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Marine Technology and Engineering – Guedes Soares et al. (Eds)© 2011 Taylor & Francis Group, London, ISBN 978-0-415-69808-5

Mathematical models for simulation of manoeuvring performance of ships

Serge Sutulo & C. Guedes SoaresCentre for Marine Technology and Engineering (CENTEC), Instituto Superior Tecnico, Technical University of Lisbon, Portugal

ABSTRACT: The review covers a number of selected topics related to ship mathematical models used in ship manoeuvring, mainly for simulation purposes. The focus is given to relatively obscure or even some-times misunderstood issues. A rather detailed analysis of the manoeuvring motion kinematics is given followed by discussion of various forms of dynamic equations and parametric representation of hydro-dynamic forces. Special attention is given to comparative evaluation of methods for estimating manoeu-vring model’s parameters including application of optimal experimental designs to captive-model tests and various system identification techniques. Comments are given on the Munk method and cross-flow drag concept. Review of the most popular database methods for estimating manoeuvring derivatives is performed. Methods for practical estimation of forces on propellers and rudders are analysed. The review is concluded with brief comments on the numerical integration of equations of motion and on the code organization.

hydrodynamics of curvilinear motion has been made and the most essential theoretical analysis of the directional stability of the straight-path motion has been completed. But it became clear that further development was conditioned by the progress in computer technology. In fact, even before reasonably powerful digital computers became common in ship dynamics, analogue com-puters were from time to time used for simulating manoeuvring motion but this technique was not sufficiently flexible and efficient. Most difficulties were related to modelling complex nonlinearities which imposed restrictions on complexity of treat-able mathematical models. As result, analogue modelling was used rarely, mostly in applications to submarine dynamics and/or design and analysis of automatic controllers.

At certain moment—apparently by the end of 70s it appeared that a manoeuvring mathematical model of any desired complexity could be relatively easily coded and made run on a common digital computer in real or even accelerated time and this changed the situation dramatically. Not only a powerful research tool became available to scien-tists but the computer simulation made the ship manoevrability theory very demanded from the side of ship operators and navigators. Problems of simulating the manoeuvring motion have become very important as this served as base for ship han-dling simulators of various complexity. The soft-ware implemented in these simulators comprised two relatively independent and equally important

1 INTRODUCTION

The theory of ship manoeuvrability was the last domain of ship hydrodynamics and dynamics to emerge and to mature. The state of the art in this area as it was consolidated by the end of 30s is well exposed by Schoenherr (1939) in the first edition of the well known Principles of Naval Architec-ture (PNA) and it apparently remained practically unchanged until the end of 40s when the last printing of the same edition appeared. The theory presented there looks elementary and many points are even not quite correct in view of the nowadays knowledge.

The situation, however, changed considerably by the beginning of 60s when the second, revised edition of PNA was published and the chapter on manoeuvring written by Mandel (1965) does not look obsolete even today. No wonder that most of the material was reproduced with minor changes in the third edition of the same reference book (Crane et al., 1989). It is interesting to note that similar level of maturity was achieved in the textbook by Fedyaevsky and Sobolev (1964) who presented accomplishments of the Russian manoeuvrability school which, due to obvious political causes, was developing relatively independently of the world mainstream.

But during next three decades, the evolution of and advance in the manoeuvring theory resulted in certain qualitative changes. Of course, by the begin-ning of 60s, substantial progress in understanding

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parts: the core mathematical model describing the ship’s dynamics together with the required environmental factors and the graphic interface necessary for realistic interaction between the operator, instructor or trainee with the simulated ship. The difference between these two compo-nents is that the graphic interface is clearly visible while the core dynamic model is hidden and only felt through its reactions to the control actions and simulated exogenous factors (wind, current, waves etc.). It is usually required that the core model correspond to a certain real ship and its responses must, in the ideal case, be identical to those of the real vessel although the complete similarity still cannot be reached.

Many practical aspects concerning various types of bridge simulators were covered by Hensen (1999). Besides visualisation issues, the emphasis is made on validation of core manoeuvring models which is in no way a simple task. The present con-tribution will, however, focus on problems related to development of dynamic models while certain visualisation and interface issues can be found in (Silva et al., 2000), (Silva & Guedes Soares 2000) and (Varela et al., 2008).

A comprehensive review of many problems related to the manoeuvring simulation was given in (Barr 1993). In some sense, the present review is complementing the cited one concentrating more on theoretical issues many of which seem, however, to be of considerable practical importance. Also, the contemporary manoeuvring theory after many dec-ades of development carried out by many scientists from different countries and with different back-ground may look somewhat eclectic and certainly contains a number of “dark spots” and obscurities sometimes misleading even relatively experienced researchers. The authors tried to reveal and clarify some of these obscurities hoping that this will help to reduce the number of not very well designed for-mulations which still appear in applications.

The scope of this paper will be restricted to “normal” surface displacement ships excluding, for instance, planing boats, hydrofoils, air-cushion vehicles, submarines etc. although in some cases these objects will be mentioned as providing help-ful comparisons.

2 DESCRIPTION OF MANOEUVRING MOTION

2.1 General remarks: rigid body concept

A fundamental fact about the ship manoeuvring theory, usually taken for granted, is that a ship can always be modelled as an absolutely rigid body. This is, however, less obvious than it could seem

bearing in mind that in the neighbouring area of ship seakeeping hydroelastic formulations are now-adays not uncommon. As one of the current trends in ship hydrodynamics is fusion of manoeuvring and seakeeping formulations (Sutulo & Guedes Soares 2009a), development of combined manoeu-vring-and-seakeeping models based on the flexible hull concept can be expected in the future. How-ever, current state of the art in ship manoeuvring does not suggest such formulations, and, definitely, they have no sense in still-water manoeuvring.

2.1.1 Frames of referenceThe main task of any ship mathematical model incorporated into a bridge simulator is to represent a time sequence of the ship’s instantaneous posi-tions modelling the motion.

Although on the theoretical level every such position can be described using coordinate-free vector-tensor techniques, in practice, an Earth-fixed Cartesian frame of reference is indispensable. There are several definitions of this frame which differ each from other by orientation of certain axes. Probably, the most common is the right-handed system Oξηζ with the origin O placed at the undisturbed water surface and fixed with respect to the ground. The ζ-axis is directed vertically downwards. Horizontal location of the origin and direction of the axis O can be chosen arbitrarily in manoeuvring problems of investigation character. For instance, O may become the starting point of the simulated manoeuvre and the direction of the ζ-axis may then correspond to the approach path. But in practical ship handling tasks typically repro-duced on simulators certain standard geographic coordinates must be assigned to the origin and the axis Oξ must be oriented along the true meridian. The Earth-fixed frame, as defined, can be treated as inertial practically without limitations in ship dynamics. The only known exception is the prob-lem of prediction of the drift of an unpowered ves-sel under action of wind and current. In this case, accounting for the Coriolis force caused by the rotation of the Earth can substantially influence the predicted location of the vessel. That is why more “global” frames like the geocentric frame (Fossen 1994), which is very important in missile dynamics and space navigation, are almost useless in dynamics of marine craft.

The second necessary frame is the body-fixed frame Cxyz. Its origin is always located in the ship’s centerplane but as to the longitudinal and vertical position, it can either coincide with the centre of mass G or be placed in the midship of the hull and on the waterplane at rest. In the first case, the frame becomes central and some equations take simpler form but here a more general second option will be preferred.

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2.1.2 Generalized coordinatesAccording to classic general mechanics (Lurie 2002), 6 generalized coordinates are required to describe position and orientation of a rigid body. The three obvious and natural linear generalized coordinates of the ship are Cartesian coordi-nates of the origin C in the Earth-fixed frame: the advance ξC, transfer ηC, and sinkage (heave) ζC.

Describing the angular orientation is more com-plicated and can be done in various ways but in ship dynamics this role is in fact monopolised by the Euler angles obtained through consecutive rotation around the axes: ζ, η and ξ. The thus obtained angular generalized coordinates are: the heading ψ, the pitch angle θ, and the roll angle ϕ. In its standard navigation version the heading angle is non-negative and ψ ∈ [0, 2π] measured, however, usually in degrees. But it can be also assumed in research applications that ψ ∈ [−π, π] and this angle can be then called alternatively the head-ing change or the yaw angle. Alternative methods of describing the angular orientation, like the Rodrigues—Hamilton parameters, did not find applications in marine dynamics. The cause of this will be explained later.

2.1.3 Generalized velocities and quasi-velocitiesA generalized velocity is simply derivative of a gen-eralized coordinate with respect to time t. The six ship generalized velocities are then ξ η ζ ϕ θcξ ηξ η ζζ,ηcηη ,ϕ , and ψ. These parameters are, however, used not often. It is much more convenient to describe the motion with the so-called quasi-velocities. Although various quasi-velocities have been used for decades and even centuries by multiple scien-tists, apparently this important concept appeared in English-language literature for the first time in the book by Lurie (2002)1. A quasi-velocity is defined as a linear form over generalized velocities with dimensionless coefficients which, however, may depend on general coordinates. Use of this term is explained by the simple fact that, unlike a usual velocity of any nature, a quasi-velocity is not time derivative of any displacement.

Although the formal definition of the quasi-velocity may seem a bit obscure, the ship qua-si-velocities are obtained in a very natural way as projections of the instantaneous ground veloc-ity of the origin V G and of the angular velocity vector Ω on the body axes Cxyz. This operation results in the well-known velocities of: surge uG, sway υG, heave w, roll p, pitch q, and yaw r (the meaning of the subscript G added to the first two

velocities is to emphasize that these are ground quasi-velocities).

Introducing the vector of generalized coordi-nates Ξ = ( , , )ξ η( , ζ ϕ, θ ψ,cη, ζζ T and the vector of quasi-velocities VG = (uG, vG, w, p, q, r)T it is possible to write relations linking the generalized velocities with quasi-velocities as

Ξ = TVGVV , (1)

where the transition matrix is:

T =⎡

⎣⎢⎡⎡

⎣⎣

⎤

⎦⎥⎤⎤

⎦⎦

T

T

1TT

2TT

0

0 (2)

and

T

c c c s s c s s s

s c s s s c s c s

s

1TT =

− +s c c s c

+ −c c s s c

−

ψ θcc ψ θss ssϕ ψs cs ccϕ ψc sc ssθ ϕcccc ψ ϕss

ψ θcc ψ θss ssϕ ψc cc ccθ ψs ss ssθ ϕcccc ψ ϕss

θ c scc c c

T

s

c s

s c

θ ϕss θ ϕcc

ϕ ϕ

ϕ ϕs

ϕ θ ϕ θc

θ θϕ

⎡

⎣

⎢⎡⎡

⎢⎢⎢

⎢⎢⎢

⎢⎣⎣⎢⎢

⎤

⎦

⎥⎤⎤

⎥⎥⎥

⎥⎥⎥

⎥⎦⎦⎥⎥

c=

⎡

⎣

⎢⎡⎡

;

tanθ cϕc

/ /c cθc ϕ

2TT

1

0

0

⎢⎢⎢⎢⎢⎢

⎢⎢⎢⎢⎢

⎢⎣⎣⎢⎢

⎤θ

⎦

⎥⎤⎤

⎥⎥⎥

⎥⎥⎥

⎥⎦⎦⎥⎥, (3)

where cψ = cos ψ, cθ = cos θ, sθ = sin θ, sϕ = sin ϕ, sψ = sin ψ, and cϕ = cos ϕ.

It is clear that, when inverted, the Equations (1) will provide the mentioned formal definition of the introduced quasi-velocities as linear forms but more important is that Eq. (1) represent the first part of equations of motion, namely, the kinematic differen-tial equations. It is obvious from (3) that the fourth and the last equations have singular behaviour as | | .π| 2 This is why, some alternative angular generalized coordinates are often preferred in aircraft and space dynamics. Fortunately, no one marine object and much less some surface displacement ship threatens to approach this pitch value in any think-able motion. This explains why any alternatives to the Euler angles are in ship dynamics unnecessary.

2.1.4 Kinematics with wind and currentIn presence of a uniform steady current with the velocity Vc and uniform steady wind with the veloc-ity Vw besides the velocity over the ground VG, it is necessary to consider also the ship’s velocity with respect to water V and the velocity with respect to air VA. Obviously,

V G = V + V c = V A + V w. (4)

The projections of the air and water veloc-ity vectors on the body axes are respectively the air linear quasi-velocities uA, υA, wA and water

1 The Russian original of this book was, however, pub-lished in 1961 and the concept itself had been introduced even earlier.

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quasi-velocities u, υ, w. The latter ones are usually notated without subscripts as they are the most important and most frequently used. In absence of wind and current all three sets of linear quasi-velocities, i.e., ground, water and air velocities, become identical.

2.1.5 Non-dimensional and alternative kinematic parameters

The primary dimensional kinematical parameters introduced above are sufficient for description of arbitrary ship motion but it is often coven-ient and natural to deal with their dimensionless analogues designated with primes: u′ = u/V, u′ = u/V, υ′ = υ/V, w′ = w/V, u′Α = uA/VA, υ′A= υA/VA,

w′A= wA/VA, p′ = pL/V, q′ = qL/V, and r′ = rL/V,

where the speed V = |V| = u w2 2 2+2υ , the air speed VA = |V A|, and L is the reference length, which is usually the length between perpendicu-lars, for water velocities and length overall for air velocities. The non-dimensional time/path t′ (being non-dimensional, it can be interpreted in any way) is defined by the equation

dtV

Ldt′ = ( )t

. (5)

The thus defined dimensionless parameters are used when the speed V in the denominators is not too small. When it may happen, preferred are the drift angle β and the attack angle α defined as an appropriate solution to the equations:

cos ,sin ,

sin .

α βcosβ υ

α βcos

= ′− sin ′

= ′

u

w (6)

The attack angle is indeed extensively used in submarine dynamics (Rozhdestvensky 1970) but pratically never in manoeuvring of surface ships. In fact, at low speed the motion is practically two-dimensional and it can be assumed α ≡ 0. Low-speed analogues of the angular quasi-velocities are defined as result of an appropriate mapping (−∞, +∞) → [−1, +1]. In two-dimensional manoeu-vring it only goes about the rate of yaw. The most convenient generalized angular rate of yaw r″ is apparently (Sutulo 1994)

′′ = ′

+ ′r

r

r1 2. (7)

An evident possible alternative is

r ′2

2ππ

arctan ( )r′2

π (8)

and a similarly defined generalized dimensional velocity of yaw is used by Chislett (1996).

The dimensionless velocity r can be interpreted as some fictitious angle and—without normaliz-ing factors π 2 and 2 π —is called the yaw rate angle in (Oltmann and Sharma 1985). Using angu-lar parameters defining orientation of the angu-lar velocity vector Ω in the body axes turned out the only consistent way for defining generalized dimensionless angular velocity parameters suitable for arbitrary 3D motion (Sutulo and Kim 1997). Those parameters may become important for underwater vehicles dynamics but in manoeuvring of surface ships the yaw rate only is used in the non-dimensionalized form.

The parametric plane defined by coordinates β and r″ is convenient to represent the hierarchy of manoeuvring domains (Fig. 1). All thinkable manoeuvres can be characterized by combinations of the drift angle and of the generalized dimension-less rate of yaw falling within the large embracing rectangle. Most of these combinations can only be reached with the help of side thrusters, azimuth-ing thrusters at low speed and thus correspond to strong or hard manoeuvres. The hatched rectangle (its boundaries are, of course, approximate and fuzzy) corresponds to all manoeuvres that can be performed at arbitrary ship speed with the help of “normal” steering devices like the common rud-der or the steering nozzle. It is appropriate to call such manoeuvres moderate. Finally, the small filled rectangle outlines the domain of weak or gentle manoeuvres presuming small rudder deflections typical in the course keeping or in slight head-ing corrections. Often (but not always!) the ship’s mathematical model can be linearized within this smallest area.

The attack and drift angle introduced here char-acterize the ship’s motion with respect to water. As to the motion with respect to the surrounding air, the air attack angle is never considered and,

Figure 1. Manoeuvring domains: entire domain—all manoeuvrs; hatched rectangle—moderate manoeuvres; filled rectangle—weak manoeuvres.

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hence, the air drift angle βA can be defined as solution to the equations

cos ,sin .

ββ υ

A A

A Aβ υβ′

′

(9)

The standard definition of both drift angles β and βA implies that there values belong to the interval [−π, + π]. The air drift angle is also often called “the relative wind angle” meaning that the relative wind velocity V wr = −V A. However, giving here priority to the motion of the ship over the wind has advantage of introducing certain uniformity in treatment of the ship’s interaction with both water and air.

Some of the parameters introduced above are illustrated by Fig. 2 where, in addition, is shown the course (over the ground) angle χ and the rudder deflection angle δR which is assumed positive when the rudder is deflected to starboard.

2.1.6 Important particular cases and “manoeuvring” frame

The complete kinematic description with 6DOF is typically used in surface ship manoeuvring when the waves-induced motions are taken into account. In still water, the manoeuvring motion is performed in the horizontal plane with 3DOF but an additional DOF related to the roll motion is often added as the back influence of the roll may be substantial for naval com-battants and also Ro-Ro ships, container carriers and ferry boats (Oltmann 1993). In this case, the so-called manoeu-vring frame is used instead of the body-fixed axes. The only difference is that the manoevring axes are not involved in the roll motion (the pitch and heave motions are supposed to be absent) which facilitates description of hydrodynamic hull forces. No spe-cial nomenclature is usually introduced and all the kinematic parameters can be used as defined above

but under the condition that w ≡ q ≡ α ≡ 0. The rate of roll p is or is not assumed to be zero depend-ing on the context which will become clarified from further developments. As all quasi-velocities are then defined in the manoeuvring axes, it must be taken ϕ ≡ 0 in the right-hand side of kinematic Equations (2) but the differential equation for the roll angle remains in a very simple form: ϕ = p. Four degrees of freedom are sufficient for most surface displacement ships but sometimes influ-ence of the dynamic trim on fast ships cannot be neglected and the 6DOF formulation in the body frame must then be used.

