J Mar Sci Technol (2013) 18:42–49 DOI 10.1007/s00773-012-0190-1

ORIGINAL ARTICLE

Black-box modeling of ship manoeuvring motion based on feed-forward neural network with Chebyshev orthogonal basis function Xin-Guang Zhang • Zao-Jian Zou

Received: 30 June 2011 / Accepted: 27 June 2012 / Published online: 3 August 2012  JASNAOE 2012

Abstract Based on polynomial interpolation and approximation theory, a novel feed-forward neural network, the feed-forward neural network with Chebyshev orthogonal basis function, is proposed for black-box modeling of ship manoeuvring motion. The neural model adopts a three-layer structure, in which the hidden layer neurons are activated by a group of Chebyshev orthogonal polynomial activation functions and the other two layers’ neurons use identity mapping as activation functions. Weight update formulas are derived by employing the standard back-propagation (BP) training method. With the simulated 158/158 zigzag test data as input and calculated values of the hydrodynamic forces and moment as output, the feed-forward neural network with Chebyshev orthogonal basis function and the BP neural network are applied to identify the nonlinear functions in the nonlinear hydrodynamic model of ship manoeuvring motion. With the simulated 208/208 zigzag test data and 358 turning test data as input, the hydrodynamic forces and moment are predicted by using the identified nonlinear functions. Comparison between the calculated and predicted hydrodynamic forces and moment shows that the feedforward neural network with Chebyshev orthogonal basis function is superior to the BP neural network in identifying the nonlinear functions of the nonlinear hydrodynamic model of ship manoeuvring motion and is an

X.-G. Zhang  Z.-J. Zou (&) School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China e-mail: [email protected] Z.-J. Zou State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

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effective method to conduct the black-box modeling of ship manoeuvring motion. Keywords Ship manoeuvring  Nonlinear hydrodynamic model  Black-box modeling  Feed-forward neural network  Chebyshev orthogonal basis function

1 Introduction Mathematical modeling of ship manoeuvring motion involves mechanism modeling and black-box modeling. Black-box modeling is concerned with the response characteristic of system output to input. How to accurately determine the mapping relationship between the system input and system output is vital to the black-box modeling. Feed-forward neural network based on error back-propagation algorithm and its variants are available for the black-box modeling of ship manoeuvring motion. For example, Hess and Faller et al., predicted ship manoeuvrability by constructing the structure of the neural model which is equivalent to the mathematical models of ship manoeuvring motion [1–5]. Ebada and Abdel-Maksoud [6], Fan etc. [7] identified the mapping relationship between the manoeuvrability parameters and the ship form parameters by using a neural network and utilized the mapping to predict ship manoeuvrability. The BP neural network proposed by Rumelhart and McClelland et al. [8], is a typical feed-forward neural network, of which the error backpropagation algorithm could be simply summarized as: goE  wðk þ 1Þ ¼ wðkÞ  DwðkÞ ¼ wðkÞ  ð1Þ w¼wðkÞ ow where w denotes a vector or matrix of neural weights; k = 0, 1, 2… denotes the iteration step; Dw(k) denotes the

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weight update at the kth iteration step; g denotes the learning rate; E denotes an error function for the learning procedure. Due to some inherent shortcomings in the BP neural network, such as relatively slow convergence and existence of local minima etc., many improved BP algorithms have been proposed. There are two types of improvement. On the one hand, the standard gradient descent method is introduced to improve the BP algorithms. On the other hand, numerical optimization techniques are adopted for network training. Generally speaking, efforts are always made to improve the training algorithm in order to ameliorate the performance of the BP neural network. Different from the aforementioned algorithmic improvement, to overcome the weaknesses of BP neural network, this paper proposes a novel feed-forward neural network with Chebyshev orthogonal basis function for the blackbox modeling of ship manoeuvring motion based on the polynomial interpolation and approximation theory. The 158/158 and 208/208 zigzag tests as well as the 358 turning test are simulated by using the Abkowitz model, in which the hydrodynamic derivatives are obtained from the PMM tests [9]. With the simulated 158/158 zigzag test data as input and the calculated hydrodynamic forces as output, the nonlinear functions in the nonlinear hydrodynamic model of ship manoeuvring motion are identified by the feed-forward neural network with Chebyshev orthogonal basis function and the BP neural network, respectively. With the simulated 208/208 zigzag test data and 358 turning test data as input, the hydrodynamic forces and moment are predicted by using the identified nonlinear functions in the nonlinear hydrodynamic model. The predicted results are compared with the calculation results of the hydrodynamic forces and moment obtained from the nonlinear hydrodynamic model. The comparison shows that the feed-forward neural network with Chebyshev orthogonal basis function is superior to the BP neural network in determining the nonlinear functions of the nonlinear hydrodynamic model of ship manoeuvring motion and is promising in identifying the nonlinear functional relationships in the mathematical models of ship manoeuvring motion.

