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.

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• About the Author xviiPreface xixList of Tables xxiPart One Marine Craft Hydrodynamics1 Introduction to Part I 31.1 Classification of Models 61.2 The Classical Models in Naval Architecture 81.2.1 Maneuvering Theory 101.2.2 Seakeeping Theory 121.2.3 Unified Theory 141.3 Fossen’s Robot-inspired Model for Marine Craft 142 Kinematics 172.1 Kinematic Preliminaries 182.1.1 Reference Frames 182.1.2 Body-fixed Reference Points 212.1.3 Generalized Coordinates 222.2 Transformations Between BODY and NED 232.2.1 Euler Angle Transformation 262.2.2 Unit Quaternions 322.2.3 Unit Quaternion from Euler Angles 382.2.4 Euler Angles from a Unit Quaternion 382.3 Transformations Between ECEF and NED 392.3.1 Longitude and Latitude Rotation Matrix 402.3.2 Longitude, Latitude and Height from ECEF Coordinates 412.3.3 ECEF Coordinates from Longitude, Latitude and Height 442.4 Transformations between ECEF and Flat-Earth Coordinates 452.4.1 Longitude, Latitude and Height from Flat-Earth Coordinates 452.4.2 Flat-Earth Coordinates from Longitude, Latitude and Height 462.5 Transformations Between BODY and FLOW 472.5.1 Definitions of Heading, Course and Crab Angles 472.5.2 Definitions of Angle of Attack and Sideslip Angle 492.5.3 Flow-axes Rotation Matrix 513 Rigid-body Kinetics 553.1 Newton–Euler Equations of Motion about the CG 563.1.1 Translational Motion About the CG 583.1.2 Rotational Motion About the CG 593.1.3 Equations of Motion About the CG 603.2 Newton–Euler Equations of Motion About the CO 603.2.1 Translational Motion About the CO 613.2.2 Rotational Motion About the CO 613.3 Rigid-body Equations of Motion 633.3.1 Nonlinear 6-DOF Rigid-body Equations of Motion 633.3.2 Linearized 6-DOF Rigid-body Equations of Motion 694 Hydrostatics 714.1 Restoring Forces for Underwater Vehicles 714.1.1 Hydrostatics of Submerged Vehicles 714.2 Restoring Forces for Surface Vessels 744.2.1 Hydrostatics of Floating Vessels 744.2.2 Linear (Small Angle) Theory for Boxed-shaped Vessels 774.2.3 Computation of Metacenter Heights for Surface Vessels 794.3 Load Conditions and Natural Periods 824.3.1 Decoupled Computation of Natural Periods 824.3.2 Computation of Natural Periods in a 6-DOF Coupled System 844.3.3 Natural Periods as a Function of Load Condition 874.3.4 Free-surface Effects 894.3.5 Payload Effects 904.4 Seakeeping Analysis 904.4.1 Harmonic Oscillator with Sinusoidal Forcing 904.4.2 Steady-state Heave, Roll and Pitch Responses in Regular Waves 924.4.3 Explicit Formulae for Boxed-shaped Vessels in Regular Waves 944.4.4 Case Study: Resonances in the Heave, Roll and Pitch Modes 964.5 Ballast Systems 974.5.1 Static Conditions for Trim and Heel 994.5.2 Automatic Ballast Control Systems 1025 Seakeeping Models 1055.1 Hydrodynamic Concepts and Potential Theory 1065.1.1 Numerical Approaches and Hydrodynamic Codes 1085.2 Seakeeping and Maneuvering Kinematics 1105.2.1 Seakeeping Reference Frame 1105.2.2 Transformation Between BODY and SEAKEEPING 1115.3 The Classical Frequency-domain Model 1145.3.1 Frequency-dependent Hydrodynamic Coefficients 1155.3.2 Viscous Damping 1195.3.3 Response Amplitude Operators 1215.4 Time-domain Models including Fluid Memory Effects 1225.4.1 Cummins Equation in SEAKEEPING Coordinates 1225.4.2 Linear Time-domain Seakeeping Equations in BODY Coordinates 1255.4.3 Nonlinear Unified Seakeeping and Maneuvering Model with Fluid Memory Effects 1295.5 Identification of Fluid Memory Effects 1305.5.1 Frequency-domain Identification Using the MSS FDI Toolbox 1316 Maneuvering Models 1356.1 Rigid-body Kinetics 1376.2 Potential Coefficients 1376.2.1 Frequency-independent Added Mass and Potential Damping 1396.2.2 Extension to 6-DOF Models 1406.3 Added Mass Forces in a Rotating Coordinate System 1416.3.1 Lagrangian Mechanics 