2.1.7 Simplest purely kinematic manoeuvring models

Most problems in navigation and guidance can be solved on the basis of simplest 3DOF kinematic models assuming that the ship is keeping certain val-ues of its quasi-velocities or—in a more complicated case—values of quasi-accelerations. The velocities in the latter case will change linearly in time. Such models were explored in detail in (Sutulo et al., 2002) and they can be used in manoeuvring simula-tors just for short-time predictions of the position of the ship while the simulation itself can only be based on dynamical ship models.

2.2 Dynamic equations

Although equations of motion of a free rigid body representing the ship are often readily discussed in articles on ship manoeuvring, in fact these do not need much attention and much less any spe-cial derivation being one of fundamental results obtained in classic general mechanics centuries ago (Lurie 2002). The only simplifying specifics stem from the centerplane symmetry of any undam-aged ship. Considering again the 6DOF motion these equations can be written in the following matrix form:

C F+ =C( )V( )VV ,0FF (10)

where M is the main inertial matrix; V = (u, v, w, p, q, r)T is the column matrix (or arithmetic vec-tor) of water quasi-velocities; C = (Cu, Cv, Cw, Cp, Cq, Cr)

T is the vector of the centripetal and Coriolis inertial forces, and F0 = (X0, Y0, Z0, K0, M0, N0)

T is the vector of all active forces applied to the ship. The inertial matrix is

MM M

M M=

⎡

⎣⎢⎡⎡

⎣⎣

⎤

⎦⎥⎤⎤

⎦⎦0 1M

1 2M, (11)

Figure 2. Main kinematic parameters in horizontal-plane motion in presence of wind and Current: all angles are positive.

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where

M diag

M

M

0

1

2

= −⎡

⎣

⎢⎡⎡

⎢⎢⎢

⎢⎣⎣⎢⎢

⎤

⎦

⎥⎤⎤

⎥⎥⎥

⎥⎦⎦⎥⎥

=

( , , );

;

m, ,

mz mx

I I−

G

G Gmx

G

xxI xI

0 0mzG

0

0 0−mxG

0 zz

yy

xz zzI Ixz z

0 0yyIy

0−

⎡

⎣

⎢⎡⎡

⎢⎢⎢

⎢⎣⎣

⎢⎢

⎤

⎦

⎥⎤⎤

⎥⎥⎥

⎥⎦⎦

⎥⎥ .

(12)

Here: m is the mass of the ship, xG, zG—coordinates of its centre, and Ixx, Iyy, Izz, Ixz are the moments of inertia of the ship.

The column matrix C is

C

mvr mx r p mwq mx qmur mwp mx pq mz qr

muq mz q

G Gr m G

G Gpq mz

G

=

−mvr + +mz prGmz− +mwp +

−muq +

2 2mwq mx q+ +m pr

2 mvmm p mv x mz pmz wp qr mz ur I pq

mx uq wq

G Gpr mz

G zwp z yy Gqr mzyy xzII

G Guq mz

+ −mx prpr+ − −mz ur

+ +mz wqGmz

2

( )I Izz yI yyIzI

(I III pr

Ix mz vr mx vpvmx ur wp pq

xxII zzI

G Gvr mx

G Gur mx yy xx

+ Ixz mz vr− +mx wpGmx +pq

)

( )p r( −p( )I IyyI xxIxxI

2 2rI qII rxzIII

⎡

⎣

⎢⎡⎡

⎢⎢⎢

⎢⎢⎢

⎢⎢⎢

⎢⎢⎢

⎢⎢⎢

⎢⎢⎢

⎢⎣⎣⎢⎢

⎤

⎦

⎥⎤⎤

⎥⎥⎥

⎥⎥⎥

⎥⎥⎥

⎥⎥⎥

⎥⎥⎥

⎥⎥⎥

⎥⎦⎦⎥⎥

.

(13)

2.2.1 Separation of unsteady effects in dynamic equations

The active forces and moments in the right-hand side of the Equation (10) depend, in the general case, on the generalized coordinates Ξ, quasi-velocities V, quasi-accelerations V, and some other parameters describing control actions and external influence. It is possible to imagine the fol-lowing separation:

F F F V0FF ( ), V, V ( ), V, V( )V V ( , ),F, V, V ( +( ), V, VF (AF (14)

where FA is the acceleration-dependent component such that FA(Ξ, V, 0) ≡ 0 and F is the quasi-steady force (in the sense that it does not depend on quasi-accelerations).

Modern ship dynamics is in fact based on the assumption that FA( ) does only depend on quasi-accelerations and, moreover, this dependence is linear i.e.,

F M V,A AF M (15)

where

MA iM ji i j= ⎡⎣⎡⎡ ⎤⎦⎤⎤μ

, 1j=

6

(16)

is the matrix of added mass coefficients.

The possibility to absorb the influence of quasi-accelerations2 with such a simple formula is not evident and, to our best knowledge, has never been proven rigorously and definitively although there are indeed many reasons in favour of this assumption. First, the quasi-accelerations do only enter linearly into the rigid-body equations. Fur-ther, the potential theory of the motion of a rigid body in an unbounded fluid or with rigid bounda-ries yields exactly Eq. (15) (Lamb 1968). This can also be associated with the principle of Newton’s determinism (Arnold 1989) according to which in this world (universum) the initial state of a mechan-ical system influencing its future does only include coordinates and velocities but not accelerations.

Meanwhile, in the real fluid rather compli-cated unsteady effects can be imagined and the assumption about the validity of Eq. (15) was, for instance, put in doubt by Grim et al. (1976) where the authors discussed a possible existence of such terms as υ υ2υ and r r2 . However, apparently no firm confirmation of presence of these or similar terms was found and models of the type (15) are commonly accepted. One of the reasons for this can be, as was established in (Ishiguro et al., 1996) and (Sen 2000), a relatively small sensitivity of most characteristics of the ship motion to the ship inertial parameters as compared to other compo-nents of hydrodynamic forces.

When Eq. (15) is assumed, the Equation (10) can be re-written as

( ) ( )V C F))VA (17)

or, further, as

V = (M M V F].+V+ MA ) ([C) ([C[C )1 (18)

In the latter form, the dynamic equations are ready for numerical integration if the quasi-steady forces F are defined. In some cases with reduced number of degrees of freedom the inertial matrix can be inversed analytically but in general numeri-cal inversion must be preferred. This can be done only once if all the inertial characteristics remain constant during the simulated manoeuvre but they may be slowly changing and then the matrix must be inversed at each integration step.

The added masses μij can be found in differ-ent ways. This can be performed with 3D panel methods (Sutulo and Guedes Soares 2008b),

2 The quasi-accelerations are not all the accelerations act-ing on the moving ship as these also include centripetal accelerations present in Eq. (13) which are not, however, associated with the unsteady motion.

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or strip methods (the coefficient μ11 cannot be determined in the latter case but it is very small anyway), or even approximating the doubled ship hull with an ellipsoide and using the classic analytic solution for the latter (Fedyaevsky and Sobolev 1964). If estimates made for seakeeping purposes are available for a given ship, they can be also used in manoeuvring taking values for the zero (or the smallest available) oscillation frequency. Finally, the added masses can be estimated experimentally. In this case, effects of viscosity and wavemak-ing neglected in most theoretical methods can be taken into account. Traditionally, experimentally determined added masses are called acceleration derivatives3 and are designated also differently. For instance, the notation YrYYYY is used instead of −μ26, NυNN instead of −μ62 etc. but the meaning of these parameters always remains the same.

In some sources e.g., (Rozhdestvensky 1970), (Fedyaevsky and Sobolev 1964), (Sen 2000) it is possible to observe dynamic equations with the added mass coefficients present also in the cen-tripetal part. These are the so-called Kirchhoff equations associated also with the names of Thomson and Tait (Lamb 1968) which completely describe hydrodynamic loads on a body moving in the unbounded perfect fluid. In particular, this theory yields the so-called Munk moment which in the equation of yaw looks like (μ11 − μ22)uv. How-ever, the practical meaning of these equation is almost illusory because in real fluid these centrip-etal hydrodynamic loads cannot be separated from the quasi-steady viscous contribution which is of the same order of magnitude. The destabilizing Munk moment does exist but the potential theory heavily overestimates it. The only possible appli-cation for such equations could be a very rough qualitative estimation of some coupling effects for which other estimates are absent.

2.2.2 Reduced and alternative forms of dynamic equations

In still-water surface ship manoeuvring, the pitch and heave modes are normally neglected. In this case, equations for the heave and pitch are dropped and in the remaining equations it is assumed that

w q w q≡q ≡q 0. If the motion is described in the manoeuvring axes, it should also be assumed

p p ≡p 0 everywhere except for the roll equation. The latter, considered as auxiliary, is also some-times heuristically simplified further stripping it of most cross-coupling terms (Inoue, Hirano, Kijima, and Takashina 1981). Strictly speaking, the thus obtained 4DOF equations in manoeuvring axes

will not be quite accurate as the moments of inertia and vertical coordinate of the centre of mass will not remain constant but, assuming the roll angles are moderate, this is typically neglected.

The traditional scalar form of the dynamic equations can be immediately restored from the matrix representation above. In the case of 4 or 3 degrees of freedom the resulting scalar equations are compact enough and are widely used in theo-retical studies. However, in general, the presented matrix form is most suitable for numerical solution in simulation systems especially when a pre-coded class of matrices is available and all matrix opera-tions can be coded directly.

Another matrix representation of the Euler equations developed primarily for robotics appli-cations and using somewhat uncommon nota-tion was proposed by Fossen (1994). Namely, the equation of the rigid body in this notation is written in the form:

Mv C v =v+ C( )v( )v ,τ (19)

where M is the inertial matrix identical to M in the present notation, v is the vector of quasi-velocities, i.e., the same as V, τ is the 6-vector of all external forces and moments, and the matrix C is defined in such a way that the vector Cv be identical to the vector C in Eq. (10). A similar matrix form but with modified matrices M and C including also added mass coefficients was also proposed as a representation of the Kirchhoff equations. These “robotics” forms of dynamic equations apparently do not possess any specific advantages in dynam-ics of marine craft but, due to historical reasons, gained extreme popularity in the literature on con-trol applications to marine systems.

The vector-tensor form of equations of motion of a rigid body is typically exposed in books on general mechanics (Lurie 2002) and is written in terms of the “physical” 3-vector V representing the velocity of the origin and of the pseudovector of angular velocity Ω defined in the same physical space.

m V r

m M

CGrr

CG

CGr

[( )rCGrr ] ,F

( )V V ,

+ Ω × + Ω ×+ Ω × ( rrr

Ω Ω ⋅+ ×mrCGrr =

V

Θ Θ⋅Ω + Ω × Ω

(20)

where rCG is the position vector from the origin C to the centre of mass G, Θ is the tensor of (rota-tional) inertia, F and M are the total force and moment acting on the body. Here the first equation is for the forces and the second—for the moments which is quite physical and they are equivalent to the classic Euler equations in the scalar form.

3 The origin of the term “derivative” in this context will be clarified later.

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While these equations are often used in teaching, they are not, however, convenient for immediate practical application.

Some authors (Ishiguro et al., 1996), (Matsunaga 1993) prefer to write scalar dynamic equations, mainly in 3DOF, in terms of the parameters V and β instead of u and v. No advantages of this refor-mulation have ever been found while the disadvan-tage is quite obvious: the equations are becoming more polluted and unnecessarily complicated. For instance, the following complication will happen:

υ β β β βd

dtβ) sββ Vββ = − in cos (21)

—the difference in the resulting complexity of the dynamic equations is evident!

It is sometimes claimed (Lewandowski 2004) that when considering motion in uniform steady cur-rent it is necessary to consider the vector VG instead of the vector V (i.e., the ground quasi-velocities) at the left-hand side of Eq. (10). In fact, any of these two vectors can be equally used here. Otherwise, the Galileo relativity principle (Arnold 1989) would have been violated but equivalence of the both vari-ants can also be verified through direct calculations. In fact, it is easier to keep with the vector V as in this case the ground quasi-velocities VG will only enter kinematic equations and relations. In practice, this “kinematic” handling of the current is often extended to non-steady and non-uniform current fields which are, however, supposed to vary slowly in time and space (Hensen 1999). But this apporo-ach is not always applicable: in a curvilinear current typically faced in river fairways additional inertial hydrodynamic forces and the so-called “slip-down” force appear and must be accounted for (Pershitz and Tumashik 1985).

2.3 Additional equations

Besides the kinematic and rigid-body dynamic equations, a ship mathematical model can and usu-ally does include some additional groups of differ-ential equations. These are, first of all, equations describing the propeller-and-engine dynamics and the steering gear dynamics. Also, if a ship is equipped with automatic controllers, these should also be simulated. But the task of simulating deci-sions and actions of a human operator is stated relatively rarely. First, adequate modelling of the commanding officer and of the helmsman is extremely difficult and the human behaviour is not always predictable. Second, interactive manoeu-vring simulators presume direct incorporation of the operator into the control loop. In this case, correctness of the operator’s behaviour and actions

strongly depends on the realism of the visual interface and on the accuracy of the implemented core mathematical model.

Although auxiliary submodels mentioned above are very important for the overall quality of sim-ulations, they will not be covered by this review. Some interesting discussion on modelling steam turbines and low-speed Diesel engines can be found in (Tanaka and Miyata 1977).

3 FORCES AND MOMENTSIN MANOEUVRING

3.1 Decomposition of active forces

The ship mathematical model can only be com-pletely defined if defined are the active forces F0 or F at the right-hand side of dynamic equations. As is typical in the ship hydrodynamics, these forces are subject to decomposition which can be performed in several steps.

The first decomposition step is:

F0 = FAH0 + FM; or F = FAH + FM, (22)

where FAH0 or FAH is the main aero- and hydrody-namic force acting upon the ship (the difference between each of them is defined by the difference between F0 and F), and FM stands for miscellane-ous forces which may also be hydrodynamic or of any other nature. These include e.g., fender reac-tions, forces from tow lines or directly from the pushing tug’s hull. Also, all hydrodynamic interac-tion (with other ships, banks, piers etc.) loads are also falling into this category.

The present review only focuses on the main aero-and hydrodynamic forces FAH0 or FAH.

In the most general formulation the ship must be treated as a body moving in a two-phase medium with the separation surface (water free surface) and if a suitable two-phase numerical method is applied, the force FAH0 will be estimated directly, there will be no need even in the added masses and no future decomposition will be required. Such ship mathematical model would automati-cally account for most subtle effects even including the sea waves generation from the specified wind which can be of variable speed and direction, and difference in velocity profiles of the true wind and of the ship-speed-induced relative wind.

Although isolated implementations of the two-phase motion CFD codes were reported, no appli-cations to manoeuvring are known to the authors. But there is no doubt that even when successfully realized, such implementations would remain pro-hibitively complicated and time-consuming to be used for any practical simulations for many years to come.

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The next natural force splitting step is to separate hydrodynamic loads (FHY or FHY0) from aerodynamic ones:

FAH0 = FHY0 + FA or FAH + FHY + FA. (23)

It is assumed in the relations above that the aerodynamic component can always be treated as quasi-steady: the aerodynamic added masses are always negligibly small due to the small density of the air and there have been no reports about sig-nificance of other unsteady aerodynamic effects on ship hulls. Apparently, this can be explained by the large ship inertia and its weak reponse to short-period gusts and wind velocity variations.

In general, aerodynamic loads are always act-ing upon the ship hull but in absence of wind they can always be neglected due to the simple fact that the air density ρA is by three orders of magnitude smaller than the water density ρ. At the same time, even a relatively moderate wind can significantly affect the ship’s trajectory. This happens because the wind has some determined direction in Earth axes and this introduces the otherwise absent dependence of the forces on the ship’s heading changing the structure of the dynamic system in consideration. Aerodynamic forces will be, how-ever, commented later and now the attention will be addressed to the hydrodynamic forces which, obviously, are most important in ship dynamics.

The most straightforward approach for estimat-ing hydrodynamic forces is to plug an online CFD procedure into the dynamic model. This procedure can be single-phase, unless the two-phase approach is used to capture the free surface and it will work out values of all hydrodynamic forces at each inte-gration step as functions of the current values of generalized coordinates, quasi-velocities and func-tionals of the motion’s pre-history.

Of course, the method implemented for this purpose must be fully unsteady and it must model turbulent flow of the real fluid as viscous contri-bution is extremely important for manoeuvring forces. Of course, in this case still there is no need in the added masses as the unsteady force FH0 is being modelled. In general, such methods account for the free surface and, if the sea waves are numer-ically introduced into the simulation domain, will model the manoeuvring motion in the seaway. Moreover, an arbitrary current field inhomoge-neous vertically and horizontally, can then be accounted for without additional difficulties while this is a very complicated problem for more sche-matized approaches described below.