2 Feasibility analysis of neural model As shown in Fig. 1, the feed-forward neural network with Chebyshev orthogonal basis function consists of three layers. The input of the neural network is denoted as xi ; i ¼ 1; 2; . . .; m. The output is denoted as y. The input layer and output layer employ, respectively, m neurons and one neuron with identity mapping. The hidden layer has n þ 1 neurons

43 Hidden layer

w10

Input layer

x1

g 0 (net0 )

c0

w11 g1 (net1 )

w1n

c1

Output layer

y

wm 0 w m1 xm

cn

wmn

g n (netn )

Fig. 1 Feed-forward neural network with Chebyshev orthogonal basis function

activated by a group of Chebyshev orthogonal polynomial P gj ðnetj Þ; netj ¼ m i¼1 wij xi ; j ¼ 0; 1; 2; . . .; n. wij ði ¼ 1; 2; . . .; m; j ¼ 0; 1; 2; . . .; nÞ denotes the weights between the input layer and hidden layer. cj ðj ¼ 0; 1; 2; . . .; nÞ denotes the weights between the hidden layer and output layer. Moreover, all the neuronal thresholds are fixed as 0, which would simplify the network model and possible hardware-implementation. The feed-forward neural network with Chebyshev orthogonal basis function can be regarded as an improved BP neural network and the standard BP training algorithm can be adopted as its training rule. The approximation capability of the feed-forward neural network with Chebyshev orthogonal basis function can be analyzed as follows. Based on the polynomial interpolation and approximation theory [10], some polynomial function g(x) can be constructed to approximate the unknown target function /(x). For this approximation, we have: Pj i Definition 1 Define that gj(x) = i=0aix (ai = 0, j =  n 0, 1, 2, …) is a polynomial of order j and gj ðxÞ j¼0 is a set of n þ 1 orthogonal-polynomial functions with respect to some weight function wðxÞ over a finite or infinite interval ½a; b, i.e., for any i; j 2 f0; 1; 2; . . .; ng, we have: Zb

( wðxÞgi ðxÞgj ðxÞdx

a

¼ 0 i 6¼ j [0 i ¼ j

ð2Þ

Definition 2 Chebyshev polynomials can be defined by the following formula: gn ðxÞ ¼

sin½ð1 þ nÞ arccos x pffiffiffiffiffiffiffiffiffiffiffiffiffi 1  x2

ð3Þ

which is an orthogonal polynomial of order n with respect pffiffiffiffiffiffiffiffiffiffiffiffiffi to weight function wðxÞ ¼ 1  x2 over interval ½1; 1. Moreover, the recurrence relation for Chebyshev orthogonal polynomial can be expressed as follows:

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8 > < g0 ðxÞ ¼ 1 g1 ðxÞ ¼ 2x > : gnþ1 ðxÞ ¼ 2xgn ðxÞ  gn1 ðxÞ;

ð4Þ n ¼ 1; 2; . . .:

From the mathematical perspective of polynomial approximation, training of the neural network is aimed at establishing the best function approximation. Therefore, the theoretical foundation about the approximation of the feed-forward neural network with Chebyshev orthogonal basis function needs to be given. Definition 3 Assume that /ðxÞ; gj ðxÞ 2 C½a; b; j ¼ 0; 1; . . .; n; /ðxÞ and gj ðxÞ are continuous over the closed interval ½a; b. Given the weight function wðxÞ on interval ½a; b, appropriate coefficients wij and cj (i ¼ 1; 2; . . .; m; j ¼ 0; 1; 2; . . .; n) can be chosen for a generalized polynomial n X

m X

9 s 1X > ð dt  y t Þ 2 > > > > 2 t¼1 > > > > n = X yt ¼ cj gj ðnetj Þ : > > j¼0 > > > m > X > > > wij xit netj ¼ ;

