System identification for nonlinear maneuvering of large tankers using artificial neural network

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Applied Ocean Research 30 (2008) 2 5 6 - 2 6 3

ELSEVIER

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Applied Ocean Research

journal homepage: www.elsevier.com/locate/apor

System identification for nonlinear maneuvering of large tankers using artificial

neural network

G. Rajesh, S.K. Bhattacharyya *

Department of Ocean Engineering, Indian Institute of Technology Madras, aennai-600036, India

A R T I C L E I N F O Article history: Received 8 July 2008 Received in revised f o r m 20 October 2008 Accepted 23 October 2008 Available online 1 January 2009

Keywords: H y d r o d y n a m i c derivatives Maneuvering Neural n e t w o r k Neurons Nonlinear Ship Simulation System i d e n t i f i c a t i o n Tanker A B S T R A C T T h i s p a p e r d e a l s w i t h t h e a p p l i c a t i o n o f n o n p a r a m e t r i c s y s t e m i d e n t i f i c a t i o n t o a n o n l i n e a r m a n e u v e r i n g m o d e l f o r l a r g e t a n k e r s u s i n g a r t i f i c i a l n e u r a l n e t w o r k m e t h o d . T h e t h r e e c o u p l e d m a n e u v e r i n g e q u a t i o n s i n t h i s m o d e l f o r l a r g e t a n k e r s c o n t a i n l i n e a r a n d n o n l i n e a r t e r m s a n d i n s t e a d o f a t t e m p t i n g t o d e t e r m i n e t h e p a r a m e t e r s (i.e. h y d r o d y n a m i c d e r i v a t i v e s ) a s s o c i a t e d w i t h n o n l i n e a r t e r m s , a l l n o n l i n e a r t e r m s are c l u b b e d t o g e t h e r t o f o r m o n e u n k n o w n t i m e f u n c t i o n p e r e q u a t i o n w h i c h are s o u g h t t o b e r e p r e s e n t e d b y t h e n e u r a l n e t w o r k c o e f f i c i e n t s . T h e t i m e s e r i e s u s e d i n t r a i n i n g t h e n e t w o r k are o b t a i n e d f r o m s i m u l a t e d d a t a o f z i g z a g m a n e u v e r s a n d t h e p r o p o s e d m e t h o d has b e e n a p p l i e d t o t h e s e d a t a . T h e n e u r a l n e t w o r k s c h e m e a d o p t e d i n t h i s w o r k has o n e m i d d l e o r h i d d e n l a y e r o f n e u r o n s a n d i t e m p l o y s t h e L e v e n b e r g - M a r q u a r d t a l g o r i t h m . U s i n g t h e b e s t c h o i c e s f o r t h e n u m b e r o f h i d d e n l a y e r n e u r o n s , l e n g t h o f t r a i n i n g d a t a , c o n v e r g e n c e t o l e r a n c e etc., t h e p e r f o r m a n c e o f t h e p r o p o s e d n e u r a l n e t w o r k m o d e l has b e e n i n v e s t i g a t e d a n d c o n c l u s i o n s d r a w n . © 2 0 0 8 E l s e v i e r L t d . A l l r i g h t s r e s e n / e d . 1. Introduction

System identification can be defined as a systematic approach to find a model of an unknown system from the given input-output data. For system identification to be successful, three items should be properly selected or designed; mathematical model of the system, input-output data and parameter estimation scheme. The area of ship maneuvenng has seen extensive application of a variety of system idenrification methods. Some of the established system identification methods in this area are indirect model reference adaptive systems [1], continuous least square estimation [2], recursive least square estimadon [3,4], recursive maximum likelihood estimation [5], recursive predicdon error technique [6], extended Kalman filter approach [7]. In recent times, various approaches and techniques of system identification that have been used in the area of ship hydrodynamics are Markov process theories, statistical linearization techniques [8] and reverse multiple input-single output methods [9,10). Recently, the neural network based identification has drawn attention in ship maneuvering [11-13]. The mathematical model of the neural network is so called because it mimics the learning process of

* Corresponding author. Tel.: +91 44 22574803: fax: +91 44 22574802. £-mai7 address: skbh<s)iitm.ac.in (S.K. Bhattacharyya).

the human brain and does not use a physical model. Because of this, it should be more robust than the classical physical model based identification techniques, especially when the physical models are complex and semi-empirical in nature. Neural network based system identification models, developed in this paper, are shown to provide an attractive alternative to the identification methods relying upon physics based mathematical models of ship maneuvering. The input-output data required for this neural network based identification method can be directly obtained either f r o m free running model tests or full-scale maneuvering trials so that the method is accurate enough for all practical simulation work.

