A LAGRANGIAN APPROACH - OSTI.GOV

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SVEIN IVAR SAGATUNNEI-NO--;

MODELING AND CONTROL OF UNDERWATER VEHICLES:

A LAGRANGIAN APPROACH

NTH DOKTOR INGENI0RAVHANDLING 1992:18 INSTITUTT FOR TEKNISK KYBERNETIKK TRONDHEIM

UNIVERSITETET I TRONDHEIM NORGES TEKNISKE H0GSKOLE ITK-rapport 1992:28-W

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DISCLAIMER

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NEI-NO-302

DE93 752999

Modeling and Control of Underwater

Vehicles :

A Lagrangian Approach

by

Svein Ivar SagatunDivision of Engineering Cybernetics

Department of Computer Science and Electrical Engineering The Norwegian Institute of Technology

Diploma Engineer from the Division of Marine Systems Design,Department of Marine Technology, Norwegian Institute of Technology (March 1988).

Submitted tothe Division of Engineering Cybernetics, Department of Electrical Engineering and Computer Science, Norwegian Institute of Technology in partial fulfillment of the requirements for the

Dr.Ing. degree in Engineering Cybernetics

at the

Norwegian Institute of Technology (March 1992)

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Summary

This thesis contains an extensive study of modeling and control of underwater vehicles. The main results of this thesis are or will be published in international journals and conferences.

The thesis concludes that there will be an emerging demand for more optimal and advanced nonlinear controllers for underwater vehicles. Minimum control effort controllers will be especially needed.

Lagrangian dynamics has successfully been employed to derive the equations of motion for underwater vehicles.The Lagrangian approach in modeling exposes the properties of the underwater vehicles’ equation of motion in a way that the more traditional Newtonian approach does not. The expressions for energy and work used in the Lagrangian mechanics are also useful in control system design when stability and optimality are proved by using nonlinear stability theory. The equations of motion are derived in both an inertial reference-frame and a vehicle-fixed coordinate system.

Results from open-water tests of the Norwegian Experimental Remotely Operated Vehicle’s (NEROV) thruster system are presented. These test-results show the nonlinear relation between the vehicle’s velocity and thrust. Test results from free-decay tests of the NEROV are also presented. These results show how the vehicles added inertia and damping coefficients vary as a function of the Keulegan-Carpenter number. These parameters’ dependencies of frequency and a perturbed vehicle geometry are also investigated. These tests clearly show that the model parameters of underwater vehicles are time- varying.

Two optimal and near-optimal nonlinear controllers are derived in minute detail. Both controllers are optimal in the sense that they minimize the generalized forces that correspond to the vehicle’s kinetic energy and the energy which dissipate from the vehicle. Adaptive versions of the controllers are also presented. Lyapunov theory and Barbalat’s lemma are used to prove stability. The expressions for energy and work derived in the mod

IV

eling chapter will be particularly useful in these proofs. This thesis shows how favourable it is to first derive the controllers in an inertial reference frame for then transforming the control scheme to a vehicle-fixed representation before implementation.

Contents

Summary iv

List of Tables ix

List of Figures xiii

Nomenclature xv

Preface xxii

1 Introduction 11.1 Background and Motivation for this Research .............................................. 11.2 Optimal Path Planning, Trajectory Generation and Tracking....................... 31.3 Why Lagrangian Mechanics............................................................................ 51.4 Contributions of this Thesis............................................................................ 51.5 A Brief Tour through this Thesis................................................................... 7

2 Kinematics 112.1 Kinematics ....................................................................................................... 112.2 Coordinate Systems........................................................................................... 112.3 Euler Angle Representation............................................................................ 12

2.3.1 The rotation matrix Ji(x£) ......................................................... 132.3.2 The angular velocity transformation matrix ^(x#) ........................ 142.3.3 The complete transformation matrix J(xe)......................................... 142.3.4 Quasi coordinates....................................... 15

2.4 Euler Parameters.............................................................................................. 152.5 Euler Parameters to Euler Angles Transformation........................................ 16

3 Modeling of Underwater Vehicles 193.1 Advantages with the Lagrangian Formulation .............................................. 19

v

VI CONTENTS

3.1.1 The Lagrangian versus the Newtonian approach.................................. 203.1.2 The Lagrangian versus the Hamiltonian approach............................... 20

3.2 Lagrangian Formulation......................................................................................213.3 Rigid Body Moving through a Liquid.................................................................22

3.3.1 Conditions on the velocity potential........................................................233.3.2 Kinetic energy of the fluid.......................................................................233.3.3 Kinetic energy of the rigid body..............................................................253.3.4 Rate of change of angular and linear impulses in a moving origin . . 263.3.5 Kirchhoff’s equations in vector form, Kirchhoff (1869)..................... 27

3.4 Rigid Body Equations of Motion..........................................................................273.5 Added Inertia.................................................................................................... 31

3.5.1 Symmetry properties of the mass matrix.............................................. 343.6 Dissipative Forces.................................................................................................. 35

3.6.1 Forces and moments from an umbilical................................................. 393.7 Restoring Forces and Moments.............................................................................403.8 Propulsion and Control Forces and Moments.................................................... 42

3.8.1 Thruster hydrodynamics..........................................................................433.8.2 The b(q,u) vector ...................................................................................453.8.3 The NEROV thruster system.................................................................453.8.4 Open water test of the NEROV thrusters.............................................. 47

3.9 Wave and Current Forces and Moments ...........................................................483.9.1 Wave models in the frequency domain ................................................. 493.9.2 State-space representations of sea waves.............................................. 513.9.3 Linear potential theory.............................................................................533.9.4 Morrison’s equation...................................................................................553.9.5 Current ................................................................................................. 553.9.6 State-space representation of sea current.............................................. 55

3.10 Underwater Vehicles - Nonlinear Equations of Motion................................. 573.10.1 Alternate representation..........................................................................573.10.2 Hydrodynamical and hydrostatistical stability analysis...................... 58

3.11 Equations of Motion Formulated in the Inertial Reference Frame................... 583.12 Model Properties................................................................................................ 60

3.12.1 Properties of the q-frame formulation.................................................... 603.12.2 Properties of the x-frame formulation.................................................... 63

3.13 Experimental Determination of Added Inertia and Damping..................... 653.13.1 Experimental setup...................................................................................653.13.2 Theory........................................................................................................ 663.13.3 Added mass and damping as a function of frequency ......................... 683.13.4 Added mass and damping as a function of the KC number................69

CONTENTS vii

3.13.5 Off-diagonal coupling terms................................................................ 703.14 Linearized Equations of Motion...................................................................... 70

3.14.1 Linearization about constant velocity q = q0..................................... 703.14.2 Linearization about zero velocity </=0................................................. 713.14.3 A state space model ............................................................................ 723.14.4 Properties of the NEROV state-space model..................................... 743.14.5 Eigenvalue analysis............................................................................... 74

3.15 Summary of Chapter 3........................................................................................ 76

4 Optimal Control Algorithms for Underwater Vehicles 794.1 Introduction........................................................................................................ 794.2 Problem Statement............................................................................................ 79

4.2.1 Small underwater vehicles - a special case ........................................ 804.3 A Brief Review of Existing Systems................................................................ 81

4.3.1 Dynamic positioning systems ............................................................. 814.3.2 Autopilots............................................................................................... 83

4.4 Design Requirements........................................................................................ 844.5 Optimalization with Respect to Minimum Control Effort................................85

4.5.1 Minimum fuel......................................................................................... 864.5.2 Minimum energy...................................................................................... 874.5.3 Other minimum control effort criteria................................................. 87

4.6 Basic Assumptions............................................................................................ 904.7 The NOpAC Control scheme................................................................................ 91

4.7.1 State-space representation................................................................... 924.7.2 The control objective................................................................................ 924.7.3 Optimality and stability...................................................................... 934.7.4 The control law..................................................................................... 984.7.5 The adaptive control algorithm.......................................................... 994.7.6 Some critical remarks...............................................................................1014.7.7 A simulation study..................................................................................1014.7.8 Implementation of the NOpAC control scheme.................................... 1024.7.9 Comments to the NOpAC control scheme............................................. 102

4.8 The OpAC Control Scheme ...............................................................................1054.8.1 State-space representation..................................................................... 1054.8.2 The control objective...............................................................................1054.8.3 Optimality and stability.........................................................................1064.8.4 The control law........................................................................................1094.8.5 The adaptive control algorithm............................................................ 1124.8.6 A simulation study..................................................................................115

CONTENTSviii

4.8.7 Implementation of the OpAC algorithm................................................1164.8.8 Comments to the OpAC control scheme................................................116

4.9 Implementational Remarks................................................................................. 1194.9.1 Selection of parameters to adapt............................................................ 1194.9.2 Saturation handling................................................................................. 1204.9.3 Integral action.......................................................................................... 1234.9.4 Modified parameter update laws............................................................125

4.10 Summary of Chapter 4..........................................................................................126

5 Conclusions and Recommendations 1275.1 Summary and Conclusions................................................................................. 1275.2 Recommendations for Future Work..................................................................128

A Proofs 131A.l Proof of Lemma 4.3............................................................................................ 131A.2 Proof of Comment 4.1 ...................................................................................... 133A. 3 Alternative proof of Theorem 4.5.................................................................... 134

B Fluid Mechanics and Hydrodynamics 135B. l Basic fluidmechanics and hydrodynamics........................................................ 135B.2 Linear wave theory............................................................................................ 137B. 3 Nonlinear effects - Stoke’s drift ....................................................................... 139

C The NEROV 141C. l NEROV.............................................................................................................. 141C.2 Background........................................................................................................ 142C.3 Program Goals and Objectives.......................................................................... 142C.4 General Arrangement......................................................................................... 143C.5 Propulsion System ............................................................................................ 143C.6 The Sensor System............................................................................................ 143C.7 The Computer System...................................................................................... 144

D Results from the Free Decay Test 149

E A Complete Mathematical Model 153E.l Complete Equations of Motion.......................................................................... 153E.2 NEROV Model Data......................................................................................... 155

References 156

Index 166

List of Tables

1.1 Subsea completions on the Norwegian continental shelf. The list is notcomplete. Fields as Gullfaks Spr, Gullfaks Gamma, Mikkel, Sleipner Vest, Tyrihans Nord, Smprbukk 0st and Balder may be completed subsea before year 2000. We can not rule out the possibility that existing fields will be expanded subsea............................................................................................ 8

1.2 A comparison between underwater and space robotics........................... 9

3.1 Nondimensional added mass coefficients and linear and quadratic dampingterms for the NEROV for an oscillation period of ^ 11 6 and KC ~ 1. . . 70

3.2 The open-loop eigenvalues for surge, sway, and heave motion for KC ~ 1 . 753.3 Estimated open-loop eigenvalues for roll, pitch and yaw motion for KC ~ 1 75

B.l Equations derived from <j> Faltinsen (1990a) T = Wave period, to — -y,A =Wavelenght, k = , (a =Wave amplitude, g = Acceleration of gravity,z positive upwards, h = waterdepth, total pressure in fluid: pp — pgz + p0 . 138

E.l NEROV masses and inertias........................................................................ 156

ix

X LIST OF TABLES

List of Figures

1.1 Hierarchical optimal trajectory planning and tracking............................. 4

2.1 Coordinate systems used in this thesis........................................................ 12

3.1 Rate of change of impulses in a moving origin with respect to a fixed reference frame....................................................................................................... 26

3.2 Steady state resistance components, Clayton and Bishop (1982) p. 200. . . 363.3 Streamlines, boundary layer and wake........................................................ 373.4 Drag coefficient Co for a cylinder as a function of the Reynolds number . . 383.5 Restoring forces and moments..................................................................... 413.6 A typical plot of Kj, Kq and t)0 against positive J0,................................ 443.7 A schematic drawing of the NEROV thruster Sagatun and Fossen (1991a). 463.8 The analog inner feedback loop...................................................................... 473.9 (a) Measured thruster force T as a function of propeller revolutions n for

different speeds of advance V^fb) Non-dimensional thrust characteristicsKt versus the advance coefficient J0 for the NEROV.................................. 48

3.10 (c) Nondimensional thrust characteristics Ktx (o marked) and Kty (x marked) in the x- and y-direction respectively as a function of the advanced coefficient J0 and angle a = 0° (dashed) and 90° (solid) between thrusterand vehicle speed................................................................................................. 49

3.11 Measured time serie from the North Sea with its corresponding power density plot................................................................................................................. 51

3.12 The JONSWAP spectrum (solid) (3.39) and the Pierson-Moskowitz spectrum (dashed) (3.38)........................................................................................ 52

3.13 A comparison between the JONSWAP spectrum (3.39) and a measuredspectrum................................................................................................................. 53

3.14 H is the wave height, D is a characteristic diameter and A is the wave length. 563.15 A typical decay................................................................................................. 663.16 Experimental setup.............................................................................................. 67

xi

LIST OF FIGURESxii

3.17 The total linear damping coefficient per unit mass p = pj + plottedas a function of the equivalent velocity ....................................................68

3.18 The two geometrical perturbation objects............................................................ 693.19 The equivalent quadratic damping coefficient Odd and the added mass co

efficient Ca (both in x-direction) plotted as a function of the Keulegan- Carpenter number, KC = where X is the amplitude of oscillationand D is a characteristic dimension for the vehicle........................................ 71

3.20 Ca versus Cdd for a KC value between 1 and 0.1....................................... 723.21 Quasi-linearization of the quadratic damping terms............................................733.22 The eigenvalues of the NEROV in surge sway and heave plotted as functions

of the KC number............................................................................................... 743.23 Open-loop poles for the NEROV vehicle for high KC values........................... 76

4.1 Block diagram showing the Kalman filter based DP system o/Balchen et al.(1980a)................................................................................................................ 82

4.2 An underwater vehicle in transit exposed for disturbances............................... 964.3 The block diagram for the NOpAC control scheme........................................... 994.4 NOpAC simulation study....................................................................................1044.5 The block diagram for the OpAC control scheme.............................................1144.6 OpAC simulation study ....................................................................................1184.7 The overall block diagram for the saturation handling scheme......................... 1214.8 Generation of the desired reference trajectory...................................................1214.9 Generation of the modified desired state Amerongen (1982)............................ 122

C.l The NEROV vehicle - general arrangement..................................................... 141C.2 The NEROV computer system.......................................................................... 145C.3 Schematic drawing of the sensor system............................................................ 146C. 4 Experimentally obtained sensor data with and without Kalman filter Fossen

and Sagatun (199lb).............................................................................................147

D. l Nondimensional free decay test-results for the x-direction. The upper leftframe contains the dimensionless linear damping coefficient , upper right frame contains the quadratic damping coefficient in dimensionless form, while the bottom left frame shows the added mass coefficients (made dimensionless such that mA = CapV ), solid = unperturbed geometry, dashed = small perturbation and dashdot = large perturbation...................................... 150

LIST OF FIGURES xiii

D.2 Nondimensional free decay test-results for the y-direction . The upper left frame contains the dimensionless linear damping coefficient , upper right frame contains the quadratic damping coefficient in dimensionless form, while the bottom left frame shows the added mass coefficients (made dimensionless such that mA = CapV ), solid = unperturbed geometry, dashed =small perturbation and dashdot = large perturbation.......................................151

D.3 Nondimensional free decay test-results for the z-direction . The upper left frame contains the dimensionless linear damping coefficient , upper right frame contains the quadratic damping coefficient in dimensionless form, while the bottom left frame shows the added mass coefficients (made dimensionless such that mA = Cap^J ), solid = unperturbed geometry, dashed = small perturbation and dashdot — large perturbation.......................................152

xiv LIST OF FIGURES

N omenclat ur e

Vectors and Matrices

Bold type is used exclusively to denote vectors and matrices. Bold uppercase denotes matrices and lowercase vectors.

Symbols

Symbol Definition

A Area, projected or characteristic areaA System matrixA{ Wave amplitudeB Buoyancy forceBt Input matrix, thruster configuration matrixB Input matrixBG Distance between CG and CB BM Distance between the vehicle’s metacentre and CB Bf Generalized inverse of Bb Vector containing nonlinear control forces and momentsC Matrix of Coriolis and centrifugal termsCam Matrix of Coriolis and centrifugal terms due to hydrodynamic added massCB Centre of buoyancyCd Drag coefficientCG Centre of gravityCij Rotation matrix describing a rotation j about the i-axis Ca Inertia coefficient

xv

XVI LIST OF FIGURES

Cdd Equivalent quadratic drag coefficientCP Centre of pressureCf nondimensional skin-friction coefficientCrb Matrix of Coriolis and centrifugal terms due to rigid body forces c Positive damping coefficientdq Vector of quadratic damping terms due to skin friction and drag dq Quadratic drag coefficientdi Linear drag coefficientD DiameterD Matrix of dissipative (damping) termsDl Matrix of linear damping terms due to skin friction and dragDq Matrix of quadratic damping terms due to skin friction and dragDu Damping matrix due to umbilicale Euler parameter vector, prediction error vector e,- Euler parameter i (i=1..3)E Energy, angular velocity to the time derivative of Euler parameters transformation matrf Force vectorfid Damping farceE Power function for dissipative forcesg Acceleration of gravityg Vector of gravitational and buoyancy forcesGM Distance between the vehicle’s metacentre and CGh Transfer functionH HamiltonianH Transfer function matrix, wave heightHi Significant wave heightk Wave numberKG Distance between the vehicle’s CG and keel line Inxn n x n Identity matrixI Inertia tensor referred to the vehicle-fixed coordinate systemIx Moment of inertia about x-axisIy Moment of inertia about y-axisIz Moment of inertia about z-axisIxy Product of inertia about x- and y-axesIXz Product of inertia about x- and z-axesIv* Product of inertia about y- and z-axesJ Cost functionJ0 Open water advance coefficientJ Linear and Angular velocity transformation matrixJ\ Linear velocity transformation matrixJi Angular velocity transformation matrixK Hydrodynamic moment component about x-axis (rolling moment)K Solution to the Riccati equation

LIST OF FIGURES xvn

KC Keulegan-Carpenter’s numberKG Distance between the vehicle’s CG and keel line Kd Regulator gain matrix, derivate actionKp Regulator gain matrix, proportional actionKg Adaption gain matrixKp Non-dimensional thrust coefficientKq Non-dimensional torque coefficientL Lagrangianl, Thruster moment arm about axis jm Mass of the vehiclem Mass of the vehicle’s displaced fluidM Hydrodynamic moment component about y-axis (pitching moment)M Inertia matrixMam Added mass matrixMpk Froude-Kriloff inertia matrixMrs Rigid body inertia matrixMw Wave-force inertia matrixn Propeller revolutionn Propeller revolution vector, vehicle-fixed unit outward normal vector N Hydrodynamic moment component about z-axis (yawing moment), integer N Matrix representing workless forcesp Angular velocity component of q about x-axis (roll), pressurep Linear and angular impulse vectorPd Dynamic pressurep0 Atmospheric pressure on the water surfaceq Angular velocity component of q about y-axis (pitch)q Vehicle-fixed vector of linear and angular velocities componentsq Vehicle-fixed quasi-vectorqd Steady state desired velocity vectorqj Fluid velocity vector referred to the vehicle-fixed coordinate systemqT Relative fluid velocity vector referred to the vehicle-fixed coordinate system, reference q Velocity tracking error vectorQ Propeller torqueQ Positive weighting matrixQi Generalized forcer Angular velocity component of q about z-axis (yaw)r Position vector in the vehicle-fixed coordinate system X0Y0Z0tq Position vector in the local coordinate system XoY0Zo to CGR Positive weighting matrixRe Reynold’s numberS Wave spectrums Measure of trackingt Time

xviii LIST OF FIGURES

T Kinetic energy, time constant, wave period, propeller thrustT0 Optimization criterion weighting matrixTP Towing pointu Linear velocity component of q in x-direction (surge)u Control input vectorV Potential energyUf Fluid motion velocity component in the x-directionur Relative fluid motion velocity component in the x-directionv Linear velocity component of q in y-direction (sway)v Vehicle-fixed linear velocity vectorVf Fluid motion velocity component in the y-directionvr Relative fluid motion velocity component in the y-directionV Volume, vehicle speed, noise covariance matrix, Lyapunov functionVa Advance velocity at the propellerw Linear velocity component of q in z-direction (heave), wake fraction numberw Wave forcesWf Fluid motion velocity component in the z-directionwT Relative fluid motion velocity component in the z-directionW Weight, workx Surge position referred to the earth-fixed reference framexb The x-coordinate of CBxq The x-coordinate of CGxp The x-coordinate of CPxc Inertial fixed current vectorx Earth-fixed vector of position and Euler angle componentsXd Desired state vectorxp Roll, pitch, yaw, Euler anglesxq Euler parametersX Hydrodynamic force component along x-axisXYZ Earth-fixed coordinate systemX0Y0Z0 Vehicle-fixed coordinate systemy Sway position referred to the earth-fixed reference framey Measurement vector, state vector for the NOpAC controllerys The y-coordinate of CBya The y-coordinate of CGxp The y-coordinate of CPV Hydrodynamic force component along y-axisz Heave position (depth) referred to the earth-fixed reference framez State vectorzb The z-coordinate of CBzq The z-coordinate of CGzp The z-coordinate of CPZ Hydrodynamic force component along z-axis

LIST OF FIGURES XIX

Greek Symbols

e Permutation symbol6 Kroenecker delta function, a Dirac pulse( Wave elevation, damping ration(a Wave amplitude($ Damping ration in pitch(4 Damping ratio in rollr]M Mechanical efficiencyrjo Thruster open-water efficiency in undisturbed water rja Relative rotative efficiency9 Angle of pitch, Euler angle0 Parameter vectorA Wave length, closed loop bandwidth, eigenvalue, forgetting factor A Unit vectorp Densitya Standard deviationr Force and moment vectorTu Force and moment vector due to umbilical forces<f> Angle of roll, Euler angle, velocity potential, power spectral density<f>7 Scattering potential<f>0 Incident regular wave velocity potential<f>j Radiation potential•P Power spectral density matrix, regressor matrix xj) Angle of yaw, Euler anglexp Regressor matrixlj Circular frequency, propeller angular velocityu;e Frequency of encounteru)p Modal or peak frequencyw„ Vehicle-fixed frequencyug Natural frequency in pitchuty Natural frequency in rollV Volume of the displaced fluid

Hydrodynamic Coefficients

The hydrodynamic forces and moments are written in accordance with the SNAME (1950) notation; e.g. the hydrodynamic added mass force Za along the z-axis due to a linear acceleration v in the y-direction is written as:

XX LIST OF FIGURES

d ZZa = Zyii where Zy = ——ov

Similarly the damping in roll Kd due to a linear velocity u in the x-direction can be expressed as:

dKKd = Kuu where Ku = —ou

Sub- and superscriptsSubscripts aq The teax The term a is formulated in an inertial reference frameaq The term a is formulated in the vehicle-fixed reference frame

Superscripts * optimal termL Liquid, the vehicle’s ambient fluid S Solid, the vehicle d desired or drag term

The q and x subscripts are omitted when it is clear from the text which reference frame the equations are expressed in.

MiscellaneousThe term diag(-) denotes a diagonal matrix with the terms (•) on the diagonal.

The term [a:x] denotes the skew-symmetric matrix formed by the 3x3 vector x.

(•), (■) and (■) denote the error (•) — (•),-, the estimated value of (•) and the time derivative of (•) respectively.

Preface

I am very grateful to my supervisor Professor Jens G. Balchen at Division of Engineering Cybernetics who gave me a scholarship when I really needed one, who has been a great inspirator and supervisor and whose knowledge and creativity have been invaluable in motivating me throughout the writing of this thesis.

I would also like to express my gratitude to my friend and colleague dr.ing., Thor Inge Fossen with whom I have been working since my early undergraduate years. His encouragement, support, and friendship throughout these years are very appreciated. I am particularly grateful for our discussions, our cooperation building the NEROV, the articles we have written together, the parties and squash matches we have had.

I am indebted to managing director D. Richard Blidberg and his staff at the Marine Systems Engineering Laboratory at University of New Hampshire for giving me the opportunity toi work with them for one year. It was a very instructive and inspiring stay. I would also like to thank Professor David Limbert at the Department of Mechanical Engineering at UNH for teaching me more control theory in one semester than I thought was possible. I am also thankful to my good friends Barry Wythoff, Paul Franz and the rest of the students at Babcock graduate dormitory for making my year in New Hampshire most enjoyable.

I would like to express my gratitude to my friends and colleagues Sverre Hendseth, Stefano Bertelli and the rest of the people at the Center of Robotic Research and on the M0- BATEL program at the Division of Engineering Cybernetics. The help from the ROV’91 group was also appreciated. I offer special thanks to 0rnulf R0dsth for invaluable help with C programming and making things work on my SUN . Also I wish to thank Professor Olav Egeland for helpful comments and discussions.

Assistant Professor Bj0rn Sortland and Erik Lehn at the Center of Marine Research has been very helpful during the design and testing of various components of the NEROV.

xxi

xxn LIST OF FIGURES

Thanks are also due to Asgeir S0rensen and Thor Inge Fossen for helping me reading through this thesis and correcting numerous errors.

Many thanks to Bjprn, Gaute and Helge for many inspiring discussions, their friendship and for throwing some great parties during the years.

Finally, I am very grateful for the financial support from the Fulbright Foundation, the Norwegian Council for Scientific and Industrial Research and the British Petroleum Norway.

Chapter 1

Introduction

1.1 Background and Motivation for this ResearchOil and gas exploration and production offshore moves towards increasingly deeper waters. The cost of building, maintaining, inspecting, and installing structures increases drastically as the depth of operation increases. If some of the equipment can be placed wet, that is on the seafloor or in the water column, substantial costs can be saved. This is mainly due to the reduced steel weight of the structures. For that reason, more production equipment is moved under water, see Table l.l1.These underwater installations need periodical inspection and maintenance.

As an example, the Norwegian Petroleum Directorate’s regulation states that primary and secondary structures have to be inspected annually , OD (1982). The annual inspection includes visual inspection of selected areas, potential measurements, determining the extent of corrosion attack, dimensional measurements of selected anodes and visual inspection for mapping type of marine growth. In addition, there are requirements to perform non-destructive testing of critical steelmembers and areas where cracks or corrosion may be expected.

The conventional method of inspecting underwater installations, e.g. bottom templates, pipe-lines and risers, is by utilizing divers and remotely operated vehicles (ROVs). Divers are much more flexible than ROVs with respect to various work tasks which they can be set to do. Divers are also inherently more adaptive to unexpected changes of the surrounding conditions. This is because humans have flexible manipulators ,i.e. the human arms and legs, and sensor system ,e.g. eyes, touch, hearing and the sense of distance, motion and orientation. The state-of-the-art work-ROV is usually equipped with two ma-

1 These data were provided by Managing Director Harry Vagseth, Racal Survey Norge A/S

1

2 CHAPTER 1. INTRODUCTION

nipulators , typically Slingsby or Shilling, with 4 to 8 degrees of freedom (DOFs) Sagatun and Fossen (1990a). The human being on the other hand, is equipped with two arms with approximately 30 dofs each Morecki et al. (1984). It is, however, desirable to limit the use of divers in water depths more than 200 m. This is due to physiological, economical, and political reasons.

There are a number of possible application areas for underwater robots. We have, for a number of years used underwater robots (ROVs) for visual inspection, light maintenance and repair in the North Sea and other areas. On the other hand, the relatively easy and repetitive task of performing nondestructive testing of nodal welds on underwater structures has been mostly performed by divers. It is believed that this is mainly due to limitations of the underwater vehicle’s and its manipulator’s performance. This has also been recognized by the oil companies and underwater contractors, e.g. Oceaneering (1988). This is a motivation for developing more advanced, optimal, and user-friendly underwater robots.We will like to add that oil- and gas-related applications of underwater robotics are only one of many areas were underwater robots are favorable compared to the use of divers. Other areas are:

• observation and inspection related to:

— pipelines— cables— geology— pollution

• location and identification of underwater objects.

• diver assistance

• installation and retrieval of underwater systems

• maintenance

• drilling support

• scientific research

• military applications

and

1.2. OPTIMAL PATH PLANNING, TRAJECTORY GENERATION AND TRACKING 3

• underwater search and identification

• under ice mapping

• long distance and deep sea underwater inspection, survey and mapping

• bottom photography and sampling

• testing of underwater technology

• mapping marine population

• long distance and long term pollution and water quality studies

• military applications

where the former list is mainly oriented to the use of ROVs, while the latter one favors the use of Autonomous Underwater vehicles (AUV).

This wide range of application areas for underwater robotics is the major motivation for this research. But what is underwater robotics ? What does the term “underwater robotics” mean ? Underwater robotics is a multidisiplinary research field. An easy way of defining it is ti differentiate it it from space robotics, a better known and, in many ways, similar research area. Table 1.2 points out the main differences between autonomous underwater robots (which can be considered to be the purest form of underwater robots) and space robots.

1.2 Optimal Path Planning, Trajectory Generation and Tracking

Trajectory planning and tracking may take place at several abstraction levels in an autonomous robot, Blidberg and Chappell (1986) and Albus et al. (1987), see also Fig. 1.1. We have, at the highest level task planning which plans which tasks the vehicle shall perform and the sequencing of those. Blackboard-based planning Haues-Roth (1985) has been successfully applied to perform task planning for autonomous underwater vehicles, Chappell (1987). All planning at this level is performed on a symbolic level, often implemented in LISP or PROLOG. Sagatun (1989) contains a situation assessment system which performs symbolic assessment and (reactive) planning at the level below. At a lower level, way-point planning takes place. This planning is more algorithmic in the sense that it is usually performed on basis on a mathematical optimization algorithm like the A* search, e.g. Sagatun (1988). The level below the way-point planner may


Figure 1.1: Hierarchical optimal trajectory planning and tracking

again consist of an optimal planner with respect to the trajectory. This trajectory planner may be a minimum-control-effort planner which optimizes an effort criterion like J = f uTu dt. Spangelo and Egeland (1992) contains an optimal minimum control effort trajectory planner for an underwater vehicle with electric thrusters. Minimum-effort controllers are almost always open-loop methods which are calculated off-line. The vehicle’s actual trajectory will always deviate from the nominal trajectory without control feedback since disturbances and model errors are always present. The principle of optimality then states: Lee and Markus (1967) p. 424.

An optimal-control policy has the property that, whatever the initial state and initial control policy, the remaining control policy (that is, the policy after a short lapse of time) must constitute an optimal policy with regard to the state that results from use of the initial policy during the short lapse of time.

This basically states that the optimal control strategy depends only on the present state t0 and the goal state and not on any states at t < t0. Deviations from the planned trajectory must be taken care of by optimal trajectory following algorithm, since it require too much computation to be practical to perform the whole planning process when there is a deviation from the nominal trajectory. This family of control algorithms is, for instance

1.3. WHY LAGRANGIAN MECHANICS 5

optimal with respect to trajectory errors and control effort, like LQ-based methods. This thesis focuses on optimization criteria which minimize tracking errors as well as the forces which contribute to the kinetic energy and the energy which will dissipate from the vehicle. Potential energy due to gravity is neglected, since potential energy is end-point dependent only. These optimization criteria are specially useful since they not only minimize tracking errors, they also include a minimum control effort term which has an obvious physical interpretation. The presented algorithms result in feedback laws which are made adaptive to correct for errors in the model, albeit only parameter errors.

