assessment of ship manoeuvrability by using a coupling between a nonlinear transient manoeuvring...

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2013,25(5):788-804 DOI: 10.1016/S1001-6058(13)60426-6

Assessment of ship manoeuvrability by using a coupling between a nonlinear transient manoeuvring model and mathematical programming techniques*

TRAN KHANH Toan, OUAHSINE Abdellatif Laboratoire Roberval, Université de Technologie de Compiègne, UTC-CNRS 7337, CS 60319, 60206 Compiègne cedex, France, E-mail: [email protected] NACEUR Hakim Laboratoire Roberval, Université de Technologie de Compiègne, UTC-CNRS 7337, CS 60319, 60206 Compiègne cedex, France Laboratoire LAMIH, Université de Valenciennes, UMR 8201 CNRS, 59313 Valenciennes, France EL WASSIFI Karima Université Cadi Ayyad, Faculté des Sciences et Techniques, BP 549, Marrakech, Morocco (Received June 17, 2013, Revised August 15, 2013) Abstract: In this paper, a numerical method based on a coupling between a mathematical model of nonlinear transient ship manoeuvring motion in the horizontal plane and Mathematical Programming (MP) techniques is proposed. The aim of the proposed procedure is an efficient estimation of optimal ship hydrodynamic parameters in a dynamic model at the early design stage. The 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 a good agreement of overall tendency in manoeuvring trajectories. Key words: ship manoeuvrability, hydrodynamics coefficients, nonlinear transient dynamics, sensitivity analysis, mathematical programming

Introduction Controlling ship manoeuvrability, especially

when approaching ports in foggy weather or by night time, is a vital concern. Nowadays, it is still tedious to perform real ship manoeuvres in an open sea or to carry out fine simulations using complex 3-D CFD calculations. In spite of their fast calculations, system- based simulations need numerous tests to adjust the manoeuvring 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.

* Biography: TRAN KHANH Toan (1979-), Male, Ph. D. Corresponding author: OUAHSINE Abdellatif, E-mail: [email protected]

Thus, enhancement of ship manoeuvrability in different approaches is introduced more and more in the early design stage of ships and vessels. It is important to understand the ship manoeuvrability in the early design stage. In order to force the vessel motion to follow a prescribed trajectory, accurate prediction of hydrodynamic forces acting on a ship hull and interaction among them, propeller and rudder is essential. A literature survey[1-3] showed that there exist three methods for ship manoeuvrability prediction: the ex- periment-based method, the system-based method and the CFD-based method. A general overview of methods for ship trajectory prediction is given in Fig.1.

Manoeuvring experiments can be accomplished in two different ways, depending on the testing purpose, facilities and equipment, as free sailing or cap- tive model tests. Objectives of manoeuvrability test are: the verification of manoeuvrability to fulfill the

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Fig.1 Overview of methods for ship trajectory prediction International Maritime Organization (IMO) criteria, and the determination of hydrodynamic manoeuvring coefficients. Although this method is effective, unfor- tunately, experimental determination of the hydrodynamic coefficients might be tedious and expensive.

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 Reynolds-Averaged Navier-Stokes (RANS) equations[3,4]. Lately, the CFD–based method has been extended for free-running simulations, including measurable data from experimental fluid dynamics for ship manoeuvres in calm sea and in the presence of waves. In their numerical investigations, Fonfach et al.[5] used CFD with both free surface model and rigid wall free surface as- sumption to study the interaction between a smaller tug and a larger tanker moving in parallel and close to each other at low speed in shallow water. For surge force, sway force and yaw moment in different water depths and Fr , they showed that CFD results give better agreement with experimental fluid dynamics than available Potential Flow (PF) results.

The system–based method is a major simulation task to predict ship manoeuvrability[2]. Computation time is much shorter than that of CFD–based method since such method needs only solve the equations of motion using a prescribed mathematical model and manoeuvring coefficients. System–based method requires approximately a few minutes of computation on a personal computer for one free-running trial while CFD method needs a few hours or even days depending on the turbulence and propulsion modeling and the mesh size.

System–based methods have been extensively in-

vestigated by researchers[6-8]. A simplified mathematical formulation can be obtained by using Taylor’s series expansion method. 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 considered. However, the manoeuvres at high rudder defle- ction 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[6].

