TRA2014_Fpaper_29026

Identification of hydrodynamic coefficients from sea trials for ship maneuvering simulation

K.T. TRAN(a,c)

a) University of Technology of Centre de recherches Royallieu BP 20529, 60206 Compiègne cedex, France

b) CETMEF, 134, rue de Beauvais , Fc) Division of Waterway Safety Port and Waterways Engineering Faculty, VMU,

Abstract

In this study, an identification procedure is proposed to estimate the hydrodynamic parameters of a range of ship maneuvering models, by coupling the dynamic ship motion model with mathematical programming techniques. In order to assess efficiently the hydrodynamic parameters, a sensitivity analysis is first performed to identify the most sensitive coefficients. Different Mathematical Programming Techniques have been used and compared in the determination of optimal hydrodynamic parameterThe proposed procedure has been validated through turning circle and zigzag manoeuvres based on experimental data of sea trials of the 190,000-dwt oil tanker. Comparisons between experimental and computed data show the RMSD (Root-Mean-Square Deviation) of ship trajectory decreases from 68.0m to 5.8m in Turning Circle test, and RMSD of ship’s heading angle decreases from 17.3deg to 6.6deg in Zigzag. Keywords: Ships; waterway; maneuverability;

Résumé

Une procédure d’identification de paramètres est proposée pour estimer les paramètres hydrodynamiques des bateaux dans la poursuite de trajectoires optimales définies au préalable pour des navires inconnues ou connus (Esso Bernicia, Mariner Class Vessel,..). Elle est baséemodèle mathématique de mouvements de navires en dans un premier temps pour identifieroptimales de ces coefficients hydrodynamiques sont La procédure proposée a été validée en utilisant des données expérimentales de190000dwt pour les tests classiques de Giration et Zigzagnumériques montrent que l'écart quadratique moyen cas du test de Giration et de 17.3deg à 6.6deg Mots-clé: Navires; voies navigables

Transport Research Arena 2014, Paris

Identification of hydrodynamic coefficients from sea trials for ship maneuvering simulation

(a,c), A. OUAHSINE(a) , F. HISSEL(b), P. SERGENT

University of Technology of Compiègne, Lab.Roberval, UMR CNRS 7337, Centre de recherches Royallieu BP 20529, 60206 Compiègne cedex, France

134, rue de Beauvais , F-60280 MARGNY LES COMPIEGNE, France Division of Waterway Safety Port and Waterways Engineering Faculty, VMU,Vietnam

, an identification procedure is proposed to estimate the hydrodynamic parameters of a range of ship maneuvering models, by coupling the dynamic ship motion model with mathematical programming techniques. In

efficiently the hydrodynamic parameters, a sensitivity analysis is first performed to identify the most sensitive coefficients. Different Mathematical Programming Techniques have been used and compared in the determination of optimal hydrodynamic parameters. The proposed procedure has been validated through turning circle and zigzag manoeuvres based on experimental

dwt oil tanker. Comparisons between experimental and computed data show Deviation) of ship trajectory decreases from 68.0m to 5.8m in Turning Circle test,

and RMSD of ship’s heading angle decreases from 17.3deg to 6.6deg in Zigzag.

maneuverability; hydrodynamic forces; parameters identification

Une procédure d’identification de paramètres est proposée pour estimer les paramètres hydrodynamiques des bateaux dans la poursuite de trajectoires optimales définies au préalable pour des navires inconnues ou connus

cia, Mariner Class Vessel,..). Elle est basée sur le couplage des techniques d’optimisation avec un vements de navires en manœuvre. Ainsi, une analyse de sensibilité,

identifier les coefficients hydrodynamiques les plus sensibles. Ensuiefficients hydrodynamiques sont identifiées en utilisant les techniques d’optimisation

La procédure proposée a été validée en utilisant des données expérimentales des essais en mer de l'Esso Bernicia les tests classiques de Giration et Zigzag. Les comparaisons entre ces données et

'écart quadratique moyen diminue de 68.0m à 5.8m pour la trajectoire du navire 17.3deg à 6.6deg pour l’angle de cap dans le cas du test de Zigzag

; manoeuvrabilité; forces hydrodynamiques ; identification de paramètres.

Transport Research Arena 2014, Paris

Identification of hydrodynamic coefficients from sea trials for

P. SERGENT (b)

Vietnam

, an identification procedure is proposed to estimate the hydrodynamic parameters of a range of ship maneuvering models, by coupling the dynamic ship motion model with mathematical programming techniques. In

efficiently the hydrodynamic parameters, a sensitivity analysis is first performed to identify the most sensitive coefficients. Different Mathematical Programming Techniques have been used and compared in the

The proposed procedure has been validated through turning circle and zigzag manoeuvres based on experimental dwt oil tanker. Comparisons between experimental and computed data show that

Deviation) of ship trajectory decreases from 68.0m to 5.8m in Turning Circle test,

ters identification.

Une procédure d’identification de paramètres est proposée pour estimer les paramètres hydrodynamiques des bateaux dans la poursuite de trajectoires optimales définies au préalable pour des navires inconnues ou connus

sur le couplage des techniques d’optimisation avec un une analyse de sensibilité, est effectuée

coefficients hydrodynamiques les plus sensibles. Ensuite, les valeurs identifiées en utilisant les techniques d’optimisation.

s essais en mer de l'Esso Bernicia ces données et les résultats

la trajectoire du navire dans le test de Zigzag.

; identification de paramètres.