1426.3.2 Kirchhoff’s Equation 1436.3.3 Added Mass and Coriolis–Centripetal Matrices 1436.4 Dissipative Forces 1486.4.1 Linear Damping 1506.4.2 Nonlinear Surge Damping 1516.4.3 Cross-flow Drag Principle 1546.5 Ship Maneuvering Models (3 DOFs) 1556.5.1 Nonlinear Equations of Motion 1556.5.2 Nonlinear Maneuvering Model Based on Surge Resistance and Cross-flow Drag 1586.5.3 Nonlinear Maneuvering Model Based on Second-order Modulus Functions 1596.5.4 Nonlinear Maneuvering Model Based on Odd Functions 1616.5.5 Linear Maneuvering Model 1636.6 Ship Maneuvering Models Including Roll (4 DOFs) 1656.6.1 The Nonlinear Model of Son and Nomoto 1726.6.2 The Nonlinear Model of Blanke and Christensen 1736.7 Low-Speed Maneuvering Models for Dynamic Positioning (3 DOFs) 1756.7.1 Current Coefficients 1756.7.2 Nonlinear DP Model Based on Current Coefficients 1796.7.3 Linear Time-varying DP Model 1807 Autopilot Models for Course and Heading Control 1837.1 Autopilot Models for Course Control 1847.1.1 State-space Model for Course Control 1847.1.2 Course Angle Transfer Function 1857.2 Autopilot Models for Heading Control 1867.2.1 Second-order Nomoto Model 1867.2.2 First-order Nomoto Model 1887.2.3 Nonlinear Extensions of Nomoto’s Model 1907.2.4 Pivot Point 1928 Models for Underwater Vehicles 1958.1 6-DOF Models for AUVs and ROVs 1958.1.1 Equations of Motion Expressed in BODY 1958.1.2 Equations of Motion Expressed in NED 1978.1.3 Properties of the 6-DOF Model 1988.1.4 Symmetry Considerations of the System Inertia Matrix 2008.2 Longitudinal and Lateral Models for Submarines 2018.2.1 Longitudinal Subsystem 2028.2.2 Lateral Subsystem 2048.3 Decoupled Models for “Flying Underwater Vehicles” 2058.3.1 Forward Speed Subsystem 2068.3.2 Course Angle Subsystem 2068.3.3 Pitch–Depth Subsystem 2078.4 Cylinder-Shaped Vehicles and Myring-type Hulls 2088.4.1 Myring-type Hull 2098.4.2 Spheroid Approximation 2108.5 Spherical-Shaped Vehicles 2149 Control Forces and Moments 2179.1 Propellers as Thrust Devices 2179.1.1 Fixed-pitch Propeller 2179.1.2 Controllable-pitch Propeller 2209.2 Ship Propulsion Systems 2259.2.1 Podded Propulsion Units 2259.2.2 Prime Mover System 2279.3 USV and Underwater Vehicle Propulsion Systems 2289.3.1 Propeller Shaft Speed Models 2299.3.2 Motor Armature Current Control 2309.3.3 Motor Speed Control 2329.4 Thrusters 2339.4.1 Tunnel Thrusters 233 9.4.2 Azimuth Thrusters 2349.5 Rudder in the Propeller Slipstream 2369.5.1 Rudder Forces and Moment 2379.5.2 Steering Machine Dynamics 2409.6 Fin Stabilizators 2439.6.1 Lift and Drag Forces on Fins 2449.6.2 Roll Moment Produced by Symmetrical Fin Stabilizers 2459.7 Underwater Vehicle Control Surfaces 2459.7.1 Rudder 2479.7.2 Dive Planes 2489.8 Control Moment Gyroscope 2499.8.1 Ship Roll Gyrostabilizer 2499.8.2 Control Moment Gyros for Underwater Vehicles 2529.9 Moving Mass Actuators 25810 Environmental Forces and Moments 26110.1 Wind Forces and Moments 26310.1.1 Wind Forces and Moments on Marine Craft at Rest 26310.1.2 Wind Forces and Moments on Moving Marine Craft 26510.1.3 Wind Coefficients Based on Helmholtz–Kirchhoff Plate Theory 26610.1.4 Wind Coefficients for Merchant Ships 26910.1.5 Wind Coefficients for Very Large Crude Carriers 27110.1.6 Wind Coefficients for Large Tankers and Medium-sized Ships 27210.1.7 Wind Coefficients for Moored Ships and Floating Structures 27210.2 Wave Forces and Moments 27410.2.1 Sea-state Descriptions 27510.2.2 Wave Spectra 27610.2.3 Wave Amplitude Response Model 28710.2.4 Force RAOs 29010.2.5 Motion RAOs 29310.2.6 State-space Models for Wave Response Simulation 29610.3 Ocean Current Forces and Moments 30010.3.1 3D Irrotational Ocean Current Model 30310.3.2 2D Irrotational Ocean Current Model 304Part Two Motion Control11 Introduction to Part II 30911.1 Guidance, Navigation and Control Systems 31011.1.1 Historical Remarks 31211.1.2 Autopilots 31411.1.3 