Generally the manoeuvring coefficients are sup- posed to be constant. Nonetheless, recently Araki et al.[3] showed a great disparity of manoeuvring coefficients when including waves in which the encounter frequency is low. The complexity of the hydrodynamic processes caused by the wide variety of ship sha- pes, sizes and motions leads to a multitude of mathematical models. Thus, to obtain an optimized trajectory and also a better understanding of ship manoeuvring it is necessary to improve the understanding of the hydrodynamic forces acting on the hull, the rudder and the propeller. Several mathematical models of dynamic ship motion have been proposed in the literature to identify the hydrodynamic parameters[9,10]. Yoon and Rhee[11] investigated a mathematical model based on the Estimation-Before-Modeling (EBM) technique, which is an identification method that esti- mates parameters in a dynamic model. The algorithm was validated using real sea trial data of a 113 K tanker. Viviani et al.[12] carried out a numerical study based on optimization techniques that used a Multi-

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Objective Genetic Algorithm (MOGA). They identified five most sensitive hydrodynamic coefficients from standard manoeuvres (specified by the IMO) for a series of twin-screw ships. Rajesh and Bhattacha- ryya[13] conducted a numerical study based on system identification for a nonlinear manoeuvring model de- dicated for large tankers by using the artificial neural network method. In this model, all nonlinear terms are set together to form one unknown time function per equations which are sought to be represented by the neural network coefficients. Seo and Kim[14], perfor- med a numerical analysis of ship manoeuvring perfor- mance in the presence of incident waves and resultant ship motion responses. To this end, a time domain ship motion program was developed to solve the wave body interaction with the ship slip speed and rotation, and coupled to a modular type 4DOF manoeuvring model. Zhang and Zou[15] analyzed the data of longitu- dinal and transverse velocity, rudder angle, etc. in the simulated zigzag test. This 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 the following 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 3-D 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 affecting the ship trajectory. The interest of app- lying sensitivity analysis is to reduce the number of coefficients to be optimized. Therefore the hydrodynamic model can be easily treated using a constrained mono-objective minimization procedure. In the present investigation, it is found that only 14 coefficients are sensitive for the prediction of the trajectory of the ESSO 199 000-dwt oil tanker benchmark[7]. The obtained results have been validated through turning circle and zigzag manoeuvres based on experimental data of sea trials of the above-cited benchmark. The last step of the proposed procedure concerns the determination of optimal hydrodynamic parameters using MP techniques[16,17]. To date, based on our knowledge, the MP techniques based identification has not been studied in the context of the manoeuvring of large tankers. The present paper makes an attempt to do so for the first time. A summary of the MP based system identification for the identification of optimal hydrodynamic coefficient is given in Table 1.

The structure of the paper is as follows. First, the dynamics of ship motion and hydrodynamic forces in coupled equations (the so-called hydrodynamic

model), are explained in Section 1. Section 2 presents the mathematical programming based system identification for manoeuvring of large tankers. In Section 3 numerical applications are exposed and the obtained results are analyzed and compared to the experimental measurements. Finally, conclusions are drawn in Sec- tion 4 to summarize the present investigation. Table 1 Summary of MP based system identification

Step 1 Step 2 Step 3

Solve with origi-nal coefficients ( ) from ship

motion equations

Select mos timpo- rtant coefficients ( S ), using sen-

sitivity analysis

Find the optimalcoefficients

( opt ), using MP

techniques

Fig.2 Ship motion in 6 DOF

Fig.3 Definition of the coordinate system, speed and direction 1. Formulation of the nonlinear transient hydro-

dynamic model 1.1 Lagrangian based dynamic equation of 2-D ship

manoeuvring motin in the horizontal plane In the manoeuvring theory, we use the coordinate

system Oxyz (see Fig.2), fixed to the rigid body of the

ship. G is the center of mass and Oxz is a plane of symmetry. The space fixed system is 0 0 0 0O x y z as

shown in Fig.3. The ship motion has six degrees of freedom (6 DOF). The Euler angles describing the position of the ship axes are the heading or yaw

and the roll . 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 0 0( , )G Gx y of the ship

center of mass in the global coordinate system

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0 0 0 0O x y z , and the orientation of the ship is obtained

by the heading angle (see Fig.3).

The transient ship manoeuvring motion in the horizontal plane can be described by the translational velocity V and the yaw rate =r of the rotational

motion around the Oz axis (see Fig.3). The instantaneous cartesian components of the velocity V in the mobile coordinate system Oxyz are u in the x -axis

direction and v in the y -axis direction, respectively.

The angle between the direction of velocity V and the ship axis Ox axis is defined as the drift angle .

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[18]

2= ( )GX m u v r x r (1a)

= ( + + )GY m v u r x r (1b)

= + ( + )z GN I r mx v u r (1c)

where m is the mass of the ship, X and Y the external forces, N the external moment, zI the moment

of inertia about the z axis, r the heading/yaw, ( ,Gx

0, )Gz are the coordinates of the center of mass G .

