Short Term Prediction of Ship Pitching Motion Based ...

Proceedings of the ASME 2016 35th International Conference on Ocean, Offshore and Arctic Engineering OMAE2016 June 19-24, 2016, Busan, South Korea

OMAE2016-54317

SHORT-TERM PREDICTION OF SHIP PITCHING MOTION BASED ON ARTIFICIAL NEURAL NETWORKS Bai-Gang Huang School of Naval Architecture, Ocean and Civil Engineering Shanghai Jiao Tong University Shanghai 200240, China Zao-Jian Zou School of Naval Architecture, Ocean and Civil Engineering; State Key Laboratory of Ocean Engineering Shanghai Jiao Tong University Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE) Shanghai 200240, China

which may prevent crash of cargo in cargo transfer, improve the landing safety of aircrafts on a carrier and fire accuracy of the ship-borne weapon systems[1]. Another important application of ship motion prediction is to extend the operational limits by forecasting the quiescent periods where the ship motions are within acceptable limits to perform a desire maritime activity[2]. The prediction of ship pitching motion is an essential part in ship motion research. The dynamics of ship pitching motion varies along with the ship’s navigational status in relation to loading conditions and ship speed, and it is also interfered by environmental disturbances in open sea, such as winds, waves and currents. Therefore, it is difficult to establish a strictly founded mathematical model for the prediction of ship pitching motion at sea. Lots of studies on ship motion prediction have been carried out by using various methods. The traditional methods, such as time series auto regression (AR) model, auto regressive moving average (ARMA) model[3, 4], etc., belong to linear analysis methods, which have not enough precision for predicting the nonlinear and time-varying ship motion. Sun and Shen established a grey metabolism model to predict ship pitching motion[5]. Gray theory mainly aims at the prediction of trend. For the ship motion with oscillation, the prediction accuracy is not high. Theoretically, artificial neural network (ANN) can approximate any nonlinear function with arbitrary accuracy; for this reason, many researchers used this intelligent technology to predict ship motion. Khan et al. used the back propagation (BP) neural network based on the singular value decomposition to give excellent predictions of ship roll motion[6]; Masi et al. used the radial basis function network (RBFN) to develop a model of

ABSTRACT Due to the random nature of ship motion in an open sea environment, the ship related maritime operations such as landing on an aircraft carrier, ship-borne helicopter recovery, cargo transfer between ships and so on, are usually very difficult. An accurate prediction of the motion will improve the operation safety and efficiency on board ships. This paper presents a research on the application of artificial neural network methods in the short-time prediction of ship pitching motion. The radial basis function (RBF) neural network is applied to develop a model for short-time prediction of ship pitching motion, and the other two kinds of artificial neural networks, i.e., back propagation (BP) neural network, Elman neural network are also applied for the same purpose. A comparative analysis among them is presented. It is shown that RBF neural network provides a more effective and accurate tool for predicting the ship pitching motion. Keywords: ship pitching motion; short-term prediction; RBF; BP; Elman; neural network INTRODUCTION Ship motion at sea is a complex nonlinear system with time-varying dynamics and strong uncertainty. Prediction of ship motion is an important issue in marine field for navigational and operational safety and efficiency, and the accurate prediction of ship motion allows improving effectively the safety and efficiency. The motion information from short-term prediction is critical for the motion compensation

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short-term prediction of vessel heave motion[7]; Yin and Wang proposed a variable-RBFN-based prediction method for ship roll motion[8]. The neural network methods demonstrate a good performance in dealing with complex nonlinear processes. Various kinds of neural network methods have been implemented in different fields and good results have been acquired. In this paper, the neural network methods are adopted to predict the short-term ship pitching motion. Three kinds of neural networks, i.e., radial basis function (RBF), BP, and Elman neural networks, are applied to establish the prediction model. The results of simulation and comparative analysis indicate that the RBF neural network-based prediction model has better efficiency and higher accuracy. The paper is organized as follows: firstly, a brief introduction of the RBF neural network (RBF-NN) is given, including the principal ideas of the RBF-NN and the basic method for training the network. Secondly, the autoregressive time series model of ship pitching motion is presented. Then the performances of the nonlinear autoregressive (NAR) models established by RBF, BP and Elman neural networks in pitching motion prediction are compared by using the simulation results. Finally, conclusions drawn from this study are highlighted in the last section.