Dynamic Positioning and Position Mooring Systems 31511.1.4 Waypoint Tracking and Path-following Control Systems 31611.2 Control Allocation 31611.2.1 Propulsion and Actuator Models 31811.2.2 Unconstrained Control Allocation 32211.2.3 Constrained Control Allocation 32412 Guidance Systems 33112.1 Trajectory Tracking 33312.1.1 Reference Models for Trajectory Generation 33412.1.2 Trajectory Generation using a Marine Craft Simulator 33912.1.3 Optimal Trajectory Generation 34012.2 Guidance Laws for Target Tracking 34112.2.1 Line-of-sight Guidance Law 34212.2.2 Pure-pursuit Guidance Law 34312.2.3 Constant Bearing Guidance Law 34412.3 Linear Design Methods for Path Following 34612.3.1 Waypoints 34612.3.2 Path Generation using Straight Lines and Inscribed Circles 34712.3.3 Straight-line Paths Based on Circles of Acceptance 34912.3.4 Path Generation using Dubins Path 35112.3.5 Transfer Function Models for Straight-line Path Following 35212.4 LOS Guidance Laws for Path Following using Course Autopilots 35312.4.1 Vector-field Guidance Law 35412.4.2 Proportional LOS Guidance Law 35612.4.3 Lookahead- and Enclosure-based LOS Steering 35912.4.4 Integral LOS 36112.5 LOS Guidance Laws for Path Following using Heading Autopilots 36312.5.1 Crab Angle Compensation by Direct Measurements 36312.5.2 Integral LOS 36412.6 Curved-Path Path Following 36512.6.1 Path Generation using Interpolation Methods 36612.6.2 Proportional LOS Guidance Law for Curved Paths 37812.6.3 Path-following using Serret–Frenet Coordinates 38012.6.4 Case Study: Path-following Control using Serret–Frenet Coordinates 38413 Model-based Navigation Systems 38713.1 Sensors for Marine Craft 38713.1.1 GNSS Position 38813.1.2 GNSS Heading 38913.1.3 Magnetic Compass 39013.1.4 Gyrocompass 39013.2 Wave Filtering 39113.2.1 Low-pass Filtering 39313.2.2 Cascaded Low-pass and Notch Filtering 39613.2.3 Wave-frequency Estimation 39713.3 Fixed-gain Observer Design 40313.3.1 Observability 40313.3.2 Luenberger Observer 40513.3.3 Case Study: Luenberger Observer for Heading Autopilot 40613.4 Kalman Filter Design 40813.4.1 Discrete-time Kalman Filter 40813.4.2 Discrete-time Extended Kalman Filter 41113.4.3 Modification for Euler Angles to Avoid Discontinuous Jumps 41213.4.4 Modification for Asynchronous Measurement Data 41513.4.5 Case Study: Kalman Filter Design for Heading Autopilots 41613.4.6 Case Study: Kalman Filter for Dynamic Positioning Systems 41913.5 Passive Observer Design 42413.5.1 Case Study: Passive Observer for Dynamic Positioning using GNSS and Compass Measurements 42413.5.2 Case Study: Passive Observer for Heading Autopilots using only Compass Measurements 43313.5.3 Case Study: Passive Observer for Heading Autopilots using both Compass and Angular Rate Sensor Measurements 44014 Inertial Navigation Systems 44314.1 Inertial Measurement Unit 44414.1.1 Attitude Rate Sensors 44614.1.2 Accelerometers 44614.1.3 Magnetometer 44914.2 Attitude Estimation 45114.2.1 Static Mapping from Specific Force to Roll and Pitch Angles 45114.2.2 Vertical Reference Unit (VRU) Transformations 45214.2.3 Nonlinear Attitude Observer using Reference Vectors 45314.3 Direct Filters for Aided INS 45714.3.1 Fixed-gain Observer using Attitude Measurements 45814.3.2 Direct Kalman Filter using Attitude Measurements 46214.3.3 Direct Kalman Filter with Attitude Estimation 46514.4 Indirect Filters for Aided INS 46714.4.1 Introductory Example 46914.4.2 Error-state Kalman Filter using Attitude Measurements 47214.4.3 Error-state Extended Kalman Filter with Attitude Estimation 48015 Motion Control Systems 49315.1 Open-Loop Stability and Maneuverability 49415.1.1 Straight-line, Directional and Positional Motion Stability 49515.1.2 Maneuverability 50415.2 Autopilot Design Using Successive Loop Closure 51615.2.1 Successive Loop Closure 51615.2.2 Case Study: Heading Autopilot for Marine Craft 51815.2.3 