Generally it is more suitable to use the non-dimensional form of the dynamic equations of motion. Dividing the first two equations of (1) by mg and the

last equation by mgL , where g is the gravity accele-

ration and L is the ship length, we obtain the linear equations of ship manoeuvring motion

2 =Gu v r x r g x (2a)

+ + =Gv u r x r gy (2b)

2 2 + ( + ) =z GL k r x v u r gLN (2c)

where

1 1/ 2= ( / )z zk L I m is the non-dimensional radius

of gyration and = /x X mg , = /y Y mg , = /N N

mgL are the non-dimensional forces and moment re-

spectively. 1.2 Formulation of the hydrodynamic forces

A survey of literature review shows that there exist commonly two formulations to give expressions of the hydrodynamics forces[17]. In the present investigation, the hydrodynamic forces are expressed as fun- ctions of the kinematic parameters u , v , r and the rudder angle in the form

= ( , , , , , , )x x u v r u v r (3a)

= ( , , , , , , )y y u v r u v r (3b)

= ( , , , , , , )N N u v r u v r (3c)

By using an expanded Taylor series to the second

order, about the steady state of forward motion, and by maintaining only most influencing physical terms, one can obtain the following expressions of the nonlinear hydrodynamic forces and moment[19]:

2= ( + + + + +u vru u v v c cx uLx u u x vrx v v x c c x

+ (1 ) + + +d uc c u uc c x gL t T u x u u x

2 2+ ) /vr vvvr x v x gL (4a)

= ( + + + + +v uv urv v c cy vLy uvy v v y c c y ur y

+ + + +s T ur uvc cc c y gT Ly ur y uv y

+ + ) /v v v c cv y v v Ly c c y gL (4b)

= ( + + + + +r uv urv r c cN r N uvN v rLN c c N urLN

2+ + + +s T ur rc cc c N gT LN ur LN r L N

3+ + ) /uv vr c cuv N vr LN c c N gL

(4c) where is the ship drift angle, , , , , ,u u uvu ux x y y

, , , ,r uv c cN N N are the non-dimensional de-

rivatives of ship hydrodynamic coefficients, which have to be identified, dt is the thrust deduction coeffi-

cient, = /v u , = /( )s sT h T where sT is the ship

draft, h is the water depth. The non-dimensional propeller thrust T is given by

2 2+ +=

uu un n nu T uLnT nL n TT

gL (5)

where uuT , unT and n nT are the hydrodynamic coeffi-

cients and n is the shaft velocity. The flow velocity c at the rudder is estimated by

2 2 2= +un nnc nuc n c (6)

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where unc and nnc are the hydrodynamic coefficients.

1.3 Numerical integration of the nonlinear transient

equations In order to solve the manoeuvring problem, the

nonlinear Ordinary Differential Equations (ODE) given in Eq.(2) have to be solved using appropriate numerical integration schemes. A stable and accurate integration of nonlinear ODE is of great importance for the solution of the transient equations of motion.

For this purpose, we first define a set of primary variables x describing entirely the manoeuvrability, which can be composed of: the ship velocity components u , v , r , the ship position components 0x , 0y ,

in the global cartesian coordinate system 0 0 0 0O x y z ,

the actual rudder angle and the shaft instantaneous velocity n . This may be written as

T = { }x u v r x y n (7)

The complete manoeuvrability of ship is carried

out by assemblying and solving the set of ODE given

by Eq.(2), the yaw rate ˆ = r and the rates ̂ , n̂ of

rudder angle and the shaft velocity, respectively.

2= + + Gu g x vr x r (8a)

+ =Gv x r gy ur (8b)

2 2 + =z G GL k r x v gLN x ur (8c)

0 = cos sinx u v (8d)

0 = sin + cosy u v (8e)

= r (8f)

= c (8g)

60( )= c

m

n nn

T

(8h)

where mT is the coefficient of propeller, c the com-

manded rudder angle, and cn is the commanded shaft

velocity. Equation (8) can be rewritten in a more sim- ple form, function of the state variables of ship motion, leading to a nonlinear time-varying system.

= ( , )x f t x (9)

with T

0 0= { }x uvrx y n the rate of the state variables

of ship motion and ( , )f t x the nonlinear time-varying

function given by

2

2 2

2 2 2

2 2 2

+ +

( + ) + ( + )

( )

( , ) = cos sin

sin + cos

60 60

G

z G G

z G

G

z G

c

c

m m

rv g x r x

L ru gy k x gLN rux

L k x

g LN yx

L k x

f t x u v

u v

r

n n

T T

(10)

with t denoting the time. Table 2 Procedure used in the implicit Euler integration

scheme

Step 1: Set = 1iter , =itertx x

% solution of the previous time step

Step 2: While iteriter N do Steps 3-7

Step 3: Calculate = ( + , )iter iter itertr x x t f t t x

Step 4: Calculate ( + , )

=iter

iter f t t xk I t

x

Step 5: Solve: =iter iterk x r

Step 6: If TOLiterr

then

Output BREAKiterx % convergence achieved Else

Step 7: Set +1 = +iter iterx x x % update solution

Set = +1 GOTOiter iter Step 3 End if

Step 8: Stop (“method failed after iterN iterations”)

1.3.1 Implicit Euler scheme

A first approach to the discrete approximate solution of Eq.(9), assuming that f is sufficiently differe-

ntiable with respect to t and x , is to express Eq.(9) at time +t t by

+ +( + , ) = 0t t t tx f t t x (11)

Then the time derivative of x at time +t t is approximated using the Euler backward finite difference formula

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++

t t tt t

x xx

t

(12)

It is noted that other more sophisticated numerical schemes may be used[18].

