Geodetic Investigation of Ship Roll Estimation for Manoeuvring Control of Cruise Liner Conveyance

Rüdiger Lehmann Dresden University of Applied Sciences Faculty of Geoinformation PF 120701 D-01008 Dresden r.lehmann@htw-dresden.de Bernd Koop HydroSupport Buxtehude Langenbeckweg 2 D-21614 Buxtehude, GERMANY Phone: +49 4161 / 554 557 Fax: +49 4161 / 554 597 berndkoop@online.de

1 Introduction The shipyard Meyer-Werft at Papenburg (Germany) builds luxury cruise liners for the international market. Two or three of such “swimming hotels” per year leave the dock at the river Ems. The latest cruise liner finalized there was the Celebrity Equinox (June 2009). This new and uniquely powersaving type of a ship has an extraordinary length of 315 metres and a beam of 37 metres. The gross tonnage is given to 122,000 tons. It shares these values with it’s sister ships Celebrity Solstice (finalized in September 2008) and Celebrity Eclipse (to be finalized in 2010). They are the biggest cruise liners ever built in Gemany so far. The order book of the Meyer-Werft contains orders for more large ships to be built during the next decade.

Unfortunately, the dock bassin of the Meyer-Werft is 36 km distant to the open ocean (North Sea), such that the ships must be conveyed via the small river Ems. Normally, this river with an average discharge of only 80 m³/s would be much to small for such large ships up to a draft of 8.50 metres. Therefore, it is necessary

In: A. Härting & J. Reinking (Eds.) Nautic Aspects of Ship Dynamics, Elsflether Schriften zur Seeverkehrs- und Hafenwirtschaft, Vol. 4, pp. 167-176, 2009

• to regularly dredge the navigation channel,

• to dam up the river before and during the conveyance, raising the water level by more than 2 metres − for this purpose, the Ems barrier has been built from 1998 to 2002 − and

• to apply advanced methods of Geodetic Navigation such that small toler-ances suffice to convey the ships savely.

The first two activities have negative effects on the environment: Dredging is bad for fishes (there are none at the moment) and damming up is bad for nesting birds along the riverbanks. (The lower Ems is therefore cynically called Meyer-Channel.) It is the task of Geodetic Navigation to minimize the necessary dred-ing and damming to an absolute minimum by optimizing their methods such that even smaller tolerances suffice to convey the ships savely.

In this paper we are mainly concerned with the problem of ship rolling and how it is optimally solved during Ems conveyances.

2 Geodetic Navigation during Ems conveyances The task is to navigate these large cruise liners savely to the North Sea. On this 36 km long journey we must pass two locks and two bridges (Fig. 1). For this purpose we install GNSS-antennas at bow and stern of the ships. In order to op-timize the signal quality they are placed on the most prominent positions, i.e. up to 50 m above water level. Using GNSS phase corrections we are able to com-pute the positions with centimetre-accuracy with a frequency of 1 Hz. This job is done since 1991 as a matter of routine by the Amt für Landesvermessung und Geobasisinformation Niedersachsen (Geodetic Authority of the German Federal state of Lower Saxony).

In order to bridge possible GNSS data gaps and interruptions, the GNSS receiv-er at the bow sends its position solutions to an inertial navigation system (INS). In conjunction with data from an inertial measuring unit the INS computes inte-grated GNSS+INS solutions with a frequency of 2 Hz. This computation is basi-cally a Kalman filter algorithm (see Grewall u.a. 2007). Different INS have been used, scientifically investigated by Röttger (2008). The current choice is HY-DRINS produced by iXSea (2008) using high precision faser optical inertial sensors. It outputs positions as well as roll, pitch and yaw angle with a standard deviation of 0.01°.

Permanently, we compute deviations of the current position of the ship from the middle line of the navigation channel. These values are presented to and used by the pilot for manoeuvring the ship.

Besides the current position, the pilot demands a prediction of the ship move-ment over the next one or two minutes. This computation is tricky because the GNSS antennas in their elevated positions strongly take part in the rolling movement of the ship. This effect is most undesired because the normal and spa-tially smooth curve of headway to be extrapolated for prediction is superim-posed by an oscillating behaviour caused by rolling. This complicates the pre-diction considerably.

We approach this problem from the viewpoint of discrete filtering. We can hard-ly rely on previous investigations of different authors because the situation of Ems conveyance is very special. The accuracy requirement is several orders of magnitude higher than in standard applications of ship dynamics. Only in hy-drographic surveying (Böder 2002, 2006) or manoeuvers in harbours (Benedict 2008) we get a similar situation.

Fig. 1: Conveyance of AIDAluna in Februar 2009, Passage of the Jann-

Berghaus-Bridge near Leer (Photograph: Heiner Unkel, Leer).

The hydrostatics of a ship is governed by two forces (Barrass et al. 2001): the force of weight FG and the force of buoyancy FB. The resultant of both forces, if not vanishing, causes a righting torque τ. In Fig. 2 we display the situation in a vertical cross section with the following notations: G = Mass centre of the ship (the centre of gravity of the body of the ship) B = Centre of buoyancy or centre of immersed bulk (the centre of gravity of the volume of water which the hull displaces) M = Metacentre (the point where all lines of buoyant force intersect)

Fig. 2: Hydrostatics of rolling (vertical cross section)

3 Data All data subsequently used were recorded during the conveyance of the Celebri-ty Solstice in September 2008.

In Fig. 3 we display a GNSS data set of one hour recorded at the bow antenna during the bridge passage of “Friesenbrücke Weener”. The high frequency con-tent of the data becomes visible after applying a high-pass filter, in this case a Gaussian filter with standard deviation (half-width of inflection points) of σ= 10s. The antenna oscillates about the smooth curve with an amplitude of several decimetres. (On a day with stronger wind we easily obtain more than one me-tre.) This effect is to a considerable degree caused by rolling and must be re-moved from the data.

A simple solution of the problem would be to take the roll angle computed by INS and to transfer the position of the antenna down to the water level. This is not possible because the location of the roll axis is unknown. This location de-pends on the cause of the rolling movement (wind, manoeuvring forces, etc.). In fact, in Fig. 4, we do not observe a striking similarity of the roll angle to the fluctuations depicted in Fig. 3. For example, the strong rolling at 13:45 causes only moderate fluctuations in the GNSS data. One explanation is that the roll

axis is elevated here. Another one is that we did not correctly split rolling and headway when tentatively applying the Gauss filter with σ=10 s in Fig. 3

Fig. 3: Conveyance of Celebrity Solstice,across track component [m] of bow

antenna,Gaussian high-pass filter with standard deviation of 10 s applied

Fig. 4: Conveyance of Celebrity Solstice, roll angle [mdeg]

4 Methods of filtering The choice of a filtering method turns out to be intricate because a method oper-ational at ship A must not necessarily work at ship B due a different ship dynam-ics or different wind conditions. This cannot be tested previous to the con-veyance. The method must be operational from the very start of the conveyance.

The following methods come into consideration:

1. Filtering in the frequency domain: This requires that the data sets are Fouri-er transformed to the frequency domain. If the typical frequency band of the rolling is known than we can remove its contribution from the data set and trans-form it back to the space domain. In Fig. 5 we display the roll angle in the fre-

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quency domain. Obviously we get signal frequencies in the range of 0.02 … 0.04 Hz. This corresponds to a roll period of 25 … 50 s, quite a reasonable re-sult. Unfortunately, when we want to predict the headway we are most interest-ed in the data at the right limit of the interval, where spectral domain techniques suffer from the leakage effect.

2. Filtering based on a wavelet-transform: This transform is better able to handle finite data sets. Up to now we did not try to apply it, but it is an interest-ing alternative to the Fourier transform.

3. Kalman-Filtering: This method is common in Geodetic Navigation, particu-larly in INS and GNSS techniques (Grewall et al. 2007). It is based on modeling the entire situation at the conveyance as a dynamical system. This system should not only contain the ship, but also wind and manoeuvring forces. Moreover, even the bathymetry should be considered because water can only flow where the river provides space. This system may contain a number of unknown state parameters to be estimated using the measurements. The method requires that the measurement errors are modelled correctly, usually by white noise. Unfor-tunely, this model would be much to complex for practical application and not fail-safe. Moreover, wind and manoeuvring forces are not measured at the mo-ment. First steps into this direction were not successful (Lehmann and Koop 2008).

Fig. 5: Fourier transform of the roll angle in Fig. 4

4. Least Squares Collocation: The follwing approach is very common in Geo-desy: A stationary stochastic process x(t) is split into three additive constituents:

• a smooth deterministic trend,

• a stochastic signal with a covariance function to be estimated and

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• a stochastic (usually white) noise.

The noise is equated with the inevitable measurement errors (GNSS/INS sensor noise). The signal is equated with the effect of rolling, and the trend remains to be equated with the normal headway of the ship.

Based on the covariance function of the signal we are able to estimate function-als of the signal. In the case of ship dynamical investigations we are interested in frequency and damping of the rolling as well as in the location of the roll axis. We also get error estimates of these estimated functionals. This method of filter-ing and prediction is due to Moritz (1973).

In this contribution we will restrict ourselves to method 4 (Least Squares Collo-cation) only. The stationary stochastic processes to be considered are the across track component x of the bow antenna provided by the GNSS receiver and the roll angle ξ. A trend function is sought in the space of polynomials in time t of degree n. For the roll angle ξ it is sufficent to set n=0 because if there would be neither rolling nor noise we would get ξ equal to a constant, further denoted by ξ0.

(1) (2)

The contributions s and n denote signal and noise respectively. Both signals sx and sξ are generated by one and the same process ψ, the true roll angle. They should be related to each other by (see Fig. 6):

(3) (4)

Dicsussion of (3) and (4):

• The measured roll angle may differ from the true one not only by noise, but also by an instrument offset .

• Additionally, it is necessary to take a time offset τ into account because the digital processes x and ξ take different paths and do not arrive exactly synchronously at the navigation computer.

Obviously, (3) and (4) mean that both signals sx and sξ are linearly depend. Therefore, we expect the empirical correlation coefficient of both signals

(5)

to be close to 1. Only “close to” because we get only discrete data and the split-ting of trend, signal and noise may not be 100% correct.

Hint: In equation (5) the noises seem to missing. This is not true because for τ≠0 the noises are neither self-correlated (white noise) nor correlated to the signal. Therefore, the noise-related terms in (5) cancel.

Fig.6:cross-section of the rolling ship for illustration of (3), (4)

Up to now we did not specify

• the degree of expansion n of the trend polynomial of x and

• the length of the interval t1-t0 of data to be considered for the determina-tion of the polynomial coefficients.(The whole set of data would not be good because this would require a large n, and polynomials do not always behave well in this case. Anyway, the older the data the more dispensible they become for prediction).

Additonally, we do not know the time offset τ yet. We exploit the fact that ρ(t) must be close to 1, when τ is correct and trend signal and noise are split success-fully. We use it as a criterium as follows:

(6)

For more mathematical derivations we refer to (Lehmann and Koop 2009).

5 Selected results We investigate the situation during the passage of “Friesenbrücke Weener” by the Celebrity Solstice at t1=13:46 UTC (see Fig. 3 and 4). Here we observe a strong rolling due to heavy manoeuvring. Simultaneously, great demands are made on the reliability of the prediction because the bridge provides just as much space as absolutely needed by such a beamy ship. For the determination of the maximum we compute the cross correlation coefficient in (5) on a grid of values (n,t0). The results are presented in Tab. 1.

Tab. 1: Maximum values of the cross correlation coefficient in (5) [percent]. („*“ means no significant correlation.) t1-t0 [s] 30 60 90 120 150 180 210 240

n=2 49 97 96 92 91 83 69 64 n=3 27 97 96 97 94 85 82 83 n=4 10 65 90 96 95 94 86 84 n=5 * 64 90 96 96 94 89 88

This shows that the splitting of trend, signal and noise can be made in a way that in fact a high correlation between the signals is achieved. Moreover, we observe no unique optimum in the sense of (6). Higher polynomial degrees n permit the approximation of longer data intervals just as well. Of course, this does not mean that the prediction is more reliable.

Tab. 2: Time offset τ [s] for which the maximum in Tab. 1 is attained. t1-t0 [s] 30 60 90 120 150 180 210 240

n=2 3 0.6 0.6 0.7 0.7 0.8 1.1 1.2 n=3 -2.2 0.6 0.8 0.8 0.8 0.8 0.8 0.8 n=4 2.0 0.8 0.9 0.8 0.8 0.8 0.7 0.8 n=5 * -1.3 0.9 0.8 0.8 0.8 0.9 0.9

Tab. 2 shows, at which values of the time offset τ the maximum value of the cross correlation is attained. For the values of t0 and n, where the filtering was successful (ρ>95%), τ is nearly constantly determined in the range of 0.6…0.8 s. The INS roll angle ξ arrives delayed at the navigation computer by 0.7 s with respect ot the GNSS solution.

The prediction of the headway track is now simply the extrapolation of the trend:

(7)

It remains to be shown, which polynomial degree n is optimal for this purpose. This can be judged by a comparison of the predicted track with the track actually observed. Though, this comparison must be regarded with suspicion because the actual track is influenced by the manoeuvres applied after time t1. Therefore, the prediction is probably better than this comparison indicates. In Tab. 3 the poly-nomial degree 4 seems to work best. This is not alway the case (see Lehmann and Koop 2009), that is why we recommend the method of cross validation (see last section).

Tab. 3: Deviation Δx of the actual track of the ship from the predicted track (7) during the next 60 s after t1=13:46 UTC. polynomial degree t1-t0 [s] ρmax [%] Δx [m]

n=2 60 97 0,5 n=3 90 96 2,2 n=4 120 96 0,2 n=5 150 96 1,0

6 Ship dynamical investigation Although the actual problem is solved, we can do more than that. The signal contains important information on the rolling behaviour like frequency and loca-tion of the roll axis. At the conveyance, the dynamical properties of the ship are as yet unknown. We determine the height of the GNSS-antenna above the roll axis by virtue of (3) and (4) as

(8)

and therefrom the location of the roll axis with respect to the ship reference frame or the water level. The problem with (8) is that the denominator can be very small or even zero for some point in time t. Consequently, it is recommend-ed to evaluate (8) only in the local extreme values of |sξ|.

We consider the case of t0=13:44 UTC, t1=13:46 UTC and n=4. The correlation coefficient assumes a reasonable value of 96% (see Tab. 1). Fig. 6 indicates that the roll axis is locally unsteady. But the result could be inaccurate due to poor error propagation. Note that the triangle in Fig. 6 is very acute-angled. Here we get |sξ|<0.40° !

Fig.7: Computed height of the roll axis above water level (49 m – e)

More results are presented in (Lehmann and Koop 2009).

7 Conclusions The derived method is based on a typical geodetic approach, called Least Squares Collocation. We point out that the whole potential of the method has not been exploited. Besides the predicted values (headway, roll axis) we also get the associated error estimates.

The method is operating automatically and in real time, apart from the choice of the polynomial degree n, where no unique optimum was found. Here we suggest the method of cross validation: From the current data set we reserve a subset to be used as test data, the rest acts as training data. The filtering and prediction is applied to the training data with various polynomial degrees n and the results are compared to the test data. The polynomial degree with the best prediction accu-racy is chosen for the processing of all relevant data.

Acknowledgement: We thank the “Landesvermessung und Geobasisinforma-tion Niedersachsen” (Geodetic Authority of the German Federal State of Lower Saxony) for providing the GNSS data.

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