3.1.

Power spectrum analysis

This paper uses the modified covariance method (MCOV) to analyse the power spectrum of the ship motion, and decomposes it into several periodic motions with different frequencies.

As a typical stationary random process, the ship-motion has same statistical characteristics in forward measurement sequences and reverse measurement sequences. The MCOV method uses the total energy of the prediction error, which is the minimum squares sum of the forward prediction error and the backward prediction error¹², to estimate the power spectral density of the ship motion.

The total energy of the prediction error is,

where, and (n) are the forward prediction error and the backward prediction error respectively, which can be calculated by

Obviously when

The total energy of the prediction error ε is the smallest, and the prediction accuracy of the model is highest. From equation (5), we can obtain:

By introducing of the vector , the linear equation (6) can be expressed in the matrix form

where

is p+1 order square matrix, and

After solving the equations (7), we can get · E_p, where A_p is the optimal prediction coefficient under the minimum error criterion.

In order to solve equation (7), Marple proposed an efficient recursive algorithm. The algorithm introduces additional error energy and , to calculate the autoregressive coefficient a_k (i = 1,2,…,p) and white noise σ^{2 13}.

After obtaining a_k (i = 1,2,…,p) and σ², the power spectrum of the ship motion can be obtained by the power spectrum density calculation formula, which is:

Taking the calculated power spectrum peak as the main periodic frequency of the ship motion, we can obtain several main periodic motion frequencies in the ship motion.

3.2.

Ship-motion prediction model

Based on the main periodic frequency of the ship motion estimated by the MCOV method, the ship- motion prediction model can be established as follows:

where, is the estimated ship motion, is the periodic motion frequency obtained by the modified covariance method, and are the amplitude and phase of the periodic motion, respectively.

By introducing , we can use the BP neural network (NN) method^{14, 15} to obtain the ship-motion prediction model parameters , and .

The BP neural network adopts a three-layer neural network structure. The input layer is the ship motion data, which takes one node. The output layer is the non-linear ship-motion prediction model, which also take one node. The hidden layer takes three nodes, to estimate the non-linear mapping from input to output:

where is the input of the BP neural network; G(x) is the unknown nonlinear mapping; f(x) is the activation function; N and M is the number of the input layer nodes and the hidden layer nodes, respectively. In the training of the BP neural networks, W_m and w_mj, the weights of the neural network, will be continuously adj usted to minimize.

where l is the number of the training data.

When the neural network training is completed, the ship-motion prediction model can be obtained.

4.1.

Ship-motion prediction experiment

This paper uses real ship roll motion data to predicate ship roll motion in advance 5 seconds and 10 seconds, respectively. The ship roll motion data is obtained by shipboard inertial navigation measurement (as shown in Figure 2), and the data’s sampling frequency is 10 Hz.

Figure 2.

40 seconds ship roll motion data.

A ship-motion prediction model is established based on 40 seconds ship roll motion data, and predict the ship’s roll angle in advance 5 seconds and 10 seconds. The prediction results are shown in Figures 3 and 4, respectively.

Figure 3.

Roll motion prediction in advance 5 seconds.

Figure 4.

Roll motion prediction in advance 10 seconds.

The experimental results show that, the ship-motion prediction method, which is based on the modified covariance method and neural network, can obtain effective prediction results in the given sea conditions. The deviation between the predicted roll angle and the real roll angle is less than 0.02 degrees.

4.2.

Comparative experiment

In order to better analyse the accuracy of the prediction method, we design four different experiments, by using 20- second and 40-second motion data to predict the ship motion in advance 5 seconds and 10 seconds, respectively. Every experiment will be carried out 50 times. After that, the average error and standard deviation of each experiment’s results will be calculated, and compared with the autoregressive sequence (AR), the neural network (NN) and the Wiener filter (Wiener). The comparison experiments’ results are shown in Tables 1 and 2.

Table 1.

Performance for four prediction methods in advance 5 seconds.

Table 2.

Performance for four prediction methods in advance 10 seconds.

The comparison results show that, when using 40-second motion data, the average error of the 5-second prediction results is 0.0030°, and the standard deviation is 0.0241°; the average error of the 10-second prediction results is 0.0060°, and the standard deviation is 0.0735°. Meanwhile, this method can obtain better prediction effect when the motion data is less than 40 seconds. The average error of the 5-second prediction results is 0.0018°, the average error of the 10-second prediction results is 0.0059°, and the standard deviations are 0.0171° and 0.0647°, respectively. Except for special indicator (20-second motion data, 5-second execution time), the average error, standard deviation and algorithm execution time of this method are better than AR, NN and Wiener. In that case, this ship-motion prediction method can obtain an effective prediction result in different and complicated sea conditions.