Case Study: Path-following Control System for Marine Craft 51915.2.4 Case Study: Diving Autopilot for Underwater Vehicles 52115.3 PID Pole-Placement Algorithms 52315.3.1 Linear Mass–Damper–Spring Systems 52315.3.2 SISO Linear PID Control 52715.3.3 MIMO Nonlinear PID Control 52915.3.4 Case Study: Heading Autopilot for Marine Craft 53215.3.5 Case Study: LOS Path-following Control for Marine Craft 53815.3.6 Case Study: Dynamic Positioning System for Surface Vessels 54015.3.7 Case Study: Position Mooring System for Surface Vessels 54616 Advanced Motion Control Systems 54916.1 Linear-quadratic Optimal Control 55016.1.1 Linear-quadratic Regulator 55016.1.2 LQR Design for Trajectory Tracking and Integral Action 55216.1.3 General Solution of the LQ Trajectory-tracking Problem 55416.1.4 Operability and Motion Sickness Incidence Criteria 56016.1.5 Case Study: Optimal Heading Autopilot for Marine Craft 56216.1.6 Case Study: Optimal DP System for Surface Vessels 56616.1.7 Case Study: Optimal Rudder-roll Damping Systems for Ships 57016.1.8 Case Study: Optimal Fin and RRD Systems for Ships 57916.2 State Feedback Linearization 58016.2.1 Decoupling in the BODY Frame (Velocity Control) 58116.2.2 Decoupling in the NED Frame (Position and Attitude Control) 58216.2.3 Case Study: Speed Control Based on Feedback Linearization 58416.2.4 Case Study: Autopilot Based on Feedback Linearization 58516.3 Integrator Backstepping 58616.3.1 A Brief History of Backstepping 58616.3.2 The Main Idea of Integrator Backstepping 58716.3.3 Backstepping of SISO Mass–Damper–Spring Systems 59416.3.4 Integral Action by Constant Parameter Adaptation 59716.3.5 Integrator Augmentation Technique 59916.3.6 Case Study: Backstepping Design for Mass–Damper–Spring 60216.3.7 Case Study: Backstepping Design for Robot Manipulators 60416.3.8 Case Study: Backstepping Design for Surface Craft 60616.3.9 Case Study: Autopilot Based on Backstepping 61016.3.10 Case Study: Path-following Controller for Underactuated Marine Craft 61116.3.11 Case Study: Weather Optimal Position Control 61616.4 Sliding Mode Control 63416.4.1 Conventional Integral SMC for Second-order Systems 63416.4.2 Conventional Integral SMC for Third-order Systems 63716.4.3 Super-twisting Adaptive Sliding Mode Control 63716.4.4 Case Study: Heading Autopilot Based on Conventional Integral SMC 63916.4.5 Case Study: Depth Autopilot for Diving Based on Conventional Integral SMC 64316.4.6 Case Study: Heading Autopilot Based on the Adaptive-gain Super Twisting Algorithm 646Part Three AppendicesA Nonlinear Stability Theory 651A.1 Lyapunov Stability for Autonomous Systems 651A.1.1 Stability and Convergence 651A.1.2 Lyapunov’s Direct Method 653A.1.3 Krasovskii–LaSalle’s Theorem 654A.1.4 Global Exponential Stability 655A.2 Lyapunov Stability of Non-autonomous Systems 656A.2 1 Barbălat’s Lemma 656A.2.2 LaSalle–Yoshizawa’s Theorem 656A.2.3 On USGES of Proportional Line-of-sight Guidance Laws 657A.2.4 UGAS when Backstepping with Integral Action 658B Numerical Methods 661B.1 Discretization of Continuous-time Systems 661B.1.1 State-space Models 661B.1.2 Computation of the Transition Matrix 663B.2 Numerical Integration Methods 663B.2.1 Euler’s Method 664B.2.2 Adams–Bashford’s Second-order Method 665B.2.3 Runge–Kutta Second-order Method 666B.2.4 Runge–Kutta Fourth-order Method 666B.3 Numerical Differentiation 666C Model Transformations 669C.1 Transforming the Equations of Motion to an Arbitrarily Point 669C.1.1 System Transformation Matrix 669C.1.2 Equations of Motion About an Arbitrarily Point 671C.2 Matrix and Vector Transformations 672D Non-dimensional Equations of Motion 675D.1 Non-dimensionalization 675D.1.1 Non-dimensional Hydrodynamic Coefficients 676D.1.2 Non-dimensional Nomoto Models 677D.1.3 Non-dimensional Maneuvering Models 678D.2 6-DOF Procedure for Non-dimensionalization 678References 681Index 701