1.3 Why Lagrangian MechanicsThe primary contribution of this thesis is that modeling and control of underwater vehicles are done by means of Lagrangian mechanics in a compiled and systematic fashion. Although Lagrangian mechanics has been known for almost 200 years, the systematic approach of using energy and work as the primary terms in modeling and control has not been done before for underwater vehicles. This thesis shows that the Lagrangian approach in modeling exposes the properties of the underwater vehicles’ equations of motion in a way where the more traditional Newtonian approach is less well suited. Lagrangian mechanics attack the mechanics from a more analytical point of view than the more geometrical approach used by the Newtonian method. The energy and work terms are also useful in control system design when stability and optimality are proved by using nonlinear stability theory.

1.4 Contributions of this ThesisThis thesis contains both theoretical and experimental contributions. The theoretical contributions are partly found in the modeling chapter (Chapter 3) and the control chapter (Chapter 4). The experimental results are from the work carried out during the design and the process of building the Norwegian Experimentally Remotely Operated Vehicle (NEROV). NEROV and its subsystems are described more closely in Appendix C and in the following reports:

• overall vehicle design , Sagatun and Fossen (1990b)

• propulsion system, Sagatun and Fossen (1991a)

• sensor system, Fossen and Sagatun (1991b)

• computer system, Sagatun and Fossen (1991b)


• mathematical model, Fossen and Sagatun (1991c)

• reports from the free decay tests, Sagatun and Fossen (I991d)

The different contributions in this thesis are presented in:

Chapter 3. This chapter contains the derivation of the complete equations of motion for an underwater vehicle from a Lagrangian point of view. The Lagrangian mechanics is well known, but the compiled work is a contribution as such.

Section 3.8.4 contains results from open water tests of the NEROV. The tests reveal that the mapping from propeller angular revolution to thrust is nonlinear. The section presents the complete nonlinear thrust vector for the NEROV. Results are published in Sagatun and Fossen (1991c) and Fossen and Sagatun (l991d).

Section 3.11 derives the equations of motion expressed in an inertial-fixed reference frame directly from the kinetic energy also expressed in the inertial-fixed reference frame, instead of going through the vehicle-fixed frame formulation.

Section 3.12 Lagrangian mechanics is used to discover the most important properties of the equations of motion for an underwater vehicle. All proofs are original contributions, except when stated otherwise. We will specially like to point to property 3.2. with remarks and property 3.7. as specially useful. This work has been published in Sagatun and Fossen (l991f).

Section 3.13 presents the results from free-decay tests on the NEROV. Of special importance are the results presented in 3.13.4. where the added inertia and damping is plotted as a function of the Keulegan-Carpenter number (KC number). The results are published in Sagatun and Fossen (l991e)

Section 3.14.5 shows how the eigenvalues of a linearized model of the NEROV varies as a function of the KC number.

Section 4.7 A near-optimal adaptive controller (NOpAC) is derived in this section. The proof of near-optimality and the solution to the Hamilon-Jacobi equation are especially interesting. This work will be published in Sagatun (1992).

Section 4.8 An optimal adaptive controller (OpAC) is derived in this section. Especially noteworthy are the control objective in section 4.8.2, the optimal feedback law in lemma 4.3, the stability proof in theorem 4.4, the modified optimal feedback law in lemma 4.4 and the adaption law in theorem 4.5. The transformation from the inertial-fixed generalized coordinates to vehicle fixed quasi coordinates in lemma 4.5 is also original work. This work is presented in Sagatun and Johansson (1992).

1.5. A BRIEF TOUR THROUGH THIS THESIS 7

1.5 A Brief Tour through this ThesisThis thesis is aimed at a control engineer with some knowledge in mathematical modelingand especially dynamics and hydrodynamics. The remaining chapters and appendicescontain the following:

Chapter 2. “Kinematics” contains a background in kinematics for modeling underwater vehicles. Both Euler parameters (quaternions) as well as Euler angles are discussed.

Chapter 3. “Modeling of Underwater Vehicles” derives the complete equations of motion for an underwater vehicle from a Lagrangian dynamics point of view. Both theoretical as well as experimental work are reported. The equations of motion are derived in both a vehicle-fixed as well as inertial reference frame. Model properties of both the nonlinear equations and a linearized model are derived.

Chapter 4. “Optimal Control Algorithms for Underwater Vehicles” starts by giving a problem statement of the underwater vehicle control problem. A review of existing systems is presented before design requirements and optimization criteria are discussed. The basic assumptions for the control schemes are presented before the two near-optimal and optimal adaptive controllers NOpAC and OpAC are derived in minute detail. The Lagrangian formalism used in the previous chapter is sucessfully employed here to construct optimization criteria and in proofs of stability and optimality.

Chapter 5. “Conclusions and Recommendation” contains what the title says, the conclusion and some recommendation for future work.

Appendix A. “Proofs” contains proofs of lemmas, a theorem and a comment from Chapter 4.

Appendix B. “Fluid mechanics and Hydrodynamics” gives the reader the necessary background in fluid mechanics and hydrodynamics to follow the derivations of Chapter 3.

Appendix C. “The NEROV” describes the Norwegian Experimental Remotely Operated Vehicle more closely.

Appendix D. “Results from the free decay test” presents results from the free decay test described in Section 3.13.1.

Appendix E. “A Complete Mathematical model” contains a complete mathematical model of the NEROV with the estimated parameters.


field operator year number of subsea wells

Troll 2 Hydro 1991 1Oseberg Gamma Hydro 1991 1Sleipner Theta Statoil 1993 3Liille Frigg EH 1993 3Embla Phillips 1993 2-3Draugen Shell 1993-96 9Tordis Saga 1994 7Mime Hydro 1994 215-5 Hydro 1994 2Statfjord Satelite Statoil 1994 24Huldra Statoil 1994 6Troll Olje Hydro 1995 6Njord Hydro 1995 3Heidrun Conoco 1995 825-5-3 EH 1995 3Troll Olje Hydro 1996 15Njord Hydro 1996 7Visund Hydro 1996 6Mjplner Hydro 1996 6Hild Hydro 1996 6Smprbukk Sor Statoil 1996 12Snphvit Statoil 1996-97 8Askeladden Statoil 1997 8Troll Olje Hydro 1997 15Njord Hydro 1997 7Visund Hydro 1997 4Hild Hydro 1997 4Trestakk Statoil 1997 6Troll Olje Hydro 1998 15Njord Hydro 1998 23Visund Hydro 1998 3Total 216

Table 1.1: Subsea completions on the Norwegian continental shelf. The list is not complete. Fields as Gullfaks Spr, Gullfaks Gamma, Mikkel, Sleipner Vest, Tyrihans Nord, Smprbukk 0st and Balder may be completed subsea before year 2000. We can not rule out the possibility that existing fields will be expanded subsea.

1.5. A BRIEF TOUR THROUGH THIS THESIS 9

under water in space

• data transmission very noisy environment with respect close to perfect for optical,to RF, optical and acoustic data very good for RF andtransmission impossible for acoustic data

• data transmission < 1 kbaud (acoustic) > 10 Mbaud (optical and RF)bandwidth

• data transmission <10 km, typical < 1 km > 1000 kmrange

• time delay < 10s Is to hours• navigation and No commercial underwater GPS GPS with an accuracy better

positioning system system is available than 1 m (in earth orbit)• disturbances waves, current and animals solar wind, meteorites

nominal operation oA and high energy particles< io-4 [&]

• launch loads low frequency < Q.3Hz high frequency 3 — 500/?zloads < 2g loads 1 — Gg

• structure mostly rigid mostly flexibleflexibility

• pressure Ap typical 300 atm. 1 atm.• visibility < 25 m > 1000 km• temperature ~ 4°C ~ 2-4K• research funding < than 1 x 109 ECU > 1 x IO10 ECU

Table 1.2: A comparison between underwater and space robotics


Chapter 2

Kinematics

This section contains the necessary kinematics for reading the rest of this thesis.

2.1 KinematicsIt is straightforward to show that a rigid body in space needs six independent or generalized coordinates to specify its configuration, i.e. orientation and position. We will in this thesis only use the Cartesian coordinate system such that three independent coordinates are used to specify position and three independent coordinates are left to specify orientation. Euler showed already in 1776 ,Euler (1776) that a set of three angles are adequate to specify the relative orientation of two orthogonal coordinate systems. This chapter will discuss Euler angles and Euler parameters (quaternions) as a way of representing the configuration of a rigid body in space. The former method is a three-parameter description, while the latter uses four parameters. Goldstein (1980) and Kane et al. (1983) contain several other ways of specifying a rigid body’s configuration.

2.2 Coordinate Systems

This thesis uses an earth-fixed coordinate system and a vehicle-fixed system. The earth- fixed coordinate system will also be called the inertial-system or the x-frame, while the vehicle-fixed system will also be denoted the q-frame, see Fig. 2.1. Notice that the 0—axis point down. This orientation of axis corresponds to the SNAME convention, SNAME (1950).Throughout this thesis earth-fixed coordinates will be denoted x 6 while we will only deal with vehicle-fixed velocities and accelerations denoted q (z $tn and q G 3Jn. n is the number of degrees of freedom (dofs) the vehicle is expressed in.

11

12 CHAPTER 2. KINEMATICS

Figure 2.1: Coordinate systems used in this thesis

2.3 Euler Angle Representation

The transformation from one Cartesian coordinate system to another can be performed by means of three successive rotations in a sequence. The Euler angles, denoted xg are defined as the three successive angles of rotation, Goldstein (1980). While there are several different ways of specifying the three successive rotations, this thesis will only use the so called “roll, pitch, yaw” convention, “X-Y-Z fixed angle” convention or “Tait-Bryant” convention. Craig (1989) contains a complete set of the Euler angle conventions. We find the new the orientation of the rotated frame B as follows:

Start with the B frame coincident with the known reference frame A. Rotate B about A’s x-axis an angle (j), then rotate the new frame an angle 9 about the new frame’s y-axis, and then rotate the resulting frame an angle about the resulting frame’s z-axis.

For an underwater vehicle the angle <f> is the angle of roll, the angle 9 is the angle of trim and the angle t/> is the angle of yaw.

Wampler (1986) p. 94 points out that any set of three orientation parameters, such as Euler angles are either discontinuous or singular. This singularity can be avoided by using the the coordinate transformation matrix.

A vector in system A can be transformed from system A to system B by premultiplying the vector Ax with the transformation matrix ^C(xe) such that Bx — bC(xe)ax. The

2.3. EULER ANGLE REPRESENTATION 13

sub- and superscripts are explained in the nomenclature.

Notice that the vector itself is unchanged; only its representation has changedsince it is formulated in two different coordinate systems. Despite this, we will write that a vector is transformed from one coordinate system to another even when we actually mean that the vector’s representation is transformed and not the vector itself.

2.3.1 The rotation matrix Ji(x£)The matrix ^C(xe) transforms a vector from a fixed coordinate system A to a rotated coordinate system B. However, we also want an expression for the transformation from the rotated system (e.g. a underwater vehicle) to the fixed system (e.g. the earth fixed system) denoted Ji(xe)- We will first derive the %C(xe) matrix and then invert the resulting matrix to obtain the Ji(xe) matrix. In the following a transformation matrix which performs a rotation 7 around the axis j is denoted Ojn. cos(^), sin(0) and tan(<ji) is denoted c(j>, s<f> and t<f> respectively. The following three matrices rotate a system around the z—, y— and z—axes respectively.

exp sxp 0 ‘ cQ 0 —sO ' 1 0 0 ‘Cr.tA — —sxp exp 0 0 II 0 1 0 0 H II 0 cp sp

0 0 1 sO 0 c6 0 —sp cpThe elements of the transformation done by the three Euler angle rotations in the previous paragraph can be obtained by writing the matrix ^C(xe) as the triplet product of the separate rotations, each of which have the form of the matrices in (2.1). Hence, the transformation matrix ^C(xe) which transforms a vector from frame A to frame B can be found by performing the successive multiplication ®C(<£, fl,^) =Remember that J\{xe) — ((f),6,ip)'1 which yields

Ji(xe) =cipcOsxpcO~sB

—stpcp + cipsOsp sipsp + cipapsd apap + spsOsip —cxpsp + sOsipcp

cOsp cOcp(2.2)

Notice that coordinate transformation follows the rules of matrix multiplications, such that coordinate transformations are associative but not commutative. A coordinate transformation matrix A is orthogonal, that is A-1 = AT and orthonormal, that is < et, ej >— 1 if i = j and 0 otherwise, e,- and ej are column and row vectors of A.

The transformation matrix ^C(xe) can also be defined in an other way. Consider two orthogonal coordinate systems defined respectively by the following two sets of unit vectors: ai, a2, 03 and b\, b2, 63. Then the transformation matrix %C(xe) consists of the elements defined as c,7 = 6, • aj, Craig (1989).


2.3.2 The angular velocity transformation matrix ^(x#)We derived in the previous paragraph a transformation matrix J\{xe) which transformed a vector from one orthogonal space (the vehicle-fixed coordinate system) to another (the earth-fixed coordinate system). We also need a transformation matrix which transform the vehicle’s angular velocities = [p, g, r]T to the time derivative of the Euler angles

= [<f>,’/,]r such that xe = */2(s£)w.

An easy approach to derive the J2{xe) transformation matrix is to first find the ^(ê)-1

matrix and then perform the matrix inversion. The vector [p, q, r]T is recognized as

’ p' 9 —

1

01____ + cl*

■ 0 ■ $ + Cl,Cle

' 0 ' 0

r . 0. . 0. . ^ .

Hence, we obtain the following kinematic differential equation after some rearranging

’ p" 9 = J2(x£) 1

' 4 ' 0

r . .a> = J2(xe) 'xe

where

J^xe)!0

0

S<j)S$

c0c<t>$4>cd

c4>s9c6

—s<f>c±c6

(2.3)

Notice that J2(*e) is singular for 6 = which correspond to a pitch angle of 90 degrees.

2.3.3 The complete transformation matrix J(x^)

The vehicle-fixed velocity vector q = [u, u, tn, p, <7, r]T can now be transformed to earth- fixed velocities and rate of change of Euler angles x = [x,y,z,4>,0,xj)]T by using the following transformation:

x = J{xE)q

where the transformation matrix J{xe), also called the Jacobian, is found to be

J(xE) = diagtJ^XE), J2(X£;))

2.4. EULER PARAMETERS 15

Hence, the transformation from the q-frame to the x-frame can be performed by employing the following relations:

q — J 1(xE)xq = J^ê) (x - j(xE)J-1(xE)x) (2.4)

for a nonsingular J{xE) matrix.

2.3.4 Quasi coordinatesThe transformation of angular velocities expressed in the vehicle-fixed reference frame to an inertial coordinate system is necessary because the components of the angular velocity vector u> cannot be integrated to obtain actual angular coordinates. The vector « is described as a nonholonomic vector by Goldstein (1980). The vector q is not the time derivative of generalized coordinates, but sometimes referred to as the time derivative of the quasi coordinates q e.g. Meirovitch (1990). Remark that q has no immediate physical interpretation. On the other hand, the vector x = [x,t/,2, 0, representsproper generalized coordinates excluding 6 — ±^. This leads to a notational conflict since the velocity vector q is not the time derivative of any proper (meaningful) vector. The dot above the q is therefore misleading. We will, however, use this notation throughout this thesis, since it is conceptually easier to relate velocities and accelerations with (•) and (•) than (•) and (•) even though it is mathematically erroneous.

2.4 Euler ParametersWhile the Euler angle representation and the angular velocity transformation matrix J2{xE) matrix have singularities, a representation with Euler parameters are not. The Euler parameters, also known as Cayley-Klein parameters or unit quaternions, is a four- parameter description of a rigid body’s configuration based on simple rotations. Euler’s theorem of rotation states that

every change in the relative orientation of two rigid bodies or reference frames A and B can be produced by means of a simple rotation of B in A.

The unit vector which B is rotated about is denoted A and the angle which is the amount frame B is rotated about A is denoted 6. The four Euler parameters xq consists of the Euler vector e and the scalar e4 such that Xq = [eT, e4J. e is defined as e = A sin | and

e4 = cos|. The four parameters are obviously not independent of each other and the


Euler parameters have to satisfy ej + + e| + = 1. The coordinate transformationmatrix J\{xe) can be written as a function of the Euler parameters Kane et al. (1983)

Ji{xQ)e\ — e\ — e.\ 2(eie2 — 6364) 2(e3Cj + 6364)

2(eie2 + 6364) —ej + 62 — 63 + 64 2(6363 — 6164)2(6361 — 6364) 2(6263 + 6164) —ej — 62 + e§ +

(2.5)

Ji(xQ) can be written in compact form as Ji(*Q),j = ^(ej—eiei)+2e,ej+2eijieiei where Einstein’s summation convention has been employed, and etJk is the permutation symbol1

Goldstein (1980).The differential equation for the integration of absolute orientation can be found to be xq — Êu>, where E is defined as

64 -63 6263 64 61

-62 6l 64. “Cl -62 -63

Notice that ETE — /sxs-

2.5 Euler Parameters to Euler Angles Transformation

The transformation from Euler parameters to Euler angles are for the roll-pitch-yaw convention, Egeland (1985).

^ =

6 =

<t> =

atari Ji(xq)2i\Ji(xQ)uJ

atan2

atan2

Ji(jcg)3icos(4’)Ji(xq)u + sin(ip)J1(xQ)2i

Sm(V))Ji(xQ)i3 - COS(0)Ji(xg)23N\ Sm(l/>)Ji(XQ)i2 + C0s(iI>)Ji(Xq)22 ) (2.6)

The rotation matrix J\{xe) can be expressed as a function of the Euler parameters such that, Salcudean (1991):

J\(xe) — ^3x3 + 2e4 (ex) + 2(ex)2

bijt is zero if any two of the indices are equal and ±1 otherwise. The sign is dependent of if the ijk is even or odd permutation of 1, 2, 3.

2.5. EULER PARAMETERS TO EULER ANGLES TRANSFORMATION 17

or the other way around (for e > 0):

1

2(1 + trace (J^xe)))1(ji(xe) — J\{xe)}ex =


Chapter 3

Modeling of Underwater Vehicles

We will in this chapter derive in detail the equations of motion for an underwater vehicle in six degrees of freedom.The equations of motion for underwater vehicles can be written as follows:

M,9 + C,(9)g + r>,(qr)9+flr(a;) = w((j>) + bq(q,n)x = J(x)q (3-1)

where q € 9?6 defined as q = [u,u,tn,p, 9,r]T. As the time derivative of w is not a physically meaningful vector, the vector q is termed a quasi vector, q is not used in the following, only the time derivatives q and q. x € Sfc6 defined as x = [z, y, z, rf), 6, tjj]T is a vector of generalized coordinates containing earth-fixed positions and Euler angles, see also subsection 2.3.4. Mq = > 0 is the inertia matrix including the hydrodynamicvirtual inertia (added mass), Cq(q)q contains the nonlinear forces and moments due to centripetal and Corolis forces and Dq(q) > 0 is the vehicle’s damping matrix. We assume that the potential damping and the viscous effects are lumped together in the Dq{q) matrix. g{x) is a vector containing the restoring terms formed by the vehicle’s buoyancy and gravitational terms. w{<f>) is the wave and current disturbance vector and fc,(q, n) is a vector containing the vehicle’s propulsion and control forces and moments. The vector n represents the thruster propellers’ angular revolution. J{x) is a velocity transformation matrix which transforms velocities in the vehicle-fixed (the q-frame) to the earth-fixed reference frame (the x-frame). Each of the elements in (3.1) will be derived in the coming chapters. The SNAME notation, SNAME (1950) is used when appropriate.

3.1 Advantages with the Lagrangian FormulationWe have used a Lagrangian approach to the derivation of the equations of motion. The Lagrangian approach involves three basic steps. First, we need to formulate a suitable ex-

19

20 CHAPTER 3. MODELING OF UNDERWATER VEHICLES

pression for the vehicle’s kinetic and potential energy, denoted T(q, q) and V(q, q) respectively, where q are generalized coordinates. Then, we want to calculate the Lagrangian denoted L(q,q) = T(q,q) — ^(9,7). Lastly,, we apply the n 2nd. order differential equations given by the Euler-Lagrange equation:

dLdin

dLdq,

i = 1..TZ (3.2)

to obtain the dynamic equations. Qi is a generalized force and n denotes the system’s degrees of freedom. It is important to note that the Euler-Lagrange equation is only valid for generalized coordinates. The Lagrangian mechanics describes the system’s dynamics in terms of energy and work, and the equations of motion may be derived almost straightforwardly when L is given. The Lagrangian approach in modeling and control of underwater vehicles has several distinc advantages over the Newtonian, the Hamiltonian, and the Euler methods of rigid body dynamics approach found in most textbooks, e.g. ref. Fossen (1991b), Roskam (1982), Allmendinger (1990) and Clayton and Bishop (1982). The Euler-Lagrange equation is valid regardless of the number of masses considered, the type of coordinates employed, the number of holonomic contstraints on the system, and whether or not the constraints and frame of reference are in motion. Hence the Lagrangian approach replaces a large set of special methods which have to be utilized for other methods. The Euler-Lagrange equation is also valid in any coordinate system, inertial or not as long as generalized coordinates are used.

3.1.1 The Lagrangian versus the Newtonian approachFirst, in the Lagrangian formulation one has only to deal with the two scalar functions T(q,q) and V(q,q). The Newtonian approach is vector oriented since everything is derived from Newton’s second law. This leads to a more cumbersome derivation of the equations of motion. See, for instance, how elegantly the equations of motion for a rigid body are derived in section 3.4. Second, the derivation of the added inertia elements and the vehicle’s rigid body’s equations of motion are done in a common framework. The added inertia is given a clear and physical meaning when we attack the “vehicle-ambient water system” from an energy point of view instead of a force-moment point of view, see also section 3.3. Third, properties of the mass matrix and the C(q) matrix are more easily seen when using Lagrangian mechanics instead of Newtonian mechanics, see section 3.12.2.

3.1.2 The Lagrangian versus the Hamiltonian approachThe Hamiltonian equations of motion constitute another way of expressing dynamic equations of motion of a rigid body. The Hamiltonian equations are 2n n-th order differential

3.2. LAGRANGIAN FORMULATION 21

equations, the Hamilton’s canonical equations, while the Lagrangian approach involves the use of n 2nd. order differential equation. The Hamiltonian point of view uses the Hamiltonian function H defined as:

Ti = pTq - L

where p is the generalized momentum p = Mq. By taking the differential of Ti and comparing terms we arrive to the 2n Hamilton’s canonical equations:

dTiP' dqi

dTi . ,9; = -a- * =dpi

Having a time-varying system, we also get the equation:

dH _ _dL dt dt

Notice that if we have energy conservation for our system, that is no dissipative forces are present, then this last equation can be written as:

This equation is sometimes referred to as Jacobi’s integral. We observe that as a mean of treating rigid bodies the Hamiltonian method is less convenient than the Lagrangian. There are, however, some cases were the Hamiltonian point of view is favorable e.g. celestial, quantum and statistical mechanics, Wells (1967).

Effects like restoring forces, viscous damping and wave load, which are not conveniently derived in the Lagrangian framework are explained in the Lagrangian framework before they are derived in the Newtonian framework.

3.2 Lagrangian FormulationIn the following, we will derive the equations of motion of a rigid body (e.g. an underwater vehicle) moving in an ideal homogeneous unbounded fluid of infinite extent. Expressions for both the rigid body’s as well as the ambient water’s kinetic energy are derived.

A rigid body moving in an unbounded fluid is holonomic by the definition of a rigid body and the unboundness and the infinite extent of the fluid. A rigid body moving without


constraints in six degrees of freedom is an ordinary Lagrangian system, i.e. a holonomic system. The infinite degrees of freedom system formed by the ambient water particles can not from a rigorous standpoint be considered to be holonomic. A line of reasoning used in Milne-Thomson (1968), pp. 99-101 will be used as an argument for this. Assume that a solid forms a holonomic system and that all the motion of the ambient water is due to the solid’s movements and that it would instantly cease when the solid is brought to rest. The motion of the water will then be irrotational and acyclic. The ambient water particles should end up in their original position if the solid were moved through a cycle of movements such that it returned to its original position. This is, however, not necessarily the case, and the ambient water particles do not meet the definition of holonomic. It is, however, common in the literature, Lamb (1932) and Birkhoff (i960), to also consider the system formed by the surrounding water particles as an Lagrangian system having six degrees of freedom. The proof that the system formed by the vehicle together with its ambient water is a Lagrangian system and that Qi is the generalized force on the water particles is rather complicated. This proof is made difficult by the fact that the total mass of the water is infinite and that the configuration space of the water particles has infinite dimension, that is the number of degrees of freedom of the ambient water is infinite. A complete proof of this was first presented in Lamb (1932) and later in Birkhoff (i960).It is desirable to express the vehicle’s equation of motion in the vehicle-fixed reference frame, that is with the time derivative q of the quasi coordinate vector q. This leads to one important problem, since the Euler-Lagrange equation is only valid for proper generalized coordinates. This problem can be avoided in two ways. Derive the equations of motion in the inertial reference frame (expressed with the generalized coordinates x = [x, y, z, <f), 6, xJj]t, see section (3.11) and then transform the resulting expression into the vehicle-fixed frame with the transformations given in (2.4). Another and more convenient approach is to use q and Kirchhoff’s equations in vector form, Kirchhoff (1869) where vehicle-fixed velocities are employed. This approach is used in this thesis.

3.3 Rigid Body Moving through a Liquid

We will in the coming discussion assume, if nothing else is stated, an ideal homogeneous fluid. This section will, in detail, derive Kirchhoff’s equations, Kirchhoff (1869), which will prove to be very useful in the derivation of a rigid body’s equations of motion and expressions for added inertia.

This section will use a similar approach as the ones in Milne-Thomson (1968) and Lamb (1932) in the derivation of the expressions of kinetic energy for a system of a rigid body moving through a fluid.

3.3. RIGID BODY MOVING THRO UGHALIQ UID 23

3.3.1 Conditions on the velocity potentialThe restrictions on the fluid motion presented in section 3.2 imply that the fluid’s motion have to be irrotational (B.4) and incompressible (B.5). The fluid’s velocity potential must satisfy Vcf> = 0 at infinity, which implies that a finite motion of the water particles generated by the solid, contains finite energy. This is often called the radiation condition. The motion of the fluid particles also has to be the same as the motion of the surface on the body, such that

- ^ = nT(v +uj x r) (3.3)

where r is a vector from the vehicle-fixed coordinate system to the vehicle’s surface, n is the unit outward normal vector of the vehicle and u> = [p,q, r]T and v = [u, v, w]T are the vehicle’s angular and linear velocity components along and about the vehicle-fixed coordinate axis. denotes the velocity of the water particles at the point r on the boundary of the solid. It can be shown that the following velocity potential satisfies the boundary condition in (3.3).

(f> = vTtp -|- u>Tx (3-4)

where cp and x are vectors of fluid velocities along the cartesian axis which meet the conditions mentioned in the beginning of this section on the motion of the fluid particles. Notice that <p and x have to satisfy

dtp—— — n and ondxdn = r x n (3.5)

Since cp and x must depend solely on the vehicle’s shape and not its motion.

3.3.2 Kinetic energy of the fluidWhen a rigid body moves in an ideal homogeneous fluid at rest, the ambient water particles will undergo an acceleration and a transformation of energy takes place.

The kinetic energy of the fluid Ti can be written as:

Tl = \J J JT‘ = -W^dS

Tl = J J {vT<P + uTx) {nT (v + io x r)) dS (3.6)


where the first transition uses Green’s first identity1 and the incompressibility condition, and the second is obtained by using (3.3) and (3.4) and assuming that the fluid density is constant.

We observe that (3.6) is a homogeneous equation of second degree of the vectors v and w. Therefore, by applying Euler’s theorem on homogeneous functions 2 on (3.6), we get:

(3'7)

We will now derive some expressions which will be useful in the discussion of added inertia in section 3.5. We find that

9 ^-U-\ = -n* + v-

by using (3.3) and (3.4). Therefore

Tsr-W/"*‘S-W/*K<S

by remembering that ip satisfies Laplace’s equation, Green's second identity3 can be used to rewrite the last integral in the previous expression to

-y 1J ,'T„dS=-y J J 4^dS=nil n4‘dS(3.3) is also used in the last transition. Expressions for and can now easily be found.

~ P J J ^ an<^ ~ P J J {r x n)<t> dS (3.8)

The last equation in the equality is found by a similar argument as above.

Notice further that the kinetic energy of the water particles generated by a moving solid can be written as

6

= ^2 Mam%} qxq3 (3.9)

'Greens’s first identity states: J J JVfVg dV — — f J/|^ dS — J f J fV2g dV2Euler’s theorem on homogeneous functions states that if /(ii, ...,xn) is a homogeneous function of

degree n, then xi^£- + •••• + xn-§T = kf.

3Greens second identity states that J f d>%£dS = f J + /// (^V2^ — t/’V2^) dV.

3.3. RIGID BODY MOVING THROUGH A LIQUID 25

where Mam,, are components of the matrix formed by the elementsMam,, = 11 f pVfa-VtfrjdV, Newman (1977) pp. 141-142. Hence, the total kinetic energyof the water particles is expressed in matrix notation

Tl = qT M am<1 (3.10)

REMARK 3.1. The matrix Mam is symmetric, that is Mam = since / / p<j>i ^tdS =//p<j>j^dS, see also Property 3.1 in section 3.12.2.REMARK 3.2. The matrix Mam is positive definite , that is Mam > 0, since the water particle’s kinetic energy Tl = qTMam<I always is greater than zero Vqr > 0.

It important to notice that the fluid’s kinetic energy, due to the acceleration of the body, is a function of the body’s velocity q = [u,v,w,p,q,r]T and not the velocity of the water particles 9/ = [u/,u/, u;/,0,0,0]7’. Notice also that the fluid particles have no angular velocity since we have assumed an irrotational fluid, (B.4).

3.3.3 Kinetic energy of the rigid body

The kinetic energy of a rigid body in motion Ts can be written as

Ts = ^JJJp(v + (uix r))T (v + (u> x r)) dV

1 ,TTs = -q MRBq (3.11)

This can be rewritten to

2Ts = vT dTg jdv

dTsdu> (3.12)

by using a similar line of reasoning as in the previous section. The expressions for and are found by comparing: (3.11) and (3.12)

7ST = J J Jffv + ^x^dv= J j j p{r x (v+ {u x r)))dV (3.13)

and denote the linear and angular impulses respectively.


3.3.4 Rate of change of angular and linear impulses in a moving origin

We recognize from (3.12) and (3.7) that ^ and have units and corresponding to the linear and angular impulse, respectively.

Remember that the motion of the fluid is due solely to the motion of the solid. The motion of the fluid can be considered to be instantaneously generated from rest by applying suitable impulses to the solid. The impulse on the solid is the impulse of the system at that instant.

We know that the time differentiated of impulse are force and moment, so let us find an expression of how linear and angular impulses, denoted $ and A respectively, change with respect to time in a coordinate system which is undergoing an infinitesimal rotation 6tu> and translation Stv. Notice that £ and A are constant in the solid-fixed coordinate system and that the rates of change are observed from an earth-fixed coordinate system.

translation

Figure 3.1: Rate of change of impulses in a moving origin with respect to a fixed reference frame.

