Thor I. Fossen is a naval architect, cyberneticist, and Professor of Guidance, Navigation, and Control at the Norwegian University of Science and Technology. He received his MS in Naval Architecture and his PhD in Engineering and Cybernetics from the Norwegian Institute of Technology. Fossen was elected to the Norwegian Academy of Technological Sciences in 1998 and became an Institute of Electrical and Electronics Engineers (IEEE) Fellow in 2016.

About the Author xvii

Preface xix

List of Tables xxiii

Part One Marine Craft Hydrodynamics

1 Introduction to Part I 3

Degrees of Freedom and Motion of a Marine Craft 5

1.1 Classification of Models 6

1.2 The Classical Models in Naval Architecture 8

1.2.1 Maneuvering Theory 10

1.2.2 Seakeeping Theory 12

1.2.3 Unified Theory 14

1.3 Fossen's Robot-inspired Model for Marine Craft 14

Component Form 14

Matrix-vector Representation 14

Component Form Versus the Matrix-vector Representation 15

2 Kinematics 17

2.1 Kinematic Preliminaries 18

2.1.1 Reference Frames 18

2.1.2 Body-fixed Reference Points 21

2.1.3 Generalized Coordinates 22

2.2 Transformations Between BODY and NED 23

2.2.1 Euler Angle Transformation 26

2.2.2 Unit Quaternions 32

2.2.3 Unit Quaternion from Euler Angles 38

2.2.4 Euler Angles from a Unit Quaternion 38

2.3 Transformations Between ECEF and NED 39

2.3.1 Longitude and Latitude Rotation Matrix 40

2.3.2 Longitude, Latitude and Height from ECEF Coordinates 41

2.3.3 ECEF Coordinates from Longitude, Latitude and Height 44

2.4 Transformations between ECEF and Flat-Earth Coordinates 45

2.4.1 Longitude, Latitude and Height from Flat-Earth Coordinates 45

2.4.2 Flat-Earth Coordinates from Longitude, Latitude and Height 46

2.5 Transformations Between BODY and FLOW 47

2.5.1 Definitions of Heading, Course and Crab Angles 47

2.5.2 Definitions of Angle of Attack and Sideslip Angle 49

2.5.3 Flow-axes Rotation Matrix 51

3 Rigid-body Kinetics 55

3.1 Newton-Euler Equations of Motion about the CG 56

Euler's First and Second Axioms 56

3.1.1 Translational Motion About the CG 58

3.1.2 Rotational Motion About the CG 59

3.1.3 Equations of motion About the CG 60

3.2 Newton-Euler Equations of Motion About the CO 60

3.2.1 Translational Motion About the CO 61

3.2.2 Rotational Motion About the CO 61

3.3 Rigid-body Equations of Motion 63

3.3.1 Nonlinear 6-DOF Rigid-body Equations of Motion 63

3.3.2 Linearized 6-DOF Rigid-body Equations of Motion 69

4 Hydrostatics 71

4.1 Restoring Forces for Underwater Vehicles 71

4.1.1 Hydrostatics of Submerged Vehicles 71

4.2 Restoring Forces for Surface Vessels 74

4.2.1 Hydrostatics of Floating Vessels 74

4.2.2 Linear (Small Angle) Theory for Boxed-shaped Vessels 77

4.2.3 Computation of Metacenter Heights for Surface Vessels 79

4.3 Load Conditions and Natural Periods 82

4.3.1 Decoupled Computation of Natural Periods 82

4.3.2 Computation of Natural Periods in a 6-DOF Coupled System 84

4.3.3 Natural Periods as a Function of Load Condition 87

4.3.4 Free-surface Effects 89

4.3.5 Payload Effects 90

4.4 Seakeeping Analysis 90

4.4.1 Harmonic Oscillator with Sinusoidal Forcing 90

4.4.2 Steady-state Heave, Roll and Pitch Responses in Regular Waves 92

4.4.3 Explicit Formulae for Boxed-shaped Vessels in Regular Waves 94

4.4.4 Case Study: Resonances in the Heave, Roll and Pitch Modes 96

4.5 Ballast Systems 97

4.5.1 Static Conditions for Trim and Heel 99

4.5.2 Automatic Ballast Control Systems 102