In this paper, the pitching angle is the unique output; hence the single-output RBF-NN is adopted. Although the scalar output case is considered here, extension to the multi-output case is straightforward. In fact, a multi-output RBF-NN can always be separated into a group of single-output RBF-NNs. In the RBF-NN, the function 𝜑( ∙ ) and the centers 𝒁𝑖 are assumed to be fixed. By providing a set of the input and the corresponding output desired, the values of the weights 𝑤𝑖 can be determined using the linear least squares (LS) method. However, 𝜑( ∙ ) and 𝒁𝑖 must be carefully chosen so that the RBF-NN is able to match closely the performance of the neural network. Theoretical investigation and practical results suggest that the choice of the nonlinear function 𝜑( ∙ ) is not crucial to the performance of the RBF-NN[10]. There is a large class of radial basis functions, for example, the thin-plate-spline function, the multi-quadric function and the Gaussian function, etc., among them the most popular and widely used one is the Gaussian function  1  (2)   X , Z   exp   2 X  Z 2 

 

where 𝜎 is the width of the radial basis function, and ‖ ∙ ‖ denotes the Euclidean norm. Then, a Gaussian RBF-NN can be expressed as n  1  (3) y  w0  wi exp   2 X  Z 2 

RADIAL BASIS FUNCTION NETWORK The basic form of the RBF-NN consists of three layers. The first layer is the input layer composed of input source nodes that connect the network to its environment. The second layer, the only hidden layer in the network, applies a nonlinear transformation from the input space to the hidden space. The last layer is the output layer composed of linear units, being fully connected to the hidden layer[9]. Figure 1 depicts the schematic of a RBF-NN with m inputs and a scalar output. Such a network implements a nonlinear mapping 𝑦: 𝑅𝑚 → 𝑅 according to

i 1

(1)

where 𝜑( ∙ ) is a given function mapping 𝑅𝑚 → 𝑅, 𝑿 ∈ 𝑅𝑚 is the input vector, 𝒁𝑖 ∈ 𝑅𝑚 ( 0 ≤ 𝑖 ≤ 𝑛) are known as the RBF centers, and n is the number of the centers; 𝑤𝑖 ∈ 𝑅 (0 ≤ 𝑖 ≤ 𝑛) are the weights.

…

…

Hidden layer



AUTOREGRESSIVE TIME SERIES MODEL OF SHIP PITCHING MOTION Ship heaving and pitching motions are usually coupled together. Assuming that the fluid is inviscid and incompressible, the waves and the oscillating motion of the ship are regular with small amplitude, the mathematical models of the heaving and pitching motions can be expressed as    m0  m33  z  N33 z  C33 z  m35  N35  C35  F3 (4)    I  m   N   C   m z  N z C z M    55 55 55 53 53 53 5  22

i 1

Input layer

 

In practice, the centers are normally chosen from the input data points. The key problem is how to select the centers appropriately from the data set. There are several methods for choosing the centers, such as random choice method, K-means cluster and orthogonal least squares (OLS) algorithm, etc. Here OLS is used to obtain the centers.

n

y  w0  wi ( X , Z i )



where 𝑧 , 𝑧̇ and 𝑧̈ denote the displacement, velocity and acceleration of the heaving motion, respectively; 𝜃, 𝜃̇ and 𝜃̈ denote the pitching angle, angular velocity and angular acceleration of the pitching motion, respectively; 𝑚0 is the mass of the ship, 𝐼22 is the ship’s mass moment of inertia; 𝑚33 , 𝑚55 , 𝑚35 , 𝑚53 are the added masses of the ship; 𝑁33 , 𝑁55 , 𝑁35 , 𝑁53 are the damping coefficients; 𝐶33 , 𝐶55 , 𝐶35 , 𝐶53 are the restoring force coefficients; 𝐹3 is the vertical force on the ship, 𝑀5 is the pitching moment on the ship. The strip theory is usually used to determine these coefficients [11]. However, a ship sailing at sea experiences a random disturbance from the environment, hence it is difficult to

