Motion Control of Marine Craft - Thor I. Fossen CRC...32.2 Marine Craft Rigid-Body Dynamics 32.2.1 Kinematics To describe the motion of a marine craft, we consider two reference frames:

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32Motion Control of

Marine Craft

32.1 System Architecture and ControlObjectives ..........................................................32-1

32.2 Marine Craft Rigid-Body Dynamics................32-4Kinematics • Equations of Motion

32.3 Maneuvering Hydrodynamics and Models ....32-8Example: Maneuvering Model of a High-SpeedVehicle--Passenger Trimaran

32.4 Seakeeping Hydrodynamics and Models ......32-12Wave Environment • Time-Domain SeakeepingModels • Frequency-Domain Seakeeping Models •Time-Domain Model Approximations •Time-Domain Wave Excitation

32.5 Models for Maneuvering in a Seaway............32-2132.6 Design Aspects of Vehicle Motion Control

Systems ............................................................32-22Observers and Wave Filtering • Control Allocation •Overview of Vehicle Motion Control Problems

32.7 Example Positioning Control of a SurfaceVessel ...............................................................32-27Unconstrained Control Allocation • ConstrainedControl via Input Scaling • Simulation Case Study

32.8 Example: Course Keeping Autopilot for aSurface Vessel..................................................32-31

32.9 Conclusion ......................................................32-34References ..................................................................32-35

Tristan PerezThe University of Newcastle and NorwegianUniversity of Science and Technology

Thor I. FossenNorwegian University of Science and Technology

Marine craft (surface vessels, underwater vehicles, and offshore rigs) perform operations that require tightmotion control. During the past three decades, there has been an increasing demand for higher accuracyand reliability of marinecraft motion control systems. Today, these control systems are an enabling factorfor single and multicraft marine operations. This chapter provides an overview of the main characteristicsand design aspects of motion control systems for marine craft. In particular, we discuss the architectureof the control system, the functionality of its main components, the characteristics of environmentaldisturbances, control objectives, and essential aspects of modeling and motion control design.

32.1 System Architecture and Control Objectives

The purpose of a marine craft motion control system is to act on the craft using force actuators such thatthe craft follows a desired motion pattern despite environmental forces. The essential components of such

32-1

thor

Typewritten Text

Perez, T. and T. I. Fossen. Motion Control of Marine Craft. Chapter 33, In "The Control Systems Handbook: Control System Advanced Methods (2nd Edition)", CRC Press (W. S. Levine, Ed.), 2011. ISBN 1420073648.

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32-2 Control System Applications

a control system are depicted in Figure 32.1. The main element is the marine craft, which incorporatessensors that provide information related to motion (position, velocity, acceleration), and actuators thatproduce forces to control the motion. The other system components can be grouped into three mainsubsystems, namely, guidance, navigation, and control:

Guidance system: The guidance system provides information on where the craft should go and how itshould get there. The guidance system generates feasible desired reference trajectories describedin terms of position, velocity, and acceleration. The trajectory may be generated by algorithms thatuse the craft’s actual and desired position and, oftentimes, a mathematical model the craft dynamicresponse to control forces. Information about missions, operator decisions, weather, other localvessels, and fleet operations have a bearing on guidance.

Navigation system: The navigation system provides information on where the craft is and how it issailing (speed and heading). The navigation system collects motion information from the variousonboard sensors, such as global navigation satellite systems (GPS, Galileo, GLONASS), speed log,compass, RADAR, and accelerometers; it performs signal quality checking; and it transforms themeasurements to a common reference frame used by the control and guidance systems.

Control system: The control system processes information from the navigation and guidance systemsand provides actuator commands to control the motion. The control system uses motion-relatedsignals to infer the state of the craft and disturbance forces. This processing often involves anobserver–model-based filtering. The motion controller generates appropriate commands for theactuators so as to reduce the difference between the actual and desired craft trajectories. Forsome craft, the actuator configuration is such that the same desired control action can be deliveredusing different combinations of actuator commands. This provides increased reliability to actuatorfaults. In this case, it is common to incorporate a control allocation function, and let the motion

Marine craftForce actuatorsControl allocation

ObserverMotion controller Navigation system

Guidance system

Motion sensors

Environmental forcesControl system

FIGURE 32.1 Marine craft motion control system.

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Motion Control of Marine Craft 32-3

controller be designed to generate desired force commands for the craft’s degrees of freedom ofinterest instead of direct actuator commands. The control allocation function then maps the desiredcontrol forces into actuator commands. Should an actuator fail, the control allocation reconfiguresthe remaining healthy actuators; and thus, the effect of the fault may not require changes in themotion controller. This provides a first level of actuator-fault tolerance.

For marine craft, the environment consists of waves, wind, and ocean currents. The environmentinduces forces on the craft that are considered disturbances to the motion control system. These forcesare conceptually separated into three components:

• Low frequency• Wave frequency• High frequency

where the descriptions low and high are relative to the wave frequency. Waves produce a pressure changeon the craft’s hull surface, which in turn induces forces. These pressure-induced forces have an oscillatorycomponent that depends linearly on the wave elevation. Hence, these forces have the same frequency asthat of the waves and are therefore referred to as wave-frequency forces. Wave forces also have a compo-nent that depends nonlinearly on the wave elevation. These nonlinear-force components have a meancomponent or wave drift force, a low-frequency oscillatory component, and a high-frequency oscillatorycomponent. The low-frequency wave forces cause the vessel to drift and the oscillatory components canexcite resonant modes in the horizontal motion of vessels that are at mooring or under positioning con-trol. The high-frequency wave forces are normally too high to be considered in motion control of marinecraft, but these forces may contribute to structural vibration of the hull. For further details about waveloads and their effects on craft motion, see Faltinsen (1990).

Wind and ocean currents also induce forces due to pressure variation on the craft structure. Wind forceshave a mean component and an oscillatory component due to gusts. In marine craft motion control, onlythe mean wind forces are compensated since the frequency of gusts is often outside the bandwidth of thevessel response—this, however, depends on the size of the craft. Current-induced forces affect marine craftrequiring positioning control and at mooring. These forces have a mean and an oscillatory low-frequencycomponent due to hull vortex shedding. Low-frequency forces on marine craft, therefore, include theeffect of waves, wind and current.

Due to the characteristics of the environmental forces mentioned above, the following problems areconsidered:

• Control only the low-frequency motion.• Control only the wave-frequency motion.• Control both low- and wave-frequency motion.

In low-to-medium sea states, the frequency of oscillations of the linear wave forces do not normallyaffect the operational performance of the craft. Hence, controlling only low-frequency motion avoidscorrecting the motion for every single wave, which can result in unacceptable operational conditions forthe propulsion system due to power consumption and potential wear of the actuators. Applications thatrequire the control of only the low-frequency motion include dynamic positioning, heading autopilots,and thruster-assisted position mooring. Dynamic positioning refers to the use of the propulsion systemto regulate the horizontal position and heading of the craft. In thruster-assisted position mooring, thepropulsion system is used to reduce the mean loading on the mooring lines. Additional applications thatrequire the control of only the low-frequency motion include slow maneuver of surface vessels that arise,for example, from following underwater remotely operated vehicles.

Operations that require the control of only the wave-frequency motions include heave compensationfor deploying loads at the sea floor as well as ride control of passenger vessels, where reducing roll andpitch motion helps avoid motion sickness (Perez, 2005). As the sea state develops, the waves grow in size

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and their frequency reduces; and thus, the wave-induced forces start to appear within the bandwidth ofthe motion control system. In this situation, the control system objective may be to control both low-and wave-frequency motion. This is particularly so for positioning of offshore vessels and oil rigs, forwhich low-frequency motion is controlled in low sea states and both low- and wave-frequency motionis controlled in severe sea states (Fossen, 2002). Another example of a control problem that requirescontrolling both low- and wave-frequency motion is that of using the rudder for simultaneous coursekeeping and roll motion reduction in surface vessels—see Perez (2005) and references therein.

32.2 Marine Craft Rigid-Body Dynamics32.2.1 Kinematics

To describe the motion of a marine craft, we consider two reference frames: Earth-fixed and body-fixed.Marine craft move at a relatively low speed, and hence considering the Earth to be an inertial frameis a good approximation. Associated with the Earth frame, we consider a local geographical coordinatesystem with origin on fixed to the mean water surface and positive directions along the North, East, anddown. This system is abbreviated NED and denoted {n}. The craft is considered to be a rigid body, andthus a reference frame. Associated with the body frame there is a coordinate system with origin at a pointob fixed to the craft and positive directions forward, starboard (right-hand side of the craft when lookingtoward the front), and down. Such a system is denoted by {b}. These frames are illustrated in Figure 32.2.The location of {b} can vary for different control applications.

The position of the craft is given by the relative position of ob with respect to on. The components ofthis vector are North, East, and Down positions and are denoted

pnb/n ! [N , E, D]T .

The lower script b/n indicates that the position refers to that of ob with respect to on. The upper scriptn indicates that components correspond to expressing the vector in {n}.

The orientation of the vessel is given by the Euler angles which correspond to three consecutive

single rotations that take {n} into the orientation of {b}. These rotations are {n} !/zn! {n"}, {n"} "/y"n! {n""},

and {n""} #/x""n! {b}, where the upper script indicates the angle rotated about the axis of rotation, and the

coordinate system on the right of the arrow is the result of the rotation. With the rotations thus defined,

Body frame(b)

Earth frame(n)

x

y

z z

ob

onx

y

FIGURE 32.2 Reference frames and coordinate systems used for marine craft motion description.

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the angles are called !–yaw, "–pitch, and #–roll. The Euler angle vector is given by

! ! [#, ", !]T .

The generalized-position vector (position-orientation) is defined by

" !!

pnb/n!

