUV. When travelling at a constant speed, the thrust produced by the motors is equalto the friction or drag of the vehicle, that is

Thrust = Drag = 0.5ρs2ACD (2.1)

whereρ is the water density,s is the speed,A is the effective surface area andCD is thedrag coefficient. Power consumption for the propulsion system increases dramatically asthe speed of the vehicle increases. This is because the thrust power is equal to the productof the thrust and the speed, meaning thrust power is a function of speed cubed,

Thrust Power = Thrust× s = 0.5ρs3ACD (2.2)

Therefore, because of an AUV’s limited energy supply, it must travel at a speed that doesnot draw too much power, but at the same time does not take too long to complete itsmission. Obtaining the ideal speed becomes an optimization problem.

2.3.4 Electric power

Electric power is commonly provided via sealed batteries. The ideal arrangement of bat-teries is to have them connected in parallel with diodes between each one to allow evendischarge and to prevent current flow between batteries. Fuses or other protective devicesshould also be used to prevent excessive current flow in case of short circuits occurring orcomponents malfunctioning. The restrictive nature of power on AUVs influences the typesof components and equipment that can be utilized. Components and equipment should bechosen so as to draw as little power as possible in order to allow the batteries to providemore than enough time for the vehicle to complete its mission.

2.4 Factors Affecting an Underwater Vehicle

Several forces that require consideration for the design process act on an underwater vehi-cle. These include buoyancy, hydrodynamic damping, Coriolis and added mass. Buoyancy,which significantly affects the vehicle’s ability to submerge as well as its stability, is oneof the most important factors. Stability is also affected by external forces. Pressure thatneeds to be taken into consideration in the design process [29] is another significant factor

18 2. REVIEW OF AUV D ESIGN AND CONTROL

for underwater vehicles.

• Buoyancy: The magnitude of the buoyant force (B) exerted on a body, which isfloating or submerged, is equal to the weight of the volume of water displaced by thatbody. The ability of an object to float depends on whether or not the magnitude of theweight of the body (W ) is greater than the buoyant force. Clearly, ifB > W , thenthe body will float, while ifB < W it will sink. If B andW are equal, then the bodyremains where it is.

• Hydrodynamic Damping: When a body is moving through the water, the main forcesacting in the opposite direction to the motion of the body are hydrodynamic dampingforces. These damping forces are mainly due to drag and lifting forces, as well aslinear skin friction. Damping forces have a significant effect on the dynamics of anunderwater vehicle which leads to nonlinearity. Linear skin friction can be considerednegligible when compared to drag forces, and therefore, it is usually sufficient to onlytake into account the latter when calculating damping forces.

• Stability: Assuming no water movement, the stability of a static body underwater ispredominantly affected by the position of the center of mass (CM ) and the center ofbuoyancy (CB). The center of buoyancy is the centroid of the volumetric displace-ment of the body. IfCM andCB are not aligned vertically with each other in eitherthe longitudinal or lateral directions, then instability will exist due to the creation of anonzero moment. IfCM andCB coincide in the same position in space, the vehiclewill be very susceptible to perturbations. Ideally, the two centroids should be alignedvertically some distance apart from each other withCM belowCB. This results inan ideal bottom-heavy configuration with innate stability. In the case of a dynamicunderwater body, stability is affected not only by the centers of mass and buoyancy,but also by factors such as external forces and centers of drag. To increase dynamicstability, the centers of drag, determined by the centroids of the effective surface ar-eas of the vehicle, should be aligned with the centers of the externally applied forces.In this manner, the vehicle will not tend to exhibit undesirable characteristics in itsmotion.

• Coriolis: Coriolis is an inertial force that acts perpendicular to the direction of motionof a body. The force is proportional to both the velocity and rotation of the coordinatesystem. The effect of the Coriolis force then, is that the path of the body is deflected.In reality, however, the path of the body is not actually deflected, but only appears tobe. This is due to the motion of the body’s coordinate system. Since the coordinatesystem of an AUV rotates with respect to another reference frame, the effect of theCoriolis force is usually taken into account and included in the equations of motion.

