Handbook of Marine Craft Hydrodynamics and Motion Control
Description
The latest tools for analysis and design of advanced GNC systems
Handbook of Marine Craft Hydrodynamics and Motion Control is an extensive study of the latest research in hydrodynamics, guidance, navigation, and control systems for marine craft. The text establishes how the implementation of mathematical models and modern control theory can be used for simulation and verification of control systems, decision-support systems, and situational awareness systems. Coverage includes hydrodynamic models for marine craft, models for wind, waves and ocean currents, dynamics and stability of marine craft, advanced guidance principles, sensor fusion, and inertial navigation.
This important book includes the latest tools for analysis and design of advanced GNC systems and presents new material on unmanned underwater vehicles, surface craft, and autonomous vehicles. References and examples are included to enable engineers to analyze existing projects before making their own designs, as well as MATLAB scripts for hands-on software development and testing. Highlights of this Second Edition include:
* Topical case studies and worked examples demonstrating how you can apply modeling and control design techniques to your own designs
* A Github repository with MATLAB scripts (MSS toolbox) compatible with the latest software releases from Mathworks
* New content on mathematical modeling, including models for ships and underwater vehicles, hydrostatics, and control forces and moments
* New methods for guidance and navigation, including line-of-sight (LOS) guidance laws for path following, sensory systems, model-based navigation systems, and inertial navigation systems
This fully revised Second Edition includes innovative research in hydrodynamics and GNC systems for marine craft, from ships to autonomous vehicles operating on the surface and under water. Handbook of Marine Craft Hydrodynamics and Motion Control is a must-have for students and engineers working with unmanned systems, field robots, autonomous vehicles, and ships.
MSS toolbox: https://github.com/cybergalactic/mss
Lecture notes: https://www.fossen.biz/wiley
Author's home page: https://www.fossen.biz
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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.
Content
About the Author xvii
Preface xix
List of Tables xxi
Part One Marine Craft Hydrodynamics
1 Introduction to Part I 3
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
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
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 119
5.3.3 Response Amplitude Operators 121
5.4 Time-domain Models including Fluid Memory Effects 122
5.4.1 Cummins Equation in SEAKEEPING Coordinates 122
5.4.2 Linear Time-domain Seakeeping Equations in BODY Coordinates 125
5.4.3 Nonlinear Unified Seakeeping and Maneuvering Model with Fluid Memory Effects 129
5.5 Identification of Fluid Memory Effects 130
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
8.4.1 Myring-type Hull 209
8.4.2 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
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 Heading 389
13.1.3 Magnetic Compass 390
13.1.4 Gyrocompass 390
13.2 Wave Filtering 391
13.2.1 Low-pass Filtering 393
13.2.2 Cascaded Low-pass and Notch Filtering 396
13.2.3 Wave-frequency Estimation 397
13.3 Fixed-gain Observer Design 403
13.3.1 Observability 403
13.3.2 Luenberger Observer 405
13.3.3 Case Study: Luenberger Observer for Heading Autopilot 406
13.4 Kalman Filter Design 408
13.4.1 Discrete-time Kalman Filter 408
13.4.2 Discrete-time Extended Kalman Filter 411
13.4.3 Modification for Euler Angles to Avoid Discontinuous Jumps 412
13.4.4 Modification for Asynchronous Measurement Data 415
13.4.5 Case Study: Kalman Filter Design for Heading Autopilots 416
13.4.6 Case Study: Kalman Filter for Dynamic Positioning Systems 419
13.5 Passive Observer Design 424
13.5.1 Case Study: Passive Observer for Dynamic Positioning using GNSS and Compass Measurements 424
13.5.2 Case Study: Passive Observer for Heading Autopilots using only Compass Measurements 433
13.5.3 Case Study: Passive Observer for Heading Autopilots using both Compass and Angular Rate Sensor Measurements 440