A CFD procedure in the loop is reality and apparently its first implementation was described in (Sato et al., 1998). In that study, as well as in most of contemporary realizations, a Reynolds-Averaged

Navier–Stokes Equations (RANSE) solver was used. For such solvers, hardware requirements are minimum compared to the Large Eddy Simulation (LES) and Direct Navier–Stokes (DNS) methods. But RANSE codes have an inherent limitation from their dependence on the used semi-empiric turbulence model. The flow structure in curvilin-ear motion with drift angle is very complicated and it may happen that models well suitable for the straight motion handled in ship resistance work not so good here.

As DNS still practically cannot be run at all on available equipment, probably LES methods, which are in some sense intermediate, have the best perspective in the not so distant future.

However, even the fastest RANSE codes are too slow to be run online in real time on reason-able hardware and it is clear that for many years to come CFD codes will be only used in the off-line pre-processing mode to create databases of hydrodynamic responses further used in online simulations. In this mode, CFD codes are in fact emulating certain exprimental facilities and in this case the ship mathematical model (17) i.e., that with the separated quasi-steady contribution is preferred although the added-mass coefficients can be computed with the same CFD code. Substantial advantage over physical experimentation is that, at least in theory, results of computations can be made free of the scale effect although the higher is the Reynolds number, the less efficient becomes the solver and natural absence of direct full-scale measurements makes difficult final validation of such results. In general, CFD codes can nowadays compete with captive-model tests as to accuracy of prediction being often more accessible (Stern and Agdrup 2009), (Stern et al., 2009), (Stern et al., 2011).

When the ship hydrodynamic characteristics are pre-computed or determined experimentally, the way how these data are represented in the manoeu-vring code is of substantial importance, and, moreover, a considerable part of the content of the ship manoeuvrability theory is in fact dedicated to this issue.

3.2 Representation and decomposition of quasi-steady hydrodynamic forces

The most evident and straightforward method for storing pre-computed hydrodynamic charac-teristics of the whole ship or of the ship’s hull is to tabulate them and memorize in arrays or other suitable numerical containers to be further used together with search and interpolation algorithms. Such a method was used e.g., in the simulator-oriented manoeuvring model described in (Jensen 1993) and (Chislett 1996). The number of one- or

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two-dimensional tables used in that model is about 300 and cubic splines are used for interpolation. This approach is flexible, results in very fast algo-rithms and does not suppose use of any a priori information. One can notice that this method is not commmented very often inpublications but, apparently, is used widely in proprietary mathe-matical models implemented in bridge simulators.

However, while this, how it is often called, grid-function method is close to be perfect in the case of two defining variables (factors) and still quite realizable with 3 factors, its implementation in the case of 4 and more factors is facing serious difficulties: first, the number of stored tabulated values is becoming very large and poorly observ-able and, second, collecting a large number of response values presumes performing an equally large number of test runs (or CFD computations) that can take a prohibitively long time. Difficulties in handling more-than-two variables grid function are recognized even by the specialists who prefer this approach (Jensen 1993).

The quasi-steady hydrodynamic force FHY is acting upon the ship which can be treated as a whole entity including the hull, the propulsors and the steering devices—the so-called integrated or holis-tic approach—or may be split into the mentioned components resulting in the modular approach:

F v p r u v p rF J u v r J

HYFF R H

P RFF J R

( ,u ,p , ) (FHFF , v , ,r )u( ) (FRFF , , ,v r , )R ,

ϕ δJ RR, ,J ϕ))δRRJJ+ FFF uu (24)

where J is the propeller’s advance ratio, and the subscripts in the right-hand side stand for: H—hull, P—propeller, R—rudder or other steer-ing device.

Such a decomposition can always be per-formed albeit sometimes artificially as happens, for instance, in the case of steered waterjets when propulsive and steering actions are naturally com-bined in one and the same unit.

The main sense of the modular decomposi-tion is that the argument list for each component becomes shorter than for the entire ship and this simplifies description of the forces. Also, possi-bility of separate developement of mathematical models for each component is often attractive as these pertial models usually can be made more “physical” which facilitates possible extensions and estimation of their parameters.

Besides the complete decomposition (24), it is possible to use a partial one:

FHY = FHR + FP, (25)

where FHR = FH + FR is the joint contribution of the hull and rudder but the propeller’s contribu-tion is separated. This kind of decomposition

makes sense when control surfaces do not work in the propeller slipstream which is typical for submarines (of course, in this case FHR includes also action of the bow and stern planes) and some naval combattants.

In general, the decomposition principle must be applied with certain care because of possible inter-actions between various contributions which must be accounted for in one or other way.

3.3 Regression models

It turned out that in ship manoeuvrability, the most common method of describing total or hull hydrodynamic forces is to apply regression models (Draper and Smith 1998). Moreover, this approach has become so popular that some peo-ple with insufficient expertize in ship manoeuvring consciously or incon-sciously see it as unseparable part of the manoeuvrability theory. However, this is not true because e.g., grid functions or solver-in-the-loop implementations successfully work without recourse to such techniques.

Application of regression models implies the following steps:

1. Model construction i.e., a certain structure of the regression model must be established.

2. Design of the response experiment (physical or numerical) which serves to estimate parameters of a regression model.

3. Carrying out the experiment mentioned above. 4. Estimation of the regression parameters

(coefficients).5. Analysis of variances (ANOVA) in course of

which significance of the regression parameters is checked and, as result, the model’s struc-ture may be modified and the previous step repeated.

6. Validation of the model, mainly through assess-ing its response.

3.3.1 Structures of regression modelsPractically all regression models ever used in ship manoeuvrability are linear with respect to the regression coefficients (a rare exception of a nonlinear regression can be found in (Khristov and Zilman 1989)). Considering, for instance, the hull quasi-steady force column matrix FH it can be approximated with the linear regression model

FH iF ii

m

F fi i ( )V=∑ ,)V

1

(26)

where Fi are generally the 6-vectors of regression coefficients and fi are the corresponding vectorial regressors which are functions of generalized coor-dinates and quasi-velocities.

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If only the motion in the unbounded fluid or a 3DOF motion over the flat bottom is considered and the position of the ship with respect to the free surface is not changing, dependence on generalized coordinates vanishes. In 4DOF models dependence on only one generalized coordinate, the roll angle, is considered.

The specific form of each regressor must be established a priori. In most cases, all regressors from some given regression model belong to the same family. Selection of the family depends pri-marily on the expected range of variations of the kinematic parameters, first of all, of the quasi-velocities (Fig. 1).

One of the most popular family of regressors suitable for moderate manoeuvres and, apparently, the first used in ship manoeuvrability consists of the following functions:

f uiffr

r

,υ ϕr rα αuuυυ υ αrrr rr αϕϕϕα αuu υ αr αϕ μ …

( )u, ,υ ϕr, , …+ +αυυ + ≤αϕ +μμ

… (27)

where μ is called the order of the polynomial regres-sion model. Practically in all existing manoeuvring polynomial regression models μ = 3.

In the typical case of hull forces on a symmetric hull in 3DOF motion the cubic regression polyno-mials are usually taken in the form:

X X X

X X r X u

X

u uX uX XX u uuuXX

rr

uX

+X Δ +uu ++ X X r Δ+

0XX 2 3X+2 2X 2

( )uΔu ( )uuΔυυ υ υXr ur XXrr υΔυ X rrrX + X rrX r υrr rrr r υ22

rrrr u r

r rrr

rr

ur X uu r r

Y Y Y Y Yr r

Y r Y

Δ +ur Δ+Y +Y

+Y +

2

3 3Y r2 2Y r+

υrrrr

υYY υυυ

υυυYY υrr

υrr

υ υrY rr+ +YυυυYY

υ υrrY rrrr+r Y rr+r υrrYY rr

;

u uuuYYYY r

uu uur

r rrr

Yu ur

Yuu Yuur r

N N N N Nr r

υ

υ

υ υυυ

υυ υrN rr Nυυυ

Δ +uυ Δ+Yuu +Yuur

+Nυυ + NN

( )uΔ ( )u ;2 2υ +Y ( )uΔ3 3N r333

2 2

2 2

+ + Δ + Δ+ +

N 22 r uΔ N uΔ r

N N rr urNN

uuNN uurNNυυrr υ υ+r Δrrr u+r NN

υ

+22 N rr+r N rrrr r υυ( )ΔuΔ ( )uΔ ,

(28)

where Δu = u − u0 and u0 = V0 is the approach speed i.e., the speed before execution of the manoeuvre. It can be seen that some expected terms are dropped: this happened due to the symmetry con-siderations and, as result, regressions for the sway force and yaw moment are identical but both dif-ferent from the surge regression.

The polynomial models (28), possibly with some terms dropped as insignificant, are usually associated with the name of Abkowitz (Crane et al., 1989) and are often referred to as “truncated Taylor expansions”. The latter fact, in its turn, inspired calling regression coefficients in there as manoeuvring derivatives. In some sense, such ter-minology is justified: a moderate manoeuvring motion can indeed be viewed as some perturbation

of the straight uniform reference motion and the Taylor expansion technique is quite adequate for describing this perturbation. Of course, even then, the constant coefficients in (28) are not exactly derivatives (which, in general, are functions, not constants!) but their values at a fixed point and part of them also incorporate factors 1 2! or 1 3! More important is, however, that all modern meth-ods of estimating these coefficients are not based on their Taylor-expansion interpretation. Once the structure of the regression model is established, it is usually much more productive and less restric-tive to forget about its origin and to treat it just as a polynomial. From this viewpoint it would be more consistent to talk about “coefficients” or “param-eters” instead of “derivatives”. However, the tra-ditional “derivative” terminology is very common and will be also sometimes used.

3.3.2 Modifications of polynomial modelsSome authors prefer for the sway force and yaw moment second-order regression models of the type

Y Y Y Y Y Y r

Y Y Yr+Y + Y +

+ +Y Δ +υ υrYY r

υuυYY

Y r+ + Yυ Y rrY rY+ + YYYY Y r+Y r YY+rrYrYY

υY u+ uυYuYY+| |υ | |rr | |υYYYYYYY

| |r

| | | |rrr | |υ| |rrr uruuYYYY urΔ .

(29)

Although such models are usually also called “polynomial”, they do not represent classic alge-braic polynomials. For instance, unlike normal polynomials, these are not infinitely differenti-able and the response surface defined by them is not smooth everywhere. It can be suggested to call such objects quasi-polynomials.

Quasi-polynomial quadratic models can be slightly more economical from the viewpoint of the number of terms and computation speed and they can behave better outside their domain of validity which sometimes encourages their application for simulating even hard manoeuvres (Ankudinov and Jacobsen 2006). However, as will be commented in more details later, the alleged stronger physical significance of quadratic terms is much less evident and much less true than it is sometimes believed. Fitting qualities of both cubic polynomial and quadratic quasi-polynomial mod-els were compared by Kose (1982) (in addition, he also studied the cross-flow drag model commented later). Although both models were found suitable for fairing a set of measurements obtained on a rotating arm facility, the cubic model turned out somewhat more adequate as it resulted in a smaller residual error. The difference is not, however, very significant and the choice of one or another mod-els is typically based more on personal preferences and traditions than on rational considerations. Another comparative analysis of quadratic and

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cubic models, which includes comparative plots of response surfaces, was made by Clarke (1982).

Sometimes mixed models containing terms of different kind are suggested. For instance, the dimensionless yaw moment in (Inoue, Hirano, Kijima, and Takashina 1981) is approximated as

′ ( )′ ′ ′ ′ ′ ′ ′ ′ ′+ ′ ′ ′ + ′ ′+

N ′ ( ′ N N′ ′ + r N′ + r

N r′ ′ N r′H r( )N ( N+ r

rr rN

)′ =′ N ′) = N υυ

υ υr NN+rN+NN υr

υNN | |′r| |r

2

2

′′ ′ ′ + ′ ′N N′′ + N ′ϕ υNN ϕ ϕrυ′N+ϕ υNN+ ϕ+ ′N r′+NrNN|ϕϕϕ |ϕϕϕϕ| |ϕ | |ϕ ,

(30)

where, without loss of consistency, the term ′ ′N r′rNN | |r | |′r could be certainly replaced with ′ ′N r′rrrNN 3

etc. Of course, the response of the thus modified regression model would be slightly different but these differences would not be substantial. How-ever, once the regression coefficients are estimated, such modifications are not very easy as they would require re-estimation of all the coefficients.

A clear warning must be given against attempts of combining similar terms of two types, like υ|υ| and υ3, in one and the same regression equation: as both terms produce similar effects, this almost certainly would result in overfitting and multicollinearity! This means that although a mixed model can fit all available data very well, extrapolation properties of the model will be, most likely, unacceptably bad. Similar effect can be triggered by introduction of terms of the higher order, like 4 and 5. Examples of such clearly overparamaterized regressions can be found in (Oltmann et al., 1980) and (Perez and Revestido-Herrero 2010).

In polynomial or quasi-polynomial models for the surge force Eq. (26), the group of terms X0 +XuΔu + Xuu(Δu)2 + Xuuu(Δu)3 or similar is often substituted with −RT (u) sign u, where RT (V ) is the standard resistance curve for the ship in concern. This curve is often known from resistance tests or it can be estimated with one of appproximate empiric methods developed in ship resistance (Van Mannen and Van Oossanen 1988b).

Some methods for estimating hull forces are using heavily reduced polynomial models. For instance, the model proposed by Pershitz (1983) can be represented like

′ ′ + ′ ′′ ′ ′ + ′ ′ + ′ ′′

X X′ = X X′ ′ +Y Y′ = Y ′ Y

r+HX X= X X+HY YY =

0XXX 2υυXX υrXX+

υ υrYY υ υ′ + ′ + ′Y r′′ Y+Y υ+r YYr+YY| |υ

| |υυ

| |′′ ,

| |′υ ,

N′NN N rH rN NN N′ ′ + ′ ′rυNN υ (31)

i.e., the both equations for the lateral motion do only contain one nonlinearity of the type υ'|υ'|. Strangely, this single term is sufficient to make the whole mathematical model qualitatively correct and the absence of nonlinearities in the yaw equation

is even somehow justified as often nonlinearity of this component is very weak.

3.3.3 About linear models and linearisationIn each of (quasi-)polynomial models, it is possible to reject all terms except those linear with respect to the kinematic parameters. It must be, however, underlined that, contrary to the sometimes still existing belief, any surface displacement ship is, even in moderate manoeuvring, a highly nonlinear object and such fully linear models cannot be used for simulation purposes. In fact it is not even appro-priate to talk about the so-called linear manoeu-vring theory although sometimes such attempts are undertaken (Clarke et al., 1982).

It can be noticed that this situation is very dif-ferent from that in ship seakeeping where the linear approximation is in most cases very consistent and robust (Beck et al., 1989). In manoeuvring, one can only hope to have sometimes adequate linear description of weak manoeuvres (Fig. 1) and this may happen to be the case when relatively fine and fast ships, like naval combattants, are con-sidered. At the same time, full-bodied slow-speed vessels, which are often directionally unstable, may be practically unlin-earisable (Sutulo and Guedes Soares 2007a), when it goes about simula-tion of motion, even within the weak manoeuvres domain.

There exists another misunderstanding regard-ing damping forces acting on the ship in dynamic positioning (DP). Sometimes these forces are assumed to be linear, probably basing on their linearity in sea-keeping problems. However, in the seakeeping the damping forces in pitch, sway and yaw are mainly radiation potential forces and thus really linear at those relatively high frequencies which are typical for sea-keeping motions. On the contrary, the DP-process is characterized by low-frequency or even aperiodic motions caused by the thrusters. Then, the wave radiation practically disappears and the damping is mainly caused by viscosity and thus is basically quadratic. The situation here is similar to that observed in the roll damping which has a strong quadratic com-ponent caused by viscosity while the potential radiation damping is relatively weak even at high frequencies.

Apparently, occasionally overestimation of linear components in manoeuvring repeatedly occuring in various publications is caused by an encountered in many books historically condi-tioned methodological practice of consecutive introduction of nonlinear contributions as some refinement and improvement to an allegedly linear basic model (Crane et al., 1989), (Fedyaevsky and Sobolev 1964) while all components, linear and nonlinear, are in fact of equal importance. As this

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is apparently confusing less experienced readers, it would be probably desirable to get rid of this rather obsolete practice.

While definitely not suitable for simulation pur-poses, linearised models can be very important for qualitative studies like stability analyisis and for synthesis of control systems (Sutulo 1998). A flexi-ble linearisation algorithm was proposed in (Sutulo and Guedes Soares 2007a). The algorithm is based on the least-square approximation of the response hypersur-face over some specified finite domain in the state space. When the domain is chosen to be very small, the classic differential linearisation in the neighbourhood of a given point is obtained as particular case.

Finally, it must be said that concerning linearis-ability of dynamical models, surface displacement ships present a rather special case. Many other moving objects, like aircraft, are highly linearisable while most vessels are not. This is partly explained by relatively slow speeds and low degree of direc-tional stability and even admissible directional instability of many surface ships.