The expressions

dtdt

dXdt

d£

d\— + ux\ + vxt dt (3.14)

are found by inspection of Fig. 3.1. Let us first ignore the rotation uht and only look on the effect of the translation vSt. The body-fixed linear momentum £ is only moved parallel to itself and undergoes no change. The angular moment A is increased by the new

3.4. RIGID BODY EQUATIONS OF MOTION 27

moment about the origin with the vector formed by the cross product vSt x £. Hence, the effect of translation of the origin is that A is increased by vSt x The other two terms in (3.14), that is the effect of rotation, are found by using the identity:

dodt inertial

d±)

dt vehicle+ w X (•)

d£ j-\Remember that -gf and ^ are the forces and moments applied to the solid.

3.3.5 Kirchhoff’s equations in vector form, Kirchhoff (1869)We are now ready to express the equations of motion for the vehicle - fluid system in termsof kiootir onprav Thp following spl. of pnnationc ic formprl hv cnKnfifnfinor for

These equations are known as Kirchhoff’s equation in vector form, Kirchhoff (1869), Lamb (1932). The left side of the equations contains the rigid body’s equations of motion. Ti..6

is a vector containing the generalized forces and moments acting on the rigid body, while the rest of the terms on the right side of the equations are forces and moments exerted on the rigid body by the fluid pressure. Kirchhoff’s equations will prove to be very useful in the derivation of the added inertia elements and the equations of motion for a rigid body. Note from the derivation of Kirchhoff’s equation that we first use kinetic energy as our leading term. Then we use the time rate of change of the impulse vectors given by (3.14) to take the rotational u> and translational effects into account.

A modified version of Kirchhoff’s equations is found in Meirovitch (1990) p. 42:

dtdL -r. ,

+ u x Ifo ~ JidLdx

£dt

+ U) X du> + » X dLdv

jT (-J*{xE)di

Tl .3

T4..6

(3.16)

3.4 Rigid Body Equations of MotionWe now have the necessary tools to derive the equations of motion for a rigid body. If we combine the elements in Kirchhoff’s equations which correspond to the rigid body


(the solid) with (3.13), the general equations of motion for a rigid body expressed in vehicle-fixed velocities become:

3.4. RIGID BODY EQUATIONS OF MOTION 29

= ^ />(» +(u> x r))^^+u» x J p(v + (uxr))dVT1..3 =

■n.,3 = m(v + u> x » + u> x re+ 1*; x (u> x re))

and

T4..6 =

T4..6 = m (re x t> + re x (id x u)) + Jid + id x Jid where

)dV

(^J P*" x (« + (“ x r)) + id x J (pr x (t) + (id x r))) dK + x J p (t) + (id x r)) dV

(3.17)

/id = / pr x (d} x r)tJv

Id X /id = Id x / p (r x (id x r)) dKJv

m(re x (id x t»)) = id x / p(r x ®)dK + « x / p{uxr)dV Jv Jv

The last equality is found by employing the equation:

a x (b x c) = (ac) b — (ab) c

fv denotes the triple integral, and the vector Tj..6 = [Nr, Yr, Zr, Kr, Mr, Nr]t is defined according to the SNAME notation. / is the inertia tensor defined as It] = Jv p(r ■ rS{j — r,r;) dV where <5,j is the Kroenecker delta function, ^ Jv Pr dV —[xc, ya, zqY is the vector from the vehicle-fixed coordinate system to the centre of gravity and m — Jv pdV is the rigid body’s mass. We have assumed that Jv prdV = 0, i.e. the vehicle’s center of gravity (CG) does not change with respect to time.

Notice how Kirchhoff’s equations allow us to elegantly and compactly derive the general equations of motion for a rigid body moving relatively to a fixed coordinate system.

(3.17) written out yields:

Xr = m[u - vr+ wq - xG(q2+ r2)+ yG{pq - r) + zG{pr + qj\


Yr - Tn[v -wp + ur- }/G(r2 + p2) + zG(qr - p) + xc(qp + r)]

ZR = m \w - uq + vp - zg{p2 + q2) + xG{rp - q) + yoirq + p)]

Kr = Ixxp + (Iz - Iy)qr - (r + pq)IX2 + (r2 - q2)Iyz + (pr - q)Ixy + m [2/g(u> — uq + vp) — zG(v — wp + ur)]

Mr = Iyq+(lx-Iz)rp-(i> + qr)lxy + (p2-r2)Izx + (qp-r)Iyz + m [zG(u — vr + wq) — xG(w — ug + vp)]

Nr = Izr + (Iy-Ix)pq-(q + rp)Iyz + (q2-p2)Ixy + (rq-p)Izx +m [xg(u — wp + ur) — yG(u — vr + tvg)] (3.18)

We are now ready to express the rigid body’s mass matrix which consists of the elements which are multiplied with the acceleration terms in (3.18)

m 0 0 0 mzG -myG '0 m 0 -mzG 0 —mxG

Mrs = 00

0—mzG

mmyG

myG/«

mxG~hy

0

mzG 0 —mxG ly -ly*. -myG mxG 0 -hz -h* h J

The lower right 3 by 3 matrix is recognized as the inertia tensor. The Mrs matrix is the rigid body’s contribution to the M matrix in (3.1). The remaining terms in (3.18) form the rigid body’s contribution to the C(q)q vector. Where the Crb((i) matrix may be written as

C Rs(q) =-m(*Gr + v)“m(VGr “ «) m(xGP + ya?)

IyZr + ISyV - lyq -lxtT — lxy<! + IxP 0

-m(yG<? + 2Gr) "»(*G9 “ u;) m(*Gr+ *)"*(VGP + u’) -m(«Gr + XGP) "»(>Gr ~ u)

00m(zGp - t>)

+ “)

•m(xGP + »G?)

"»(l/G'J + *Gr) -m(yGP + «#)-(miGp - v)

0lyzl + IxzP — Iz* — lyzr — JxyP + Iy<l

-m(xGq - ti/)+ xGp) -m(zGq + u) -lyzq — *xzP + ^rr 0

Ixzr + — IxP

(3.20)

Notice that the CRR(q) matrix is not unique. See section 3.12.2.

3.5. ADDED INERTIA 31

3.5 Added InertiaAlmost everyone has experienced the phenomenon of added inertia. For example, dip your outstretched hand into still water and then suddenly give it a rapid acceleration broadside. The apparent inertia of your hand will be felt as it was greatly increased. This increased inertia due to the acceleration of the hand’s ambient water particles is called added mass or added inertia.

Section 3.3 arguments for that a solid moving in an ideal homogeneous fluid causes an increase in the fluid’s kinetic energy. Kirchhoff’s equations prove that this effect can be represented by an addition to the inertia (an added mass) of the solid. This section is going to derive the expressions for these added inertia elements.

Written out the expression for the fluid’s kinetic energy, (3.10) becomes, Lamb (1932),

2Ti = —Xû2 — Yi,v2 — Z^v2 — 2Y^,vw — 2XyjWu — 2Xi,uv — (3-21)ATpP2 - Mgq2 - N+r2 - 2Mfqr - 2Kipr - 2K,pq - 2p{Xj,u + Yj,v + Zj,w) — 2q(Xg\i + Y^v + Z^w) — 2r(XrU + Y+v + Z+w)

where the twenty-one parameters in (3.22) are certain constants determined by the velocity potential in (3.4) and the solid’s shape. Thus, for example,

-Xu = p JJ ifx'^-dS = P JJ Vxdydz

—Xp = p JJ (px^-dS - p JJ ipxy dxdy ~ P J j V*2 dzdx

where ^ denotes the normal velocity component of the solid under unit velocity along or around the j—axis. The added mass parameters are defined according to the SNAME conversion such that Xu, denotes a force in the x—direction caused by a unit acceleration from rest in the z—direction. Notice that the added inertia elements are, by conversion, defined negative. We observe from Kirchhoff’s equations (3.16), that the forces and moments exerted on the moving solid by the fluid pressure are written as

XA

Ya

Za

d_ fdTA dTL drL dt \ du J dv ^ dw

d fdTA dTl _ dll dt \ dv J ^ dw r du

d fdTL \ dTL dTLdt y dw J ^ du P dv


Ka =d (dTA ^ dTl dTi , dTi dTidt

Up]dv ’v dw dq

Ma =d fdrL\ , dTL dTL . dTi dTidt

UJW du +par -~r dP

Na =d fdTA , dTl dTi , dTi dTidt

Ur)U dv + ^-

(3.22) combined with (3.22) gives the resulting expressions for the forces and moments on the solid by the fluid pressure, i.e. the added inertia terms

Xa = Xîi + X*(w + uq) + Xgq + Z^wq + Z^q2 +Xi,v + Xpj) + X+r - Yi,vr - Yj,rp - Yfr2 —XiW— Y^wr +YvVq + Zj,pq - (Yg - Zi)qr

Ya = XiU + Y^w + Ygq+Yii) + YpP + Yrf + Xyvr - Y„vp + Xfr2 + (Xp — Zf)rp — Zpp2 —Xu,(up — wr) + Xûr — Z^,wp -Zgpq + Xjqr

Za = Xti(u — wq) + Z^w + Zgq — Xûq — Xgq2 +Y^v + Zpp + Zfr + Yyvp + Y+rp + Ypp2 +Xiup + Y^wp -Xi,vq - (Xfi - Yj)pq - X+qr

Ka = X,;M + ZpW + Kgq - Xi,wu + Xruq — Y^w2 - (Yg — Z+)wq + Mfq2

+y-n + Kpp + Ktr + Y*v2 - (Yg - Zf)vr + ZpVp - M+r2 - K^rp +Xu,uv — (Yi — Z^)vw - (V) + Zg)wr — Ypwp — Xgur +(?) + Zg)vq + Kfpq - (Mg - Nr)qr

Ma = X$(u-\-wq)-^-Zg(w — uq) + Mgq - Xw(u2 - w2) — (Zu - Xu)wu

+Ygi + Kgj> + M+r + Ypvr - Y^vp - Kr(p2 - r2) + (Kj, — Nj)rp

—Yûv + Xi,vw — (Xr + Zp)(up — wr) + (X^ — Zr)(wp + ur)

-Mrpq + K$qrNa — Xfii + ZfW + Mfq + Xf,u2 + Y^wu — (Xp — Yg)uq — Z^wq — Kgq2

+yrri’ + KfP + Nff — Xi,v2 - Xfvr — (Xp - Yg)vp + Mfrp + Kgp2

—(Xu — Yi,)uv — X^jvw + (Xg + Yp)up + Yfur + Zgwp —(Xg + Yp)vq - (Xp - MJpq - Kfqr (3.23)

3.5. ADDED INERTIA 33

The added inertia elements are grouped as in Imlay (1961), where the first row for each degree of freedom contains longitudinal components of motion. The second line contains lateral components of motion, while the third row contains mixed term involving u or w. The fourth row contains coupling terms of such magnitude that they usually can be neglected in most cases.

Notice how easily the added inertia elements are derived by using the Lagrangian approach.

We are now ready to regroup the terms in (3.23) into one added inertia matrix which correspond to the rigid body’s mass matrix (3.19) and one matrix corresponding to the rigid body’s nonlinear matrix C(c[)am, (3.20).

' Xi X, x<, x? Xi Xr 'Xi n Y«, Yp Yi Yr

Zw Z* Zi ZrX, Zj, Kp Xi XrXi Yi Zi Xi Mi MiL Xt Yi Zr Kr Mr Nr

(3.24)

This matrix is the same matrix as the one defined in section 3.3.2. Triantafyllou and Amzallag (1984) contains a discussion of how to calculate the various elements in Mam for different geometrical bodies.The complete Cam(q) matrix is rather complex and, for brevity, is not included in the text. If the off-diagonal coupling terms are neglected, the simplified Cam{<i) matrix is

Cam{q) -

0 0 0 0 Z^w -Yiv '

0 0 0 -Z^w 0 Xû0 0 0 YiV Xû 00 ZwW -Yiv 0 N+r -Miq

—Z^w 0 X{,u —N+r 0 XpPYiv -XuU 0 M4q -XpP 0

(3.25)

We notice that the Cam{<i) matrix is skew-symmetric. It is, however, possible to show that the complete Cam(q) matrix also has this property, see section 3.12.2.It is important to notice that the Mam and the Cam{q) matrices are formulated on the right side of Kirhhoff’s equations while the Mrs and the CRg(q) matrices are formulated on the left-hand side of the equations.


3.5.1 Symmetry properties of the mass matrixThe total mass matrix M = Mrb — Mam, which is symmetric, consists in general of 21 inertial coefficients. This can be reduced to 15 coefficients by an appropriate choice of coordinate axes such that the matrix is simplified to the following form .

' mu 0 0 mi4 mis mie0 m22 0 mis m25 m260 0 m33 mie m26 77736

mj 4 mis mie m44 m4s 77746mis m25 m26 77145 mss mse

. mi6 m26 mse m46 mse m66This matrix can be reduced further if the solid has any plane of symmetry.

xy-plane symmetry xz-plane symmetry

' mn mi 2 0 0 0 mie ' ' mil 0 777 1 3 0 mis 0 '777 l2 77722 0 0 0 m26 0 77722 0 77724 0 77726

00

00

m33m34

m347Tl4<

mss77745

00 MXZ —

mis0

077724

777330

077144

mss0

077746

0 0 m35 m45 mss 0 mis 0 77135 0 mss 0. mi6 m26 0 0 0 mee . 0 77726 0 77146 0 77766 .

yz- plane symmetry

' mn 0 0 0 mis 77716 '0 77722 m23 m24 0 0

Myz = 00

m23 m33m24 77734

7713477144

00

00

mis 0 0 0 mss 77756. mi6 0 0 0 mse 777-66 .

If two or more symmetry planes exist the resulting mass matrix is formed by the intersection of the corresponding symmetry matrices. An example is if a vehicle is both yz- and zz-symmetric. The resulting mass matrix Myz^xz is then formed by the intersection Af Vzaxz = Myz D Mxz of the two matrices.

REMARK 3.3. Notice the special case when the vehicle is symmetric in three perpendicular axes. The mass matrix is then the diagonal matrix formed by diag(m„) i — 1..6.

REMARK 3.4. Notice that m14, m2s and m36 is zero when there is any plane of symmetry.

3.6. DISSIPATIVE FORCES 35

3.6 Dissipative ForcesThus far in this chapter, we have only discussed a solid moving in a nonviscous fluid. D’Alambert’s paradox states that no hydrodynamic forces act on a solid moving completely submerged with constant velocity in a nonviscous fluid. In a viscous fluid, frictional forces are present such that the system is not conservative with respect to energy. Frictional forces can from a Lagrangian point of view be derived from a scalar dissipative function ^ defined such that the rate of energy dissipating from the system can be expressed as:

dEdt

were Ti is a power function, Ti has dimension of power. The force Ti works against the motion of the solid and is in general a function of the velocity. We can let

in+l

where c is a positive damping coefficient, Irgens (1990) and a; is a vector containing the generalized coordinates. Ti for n = 1 is sometimes known as Rayleigh’s dissipation function. We then notice that the generalized damping force becomes:

Qi, = --jrr- = -Ci | ii p-1 iiOXi

Notice that ti = 0 corresponds to dry friction or Coulomb damping, n = 1 corresponds to linear viscous friction or Newtonian damping and n = 2 gives the expression for quadratic damping. The n 2nd order Euler-Lagrange equations for the vehicle can now be written as:

d_ (dT\ dT dTddt l dx,) dx, dx, (3.26)

We see from (3.26) that the dissipative forces represented by give a larger Qi to maintain the same level of kinetic energy T. It is, however, easier to determine ^ directly rather than first find the power function and next perform the differentiation. This leads to the Newtonian discussion which the rest of this section will contain.

Steady motion is an ideal case which cannot be achieved in practice even for an ideal fluid when the propulsion is from propellers, Clayton and Bishop (1982). It is, however, realistic to assume steady motion when the motion of a solid can be expressed by a sum of a time average component v and a fluctuation component v such that v = v v and v v.


Two main types of forces are exerted by the fluid on a neutrally buoyant vehicle when the vehicle moves with a constant speed relative to the fluid, denoted qr = (ur,vr, wr,p, q, r)T. First, we have those forces which are distributed over the complete wetted surface of the vehicle, often called drag forces, and second, we have the propulsion and control forces from e.g. propellers and rudders. This section will discuss the former type of forces, while the latter ones are dealt with in a later section.

Total resistance Ry.i

£8.I

I

Pressure resistance Rp Skin friction resistance Rp

Wave-making resistance Rw

Viscous pressure resistance RPV

Viscous resistance Rv

StB8

Energy in wave pattern well away from vehicle Energy in wake

Total energy loss

Figure 3.2: Steady state resistance components, Clayton and Bishop (1982) p. 200.

We observe from Fig. 3.2 that the total resistance can be broken down to three components. One component Rp, arises from the presence of shear stress between the solid’s surface and the ambient fluid. This shear stress takes place inside the boundary layer on the solid’s surface, see Fig. 3.3. A second component, the pressure resistance Rpr, is cause by the boundary layer. The boundary layer prevents a complete pressure recovery at the aft of the solid. The energy loss due to these two viscous effects can be observed as a wake behind the solid. Both these forces are, in general, frequency and Keulegan- Carpenter number dependent. A third effect, the wave-making resistance Rw, arises if the solid is moving close to the surface. A wave pattern will be generated on the surface of the fluid if the solid moves sufficiently close to the surface. The energy loss due to this effect is the energy contained in this wave pattern. This last effect can be looked upon as a potential damping, since it generates a wave pattern with a corresponding wave potential. A similar effect is introduced if a large underwater vehicle is in the wave-affected zone. The wave potential will then be altered as it moves over the vehicle. This change


in the wave potential corresponds to a loss or gain of energy for the vehicle. Large is, in this context a relative term which describes the ratio of the wavelength A to the vehicle’s characteristic dimension d. A ratio of less than approx. 5 denotes a large vehicle. A fourth effect, not shown in Fig. 3.2, may occur if the solid moves sufficiently close to a solid boundary such as the sea floor. This effect can be observed when a vessel is moving in shallow water. Seaweed on the seafloor will then be given a motion which can be looked upon as a loss of energy for the vessel. The rest of this section will group together the bounded fluid component, the wave-making and the wave-changing component, and with the pressure resistance, since they are difficult to determine separately, both experimentally and analytically. The total resistance force will be denoted drag force in coming discussion.

streamline

Figure 3.3: Streamlines, boundary layer and wake

The point on the vehicle on which the drag forces act is called the center of pressure, denoted CP = [xp, yp, zp]T . This center of pressure is a function of the vehicle’s geometry, velocities, the sign of the velocities and the vehicle’s angle of attack and drift.

The drag force in x—direction due to a surge motion can be calculated as follows:

Aa = ~^pCdA |u | u

where Co is a dimensionless resistance coefficient defined such that Co = Cpp -|- Cp and A is a characteristic area of the solid, e.g. the wetted surface of the body or the projected area in the x—direction. The above equation can be derived by using Bernoulli’s equation, Faltinsen (1990a). Cp can usually be neglected for bluff body’s since Cpp Cp for this set of geometries. This is because the boundary-layer separation produces a large wake behind bluff bodies. The wake indicates that most of the energy dissipation is due to


the pressure resistance. The Cp is, however, significant for large streamlined bodies, e.g. military submarines. The following expression suggests one way of calculating the Cp coefficient Sarpkaya (1981)

Cp = 0.075(log10 Re - 2)2

Fig. 3.4 shows in principle the Co coefficient for a circular cylinder as a function of the Reynold’s number 71] = ETl where u is the kinematic viscosity.

100

ci

10

1

0.1

io° n? i<? n? io4 k? iof io7Re

Figure 3.4: Drag coefficient Co for a cylinder as a function of the Reynolds number

It is common engineering practice to assume a linear and quadratic damping term such that the total damping for the x—is modeled as

\\

.... ,.v.

______ ______ ______ ______ ______

.—r;

i

fud = —dû — dq \ u\ u (3.27)

where the damping coefficients dp and dq are found from free decay tests, Sagatun and Fossen (l991d) or parameter identification experiments, Zhou (1987). (3.27) can in its most general form be written in matrix notation as

fd = ~DLqr - dQ{qr) - h.o.t.

where Dp is a 6 x 6 positive definite matrix, and the dQ(q) vector is on the form [qJXq |<7r|, qjYq |gr|, ... ,qjNq |9r|]r. The matrices Xq,...,Nq are all positive definite 6x6 matrices and | (jrr | is defined as the absolute value of each element in


the qr = q — qj vector. The qj vector denotes the fluid particle’s velocities such that q j = [it/, Vf,wj, 0,0,0]T. This expression is unnecessarily complicated for most practical purposes and may be simplified to

■ fxvi fxzi 0 0 0 'fv^t fyw, 0 0 0

= -D(qr)qr = - fzui0

fzvi-zpf:d

fwdypfwd

0rjpd

00

00 (3.28)

0 ~XPfwd 0 r 0. -ypfud zpf:d 0 0 0 /;J

where the terms on the form is defined according to Newman (1977, P-21, such that/«., = \PCJA that is fzu ur is the force in z—direction due to a relative velocity inthe x—direction. The subscript l denotes a lift force and corresponds to the lift force in aerospace terminology and is zero for a symmetric vehicle with zero angle of attack. The f*d terms are defined such that f*du equals to (3.27) and the xp x f* terms correspond to moments which will act on a vehicle with a CP that is not aligned with CG . For simplicity, we have ignored that the center of pressure may changes when the sign of the velocities change and that the drag forces in general also are a function of the vehicle’s angle of attack and drift. Notice also that the D(qr) matrix is still positive definite. A further simplification is to assume a symmetric vehicle and that the centre of pressure CP lays in the centre of gravity CG. Hence, the matrix in 3.28 is reduced to the familiar diagonal matrix

D(qr) = diag(dii - diQ | qri |); i - 1..6 (3.29)Experiments indicate that this form is sufficient for most practical purposes for almost symmetric and bluff-bodied vehicles.

It is important to notice that a linear and quadratic damping term are not an exact description of the damping, but only an engineering approximation. The skin frictionmay, for example, dominate for large, streamlined vehicles such that the damping becomes proportional to (loffioii)-1. It is also necessary to remember that the damping terms are very sensitive to the Keulegan-Carpenter number, see section 3.13.5, and the Reynold’s number, see Fig. 3.4.

3.6.1 Forces and moments from an umbilicalForces and moments from an umbilical are mainly due to drag forces. The drag forces on the umbilical will in extreme cases account for more than 90% of the total drag. Consider, for example, a 3000 [m] long 2.5 [cm] thick umbilical. This cable will have a total surface area of 72m2 compared to a medium-sized ROV which has a surface less than 2m2.


The forces attack at the connection point TP = [xu, j/u, zu]T on the vehicle. The TP vector is relative to the CG. Additional weight or buoyancy must also be added to the TP point if the umbilical is not neutrally buoyant. The forces and moments on an underwater vehicle are generally velocity dependent and may be written as

Tu = -Du(qr)qr (3.30)

where the components in the D matrix can be calculated on basis of equations given in Faltinsen (1990a) and Dand and Every (1983). Lie et al. (1989) show how the umbilical can be shaped to minimize the drag.

3.7 Restoring Forces and Moments

This chapter derives the forces and moments that act on the vehicle due to gravity, socalled conservative forces. This chapter starts by attacking the phenomenon from a Lagrangian point of view. The resulting expression is, however, derived in the Newtonian framework, since this is much simpler for this case.

Restoring forces for underwater vehicles are mainly due to gravity. The gravity can be looked upon as a scalar force field expressed by the scalar function V(x,t) which may be a function on both the system’s generalized coordinates x and time t. An appropriate choice of generalized coordinates for this case is earth-fixed positions. The forces and moments, due to gravity acting on the vehicle, can now be found as the gradient of the scalar function V{x) along and about the generalized coordinate axis.

Qi =dV{x)

dxi (3.31)

It is well known that the change in potential energy is only end-point dependent and not a function of the path traveled from point 1 to 2. A necessary and sufficient condition for this is that the force Q is the gradient of the scalar function V/(*) of position only such that (3.31) can be expressed as, Goldstein (1980):

Q = — W(x)

The work necessary to travel from 1 to 2 can then be expressed as

W,12dV{x) dV(x)

dz dz

3.7. RESTORING FORCES AND MOMENTS 41

or

W12 =

W12 =

dV(x)dx dV(x)dy dV(x)dzdx dt dy

V(x(2)) - V(*(l))dt dz dt )*-£* (V(x(t)))dt

Hence, end point dependency is proved. These type of forces are called conservative forces.

Notice also that we assume that V[x) is not a function of time. The n 2nd order Euler- Lagrange equations given in (3.26), when we neglect the dissipative forces, can now be rewritten to:

n =- ^ dT~V’ dt \dxi) dxi

orddt

dl_dxi

where the new Lagrangian L is T — V. Notice also that the kinetic energy may be a function of the position vector since we have changed generalized coordinates.

Gravity forces work on the vehicle because the submerged vehicle has weight and buoyancy. The restoring forces denoted £f(x) form in general a 6 x 1 dimensional nonlinear vector where the elements are functions of the vehicle’s orientation parameters, e.g. Euler parameters or Euler angles and the vehicle’s weight and buoyancy. The weight force W = mg acts in the vehicle’s center of gravity, CG = (xg, 2/g, zg)t, while the buoyancy force B = pjVg acts in the vehicle’s center of buoyancy, CB = (xg, ys, zb)T■ m is the vehicle’s mass including water in free-floating spaces, g is the gravitational constant, p; is the density of the ambient fluid and V is the volume of fluid displaced by the vehicle.

Figure 3.5: Restoring forces and moments


The W and B forces also produce moments, since the center of gravity and centre of buoyancy are, in general, not placed in the same place in the vehicle, see Fig. 3.5. The vector g(x), expressed in the earth-fixed coordinate frame, looks as follows:

9e(*)

0

0

W - BVbB — yGW xqW — xbB

0

fw ~ fa*G * fw ~ XB * fB

The same vector, expressed in the vehicle-fixed coordinate frame, is found by premultiplying the vector with the transformation matrix in (2.2), such that g(x) =Ji{<t>,6,xl))gE(x). Thus vector g(x) written out becomes

(W - B)sm0 —{W — B) cos 0 sin <f>—{W — B) cos 0 cos 4>

—(yoW — ysB) cos 0 cos <j> -f {zqW — zbB) cos 0sin <j> {xcW — xbB) cos 6 cos <f> + (zqW — zbB) sin 0

—{xqW — xbB) cos 0 sin ^ — {yoW — yaB) sin 0

(3.32)

The complexity of vector g(x) is reduced when the vehicle is neutrally buoyant and when the vehicle-fixed coordinate system is placed in CG or CB. Hence, the g{x) vector becomes

0

0

( \- 0—BGyW cos 0<j> + BGZW cos 0 sin <f>

BGZW sin 0 + BGXW cos 0 cos (j)—BGXW cos 0 cos <f> — BGyW sin 6

if the vehicle-fixed coordinate system is placed in CG for a neutrally buoyant vehicle. The BG vector is defined according to the SNAME convention SNAME (1950), such that BG denotes the the vector from CG to CB.

3.8 Propulsion and Control Forces and MomentsThe Qi on the left hand side of (3.26) consists of the driving forces on the solid. These forces consist of the control and propulsion forces and the disturbances forces which are discussed in Chapter 2 and the next section.

3.8. PROPULSION AND CONTROL FORCES AND MOMENTS 43

The vehicle’s control and propulsion forces will be denoted

r = b(q,u)

where r € 3?", u £ and the nonlinear function b(q, u) is defined such that b(q, u) : 3?m i-» 3tn.

This section denotes control forces as forces which are used to control the vehicle’s orientation and propulsion forces as forces which are used to control the vehicle’s velocity and acceleration.

Steering fins or rudders are the most common control devices on large submarines, while small underwater vehicles like ROVs usually use ducted propellers. Propellers are by far the most common propulsion device under water. We have here excluded fish which use active foils MITSeaGrant (1991). Other propulsion devices are water jets, Bratland (1989) and different variable ballast-buoyancy systems Kleppaker et al. (1986) and Ura and Otsubo (1988). Small underwater vehicles usually use the same actuators to control their orientation as well as velocity and acceleration. This is because fins and rudders are most effective under relatively high and steady speed. Rudders and fins lose most of their control effect when the vehicle operates in its own wake, a situation that is common in dynamic positioning mode and when the vehicle performs finer maneuvering.

This section will only discuss actuators where the propulsion and control are provided by screw propellers enclosed in a duct.

3.8.1 Thruster hydrodynamics

In this section we assume that the ducted propeller is isolated from the vehicle such that all vehicle-thruster interaction is neglected. We can now analyze the thruster’s dynamic from an “open-water” (alone without a hull) perspective. Dimensional analysis of the thrust force from propellers reveals that the thrust is a function of the propeller diameter D, propeller revolution n and the propeller’s speed through the water Va, Walderhaug (1990). Va is defined such that V4 = (1 — w)V where V is the vehicle’s total speed and w is the wake fraction number. The wake fraction number represents the reduction of velocity of the incoming water to the propeller. V = Va for the open water test condition. These parameters define the non-dimensional coefficient, the advance coefficient Ja:


Similar non-dimensional coefficients for the propeller’s torque Kq and thrust Kr can be achieved such that

= —Tth Kq = —yjrj (3.33)pnlD* pn2Db

The thruster’s open water efficiency coefficient r)0 is defined as r?0 = A typical plotof Kr, Kq and r/0 against J0 is shown in Fig. 3.6. Notice that maximum thrust is not

Figure 3.6: A typical plot of Kr, Kq and r)0 against positive J0,

necessarily found where the Kr is at its maximum since Kr is a nonlinear function of J0.

The overall efficiency of the thruster system T)r is defined as the ratio of efficient power of thruster mounted on the vehicle divided by the power supplied to the thruster system.

VT = VMrioVH^R

where i]m is the mechanical efficiency, t}r is the “behind-to-open” efficiency, Clayton and Bishop (1982) p. 370, and r)n is the “hull efficiency”, Clayton and Bishop (1982) p. 371. VhVR's experimentally determined to be 0.85 by Parfitt and Watters (1990) for the BP’s DISPS vehicle.

Duct

A duct around a screw propeller has two functions. First, it is supposed to improve the thruster’s efficiency at high trust loads on low speed, and second, it protects the propeller from entanglement in the umbilical, cables and collisions. A very good article on duct design and performance is Oosterveld (1973).


Disturbances

A thruster mounted on a vehicle may lose efficiency due to several type of disturbances. Cavitation Clayton and Bishop (1982) p. 380, momentum drag Dand and Every (1983) p. 32, thrusters-hull effects e.g. the Coanda effect, loss of efficiency due to current, free surface effects on thrusters and thruster-thruster interactions Faltinsen (1990a) pp. 270- 281 are the most important effects.

For the interested reader we would like to recommend Blanke (1981) which contains a detailed model of propulsion dynamics from a control engineer’s point of view.