Tran K., Ouahsine A., Hissel F. and Sergent P./ Transport Research Arena 2014, Paris

Nomenclature

x, y ship’s position u velocity in the x-axis v velocity in the y-axis r= ψ heading (yaw) φ roll c flow velocity n shaft velocity X external force in the x-axis Y external force in the y-axis G center of mass δ rudder angle β the ship drift angle τ non-dimensional propeller thrust

objF function objective

1. Introduction

It is still tedious to perform real ship manoeuvres in an open sea or to carry out fine simulations usingCFD calculations. Even though their fast calculations, systemmanoeuvring hydrodynamic coefficients (hull, rudder, propeller,...) in order to achieve quantitative agreement with the experimental measurements. In this context, manoeuvrability turns out to be an essential ability to perform a safe navigation of any ship against the danger of collisions and stranding.A literature review (Rawson and Tuppermanoeuvrability prediction: the experimental based method, the Computational Fluid Dynamics (CFD) based method and the system based method.manoeuvrability tests to fulfill the International Maritime Organization (IMO) criteria, the determination of hydrodynamic manoeuvring coefficients. Unfortunately experimental determination of the hydrodynamic coefficients can be tedious and expensive. the ships but also the flow field around ships by solving a set of RANS (Reynoldsequations (Ji et al., 2012). Lately, CFD based method has been extended for freemeasurable data from Experimental Fluid Dynamics for ship manoeuvres in calm seawaves. However, CFD method needs few hours or even days depending on the turbulence and propulsion modeling and the mesh size. (iii) Elsewhere, manoeuvrability. Computation time is much smaller than that of CFD based methodfew minutes of computation on a Personal Computer for one freeSystem-based methods have been extensively investigated by researchersRodrigues, 2009). The course keeping stability of the ship can be investigated on the basis of the stability of the solutions of the linear equations of mmanoeuvres at high rudder deflection angles require the consideration of nonlinear hydrodynamic and inertial components. This leads to the utilization of nonlinear hydrodynamicTaylor’s series expansion of the hydrodynamic external forces and moments

The herein proposed paper presents an efficient procedure to determine optimal hydrodynamic coefficients by using a mono-objective optimization based on a mathematical programming (MP) technique suitable for highly nonlinear problems such us ship manoeuvring simulation from sea trials. Accurate modeling of a ship trajectory is achieved effectively using three steps. The firstmotion, to this end a 3DOF mathematical model based on the Lagrangian dynamic motion of a 3D rigid body has been developed taking into account the nonlinear hydrodynamic forces acting on the ship rudder. Then a sensitivity analysis is carried out to identify the most significant hydrodynamic coefficients which affect the ship trajectory. The interest of applying sensitivity analysis is to reduce the number optimized. In the present investigation, it has been found that only 14 coefficients were sensitive for the prediction of the trajectory of the ESSO 199,000

The structure of the paper is as follows. Tpresents the mathematical programming based system identification for manoeuvring of large tankers. devoted to the numerical results and discussions


r thrust

t is still tedious to perform real ship manoeuvres in an open sea or to carry out fine simulations usingCFD calculations. Even though their fast calculations, system-based simulations need numerous tests to adjust the manoeuvring hydrodynamic coefficients (hull, rudder, propeller,...) in order to achieve quantitative agreement

al measurements. In this context, manoeuvrability turns out to be an essential ability to perform a safe navigation of any ship against the danger of collisions and stranding.

and Tupper, 2001),( Arak et al., 2012 showed that there exist three methods for ship prediction: the experimental based method, the Computational Fluid Dynamics (CFD) based

and the system based method. (i) Manoeuvring experiments are based on the verification of to fulfill the International Maritime Organization (IMO) criteria, the determination of

hydrodynamic manoeuvring coefficients. Unfortunately experimental determination of the hydrodynamic coefficients can be tedious and expensive. (ii) The CFD based method allows determining not only the motion of the ships but also the flow field around ships by solving a set of RANS (Reynolds-Averaged Navier

. Lately, CFD based method has been extended for free-running simulations, inclmeasurable data from Experimental Fluid Dynamics for ship manoeuvres in calm sea

However, CFD method needs few hours or even days depending on the turbulence and propulsion . (iii) Elsewhere, system-based method is a major simulation task to predict ship

manoeuvrability. Computation time is much smaller than that of CFD based method, and few minutes of computation on a Personal Computer for one free-running trial while.

based methods have been extensively investigated by researchers (Sutulo et al.,The course keeping stability of the ship can be investigated on the basis of the stability of the

solutions of the linear equations of motion, if only first order terms of this expansion are consideredmanoeuvres at high rudder deflection angles require the consideration of nonlinear hydrodynamic and inertial components. This leads to the utilization of nonlinear hydrodynamic models, which include higher order terms of Taylor’s series expansion of the hydrodynamic external forces and moments (Dan, 20010).

The herein proposed paper presents an efficient procedure to determine optimal hydrodynamic coefficients by bjective optimization based on a mathematical programming (MP) technique suitable for highly

nonlinear problems such us ship manoeuvring simulation from sea trials. Accurate modeling of a ship trajectory is achieved effectively using three steps. The first step deals with the modeling of the nonlinear dynamic ship motion, to this end a 3DOF mathematical model based on the Lagrangian dynamic motion of a 3D rigid body has been developed taking into account the nonlinear hydrodynamic forces acting on the ship rudder. Then a sensitivity analysis is carried out to identify the most significant hydrodynamic coefficients which affect the ship trajectory. The interest of applying sensitivity analysis is to reduce the number

. In the present investigation, it has been found that only 14 coefficients were sensitive for the prediction of the trajectory of the ESSO 199,000-dwt oil tanker benchmark (Manoeuvring committee

The structure of the paper is as follows. The nonlinear hydrodynamic model in givenpresents the mathematical programming based system identification for manoeuvring of large tankers. devoted to the numerical results and discussions. Conclusions are drawn in section 5.