E¼

ð7Þ

i¼1

Weight update formulas are derived by employing the standard BP training algorithm and described by the following two theorems. Theorem 2 The weight update formula between the input layer and the hidden layer for the feed-forward neural network with Chebyshev orthogonal basis function can be expressed as: wij ðk þ 1Þ ¼ wij ðkÞ þ gðdt  yt Þcj g0j ðnetj Þxit

ð8Þ

ð5Þ

where i ¼ 1; 2; . . .; m; j ¼ 0; 1; 2; . . .; n ; t ¼ 1; 2; . . .; s; k ¼ 0; 1; 2; . . .; learning rate g [ 0.

so as to minimize a wðxÞ½/ðxÞ  gðxÞ2 dx, where gðxÞ is called the least-square approximation of /ðxÞ with respect to wðxÞ over interval ½a; b.

Theorem 3 The weight update formula between the output layer and the hidden layer for the feed-forward neural network with Chebyshev orthogonal basis function can be expressed as:

gðxÞ ¼

cj gj ðxÞ; x ¼ netj ¼

j¼0

wij xi ;

i¼1

Rb

Theorem 1 For the unknown target function /ðxÞ 2 C½a; b, its least-square approximation gðxÞ (as described in the above definition) exists uniquely. Note that, when the feed-forward neural network with Chebyshev orthogonal basis function is used to approximate the unknown target function /ðxÞ, its input-output relation can be given as follows: y^ ¼ gðxÞ ¼

n X

cj gj ðxÞ; x ¼ netj ¼

j¼0

m X

wij xi ;

ð6Þ

Proof

3 Weight update formulas As discussed above, weights wij between the input layer and the hidden layer and weights cj between the output layer and the hidden layer (i ¼ 1; 2; . . .; m; j ¼ 0; 1; 2; . . .; n) can be generated so as to approximate effectively the unknown target function by training the given data samples. Take fðXt ; dt Þ; Xt ¼ ðx1t ; x2t ; . . .; xmt Þ; t ¼ 1; 2; . . .; sg as the training data set, where Xt denotes the input of the neural network, and dt is the output of the neural network, and define the batch-processing error function E as:

ð9Þ

Based on Eq. (7), we have:

oE ¼ ðdt  yt Þcj g0j ðnetj Þxit ; i ¼ 1; 2; . . .; m; owij j ¼ 0; 1; 2; . . .; n; t ¼ 1; 2; . . .; s

9 > > > > > =

> > > oE > ; ¼ ðdt  yt Þgj ðnetj Þ; j ¼ 0; 1; 2; . . .; n; t ¼ 1; 2; . . .; s > ocj

:

ð10Þ

i¼1

which is exactly the least-square approximation to /ðxÞ as described and guaranteed by definitions 1 to 3 and theorem 1.

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cj ðk þ 1Þ ¼ cj ðkÞ þ gðdt  yt Þgj ðnetj Þ:

By using the standard BP training algorithm, we have: 9 oE  > > wij ðk þ 1Þ ¼ wij ðkÞ  g w¼wðkÞ ; > > owij > = : i ¼ 1; 2; . . .; m; j ¼ 0; 1; 2; . . .; n > > >  oE  > > cj ðk þ 1Þ ¼ cj ðkÞ  g c¼cðkÞ ; j ¼ 0; 1; 2; . . .; n ; ocj ð11Þ By substituting Eq. (10) into the Eq. (11), Eq. (8) and Eq. (9) can be obtained. 4 Neural network training algorithm Based on the aforementioned weight update formulas, the network training algorithm can be further designed as follows:

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Step 1: Choose randomly the number of hidden layer neurons, learning rate 0\g\1, tolerance e [ 0, training time K. Step 2: Obtain the training sample set fðX t ; dt Þ; Xt ¼ ðx1t ; x2t ; . . .; xmt Þ; t ¼ 1; 2; . . .; sg: Step 3: Let E ¼ 0, t ¼ 1; k ¼ 0. Step 4: Initialize the weights wij ð0Þ (i ¼ 1; 2; . . .; m; j ¼ 0; 1; 2; . . .; n) between the input layer and the hidden layer and the weights cj ð0Þ between the output layer and the hidden layer. Step 5: Compute the input value of hidden layer neurons P netj ¼ m i¼1 wij ðkÞxit , the output value of hidden layer neuron gj ðnetj Þ, the output value of network yt ðkÞ ¼ Pn j¼0 cj ðkÞgj ðnetj Þ and prediction error et ¼ dt  yt ðkÞ. Step 6: Let E þ 0:5e2t ! E. Step 7: Adjust, respectively, wij and cj by using the Eq. (8) and the Eq. (9). Step 8: Let t þ 1 ! t and if t\s, go back to step 5; otherwise, continue to step 9. Step 9: If E  e, terminate the algorithm; otherwise, continue to step 10. Step 10: If E [ e and k\K, let E ¼ 0; t ¼ 1; k þ 1 ! k and go back to step 5.