3.8.2 The b(q,u) vectorWe will now show that the thrust T is as a nonlinear function of the propeller revolution and that the thrust is a function of the vehicle’s speed. We see from (3.33) that the thrust T is related to the Kt coefficient such that T = Kt(Jo)pD4ti \ n | We observe from Fig. 3.6 that Kt can be approximated by a linear interpolation Kj = a + ftj0, Fossen (1991c) Sagatun and Fossen (I991d). Substituting this approximation for Kt into (3.33) together with the expression for J0 yields

r w &iu | u | +&2 I 9 b | u | (3.34)

where 6j = pD4a and 62 = pD3(l — w)P- | q I2 denotes the total speed which simply is the Euclidean length of the velocity vector q. Remember that the input u is simply n. This is clearly verified by Fig. 3.9 (a) where the thrust is plotted as a function of different speeds of advance and propeller revolutions. Hence, the b(q,u) vector can for the NEROV be written as

bini | ni | +62 | 9 I2 | | +M2 | ^2 | +&2 | 9 I2 | «2 I

t>in3 | n3 | +&2 | 9 I2 | ns | +6jn4 | n4 | +&2 | 9 I2 I |

bin5 | n5 | +62 I 9 I2 | ns | +^>1^6 | | +&2 I 9 I2 | ^6 j

(6i«3 I 713 I +&2 I 9 I2 I 713 |)fr

— (bins | ris | -f62 | 9 I2 | Tis |)/y + (bins | Tig | +62 I 9 I2 | Tig |)/j,

. (&1711 | Til I +62 | 9 |2 | Til |)/2 - (&1712 I 712 I +&2 | 9 I2 | 7l2 |)/*

(3.35)

where ti, is the propeller revolution of thruster i and /, is the momentum arm around the i axis.

3.8.3 The NEROV thruster systemThe NEROV propulsion system consists of six ducted thrusters and their control electronics. The most important design criterion for the thruster system was the price. The


whole system, including its control electronics costs in the area of than NOK 100,000.—, ECU 12,500 (approx). All thrusters have a maximum thrust of 78 N in bollard pull. The thruster force is reversible and almost symmetrical, see Fig. 3.9 (b). The thruster bandwidth is 2 — 3 All the thrusters are equipped with propeller angular velocity measurements for inner loop feedback. This is obtained by mounting tachogenerators on the motor shaft. Fig. 3.7 shows a schematic drawing of the thruster with its main dimensions.

Figure 3.7: A schematic drawing of the NEROV thruster Sagatun and Fossen (1991a).

The thruster’s momentum drag were not found to be significant compared to the vehicle’s total drag with the current maximum thrust, see Fig. 3.10.

The motor

The NEROV thruster is constructed and built at the Division of Engineering Cybernetics. The thruster motor is a 24 V DC 400 W permanent magnet motor made by Outboard Marine Corporation. The motorhouse is modified to fit the tachometer.

The duct

The diameter of the duct is 244 mm and the length is 180 mm. The duct is made of PVC. The duct profile is symmetrical to obtain the same amount of thrust in both directions. This was an important design criterion in the design of the duct profile. The duct profile is shaped approximately like nozzle no. 37 described in Oosterveld (1973).


The propeller

The propeller is a three-bladed Kaplan type propeller with radial pitch distribution. The propeller is made of plastic. The main data for the propeller are; diameter 241 mm, the pitch P/d at 0.7d is 0.89, the chord length at 0.7d is 93 mm and the thickness of propeller blade at 0.7d is 4.5 mm.

The control electronics

The analog control board consists of one 24 V DC pulse-width modulator and one monovariable PID-type controller. The input to the PID-controller is the commanded propeller revolution from the VME rack and the measured propeller revolution from the tachometer. The PID controller works as an inner servo loop which improves the thruster bandwidth as well as reducing effects from nonlinearities in the control loop, e.g. friction. The thrusters currently have a bandwidth of 2 Hz. The pulse-width modulator is used to control the propeller angular revolution.

Figure 3.8: The analog inner feedback loop.

The NEROV propulsion system and the actuator dynamics are described in detail in Sagatun and Fossen (1991a). Recent tests of the PID regulator suggest the removal of the derivate action. This is due to significant noise form the tacho-generator. A redesign of the thrusters should involve the use of a pulse-counters instead of tacho-generators. The pulses could then be integrated by analogue electronics and used in the servo-loop as feedback.

3.8.4 Open water test of the NEROV thrustersThe open water test of the NEROV thrusters was done full scale with Va — V i.e. a wake fraction number u> = 0. The test was performed in the towing tank at the Norwegian Marine Research Institute (MARINTEK). Fig. 3.9 and 3.10 present the test results.On basis on Fig. 3.9, we can now find the parameters in (3.35). We find that Kt can be approximated by the function Kj = —0.405Jo + 0.225. Eq. can now be written as


T[N3lOOi/. .9po/»._80 -■

20 -

-t-VaXm/sJ

-40 -

-60 -iXy.Avafs^tl -0.2m/»'/

-80 - / OnVs •!

-lOO '---- ---- L-20 -1C

rev/s

ktVaX)nkO

-0.1 - i

-0.2 - |

-0.3-0.5

(a) (b)

Figure 3.9: (a) Measured thruster force T as a function of propeller revolutions n for different speeds of advance Va-(1>) Non-dimensional thrust characteristics Kt versus the advance coefficient J0 for the NEROV.

b(q,u)

O.Sri! | uj | -4.71 | 9 b I | +0.8n2 | n2 | +4.71 | 9 I2 | «2 |0.8n3 j n3 j -4.71 j 9 I2 | «3 j +0.8n4 j n4 j +4.71 | 9 (2 | «4 |0.8n5 | n5 | -4.71 | 9 I2 I | +0.8n6 | n6 j +4.71 | 9 I2 | ^6 |

0.33n3 | n3 | -1.92 | 9 |2 | n3 |-0.32n5 | n5 | +1.88 | 9 I2 | ^5 I +0.32n6 | n6 | —1.88 | 9 I2 | ^6

0.34ni | ni | —1.98 | 9 I2 | | —0.34n2 | n2 \ +1.98 | 9 I2 | ”2 |

(3.36)

where we have used a wake fraction number w = 0.2 and lx = 0.41 [m], lx = 0.40 [m] and lz = 0.42 [m].

3.9 Wave and Current Forces and Moments

This chapter will deal with the other driving term on the left-hand side of the Euler- Lagrange equation. This term is due to wave forces and will be denoted u)(<f>) where <j> is the wave potential.

3.9. WAVE AND CURRENT FORCES AND MOMENTS 49

Figure 3.10: (c) Nondimensional thrust characteristics Kjx (o marked) and Kty (x marked) in the x- and y-direction respectively as a function of the advanced coefficient J0 and angle a = 0° (dashed) and 90° (solid) between thruster and vehicle speed.

In the simplest case we assume that the sea waves are regular plane progressive waves of small amplitude, with sinusoidal time dependence. This is a crude approximation since the actual ocean waves are most accurately described as a random process, which causes random responses of a body in the wave-affected zone. Why do we study regular wave theory at all ? Fortunately, the random response of a body in random sea can be described by a linear superpositioning of waves with sinusoidal components.

This section presents first a stochastic model of waves. Different state-space realizations of sea waves are then presented, before Morrison’s equation is introduced. The section ends with a brief discussion of current and current induced forces.

It might be uesful to read Appendix B for background in hydrodynamics.

3.9.1 Wave models in the frequency domain

The ocean waves are best described as random processes. It is convenient in the further discussion to regard ((t,x,y,z) as a continuous, stationary and ergodic process. In prac- tic, irregular sea is simulated with a sum of a large number of linear wave components,


Faltinsen (1990a):

N£ = sin(w,t — kii + e,) (3.37)

i=l

Here u>,, and e, denote wave amplitude, circular frequency, wave number and random phase angle respectively. The random phase angles c, are uniformly distributed between 0 and n. The wave amplitude Ai can be expressed by a wave spectrum S(lo) as Â? = S(u)j)Au. The instantaneous wave elevation ( is considered Gaussian with zero mean and variance a2 = S(tj)du) where 5(w) is the wave spectrum. This is derived by noticing from (3.37) that tr2 = ^?/2, by letting iV —► oo and Aw —» 0, a2 becomes/0“5(w)dw.

The wave spectrums are usually estimated on basis on wave measurements taken over a short period of time e.g. | hour - 10 hours. This is what the literature describes as short term wave description. The spectrum S(u>) is then found by performing the discrete fast fourier transformation FFT on the sequence of wave measurements. The discrete Fourier transform pair which is applied is given by: F(u) = j$Ylx=o f(x)e~*2™x/N and f(x) = Eu=o F{u)e,2ûxlN where Au =

Fig. 3.11 shows a plot of a part of a time series measured by a wave buoy4 outside the Norwegian coast, together with its corresponding wave spectrum. The spectrum is based on 2048 samples. Significant wave height is found to be Hi = 1.22m and mean wave period 7\ = 7.24s.It is common to use recommended sea spectra from ISSC (International Ship and Offshore Structure Congress) and ITTC (international Towing Tank Conference) in engineering design. For open-sea condition, the Pierson-Moskowitz spectrum (3.38) is recommended,

S(w) = 0.0081-^ exp (3.38)

while a JONSWAP(Joint North Sea Wave Project) -type spectrum is recommended for limited fetch eq. 3.39.

4provided by the Trondheim-based company OCEANOR


Figure 3.11: Measured time serie from the North Sea with its corresponding power density plot.

SM = 0.0081exp (-1 • • yexp(-5^ )

- Ti (3.39)_ j 0.09 if w > wp

<T~\ 0.07 if w < wp Y is usually between 1 and 7

The JONSWAP and the Pierson-Moskowitz spectrum are shown in Fig. 3.12. Fig. 3.13 compares a measured wave spectrum and the JONSWAP spectrum for the corresponding sea state. Notice that the measured spectrum is more narrow banded than the JONSWAP spectrum. The JONSWAP spectrum’s modal frequency lop (peak frequency) is at a higher frequency than the measured spectrum. This because the measured waves have had time to develop over a longer period than assumed for the JONSWAP case.

3.9.2 State-space representations of sea wavesWe know that we can generate a stochastic process with an almost arbitrary power spectrum by sending white noise through a dynamic system with the appropriate transfer function. The JONSWAP power spectrum in Fig. 3.12 can be approximated by sending

52 CHAPTERS. MODELING OF UNDERWATER VEHICLES1

SP)

0.8 -

Figure 3.12: The JONSWAP spectrum (solid) (3.39) and the Pierson-Moskowitz spectrum (dashed) (3.38)

white noise with unitary power spectrum Q2S(t), Q — l through a second order filter on the form

h(s) = k--------------l+2<£ +

(3.40)

Notice that the power spectrum of this filter (jui) has zero energy for zero frequency, i.e. w = 0 —» 4>yy(0) = 0. This is of course a desirable behavior for this power spectrum. The parameters k — 0.22 and ( = 0.03 in the resulting power spectrum formed by

$yy(Jw) = hUu) ■ Q ■ H~ju) (3.41)

were found by minimizing

rooj — (^yy0w) — $jonswapUu)) ^ (3.42)Jo

Saelid et al. (1983b). w0 = 0.87 was found by letting u»0 = ^ where Ti is the mean wave period. The corresponding state space representation of the filter in (3.40) is realized by the 4-tuple (A,b,c,d) where


0.05 0.1 0.15 0.2 0.25 0.3

Figure 3.13: A comparison between the JONSWAP spectrum (3.39) and a measured spectrum

c = [0 1 ] d = Q

(3.43)

and a — \Juj0 and b = 2\uj0. The corresponding power spectrum is formed as follows:

fcyyO’w) = [c(jw/ - A)~'b + d] Q [c(-jw/ - A)~lb + d]T (3-44)

where Q = \.

3.9.3 Linear potential theory

The analysis in this section will use the “long wave approximation” Newman (1977), that is the wavelength A is much greater than the underwater vehicle’s characteristic dimension D, X^> D. The section will only discuss the case of a fully submerged body.


Consider a fully submerged vehicle which is exposed to incident waves. The velocity potential can then be written as, Newman (1977) pp. 289-290:

<t>(x,y,z,t) = Re y, z) + A(<t>0{x, y, z) + Mx> y. z)\/=1 RADIATION DIFFRACTION t

e‘“‘ (3.45)

where A is the wave amplitude and is the three translational and three rotational motions along and about the vehicle-fixed axis. The first term in this equation, called the radiation potential, corresponds to a rigid body forced to move by an external mechanism in still water. This effect corresponds to the added inertia effects discussed in Section 3.3 and Section 3.5.

The second term, the wave diffraction problem, represents the incident waves and their interaction with the rigid body. <j>0 is the incident wave potential, see Table B.l and fa is the scattering potential which represents the disturbance on the incoming wave train.

fa may be evaluated at one point if we assume D. A physical interpretation of this is that the pressure field of the incoming wave train does not change over the vehicle’s length. This approach is known as the Froude-Krylov hypothesis. The diffraction term in (3.45) may be divided into one Froude-Krylov term and one diffraction term. The exciting forces and moments can now for a completely submerged body be written as:

u(<t>)= pVqj, + + MAM,2Vf + Mam,3wj ;i = 1..3 (3.46)F-K FORCES DIFFRACTION FORCES

where ijj = [uf,Vf,Wf,0,0,0]T is the fluid particles’ velocities and V is the volume of the submerged body. Expressions for is found in Table B.l and the Mam matrix is given in (3.24). Written in vector form the Froude-Krylov and the diffraction forces may be expressed as

u{<f>) = Mwqf (3.47)where Mw is defined as

Mw =M i_3i_3 03x3

03x3 03><3 (3.48)

where Misa-s is the sum of the upper left 3x3 matrices of Mrs and Mam- Notice that (3.46) does not include any moments or centripetal and Corolis forces. This is because we have assumed that the pressure is constant over the vehicle’s surface, i.e. A D.

Potential damping has been neglected completely in this section. The potential damping is grouped with the rest of the dissipative forces and treated as one set of forces in Section


(3.6.1). Equation 3.46 is based on potential theory and does not account for viscous forces. These forces can be accounted for by adding a drag term to eq. 3.46.

3.9.4 Morrison’s equationMorrison’s equation is a semi-heuristic extension of 3.46 where the dissipative forces also are taken into account.

= pVCaqTi + ^pCD I | 9r, (3.49)

The added mass coefficient Ca and the drag coefficient Cd are empirically determined parameters which are dependent on many parameters e.g. the Keulegan-Carpenter number and the Reynold’s number. Fig. 3.14 contains some useful rules of thumb when Morrison’s equation is applicable and when the drag- and mass-forces are dominating.

3.9.5 CurrentCurrent is defined as slowly varying or constant movement of water.The surface current U is divided into the following components, Faltinsen (1990a):

U = Ut + Uv> + U. + Um + U,'t-up + Ud (3.50)

where Ut is the tidal component, Uw is the wind generated component, Ua is the Stoke’s drift, Um is the component from major ocean circulation, Uset-up is the component due to set-up phenomena and storm and Ud is the current component due to difference in the water density.

In control system design, the current is usually modeled as

U = Ut + Urest (3.51)

where the Ut component is considered known and the Ure,t is modeled as a stochastic process. A steady state current of 0.5y is not unusual in the North Sea.

3.9.6 State-space representation of sea currentThe current to ship transferfunction can be modelled as a constant or slowly varying process with states:

xcwxcN

WiU>2


Dragforces are dominatingNo waves possible

Massforces are dominatings' Reflection

and diffraction are important

large-volume small-volume

Mac-Camy and Fuchs Morrison’s equationtheory

Figure 3.14: H is the wave height, D is a characteristic diameter and A is the wave length.

where Wj is a zero mean white noise process and xcn and vcw are the inertial north and west component of the current. Vehicle-fixed velocities due to the current can now be found by using the rotation matrix given by (2.2) such that:

uc — apwi — srl>W2 vc = SljjWi + sV>U>2

uc and vc can be equated with white noise processes directly, when t/> is small.

The resulting expression for vehicle-fixed current disturbances should be used together with a state estimator to estimate the bias force from the current. See also subsection 4.9.1 which discuss estimation or adaption of current.

3.10. UNDERWATER VEHICLES - NONLINEAR EQUATIONS OF MOTION 57

3.10 Underwater Vehicles - Nonlinear Equations of Motion

We have now, in detail, derived each term in the equation below The terms are explained from an energy point of view and derived in the Lagrangian or Newtonian framework when appropriate.

Mq + C(q)q + D(q)q + g(x) = w(<f>) + b(q,u)

x = J(x)q

We can simplify this equation to

Mq + C(q)q + D(q)q - r

x = J(x)q (3.52)

where we have assumed still water, included the effect of gravity in r such that

t = b{q,u) - g(x). (3.53)

The gravity term is completely known and may be compensated for in an inner linearization feedback loop.

3.10.1 Alternate representationUp until now we have derived the equations of motion for an underwater vehicle from a physical point of view; however, another approach is also much in use. This approach uses a row expansion of the hydrodynamical forces on the vehicle. The number of terms used in the equations and the magnitude and sign of these terms have to be decided experimentally. The following equation for an underwater vehicle’s surge motion is included, Kalske (1989):

(m — Au)u — X0 + Xuu + Xuliu2 + Xuuuii3 + Xvvv2 + (XTTr2 + mxg)r2 +XsrsTSr2 + (XVTvr + m)vr + XvSrvSr

as a comparison with our approach. The notation is similar to the one used in SNAME (1950).


3.10.2 Hydrodynamical and hydrostatistical stability analysisHydrostatical analysis involves static stability analysis of the vehicle in a fluid. This stability analysis involves a surface stability, a transition stability, and a submerged stability analysis all of which use the vector CG and the metacenter height as their main object of study. The interested reader is referred to Allmendinger (1990).

Dynamic stability criteria include both nonlinear analysis and a linear treatment of the subject. The linear analysis is useful to investigate terms like position stability, straight- line stability and directional stability. An eigenvalue analysis may be useful in this context, see section 3.15. Nonlinear analysis of, e.g., stick-free and stick-fixed stability, can be performed by using for instance Lyapunov stability theory. As Fossen (1992) provides an excellent treatment of this subject, the results are not repeated in this text.

3.11 Equations of Motion Formulated in the Inertial Reference Frame

While it is most common to express the equations of motion in the q-frame, the x-frame formulation has some advantages. One is, while the quasi coordinates represented by the q vector have no physical interpretation, both the x and the x vectors are easily given a meaningful interpretation. This implies that the differentiation of position, with respect to time, yields velocity directly without going through the Jacobian matrix J(x). This will prove to have a great advantage in proofs of stability. Remember that we assume throughout this thesis that an appropriate nonsingular J(x) transformation matrix is used.

The transformation from the q-frame to the x-frame can be done by using the transformations given by (eq:transforml) on the set of equations given by (3.52). Another approach is to derive the equations directly in the x-frame formulation by using the kinetic energy of the system expressed in the x-frame. This kinetic energy is then given by:

Tx = ^xtMx(x)x

Mx(x) = J_T(*)M?J_1(a:) and the generalized coordinate vector* is a: = {x,y,z,<j>,0,ip)T. We can now formulate the Euler-Lagrange equations for the “rigid body-ambient water” system in the x-frame:

3.11. EQUATIONS OF MOTION FORMULATED IN THE INERTIAL REFERENCE FRAMl 59

which yields

tx = Mx(x)x + MT(x,x)x - ]--^-(xTMX(x)x) + Dx(x,x)xL OX

where

l-^-(iTMx(x)i) = -J-T(x)(Cq(i) + MgJ-\x)j(x)-Mq)j-\x)x

Mx(x, x)x = J-T{x)(Mq-2MqJ-\x)j{x))j-1{x)x (3.55)

andDx(x,x)x = = J~T (x)Dq(x)J~'i (x)x

Recall that Mq is 0. It is now useful to define the matrix Cx(x,x) as:

Cx(x,i) = ^Mx(x) + Nx(x,i) (3.56)

where Nx(x,x) is defined such that

1 1Nx(x, x)x = -Mx(x)x - - — (xtMx(x)x) (3.57)

Consequently, the underwater vehicle’s equations of motion expressed in the x-frame formulation now becomes:

Mx(x)x A Cx{x, x)x + Dx(x, x)x — tx (3.58)

It is easy to prove that Nx(x,x) is such that xTNX(x,x)x — 0 Vx(t). The physical interpretation of this is that the Nx(x,x) matrix represents the workless forces of the vehicle, see also section 4.5.3. It is also straightforward to show that both xT(Mx — 2Cx(x,x))x = 0 \/x(t) and qTCq(q)q = 0 Vq(t), Sagatun and Fossen (l991f) and section 3.12.2.

We will, in the rest of this thesis, assume that potential energy, i.e. the effect of gravity, is included in the tx vector such that

OXtx = r,dy_

dx (3.59)


3.12 Model PropertiesThis section derive some properties of the models given by (3.52) and (3.58). The first subsection derive properties for the model in vehicle-fixed reference frame, while the next section does the same for the inertial-fixed reference frame.

While some of the properties may seem trivial, all the proofs are derived independently of any other text, except where otherwise stated.

3.12.1 Properties of the q-frame formulationProperty 3.1.The inertia matrix M = Mrb + Mam is symmetric, constant and positive definite such that M = Mt > 0.

proof: Mrb is found from (3.19) to be:

MrbdiagaxaM -m[rGx]

m[rGx] /

where the elements in the inertia tensor I is given by Lj = p(r ■ rSij — rp-j) dV Mrb is clearly symmetric. Mam is also symmetric since = Tr { because 2 //= | //p<t>i^dS (Green’s second identity) , see also (3.6). The positive definiteness is found by the definition of kinetic energy, that is T is greater than zero for > 0.□

Remark 3.5.It should be noted that the symmetric property is only valid for still water or for a zero velocity. The elements in the M matrix will otherwise be velocity dependent and the matrix will in general not be symmetric, Faltinsen (1990a) p. 56.

Property 3.2.The product C(q)q may be parameterized such that C(q) becomes skew symmetric, that is CT + C = 0.

proof: The C(q)q vector can be parameterized to

C{q)q = (3.60)

3.12. MODEL PROPERTIES 61

Hence it is necessary to prove that An and A22 are skew symmetric or nil matrices and sufficient to prove that Au — —An to prove that C(q) can be parameterized such that C(q) is skew symmetric.The C(q) matrix is formed by the cross product terms in Kirchhoff’s equation such that

C(q)q =u x

3TS

dTsdV

97Vw x aw + ^ x atf + ^ x gu v x dv j

97V

Ti and Ts are expressions for the solid’s and the ambient water particle’s kinetic energy. The above expression for the C{q)q vector can be rewritten to

C{q)q =03x3

[(Hf+fSfH -V (3.61)

by recognizing the identity a x Ab = —Ab x o = — [>16x]a, where the matrix [Ab x] is skew symmetric. We observe that (3.61) is now on the form of (3.60). It is now trivial to see that Ai2——A^, A22 is skew symmetric and that An is the nil matrix. Hence we have proved that there exists a parameterization of C(q)q such that C(q) is skew symmetric.□

Remark 3.6.It is important to note that the skew symmetric O(q) matrix in (3.61) is not unique.Remark 3.7.Notice that all direct coupling terms among the linear velocities are removed with this parameterization.Remark 3.8.Another parameterization than (3.61) can be obtained when the rigid body’s and the added inertia’s kinetic energy are treated separately.

Proof: Below is a different parameterization of the C(q) matrix.- [(Mi2u> + x]

- [{^22^ + ff#) x]C{q)q =

[Mnwx]

+ A1\2V + 97VdV H

vU) (3.62)

where M\\ >s the 3x3 diagonal matrix of m, M\2 is the skew symmetric matrix formed by m[xcx] and M22 is the inertia tensor I. The identities a x (6 x c) = (a • c)b — (a ■ b)c and (acT)b — (a • b)c can be used to show that An^—A^.□


Property 3.3.The property qT (M, — 2C(g)) q = 0 is always true for any parameterization of C(q).q is the time derivative of the vehicle-fixed quasi coordinates.

proof: The property is proven for the x-frame formulation in property 3.7. The extension to the q-frame formulation is trivial by using (3.55) and (3.56).

Property 3.4.The dissipative damping matrix D(q) is positive definite that is D(q) > 0 Vq > 0.

proof: The rate of energy dissipation from a system is defined as ^ — —Ti where the dissipative forces working on the system is defined as /, = The energydissipation from the system can then be written as | qTDq > 0V<7>0 since Td>d if we, for argument’s sake, assume that the dissipative forces are linear in q. Hence, the property is proved. The line of reasoning is analogous if we assume another structure of the energy dissipation.□

Property 3.5.The dynamic equations for an underwater vehicle define a passive mapping between rand q written t q.

proof: Define the Lyapunov function V — qTMq. The function V is clearly lower bounded. By differentiating V with respect to time we get

V - qTMq = qTT - qTDq (3.63)

which prove that the system is passive.

Remark 3.9.The system is also dissipative (strictly passive since /“ qTDqdr > 0 for <7 ^ 0.□

Property 3.6.The thrust r is a nonlinear function of the control input u and a function of the velocitysuch that r = 6(u, q).

proof: See Section 3.8.4.□

3.12. MODEL PROPERTIES 63

3.12.2 Properties of the x-frame formulationThroughout this section we assume that the matrix J(x) is nonsingular. This can always be achieved by using an appropriate representation of orientation.

It is straightforward to extend properties 3.1, 3.4, 3.5 and 3.6 to the x-frame formulation by using the identities:

Mx = J-t(x)MJ-1(x) ;Dx = J-t(x)DJ~1(x)

We observe that the new M matrix is positive and symmetric since Af J = Mx and

xM~lx = xtJ(x)M~1Jt(x)x — yTM~1y >0 Vj/ > 0

The Dx matrix is still positive for the same reason as cited above. It is now clear that the positive definite properties and the symmetric properties are preserved by going from one coordinate system to another. Property 3.2. is not valid in the x-frame formulation while property 3.3. is.

Property 3.7.The expression: xT (Mx(x) - 2CI(x,i)) i = 0 is always true for any parameterization of C(q). x is the vector of generalized coordinates for the system.

proof: This proof follows the same line of reasoning as the one employed in Ortega and Spong (1988). The Euler-Lagrange equations for an solid moving in an ideal unbounded fluid can be written as an inertial Lagrangian system

_ d_dT _ dT T dt dx dx

where the potential energy, i.e. the effect of gravity, is neglected by using Avanzini’s Theorem, Birkhoff (i960), such that L(x,x) = T(x). The Euler-Lagrange equation describe n 2nd order equations. We can transform this to 2n first order equations the so-called Hamilton’s equation by defining the Hamiltonian H and using the Legendre Transformation such that

H = pTx - L(x,x)

p is the generalized momentum defined as p = J^. H is, on the other hand, found to be the kinetic energy such that

H = ^xtMx(x)x (3.64)


The 2n first order Hamilton’s equations are found by employing Lagrange’s equations on the Hamiltonian, which yields

dHX ' dpp = -|? + t (3.65)

The rate of change of the energy in the system defined by the Hamiltonian can now be found from (3.65):

dH_dt

dH . dH .----x -|------ pdx dp

■ TX T

If dissipative forces, like viscousus friction, are present, the right-hand side of the last equation may be subtracted by a positive term xTDx(x,x)x > 0 which accounts for this reduction in kinetic energy with respect to time due to these dissipative forces, thus

dH _ dt

(r - Dx{x,x)x) (3.66)

However, if we use (3.64), we obtain:

= xTMx(x)x+^xTMX(x)x — xtt-xt (^Mx(x) - Cx(i,x)j x-xTDx(x,x)x

(3.67)where the last transition uses (3.58). We now see that xT (jMx(x) — 2C'x(*,a;)) x is always zero by comparing (3.67) and (3.66).□

Remark 3.10.Observe the similarity between (3.63) and (3.66). This is because both equations express the rate the energy which dissipate from a mechanical system. The reference frame the system is expressed in does not affect the basic properties of the system.

Remark 3.11.A physical intepretation of (3.67) is that the terms xT (^Mx(x) — Cx(x, a;)) x represents workless forces.

Property 3.8.The eigenvalues of the inertial-fixed linearized system M~lDx are the same as the ones of the linearized vehicle-fixed system M~xDg.

3.13. EXPERIMENTAL DETERMINATION OF ADDED INERTIA AND DAMPING 65

proof: The MX^DX matrix product can be written as:

M-?Ds = J(x)M?DqJ-\x)

Hence we see that Dx is a similarity transformation of the corresponding matrices formulated in the vehicle-fixed reference frame. This transformation preserve the eigenvalue properties of the system. The rest of the proof follows the exact same line of reasoning as property 3.6.□

Remark 3.12.That the eigenvalue properties are the same in both reference frames should not come as a surprise to the reader, since the stability properties of a mechanical system are inherently a function of the physical system and not the reference frame in which it is represented.

3.13 Experimental Determination of Added Inertia and Damping

This section contains the results of free-decay tests performed on the NEROV. The added inertia and damping coefficients’ dependency on the KC number and frequency are presented. Sensitivity to perturbations of the NEROV’s geometry is also investigated. The section contains a discussion of the use of both linear and quadratic drag terms. Off- diagonal coupling terms are also discussed.

3.13.1 Experimental setup

Fig. 3.16 shows the experimental setup. The vehicle’s mass in air and deplacement are known in advance. The vehicle is suspended under water by a spring with a known spring constant (k — ^). The spring is hanged from a force ring (a force sensor). The vehicle is moved out of its equilibrium point and then released. The extension of the spring is measured with a force ring with a known force constant (c = •^). The movement, i.e. the decaying behavior of the vehicle, is then calculated by measuring the voltage over the force ring. This signal is passed through a fourth-order analog Butterworth filter with a bandwidth of 4Hz and a signal amplifier before it is digitalized and stored on a PC. Notice that the vehicle can move in only the desired direction. This implies that the added mass and damping terms are found for one degree of freedom at the time.Fig. 3.15 shows a typical decay.


a typical decay

t

Figure 3.15: A typical decay

3.13.2 Theory

This section is based on the theory presented in Lehn (1990).The equation of motion for the experiment can be expressed as

mxi + Biii + B2 | ii | x-i + cXi — 0 (3.68)

where c is the spring constant used in the free decay test and m denotes the total mass. Dividing on both sides with m yields

Xi + piii + p2 | ii | ii + pzXi - 0 (3.69)

This equation can, in each cycle of the oscillation, be linearized to

Xi + pii + p3x, - 0

where p = jtq + P denotes the total linear damping coefficient per unit mass. Thefactor Vpp- is the equivalent velocity found from the linearization of the quadratic damping term, X is the amplitude of the oscillation and T is the period. The energy in this linearized equation is the same as the energy in (3.69) in each oscillation cycle.


Figure 3.16: Experimental setup.

The parameters in (3.68) can be made dimensionless such that

hi = SBitt ;&

2

andbo =

2pAD

B2\pA

The dimensionless added mass coefficient is found to be^(1 - fc2) - M_ P32_____ '____

pV

where k = pCT — 2u>0 and lo0 ~ . Z) is a characteristic height in the directionof oscillation (1m for the NEROV), /f is a characteristic cross flow area (0.64m2 for the


NEROV), g is acceleration of gravity, c is the spring constant, p is the density of water, V is a characteristic volume (0.182m3 for the NEROV), M is equivalent oscillatoric mass in air (183% for the NEROV) and m is total oscillatory mass.

A least square approach is used when fitting the decay curve to (3.69).

Fig. 3.17 justifies the use of a linear damping term. The figure shows the total linear damping coefficient per unit mass plotted versus the equivalent velocity.

0.16

0.14

DO

I 0.1

■O| 0.08

0.06

0.04

o.o2-.... ............................... i....... ........ ..... i....... ;..pi

ol- - - - - - - ;- - - - - - - - i- - - - - - - i- - - - - - - - i- - - - - - - i- - - - - - - i- - - - - - - 1- - - - - - - i- - - - - - - -0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

equvivalent velocity

Figure 3.17: The total Imear damping coefficient per unit mass p = pi + P2'^' plotted as a function of the equivalent velocity

3.13.3 Added mass and damping as a function of frequencyThe free decay test allows us to calculate the added mass coefficients and the linear and quadratic damping terms for the vehicle’s x-, y- and z-direction. We performed the test for three frequencies (« 1.3, « 0.8 and « 0.4 ~). We also investigated how sensitive


these parameters are to perturbations of the vehicle’s geometry. We performed the test for two different perturbation objects for all three frequencies. The largest perturbation object was a second battery-container with a deplacement of 0.042 m3 and the small object was four cylinders, simulating a manipulator, with a deplacement of 0.009 m3. See also Fig. 3.18.

