On-line prediction of ship roll motion during maneuvering using sequential learning RBF neuralnetworks

Ocean Engineering 61 (2013) 139–147

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

On-line prediction of ship roll motion during maneuvering using sequential learning RBF neural networks Jian-chuan Yin a,b, Zao-jian Zou a,c,n, Feng Xu a a

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China Navigation College, Dalian Maritime University, Dalian 116026, Liaoning, China c State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China b

a r t i c l e i n f o

abstract

Article history: Received 21 July 2012 Accepted 5 January 2013 Available online 9 February 2013

The on-line prediction of ship roll motion during maneuvering plays an important role in navigation safety and ship control applications. This paper presents an on-line prediction model of ship roll motion via a variable structure radial basis function neural network (RBFNN), whose structure and parameters are tuned in real time based on a sliding data window observer. The RBFNN is sequentially constructed by adding the new sample in the hidden layer and pruning the obsolete hidden units at each epoch, with the connecting parameters adjusted simultaneously. Gaussian functions with multi-scale kernel width are adopted to provide more flexible representations of model input terms and to achieve better generalization capability. Simulation study of ship roll motion prediction is conducted with measurement data of turning circle test and zigzag test in full-scale sea trial. Results demonstrate that the proposed neural network predictive model can on-line predict the roll angle with high accuracy. The predictive model is also featured with its compact network structure and fast computational speed. & 2013 Elsevier Ltd. All rights reserved.

Keywords: On-line prediction Ship roll motion Sequential learning Variable structure radial basis function neural network Sliding data window

1. Introduction Many research works have been conducted on system identification and prediction for ship motions. A large proportion of such researches are focused on ship roll motion, since it is the most important motion in relation to ship stability and is closely related to ship navigation safety and comfortableness of passengers. Ship’s roll motion at sea is difficult to predict because it is a complex nonlinear system with time-varying dynamics (Fossen, 2011). Its dynamics varies along with navigational status such as loading conditions, trim and speed (Vidic-Perunovic and Jensen, 2009); similar changes also occur resulting from environmental disturbances such as wind (Bacˇkalov et al., 2010) and waves (Bulian and Francescutto, 2011), etc. Besides the ship’s roll motion itself, heave (Yu et al., 2006) and pitch (Zhou and Chen, 2008) motions and their coupling with roll motion have been of great research interest (Haddara and Xu, 1999, Chang, 2008, Neves and Rodrı´guez, 2006). Based on the research of ship roll motion, the roll reduction control techniques are developed to ensure the safety and ship operation (Fang et al., 2010) by means of anti-roll tanks (Youssef et al., 2002), fin/rudder (Perez, 2005), fin (Koshkouei et al., 2007) and rudder (O’Brien, 2009). n Corresponding author at: School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China. Tel./fax: þ 86 21 34204255. E-mail address: [email protected] (Z.-j. Zou).

0029-8018/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2013.01.005

Neural networks are capable of taking into account more complex relations existing among the analyzed data, so they are capable of generating more accurate prediction attributing to such feature. The application of neural networks in the identification and prediction of nonlinear system has been intensively researched in recent years, such as the radial basis function neural network (RBFNN) (Haykin, 1998), multilayer perceptron (MLP) (Mahfouz, 2004) and support vector machine (SVM) (Mahjoobi and Mosabbeb, 2009). Featured by its local response property and fast convergence speed, RBFNN has been intensively studied and various applications can be found in ocean engineering field. It can be used to predict the main design parameters in oil/chemical tanker design (Ekinci et al., 2011), to anticipate the damage ratio in break water design process (Yagci et al., 2005), to obtain significant wave heights at a specified coastal site from their values gathered by a satellite at deeper offshore locations (Kalra et al., 2005), to on-line predict tidal level with variable network structure (Yin et al., 2013), to represent the ship’s dynamic behavior in different control schemes (Wu et al., 2012; Dai et al., 2012). Due to the complexity of the time-varying dynamics of ship roll and its underlying processes, it is hard to obtain a suitable prediction model by static structure neural network. Sequential learning algorithms, which are oriented to on-line constructing neural networks, have attracted much attention over the last decade. Since a fixed mathematical model can never be exhaustive enough to cover all possible sailing and environmental conditions,

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the network structure and connecting parameters should be adjusted on-line to adapt to the time-varying variations of ship dynamics. In this paper, a RBFNN constructed by sequential learning is employed for real-time representing time-varying dynamics and predicting ship’s motion. The resulting variable-structure neural networks are capable of representing time-varying dynamics of nonlinear systems and their applications can be found in fields of communication and control. To validate the efficiency of the proposed RBFNN roll predictive model, simulation experiments are conducted using the measurement data of ship roll motion in turning circle test and zigzag test of full-scale sea trial. Simulation results of ship roll motion prediction are presented and analyzed in the paper.