By substituting formula (12) into formula (11), we obtain the final implicit Euler prediction/correction procedure.

+ += + ( + , )t t t t tx x t f t t x (13)

Equation (13) represents a system of nonlinear equations in +t tx , and a Newton-Raphson (NR) procedure

should[20] be used at every time step to find +t tx . By

denoting + += ( + , )t t t t tr x x t f t t x the residual

vector from Eq.(13), the general NR iterative procedure is shown in Table 2.

It is worth noting that the proposed NR iterative procedure given in Table 2 converges in 3 to 4 iterations, which is no time consuming, and that the Euler implicit scheme is unconditionally stable. 1.3.2 Runge-Kutta scheme

The Euler scheme is not so useful because of its low accuracy. It is possible to use one step methods that match the accuracy of the higher-order Taylor series expansions by sequentially computing the function ( , )f t x at several points within the time interval.

One of the most widely used Runge-Kutta (RK) methods is the fourth order method which requires four function evaluations per time step.

1 2 3 4= + ( + + + )6t t t

tx x k k k k

(14a)

1 = ( , )t tk f t x (14b)

2 + 1= + , +2 2t t

t tk f t x k

(14c)

3 + 2= + , +2 2t t

t tk f t x k

(14d)

4 + 3= ( + , + )t tk f t t x t k (14e)

This method is called the 4th order implicit RK method, because all the ik ’s depend on unknown

values of +t tx not yet calculated. Solving for +t tx at

each time step requires an NR iterative procedure for the solution of the set of nonlinear equations. This can be achieved in the same manner as that shown in Table 2.

2. Mathematical programming based system identification

2.1 Statement of the optimization problem

The optimization (or the mathematical programming problem) can be stated as follows[21]: Find T

1 2= { , , , }N ,

which minimizes ( )objF (15)

Subject to the constraints

min maxi i i , = 1, 2, ,i N (16a)

( ) 0jg , = 1, 2, , gj N (16b)

( ) = 0kl , = 1, 2, , lk N (16c)

where is an N -dimensional vector called the design vector which contains the Design Variables (DV), in our case they are represented by the ship hydrodynamic coefficients to be determined, and ( )objF is

called the Objective Function (OF), and ( )ig and

( )kl are known as inequality and equality constrai-

nts, respectively. The number of variables N and the number of constraints gN and/or lN need not be rela-

ted in any way. The problem formulated using the condition (15) and the constraints (16) is called a constrained optimization problem, which can be solved using different algorithms.

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

1/ 2

2

=1

=p

obj ii

F S

(17)

where p is the number of sampling points, 2iS is

the square root of the difference between the computed and the experimental ship positions on the trajectory, which depends on ship hydrodynamic coefficients .

2 num. exp. 2 num. exp. 2= ( ) + ( )i i i i iS x x y y (18) where the superscripts num. and exp. indicate the computed and experimental data respectively, and ( , )i ix y are the coordinates of the point i . For the case

of a zigzag test, the expression of objF follows

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1/ 2

num. exp. 2

=1

= ( )p

obj i ii

F

(19)

where i is the ship’s heading angle, which depends

also on ship hydrodynamic coefficients , p is the

number of sampling points where numerical solution has to be compared to the experimental measurements. The optimization problem stated in the condition (15) and the constraints (16) can be solved using different approaches. One of the most efficient approaches is related to the so-called mathematical programming techniques, where not only information on the OF are necessary but also values of the gradients of the OF (and the constraints) with respect to the vector of design parameters are required to update the solution from one iteration to the other.

Among all optimization algorithms related to the MP techniques, the Sequential Quadratic Progra- mming (SQP) algorithm[17] is a widely applicable in various optimization fields to solve constrained problems in engineering science. The Broyden-Fletcher- Goldfarb-Shanno (BFGS) algorithm[16] is a part of the family of quasi-Newton algorithms, which may be used for the unconstrained optimization problems. 2.2 Normalization and sensitivity analysis

Normalization of DV plays an important role in the convergence of the optimization algorithm and in the quality of the optimal solution. It consists in a linear transformation of the original variables into new transformed variables , which is given by

= +A B (20) where A and B are constant diagonal matrix and vector respectively. Generally, the most frequently used variables normalization uses the lower and upper bounds such as

max min

1diag ( ) = , ,

i i

A

, = 1, ,i N (21a)

min

max min= , ,i

i i

B

, = 1, ,i N (21b)

The normalization of the objective function is easily accomplished by dividing the objective function at each iteration by 0 0( = / )obj obj obj objF F F F , where 0

objF is

the value of the objective function at the first iteration. Caution is to be taken if 0

objF is very low (e.g., 1×10–6)

which may affect the values of the gradients and cause instability of the optimization algorithm. Constraints may be normalized using the same procedure.