"

= [N , E, D, #, ", !]T . (32.1)

The velocities are expressed in terms of body-fixed coordinates and are denoted by the generalizedvelocity vector (linear-angular),

# !!

npbb/n

$bb/n

"

= [u, v, w, p, q, r]T . (32.2)

The linear-velocity vector npbb/n = [u, v, w]T is the time derivative of the position vector as seen from

the frame {n}, and the components correspond to expressing the vector in {b}. These components are thesurge, sway, and heave velocities, respectively. The vector $b

b/n = [p, q, r]T is the angular velocity of thebody with respect to the {n} frame with components corresponding to expressing the vector in {b}. Thesecomponents are the roll, pitch, and yaw rates respectively. Figure 32.3 shows the positive convention forthe generalized velocities, and Table 32.1 summarizes the notation.

The trajectory of the craft is given by the time evolution of the generalized positions " defined inEquation 32.1. The time derivative of the positions is related to the body-fixed generalized velocities viaa kinematic transformation,

" = J(") #, (32.3)

where

J(") =#

Rbn(!) 00 T(!)

$.

xb

yb

zb

u(surge)

r(yaw)

v(sway)

w(heave)

p(roll)

q(pitch)

FIGURE 32.3 Positive convention of generalized velocities.

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TABLE 32.1 Summary of Marine Craft Motion Variables

Variable Name Frame Units

N North position Earth-fixed m

E East position Earth-fixed m

D Down position Earth-fixed m

! Roll angle – rad

" Pitch angle – rad

# Yaw angle – rad

u Surge speed Body-fixed m/s

v Sway speed Body-fixed m/s

w Heave speed Body-fixed rad/s

p Roll rate Body-fixed rad/s

q Pitch rate Body-fixed rad/s

r Yaw rate Body-fixed rad/s

pnb/n = [N , E, D]T Position vector Earth-fixed

npbb/n = [u, v, w]T Linear-velocity vector Body-fixed

! = [!, ", #]T Euler-angle vector –

"bb/n = [p, q, r]T Angular-velocity vector Body-fixed

# = [(pnb/n)T , !T ]T Generalized position vector –

$ = [(npbb/n)T , ("b

b/n)T ]T Generalized velocity vector Body-fixed

The rotation matrix Rnb (!) is given by

Rnb (!) =

!

"c#c" !s#c! + c#s"s! s#s! + c#c!s"s#c" c#c! + s!s"s# !c#s! + s#c!s"!s" c"s! c"c!

#

$ ,

where sx " sin(x) and cx " cos(x). Note that rotation matrices are orthogonal, that is, R(!)!1 = R(!)T

and also det R(!) = 1. The transformation from the angular velocity expressed in the body-fixed coor-dinates to the time derivatives of the Euler angles is given by

T(!) =

!

%%"

1 s!t" c!t"0 c! !s!

0s!c"

c!

c"

#

&&$ , t" " tan("), cos(") #= 0.

Note that T(!) is not orthogonal. Also, T(!) and its inverse are singular for " = ±$/2—known as theEuler-angle singularity. This is not usually a problem for marine surface vessels, but it may be an issue forunderwater vehicles. In such cases, alternative representation of the kinematic transformations withoutsingularities can be obtained in terms of quaternions. For further details about marine craft kinematics,see Fossen (2002).

Note that there is no physical vector whose time-derivative gives $ (Goldstein, 1980). Note also that inthe literature of analytical mechanics, the term generalized velocity usually refers to #; however, in thecontext of this chapter, we use the term in relation to the body-fixed velocity vector $.

32.2.2 Equations of Motion

The equations of motion of an unconstrained rigid body can be derived using either vectorial or analyticalmechanics. Here, we will outline the second approach, and in particular the work of Kirchhoff, in whicha rigid body moving in a fluid and the fluid are teated as a single dynamic system (Lamb, 1932).

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The kinetic energy of the craft due to its rotation and translation (without interacting with the fluid) canbe expressed in terms of body-fixed generalized velocities (Egeland and Gravdahl, 2002; Fossen, 2002),

T = 12!T MRB !, (32.4)

where the craft rigid-body generalized mass matrix is of the form

MRB =!

mI3!3 "mS(pbg/b)

mS(pbg/b) Ib

b

"

,

in which, m is the mass of the craft, pbg/b is the position of the craft’s center of gravity (CG) relative to ob

in {b}, and S(a) is, by definition, the skew-symmetric matrix form of any vector a = [ax , ay , az]T , that is

S(a) =

#

$0 "az ayaz 0 "ax

"ay ax 0

%

& .

The inertia matrix Ibb about the point ob can be expressed using the parallel-axis theorem, namely

Ibb = Ib

g " mS(pbg/b)S(pb

g/b) =

#

'$Ibxx "Ib

xy "Ibxz

"Ibyx Ib

yy "Ibyz

"Ibzy "Ib

zy Ibzz

%

(& ,

where pbg/b is position of the CG in {b}, and Ib

g is the inertia matrix about CG in {b}. The inertia matrixabout CG can be computed using the following sum over the vessel mass particles mi :

Ibg =

)

i

mi ST (pbi/g ) S(pb

i/g ),

where pbi/g represents the position of the mass particle i with respect to the CG.

Let the body-fixed generalized forces (forces and moments) be

" = [X, Y , Z, K , M, N]T ,

where

• X, Y , Z are the surge, sway and heave forces respectively.• K , M, N are the roll, pitch, and yaw moments respectively.

Then, we can use Kirchhoff’s equations to derive a dynamic model that relates the forces to the velocities(Kirchhoff, 1869):

ddt

*!T!!1

++ S(!2)

!T!!1

= "1, (32.5)

ddt

*!T!!2

++ S(!1)

!T!!1

+ S(!2)!T!!2

= "2, (32.6)

where,

!1 = [u, v, w]T , !2 = [p, q, r]T ,

"1 = [X, Y , Z]T , "2 = [K , M, N]T .

Substituting Equation 32.4 into Equations 32.5 and 32.6 and manipulating the latter, a general modelstructure for craft’s rigid-body dynamics can be expressed as

MRB! + CRB(!)! = ". (32.7)

The first term on the left-hand side of Equation 32.7 is the product of the generalized mass matrix andthe acceleration. The second term represents forces due to Coriolis and centripetal accelerations. These

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accelerations are due to the rotation of the the body frame relative to the local geographical frame {n}.Note these Coriolis forces are different from the Coriolis forces due to the rotation of the Earth. The latterare ignored since marine craft move at low speeds; and therefore, we can assume that the Earth frame isinertial.

The Coriolis and centripetal acceleration matrix in Equation 32.7 can be expressed in different ways;one of such representations is

CRB(!) =!

mS(!2) !mS(!2)S(pbg/b)

mS(pbg/b)S(!2) !S(Ib

b!2)

"

. (32.8)

For alternative representations of Equation 32.8 see Fossen (2002). Note that

MRB = 0, MRB = MTRB, CRB = !CT

RB.

The kinematic transformation Equation 32.3 together with the kinetic model (Equation 32.7) providea dynamic model for rigid-body motion of the craft without interaction with the fluid. To describe suchinteraction, we need to separate the generalized forces on the right-hand side of Equation 32.7 into

" = "hyd + "ctrl + "env , (32.9)

where "hyd describes fluid pressure-induced forces due to the motion of the craft, "ctrl the control forcesdue to actuators, and "env the environmental forces due to waves, wind and current.

32.3 Maneuvering Hydrodynamics and Models

The study of marine craft dynamics has traditionally been covered by two main theories: maneuveringand seakeeping. Maneuvering refers to the study of craft motion in the absence of wave excitation (calmwater). Seakeeping, on the other hand, refers to the study of motion when there is wave excitation andwhile the vessel keeps its course and speed constant (which includes the case of zero speed). Althoughboth areas are concerned with the same issues: study of motion, stability and control, the separation allowsmaking different assumptions that simplify the study hydrodynamic forces in Equation 32.9.

In maneuvering theory (in calm water), the hydrodynamic forces in Equation 32.9 can be expressed as

"hyd = !MA! ! CA#!$! ! D(!)! ! g(#). (32.10)

The first two terms on the right-hand side of Equation 32.10 can be explained by considering themotion of the craft in an irrotational flow and for ideal fluid (no viscosity). As the craft moves, it changesthe momentum of the fluid. The kinetic energy of the ideal fluid due to the motion of the craft can beexpressed as

TA = 12!T MA!, (32.11)

where the constant matrix MA is called the matrix of added mass coefficients,

MA = !

%

&&&&&&'

Xu Xv Xw Xp Xq XrYu Yv Yw Yp Yq YrZu Zv Zw Zp Zq ZrKu Kv Kw Kp Kq KrMu Mv Mw Mp Mq MrNu Nv Nw Np Nq Nr

(

))))))*, MA = 0.

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The notation used for the coefficients is related to the forces. For example,

• The product Xu u gives the force in surge due to surge acceleration,• The product Yr r is the sway force due to the yaw angular acceleration.

Note that not all the coefficients have units of mass, and that they have signs. For example Xu < 0, and thesign of Yr depends on the extent of fore–aft symmetry of the submerged hull. Note also that dependingon the symmetry of the hull, some of the added mass coefficients can be zero. For vessels maneuvering atlow speeds the added mass matrix is positive definite and symmetric. For surface vessels in waves sailingat forward speed symmetry may be lost (Faltinsen, 1990).

Using the fluid kinetic energy (Equation 32.11) in Kirchhoff’s equations 32.5 and 32.6, we can obtainthe forces on the vessel due to the change in the energy of the ideal fluid (Lamb, 1932) (p. 168). Withsome elementary algebraic work, this gives the first two terms in Equation 32.10 (Fossen, 2002). The firstterm represents pressure-induced forces proportional to the accelerations of the craft. The second termcorresponds to Coriolis and centripetal forces due to the added mass. The Coriolis-centripetal matrix canbe expressed as

CA(!) =!

03!3 "S(A11!1 + A12!2)"S(A11!1 + A12!v) "S(A21!1 + A22!2)

",

where

A = 12

(MA + MTA).