5 Mathematical model of ship manoeuvring motion There are two kinds of mathematical model of ship manoeuvring motion, i.e., the hydrodynamic model including the Abkowitz model and the Manoeuvring Mathematical Modeling Group (MMG) model [11, 12] and the response model [13]. In this paper, the Abkowitz model is studied in order to verify whether the feed-forward neural network with Chebyshev orthogonal basis function can identify successfully the nonlinear functions in the nonlinear hydrodynamic model of ship manoeuvring motion. Ship manoeuvring motion in the horizontal plane includes the longitudinal motion, lateral motion and yaw motion. It is described by the surge speed u, the sway speed v and the yaw rate r in the body-fixed coordinate. The rudder angle is d. The corresponding variables at the state of straight forward motion with constant speed are u0 , v0 , r0 and d0 . Since v0 ¼ 0, r0 ¼ 0 and d0 ¼ 0, we have u ¼ u0 þ Du; v ¼ Dv; r ¼ Dr; d ¼ Dd, where Du; Dv; Dr and Dd; are the disturbing quantity of speed (angular speed) and rudder angle due to manoeuvring motion. The nondimensional Abkowitz model for ship manoeuvring motion in the horizontal plane can be written in the following form [14]:

45

2

m0  Xu0_ 4 0 0

0 m0  Yv0_ m0 x0G  Nv0_

32 0 3 2 0 3 0 Du_ X m0 x0G  Yr0_ 54 Dv_ 0 5 ¼ 4 Y 0 5; Iz0  Nr0_ Dr_0 N0 ð12Þ

where m xG Iz Du_ ; ; x0G ¼ ; Iz0 ¼ 1 5 ; Du_ 0 ¼ 2 3 ðU L =LÞ qL qL 2 2 Dv_ Dr_ ; Dr_0 ¼ 2 2 ; Dv_0 ¼ 2 ðU =LÞ ðU =L Þ

m0 ¼ 1

Xu_ Yv_ Yr_ ; Yv0_ ¼ 1 3 ; Yr0_ ¼ 1 4 ; 3 2 qL 2 qL 2 qL N N v_ r_ Nv0_ ¼ 1 4 ; Nr0_ ¼ 1 5 ; 2 qL 2 qL

Xu0_ ¼ 1

m is the mass of ship; xG is the longitudinal coordinate of the ship gravity center; Iz is the moment of inertia about z axis; q is the mass density of fluid; L is the ship length; U is the resultant speed in the horizontal plane, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Du_ ¼ u_  u_ 0 ; Dv_ ¼ v_  v_0 ; D U ¼ ðu0 þ DuÞ2 þDv2 ; _ v_ and r_ denote the surge r_ ¼ r_  r_0 , where u, acceleration, sway acceleration and yaw angular acceleration, respectively, u_ 0 , v_ 0 and r_0 are the corresponding state variables during the straight forward motion with constant speed. Since u_ 0 ¼ 0, v_ 0 ¼ 0 and _ Dv_ ¼ v; _ Dr_ ¼ r. _ Xu_ ; Yv_ ; Nr_ r_0 ¼ 0, we have Du_ ¼ u; etc., are the hydrodynamic derivatives due to accelerations. The nondimensional hydrodynamic forces and moment X 0 ;Y 0 and N 0 can be defined as [14]: 0 0 0 Du02 þ Xuuu Du03 þ Xvv Dv02 þ Xrr0 Dr 02 X 0 ¼ Xu0 Du0 þ Xuu 0 0 0 þ Xdd Dd02 þ Xddu Dd02 Du0 þ Xvr Dv0 Dr 0 0 0 0 0 0 0 0 þ Xvd Dv Dd þ Xvdu Dv Dd Du