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

0

1=obj obj obj

obj

F F FA

F

(22)

As was mentioned before, the optimization pro-

blem involving time-varying hydrodynamic coefficients is highly nonlinear, so the authors chose to pro- ceed with the Finite Difference (FD) technique for the estimation of the gradients /objF

( + ) ( )

2obj obj objF F F

(23)

In the present investigation was fixed to 0.001. 2.3 Brief recall of the BFGS method

Both the BFGS and SQP techniques belong to the Newton-like methods which are based on a quadratic approximation, more exactly in the second- order Taylor approximation of ( )objF x about ( )k .

( ) ( ) T ( ) T ( )1[ + ] + +

2k k k k

obj obj objF F F G (24)

with ( )k

objF the gradient vector and ( )kG the Hessian

tensor of the OF at iteration k . The stationary point of this approximation is a solution of a linear system of equations

( ) ( )=k kG F (25)

It is unique if ( )kG is non-singular and corresponds to

a minimum if ( )kG is positive definite. In the BFGS

method ( )kG are approximated by the symmetric ma-

trices ( )kH , which are updated from iteration to iteration using the most recently obtained information. By considering ( ) ( +1) ( )=k k k

obj objy F F , the BFGS upda-

ting formula is given by[20]

T T+1

T T= + 1+

kk k y H y

H Hy y

T T

T

+k ky H H y

y

(26)

Global convergence has been proved for the BFGS method with inexact line searches, applied to a convex

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Fig.4 Flowchart of the optimization procedure for the identification of hydrodynamic coefficients objective function[16]. The BFGS method with inexact line searches converges linearly if ( )kG is positive definite. The contemporary optimization literature[19] suggests the BFGS method is a preferable choice for general unconstrained optimization based on a line search prototype algorithm. Table 3 Parameters of the ESSO 190 000-dwt oil tanker[7]

Physical and geometrical parameters Value

Length between perpendicular ( )ppL 304.8 m

Beam ( )B 47.17 m

Draft to design waterline ( )T 18.46 m

Displacement ( ) 220 000 m3

/ppL B 6.46

/B T 2.56

Block coefficient ( )BC 0.83

Design speed 0( )U 16 knots

Nominal propeller ( )n 80 rpm

2.4 Brief recall of the SQP method

The SQP algorithm belongs to the so-called constrained optimization methods, which are much more complex to formulate[17]. Many algorithms for their solution are based on transformation of the constrained problem to a sequence of unconstrained optimization subproblems whose solutions converge to the solution of the constrained problem. For instance, for a typical constrained optimization problem given by the condition (15) and the constraints (16), it can be

written in a more general form

min ( )objF

, subject to ( ) 0c (27)

where , objF are already defined and c represents

the nonlinear constraints ( ( )ig , ( )kl in formula

(22)). Using the Lagrangian function ( , ) =La T( ) + ( )objF c one can derive first and higher order

optimality conditions. Table 4 Input data for turning circle and zigzag (ESSO

190 000-dwt oil tanker model)

Input data Turning circle test zigzag test

Initial ship’s position

0 0( , )x y (0, 0) m (0, 0) m

Initial heading angle

0( ) 0o 0o

Initial advance velocity of ship 0( )U

5.3 m/s 7.5 m/s

Initial of rudder angle

0( ) 0o 0o

Maxi rotation velocityof rudder max

2.7 o/s 2.7 o/s

Initial shaft velocity

0( )n 57 rpm 80 rpm

Shaft velocity command ( )cn

57 rpm 80 rpm

Rudder command ( )c

–35o [–20, +20]o

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Table 5 Reference hydrodynamic coefficients of the ESSO 1190 000-dwt oil tanker model

Id. Coefficient Reference Id. Coefficient Reference

1 ux –0.0500 19 vrN –0.1200

2 vrx 1.0200 20 c cy 0.2080

3 vy –0.0200 21 uvy 0

4 c cy –2.1600 22 uvN –0.2410

5 Ty 0.0400 23 c cx 0.1520

6 TN –0.0200 24 c cN –0.0980

7 rN –0.0728 25 vvx 0.0125

8 v vy –2.4000 26 c cy –2.1600

9 v rN –0.3000 27 c cN 0.6880

10 v vx 0.3000 28 c cy –0.1910

11 uvy –1.2050 29 c cN 0.3440

12 uvN –0.4510 30 ury 0.2480

13 ux –0.0500 31 urN –0.2070

14 vy –0.3780 32 u ux –0.0377

15 ury 0.1820 33 rN –0.0045

16 urN –0.0470 34 u ux –0.0061

17 vrx 0.3780 35 c cx –0.0930

18 v vy –1.5000

Fig.5 Sensitivity analysis

We note that all vectors and matrices depend on the optimization variables or the Lagrange multi- pliers or both. For clarity, we suppress this depe- ndence.