14 Inertial Navigation Systems 443
14.1 Inertial Measurement Unit 444
14.1.1 Attitude Rate Sensors 446
14.1.2 Accelerometers 446
14.1.3 Magnetometer 449
14.2 Attitude Estimation 451
14.2.1 Static Mapping from Specific Force to Roll and Pitch Angles 451
14.2.2 Vertical Reference Unit (VRU) Transformations 452
14.2.3 Nonlinear Attitude Observer using Reference Vectors 453
14.3 Direct Filters for Aided INS 457
14.3.1 Fixed-gain Observer using Attitude Measurements 458
14.3.2 Direct Kalman Filter using Attitude Measurements 462
14.3.3 Direct Kalman Filter with Attitude Estimation 465
14.4 Indirect Filters for Aided INS 467
14.4.1 Introductory Example 469
14.4.2 Error-state Kalman Filter using Attitude Measurements 472
14.4.3 Error-state Extended Kalman Filter with Attitude Estimation 480
15 Motion Control Systems 493
15.1 Open-Loop Stability and Maneuverability 494
15.1.1 Straight-line, Directional and Positional Motion Stability 495
15.1.2 Maneuverability 504
15.2 Autopilot Design Using Successive Loop Closure 516
15.2.1 Successive Loop Closure 516
15.2.2 Case Study: Heading Autopilot for Marine Craft 518
15.2.3 Case Study: Path-following Control System for Marine Craft 519
15.2.4 Case Study: Diving Autopilot for Underwater Vehicles 521
15.3 PID Pole-Placement Algorithms 523
15.3.1 Linear Mass-Damper-Spring Systems 523
15.3.2 SISO Linear PID Control 527
15.3.3 MIMO Nonlinear PID Control 529
15.3.4 Case Study: Heading Autopilot for Marine Craft 532
15.3.5 Case Study: LOS Path-following Control for Marine Craft 538
15.3.6 Case Study: Dynamic Positioning System for Surface Vessels 540
15.3.7 Case Study: Position Mooring System for Surface Vessels 546
16 Advanced Motion Control Systems 549
16.1 Linear-quadratic Optimal Control 550
16.1.1 Linear-quadratic Regulator 550
16.1.2 LQR Design for Trajectory Tracking and Integral Action 552
16.1.3 General Solution of the LQ Trajectory-tracking Problem 554
16.1.4 Operability and Motion Sickness Incidence Criteria 560
16.1.5 Case Study: Optimal Heading Autopilot for Marine Craft 562
16.1.6 Case Study: Optimal DP System for Surface Vessels 566
16.1.7 Case Study: Optimal Rudder-roll Damping Systems for Ships 570
16.1.8 Case Study: Optimal Fin and RRD Systems for Ships 579
16.2 State Feedback Linearization 580
16.2.1 Decoupling in the BODY Frame (Velocity Control) 581
16.2.2 Decoupling in the NED Frame (Position and Attitude Control) 582
16.2.3 Case Study: Speed Control Based on Feedback Linearization 584
16.2.4 Case Study: Autopilot Based on Feedback Linearization 585
16.3 Integrator Backstepping 586
16.3.1 A Brief History of Backstepping 586
16.3.2 The Main Idea of Integrator Backstepping 587
16.3.3 Backstepping of SISO Mass-Damper-Spring Systems 594
16.3.4 Integral Action by Constant Parameter Adaptation 597
16.3.5 Integrator Augmentation Technique 599
16.3.6 Case Study: Backstepping Design for Mass-Damper-Spring 602
16.3.7 Case Study: Backstepping Design for Robot Manipulators 604
16.3.8 Case Study: Backstepping Design for Surface Craft 606
16.3.9 Case Study: Autopilot Based on Backstepping 610
16.3.10 Case Study: Path-following Controller for Underactuated Marine Craft 611
16.3.11 Case Study: Weather Optimal Position Control 616
16.4 Sliding Mode Control 634
16.4.1 Conventional Integral SMC for Second-order Systems 634
16.4.2 Conventional Integral SMC for Third-order Systems 637
16.4.3 Super-twisting Adaptive Sliding Mode Control 637
16.4.4 Case Study: Heading Autopilot Based on Conventional Integral SMC 639
16.4.5 Case Study: Depth Autopilot for Diving Based on Conventional Integral SMC 643
16.4.6 Case Study: Heading Autopilot Based on the Adaptive-gain Super Twisting Algorithm 646
Part Three Appendices
A Nonlinear Stability Theory 651
A.1 Lyapunov Stability for Autonomous Systems 651
A.1.1 Stability and Convergence 651
A.1.2 Lyapunov's Direct Method 653
A.1.3 Krasovskii-LaSalle's Theorem 654
A.1.4 Global Exponential Stability 655
A.2 Lyapunov Stability of Non-autonomous Systems 656
A.2 1 Barbalat's Lemma 656
A.2.2 LaSalle-Yoshizawa's Theorem 656
A.2.3 On USGES of Proportional Line-of-sight Guidance Laws 657
A.2.4 UGAS when Backstepping with Integral Action 658