2. On-line sequential learning RBFNN 2.1. Algorithm description The on-line construction of variable-structure neural network is a research focus in recent years. Among various implementing network types, RBFNN is the most popular network attributing to its merits such as local response nature, simple topology, fast convergence speed and no local minima. Sequential learning algorithms are designed for constructing variable RBFNN for online identification and prediction applications, which learn samples one by one and do not need retraining whenever a new observation is received. They possess merits of low computation burden and adaptive capability, and are generally preferred over batch learning algorithms in learning speed and network dimension. A number of sequential learning algorithms have been proposed and the most widely used sequential learning algorithms are resource allocation network (RAN) (Platt, 1991), RAN with extended Kalman filter (RANEKF) (Kadirkamanathan and Niranjan, 1993) and minimal RAN (MRAN) (Lu et al., 1997) algorithms and their variations (Huang et al., 2005, Vigneshwaran et al., 2007, Lim and Rao, 2009, Suresh et al., 2010). To overcome the drawbacks of static structure neural network, Platt (1991) developed the RAN algorithm which adds hidden units to the network based on the novelty of the new data. The algorithm is improved by adopting extended Kalman filter (EKF) instead of the conventional least squares method (LSM), which is referred to as RANEKF algorithm. To delete the inactive hidden units and reduce the network dimension, Lu et al. (1997) developed MRAN algorithm and many applications can be found based on it (Vigneshwaran et al., 2007, Lim and Rao, 2009, Suresh et al., 2010). However, there are too many parameters to be adjusted and the tuning of parameters becomes a time-consuming task. Another drawback of the abovementioned sequential learning algorithm is that there are only one sample learned at each step, the resulting network may be severely affected by the sample and lead to the instability of the network. Gradient orthogonal model selection (GOMS) algorithm is a sequential learning algorithm proved to be effective in planar ship motion prediction (Yin et al., 2012) and on-line tidal level prediction (Yin et al., 2013) applications. In this study, the algorithm is employed for on-line constructing variable-structure RBFNN to estimate the ship roll motion. The most important feature of variable-structure neural network is that the number and location of hidden units are adjusted at each step according to the sequential learning result, thus the variable structure can capture the changes of time-varying system dynamics. However, if the algorithm learns the samples one by one, as in the conventional sequential learning algorithm, the resulting network may be apt to be influenced too much by the particular sample; if there are too much samples to be learned at one step, the resulting network can only reflect the holistic characteristics of the system and cannot reflect the changes

in system dynamics in local time domain, the computational burden will also be aggravated. In this algorithm, therefore, a first-in-first-out sliding data window is employed as a system dynamics observer. The newly received sample is added in the window and the foremost one is removed from the window simultaneously. The variablestructure RBFNN is on-line constructed by learning samples in the sliding data window. The structure adjustment strategy of GOMS algorithm is that the newly received sample is added as a new hidden unit, while units contributing less to the output are excluded from the network. The contribution of a hidden unit is evaluated by an index referred to as normalized error reduction ratio (nerr). The learning process of the algorithm is described hereunder. The sliding data window is described as: W SD ¼ ½ðxtN þ 1 ,ytN þ 1 Þ,. . .,ðxt ,yt Þ

ð1Þ

where WSD denotes the sliding data window, N denotes the window width, t is the current time instant, x and y denote input and output, respectively. The current system dynamics is represented by the input matrix X ¼ ½xtN þ 1 ,. . .,xt  A RnN and the output matrix Y ¼ ½ytN þ 1 ,. . .,yt T A RmN , with n and m being the dimensions of input and output, respectively. The centers can be represented by matrix C ¼ ½c1 ,. . .,cM  A RnM , where M is the number of hidden units, which is variable during learning process. The tendency of the system changes is described by the derivatives of input and output. For instance, the first-order description of the input and output are: X 0t ¼ ½xtN þ 1 xtN ,. . .,xt xt1 , Y 0t ¼ ½ytN þ 1 ytN ,. . .,yt yt1 T