The results from these tests are shown in Appendix D and Table 3.1. Notice how little the parameters change when the geometry is perturbed. This may be explained by noticing that the NEROV can be considered to be a bluff body such that we have an almost immediate boundary-layer separation when the streamlines hit the vehicle.

small pertubation V= 0.009 m3

NEROV computersensor

large pertubation V = 0.042 m3

Figure 3.18: The two geometrical perturbation objects .

3.13.4 Added mass and damping as a function of the KC number

We investigated how the added mass parameters and damping terms changed with respect to the KC number, (KC — = 27t-^, Um is maximum speed in an oscillation period,T is the period, X is the amplitude and £) is a characteristic length). Fig. 3.19 shows


no pert. small pert. large pert.

axis CA bi b2 CA bi b2 CA 6i b2

x-axis 0.9 0.04 2.7 0.9 0.07 2.9 0.9 0.09 2.4y-axis 1.3 0.11 1.9 1.3 0.07 3.0 1.2 0.07 3.1z-axis 1.2 0.05 3.2 1.2 0.04 3.6 1.3 0.05 3.6

Table 3.1: Nondimensional added mass coefficients and linear and quadratic damping terms for the NEROV for an oscillation period of k \ l s and KC ~ 1.

very strong dependencies between the vehicle’s hydrodynamical coefficients and the KC number. The KC number indicates the relative importance of drag and inertia forces. The numbers for high KC values correspond with experimentally obtained data for the BP’s DISP vehicle where the corresponding values where Cm — 3.5 and Ca = 1.1 ,Parfitt and Watters (1990). The difference is mainly due to the fact that the DISPS vehicle Moore et al. (1990) is more compact relative to its surface area.Notice how the KC number may be used to characterize the vehicle’s cruise condition. A constant velocity corresponds to a infinite KC number, while an ROY or an A1JV in a positioning mode correspond to a very small KC. Fig 3.20 shows the relationship between the Ca and the equivalent quadratic drag coefficient Cdd defined as Cdd — t ....

The relationship between the Ca and the Cdd agrees with observations made by Sarpkaya (1981) pp. 102-103 where the same dependency is observed for some higer KC numbers, KC > 10.

3.13.5 Off-diagonal coupling termsVery small coupling effects were observed during the tests. It is also very difficult to measure these effects with the experimental setup that was used. We did witness some small coupling effects between the sway and yaw and heave and pitch. It is, however, difficult to find out if these effects are acceleration or velocity dependent or effects due to an unbalanced vehicle.

3.14 Linearized Equations of Motion

3.14.1 Linearization about constant velocity q = q0

(3.1) can be written asMv = f(v,u)

3.14. LINEARIZED EQUATIONS OF MOTION 71

KC

0.85

0.75

0.65

Figure 3.19: The equivalent quadratic damping coefficient Cdd and the added mass coefficient Ca (both in x-direction) plotted as a function of the Keulegan-Carpenter number, KC = where X is the amplitude of oscillation and D is a characteristic dimensionfor the vehicle.

Neglecting higher order terms this equation can be linearized around the nominal point f(v0, u0) as follow:

Mv0 + MAv = f(v0, u0) + dJÂ[v _ _ jov ou

Defining An — (v — v0) and Aw = (w — u0) and recognizing that f(v0,u0) = Mv we obtain:

MAv -OV ou

ifAn = AAv + BAu (3.70)

where the A and B matrices are M~l dfivû°) ancl M-1 respectively. Milliken(1984) contains the complete equations of motion of a submarine linearized around the velocity n0 = [u0,v0,w0,p0,q0,r0]T.

3.14.2 Linearization about zero velocity g=0This section contains a linear state-space model of an underwater vehicle linearized about zero velocity. This requires that the C(q) matrix is set to zero, that the velocity transformation matrix J(x) is the identity matrix, and that we assume diagonal added inertia


Figure 3.20: Ca versus Ca for a KC value between 1 and 0.1

and damp damping matrices. Calculations made in Humphreys and Watkinson (1982) and observations documented in Sagatun and Fossen (l991d) indicate that these requirements are not harsh for symmetric vehicles with small velocities. We have further utilized only linear damping terms. That is the quadratic damping terms are made linear so that the error of the linearization is minimized in a ball around origin , see Fig. 3.21. This linearizations for the surge, heave, and sway degrees of freedom are done with respect to the energy in each cycle period in the added mass analysis, Lehn (1990). The linear damping terms for the roll, pitch, and yaw motion are found by studies of similar vehicles, observations made of the NEROV in water and engineering guesses.The resulting linear D matrix becomes:

Dl = -diag(82,156,88,18,30,30)

where the dj terms is the sum of the above linearization and the linear drag terms.

3.14.3 A state space modelThe state variables

We have assumed small velocities and small angles such that vehicle-fixed velocities and accelerations are the same as the corresponding inertia-fixed vectors, that is the transfer-


where JBi is

S, =

£>i b! 0

0 0

0 0 00 0 • 40 0 0

'i • h —bi ■ 4 0

0 0 00 0

0 &! 6,0 0 00 —bl'ly bl • ly

0 0 0The &i and lj terms are described in section 3.8.4.

(3.72)

3.14.4 Properties of the NEROV state-space modelThe system described by (3.71) and (3.72) is both observable and controllable. Note also that Bi is invertible.

3.14.5 Eigenvalue analysisBelow is plots of the eigenvalues of the NEROV vehicle for surge, heave, and sway plotted as functions of the KC number. The eigenvalues are determined on basis on the free-decay test documented earlier in this thesis. This implies that we have neglected all off-diagonal terms in both the damping and mass matrix. Hence, the eigenvalues become A; = n+AM • We observe that the mass-damping dependent eigenvalues vary with a factor of more than 5 for each degree of freedom. The integral terms are, of course, unchanged.

lm

-0.1 . .-°A .Re *** r

KG= 0.9 ( 0.04J if l i I

Iz

Figure 3.22: The eigenvalues of the NEROV in surge sway and heave plotted as functions of the KC number.


[N] or [Nm]

(m/s] or [1/s]

Figure 3.21: Quasi-linearization of the quadratic damping terms.

mation matrix J{x) = lexe- Hence the state variables z € 9?2n becomes:

_ _ r-T _TiT2 = 19 ? * J

where q £ dln defined as qr = [it, v, w, p, q, r]T is the time derivative of the quasi coordinate vector in the vehicle-fixed reference frame and at £ defined as x = [i, y, z, <}>, 9, ip}T is earth-fixed positions and Euler angles.

The A, B, C and D matrices

The 4-tuple (A, B, C, D) of real matrices which is a realization of the linearized system of(3.1) and which-are related to the transfer function H{s) — C (si — A)~l B-\-D between input signal | n | n (3.34) and output z is found to be:

A = —M~XD 0nxnInxn Onxn

C = I2nx2n

Onxn 2nxnB = D = 0

(3.71)


Ax = -0.25 0

Ay = -0.31 0

Az = -0.20 0

Table 3.2: The open-loop eigenvalues for surge, sway, and heave motion for KC

The damping and added inertia are not determined experimentally for the roll, pitch, and yaw degrees of freedom, but observations of the NEROV vehicle in water and by comparisons with similar vehicles, e.g. the MSEL’s EAVE vehicle Humphreys and Watkinson (1982) and MSEL (1988), allow us to make engineering guesses for KC ~ 1, see Table 3.3: The corresponding poles for the roll motion for the the DRAPER’S UUV are 0.074±1.102i

XK = -0.11+0.711 -0.ll-0.71iAm = -0.15+0.79i -0.15-0.79iAjv — -0.33 0

Table 3.3: Estimated open-loop eigenvalues for roll, pitch and yaw motion for KC ^ l

Koenig (1991). The complex conjugated poles for the roll and pitch motion are found by looking on the differential equations:

(Iy - Mi)9 - Mg6+ BG2pgV9 = 0

(Ix-Kr)d>-Kp4> + BGzPgV<f> = 0 (3.73)

where BGZ is the distance between the center of gravity and center of buoyancy. lx, Iy and Iz are 25[kgm2}, 29[A;<7m2] and 28[kgm2}, respectively. The current NEROV has a BGZ — 0.03[m] and a V = 0.183[m3]. The added inertia coefficients Caj are estimated to be approximately 2.5 about the x, y and z axes. If we now plot all eigenvalues for high KC values we obtain the typical plot of the open loop poles: Each equation in (3.73) may be rewritten to

A2 + 2(wA + ui2 = 0It is now easy to see that the natural frequency and damping ratio for the roll and pitch motions becomes:

BGzpgV\| /x - Kp C* = -Kv

2yfBGzpgV (Ix - Kj)U4, =


x X

• oa • 0.2 • o.i. \/W \/Mf 1 S

bn

-0.7

4*

" JRt

x x - -0.7$

Figure 3.23: Open-loop poles for the NEROV vehicle for high KC values.

U!$ =BGzpgVIy Mq Ce = -M.

2^BGzPgV(Iy-M4)

Hence, the natural frequencies and damping ratios for the NEROV is 0.72 [i], 0.80 [^] and 0.15 [—], 0.19 [—] for roll and pitch respectively.

3.15 Summary of Chapter 3.This section has successfully applied Lagrangian mechanics to derive the equations of motion for an underwater vehicle. The equations are first derived in the common vehicle-fixed reference frame before they are derived in an inertial reference frame. This latter model will be very useful in the next chapter. The vehicle, with its ambient water, is treated in a common framework where the added inertia is given a clear and physical explanation. Restoring forces and the dissipative forces are also explained from a Lagrangian point of view before a Newtonian approach is used to derive the specific terms. Propulsion forces, wave loads, and experimental results from both open water tests of the NEROV thruster system as well as a free decay test of the NEROV are discussed. A decoupled linearized model of the NEROV is also presented with experimentally obtained data.

The Lagrangian approach is especially useful when properties of the equations of motions are derived. The use of energy as the leading factor in the discussion will also be of great importance in the next section where stability is proved by Lyapunov theory.

3.15. SUMMARY OF CHAPTER 3. 77

This chapter has clearly shown that the equations of motion for an underwater vehicle is time-varying (a function of the Reynold’s number and the Keulegan-Carpenter number) and nonlinear. This is a strong motivation to derive nonlinear and adaptive control algorithms. The next chapter will discuss this topic in detail.


Chapter 4

Optimal Control Algorithms for Underwater Vehicles

4.1 IntroductionThis chapter presents an optimal and a suboptimal adaptive control algorithm for small underwater vehicles. The control schemes are minimum effort controllers which result in (sub)optimal feedback control laws.

The chapter starts with a definition of the problem of controlling a small underwater vehicle. Differences between controlling surface vessels and small underwater vehicles are pointed out. A brief review of existing guidance and control systems for both surface ships and underwater vehicles are presented. Design requirements and basic assumptions for the derivation of the control algorithms are then discussed before the two control schemes are derived in detail. The final section of this chapter comes with some implementational remarks.

4.2 Problem StatementBalchen (1991) contains the following specification for a DP system:

A good system for DP should keep a vessel within specified position and heading limits with minimal fuel consumption and wear on the propulsion equipment. Furthermore, the system should tolerate transient failures in the measurement and propulsion systems.

This specification may be applied to all control and guidance systems for marine vehicles if “position and heading” were exchanged with “trajectory” in the above quote. There are

79

80 CHAPTER 4. OPTIMAL CONTROL ALGORITHMS FOR UNDERWATER VEHK

two extreme operation modes, with respect to the KC number, in control and guidance of marine vessels. One is a DP system which operates in the extreme lower KC number range and secondly we have the course-keeping modus for an autopilot which operates with infinite KC number. A high performance guidance system which is supposed to control a vehicle in both modes must be able to account for these two very different situations, either explicitly, such as gain scheduling as a function of the KC number, or implicitly as a direct adaptive control scheme. Furthermore, the controller should be optimal with respect to some minimum effort criterion and tolerant to system failures, e.g. failure in the sensor or propulsion system.

4.2.1 Small underwater vehicles - a special caseMost automatic control and guidance systems for marine vehicles are developed for surface vessels. We will mainly focus on automatic control systems for small unmanned underwater vehicles which may be regarded as a special case of the larger class of vehicles, i.e. marine vehicles. Automatic control systems for small unmanned underwater vehicles have several distinct features which separate them from traditional marine control and guidance systems. The most important one is:

It is desirable to control a small underwater vehicle with relatively high velocity along or about two or more axis at the same time. This leads to stronger coupling, larger nonlinearities, more states in the equations and more unknown parameters in the vehicle’s equations of motion. This fact is the single most important item which makes it so much more difficult to control small underwater vehicles compared to surface vessels.

Other factors are:

• While most surface vessels are controllable in maximum 3 dofs, a small underwater vehicle may be controllable in 6 dofs.

• A small underwater vehicles natural frequency is usually one order of magnitudehigher than most surface vessels. Most ships have their natural frequencies in the range of | ^ [Hz] and semisubmersibles usually have their natural periods above^5 [Hz]. The NEROV, on the other side, has its natural frequencies at 0.7 —0.9 [Hz].

• The actuator dynamics on a small underwater vehicle are much faster than on most surface vessels. The bandwidth on the NEROV’s thrusters is 2 — 3[//z], while the bandwidth on the rudder of a merchantman may be in the range of 0.01 — QA[Hz].

• Most autopilots and DP systems contain one high- and one low- frequency model where the control actions usually are determined on basis on the low-frequency

4.3. A BRIEF REVIEW OF EXISTING SYSTEMS 81

model. The high-frequency model is used to filter out the relatively high-frequency oscillatoric wave motion from the sensor data. Small underwater vehicles have a much higher bandwidth and may compensate for the first order wave disturbances by means of feedforward and feedback control laws. Hence, the usefulness of dividing the system model into one high- and one low-frequency part is not present.

4.3 A Brief Review of Existing Systems

I have restricted my discussion to including only control schemes which are actually implemented on full-scale vehicles.

4.3.1 Dynamic positioning systems

OD (1983) defines a dynamic positioning system as:

All equipment and components involved in retaining the vessel in its required position.

This definition will be used in the coming brief review of DP systems.

Conventional DP systems

The first modern DP system was first introduced in 1976 by Balchen et al. (1976) where a Kalman filter approach for solving the wave noise filtering problem was employed, see Fig. 4.1. DP systems, before this new generation, were mainly based on PID-controllers and matching notch-filters. Kalman filter based systems are now dominating the DP marked. Relevant references for Kalman filter based DP systems are Balchen et al. (1976), Jenssen (1980), Balchen et al. (1980a), J. G. Balchen et al. (1980b), Saelid and Jenssen (1983a), Saelid et al. (1983b), Fung and Grimble (1983a) and Triantafyllou et al. (1983). A selftuning based DP system is found in Fung and Grimble (1981).

As mentioned earlier, the Kalman filter based DP systems usually consist of one high- and one low-frequency model where the control actions usually are determined on basis on the low frequency model. The high frequency model is used to filter out the relatively high frequency ocsillatoric wave motion from the sensor data. This would not be not necessary for small underwater vehicles if it were possible to compensate for the first-order wave disturbances, at least the waves with longest wavelength.


thrust

setpoint

thrustercalc.

highfrequencymodel

currentmodel

lowfrequencymodel

model

physicalsystem

Figure 4.1: Block diagram showing the Kalman filter based DP system o/Balchen et al. (1980a)

DP systems for small underwater vehicles

Although there exist several commercial available DP systems for surface vessels, for instance Simrad-Albatross (1990), the author is aware of only one general purpose and commercially available DP system for underwater vehicles is presently on the market. That is the Marquest’s ROV-DP system , Marquest (1991), which is based on a model- based Kalman filter to produce smooth estimates of the vehicle’s position and velocity and PID controller to control the vehicle. The sensor system consists of a hydroacoustic positioning system, two transeivers mounted on the vehicle and an onboard heading rate sensor.

Most ROVs designated for off-shore use are, however, equipped with functions like auto heading and auto depth. Not much information is available on how they are implemented, but they are usually based on measurements from a pressure meter and a magnetic com

4.3. A BRIEF REVIEW OF EXISTING SYSTEMS 83

pass.

There exist few articles which explicitly discuss DP of small underwater vehicles. Relevant references on the topic of control of underwater vehicles, in general, will be given in the next paragraph.

4.3.2 AutopilotsAutopilots or automatic steering machines can be defined as:

An autopilot is a device which automatically steer a vehicle from point A to point B in an optimal manner.

Conventional autopilots

While it is customary in DP system to explicitly model the relatively high frequency motions induced by wave disturbances, this is less common in autopilots. The suppression of high-frequency motions induced by the wave forces are commonly done implicitly by including states like the rudder angle, the rudder angle velocity or states which are function of both in a quadratic optimization criterion, see e.g. Norrbin (1970) and Kallstrom et al. (1979). Amerongen (1984) contains an adaptive Kalman filter where the gain is calculated as a function of some heuristically estimated high- and low-frequency component of the rate of the heading error.

As for DP systems there exist numerous commercial available autopilots for surface vessels, e.g. Simrad-Albatross (1990) and Robertson (1991). Several different control schemes are tested on full scale vessels like self-tuning based algorithms Mort and Linkens (l98l)and Kallstrom et al. (1979), Model Reference based control schemes Amerongen and ten Cate (1975), Amerongen (1981) and Amerongen (1984), Kalman-filter based systems Kallstrom et al. (1979) and autopilots based on optimal theory Ohtsu et al. (1979) and Burns (1990).

Autopilots for small underwater vehicles

There is little difference between a DP system and a autopilot system for small underwater vehicles, since it is possible to compensate for the first wave disturbances. The DP case becomes a special case of the general control problem where the desired trajectory’s velocities are zero. Therefore, both kinds of control systems are treated in this section.

The first implementation of a control and guidance system on a small unmanned underwater vehicle is the SISO PID-based regulators on MSEL’s EAVE vehicles, Venkatachalam

84 CHAPTER 4. OPTIMAL CONTROL ALGORITHMS FOR UNDERWATER VEHh

et al. (1985) and Franz and Limbert (1987). The position feedback is obtained from a long base line system (LBL) a depth sensor and a magnetic compass. The vehicles are consequently restricted to operate in the area covered by the LBL system. Yoerger et al. (1986) describe the success with which SISO sliding mode controllers have been implemented on the Jason ROV. The state feedback is obtained in the same way as for the EAVE vehicle. Recursive parameter identification based control schemes have been successfully implemented in Goheen (1986). Dougherty et al. (1998) and Dougherty and Woolweaver (1990) show how Martin Marietta’s MUST vehicle is controlled by SISO sliding mode controllers. Feedback is obtained from an angular rate sensor, inclinometers, a magnetic compass, a depth sensor and a water relative forward speed sensor. They also plan to use the SHARP hydroacoustic positioning system Marquest (1991) to obtain position feedback. LQ-based methods have been implemented to control the Naval Postgraduate’s AUV, the NPS AUV-1, in the dive plane, Healey et al. (1989) while Koenig (1991) has implemented a multivariable LQG/LTR controller for headway control of DRAPER’S UUV (Unmanned Underwater Vehicle). The sensor system of both the two latter implementations are equal to the MUST vehicle’s sensor system. Lastly Healey and Marco (1992) describe a hybrid nonlinear controller where the steering system states, speed control states and diving system states are controlled separately by means of single input multiple-state (SIMO) modified sliding controllers. The vehicle is the NPS AUV-2 which has a sensor suite of a 3-axis rate gyro, a paddle-wheel speed sensor, a pressure-cell depth sensor and 4 sonar range sensors. Even though it does not report from any actual implementations, we would like to mention Fossen (1991c) which contains an extensive review of nonlinear control schemes for underwater vehicles.

It is worth noting that all the systems mentioned above are either SISO systems or control schemes for reduced order models of the vehicle, e.g. for headway control or controller for the vehicle’s dive plane.

4.4 Design RequirementsThere are numerous factors which have to be considered in the design of a control system for implementation onboard a commercial ROV. Some of the most important ones are:

• The bandwidth is usually limited by the control system’s sampling rate, time delay in the communication channel or structural resonance. The two former factors are most important for the case of underwater vehicles since most of the today’s ROVs’ structures are very rigid. The time delay factor is especially important when the control law is implemented outside the vehicle, thus the measured signals have to be sent to the surface through an umbilical or by means of a hydroacoustic link.

4.5. OPTIMALIZATION WITH RESPECT TO MINIMUM CONTROL EFFORT 85

• The control system’s sensitivity to noise has to be considered, because it determines how good a sensor system is and how robust the control algorithm is with respect to noisy measurements.

• Overshoot is an important property for control algorithms for moving vehicles. An overdamped system is usually desirable, but some overshoot may be tolerable if this is taken into account when the trajectory is planned.

• The desired accuracy of the system is usually a function of the commanded tasks, hence it is not meaningful to discuss this factor before a good task specification is made.

• Stability is also of utmost importance. For linear SISO systems, stability or phase margins are common criteria, singular values analysis may turn out to be useful for MIMO linear systems, while terms like asymptotically stability and exponentially stability are especially useful for nonlinear systems. The magnitude of the differentiated normalized Lyapunov function V may also be a useful criterion for such systems.

• Fail-safe requirements involve requirements on graceful handling of system failure, e.g. sensor failure, actuator failure and even computer failure , DnV (1990). These requirements are typically operational requirements which have to be met during the engineering phase of the system’s implementation.

• Parameter variation is also an important aspect of the robustness analysis of the control system. Parameter variations can be taken care of by well-designed adaptive algorithms or by employing a control scheme which is robust with respect to these variations. For linear systems, singular value or fi analysis may turn out to be useful, while an analysis of the behavior of the differentiated Lyapunov function candidate V for all possible perturbations of the nominal system is a common for nonlinear system.

While all the factors listed above are important in a real implementation of a control system, this thesis will only in detail discuss factors like stability and parameter variation. The other items will be commented on briefly when appropriate.

4.5 Optimalization with Respect to Minimum Control Effort

Optimization is in Gay et al. (1984) defined as: ”To make as perfect, effectiveor functional as possible”. Formally speaking, the optimal-control problem is to find the optimal control


«* which minimizes a given performance functional J(u). This functional is usually written as:

J(z,u) = S(z(t/)) + f ’ L(z(t),u{t)) dr (4.1)JtQ

where z is the system state we wish to control. This system will only discuss so-called Lagrangian system, i.e. S(z(tf)) = 0. It is well known from the literature, e.g. Athans and Falb (1966) or Balchen et al. (1978) , that the optimal control «*(t) for the state space we want to control with the performance measure in (4.1) is found by solving the following equation:

dJ'(z,t) . “ft L{z(t),ux(t)) + dJ*(z,t)T,dz 0 for Vt €[*„,</) (4.2)

The problem is now converted to finding the function J*(z, t) which is called the Hamilton principle function of optimization. Eq. (4.2) is called the Hamilton-Jacobi equation and the terms between the parentheses in (4.2) is the Hamiltonian H. The analytical solution to this equation is, however, often difficult if not impossible for a nonlinear system. There exists no general algorithm to obtain a closed-form solution to the nonlinear and multivariable case or if the cost functional is not quadratic in z and u.

A natural question is: With respect to what do we want to optimize the control system ? The most traditional optimization criterion for linear systems is the linear quadratic (LQ) criterion where the Lagrangian is on the form

L(z(t),u(t)) = ^zT(t)Qz(t) -f ûT(t)Ru(t)

where R > 0 and Q > 0 are weighting matrices, u(t) is the control input and z is the state errors to be minimized. This section will, however, only consider minimum effort controllers where either the Lagrangian is only a function of u(t) or on the above form, but where u(t) is a cost function representing energy or fuel. The result of a solution to fuel- or energy-optimal control of a nonlinear system, where a solution is found, is almost always open-loop and for the minimum fuel case often on the “bang-off-bang” or “bang-bang”, where the “bang” represents a switching term Kirk (1970) p. 290. Relevant references to this section are e.g. Athans and Falb (1966), Lee and Markus (1967) or Kirk (1970).

4.5.1 Minimum fuelThe minimum fuel control problems usually assume that the state equations of the system are in the form

x(t) = A(x,t)x + B(x,t)u(t)


with a performance measure to be minimized on the form

r(/ f mJ(“) = / 53 I ftui(f) dr

where /3, is a nonnegative weighting factor, Kirk (1970). This control problem is very difficult to solve on a closed form for the multivariable nonlinear case. Athans and Canon (1964) include the solution to the minimum fuel problem for the second order nonlinear differential equation:

y(<) + ay(t) | y(t) |= u(t)Only an open-loop control law is presented.

4.5.2 Minimum energyThe form on the state space representation for minimum energy problems are usually the same as for minimum fuel problems, but with the following performance measure to be minimized, Kirk (1970):

riuf(t) dr

where r,- is a nonnegative weightning factor. Spangelo and Egeland (1992) have showed that the performance index

J{u) = fJt0

Ei=i

Ui(t) dr

is an approximation to the minimum control energy problem for an underwater vehicle with electric DC motors. They assume that the power dissipation due to the interaction between propeller and water is the dominating term and that each degree of freedom is controlled independently. However, they only present an open-loop control law which was obtained numerically by using a conjugate-gradient method.

4.5.3 Other minimum control effort criteriaLee and Chen (1983)

Lee and Chen (1983) present a suboptimal controller for a mechanical manipulator which consists of a feedback linearization term and one optimal feedback term where the following performance index is used:

rocJ(u) = / xt(t)Qx(t) -I- uT(r)Ru(T)dT


The state variable x is defined as x = [pT, qT]T, where p is the generalized momenta defined as p = M(q — qref)q, where q and qTCj are the actual and desired link positions and q is the corresponding angular velocities. The control variable u is:

dUi~Jt £ _ Qre/Hk

J:=lThe condition </ = oo is used to simplify the Hamilton-Jacobi equations and the optimal solution is a closed-loop control law on closed form.

Johansson (1990a)

Johansson (1990a) presents an adaptive and optimal controller for a mechanical manipulator which minimizes the tracking errors and the forces which are needed to correct these errors. Only the forces which contribute to the manipulator’s kinetic energy are minimized. There is no point in optimizing the forces that contributes to the potential energy since the potential energy is end-point dependent only. We will, therefore, use the expressions for r given by (3.53) or (3.59) where the forces due to potential energy are subtracted from the vector r. The work done by the system when subtracting the potential energy is

W = f rTq dt Jto

where the torque r is found from the n 2nd order Euler-Lagrange equations (3.54) for the manipulator system such that

r = M{q)q + (9, q)q + N(q, q)q

where q is the joint angles. The work due to the manipulator’s kinetic energy can now be formulated as:

W = J/ iT (M(9)9 + ^(9,9)9) dt

Johansson suggests the following control variable to be minimized

u = M(q)Tii + | Af (9,9)T,i (4.3)

where the vector z is defined as 2 = [(9 — 9r)T! (9 ~ 9r)T]T and the matrix T\ € 3?nx2" will be defined in the next section. Hence, he use the following performance index:

J{z,u) = f 1 }-zT(t)Qz(t) -(- ûT(t)Ru(t)dt Jto l J

where R> 0 and Q >0 are weighting matrices.

(4.4)


Sagatun and Johansson (1992)

Sagatun and Johansson (1992) contains an adaptive and optimal controller for underwater vehicles. The article presents several modifications of the control scheme introduced by Johansson (1990a). One of these is the performance measure which in addition to the kinetic energy also includes the energy which dissipate away from the vehicle due to damping effects from the water, e.g. viscous friction. The work done by the system when subtracting potential energy becomes:

W, = It’ ^ = / / *T + JV,* + Dxi^ dt

The term xTNxx is evaluated to zero which implies that the term corresponds to the workless forces of the system. Johansson (1990b) suggests the control variable in (4.3) to be minimized, while a more natural choice for underwater robots is:

u = M^z + (Dx + ^Mx)Tii (4.5)

' Tj ' ' Tn T12. T2 . Onxn ■^nXn

since the dissipative effect (i.e. viscous damping) is very important for marine vehicles, ti € 3?” and Ti is the upper n x2n matrix of T0 defined as

T„ —

The introduction of the Ta matrix results in a non-physical interpretation of the u vector since the matrices now are multiplied with a linear combination of acceleration and velocity, and velocity and position. An advantage with the use of the Ta matrix is that the new control variable also is a function of the position. This becomes more clearer in section 4.8.4. Sagatun and Johansson (1992) present two different Lagrangians. Both will be discussed in detail in the next section.

Sagatun (1992)

Sagatun (1992) derives an adaptive and near-optimal control algorithm. The algorithm is near optimal only in velocity control. This control scheme also minimizes the forces which contribute to the kinetic energy but uses the vehicle-fixed reference frame in the derivation of the algorithm. The work done by the system when the potential energy is subtracted from the force ,(3.53) is in the q-frame formulation:

Wa = f S rTq dt = f , qT (Mq + D(q)q) dtJto Jto

(4.6)

90 CHAPTER 4. OPTIMAL CONTROL ALGORITHMS FOR UNDERWATER VEHIC

since qTC(q)q = 0. The control variable to be minimized is simply, Sagatun (1992):

u = Mq + D{q)q (4.7)

with the performance index. too . t . _

J{q)= I 9 Qq + uTRud,T (4.8)Jto

where R> 0 and Q > 0 are weighting matrices and </ = oo. Remark that we only minimize velocity tracking errors and the forces which correct the deviations from the desired velocities. Hence, the method optimize the forces which correspond to the disspation of energy and kinetic energy which are necessary to employ to correct the deviations from the desired velocities, see also Section 1.2.

4.6 Basic AssumptionsThe following assumptions are made in the derivation of the OpAC and NOpAC control schemes. Assumption A8 is relaxed when the adaptive controllers are derived.

Al: The motion is governed by the equations (3.52) (q-plane formulation) or (3.58) (x- frame formulation).

A2: The reference trajectory is smooth, bounded and within kinematic and physical limits. This can be achieved by assuming that the desired reference trajectory is generated by the strictly stable reference model

Xr + KdXr + KpXr = Kpxd (4.9)

The n X n matrices Kd and Kp are define such that xr, xr and xr are within the physical limits of the vehicle. This can be taken care of by employing the reference model adjustment (RMA) technique presented in Amerongen (1982), utilized in Fjellstad et al. (1992) and described in subsection 4.9.2. Physical limited accelerations and velocities imply that xT, xr and xr are € L°° and xr € C1. The transformation in (2.4) is used when a vehicle-fixed reference trajectory is used. The following relation is used when we operate with only velocity control:

qr + Kdqr = Kdqd (4.10)

A3: The state variables we use in the OpAC control scheme are defined by the vector z = [xT,xT]T and the tracking error vector z = [(i — xt)t,(x — zr)T]T, z 6 3J2”.

4.7. THE NOP AC CONTROL SCHEME 91

It will also be useful to define the vector zT = [xJ,xJ]T. This implies that the z vector becomes z = [(se — xt)t, (x — xr)TY■ The state variables for the NOpAC control scheme are y = [qT, xT]T, since the velocities are decomposed in the vehicle fixed reference frame. The yT and y vectors are altered accordingly.

A4: The control variable to be minimized is given by (4.5) or (4.7).