B Numerical Methods 661
B.1 Discretization of Continuous-time Systems 661
B.1.1 State-space Models 661
B.1.2 Computation of the Transition Matrix 663
B.2 Numerical Integration Methods 663
B.2.1 Euler's Method 664
B.2.2 Adams-Bashford's Second-order Method 665
B.2.3 Runge-Kutta Second-order Method 666
B.2.4 Runge-Kutta Fourth-order Method 666
B.3 Numerical Differentiation 666
C Model Transformations 669
C.1 Transforming the Equations of Motion to an Arbitrarily Point 669
C.1.1 System Transformation Matrix 669
C.1.2 Equations of Motion About an Arbitrarily Point 671
C.2 Matrix and Vector Transformations 672
D Non-dimensional Equations of Motion 675
D.1 Non-dimensionalization 675
D.1.1 Non-dimensional Hydrodynamic Coefficients 676
D.1.2 Non-dimensional Nomoto Models 677
D.1.3 Non-dimensional Maneuvering Models 678
D.2 6-DOF Procedure for Non-dimensionalization 678
References 681
Index 701
Chapter 1
Introduction to Part I
The subject of this book is motion control and hydrodynamics of marine craft. The term marine craft includes ships, high-speed craft, semi-submersibles, floating rigs, submarines, remotely operated and autonomous underwater vehicles, torpedoes, and other propelled and powered structures, for instance a floating air field. Offshore operations involve the use of many marine craft, as shown in Figure 1.1. Vehicles that do not travel on land (ocean and flight vehicles) are usually called craft, such as watercraft, sailcraft, aircraft, hovercraft and spacecraft. The term vessel can be defined as follows:
Vessel: "hollow structure made to float upon the water for purposes of transportation and navigation; especially, one that is larger than a rowboat."
The words vessel, ship and boat are often used interchangeably. In Encyclopedia Britannica, a ship and a boat are distinguished by their size through the following definition:
Ship: "any large floating vessel capable of crossing open waters, as opposed to a boat, which is generally a smaller craft. The term formerly was applied to sailing vessels having three or more masts; in modern times it usually denotes a vessel of more than of displacement. Submersible ships are generally called boats regardless of their size."
Similar definitions are given for submerged vehicles:
Submarine: "any naval vessel that is capable of propelling itself beneath the water as well as on the water's surface. This is a unique capability among warships, and submarines are quite different in design and appearance from surface ships."
Underwater vehicle: "small vehicle that is capable of propelling itself beneath the water surface as well as on the water's surface. This includes unmanned underwater vehicles (UUV), remotely operated vehicles (ROV), autonomous underwater vehicles (AUV) and underwater robotic vehicles (URV). Underwater vehicles are used both commercially and by the navy."
Figure 1.1 Marine craft in operation. Source: illustration by B. Stenberg.
From a hydrodynamic point of view, marine craft can be classified according to their maximum operating speed. For this purpose it is common to use the Froude number
(1.1)where is the craft speed, is the overall submerged length of the craft and is the acceleration of gravity. The pressure carrying the craft can be divided into hydrostatic and hydrodynamic pressure. The corresponding forces are:
- Buoyancy force due to the hydrostatic pressure (proportional to the displacement of the ship).
- Hydrodynamic force due to the hydrodynamic pressure (approximately proportional to the square of the relative speed to the water).
For a marine craft sailing at constant speed , the following classifications can be made (Faltinsen 2005):
- Displacement vessels (): The buoyancy force (restoring terms) dominates relative to the hydrodynamic forces (added mass and damping).
- Semi-displacement vessel (0.4--1.2): The buoyancy force is not dominant at the maximum operating speed for a high-speed submerged hull type of craft.
- Planing vessel (-1.2): The hydrodynamic force mainly carries the weight. There will be strong flow separation and the aerodynamic lift and drag forces start playing a role.
Figure 1.2 Displacement vessel.
In this book only displacement vessels are covered; see Figure 1.2.