ð2Þ

where the superscript ‘‘0 ’’ denotes differential. After the learning of mapping between input and output, the final output can be integrated as: 0 Y^ t ¼ FðX 0t Þ ¼ f ðX 0t Þ þY t1 ¼ Y^ þ Y t1

ð3Þ

Higher order description can be extended by analogy. The order of the RBFNN is decided by the differential order of the sliding data window. The incorporation of the higher order information improves the predictive capability of the proposed algorithm. The variable structure is constructed by adding the new sample in the network and pruning the units which contribute less to the output at the same time. After the hidden units are adjusted, the tuning parameters between the hidden layer and output layer are updated. The contribution of hidden units is measured by the index nerr defined as: err nerr k ¼ PM k k ¼ 1 err k

ð4Þ

where M is the number of current hidden units, errk denotes error reduction ratio (err) of the k-th unit, which is calculated according to: err k ¼

ðwTk yðnÞ Þ2 T ðwk wk ÞðyðnÞT yðnÞ Þ

ð5Þ

where n is the order of the RBFNN, wk is the k-th vector of the orthogonalized response matrix of input matrix with regard to the hidden units matrix. The concept of error reduction ratio (err) is proposed in the well-known orthogonal least squares algorithm (Chen et al., 1991).

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The response matrix F ¼ ½f1 ,. . ., fM  A RNM , with elements ! :xðj nÞ ck : fj,k ¼ exp  , 1r j r N, 1 rk rM ð6Þ 2s2 where ck is the center of the k-th hidden unit and s is the center width of the k-th Gaussian function, and :  : denotes the Euclidean distance. The contribution of each center to the output is coupled, so the response matrix F is orthogonalized using Gram–Schmidt method:

F ¼ WA

ð7Þ

where A is a upper triangle matrix and the orthogonalized matrix satisfies that WTW¼{wT1w1, wT1w1,y, wTMwM}. The space spanned by wk is the same as that spanned by fk. For a multi-in-multi-output (MIMO) process, the values of err are calculated by: Pm T ðnÞ 2 i ¼ 1 ðwk yi Þ ð8Þ err ki ¼ ðwTk wk Þ trace ðY ðnÞT Y ðnÞ Þ Index nerr is the generalized form of err, the generalization makes that the summation of the nerrk (k¼1, 2,y, M) is 1: XM nerr k ¼ 1 ð9Þ k¼1 which enables the intuitionistic evaluation of the contribution of the hidden units and consequently the direct application of the index nerr.

j€ ¼

Assume that there are m different scales of input and each contains dk (k¼1, 2,y, m) terms in the prediction model, we have: " (    x1 ci,d1 2 1 x1 ci,1 2 fi ¼ exp  þ. . .þ 2 s1 s1  2   #) xm ci,d1 þ :::dk þ ::: þ dm1 þ 1 xm ci,d1 þ :::dk þ ::: þ dm 2 þ þ. . .þ

sm

where p is the number of hidden units those are selected and considered as those contributing little to the overall output. If certain units have been selected for l times successively, they will be pruned from the network. After the hidden units are adjusted at each step, the connecting weights Y of the network will be updated using conventional least squares method:

Y ¼ F þ Y ðnÞ ¼ ðFT FÞ1 FT Y ðnÞ

sm

ð12Þ where fi is the response of i-th center corresponding to the input, and s1,y,sk,y,sm are center widths determined for each category of input, respectively. Therefore, in GOMS algorithm, the vectors of centers and sliding data window input are divided into categories. The implementation of multi-scale improves the fitness of the hyperplane to the input–output mapping, can reduce the dimension of hidden units in identification and prediction applications.

3. RBFNN prediction model It is difficult to measure the exciting roll moment acting on a ship at sea; therefore, roll motion parameters have rarely been studied using full scale ships at sea. In this study, we give predictions of roll based on time series of relevant variables. Here the following nonlinear motion equation is used for the determination of rolling (Chang, 2008):

€ þ cj _ 2 Þsin jðc _ 2 Þcos j þM wind þ M cd þM sy þM waves g fM d mðgz€ Þhs Ixz ½ðy€ þ yj fIxx Ixz ðcsin j þ ycos jÞg

At each step, the nerr of each hidden unit is calculated and the units with small nerr are selected in ascending order. The selection process is terminated, once the summation of nerr reaches the preset accuracy threshold r: Xp nerr Z r ð10Þ j¼1

141

ð13Þ

where the dots denote derivatives with respect to time; j, y and c are roll, pitch and yaw angle, respectively; z is heave displacement; m is the mass of ship; g is the gravitational acceleration; hs is the righting arm; Msy is the moment due to yaw and sway motions; Md, Mwind, Mwave and Mcd are moments attributing to damping, wind, waves and water motion in cabins, respectively; Ixx is the moment of inertia about longitudinal axis through the center of gravity of the ship, including added inertia due to the outside water; and Ixz is the product of inertia relating to the center of gravity of the ship, including added inertia due to the outside water.