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forces and moments acting on tlie tanker that appear in its ma-neuvering equations, (iii) shallow water effect on mama-neuvering. To date, neural network based identification has not been studied in the context of the maneuvering of large tankers. The present pa-per makes an attempt to do so for the first time.

2. Nonlinear equations of motion

The nondimensional surge, sway and yaw equations of large tankers [15,16] are considered in this paper. These are:

u-vr=gX (1) V + ur = gY (2) (Lk.fr + Lxcur = gLN (3)

where

gX = Xi,ü + L-'XuuU^+X,rVr + L-'X,y + L-'XcM,McS'

+ r'x,,y^'+gT{-i-i) (4) gY = YiV + L-'Y,,uv + L-'Y,\,^v\v\+r'Y^,^cs\c\cS + Y,,ur

+ L - ' y | c | c | / j | / i | s | | c | c | / ] | / J | 5 | + y , r f U r f + L - ' y , „ f U i ; r

+ i"'y|c|c|/.|/im<|c|c|/J|/J \S\ tj + Y.^hij + YjgT (5) glN = l\Ntr + Nt^rt;)+Nu„m+lN\,^r\v\r + N^c\c!.\c\c&

+ lN,rUr + N^c\cmm\^\c\P\P\&\

+ L N | „ , r f | H ' t +NMc\ms\(\c\c\P\Pm+LNTgT (6) gT = L ^Tuuu'^+ Tu„un + LTi„,„\n\n

d \nln c„„un + c„„n^ h-d v/u. (7) (8) In the above, u and v are the velocities along X (towards forward) and Y axis (towards starboard) respectively, r is the yaw rate ( = i/f, where f is the yaw angle in the horizontal plane), an overdot denotes time (t) derivative, L is length of the ship, d is the draft of the ship, l<z is the nondimensional radius of gyration of the ship i n yaw, m is the mass of the ship, 1^ is its mass moment of inertia about Z axis (vertically downward w i t h axis

origin at free surface), XQ is the nondimensional X coordinate of

ship's centre of gravity {Y coordinate of ship's centre of gravity yc is taken as zero), g is acceleration due to gravity, X, Y and N are the nondimensional surge force, sway force and yaw moment respectively, S is the rudder angle, c is the f l o w velocity past rudder, f is the water depth parameter, Cu„ and c„„ are constants, T is the propeller thrust, h is the water depth, f is the thrust deduction factor and n is the r p m of the propeller shaft. All other quantities are constant hydrodynamic derivatives. All quantities in the above equations are nondimensional and may be related to their dimensional counterparts (denoted by an overbar) using BIS system given by [15] according to

(u, V) = (Ü, V) / / L ^ r = f / ^ L (u, v) = (u, v)/g r = f / t e / L ) (9) (xG,yc) = ixG,yc)/L co = cb/^/gJl m = m / ( p V ) ; /, = / . / ( p V L ^ ) .

In the above equations w is the nondimensional circular frequency, p is the sea water density and V is the volumetric displacement of the hull. The system of three equations, as represented by (1), contains 10 hydrodynamic derivatives in the X-equation (surge) and 12 in both y-(sway) and N-(yaw) equations, a total of 34 hydrodynamic derivatives.