Tran K., Ouahsine A., Hissel F. and Sergent P./ Transport Research Arena 2014, Paris 2

t is still tedious to perform real ship manoeuvres in an open sea or to carry out fine simulations using complex 3D based simulations need numerous tests to adjust the

manoeuvring hydrodynamic coefficients (hull, rudder, propeller,...) in order to achieve quantitative agreement al measurements. In this context, manoeuvrability turns out to be an essential ability to

ere exist three methods for ship prediction: the experimental based method, the Computational Fluid Dynamics (CFD) based

Manoeuvring experiments are based on the verification of to fulfill the International Maritime Organization (IMO) criteria, the determination of

hydrodynamic manoeuvring coefficients. Unfortunately experimental determination of the hydrodynamic d allows determining not only the motion of

Averaged Navier-Stokes) running simulations, including

measurable data from Experimental Fluid Dynamics for ship manoeuvres in calm sea and in the presence of However, CFD method needs few hours or even days depending on the turbulence and propulsion

based method is a major simulation task to predict ship , and requires approximately

et al., 2002, Neves and The course keeping stability of the ship can be investigated on the basis of the stability of the

otion, if only first order terms of this expansion are considered. However, the manoeuvres at high rudder deflection angles require the consideration of nonlinear hydrodynamic and inertial

models, which include higher order terms of Dan, 20010).

The herein proposed paper presents an efficient procedure to determine optimal hydrodynamic coefficients by bjective optimization based on a mathematical programming (MP) technique suitable for highly

nonlinear problems such us ship manoeuvring simulation from sea trials. Accurate modeling of a ship trajectory is step deals with the modeling of the nonlinear dynamic ship

motion, to this end a 3DOF mathematical model based on the Lagrangian dynamic motion of a 3D rigid body has been developed taking into account the nonlinear hydrodynamic forces acting on the ship hull, propeller and rudder. Then a sensitivity analysis is carried out to identify the most significant hydrodynamic coefficients which affect the ship trajectory. The interest of applying sensitivity analysis is to reduce the number of coefficients to be

. In the present investigation, it has been found that only 14 coefficients were sensitive for the prediction anoeuvring committee, 2005).

hydrodynamic model in given in section 2. Section 3 presents the mathematical programming based system identification for manoeuvring of large tankers. Section 4 is

Tran K., Ouahsine A., Hissel F. and Sergent P / Transport Research Arena 2014, Paris

2. Nonlinear transient hydrodynamic model

2.1. Lagrangian dynamic equation of 2D ship manoeuvring

Let 0 0 0 0O x y z the reference frame , Oxyz

is a plane of symmetry and (Fig 1). The Euler angles describing the position of the ship axes are the heading or

yaw ψ and the roll φ . The angle between the directions of

During manoeuvring, the position of the ship can be obtained by the coordinates

mass in the global coordinate system

(Fig.2).

Fig.1 Ship motion in 6 DOF

According to the Newton second law, the nonlinear transient equations of motion in the ship moving coordinate system Oxyz can be written in the form (

( )

2

2 2

=

=

=

G

G

z G

u v r x r g

v u r x r g

L k r x v u r gL

− − Χ+ + ϒ

+ + Ν

ɺ

ɺ ɺ

ɺ ɺ

=X

mgΧ , =

Y

mgϒ , =

N

mgLΝ are the non

ship; X and Y the external forces; N

is heading/yaw, (u,v) are the components of the velocity in the x

= zz

Ik L

m is the non-dimensional radius of gyration

2.2. Formulation of the hydrodynamic forces

A survey of literature review shows that there exist commonly two formulations to generate expressions of the hydrodynamics forces. In the present investigation, the hydrodynamic forces are expressed as functions of the kinematic parameters rvu ,, and the rudder angle

(

(

=

(1 )

=

=

u vru u v v c c c c

d u vr vv

v uv ur

s T ur uv v

u L u u vr v v c c c c

gL t u u u vr v gL

v L uv v v c c ur c c

g L ur uv v v v L c c gL

r

ξ ξ ξξ

ξ ξ ξ

ξ ξ ξ ξ

ξ ξ ξ ξ β βδξ

Χ Χ + Χ + Χ + Χ + Χ + Χ

+ − Χ + Χ + Χ + Χ + Χ

ϒ ϒ + ϒ + ϒ + ϒ + ϒ + ϒ

+ ϒ ϒ + ϒ + ϒ + ϒ + ϒ + ϒ

Ν Ν

ɺ

ɺ

ɺ

ɺ

ɺ

ɺ

ɺ

ɺ( r uv ur

s T ur r uv vr

uv v rL c c urL c c

g L ur L r L uv vr L c c gLξ ξ ξ ξξ ξ ξ ξ β β δ ξ

+ Ν + Ν + Ν + Ν + Ν

+ Ν Ν + Ν + Ν + Ν + Ν + Ν


transient hydrodynamic model

Lagrangian dynamic equation of 2D ship manoeuvring motion

Oxyz the frame fixed to the rigid body ( ship), G is the center of mass and

. The Euler angles describing the position of the ship axes are the heading or

. The angle between the directions of 0x axis and x axis is defined as the heading angle

During manoeuvring, the position of the ship can be obtained by the coordinates Gx0( , y

mass in the global coordinate system 0000 zyxO , the orientation of the ship is obtained by the heading angle

Fig.1 Ship motion in 6 DOF Fig.2 Definition of the coordinate system

According to the Newton second law, the nonlinear transient equations of motion in the ship moving coordinate can be written in the form (Tran, 2012).

L k r x v u r gL

− − Χ+ + ϒ

+ + Ν

are the non-dimensional forces and moment respectively, where m is the mass of the

N is the external moment;zI is the moment of inertia about the

components of the velocity in the x-axis and y-axis directions,

dimensional radius of gyration.