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.

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• About the Author xviiPreface xixList of Tables xxiPart One Marine Craft Hydrodynamics1 Introduction to Part I 31.1 Classification of Models 61.2 The Classical Models in Naval Architecture 81.2.1 Maneuvering Theory 101.2.2 Seakeeping Theory 121.2.3 Unified Theory 141.3 Fossen’s Robot-inspired Model for Marine Craft 142 Kinematics 172.1 Kinematic Preliminaries 182.1.1 Reference Frames 182.1.2 Body-fixed Reference Points 212.1.3 Generalized Coordinates 222.2 Transformations Between BODY and NED 232.2.1 Euler Angle Transformation 262.2.2 Unit Quaternions 322.2.3 Unit Quaternion from Euler Angles 382.2.4 Euler Angles from a Unit Quaternion 382.3 Transformations Between ECEF and NED 392.3.1 Longitude and Latitude Rotation Matrix 402.3.2 Longitude, Latitude and Height from ECEF Coordinates 412.3.3 ECEF Coordinates from Longitude, Latitude and Height 442.4 Transformations between ECEF and Flat-Earth Coordinates 452.4.1 Longitude, Latitude and Height from Flat-Earth Coordinates 452.4.2 Flat-Earth Coordinates from Longitude, Latitude and Height 462.5 Transformations Between BODY and FLOW 472.5.1 Definitions of Heading, Course and Crab Angles 472.5.2 Definitions of Angle of Attack and Sideslip Angle 492.5.3 Flow-axes Rotation Matrix 513 Rigid-body Kinetics 553.1 Newton–Euler Equations of Motion about the CG 563.1.1 Translational Motion About the CG 583.1.2 Rotational Motion About the CG 593.1.3 Equations of Motion About the CG 603.2 Newton–Euler Equations of Motion About the CO 603.2.1 Translational Motion About the CO 613.2.2 Rotational Motion About the CO 613.3 Rigid-body Equations of Motion 633.3.1 Nonlinear 6-DOF Rigid-body Equations of Motion 633.3.2 Linearized 6-DOF Rigid-body Equations of Motion 694 Hydrostatics 714.1 Restoring Forces for Underwater Vehicles 714.1.1 Hydrostatics of Submerged Vehicles 714.2 Restoring Forces for Surface Vessels 744.2.1 Hydrostatics of Floating Vessels 744.2.2 Linear (Small Angle) Theory for Boxed-shaped Vessels 774.2.3 Computation of Metacenter Heights for Surface Vessels 794.3 Load Conditions and Natural Periods 824.3.1 Decoupled Computation of Natural Periods 824.3.2 Computation of Natural Periods in a 6-DOF Coupled System 844.3.3 Natural Periods as a Function of Load Condition 874.3.4 Free-surface Effects 894.3.5 Payload Effects 904.4 Seakeeping Analysis 904.4.1 Harmonic Oscillator with Sinusoidal Forcing 904.4.2 Steady-state Heave, Roll and Pitch Responses in Regular Waves 924.4.3 Explicit Formulae for Boxed-shaped Vessels in Regular Waves 944.4.4 Case Study: Resonances in the Heave, Roll and Pitch Modes 964.5 Ballast Systems 974.5.1 Static Conditions for Trim and Heel 994.5.2 Automatic Ballast Control Systems 1025 Seakeeping Models 1055.1 Hydrodynamic Concepts and Potential Theory 1065.1.1 Numerical Approaches and Hydrodynamic Codes 1085.2 Seakeeping and Maneuvering Kinematics 1105.2.1 Seakeeping Reference Frame 1105.2.2 Transformation Between BODY and SEAKEEPING 1115.3 The Classical Frequency-domain Model 1145.3.1 Frequency-dependent Hydrodynamic Coefficients 1155.3.2 Viscous Damping 1195.3.3 Response Amplitude Operators 1215.4 Time-domain Models including Fluid Memory Effects 1225.4.1 Cummins Equation in SEAKEEPING Coordinates 1225.4.2 Linear Time-domain Seakeeping Equations in BODY Coordinates 1255.4.3 Nonlinear Unified Seakeeping and Maneuvering Model with Fluid Memory Effects 1295.5 Identification of Fluid Memory Effects 1305.5.1 Frequency-domain Identification Using the MSS FDI Toolbox 1316 Maneuvering Models 1356.1 Rigid-body Kinetics 1376.2 Potential Coefficients 1376.2.1 Frequency-independent Added Mass and Potential Damping 1396.2.2 Extension to 6-DOF Models 1406.3 Added Mass Forces in a Rotating Coordinate System 1416.3.1 Lagrangian Mechanics 