3.3.4 Force coefficients and dimensionless regressions

Except for Eq. (30), all examples of regressions given so far were written for the dimensional forces and moments and presented as functions of dimensional (natural) generalized coordi-nates and quasi-velocities. It is often more con-venient to use dimensionless hull force/moment coefficients which in case of moderate manoeu-vring are introduced in accordance with the representation

F Fn WH HF lFn WnV

LTl′ ( ,RRn , ,WWn , , )r ,κ ϕ, υ ρVV′ ′

2

2 (32)

where ′FHF is the dimensionless force component or the force coefficient; Rn =VL v is the Reynolds number; Fn = gV L —the Froude number; Wn—the Weber number; κ is the cavitation number, l = 1 for the forces and l = 2 for the moments; T is the ship draught. In the case of arbi-trary manoeuvres:

F Fn WH HF rFn WnV

′′″ ( ,RRn , ,WWn , ) ,V

L TmoVVVV dκ β,ρVVV 2

2LL (33)

where ′′FHF is the modified force coefficient;

V V r LmoVV d2 2V 2 2LL+V 2V is the modified reference

velocity. Instead of the reference area LT, which is roughly the submerged centerplane area and this choice is inspired by wing hydrodynamics, a more artificial reference area’s definition as L2 is often used.

Obviously, all hydrodynamic force coefficients depend on several dimensionless similarity param-eters. However, when goes about hull forces, influence of the Weber number is practically always insignificant: the surface tension can only become tangible for very small toy models never used in research. Cavitation on the hull of a dis-placement ship is also an extremely rare event and its influence on forces is even less likely. But both these parameters can become important when considering propeller and rudder forces: the Weber number is connected with aeration and the rud-der forces can drop at low cavitation numbers on high-speed ships.

The Froude number can be and usually is modelled in physical experiments. However, in many cases the Froude number is assumed zero and not influencing the coefficients. This assumption looks quite justified when Fn ≤ 0.2 but often, with some-what increased tolerance, the force coefficients are in practice considered Fn-independent even for its values reaching 0.4–0.5.

The Reynolds number is the main factor respon-sible for the scale effect. Its variations, say, in the full-scale ship apparently do not influence the force coefficients considerably but the same can-not be said about the transition from the model to the full scale. Publications on the scale effect in manoeuvring are scarce and are mainly based on experiments with geosim models. For instance, it was discovered in (Nikolaev and Lebedeva 1980) that while 4-, 6-, and 8-meter long self-running models showed more or less similar characteristics, 2 m models brought substantially different results. Additional tests with bare hulls on the rotating arm proved that the effect was caused by changes in hull forces, especially at large path curvatures, and could be substantially reduced by sand-roughening the surface of the model.

Unfortunately, extrapolation methods similar to those developed in ship resistance do not exist and the Reynolds-number dependence in manoeu-vring is typically neglected (except for the resist-ance force if it enters the model explicitly) but it is usually recommended to use larger models and to stimulate turbulence4. Oltmann (1993) discov-ered no scale effect comparing full scale data with results of simulation with rather reliable math-ematical models obtained from high-precision CPMC model tests.

When all the mentioned provisions are in effect, the 3rd order dimensionless regression model for

4 It turned out easier to account for other scale effects like alteration in the rudder stall angle, propeller loading and wake fraction (Oltmann et al., 1980).

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the hull surge and sway force coefficients defined by Eq. (32) will become:

′ ′ + ′ ′ ′ ′ ′′ ′ ′ + ′ ′ + ′ ′ +

X X′ = X ′ ′ X+ r

Y Y′ = Y ′ Yr+ XHX X= r r

H rY YY =0XXX 2 2′ ′X+ ′

3υυ υrr

υYY υυυ

υ′ ′′X′′ + r X+rX+ r rr X+′X+ r′ υrr

υ υ′′ + ′ + ′ ′′Y r′ Yr+YY υυυYY

,

′′ ′+ ′ ′

′ ′ ′ + ′ ′ + ′ ′ +

Y r′′Y ′ ′ r

N N′ = N ′ N

rrrYY

r rr

H rN N=

3

2 2′ + ′ ′ ′Y r′ ′3

,

,υυYY rr υrr

υ υυυ

υ+r′′ Y r+r Y r′ +r Y ′ ′′YY rrYY

υ υ′′ + ′ + ′ ′′N r′ Nr+ N υυυ ′′ ′+ ′ ′

N r′′N ′ ′ r

rrrNN

r rr

3

2 2′ + ′ ′ ′N r′ ′υυrr υrr υ+r′′ N r+r N r′ +r N ′ ′′rr .

(34)

As compared to the dimensional regressions (28), the non-dimensional ones look simpler due to absence of Δu-terms.

In addition, the following important comments must made on the polynomial regressions:

1. These regressions are not orthogonal, all regres-sion coefficients are interdependent and after the coefficients are estimated and the response of the regression is validated, no one term can be dropped or substituted without danger to ruin the regression. In particular, it is not appropriate to blend regressions. For instance, taking the linear part from one source and the nonlinear part—from another must be avoided: even if the resulting regression looks accept-able, this must be rather considered as occa-sional luck.

2. The regression coefficients (manoeuvring deriv-atives) per se are not so important. Important is the response of the regression which must be the ultimate criterion of validity of this or that regression model and the response must be checked after every estimation and approxi-mation. This is especially important when the regression coefficents are, in their turn, approxi-mated by regressions over, say, hull shape parameters: the resulting compound regression may become highly inaccurate.

3. Instead of the dimensionless sway velocity υ' often the drift angle β is used in polyno-mial regressions. As for moderate manoeuvres

⎛⎝⎛⎛⎝⎝

⎞⎠⎞⎞⎠⎠

υ β′ = − β β ,⎞⎠⎟⎞⎞⎠⎠

≈ −⎛⎝⎜⎛⎛⎝⎝

β β β− 1

6

3β the difference

between them is very small and they are prac-tically interchangeable. However, if desired, appropriate corrections can be easily introduced when transforming a regression from one set of variables to another.

In the case of arbitrary manoeuvres, even dimensionless regressions are more complicated and, because of easily establishable periodic prop-erties, must include trigonometric functions that makes them remind truncated Fourier expansions. For instance, such trigonometric model may look as (Varela et al., 2008).

X X r

X X r

HX " " " "

" "

" cos

i cX os "

X cos r

+ X +

0 1cos cosXX cos r

2 3s XXXX

2

2

π β πX rr" sX in1XX 2 β

β" i"rr sinsr sinπ

cocc s ,

" i cos "sin

cos "si

" " "

"

3

2 2

2

0 1cos s n2

3

25

β,,

π β π βπ β

Y Y Y" i β r

Y

HYY cos0 cosY0 "sinsinβ

+s "sin+ 5 βY cos2 YY

Y Y

Y

3YYYY

43

52

6YY 2

" "

" "3

"

si

si 3 sY5YY i

sin "r sin

π β" crr os

π βcosβ" π βrr"sin

πrr

++YYY π β"rr cos

+ ββπ π

π β π

,β" sin " i "

"π

i cos

" " " "

" "π β

N N " ππ " rππ

Nπ

N

HN N r"

+ "sinN cos +ββ0 1NN r NNN r 2

3

3 4βsin βN rcos NβsinNN cos2

22

rr

N N

"sin

i N i ," "

3

5N 6NNNN 2

2

NN iNN 2

β

π βs β" i 2 2 π β"si ,2rr"sin+ NNN "rr sin2 2

(35)

In theory, coefficients in regressions of this type should be estimated similarly to coefficients in the polynomial regressions i.e., after specially planned experiments including tests at large drift angles and rotations on the spot. However, it may happen in practice that it is impossible to carry out the full set of experiments while moderate-manoeuvring coefficients from Eq. (34) are somehow known. Then, connection between the “prime” and “bis” regression coefficient can be established from the condition of asymptotic equivalence of the both models at small β and r˝, see (Varela et al., 2008) for details.

The model (35) is close to the simplest of the kind: the ship is suppposed to be symmetric not only with respect to the centerplane but is also midship-symmetric and possible “higher harmon-ics” are not captured. Fig. 3–5 demonstrate an example of this model’s reponses.

Driftangle,deg

X″

-200 -100 0 100 200-0.3

-0.2

-0.1

0

0.1

0.2

0.3

r″=-1

r″=-0.8

r″=-0.6

r″=-0.4

r″=-0.2

r″=0

r″=0.2

r″=0.4

r″=0.6

r″=0.8

r″=1

Figure 3. Dimensionless surge force.

675

Another, somewhat more sophisticated exam-ple of the “four-quadrant” trigonometric model can be found in (Sutulo 1994) and (Sutulo and Guedes Soares 2005c). Under any circumstances, coefficients of trigonometric regression models hardly can be called “hydrodynamic derivatives” and this traditional terminology is not applicable here at all.

3.4 Estimation of regression coefficients and experimental designs

Once the structure of a linear regression model is established, it is necessary to estimate the regres-sion coefficients. This can be done on the basis

of one of the following procedures supplying the neccessary information:

1. Perform rotating arm or circular motion captive-model tests.

2. Execute unsteady captive-model tests.3. Perform free-running model tests.4. Perform full-scale manoeuvring trials.5. Carry out offline CFD computations.

Circular motion tests can be realized either in circulating tanks or with computerized planar motion carriages (CPMC). In the latter case, the circle may be not complete but it must be sufficient for the steady circular motion to materialize during a sufficient time interval. The tests are carried out at various combinations of defining parameters (factors) and the quasi-steady forces and moments on the model are measured almost directly5. The minimum number of test runs N is, obviously, equal to the number of regression coefficients m to be determined although in practice N is greater than m and often substantially.

For instance, if the dimensionless sway force Y'(υ', r') is defined by the regression (34) and its measured values are Y r i Ni irrmeasYY ured

′g

′g

′υ ,irr , , ,(gggg

) 1 the following linear algebraic set of equations can be written for the unknown regression coefficients:

Y Y Y Y r

Y ri r i i rrrYY irr

r i i i irrυYY υυυ

υυrr

υ υYi r i iiYYY

Yi i YYY

′ ′ ′ ′ ′ ′ ′ ′

′ ′ ′ ′ ′YY rrYY rr +

+YYY rirr

3 3Y r′ ′+2 2Y rY′ ′ ′ ′+r == ( )Y (

i N=measYY ured′ ( ,)

, , .1

(36)

This set, obviously overdefined at N > 6, is usu-ally solved in the least square sense (Draper and Smith 1998) and besides the estimates of the coef-ficients per se, some statistics can be obtained which make possible analysis of the significance of any regressor. As result, some of the regressors can be excluded from the final model.

The set of points {(υ'i, ri')}, i = 1, …, N in the so-called factor space, at which the measurements are taken, defines an experimental design. The accuracy of final estimates of the regression coef-ficients and the quality of the regression depend on the number and location of these points within the experimental domain. The problem of optimal experimental design in application to demands of ship manoeuvring was investigated in detail by Sutulo and Guedes Soares (2002) and (2005e) where it was demonstrated that synthesis and application of so-called D-optimal designs mini-mizing the variance of estimates of the regression

Driftangle,deg

Y″

−200 −100 0 100 200−4

−2

0

2

4r″=-1

r″=-0.8

r″=-0.6

r″=-0.4

r″=-0.2

r″=0

r″=0.2

r″=0.4

r″=0.6

r″=0.8

r″=1

Figure 4. Dimensionless sway force.

Driftangle,deg

N″

−200 −100 0 100 200−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

r″=-1

r″=-0.8

r″=-0.6

r″=-0.4

r″=-0.2

r″=0

r″=0.2

r″=0.4

r″=0.6

r″=0.8

r″=1

Figure 5. Dimensionless yaw moment.

5 Except for the centrifugal force which must be appropriately subtracted from the measured values.

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coefficients can save much of experimentation time without substantial loss of accuracy. For instance, both 3-factor designs shown on Fig. 6 would result in almost identical estimates of parameters of cubic polynomial regression models (the variables x1, x2, and x3 are normalized values of υ', r’, and δR respectively) (Sutulo and Kim 1998). However, in this somewhat artificial example the second design is saturated i.e., the number of test runs was equal to the number of unknown regression coefficients. In practice, certain redundancy is necessary: the number of test runs must exceed the number of parameters to estimate by the factor from 3 to 5. Also, it still makes little practical sense to devise economical optimized designs with only three fac-tors as corresponding full-factorials are still not prohibitively bulky. However, when it is necessary to deal with 5 factors, full-factorial plans become unacceptable and optimized economical designs become the only viable option. An example of suc-cessful construction of 5-factor (besides the veloci-ties of sway and yaw, considered were also the sinkage, pitch and roll angles) polynomial regres-sion models for a catamaran from an optimized CPMC experiment is given by Sutulo and Guedes Soares (2006a). The resulting models were used for simulation of the manoeuvring motion of a fast catamaran (Sutulo and Guedes Soares 2005d). The estimated manoeuvring characteristics were in accordance with the results of full-scale trials of the same catamaran (Guedes Soares et al., 1999) and it can be thus stated that the thread: regres-sion model construction—experimental design—CPMC captive-model tests regression parameter estimation—computer simulation outlined above does really represent a reliable tool for working out rather complex and usable ship mathematical models.

Both rotating arm and CPMC facilities are extremely expensive in manufacturing and opera-tion and, as alternative, simpler planar motion mechanisms (PMM) representing an attach-ment to plain towing carriages are also used as alternative to estimate hydrodynamic derivatives.

These mechanisms are capable to generate sinusoidal motions in sway and yaw and sometimes also in surge. Different regimes are implemented through appropriate phase shifts between sway and yaw motion. The forced motion of a model is obviously unsteady but the Strouhal number is assumed to be low enough for the quasi-steadiness hypothesis to be applicable. From this viewpoint, PMMs with larger amplitudes and lower frequen-cies are preferrable. As the motion is unsteady, the recorded time histories of hydrodynamic loads contain more information than in rotating arm tests but the required quasi-steady values are no longer measured directly but through some simple identification procedures involving Fourier analysis of the records resulting in the mean, in-phase and out-of-phase amplitudes of the first and second order. The in-phase output is used for experimen-tal estimation of the added mass coefficients while all the remaining readings serve to estimate quasi-steady derivatives similarly to how it is done with the rotating-arm measurements.

It was demonstrated by Sutulo and Guedes Soares (2007b) that the so-called combined sway-yaw tests (oscillatory in yaw with fixed drift angle) are quite sufficient for estimating all quasi-steady manoeuvring derivatives. Moreover, it was also shown that mean and first-order out-of-phase val-ues are sufficient for estimation of all regression coefficients. Amplitudes of second harmonics can be also involved augmenting the number of virtual or effective test runs but apparently this does not improve final estimates as the higher is the har-monic, the more prone it is to noise penetration.

Hence, evaluation of second-order Fourier coef-fcients typically performed after each oscillatory tests is nowadays more a tribute to tradition: when these values are available, numerically simpler consecutive procedures based on various kinds of simplified tests (oblique towing, pure sway, pure yaw) may be followed. However, results obtained by Sutulo and Guedes Soares (2007b) rather clearly indicate that such practice is obsolete and inferior from the viewpoint of the quality of final estimates.

Oscillatory tests turned out very conven-ient and they served also as main tool in CPMC experimenting although in this case arbitrary motions can be modelled. There were, however, suggestions to apply forced motions with wider spectrum, like step excitation. The problem of restoring parameters of regression models becomes then more complicated, requiring application of rather sophisticated system identification meth-ods (Rhee et al., 1993), (Sung et al., 1998), (Eloot and Vantorre 1998) while no tangible advan-tages as compared to classic oscillatory tests were revealed.

x 1

-1

-0.5

0

0.5

1

x_2

-1

-0.5

0

0.5

1

x3

-1

-0.5

0

0.5

1

x 1

-1

-0.5

0

0.5

1

x2

-1

-0.5

0

0.5

1

x3

-1

-0.5

0

0.5

1

Figure 6. Three-factor experimental designs: left—full-factorial with 216 points; right—D-optimized with 20 points.

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However, the system identification approach is mostly associated with free sailing tests no mat-ter with self-running models or full-scale ships. In this case only kinematic output is available, no force measurements. Some structure of the ship mathematical model is, as a rule, assumed a priori and the parameters of the model are adjusted to minimize, in some metrics, the difference between the observed and simulated outputs. This prob-lem remains ill-posed every time when hydro- and aerodynamic forces are estimated from kinemati-cal measurements and it is known that even iden-tification of parameters of mathematical models of submarines presented substantial problems although the full-scale measurements were taken at deep submergence in perfect conditions.

But it looks like the problem of parameter esti-mation turned out especially difficult in application to surface displacement ships for which the dis-turbance energy level, as compared to the proper kinetic energy of the moving object, is the highest.

Probably, for the first time the problem of esti-mating parameters of ship mathematical mod-els from kinematic measurements was stated by Nomoto et al. (1957) and the now famous Nomoto equations were devised aiming primarily at that task although they found later many other applica-tions. The salient feature of the Nomoto equations is that they contain very few parameters having more or less transparent dynamic and even kin-ematic meaning: in the case of the dimensionless second order equation

r r r K R R1 2T TT T 1 2 3′ ′ ′ ′ ′′ ′ ′ ′K+ =r( )T1 2TTT T+T1TTT ( )R Rδ δTR RR RT3TTTT+ (37)

these are the three time lags T1TT ′ , T2TT ′ , T3TT ′ and one gain K’ i.e., in total only 4 parameters. This number can be further reduced by two through simply res-caling the time and the output variable (Sutulo and Guedes Soares 2005b).

The first-order Nomoto equation

T r r K R′ ′ ′ ′K ′+ =r δRR ( )t′ , (38)

when appropriately scaled, does not contain any parameters at all. Identification of these simplistic models does not present major problems although, due to the oversimplified structure, good reproduc-tion of the observed motion can rarely be achieved. A later—though not very successful—example of identification of the Nomoto model is described in (Tiano 1976). In terms of the control theory and general theory of system identification, this approach, at least in the linear case, is equivalent to identification of the transfer function.