T+= = 0obj

La F c

La c

(28)

This expression represents a system of nonlinear

equations, and the Jacobian of this system of equations is called the Karush-Kuhn-Tucker (KKT) matrix of the optimization problem. A Newton step on the optimality conditions is given by

T

= = 00

Tobj

pW c F c

pc c

(29)

where c is the Jacobian of the constraints, =W

+obj i iiF c is the Hessian of the Lagrangian,

and p and p are the updates of and from

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Table 6 Most sensitive coefficients of the ESSO 190 000-dwt oil tanker model

Id. Turning circle test Id. Zigzag test

6 TN 6 TN

15 ury 15 ury

16 urN 16 urN Common

22 uvN 22 uvN Coefficients

24 c cN 24 c cN

34 u ux 34 u ux

20 c cy 5 Ty

23 c cx 7 rN

31 urN 32 u ux

35 c cx 33 rN

current to next iterations. The SQP algorithm proceeds by a block elimination on the KKT system[18], by first solving p , and then substituting to find p

[21].

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

(1) Calculate the ship trajectory with original coefficients from dynamic ship motion equations given by (8).

(2) Filter the most sensitive coefficients S

among all others , based on the sensitivity analysis. (3) Calculate optimal values opt for only sensi-

tive coefficients S , by carrying out the optimization

procedure using SQP or BFGS algorithms. The flowchart of the optimization procedure is

shown in Fig.4.

Fig.6 Ship trajectory using initial reference hydrodynamic coe-

fficients

3. Numerical applications, results and discussion The proposed procedure has been validated

through turning circle and zigzag manoeuvres accordi- ngly to the IMO and based on experimental data of sea trials of the ESSO 190 000-dwt oil tanker[7]. The associated physical and geometrical parameters related to the ship are given in Table 3. Table 7 Summary of final optimal solutions obtained by

both SQP and BFGS in turning circle test

SQP BFGS

Total nb of iterations 21 10

Final error on objF 1×10–4 1×10–4

Final optimal value of objF 0.084 0.12

Total cumulative error on trajectory S (m)

5.8 8.0

Fig.7 Evolution of the OF during the optimization process in

turning circle manoeuvre

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Table 8 Optimal hydrodynamic coefficients in turning circle test

Design variables Hydrodynamic coefficients Initial values Optimal values (SQP) Optimal values (BFGS)

1 TN –0.0200 –0.0240 –0.0207

2 ury 0.1820 0.1598 0.1822

3 urN –0.0470 –0.0416 –0.0533

4 c cy 0.2080 0.1761 0.2052

5 uvN –0.2410 –0.2823 –0.2400

6 c cx 0.1520 0.1684 0.1519

7 c cN –0.0980 –0.0805 –0.0942

8 urN –0.2070 –0.2105 –0.2096

9 u ux –0.0061 –0.0073 –0.0065

10 c cx –0.0930 –0.1000 –0.0936

The associated input data related to the turning

circle test and to the zigzag manoeuvre are presented in Table 4.

In the present application, it is found that 35 hydrodynamic coefficients are necessary to control the manoeuvrability of the ESSO 190 000-dwt oil tanker model[7]. Thus, these coefficients (see Table 5) are used as as initial guess for the numerical processing.

The numerical identification procedure starts from original reference values of all hydrodynamic coefficients ( ) in the dynamic ship motion equations

given in Eq.(8), which will be firstly analyzed through a sensitivity analysis procedure to select the most important parameters.

The sensitivity analysis will show how important is the relative gradient value of objF response for a

small variation of the hydrodynamic coefficient i .

The gradient of objF at coefficient i is computed

using formula (23) and formula (24). This analysis consists of filtering the largest gradient corresponding to each i , which is accomplished using the following

criteria

obj

i

F

(30)

where = 0.1 and / = 1obj iF , is the normalized

gradient value, which corresponds to the largest value of the gradient objF . By using the developed ma-

noeuvring model, the sensitivities are computed and filtered using the above criterion. Variations of gradients are shown in Fig.5 for the turning circle and zigzag manoeuvres. As we can observe from Fig.5, only cer-

tain coefficients are of a great sensitivity (greater than 10% ), and these sensitive hydrodynamic coefficients are identified and summarized in Table 6. We can also see that only 6 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.

Fig.8 Comparison of ship trajectories after optimization

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Fig.9 Comparison of hydrodynamic forces before and after optimization (SQP algorithm) 3.1 Identification of hydrodynamic coefficients using a

turning circle test The proposed procedure has been validated using

a turning circle test. For this purpose, we define an OF given by Eq.(17). This OF is considered to be suffi- cient for the case of a turning circle test, since no big variations arise in the ship dynamic motion.

At first the initial reference values of hydrodynamic coefficients are used and the developed ship manoeuvring model is used to evaluate the predicted trajectory. Figure 6 shows the calculated ship trajectory using our manoeuvring model. It is interesting to notice that before optimization, i.e., when starting with the initial reference values of hydrodynamic coefficients[19], the total cumulative error on the ship trajectory was =S 68 m.