1426.3.2 Kirchhoff’s Equation 1436.3.3 Added Mass and Coriolis–Centripetal Matrices 1436.4 Dissipative Forces 1486.4.1 Linear Damping 1506.4.2 Nonlinear Surge Damping 1516.4.3 Cross-flow Drag Principle 1546.5 Ship Maneuvering Models (3 DOFs) 1556.5.1 Nonlinear Equations of Motion 1556.5.2 Nonlinear Maneuvering Model Based on Surge Resistance and Cross-flow Drag 1586.5.3 Nonlinear Maneuvering Model Based on Second-order Modulus Functions 1596.5.4 Nonlinear Maneuvering Model Based on Odd Functions 1616.5.5 Linear Maneuvering Model 1636.6 Ship Maneuvering Models Including Roll (4 DOFs) 1656.6.1 The Nonlinear Model of Son and Nomoto 1726.6.2 The Nonlinear Model of Blanke and Christensen 1736.7 Low-Speed Maneuvering Models for Dynamic Positioning (3 DOFs) 1756.7.1 Current Coefficients 1756.7.2 Nonlinear DP Model Based on Current Coefficients 1796.7.3 Linear Time-varying DP Model 1807 Autopilot Models for Course and Heading Control 1837.1 Autopilot Models for Course Control 1847.1.1 State-space Model for Course Control 1847.1.2 Course Angle Transfer Function 1857.2 Autopilot Models for Heading Control 1867.2.1 Second-order Nomoto Model 1867.2.2 First-order Nomoto Model 1887.2.3 Nonlinear Extensions of Nomoto’s Model 1907.2.4 Pivot Point 1928 Models for Underwater Vehicles 1958.1 6-DOF Models for AUVs and ROVs 1958.1.1 Equations of Motion Expressed in BODY 1958.1.2 Equations of Motion Expressed in NED 1978.1.3 Properties of the 6-DOF Model 1988.1.4 Symmetry Considerations of the System Inertia Matrix 2008.2 Longitudinal and Lateral Models for Submarines 2018.2.1 Longitudinal Subsystem 2028.2.2 Lateral Subsystem 2048.3 Decoupled Models for “Flying Underwater Vehicles” 2058.3.1 Forward Speed Subsystem 2068.3.2 Course Angle Subsystem 2068.3.3 Pitch–Depth Subsystem 2078.4 Cylinder-Shaped Vehicles and Myring-type Hulls 2088.4.1 Myring-type Hull 2098.4.2 Spheroid Approximation 2108.5 Spherical-Shaped Vehicles 2149 Control Forces and Moments 2179.1 Propellers as Thrust Devices 2179.1.1 Fixed-pitch Propeller 2179.1.2 Controllable-pitch Propeller 2209.2 Ship Propulsion Systems 2259.2.1 Podded Propulsion Units 2259.2.2 Prime Mover System 2279.3 USV and Underwater Vehicle Propulsion Systems 2289.3.1 Propeller Shaft Speed Models 2299.3.2 Motor Armature Current Control 2309.3.3 Motor Speed Control 2329.4 Thrusters 2339.4.1 Tunnel Thrusters 233 9.4.2 Azimuth Thrusters 2349.5 Rudder in the Propeller Slipstream 2369.5.1 Rudder Forces and Moment 2379.5.2 Steering Machine Dynamics 2409.6 Fin Stabilizators 2439.6.1 Lift and Drag Forces on Fins 2449.6.2 Roll Moment Produced by Symmetrical Fin Stabilizers 2459.7 Underwater Vehicle Control Surfaces 2459.7.1 Rudder 2479.7.2 Dive Planes 2489.8 Control Moment Gyroscope 2499.8.1 Ship Roll Gyrostabilizer 2499.8.2 Control Moment Gyros for Underwater Vehicles 2529.9 Moving Mass Actuators 25810 Environmental Forces and Moments 26110.1 Wind Forces and Moments 26310.1.1 Wind Forces and Moments on Marine Craft at Rest 26310.1.2 Wind Forces and Moments on Moving Marine Craft 26510.1.3 Wind Coefficients Based on Helmholtz–Kirchhoff Plate Theory 26610.1.4 Wind Coefficients for Merchant Ships 26910.1.5 Wind Coefficients for Very Large Crude Carriers 27110.1.6 Wind Coefficients for Large Tankers and Medium-sized Ships 27210.1.7 Wind Coefficients for Moored Ships and Floating Structures 27210.2 Wave Forces and Moments 27410.2.1 Sea-state Descriptions 27510.2.2 Wave Spectra 27610.2.3 Wave Amplitude Response Model 28710.2.4 Force RAOs 29010.2.5 Motion RAOs 29310.2.6 State-space Models for Wave Response Simulation 29610.3 Ocean Current Forces and Moments 30010.3.1 3D Irrotational Ocean Current Model 30310.3.2 2D Irrotational Ocean Current Model 304Part Two Motion Control11 Introduction to Part II 30911.1 Guidance, Navigation and Control Systems 31011.1.1 Historical Remarks 31211.1.2 Autopilots 31411.1.3 