ð11Þ

where F is the response matrix of input layer with regard to hidden layer, superscript ‘‘ þ ’’ denotes the pseudo-inverse calculation. The detailed expression and examples of the algorithm can be found in Chen et al. (1991, 1992).

2.2. Multi-scale RBF neural network As the inputs of identification or prediction model might be multiscale, for instance, belong to different categories and value ranges, it is reasonable to employ different center width in activation functions, which is different from a common center width in conventional single scale RBFNN. The resulting network can be referred to as generalized multi-scale RBFNN, which provides more flexible representations of system dynamics and shows better generalization properties (Billings et al., 2007). In this application, the GOMS algorithm is improved by adopting multi-scale RBFs in the network.

Fig. 1. Schematic of online RBFNN prediction model (learning process).

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Discretize (13) into a difference form and neglect the terms which are hard to achieve precisely in this study, the roll angles can then be expressed as the nonlinear mapping of j, y, c:

jðt þ 1Þ ¼ f ðjðtÞ, jðt1Þ, yðtÞ, yðt1Þ, cðtÞ, cðt1ÞÞ

ð14Þ

Various algorithms can be implemented for Eq. (14) to give online predictions. Considering the nonlinearity of ship roll motion and the nonlinear nature of neural network, the sequential learning RBFNN is used in this study, whose location and number of hidden units and the connection parameters between the hidden layer and output layer are online tuned by the GOMS algorithm. The learning process and prediction process of the RBFNN-based prediction model are depicted in Figs. 1 and 2, respectively. As revealed in Figs. 1 and 2, the RBFNN employs three layers for the architecture: an input layer which transfers the input signal to the hidden layer directly; one hidden layer which produces the response to the input; an output layer which linearly combines the results of hidden units. In this paper, Gaussian function is employed as the active function of hidden unit and the RBFNN is online constructed by the GOMS algorithm. The processes of learning and prediction are conducted at each step. It is shown in Fig. 1 that at the t-th step of learning process, the inputs to the RBFNN are {j(t1), j(t 2), y(t 1), y(t 2), c(t1), c(t 2)} and the output is j(t). By learning the samples in the sliding data window sequentially, the structure and parameters of network are on-line adjusted. After the network adjusted at each step, the output j(tþ1) is consequently achieved by substituting {j(t), j(t1), y(t), y(t1), c(t), c(t 1)} as inputs to the adjusted network, as shown in Fig. 2.

Fig. 2. Schematic of online RBFNN prediction model (prediction process).

Table 1 Principal particulars of Yu Kun. Description

Values

Loa (Length overall) Lpp (Length between perpendiculars) B (Breath molded) D (Depth to main deck) T (Design draft) GT (Gross tonnage) V (Design speed)

116 m 105 m 18 m 8.35 m 5.4 m 6106 16.9 kn

4. On-line ship roll prediction using variable-structure RBFNN

4.1. Ship roll prediction based on turning test of full-scale trials The measurement data to be analyzed in this section is obtained during the turning test at sea from vessel Yu Kun, a scientific research and training ship of Dalian Maritime University. The main particulars of Yu Kun are given in Table 1 (Cao et al., 2009). Turning circle test is the conventional maneuvre used to determine ship maneuverability. The motion status of Yu Kun during sea trials is measured by the shipboard Ashtech ADU2, the most precise GPS-based three-dimensional position and attitude determination system available. The heading, pitch and roll measurements with accurate position and velocity for static and dynamic platforms is provided in real-time. Fig. 3 shows the ship’s track in turning circle test which was performed at initial speed of 16 kn, with rudder angle of 101 to the starboard side. The turning circle test was conducted under sea state of 5. ADU2 is a ship attitude measurement device, it can measure the values of j, y and c, as well as the information of position, vertical speed, course and speed over ground, etc. Therefore, in learning process, inputs are set as {j(t  1), j(t  2), y(t 1), y(t  2), c(t 1), c(t 2)} and the output is set as j(t). The parameters of GOMS algorithm are set as: N ¼50, r ¼0.1, l ¼2. At each step, the prediction is conducted once the network is on-line constructed in learning process. The values of j, y and c are depicted in Fig. 4.