The third term on the right-hand side of Equation 32.10 corresponds to damping forces, which havethe following origins:

Potential damping: This damping force is the result of a body passing through a fluid and wake wavemaking. The word potential indicates that this damping force can be obtained from a study ofirrotational flow in an ideal fluid (no viscosity). In such a fluid, one can define a potential functionof the space, such that its gradient gives the vector field of flow velocity, and the pressure can becomputed from the potential function (Lamb, 1932; Newman, 1977; Faltinsen, 1990). As the fluidis displaced, there is a buildup of pressure in front, and a decrease of pressure behind the body.This pressure difference induces a force that opposes the motion.

Skin friction: This viscous effect is caused by the passage of water over the wetted surface of the hull.At low speeds, it is dominated by a linear component, while at higher speeds, nonlinear termsdominate.

Vortex shedding: This damping force arises from the vortex generation due to flow separation (viscouseffects), which occurs at the sharp edges commonly found at the bow and stern of a marine craft,or the control surfaces.

Lifting Forces: The hull of a craft can be modeled as a low aspect ratio wing (Blanke, 1981; Ross et al.,2007; Ross, 2008). The forces generated by the hull at some angle of attack due to a maneuver canbe resolved into two components: lift and drag. The former acts perpendicular to the direction ofmotion of the vessel, while the latter acts in the opposite direction of the motion. This is depictedin Figure 32.4. Note that the lift force is not actually a damping force as it is oriented perpendicularto the motion of the vessel, it does not serve to dissipate energy. With an abuse of terminology, alllift forces are generically grouped into a damping term. The drag component that arises due to liftis known as parasitic drag, or lift-induced drag. This force acts directly in opposition to motion,and is properly referred to as a damping force.

For vessels maneuvering such that the athwart component of the flow velocity is larger than theforward component, viscous damping effects due to skin friction and vortex shedding are generallygrouped into what is called cross-flow drag (Faltinsen, 1990). This is particular of low-speed maneuveringand positioning applications.

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! !


L D

Ob

xb

U v

u

yb tan β = vu

FIGURE 32.4 Lift and drag forces experienced during maneuvering.

Due to the complex interaction of different effects, the hydrodynamic damping forces of maneuveringvessels are generally modeled by a series expansion, and the coefficients of the model are obtained viaregression analysis of data measured from scaled model tests or by system identification. Two types ofmathematical models are commonly used. The first one consists of representing the complex interac-tions by a multivariable Taylor expansion with only odd terms generally up to third order. This modelwas proposed by Abkowitz (1964) taking into consideration port-starboard symmetry, and it has themathematical elegance of the Taylor terms, but they have no inherent physical meaning. The secondmodel commonly used is an expansion in terms of second-order modulus terms (|x|x). This approachwas introduced by Fedyaevsky and Sobolev (1964) to capture cross-flow drag effects at large angles ofincidence !, see Figure 32.4. It should be mentioned that the differences between these two type of modelof are quite fundamental, and the different coefficients are irreconcilable. Ross et al. (2007), see also Ross(2008), used a first-principle approach and considered the different theories that explain the phenomenainvolved to a great extent and derived a comprehensive model.

The last term in Equation 32.10 represents forces due to gravity and buoyancy. These forces tendto restore the up-right equilibrium of the vessel; and therefore, are called restoring forces. These forcesdepend on the displacement volume of the vessel, its shape and heave, pitch, and roll angles:

g(!) = [0, 0, Zg (!), Mg (!), Kg (!), 0]T .

Depending on the symmetry of the vessel, the forces may be coupled. For example there is usually aheave force due to pitch angle, which is a consequence of the fore–aft asymmetry of the hull.

Replacing Equation 32.10 into Equation 32.9 and combining the latter with Equations 32.7 and 32.3, amaneuvering model takes the following form:

! = J(!)", (32.12)

(MRB + MA)" + CRB(")" + CA(")" + D(")" + g(!) = #ctrl + #env . (32.13)

If there is ocean current, the model must be modified according to

! = J(!)", (32.14)

(MRB + MA)" + CRB(")" + CA("rc)"rc + D("rc)"rc + g(!) = #ctrl + #env , (32.15)

where "rc is the craft velocity relative to the current (seen from the body-fixed frame):

"rc = " ! "c ,

where "c = [uc , vc , wc , 0, 0, 0]T . Current forces can be separated into a potential component due to irro-tational flow in an ideal fluid and a viscous component (nonideal fluid). The potential part in the model

!!

!!

! !


Equation 32.15 is represented by the Coriolis and centripetal term CA(!rc)!rc due to added mass, whereasthe viscous part is incorporated in the damping term D(!rc)!rc . Then, "env in Equation 32.15 accountsfor environmental forces other than ocean currents—for example, wind in the case of surface vessels. Wenext present an example of a maneuvering model for a surface vessel.

32.3.1 Example: Maneuvering Model of a High-Speed Vehicle--PassengerTrimaran

In this section, we consider an example of a 4-degree-of-freedom maneuvering model of high-speedtrimaran adapted from Perez et al. (2007). Figure 32.5 shows a picture of the vessel.

The model considered is given by Equations 32.12 and 32.13 for the degrees of freedom of surge, sway,roll, and yaw, that is,

# = [x, y, !, "]T

! = [u, v, p, r]T

" = [X, Y , K , N]T .

The rigid-body mass and Coriolis-centripetal matrices are given by

MRB =

!

""#

m 0 0 !myg0 m !mzg mxg

0 !mzg Ibxx !Ib

xz!myg mxg !Ib

zx Ibzz

$

%%& ,

and

CRB'n(=

!

""#

0 0 mzg r !m'xg r + v

(

0 0 !myg p !m'yg r ! u

(

!mzg r myg p 0 Ibyzr + Ib

xypm

'xg r + v

(m

'yg r ! u

(!Ib

yzr ! Ibxyp 0

$

%%& ,

FIGURE 32.5 Austal’s hull H260 “Benchijigua Express.” (Courtesy of Austal Ships, http://austal.com.)

http://www.crcnetbase.com/action/showImage?doi=10.1201/b10382-40&iName=master.img-000.jpg&w=312&h=208

!!

!!

! !


where m is the mass of the vessel, pbg/b = [xg , yg , zg ]T gives position the CG relative to ob, and Ib

ik are themoments and products of inertia about ob.

The kinematic transformation (Equation 32.12) reduces to

J!!"=

#

$$%

cos(!) ! sin(!) 0 0sin(!) cos(!) cos(") 0 0

0 0 1 00 0 0 cos(")

&

''( .

The added mass matrix and the Coriolis-centripetal matrix due to added mass are given by

MA = MTA = !

#

$$%

Xu 0 0 00 Yv Yp Yr0 Kv Kp Kr0 Nv Np Nr

&

''( ,

CA!""=

#

$$%

0 0 0 Yvv + Ypp + Yrr0 0 0 !Xuu0 0 0 0

!Yvv ! Ypp ! Yrr Xuu 0 0

&

''( .

The adopted damping terms take into account the lift, drag, and viscous effects.

D(") = DLD(") + DVIS("),

where

DLD(") =

#

$$%

0 0 0 Xrv v0 Yuv u 0 Yur u0 Kuv u 0 Kur u0 Nuv u 0 Nur u

&

''( . (32.16)

DVIS(") =

#

$$%

Xu|u| 0 0 00 Y|v|v |v| + Y|r|v |v| 0 Y|v|v |v| + Y|r|r |r|0 0 Kp|p| + Yp 0 00 N|v|v |v| + N|r|v |v| 0 N|v|v |v| + N|r|r |r|

&

''( . (32.17)

The lift-drag representation in Equation 32.16 is consistent with taking only the first order terms derivedin Ross et al. (2007)—see also Ross (2008), whereas the viscous damping representation in Equation 32.17follows from Blanke (1981). Finally, the restoring term reduces to

g(!) = [0, 0, Mg (!), 0]T .

Figure 32.6 shows model validation data for the velocities based on a full scale zig-zag sea trial.The hydrodynamic parameters of the model were partially obtained from computational fluid dynamicsoftware, and partially by optimizing model prediction errors using full-scale trial data. For further detailssee Perez et al. (2007).

32.4 Seakeeping Hydrodynamics and Models

Seakeeping theory of marine craft motion refers to the study of motion in waves while the vessel keeps aconstant course and speed. The equations of motion are described within the linear framework, whichallows considering the problem in the frequency domain. The latter, in turn, enables one to compute

!!

!!

! !


0 50 100 150 200 250 300 350

20

u (m

/s) TrialModel

0 50 100 150 200 250 300 350!10

0

10

t (s)

v (m

/s) EstimModel

0 50 100 150 200 250 300 350!2

0

2

p (de

g/s) Trial

Model

0 50 100 150 200 250 300 350!2

0

2

r (de

g/s) Trial

Model

FIGURE 32.6 Model validation against a zig-zag sea trial data for a high-speed trimaran ferry.

wave-to-force and force-to-motion frequency response functions. The waves are described as realizationsof stationary stochastic processes. When the vessel frequency responses are combined with the waveelevation power spectral density (PSD), one can compute the response, from which several statisticsof motion are derived. The approach just described is commonly used by naval architects to comparedifferent hull forms at a craft-design stage.

The frequency-domain models used in seakeeping theory provide valuable information for marine craftmotion control design. In this section, we discuss models for wave elevation, vessel frequency responsewith particular parameterizations used in naval architecture, and time-domain simulations. Finally, wediscuss how the seakeeping models can be combined with maneuvering models for control system design.

32.4.1 Wave Environment

Ocean waves are random in both time and space. These characteristics are often summarized by the termirregular in the marine literature. The stochastic description is, therefore, the most appropriate approachto characterize them. The following simplifying assumption regarding the underlying stochastic modelare usually made: The observed sea surface elevation !(t), at a certain location and for short periods of time,is considered a realization of a stationary and homogeneous zero mean Gaussian stochastic process. Theperiod for which ocean waves can be considered stationary can vary between 20 min and 3 h. For deepwater, wave elevation tends to present a Gaussian distribution, as the water becomes shallow nonlineareffects dominate, and the waves become non-Gaussian (Ochi, 1998).