Even extended nonlinear Nomoto equa-tions (called sometimes Norrbin equations) are

relatively easily identifiable. In the case of rescaled dimesionless variables r and δRR the second-order Norrbin equation is

Pr r F R RQ+ =r ′ ( )r ),δ δQ RRQ (39)

where P and Q are constant parameters, δRR F= ( )r describes the rescaled spiral curve of the ship. If the nonlinear function F () is appropriately parameterized with a suitable polynomial or spline, it can be easily restored using results of the spiral test or several turning circles. Then, any zigzag test can be used to estimate P and Q. Thanks to low dimension of the factor space, even the exhaustive search can be applied here providing ultimately best estimates of the parameters.

Unfortunately, no matter how well the param-eters are estimated, these ”black box” ship mathe-matical models based on Nomoto equations rarely, and only for very specific purposes, can serve as adequate simulation models. For instance, it is clear that the ship trajectory can never be repro-duced correctly on the basis of any of these model because they do not provide any information on the velocity of sway or drift angle, the trajectory depends on. Of course, an isolated equation for the velocity of sway can be obtained (Sobolev 1976):

T T TR RT4 5T TT T 4 5 6TTT′ ′ ′ ′ ′′ ′υ υ′ δυ L′ ′L δ+ υ′ ( )( )4 5T T υυT4 5TTTTTTTT (40)

where all parameters T T4 6T TT T′ ′ and L′ are similar to those in Eq. (37) but they are not identical to them!

Solution of this equation together with, say, (37) could provide information necessary for plotting the trajectory but, apart from absence of nonlinear extensions of the sway Equation (40), the result-ing set would be even more complicated that the standard sway-yaw equations and such approach makes no sense. Also, it must be emphasized that nonlinear extensions of Nomoto equations cannot be devised rigorously from the standard set of non-linear sway-yaw equations but they are heuristic only reproducing correctly the stedy turn charac-teristic of the ship and, as they are asymptotically equivalent to the linear Nomoto equations at slightly perturbed motion, become adequate in weak manoeuvres. One can hope that time histo-ries in unsteady moderate manoeuvres are repro-duced reasonably well but so far it has not been demonstrated directly.

Also, reduced number of parameters in equations of the Nomoto family does not permit accurate description of dynamics of most ships. Even if only moderate manoeuvres are considered, appropriate simulation of manoeuvring motion

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requires application of much more complex mod-els which may contain up to 30 parameters and even more.

Some authors (Gill 1976) tried to perform identification of complex models still applying elementary techniques. However, in the cited study it appeared, first, necessary to base on the spiral manoeuvre which had to be necessarily Bech’s manoeuvre in the case of a directionally unstable ship or even in case of any doubts in this respect. But spiral manoeuvres are not convenient for tri-als as they require much of dedicated time and of aquatic space. A more suitable zigzag manoeuvre was only used here to determine one cross-coupling added-mass coefficient while the remaining added masses and all propulsion coefficients were sup-posed to be known a priori. Even so, the identifica-tion process could not be carried out automatically but had to be performed interactively. Also—and this is the most important drawback the model was to be heavily simplified only leaving two nonlinear terms (δR

2δ and υ'r') in the X-equation and R and only one nonlinear term υ'2 r' in the Y-equation.

In general, however, more sophisticated meth-ods were applied to identification of complex manoeuvring models. Specific methodology of system identification had been developed in the control theory (Speedy et al., 1970), (Eykhoff 1974), (Ljung 1978) and has been actively applied to ship manoeuvring since 70s. Its application to flight dynamics, see e.g., (Ross 1979), probably started a little earlier.

It is possible to notice a considerable number of studies initiated at the Massachusets Institute of Technology and based on the Extended Kalman Filter (EKF) technique (Brinati and Rios Neto 1975), (Brinati 1979), (Abkowitz 1980), (Abkowitz 1988). The Kalman filter (the specifier “extended” means its generalization to nonlinear systems) was primarily developed to provide optimal filtering, estimation and prediction in the state space (Fried-land 1986) when the process was contaminated with a Gaussian white noise and so were the observable variables. However, it was noticed that when the system’s parameters are interpreted as additional states satisfying homogenous linear differential equations, the Kalman filter can also be used to identify these constant parameters in respect to which convergence to steady values was expected.

In practice, this convergence is only defi-nitely observed when the algorithm is applied to simulated—using the known model—observations with white Gaussian noise artificially added. When the method is applied to real-world data, the con-vergence becomes uncertain and the cancella-tion effect with drifting estimates often happens (Abkowitz 1980) forcing deliberate simplification of the model. For instance, in the cited study by

Abkowitz (1980) the number of regressors involving the rudder angle was reduced through introduction of the geometric (not real!) rudder attack angle α and it was supposed that regres-sors of the type α, α2, α3 would suffice. Only terms of the type υ'r'2 were retained in the sway and yaw equations. But even with these simplifications many additional techniques were necessary to provide resonable estimates of the parameters on the basis of stand-ard and modified Z-tests.

It must be noted that simplifications of the regression models mentioned above are more or less acceptable when the ship is performing nor-mal manoeuvres without wind: in this case the yaw and drift motions are strongly coherent as demonstrated by Lebedeva et al. (2006). But such models are hardly usable in versatile ship handling simulators which must simulate motion under all environmental conditions. Sometimes, the observed annoying tight connection between the drift and yaw motion was attributed to the stand-ard Z-manoeuvres and Pseudo-Random Binary Signals (PRBS) were sometimes used to generate rudder deflections (Källström 1982) but there is no evidence that this could dramatically or even at all improve the estimates. At the same time, the EKF-algorithm, being implicit or tuning algorithm, can be used online and in real time supporting con-tinuous estimation of the states and of varying model parameters which is important for applica-tions to adaptive autopilots and for the motion prediction. In general, the EKF tends to generate models providing good prediction within some set of manoeuvring conditions but not always guar-anteeing good estimates of the parameters. This property is characterized in (Bozzo 1972) as “bad distance of the structure of the model and good distance for the state”. Although sometimes this could seem acceptable (indeed, the ship operator has no information about parameters of the ship manoeuvring model but can easily observe the output!), it is dangerous as can sometimes lead to unexpected responses of the model.

A modification of the EKF-method called the Nonlinear Recursive Prediction Error method was developed by Zhou and Blanke (1987).

The adaptive backstepping method was used by Casado et al. (2007) but only for a highly simplified ship mathematical model (1st order Nomoto equa-tion with a simple nonlinearity added).

However, identification aiming at develop-ing manoeuvring models for simulators does not require online estimation which makes possi-ble application of more complicated and slower algorithms. A good exposure of such approaches is presented in (Silman 1988) where two efficient offline methods are proposed. The first method is based on minimization of the error in observed and

679

simulated variables in L2-metrics with the Gauss—Newton method (Eykhoff 1974). This method was enhanced through regularization of the opti-misation task by using regularizing functionals (Tikhonov and Arsenin 1977), decomposing the problem and carrying out sequential identifica-tion, realizing multi-criterial optimization with creation of a Pareto set. The second method was based on the global optimization by means of the quasi-random search in the parameter space with a number of a priori constraints. For instance, one of the constraints was the requirement that the ship’s eigenvalues be real and only one of them could be slightly positive. This requirement is beyond any doubts when it goes about the base mathemati-cal model but could diverge from reality if the processed sea trials were carried out at substantial wind as in this case conplex conjugate eigenvalues may appear, see e.g., (Sutulo 1985). Both methods were used successfully for practical identification of simulator models from full-scale trials but the identification process was interactive and required a number of tries and decisions from the user.

Another offline identification method suggested by Di Mascio et al. (2011) is based on the genetic algoritm (GA). The merit functions were con-structed as sums of discrepancies of such measures as the advance, transfer tactical diametre, zigzag overshoot angles etc. instead of typically applied functionals of the response difference. Compari-son with another method based on a modification of the extended Kalman filter showed clear inferi-ority of the latter. At the same time, the cancella-tion problem still was present also in the GA-based method which forced the authors to use somewhat simplified regression models.

As was already mentioned, many research-ers came to conclusion that usual manoeuvres could not provide a good base for the identifica-tion of manoeuvring parameters and PRBS were thought to be a better choice. An attempt to create a D-optimal manoeuvre was undertaken by Rhee et al. (2004) and what resulted looks close to a frag-ment of a very hard (like 35 deg–60 deg) zigzag manoeuvre. Unfortunately, claimed advantages of this manoeuvre look somewhat uncertain.

Identification in the frequency domain, once used in submarine dynamics, was recently pro-posed in application to surface ships (Panneer Selvam et al., 2005), (Bhattacharyya and Haddara 2006). Although test results on generated data look encouraging, efficacy of the method in real-life identification was not demonstrated. The same statement is true for the study applying support vector machines (Luo and Zou 2009).

Although the identification problem is consid-ered as stochastic, there are relatively few studies where the maximum likelyhood (ML) method

was used (Källström 1982), (Tiano 1976) and no advantages of this approach have ever been demonstrated. The ML-method presumes that the distribution function or density of the observed process is known which is rarely the case. Usually, the process is assumed to be Gaussian with, maybe unknown, parameters which can be also estimated. However, as the assumed mathematical model of the object is never exact, part of the error will be caused by this discrepancy which is usually treated as additional noise. But this noise is obviously non-Gaussian and biased estimates of the param-eters are almost inevitable. Under these circum-stances, it is difficult to expect any advantages of the ML-method over much simpler deterministic approaches. This becomes even more true for the most sophisticated Bayesian estimation which requires also a priori information about distribu-tion of unknown parameters viewed as random variables and knowledge of the error penalty function.

Summing up the available information about practice of applying system identification meth-ods to ship dynamics it can be concluded that, in spite of substantial progress, there are no available robust and accurate algorithms and approaches which make possible reliable estimation of the parameters of arbitrarily developed and com-plicated manoeuvring models from tests with free-sailing models and, especially, from sea trials carried out in non-ideal conditions.

A unique publication unconditionally report-ing about success in identifynig 41 parameters of a complex mathematical models from various Z-manoeuvres is (Oltmann et al., 1980). It was even stated that this approach makes experiments with captive-model tests not necessary. However, in this case free-running model tests were carried out in a tank and the model was followed by the CPMC facility in the tracking mode. As result, all state variables could be continuously measured with a very high accuracy unreachable in any other conditions. Later, similar success was repeated for an even more complex 4DOF model with 65 parameters (Oltmann 1993) but apparently no one else has ever reported a similar achievement.

But this isolated success cannot shadow the truth that in general the problem of identifica-tion of hydrodynamic characteristics of ships in manoeuvring motion is definitely ill-posed due to large number of involved parameters and cancel-lation effect. Additional difficulty stems from the fact that the “true” model structure is never known for sure although often the “assumed” structure occurs to be rather close to a fully adequate. But this is obviously not true for simplified “black box” models based on nonlinear extensions of Nomoto equations while identifying all their parameters

680

may become a “well posed” and relatively easy problem. In general, here can be appreciated two comments by Ljung (1978): (1) independent on the quality and quantity of information available, a good model can never be obtained if its structure is poor and (2) in more or less realistic cases the iden-tification process can never be fully automated and final decisions must be taken by the researcher.

3.5 Simple practical theoretical models for slender ship hulls

Although the flow around a ship’s hull perform-ing curvilinear motion with nonzero drift angle is extremely complex, some simple flow models qualitatively capturing main effects with greater or smaller success have been created. These models are based on the perfect fluid concept and are intended to predict the sway force and yaw moment. The main idea behind such models was inspired by the effectiveness of the classic wing theory which is also based on the perfect fluid model. Viscosity respon-sible for appearance of the non-zero lift is only accounted for indirectly by means of the famous Kutta—Zhukovsky condition on the trailing edge.

However, while the classic wing theory proved to be very successful and accurate when applied to lifting bodies specially shaped to generate high and stable lift, its applications to the bodies like ship hulls, for which lift generation is nothing more than a side effect, produced much less impressive results: 50 percent errors in estimating sway forces and yaw moments are common. In spite of this, such models still have some substantial theoretical value and often could serve as a good platform for introducing various empiric corrections imroving agreement with experimental data.

When considered as a wing, the ship hull has substantial thickness (breadth B) and a very small aspect ratio kH = 2T /L, where T is the ship draught and L—its length; the factor 2 comes from the mirror-image principle applicable at low Froude numbers. It was known from multiple experi-ments that hydrodynamic characteristics of wings with aspect ratio values below 1.0–1.5 are highly nonlinear and any viable theory must take this into account. Of all possibilities for constructing nonlinear models of small-aspect-ratio wings only one found wide use in ship manoeuvring: this is the model based on separation of linear and non-linear parts of the load. The linear part is esti-mated with the Munk method while the nonlinear part is described with the cross-flow drag model (Fedyaevsky and Sobolev 1964).

This approach presumes that the linear and nonlinear parts of the load co-exist without interfering with each other and can be estimated independently. To some extent, this viewpoint was

justified by measurements of pressure distribution on a low-aspect ratio flat plate placed into the flow at small and moderate incidence. The measure-ments showed (Fedyaevsky and Sobolev 1964) that the “linear” part of the load is mainly concentrated near the plate’s leading edge while the nonlinear part is caused by smaller peaks along the wing tips (side edges corresponding to the ship’s bottom and its mirror image). Then, the load distribution in each section perpendicular to the chord has some similarities with that on a flat plate set perpendicu-lar to the flow and this inspired assumption that the nonlinear load is the drag load caused on each strip by the flow normal to the wing while the cir-culation flow associated with the linear load is here completely ignored. Obviously, this “separation principle” is highly crude as the actual flow pattern shown on Fig. 7 reproduced from (Nikolaev 1990) is very complicated and depends on conditions of motion. This is one of causes of substantial errors often shown by simplified flow schemes although main effects still can be roughly captured.

Figure 7. Sketches of vortex patterns around the ship hull: from captive model test runs on a rotating arm.

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3.5.1 Main results of the Munk methodThe Munk method was primarily developed for estimating transverse aerodynamic loads on airships (Munk 1924) but was later somewhat adapted to ship forms and given a much better exposure (Fedyaevsky and Sobolev 1964). The main idea behind the method is the application of the momentum theorem to the fluid surrounding a hull strip with further integration of obtained 2D inertial loads along the ship’s length with introduc-tion of certain correction factors accounting for the influence of viscosiy.

Evaluations, which can be found in multiple sources, being omitted, the final general formulae for linear hydrodynamic derivatives is here pre-sented in the form slightly different from common and apparently more convenient for handling:

FkH

w T dxdd

w

ωFπkk μ

λ

′ ′kHπk′ ′μμ ′ ′T ′

′ ′λ

=

⎡

⎣

⎢⎡⎡

⎢⎢⎢

⎢⎢⎢

⎢⎣⎣⎢⎢

+∂

∂

−

∫21 2μ 2

2

1

2

1

2

2

( )x′x ( )x′ ( )x

( )x′ ( )xλλ ′λλxx

x x dxdd′

′ ′ ′ ′d′[ (′ ) (T )] ,μ′′222

1

2

1

2

−

∫⎤

⎦

⎥⎤⎤

⎥⎥⎥

⎥⎥⎥

⎥⎦⎦⎥⎥

(41)

where F = Y, N is the generalized notation for the force and moment, ω = υ, r—the same for quasi-velocities; w1,2 are the weighting factors defined as follows:

F w w

Y

Y r x

N x

N r x x

ωυ

υ

1 2w

2

0 1

1

0

′′

′ ′

and x x T xx′ ′′ ′x / ,LLL )′ ( )x /( (( ))μ μ=( )x′x ρπTT22μμ 22μ 22 is the sectional sway-sway added mass coefficient non-dimensionalized by the added mass of an elliptic section with the same local draught T (x).

Of special nature is the factor λ(x) which serves to introduce into the theory an analogue of the Kutta condition which is necessary to account for the viscosity effects and to obtain realistic esti-mates of hydrodynamic derivatives. The λ-factor is supposed to emulate the experimental fact that the transverse hydrodynamic load is satisfactorily predicted by the momentum theory in the fore part of the hull while is heavily overestimated it in the aft part. On bodies of revolution, like airship hulls, it was recommended (Munk 1924) to apply the so-called von Kármán correction factor gradually

reducing the aft load in accordance with reduction of the aft cross sections. On ship-like forms, how-ever, that function is always taken in the form (Fedyaevsky and Sobolev 1964).

λ( )λλ ,d)x xx x

e

e

10

atat

><{ (42)

where xe is the abscissa of the section behind which the load caused by the added mass gradient is sup-posed to be absent. Recommendations for a pri-ori choosing xe are somewhat fuzzy and in most cases it goes about the section with the maximum added mass or section with the maximum draught. Although these recommendations provide reason-able estimates, inaccuracies may be substantial and the λ-function can be regarded as the main source of uncertainties of the Munk method.

Hearn et al. (1996) have undertaken an attempt to reach agreement between the measured and esti-mated transverse load at the stern region by add-ing the momentum of the fluid motion induced by free trailing vortices to the 2D potential flow momentum considered for each strip in the Munk method. The separation point and the strength of the vortices were determined experimen-tally making direct application of this approach impossible.

The so-called canonic values of the four lin-ear derivatives correspond to the case when the center-plane is rectangular and all sections are elliptic6. In this case, μ22 1′ ′ ′T( )′x′ ≡T ( )x( )′xx inside the integration interval, and ∂/∂x′[μ′22(x′)] = δ(x′ + 1/2) − δ(x′ − 1/2) where δ(x') is the Dirac delta-function. The Equation (41) will then yield: Yr' = −πkH/2,Yr' = πkH/4, Nr' =−πkH/4, and N kr HN k′ / . These values notonly have theoreti-cal meaning but they also serve, augmented with correction factors and terms, as basis for many empiric methods like that described in (Inoue, Hirano, and Kijima 1981).