5 Seakeeping Models 105

5.1 Hydrodynamic Concepts and Potential Theory 106

5.1.1 Numerical Approaches and Hydrodynamic Codes 108

5.2 Seakeeping and Maneuvering Kinematics 110

5.2.1 Seakeeping Reference Frame 110

5.2.2 Transformation Between BODY and SEAKEEPING 111

5.3 The Classical Frequency-domain Model 114

5.3.1 Frequency-dependent Hydrodynamic Coefficients 115

5.3.2 Viscous Damping 118

5.3.3 Response Amplitude Operators 122

5.4 Time-domain Models including Fluid Memory Effects 122

5.4.1 Cummins Equation in SEAKEEPING Coordinates 123

5.4.2 Linear Time-domain Seakeeping Equations in BODY Coordinates 126

5.4.3 Nonlinear Unified Seakeeping and Maneuvering Model with Fluid Memory Effects 129

5.5 Identification of Fluid Memory Effects 131

5.5.1 Frequency-domain Identification Using the MSS FDI Toolbox 131

6 Maneuvering Models 135

6.1 Rigid-body Kinetics 137

6.2 Potential Coefficients 137

6.2.1 Frequency-independent Added Mass and Potential Damping 139

6.2.2 Extension to 6-DOF Models 140

6.3 Added Mass Forces in a Rotating Coordinate System 141

6.3.1 Lagrangian Mechanics 142

6.3.2 Kirchhoff's Equation 143

6.3.3 Added Mass and Coriolis-Centripetal Matrices 143

6.4 Dissipative Forces 148

6.4.1 Linear Damping 150

6.4.2 Nonlinear Surge Damping 151

6.4.3 Cross-flow Drag Principle 154

6.5 Ship Maneuvering Models (3 DOFs) 155

6.5.1 Nonlinear Equations of Motion 155

6.5.2 Nonlinear Maneuvering Model Based on Surge Resistance and Cross-flow Drag 158

6.5.3 Nonlinear Maneuvering Model Based on Second-order Modulus Functions 159

6.5.4 Nonlinear Maneuvering Model Based on Odd Functions 161

6.5.5 Linear Maneuvering Model 163

6.6 Ship Maneuvering Models Including Roll (4 DOFs) 165

6.6.1 The Nonlinear Model of Son and Nomoto 172

6.6.2 The Nonlinear Model of Blanke and Christensen 173

6.7 Low-Speed Maneuvering Models for Dynamic Positioning (3 DOFs) 175

6.7.1 Current Coefficients 175

6.7.2 Nonlinear DP Model Based on Current Coefficients 179

6.7.3 Linear Time-varying DP Model 180

7 Autopilot Models for Course and Heading Control 183

7.1 Autopilot Models for Course Control 184

7.1.1 State-space Model for Course Control 184

7.1.2 Course Angle Transfer Function 185

7.2 Autopilot Models for Heading Control 186

7.2.1 Second-order Nomoto Model 186

7.2.2 First-order Nomoto Model 188

7.2.3 Nonlinear Extensions of Nomoto's Model 190

7.2.4 Pivot Point 192

8 Models for Underwater Vehicles 195

8.1 6-DOF Models for AUVs and ROVs 195

8.1.1 Equations of Motion Expressed in BODY 195

8.1.2 Equations of Motion Expressed in NED 197

8.1.3 Properties of the 6-DOF Model 198

8.1.4 Symmetry Considerations of the System Inertia Matrix 200

8.2 Longitudinal and Lateral Models for Submarines 201

8.2.1 Longitudinal Subsystem 202

8.2.2 Lateral Subsystem 204

8.3 Decoupled Models for "Flying Underwater Vehicles" 205

8.3.1 Forward Speed Subsystem 206

8.3.2 Course Angle Subsystem 206

8.3.3 Pitch-Depth Subsystem 207

8.4 Cylinder-Shaped Vehicles and Myring-type Hulls 208

Myring-type Hull 209

8.4.1 Spheroid Approximation 210

8.5 Spherical-Shaped Vehicles 214

9 Control Forces and Moments 217

9.1 Propellers as Thrust Devices 217

9.1.1 Fixed-pitch Propeller 217

9.1.2 Controllable-pitch Propeller 220

9.2 Ship Propulsion Systems 225

9.2.1 Podded Propulsion Units 225

9.2.2 Prime Mover System 227

9.3 USV and Underwater Vehicle Propulsion Systems 228

9.3.1 Propeller Shaft Speed Models 229

9.3.2 Motor Armature Current Control 230