Now, substituting (4) in (1), we get

(1 - X , i - X i f f )ü = g,(i(, V, r, T, f , c, (10)

Similarly, substituting (5) in (2) and (6) in (3), we get

{\ - Yi - Yi,^t;)h = g2{,u,v,rj,tj,c,8) (11) Qi] - N r - Nf^Or = g3(u, V, r, T, f , c, S) (12) where

g, = r H x „ „ + X „ „ ^ f ) i ' ' + ( l + X „ r + X „ r f O i ^ r + L-\X„, + X , , „ f + ri(X|c|c,s,5|c|c52)

+ L-\Xicicps\c\cp8)+gT0-i) (13) g2 = L-'YuuUv + L-'Y\,iMv + L-^Y^,^a\c\cS + (Yur--^)ur

+ L-V|„|„j|u|uf + r'y|,|,|;,|;,|,|^|c|c|/J|/515| ? + YjgT (14) g3 = L-^{N,,uv+LNi,ir\v\r + NMa\c\cS + L{Nur-xc)ur

-t-N|c|c|/j;/i|i||c|c \p\p \S\ + INur^url; + N„,^uvi;

+ L N | „ | , ^ | i ; | r f + N | c | c | / i | / i | S | d c | c | / ? | / S | 5 | f + L N r g T } . (15) 3. Model for neural network

From the maneuvering equations given by (10)-(15), i t may be observed that the inertia terms contain linear hydrodynamic derivatives (one in each equation) and their corrections (one in each equation) and the functions gi,g2 and g->, contain only nonlinear hydrodynamic derivatives. System identification requires knowledge of at least some of these derivatives and in the present model it is obviously acceleration derivatives (first order) which are relatively easy to estimate. Therefore in this w o r k we have chosen a model for system identification where the three linear hydrodynamic derivatives and their water depth dependent corrections alone are assumed known as given in (10)-(12). Thus, this model requires knowledge of three acceleration derivatives Xu, 7;, and Nf and their corrections due to water depth, i.e. Xi,f, Yi^l;

and N f f , a total of six constants. 4. Neural network formulation

The unknown nonlinear functions g i , gj and g3 are simply the sum of all nonlinear terms in(13)-(15) and hence to be determined by a neural network model. A three layer neural network model is used i n the present work to represent the unknown functions g^, g2 and g3 as shown in Fig. 1. The input layer has time functions surge velocity u(t), sway velocity v{t), yaw velocity r ( f ) , rudder angle i5(f) and a bias w i t h value of unity, i.e. a total of five neurons. The middle layer has m neurons where the value of m has to be found by numerical trials. The output layer consists of the functions gi, g2 andg3. Denoting

Xi = 1, X2 = u ( f ) , X3 = u ( f ) , X4 = r ( t ) and

X 5 = 5 ( f ) (16) we relate the input and middle layer neurons as

Z i = ' ^ W i j X j (! = l , . . . , m - 1 )

and then transform them using a squashing function as ( l + e - " ' ) " ' , i = l , .

1, i = m.

m - 1

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258 G. Rajesh, S.K. Bhattacharyya /AppUcd Ocean Research 30 (2008) 256-263

gl g2 g3

X l = 1 X2 = U X 3 = V X4 = r X 5 = 8

Fig. 1. Neural n e t w o r k model.

Finally, the output neurons are obtained as

gk I^B^ia,- (fc = l , 2 , 3 ) . (19)

In the above, wy are the weight functions relating the input layer and the middle (hidden) layer and BM are the weight functions relating the middle and output layers.

Suppose the training time series data of u, v, r, ii, u, r and S are known either from experiments or f r o m simulation, then one can obtain the time series of the functions g i , g 2 and gs f r o m (10)-(12). Representing the discrete time series at the time instances t„{n = 1, 2 , . . . , N) these functions may be denoted as g^l^ (k = 1,2,3; n = 1, 2 , N ) where the superscript (1) indicates that these are target functions. Now using the training time series data and assuming the trial values of and B^i one can obtain the functions gx,g2,g3 by successively using (16) to (19). These functions are the output functions of the neural network as shown in Fig. 1 and denoted g^JJ" (k = 1, 2, 3; n = 1, 2 , . . . , N) where the superscript (N) indicates that these are output functions f r o m the neural network. Denoting the weights (including biases) of the network by vector W one can define the error function as

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k = l n=l

The Levenberg-Marquardt (LM) backpropagation algorithm, available as a TRAINLM function in Matlab, has been used to

minimize the error function in (20). The LM algorithm [17, 18] is a popular and successful backpropagation algorithm for neural network training and had been investigated by many researchers [19,20]. The Matlab manual may be consulted for implementation details and syntax. In order to start the algorithm, one has to specify the trial values of wu and Bw, and this has been done using a random number generator available in Matlab as function RAND which generates random numbers between - 1 to + 1 . The TRAINLM function then updates the weight and bias values according to the Levenberg-Marquardt optimization. Several control parameters are required to run this optimization and the typical values of these parameters that have been used in most calculations are given below so that similar results as obtained in this paper are reproducible:

(a) Maximum number of epochs (i.e. iterations), "net.train-Param.epochs" (30000)

(b) Performance goal, "net.trainParam.goal" (10 '°)

(c) Maximum validation failures, "net.trainParam.maxJail" (5) (d) Factor to use for memory/speed trade off, "net.trainParam. mem_reduc" (1)

(e) M i n i m u m performance gradient, "net.trainParam.min_grad" ( 1 0 - " )

(f) Initial learning rate, "net.trainParam.mu" (0.001) (g) M u decrease factor, "net.trainParam.mu_dec" (0.1) (h) Mu increase factor, "net.trainParam.muJnc" (10) (i) Maximum Mu, "net.trainParam.mu.max" (lO"^")

(j) Maximum time to train in seconds, "net.trainParam.time" (set usually to inf).

Upon training, the best estimates of connection-weights tuy and Bfei are stored for future use in simulation. The estimates of the functions g , , g2 and gs are substituted in (10)-( 12) to get the surge, sway and yaw accelerations i.e., u, v and r respectively and then these equations are integrated numerically using a fourth order Runge-Kutta technique to obtain the corresponding estimates of the average values for the surge, sway and yaw velocities. This second phase of the network operation is called testing and identification of the model.

5. Numerical example, results and discussion

5.J. Trainmgdata

The system identification of the maneuvering equations using the neural network approach has been applied to Esso 190 000 dwt tanker [14-16]. The principal particulars of the ship are: length between perpendiculars (L) = 304.8 m, breadth (B) = 47.17 m,

2000 1000 -1000 -2000 -3000 1000 2000 3000 4000 5000 6000 7000 X ( m ) 1500 t ( s )

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C. Rajesfi, 5.K B/iattachao'ya//lpp/ied Ocean Research 30 (2008) 256-263 259

Table 1

Nondimensional hydrodynamic derivatives and propulsion system constants for 190 000 d w t tanker.

Surge h y d r o d y n a m i c coefficients Reference values Sway h y d r o d y n a m i c coefficients Reference values

1 - X i 1.050 1 - Y i , 2.020 - 0 . 0 3 7 7 - 0 . 7 5 2 2.020 YN» - 1 . 2 0 5 x.„ 0.300 - 2 . 4 0 0 Xc/c/lS - 0 . 0 9 3 0.208 0.152 y/c/c/fi/n/s/ - 2 . 1 6 Xiif - 0 . 0 5 Vr 0.04 Xuuf - 0 . 0 0 6 1 - 0 . 3 8 7 0.387 0.182 X i , „ f f 0.0125 Yiiuf O f o r f < 0.8 a n d - 0 . 8 5 ( 1 - 0 . 8 / f ) f o r i ; > 0.8 Yu/u/t - 1 . 5 0 - 0 . 1 9 1 Yaw h y d r o d y n a m i c coefficients Reference values Thrust coefficients Reference values

( f c z ) ' - N ; 0.1232 T„„ - 0 . 0 0 6 9 5

- 0 . 2 3 1 Tun - 0 . 0 0 0 6 3 0

- 0 . 4 5 1 T/n/n 0.0000354

- 0 . 3 0 0 f 0.22

- 0 . 0 9 8 Flow velocity Coefficients Reference Values

^/c/c/fl/p/SJ 0.688 Cun 0.605 NT - 0 . 0 2 Cnn 38.2 - 0 . 0 0 4 5 - 0 . 0 4 7 Nuv{ - 0 . 2 4 1 - 0 . 1 2 0 0.344

design draft (T) = 18.46 m, displacement ( V ) = 220 000 m ^ block coefficient (Cg) = 0.83, design speed (UQ) = 16 kn, nominal propeller speed (n) = 80 rpm and rudder rate (5) = 2.33 deg/s. The values of hydrodynamic coefficients and other data are shown in Table 1. Using these data, surge, sway and yaw equations are solved using Matlab ODE45 routine and these solutions have been used to train the network. In what follows, the results of simulation of (1 )-(6) directly by Matlab ODE45 routine are denoted as 'reference' results w i t h which the results of simulation by neural network model are compared. In all results of this work, deep water condition ( f = 0) has been assumed.