Output layer

Fig. 1 Schematic of a RBF-NN with m inputs and a scalar output

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measure and calculate these coefficients. On the other hand, in a relatively short time, the marine environment can be considered as stationary; therefore, these coefficients can be assumed to be constant. Under the conditions of zero input and zero state, the Laplace transformation of Eq. (4) can be written as 2 2   m0  m33  S  N33 S  C33  Z  S   (m35 S  N35 S  C35 )  S   F3  S  (5)  2 2  I 22  m55  S  N55 S  C55    S   (m53 S  N53 S  C53 )Z  S   M 5  S 

enough accuracy for the problem. In this study, Akaike Information Criterion (AIC) is introduced to determine the number of nodes in the hidden layer.

T

T …

S

4



 aS 3  bS 2  cS  d   S   F  S 

T

(6)

where 𝑎, 𝑏, 𝑐 and 𝑑 are constant coefficients, 𝐹(𝑆) is the Laplace function. Then, the Laplace inverse transformation of Eq. (6) can be expressed as

  4  a  b c  d  f  t 

…

Apparently, by analyzing Eq. (5), it can be seen that   S  satisfies the following equation:

Neural Network

Fig. 2 Neural network NAR model Determining the Number of Nodes in the Hidden Layer AIC is a measure of the relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Hence, AIC provides a means for model selection. AIC can be expressed as (10) AIC  2log  ˆ |data    2K

(7)

where 𝑓(t) is a continuous function. Discretizing Eq. (7) by using the forward difference formula, for example,   t     t  1 (8)  t   t where 𝛥𝑡 is the data sampling interval, Eq. (7) can be rewritten as a nonlinear equation (9)  (t )   ( (t  1),  (t  2),  (t  3),  (t  4)) where 𝜓( ∙ ) is a nonlinear autoregressive function. The deduced Eq. (9) demonstrates that the ship pitching motion has relation to the state of the pitching motion in the past. It can be considered as an autoregressive time series model. There are a variety of ways to build the time series models, for example, AR, ARMA. In this paper, neural network is used to build the model to fit the function of 𝜓( ∙ ) . Obviously, the number m of neural network input layer is 4.

where ℒ( ∙ ) is the likelihood function of the assumed model, 𝜔 ̂ denotes the maximum likelihood estimator (MLE) of parameter ω based on the assumed model and data; K denotes the total number of parameters in the model. In the special case of least squares (LS) estimation with normally distributed errors, AIC can also be expressed as (11) AIC  nlog(ˆ 2 )  2 K where n is the number of sample data, and

ˆ

2

 

n i 1

(î )2

n

(12)

î (i=1,2,…,n) are the estimated residuals from the fitted

SIMULATION RESULTS AND ANALYSIS This section presents the simulation results of ship pitching motion by the neural network NAR models, i.e., the NAR models established by RBF-NN, BP and Elman networks; and a comparison among them is given. Structure of Neural Network NAR Model A structure diagram of the neural network NAR model is depicted in Fig. 2, where “T” in the square box represents the time delay unit. According to Eq. (9), four time delay units are needed to simulate the pitching motion. Three kinds of neural network, i.e., RBF-NN, BP and Elman, are used to build the NAR model. BP is a typical multilayer feedforward neural network, which is commonly used in many fields. Elman is a global feedforward local recurrent neural network, which is a dynamic network and often used in the prediction of stock index and chaotic time series[12]. To determine the structure of the neural network, the key problem is how to choose the number of nodes in the hidden layer. An excessive node number may make the network over fitting to the training data and lead to a poor generalization ability, while an insufficient node number cannot provide