Formulation of the hydrodynamic forces

A survey of literature review shows that there exist commonly two formulations to generate expressions of the In the present investigation, the hydrodynamic forces are expressed as functions of the

and the rudder angle δ , thus

)2

2 2

u vru u v v c c c c

d u vr vvu u

v uv urv v c c c c

s T ur uv v v v c c

u L u u vr v v c c c c

gL t u u u vr v gL

v L uv v v c c ur c c

g L ur uv v v v L c c gL

δδ βδ

ξ ξ ξξξ

δ β βδ


δ βδ

ξ ξ ξ ξ

δ β βδ


Χ Χ + Χ + Χ + Χ + Χ + Χ

+ − Χ + Χ + Χ + Χ + Χ

ϒ ϒ + ϒ + ϒ + ϒ + ϒ + ϒ

+ ϒ ϒ + ϒ + ϒ + ϒ + ϒ + ϒ

ɺ

ɺɺ

2 3

r uv urv r c c c c

s T ur r uv vr

uv v rL c c urL c c

g L ur L r L uv vr L c c gL

δ β βδ

ξ ξ ξ ξ

δ β βδ

ξ ξ ξ ξ β β δ ξ

+ Ν + Ν + Ν + Ν + Ν

+ Ν Ν + Ν + Ν + Ν + Ν + Νɺ

ɺ


G is the center of mass and Oxz

. The Euler angles describing the position of the ship axes are the heading or

axis is defined as the heading angle ψ .

)0Gy of the ship center of

, the orientation of the ship is obtained by the heading angle ψ

of the coordinate system

According to the Newton second law, the nonlinear transient equations of motion in the ship moving coordinate

(1)

where m is the mass of the

is the moment of inertia about the z axis; ψɺ=r

axis directions, respectively.

A survey of literature review shows that there exist commonly two formulations to generate expressions of the In the present investigation, the hydrodynamic forces are expressed as functions of the

)g L ur uv v v v L c c gLξ β βδξ

)2 3c cg L ur L r L uv vr L c c gLβ β δ ξ+ Ν Ν + Ν + Ν + Ν + Ν + Ν (2)


where β is the ship drift angle, Χ Χ ϒ ϒ Ν Ν Ν

ship hydrodynamic coefficients, which have to be identified.

= /( )S Shξ Τ − Τ with SΤ is the ship draft,

given by: ( 2 2= | | /uu un n nu uLn nL n gLτ τ τ τ+ +is the shaft velocity. The flow velocity

2 2 2= un nnc nu c n c+

Where cun and cnn are the hydrodynamic coefficients.

2.3. Numerical integration of the nonlinear transient equations

The set of nonlinear ordinary differential equations (ODE) (integration schemes. Thus, we first define a set of primary variables

{ }nyxrvuxT δψ=

where , ,u v r are the ship velocity components

0000 zyxO , δ is the actual rudder angle

The complete manoeuvrability of ship is carried out by assembly and solving the set of ODE given by (

rate r=ψɺ and the rates , nδɺ ɺ of rudder angle and the shaft velocity, respectively.

2 2

0

0

=

=

=

= cos sin

= sin cos

=

=

60 ( )=

G

z G G

c

c

m

u g v r x r

v x r g u r

L k r x v gL x u r

x u v

y u v

r

n nn

T

ψ ψψ ψ

ψδ δ δ

Χ + ++ ϒ −

+ Ν −−+

−−

ɺ

ɺ ɺ

ɺ ɺ

ɺ

ɺ

ɺ

ɺ

ɺ

mT is the coefficient of propeller, δequation (6) can be rewritten in a more

( )= ,Tx f t xɺ

where { }0 0=Tx u v r x y nψ δɺɺɺ ɺ ɺ ɺ ɺ ɺ ɺ and

By assuming that f is sufficiently differentiable

t t tt t

x xx

t

•+∆

+∆−≈

∆ the discret implicit Euler

( )ttttt xttftxx ∆+∆+ ∆+∆+ ,=

The latter system of of nonlinear equations and the 4th order implicit Runge-Kutta scheme .


, ,.., , ,..., , ,...,u v uv r uvu u c cβ β δ ξΧ Χ ϒ ϒ Ν Ν Νɺ ɺ ɺ

are the non-dimensional derivatives of

ship hydrodynamic coefficients, which have to be identified. dt is the thrust deduction coefficient,

is the ship draft, and h is the water depth. The non-dimensional propeller thrust

)| |= | | /uu un n nu uLn nL n gLτ τ τ τ , where uuτ ,

unτ and | |n nτ are the hydrodynamic coefficients and

is the shaft velocity. The flow velocity c at the rudder is given by:

hydrodynamic coefficients.

Numerical integration of the nonlinear transient equations

ordinary differential equations (ODE) (1) have to be solved using appropriate numerical we first define a set of primary variables x , given by:

the ship velocity components , 0 0, ,x y ψ the ship position components in the

the actual rudder angle and n is the shaft velocity.

The complete manoeuvrability of ship is carried out by assembly and solving the set of ODE given by (

of rudder angle and the shaft velocity, respectively.