The same values of these coefficients are sup-plied by the asymptotic low-aspect-ratio wing the-ory by Jones (1946) and can be also obtained after asymptotic transformation of the lifting surface integral equation. Sometimes the latter approach is preferrrable to the Munk method as, for instance, it permits generalization to the case of finite Froude numebrs (Sobolev 1976) although results of such generalisation remained inconclusive and have never come to practical applications.

6 Pay attention that in case of elliptic sections the sectional added mass does not depend on the local breadth and this case includes also the thin flat plate.

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3.5.2 Cross-flow modelAs was already mentioned, the Munk method is only providing the linear part of the load and it must always be combined with the cross-flow drag model at least when it goes about simulation applications. According to this method, the transverse load is proportional to the squared transverse velocity υ(x) and always has the opposite sign which is natural for the local drag. The main formula for nonlinear parts of the sway force YCF and yaw moment NCF is then (Fedyaevsky and Sobolev 1964):

F w C T r dxddCFFF DCL

L

−

∫ρ2

2

2

( )x ( )x ( )x ( )+υ x+ xr | |xr+υ ,

(43)

where w(x) = 1 for F = Y and w(x) = x for F = N, and CD(x) is the function describing distribution of the local transverse drag coefficient.

This dimensional formula can be applied to any manoeuvre including hard manoeuvres at zero or very low speed of advance. The latter situa-tion is typical for dynamic positionning processes and it can be noticed that the linear part of the load is then absent and no linearized model can be applied7. A unified ship model combining the modified formula (43) with empiric linear terms and valid for arbitrary manoeuvres was proposed by Yoshimura and Nakao (2009). Modification consisted in replacing υ + xr with υ + CrY,N, where CrY and CrN are adjustable empiric coefficients.

When only moderate manoeuvres are consid-ered, the non-dimensional form of the CD-model may be preferred:

F w C T

dx

CFFF DC′ ′ ′ ′T ′

′ ′ ′ ′ ′ ′ ′×−

∫ ( )x′ ( )x′ ( )x′

( )′ ′ ′′ | |x r′ ′ ′+ .

1

2

1

2

)x r′ ′+′′ | ′′

(44)

Contrary to a rather common belief, estimation of the function CD(x) from resistance data for series of cylinders with cross sections identical to cor-responding ship sections, besides being very time cosuming, will not bring good results. This happens because the cross-flow model is a very approximate

concept and the actual flow is different from that during wind tunnel tests with cylinders. That is why, the function CD(x') is usually approximated with some polynomial whose coefficients must be identified from rotating-arm or PMM captive-model tests or even after free-running model tests. Some time ago, many researchers believed that due to a seemingly better physical soundness of the CD-model as compared with formal polynomial or quasi-polynomial models, a smaller number of parameters could be required for regression models of equal accuracy. However, it was shown by Kose (1982) that when the 3-parameter approximation CD(x)T(x) = c0 + c1x + c2x

2 was used, the fitting quality of the CD-model was substantially inferior to that of (quasi)polynomial models. When the degree of the polynomial was increased, better fit could be formally reached but the product CD(x)T(x) was becoming negative for some x' which is physically meaningless and was considered unacceptable. Khristov and Zilman (1989) showed that somewhat better results could be obtained when an exponential approximation of the func-tion CD(x') was used instead of the polynomial one which permitted peaks of the local drag coef-ficients close to the ship’s extremities. However, the regression model becomes than nonlinear which makes estimation of its parameters much more tedious.

In practice, the transverse drag coefficient is often assumed constant along the hull i.e., CD(x) = CD0. Of course, such a model cannot be accurate but it contains only one parameter and is reasonably suitable for rough estimates. In this case, the integral in Eq. (44) can be evaluated analytically8:

YC

vr

vr

NC v r

CFYY DC

CFNN DC

′

′′ ′

′′

′′

′= − + − −

⎡

⎣⎢⎡⎡

⎣⎣⎢⎢

⎤

⎦⎥⎤⎤

⎥⎦⎦⎥⎥

= − +⎛

⎝⎜⎛⎛

⎝⎝

03 3′

02 2r′

23r′ ⎣⎢⎣⎣⎢⎢

2

4 2⎝⎝⎝ 16

⎞⎞

⎠⎟⎞⎞⎞⎞

⎠⎠+

⎡

⎣⎢⎡⎡

⎣⎣

⎤

⎦⎥⎤⎤

⎥⎦⎦⎥⎥− ++( )+ − ( )++ −+ −

v r++

′ ′3

(45)

where s r± ±sig ((υ ′ ′±1 2 ) It is clear that, contrary to often expressed statements (see e.g., (Clarke 2003)), the result does not have the structure of the quasi-polynomial (29). That is why, it is difficult to say that the latter is better physically based than the normal cubic polynomial model!

Regarding that the primary objective of intro-ducing the cross-flow drag model was explanation

7 It must be emphasized, however, that this is only true for motions of manoeuvring type characterized by low Strouhal numbers: when ship motions in sway and yaw caused by waves are considered, Strouhal numbers are much higher and then the non-steady linear wave damp-ing is dominating.

8 This can be done using the identity | | x| 2 valid for

all real numbers and some evident substitutions. However, Mathematica also handles this quadrature.

683

of nonlinear hydrodynamic characteristics, it could be suggested to replace in that model, at least in the non-dimensional case, the nonlinearity of the type v(x)|v(x)| with v3(x). Then, the Equation (44) will be replaced with

F w C T dxCFFF DC′ ′ ′C ′ ′ ′ ′ ′d′−

∫ ( )x′ ( )x′ ( )x′ ( )v x r′ ′ ′+ ,33

1

2

1

2

(46)

where CD3(x') is the modified cross-flow drag coefficient.

When CD3(x') = CD30, the integrals become elementary and then (compare these formulae with (45)):

Y C

N C

CFYY DC

CFNN DC

′ ′ ′

′ ′ ′′

303 2′′

302 3′

1

41

4

1

90

( )v v r3 2v r′+4

,

( )v r r′ +2 3r+1

4 90.

(47)

These are incomplete cubic polynomials depend-ing on only one parameter. No doubt, it is difficult to expect an accurate description of the actual loads from such a simple model but it can serve as a rough estimate apparently not worse than that provided by the model (45). Relative values of nonlinear hydrodynamic derivatives defined by Eq. (47) are not always confirmed by experimental data (Crane et al., 1989), (Oltmann et al., 1980) but it cannot be also said that they definitely contra-dict them and for some coefficients the agreement can be very good.

Strangely, there exists a direct experimental confirmation of the cubic dependence of the local loads on the transverse velocity obtained with the help of segmented models (Beukelman 1989). Sticking to the cross-flow drag concept this can be explained by the transverse wave resistance which is typically increasing faster than the speed squared but this can be also due to more complicated effects related to the actual flow pattern.

3.5.3 Some generalizationsBoth the Munk method and the cross-flow drag method are also applicable to 3D motions of the ship hull. In application to submarines this was implemented by Bohlmann (1990). For surface ships, Soding extended the theory, first, to include 3D corrections (Söding 1982) and also the influ-ence of the instantaneous ship roll (Söding 1984).

As was said before, the flow pattern assumed in the classic CD-model is very different from that observed in reality. An original attempt to make the modelled flow more realistic although still retain-ing the simplicity of the model was undertaken

by Faltin-sen (2005) who developed the so-called 2D + t theory meaning that it is based on the 2D flow in each strip but is viewed as developing in time t. The first bow section correspond then to a cylinder starting its motion from rest while further sections correspond to later moments from the start. Using experimental or computational data on the drag of cylinders in unsteady transverse motion closes the theory and paves way to estima-tion of the sway force and yaw moment. Although in some cases this method leads to better estimates, this improvement is not definite and stable and the longitudinal distribution of CD(x) predicted by the 2D + t-method (smaller values at the bow with growth to the aft region) does not agree with meas-urements performed by Beukelman (1989) and with estimates obtained by Khristov and Zilman (1989) and by Oltmann and Wolf (1990).

In spite of all improvements and modifications, combination of the Munk and CD methods gives, in general, only qualitatively consistent results and they cannot be used in practical simulation sys-tems without extensive tuning but then they can only rarely compete with simpler and more flexible models based on direct linear regressions.

The Munk method can also serve as a conven-ient tool for modelling manoeuvring in waves. When applied correctly, it provides simultaneous estimation of the linear part of forces caused by manoeuvring actions as well as by waves-excited motions. It is also possible to combine the Munk model for wave-excited motion with manoeuvring forces estimated from experiment or an empiric database method but in this case special care must be taken against double account of linear part of the loads in slow manoeuvring motion. A variant of such an approach was suggested by Sutulo and Guedes Soares (2006b), (2006c), (2008a), (2009a).

3.6 Empiric methods based on serial model tests

Such methods are also called database methods. The idea behind this approach is quite transpar-ent: first, it is supposed to carry out model tests with varying parameters describing the hull shape and then obtain regressions not only over kin-ematic but also over form parameters. As a mat-ter of fact, such compound regressions usually are not considered but separate approximations for hydrodynamic derivatives are constructed as func-tions of the form parameters. In general, such two-step procedures are somewhat risky as the error of the resulting response may become unpredict-able. However, it can be assumed that when it goes about well tested published methods, all necessary checks and adjustments were performed.

The database method can be viewed as natural generalization of the well-know residual resistance

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series alhough in manoeuvring the similar task becomes much more complicated which is due to the increased number of kinematic parameters and also because of necessity to use less common and more complicated experimental facilities. As result, there are very few “manoeuvring series” known so far, and their prediction certainty is much inferior to that in ship resistance.

Mathematical models based on serial tests rarely can be used in simulators without adjustments as errors can be substantial for any specific vessel but appropriate tuning can make them acceptable. Data-base data can also be used as first approximation in system identification procedures. Most of these methods are predicting not only hull forces but are also estimating contribution of the rudder although the latter is not considered in this subsection.

Apparently, the first manoeuvring prediction method of this kind was developed in 50s by Per-shitz (1983). The method was initially based on results of rotating-arm tests performed with models of light cruisers and its first version was published by Voytkunsky et al. (1960). Later, the method was somewhat updated (its last version was published in (Voytkunsky 1985)) and allegedly data on cargo ships were accounted for but it still serves the best just for naval combattants.

Dimensionless hull forces in Pershitz’ method were described by the regressions (31) whose coefficients were represented as functions of the following dimensionless form parameters: L/B, T/L, the prismatic coefficient CP, and the “effec-tive centerplane area coefficient” CLe. The latter was supposed to depend on the aft cut off area of the centerplane and on the parameter i which is the number of the section (not necessarily integer, based on 21 theoretical sections), corresponding to the front point of the skeg or to the transition from U- to V-shaped sections forming an implicit skeg.

Strong dependence of the sway force and yaw moment on the form of the after body is a well-established fact and from this viewpoint the method was quite consistent. On the other hand, the oversimplified structure of the regressions (31) is hardly sufficient for adequate modelling of arbi-trary moderate manoeuvres. In fact the method was not developed for simulation purposes but mainly for predicting the spiral curve in ship design and was intentionally simplified to make it suitable for hand calculations.

Another popular method, or rather collection of related methods, was developed much later by a group of Japanese experts and all these methods are known as MMG methods according to the des-ignation of a workgroup group organized by the Manoeuvrability Subcommittee of the Japanese Towing Tank Conference. General principles laid in the foundation of the MMG method, i.e., the

modular principle, a semi-empirical rudder model based on the actuator disk theory, representation of the hull forces etc. are well commented by Ogawa and Kasai (1978) although in this publication it was still supposed to describe the nonlinear part of the hull forces with the cross flow-drag model which finally was rejected in favour of a relatively full polynomial regression (Inoue et al., 1979). Ini-tially 3DOF, the model was soon augmented to include the roll equation and the regression for the yaw moment obtained additional terms depending on the roll angle (Hirano 1980).

The most popular variant of the MMG model, which even was included into the third edition of PNA (Crane et al., 1989), was described with suffi-cient completeness in (Inoue, Hirano, Kijima, and Takashina 1981) and (Inoue, Hirano, and Kijima 1981) and it was independently implemented by many reserchers and engineers around the world including the authors of this contribution. The structure of the sway force and yaw moment coef-ficients was assumed in the form:

Y Y r v v v r Y r r

N N N r N

v rv YY v v v rY r rYY

v rN v NN

′ ′Y ′ ′Y ′ ′Y ′ ′Y ′′ ′N ′

′ ′ ′ ′ ′

′ ′

Y vvYY vY +Y v v Y v rv vYY YY′Y ′ ′Y

N vvNN vN

,

vvrvvNNNN vvr r r

r

v r N vvvr r Nr r r

N r

′ ′ ′ ′ ′

′ ′ ′

′ ′′ ′

2 2r ′ ′′ ′N+ N vvvr rN v rN

+ Nϕ υ ϕυ ϕϕϕvϕ υ ϕυ ϕ ϕϕN N′ +′v′ ,

(48)

where ϕ is the roll angle.The regression coefficients in (48) depended

on the following form parameters: T/L, CB, B/L, and B/T. Correction for the trim was devised on the basis of the Munk theory and only applied to linear derivatives. Strangely, there were no param-eter describing the shape of the after body and dimensions of the skeg. However, the method provided reasonable predictions for most cargo ships and, apparently this was the first practical 4DOF method although the phenomenon of the roll influence and its importance for relatively fast ships with moderate metacentric height was in detail studied by Eda (1980).

A very similar model was described by Matsunaga (1993). The difference is in the struc-ture of the sway force regression which is here iden-tical to that for the yaw moment in Eq. (48). The regressions for nonlinear derivatives are somewhat different although depending on the same form parameters. Another difference is that the trim cor-rections are also applied to nonlinear derivatives. The method includes corrections for the shallow water but, on the other hand, it is purely 3DOF i.e., dependence on the roll is dropped.

Neglection of the afterbody’s shape was remain-ing a major weak point of MMG methods until it was handled by Kijima et al. (1993). As presented

685

by Fujino (1996) and by Ishiguro et al. (1996), the sway and yaw derivatives depend on the parameter, σa W a PaCW a P= 1 1CW a− where CWa and CPa are respec-tively the water plane and prismatic coefficients for the aft CP a half of the hull. The full description of the improved MMG method appeared in (Kijima 2003). The structure of regressions for the sway force and yaw moment are here similar to those in (Matsunaga 1993) but most of approximations for the derivatives are different. Also trim correc-tions apply to all derivatives and the roll equation is absent. It is still not clear how the parameter σa relates to the earlier Pershitz’ parameter CLe and these two are in no way identical although certain correlation seems to be evident.

Besides the database methods reviewed above, there were some other suggestions. For instance, an interesting attempt to build a method based on 160 data sets accumulated during captive-model tests in Hamburg Model Basin (HSVA) was under-taken by Oltmann (2003). Unfortunately, this method remained incomplete as regressions were established and validated only for the linear part of the model.

Kose et al. (1996) proposed the so-called Type Ship or TS-method presuming availability of credible experimental data for some (proto)type ship TS. For any ship under study it is supposed to fix differences in its geometric parameters with respect to the type ship. After that, some database or even theoretical or numerical method is used to find differences in predicted values of the hydro-dynamic derivatives between the studied and type ship. These differences are used to extrapolate val-ues of the manoeuvring derivatives from the type ship to the actual ship of interest. The reason for this approach is that typically most methods bet-ter predict trends in dependencies of the hydro-dynamic derivatives on the geometric parameters than their actual values. Success in using this method depends, however, on how close the type ship is to the vessel under study.

Another useful review of database method sup-plementing that presented here above can be found in (Di Mascio et al., 2011).

3.7 Models for propulsion and steering devices

Regarding main propulsion and steering devices9 three different situations can be discriminated:

1. Propulsion and steering functions are fully separated, realized with different devices with

negligible hydrodynamic interaction between them. Example: screw propellers and a rudder outside the slipstream(s) that is typical for many naval ships especially submarines.

2. The same as above, but the steering device is hydrodynamically influenced by the propulsor with much smaller back influence. Examples: a rudder in the slipstream (apparently the most typical arrangement on most ships) or a steer-ing nozzle.

3. Propulsion and steering functions are sup-ported by the same integrated device. Examples: azimuthal thrusters and waterjets.

The first case is an obvious simplification of the second one and the third case will not be commented here as currently there have been not so many implementations (for a possible water-jet model see e.g., (Sutulo and Guedes Soares 2005d)) and direct empiric descriptions are often preferred: evidently, in the general case it is not possible to assume that the thrust vector is sim-ply deflected together with the thruster and there are too few systematic data for screw propellers in oblique flow.

Hence, the emphasis will be given here to the classic case of a rudder behind the propeller. When an integrated polynomial regression model is used, the rudder action is accounted for by regressors of the type δR, v'δR, v'2 δR, v'γ'δR etc. As to the influ-ence of the propeller, introduced is the parameter (ship propulsion ratio) η = 1− J/J0, where J is the current advance ratio and J0 is the advance ratio corresponding to the propulsion point in approach phase, and regressors of the type η, v'η, δRη etc. are added to the model.

However, more promising and physically more consistent are modular submodels for the propeller and the rudder which are briefly described below.