model. Thus, AIC is easy to compute from Eq. (11)[13]. Usually, the preferred model is the one with the minimum AIC value. In this study, the pitching data taken from the experimental data of ship model test carried out in the towing tank at China Ship Scientific Research Center[14] is used to train the neural network with different number of nodes in the hidden layer. The data sampled every second for 2 minutes is a total of 120. The first 103 data are used to train the neural network, while the rest are supplied to the neural network one by one for one-step forward predicting test. Since there are four time delay units in the NAR model, the first 103 data constitute 99 input-output samples. Figure 3 shows the result of AIC value in RBF-NN. It is obvious that the AIC reaches its lowest value when the node number is 4, so the node number of the hidden layer is set as 4.

3


experimental data training data predicting data error

-350

AIC value

1.0

-400 -450

0.8

0.6

-550

pitch/(°)

AIC value

-500

-600 -650

0.4

0.2 -700 -750

0.0 0

5

10

15

20

25

30

-0.2

Number of hidden layer node

0

Fig. 3 AIC value in RBF-NN with different node numbers of the hidden layer

40

60

80

100

120

t (s)

Fig. 5 Simulation result of Elman-based NAR model

Simulation Results and Analysis Using the experimental data of pitching motion taken from the model test[14] to train the RBF-NN, BP and Elman networks and using the established NAR models to predict the pitching motion, simulation study is carried out. All the simulations are executed in MATLAB R2015a environment under WINDOWS 7 (64bit) operating system with Intel® Core™ i5-3230 (CPU) and 8.00 GB memory (RAM). Figures 4, 5 and 6 show the prediction results with the NAR models established by BP network, Elman network and RBF-NN, respectively. The performances of the different models are compared in Table 1, where the root mean square error (RMSE) is defined as

 i 1 (i  î )2

20


1.0

0.8

pitch/(°)

0.6

0.4

0.2

0.0

n

RMSE 

0

n

1.2

1.0

pitch/(°)

0.4

0.2

0.0 80

100

100

120

Neural Network NAR models

BP

Elman

RBFN

Node number of hidden layer

10

3

4

RMSE of training data / (%)

2.46

3.56

2.24

RMSE of predicting data / (%)

2.07

7.84

1.74

Time of training NN / (s)

0.3676

0.5528

0.0850

CONCLUSIONS Artificial neural network methods provide an important tool for predicting ship motion, and different neural networks have different prediction accuracy. In this paper, three kinds of neural network, BP network, Elman network and RBF-NN, are applied to establish the NAR model for ship pitching motion prediction. It is shown that the NAR models established by these neural networks have the ability to predict the ship pitching motion, but RBF-NN-based NAR model has a better performance in prediction with less computation time and higher accuracy.

0.6

60

80

Table 1 Performance comparison of different Neural Network NAR models

0.8

40

60

Fig. 6 Simulation result of RBF-NN-based NAR model


20

40

t (s)

where  i and î are the real pitching angle and predicted pitching angle by the models, respectively. In Table 1, the numbers of nodes in BP and Elman networks correspond to the best performance and are determined by simulation experiments. It can be seen from the comparisons, RBF-NN has higher accuracy than the other two neural networks; and it takes less time to train the neural network.

0

20

(13)

120

t (s)

Fig. 4 Simulation result of BP-based NAR model

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[7] Masi G D, Gaggiotti F, Bruschi R, et al. (2011). “Ship

The present study provides a foundation for real-time prediction and control of ship pitching motion. It may be extended to the multi-output neural network, so it can be used to predict simultaneously the multi degrees of freedom (DOF) ship motion. This will be the objective of the further study. Moreover, intelligent optimization methods will be applied to determine the number of hidden neurons and improve the convergence rate and modeling accuracy in the further study.

[8] [9]

ACKNOWLEDGMENTS This work is financially supported by the National Natural Science Foundation of China (Grant No: 51279106).

[10]

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