The implementation of a ship manoeuvring model in an integrated navigation system

0

THE IMPLEMENTATION OF A

SHIP MANOEUVRING MODEL

IN

AN

INTEGRATED

NAVIGATION SYSTEM

ÀL

TEcmIIS cEE UNIVERS ITEIT

Scheepsliydromechan.ica

Archief

Mekelweg 2, 2628 cD Delf t

Tel: 015-786873/Fax:781836

In de literatuur over geïntegreerde navigatiesystemen wordt regeirnatig de meetruis wit verondersteld zonder dat gebruik gernaakt wordt

van een vormend filter in het

systeemmodel. Deze veronderstelling

is voor praktische toepassingen niet te

verdedigen. Het weergeven van de nauwkeurigheid van een navigatiesysteem door middel

van

één statistische parameter, de standaardafwijking, is

een verkeerde methode, welke

suggereert dat de meetruis onafhankclijk van de omgevingscondities

is.

Een

representatiever beeld zou een beschrijving zijn waarbij het energiespectrum van de meetruis, de dynamica van het voertuig, het type omgeving en liet weer

gegeven

worden. De huidige generatie navigatiesensoren aan boord van schepen is ontworpen

voor

stand-alone gebruik. Deze sensoren worden als een black-box

aan de gebruiker

verkocht. 0m de navigatiedata correct te integreren dienen de toekomstige

sensoren

zo ontworpen te worden dat deze ook het onbewerkte meetsignaal kunnen geven. Als dit niet mogelijk is, dient in elk geval een volledig overzicht van de intern gebruikte filtertechnieken gegeven te worden.

4.

Het ontwerp van Sensoren die gebruik maken van de zwaartekracht, zoals het gyrokompas en de VRU, kan waarschijnlijk aanzienlijk vereenvoudigd worden indien deze sensoren voor een geintegreerd navigatiesysteein gebruikt worden, daar het dan mogelijk wordt de versnellingsfouten mathematisch te corrigeren in plaats

van deze

door middel van een mechanische constructie te reduceren.

Modern optimization, estimation and control theory have been applied to ship operations with remarkable success

in

track keeping,

heading/eiigine

control,

weather routing, etc.. Further computerization will increase their importance. These subjects should be compulsory for students specializing in Maritime Operations. (Hideki Hagiwara, Weather routing of (sail-assisted) motor vessels, PhD-thesis TU Delft, 7 November 1989) Het valideren van theoretische modellen of methoden aan de hand van veldmetingen is

een noodzaak orn tot verantwoorde conclusies

te komen over de praktische

bruikbaarheid van een model of methode, daar gesimuleerde resultaten en

laboratori umopstelli ngeii aanzienlijk vaii de werkehijkheid kunnen afwijken. Daarom dienen veldmetingen een essentieel onderdeel voor een promotieonderzoek in de ingenieurswetenschappen te zijn. De begeleiding

die een beginnend zeilinstructeur/trice

krijgt

voordat

hij/zij

zelfstandig les mag geven op een ANWB-erkende zeilschool is professioneler geregeld dan die van een beginnend docent aan de TU Delft, terwijl de eerste meestal lesgeeft uit vrïjetijdsbesteding.

9.

De kosten die de VN moet maken ter uitvoering van een resolutie tegen een

agressieve mogendheid zouden ten gevolge van het profijtbeginsel verhaald moeten worden op de wapenindustrie die de desbetreffende mogendheid wapens geleverd heeft.

5.

Good filter design is a mixture of both art and science.

10. Het gebruik van zwaardboten in plaats van kielboten tijdens zeilles bevordert,

(A.Gelb, 1974. Applied optimal estimation)

11. Indien men de traditionele navigatietecliniek van de bewoners in de West Pacific

toetst aan de vijf criteria genoemd in dit proefcIìrift, nauwkeurigheid, robuustheid,

kosten, implementatie en integriteit, blijkt dat de huidige navigatiemetlioden alleen

op het criterium nauwkeurigheid beter zijn, daarentegen zijn de kosten veel lioger.

12. Bij de aankoop van navigatieapparatuur zou

uit veiligheidsoverwegingen een watersporter verplicht moeten worden een cursus te volgen betreffende de

nauwkeurigheid van navigatiemethoden en de consequenties daarvan tijdens het

gebruik. Uit menig gesprek in watersportcafe's blijkt telkens weer dat bij de meeste

watersporters een realistische kennis over dit onderwerp afwezig is.

Het copiëren van coinputerspelletjes door acadernici gebeurt meestal voor "hun

kleine neefje".

Het flamenco-compas is ongeschikt voor navigatiedoeleinden, desondanks kan een

nauticus er veel plezier aan beleven.

STELLINGEN

behorende bij het proefschrift

THE IMPLEMENTATION OF A SHIP MANOEUVIUNG MODEL IN

AN INTEGRATED NAVIGATION SYSTEM,

Johan H. Wulder. TECHNISCHE UNIVERSITEIT Laboratorium voor Scheepshydromechanlca Archief Mekoweg 2, 228 CD Deift TeL; 015- 786373. Fax; 015 781C38

Al slag DeÌft-Pijnacker A13

-J' ¡'

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Yau..Ll j AULA //./ -&__'l

_Isi!

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Den Haag L -'

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-- [

It il i [ i Vanat NS station: bu 128 richting Tanthof

bus 129 richting Rotterdam bus 63 richting Tanthof

taxi

wandelen ca. 15 min.

Technische Universiteit Deift

Atsiag Doit t-Zuid

UITNODIGING

Op maandag 22 juni 1992 te 10.00 uur za! ik mijn

proefschrift} getite!d

The implementation of

a ship manocuvring mode! in

an integrated navigation system,

in het openbaar verdedigen.

Voorafgaand aan de verdediging, orn 9.30 uur za! ik

een

eenvoudige

samenvatting

geven

van

het

proefschrift. Na afloop is er koffie.

U bent

van harte welkorn in de

aula van de

Technische Universiteit De!ft Meke!weg 5

Delft.

Johan H. Wulder

TECHNISCHE UNIVERSITEIT Laboratorium voor Scheepshydromechanica Archief Mekelweg 2, 2628 CD Deift Tel.: 015-786873- Fax: 015- 781338

THE IMPLEMENTATION

OF

A SHIP MANOEUVRING MODEL

IN

CIP-DATA KONINKLUKE BH3LIOTHEEK, DEN HAAG

Wulder, Johan Hendrik

The implementation of a ship manoeuvring model in an integrated navigation system. / Johan Hendrik Wulder.

-Delft : Technische Universiteit -Delft,

Faculteit der Werktuigbouwkunde en Maritieme Techniek. - Ill. Thesis Technische Universiteit Delft. With ref.

-With summary in Dutch.

ISBN 90-370-0063-O

Subject headings: navigation / Kalman Filter.

Copyright 0 1992,

Faculteit der Werktuigbouwkunde en Manitieme Techniek, Technische Universiteit Deift.

Alle rechten voorbehouden.

Niets uit dit rapport mag op enigerlei wijze worden verveelvoudigd of openbaar gemaakt sonder schniftelijke toestemming van de auteur.

All rights reserved.

No part of this book may be reproduced by any means, or transmitted without the written permission of the author.

Gebruik of toepassing van de gegevens, methoden en/of resultaten enz., die in dit rapport voorkomen, geschiedt geheel op eigen risico. De Technische Universiteit Deift, Faculteit der Werktuigbouwkunde en Maritieme

Techniek, aanvaardt geen enkele aansprakelijkheid voor schade, welke uit gebruik of toepassing mocht voortvloeien.

Any use or application of data, methods and/or results etc., occurring in this report will be at user's own risk. The Delft University of Technology, Faculty of Mechanical Engineering and Marine Technology, accepts no liability for damages suffered from the use or application.

THE IMPLEMENTATION

OF

A SHIP MANOEUVRING MODEL

IN

AN INTEGRATED NAVIGATION SYSTEM

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Deift,

op gezag van de Rector Magnifi cus, prof.drs. P A. Schenck, in het openbaar te verdedigen ten overstaan van een commissie aangewezen door het College van Dekanen

op maandag 22 juni 1992 te 10.00 uur door

Johan Hendrik Wulder, geboren te Landsmeer, Scheepsbouwkundig Ingenieur.

Dit proefschrift is goedgekeurd door de promotoren: Prof. dr.ir.H. G. Stassen,

ACKNOWLEDGEMENTS

The work described in this thesis is achieved with the stimulus and assistance of a great number of people. I would like to mention the contribution of:

- both the thesis supervisors Jacques Spaans and Henk Stassen, with their never failing interest in the project. Their supervision was complementary, which resulted in an almost complete coverage of the research field, but also in extra work;

- the staff of the Ship Hydromechanics Laboratory, especially Johan Journée for the

technical discussions and the use of his computer program Seaway, and Hans Ooms for his advice and assistance during the experiments;

- the employees of the Laboratory of Measurement and Control who created a very stimulating research environment;

-

the assistance of Leo Beckers, Fred den Hoedt, and Piet Teerhuis during the

preparation of the sea trials;

- Ben Wenneker for his assistance during the sea trials, his creativity prevented and

solved a lot of problems;

- Maarten de Vries, Dik Engeibracht and Harry Hessel of The Royal Netherlands

Naval College for their support during the sea trials;

- the crew of the "nv Zeefakkel" for their cooperation during my stay at their ship.

The installation of my measurement equipment was not a problem for them, although the ship was almost rebuilt by it;

- Datawell by, who made available the VRU;

- the survey department of Rijkswaterstaat for the use of their GPS reference station;

- Frank Soonieus, who designed the first VRU filter for his Master thesis;

Aad Gutteling for making the drawings;

- the Deift University Fund for their financially support; and

- especially my parents for their confidence, and their never failing support.

TABLE OF CONTENTS

i INTRODUCTION.

. . . i

I 1.1 Integrated ship navigation i i 1.2 The implementation of the system dynamics 3 1.3 Problem definition 6 1.4 Preview 7 2 MODELLING THE SmP DYNAMICS . 9 2.1

Introduction ...9

2.2

The ship manoeuvring model ...il

2.2.1 The choice of a ship manoeuvring model ...11

2.2.2 The adjustment of the moue model ...19

2.2.3 The estimation of the hydrodynamic coefficients

...25

2.2.4 Conclusions...32

2.3

The disturbances acting on a ship ...32

2.3.1 The waves ...32

2.3.2 The wind ...38

2.3.3 The current...41

2.4

The complete system model ...43

3 MODELLING THE SENSOR DYNAMICS. 47 3.1 Introduction . S . . 47

3.2

Navstar/GPS ...48

3.3 The gyrocompass 51

3.4 The speed log 53

3.5 The vertical reference unit 56

3.5.1 Introduction 56

3.5.2 The VRU system model 56

3.5.3 Adaptation of the dynamic model by experimental data 65

3.6 Conclusions 69

V

viii Table of contents

4 THE DESIGN OF AN INTEGRATED NAVIGATION SYSTEM 71

4.1 Introduction 71

4.2 The Kalman Filter . . . 71

4.3 The design philosophy . . . S 75

4.4 The positioning filter . . O . 77

4.5 The VRU filter 82

4.6 Conclusions 89

5 THE SEA TRIALS WITH THE "ny ZEEFAKKEL" . . 91

5.1 Introduction . . . . 91

5.2 The measurement plan 91

5.2.1 Introduction 91

5.2.2 The sea trials required for the identification procedure . . 93

5.2.3 The sea trials required for the validation of the Kalman Filter . 94

5.2.4 The measurement equipment 95

5.3 The sea trials 99

5.3.1 Performing the sea trials 99

5.3.2 Acquisition of data after the sea trials 100

5.4 Results 101

5.4.1 Introduction 101

5.4.2 Discussion of the sea trials required for the identification purposes 101

5.4.3 Discussion of the sea trials required for the validation of the

Kalman Filter 102

5.5 Conclusions . . . O

. 103

6 THE VALIDATION OF THE DYNAMIC MODELS 105

6.1 Introduction 105

6.2

The ship dynamics ...

105

6.3

The gyrocompass ...

109

6.4 Navstar/GPS 113

6.5

The vertical reference unit ...

115

ix

7 PERFORMANCE OF THE NAVIGATION SYSTEM 119

7.1 Introduction 119

7.2 The VRU filter 119

7.3 The navigation filter 120

7.3.1 Introduction 120

7.3.2 The accuracy of the navigation filter 123

7.3.2.1 Introduction 123

7.3.2.2 The accuracy of the estimated state variables . 123

7.3.2.3 The influence of errors on the initial state vector . 127

7.3.3 The robustness of the navigation filter 134

7.3.3.1 Introduction 134

7.3.3.2 The failure of the gyrocompass 134

7.3.3.3 The failure of GPS . 139

7.4 Discussion of results 146

8 DISCUSSION OF THE INTEGRATED NAVIGATION SYSTEM. . . . 149

8.1 Introduction 149

8.2 The ship dynamics 149

8.3 The sensor dynamics 150

8.4 The integrated navigation system 152

8.5 Concluding remarks 154 REFERENCES 155

SYMBOL LIST ...

. 161

SUMMARY ... 167

SAMEN VATTING ... 169

CURRICULUM VITAE . 171

i INTRODUCTION

1.1 INTEGRATED SHIP NAVIGATION

Throughout the ages man has been sailing over the oceans to trade, to fish or to make

war. In the beginning navigation was based on knowledge of waves, current, colour and taste of the sea bottom. Some of the seafarers were assumed to have a cognitive map or

internal representation as a reference system for their navigation [Lewis and

George,1991J. The knowledge needed for the navigation of ships was passed on from

generation to generation. In Western Europe the need for a more scientific approach was

felt in the 15th century when, starting with Prince Henry the Navigator in Portugal

(1394-1460), the great voyages of discovery began. The determination of longitude was

such a problem that on the ist of April i600 the States-General in the Netherlands

promised a premium of 5000 Dutch guilders and a yearly stipend for the discovery of a reliable method to determine longitude [Davids,1986]. Today, in contrast, for almost the

same sum it is possible to buy a hand-held GPS receiver which gives a 3 dimensional

position worldwide within an accuracy of loo m.

Today also ships sail over the ocean for different purposes, each type with a different

kind of navigational requirement. A merchant vesel which sails on the open ocean does not require highly accurate positioning, but when the same vessel approaches the coast a higher degree of accuracy is needed. A fisherman uses a data set containing the positions

of the good fishing grounds. To return to these grounds he requires at least equipment which has a high repeatability, or in other words a navigation system with systematic errors. A yacht owner would like to have equipment which is easy to use, which is

compact and which uses little energy. So, every sailor makes his own specific demands on his navigation equipment.

Three groups of navigators make very high demands upon the performance of their

navigation system. First, in the field of hydrography the position data which are used for

a nautical chart need to be measured with more accuracy than the generally available navigation systems are able to provide. The second group of demanding users can be

found in the offshore oil industry. An example of their requirements is a seismic survey

which is normally executed to get an impression of the geological structure of the sea

bottom. In order to process the seismic data the positions of the hydrophones used have to be known very exactly. Finally, military applications usually require a high degree of accuracy, for example during mine hunting. Here the position of every object on the sea

2 In.roducLzoi

bottom is screened and catalogued. At regular intervals a part of the sea bottom is

re-examined, and if a new object is detected it must be identified. To distinguish old and new objects the position of the object has to be measured with an accuracy better then 2 metres. To obtain such a high navigation performance several kinds of sensors are often

used of which the position data are simultaneously processed to provide integrated navigation.

In the literature on this subject several integrated navigation systems

have been

described, each of which is based on different combinations of positioning sensors. These

systems can be distinguished by means of the kind of data and knowledge used to estimate the ship's position. Fig.1.1 shows all the system dynamics involved in the

navigational process, providing an overview of the integrated navigation systems which

are described in literature.

In Fig.1.1 it can be seen that the ship is controlled by the input i(t) which represent the

rudder angle and the propeller propulsion. The ship is also influenced by the system noise w(t) such as the wind, waves, and the current. The ship will respond to i(t) and w(t), which results in a position, velocity and heading of the ship: the state x(t). This

state is measured by different kind of sensors like a speed log, a compass, and positioning

systems. Each sensor has its own dynamics and noise

characteristics

s(t), so the

measurement vector y(t) is disturbed. Hence, the final measured position of the ship is

influenced by the ship dynamics, the sensor dynamics, the system

noise, and the

measurement noise.

1(t)

w(t)

ship

_x(t)

sensors

-J

Figure 1.1: Block diagram of the system dynamics involved in the navigation process, with

w(t) the system noise;

s(t) the measurement noise;

x(t) the state of the ship;

y(t) the measured output; and

i(t) the input.

1.2 The implernentat2on of the sistem dynamecs 3

From a study of the literature three categories of integrated navigation systems can be discerned. Each category uses a different combination of the four points mentioned

earlier: the ship dynamics, the sensor dynamics, the system noise, and the measurement noise. These three categories are the systems which make use of either:

- The measurement noise statistics; if two types of navigation equipment, each having

different noise characteristics, are integrated, an improvement in the navigation

performance can be expected. In most cases one system has high frequency errors and

the other possesses a slowly varying error as, for instance, occurs in the integration

of GPS with

an inertial navigation system {Napier,1990; Diesel,1987]. Other combinations also occur {Polhemus,1988; Maenpa,1978].

The sensor dynamics and the measurement noise statistics; besides the positioning equipment a speed sensor, a roll/pitch sensor, and a heading sensor are also used,

which makes it possible to model the ship's influence on the sensor, for example the

deviation of the compass caused by the speed and the accelerations of the ship, the

error of the speed log caused by the pitch and the roll, and the position error caused

by the roll of the ship. Examples of this type of integrated navigation system are

given by [Meyer,1984; Spaans,1988; Cross and Pritchett,1986; McMiIlan,1987].

- The ship dynamics, the system noise statistics, and the measurement noise statistics;

the ship's state is estimated by means of a Kalman Filter, which makes use of the ship dynamic model, and the statistics of the system noise and the measurement

noise. The measurement noise is often modelled as white noise and the sensor

dynamics as indicated in Fig.1.1 are neglected. This type of system is often used for

special applications of navigation techniques, like dynamic positioning [Balchen

et al.,1980], track prediction {Passenier,1989] or automatic steering [Amerongen,

1982].

In Dove and Miller [1989] an overview of the available literature on this subject is given. None of the navigation systems described above integrate all the available information

which is depicted in Figli. They may omit the information from the sensor dynamics

or the ship dynamics or both these sources. So, if both the dynamics of ship and sensors are used an improvement in the navigation performance may be expected.

L2 THE IMPLEMENTATION OF THE SYSTEM DYNAMICS

In the previous section it was reasoned that the use of the system dynamics could lead to

4 Introd1iciion

developed. Firstly the technique which can be used to design an integrated navigation

system is considered after which what can be expected of the navigational performance of such a system will be discussed.

A technique which can be used to integrate all the available navigation data (the ship

dynamics, the sensors dynamics, the system noise statistics and the measurement noise

statistics) is the Kalman Filter, KF. As a navigation filter the KF has some interesting properties: the filter can generate a real-time estimate of the state of the ship; the filter

is based on a recursive algorithm, which means that the calculation time required for a

state update

is

independent of the number of samples;

a KF gives an optimal

reconstruction of the state of the ship under certain conditions; the KF can reconstruct

states which are not measured; and a KF can handle the different kind of data described previously. Here the main principles of the Kalman Filter are discussed, Fig.l.2, for more detail reference should be made to [Anderson and Moore,1979; Gelb,l974.

w0(t) SF'

w(t)

ship _{x (t)} sensors

model of ship and sensors

S 2

(t)

(t)

-

y(t)

- (t)

Kalman Filter

Figure 1.2: The Kalman Filter, with w0(t), s0(t) the white noise sources;

SF' the shaping filter of the system noise;

SF2 the shaping filter of the measurement noise;

e(t) the innovation sequence;

i(t) the estimated output;

(t) the estimated state; and

K(t) the Kalrnan gain.

1.2 The implementation of the system dynamics 5

Fig.1.1 illustrates that a ship can be represented as a system with an input vector i(t)

which contains the rudder angle and the propulsion force. The response of the system is

described by the state vector x(t) which can be measured by certain sensors finally

resulting in the measured output vector y(t). This process is reconstructed as part of the

Kalman filter, the input i(t) is measured and fed to a mathematical model of the ship and the sensor dynamics, which results in an estimated output 5(t). The estimated

output (t) differs from the measured output y(t) because the measurement noise s(t) is

absent and the system disturbances are not acting on the model of the ship. The

difference between y(t) and (t), the signal e(t), is weighted by the Kalman gain K(t)

and fed back in the model. The value of K(t) depends on the statistics of w(t) and s(t). So for a relatively large s(t) in relation to w(t), the value of K(t) is small; the KF gives

less attention to the measurements and more attention to the system dynamic model output. With a relatively small s(t) in relation to w(t) the KF gives more attention to the measurements and less to the output of system dynamic model. If the system is

linear it leads to an optimal reconstruction, *(t), of the state of the ship x(t). The theory

of the KF supposes also that the noises acting on the ship and sensors are white noise.

Under real circumstances this condition for the noises will never be fulfilled, but it can be achieved by modelling the noises by using two shaping filters, SF' and SF2, which are fed by white noise.

Until now it has been reasoned that it is technically possible to design an integrated

navigation system based on a KF which uses all the available navigation data: the ship

dynamics, the sensor dynamics, the statistical quantities of the system noise and the measurement noise. The next step is to see whether the performance of such a system will improve in relation to a system which does not use the system dynamics. The

performance of navigation equipment can be evaluated in relation to:

- accuracy, which is described by the statistical quantities of the position error;

robustness, which is the sensitivity of the navigation system to unpredicted events,

like the failure of a sensor;

- integrity, which is defined as the certainty that the user will be informed promptly

when the navigation system fails. The failure of the system could for instance result from the malfunction of a beacon;

- cost of the system; and

- implementation consequences, like space required, energy consumptions, etc.

These five points can be evaluated for the concept on the KF based integrated

navigation system. It is likely that the accuracy of the system will increase when more

6 Introduc,ion

Moreover, the KF will give an optimal reconstruction of the state of the ship under certain conditions. The robustness of the KF based integrated navigation system will also be better than that of a system which does not integrate the system dynamics; the

theory of the KF shows that under certain conditions it is possible to reconstruct a state

which has not been measured. So, the consequences of a failure of a sensor are not disastrous because the KF can also estimate the same variable. It is probable that the integrity of the system will increase when more is known about the behaviour of the system dynamics it is easier to detect a deviation in the measurement vector and, with that, the failures of the navigation equipment. The cost of the system is determined by

the processing unit of the KF, usually a PC which is already installed on board, and, in

addition, the cost involved in the identification of the system model. The

implementation of a KF based navigation system is less flexible than that of navigation

systems which do not require the system dynamics. The need for a model of the ship

dynamics means that for every ship on which the KF based navigation system is

installed, the system dynamics have to be modelled. On the other hand it becomes possible to use all the available sensors with the KF based navigation system. For the performance of the KF based navigation system it could be concluded that accuracy,

robustness and integrity will increase at the cost of the flexibility of the system.

In this section it is shown that the implementation of the dynamic models of the ship

and the sensors is a logical step in the development of an integrated navigation system, following on from the present systems which integrate two sensors or make use only of

the sensor dynamics. The KF can be the basis for an integrated navigation system which may use different kinds of sensors, depending which are available; its final performance will depend on the type and the quality of the sensors used. So, the improvement in the navigation performance will be the result of a better processing of the data measured by the available sensors.

1.3 PROBLEM DEFINITION

In the previous section it was reasoned that the use of the system dynamics of the ship

and the sensors in an integrated navigation system can lead to a better

navigation

performance. This thesis investigates whether the implementation of such a

ship

dynamic model in an integrated navigation system is feasible. Feasibility implies that the installation of such an integrated system on board of a ship does not lead to high costs. It must be possible to obtain information relating to the sensor and ship dynamics

1. Preview 7

without too much difficulty and the integrated navigation system itself must be able to deal with different kinds and types of sensors because these differ for every individual

ship. To obtain such an integrated navigation system the ship and sensors dynamics have to be analysed and modelled. These system models have to be evaluated with

regard to their usefulness as a system model in a KF. Further, the system models of ship and sensors have to be implemented in a KF, which results in the integrated navigation system. Finally the performance of the integrated navigation system must be validated on board a ship.

1.4 PREVIEW

The knowledge of the ship dynamics is the basis of an integrated navigation system. In Ch.2 a ship dynamic model will be derived. It will be shown that it is possible to fit the model to a specific ship by just five manoeuvring trials. The influence of the disturbances acting on the ship are included in the model.

Ch.3 gives an overview of a series of possible sensors on board of the ship. The dynamics

of most of the sensors are given in the literature, the dynamics of the other sensors are

derived on the basis

of the construction of the sensors. The behaviour of the

measurement noise of the sensors is estimated.

The design of the integrated navigation system is shown in Ch.4. The implementation of

the non-linear ship dynamic model in the KF and the choice of the state vector will be

discussed. Finally the integrated navigation system is given.

In Ch.5 the sea trials which were used to validate the system model and the Kalman

Filter are described.

In Ch.6 the theoretical models of Chs 2 and 3 are verified by means of sea trials.

Whether it is possible to identify the ship dynamic model in practice is verified. The accuracy of the model is determined to evaluate whether it is useful for an integrated navigation system. The sensor dynamics are compared with the theoretical dynamics

described in Ch.2 and Ch.3.

In Ch.7 the performance of the integrated navigation system is tested by means of sea trials. The criteria, accuracy and robustness, will be discussed. During the sea trials a

limited number of sensors was available. In this chapter a few sensors for which the data is simulated are added, to test their influence on the system.

In the last chapter, Ch.8, the conclusions will be given. The properties of the navigation system will be discussed. Some suggestions for further research are given.

8

In this

thesis theories from various disciplines

are used to obtain the integrated

navigation system: navigation, measurement and control, and ship hydrodynamics. This has two consequences:

The reader is probably not familiar with all of these fields, and does not yet have a complete overview of the theories used. The basic theories are therefore explained briefly and extensive references to the literature are given. Only the properties of a

model or a theory are given in the main text.

- The symbols of the different disciplines may conflict. The choice has been made to

use the symbols used in ship hydrodynamics literature and change the symbols used in the control theory in so far as these conflict.

2 MODELLING THE SHIP DYNAMICS

21 INTRODUCTION

An overview of definitions and assumptions which are used in the modelling of ship dynamics is given. In Fig.2.1, a block diagram of the ship, it can be seen that three

quantities are of importance in describing the ship dynamics:

- the input i(t) of the ship, it is assumed that a general survey vessel is used, with the

rudder and the propeller as input;

- the disturbances w(t), which are the external forces acting on the ship, include the influence of wind, waves, current, loading of the ship, and restrictions of the water like depth or shore. In this thesis it is supposed that the ship sails in unrestricted

water, and that the draught of a survey vessel is constant; and

- the state of the ship x(t).

The ship dynamic model in Fig.2.1 is assumed to describe a ship as a rigid body with six degrees of freedom, DOF (Fig.2.2). The DOF are:

- surge the longitudinal motions of the ship (x-axis direction);

- sway the transverse motion of the ship (y-axis direction);

- heave the vertical motion of the ship (z-axis direction);

- roll the rotation around the x-axis;

- pitch the rotation around the y-axis; and

- yaw the rotation around the z-axis.

These ship motions are described by the theory of ship hydromechanics. The theory of ship hydromechanics distinguishes:

- Manoeuvring describes the motions which are caused by the propeller, the rudder, the

wind and the current. Those motions have a low frequency character (<'O.2 rad/s).

Three DOF are of interest for manoeuvring: the horizontal translations and the

w(t)

i(t)

ship x (t)

Fzgure 2.1: Block diagram of the ship dynamics.

Figure 2.2: The definition of the coordinates, with

u(t) the longitudinal velo city;

v(t) the transverse velocity;

r(t) the rate of turn, b(t);

x,y,z the ship fixed coordinate system;

the earth fixed coordinate system;

«t)

the heading;

(t) the roll angle;

t9(t) the pitch angle; and

8(t) the rudder angle.

FRONT VIEW

3t)

X

2.2 The ship manoeuuring model 11

heading. Sometimes the heel angle is included too.

Seakeepin describes the motions of the ship which are caused by the waves. The

waves act on all the six DOF of the ship. The motions caused by the waves are

mainly in the frequency band of the waves (>±0.4 rad/s).

To model the ship dynamics a manoeuvring model, which describes the relation between the input i(t) and the state x(t) of the ship (Section 2.2), is used. The motions caused by the waves are modelled as system disturbances (Section 2.3.1). During the navigation of a ship both these motions occur; it is assumed that the high and low frequency motions

can be superimposed. In the literature a few publications on this subject can be found, mostly as part of a model which is used for control purposes [Balchen et al.,1980].

The next section describes a manoeuvring model which can be used in an integrated

navigation system. The third section deals with the disturbances acting on the ship.

2.2 THE SHIP MANOEUVRTNG MODEL

2.21 The choice of a ship manoeuvring model

In the literature several kinds of manoeuvring models are described [Abkowitz,1969;

moue et al.,1981:2; Nomoto,1978]. The manoeuvring models are designed for purposes like the design of a steering controller, a ship bridge simulator or the prediction of ship dynamics in the initial design stage. To choose one of the models for implementation in a navigation system, a comparison has been made between the properties of three different manoeuvring mode]s which are representative of the present manoeuvring models found in the literature. The final choice of model depends on the demands that the navigation system makes on its system model. First the three manoeuvring models are discussed. The model of Abkowitz [1969]. Abkowitz designed a manoeuvring model to investigate

the steering qualities of a ship, the manoeuvring performance of large tankers in the

initial design stage, and the design of control systems. The ship dynamics are modelled

on the basis of the Newtonian force equation. The forces which act on the ship are

assumed to be functions of the speed of the ship, the accelerations, and the rudder angle. Those functions are developed in a Taylor expansion of the third-order, in formula:

12 Modelling the ship dynamics

X(t) f{u(t),v(t),r(t),û(t),'i(t),(t),8(t)}

= X0 + Xu(t) + Xv(t) +

+ X558t5(t) (2.2.1)

with X(t) the hydrodynamic forces in the longitudinal direction of the

ship; and

X0, X, Xx,, the Taylor expansion terms with X= , etc. in the starting

point X.

Similar equations can be given for the transverse hydrodynamic force and the moment around the z-axis [Abkowitz,1969]. Abkowitz showed that some of the Taylor terms (also called the hydrodynamic coefficients) are equal to zero. After eliminating the coefficients which equal zero, 63 coefficients remain. According to Abkowitz [1980] it is possible to estimate these coefficients for different kinds of ships by means of scale model tests or full scale trials. The hydrodynamic coefficients for one tanker, the Esso Osaka, are given.

These can be scaled for other comparable ships. In Abkowitz [1980] the model is

expanded by some terms to include the number of propeller revolutions in

the manoeuvring model.

In [Abkowitz,1980] the model for the 280.000 dwt tanker, the Esso Osaka, is evaluated.

This tanker has a length of 343 m. The hydrodynamic coefficients of that ship were

estimated by means of four sea trials, which were a turning circle (350) and three zigzag manoeuvres (100/100, 200/200, 50_250_50) at a speed of about seven knots. The turning

circle and the zigzag manoeuvre were two of a series of standard manoeuvringtrials which are often used for navigation purposes [NMI,1977; Glas and Schutte,1976]. The turning circle is initiated by a constant rudder order, here 350, which causes the ship to sail in a circle. The zigzag manoeuvre results in a sinus shaped track which is the result

of the successive rudder orders over port and starboard. The criterion for giving the

opposite rudder order, 6=x, is the deviation of the heading to the ship's average

heading, ¿=x°. The notation is often given as (S=100/1'=100) or

(100/100). The hydrodynamic model obtained is validated by means of simulation of one of the turning

circles (350) and a zigzag manoeuvre (100/100), which were also used for the

identification of the hydrodynamic coefficients. The simulation results of the trials are presented in time series for '(t),

r(t), u(t) and v(t). The accuracy obtained for the

velocity is about 0.15 rn/s and for the rate of turn 0.03°/s; the simulated headingdrifts away from the measured value, and the maximum drift is about 200/30 minutes. In

2.2 The ship monoeuvr2ng model 13

The model of moue et al. [1981:2]. moue designed a manoeuvring model intended to predict the ship manoeuvring performance in the initial design stage for different types of

ships. The basis of the model is again the Newtonian force equation. The way the

hydrodynamic forces are estimated differs from that used by Abkowitz. The forces are directed to their origins: the hydrodynamic forces on the hull, the rudder forces, and the thrust given by the propeller. In the formula

X(t) = X11(t) + X(t) + XR(t),

(2.2.2)

with XH(t) the hydrodynamic forces on the hull;

X(t) the thrust delivered by the propeller; and XR(t) the rudder force.

The forces XH(t), X(t), and XR(t) are modelled on physical principles, based on fluid mechanics theory. If the real forces differ from the theoretical forces, such as those

caused by turbulence, correction factors are implemented in the force model. Some forces

on the hull, which are assumed to be a function of u(t) ,v(t), r(t), u(t),

'(t), t(t) and

are modelled by Taylor expansions. The number of Taylor terms was reduced as far

as possible in an empirical way [moue et

al.,1981:l] which resulted in about 25 coefficients to describe the behaviour of the ship. The exact number of coefficients

depends on the propeller model. The values of the coefficients are estimated empirically;

each coefficient is described as a function of the mean dimensions of the ship (length,

breadth, etc.). Those functions are fitted to the known coefficients of a series of ships. In this way, it is possible to estimate the hydrodynamic coefficients for a particular ship if the mean dimensions of that ship are known.

The model is tested for different kinds of ships, such as a container ship, a general cargo

boat, a bulk carrier or a tanker; all these ships were longer than 160 m. Sea trials

performed with these ships were compared with the predictions of the manoeuvring

model. The manoeuvres used were a turning circle (350), a zigzag manoeuvre (100/loo), and a spiral manoeuvre. The spiral manoeuvre gives information on the relation between

the rudder angle and the rate of turn of the ship. The speed of the ships used was

between 15 and 25 knots, dependillg on the type of ship. The results of the manoeuvring model are given in three different forms: the sailed track {x0(t),y0(t) }; the time series

for v(t) and (t); and the relation between the rudder angle 8(t) and r(t) during a steady

turn.

The accuracy of the results given is comparable with the model of Abkowitz. In

14 Modelling he ship dynamics

six different turning circles (100, 20°, 30° starboard and 100, 20°, 300 port), a spiral test,

and three zigzag manoeuvres (100/100, 200/200 and 300/300) are comparable with the

results previously discussed. The speed of the vessel during these manoeuvres was 11

knots.

The model of Nomoto [1978]. Nomoto designed a manoeuvring model which is used to

investigate the behaviour of human control by means of a bridge simulator, which

became important with the increasing size of the tankers used. Nomoto started with two uncoupled differential equations which describe the yaw and surge motions of the ship.

To simulate realistic behaviour of the ship, the yaw equation was extended with a term

which described the influence of the change of speed on the rudder. The surge was extended with a term to include the speed reduction caused by yaw. The interaction

effects between sway and yaw are not included, and also the interaction effect between surge and yaw caused by the hull is not modelled. The ten parameters of the final model

can be estimated with tests with a free sailing scale model or by sea trials. The

manoeuvring model is validated by means of a free sailing scale model. The coefficients

of the manoeuvring model are estimated by a

spiral

test and a

series of zigzag

manoeuvres. Afterwards one of these zigzag manoeuvres and a stop manoeuvre are

simulated by the manoeuvring model. The results of the zigzag manoeuvre are presented

as a time series of «t), r(t) and i(t). The results of the stop manoeuvre are presented as

the track sailed, and the time series of the velocity are given. The accuracy obtained is of the same order as that of the Abkowitz model.

The properties of the existing manoeuvring models are discussed. The next step in

choosing a manoeuvring model is a discussion about the demands which have to be made

for a navigation system. These demands can be divided into two categories: First the demands which the KF makes on its system model to fulfill the assumptions in the

theory of the KF, and second the practical consequences of the manoeuvring model for use in a navigation system.

To judge a model on its usefulness as system model in a KF, three items have to be

taken into account:

- The linearity of the model; the theory of the KF is based on linear system theory, otherwise the KF performance is not optimal. This raises a problem, because none of

the three manoeuvring models discussed is linear. In Ch.4 it will be shown that this

problem can be solved in practice.

.2 The ship manoeuvrtng model 15

time available is limited. Thus the model which requires the lowest calculation time, the model of Nomoto, is favoured.

The knowledge of the ship dynamics; the KF theory supposes that the system model

is an exact representation of the ship dynamics. This can be judged on basis of the results presented in the literature which are the same for the three manoeuvring

models discussed. Therefore, none of the previously described models is favoured at

that point. A second criterion to judge the system representation is the way the model is related to the physics in reality. Then, it appeared that the Nomoto model

neglects the interaction forces between the three motions in the horizontal plane. The model of Nomoto is at a disadvantage here, as the other two models (Abkowitz and moue) use fully coupled equations.

The next step to discuss is the manoeuvring model in relation to the navigation process. Here the practical implications of the choice of model are listed.

The complete manoeuvring range. During the navigation of a survey ship different

kinds of tracks are sailed; sometimes a straight track is sailed full speed ahead,

sometimes curves are sailed at low speed. The manoeuvring model must give a good

prediction in both the situations. It is difficult to compare the three manoeuvring models on this point, as all three the models are validated for only one or two

situations, and all are tested for a relatively high speed of the vessel. No conclusion can be given here.

The type of ship. The navigation system has to be used on different kinds of ships. In

hydrography usually ships with a length less then 100 m are used. Results of the

manoeuvring models of Abkowitz and Nomoto are given for one ship, a tanker, but both the models were designed for different kinds of ships. ¡noue tested the model for different kinds of ships, all with a length greater than 160 m. The model of ¡noue was tested by Boer [1983] for a vessel of 46 m. So, the model of moue has an advantage in relation to the other two models; moue [1981:2] shows that the manoeuvring model can be used for different types of ships.