ð13Þ

0 0 0 Dv03 þ Yvvr Dv02 Dr 0 þ Yvu Dv0 Du0 Y 0 ¼ Yv0 Dv0 þ Yr0 Dr 0 þ Yvvv 0 0 0 þ Yru Dr 0 Du0 þ Yd0 Dd0 þ Yddd Dd03 þ Ydu Dd0 Du0 0 0 0 þ Yduu Dd0 Du02 þ Yvdd Dv0 Dd02 þ Yvvd Dv02 Dd0 þ Y00 0 0 þ Y0u Du0 þ Y0uu Du02

ð14Þ

0 0 Dv03 þ Nvvr Dv02 Dr 0 N 0 ¼ Nv0 Dv0 þ Nr0 Dr 0 þ Nvvv 0 0 0 þ Nvu Dv0 Du0 þ Nru Dr 0 Du0 þ Nd0 Dd0 þ Nddd Dd03 0 0 0 þ Ndu Dd0 Du0 þ Nduu Dd0 Du02 þ Nvdd Dv0 Dd02 0 0 0 þ Nvvd Dv02 Dd0 þ N00 þ N0u Du0 þ N0uu Du02 :

ð15Þ

Expressing the variables in manoeuvring motion in dimensional form and the hydrodynamic derivatives in nondimensional form, we obtain the training sample couples from Eqs. (13), (14) and (15): The input data: DuðkÞ=UðkÞ, DvðkÞ=UðkÞ, DrðkÞL=UðkÞ, DdðkÞ:

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The output data: X 0 ðkÞ, Y 0 ðkÞ, N 0 ðkÞ where k is the sampling time.

6 Modeling results Taking the Mariner class vessel as the object model [9], 15/15 zigzag test, 20/20 zigzag test and 358 turning test are simulated firstly by using the Abkowitz model, i.e., Eq. (12), where the hydrodynamic derivatives are obtained from PMM tests The main data and dimensions of the corresponding full-scale ship are shown in Table 1. The surge speed u, sway speed v, yaw rate r and the resultant speed U in the horizontal plane, etc., are obtained from the simulated tests. During the 15/15 zigzag test, the sampling interval is 0.5 s and the length of sampling is 563 s, while in the 20/20 zigzag test and the 358 turning test, the sampling interval is 0.5 s and the length of sampling is 1,200 s. With the first 300 data of the 15/15 zigzag test as input and the first 300 data of hydrodynamic forces and moment calculated by using the nonlinear hydrodynamic model as output, the feed-forward neural network with Chebyshev orthogonal basis function and the BP neural network are applied to identify the nonlinear functions in the nonlinear hydrodynamic model of ship manoeuvring motion. For the feed-forward neural network with Chebyshev orthogonal basis function, the hidden layer has two neurons. The numbers of input layer and output layer neurons can be determined, respectively, by the training samples. The initialization of weight matrix between the input layer and hidden layer, and that between the output layer and hidden layer, as well as the learning rate and learning time are chosen by experience. The parameters of the feedforward neural network with Chebyshev orthogonal basis function are shown in Table 2, in which NIL denotes the number of input layer neurons; NOL denotes the number of output layer neurons; NHL denotes the number of hidden layer neurons; CNN denotes the feed-forward neural network with Chebyshev orthogonal basis function; WM1 denotes the initialized weight matrix between the input layer and hidden layer; WM2 denotes the initialized weight matrix between the hidden layer and output layer; ones(M, N) is a M-by-N unit matrix. To verify the generalization performance of the nonlinear functions identified by using the feed-forward neural network with Chebyshev orthogonal basis function, the prediction results of the surge force, sway force and yaw moment with the 20/20 zigzag test data and the 358 turning test data as input are compared with the calculated hydrodynamic forces and moment obtained from the nonlinear hydrodynamic model, as shown in Figs. 2 and 3. For the BP neural network, a three-layer BP neural network is constructed by employing the MATLAB neural