Under the stationary and Gaussian assumptions, the sea surface elevation is completely characterizedby its PSD S!!("), commonly referred to as the wave spectrum,

E[!(t)2] =! !

0S!!(") d".

The wave spectrum can be estimated from data records. However, to study the response of marinestructures, a family of idealized spectra is commonly used. One commonly used family is the modified

!!

!!

! !


Pierson-Moskowitz or International Towing Tank Conference (ITTC) spectrum—which was recom-mended by the ITTC in 1978:

S!!(") = A"5 exp

!!B"4

"(m2s). (32.18)

The parameters A and B are given by

A =173H2

1/3

T41

, B = 691

T41

,

where H1/3 is the significant wave height (average of the highest one third of the waves) and T1 is theaverage wave period. These two parameters are referred to as long-term statistics. There are wave atlaseswith scatter diagrams of H1/3 and T1 for specific locations on the globe and time of the year. Figure 32.7shows a plot of the ITTC wave spectrum (Equation 32.18) for 4 m significant wave height and threeaverage wave periods. This type of spectrum is used to describe fully developed seas in deep water. Fora thorough introduction to the modeling of ocean waves see Ochi (1998), and for the models used inmarine craft motion control see Perez (2005).

When a marine craft is at rest, the frequency at which the waves excite the craft coincides with the wavefrequency; and thus, the previous description is valid. However, when the craft moves with a constantforward speed U , the frequency observed from the craft differs from the wave frequency. The frequencyexperienced by the craft is called the encounter frequency. The encounter frequency depends not only onthe speed of the craft, but also on the angle the waves approach:

"e = " ! "2Ug

cos(#). (32.19)

where, the encounter angle # defines the sailing condition, namely,

• Following seas (# = 0 or 360 deg)• Quartering seas (0 < # < 90 deg or 270 < # < 360 deg)• Beam seas (# = 90 deg—port or 270 deg—starboard)• Bow seas (90 < # < 180 deg or 180 < # < 270 deg)• Head seas (# = 180 deg)

The encounter frequency captures a Doppler effect. Figure 32.8 shows a schematic representation ofthe transformation Equation 32.19. From this figure, we can see that when the vessel is sailing in bow orhead seas, the wave frequencies are mapped into higher frequencies. In beam seas, however, there is nochange and both " and "e are the same. In following and quartering seas, the situation becomes moreinvolved as different wave frequencies can be mapped into the same encounter frequency. In deep water,

0 0.5 1 1.5 2ω (rad/s)

S ζζ(ω

) (m

2 s)

2.5 3 3.500.5

11.5

22.5

3T1 = 7 s

Top of sea state 5 : H1/3 = 4 m

T1 = 9 sT1 = 10 s

FIGURE 32.7 Example of ITTC PSD for the wave elevation and wave slope for H1/3 = 4 m and T1 = 7, 9 and 10 s.

!!

!!

! !


Following seasQuartering seas

ω =

ωe =

g2U cos(χ)

ωe

g4U cos(χ)

ω = gU cos(χ)

Beam seas

Bow seasHead seas

ω

FIGURE 32.8 Transformation from wave to encounter frequency under different sailing conditions for U fixed.

long waves travel faster than short waves. Hence, in following and quartering seas, long waves overtakethe vessel whereas short waves are overtaken by the vessel. Indeed, for 0 < ! <

gU cos(") the waves overtake

the vessel. The wave frequency ! = gU cos(") , at which !e = 0, corresponds to the situation in which the

component of the craft velocity in the direction of wave propagation is the same as the wave celerity. Inthis case, the wave pattern observed from the craft remains stationary and travels along with the craft.Finally, for high-frequency waves, the encounter frequency is negative, meaning that the craft overtakesthe waves.

Since the power of any magnitude is invariant with respect to the reference frame from which it isobserved, for any PSD the following holds:

S(!e) d!e = S(!) d!.

From this, it follows that

S(!e) = S(!)!!!!d!

d!e

!!!!

= S(!)!!!!1 ! 2!Ug

cos(")!!!!

.

For beam seas, the transformation is trivial, that is since cos(#/2) = 0, then S(!e) = S(!). In bow seas,the encounter spectrum is a spread version of the wave spectrum shifted toward higher frequencies. Forquartering and following seas the situation becomes complex since expression (Section 32.4.1) is singularat !w = g/(2U cos ") where the denominator vanishes. This is an integrable singularity, and the varianceof the process remains the same in both wave- and encounter-frequency domains (Price and Bishop,1974).

The motion of a marine craft in waves is the result of the wave excitation due to the varying distributionof pressure on the hull. Therefore, the wave excitation, as well as the vessel response, will depend notonly on the characteristics of the waves—amplitude and frequency—but also on the sailing conditions:encounter angle and speed. The wave spectrum, and thus the wave-induced forces can change significantlywith the sailing conditions for a given sea state. These changes have a significant bearing on control systemdesign.

32.4.2 Time-Domain Seakeeping Models

For simplicity, in this and the next section, we will only consider the case of zero forward speed and thencomment on the extensions to the forward speed case in Section 32.4.4.

!!

!!

! !


For the zero-speed case, the models (32.3) and (32.7) can be linearized about the equilibrium point(! = 0, " = 0) and expressed as

!! = !",

MRB!" = !#,

where !! represents the generalized perturbation position-orientation vector and !" the generalizedperturbation body-fixed velocity vector.

For a vessel in waves, the generalized pressure force !# vector can be separated into two components:

!# = #rad + #exc .

The first component corresponds to the radiation forces arising from the change in momentum of thefluid due to the oscillatory motion of the craft at the wave frequency. The radiation forces are the resultof the waves being radiated from the hull as a consequence of its motion. The excitation represents thepressure forces due to the incoming waves. Both radiation and excitation forces in sinusoidal waves canbe studied using potential theory, that is, irrotational flow of an ideal fluid (Newman, 1977; Faltinsen,1990).

In seakeeping theory, the forces are computed in a reference frame fixed to the equilibrium position ofthe vessel—called equilibrium or seakeeping frame, to which we associate a coordinate system {s}. Hence,the forces require a kinematic transformation. For the case where the seakeeping frame is stationaryrelative to the local Earth frame {n} (zero speed case), these kinematic transformations can be neglectedunder the small angle assumption. Then, for zero speed,

!! ! !,

!" ! ",

!# ! #.

Cummins (1962) studied the radiation hydrodynamic problem in an ideal fluid and found the followingrepresentation for linear hydrodynamic forces:

#rad = "A " "! t

0K(t " t#)"(t#) dt#. (32.20)

The first term in Equation 32.20 represents forces due the accelerations of the structure, and A is theconstant positive-definite added mass matrix$. The second term represents fluid-memory effects thatincorporate the energy dissipation due the radiated waves as a consequence of the motion of the vessel.The kernel of the convolution term, K(t), is the matrix of retardation or memory functions (impulseresponses).

By renaming the variables, combining terms, and adding the hydrostatic restoring forces due to gravityand buoyancy (#hs = "G!), we obtain the Cummins Equation for zero forward speed:

(M + A)" +! t

0K(t " t#)"(t#) dt# + G! = #exc . (32.21)

Equation 32.21 describes the motion of a vessel at zero speed for any wave excitation #exc(t) providedthe linearity assumption is satisfied. In the case of forward speed, additional linear terms appear inEquation 32.21. This is further discussed in Section 32.4.4.

$ Note that this added mass matrix is different than the one used in maneuvering—Section 32.3. We will explain thedifference in Section 32.4.4.

!!

!!

! !


32.4.3 Frequency-Domain Seakeeping Models

When Equation 32.21 is considered in the frequency domain, it can be expressed in the following form(Newman, 1977; Faltinsen, 1990):

(!!2[M + A(!)] ! j!B(!) + G)!( j!) = "exc( j!), (32.22)

where !( j!) and "exc( j!) are the complex response and excitation variables:

"i(t) = "i cos(!t + #i) =" "i( j!) = "i exp( j#i)

$i(t) = $i cos(!t + %i) =" $i( j!) = $i exp( j%i).

The parameters A(!) and B(!) are the frequency-dependent added mass and damping respectively.Equation 32.22 is also commonly written in a mixed frequency–time–domain form:

[M + A(!)]! + B(!)! + G! = "exc . (32.23)

This unfortunate form is rooted deeply in the literature of marine hydrodynamics and the abuse ofnotation of this false time-domain model has been discussed eloquently in the literature (Cummins,1962). The reader is warned that Equation 32.23 is not a time-domain model, rather a different way orwriting Equation 32.22, which is a frequency response function. The corresponding time–domain modelis given by Equation 32.21.

Expression 32.22 provides the frequency response from force to displacement,

!( j!) = G( j!)"exc( j!),

that is,

G( j!) = [!!2(MRB + A(!)) + j!B(!) + G]!1 =

!

"#

G11( j!) · · · G16( j!)...

...G61( j!) · · · G66( j!)

$

%& .

Similarly, there exists a frequency response from wave elevation to wave-excitation force,

"exc( j!) = F( j!, &) '( j!),

where & is the angle at which the waves approach the vessel—see Section 32.4.1, and

F( j!, &) ='F1( j!, &) · · · F6( j!, &)

(T.

The latter frequency response is known as the force response amplitude operator (FRAO) in the navalarchitecture literature. By combining the above frequency responses we obtain the wave-to-motion fre-quency response, which is known as the motion response amplitude operator (MRAO),

!( j!) = H( j!, &)'( j!),

whereH( j!, &) = G( j!) F( j!, &).

Hydrodynamic codes based on potential theory are nowadays readily available for the computation ofthe frequency-dependant added mass, A(!), and potential damping, B(!), and therefore the force andmotion RAO. These codes use information about hull geometry and weight distribution to compute thecoefficients and responses for finite set of frequencies.

If we combine the RAO with the wave spectrum, we can compute the force and motion spectra:

S$$( j!) = |F( j!, &)|2S''( j!), (32.24)

S""( j!) = |H( j!, &)|2S''( j!). (32.25)

These spectra can be used to compute wave force and motion statistics and also the time series forsimulations.