• Added Mass: Another phenomenon that affects underwater vehicles is added mass.When a body moves underwater, the immediate surrounding fluid is accelerated alongwith the body. This affects the dynamics of the vehicle in such a way that the force

2.5. PREVIOUS WORK ON CONTROL OFUNDERACTUATED AUV S 19

required to accelerate the water can be modelled as an added mass. Added mass is afairly significant effect and is related to the mass and inertial values of the vehicle. Itis greatly influenced by the shape of the vehicle. Added mass coefficient is extractableusing empirical formulas, the simple analytical relations and numerical methods suchas Strip theory and lab tests. In [44], the formulas for the elements of the addedmass and inertia matrices for a submerged ellipsoidal body is discussed while in [27],the added mass coefficient has been examined for a sphere and ellipsoid body usingnumerical boundary element method.

• Environmental Forces: Environmental disturbances can affect the motion and stabil-ity of a vehicle. This is particularly true for an underwater vehicle if waves, currentsand even wind can perturb the vehicle. When the vehicle is submerged, the effectof wind and waves can be largely ignored. The most significant disturbances thenfor underwater vehicles are currents. In a controlled environment such as a pool, theeffect of these environmental forces is minimal.

• Pressure: As with air, underwater pressure is caused by the weight of the medium,in this case water, acting upon a surface. Pressure is usually measured as an abso-lute or ambient pressure; absolute denoting the total pressure and ambient being ofa relativistic nature. At sea level, pressure due to air is 14.7 psi or 1 atm. For every10 m of depth, pressure increases by about 1 atm and hence, the absolute pressure at10 m underwater is 2 atm. Although it is linear in nature, the increase in pressure asdepth increases is significant and underwater vehicles must be structurally capable ofwithstanding a relatively large amount of pressure if they are to survive [29].

2.5 Previous Work on Control of Underactuated AUVs

This section starts with a basic principle of the control system design for marine systems,followed by a brief review on the control of nonholonomic systems, due to their relevance tothe control of underactuated AUV. Next, the existing methods on control of underactuatedAUV are reviewed.

2.5.1 Basic principle for the development of marine control system

A basic principle of the control system design for marine systems is described in this sec-tion. According to Yamamoto experience [80], an index map for the control system designhas been proposed in Figure 2.7. The horizontal line represents the scale and complexityof requested system. The vertical line represents the grade of difficulty of the controlledsystem, which means the degree on nonlinearity.

In terms of the model involved, the control design can be categorized into three dif-ferent approaches [13][12].

1. Model-based nonlinear control. A nonlinear model is first derived using the Newton-Euler equation of motion for underwater vehicle moving in 6-DOFs. The forces andmoments working on Unmanned Underwater Vehicle (UUV) are to be formulatedby using hydrostatic (gravity and buoyancy) and hydrodynamic (added mass, steady-state and thruster) components. Using this approach, only one model is required

20 2. REVIEW OF AUV D ESIGN AND CONTROL

　

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Scale & Complexity of Requested System (Dimension, Number of I/O Units)

ADAPTIVE CONTROL

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ROBUST

CONTROL CLASSICAL CONTROL

Rule-Based

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-Synthesis Control

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Feedforward Control

Autotuning Control

PID Control

Decoupling PID Control

Dead Time Compensation Control

I-PD, Two-Degree-of

Freedom PID Control

Figure 2.7: Map of the control system design [80]　　

to represent the whole underwater vehicle envelope of the UUVs. Nonlinear opti-mization techniques are used as tool for system identification. Appropriate nonlinearcontrol can then be designed using the model.

2. Model-based linear control. The method starts by defining a number of trim condi-tions associated with representative UUVs operation. Linearization procedure is con-ducted based on the principle of small perturbation around trimmed points. Linearcontrol synthesis can then be applied to each linear model. To cover the entire UUVenvelope, a gain-scheduled controller is applied based on the interpolation betweenfamilies of linear models.