9.3.3 Motor Speed Control 232

9.4 Thrusters 233

9.4.1 Tunnel Thrusters 233

9.4.2 Azimuth Thrusters 234

9.5 Rudder in the Propeller Slipstream 236

9.5.1 Rudder Forces and Moment 237

9.5.2 Steering Machine Dynamics 240

9.6 Fin Stabilizators 243

9.6.1 Lift and Drag Forces on Fins 244

9.6.2 Roll Moment Produced by Symmetrical Fin Stabilizers 245

9.7 Underwater Vehicle Control Surfaces 245

9.7.1 Rudder 247

9.7.2 Dive Planes 248

9.8 Control Moment Gyroscope 249

9.8.1 Ship Roll Gyrostabilizer 249

9.8.2 Control Moment Gyros for Underwater Vehicles 252

9.9 Moving Mass Actuators 258

10 Environmental Forces and Moments 261

10.1 Wind Forces and Moments 263

10.1.1 Wind Forces and Moments on Marine Craft at Rest 263

10.1.2 Wind Forces and Moments on Moving Marine Craft 265

10.1.3 Wind Coefficients Based on Helmholtz-Kirchhoff Plate Theory 266

10.1.4 Wind Coefficients for Merchant Ships 269

10.1.5 Wind Coefficients for Very Large Crude Carriers 271

10.1.6 Wind Coefficients for Large Tankers and Medium-sized Ships 272

10.1.7 Wind Coefficients for Moored Ships and Floating Structures 272

10.2 Wave Forces and Moments 274

10.2.1 Sea-state Descriptions 275

10.2.2 Wave Spectra 276

10.2.3 Wave Amplitude Response Model 287

10.2.4 Force RAOs 290

10.2.5 Motion RAOs 293

10.2.6 State-space Models for Wave Response Simulation 296

10.3 Ocean Current Forces and Moments 300

10.3.1 3D Irrotational Ocean Current Model 303

10.3.2 2D Irrotational Ocean Current Model 304

Part Two Motion Control

11 Introduction to Part II 309

11.1 Guidance, Navigation and Control Systems 310

11.1.1 Historical Remarks 312

11.1.2 Autopilots 314

11.1.3 Dynamic Positioning and Position Mooring Systems 315

11.1.4 Waypoint Tracking and Path-following Control Systems 316

11.2 Control Allocation 316

11.2.1 Propulsion and Actuator Models 318

11.2.2 Unconstrained Control Allocation 322

11.2.3 Constrained Control Allocation 324

12 Guidance Systems 331

12.1 Trajectory Tracking 333

Trajectory-tracking Control 333

12.1.1 Reference Models for Trajectory Generation 334

12.1.2 Trajectory Generation using a Marine Craft Simulator 339

12.1.3 Optimal Trajectory Generation 340

12.2 Guidance Laws for Target Tracking 341

12.2.1 Line-of-sight Guidance Law 342

12.2.2 Pure-pursuit Guidance Law 343

12.2.3 Constant Bearing Guidance Law 344

12.3 Linear Design Methods for Path Following 346

12.3.1 Waypoints 346

12.3.2 Path Generation using Straight Lines and Inscribed Circles 347

12.3.3 Straight-line Paths Based on Circles of Acceptance 349

12.3.4 Path Generation using Dubins Path 351

12.3.5 Transfer Function Models for Straight-line Path Following 352

12.4 LOS Guidance Laws for Path Following using Course Autopilots 353

12.4.1 Vector-field Guidance Law 354

12.4.2 Proportional LOS Guidance Law 356

12.4.3 Lookahead- and Enclosure-based LOS Steering 359

12.4.4 Integral LOS 361

12.5 LOS Guidance Laws for Path Following using Heading Autopilots 363

12.5.1 Crab Angle Compensation by Direct Measurements 363

12.5.2 Integral LOS 364

12.6 Curved-Path Path Following 365

12.6.1 Path Generation using Interpolation Methods 366

12.6.2 Proportional LOS Guidance Law for Curved Paths 378

12.6.3 Path-following using Serret-Frenet Coordinates 380

12.6.4 Case Study: Path-following Control using Serret-Frenet Coordinates 384

13 Model-based Navigation Systems 387

13.1 Sensors for Marine Craft 387

13.1.1 GNSS Position 388

13.1.2 GNSS...