For training the neural network, a zigzag maneuver simulation w i t h 20° rudder angle and a heading angle (i/') limit of 20° w i t h approach speed of 16 knots (denoted z z - 2 0 ° - 2 0 ° - 1 6 kn) is carried out. The results of this maneuver have been used as the data to train the neural network. The trajectory, yaw angle and rudder angle data of this maneuver are shown in Fig. 2.

5.2. Determination of training parameters 5.2.1. Number of neurons in the hidden layer

Number of neurons in the hidden layer is denoted b y ' m ' . It is known f r o m the neural network literature that m should be in an opUmum range. The value of m higher than this optimum range causes over specification of input layer-hidden layer relations and this leads to over fitting of the model. Training of the network using z z - 2 0 ° - 2 0 ° - 1 6 kn was conducted for model (see (10)-(12)) using m = 11, 15, 16 and 26 and in all trainings, a sample length (or training time series length) of 1500 s was used. W i t h the trained weights { W i j and B^,-; / = 1 to m, j = 1 to 5, k = 1 to 3), a zigzag maneuver z z - 1 5 ° - 1 5 ° - 1 6 kn (which is different from that used for training the network) was simulated and compared w i t h its reference maneuvers for m = 11,15,16 and 26. For the trajectory alone, these comparisons are shown in Fig. 3. The number of epochs or iterations (Ng) and convergence tolerance (tol) achieved in these comparisons are as follows: (a) m = 11, tol = 3.77 x lO"^", NE = 17077; ( b ) m = 15, tol = 7.5276 x 1 0 " ' ° , NE = 30000; (c) m = 16, tol = 3.613 x 1 0 " " , NE = 30000; (d) m = 26, tol = 1 X 1 0 " " , NE = 1789. From these results, the following

500

xcoordinate (m)

Fig. 3. Comparison of z z - 1 5 ° - 1 5 ° - 1 6 kn trajectory for d i f f e r e n t values of m using coefficients trained w i t h z z - 2 0 ° - 2 0 ° - 1 6 kn.

conclusions are drawn: (a) m = 15 gives reasonably accurate comparison than m = 16; (b) m = 11 performs badly and (c)m = 26 performs worst. Based on these numerical experiments, the number of neurons in the hidden layer is chosen as 15 ( = m). A typical set of neural network coefficients is given in Table 2. 5.2.2. Length of training data

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260 G. Rajesh, S.K. Bhattacharyya/Applied Ocean Research 30 (2008) 256-263