A5: All states are measured, that is vehicle fixed velocities q and inertial positions and orientations x.

A6: In the discussion of the OpAC controller, the y vector is reffered to as the measurement vector, see also A5. A useful transformation in the coming discussion is z = Ty where the transformation matrix T is given as

T = diag(J(x),Inxn) (4.11)

A7: The structure of the equations of motion are completely known.

A8: The parameters in the equations of motion are completely known.

4.7 The NOpAC Control schemeI have in Chapter 1 shown that limitations on work and energy are important constraints for effective operation of completely autonomous underwater vehicles. Chapter 3 shows that the dynamics of underwater vehicles are multivariable, nonlinear and time-varying. These factors combined are the motivation for deriving adaptive and optimal control algorithms for small underwater vehicles. This section presents the first near-optimal and adaptive control scheme, named the NOpAC control scheme (Near-Optimal and Adaptive Control), while the next section presents an optimal adaptive control scheme named the OpAC control scheme.

The NOpAC algorithm is optimal in the sense that it minimizes velocity tracking errors and the forces which contribute to the vehicle’s kinetic energy and the effort to overcome dissipative forces that is spend to correct these errors. The near-optimal system is proved to be globally asymptotically stable with respect to velocity tracking errors in the case of a perfectly known vehicle mode. However, this is rarely the case for underwater vehicles, so an adaptive version of the algorithm is also presented. Globally asymptotically stability with respect to tracking errors and bounded parameter errors are proved in the adaptive case. A globally asymptotically stable position feedback loop is also added to the feedback law. The system is not optimal when the position feedback loop is employed. The results in this section are published in Sagatun (1992).


4.7.1 State-space representationThe NOpAC algorithm is derived in the y state-space where y € 9i2n where y is defined as y — [qT, xT]T. A state-space representation of (3.52) expressed in the y state-space is

-m;' (c,(*)+ £>,(*)) Onxl ■1/ | -9r - Mg1 (Ci(9) + Dq{q)) qT4_ 7nxn

J(xe) Onxny-r Onxn i Onxnunxn

(4.12)were yT can be generated as described in A3.

4.7.2 The control objectiveThis algorithm minimizes the thrust forces which correct the deviations between the vehicle’s actual and desired velocities. We only consider the forces which contribute to the underwater vehicle’s kinetic energy and the dissipative forces. The optimization criterion does also contains a term which is quadratic in the velocity error. The work done by the system when using subtracting the effect of potential energy from the thrust forces is given by (4.6). Therefore, we choose to minimize the control variable in (4.7). We can rewrite (4.7) such that we obtain the following state space description:

q = —M~l Dq(q)q + M~ru_ tq — A(q,t)q + Bu (4-13)

Notice that (4.13) is a nonautonomous system since A(q) = A(q + qr(t)) — A(q,t). The entries in A(q, t) are continuous and the system [A, B] is controllable except for the trivial singularity <jr = 0.

We wish to minimize the quadratic performance indextoo

J{q,t) = / L(q,u) drJto

(4.14)

with the LagrangianL(q,u) = q Qq + uTRu (4.15)

where R — RT > 0 and Q = Qt > 0 and u is given by (4.7). We will without lossof generality restrict the coming discussion to discuss problems with B. matrices on the form R — r-i ■ Inxn- This is realistic since energy spent to correct tracking errors in one direction is assumed to be as valuable as in another one.

We are now able to find an optimal control u = u” that moves the system represented by (4.13) from an arbitrary initial state to the origin while minimizing J(q,t).

4.7. THE NOPAC CONTROL SCHEME 93

4.7.3 Optimality and stability

The Hamilton-Jacobi equation can be reformulated to:

-Tn V dj dt(4.16)

where aq(4.16)

q is the gradient of J* with respect to the error state q and is

combined with (4.13) written out yields

written

- T {2VJ*(^ t)TA(i, t) + 2VJ*(4, tfBu + utRu + (4.17)

We minimize this blend of a functional and partial differential equation by differentiating the left-hand side with respect to u, which yields:

u* = (4.18)

Notice that we have assumed that

d_ fdPdu 1 dt 11=11’ — 0

Finding an appropriate J* for the general nonlinear case is, if possible, very difficult, so a suboptimal approach is utilized. Notice that J* is a homogenous function of degree two of the vector q such that 2J* = q- It is now possible to show by using induction

on Euler’s theorem on homogenous functions that the solution of (4.17) is quadratic in q such that the optimal cost function becomes:

J'{q, t) = qT{t)K{q, t)q(t) (4.19)

where K(q,t) is symmetric and positive definite, Anderson and Moore (1989) pp. 21-27. Anderson and Moore (1989) prove the existence of a finite K(q,t) for the infinite time problem (tf — oo). Hence, we obtain the following Riccati equation:

qT (-K{q,t)BR-'BTK(q,t) + K(q,t)A(q,t) + AT(q,t)K(q,t) + Q + K(q,t)) q = 0(4.20)


Theorem 4.1

The following feedback law gives a close to optimal solution to the optimization problem formulated by (4.2) with the Lagrangian in (4.15).

u* = -R-1BTK{q,t)li (4.21)

where K(q,t) € 3?nxn is the symmetric positive definite solution to the Riccati equation given by (4.20)

Proof: The optimal feedback is verified by inserting (4.21) and (4.19) in (4.17). The solution to the Riccati equation is not unique, but if we restrict ourselves to symmetric positive definite solutions, one unique solution is found, see e.g. (Barnett, 1971). Hence, we have proved that (4.21) is the optimal feedback law.□

It is worth while noting that the index * on the control feedback law is a misnomer, since J* will eventually become infinite for non-zero qd since t-final is infinite and steady state u* is not zero for non-zero qr, K(q , too) will still be finite.

The system in (4.13) becomes autonomous for ty —» oo when qd is constant, but it is still necessary to use nonautonomous stability theory, since stability in the transient phase must also be proven.

Notice that when qd is constant and the system has arrived to a steady state, then^ = 0- This implies that K{q,t) = 0, thereby easing the computational effort

needed to solve the Riccati equation. The optimal feedback law is still given by (4.21), where K(q,t) is found by solving:

-K(q,t)BR'lBTK(q,t) + K{q,t)A(q,t) + AT(q,t)K(q,t) + Q = 0

Hence, a new feedback matrix K(q,t) can be calculated every time the condition, i.e. the velocity, changes are substantial. One criterion for starting the gain scheduling can be:

II I a»(9><) ~ a«(9,< + 1) l> «i=l

where a is a positive design parameter.


Lemma 4.1

The nonautonomous system in (4.33) controlled with the feedback in (4.18) is always uniformly globally asymptotically stable with respect to q under the above assumptions.

Proof: Throughout this section we will use Barbalat’s lemma for global stability, (Barbalat, 1959) and a method found in (Popov, 1973) pp. 210-213 which states: Assume that the Lyapunov function candidate V(q,t) satisfies:

1. V(q,t) is lower bounded2. V(q,t) is negative semi-definite3. V(q, t) is uniformly continuous in time

then V^q,^) —> 0 as < —► oo. We suggest to use the following Lyapunov function candidate.

v(ii,t) = iiTK(ii,t)ii (4.22)

Differentiating V with respect to time and using (4.20) yields:

v{Q,t) = -qTQq < o

Hence V(q,t) < 0. This implies that V(q,t) < V(q, 0) which in turn implies that q is bounded. Bounded reference trajectories and the fact that Dq{q), Mq and K(q,t) are continuous bounded functions ensures that V(q,t) is bounded; consequently, V(q,t) is uniformly continuous in time. Finally, application of Barbalat’s lemma shows that V(q, f) —► 0 which again implies that q converges asymptotically towards zero. This concludes the proof.□

Lemma 4.1 guarantees that errors in velocities will converge to zero. Thus far we have not included any position feedback to the controller. This is necessary if we want accurate automatic positioning of an underwater vehicle but not necessary for teleoperated vehicles (ROVs) or underwater vehicles in autopilot mode. The OpAC control scheme presented in the next section includes position feedback by using a non-physical control variable u where the pseudo-kinetic energy and pseudo-work to overcome dissipative forces consist of a linear combination of acceleration, velocity, and position errors. This control scheme uses the control variable in (4.5) which has a more physical interpretation, with the extra cost of having to add a position feedback loop to the system. It is worth noting that the Riccati equation is no longer satisfied when this additional term is added to the control input. This results in a near-optimal control law when small deviations from the desired trajectory are present. This scenario will often occur when an underwater vehicle is in


transit and exposed for current disturbances, see Fig. 4.2. This position feedback loop is presented in the next lemma.

Assumption A9:The proof of stability for the position control assumes a constant commanded velocity. This implies that the system becomes autonomous.

Preliminary assumption A10:We will first discuss the case when we have diagonal Mq and Dq(q) matrices which results in a diagonal K(q) matrix and then extend our proof to the general case when stability is proved.

inertial-fixed reference frame

current

\ desired trajectory

\

Figure 4.2: An underwater vehicle in transit exposed for disturbances


Lemma 4.2

Under the above assumptions, the autonomous system in (4.33) controlled with the augmented feedback in (4.23) is globally asymptotically stable with respect to g and x.

u = u* — JT(x)Kpx Kp — Kj, > 0 (4.23)

Proof: A standard Lyapunov proof (Lyapunov’s direct method) may be applied to this lemma, since we asumme that the system is autonomous, i.e. constant commanded velocity. Consider the following Lyapunov function candidate:

Nfq OnxnOnxn Kp y

which can be looked upon as a combination of the vehicle’s kinetic energy and virtual potential energy where the Kp matrix acts like a virtual spring constant. Differentiating V with respect to time and using the augmented feedback in (4.23) we get:

V(y) = (Dq(q) + ±M;lK(qj) $ < 0

Hence, globally asymptotically stability is proved for q. Note that the rate of convergence increases with increased damping. This is as expected since the system becomes more dissipative with more damping. That is U = TTq — qTDqq for an underwater vehicle. There remains to prove that also x converges to zero. This can be done by using the global invariant set theorem, Salle and Lefschetz (1961) pp. 56-59. We need to check if the system can get stuck at a position where <7 = 0, but where i ^ 0. This, however, is not the case here since when g = 0 we get

q = —M~1JT(x)KPx

which is non-zero for non-zero x, since JT(x)Kp is non-singular.□

Note that the position errors remain bounded for the nonautonomous case, since we then can apply Barbalat’s lemma with the same Lyapunov function candidate. The position errors would then converge asymptotically to zero when the vehicle’s velocity again returns to steady state, i.e. constant velocity.

Relaxation of preliminary assumption A10.

The K{q) matrix is usually diagonally dominant since both the Mq and D{q) matrices are strongly diagonally dominant (a„- >> Y.iaij i ^ j h an » Y,jaij * 7^ j) for most


marine vehicles. This, in turn, implies that the matrix product is also diagonaldominant and usually positive. When the product qT (^D(q) +-AM~xK(q)j q < 0 a

switching term, the scalar function 7(9,9) is multiplied to the K(q) matrix. 7(9,9) is defined as:

7i(9,9)

1 r * ?T (±M;'K(q) + D(q)) 9 > 0

where > 0 is a design parameter. Notice that a more conservative version of the 'y(t) function is :

7(9,9) =sgn(qT—M;1K(q)q) r\

however this term is more convenient to use in stability analysis. Remark that a nondiagonal D(q) matrix does not cause instability, since addition of damping only makes the system more dissipative.

4.7.4 The control lawA combination of the control variable in (4.23) with the state-space representation in (4.12) yields the following expression for the resulting thruster forces:

r; = M ,9r + Dq(q)qr + Cq(q)q + u*t= V’«(9r,9r,9)®1) + V’90(9r,9r,9)+^(0 (4-24)

whereu* = --Mr1 jr(g)i - Jt(x)kpx

riWe observe that the model is linear in the parameters. Note that we have used the notation it’ even though the system is only near-optimal in velocity control. This is for notational convenience, and the reason should be evident in the discussion of the adaptive controller. The corresponding commanded angular velocities nj to each thruster can be computed from:

nd = b} (9)t* (4.25)

where B^g) = W-1 Bj{BjW*1 Bt)*1 for m > n and b] = B^l(q) when Bj(q) is non-singular. We have used a positive and diagonal IV which distributes energy between the different thrusters after a quadratic cost criterion.


UnderwaterVehicle

Figure 4.3: The block diagram for the NOpAC control scheme

4.7.5 The adaptive control algorithmThis paragraph derives an adaptive version of (4.24). Rewrite (4.24) such that it is expressed as a function of the present estimate of the parameters:

r = (MqqT + Cg(q)q + Dq(q)qr) + u*(t)

$T = V,,(9r,9r,9)^ + V’?o(9r,9rI9) + <(0 (4-26)

where tf>0 represents the part of the model which is completely known and V*? £ 3?nxp, 9 G 5ip. The new control variable (with model errors present) uq is now found by subtracting (4.24) from (4.26):

uq = u*q + il>q{qr,qT,q)eq (4.27)

where 9 =9 — 9. Notice that optimal control is achieved when the parameter errors, that is <? = 0. We can now define the augmented error vector e G 5R2n+p such that e = {yT,0]T and introduce the new Lyapunov function candidate:

Vc(e,t) = V(y,t) + Ve(9) (4.28)

V,(0) = ^9TKt0where

Kt = Kj>0 (4.29)


and V(y,t) is found in lemma 4.2 We are now ready to formulate the following theorem:

Theorem 4.2

The system described by (4.26) with unknown parameters and controlled by (4.23) is globally asymptotically stable with respect to y with the following adaption law:

^ = -Ke'tf'q Ke = Kj>0 (4.30)

Furthermore, the adaptive control law in (4.30) is stable in the sense that all parameters remain bounded for all t. The adaptive controller will be a suboptimal adaptive control system for constant parameters when 0 — 0 and under the assumptions made in the previous section.

Proof: It is easiest to show that V(e,t) is negative semi-definite by employing the Lyapunov function candidate in lemma 4.2 with the signum version of the 7(9,9) multiplier defined in the previous subsection. If we differentiate (4.28) with respect to time and assume constant 0 we get:

Ve(e,f) = V(y,*) + V#(d) (4.3!)

A combination of (4.31) with the control law in (4.27) and the results in lemma 4.2 yields:

Ve = ~qD(q)k ~ ~ I qM~qlK{qYq \ri

We recognize the two first terms as negative from lemma 4.2 and that the two last ones cancel each other when 0 is defined as in (4.30) such that:

I/e = --qTD{qyq--\qM~'K{qyq\n

Ve < 0

Hence, we have proved that V(e,t) < 0. This implies that V{t) < V(0) which in turn implies that e is bounded. Bounded reference trajectories and the fact that ip, 0 and J{x) are continuous bounded functions ensure that V(e,t) is bounded, consequently V(e,t) is uniformly continuous in time. Finally, application of Barbalat’s lemma shows that V(e,t) —» 0 which again implies that 9 converges to zero and that 0 remains bounded for all t. We have u = u* in the case of no parameter errors, and hence, we have a near-optimal controller with respect to the performance criterion in (4.14).□


4.7.6 Some critical remarksA drawback to the results of theorem 4.2 is the requirement that Mq must remain positive definite. Spong and Ortega (1990) discuss briefly how to implement algorithms which involves the inversion of the adapted inertia matrix and Craig (1988) discusses this topic in detail, see also subsection 4.9.4. Another simpler modification to avoid the inversion of the Mq matrix involves a fixed a priori estimate of Mg denoted Ma in the calculation of the near-optimal control law and the adapted version of Af,, Mq in the feedforward linearization in (4.24). This approach is used in the simulation study. It is useful to remember that the Mq matrix is usually strongly diagonal dominant and that we have a good a priori estimate of the inertia matrix, see e.g. Humphreys and Watkinson (1982) and Sagatun and Fossen (l991d). A block diagram of the complete control algorithm is found in Fig. 4.3.

4.7.7 A simulation studyThe following simulation study is based upon the same 3DOF model as the one used in the OpAC simulations. The Q, R and Kp matrices were chosen to 10 • isxs ,10-4 • Isxs and 300 • 13x3 respectively. We have here neglected off-diagonal added inertia and damping terms. We also assume (not necessarily realistically) that all the damping terms are fully known such that 0 € Jf3, hence 0 becomes 0 = N^) with an initial estimate0 — (—m, —m, —/z). The actual values are 0T = (0.8m, 1.1m, 0.8/*). The parameterupdating gain matrix Kq were chosen to 0.01 • /axs- The V’ matrix and the V’o vector are found to be

f m (ur — vr) + I u m (vT + ur) + y|„|„ | v

\ IzrT + Af|r|r | r | rv f —ur vr 0 \

—ur —vr 0

uv —uv —rr

The desired state vector was yj — (0,0,0,0,0,0)T while the initial positions and velocities were yT = (1,1,1,1,1,1). The sampling rate was set at 30 Hz A simulation of the controller with the adaption is shown in Fig. (4.4).

We notice that the tracking is good from the two upper frames. The lower right frame shows the principal function of optimization plotted as a function of time. We observe that = 0 after approximately 4s. Jm converges to a constant cost since qd = 0. The lower left frame shows the V(y,t) and Vê,^) functions plotted versus time. We

102 CHAPTER 4. OPTIMAL CONTROL ALGORITHMS FOR UNDERWATER VEHl

notice that V(e, <) is negative semi-definite which implies that all parameter estimates are bounded, while V(y,t) is after a close look negative definite which implies that the system is globally asymptotically stable. The adaption of the three unknown parameters together with their actual values is shown in the middle right frame. We observe that no parameters converge to their correct values. This can be explained by looking on the system input (i.e. the thrust) in the middle left frame which does not contain enough information, i.e. the system input is not sufficiently persistent excitative, e.g. Ortega and Tang (1989) or Craig (1988).

4.7.8 Implementation of the NOpAC control schemeThe complete control algorithm consists of the nonlinear feedback term CQ(q)q, the nonlinear feedforward terms MqqT+Dq(q)qT, the suboptimal feedback term Mq'K(q)qand the proportional term represented by JT(x)Kpx. In addition we also have the adaption mechanism in (4.30).

The overall control can be computed from the following steps.: Calculate the desired trajectory off line.

step 1. Read the measured positions x and velocities q from the sensors,

step 2. Calculate the J(x) matrix and the 7(9,9) switching term,

step 3. Compute the matrix and the il>0{qT,qr,q) vector,

step 4. Use the adaption law given by (4.30).

step 5. If changed conditions, solve the Riccati equation for the new values.

step 6. Determine the optimal feedback term given by (4.21).

step 7. Compute the nonlinear feedforward, feedback and proportional terms.

4.7.9 Comments to the NOpAC control scheme

We have derived an optimal controller which minimizes the tracking error and the work that corresponds to the vehicle’s kinetic energy and the dissipative forces that are used to correct these tracking errors. The controller consists of one feedforward linearization term, one feedback linearization term, the proportional term and the optimal feedback

4.7. THE NOP AC CONTROL SCHEME 103

law. This is most easily seen by conferring (4.24) with u written out:

t= Mqqr + Dq(q)qr+ Cq(q)q - _ fr(m)KPx'---------- v----------' '--- v--- ' Tl '------ V--------feedforward terms feedback term TT ” proportional termoptimal feedback term

The resulting control algorithm utilize the desired trajectory information instead of the actual measurements in a large degree. This robustifies (with respect to performance) the regulator as well as makeing the actual implementation easier.Barbalat’s lemma has successfully been employed to prove global asymptotically stability for both the completely known as well as the adaptive controller. We have also presented techniques to avoid the problem of inverting the adapted inertia matrix. The simulation study indicates that the NOpAC controller may turn numerically unstable for bad choices of gain matrices (e.g. large R and Kg matrices) together with a small sampling frequency when Euler integration is employed.

The next section will derive an optimal controller which is optimal even through the transient phase of the tracking.


velocities

normalized 8

-500-1000

-2000

-0.5 -

-1500 -

0.8 -/

0.2 -

Figure 4.4: NOpAC simulation studyThe two upper plots show the vehicles positions and velocities, (solid — x-direction, broken = y-direction and dotted — about the z-axis). The two plots in the middle show the thrust forces in and about the x-, y- and z-axis and 6 plotted versus time. The bottom left plot shows V(e,t) and V(y,t) while the bottom right plot shows J(u*,y) plotted versus time.

4.8. THE OPAC CONTROL SCHEME 105

4.8 The OpAC Control SchemeThis section presents extensions of the algorithm presented in Johansson (1990b) and the control scheme presented in the previous section. The algorithm is a continuous-time optimal control algorithm for underwater vehicles. The algorithm is optimal in the sense that it minimizes the state errors and the forces and moments that correspond to the vehicle’s kinetic energy that is spent to correct these errors. The performance measure also contains a term which penalizes the quadratic tracking errors proportional to the rate of energy required by the controller due to dissipative forces. Alternative optimization criteria are presented with their control laws. Global uniformly asymptotic stability is proven for the case of a perfectly known vehicle model. However, this is rarely the case for underwater vehicles, so an adaptive version of the algorithm is also presented. Global asymptotic stability is proven for the tracking errors and a bounded parameter estimate is guaranteed in the adaptive case. The results in this section are going to be published in Sagatun and Johansson (1992).

We will first derive the controller for the completely known model in the x-frame formulation. The controller will be transformed to the vehicle-fixed reference frame. An adaptive version is then derived.

4.8.1 State-space representationA state-space description of (3.58) expressed in the z space can now be found:

-M,-1(C, + D,) 0nxnInxn Onxn

i + -xr — Mx 1 (Cx + Dx) irOnxn

lnxn I n* —1^.n | Mx rxunxn

(4.32)

The matrices’ explicit dependency on the x and the x vectors are skipped for brevity. Eq. (4.32) is written in compact form as

z = A(x, x)z + n(x, x, xr, xr) + EMX 1tx

where E is defined as E = [Jnxn, Onxn]T- Notice that (4.32) represents a nonautonomous system since, for instance, the D(z) matrix can be written as

D(z) = D(z + zT{t)) = D(z,t)

4.8.2 The control objectiveEqs. (4.32) combined with the control variable in (4.5) yield the following state space description:

-Mx-'iCx + Dx) On>11 J 12 J

T0i + r; ■i [ Mx-1 ' [ Onxn (4.33)

106 CHAPTER 4. OPTIMAL CONTROL ALGORITHMS FOR UNDERWATER VEHH

The control objective is to minimize the quadratic performance index given by

J(z,ux) = f ’ L(z(t),ux(t)) dr (4.34)Jt0

with the Lagrangian inL(z(t), ux(t)) = X-zT{t) (Q + Tt0ED{z)EtT0) z(t) + ^(t)Rux(t) (4.35)

where R = ri-Imxm = RT > 0, Q = QT > 0 and f/ is not fixed. Notice that we have both a penalty on the tracking error as well as the applied work that is needed to correct the state errors. This allows us to derive a feedback solution for the minimum control effort problem. Another important detail is that (4.35) also contains a term which penalizes the quadratic position and velocity errors proportional to the rate of energy which dissipate from the vehicle due to dissipative forces, e.g. viscous friction. A diagonal structure of the matrix R given above is assumed since energy spent to correct tracking errors in one direction is as valuable as in another one. The weighting of the distribution of energy to each of the vehicle’s thrusters is taken care of in (4.25). It is important to realize that we minimize the velocity and thrust in the inertial coordinate system. The transformation to a vehicle-fixed coordinate system will come at a later stage of this control system design.

We are now able to find an optimal control ux(t) — ux(t)m that moves the system represented by (4.33) from an arbitrarily initial state to the origin while minimizing J(z(t)) under the assumptions made in section 4.6.

4.8.3 Optimality and stabilityThis section utilizes the same line of reasoning as employed in Johansson (1990b) to prove optimality and stability. The new system (an underwater vehicle instead of a manipulator), a new set of equations of motion, the inclusion of dissipative forces and the modified performance index result in a different solution to the Hamilton-Jacobi equation and a different control law.

It is well known from the literature, e.g. Athans and Falb (1966), that the optimal control «*(<) for the state equation in (4.33) with the performance measure in (4.1) can be found by solving the Hamilton-Jacobi equation given by (4.2). One way of doing this is to find the function J*(z,t) which is called the Hamilton principle function of optimization.

Lemma 4.3

This lemma gives the optimal feedback law for controlling an underwater vehicle.The following function J* satisfies the Hamilton-Jacobi equation and constitutes a Hamil


ton’s principal function for the optimization problem formed by (4.2), under the assumptions made above:

iTo (4.36)

where AT is a positive definite symmetric matrix K £ 3?nxn and for K and Ta solving the matrix equation

+ Q- TtoER-1EtT0 = 0 (4.37)

The optimal feedback law u* that minimizes (4.1) with the Lagrangian in (4.35) is

<(t) = -R-'ETT0z{t) (4.38)

Proof: See Appendix A.l□

Note that (eq:riclike2) is quadratic in z\ this is as expected since the performance measure in (4.34) is a homogenous function of degree two of z, see also subsection 4.7.3.

Theorem 4.3 (Johansson (1990b))

Subject to the assumptions made above, the optimal control solutions of (4.1) result in an L2 stable closed-loop system controlled by the optimal feedback given by (4.38). The following choices of the weighting matrices T0 and K are necessary:

Tn T12 _ R-jQi RjQ2

andjr = a:t = 1 (q[q2 + Q2Q1) - \ (Qi2 + q£) > o

where Qt and Hi are the Cholesky factors of Qtt and R respectively.

Proof: Johansson (1990b).□


Theorem 4.4

The system described by (4.33) and controlled by (4.38) always globally uniformly asymptotically stable with the choice of the weighting matrices T0 and K given in theorem 4.3 and the above assumptions.

Proof: The theorem is proved if we can find a suitable Lyapunov function candidate V(z, t) for the nonautonomous system described by (4.33) satisfying, Lee and Markus (1967):

i) V(z,t) is continuous at z = 0 Vt,n) V(z,t) is positive radially growing with ||z||,in) V(z,t) has a unique minimum at the origin of the error space

andiv) V(z,t) is negative definite along z and t.

Mz(x) 0nxn

Onxn K T~zIt is straightforward to show that V(z,t) = J*(z,t) — rosatisfies the three first requirements. The last requirement is also easily proven since(4.2) states that

dVdt

dVdt

dJm(i,t) dJ'(i,t)T, _dt dz Z -L(2(<), MO)

< 0 Vz(<) # 0

Hence we have shown that the system described by (4.33) and controlled by (4.38) is always globally uniformly asymptotically stable. This concludes the proof.□

A similar proof is presented in Johansson (1990b) for a system without dissipative forces and with the Lagrangian in Lemma 4.4.

Comment 4.1. The stability properties proved in theorem 4.4 can also be proven by differentiating V{z,t) explicitly to show that

^<0 Vi(t)#0

This is proved in Appendix A.2.□

Comment 4.2. Remark that the dissipative forces in (A.8), represented by the D(z) matrix, increase the stability of the system. This can also been seen when we look

4.8. TEE OPAC CONTROL SCHEME 109

on the mapping from the vehicle’s thrust r to velocity x written as r >—> i which is strictly passive (dissipative) since V = ttx — xTD(z)x, Sagatun and Fossen (l991f). This corresponds to a strictly positive real system in the linear time invariant framework. We observe from the expression of V that the gradient of V becomes more negative, that is V converges faster to zero, with increased damping. This is, of course, expected i.e. compare with the SISO case where the phase margin increases with increased damping. The positive term z{t)TTÊD{z)ErT0z(t) with T12 = 0 Tn = Jnxn corresponds to the rate the energy dissipates from the vehicle, i.e. the dissipative character of the equations given by (3.1) or (3.58) increases with increased D{z).□

Lemma 4.4

The system in (4.33) with the following Lagrangian:

L(z{t),ux(t)) = ^zT{t)Qz(t) + ûl(t)Rux(t)

is stable and optimal with the optimal feedback law ul(z,t) given by:

u;(z,t) = -R-'(z)ETT0z(t) (4.39)

where R(z) is formed by the product of the two Cholesky factors RlT(z) and R11(z) such that

R-'(z) = R-t(z)R^\z) = R-' + Dx(z) (4.40)

Proof: The proof is straightforward by inserting (4.40) in (A.8). Notice that it is always possible to perform the Cholesky decomposition in (4.40) since R-1 > 0 and Dx(z) > 0 □

Notice that the dissipative forces are now represented explicitly in the feedback law. This has been obtained by adding the damping matrix Dx(z) to the feedback gain matrix R_I.

4.8.4 The control lawA combination of the control variable in (4.5) with the state-space representation in (4.33) yields the following expression for the resulting thruster forces:

T* = Mr [*r - TulTl2i - T^Mx-1 {(Cx + Dr) ETT0i - <(<))] + Cri + Dri


where u* is given by (4.38). This expression is considerably simplified if we also assume that the Q matrix can be written on the form Q = diag[q\ • Inxn, <?2 • /nxn]- Tn and Tu will then become diagonal matrices such that Tn = tn ■ /„xn and Tn = tn • /nxn The new control law then becomes

'ri = ~ [Mx (tn&r ~ hix) + (Cx + Dx) (tuxT - t12x) + <(<)] (4.41)

This expression can be simplified even more by defining the signals

MO = <ll*r ~ <12*

«*(<) = tnxr - tnx (4.42)

Notice that MO's no< a time derivative of any meaningful vector. The dot notation is used such that s multiplied with the mass matrix denotes a quasi-acceleration dependent force and s denotes a corresponding dissipative force. The control law in (4.41) can now be written as

T*x = T~ {Mxsx + Cxsx + Dxsx + «*(0) (4.43)tn

or in a more general form

K = 4>x(sx, sx, x, x)ex + <tn(0 (4-44)

whereV^x(^x) 8X) x, x^0x — ~ (Md- GX8X -j- Dxsx)

tnand Ux,u(t) = ^-ux(0 Notice the similarity between the error signal s(t) and the one employed by Fossen and Sagatun (1991a). The control law in (4.43) and (4.44) is of no practical use, since the resulting thrust forces are calculated in an inertial reference frame. Eqs (4.43) and (4.44) are also impractical in the sense that they use velocities decomposed in an inertial reference frame while we measure the vehicle-fixed velocities. A last drawback with (4.43) and (4.44) are that they are much more complicated when they are formulated in the x-frame than in the q-frame. It is, however, possible to transform equations (4.43) and (4.44) to the q-frame by using the following lemma.

Lemma 4.5

The equation described by (4.44) can be transformed to a vehicle-fixed coordinate system by employing the following transformation:

JT{x)il>x(sx, M X, x)0x = ikq(8q, M q)0g


where the regressor matrix ‘4>q is found from

9)e<> = T~ + <?,($)*, + (4.45)*11

where 5, and s, is defined in (4.48). The resulting control law formulated in the vehicle fixed coordinate system then becomes:

T*,='l’',{8i>8q,q)6q + u* (4.46)

The feedback control law formulated in the q-frame u*(<) is found from

u!(f) = -^-JT(x)R-1ETT0Ty (4.47)tn

where y is the measurement vector defined in A6.

Proof: The new signals sq and sq are found by using (2.4) on each term in (4.42) such that:

sq{t) = tnqrsq(t) = tuqT - <12 ^j-1(x)x -(- q^j (4.48)

The rest of the proof is straightforward. Noticethat j '(a:) = —J~1(x)j(x)J~1(x). □

Comment 4.3. If lemma 4.4 is applied, the new control law becomes:

«;(*,9) = - (jT(x)il_1J(*) + D,(g)) q-j^ ^{x)^1 J(x) + Dq{q)) J_1(a;)*11 (4.49)

□

Note that the first term in sq and sq are feed-forward terms while the latter are feedback terms. The reparameterization, by using a virtual vector in (4.45), is similar to the one employed in Fossen and Sagatun (1991a). Notice also that all signals in (4.48) are measurable or preprogrammed. The form of the rpq vector is also much simpler than the ipx vector.

