Handbook of Marine Craft Hydrodynamics and Motion Control

This book is dedicated to my parents Gerd Kristine and Ole Johan Fossen and my family Heidi, Sindre and Lone Moa who have always been there for me.

Thor I. Fossen

About the Author

Professor Thor I. Fossen received an MSc degree in Marine Technology in 1987 from the Norwegian University of Science and Technology (NTNU) and a PhD in Engineering Cybernetics from NTNU in 1991. In the period 1989–1990 he pursued postgraduate studies in aerodynamics and flight control as a Fulbright Scholar at the University of Washington, Seattle. His expertise is in the fields of hydrodynamics, naval architecture, robotics, marine and flight control systems, guidance systems, navigation systems and nonlinear control theory. In 1993 he was appointed as a Professor of Guidance and Control at NTNU. He is one of the founders of the company Marine Cybernetics where he was the Vice President R&D in the period 2002–2007. He is the author of Guidance and Control of Ocean Vehicles (John Wiley & Sons, Ltd, 1994) and co-author of New Directions in Nonlinear Observer Design (Springer Verlag, 1999) and Parametric Resonance in Dynamical Systems (Springer Verlag, 2011). Professor Fossen has been instrumental in the development of several industrial autopilot, path-following and dynamic positioning (DP) systems. He has also experience in nonlinear state estimators for marine craft and automotive systems as well as strapdown GNSS/INS navigation systems. He has been involved in the design of the SeaLaunch trim and heel correction systems. He received the Automatica Prize Paper Award in 2002 for a concept for weather optimal positioning control of marine craft. He is currently head of automatic control at the Centre for Ships and Ocean Structures (CESOS), Norwegian Centre of Excellence, and a Professor of Guidance and Control in the Department of Engineering Cybernetics, NTNU.

Preface

The main motivation for writing this book was to collect new results on hydrodynamic modeling, guidance, navigation and control of marine craft that have been developed since I published my first book:

Fossen, T. I. (1994). Guidance and Control of Ocean Vehicles. John Wiley & Sons, Ltd. Chichester, UK. ISBN 0-471-94113-1.

The Wiley book from 1994 was the first attempt to bring hydrodynamic modeling and control system design into a unified notation for modeling, simulation and control. My first book also contains state-of-the-art control design methods for ships and underwater vehicles up to 1994. In the period 1994–2002 a great deal of work was done on nonlinear control of marine craft. This work resulted in many useful results and lecture notes, which have been collected and published in a second book entitled Marine Control Systems: Guidance, Navigation and Control of Ships and Underwater Vehicles. The 1st edition was published in 2002 and it was used as the main textbook in my course on Guidance and Control at the Norwegian University of Science and Technology (NTNU). Instead of making a 2nd edition of the book, I decided to write the Handbook of Marine Craft Hydrodynamics and Motion Control and merge the most important results from my previous two books with recent results.

Part I of the book covers both maneuvering and seakeeping theory and it is explained in detail how the equations of motion can be derived for both cases using both frequency- and time-domain formulations. This includes transformations from the frequency to the time domain and the explanation of fluid-memory effects. A great effort has been made in the development of kinematic equations for effective representation of the equations of motion in seakeeping, body, inertial and geographical coordinates. This is very confusing in the existing literature on hydrodynamics and the need to explain this properly motivated me to find a unifying notation for marine and mechanical systems. This was done in the period 2002–2010 and it is inspired by the elegant formulation used in robotics where systems are represented in a vectorial notation. The new results on maneuvering and seakeeping are joint work with Professor Tristan Perez, University of Newcastle, Australia. The work with Professor Perez has resulted in several joint publications and I am grateful to him for numerous interesting discussions on hydrodynamic modeling and control. He should also be thanked for proofreading parts of the manuscript.

Part II of the book covers guidance systems, navigation systems, state estimators and control of marine craft. This second part of the book focuses on state-of-the-art methods for feedback control such as PID control design for linear and nonlinear systems as well as control allocation methods. A chapter with more advanced topics, such as optimal control theory, backstepping, feedback linearization and sliding-mode control, is included for the advanced reader. Case studies and applications are treated at the end of each chapter. The control systems based on PID and optimal control theory are designed with a complexity similar to those used in many industrial systems. The more advanced methods using nonlinear theory are included so the user can compare linear and nonlinear design techniques before a final implementation is made. Many references to existing systems are included so control system vendors can easily find articles describing state-of-the art design methods for marine craft.

The arrangement of the subject matter in major parts can be seen from the following diagram:

Most of the results in the book have been developed at the Department of Engineering Cybernetics and the Centre of Ships and Ocean Structures, NTNU, in close cooperation with my former doctoral students, Ola-Erik Fjellstad, Trygve Lauvdal, Jann Peter Strand, Jan Fredrik Hansen, Bjϕrnar Vik, Svein P. Berge, Mehrdad P. Fard, Karl-Petter Lindegaard, Ole Morten Aamo, Roger Skjetne, Ivar-Andre Flakstad Ihle, Andrew Ross, Gullik A. Jensen and Morten Breivik, in the period 1991–2010. We have been a productive team, and have written hundreds of international publications in this period. Our joint efforts have resulted in several patents and industrial implementations. Morten Breivik has contributed with many important results on guidance systems (Chapter 10) and he should also be thanked for proofreading parts of the manuscript. Bjarne Stenberg should be thanked for creating the artistic front and back covers of the book and many other graphical illustrations. Finally, Stewart Clark, Senior Consultant, NTNU, should be thanked for his assistance with the English language. The book project has been sponsored by The Norwegian Research Council through the Center of Ships and Ocean Structures, Norwegian Center of Excellence at NTNU.

Thor I. Fossen

www.wiley.com/go/fossen_marine

List of Tables

2.1 The notation of SNAME (1950) for marine vessels

2.2 WGS-84 parameters

7.1 Parameters for a cargo ship and a fully loaded oil tanker

7.2 Normalization variables used for the prime and bis systems

7.3 6 DOF normalization variables

8.1 Air density as a function of temperature

8.2 Definition of Beaufort numbers (Price and Bishop, 1974)

8.3 Coefficients of lateral and longitudinal resistance, cross-force and rolling moment (Blendermann, 1994)

8.4 Wind force parameters in surge, sway and yaw (Isherwood, 1972)

8.5 Definition of sea state (SS) codes (Price and Bishop, 1974). Notice that the percentage probability for SS codes 0, 1 and 2 is summarized

11.1 Discrete-time Kalman filter

11.2 Continuous-time Kalman filter

11.3 Discrete-time extended Kalman filter (EKF)

11.4 Alternative choices of attitude update laws

12.1 Routh array

12.2 PID and acceleration feedback pole-placement algorithm

12.3 Definition of actuators and control variables

13.1 Eigenvalues, damping ratios and frequencies for the RRD control system

13.2 Criteria for effectiveness of the crew (Faltinsen, 1990)

A.1 Classification of theorems for stability and convergence

Part One

Marine Craft Hydrodynamics

De Navium Motu Contra Aquas

Chapter 1

Introduction

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.

Figure 1.1 Marine craft in operation. Illustration Bjarne Stenberg/Department of Marine Technology, NTNU.

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 500 tons 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.

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) equation

where U is the craft speed, L is the overall submerged length of the craft and g 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 U, 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 ( : The buoyancy force is not dominant at the maximum operating speed for a high-speed submerged hull type of craft.

Planing Vessel ( ): The hydrodynamic force mainly carries the weight. There will be strong flow separation and the aerodynamic lift and drag forces start playing a role.

In this book only displacement vessels are covered; see Figure 1.2.

Figure 1.2 Displacement vessel.

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 to use two- and three-dimensional potential theory programs 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 6 degrees of freedom (DOFs); see Section 9.4. 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.

Figure 1.3 Motion in 6 degrees of freedom (DOF).

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 DOF. This is usually done by decoupling the motions of the craft according to:

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 DOF.

1.1 Classification of Models

The models in this book can be used for prediction, real-time simulation and 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):

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 DOF 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 controller model is a reduced-order or simplified version of the simulation model that is used to design the motion control system. 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 use a dynamic model to generate feedforward and feedback signals. This is referred to as model-based control. 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 to limit noise amplification. Consequently, setpoint regulation in 6 DOF can be implemented by using 12 ODEs. However, trajectory-tracking controllers require additional states for feedforward as well as filtering so higher-order control laws are not uncommon.

Observer Design Model: The observer model will in general be different from the model used in the controller since the purpose is to capture the additional dynamics associated with the sensors and navigation systems as well as disturbances. It is a simplified version of the simulation model where attention is given to accurate modeling of measurement noise, failure situations including dead-reckoning capabilities, filtering and motion prediction. For marine craft, the model-based observer often includes a disturbance model where the goal is to estimate wave, wind and ocean current forces by treating these as colored noise. For marine craft the number of ODEs in the state estimator will typically be 20 for a dynamic positioning (DP) system while a basic heading autopilot is implemented with less than five states.

Figure 1.4 Models used in guidance, navigation and control.

1.2 The Classical Models in Naval Architecture

The motions of a marine craft exposed to wind, waves and ocean currents takes place in 6 DOF. The equations of motion can be derived using the Newton–Euler or Lagrange equations. The equations of motion are used to simulate ships, high-speed craft, underwater vehicles and floating structures operating under or on the water surface, as shown in Figure 1.5. In Section 3.3 it is shown that a rigid body with constant mass m and center of gravity relative to a fixed point on the hull can be described by the following coupled differential equations:

(1.2)

where and N denote the external forces and moments. This model is the basis for time-domain simulation of marine craft. The external forces and moments acting on a marine craft are usually modeled by using:

Maneuvering Theory: The study of a ship moving at constant positive speed U in calm water within the framework of maneuvering theory is based on the assumption that the maneuvering (hydrodynamic) coefficients are frequency independent (no wave excitation). The maneuvering model will in its simplest representation be linear while nonlinear representations can be derived using methods such as cross-flow drag, quadratic damping or Taylor-series expansions; see Chapter 6.

Seakeeping Theory: The motions of ships at zero or constant speed in waves can be analyzed using seakeeping theory where the hydrodynamic coefficients and wave forces are computed as a function of the wave excitation frequency using the hull geometry and mass distribution. The seakeeping models are usually derived within a linear framework (Chapter 5 while the extension to nonlinear theory is an important field of research.

Figure 1.5 Ship and semi-submersibles operating offshore. Illustration Bjarne Stenberg/MARINTEK.

For underwater vehicles operating below the wave-affected zone, the wave excitation frequency will not affect the hydrodynamic mass and damping coefficients. Consequently, it is common to model underwater vehicles with constant hydrodynamic coefficients similar to a maneuvering ship.

1.2.1 Maneuvering Theory

Maneuvering theory assumes that the ship is moving in restricted calm water, that is in sheltered waters or in a harbor. Hence, the maneuvering model is derived for a ship moving at positive speed U under a zero-frequency wave excitation assumption such that added mass and damping can be represented by using hydrodynamic derivatives (constant parameters). The zero-frequency assumption is only valid for surge, sway and yaw since the natural periods of a PD-controlled ship will be in the range of 100–150 s. For 150 s the natural frequency is close to zero, that is

(1.3) equation

This clearly gives support for the zero-frequency assumption. The natural frequencies in heave, roll and pitch are much higher so it is recommended to use the zero-frequency potential coefficients in these modes. For instance, a ship with a roll period of 10 s will have a natural frequency of 0.628 rad/s which clearly violates the zero-frequency assumption. This means that hydrodynamic added mass and potential damping should be evaluated at a frequency of 0.628 rad/s in roll if a pure rolling motion is considered. As a consequence of this, it is common to formulate the ship maneuvering model (1.2) as a coupled surge–sway–yaw model and thus neglect heave, roll and pitch motions:

(1.4) equation

The rigid-body kinetics (1.4) can be expressed in vectorial form according to (Fossen, 1994)

(1.5) equation

(1.6) equation

where is the rigid-body inertia matrix, is a matrix of rigid-body Coriolis and centripetal forces and is a vector of generalized forces.

The generalized velocity is

(1.7) equation

where the first three components are the linear velocities in surge, sway and heave and are the angular velocities in roll, pitch and yaw. The generalized force acting on the craft is

(1.8)

where the subscripts stand for:

Hydrodynamic added mass, potential damping due to wave radiation and viscous damping

Hydrostatic forces (spring stiffness)

Wind forces

Wave forces (first and second order)

Control and propulsion forces

This model is motivated by Newton's second law: , where F represents force, m is the mass and a is the acceleration. The Coriolis and centripetal term is due to the rotation of the body-fixed reference frame with respect to the inertial reference frame. The model (1.5) is used in most textbooks on hydrodynamics and the generalized hydrodynamic force can be represented by linear or nonlinear theory:

Linearized Models: In the linear 6 DOF case there will be a total of 36 mass and 36 damping elements proportional to velocity and acceleration. In addition to this, there will be restoring forces, propulsion forces and environmental forces. If the generalized hydrodynamic force is written in component form using the SNAME (1950) notation, the linear added mass and damping forces become:

(1.9)

(1.10)

where are the linear damping coefficients and represent hydrodynamic added mass.

Nonlinear Models: Application of nonlinear theory implies that many elements must be included in addition to the 36 linear elements. This is usually done by one of the following methods:

1. Truncated Taylor-series expansions} using odd terms (first and third order) which are fitted to experimental data, for instance (Abkowitz, 1964):

(1.11)

(1.12)

In this approach added mass is assumed to be linear and damping is modeled by a third order odd function. Alternatively, second-order modulus terms can be used (Fedyaevsky and Sobolev, 1963) for instance:

(1.13)

(1.14)

This is motivated by the square law damping terms in fluid dynamics and aerodynamics. When applying Taylor-series expansions in model-based control design, the system (1.5) becomes relatively complicated due to the large number of hydrodynamic coefficients on the right-hand side needed to represent the hydrodynamic forces. This approach is quite common when deriving maneuvering models and many of the coefficients are difficult to determine with sufficient accuracy since the model can be overparametrized. Taylor-series expansions are frequently used in commercial planar motion mechanism (PMM) tests where the purpose is to derive the maneuvering coefficients experimentally.

2. First principles where hydrodynamic effects such as lift and drag are modeled using well established models. This results in physically sound Lagrangian models that preserve energy properties. Models based on first principles usually require a much smaller number of parameters than models based on third order Taylor-series expansions.

1.2.2 Seakeeping Theory

As explained above, maneuvering refers to the study of ship motion in the absence of wave excitation (calm water). Seakeeping, on the other hand, is the study of motion when there is wave excitation and the craft keeps its heading and its speed U constant (which includes the case of zero speed). This introduces a dissipative force (Cummins, 1962) known as fluid-memory effects. Although both areas are concerned with the same issues, study of motion, stability and control, the separation allows different assumptions to be made that simplify the study in each case. Seakeeping analysis is used in capability analysis and operability calculations to obtain operability diagrams according to the adopted criteria.

The seakeeping theory is formulated using seakeeping axes where the state vector represents perturbations with respect to a fixed equilibrium state; see Figure 1.6. These perturbations can be related to motions in the body frame and North-East-Down frame by using kinematic transformations; see Section 5.2. The governing model is formulated in the time domain using the Cummins equation in the following form (see Section 5.4):

(1.15)

where is the perturbed control input due to propulsion and control surfaces, is the infinite-frequency added mass matrix, is the infinite-frequency damping matrix containing potential and viscous damping terms, is the spring stiffness matrix and is a time-varying matrix of retardation functions given by

(1.16) equation

The frequency-domain representation of (1.15) is (Newman, 1977; Faltinsen, 1990)

(1.17)

where is a complex vector with components:

(1.18) equation

Similarly, the external signals and are complex vectors.

Figure 1.6 Coordinate systems used in seakeeping analysis.

Naval architects often write the seakeeping model as a pseudo-differential equation:

(1.19)

mixing time and frequency. Unfortunately this is deeply rooted in the literature of hydrodynamics even though it is not correct to mix time and frequency in one single equation. Consequently, it is recommended to use the time- and frequency-domain representations (1.15) and (1.17). Computer simulations are done under the assumptions of linear theory and harmonic motions such that the resulting response is linear in the time domain. This approach dates back to Cummins (1962) and the necessary derivations are described in Chapter 5.

1.2.3 Unified Theory

A unified theory for maneuvering and seakeeping is useful since it allows for time-domain simulation of a marine craft in a seaway. This is usually done by using the seakeeping representation (1.19) as described in Chapter 5. The next step is to assume linear superposition such that wave-induced forces can be added for different speeds U and sea states. A similar assumption is used to add nonlinear damping and restoring forces so that the resulting model is a unified nonlinear model combining the most important terms from both maneuvering and seakeeping. Care must be taken with respect to double counting. This refers to the problem that hydrodynamic effects can be modeled twice when merging the results from two theories.

1.3 Fossen's Robot-Like Vectorial Model for Marine Craft

In order to exploit the physical properties of the maneuvering and seakeeping models, the equations of motion are represented in a vectorial setting which dates back to Fossen (1991). The vectorial model is expressed in and so appropriate kinematic transformations between the reference frames and must be derived. This is done in Chapters 2 and 5. The vectorial model is well suited for computer implementation and control systems design.

Component Form

The classical model (1.2) is often combined with expressions such as (1.9)–(1.10) or (1.11)–(1.14) to describe the hydrodynamic forces. This often results in complicated models with hundreds of elements. In most textbooks the resulting equations of motion are on component form. The following introduces a compact notation using matrices and vectors that will simplify the representation of the equations of motion considerably.

Vectorial Representation

In order to exploit the physical properties of the models, the equations of motion are represented in a vectorial setting. It is often beneficial to exploit physical system properties to reduce the number of coefficients needed for control. This is the main motivation for developing a vectorial representation of the equations of motion. In Fossen (1991) the robot model (Craig, 1989; Sciavicco and Siciliano, 1996)

(1.20) equation

was used as motivation to derive a compact marine craft model in 6 DOFs using a vectorial setting. In the robot model is a vector of joint angles, is the torque, while and denote the system inertia and Coriolis matrices, respectively. It is found that similar quantities can be identified for marine craft and aircraft. In Fossen (1991) a complete 6 DOF vectorial setting for marine craft was derived based on these ideas. These results were further refined by Sagatun and Fossen (1991), Fossen (1994) and Fossenand Fjellstad (1995). The 6 DOF models considered in this book use the following representation:

(1.21)

where

(1.22) equation

(1.23) equation

are vectors of velocities and position/Euler angles, respectively. In fact and are generalized velocities and positions used to describe motions in 6 DOF. Similarly, is a vector of forces and moments or the generalized forces in 6 DOF. The model matrices , and denote inertia, Coriolis and damping, respectively, while is a vector of generalized gravitational and buoyancy forces. Static restoring forces and moments due to ballast systems and water tanks are collected in the term .

Component Form versus Vectorial Representation

When designing control systems, there are clear advantages using the vectorial model (1.21) instead of (1.5)–(1.6) and the component forms of the Taylor-series expansions (1.11)–(1.14). The main reasons are that system properties such as symmetry, skew-symmetry and positiveness of matrices can be incorporated into the stability analysis. In addition, these properties are related to passivity of the hydrodynamic and rigid-body models (Berge and Fossen, 2000). The system properties represent physical properties of the system, which should be exploited when designing controllers and observers for marine craft. As a consequence, Equation (1.21) is chosen as the foundation for this textbook and the previous book Guidance and Control of Ocean Vehicles (Fossen, 1994). Equation (1.21) has also been adopted by the international community as a standard model for marine control systems design (controller and observer design models) while the classical model (1.5)–(1.6) is mostly used in hydrodynamic modeling where isolated effects often are studied in more detail.

It should be noted that the classical model with hydrodynamic forces in component form and the vectorial model (1.21) are equivalent. Therefore it is possible to combine the best of both approaches, that is hydrodynamic component-based modeling and control design models based on vectors and matrices. However, it is much easier to construct multiple input multiple output (MIMO) controllers and observers when using the vectorial representation, since the model properties and model reduction follow from the basic matrix properties. This also applies to system analysis since there are many tools for MIMO systems. Finally, it should be pointed out that the vectorial models are beneficial from a computational point of view and in order to perform algebraic manipulations. Readability is also significantly improved thanks to the compact notation.

Chapter 2

Kinematics

The study of dynamics can be divided into two parts: kinematics, which treats only geometrical aspects of motion, and kinetics, which is the analysis of the forces causing the motion. In this chapter kinematics with application to local and terrestrial navigation is discussed. Kinetics is dealt with in Chapters 3–8.

The interested reader is advised to consult Britting (1971), Maybeck (1979), Savage (1990), Forssell (1991), Lin (1992), Hofmann-Wellenhof et al. (1994), Parkinson and Spilker (1995), Titterton and Weston (1997), and Farrell and Barth (1998) for a discussion of navigation kinematics and kinematics in general. The development of the kinematic equations of motion are also found in Kane et al. (1983) and Hughes (1986). Both of these references use spacecraft systems for illustration. An alternative derivation of the Euler angle representation in the context of ship steering is given by Abkowitz (1964). A more recent discussion of quaternions is found in Chou (1992). An analogy to robot manipulators is given by Craig (1989) or Sciavicco and Siciliano (1996), while a more detailed discussion of kinematics is found in Goldstein (1980) and Egeland and Gravdahl (2002).

6 DOF Marine Craft Equations of Motion

The overall goal of Chapters 2–8 is to show that the marine craft equations of motion can be written in a vectorial setting according to Fossen (1991):

(2.1) equation

(2.2)

where the different matrices and vectors and their properties will be defined in the forthcoming sections. This model representation is used as a foundation for model-based control design and stability analysis in Part II.

Motion Variables

For marine craft moving in six degrees of freedom (DOFs), six independent coordinates are necessary to determine the position and orientation. The first three coordinates, and their time derivatives, correspond to the position and translational motion along the x, y and z axes, while the last three coordinates and their time derivatives are used to describe orientation and rotational motion. For marine craft, the six different motion components are conveniently defined as surge, sway, heave, roll, pitch and yaw (see Figure 2.1 and Table 2.1).

Figure 2.1 The 6 DOF velocities and in the body-fixed reference frame .

Table 2.1 The notation of SNAME (1950) for marine vessels.

2.1 Reference Frames

When analyzing the motion of marine craft in 6 DOF, it is convenient to define two Earth-centered coordinate frames as indicated in Figure 2.2. In addition several geographic reference frames are needed.

Figure 2.2 The Earth-centered Earth-fixed (ECEF) frame is rotating with angular rate with respect to an Earth-centered inertial (ECI) frame fixed in space.

Earth-Centered Reference Frames

ECI: The Earth-centered inertial (ECI) frame is an inertial frame for terrestrial navigation, that is a nonaccelerating reference frame in which Newton's laws of motion apply. This includes inertial navigation systems. The origin of { } is located at the center of the Earth with axes as shown in Figure 2.2.

ECEF: The Earth-centered Earth-fixed (ECEF) reference frame has its origin fixed to the center of the Earth but the axes rotate relative to the inertial frame ECI, which is fixed in space. The angular rate of rotation is rad/s. For marine craft moving at relatively low speed, the Earth rotation can be neglected and hence can be considered to be inertial. Drifting ships, however, should not neglect the Earth rotation. The coordinate system is usually used for global guidance, navigation and control, for instance to describe the motion and location of ships in transit between different continents.

Geographic Reference Frames

NED: The North-East-Down (NED) coordinate system with origin is defined relative to the Earth's reference ellipsoid (World Geodetic System, 1984). This is the coordinate system we refer to in our everyday life. It is usually defined as the tangent plane on the surface of the Earth moving with the craft, but with axes pointing in different directions than the body-fixed axes of the craft. For this system the x axis points towards true North, the y axis points towards East while the z axis points downwards normal to the Earth's surface. The location of relative to is determined by using two angles and denoting the longitude and latitude, respectively.

For marine craft operating in a local area, approximately constant longitude and latitude, an Earth-fixed tangent plane on the surface is used for navigation. This is usually referred to as flat Earth navigation and it will for simplicity be denoted by . For flat Earth navigation one can assume that is inertial such that Newton's laws still apply.

BODY: The body-fixed reference frame with origin is a moving coordinate frame that is fixed to the craft. The position and orientation of the craft are described relative to the inertial reference frame (approximated by or for marine craft) while the linear and angular velocities of the craft should be expressed in the body-fixed coordinate system. The origin is usually chosen to coincide with a point midships in the water line. This point will be referred to as CO (see Figure 2.3). For marine craft, the body axes , and are chosen to coincide with the principal axes of inertia, and they are usually defined as (see Figure 2.3):

Figure 2.3 Body-fixed reference points.

- longitudinal axis (directed from aft to fore)

- transversal axis (directed to starboard)

- normal axis (directed from top to bottom)

In addition to the body-fixed coordinate system , it is convenient to define other body-fixed coordinate systems when performing hydrodynamic computations. This includes a system using flow axes (see Section 2.4) and the seakeeping reference frame (see Section 5.2).

Body-Fixed Reference Points

The following reference points are defined with respect to CO:

CG - center of gravity

CB - center of buoyancy

CF - center of flotation (located a distance LCF from CO in the -direction

The center of flotation is the centroid of the water plane area in calm water. The craft will roll and pitch about this point. Consequently, this point can be used to compute the pitch and roll periods. The eigenvalues of the 6 DOF linear equations of motion are independent of the reference point but the decoupled equations will produce incorrect results if they are formulated in a point different from CF (see Section 4.3).

6 DOF Vectorial Notation

We will use the notation to refer to a coordinate free vector, that is a directed line segment. When a vector is described relative to a coordinate system , the following notation will be used:

(2.3) equation

where are the unit vectors that define , are the measures of along and are the components of in . We will also use the coordinate form of in which is represented by a column vector in :

(2.4) equation

For marine craft the following notation will be adopted for vectors in the coordinate systems and :

The different quantities in Table 2.1, as defined by SNAME (1950), can now be conveniently expressed in a vectorial setting according to:

where is the Euclidean space of dimension three and denotes a torus of dimension two (shape of a donut), implying that there are two angles defined on the interval . In the three-dimensional (3-D) case the set is a sphere. Hence, the general motion of a marine craft in 6 DOF with as coordinate origin is described by the following vectors:

(2.5)

where denotes the position and orientation vector where the position vector is the distance from NED to BODY expressed in NED coordinates, is a vector of Euler angles, denotes the linear and angular velocity vectors that are decomposed in the body-fixed reference frame and is used to describe the forces and moments acting on the craft in the body-fixed frame.

In many applications, such as flat Earth navigation, the position vector from NED to BODY is expressed in NED coordinates. For global navigation it is convenient to express the position of the BODY origin with respect to ECEF, that is . The orientation of the marine craft with respect to NED will be represented by means of the Euler angles or the quaternions . In the next sections, the kinematic equations relating the BODY, NED and ECEF reference frames will be presented.

2.2 Transformations between BODY and NED

The rotation matrix between two frames a and b is denoted as , and it is an element in , that is the special orthogonal group of order 3:

(2.6)

The group is a subset of all orthogonal matrices of order 3, that is where is defined as

(2.7) equation

Rotation matrices are useful when deriving the kinematic equations of motion