Table 2

Neural n e t w o r k coefficients trained w i t h z z - 2 0 ° - 2 0 ° - 1 6 kn. 1 J = l J = 2 J = 3 j = 4 J = 5 1 - 3 . 5 6 8 9 6 2 - 4 . 0 0 5 3 4 5 - 4 . 0 1 7 6 9 9 0.2009315 1.182008 2 3.871750 3.661926 3.379155 - 0 . 1 6 5 6 1 2 1 - 1 . 2 1 5 2 8 9 3 3.659958 3.974254 3.862647 - 0 . 1 9 0 5 6 8 3 - 1 . 1 9 0 2 1 5 4 - 0 . 6 3 2 8 7 5 3 - 6 . 2 3 4 6 9 5 2.215056 0.1255864 0.7215229 5 - 3 . 7 8 6 1 1 5 - 3 . 8 3 6 9 1 5 - 3 . 6 0 1 6 3 9 0.1759966 1.203865 6 - 0 . 1 . 2 2 7 3 0 0 4.008568 - 3 . 8 3 0 1 4 5 - 0 . 3 6 1 4 8 2 9 0.6761313 7 - 2 . 4 4 7 5 4 9 - 9 1 . 8 8 0 0 7 19.64835 4.925855 0.6008219 8 13.94800 - 4 7 . 2 7 7 4 0 1.485011 - 0 . 9 0 8 8 7 1 3 1.500057 9 - 0 . 7 4 5 4 3 1 0 - 4 4 . 6 3 6 5 4 - 1 . 2 6 0 4 8 5 0.5852645 0.3.067825 10 89.64103 - 1 3 6 . 6 1 2 2 25.93242 - 1 3 . 1 1 6 2 7 - 1 0 . 0 5 3 6 2 11 42.05385 142.1548 13.70088 4.052687 - 2 . 8 4 8 3 9 7 12 - 3 1 . 5 3 8 0 8 - 1 5 9 . 9 1 7 9 - 3 6 . 1 3 6 6 8 - 4 . 8 4 0 9 5 0 1.952682 13 - 2 7 . 8 7 4 5 1 - 1 2 7 . 3 0 5 6 49.91106 14.46869 - 1 . 5 0 8 7 1 5 14 37.04911 - 5 4 . 1 2 6 9 0 27.78480 7.815652 0.5351363 Bki k= 1 fc = 2 k = 3 1 26.992349 - 1 8 . 0 0 9 6 3 0 23.188122 2 21.030902 - 9 . 4 2 6 3 2 0 21.292630 3 60.729661 - 3 7 . 5 5 6 6 5 0 54.689902 4 0.035510 0.027862 - 0 . 0 0 8 0 4 3 5 54.760059 - 2 8 . 9 7 4 2 8 1 52.767782 6 0.022237 - 0 . 0 0 2 3 6 2 0.000548 7 0.000025 0.000607 - 0 . 0 0 0 2 7 7 8 - 0 . 0 0 0 8 1 0 0.012706 - 0 . 0 0 3 9 5 6 9 0.000237 - 0 . 0 0 5 7 9 6 0.001517 10 - 0 . 0 0 0 0 1 4 - 0 . 0 0 0 1 9 5 0.000069 11 - 0 . 0 0 0 0 2 0 - 0 . 0 0 0 9 6 2 0.000269 12 - 0 . 0 0 0 0 6 1 - 0 . 0 0 1 7 3 3 0.000561 13 0.000004 - 0 . 0 0 0 1 9 0 0.000032 14 0.000072 0.000919 - 0 . 0 0 0 2 7 6 15 - 0 . 0 2 5 9 8 3 - 0 . 0 2 1 5 7 9 0.030397

the reference curves for t > 1200 s, whereas simulations w i t h neural network coefficients using Ti = 1200 s leads to divergence f r o m the reference curves for f > 1400 s. However, simulation w i t h neural network coefficients using Ti = 1500 s yield a good comparison up to f = 2000 s, a simulation time length much greater than the training data length and this simulation dme length consists of 5 cycles in zigzag maneuver trajectory, which is more than the usual 3 to 4 cycles for which practical zigzag maneuvers are carried out.

Based on the above, a 1500 s length of training data (equivalent to 4 zigzag cycles consisting of 7 rudder executes) has been found to be adequate and used in all simulations reported here.

5.3. Ctioice of maneuver for training of neural network

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G. Rajesh, S.K. Bhattacharyya / Applied Ocean Research 30 (2008) 256-263 ₂₆₁ 500 1000 1500 2000 2500 3000 3500 4000 x( m ) ( a ) z z - 2 5 ° - 2 5 ° - 1 6 k n . 1000 (b) z z - 1 5 ° - 1 5 ° - 1 6 kn. 2000 3000 x( m ) 4000 5000 6 0 0 0 2000 1000 -1000 -2000 -3000 t 1000 2000 3000 4000 5000 GOOO 7000 X (m) ( c ) z z - 1 0 ° - 1 0 ° - 1 6 k n . 500 1000 x ( m ) 1500 ( d ) t c - 1 5 ° - 1 6 k n . -200 -400 Ê -600 -800 -1000 -1200 0 200 4 0 0 600 8 0 0 ' 1000 1200 1400 0 200 400 600 800 1000 1200 x ( m ) x ( m ) ( e ) t c - 2 0 ° - 1 6 k n . ( 0 t c - 2 5 ° - 1 6 k n .