The corresponding commanded propeller angular velocities are found by employing (4.25).

112 CHAPTER 4. OPTIMAL CONTROL ALGORITHMS FOR UNDERWATER VEH

4.8.5 The adaptive control algorithmMany of the parameters in the M, D(q) and C(q) matrices are unknown and time- varying for marine vehicles and especially for small underwater vehicles. This motivates us to use an adaptive version of (4.46).Rewrite (4.46) such that it is expressed as a function of the current estimate of the parameters

Ti = J-(Mq$q + Cq{q)kq + Dq{q)k„)+u*{t)Ml

trq = tl>q(sq,sq,q)0q + iJ>qo(sq,sq,q) + u'{t) (4.50)

where (•) denotes the estimated value of (•) and ip0 represents the part of the model which is completely known. Notice that we are not applying the optimal thruster forces r* anymore since we have model errors. The effective control variable uq is now found by comparing (4.50) and (4.46) to be:

uq = u* + i/>q(sq,sq,q)dq (4.51)

where V, € 3?nxp and 6q € 3fp, p is the number of parameters we want to adapt. The corresponding new feedback law formulated in the x-frame becomes

«*.„(<) = + V’x(«x, Sx, X, X)0x (4.52)__ y

We can now define the augmented error vector e € 5J2n+p such that e = [z7", 6 ]T and introduce the new Lyapunov function candidate, Johansson (1990b):

Ve(e,t)=V(z,t) + Veq(9q) (4.53)

whereVgq(0q) = 0TqKe0q Kg = Kj>0 (4.54)

and V(z,t) is found in theorem 4.4. We are now ready to formulate the following theorem:

Theorem 4.5

The system described by (4.50) with unknown parameters and controlled by (4.51) is always globally asymptotically stable with respect to y with the following adaption law:

0q = -K^xl>Tj-'(x)ETToTy (4.55)


Furthermore, the adaptive control law in (4.55) is stable in the sense that all parameters remain bounded for all t. The adaptive controller will be an optimal adaptive control system for constant parameters when = 0 and under the assumptions made in the previous section.

proof: It is easiest to prove this theorem by using the results from lemma and applying Barbalat’s lemma for global stability, Barbalat (1959). It is easiest to show that V{e,t) is negative semi-definite by employing the Lyapunov function candidate in (4.53) in the x-frame formulation and then transform the solution to the q-frame. If we differentiate (4.53) with respect to time we get:

Ve(e,0 = V(i,f) + V,f(0,) (4.56)

A combination of (4.56) with the control law in (4.52) and the results in theorem4.4 yields:

\t)TT0E (R-1 + D{»)) ETT0z(t)+^K»^,+-zT(t)TÊrl>xei

where constant parameters are assumed, i.e. 0 = 0. This can be rearranged to

= -\yT(t)TT ( ZTu { -K +2:

(R-1+ £>(*)) Tt —K + 2Tjl (R-1 + D(z)) Tn2T[2 (R-1 + D{z)) Tu 2TTn (R-1 + D{z)) Tn Ty{t)

+»: + V1 {t)TTEJ~1 .61 1

V* = -\yT(t)T' ( 2TTn{R-1 + D{z))Tn V -K + 2Ttu{R-1 + D{z))T1

-K + 2TTn{R-' + D(z))Tu \2Tj2(R-1 + D(z))T12 )Iy()

Ve < 0

where we have used the adaption law in (4.55) to eliminate the terms containing 0 and 0. We recognize the matrix on the right-hand side as positive definite from comment 1. to theorem 4.4 (see also appendix A). Hence we have proved that l^(e)f) < 0- This implies that V(t) < V(0) which, in turn, implies that z = Ty is bounded. Bounded reference trajectories and the fact that ipq, 0 and J(x) are continuous bounded functions ensure that V{e,t) is bounded, consequently V(e,t) is uniformly continuous in time. Finally, application of Barbalat’s lemma shows that V(e,t) —> 0 again implying that z = Ty converges to zero and that 0, remains bounded for all t. We have ug = u* in the case of no parameter errors and, hence, we have an optimal controller with respect to the performance criterion in (4.1).

114 CHAPTER 4. OPTIMAL CONTROL ALGORITHMS FOR UNDERWATER VEHI

Comment 4.4. Remark that if lemma 4.4 is applied we must use the estimate of the damping matrix in the feedback law such that

R~T(z)R~\z) = JT1 + £)x(z)

where Dx{z) is a conservative (underestimated) estimate of Dx(z) such that Dx(z) = Dx(z) + JDx(z) and Dx{z) > 0 Vz AVt. The Dx(z) matrix can be kept conservative by finding a lower bound on Dx(z) and employing the technique in Craig (1988) pp. 55-58 to ensure that Dx(z) stays positive semidefinite. An underestimated damping matrix results in a suboptimal controller, but the controller will still be stable. This follows directly from comment 4.2.□

Comment 4.5. An alternative proof of theorem 4.5 which uses Lyapunov theory is found in Appendix A3.□

A block diagram of the complete control algorithm is presented in Fig. 4.5.

UnderwaterVehicle

Figure 4.5: The block diagram for the OpAC control scheme


4.8.6 A simulation studyThe following simulation study is based upon a 3 DOF nonlinear model of the NEROV vehicle (see Fig. C.l) moving in the horizontal plane. The model given by (3.1) expressed in the q-frame becomes:

M,Dq(q)

Cg(q)

diag(m — Xu,m — Yi, Iz — Nr) diag(XMu \ u |,yj„|„ | v |,Ar|r|r | r |)

^ 0 —mr —YuV \ mr 0 —XuU

K YyV Xû 0 j

The Q and R matrices were chosen to 10 2 • /exe and 0.005 • /sxs respectively. We assumed that all added mass terms and damping terms were unknown such that 6 €

* A AâaAAAA * T3? , hence 6 becomes 6 = (Xi, 1^,, TVr, X|u|u, yj„|u, Aîrir) with an initial estimate 6 =(—m, —m, —/2, lîr). The actual values are = (320,380,49.2,300,180,45)The parameter updating gain matrix Kg were chosen to 0.005 • /exe- The ^(^ *cj)9) matrix and the tl>qa{sq, sq,q) vector are found to be

i ( -s'D)? vs(3), 0 i(l), | u |V’,(«!,«?.9) = 7- -us(3), -s(2), 0 0

11 \ «(2)f -»i(l)f -5(3), 0j f m(s(l)? - r*(2),) \

i>qSi>qCsq,q) - — m(s(2), + ri(l)?)V /*5(3), J

The desired state vector was = (0,0,0,0,0,0)T while the initial positions and velocities were zT = (1,1,1,1, l,25r). The sampling rate was set at 10 Hz A simulation of the controller with the adaption is shown in Fig. (4.6).

0 0i(2), | v | 0

0 i(3), | r |

We notice that the tracking is good from the two upper frames. The lower-right frame shows the principal function of optimization plotted as a function of time. We observe that = 0 after approximately 2.5s. The lower-left frame shows the V(z, t) and V(e, t) functions plotted versus time. We notice that V(e, t) is negative semi-definite which implies that all parameter estimates are bounded, while V(z,t) is, after a closeer examination, negative definite, implying that the system is globally asymptotically stable. The adaption of the six unknown parameters are shown in the middle-right frame. None of the parameters converge to their correct values. This can be explained by looking on the system input, i.e. the thrust, in the middle-left frame which does not contain enough information i.e. the system input is not sufficiently persistent excitative, e.g. Ortega and Tang (1989) or Craig (1988).

116 CHAPTER 4. OPTIMAL CONTROL ALGORITHMS FOR UNDERWATER VEHl

4.8.7 Implementation of the OpAC algorithm

The overall optimal control scheme may look complicated, but is easy to implement on a computer, at least if have good sensor data. The complete control algorithm consists of one nonlinear feedback and feedforward term given by (4.46), one optimal feedback term given by (4.47) and the adaption term given by (4.55).

The overall control can be computed from the following steps.: Calculate the desired trajectory off line.

step 1. Read the y vector from the sensors,

step 2. determine the J(x) and T matrices,

step 3. Calculate the s, and s, vectors.

step 4. Compute the rjj{sq,8q^q) matrix and the ij}0(8q,8q,q) vector.

step 5. Use the adaption law given by (4.55).

step 6. Compute the optimal feedback term given by (4.38).

step 7. Compute the nonlinear feedback and feedforward terms given by (4.46).

4.8.8 Comments to the OpAC control scheme

We have derived an optimal controller which minimizes the tracking error and the work that corresponds to the vehicle’s kinetic energy that is used to correct these tracking errors. The cost function also contains a term which penalizes the quadratic tracking errors proportional to the rate of energy which dissipate from the vehicle due to dissipative forces. As an alternative, we present a feedback law which explicitly compensates for the dissipative forces which work on the underwater vehicle. This feedback law is optimal with respect to the quadratic tracking errors and the vehicle’s kinetic energy as well as the energy expended to compensate for the dissipative forces which act on the vehicle. Global uniformly asymptotically stability is successfully proven by employing Lyapunov theory. An adaptive version of the optimal controller is also derived. Stability, in the sense of Lyapunov, is proven for this case, that is all parameters remain bounded for any t > t0 while the tracking errors are proven to be globally asymptotically stable around the origin. The controller consists of one feedforward linearization term, one feedback linearization term, and the optimal feedback law. This is easily seen by conferring (4.50)


and (4.42):

r9 = ^ (A^9s, + C,(q)s, + i>,(q)s,) + g(Q

^7 TT \ TT^jr 7~ ^ optimal feedback termfeedback and feedforward terms r

The algorithm is initially derived in an inertial coordinate frame and the transformed to the vehicle-fixed frame. This transformation is simplified by defining the augmented signal vectors sx, sx, 8q and sq which can be looked upon as a measure of tracking. The performance criterion is formulated in the inertial coordinate system while the controller operates in a vehicle-fixed system. This is not in conflict with respect to the minimization of the energy. The total energy spent by the vehicle will still be the same, but the distribution of thrust in the ux versus the uq vector will be different. The controller is initially derived in the x-frame formulation, since both the inertia-fixed position and velocity (the x and the x vectors) are given an meaningful interpretation, while vehicle-fixed position (the q vector) has no physical interpretation. This approach has a great advantage over deriving the controller in the vehicle-fixed reference frame, since it avoids the problem of working with the velocity transformation matrix J(x).

The Hamilton’s principal function of optimization given by (4.36) contains nice analogies to the physical system i.e. the vehicle moving in a fluid. The first part of the equation can be looked upon as the scaled virtual energy from an artificial spring and as the vehicle’s scaled kinetic energy. It also makes sense to use the dissipative matrix D(z) in the cost function, since the velocity errors should be penalized more harshly for vehicles with large damping coefficients.


positions

Xu Yv

Figure 4.6: OpAC simulation studyThe two upper plots show the vehicles positions and velocities, (solid = x-direction, broken = y-direction and dotted = about the z-axis). The two plots in the middle show the thrust forces in and about the x-, y- and z-axis and 6 plotted versus time. The bottom left plot shows V(e,<) and V(z,t) while the bottom right plot shows J(u*,z) plotted versus time.

4.9. IMPLEMENTATIONAL REMARKS 119

4.9 Implementational RemarksThis section contains some useful remarks for the implementation of the two algorithms discussed in this chapter.

4.9.1 Selection of parameters to adaptAdaption of input uncertainty

The equation of motion of an underwater vehicle in the surge motion can be written as

mil + d | tt | u = r (4.57)

where the expression for r is in (3.34) found to be

r = | n | n + b2u | n |

where &i = pD4a and 62 = pD3(l — w)S and a and j3 are the parameters in the linear approximation of Kt such that Kt = a + /9J0. The expression for r can be rearranged to:

r =| n | n

such thatM » = . . (4-58)

02- + biWe notice that the above equation contains an algebraic loop in n. This loop can be broken by approximating the n on the right-hand side of the equation with nk~\ which is the n measured in the previous sample. Eq. (4.58) can, however, be simplified further by looking on the magnitude of the parameters bj and b2. and b2 are related to each other by the following relation

— k • D where fc ~ < —1.4, —0.5 > b2This relation is valid for most marine vehicles which operate in the first quadrant of Fig. 3.6. k is approximately —0.6 for the NEROV thrusters. Note also that is typically less than 0.01 for most modes of operation such that (4.58) can be simplified to:

n — —r 01

(4.59)

where the new control variable n is defined such that n =| n | n or n = sgn(n) (4.59) combined with (4.57) yields:

h = m'u + d' \ u \ u

120 CHAPTER 4. OPTIMAL CONTROL ALGORITHMS FOR UNDERWATER VEHI

where (•)' = Hence we have shown that input uncertainty can be corrected for byadaption of the damping and inertia parameters. This line of reasoning also applies when linear drag coefficients are present.

Estimation of current - adaption of the damping parameter

The equation of motion of an underwater vehicle in surge with current denoted uc can be written as

mil + d(u + uc) = t iic = 0

which can be rewritten to

r = dTi]) = [m, d, uc]

We do now achieve integral effect for free if we have an estimate of the current uc and the parameter vector 9 is adapted on a standard manner like

iiu1

9 = -rv>e

A more naive approach would be to try to adapt the current forces as an increase in the drag. Assume, for argument’s sake, that the adapted parameters converge to their true value. The equation of motion above can be rewritten to

mil + du + duc = mil + (d + dc) u = mil + d'u = r

where the magnitude of the increased drag dc is dc = d^. This system can be adapted as above with the regressor ip = [u, u]T and parameter vector 9 = [m,d']T

There are two major disadvantages with this approach. Firstly, we notice that no integral effect is present so this effect must be added explicitly, and secondly, we must adapt a varying parameter which may cause instability.

Hence, we have shown that current should be estimated and the estimated value can then be used in the adaption algorithm to gain integral effect.

4.9.2 Saturation handlingThis section presents a method to avoid saturation in velocities i, propeller angular velocities n and the change in propeller angular velocities n. Saturation is specially a problem in direct adaptive control where it may lead to instability. Consider Fig. 4.7. The block “reference generation” may be thought of as generating the strictly stable reference


trajectory introduced in A2 in section 4.6. The problem is now twofold: firstly, how do we generate a strictly stable reference trajectory which is whitin the physical and kinematical limits of the vehicle, and secondly, how do we avoid saturation of the thrusters.

vehiclecontrolsystem

trajectorygenerator

desired state generator

Figure 4.7: The overall block diagram for the saturation handling scheme

The reference trajectory

The equation of motion for an underwater vehicle can, for the surge, sway, heave, and yaw direction, be written as:

1 . K xT + —xT = —r

where the subscript r denotes the desired reference trajectory and the T and K values are unknown. However, we usually have an a priori upper- and lower-bounded estimate on the parameters such that e.g. T £ (Tmin,Tmax)- The states in the desired reference trajectory that can be generated by using the modified desired position x'd as an input to the linear filter given in Fig. 4.8.

Figure 4.8: Generation of the desired reference trajectory

Fossen (1991a) suggests the following values of Tm and Km:


Tm < -^TmIN Km1

4(2Tm

where £ > 1 to obtain an overdamped system.

Generation of the desired steady state

The problem is now converted to find a method to compute a desired state xjj which prevents saturation of the vehicle’s actuators and which ensures that the calculated reference trajectory stays within the physical and kinematic limits of the vehicle. Consider Fig. 4.9. The nonlinear block SAT is defined as Amerongen (1982):

Figure 4.9: Generation of the modified desired state Amerongen (1982).

SAT _ / ^ ’ i+fc if I T‘ l> ™AX l i+fe if I * l< ^

where tc is the commanded thrust if the saturation avoidance scheme in Fig. 4.9 were not present. Remember that tmax = Umax | Umax |- Amerongen (1982) suggests the following heuristic estimate of the time constant Ta in the low-pass filter.

Tatmax

Ta is usually considered to be constant, tmax is the maximum rate of change of thrust. We notice from Fig. 4.9 that the desired state generator has three inputs: tmax, tMAX

and xmax which is a physical limitation of the vehicle’s velocity.


4.9.3 Integral actionNone of the control schemes presented in the previous sections have integral action. This may be desirable when the nonadaptive methods are used and no steady-state errors are specified.

There have been several methode presented to include integral action into nonlinear ,multivariable controllers for a system on the form given by (3.52) or (3.58). This section discusses three of them.

Perfect decoupling

Perfect decoupling assumes perfect knowledge of the model such that (3.52) can be decoupled to a set of perfectly decoupled double integrators.

ij>x6x = TX Tx = Mx(x)ux -1- Cr(*, x)x -(- Dx(x, x)xV

X - ux

The new control variable u can now be chosen to= koZi + kpii + ki f Xi{T)d,T i = l..n (4.60)

JtoThe gains for the perfectly decoupled case can now be chosen from e.g. a pole placement.

Use of a virtual measure of tracking

The use of a virtual error signal vector s(t) is employed in e.g. Slotine and Li (1987). The trick is now to find a control law, adaptive or not, which makes the signal vector s(t) go to zero. Consider for instance, the system given above with the following control law:

T? = V’A - JT(x)Kds

and definition of the signal s(t):

S{ — x, + 2Axi + X2 f Xi(r)dT i = \..n Jto

where A = if we compare with (4.60). This system represents a stable multivariable controller witfi integral action since the error signal s is a linear combination of the integral of the tracking error, the derivative of the tracking, and the error itself. The stability of this system can easily be proven by applying Barbalat’s lemma with the following Lyapunov function candidate, Fossen (1991c):

V(s,t) = ±(sTMxS + 6TKed')


Nonlinear multivariable PID

Arimoto and Miyazaki (1984) prove stability of a multivariable PID controller for a robot manipulator. This control scheme, modified to fit an underwater robotic vehicle, is presented below.

The equation in (3.58) can be written as

P = -^-*?(*,*) + «

* = M~'(x)p

where the vector p is the momentum vector defined as p = Mx(x)x and the tj vector consists of model errors and the damping matrix such that

ij(x, x) = ij>xOx - i>x6x -(- Dx(x, x)x

where we assume that r) can be written as

r}(x,x) = d + fc(p,*)

I *7(*,*) I < I * i +^2 |p |

in a domain 17 which will be defined later. Stability can now be proved for the following control law:

“ii = koii + kpii + kj I Xi(r)dT t = l..n Jto

under the following assumptions, firstly we restrict ourselves to discuss only the DP case, that is xT and xT is zero, secondly, stability is proved only for z = [pT(t), i^(f), ^^(f)]7 6

J? where the set 17 is defined as

f7 = {z(f) = [pT(t),xT{t),ZT{t)}T :| Xi - xTi |< jti, | Pi |< 7r2, | £ |< 7r3}

where 7ri..3 are positive constants such that z € 17 and the vector £ is such that £ = x. The following restrictions apply on the Kp = diag(kp), Kp, = diag{ki>) and K[ = diag(kj) matrices:

Ki >0 Kd > Mx(x) KP > KD + 2a~1 Ki

Kd a >0

where a is such that


The stability is proven locally under the above assumptions by using Lasalles theorem, Salle and Lefschetz (1961) with the following Lyapunov function:

i / JW* (*) aInx.n 0nxn \V{z) = -ZT\ alnxn Kp Kj \z

1 \ 0nXn Kj aKj )

The complete proof is a straightforward extension of the proof for a robot manipulator in Arimoto and Miyazaki (1984).

4.9.4 Modified parameter update laws

Investigations in the literature, e.g. Reed and loannou (1988) and Narendra and An- naswamy (1989), show that stable adaptive and linear controller may turn unstable in present of unmodeled dynamics, bounded input disturbances and time-varying parameters. These effects can be compensated for by using modified parameter update laws like the <x modification scheme loannou (1984), the | e | modification Narendra and An- naswamy (1987), bound on the | 9 | vector Craig (1988) pp. 55-58 and the dead-zone modification. There exists very little theory on robustness of adaptive and nonlinear controllers. Most of the present analysis tools are only valid for linear systems with MRAC controllers.

Of the modification schemes above, only the bound on the j 9 | vector scheme is analyzed for nonlinear system.

Bound on the parameter vector

This robustness scheme restricts the estimate of the parameters to lie within known bounds such that

®'MIN ~ < @i{t) < @iMAX A ^

If the estimated parameter is on the boundary, the following update law is used:

f O'MIN Oi(t) < 0iMIN $

l 6'maX ^ kt)>0iMAX+8

This modification scheme is attractive if we have a priori knowledge of the bounds of the parameters in the actual system.


Dead-zone

Even though the stability improvements of the dead-zone is not proven for the general nonlinear case, it contains some heuristically attractive properties which make it useful. The dead-zone scheme is based upon the observation that most of the high frequency and small tracking errors are due to noise. It is, therefore, desirable to turn off the adaption when the errors are below some value. This is exactly what the dead-zone scheme does:

kt+) { 0 if | e |< £ (AH, if | e |> <5

The dead-zone technique is especially attractive because of due to its simple structure and easy implementation.

4.10 Summary of Chapter 4.This chapter started by defining the problem of controlling a small underwater vehicle. The differences between small underwater vehicles and surface vessels were discussed, before a brief review of existing control and guidance systems applied to marine vessels were given. Some basic design requirements for control systems for marine vehicles were discussed before providing a short review of minimum control effort optimization criteria. We will specially like to call the reader’s attention to the minimum effort criteria which contain control variables which are derived on the basis of minimum work or minimum kinetic energy. These criteria have been used successfully in the two control schemes presented in this chapter. Basic assumptions for the controllers derived in this section were presented before two (near)optimal and adaptive controllers for underwater vehicles were derived. The controllers are optimal in the sense that they minimize the forces which contribute to the vehicle’s kinetic energy and the energy which dissipates from the vehicle. Stability proofs for both the adaptive as well as the completely known cases are proven with Lyapunov theory or Barbalat’s lemma. The different terms in the Lyapunov functions correspond to a vehicle’s kinetic and virtual potential energy. This is not a surprise since Lyapunov’s direct method of stability was originally derived for motion stability of mechanic systems. The chapter ends with some remarks concerning implementation of nonlinear and adaptive controllers.

Chapter 5

Conclusions and Recommendations

5.1 Summary and ConclusionsThis thesis has presented and shown:

• That the demand for underwater robots will increase and that it is a requirement for optimal minimum control effort controllers.

• A model of underwater vehicles based on Lagrangian mechanics. Each term in the model has been derived in detail. Lagrangian mechanics have proved to be particularly useful in explaining the phenomenon of added inertia and to revealing the inherent properties of these equations. Model properties of both the nonlinear equations of motion as well as for a linearized model are presented. The problem of controlling an underwater vehicle is a nonlinear and multivariable control problem. The thesis shows that an underwater vehicle’s parameters are unknown and time- varying, hence a need for adaptive controllers is indicated.

• That the Lagrangian formalism used in the derivation of the equations of motion eases the design of minimum control effort criteria.

• A near-optimal adaptive and nonlinear controller called NOpPAC. The control scheme is optimal with respect to minimum control effort criteria which take into account forces due to the dissipation of energy and the kinetic energy. A near-optimal velocity feedback law is obtained by using an infinite-time approach.

• An optimal adaptive and nonlinear controller called OpPAC. The controller is optimal with respect to minimum control effort criteria which take into account forces due to the dissipation of energy and the kinetic energy. The derivation of this control scheme shows how beneficial it can be to initially express the equations of

127

128 CHAPTER 5. CONCLUSIONS AND RECOMMENDATIONS

motion in an inertial reference frame, transforming the resulting control law to the vehicle-fixed frame for implementation. Both proofs of optimality and stability have been more easily achieved in the x-frame formulation.

• How the Lagrangian formalism and the expressions for work and energy can be used in proofs of stability and optimality.

Energy and work have been the leading terms in both the derivation of the vehicle model as well as in proofs of stability and optimality. There are many advantages to using the Lagrangian formulation. The equations of motion may be derived almost automatically by applying the Euler-Lagrange equation on the Lagrangian. The Euler-Lagrange equation is valid regardless of the number of masses considered, the type of coordinates employed, the number of holonomic constraints on the system, and whether or not the constraints and frame of reference are in motion. Hence, the Lagrangian approach replaces a large set of special methods that must be utilized for other methods. The Euler-Lagrange equation is also valid in any coordinate system, inertial or not, as long as generalized coordinates are used. Chapter 4 shows that the Lagrangian formulation is also very useful in the formulation of minimum control effort optimization criteria, proofs of stability (by using Lyapunov theory and Barbalat’s lemma) and proofs of optimality.

5.2 Recommendations for Future Work

Theoretical fields of interest for future studies are:

• Proofs of stability of nonlinear and adaptive controllers with state estimators.

• The study of energy-based adaptive controllers with exponential stability, such that the rate of convergence a > 0 in V^z, t) = e~atV(z0,t0) can be chosen. A particularly attractive feature would be to calculate a from an optimization criterion.

• The design of adaptive controllers which are only a function of the reference trajectory zT = [xJ,xJ]T, and zr and an augmented error signal such that 9 = —Kgip(zT, zr)e. This will cause a more robust and efficient adaption law since the noise is completely removed from the off-line computed regressor.

• The study of robust optimal nonadaptive nonlinear controllers. This family of controllers is particularly attractive when we have a relatively good a priori knowledge of the process’ parameters. Many problems with the implementation of adaptive controllers can therefore be avoided.

5.2. RECOMMENDATIONS FOR FUTURE WORK 129

We are looking forward to the first tests of the completed NEROV in the Ocean Basin at the Department of Marine Engineering and Naval Architecture. We are hopeful that the vehicle will be ready for these tests by Winter 1992. Three tasks have to be completed before we can say that we have an operational vehicle:

• We must integrate the Kearfoot accelerometer with the rest of NEROV’s sensor suite to obtain a better velocity estimate. This is particularly necessary when adaptive controllers are to be tested on the vehicle.

• We should install the multiprocess multiprocessor real-time operating system Vx- Works on the vehicle’s VME rack. The system is already purchased and will be installed the coming spring. We have so far used a small OS-kernel (PEPBug) and some administrative routines written by the undersigned.

• The new modified analogue thruster control cards should be completed. The derivative effect on the old cards is removed, since we have too much noise from the tachogenerators. Hence, the new cards have only PI action.

Relative navigation and dynamic positioning of an underwater vehicle by means of sensor data from a camera together with a sonar is a particularly useful and interesting field of research. The NEROV can be very useful in this work. The NEROV should also be a vital part of the telerobotics laboratory which is under development at the Division for Engineering Cybernetics under the MOBATEL program.

130 CHAPTER 5. CONCLUSIONS AND RECOMMENDATIONS

Appendix A

Proofs

This appendix contains the proofs of lemma 4.3, comment 4.1 to theorem 4.4 and an alternative proof of theorem 4.5.

Notice that (4.32) represents a nonautonomous system since a matrix -A(z) can be written as

A(z) - A(z + zr(t)) = A(z,t)

A result of this is that the gradient of A with respect to the state z becomes

dA(z,t) dA{z,t) i dA(z,t)dt dz ~ ^ dt (A.l)

wheredA(z,t) dA{z,t) _■

dt dzr(A.2)

and denotesoZtdA(z, t) ,• £\dA(z + zT).

d*r ~T~h 9zri ZT' (A.3)

A.l Proof of Lemma 4.3Eq. (4.2) constitutes a necessary condition for optimality of the cost function in (4.1) with the Lagrangian in (4.35). The feedback law in (4.38) is found by minimizing the Hamiltonian H in (4.2) such that

dHdu

d_du

L(z(t),ux(t)) + dJ*(z,t)T,dz (A.4)

131

132 APPENDIX A. PROOFS

We need first to evaluate the gradient of J* with respect to the error state z.

iT* = iTrr (S'1 + 0,„ |r.;Onxn

dJ*T,

di( ^Xn oT ')i-*TT?E(D(z) + N(z)ETT°* + *TT°Eu

-iTTT ( E1?=1 9~Mf^^-Mr(z) 0nxn+ 25 ^ dxk

On xn On xnT0z

(A.5)

This can be simplified to:r.T

If* = rT(SrtiiTr f s-i - «-w ,j.....

^ V Onxn Onxnr„z

(A.6)

By remembering that N{z) is skew symmetric. The expression for z is found in (4.32). Eqs. (A.5) and the Lagrangian given by

L(z(<),ux(0) = \zT{t) (Q + Tt0ED{z)EtT0) z{t) + l-uTx(t)Rux(t) (A.7)

inserted into the Hamiltonian (A.4) results in the control law in (4.38). We can now use the Hamilton-Jacobi equation (4.2) to check if the found control law is optimal. ^ is

1 ~TmT I dM(* + x<.>)') o

-------= -z‘ T1 at Undt 2 0 l 0nXnxn I T r — -zi T0„x„ j ° 2

? — _ ** tT I 2^* = 1 drrL Xr* unxndxrOnxn

T0z

hence

dJ*(z,t) + dJ*{z,t) \

zT{t)

dt

Onxn K K 0nXn

z + L(z(t),u*(t)) = 0

z(t) --- 0

dz

+ Q-TT0ER'ETTi

The last transition uses the fact that

A.2. PROOF OF COMMENT 4.1 133

This concludes the proof.

□A proof following the same line of reasoning is found in Johansson (1990b) for a manipulator without friction i.e. dissipative forces. The Hamilton’s principal function of optimization is the same, but this article’s cost function is altered to account for the energy spend to overcome dissipation of energy from the vehicle. The equations of motion are, of course, also different.

A.2 Proof of Comment 4.1

It is sufficient to prove that comment 4.1 is valid without the dissipation matrix, since comment 4.2 shows that the inclusion of damping only increase the stability of the system. Remember also that we can, without loss of generality, assume that the R matrix is on the form R — r ■ Imxn

dV(i,t)dt

dV(2,t)dt

dV(i)dt

dV(~z)dt

dV(~z)dt

= ) i(t) - z(t)TTÊ (R-' + D(z)) ETT0z(t)

11

Z

2

-5*T(0

drop the dissipation term

5(0 - iitfTÊRÊ7^^)£»)

l.T„, / 2Tf1ii-1Tn -K + QThR-'Tu){ -K + 2T'f2R-lTn 2Tf3R-1T13 J U

Qi2*T(0 ( _I (qTq2 + qTqJ + 1 (qT + qt'j + 2QtQi

-5 (QlQi + Q2Q1) + | (Qfa + Q?i) + 2QTQ2 \ i(t)

<h J11

< o v*(o / o (A.8)

Hence we have shown that the system described by (4.33) and controlled by (4.38) is always globally uniformly asymptotically stable by explicitly differentiating V.□

134 APPENDIX A. PROOFS

A.3 Alternative proof of Theorem 4.5We chose the Lyapunov function candidate in (4.53). It is easiest to show that Vê, t) is negative semi-definite by expressing the Lyapunov function candidate in the x-frame formulation and then transform the solution to the q-frame. V(e, t) is clearly positive definite and radially unbounded in z. If we differentiate (4.53) along z and time t we get:

V'(e,t) = V(z,t) + Vu(6q) (A.9)

A combination of (A.9) with the control law in (4.52) and the results in theorem 2 yields:

^ ~T d\ ( xn K I rT'T ip i 7>~ l i ipT v ^v, = (<)£Xn ) Ht) - i(t)TTjE(R-1 + D(z)) ETT0~z(t) + + iT(t)T?Eil>x8z

V = _Vmrr/ 2Tl(R-l+p(z))Tn -K+ 2^ (R'1 + D(z)) T12 \ T-(t)* 2P u 1,-Ar + 2rf2(ii-1 + £>(z))r11 2tJ2(r~1+ d(z))t12 JTp{)

+6*Kiq9q + pT{t)TTTT0 EJ-T(*)V>,e, assuming 0 = 011

v; < o

We recognize the first parenthesis as positive definite from theorem 2 and that the two last terms cancel each other out when 0q is defined as in (4.55). Hence, we have proved that V(e,<) < 0. This implies that V{t) < 1/(0) which, in turn, implies that p is bounded for all t > t0. Consequently, V(e, t) is a valid Lyapunov function and the system is globally asymptotically stable. We have uq = u* in the case of no parameter errors and, hence, we have an optimal controller with respect to the performance criterion in (4.1) with the Lagrangian in (4.35)□

Appendix B

Fluid Mechanics and Hydrodynamics

This appendix introduces the reader to some fundamental hydrodynamic properties which will be needed in Chapter 2 when we introduce the term velocity potentials. This theory will be useful for the explanation of added mass in the next chapter. Most of this appendix is taken from Sarpkaya (1981), Newman (1977), and Faltinsen (1990a). The first section introduce to the reader some basic hydrodynamics and fluid mechanics, the next section derives the equation for the linear wave potential, while the third section expands the linear theory to also include the nonlinear Stoke’s drift.