2

= cos sin

= sin cos

60 ( )

G

z G G

u g v r x r

v x r g u r

L k r x v gL x u r

x u v

y u v

n n

ψ ψψ ψ

Χ + +

−+

−

cδ is the commanded rudder angle, cn is the commanded shaft velo

) can be rewritten in a more convinient form, as:

( ),f t x is the nonlinear time-varying function .

is sufficiently differentiable, and by using the Euler backward finite difference formula

implicit Euler solution of (7), with respect to t and x , reads:

of nonlinear equations (10) was solved using a Newton-Raphson (NR) Kutta scheme .


dimensional derivatives of

is the thrust deduction coefficient, = /v uβ − ,

dimensional propeller thrust τ is

hydrodynamic coefficients and n

(3)

) have to be solved using appropriate numerical

(4)

the ship position components in the reference frame

The complete manoeuvrability of ship is carried out by assembly and solving the set of ODE given by (1), the yaw

(5)

is the commanded shaft velocity. The

(6)

finite difference formula:

(7)

Raphson (NR) iterative procedure


3. Mathematical Programming Based System Identification

3.1. Statement of the optimization problem

The optimization (or the mathematical programming problem) can be stated as follows

{( )α

α

objFminimizeswhich

Find αα ,...,,= 21

Subject to the constraints

( )( )k

j

maxii

mini

Nkl

Njg

Ni

1,2,...,=0,=

1,2,...,=0,

1,2,...,=

α

α ≤≤≤ ααα

where α is an N -dimensional vector called the our case they are represented by the ship hydrodynamic coefficients to be determined,

Objective Function (OF), and (αjg

The number of variables N and the number of constraints

In this work, a single OF is used in the optimization problem in order to identify ship hydrodynamic coefficients. Therefore, for the turning circle manoeuvre for instance, the OF is chosen a

1/2

2

=1

=p

obj ii

F S

∆ ∑

where p is the number of sampling points,

( ) (22 = expi

numi

expi

numii yyxxS −+−∆

Superscripts num and exp indicate the computed and experimental data respectively,

coordinates of the point i . For the case of a zigzag test, the expression of

( )1/2

2

1=

=

−∑ expi

numi

p

iobjF ψψ

where iψ is the ship’s heading angle, which depends also on ship hydrodynamic coefficients

optimization problem stated in (8techniques. In the present investigations we use (Zhang. J et al. 2003) and the BFGS (Broyden

3.2. Normalization and sensitivity analysis

Normalization of DV consists in a linear transformation of the original variables

variables α , given by:

BA +αα =

where A and B are constant diagonal matrix and vector respectively.


Mathematical Programming Based System Identification

Statement of the optimization problem

(or the mathematical programming problem) can be stated as follows

}TNα,...,

l

g

N

N

N

dimensional vector called the design vector which contains the Design case they are represented by the ship hydrodynamic coefficients to be determined,

)α and ( )αkl are known as inequality and equality constraints

and the number of constraints gN and/or

lN need not be related in any way.

In this work, a single OF is used in the optimization problem in order to identify ship hydrodynamic coefficients. Therefore, for the turning circle manoeuvre for instance, the OF is chosen a

is the number of sampling points, 2iS∆ depending on ship hydrodynamic coefficients

)2exp

indicate the computed and experimental data respectively,

. For the case of a zigzag test, the expression of objF reads:

is the ship’s heading angle, which depends also on ship hydrodynamic coefficients

8) and (9) is solved by using the so-called mathematical programming In the present investigations we use the Sequential Quadratic Programming (SQP) algorithm

BFGS (Broyden-Fletcher-Goldfarb-Shanno) algorithm (

Normalization and sensitivity analysis

Normalization of DV consists in a linear transformation of the original variables α

are constant diagonal matrix and vector respectively.


(or the mathematical programming problem) can be stated as follows (souli et al. 1999):

(8)

(9)

esign Variables (DV), in case they are represented by the ship hydrodynamic coefficients to be determined, ( )αobjF is called the

constraints, respectively.

need not be related in any way.

In this work, a single OF is used in the optimization problem in order to identify ship hydrodynamic coefficients. Therefore, for the turning circle manoeuvre for instance, the OF is chosen as

(10)

on ship hydrodynamic coefficients α , reads:

(11)

indicate the computed and experimental data respectively, ),( ii yx are the

(12)

is the ship’s heading angle, which depends also on ship hydrodynamic coefficients α . The

mathematical programming the Sequential Quadratic Programming (SQP) algorithm

(Dai, 2002).

α into new transformed

(13)


The normalization of the objective function

iteration by 0objF

0=

obj

objobj

F

FF , where

normalization is applied, the gradients of the OF have to be adapted to the new set of design variables.

ααα ∂∂

∂∂→

∂∂

obj

objobj FA

F

FF0

1=

The numerical procedure for the identification of hydrodynamic coefficients based on MP techniques, can be summarized into three main steps:

1- Compute the ship trajectory with original coefficients 2- Filter the most sensitive coefficients

3- Calculate optimal values α

procedure using SQP or BFGS algorithms.

4. Numerical Applications, Results and Discussion

The proposed procedure has been validated through turning circle and zigzag manoeuvres accordingly to the IMO and based on experimental data of sea trials of the ESSO associated physical and geometrica

Table 1. Parameters of the ESSO 190,000

Physical and geometrical parameters

ppL : length between perpendicular

B : beam

T : draft to design waterline

∇ : displacement

BLpp/

TB/

BC : block coefficient

0U : design speed

n : nominal propeller

The associated input data related to the turning circle the zigzag manoeuvre

Table 2. Input data for turning circle and zigzag (ESSO 190,000Input data

00, yx : initial ship’s position

0ψ : initial heading angle

0U : initial advance velocity of ship

0δ : initial of rudder angle

maxδɺ : maxi rotation velocity of rudder

0n : initial shaft velocity

cn : shaft velocity command

cδ : rudder command


The normalization of the objective function (OF) is accomplished by dividing the objective function at each

, where 0objF is the value of the objective function at the first iteration. Once the


α

objF

he numerical procedure for the identification of hydrodynamic coefficients based on MP techniques, can be

the ship trajectory with original coefficients α from dynamic ship equations given by (Filter the most sensitive coefficients Sα among all othersα , based on the sensitivity analysis.

optα for only sensitive coefficients Sα , by carrying out the optimization

procedure using SQP or BFGS algorithms.