!!

!!

! !


32.4.4 Time-Domain Model Approximations

By taking the Fourier transform of Equation 32.21 and comparing it with Equation 32.22, it can be shownthat

A(!) = A ! 1!

! "

0K(t) sin(!t) dt,

B(!) =! "

0K(t) cos(!t) dt.

From these expressions, it follows

A = lim!#" A(!). (32.26)

Expression 32.26 indicates that A is the infinite-frequency added mass matrix. The models used formaneuvering—see Section 32.3—are low-frequency models. Hence, the added mass MA in maneuveringmodels is related to A(0) $= A(").

It also follows from the Fourier transform that

K( j!) = B(!) + j![A(!) ! A]. (32.27)

As commented in the previous section, hydrodynamic codes can be used to compute A(!), andpotential damping, B(!) for a discrete set of frequencies. This information can be used togetherwith Equation 32.27 to obtain rational transfer functions that approximate the convolution integralin Equation 32.21, that is

µ =! t

0K(t ! t%)!(t%) dt% ! K(s) & x = A%x + B% !

µ = C%x(32.28)

From potential theory, it can be shown (Perez and Fossen, 2008b) that

lim!#0

K( j!) = 0,

lim!#" K( j!) = 0,

limt#0+

K(t) $= 0,

limt#"

K(t) = 0,

! '# µ is passive.

These properties translate into the following constraints on the rational approximations Kik(s) =Pik(s)/Qik(s):

Kik(s) has a zero at s = 0, (32.29)

Kik(s) has relative degree 1, (32.30)

Kik(s) is stable, (32.31)

Kik(s) is at least of order 2, (32.32)

Kik(s) is positive real for i = k. (32.33)

The constraints (Equations 32.29 through 32.33) can be easily enforced if one performs the identifica-tion in the frequency domain using the nonparametric data (Equation 32.27). Figure 32.9 shows the bowhalf-hull of a semisubmersible offshore rig used to compute hydrodynamic data with the code WAMIT.These data are part of a demo of the Marine Systems Simulator (MSS) available at www.marinecontrol.org.Figure 32.10 shows the results of frequency-domain identification incorporating constraints for the cou-

!!

!!

! !


!200

2040

60!40

!200

2040

!30

!20

!10

0

10

Y-axis (m)X-axis (m)

Z-ax

is (m

)

FIGURE 32.9 Hull geometry of a semisubmersible. (Data from www.marinecontrol.org.)

10!2 10110010!1 10!2 10110010!1

10!2 10110010!1 10!2 10110010!1

100

110

120

130

140

150

160

Frequency (rad/s)

Frequency (rad/s)Frequency (rad/s)

Frequency (rad/s)

|k(jω

)|

Convolution model DoF 22

!100

!50

0

50

100

Phas

e k(jω

) (de

g)

!1

0

1

2

3

5" 107

" 107

B (k

g/s)

Potential damping DoF 22

4

5

6

7

A (k

g)

Added mass DoF 22

k ( jω)khat ( jω), order 7

BBest FD ident, order 7

AAest FD indet, order 7Ainf

FIGURE 32.10 Data fit corresponding to the coupling 2-2 of a semisubmersible. The left-hand side plots show themagnitude and phase of K22( j!) and K22( j!) for a parametric approximation of order 7. The right-hand side plotsshow damping and added mass computed by the code and the approximations based on K22( j!).

!!

!!

! !


plings 2-2 (sway-sway). The left-hand side plots show the magnitude and phase of the convolutionfrequency response and the right-hand side plots show the potential damping added mass. For furtherdetails about identification of radiation force models, see Perez and Fossen (2008a,b 2009).

Time-domain models can be obtained from Equation 32.21 and replacing the convolution by anapproximating LTI system Equation 32.28. This model, however, needs to be augmented with viscousdamping:

(M + A) ! + Dvis ! + µ + G " = #exc , (32.34)

x = A!x + B! !

µ = C!x.(32.35)

Potential theory only gives the radiation damping, which reflects the energy carried away by the wavesthat are generated as a consequence of craft motion—this is captured in the model by Equation 32.35.The radiation damping is only a part of the total damping. Unfortunately, there are no empirical rulesto incorporate damping, except for the degree of freedom of roll (Ikeda, 2004). Hence, one needs to useexperimental data to compute such damping.

If the vessel has forward speed U , the seakeeping model becomes:

(M + A) ! + Dvis ! + C" ! + D" ! + µ + G " = #exc , (32.36)

x = A!x + B! !!,µ = C!x,

(32.37)

where C" and D" are linear additional damping and Coriolis-centripetal terms that appear due to for-ward speed U and the kinematic transformation between the equilibrium frame in which the forces arecomputed and the body frame. For further details, see Perez and Fossen (2007) and Fossen (2002).

32.4.5 Time-Domain Wave Excitation

In order to generate realizations of the wave-excitation forces, the spectrum (Equation 32.24) can beused. Indeed, since the wave elevation is Gaussian and considered stationary, and the force responsebeing considered is linear, the response is also Gaussian and stationary. There are different approachesto generate realizations from the spectrum. One approach consists of making a spectral factorization ofEquation 32.24 and approximating the realizations as filtered white noise. This approach is commonlyused in stochastic control theory. Another approach, commonly used in naval architecture, consists ofusing a multisine signal. For example, for any component of #, we can generate realizations via

"i(t) =N!

n=1

"n cos(#nt + $n),

with N being sufficiently large, where "n are constants, and the phases $n are independent identicallydistributed random variables with uniform distribution in [0, 2%]. This choice of random phases ensuresthat "i(t) is a Gaussian process, and for each realization of the phases, we obtain a realization of theprocess (St Denis and Pierson, 1953).

The autocorrelation of the process for lag zero satisfies

" #

0S""(#) d# !

N!

n=1

"2n

2,

from which we can take

"n =#

2S""(#")&#,

where #" is chosen randomly within the interval [#n $ &#2 , #n + &#

2 ].

!!

!!

! !


0 1 2 30

0.02

0.04

0.06

0.08

ωe (rad/s)

ωe (rad/s)

ωe (rad/s)

Roll

Sway

Yaw

0 20 40 60 80 100

0 1 2 3 0 20 40 60 80 100

0 1 2 3 0 20 40 60 80 100

!20

!10

0

10

20

t(s)

t(s)

t(s)

Roll (

deg)

Sway

(m)

Yaw

(deg

)

0

0.5

1

1.5

!2

!1

0

1

2

0

2

4"10!4

!2

!1

0

1

2

ITTC Hs (m): 2.5, T (s): 7.5, Speed (kt): 15, Enc. Angle (deg): 90

FIGURE 32.11 Roll sway and yaw motion power spectral densities and time series of a navy vessel for beam seas at15kts. The wave spectrum used is ITTC with Hs = 2.5 m and T = 7.5 s.

Note that this method can be used not only for wave forces but also for the wave elevation and for themotion of the vessel itself. For example, Figure 32.11 shows the spectra of motion for roll, sway and yawof a navy vessel in a beam sea sailing condition, and also particular realizations of motion simulated usingthe multisine. For further details, see Perez (2005).

32.5 Models for Maneuvering in a Seaway

There are applications in which it is necessary to consider models for maneuvering in waves. The hydro-dynamic interactions in these situations are rather complex, and to date there is no theory that unifiesmaneuvering and seakeeping. Models for control design and testing can be built in different ways depend-ing on the data available. One can consider the following options:

1. Linear seakeeping model augmented with nonlinear damping terms.2. Maneuvering model with wave force excitation (force superposition, input disturbance).3. Maneuvering model with wave motion excitation (motion superposition, output disturbance).

The first option consists of using (Equations 32.36 through 32.35) and augmenting it with nonlineardamping terms. This option is used when there is no data from the vessel other than the frequency

!!

!!

! !


responses computed by the hydrodynamic codes—the viscous damping has to be added empirically. Thisapproach can give the control designer an initial model to start a design. Though the seakeeping modelis not for maneuvering, the Coriolis and centripetal terms will be incorrectly represented. Hence, oneshould be aware of the validity of the model—only mild maneuver should be attempted with this model.

The second and third option can be used when a maneuvering model is available (Equations 32.14and 32.15), which has been obtained from either model scale or full scale trials, and we also have accessto frequency responses computed by hydrodynamic codes. The frequency responses can be used inconjunction with the adopted wave spectrum to compute either the force or the motion response spectra(Equations 32.24 and 32.25), from which we can simulate realizations of either the wave-induced forcesor motion. Some of the literature argues that using force superposition is a more natural approach thanusing motion superposition. However, the motion superposition model is more accurate in terms ofdescribing the wave-induced motion.

32.6 Design Aspects of Vehicle Motion Control Systems

In this section, we discuss aspects that are fundamental to marine system control design. We first lookat the observer design and the control allocation, and then describe the main characteristics of the mostcommon motion control problems.

32.6.1 Observers and Wave Filtering

As discussed in Section 32.1, the frequency of oscillations of the linear wave forces do not normallyaffect the operational performance of vessels. Hence, controlling only low-frequency motion avoidscorrecting the motion for every single wave, which can result in unacceptable operational conditions forthe propulsion system due to power consumption and potential wear of the actuators.

The control of only low-frequency motion is achieved by appropriate filtering of the wave-frequencycomponents from the position and heading measurements and estimated velocities before the signals arepassed on to the controller. This filtering process is known as wave filtering.

Early course-keeping autopilots used a proportional (P) controller with a deadband nonlinearity.The deadband provided an effect similar to wave filtering since it delivered a null control action untilthe control signals were large enough to be outside the deadband. The amount of deadband in theautopilot could be changed, and this setting was called weather since the size of the deadband wasselected by the operator based on weather conditions. Other systems used lowpass and notch filters,which introduced significant phase lag, and thus performance degradation when a high-gain controlis required. An alternative to traditional filtering consists of using a wave-induced motion model andan observer to separate the wave motion from the low-frequency motion. This is depicted in the blockdiagram shown in Figure 32.1.