3.7.1 Models for screw propellersMathematical models for screw propellers used in manoeuvring simulation are or can be basically the same as those considered in the ship propul-sion area but certain peculiarities can be noticed. Namely:

1. Implemented models must be fast enough for online computations which excludes vortex the-ories and CFD codes giving absolute preference to simple empiric methods.

2. A propulsor’s model suitable for manoeuvring simulations must remain valid and consistent in curvilinear motion with non-zero sidewash angle.

3. In most cases it is required that the propeller model be valid for all possible combinations of the rotation frequency and of the propel-ler’s advance velocity (the so-called 4-quadrant

9 As opposed to auxiliary steering devices such as side thrusters which are mainly used in slow speed and will not be considered here.

686

model). If only moderate manoeuvres are simulated, this requirement may be dispensed.

4. While in ship propulsion only the propeller thrust and torque are of interest, estimation of the lat-eral force may be necessary in manoeuvring.

5. Approximate nature of contemporary manoeu-vring models containg a number of empiric correction factors permits to ease accuracy requirements of used propeller models as com-pared to those typically used in ship propulsion problems.

The propeller surge force XP is present in any simulation model and is usually represented as

XP = TP(1−tP). (49)

where TP is the propeller thrust, positive when directed ahead, and tP is the thrust deduction coef-ficient depending on the propeller slip ratio s.

Typically, in normal conditions i.e., from the bollard regime to the free run ahead the thrust deduction coefficient can be estimated by

t tJ J

J JP Pt

dJ0

0JJ

0JJ, (50)

where tP0 is the design regime thrust deduction coefficient which can be estimated using various empiric methods (Van Mannen and Van Oossanen 1988a), J = uPA/(nDP) is the current advance ratio, uPA is the propeller advance velocity with respect to water, n is the propeller rotation frequency, DP is the propeller diameter, J0 is the zero thrust advance ratio, and Jd is the advance ratio corresponding to the design propulsion point which is usually close to the propeller’s optimal regime. The formula (50) results from the observed equidistance of the curves of the open-water and effective thrust coef-ficients with an additional assumtion that they can be approximated by straight lines. According to this formula, the thrust deduction coefficient is increasing when the advance ratio decreases down to the bollard pull regime.

Very little information about the thrust deduc-tion is available for J < 0 and J > J0. Harvald (1977) obtained experimental results on the trust deduc-tion and wake fraction with a 7.5 m model of a bulk carrier for all possible regimes of rectilinear motion: (1) u > 0, n > 0; (2) u < 0, n > 0; (3) u > 0, n < 0; (4) u < 0, n < 0 defining 4 quadrants on the u—n plane. These data in infinite depth as well as in shallow water did not confirm the equidistance of the thrust coefficient curves and dependence of the trust deduction coefficient even in the first quadrant is very different from that defined by Eq. (50) but also from earlier Harvald’s results (Van Mannen and Van Oossanen 1988a). However, the model tested in

(Harvald 1977) had an extremely high block coeffi-cient CB = 0.805 and the scatter of values of tP cor-responding to different J is substantial. It is unclear whether these data are applicable to finer ship forms and further collection of such data is desirable.

The open-water thrust TP and torque QP are represented either as

T n D KT

Q n D KP PT nT D

P PQ n D QK

n

n

2 4D2 5D

( )J ,

( )J , (51)

or as

T A C V

Q A D C V

P dT AT TCC BVV

P dQ AA P QD CC BVV

( )B

( )B

ρ

ρ (2

2

2

2

,

,

(52)

where KT, CT, KQ, CQ are the thrust and torque coefficients, d PA D2

QQ

4/ is the disk area, γB is the effective blade advance angle, VB is the effective total blade velocity.

The Equations (51) are only applicable when J ∈ [0, J0] while (52) cover all (4 quadrant) propeller regimes. It means that the representation (51) can be used for simulating moderate manoeuvres (turning, zigzags etc.), but cannot describe the propeller loads at any reversing including the stopping manoeuvre. That is why, the representation (52) or similar is the only option for interactive manoeuvring simulators.

However, the advantage of Eq. (51) is that there are much more available data on the thrust and torque curves in the first quadrant KT (J) and KQ(J). In particular, they cover propellers of the B and BB series with the number of blades from 2 to 7, disk area ratio from 0.3 to 1.05, pitch ratio from 0.5 to 1.4 (Kuiper 1992). Published approximating poly-nomials (Oosterveld and van Oossanen 1975) not only embrace all tested propellers but also permit reasonable interpolation over geometric character-istics. Unfortunately, available 4-quadrant data for the characteristics CT (γB) and CQ(γB) represented in the form of trigonometric polynomials cannot always be used directly as they are only complete for 4-blade propellers and some isolated propellers with different number of blades (Assenberg and Sellmeijer 1984).

Oltmann and Sharma (1985) have proposed a very simple 4-quadrant model for approximating the coefficients CT and CQ

C

C C

C

CT QCC

T QCC T QCCcB

T QC sB B

T QCCccB

Q T cos

sin .

| cos |

0

0 9336+ ≥CT QCC sBsin

γ B

γ B BB BcosB

γ B cocc s

| sin |

γγ γ|

Bγ

Tss

B Bγ γγ |CT+

⎧

⎨⎪⎧⎧

⎪⎨⎨⎪⎪

⎩⎪⎨⎨

⎪⎩⎩⎪⎪

otherwise,

(53)

687

where CT, Q0, …, CT QCC ss are constant parameters whose values are only given in the cited reference for a 5-bladed propeller with the pitch ratio 1.2 and area ratio 0.86.

It is expected that a simulated ship in straight run reaches a certain specified (design) speed at a specified rotation frequency of the propeller. How-ever, even if the most accurate methods are used for predicting the resistance and the propeller thrust, perfect match is not achieved and some correction is required. One of the correction factors is needed for matching the efficiency maximum as provided by the used propeller model with the design pro-pulsion point. The most logical is to correct with the coefficient ku the propeller advance velocity:

uPA = kuu(1−Wp), (54)

where wP is the estimated wake fraction. In gen-eral, this parameter depends on the local sidewash angle βP. There are not many data on the wake fraction in oblique flow and it is typically estimated with simple empiric formulae, like (Inoue, Hirano, Kijima, and Takashina 1981):

w w eP Pw K P−0

1 2β2

, (55)

where wP0 is the straight-run wake fraction esti-mated by the ship propulsion methods (Van Mannen and Van Oossanen 1988a) and K1 > 0 is an empiric constant. When the absolute value of the sidewash angle exceeds 90 degrees, it is usu-ally assumed wP = 0 unless more accurate data are available. This assumption approximately agrees with data from (Harvald 1977) when n > 0 and |βP | = π but it can be of substantial magnitude at n < 0 even if the ship is backing. The same as with the thrust deduction, accumulation of new data is highly desirable.

The advance velocity correction factor can be estimated using the formula:

kJ n D

Vdukk

d PD

P

= optJ

( )wP−,

0

(56)

where Jopt is the maximum eficiency advance ratio for the used propeller model, nd is the propel-ler design rotation frequency, and Vd is the ship’s design speed.

To match the modelled propeller thrust with the estimated ship resistance at the design point, it is resonable to apply an additional term ΔKT or ΔCT so that

K K K orC C C

T TKK TKK

T BCC T B TCC( )JJ ( )J( ) ( )B ,

= +KKK )J+C ( )B

0

0

ΔΔCB TC)B CTCC (

(57)

where KT0 and CT0 are thrust coefficients from the propeller model.

The correction term ΔKT is defined as

ΔKR

n DKTKK T dRR

d P PDTKK= − ( )J

( )VdVV

( )tPtρnn20

4 0 J (58)

(ΔCT can be defined similarly).Choosing an additive correction instead of a

multiplicative one is driven by the properties of the curves KT (J) which are mostly slightly nonlinear and can be transformed one to another with fair accuracy through simple translation which is illus-trated by Fig. 8 where thrust coefficients curves for propellers with various pitch ratio were brought together according to the correction method described above showing relatively small diver-gences. Of course, in 4 quadrants and with other propellers the differences can be greater but remain reasonable. This transformation does not work so well for the torque (Fig. 9) but larger residual dis-crepancies between the propeller and the engine torque can be usually handled by adjusting the latter.

It must be finally noted that although the out-lined approach for handling hull–propeller–engine mismatches can reasonably deal even with large discrepancies, it is desirable to use the most ade-quate models available to bring the corrections to minimum. Regarding the 4-quadrant propeller data it is now possible to use neural-network extrapola-tion assimilating data from both (Kuiper 1992) and (Assenberg and Sellmeijer 1984). Such a method

J

KT

0 0.2 0.4 0.6 0.8 1 1.2 1.4–0.3

–0.2

–0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 8. Shifted thrust coefficient curves KT (J) for 4-bladed propellers with the disk area ratio 0.85 and pitch ratio varying from 0.5 to 1.4: a thicker line corre-sponds to P/DP = 0.9.

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was developed with great care by Roddy et al. (2007) and provides reasonable response for every Wageningen propeller although direct experimen-tal confirmation for most parameter combination is absent.

3.7.2 Rudderkinematicsandhydrodynamics

All mathematical models of rudders used in prac-tical simulation models are based on a number of rather crude assumptions and simplifications. At the same time, it was found necessary to roughly cap-ture most of interaction phenomena accompanying the rather complicated flow around the rudder.

A rudder blade is, from the hydrodynamical view-point, a wing with a symmetric profile. In general, a number of reliable methods of different complex-ity have been developed in hydro- and aerodynam-ics for predicting lift and drag of wings of various geometries. The following circumstances, however, seriously complicate the prediction task when these methods are applied to the ship rudder:

1. The proper hydrodynamic characteristics of the isolated rudder may be altered by the presence of the ship hull, other structures like the rudder horn, and by proximity of the free surface.

2. The oncoming flow is not homogenous and its parameters are not easy to determine. Not only they depend on the kinematics of the manoeu-vring motion but also on the hull wake.

3. When the rudder blade or its part is washed by the propeller race, additional complicating fac-tors appear: influence of the boundary of the jet; deflection of the jet caused by the rudder lift, tangential velocities in the slipstream. The jet can be also deflected because of the sidewash.

4. In some situations, like the pull-out manoeuvre, the rudder attack angle can exceed the stall value and in hard manoeuvres it can vary from −180 to 180 degrees.

At the same time, it can be stated with consid-erable certainty that unsteady effects are insignifi-cant in rudder hydrodynamics, at least with realistic deflection rates.

Rudder characteristics in open water. If the effec-tive rudder velocity with respect to water VR is defined (this is, however, the most difficult part of the problem!) the hydrodynamic characteristics of the rudder can be estimated relatively easily. Besides the stock moment, which is important for the rudder design but of minor interest for simula-tion systems, two force components can be defined. In the local velocity axes these are: the lift L and the drag D. Good approximations for the lift coefficient CL(αR) and for the drag coefficient CD(αR) valid for only the pre-stall region can be found in (Crane et al., 1989). Here αR = δR − βR is the rudder attack angle, δR is, as before, the rudder deflection angle which is always known exactly, and βR is the rudder sidewash angle whose estimation may be difficult. Besides the linear part, the approximation for the lift has also a quadratic contribution accounting for the nonlinearity which is rather pronounced on wings with low aspect ratio. It must be emphasized that in general both lift and drag can contribute sig-nicantly to the rudder surge and sway forces and to the rudder yaw moment, so the rudder drag cannot be neglected except for few very special cases.

The following formulae are recommended by Mandel (1965) and by Crane et al. (1989):

C C CRLC RLC R RL R R+CRLCα α αCR RC L R+Rαα |αα | |RαR , (59)

C CC

akRRDC RDC RLC+CRDC 0

2

π, (60)

where

Cak

k aRLC Rk

Rk

α π= 2

2 (61)

is the linear lift coefficient slope, a is the empiric correction factor, and kR is the rudder’s aspect ratio;

C

c

c

kc

c

k

RLC

t

Rk

t

Rk

α

γ

γ

| |α

. .

. .

=

+

+

0 075 0 735

0 075 1 6. 5

with faired tips

with fhh aireff d tips.

⎧

⎨

⎪⎧⎧

⎪⎪⎪

⎪⎨⎨

⎪⎪

⎩

⎪⎨⎨

⎪⎪⎪

⎪⎩⎩

⎪⎪

(62)

J

10K

Q

0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.5

0

0.5

1

Figure 9. Shifted torque coefficient curves KQ(J): con-ditions as in the previous figure.

689

These formulae demonstrate strong dependence of the nonlinear part of the lift on the taper ratio c ct γ which effect was not, however, reflected in the similar scheme developed by Söding (Brix 1993). The correction factor a must account for the influence of viscosity and of the profile’s thickness as 2π is the theoretical lift gradient of a thin flat plate. Crane et al. (1989) recommend to assume a = 0.9 for all typical profiles which agrees well with multiple experimental data in the field of wing aerodynamics. However, some more recent results (Söding 1998) including those obtained with CFD codes indicate that for some “good” moderately thick profiles a can reach values up to 1.17.

Instead of the lift and drag, rudder force com-ponents defined in the rudder-fixed axes can be used alternatively: the normal rudder force NR and the tangential rudder force TR. As rudder characteristics can be easily recomputed from one frame to another, the both description methods are equivalent but the latter one has the advantage that the tangential force can be neglected practi-cally always. The normal force coefficient can be approximated similarly to the lift i.e., it contains a sensible nonlinear component for low-aspect-ratio rudders. Meanwhile, most practical models still use linearized models for the rudder normal force. For instance, in (Inoue, Hirano, Kijima, and Takashina 1981) and many others the following for the nor-mal force coefficient CRN is used:

Ck

kRNC Rk

RkR=

+6 13

2 25

.

.sin ,RαR (63)

where using sin αR instead of just αR is not essen-tial at all in the pre-stall range. This formula was proposed in 1961 by Fujii and Tsuda (Ogawa and Kasai 1978) and is a slight modification of the classic Prandtl formula which is known to overes-timate the lift gradient for the aspect ratio values typical for ship rudders. This overestimation is in fact partly compensated by rejection of the non-linear part resulting in a reasonable linearisation in the pre-stall region (Fig. 10). It is clear that the lin-earized formula overestimates the normal rudder force at small attack angles and underestimates it at large angles providing, however, reasonable approximation in the whole.

Simulation of arbitrary manoeuvres requires knowledge of circular rudder characteristics i.e., those for αR ∈ [−π, +π]. Unfortunately, more or less systematic circular data are only available in the literature for 2D NACA profiles (Sheldahl and Klimas 1981). These data cannot be applied directly to lowaspect ratio wings but apparently can help to create reasonable approximations of the normal force. However, in some cases, e.g., when the rudder is attached to the skeg as may happen on twin-screw ships, it may become totally ineffective in astern motion (Brix 1993) working as a small leading-edge flap on a large steady wing. In such cases it may become most reasonable to drop any dependence of the rudder force on the deflection angle which means full loss of control-lability when the ship is backing.

Hull Influence. There are several aspects of the rudder-and-hull interaction:

1. Proximity of the hull to the rudder root affects the load distribution and increases the effective aspect ratio. The latter is theretically doubled when the root is very close to the surface of the hull but the effect can be reduced because of the boundary layer and of the hull curvature. The most consistent methods for accounting for the latter factor were developed in hydrodynam-ics of submarines (Aucher 1981) where, how-ever, the task is somewhat simpler as the hull is a body of revolution.

2. Besides the force applied to the rudder blade and to the horn, if present, a sway force is act-ing upon the aft part of the hull. This additional force can be substantial, is applied much forward of the rudder itself thus displacing considerably the centre of application of the whole sway force caused by the rudder. According to (Inoue, Hirano, Kijima, and Takashina 1981) the force augmentation can reach 35 percent and, accord-ing to (Brix 1993), can be displaced up to 0.65 × draught forward of the rudder itself.

3. Another aspect of the rudder–hull interaction is qualitatively similar to the wake fraction effect in

αR,

deg

CN

0 5 10 15 20 25 30 350

0.5

1

1.5

2

Nonlinearmodel

Linearmodel

Figure 10. Rudder normal force coefficient in the prestall region: aspect ratio kR = 1.0, rectangular form, squared tips.

690

ship propulsion. The difference is that here two rudder velocity components must be considered i.e., the rudder quasi-velocities with respect to water uR and vR. Assuming that these are lin-early depending on the ship quasi-velocities of surge, sway and yaw, it is possible to write:

u

u

wu

v

r

R

R

R uv ur

vu vv vr

⎛⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

=−⎛

⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

⎛

⎝

⎜⎛⎛

⎜⎝⎝

⎜⎜⎞

⎠

⎟⎞⎞

⎟⎠⎠

⎟⎟1 0 κ κuv u

κ κvu v κv

, (64)

where wR0 is the rudder wake fraction in straight run, κuv and κur account for the dependence of the wake fraction on the sway and yaw, and κvu κvv and κvr constitute the set of flow-straighten-ing factors.

At present, there are not enough data to estimate each component of the hull-inluence matrix and some simplifications are usually introduced. For instance, it is typically assumed that κuv = κur = κvu = 0 but, on the other hand, wR0 is replaced with, say, w w eR Rw K R−

01 2β2

, where βR is the rudder sidewash angle. Empiric data for the remaining straightening factors can be found in the literature and sometimes they are represented not like constants but as some functions of βR (Inoue, Hirano, Kijima, and Takashina 1981).