Numerical Applications, Results and Discussion

The proposed procedure has been validated through turning circle and zigzag manoeuvres accordingly to the IMO and based on experimental data of sea trials of the ESSO 190,000-dwt oil tankerassociated physical and geometrical parameters are given in Table 1.

. Parameters of the ESSO 190,000-dwt oil tanker (ITTC 2005) Physical and geometrical parameters Value

: length between perpendicular 304.8 m

47.17 m

: draft to design waterline 18.46 m

: displacement 220000 m3

6.46

2.56

: block coefficient 0.83

: design speed 16 knots

: nominal propeller 80 rpm

The associated input data related to the turning circle the zigzag manoeuvre tests are presented in Table

. Input data for turning circle and zigzag (ESSO 190,000-dwt oil tanker model).Turning circle test zigzag test

: initial ship’s position (0,0) m (0,0) m

: initial heading angle 0 deg 0 deg

: initial advance velocity of ship 5.3 m/s 7.5 m/s

: initial of rudder angle 0 deg 0 deg

: maxi rotation velocity of rudder 2.7 deg/s 2.7 deg/s

: initial shaft velocity 57 rpm 80 rpm

: shaft velocity command 57 rpm 80 rpm

: rudder command -35 deg [-20,+20] deg


is accomplished by dividing the objective function at each

is the value of the objective function at the first iteration. Once the


(14)

he numerical procedure for the identification of hydrodynamic coefficients based on MP techniques, can be

from dynamic ship equations given by (6). , based on the sensitivity analysis.

, by carrying out the optimization

The proposed procedure has been validated through turning circle and zigzag manoeuvres accordingly to the dwt oil tanker (ITTC 2005). The

are presented in Table 2.

dwt oil tanker model). zigzag test

(0,0) m

0 deg

7.5 m/s

0 deg

2.7 deg/s

80 rpm

rpm

20,+20] deg


In the present application, there is 190,000-dwt oil tanker model (ITTC

the numerical processing. Hence, thehydrodynamic coefficients α in the dynamic ship motion Eqs.(sensitivity analysis procedure to select the most important parameters.filtering the largest gradient corresponding to each

ηα

>i

objF

∂∂

where 0.1=η and = 1obj

i max

F

α∂∂

, is the normalized gradient value, which corresponds to the largest value of

the gradient objF∇ . By using the developed manoeuvring model, the sensitivities are computed and filtered

using the above criterion. Variations of gradients are manoeuvres. Tthis figure shows that shows also, that only 6 coefficients are common for the two turning circle 5); this indicates that it is important to include the physical knowledge of the hydrodynamic problem, in the identification process in order to insure a good result.

Fig.3 Sensitivity analysis. a) turning circle test, b)

4.1. Identification of hydrodynamic coefficients using a turning circle test

The proposed procedure has been validated using the computed ship trajectory before optimization, hydrodynamic coefficients, the total cumulative error on the ship trajectory was

Fig.4 Left: Ship trajectory using initial reference hydrodynamic coefficients. Right: Comparison of ship trajectories after optimization: a) SQP algorithm, b) BFGS algorithm.

We defined p =40 as the total number of experimental procedure has been carried out by using SQP and BFGS algorithms with a convergence criterion based on the error of the OF as 041 −E and a maximum number of iterations of


35 hydrodynamic coefficients to control the manoeuvrability of the ESSO ITTC 2005). These coefficients (see Table 5) were used as as

Hence, the identification procedure starts from original reference values of all in the dynamic ship motion Eqs.(7), which will be firstly analyzed through a

sensitivity analysis procedure to select the most important parameters. The sensitivity analysis consists of filtering the largest gradient corresponding to each

iα , which is accomplished using the followin


. By using the developed manoeuvring model, the sensitivities are computed and filtered

using the above criterion. Variations of gradients are shown in Fig.3 for the turning circle and zigzag shows that only a few coefficients are of a great sensitivity (greater than

coefficients are common for the two turning circle and zigzag manoeuvres; this indicates that it is important to include the physical knowledge of the hydrodynamic problem, in the

identification process in order to insure a good result.

Sensitivity analysis. a) turning circle test, b) zigzag test.

Identification of hydrodynamic coefficients using a turning circle test

The proposed procedure has been validated using Eq.(12) for the turning circle test. ship trajectory before optimization, i.e. when starting with the initial reference values of

hydrodynamic coefficients, the total cumulative error on the ship trajectory was S=∆

Ship trajectory using initial reference hydrodynamic coefficients. Comparison of ship trajectories after optimization: a) SQP algorithm, b) BFGS algorithm.

as the total number of experimental sampling points in the Eq.(1procedure has been carried out by using SQP and BFGS algorithms with a convergence criterion based on

and a maximum number of iterations of 50. Convergence has been ach


hydrodynamic coefficients to control the manoeuvrability of the ESSO were used as as initial guess for

identification procedure starts from original reference values of all be firstly analyzed through a

The sensitivity analysis consists of , which is accomplished using the following criteria:

(15)


. By using the developed manoeuvring model, the sensitivities are computed and filtered

for the turning circle and zigzag coefficients are of a great sensitivity (greater than 10%). It

and zigzag manoeuvres (see table ; this indicates that it is important to include the physical knowledge of the hydrodynamic problem, in the

turning circle test. Figure 4 (left) shows when starting with the initial reference values of

m68 .

Comparison of ship trajectories after optimization: a) SQP algorithm, b) BFGS algorithm.

(12). The identification procedure has been carried out by using SQP and BFGS algorithms with a convergence criterion based on

Convergence has been achieved in


21 iterations for the SQP algorithm with a final optimal relative value of the BFGS algorithm with a final optimal relative value of

Table 3. Summary of final optimal solutions obtained by both SQP and BFGS in turning circle test.