B.l Basic fluidmechanics and hydrodynamics

Velocity potential

A velocity potential <j> can be used to describe the fluid velocity vector r (x, y, z) = (u, v, w) at time t at the point x = (x, y, z) in a cartesian coordinate system. This implies that

„ , _ .d<t> .d<j> d<t> ,V = ^* = ,di+,Yy+kdi (EU)

where i, j and k are unit vectors along the x—, y— and 2—axes. Notice also that the dynamic pressure po is

(B.2)

135

136 APPENDIX B. FLUID MECHANICS AND HYDRODYNAMICS

Equations of motion

The equation of motion for an incompressible Newtonian fluid known as the Navier-Stokes equations may be written as

D'v di) 1= -57 + (» ffrad) v = F----Vp + i/V2v (B.3)

Dt ut p

in which v represents the velocity vector of the water particles in x, y and z directions respectively, F the components of the body force per unit mass in the corresponding directions, p the pressure and u the kinematic viscosity of the fluid. The term ^ denotes the substantive acceleration, also known as the material derivate or Eulerian derivate. The above equation consists of a local acceleration due to the change of velocity at a given point in time and a convective acceleration due to translation. A velocity potential is useful in analysis of irrotational fluid motions.

Irrotational fluid

A fluid is said to be irrotational when the vorticity vector

u = V x v (B-4)

is zero everywhere in the fluid.

Incompressible fluid

Since an incompressible fluid satisfies V • w = 0, it follows that the velocity potential satisfies the Laplace equation

d2<j> d2<f> =dx2 + dy2 + dz2 (B.5)

Bernoulli’s equation

Below is Bernoulli’s equation which is valid for unsteady, irrotational and invicid fluid motions.

9<t> P „p + pgz + p— + -v -v = C

C is a constant which can be related to the ambient or atmospheric pressure.

(B.6)

B.2. LINEAR WAVE THEORY 137

B.2 Linear wave theoryIn this appendix we present the linear velocity potential for regular sinusoidal propagating waves on finite and infinite water depth. Linear theory means that the velocity potential is proportional to the wave amplitude. The theory is valid when the wave amplitude is small relative to the wave length and the still-water depth. This linear wave theory is also known as small amplitude wave theory, sinusoidal wave theory or Airy theory.We use the following assumptions to derive the equation for the velocity potential.

Incompressible fluidThat is a fluid which satisfies Laplace equation, B.5

Sea bottom conditionThe velocity of the water particles is zero in the z—direction at the seabottom i.e.

86— = 0 on z = —h (B-7)

Kinematic surface conditionThis condition is based on the assumption that a fluid particle on the surface is assumed to stay on the surface. First we define the free-surface by the equation z = C(x, y, t) where £ is the wave elevation. Secondly, we define the function F(x,y,z,t) = z — ((x,y,t). The assumption that a fluid particle on the surface stays on the surface implies that

DFDt = 0 ^+r-VF = 0

dt (B.8)

Evaluating eq. B.8 and keeping only surface condition.

8( 8<i>dt dz

the linear terms we obtain the kinematic

on z = —h (B-9)

Dynamic free surface conditionBy inserting eq. B.l and eq. B.2 in eq. B.6 and choosing the constant C as p0/p, where p0 is the atmospheric pressure, we arrive to the following expression:

9(d<j) 1

+ W + 2 = 0 on z C{x,y,t) (B.10)

Keeping only linear terms in B.10, the expression is simplified to

9( +d<j>It = 0

which is the dynamic free surface condition.

(B.ll)


By combining B.5 B.7, B.9, and B.ll we arrive to the following equation for the velocity potential for finite water depth Walderhaug (1990):

4g( cosh k(z + h)= — --------- 7—r-.----cosfurt — kx)oj cosh kh

(B.12)

where k can be found from

— = k tanh kh9

(B.13)

Finite water depth Infinite water depth

velocity potential <f> = êkz cos(wt — kx)

wave number k and circular frequency ui

— = k tanh kh9 f-*

wave length A and wave period T

A = £T2 tanh Ifh * = £T2

wave profile c = Ca sin(wt - kx) = (fa sin(wt — kx)

dynamic pressure PD - P9(a c<Jlt } smM kx) PD = P9(aekz sin(wt - kx)

x-component of velocity u = U(a^^^t-kx) u — u(aekz sin(wt — kx)

z-component of velocity w - wCa -sJiu1 cos(wt kx) w = u)(aekz cos(wt — kx)

x-component of acceleration U - w2Ca - sidlt ) C0s(wf kx) ii = u)2(aekz cos(u;t — kx)

z-component of acceleration W- kx) w — —Lo2C,a.ekz sin(wt — kx)

Table B.l: Equations derived from <j> Faltinsen (1990a) T = Wave period, ui =A =Wavelenght, k = , (a = Wave amplitude, g = Acceleration of gravity, z positiveupwards, h = waterdepth, total pressure in fluid: po — pgz + p0

Table B.l contains some useful equations which are derived from the velocity potential. It is common to divide the range of water depth into shallow water, intermediate water and deep or infinite water where:

B.3. NONLINEAR EFFECTS - STOKE’S DRIFT 139

shallow water waves m > I

intermediate depth water ^ f < f

deep water waves I > 2

B.3 Nonlinear effects - Stoke’s driftIt is relatively easy to show that the first order wave potential is also a solution to the second-order kinematic surface condition eq. B.9 and the second-order dynamic free surface condition eq. B.ll, Newman (1977) pp. 247-248. Therefore, the first-order wave potential (for infinite water depth) can be written as

<f> = y^-ek2 cos(ujt — kx) + 0((3) (B.14)OJ

The free-surface elevation must, however, be corrected for second-order effects by including the second-order terms in eq. B.10. Eq. B.10 can be rearranged to:

(bi5)

We assume that £ can be written in the form ( = e<j>i + e2^2 + By writing ( eq. B.15 in this power series expansion form up to second-order terms, we obtain the following expression for (, Newman (1977):

c= - 1

9 dt + -V^ • 1 d(f> d2<j>g dt dzdt z=0

(B.16)

Combining eq.B.14 and eq.B.16 we get the following second-order expression for the free- surface elevation:

C = Ca sin(a;< — kx) — ^Co + ^Ca sin2(wt — kx)

C = Ca sin(wt — kx) — T^Ca cos2(wt — kx) (E-IT)

We see from this equation that the crests have become steeper and the troughs flatter as a result of the nonlinear terms. This asymmetry cause a second order mean drift of the


fluid particles in the same direction as the wave propagation. This mean drift is a net flux of water in the same direction as the waves.

We will now calculate the velocity components of a given fluid particle. Consider a particle at the position (x0,z0). This particle must satisfy the relations ^ = u(x0,zo,t) and ^ = w(x0,z0,t). We can expand these expressions by a Taylor series to

^ = u(x0,z0,t) + (x0-x)^ + (z0-z)^ + 0(O (B.18)

= w(x0,z0,t)+ (x0-x)^+ (z0-z)^+0(fi)

We have in Table B.l (column 2, row 6 and 7) found an expression for the velocity of the given water particle on infinite water depth for the x— and z—direction. If we integrate these expressions with respect to time, we get the first-order trajectories

(x — x0) = J u dt = —(aekz cos(wt — kx) (B.19)

(z — Zg) — J w dt = C,aekz sin(uit — kx)

If we combine eq.B.20 and the equations for water particle velocity in Table B.l with eq. B.19 we obtain the following:

u>(aekz sin(wt — kx) + + O(C^) (B.20)

u>(aekz cos(ut — kx) + 0(Cf) (B.21)

We see that the second-order vertical motion of the given water particle is periodic while the horizontal velocity contains a steady drift term, called the Stoke’s drift.

This horizontal drift influences on the motion of marine vessels on and in the water column. This drift is an important factor to take into account when disturbances are modeled for marine vessels. Notice that the average velocity at a fixed point is zero according to the second-order wave theory, while the velocity of a specific wave particle is the same as the constant drift term in eq. B.21. This is important when we want to model the current. If we want to model the current at a fixed point in space, the Stoke’s drift should not be taken into account when there are waves present Faltinsen (1990a).

dx0dtdz0dt

Appendix C

The NEROV

This appendix is outlined as follows. The next section gives the background, program goals and objectives for the NEROV project. The third section presents NEROV’s general arrangement, while sections four, five, and six, respectively, describe the propulsion system, sensor system, and computer system.

C.l NEROV

FORWARD

NEROV computer

Figure C.l: The NEROV vehicle - general arrangement

141

142 APPENDIX C. THE NEROV

C.2 BackgroundThe NEROV vehicle is a part of the ongoing effort at the Division of Engineering Cybernetics - Norwegian Institute of Technology in the field of underwater robotics. The NEROV vehicle will be particularly useful for in research in teleoperation and telemanipulation.

The NEROV vehicle is a testbench for testing intelligent and traditional control algorithms for autonomous and remotely operated robots with special focus on underwater vehicles. Automatic modes in the operation of remote manipulator systems and problems related to time delays in the communication channel between the operator and the robot are of particular interest.

The NEROV vehicle will be a useful simulation and research tool for the underwater industry as well as the oil companies. This is especially true in the context of advanced operation of ROVs.

C.3 Program Goals and ObjectivesThe primary goal was to design and develop a vehicle with state-of-the-art processing and data storage capability, navigation, and sensor systems.

Computer hardware and software architecture should be designed so that both traditional and intelligent control schemes can be developed and tested on the vehicle. The system should be totally autonomous with respect to both energy and control, but the vehicle has, in the first stage of the project, a communication cable from the vehicle to an onshore computer monitoring system.

The specific objectives of the study includes:

• Designing and developing a controllable underwater vehicle with six degrees of freedom.

• Integrating a suite of sensors to the vehicle.

• Conducting performance tests of the vehicle.

• Using the vehicle as a general purpose testbench in research on underwater robotics.

• Using the vehicle as a laboratory in graduate courses and research.

CA. GENERAL ARRANGEMENT 143

Using the vehicle as a simulation and research tool for the underwater industry.

C.4 General Arrangement

Figure C.l shows the general arrangement of the NEROV vehicle. The vehicle’s main dimensions are 1.065 x 0.945 x 1.005 [m] [L x B x H]. The vehicle has a deplacement of 0.183 m3 and a positive buoyancy of about 20./V. The vehicle’s six thrusters have a maximum thrust of about 78N each.The vehicle is controllable in six degrees of freedom and is stable in both pitch and roll. The limitations in the pitch and roll are with the current BG ±40° in pitch and ±20° in roll respectively. Its maximum speed in the x, y and z directions is 0.8—, 0.7— and 0.7y respectively.The supporting frame is made of stainless steel tubes. The cylindrical containers are made of PVC plastic and are rated for a depth of 100 m. The energy source which is two 12V batteries with a capacity of 69Ah each, is located in the lower cylindrical container. The computer hardware and the sensor system are located in the two upper containers. The NEROV’s general arrangement is described in more detail in Sagatun and Fossen (1990b)

C.5 Propulsion System

The NEROV thruster system is described in section 3.8.3.

C.6 The Sensor System

The NEROV vehicle is well instrumented. We have currently instrumented the vehicle as follows: two Schaevitz LSOP-90 inclinometers, one Robertson RFC250 fluxgate compass, a three-axial angular rate sensor from Watson Industries, one Keller EI-72 pressure meter and the UPOS hydroacoustic positioning system. The NEROV sensor system is described in Fossen and Sagatun (1991b). Figure C.3 is a schematic drawing of the NEROV’s sensor system with its Kalman filter based state estimators. As the figure illustrate, the linear and angular velocities are estimated independently of each other. We are currently working with integrating a 2-axis Kearfoot accelerometer to the vehicle’s sensor suite. J\{x2) and J2(x2) represent the velocity Jacobian and angular velocity Jacobian matrices. The Jf1^) matrix is singular for 0 = |, that is for a pitch angle of 90°. Experimentally obtained sensor data with their Kalman-filter estimate is shown in Fig. C.4.


C.7 The Computer SystemThe computer system is based on the VMEbus and is located in the vehicle’s upper right container. Fig. C.2 contains a schematic drawing of the NEROV computer system. We had to use the single height Eurocard format due to the size of the vehicle’s containers. The processor board consists of one Motorola MC68020 32 bits 16 MHz microprocessor, one MC68881 floating point co-processor, 1 MByte of static RAM and 512 KBytes of ROM. The microprocessor card is fitted with two RS232 serial ports. These ports are used to connect a terminal and a monitor to the system while programming, testing, and operating the computer system.

We also have two computer cards which take care of the I/O to and from the sensors and the propulsion system. We currently have sixteen 12 bits A/D input channels, eight TTL I/O ports and twelve A/D output channels. All I/O ports are programmable.

The software is interrupt driven and is programmed in C on a SUN workstation and downloaded through a serial port to the VME rack. The NEROV computer system is described in Sagatun and Fossen (1991b).

We are planning to install the real-time operating system VxWorks on the vehicle the coming season.

C.7. THE COMPUTER SYSTEM 145

The NEROV computer system

VMEbus

sensorsystem

thrustersystem

RS232UPOS transponders

RS232RS232

Joystick

VMOD 8 x DA ports

UPOS rackLandbased

VDAD16 x AO ports 4 x DA ports

MC68020 uP MC68881 f.coP1 Mbyte SRAM

-512 Kbyte ROM1B7MHZ2 x RS222 ports

Figure C.2: The NEROV computer system


Linear velocities and position (surge, sway and heave)

Keller EI-72 pressure meter

UPOSpositioning system

Angular velocities and Euler angles (roll, pitch and yaw)

theta

Robertson fluxgate compass RFC250

Watson Ind. Inc. 3-axial angular rate sensor

Schaevitz LSOP-90 Inclinometer

Schaevitz LSOP-90 Inclinometer

Figure C.3: Schematic drawing of the sensor system.

C.7. THE COMPUTER SYSTEM 147

Actual (dotted) and estimated (solidl roll angle Ide^l

Actual (dotted^) and estimated (solid) angular rate in roll Ideg/sl

time [s]

Roll angle estimation error Ideal

time [s]

Angular rate estimation error in roll fdeg/s 1

Figure C.4: Experimentally obtained sensor data with and without Kalman filter Fossen and Sagatun (199lb).


Appendix D

Results from the Free Decay Test

This appendix presents the results from the free-decay test. The test gave us the added mass terms for the vehicle’s x-, y- and z-direction as well as the linear and quadratic damping terms. We performed the test for three frequencies (« 1.3, ~ 0.8 and « 0.4 r^). We also investigated how sensitive these parameters are to perturbations of the vehicle’s geometry. We did the test for two different perturbation objects for all three frequencies. The largest perturbation object was a second battery-container with a deplacement of 0.042 m3 and the small object was four cylinders, simulating a manipulator, with a deplacement of 0.009 m3. See also section 3.13.5.

The results from these tests are shown in Table 3.1 , and Figs. D.l, D.2 and D.3.

149

150 APPENDIX D. RESULTS FROM THE FREE DECAY TEST

linear damping term

0)[rad/s]

*9

4

3

2

1

0

0.5 1 1.5(0 [rad/s]

(juadratic damping term— i

—&-1

___________ 1____________________ 1____________________ i_______

6

2

1.5

1

0.5

0

0 0.5 1 1.5CO [rad/s]

added mass coeffecient------------------------1------------------------------- 1 ------------------------------- 1

' +■

____________________ i ____________________ i

Figure D.l: Nondimensional free decay test-results for the x-direction. The upper leftframe contains the dimensionless linear damping coefficient , upper right frame containsthe quadratic damping coefficient in dimensionless form, while the bottom left frame showsthe added mass coefficients (made dimensionless such that ttia = CapX J, solid = unperturbed geometry, dashed — small perturbation and dashdot = large perturbation.

151

*cs£>

0.1-

0.05-

—

.......—*

y*

) Aft*.....s

S'

........—

___________ 1

*w>>X)

0.5

quadratic damping term

~ ^ ~ i _r_'_ ■- ■'•

0.5

added mass coeffecient

Figure D.2: Nondimensional free decay test-results for the y-direction . The upper leftframe contains the dimensionless linear damping coefficient , upper right frame containsthe quadratic damping coefficient in dimensionless form, while the bottom left frame showsthe added mass coefficients (made dimensionless such that mA = CapV ), solid = unperturbed geometry, dashed = small perturbation and dashdot = large perturbation.

152 APPENDIX D. RESULTS FROM THE FREE DECAY TEST

linear damping term quadratic damping term

--f

-----------------—1

______+---•

r ~~1

________105 1

added mass coeffecient

Figure D.3: Nondimensional free decay test-results for the z-direction . The upper leftframe contains the dimensionless linear damping coefficient , upper right frame containsthe quadratic damping coefficient in dimensionless form, while the bottom left frame showsthe added mass coefficients (made dimensionless such that = CapV ), solid = unperturbed geometry, dashed = small perturbation and dashdot = large perturbation.

Appendix E

A Complete Mathematical Model

This appendix contains the complete equations of motion for an underwater vehicle as well as the corresponding numerical values for the NEROV.

E.l Complete Equations of MotionThe following limitations applly to the model presented in this section:

• Environmental forces as wave and current loads are not included.

• Some cross-coupling terms for the dissipative forces are neglected.

• A rigid body with constant mass and deplacement is assumed.

• Off-diagonal added-inertia terms are neglected in the nonlinear matrix.

• The control forces consist of only thrusters placed in a configuration similar to the NEROV.

The following model will be used:

Mq + C(q)q + D(q)q + g(x) Ter + b{q,u)

x = J(x)qwhere the vectors and matrices are written out:

M -

m- Xi -Xi -Xu,

-Xpmza - Xq

-mya - Xr

-Xi m — Yi-Y*

-mza - Yp -Yi

mxa - Yr

-Yi,m — Xu,

myc - Zp -mxa — Zq

-Zi

-Xp

-mzG - Yp mya — Zph-Kp

-by - Kq

-It: - Ki

mza - Xq

~Yimxa — Zq -hy - Kq ly-Mq

-lyz - Mr

-myG - Xr -mxG - Yr

-Zr-Izz - Kr

fyz Mr h ~ Ni

153

154 APPENDIX E. A COMPLETE MATHEMATICAL MODEL

000

-m(yaq + zar) m(xaq - w) + Z^w m(xar + v)-YiV

000

m(!/GP+ w)~ Zu,w -m(zar + xap)

m(yar- ul + Aû

000

m(zGp — v) + YiV m(zaq + u) — X^xi -m(xap + yaq)

m(ycq + ZGr) —m(xGq -w)-Z* -m(xGr + v) + YiV-Tn(yap+ w) + Z^w /n(zGr + xap) —m(yGr — u) - Xû-m(zGP-v) - YiV -m(zGq + u) + XiU mixaP + yaq)

0 -IyZq ~ h*P+ hr - Nir IyZr + hyP - lyQ + Mjqlyzq + EtP- hr + NiV 0 -hzr - hyq + hp - Kpp-Iy*r - hyP+Iyq ~ Mjq hzr + hyq - hp+Kfp 0

■ /;< frvt fxzi 0 0 0 ■/yuj fyw, 0 0 0

D(qT)qT = fzui0

fzvi-Zpfvd

rJ wdypfwd

0

/;„0

0

0

0 9r

Zpfud 0 ~Xpfwd 0 ru 0

. -ypftd *pf:d 0 0 0/;J

(3.1)

where the terms on the form /t^ is defined according to Newman (1977) p.21, such that fzu, = \pCfA |ur|, that is fzxiltur is the force in z—direction due to a relative velocity in the x—direction.

(W-B)smO —{W — B) cos 6 sin <f>— (W — B) cos 9 cos <f>

— — VbB) cos 5 cos <j> + (zoW — zbB) cos 0 sin {xqW — xgB) cos 9 cos <j> + (zqW — zgB) sin 9

—(xgW — XBB)cos9sm(f> — (yGW — yBB)sin9

b\rii | n\ | +62 | 9 I2 I I +(>1^2 | ^2 I +(*2 I 9 I2 | ^2 |b\nz | n3 | +62 j 9 I2 I ”3 I +61^4 | | +62 | 9 b j «4 |

b\ns | n5 | +62 | 9 b I n5 | +61 n6 | n6 j +62 j 9 b I n6 j(61/23 | ns | +62 | 9 I2 I n3 |)/«

— (61/I5 | 2^5 | +62 | 9 I2 | n5 |)(y + (61/I6 I ne I +62 I 9 I2 | |)fy

. (6ini | nj | +62 | 9 I2 | ni \)lz - (b^ | n2 | +62 | 9 |2 | n2 |)(,

E.2. NEROV MODEL DATA 155

TU

TV;TuyTU,

VuTU, — ZuTrjy

Z-uTu, ~ XUTU, Z'uTUy VuTU, .

And finally the kinematic relation:

x = J(xE)q

X apcO — sx[>c<l> + ClpS0S<j> sxl>s<j> + ctjjc<j)s9 u

y sipcO Clfrctj) + S<j>S0SlJ> —cifts4> + s9sij>c<j> 03x3 Vz —sO c6s<j> c9c<j> w<t> 1 s<t>s9 c<t>s$

1 c$ cB p6 03x3 1

o

<1

. ^ . .n M c±U C0 cfl J

r

E.2 NEROV Model Data

The following assumptions are made for the NEROV nonlinear equation of motion in addition to the ones cited above:

• The vehicle-fixed coordinate system is placed in the center of gravity.

• We assume xy- and yz-symmetry.

• We neglect the effect of an umbilical.

• Only the experimentally determined or estimated hydrodynamic values are included in the model.

Kp, and N+ is estimated to have the same magnitude as the corresponding inertias. The damping terms are found in the previous appendix. An equvivalent linearized dampping matrix is found in chapter 3:

156 APPENDIX E. A COMPLETE MATHEMATICAL MODEL

parameter values unitm = 183 [Mfn = 192.5 [MV = 0.187 K1B = 1810 [WW = 1795 [N]~BG = [-0.014, 00.053]t [m]KG = [0.020, 0,0.461]t [m]

' 20.6085 0 -0.1806 'I = 0 24.1959 -0.1427 [kgm2]

-0.1806 -0.1427 18.9706144 [M240 [%]

z ~ 211 MK- ~ 20 [kgm2}Mi ~ 24 [kgm2]Ni ~ 18 [fcg'm2]

Table E.l: NEROV masses and inertias

0.8ni | nx | -4.71 | 9 I2 | I +0.8n2 | n2 | +4.71 | 9 I2 | «2 | 0.8n3 | n3 | -4.71 | 9 (2 j ra3 | +0.8ri4 | n4 | +4.71 | 9 b I ^4 |

... . _ 0.8ns j n5 j -4.71 | 9 I2 I | +0.8n6 j n6 | +4.71 | 9 I2 | "e |[q,U) 0.33n3 | n3 |-1.92 | 9 I2 | n3 |

—0.32ns | ns | +1.88 | 9 I2 I ns | +0.32ng | n© | —1.88 | 9 I2 I jQ.34ni | nx \ —1.98 | 9 I2 | I —0.34n2 | n2 | +1.98 | 9 |2 | n2 |

Notice that we have used the simplification m = m.

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Index

actuator dynamics, 78adaption law, 97, 109adaptive control algorithm, 89, 96, 109added inertia, 25, 35, 67advance coefficient, 48application areas, 2asymptotically stable, 62autonomous, 92autopilots, 81AUV, 3

bandwidth, 82 Barbalat, 92, 97, 110 basic assumptions, 88 Bernoulli’s equation, 42, 134 bound on the parameter vector, 122 boundary condition, 29 boundary-layer separation, 42 Butterworth filter, 65

center of buoyancy, 46 center of pressure, 42, 44 center of gravity, 34, 44, 46 centripetal, 25Cholesky decomposition, 106, 104 commanded angular velocities, 95 contributions, 5control and guidance systems, 78control electronics, 52control forces, 47control law, 95control objective, 90, 102control variable, 87, 89coordinate systems, 11

Corolis, 25Coulomb damping, 40 current, 23

D’Alambert’s paradox, 40damping matrix, 25damping ratio, 75dead-zone, 123deep water waves, 136design requirements, 82desired accuracy, 83dissipative forces, 40, 61, 94, 105disturbances, 50divers, 1DP system, 77drag forces, 41duct, 49, 51dynamic free surface condition, 135 dynamic positioning systems, 79 dynamical stability, 57

efficiency, 49 eigenvalue analysis, 74 end-point dependent, 45 energy conservation, 27 energy dissipation, 40, 42 equations of motion, 25, 151 ergodic process, 17 estimation, 21Euler angle representation, 11, 12 Euler parameters, 11, 15 Euler’s theorem, 30Euler-Lagrange equations, 26, 40, 46, 58, 63,

86

167

168 INDEX

experimental setup, 65

fail-safe requirements, 83 fast fourier transformation, 18 feedback linearization, 99, 113 feedforward linearization, 99, 113 fluid’s kinetic energy, 36 fluidmechanics, 133 frictional forces, 40 Froude-Krylov hypothesis, 55

gain scheduled parameters, 21, 92 Gauss’ theorem, 30 generalized coordinate, 15, 25, 26, 58 generalized forces, 28 generalized momentum, 27, 63 global uniformly asymptotic stability, 102,

105globally asymptotically stable, 89gravity forces, 46Green’s second identity, 30

Hamilton principle function of optimization, 84

Hamilton’s canonical equations, 27Hamilton-Jacobi equation, 84, 90, 103, 130Hamiltonian, 26, 27, 63, 84, 129high frequency model, 79holonomic, 28hydrodynamics, 133 .hydrostatical analysis, 57

implementation, 99, 112incompressible fluid, 134, 135inertia matrix (tensor), 25, 34, 59, 61inertial reference frame, 58inertial-system, 11input uncertainty, 117integral action, 120intermediate depth water, 136irrotationa! fluid, 134ISSC sea spectrum, 18ITTC sea spectrum, 18

Jacobi’s integral, 27 Jacobian matrix, 14, 58 JONSWAP sea spectrum, 19

Kalman filter, 21, 79, 141 Keulegan-Carpenter (KC) number, 55, 68,

74, 78kinematic surface condition, 135 kinematics, 11, 153 kinetic energy, 26, 58 kinetic energy of the fluid, 29 kinetic energy of the solid, 31 Kirchhoff’s equations, 28, 33, 60

Lagrangian, 26, 46, 84, 90, 103, 106 Lagrangian formulation, 25, 27 Lagrangian mechanics, 5 Lagrangian system, 84 Laplace’s equation, 30 Legendre transformation, 63 linear quadratic criteria, 84 linear damping, 44, 65 linear velocity potential, 135 linear wave potential, 133 linear wave theory, 135 linearized equations of motion, 70 low frequency model, 79 lower bounded, 92 Lyapunov function, 105, 109, 132 Lyapunov theory, ,92 122

mass matrix, 38 measurement vector, 89 minimum effort controllers, 4 minimum control effort, 4 minimum control effort criterias, 85 minimum energy, 85 minimum fuel, 84 model properties, 59 modified parameter update laws, 122 Morrisons equation, 55

natural frequency, 75, 78

INDEX 169

Navier-Stokes equations, 134 near optimal, 89 negative semi-definite, 92 NEROV, 66, 139NEROV nonlinear equation of motion, 153 NEROV background, 140 NEROV computer system, 5, 141 NEROV free decay tests, 6 NEROV general arrangement, 141 NEROV mathematical model, 5 NEROV model data, 153 NEROV overall vehicle design, 5 NEROV propulsion system, 5, 50 NEROV sensor system, 5, 141 NEROV state-space model, 73 Newtonian, 26 Newtonian damping, 40 nonautonomous, 90, 102, 129 nonholonomic, 15 nonlinear multivariable PID, 121 NOpAC, 88, 89, 99

off-diagonal coupling terms, 70 open water test, 52 open-loop poles, 74 optimal feedback law, 91, 104, 106 optimal planning, 3 optimization, 83 optimization criterias, 5 overshoot, 83 OpAC, 88, 102

parameter variation, 83 passive, 61perfect decoupling, 120 performance index, 85, 86, 90, 103 periodical inspection, 1 PID, 52, 121Pierson-Moskowitz spectrum, 18 position feedback loop, 93 positive definite, 31, 98 potential damping, 25, 41, 55

potential energy, 26, 45 prediction, 22 principle of optimality, 4 propeller, 52propellers’ angular revolution, 25 propulsion and control forces, 25, 47

q-frame, 11, 58 quadratic damping, 44 quasi coordinates, 15, 25 quaternions, 11

Rayleigh’s dissipation function, 40 reference trajectory, 88, 118 regular wave theory, 17 reparameterization, 108 restoring terms, 25 Reynold’s number, 43 Riccati equation, 91, 93 rigid body’s nonlinear matrix, 38 rigid body equations of motion, 33 rotation matrix, 13 ROV, 1 rudders, 48

saturation handling, 117 sea bottom condition, 135 sensitivity to noise, 82 shallow water waves, 136 ship-wave transfer functions, 21 similarity transformation, 64 simulation study, 98, 111 singular, 13 skew symmetric, 60 skin friction, 44 small underwater vehicles, 78 stability, 83 state space model, 72state space representation of sea current, 24 state variables, 88 state-space realizations, 17 state-space representation, 71, 89, 102 state-space representations of sea waves, 20

170 INDEX

steady state, 92 steering fins, 48 Stoke’s drift, 133, 137 strictly passive, 62 structure, 89

tachogenerator, 51 task planning, 3The angular velocity transformation matrix,

14the state variables, 72 tracking, 98, 112 trajectory planning, 3 transformation matrix, 13 transient, 92

umbilical, 44uniformly continuous, 92uniformly globally asymptotically stable, 92

vehicle-fixed system, 11 velocity potential, 29, 133 virtual measure of tracking, 120 viscousus effects, 25 VME, 52, 141 VxWorks, 142

wake, 42wake freaction number, 48 wave and current disturbance, 25 wave diffraction, 54 wave forces, 53 wave spectrum, 17 waves, 17way-point planning, 3 work, 86, 87 workless forces, 64

x-frame, 11, 58, 62

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