123

network tool box [15]. As for the structure of such a BP neural network, the hidden layer has two neurons. The numbers of input layer and output layer neurons can be determined respectively by the training samples. The initialized weight matrix between the input layer and hidden layer, and that between the output layer and hidden layer are randomly generated. Learning rate, learning time and momentum coefficient are chosen by experience. The tansig and purelin functions are employed respectively as the transfer functions of the hidden layer and output layer. According to the improved gradient descent approach, the traingdm function is employed as the back-propagation network training function and the learngdm function is employed as the back-propagation weight learning function. Moreover, all the neuronal thresholds are fixed as 0. The parameters of BP neural network are shown in Table 3, in which rand (M, N) is a M-by-N stochastic matrix. To verify the generalization performance of the nonlinear functions identified by using the BP neural network, the prediction results of the surge force, sway force and yaw moment with the 20/20 zigzag test data and the 358 turning test data as input are compared with the calculated hydrodynamic forces and moment obtained from the nonlinear hydrodynamic model, as shown in Figs. 4 and 5. Comparing Fig. 2 with Fig. 4, and Fig. 3 with Fig. 5, it can be seen that the feed-forward neural network with

Table 1 Main data and dimensions of Mariner class vessel Parameters

Magnitude

Length overall (Loa)

171.8 (m)

Length between perpendiculars (Lpp)

160.93 (m)

Maximum beam (B)

23.17 (m)

Design draft (T)

8.23 (m)

Design displacement (r)

18,541 (m3)

Design speed (u0)

15 (knots)

Table 2 Parameters of the feed-forward neural network with Chebyshev orthogonal basis function Parameters

CNN (surge force)

CNN (sway force)

CNN (yaw moment)

NIL

10

15

15

NOL

1

1

1

NHL

2

2

2

Learning rate

0.05

0.05

0.05

Learning time

2,000

2,000

2,000

WM1

ones(10,2)

ones(15,2)

ones(15,2)

WM2

ones(1,2)

ones(1,2)

ones(1,2)

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Fig. 2 Comparison between the calculated and predicted results with 20/20 zigzag test data as input (Feed-forward neural network with Chebyshev orthogonal basis function)

47

Fig. 3 Comparison between the calculated and predicted results with 358 turning test data as input (Feed-forward neural network with Chebyshev orthogonal basis function)

123

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Table 3 Parameters of BP neural network Parameters

BP (surge force)

BP (sway force)

BP (yaw moment)

NIL

10

15

15

NOL

1

1

1

NHL

2

2

2

Learning rate

0.05

0.05

0.05

Learning times

5,000

5,000

5,000

WM1

rand(2, 10)

rand(2, 15)

rand(2, 15)

WM2

rand(1, 2)

rand(1, 2)

rand(1, 2)

Momentum coefficient

0.9

0.9

0.9

Chebyshev orthogonal basis function is superior to the BP neural network in identifying the nonlinear functions of the nonlinear hydrodynamic model of ship manoeuvring motion and the prediction results of the surge force, sway force and yaw moment obtained by using the feed-forward neural network with Chebyshev orthogonal basis function are in good agreement with the calculated hydrodynamic forces and moment. This shows that the feed-forward neural network with Chebyshev orthogonal basis function is a promising method to conduct the black-box modeling of ship manoeuvring motion.

7 Conclusions In this article, the feed-forward neural network with Chebyshev orthogonal basis function is proposed for blackbox modeling of ship manoeuvring motion. With the 158/ 158 zigzag test data as input and calculated values of the hydrodynamic forces and moment as output, the feed-forward neural network with Chebyshev orthogonal basis function and the BP neural network are applied to identify the nonlinear functions in the nonlinear hydrodynamic model of ship manoeuvring motion. With the 208/208 zigzag test data and 358 turning test data as input, the hydrodynamic forces and moment are predicted by using the identified nonlinear functions. The comparison between the calculated and predicted hydrodynamic forces and moment shows that the feed-forward neural network with Chebyshev orthogonal basis function is superior to the BP neural network in identifying the nonlinear functional relationships in the mathematical models of ship manoeuvring motion. However, further work is needed to improve and verify this method to make it a powerful one. Firstly, future research can be focused on how to determine the optimal number of hidden layer neurons in an easy way; secondly, a theory has to be presented in order to effectively choose the parameters of the feed-forward neural network with

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Fig. 4 Comparison between the calculated and predicted results with 20/20 zigzag test data as input (BP neural network)

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Chebyshev orthogonal basis function, such as learning rate, learning time, etc. Acknowledgments Supported by the National Natural Science Foundation of China (Grant Nos. 50979060, 51079031) and the Foundation of National Science and Technology Key Laboratory of Hydrodynamics (Grant No. 9140C2201091001).

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