Total nb of iterations

Final error on objF

Final optimal value of objF

Total cumulative error on trajectory ∆ From Table 3, we can see that the SQP algorithm is more precise than the BFGS algorithm because it led to a minimal cumulative error on the trajectory of

4.2. Identification of hydrodynamic coefficients using a zigzag test

The second application used to show the efficiency of the proposed procedure is a zigzag test. In this case, we define an OF based on the Eq.(present application because in the zigzag test there will be large variations in the heading angle during the dynamic ship motion. As previously, at first thin Table 5, are used in the developed ship manoeuvring model to evaluate the predicted heading angle. Figure 10 shows the calculated ship heading angle using our manoeuvring model. It is interethat before optimization, i.e. when starting with the initial reference values of hydrodynamic coefficientstotal cumulative error on the ship trajectory was

Fig.5 Left: Comparison of the heading angle Right: Comparison of the heading angle after optimization

In this application p=53 is chosen identification procedure has been carried out by using SQP and BFGS algorithms with a convergence criterion based on the error on the OF as has been achieved in 15 iterations for the SQP algorithm with a final optimal OF relative value ofonly in 7 iterations for the BFGS algorithm with a final optimal relative value of the OF of summary of the final optimal solutions obtained by both SQP and BF

Table 4. Summary of final optimal solutions obtained by both SQP and BFGS in zigzag test

Total nb of iterations

Final error on objF

Final optimal value of objF

Total cumulative error on trajectory

Figure 5 (right-a,b) shows the optimal solution4 shows that minimal cumulative error on the heading angle for the BFGS algorithm. The optimal hydrodynamic coefficients obtained at the end of the optimization process are summarized in Table 5


iterations for the SQP algorithm with a final optimal relative value of 0.084 and only in the BFGS algorithm with a final optimal relative value of 0.12 (see Table 3).

. Summary of final optimal solutions obtained by both SQP and BFGS in turning circle test.SQP 21

1E-04

0.084

S∆ (m) 5.8

, we can see that the SQP algorithm is more precise than the BFGS algorithm because it led to a minimal cumulative error on the trajectory of 5.8m only. The results are shown in Fig.

Identification of hydrodynamic coefficients using a zigzag test

The second application used to show the efficiency of the proposed procedure is a zigzag test. In this case, ine an OF based on the Eq.(12). The OF (10) used in the turning circle test, cannot be used in the

present application because in the zigzag test there will be large variations in the heading angle during the dynamic ship motion. As previously, at first the initial reference values of hydrodynamic coefficients, given in Table 5, are used in the developed ship manoeuvring model to evaluate the predicted heading angle. Figure 10 shows the calculated ship heading angle using our manoeuvring model. It is intere

when starting with the initial reference values of hydrodynamic coefficientstotal cumulative error on the ship trajectory was deg17.3=ψ∆ .

Comparison of the heading angle using initial reference hydrodynamic coefficients.

Comparison of the heading angle after optimization by: a) SQP algorithm, b) BFGS algorithm.

is chosen as the total number of experimental sampling points inidentification procedure has been carried out by using SQP and BFGS algorithms with a convergence criterion based on the error on the OF as 041 −E and a maximum number of iterations of

iterations for the SQP algorithm with a final optimal OF relative value ofiterations for the BFGS algorithm with a final optimal relative value of the OF of

summary of the final optimal solutions obtained by both SQP and BFGS algorithms is given in Table

. Summary of final optimal solutions obtained by both SQP and BFGS in zigzag testSQP 15

041 −E

obj 0.365

ψ∆ (deg) 6.6

shows the optimal solution obtained at the end of the optimization processminimal cumulative error on the heading angle is 6.6deg for the SQP algorithm

optimal hydrodynamic coefficients obtained at the end of the optimization e summarized in Table 5. We can observe that both SQP and BFGS algorithms predicted correctly


and only in 10 iterations for

. Summary of final optimal solutions obtained by both SQP and BFGS in turning circle test. BFGS

10 1E-04

0.12

8.0

, we can see that the SQP algorithm is more precise than the BFGS algorithm because it led to The results are shown in Fig.4 (right-a, b).

The second application used to show the efficiency of the proposed procedure is a zigzag test. In this case, ) used in the turning circle test, cannot be used in the

present application because in the zigzag test there will be large variations in the heading angle during the e initial reference values of hydrodynamic coefficients, given

in Table 5, are used in the developed ship manoeuvring model to evaluate the predicted heading angle. Figure 10 shows the calculated ship heading angle using our manoeuvring model. It is interesting to notice

when starting with the initial reference values of hydrodynamic coefficients, the

using initial reference hydrodynamic coefficients.

: a) SQP algorithm, b) BFGS algorithm.

as the total number of experimental sampling points in the Eq.(12). The identification procedure has been carried out by using SQP and BFGS algorithms with a convergence

and a maximum number of iterations of 50. Convergence iterations for the SQP algorithm with a final optimal OF relative value of 0.365 and

iterations for the BFGS algorithm with a final optimal relative value of the OF of 0.389. A GS algorithms is given in Table 4.