The Froude number has influence on the hydrodynamic analysis. For displacement vessels, the waves radiated by different parts of the hull do not influence other parts of the hull. For semi-displacement vessels, waves generated at the bow influence the hydrodynamic pressure along the hull towards the stern. These characteristics give rise to different modeling hypotheses, which lead to different hydrodynamic theories.
For displacement ships it is widely accepted that two- and three-dimensional potential theory programs are used to compute the potential coefficients and wave loads; see Section 5.1. For semi-displacement vessels and planing vessels it is important to include the lift and drag forces in the computations (Faltinsen 2005).
Degrees of Freedom and Motion of a Marine Craft
In maneuvering, a marine craft experiences motion in six degrees of freedom (DOFs). The DOFs are the set of independent displacements and rotations that specify completely the displaced position and orientation of the craft. The motion in the horizontal plane is referred to as surge (longitudinal motion, usually superimposed on the steady propulsive motion) and sway (sideways motion). Yaw (rotation about the vertical axis) describes the heading of the craft. The remaining three DOFs are roll (rotation about the longitudinal axis), pitch(rotation about the transverse axis) and heave (vertical motion); see Figure 1.3.
Roll motion is probably the most influential DOF with regards to human performance, since it produces the highest accelerations and, hence, is the principal villain in seasickness. Similarly, pitching and heaving feel uncomfortable to people. When designing ship autopilots, yaw is the primary mode for feedback control. Stationkeeping of a marine craft implies stabilization of the surge, sway and yaw motions.
When designing feedback control systems for marine craft, reduced-order models are often used since most craft do not have actuation in all DOFs. This is usually done by decoupling the motions of the craft according to:
Figure 1.3 Motion in six degrees of freedom (DOFs).
- 1-DOF models can be used to design forward speed controllers (surge), heading autopilots (yaw) and roll-damping systems (roll).
- 3-DOF models are usually:
- Horizontal-plane models (surge, sway and yaw) for ships, semi-submersibles and underwater vehicles that are used in dynamic positioning systems, trajectory-tracking control systems and path-following systems. For slender bodies such as submarines, it is also common to assume that the motions can be decoupled into longitudinal and lateral motions.
- Longitudinal models (surge, heave and pitch) for forward speed, diving and pitch control.
- Lateral models (sway, roll and yaw) for turning and heading control.
- 4-DOF models (surge, sway, roll and yaw) are usually formed by adding the roll equation to the 3-DOF horizontal-plane model. These models are used in maneuvering situations where it is important to include the rolling motion, usually in order to reduce roll by active control of fins, rudders or stabilizing liquid tanks.
- 6-DOF models (surge, sway, heave, roll, pitch and yaw) are fully coupled equations of motion used for simulation and prediction of coupled vehicle motions. These models can also be used in advanced control systems for underwater vehicles that are actuated in all DOFs.
1.1 Classification of Models
The models in this book can be used for prediction, real-time simulation, decision-support systems, situational awareness as well as controller-observer design. The complexity and number of differential equations needed for the various purposes will vary. Consequently, one can distinguish between three types of models (see Figure 1.4):
Figure 1.4 Models used in guidance, navigation and control systems.
- Simulation model: This model is the most accurate description of a system, for instance a 6-DOF high-fidelity model for simulation of coupled motions in the time domain. It includes the marine craft dynamics, propulsion system, measurement system and the environmental forces due to wind, waves and ocean currents. It also includes other features not used for control and observer design that have a direct impact on model accuracy. The simulation model should be able to reconstruct the time responses of the real system and it should also be possible to trigger failure modes to simulate events such as accidents and erroneous signals. Simulation models where the fluid-memory effects are included due to frequency-dependent added mass and potential damping typically consist of 50-200 ordinary differential equations (ODEs) while a maneuvering model can be represented in 6 DOFs with 12 ODEs for generalized position and velocity. In addition, some states are needed to describe the environmental forces and actuators, but still the number of states will be less than 50 for a marine craft.
- Control design model: The motion control system is usually designed using a reduced-order or simplified version of the simulation model. In its simplest form, this model is used to compute a set of constant gains for a proportional, integral, derivative (PID) controller. More sophisticated control systems such as model-based control systems use a dynamic model to generate feedforward and feedback signals. The number of ODEs used in conventional model-based ship control systems is usually less than 20. A PID controller typically requires two states: one for the integrator and one for the low-pass filter used...