The design of observers for positioning and course keeping is normally approached within the linearframework by augmenting the model with an output disturbance model that represents the wave-inducedmotion. Let us consider here the problem for positioning of surface vessels, that is, we consider maneu-vering at low speed in the degrees of freedom of surge, sway, and yaw,

! = [N , E, !]T ,

" = [u, v, r]T .

Because of the slow maneuvering assumption, the nonlinear damping and Coriolis and centripetalterms in Equations 32.14 and 32.15 can be neglected; thus,

! = R(!) ", (32.38)

(MRB + MA)" + D"rc = #ctrl + #env , (32.39)

!!

!!

! !


where the kinematic transformation reduces to a rotation matrix

R(!) =

!

"cos (!) ! sin (!) 0sin (!) cos (!) 0

0 0 1

#

$ , R!1(!) = RT (!).

The model (Equations 32.38 through 32.39) is still nonlinear due to the kinematic transformation. Asdiscussed in Fossen (2002), this model can be linearized dynamically by introducing the vessel parallelcoordinates, which are defined in a reference frame fixed to the vessel with axes parallel to the Earth-fixedreference frame. The vessel parallel position coordinates !p are defined by using the transformation

!p = RT (!)!, (32.40)

where !p is the position-attitude vector expressed in body coordinates. For positioning control applica-tions, rotation about the z-axis is often slow. Therefore, r " 0 and R(!) " 0 are good approximations.Consequently, the time-derivative of Equation 32.40 leads to

!p = RT (!)! + RT (!)!,

= RT (!)! + RT (!)R(!)",

" ".

(32.41)

Using the vessel parallel coordinates the kinematics are linearized,

!p = ",

(MRB + MA)" + D"rc = #ctrl + #env .

The wave-frequency forces induce motion, and due to the linearity of the model, we can consider thewave-induced motion as an output disturbance. This disturbance can be modeled as filtered white noise,

$ = Aw$ + Eww,

!w = Cw$,

which in transfer-function form results

!w(s) =

!

"Gxw(s) 0 0

0 Gyw(s) 00 0 G!w(s)

#

$ w(s),

with

Giw(s) = "2i s

s2 + 2 #i "is + "2i

.

The low-frequency wave forces, wind, and the viscous component of the current forces, can all bemodeled as constant input disturbances. This leads to the following model,

$ = Aw$ + Eww1, (32.42)

!p = ", (32.43)

(MRB + MA)" + D" = #ctrl + b + w2. (32.44)

b = w3, (32.45)

with measurement!tot = !p + Cw$ + n. (32.46)

The state-noises w1, w2, and w3 represent model uncertainty, and the noise n is due to measurementinstrumentation.

!!

!!

! !


By combining Equations 32.42 through 32.46 we obtain a state-space model of the form

x = Ax + B !ctrl + Eobsw,

"tot = C x + n,

wherex = [#T , "T

p , $T , bT ]T .

With this model one can design an observer, and estimate the state. Then pass to the controller onlylow-frequency state variables, namely, "p, $, and b.

Figure 32.12 shows simulation data of a Kalman wave filter design for fishing vessel under positioningcontrol (Fossen and Perez, 2009). In this simulation, the wave filter is switched on 120 s after the vessel isunder positioning control. Then, at 200 s the vessel position is changed 10 m forward. Figure 32.12a showsthe measured and wave-filtered surge position. Figure 32.12b shows the measured and wave-filtered surgevelocity. Figure 32.12c shows the force generated by the controller. During the first 120 s, while the wavefilter is switched off, the wave-induced motion produces significant control action. Once the wave filteris switched on, the control action at wave frequencies is reduced, which is the effect sought.

0

5

10(a)

(b)

(c)

Posit

ion

surg

e (m

)

Surge

!0.5

0

0.5

1

Velo

city s

urge

(m/s)

!2000

!1000

0

1000

2000

Surg

e con

trol f

orce

(N)

0 50 100 150 200 250 300 350 400Time (s)

0 50 100 150 200 250 300 350 400Time (s)

0 50 100 150 200 250 300 350 400Time (s)

MeasuredFiltered

MeasuredFiltered

DemandedActual

FIGURE 32.12 Wave filter performance for a 15-m fishing vessel under positioning control. The wave filter isswitched on at 120 s.

!!

!!

! !


32.6.2 Control Allocation

In order to increase reliability, some marine vehicles are usually equipped with more actuators than theminimum required to control the motion in the desired degrees of freedom. Then, the forces used tocontrol the motion can be produced by different combinations of the forces produced by the actuators.This practice enables the vehicle to continue operation or to start a safe shutdown in case of an actuatorfailure by reconfiguring the forces produced by the remaining actuators. Within this framework, the craftmotion control system is separated into two main components:

• Motion controller• Control allocation mapping

This is depicted in the block diagram shown in Figure 32.1. The motion controller generates demandsfor generalized forces in the degrees of freedom in which the vehicle is controlled. Since the motioncontroller operates in terms of generalized forces, the control design and tuning can be done independentlyof the actuator configuration to a certain extent. The control allocation mapping then transforms thecontroller demands into individual actuator commands, such that the demanded generalized forces areimplemented.

Each actuator produces a bounded force vector; that is, a vector with a prescribed line of action andpoint of application:

Ti ! Si , i = 1, 2, . . . , N .

Here N identifies the number of actuators. Figure 32.13 shows a schematic diagram of a vehicle withfour actuators.

The force vectors take values in the sets Si , which represent the constraints due to limited forcemagnitude and direction that each actuator can produce at a particular time.

Due to the location of the actuators on the craft, the forces are mapped into the craft generalized forcesvia a thrust configuration matrix,

! = B(") T,

where" = [!1, . . . , !N ]T,

andT = [|T1|, . . . , |TN |]T.

The control allocation problem can then be posed as a constrained optimization problem:

(T!, "!, s!) = arg minT,",s

V (T, ", "0, s) (32.47)

α3α1

α2

α4x

y

VehicleT3

T4

T2

T1

FIGURE 32.13 Vehicle actuator forces.

!!

!!

! !


subject to

B(!) T = "d ! s (32.48)

Tmin " T " Tmax , (32.49)

!Tmin " T ! T0 " !Tmax , (32.50)

!min " ! " !max , (32.51)

!!min " ! ! !0 " !!max , (32.52)

s # 0. (32.53)

The scalar-valued objective function V (·, ·, ·, ·) in Equation 32.47 relates to the control effort and maycontain a barrier term that avoids singular actuator configurations, that is, the configurations in whichthe capacity to generate forces in some degrees of freedom are lost. For example, if in the vessel depictedin Figure 32.13 all the actuators direct their forces forward, then it is not possible to generate either sideforce or a turning moment; hence this is a singular configuration (in this case, the thrust configurationmatrix loses rank).

The constraint (Equation 32.48) ensures that the desired generalized forces demanded by the controller,"d , are implemented. The remaining constraints are related to magnitude and angle of the forces and alsotheir rates, which are set in terms of differences from given values T0 and !0.

Due to the limited authority of the actuators, that is, the constraints (Equations 32.49 through 32.52),it may happen that the generalized force vector demanded by the controller is not feasible. To avoidinfeasibility of the optimization problem, the slack variable s is included in Equation 32.48 with theconstraint (Equation 32.53). If the demanded control vector is feasible, then s! = 0.

The above optimization problem is solved online at each sampling instant; hence, the values T0 and !0correspond to the solution T! and !! obtained in the preceding sampling instant. For further details oncontrol allocation for marine vehicles, see Fossen et al. (2009) and references therein.

32.6.3 Overview of Vehicle Motion Control Problems

In the following we describe the main marine vehicle motion control problems and their characteristics.

32.6.3.1 Dynamic Positioning and Thruster-Assisted Position Mooring

Dynamic positioning refers to the use of the propulsion system to hold the vehicle’s position despiteenvironmental disturbances. For surface vessels, this problem involves horizontal position and headingregulation. that is surge, sway, and yaw. This type of control problem is common in offshore vessels andoil rigs. The objective is to control only the low-frequency motion. Therefore, the control system requireswave filtering. Since offshore vessels perform critical operations, they are over-actuated. Therefore, thecontroller generates generalized force commands and there is a control allocation mapping. The structureof the controller often consists of a velocity and a position loop. The control design can be approached asan optimal control problem for set point regulation. The implementation of the controller is done usingproportional-integral (PI) and proportional–integral–derivative (PID) controllers. In Section 32.7, weprovide an example. For underwater vehicles, the positioning problem extends to auto-depth. Since thevertical plane motion is often decoupled from the horizontal plane motion and there are specific actuatorsto control each type of motion, the two positioning problems can often be designed independently. Ina thruster-assisted position mooring, the propulsion system is used to compensate the mean loading onmooring lines. This control problem is similar to that of dynamic positioning.

32.6.3.2 Autopilots

Autopilots used in surface and underwater vehicles are alike. Their degree of sophistication can varyfrom simple course keeping to maneuvering control and can sense and avoid functionality. For surface

!!

!!

! !


vessels, the problem is considered in one degree of freedom (yaw) for a course autopilot controller,but the guidance system takes into account the position as well as the heading angle. Typical controllerimplementations consist of PID controllers with feedforward functionality. A turn rate loop is incorpo-rated. The reason for it is that the designs should avoid direct PID control from heading angle since theresulting loop transfer function would have a double integrator, and therefore a step response will alwaysovershoot. Autopilots for surface vessels do not normally require control allocation. Wave filtering is afeature of autopilots for large ocean-crossing vessels, hence there is no rudder correction for every singlewave. Autopilots for maneuvering control require nonlinear control designs.

32.6.3.3 Ride Control

Ride control refers to the use of motion control to reduce the wave-induced motion in roll and pitch.These control systems are characteristics of vessels that carry passengers and goods. Local accelerationsdue to roll and pitch are the main causes of motion sickness, which can produce cargo damage, andprevent the use of equipment on board. The control objectives are to reject the wave-frequency motionwithout affecting the steering capability of the vessel. Since the spectrum or roll and pitch can varysignificantly with the sea state and sailing condition, the control systems often have to be adaptive tomaximize performance over a wide envelope of conditions. When combined roll and pitch control isrequired, control allocation may be used as some of the specific actuators used can generate both pitchand roll, such as T-foils, interceptors and trim flaps.