Dynamic Positioning and Position Mooring Systems 31511.1.4 Waypoint Tracking and Path-following Control Systems 31611.2 Control Allocation 31611.2.1 Propulsion and Actuator Models 31811.2.2 Unconstrained Control Allocation 32211.2.3 Constrained Control Allocation 32412 Guidance Systems 33112.1 Trajectory Tracking 33312.1.1 Reference Models for Trajectory Generation 33412.1.2 Trajectory Generation using a Marine Craft Simulator 33912.1.3 Optimal Trajectory Generation 34012.2 Guidance Laws for Target Tracking 34112.2.1 Line-of-sight Guidance Law 34212.2.2 Pure-pursuit Guidance Law 34312.2.3 Constant Bearing Guidance Law 34412.3 Linear Design Methods for Path Following 34612.3.1 Waypoints 34612.3.2 Path Generation using Straight Lines and Inscribed Circles 34712.3.3 Straight-line Paths Based on Circles of Acceptance 34912.3.4 Path Generation using Dubins Path 35112.3.5 Transfer Function Models for Straight-line Path Following 35212.4 LOS Guidance Laws for Path Following using Course Autopilots 35312.4.1 Vector-field Guidance Law 35412.4.2 Proportional LOS Guidance Law 35612.4.3 Lookahead- and Enclosure-based LOS Steering 35912.4.4 Integral LOS 36112.5 LOS Guidance Laws for Path Following using Heading Autopilots 36312.5.1 Crab Angle Compensation by Direct Measurements 36312.5.2 Integral LOS 36412.6 Curved-Path Path Following 36512.6.1 Path Generation using Interpolation Methods 36612.6.2 Proportional LOS Guidance Law for Curved Paths 37812.6.3 Path-following using Serret–Frenet Coordinates 38012.6.4 Case Study: Path-following Control using Serret–Frenet Coordinates 38413 Model-based Navigation Systems 38713.1 Sensors for Marine Craft 38713.1.1 GNSS Position 38813.1.2 GNSS Heading 38913.1.3 Magnetic Compass 39013.1.4 Gyrocompass 39013.2 Wave Filtering 39113.2.1 Low-pass Filtering 39313.2.2 Cascaded Low-pass and Notch Filtering 39613.2.3 Wave-frequency Estimation 39713.3 Fixed-gain Observer Design 40313.3.1 Observability 40313.3.2 Luenberger Observer 40513.3.3 Case Study: Luenberger Observer for Heading Autopilot 40613.4 Kalman Filter Design 40813.4.1 Discrete-time Kalman Filter 40813.4.2 Discrete-time Extended Kalman Filter 41113.4.3 Modification for Euler Angles to Avoid Discontinuous Jumps 41213.4.4 Modification for Asynchronous Measurement Data 41513.4.5 Case Study: Kalman Filter Design for Heading Autopilots 41613.4.6 Case Study: Kalman Filter for Dynamic Positioning Systems 41913.5 Passive Observer Design 42413.5.1 Case Study: Passive Observer for Dynamic Positioning using GNSS and Compass Measurements 42413.5.2 Case Study: Passive Observer for Heading Autopilots using only Compass Measurements 43313.5.3 Case Study: Passive Observer for Heading Autopilots using both Compass and Angular Rate Sensor Measurements 44014 Inertial Navigation Systems 44314.1 Inertial Measurement Unit 44414.1.1 Attitude Rate Sensors 44614.1.2 Accelerometers 44614.1.3 Magnetometer 44914.2 Attitude Estimation 45114.2.1 Static Mapping from Specific Force to Roll and Pitch Angles 45114.2.2 Vertical Reference Unit (VRU) Transformations 45214.2.3 Nonlinear Attitude Observer using Reference Vectors 45314.3 Direct Filters for Aided INS 45714.3.1 Fixed-gain Observer using Attitude Measurements 45814.3.2 Direct Kalman Filter using Attitude Measurements 46214.3.3 Direct Kalman Filter with Attitude Estimation 46514.4 Indirect Filters for Aided INS 46714.4.1 Introductory Example 46914.4.2 Error-state Kalman Filter using Attitude Measurements 47214.4.3 Error-state Extended Kalman Filter with Attitude Estimation 48015 Motion Control Systems 49315.1 Open-Loop Stability and Maneuverability 49415.1.1 Straight-line, Directional and Positional Motion Stability 49515.1.2 Maneuverability 50415.2 Autopilot Design Using Successive Loop Closure 51615.2.1 Successive Loop Closure 51615.2.2 Case Study: Heading Autopilot for Marine Craft 51815.2.3 