3. Control without system model. Conventionally, the method is based on classicalsingle-input single-output (SISO) PID. The feedback control system is designed inwhich the controller parameters are tuned empirically to acquire acceptable controlsystems. The approach is predicated to the assumption that couplings between ve-hicle modes are negligible. In general the SISO approach however is not agreeablewith complex UUV vehicles with sophisticated performance criteria. To make a de-cent controller, more advanced multivariable controller synthesis approaches requireaccurate models of the dynamics.

2.5. PREVIOUS WORK ON CONTROL OFUNDERACTUATED AUV S 21

As pointed out by [80], a model-based design method is more effective for the de-sign of control system if the vehicle dynamics is modeled to some extent. For vehicles,hydrodynamic coefficients are main parameters of the vehicle dynamics which can be es-timated to a great extent by real experiments and theoretical estimation techniques. In thecase of an AUV, the complexity of its dynamics makes linear control methods difficult toachieve satisfactory results. The dynamics cannot be linearly approximated accurately andoften result in undesired behavior when linear control methods are used. One of the ear-lier nonlinear control schemes is model-based control. However, model-based nonlinearcontrollers often fail to achieve desired performance in the presence of modeling errors,parameter uncertainties and unknown disturbances. Parameters of the AUV system dependon operational conditions and may not be precisely known in advance. Therefore, manyresearchers have studied nonlinear control methods, and various schemes have been pro-posed to control nonlinear and time-invariant system with uncertainties in parameters anddisturbances.

2.5.2 Control of nonholonomic systems

The term “nonholonomic system” originates from classical mechanics and has its widelyaccepted meaning as a “Lagrange system with linear constraints being nonintegrable”. Amechanical system is said to be nonholonomic if its generalized velocity satisfies an equal-ity condition that cannot be written as an equivalent condition on the generalized position,see [75]. Control of nonholonomic dynamic systems has formed an active area in the con-trol community, see surveys by Kolmanovsky and McClamroch in [38], Murray and Sastryin [48], and references therein for an overview and interesting introductory examples inthis expanding area.

Nonholonomic systems have inherent difficulties in feedback stabilization at the ori-gin or at a given equilibrium point because the tangent linearization of these systems isuncontrollable. In fact, a direct application of Brockett’s necessary condition, see subsec-tion (1.2.6) for more details, for feedback stabilization implies that nonholonomic systemscannot be stabilized by any stationary continuous state feedback, though they are open loopcontrollable. As a consequence, the classical smooth control theory cannot be applied. Thismotivates researchers to seek novel approaches. These approaches can be roughly classi-fied into discontinuous feedback, see for example [7][8] and time-varying feedback, see forexample, [48]. The discontinuous feedback approach often uses the state scaling originatedfrom theσ-process [5] and a switching control strategy to overcome the difficulty due tothe loss of controllability. This approach results in a fast transient response and usually anexponential convergence can be achieved. The drawback is discontinuity in the control in-put. On the other hand, the time-varying feedback approach provides a smooth/continuouscontroller, i.e., no switching is required, however the price is slow convergence. The sta-bility analysis is often based on linear time-varying system theory and Barbalat’s lemma.The backstepping technique [39] is usually used for high-order chained form systems inboth discontinuous and time-varying approaches. Those aforementioned systems are ei-ther driftless or have weak nonlinear drifts. When nonholonomic systems are perturbedby drifts with uncertainties, robust and adaptive control approaches are often applied. Therobust control design schemes are based on the size domination concept [61]. The control

22 2. REVIEW OF AUV D ESIGN AND CONTROL

is conservative when a priori knowledge of uncertainties is poor. A class of nonholonomicsystems with strong nonlinear uncertainties was recently considered in [35]. Discontinuousstate feedback and output feedback controllers were designed to achieve global exponentialstability. The adaptive approach [71] provides less conservative control input but increasesthe dynamics of the closed-loop system. For a solution of the stabilization of nonholonomicsystems in a chained form with strong nonlinear drifts and unknown parameters, the readeris referred to [18].