As a final general remark on control design, we should mention that the models of marine vehiclespresent a significant degree of uncertainty, and the dynamic response changes with sea state, velocity,wave direction, water depth, and other vessels in close proximity. Since vehicle motion follows physicallaws of energy, passivity-based control designs have been very successful. A controller designed suchthat the stability depends only on dissipativity properties can result in closed-loop stability even underlarge parametric uncertainty—even changes in model order may be tolerated provided that dissipativityproperties remain unchanged. For further details, see Fossen (2002). We next present two control designexamples.

32.7 Example Positioning Control of a Surface Vessel

We consider the positioning problem in the horizontal plane of a surface vessel, that is, the generalizedposition, velocity, and force vectors are

! !

!

"NE!

#

$ , " !

!

"uvr

#

$ , # !

!

"XYN

#

$ ,

and the low-frequency control design model is given by

!p ! ",

" = M"1D" + M"1(#ctrl + #env),

where M = MRB + MA.The control system block diagram is shown in Figure 32.14. The measured generalized position vector,

thus, has a wave- and low-frequency component, which is transformed to vessel parallel coordinates,

!pm = !p,WF + !p.

A wave-filter observer is used to estimate the low-frequency position and the velocity vector usedto implement the controller. The motion controller consists of a velocity loop implemented with a PI

!!

!!

! !


Vesseldynamics

Environmentalforces

Navigationsystem

AllocationController

Manual

Observerwave filter

Actuators

RT(ψ)

RT(ψ)

τenv

τctrl

τpm ηm

ηd

υ, ηpηp,d

τ ηu

FIGURE 32.14 Block diagram of a dynamic positioning control system.

controller and a position loop implemented with a P controller. The structure of the controller is illustratedin the block diagram shown in Figure 32.15. This controller implements integral action for both positionand velocity, which rejects low-frequency force disturbances (Perez and Donaire, 2009).

32.7.1 Unconstrained Control Allocation

The force vectors produced by the actuators can be decomposed into rectangular coordinates (along thelongitudinal and transverse direction of the vessel) and combined into a single vector T,

T =!TX1 TY1 TX2 TY2 . . . TXN TYN

"T,

where N is the number of actuators. This vector is mapped into the generalized forces via the actuatorconfiguration matrix B (which for the sake of simplicity we had fixed):

! = B T. (32.54)

Since T has more components than !, there are different vectors T satisfying (Equation 32.54) a givenvalue of !. In order to limit the number of solutions, one can pose the problem as an optimization problem,for example,

T! = arg minu

(TT WT)

subject to !d = BT.(32.55)

The objective function TT WT is representative of the total energy or control effort, where W is a positivedefinite matrix weighting the relative cost of using different actuators. Thus, the control allocation seeks

ηp

τd

υ

ηp, dP

– –Pl

FIGURE 32.15 Proposed positioning controller.

!!

!!

! !


the solution that implements the desired generalized force !d whilst minimizing the control effort. Asshown in Fossen (2002), the solution of the above problem is given by

T! = B†!d , B† = W!1BT (BW!1BT )!1. (32.56)

Note that since B depends only on the location of the actuators on the vehicle, the right inverse B† can beprecomputed.

The optimization problem Equation 32.55 does not take into account the fact that the vector T mustbelong to a constraint set due to the maximum force that the various actuators can produce. Addingthis constraint to Equation 32.55 requires on-line numerical optimization as discussed in Section 32.6.2,and the solution (Equation 32.56) is no longer the optimal solution. An alternative to this approachis to constrain the desired generalized force !d , such that the constraints on T are always satisfied. Bydoing so, the force controller is also informed about reaching constraints, which prevents performancedegradation due to the combination of actuator saturation and integral action. This control method canbe implemented using multivariable anti-wind up techniques as discussed in the next section.

32.7.2 Constrained Control via Input Scaling

One of the key issues in control design for systems that require integral action and present a potentialfor actuator saturation is that of an integrator windup. That is, if the actuators saturate and the integralcontroller is not informed about the saturation, the integrators continue integrating the error signalsbut the control action is not seen by the system. This often produces a degradation of performance interms of undesirable oscillations and even instability. Control schemes that deal with this effect are calledantiwindup schemes.

If a linear controller C(s) is in minimum phase and bi-proper (as in the case of a PI and PID con-troller), then antiwindup can be achieved simply by implementation. Goodwin et al. (2001) proposed theimplementation shown in Figure 32.16. In this figure, Lim represents a saturation (magnitude, rate, or acombination of both), and the gain

c" = lims#" C(s).

Note that if the limitation is not active, the loop of Figure 32.16, reduces to the controller:

C(s) =!I + c"

"C(s)!1 ! c!1

"#$!1

c".

When the limitation becomes active, it prevents the control signal !d from exceeding its limits, and theconstrained signal drives the states of the controller, which are all on the feedback path.

e

y

–

[I + c∞ (C(s)–1 – c∞–1) ]–1c∞

Limc∞ τd

FIGURE 32.16 Block diagram of an anti-wind-up implementation of a strictly proper controller.

!!

!!

! !


The antiwindup scheme described above can be applied to the velocity PI controllers of the velocityloops of the vehicle positioning controller:

Cvel(s) =

!

"Cu(s) 0 0

0 Cv(s) 00 0 Cr(s)

#

$ ,

where, for i = u, v, r,

Ci(s) = Kip

TiI s + 1

TiI s

.

In order to constrain !d we can construct a set S!, such that

!d ! S! " T ! ST .

One way of enforcing the constraint is by computing an unconstrained control !uc and then scaling itdown if it is outside the constraint set, that is,

!d =%

!uc if !uc ! S!

" !uc if !uc /! S!, " < 1 : " !uc ! !S!,(32.57)

where !S! denotes the boundary of the set S!.By scaling the vector !uc , we preserve its direction. By this way of limiting the control command, the

block Lim in Figure 32.16 can be replaced by

Lim # #(t) I.

In order to implement Equation 32.57, we need to compute !uc and then obtain ". An algorithm toevaluate Equation 32.57 is given in Perez and Donaire (2009).

In the above proposed control system, the constraint set of actuator forces due to the limited authorityof the actuators are mapped into a constraint set for the generalized forces. This approach results ina constrained control design with a simple, unconstrained, control allocation problem. The stability ofthe closed-loop system can be analytically proven by rearranging the closed-loop system with controlallocation into a Lure system and application of Lyapunov and passivity theories (Perez and Donaire,2009).

32.7.3 Simulation Case Study

To illustrate the performance of the controller, we consider a model of an offshore vessel from Fossen(2002), and we consider the actuator configuration depicted in Figure 32.13, that is,

• Two stern azimuth thrusters, which are set at angles of #1 = 135$, and #2 = 225$.• One bow tunnel thruster, #3 = 90$.• One bow deployable azimuth thruster, which is fixed to provide force only along the longitudinal

axis of the vessel, #4 = 0$.

Figure 32.17 shows the results of a simulation experiment, in which the vessel is in hold position(regulation), then after 20 s we set a reference set-point change in surge, followed by a set point in sway,and finally one in yaw. This figure shows the demanded and actual surge, sway, and yaw positions,velocities and generalized forces. As we can see from this demanded and actual generalized forces, theantiwindup scheme works such that the demands are feasible. Due to the saturation of the actuators, thevelocity demands cannot be followed. Figure 32.18 shows the corresponding forces of the four actuators.As we can see from the latter figure, the forces remain within the constraints on the maximum forcemagnitude. The control system presents a good performance despite the actuators reaching the saturationlevels due to the antiwindup implementation.

!!

!!

! !


0

20

40

60

Surg

e pos

ition

(m)

0

2

4

6

Sway

pos

ition

(m)

Surge Sway Yaw

0

10

20

30

40

Yaw

angle

(deg

)

00.10.20.30.4

Surg

e velo

city (

m/s)

!0.050

0.050.1

0.15

Sway

velo

city (

m/s)

0

1

2

3

4

Yaw

angle

(deg

/s)

!101234 " 105 " 104 " 106

0 500 1000 0 500 1000Time (s)

0 500 1000Time (s)Time (s)

0 500 1000 0 500 1000Time (s)

0 500 1000Time (s)Time (s)

0 500 1000 0 500 1000Time (s)

0 500 1000Time (s)Time (s)

Surg

e con

trol f

orce

(N)

ActualDemanded

!5

0

5

10

15

Sway

cont

rol f

orce

(N)

!2

0

2

4

6

Yaw

cont

rol m

omen

t (N.

m)

ActualDemanded

ActualDemanded

ActualDemanded

ActualDemandedActual

Demanded

FIGURE 32.17 Performance of vehicle position regulation controller with a position set point change.

32.8 Example: Course Keeping Autopilot for a Surface Vessel

The response in yaw rate due to a small deviation in the angle of a control actuator, such as a rudder orthe steering nozzle of a water jet, can be derived from Equation 32.13 by isolating the yaw motion, whichis given by

(Ibzz ! Nr)r ! Nrr = N!!,

where Ibzz is the moment of inertia in yaw, Nr , Nr , and N! are hydrodynamic coefficients, r is the yaw rate,

and ! is the actuator angle (rudder or the steering nozzle of a water jet). This model, which is known asthe first-order Nomoto model, can be written as the transfer function

r(s)!(s)

= K1 + Ts

.

The time constant and low-frequency gain are given by

T = Ibzz ! Nr

!Nr,

K = !N!

Nr,

which can be estimated from trials in calm water.

!!

!!

! !


!1

0

1 " 105

" 105

" 105

" 105

Actuator forces

Tunn

el-bo

w (N

)

!1

0

1

Azim

uth-

bow

(N)

!4!2

024

Ster

n-sta

rboa

rd (N

)

!4!2

024

0 100 200 300 400 500 600 700 800 900 1000Time (s)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Ster

n-po

rt (N

)

FIGURE 32.18 Actuator forces for a vehicle change in position.