. Summary of final optimal solutions obtained by both SQP and BFGS in zigzag test. BFGS

7 041 −E

0.389

7.1

obtained at the end of the optimization process, where Table the SQP algorithm and is7.1deg

optimal hydrodynamic coefficients obtained at the end of the optimization . We can observe that both SQP and BFGS algorithms predicted correctly


the experimental ship heading angle. obtained using the SQP algorithm is more precise than the between the experimental trajectory and the predicted one is very small relatively compared to the length of the ESSO 190,000-dwt oil tanker. From the result of coefficient identification for turning circle and zigzag manoeuvres, we can choose finally a set of identified hydrodynamic coefficients for ESSO for both two tests as well as for other manoeuvring simulations. This set has 35 hydrodynamic coefficients, which includes the mean optimal values of 6 shared most sensitive coefficients, the optimal values of 8 independent most sensitive coefficients of each test, and 21 other origithe gradient of the objective function. Table 5. Reference and Identified hydrodynamic coefficients for the ESSO 190,00

Design Variables

Hydrodynamic coefficients

1 uΧɺ

2 vrΧ

3 vϒɺ

4 c cβ βδϒ

5 Tϒ

6 TΝ

7 rΝɺ

8 v vϒ

9 v rΝ

10 v vΧ

11 uvϒ

12 uvΝ

13 uξΧɺ

14 vξϒɺ

15 urξϒ

16 urξΝ

17 vrξΧ

18 v vξϒ

19 vrξΝ

20 c cδϒ

21 uvξϒ

22 uvξΝ

23 c cβδΧ

24 c cδΝ

25 vvξξΧ

26 c cβ βδϒ

27 c cβ βδΝ


the experimental ship heading angle. The comparison of the obtained heading angle shows that the result obtained using the SQP algorithm is more precise than the BFGS one, since the total cumulative error between the experimental trajectory and the predicted one is very small relatively compared to the length of

dwt oil tanker. From the result of coefficient identification for turning circle and zigzag manoeuvres, we can choose finally a set of identified hydrodynamic coefficients for ESSO 190,000-dwt oil tanker model, which may be used

s for other manoeuvring simulations. This set has 35 hydrodynamic coefficients, which includes the mean optimal values of 6 shared most sensitive coefficients, the optimal values of 8 independent most sensitive coefficients of each test, and 21 other original coefficients that influence weakly

the objective function..

hydrodynamic coefficients for the ESSO 190,00-dwt oil tanker model.

Original values (Reference)

Optimal Values (turning test)

Optimal Values (zigzag test)

-0.0500 - -

1.0200 - -

-0.0200 - -

-2.16 - -

0.0400 - 0.0300

-0.0200 -0.0240 -0.0160

-0.0728 - -0.0878

-2.4000 - -

-0.3000 - -

0.3000 - -

-1.2050 - -

-0.4510 - -

-0.0500 - -

-0.3780 - -

0.1820 0.1598 0.1420

-0.0470 -0.0416 -0.0380

0.3780 - -

-1.5000 - -

-0.1200 - -

0.2080 0.1761 -

0.0000 - -

-0.2410 -0.2823 -0.2910

0.1520 0.1684 -

-0.0980 -0.0805 -0.0800

0.0125 - -

-2.1600 - -

0.6880 - -


comparison of the obtained heading angle shows that the result BFGS one, since the total cumulative error

between the experimental trajectory and the predicted one is very small relatively compared to the length of

From the result of coefficient identification for turning circle and zigzag manoeuvres, we can choose finally dwt oil tanker model, which may be used

s for other manoeuvring simulations. This set has 35 hydrodynamic coefficients, which includes the mean optimal values of 6 shared most sensitive coefficients, the optimal values of 8

nal coefficients that influence weakly

dwt oil tanker model.

Values (zigzag test)

Identified values -0.0500

1.0200

-0.0200

-2.16

0.0300

-0.0200

-0.0878

-2.4000

-0.3000

0.3000

-1.2050

-0.4510

-0.0500

-0.3780

0.1509

-0.0398

0.3780

-1.5000

-0.1200

0.1761

0.0000

-0.2867

0.1684

-0.0803

0.0125

-2.1600

0.6880


28 c cβ βδξϒ

29 c cβ β δ ξΝ

30 urϒ

31 urΝ

32 u uΧ

33 rξΝɺ

34 u uξΧ

35 c cδδΧ

5. Conclusion

A model based on the coupling between ship manoeuvring simulation model and mathematical programming techniques by the use of SQP and BFGS algorithmsusing experimental data of sea trials of the ESSO 190,000zigzag manoeuvres, using 35 hydrodynamic coefficientshydrodynamic parameters were identified to be sensitive in the turning circle and zigrespectively, and that 14 hydrodynamic other manoeuvring simulations. experimental trajectories for the turning test, from an initial error of 69 m (before minimization). In the zigzag test of ship heading, the SQP algorithm gave a final cumulative error 6.6 deg, compared to the starting initial error of 17.3 deg.

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-0.1910 - -

0.3440 - -

0.2480 - -

-0.2070 -0.2105 -

-0.0377 - -0.0457

-0.0045 - -0.0054

-0.0061 -0.0073 -0.0073

-0.0930 -0.1000 -

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from an initial error of 69 m (before minimization). In the zigzag test of ship heading, the SQP algorithm gave a final cumulative error 6.6 deg, compared to the starting initial error of 17.3 deg.

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-0.1910

0.3440

0.2480

-0.2105

-0.0457

-0.0054

-0.0073

-0.1000

based on the coupling between ship manoeuvring simulation model and mathematical The model was validated

del for the turning circle and . The results show that only 10 distinct

were identified to be sensitive in the turning circle and zig-zag manoeuvres may be used for both the turning circle, zig-zag tests and

the SQP algorithm predicted accurately the rajectory of 5.8 m, starting

from an initial error of 69 m (before minimization). In the zigzag test of ship heading, the SQP algorithm gave a final cumulative error 6.6 deg, compared to the starting initial error of 17.3 deg.

Hosseini. H., Sanada. Y., Tanimoto. K., Umeda N. and Stern F.. Estimating maneuvering based, and CFD free-running trial data

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Zhang. J. and Zhang. X.. A Robust SQP Method for Optimization with Inequality Constraints [J], Computational

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