2.5.3 Control of underactuated ships and AUVs

Control of underactuated ships and AUVs is an active field due to its important applicationssuch as passenger and goods transportation, environmental surveying, undersea cable in-spection, and offshore oil installations. Based on its practical requirement, motion controlof underactuated ocean vessels has been divided into three areas: Stabilization, trajectory-tracking, and pathfollowing. These control problems are challenging due to the fact thatthe motion of underactuated surface ships and AUVs possesses more DOFs to be controlledthan the number of the independent control inputs under some nonintegrable second-ordernonholonomic constraints [29][53][54]. In particular, underactuated ships do not usuallyhave an actuator in the sway axis while in the case of AUVs there are no actuators in thesway and heave directions. This configuration is by far the most common among ma-rine vessels. Therefore, Brockett’s condition indicates that any continuous time-invariantfeedback control law does not make a null solution of the underactuated surface ship andAUV dynamics asymptotically stable in the sense of Lyapunov. Furthermore as observedin [22][54], the underactuated ship and AUV system is not transformable into a standardchain system. Consequently, existing control schemes [15][32][48] developed for chainedsystems cannot be applied directly. Nevertheless, in the past decade, stabilization, trajec-tory tracking control, and path-following of underactuated ocean vessels have been studiedseparately from different viewpoints.

2.5.4 Stabilization control of an underactuated AUVs

The problem of steering an underactuated AUV to a point with a desired orientation hasonly recently received special attention in the literature. This task raises some challengingquestions in control system theory, because the vehicle is underactuacted. Furthermore, aswill be shown, its dynamics are complicated due to the presence of complex hydrodynamicterms. This rules out any attempt to design a steering system for the AUV that would relyon its kinematic equations only. Pioneering work in this field is reported in [44], whereopen-loop small-amplitude periodic time-varying control laws are used to re-position andre-orient underactuated AUVs. A feedback control law that gives exponential convergenceof a nonholonomic AUV to a constant desired configuration is introduced in [19]. Thedesign of a continuous, periodic feedback control law that asymptotically stabilizes anunderactuated AUV and yields exponential convergence to the origin is described in [56].See also [57] for an extension of these results to address robustness issues. In [58], atime-varying feedback control law is proposed to yield global practical stabilization andtracking for an underactuated ship using a combined integrator backstepping and averagingapproach. Practical applications of these results can be found in [59]. More recent work

2.5. PREVIOUS WORK ON CONTROL OFUNDERACTUATED AUV S 23

is described in [2], where the problem of regulating a dynamic model of a nonholonomicand underactuated AUV to a desired point with a given orientation is addressed and solved.This is performed by using a discontinuous, nonlinear adaptive state feedback controller,and it yields a convergence of the trajectories of the closed-loop system in the presence ofparametric modeling uncertainty.

In fact, point stabilization of an underactuated AUV poses considerable challengesto control system designers, because the models of those vehicles typically include a driftvector field that is not in the span of the input vector fields, thus precluding the use of inputtransformations to bring them to driftless form.

The control of the position and orientation of underwater vehicles has received muchattention in recent years [55][62] due to various applications. Stabilization of this systemhas been studied by many authors from the point of time-varying, discontinuous and hybridcontrollers. We briefly summarize the previous work in this area. Asymptotic stabilizationof the surface vessel’s position but not orientation using a continuous feedback controller ispresented in [78]. Time-varying controllers are synthesized to stabilize the surface vesselin [47][55][59]. Exponential stabilization is presented in [55], a semi-global exponentialstabilization in [59] and global asymptotic stabilization in [47]. As mentioned in the previ-ous section, the time-varying controllers suffer from low convergence rate and oscillatingtrajectories because these controllers employ a sinusoidal term that generates an oscilla-tory motion. Recently a time-varying controller is designed for stabilizing a nonholonomicsystem without a sinusoidal function in the controller and has been applied to a surfacevessel [73]. In [62], a discontinuous control law has been presented to achieve asymptoticstabilization to an equilibrium configuration with exponential convergence rates. The dis-continuous nature of the control law is due to the rational transformation introduced for thestates of the system. The control law has atmost one switching and is valid for a restrictedset of initial conditions. A discontinuous controller is developed for a higher-order chainednonholonomic system and then applied to a surface vessel/underwater vehicle in [40] witha restriction on the initial condition that is less restrictive than [62].