Using the motion superposition assumption, as in the case of positioning control design, we modellow-frequency environmental disturbances with a bias moment term in the equation of motion. Then,the state-space model can be written as

! = r,

r = ! 1T

r + 1m

"N + b,

b = 0,

where m = Izz ! Nr and

"N = mKT

# = N# #

denotes the control yaw moment.For autopilot control, it is common to design a PID controller with feedforward from wind and a

smooth time-varying reference signal !d(t) according to

"N(s) = !"wind + m!

rd ! 1T

rd

"

# $% &"FF

! m!

Kp! + Kdr + Ki

' t

0!(")d"

"

# $% &"PID

, (32.58)

where "N is the controller yaw moment and "FF is a feedforward term using the reference signal rd = !d .The heading and yaw rate errors are denoted by ! = ! ! !d and r = r ! rd , respectively. The control

!!

!!

! !


gains Kp, Kd , and Ki must be chosen such that the third-order linear error dynamics

˙r +!

1T

+ Kd

"r + Kp! + Ki

# t

0!(")d" = 0

are asymtotically stable. The control law (Equation 32.58) depends on the wind-yaw moment estimate"wind , which is used as a feedforward term. This accelerates the response of the autopilot to changes inwind direction and intensity. The wind-yaw moment is modeled as

"wind = 12#aU2

rwALw LoaCNw ($rw), (32.59)

where #a is the air density, Urw and $rw are the speed and direction of the wind relative to the vessel, Loais the overall length of the vessel, and ALw is a characteristic area exposed to the wind, and CNw ($w) is ayaw moment wind coefficient. The uncertainty in Equation 32.59 has a low-frequency content, which iscompensated by the integral action of the controller (Fossen, 2002). The rudder command is computedfrom the control input "N as

% = 1N%

"N.

0

10

20

30

40(a)

(b)

(c)

Head

ing (

deg)

!0.2

0

0.2

0.4

0.6

Head

ing r

ate (

deg/

s)

!4

!2

0

2

4

0 50 100 150 200 250Time (s)

0 50 100 150 200 250Time (s)

0 50 100 150 200 250Time (s)

Head

ing (

deg)

Low-frequency motionKF estimate

Low-frequency motionKF estimate

Wave-frequency motionKF estimate

FIGURE 32.19 Performance of a wave filter for heading autopilot for a mariner cargo ship; (a) the true low-frequencyheading ! and Kalman filter estimate !; (b) the true low-frequency heading rate r and its Kalman filter estimate r;(c) the wave-frequency component of the heading !w and its Kalman filter estimate !w . The Kalman filter usesmeasurements of the sum of low- and wave-frequency heading and estimates these states and the rate using a modelof the vessel and a model of the wave-induced motion.

!!

!!

! !


!10

0

10

20

30

40(a)

(b)

0 50 100 150 200 250Time (s)

0 50 100 150 200 250Time (s)

Head

ing (

deg)

!5

0

5

10

15

Rudd

er an

gle (d

eg)

VesselDesired

FIGURE 32.20 Performance of a heading autopilot for a mariner cargo ship with a wave filter; (a) the time seriesof the desired heading !d and the actual vessel heading !; (b) the rudder angle ". As the low-frequency estimates ofthe heading angle and rate produced by the Kalman filter alone are passed to the controller, the motion of the rudderdoes not respond to the wave motion, and thus the wave filtering is achieved.

In order to implement the control law, both ! and r are needed. Most of the marine crafts have onlycompass measurements !, and thus the turning rate r must be estimated. In addition, it is necessary toperform wave filtering such that the oscillatory wave-induced motions are avoided in the feedback loop.

To illustrate the performance of a course keeping autopilot with wave filtering, we consider the caseof an autopilot application taken from the MSS (MSS, 2010). This simulation package implemented inMATLAB! and Simulink! provides models of vessels and a library of simulink blocks for headingautopilot control system design and blocks for a Kalmanfilter-based wave filter from heading-only mea-surements. The vessel considered is a 160-m mariner class vessel. From the step tests performed on thenonlinear model, a first-order Nomoto model (Equation 32.8) is identified. Figures 32.19 and 32.20 showthe performance of the wave filter. Figure 32.19a and b show the true low-frequency heading angle andrate together with the Kalman filter estimates. Figure 32.19c shows the first-order wave-induced headingangle component and its estimate. Figure 32.20 shows the performance of the control loop. Figure 32.20ashows the desired and the actual heading angle of the controlled vessel. Figure 32.20b depicts the rudderangle. In this figure, we can appreciate the effect of the wave filtering since the rudder angle has no motionat the wave frequency.

32.9 Conclusion

Marine craft perform operations that require tight motion control. During the past three decades, there hasbeen an increasing demand for higher accuracy and reliability of marine craft motion control systems.Today, these control systems are an enabling factor for single and multicraft marine operations. Thischapter provides an overview of the main aspects of marine craft motion control systems and their

!!

!!

! !


designs. In particular, we discuss the architecture of the control system, the functionality of its maincomponents, the characteristics of environmental disturbances and their influence on control objectives,and the essential aspects of modeling and motion control design. For further details on marine craftmotion control, see Fossen (2002) and Perez (2005).

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Abkowitz, M., 1964. Lecture notes on ship hydrodynamics—Steering and manoeuvrability. Technical reportHy-5, Hydro- and Aerodynamics Laboratory, Lyngby, Denmark.

Blanke, M., 1981. Ship propulsion losses related to automatic steering and prime mover control. Ph.D. thesis,Servolaboratory, Technical University of Denmark.

Cummins, W., 1962. The impulse response function and ship motion. Technical Report 1661, David TaylorModel Basin—DTNSRDC.

Egeland, O. and J. Gravdahl, 2002. Modeling and Simulation for Automatic Control. Marine Cybernetics,Trondheim.

Faltinsen, O., 1990. Sea Loads on Ships and Offshore Structures. Cambridge University Press, Cambridge.Fedyaevsky, K. and G. Sobolev, 1964. Control and Stability in Ship Design. State Union Shipbuilding,

Leningrad.Fedyaevsky, K. and G. Sobolev, 1964. Control and Stability in Ship Design. State Union Shipbuilding, Leningrad.Fossen, T., 2002. Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater

Vehicles. Marine Cybernetics, Trondheim.Fossen, T. I., T. A. Johansen, and T. Perez, 2009. A Survey of Control Allocation Methods for Underwater Vehicles,

chap. 7, pp. 109–128. In-Tech, Vienna, Austria.Fossen, T. and T. Perez, 2009. Kalman filtering for positioning and heading control of ships and offshore rigs.

IEEE Control Systems Magazine 29(6):32–46.Goldstein, H., 1980. Classical Mechanics. Addison-Wesley, Reading, MA.Goodwin, G. C., D. E. Quevedo, and E. L. Silva, 2001. Control System Design. Prentice-Hall, Inc., New Jersey.Ikeda, Y., 2004. Prediction methods of roll damping of ships and their application to determine optimum

stabilisation devices. Marine Technology 41(2):89–93.Kirchhoff, G., 1869. Uber die Bewegung eines Rotationskorpers in einer Flüssigkeit. Crelle 71:237–273.Lamb, H., 1932. Hydrodynamics, 6th ed, Cambridge University Press, Cambridge.MSS, 2010. Marine Systems Simulator. Viewed 29/1/2010, http://www.marinecontrol.org.Newman, J. N., 1977. Marine Hydrodynamics. MIT Press.Ochi, M., 1998. Ocean Waves: The Stochastic Approach. Ocean Technology Series. Cambridge University Press,

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Perez, T. and T. Fossen, 2007. Kinematic models for seakeeping and manoeuvring of marine vessels at zeroand forward speed. Modelling Identification and Control, The Norwegian Society of Automatic Control,28(1):19–30, doi: 10.4173/mic.2007.1.3.

Perez, T. and T. Fossen, 2008a. Joint identification of infinite-frequency added mass and fluid-memory modelsof marine structures. Modelling Identification and Control, The Norwegian Society of Automatic Control,29(3):93–102, doi:10.4173/mic.2008.1.1.

Perez, T. and T. I. Fossen, 2008b. Time-domain vs. frequency-domain identification of parametric radiationforce models for marine structures at zero speed. Modelling Identification and Control, The NorwegianSociety of Automatic Control, 29(1):1–19, doi:10.4173/mic.2008.3.2.

Perez, T. and T. I. Fossen, 2009. A MATLAB toolbox for parametric identification of radiation-force modelsof ships and offshore structures. Modelling Identification and Control, The Norwegian Society of AutomaticControl, 30(1):1–15, doi:10.4173/mic.2009.1.1.

Perez, T., T. Mak, T. Armstrong, A. Ross, and T. I. Fossen, 2007. Validation of a 4-DOF manoeuvring modelof a high-speed vehicle-passenger trimaran. In 9th International Conference on Fast Sea Transportation(FAST), Shanghai, China, September.

Price, W. and R. Bishop, 1974. Probabilistic Theory of Ship Dynamics. Chapman & Hall, London.

!!

!!

! !


Ross, A., 2008. Nonlinear maneuvering model based on low-aspect ratio lift theory and lagrangianmechanics. PhD thesis, Department of Engineering Cybernetics.

Ross, A., T. Perez, and T. Fossen, 2007. A novel manoeuvring model based on low-aspect-ratio lift theory andLagrangian mechanics. In Proceedings of the IFAC Conference on Control Applications in Marine Systems(CAMS). Bol, Croatia, September.

St Denis, M. and W. Pierson, 1953. On the motion of ships in confused seas. SNAME Transactions 61:280–332.

Documents

Motion Control of Marine Craft - Thor I. Fossen CRC...32.2 Marine Craft Rigid-Body Dynamics 32.2.1 Kinematics To describe the motion of a marine craft, we consider two reference frames: