Ship Motion Data and Prediction

A particular set of roll motions recorded during sea trials was chosen as representative of typical service conditions. The choice was guided by advice from Royal Navy engineers. The data was from a Leander class frigate moving at 15 knots in quartering seas from the port side. Rudder and stabilizer fin angles, yaw motion and sway acceleration were also available for use as auxiliary variables. The data consisted of 5266 points sampled at half second intervals. Figure 5.1 shows a section of the time history.

The power spectrum plot Figure 5.2 shows the main frequency component at about 0.07 Hz. The autocorrelation function is shown in Figure 5.3. The cross correlations of the roll with the other available data are shown in Figure 5.4. Cross-correlation is sometimes a useful exploratory tool that can highlight cause effect relationships. The cross-correlation curve at positive lags represents the cause to effect, in this case how roll motion

is affected by the related variable. The negative lags give the reverse correlation of how roll affects the related variable. For roll-rudder cross correlation the most significant peak in terms of magnitude lies at a negative lag. The source of the peak is probably the feedback introduced by the helmsman. The feedback loop is also apparent in the roll-stabilizer plot.

Roll motion is affected by wave motion,stabiliser action and coupling with the other antisymmetric motions of the vessel yaw and sway. Additionally the roll equations contains nonlinearities such as v|v| damping and cubic spring that may be significant in certain circumstances. Rudder action excites both roll and antisymmetric modes which couple into the roll motions (Newman 1978) . It can be argued that rudder and stabilizer angles can be ignored since they are generated by the control systems aboard ship. Since the control laws are synthesised on the same motion measurements available to the predictor no additional useful information is contained in the rudder and stabilizer angle measurements. In the ARMAX model representation all the control terms u can be expressed as linear combinations of past and present outputs y . However time varying feedback may be useful, the issue is closely related to identification in closed loop

(Gustavsson et al 1977).

The rudder angle shown plotted in Figure 5.5, is apparently on manual control.

In this study the short term ship roll motion predictions will be used to generate advance warning of dangerous roll displacements.

Model order detection was based on the calculation of the F and Akaike statistics as the ARMA model order size was increased. Table 5.1 displays the results. Both tests favours an ARMA (6,6) m o d e l . However with the large data sets these test are prone to

choose larger than necessary models. A plot of prediction mean squared errors using the different ARMA models is more revealing.

Figure 5.6, and shows that the A R M A (4,4) or A R M A (5,5) are very close to a predictor based on a A R M A {6,6) model. The A R M A (5,5) model was selected as being satisfactory

The explicit multistep predictor was used to generate predictions up to ten seconds ahead, using the iterated forgetting factor discussed in Chapter 4. Initial parameters were set at zero. To assess prediction quality the signal was conditioned by subtracting its mean value. Figure 5.7 shows the normalized mean square prediction errors up to ten seconds ahead. A sampling period of one second was chosen;the data had been recorded at half second intervals. The evolution of the errors is interesting and in particular the sharp increases in prediction accuracy that happens at about 600 and 850 seconds. From the time history plot of the roll motion the reason for these dramatic changes in performance are not apparent. However the associated changes in the parameter estimates are clearly seen in Figures 5.8 and 5.9 which shows the parameter estimate evolution. Figure 5.10 shows a plot of the predictions at various distances superimposed upon the actual roll motion.

To see how close the predictions are to minimum variance the correlations of the prediction errors at distance up to ten seconds ahead were plotted and are shown in Figure 5.11 and 5.12. The quality is reasonable with correlations falling close to the two standard error limits after the prediction distance, however at greater lag distances the correlation curve shows significant

"wandering" outside the two standard error limits.

The effect of sampling time on prediction accuracy is displayed in Figure 5.13. Sampling times of 0.5,1,1.5 and 2 seconds were considered. The predictions are all based on a 5th order ARMA model. A sampling time of one second gives the best predictions up to ten seconds ahead.

It is disappointing to find that the prediction accuracy deteriorates rapidly at prediction distances greater than 4

seconds. The ship roll is however affected by both the Gaussian waves and a very non-Gaussian helmsman generating rudder action. It seemed prudent therefore to consider auxiliary signals that may improve the predictions.

5.3.1 Use of Auxiliary Signals

The roll response due to rudder angle needs to be identified to improve roll predictions. However since the rudder signal, Figure 5.6 is not persistently exciting supervision functions have to be included in the prediction algorithm to combat non persistent data (Astrom and Bohlin 1966,Isermann 1980) . With non persistent data no usable information about the process may be gained from the input and output signals. Linear dependent rows arise in the covariance matrix so that the identification problem becomes unsolvable.

The instability can be observed in the trace of the covariance matrix and the "bursting" of parameter estimates,caused by a rapid increase in the variance of the parameter estimates. Figure 5.14 shows the trace of the covariance matrix when the roll rudder sequence is fitted by a low order ARMAX model using Extended Least Squares. Once a non-persistent excitation has been identified by monitoring the trace, the remedy is the automatic switch-off of the parameter estimation routine. In general the recursive estimator becomes

0(t) = 0(t-l) + j(t) [F(t)] (5.1)

where j (t) is either 1 (estimation enabled) or 0 (inhibit) according to whether the data is exciting or not. In Figure 5.14 the trace is very irregular and it is not an easy matter to decide when to switch the estimator on or off. Various combinations using moving averages were tried and Figure 5.15 shows the "supervised " trace when one of these procedures was implemented.

algorithm and the roll signal predicted with the rudder signal as an auxiliary input. The auxiliary input was introduced as an additional 5 element vector in the data vector,appropriately scaled. Elements of the auxiliary vector are present and past values of the rudder signal. The predictor however converged to the same parameter values totally ignoring the rudder input signal. The rudder signal in this case is of no use to the predictor.

Stabilizer action ,Figure 5.16, should simply modify the vessels roll dynamics since it is based on the roll motion of the vessel. However the stabilizer forces may be a nonlinear function of roll velocity so that explicit knowledge of the stabilizer action would assist the predictor. The use of the stabilizer angle improved the predictions a small amount ,Table 5.2. Different auxiliary signals will in general be significant according to conditions and vessel headings.

The significance of the stabilizer angle is highlighted even further if it is assumed that the stabilizer signal is available for 5 seconds beyond the present time. Prediction quality once again significantly improves.

Prediction of Roll in Different Sea States

Lincoln (1983) investigated how the centre frequencies of frigate motion are affected by variations in sea conditions using data measured on board a Tribal class frigate during sea trials and concluded that (unlike aircraft carriers) the centre frequencies of frigate motion are affected significantly by variations in sea conditions.

For the Leander class frigate a similar analysis was carried out for roll motion. The power spectrum of roll under contrasting sea conditions were calculated and the results are shown in Figure 5.17 and 5.18, the variability is apparent.

Naturally the different spectrums translate in differing degrees of predictability. Figure 5.19 shows how prediction quality is affected by sea direction. The broadband spectrum of the head seas and quartering seas from starboard , where there is an absence of a sharp pronounced peak, give a difficult to predict signal. Even at short prediction distance of three seconds the prediction quality is poor.