Control of underwater vehicles/surface vessels using various techniques such as dis-continuous and time-varying controllers are presented in [47][55][59][62][64][65]. Time-varying controllers for stabilizing a surface vessel have been proposed in [47][55][59] fromdifferent aspects such as exponential stability, state feedback and output feedback. Stabi-lization of a surface vessel using a discontinuous control law has been studied by Rey-hanoglu [62], in which almost exponential stability is achieved.

Chapter 3

Modeling of an UnderactuatedX4-AUV

The existence of several complex and nonlinear forces acting on an underwater vehicle,makes the control of AUVs trickier. Such forces are, for example, hydrodynamic drag,damping, lift forces, Coriolis and centrifugal forces, gravity and buoyancy forces, thrusterforces, and environmental disturbances [34]. In this chapter, we review the modeling ofmarine vehicles, classify the basic motion tasks for underwater vehicle, describe a type ofan X4-AUV design and their mathematical models, which will be used for the design ofvarious control systems in the subsequent chapters.

3.1 Introduction

Modeling of marine vehicles involves the study of statics and dynamics. Statics is con-cerned with the equilibrium of bodies at rest or moving with constant velocity, whereasdynamics is concerned with bodies having accelerated motion. The foundation of hydro-static force analysis is the Archimedes’ principle. The study of dynamics can be dividedinto two parts: kinematics, which treats only geometrical aspect of motion, and kinetics,which is the analysis of the forces causing the motion [24]. The increasing needs for AUVhave brought about corresponding demands of accurate control of AUV and consequently,models which control laws are based on Abkowitz [1] addressed issues pertaining to thestability and motion control of marine vehicle. He derived the dynamics of marine vehicles,and also studied and analyzed the external forces and moments acting on the vehicles. Shiphydrodynamics, steering and maneuverability are well discussed. Fossen [24] has also de-scribed the modeling of marine vehicles. He described the details of vehicles’ kinematicsand rigid body dynamics. Based on these, the compact forms of equations of vehicle motionwere explained specifically. In addition, he divided the hydrodynamic forces and momentsinto two parts: radiation-induced forces and Froude-Kriloff and diffraction forces.

The equations of motion are nonlinear. The forces and moments acting on a vehiclemoving through a fluid medium are dependent on many factors. These include the prop-erties of the vehicle (length, geometry, etc.), the properties of motion (linear and angularvelocities, etc.), and the properties of the fluid (density, viscosity, etc.). Among these forcesand moments, the hydrodynamic forces are the most difficult part to model. Newman [52]has presented the marine hydrodynamics in detail, especially the derivation of the addedmass. While many literature deal with surface ships, articles pertaining to AUVs are not ascommon. Yuh [83] is one of the earliest to describe AUV modeling. He re-emphasized the

24

3.2. BASIC MOTION TASKS 25

importance of added mass and introduced functional terms which are essential in describ-ing the equations of motion of an AUV. Since then, many papers and books which furtherextend this work have appeared.

While almost all reports on control of AUVs invariably list all or part of the 6-DOFsequations of motion, any newcomer to the topic will most likely be unable to decipher thevarious terms involved. Fossen offers the most comprehensive treatment on AUV modelingin [1][25][26]. Interested readers can find detailed explanations of the various terms thatform the equations of motion. After deriving general equations of AUV motion, the nextstep is to determine the relevant coefficients in these equations and then obtain the wholedynamics model. In these coefficients, the hydrodynamic derivatives are the most difficultterms to model. Therefore, according to the methods of modeling hydrodynamic forces,Goheen [28] has categorized two methods of modeling AUV dynamics: test-based methodand predictive method.

The test-based method requires dir