800 Distance (m)

In this section, the effectiveness of the developed algorithm will be illustrated using the measured rolling angle from full-scale trials at sea. All the simulations for the proposed algorithm are carried out in MATLAB 7.4 environment running at 2.80 GHz (CPU) and 3.25 GB memory (RAM).

600 400 200

0

0

400

800

1200

1600

2000

2400

2800

3200

Distance (m) Fig. 3. Track of ship turning test with 101 rudder angle to the starboard side.

In this study, the sampling interval is 1.0 s and the experiment was conducted for 800 steps for ship roll identification and prediction. The predicted results of roll motion compared with the full-scale data are shown in Fig. 5. It is shown in Fig. 5 that the measured roll angle during the turning circle test in sea trial is highly corrupted by strong noises which are caused by the environmental disturbances and sensor noise. Whereas, the predicted output can track the change tendency of roll angle precisely. This shows that the variable structure RBFNN constructed by GOMS algorithm can on-line track the changes of ship roll dynamics efficiently. The identification and prediction error are depicted in Figs. 6 and 7, respectively. It can be seen from these figures that the prediction error is close to the identification error, which demonstrates the satisfactory generalization capability of the algorithm. The evolution history of the number of hidden units is illustrated in Fig. 8. The number of hidden units changes within

J. Yin et al. / Ocean Engineering 61 (2013) 139–147

143

φ ( °)

5 0 -5 0

100

200

300

0

100

200

300

400 Time (s)

500

600

700

800

400

500

600

700

800

500

600

700

800

θ (°)

5

0

Time (s)

ψ (°)

400 200 0

0

100

200

300

400 Time (s)

Fig. 4. Values of j, y and c in ship turning test (101 rudder angle to starboard side).

4 Predicted Value Real Value

3

Roll Angle (°)

2 1 0 -1 -2 -3 -4 -5 0

100

200

300

400 Time (a)

500

600

700

800

Fig. 5. Predicted results of ship roll motion compared with the full-scale data (turning circle test).

2

Identification Error (°)

1.5 1 0.5 0 -0.5 -1 -1.5

0

100

200

300

400 Time (s)

500

600

Fig. 6. Identification errors of ship roll motion (turning circle test).

700

800

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2.5 2

Prediction Error (°)

1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 0

100

200

300

400 Time (a)

500

600

700

800

Fig. 7. Prediction errors of ship roll motion (turning circle test).

9

450

400

7 6

Distance (m)

Number of Hidden Units

8

5 4

350

300

3

250

2 1 0

100

200

300

400

500

600

700

800

Time (s)

200 0

500

1000

1500

2000

2500

3000

Distance (m) Fig. 8. Evolution history of the number of hidden units (turning circle test). Fig. 9. Track of ship during 201/201 zigzag test.

the range of [3, 6] during the learning process. This is a rather small number of hidden units, whereas with regard to prediction error and learning speed, the algorithm’s prediction performance is satisfactory under circumstances of highly corrupted roll motion measurements.

4.2. Ship roll prediction based on zigzag test of full-scale trials Zigzag test is another conventional maneuvre used to evaluate ship maneuverability. To further validate the effectiveness of the proposed prediction approach, the full-scale experimental data of 201/201 zigzag test of Yu Kun from sea trial is adopted. During the learning process, the input terms of prediction model are similarly set as {j(t  1), j(t  2),y (t 1), y (t 2), c (t  1), c(t  2)} and the output is j(t). The parameters of GOMS algorithm are set as: window width N ¼80, accuracy threshold r ¼0.01 and consecutive time span l¼ 2. The scales of different types of inputs are set as 101, 51 and 401, which are roughly the range of corresponding inputs of j, y and c, respectively. The sampling interval is 1 s and 400 epochs are processed. Fig. 9 shows the ship’s track when performing 201/201 zigzag test at initial speed of 14.7 kn. The sea trial was conducted under sea state of 4, with wave height of 1.8 m. To show the changes of individual variable clearly, the values of j, y and c are depicted in Fig. 10.