Case Study: Path-following Control System for Marine Craft 51915.2.4 Case Study: Diving Autopilot for Underwater Vehicles 52115.3 PID Pole-Placement Algorithms 52315.3.1 Linear Mass–Damper–Spring Systems 52315.3.2 SISO Linear PID Control 52715.3.3 MIMO Nonlinear PID Control 52915.3.4 Case Study: Heading Autopilot for Marine Craft 53215.3.5 Case Study: LOS Path-following Control for Marine Craft 53815.3.6 Case Study: Dynamic Positioning System for Surface Vessels 54015.3.7 Case Study: Position Mooring System for Surface Vessels 54616 Advanced Motion Control Systems 54916.1 Linear-quadratic Optimal Control 55016.1.1 Linear-quadratic Regulator 55016.1.2 LQR Design for Trajectory Tracking and Integral Action 55216.1.3 General Solution of the LQ Trajectory-tracking Problem 55416.1.4 Operability and Motion Sickness Incidence Criteria 56016.1.5 Case Study: Optimal Heading Autopilot for Marine Craft 56216.1.6 Case Study: Optimal DP System for Surface Vessels 56616.1.7 Case Study: Optimal Rudder-roll Damping Systems for Ships 57016.1.8 Case Study: Optimal Fin and RRD Systems for Ships 57916.2 State Feedback Linearization 58016.2.1 Decoupling in the BODY Frame (Velocity Control) 58116.2.2 Decoupling in the NED Frame (Position and Attitude Control) 58216.2.3 Case Study: Speed Control Based on Feedback Linearization 58416.2.4 Case Study: Autopilot Based on Feedback Linearization 58516.3 Integrator Backstepping 58616.3.1 A Brief History of Backstepping 58616.3.2 The Main Idea of Integrator Backstepping 58716.3.3 Backstepping of SISO Mass–Damper–Spring Systems 59416.3.4 Integral Action by Constant Parameter Adaptation 59716.3.5 Integrator Augmentation Technique 59916.3.6 Case Study: Backstepping Design for Mass–Damper–Spring 60216.3.7 Case Study: Backstepping Design for Robot Manipulators 60416.3.8 Case Study: Backstepping Design for Surface Craft 60616.3.9 Case Study: Autopilot Based on Backstepping 61016.3.10 Case Study: Path-following Controller for Underactuated Marine Craft 61116.3.11 Case Study: Weather Optimal Position Control 61616.4 Sliding Mode Control 63416.4.1 Conventional Integral SMC for Second-order Systems 63416.4.2 Conventional Integral SMC for Third-order Systems 63716.4.3 Super-twisting Adaptive Sliding Mode Control 63716.4.4 Case Study: Heading Autopilot Based on Conventional Integral SMC 63916.4.5 Case Study: Depth Autopilot for Diving Based on Conventional Integral SMC 64316.4.6 Case Study: Heading Autopilot Based on the Adaptive-gain Super Twisting Algorithm 646Part Three AppendicesA Nonlinear Stability Theory 651A.1 Lyapunov Stability for Autonomous Systems 651A.1.1 Stability and Convergence 651A.1.2 Lyapunov’s Direct Method 653A.1.3 Krasovskii–LaSalle’s Theorem 654A.1.4 Global Exponential Stability 655A.2 Lyapunov Stability of Non-autonomous Systems 656A.2 1 Barbălat’s Lemma 656A.2.2 LaSalle–Yoshizawa’s Theorem 656A.2.3 On USGES of Proportional Line-of-sight Guidance Laws 657A.2.4 UGAS when Backstepping with Integral Action 658B Numerical Methods 661B.1 Discretization of Continuous-time Systems 661B.1.1 State-space Models 661B.1.2 Computation of the Transition Matrix 663B.2 Numerical Integration Methods 663B.2.1 Euler’s Method 664B.2.2 Adams–Bashford’s Second-order Method 665B.2.3 Runge–Kutta Second-order Method 666B.2.4 Runge–Kutta Fourth-order Method 666B.3 Numerical Differentiation 666C Model Transformations 669C.1 Transforming the Equations of Motion to an Arbitrarily Point 669C.1.1 System Transformation Matrix 669C.1.2 Equations of Motion About an Arbitrarily Point 671C.2 Matrix and Vector Transformations 672D Non-dimensional Equations of Motion 675D.1 Non-dimensionalization 675D.1.1 Non-dimensional Hydrodynamic Coefficients 676D.1.2 Non-dimensional Nomoto Models 677D.1.3 Non-dimensional Maneuvering Models 678D.2 6-DOF Procedure for Non-dimensionalization 678References 681Index 701