As for other manoeuvring forces, the greatest difficulties emerge here in less usual regimes of motion like the ship backing. There are virtually no published data on the rudder–hull interaction with large absolute values of the sidewash angle especially when the flow is coming from the trail-ing edge. It is often assumed that in such situations the influence of the hull on the flow is weaker and can be neglected.

Rudder in the slipstream. Virtually all practical models of the rudder working in the propeller slip-stream (Brix 1993), (Inoue, Hirano, Kijima, and Takashina 1981) are based on the so-called “load separation principle” presuming the following:

1. The whole rudder area AR = AR0 + ARP, where ARP is the area washed by the slipstream and AR0 is the remaining area outside the jet.

2. If the rudder speed with respect to water and its attack angle are VRP and αRP inside the slip-stream, and VR0 and αR0 outside it, the total nor-mal rudder force FRN can be estimated as

F CV

A

CV

A

RNFF RNC RPVVRPA

RNC RVVRA

( )RP

+ ( )R

R

ρVV

R

ρVV

2

02

0

2

2,

(65)

where the function CRN(α) is assumed to be identical for the both parts of the rudder and to be the same as for the same whole rudder in a uniform flow.

It is evident that the formula (65) cannot be accu-rate: the jet’s boundaries will influence the function CRN(α) which will be different for each part of the rudder. However, it is easy to view Eq. (65) is a lin-ear interpolation equation giving exact results in two extreme cases: (1) when the slipstream is absent and (2) when the rudder is completely submerged into the infinite diameter slipstream. This explains the fact that so far the load separation principle has not lead to substantial errors and has become so popular.

When the separation principle is adopted, its effective application depends on the knowledge of the quantities VRP and αRP depending on the parameters of the propeller slipstream. In practi-cal rudder models these parameters are estimated on the basis of the actuator disk theory. It is clear that V u vRPVV RP RP

2 2u 2+uRP2u , where uRP and vRP are respec-

tively: the longitudinal and transverse components of the slipstream velocity. When these are known, the corresponding sidewash angle βPR can be restored and then αRP = δR − βRP.

The longitudinal slipstream velocity

uRP = uPA + wa, (66)

where wa is the axial induced velocity, and, in the first approximation, at a substantial distance from the propeller vRP = vPA although it is close to zero in immediate vicinity of the propeller.

The axial induced velocity depends on the rela-tive, i.e., divided by the propeller diameter, dis-tance x between the rudder and the propeller disk: wa = kw( )wa∞, where the theoretical axial induced velocity at infinity

wa∞ = u∞ − uPA, (67)

where

u u wPA a+ ∞2

02 , (68)

and the squared axial induced velocity in bollard pull regime:

wT

aA

02

0

2∞ =

ρ, (69)

where T is the propeller thrust and A DPD0AA 2π / .4The relations written above are approximately

valid for all regimes including the bollard pull and even—with appropriate definition of the function kw( )—when the ship is backing. For instance,

691

a direct generalization of the formula suggested by Fedyaevsky and Sobolev (1964) is

k xx

xK Twkk ( )x ( )T sin ,T= +

+

⎛

⎝⎜⎛⎛

⎝⎝

⎞

⎠⎟⎞⎞

⎠⎠

⎡

⎣⎢⎡⎡

⎢⎣⎣⎢⎢

⎤

⎦⎥⎤⎤

⎥⎦⎦⎥⎥1

1 2gnn (70)

k TxT

xT( )T

. ,xT=

≥≥

⎧⎨⎧⎧

⎩⎨⎨1 0. 0

0

at

0.7 at (71)

is an empiric correction factor.In general, however, forces on the rudder when

the propeller is reversed and in astern motion are poorly investigated. Some experimental data can be found in (Tanaka and Miyata 1977) but it is often simply assumed that when the ship is mov-ing ahead and the propeller is reversing, the rud-der becomes ineffective. More important is then the so-called Hovgaard force typically realized on the afterbody of a single-screw ship washed by the slipstream of the reversed highly loaded propeller (Sobolev 1976).

In fact, the formulae above constitute the basis for most models of the rudder in the slipstream but, when applied only to normal ahead motion, their specific form can be different and it can be based on such a parameter as the thrust load-ing coefficient C TTACC PA T( )2T 2

02/ ( V( /K) 8A)2 ( )J 2 ,πTρVPAVV ) 0V KVV KK8AVVV =0AA (

where the advance ratio J can be replaced with the slip ratio s = 1—J/J0, where J0 is the zero-thrust advance ratio.

Such rudder models can also contain a number of correction factors and terms, partly purely empiri-cal and partly based on simple theories accounting for a number of secondary effects neglected in the simplest models. For instance, the rudder model by Söding (1982), (1984), see also (Brix 1993), is espe-cially rich in corrections of the second kind taking into account:

1. Contraction and turbulent mixing of the slip-stream behind the propeller. This modifies the estimated area ARP.

2. Influence of the jet’s boundaries.3. Nonuniformity of the velocity along the span

of the rudder.

Söding’s basic idea was to create a method as free of empiric corrections as possible. Hence, within the potential theory the shape of the slip-stream was obtained with the help of a simple vor-tex tube model through an iteration process and numerical results were approximated with a simple formula different from (70) but providing similar numeric results. Correction on the turbulent mix-ing is, however, purely empirical.

To account for the jet boundaries, a linearized two-dimensional problem with the flat plate inside

the jet was solved also with the vortex method and the results were approximated as function of the ratios uPA/uRP and r/cR, where r is the slipstream’s radius and cR is the plate chord. In fact, this correc-tion accounts for the deflection of the jet caused by the rudder albeit with some overestimation.

Then, a generalization of the lifting-line theory was used to account for the spanwise non-uniformity of the flow on the rudder. In this case, no simple approximations were suggested but calculations showed that this factor alone can modify the lift coefficient by up to 20% although in most cases the correction will be smaller.

Inevitably, all methods of the described type will remain approximate as influence of the propeller is more complicated than can be described with the actuator disk theory. For instance, Laurens and Grosjean (2002) performed simulation of the slip-stream behind real propellers with the boundary-element method and discovered that corrections to the rudder’s lift and drag strongly depend on the propeller’s geometric particulars even at equal val-ues of KT/J2.

The preceding exposure of some issues related to simulation of the rudder forces, mainly in appli-cation to modular models, shows that while hydro-dynamically the problem is extremely complicated, it is possible to construct qualitatively relevant mathematical models based on schematized flow representation. In practical applications for bridge simulators a certain degree of quantitive accu-racy is required and can only be achieved through appropriate tuning of such models which must then contain a number of adjustable parameters. Example of such a parameter is the rudder lift slope correction factor a present in Eq. (61).

3.8 Additional forces in real-environment simulations

All forces and moments considered above were of hydrodynamic nature and are always present in manoeuvring mathematical models. They are suf-ficient for studying inherent manoeuvring qualities of ships and their behaviour in unbounded water. But if it goes about manoeuvring simulations in training purposes, many additional effects can be of equal or comparable importance. These effects include: influence of the bottom (shallow water), hydrodynamic interaction with other vessels and banks, reaction of fenders, wind loads. Detailed analysis of these forces being often of very specific nature lies beyond the scope of the present review mainly focusing on more fundamental issues. However, concerning the hydrodynamic interac-tion, it can be suggested to consult the authors’ publications (Sutulo and Guedes Soares 2008b), (Zhou et al., 2010), and (Fonfach et al., 2011) where

692

the problem is analyzed and a detailed review of former results is given.

4 IMPLEMENTATION OF CORE SIMULATION MODELS

4.1 State space representation of the manoeuvring model

Most practical manoeuvring models are repre-sented mathematically by a set of ordinary differ-ential equations. Sometimes, when manoeuvring in waves is described, the equation can become inte-gro-differential but even then they can be reduced to an extended set of ordinary differential equa-tions of increased order (Sutulo and Guedes Soares 2005a). Then, in the general case, the set of these equations and the corresponding initial conditions can be represented in the canonic matrix form:

XX X

( )X, U ,( ) ,0

(72)

where X(t) is the arithmetic vector of state vari-ables, Φ is the vector of all functions defining kin-ematics and dynamics of the vessel and its relevant subsystems, and U is the control vector.

The state vector X is created as concatenation of the vector of generalized coordinates Ξ, vector of quasi-velocities V and typically includes also the actual rudder angle δR and the rotation frequency n of each screw propeller. In the case of the extended model mentioned above the auxiliary state varia-bles are also added. The control vector U typically contains the required (ordered) rudder angle δR

*yy

δ and the ordered propeller rotation rate n*. In the case of a simulated automatic controller these two parameters do not explicitly enter the control vector but instead can be used, for instance, the required heading ψ* and the required ship speed V*. On the other hand, in some simplified models used for research-oriented simulations it can be assumed

R*δδ ≡ δR (ideal steering gear) and/or n* ≡ n (engine

dynamics neglected). The corresponding variables are then removed from the state vector.

4.2 Numerical integration

Each simulation represents a numerical solution to the initial-value problem (72) which requires application of this or that integration method. The problem of selection of some particular method is rarely discussed in the literature: in most cases, at best, the used method is just mentioned explicitly. In some sense, such an attitude is justified as inte-gration of manoeuvring equations typically runs smoothly and many methods can be successfully

used although the integration speed and accuracy can be somewhat different. However, our personal experience indicates that certain peculiarities could be commented.

First, it is almost self-evident that adaptive algorithms with automatic adaptation of the inte-gration step must be avoided as being too slow. In practice, several test simulation runs are always sufficient to select a suitable constant integration step to be used in all live simulation. Roughly, the integration step δt = 0.1L/V can serve as a good initial guess if the 3DOF motion in still water is considered and the mathematical model does not contain any especially “quick” elements. Other-wise, some of the equations constituting the set are somewhat stiff and the required integration step can become much smaller than indicated above. Examples of such relatively stiff submodels are:

1. Degrees of freedom in vertical planes i.e., roll, heave and pitch.

2. Second-order model of the steering gear (sel-dom used).

3. Submodel of the propulsion plant or its elements.

4. Some types of automatic controllers.

Presence of stiff submodels requires reduc-tion of the integration step δt and, consequently, decreases maximum simulation speed. This can remain insignificant in real-time simulation when the process is artificially slowed down but fast simulations are usually also required and some spe-cial measures against the stiffness can be recom-mended. Apparently, the stiffness problem in ship manoeuvring still does not require application of special implicit integration methods and more effi-cient could be application of the reduced step only to “quick” submodels (like the propulsion plant, roll motion etc.).

In some other cases, the stiffness problem can be avoided simply through application of simpler but still adequate models. For instance, the popular 1st order model of the steering gear, when all nonlin-earities are neglected, takes the form

TR RTT Rδ δR δ−δ*δ , (73)

where TR is the time lag, usually equal to several sec-onds and δ∗ is the required (ordered) rudder angle. It is clear that the right-hand side of Eq. (73) is, in general, discontinuous function of time and can change stepwise at any moment thus resulting in an instantaneous change of the rudder deflection rate meaning infinite value of the angular accelera-tion of the rudder which is physically impossible!

However, any attempt to improve this model and to account for the rudder’s inertia would result

693

in an aperiodic second-order plant containing an additional very small time lag. The model would then become stiff and much more difficult to inte-grate. At the same time, the model (73) augmented with appropriate nonlinearities proved to be, in spite of its certain physical deficiency, quite realistic and suitable for simulation of rudder deflections.

Further discussion of the issues related to the integration of the manoeuvring equations requires certain insight into the properties of these ordi-nary differential equations. First, it is worthwhile to recall that most of the results of the classic theory of such equations (Tikhonov and Arsenin 1977) were established under certain assumptions imposed on the right-hand side of the equations set. Namely, according to the Peano theorem the solution to the initial-value problem exists if the right-hand side is continuous in some neighbour-hood of the initial point. For uniquiness of the classic solution (the Picard–Lindelöf theorem), the right-hand side must satisfy a somewhat stronger Lipschitz’ condition.

Even simple continuity is not guaranteed in manoeuvring equations as e.g., this will not happen if a bang-bang control is applied. This, however, does not lead to any “catastrophy” as the solution still exists in Carathéodory’s sense (Filippov 1985). Conditions for existence and uniqueness of the generalized Carathéodory solution are rather weak and are met practically always. When the right-hand side is continuous, this generalized solution becomes identical to the classic one and this fact makes application of numerical methods quite consistent.

At the same time, the degree of continuity of the right-hand side may affect efficiency of this or that numerical method. For instance, our own practice showed that, contrary to what is often stated in the literature on nunerical methods, the simplest Euler method can provide in 3DOF manoeuvring simulation almost the same accuracy at the same integration step as a more complicated and very popular Runge–Kutta 4th order method (RK4). In fact, as the RK4 method requires 4 computa-tions of the right-hand side at each step instead of only one, its overall efficiency can be even inferior!

The reason for such results is that the RK4 can only fully realize its accuracy potential if the right-hand side is much smoother than just con-tinuous: it must possess continuous 4th-order derivatives. It is obvious that often this property is not kept in manoeuvring equations. For instance, quasi-polynomials of the type (29) are continuous but their first derivative has discontinuities. As the Euler method does not require strong smoothness of the right-hand side, it may become more suitable.

At the same time, the RK4 method can still be preferable and give better results if the model

contains poorly damped oscillatory plants, like the roll equation. Hence, in practical terms, it is desir-able to have at least two (Euler and RK4) methods implemented choosing one or another depending on the specific ship model. It is also clear that the Euler method is especially suitable for debugging the code.

An interesting analytic investigation of the sta-bility of various numerical integration schemes in application to linear integro-differential sea-keeping equations was undertaken by Kring and Sclavounos (1995). Only continuous right-hand sides were considered and it was found that some methods can become numerically unstable and even not converging as the integration step is being reduced.

4.3 Code organization

Organization and structure of a manoeuvring sim-ulation code is rarely commented in the literature. However, although simulation programs, at least in their dynamic core part, do not involve sophisti-cated mathematical methods and algorithms, they can become large enough to make navigation in their source codes a not very easy task. Only well-organized codes are acceptable from the viewpoint of reliability and sustainability.

In spite of certain extra costs, an object-oriented code seems to be the most attractive and, in a sense, the most natural choice. For instance, modelling a fleet of any number of identical or dif-ferent vessels becomes an almost trivial task within this paradigm.

There are two basic principles which can be laid in the foundation of a system of classes necessary to model the ship motion: (1) Inheritance and (2) Containing.

The inheritance principle (Lee 2003), (Kopp 2003) presumes creation of a hierarchy of classes of the type VehicleBase → VehicleMotion → VehicleDynamics → VehicleSpecific, where each preceding class is parent for the follow-ing one.

The alternative container approach apparently was proposed in application to ship dynamics for the first time by Baker (1992) and practically it is following the structure of the modular dynamic model. There, the main class Ship contains mem-bers of the class types Hull, Rudder, Engine etc. As hydrodynamic interaction exists, these classes must appropriately reference to each other.

The second concept seemed more natural as creating a kind of image of a real ship and it was used by the authors (Sutulo and Guedes Soares 2005c) for developing an object-oriented code consecutively encapsulating the ship as a manoeu-vring object. Later, however, it appeared that the

694

class Ship is not sufficient to cover all the practical cases. For instance, for appropriate description of hydrodynamic interaction forces, it was necessary to devise the additional class HydrodynamicIn-teraction (Sutulo and Guedes Soares 2009b) which is not contained in the class Ship. On the contrary, this class served as an enhanced container for the considered group of interacting ships, so the basic container principle still remained conserved.

5 CONCLUSIONS

The review presented above has covered a number of selected topics of the theory of ship manoeuvring mainly in view of applications to practical manoeuvring simulation. During their previous research activities, the authors had faced a number of relatively obscure issues and evalua-tions sometimes misinterpreted in the literature. The authors tried to comment such issues although it was not possible to avoid presenting also com-monly recognized and doubtless theories and concepts for the sake of certain continuity of the exposure. At the same time, the intention to stay within reasonable length of the paper forced the authors to assume that the readers are experts in ship manoeuvrability and can easily restore the issues that are here commented only briefly or even taken for granted.

A considerable amount of relevant literature available to the authors including their own contributions was used in preparation of this article. Whenever possible, attention was given to comparative analysis of various solutions and approaches. Some minor results of such analysis apparently are new and presented here for the first time.

The main practical conclusion which can be drawn from such analysis is that at present the task of constructing adequate and efficient math-ematical models for manoevring simulation can only be solved through reasonably selected appli-cation of various submodels and methods of dif-ferent complexity. A characteristic example of this strategy was presented by Di Mascio et al. (2011). It is hoped that the present review will somehow facilitate orientation in the available spectrum of manoeuvring mathematical models, solutions and methods thus contributing to further in the manoeuvrability theory and manoeuvring simula-tion practice.

ACKNOWLEDGEMENTS

Most of original results belonging to the authors of this review were obtained during work on

two research projects funded by the European Commission: “Effective Operations in Ports” (EFFORTS), contract No. FP6-031486 and “Safe Offloading from Floating LNG Platforms” (SAFEOFFLOAD), GROWTH programme, con-tract TST4-CT-2005-012560 and on the follow-ing research projects financed by Fundação para a Ciência e a Tecnologia (FCT), Portugal: POCI/TRA/61804/2004 “Intelligent ship manoeuvring simulating system”, PTDC/ECM/65806/2006 “Dynamics and Hydrodynamics of Ships in Approaching Fairways”, PTDC/TRA/74332/2006 “Methodology for ship maneuvering tests using self-propelled models”, and PTDC/ECM/100686/2008 “Towing Dynamics of Ships in Harbour Areas”.

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