The comparison of the predicted result with the measured roll angle is depicted in Fig. 11. It is noticed that the prediction results are in accordance with the measured data with rather small prediction errors. The variable RBFNN is online achieved by identification of data in the sliding data window. The evolution of identification error within sliding data window during the learning process is illustrated in Fig. 12. As can be seen from Fig. 12, the identification error ranges within [ 1.5, 2], which means that the network can accurately express the ship motion dynamics expressed by the sliding data window. The prediction process is simultaneously conducted soon after the identification process at each step. The evolution of prediction error at each step is shown in Fig. 13. It can be noted from Fig. 13 that the prediction error level is slightly higher than that of identification error, which shows the satisfactory generalization capability of the achieved variable RBFNN. The learning of GOMS algorithm is a dynamic process with the hidden units added and pruned simultaneously at each step. Fig. 14 shows the evolution history of the number of hidden units. The average number of hidden units is 5.715. It is noted from the results illustrated above that the RBFNN performs well in ship roll prediction with small identification and prediction errors, and the number of hidden units is small, which reduces the computational burdens and ensures its online applications.

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145

φ ( °)

10 0 -10

0

50

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150

200 Time (s)

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400

0

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100

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200 Time (s)

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350

400

0

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100

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200 Time (s)

250

300

350

400

θ (°)

5

0

ψ ( °)

50 0 -50

Fig. 10. Values of j, y and c in ship’s 201/201 zigzag test.

6

4

Roll Angle (°)

2

0

-2

-4 Predicted Value Real Value -6 0

50

100

150

200 Time (s)

250

300

350

400

Fig. 11. Predicted results of ship roll motion compared with the full-scale data (201/201 zigzag test).

2

Identification Error (°)

1.5 1 0.5 0 -0.5 -1 -1.5 0

50

100

150

200 Time (s)

250

300

Fig. 12. Identification errors of ship roll motion (201/201 zigzag test).

350

400

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2

Prediction Error (°)

1.5 1 0.5 0 -0.5 -1 -1.5

0

50

100

150

200 Time (s)

250

300

350

400

350

400

Fig. 13. Prediction errors of ship roll motion (201/201 zigzag test).

9

Number of Hidden Units

8 7 6 5 4 3 2 1

0

50

100

150

200 Time (s)

250

300

Fig. 14. Evolution history of the number of hidden units (201/201 zigzag test). Table 2 Experiment performance. Experiments

Execution epochs

Execution time (s)

Execution time per second (s)

RMSEIden (1)

RMSEPred (1)

Average number of hidden units

Turning Circle 201/201 Zigzag

800 400

2.5293 1.2512

0.0032 0.0031

0.4933 0.4307

0.5168 0.4909

4.125 5.715

The indices of root mean square error (RMSE) are employed to represent the effectiveness of identification and prediction, with RMSEIden and RMSEPred, respectively: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN PN ^ ^ i ¼ 1 ðyyiden Þ i ¼ 1 ðyyPred Þ RMSEIden ¼ , RMSEPred ¼ N N

ð15Þ

where N is the running epochs; y is the real measurement value; y^ Iden and y^ Pred are identified and predicted results of roll angle, respectively. The performances of proposed algorithm in the above experiments are listed in Table 2. It is noted that under condition of measurement data from fullscale sea trials, the algorithm operates at extreme high speed. The resulting RBFNN possesses parsimonious network structure at each step, which is attributed to the fact that the on-line predictive model focuses on the identification and prediction of system characteristics

in local time domain. The prediction error is quite close to the identification error, which shows good generalization capability and the ability to track the changes of time-varying system dynamics. It should be noted that due to the restrictions of measuring instruments, we cannot acquire enough information from the measurement data of full-scale sea trial, so the inputs information are insufficient for the proposed roll motion prediction model. The efficiency and robustness of the proposed prediction model can be improved by incorporating more useful information.

5. Conclusions A variable structure RBFNN is proposed for real-time ship roll prediction, which is on-line constructed by a sequential learning algorithm referred to as GOMS algorithm. The algorithm is based

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on the data in a sliding data window. The measurement of rolling angle for full-scale ship is employed to validate the efficiency of the proposed algorithm. Results demonstrate its effectiveness and efficiency in on-line identification and prediction applications. The future research works will focus on the incorporation of more useful information in the prediction model and improvement of the generalization capability and stability of the on-line neural prediction model.

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