Fig. 5. Comparison of trajectory w i t h reference trajectory w i t h m = 15 using z z - 2 0 ° - 2 0 ° - 1 6 k n trained coefficients.

angle) in Fig. 5(a), (ii) z z - 1 5 ° - 1 5 ° - 1 6 kn (high speed w i t h moderate rudder angle) in Fig. 5(b), (iii) z z - 1 0 ° - 1 0 ° - 1 6 kn (high speed w i t h low rudder angle) in Fig. 5(c), (iv) tc-15°-16 kn (turning circle maneuver w i t h 15° rudder angle and 16 knot speed, high speed w i t h moderate rudder angle) in Fig. 5(d), (v) tc-20°-16 kn (high speed w i t h moderate rudder angle turning circle maneuver) in Fig. 5(e) and (vi) tc-25°-16 kn (high speed w i t h high rudder angle turning circle maneuver) in Fig. 5(f). The comparisons of trajectories only are shown in the figures. From Figs. 5(a) to 5(c)

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262 C. Rajesh, S.K. Bhattachaiyya /Apphed Ocean Research 30 (2008) 256-263 1000 500 -500 -1000 - 1 5 0 0 L 500 1000 1500 2000 2500 3000 3500 4 0 0 0 500 1000 1500 2000 2500 3000 3500 X (m) X (m) (a) z z - 2 0 ° - 2 0 ° - 1 0 k n . (b) z z - 2 0 ° - 2 0 ° - 5 kn. 0 -100 -200 -300 -400 ^ -500 -600 -700 -800 0 200 400 600 800 1000 -300 -200 -100 0 100 200 300 400 500 600 x (m) X (m) ( c ) t c - 2 0 ° - 1 0 k n . ( d ) t c - 2 0 ° - 5 k n .

Fig. 6. Comparison of trajectory w i t h reference trajectory at d i f f e r e n t speeds w i t h m = 15 using z z - 2 0 ° - 2 0 ° - 1 6 k n trained coefficients.

(Fig. 5(f)), the comparison Qudged on the basis of maximum deviation f r o m the path) is relatively poor.

Now in order to study the prediction characteristics for low speed maneuvers using coefficients trained at high speed i.e. z z - ö - 0 - U o where Uo = 16 knots, various simulations have been carried out and compared w i t h the reference simulations and only a few of the significant results are presented. These are (i) z z - 2 0 ° - 2 0 ° - 1 0 kn (moderate speed w i t h high rudder angle) in Fig. 6(a) (ii) z z - 2 0 ° - 2 0 ° - 5 kn (low speed w i t h high rudder angle) in Fig. 6(b) (iii) tc-20°-10 kn (moderate speed w i t h high rudder angle turning circle maneuver) in Fig. 6(c) (vi) tc-20°-5 kn (low speed w i t h high rudder angle turning circle maneuver) in Fig. 6(d).

From the above comparisons it is clearly evident that the z z - 2 0 ° - 2 0 ° - 1 6 kn trained network gives a good prediction of z z - 2 0 ° - 2 0 ° - 1 0 kn (Fig. 6(a)) and z z - 2 0 ° - 2 0 ° - 5 kn (Fig. 6(b)). In other words, the quality of zigzag maneuver simulation does not degrade w i t h speed and this certainly is reassuring. Now f r o m the comparisons of trajectories of tc-20°-16 kn (Fig. 5(e)), tc-20°-10 kn (Fig. 6(c)) and tc-20°-5 kn (Fig. 6(d)), one can conclude that the quality of turning circle simulation degrades w i t h speed more significantly than the quality of the zigzag maneuver. However, for a 5 knot difference in speed f r o m the training maneuver (zigzag), the trajectory of turning circle maneuver is acceptable for about 180° change in heading and this may be sufficiently accurate for most practical maneuvers where a change of direcrion is not substantial. From many numerical studies, it has been concluded that the zigzag maneuver is a superior training maneuver than a turning circle or a spiral maneuver for large tankers. The results for

a larger than 5 knot difference in speed (say 10 knot) w i l l , however, give inaccurate results. Further study is required to make practical recommendations on training maneuvers of large tankers for all types of maneuvers at all speeds.

6. Conclusions

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C. Rajesh, S.K. Bhattachaiyya /Applied Ocean Research 30 (2008) 256-263 ₂₆₃

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