Fig.10 Comparison of the heading angle using initial reference

hydrodynamic coefficients

Table 9 Summary of final optimal solutions obtained by both SQP and BFGS in zigzag test

SQP BFGS

Total nb of iterations 15 7

Final error on objF 1×10–4 1×10–4

Final optimal value of objF 0.365 0.389

Total cumulative error on

trajectory o 6.6 7.1

Fig.11 OF evolution during the optimization process in the

zigzag manoeuvre

We define =p 40 as the total number of experi-

mental sampling points in the Eq.(17). The identification procedure has been carried out by using SQP and

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Table 10 Optimal hydrodynamic coefficients in zigzag tes

Design variables Hydrodynamic coefficients Initial values Optimal values (SQP) Optimal values (BFGS)

1 Ty 0.0400 0.0300 0.0356

2 TN –0.0200 –0.1600 0.0006

3 rN –0.0728 –0.0878 0.0813

4 ury 0.1820 0.1420 0.1783

5 urN –0.0470 –0.0380 –0.0383

6 uvN –0.2410 –0.2910 –0.2434

7 c cN –0.0980 –0.0800 –0.0941

8 uux –0.0377 –0.0457 –0.0395

9 rN –0.0045 –0.0054 –0.0254

10 u ux –0.0061 –0.0073 –0.0104

BFGS algorithms with a convergence criterion based on the error on the OF as 1×10–4 and a maximum number of iterations of 50 . Convergence has been achieved in 21 iterations for the SQP algorithm with a final optimal OF relative value of 0.084 and only in 10 iterations for the BFGS algorithm with a final optimal relative value of the OF of 0.12 . A summary of final optimal solutions obtained by both the SQP and BFGS algorithms is given in Table 7.

Figure 7 shows the evolution of the OF for both the SQP and BFGS algorithms during the optimization process. We can observe clearly the good convergence of both algorithms, which is mainly due to the normalization procedure affected to the DV and to the OF. We can notice also that the BFGS algorithm is faster than the SQP algorithm, which is already predictable since the first algorithm does not take into account the nonlinear constraints in the optimization process.

From Table 7, we can see that the SQP algorithm is more accurat than the BFGS algorithm because it leads to a minimal cumulative error for the trajectory of 5.8 m only. The optimal hydrodynamic coefficients obtained at the end of the optimization process are summarized in Table 8.

Figure 8 shows the optimal solutions obtained at the end of the optimization process. We can observe that both the SQP and BFGS algorithms predict corre- ctly the experimental ship trajectory. A detailed comparison of the obtained trajectories shows that the result obtained using the SQP algorithm is more accurate 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 the ESSO 190 000-dwt oil tanker.

Figure 9 shows the evolutions of the surge and the sway forces as well as the yaw moment before and after optimization using the SQP algorithm. From

Fig.12 OF comparison of the heading angle after optimization

(BFGS algorithms)

Fig.9 it appears that at the beginning of turning circle test on the port side of the ship, absolute values of the hydrodynamic forces are reduced, therefore the gyration radius of the predicted trajectory obtained after optimization is larger than the gyration radius of the initial trajectory. We can explain this phenomenon by comparing the values of the hydrodynamic coefficients before and after using the SQP method directly from Table 7. At first, among all components of surge

801

Fig.13 OF Comparison of hydrodynamic forces before and after optimization (SQP algorithm) forces, c cx appears to be the most important compo-

nent because this component acts on the rudder surface axis when the rudder rotates. c cx increases

from –0.093 to –0.1, which means that the resistance of rudder in x axis is increased, and therefore the ship turns more difficultly towards the port side. We can observe also from Table 7 that c cy is the most

important component which acts on the rudder surface in the y axis. Then c cy is reduced from 0.208 to

0.1761, which means that the ship turns more slowly towards the port side. Finally, we can remark from Table 7 that c cN is the biggest component among all

yaw moments, which is basically caused by rudder forces when the last rotates. Then c cN is reduced

from –0.098 to –0.0805, which indicates that the yaw moment is reducing when the ship turned towards the port side. 3.2 Identification of hydrodynamic coefficients using a

zigzag test The second application used to show the efficie-

ncy of the proposed procedure is a zigzag test. In this case, we define an OF based on the Eq.(19). The OF Eq.(17) 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 in the previous

case, 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 interesting to notice that before optimization, i.e., when starting with the initial reference values of hydrodynamic coefficients[18], the total cumulative error on the ship trajectory is = 17.3o.

We define in this application =p 53 as the total

number of experimental sampling points in the Eq.(19). The identification procedure has been carried out by using the SQP and BFGS algorithms with a convergence criterion based on the error on the OF as 1×10–4 and a maximum number of iterations of 50. Convergence has been achieved in 15 iterations for the SQP algorithm with a final optimal OF relative value of 0.365 and only in 7 iterations for the BFGS algorithm with a final optimal relative value of the OF of 0.389. A summary of the final optimal solutions obtained by both the SQP and BFGS algorithms is given in Table 9.

Figure 11 shows the evolution of the OF for both the SQP and BFGS algorithms during the optimization process.

We can observe clearly the good convergence of both algorithms, which is mainly due to the normalization procedure affected to the DV and to the OF. We can also notice that the BFGS algorithm is faster than the SQP algorithm, which is already predictable

802

Table 11 Identified hydrodynamic coefficients for the ESSO 19 000-dwt oil tanker model

Design variables Hydrodynamic coefficients

Original values Optimal values (turning test)

Optimal values (zigzag test)

Identified values

1 ux –0.0500 - - –0.0500

2 vrx 1.0200 - - 1.0200

3 vy –0.0200 - - –0.0200

4 c cy –2.1600 - - –2.1600

5 Ty 0.0400 - 0.0300 0.0300

6 TN –0.0200 –0.0240 –0.0160 –0.0200

7 rN –0.0728 - –0.0878 –0.0878

8 v vy –2.4000 - - –2.4000

9 v rN –0.3000 - - –0.3000

10 v vx 0.3000 - - 0.3000

11 uvy –1.2050 - - –1.2050

12 uvN –0.4510 - - –0.4510

13 ux –0.0500 - - –0.0500

14 vy –0.3780 - - –0.3780

15 ury 0.1820 0.1598 0.1420 0.1509

16 urN –0.0470 –0.0416 –0.0380 –0.0398

17 vrx 0.3780 - - 0.3780

18 v vy –1.5000 - - –1.5000

19 vrN –0.1200 - - –0.1200

20 c cy 0.2080 0.1761 - 0.1761

21 uvy 0 - - 0.0000

22 uvN –0.2410 –0.2823 –0.2910 –0.2867

23 c cx 0.1520 0.1684 - 0.1684

24 c cN –0.0980 –0.0805 –0.0800 –0.0803

25 vvx 0.0125 - - 0.0125

26 c cy –2.1600 - - –2.1600

27 c cN 0.6880 - - 0.6880

28 c cy –0.1910 - - –0.1910

29 c cN 0.3440 - - 0.3440

30 ury 0.2480 - - 0.2480

31 urN –0.2070 –0.2105 - –0.2105

32 u ux –0.0377 - –0.0457 –0.0457

33 rN –0.0045 - –0.0054 –0.0054

34 u ux –0.0061 –0.0073 –0.0073 –0.0073

35 c cx –0.0930 –0.1000 - –0.1000

803

since the first algorithm does not take into account the nonlinear constraints in the optimization process.

From Table 9, we can see that the SQP algorithm is more precise than the BFGS algorithm because it led to a minimal cumulative error on the heading angle of 6.6o only. The optimal hydrodynamic coefficients obtained at the end of the optimization process are summarized in Table 10.

Figure 12 shows the optimal solutions obtained at the end of the optimization process. We can observe that both the SQP and BFGS algorithms predict corre- ctly the experimental ship heading angle. A detailed comparison of the obtained heading angle shows that the result obtained using the SQP algorithm is more accurate 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 the ESSO 190 000-dwt oil tanker. Figure 13 shows the evolutions of the surge and the sway forces as well as the yaw moment before and after optimization using the SQP algorithm.

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 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 original coefficients that influe- nce weakly the gradient of OF. For more details, we summarize all of coefficients before and after optimization in Table 10. 4. Conclusions

This investigation proposes, for the first time, a mathematical programming based on system identification for manoeuvring of large tankers that are gove- rned by a well-established set of nonlinear equations of motion. The identification procedure is based on the coupling between ship manoeuvring simulation model and mathematical programming techniques by the use of the SQP and BFGS algorithms.

In the implemented manoeuvring model, the propulsion hydrodynamics, rudder hydrodynamics and manoeuvring hydrodynamics are strongly coupled. The model has been validated using experimental data of sea trials of the ESSO 190 000-dwt oil tanker model for the turning circle and zigzag manoeuvres, using 35 hydrodynamic coefficients. In the turning circle test, it is found that the SQP algorithm predicted accurately the experimental trajectories, with a total cumulative error on trajectory of 5.8 m, starting from an initial error of 69 m (before minimization). In the zigzag test of ship heading, the SQP algorithm gives a

final cumulative error 6.6o, compared to the starting initial error of 17.3o. In the proposed mathematical programming based system identification, it is found that only 10 distinct hydrodynamic derivatives are identified to be sensitive in the turning circle and zig-zag manoeuvres respectively. With these known parameters, the proposed system identification technique ob- viates the need to know 25 other parameters. Finally, through the application of the ESSO 190 000-dwt oil tanker, we show that it is possible to use a combina- tion of 14 hydrodynamic derivatives from both turning circle and zigzag manoeuvres, which may be used for both the turning circle, zig-zag tests and other manoeuvring simulations. Further developments of the presented procedure by introducing meta-modelling techniques based on response surfaces models and design of experiments are undertaken by the authors in order to assess the robustness and efficiency for a wider range of manoeuvring simulations. Acknowledgements

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