The cost. The only uncertain factor about the cost of a manoeuvring model is the costs involved in the estimation of the hydrodynamic coefficients. For the three

manoeuvring models

three methods are

given to estimate the hydrodynamic

coefficients:

* Model basin tests; the costs of these are comparable to the addition of a second positioning sensor like an inertial navigation system. So, one advantage of the

proposed navigation system, low cost, will be lost.

required equipment is already installed, and only one day of manoeuvring is

required. If it is possible to plan that manoeuvring day on a day that the ship is not used for other activities the cost will be only the fuel costs and perhaps the labour.

* Empirical data; moue showed that it is possible to use known coefficients from

other ships to estimate the hydrodynamic coefficients of a particular ship. The cost of that method are just the use of a computer program.

The model of moue is favoured because the hydrodynamic coefficients can be

obtained at the lowest cost.

The physical interpretation of the model. The application to different kinds of ships results in the use of different kinds of propulsion systems, one or two propellers, one

or two rudders, a fixed or variable propeller pitch, etc. The model of moue has the

advantage that these forces are implemented as modules in the manoeuvring model. For the other two models it is more difficult to show a relation between a rudder and

a hydrodynamic coefficient. Probably, it will be easier to adapt the moue model for

different kinds of propulsion systems and number of rudders than the other two

models.

In Table 2.1 an overview of the properties of the different manoeuvring models is listed. The model of Inoue is shown to be the best choice as system model for a KF. Only the

behaviour of this model over the complete manoeuvring range is uncertain. The

hydrodynamic coefficients can be estimated at low cost by a simple computer program. The model of Nomoto has the disadvantage that the coupling between the motions of the vessel are almost absent, so it gives a less exact representation of the ship dynamics; this can result in a less accurate performance of the KF.

model Abkowitz moue Nomoto

linear computation time

-

-

-

±

-+ system representation manoeuvring range + ? +

-type of ship + ? lowcost + ± physical relation +

-with + good ± acceptable

- insufficient ? unknown

16 Modelling the ship dynamics

2.2 The ship manoeuvrsng model

tact i cal diameter

u C 900 600 ç., C-) final diameter

h

II

start of trial

O O transfer 5 10 15 velocity (knots) 5 10 15 20 velocity (knots) o- _o... 20 300 2OO 100 O O 17 Ship HN1MS Tydeman Length

8.5m

Breadth 1..1 m Draught

i.75m

Displacement 2977 ton 5 10 15 20 velocity (knots) 600

f400

. 200 o O 5 10 15 20 velocity (knots)

Figure 2.3: The characteristic values of the turning circle of a hydrographic survey vessel at different velocities and rudder angles, with

the measured values;

the values calculated by the moue model;

the values calculated by the move model with the transverse hull force,

H' and

the moment acting on the hull, NH, reduced with 20%,

The values of the corresponding radder angles are indicated by: o for 5=10, * for 5=2O,

o io 0.05

o 0.05 0.10 10.15

Figure 24: A comparison between the measured values of the hydrodynamic coefficient

and the values according to [noue [1 981:1] (reprinted by permission of the publisher),

with

the hydrodynamic coefficient;

CB the block coefficient (a form parameter of the ship 's hull);

B the ship breadth;

d the ship draught;

+ o i the measured coefficients; and

- the coefficients

values according to moue.

18 Modelling the ship dynamics

The model of Abkowitz is more complex than the moue model, which means that more

coefficients have to be estimated. The model structure is less modular than the model structure of moue; the adaptation of the model for another type of propeller is more

difficult.

It is uncertain whether the model of Inoue can be used over the full manoeuvring range.

To check this point a set of manoeuvring data of an 85 m oceanographicresearch vessel,

the "HNIMS Tydeman', has been compared with simulated data using the moue model. The manoeuvring data consist of the characteristic distances of the turning circles which

were performed with the ship in a calm sea. The rudder angles and speed were varied systematically. The characteristic distances of the turning circles were plotted for both the measured distance and the simulated distances in Fig.2.3.

Unfortunately, it appeared that the performance of the moue model was not very

accurate, the distances calculated were up to 20% too large. The cause of that problem may be the method moue used to estimate the hydrodynamic coefficients of the ships. In

[moue et al.,1981:1] the measured hydrodynamic coefficients and the values of the

2.2 The ship manoeuvring model 19

it can be seen that the hydrodynamic coefficients could differ by about 30% from the

values given by the empirical function. The consequence of the error in the coefficients is

that the forces on the hull could be also 30% in error. To investigate the reason for the

deviation between the measured and simulated turning circles of the "HNlMS Tydeman" a sensitivity analysis was carried out with the manoeuvring model. The different forces acting on the ship were increased by 10% and 20% of their original value. The influence of the bilge-keels was also tested, as the moue model did not include the presence of such

keels; the forces applied by these keels were added to the model. From the sensitivity

analysis it appeared that:

- The bilge-keels had little influence on the manoeuvring behaviour of the ship.

- The increase in the rudder force of 20% decreased the deviation of the simulated and

measured characteristics distances by about 50%.

- The decrease in the hull force, _H' and moment, NH, with 20% almost eliminated

the deviation between the measured and the simulated data (Fig 2.3).

- The variation of the mass term in the equation hardly affected the model results.

So, the difference between the manoeuvring model results and the sea trials seems to be

the result of inaccuracy of the hydrodynamic coefficients estimated by the method of moue. If these coefficients were adapted the model results improved, Fig.2.3. So, the structure of the moue model can be used for the complete manoeuvring range if the

correct hydrodynamic coefficients can be obtained.

In the next section the possibility of estimating the hydrodynamic coefficients by sea

trials is discussed. This removes the advantages of low cost of the Inoue model but the other advantages, such as the clear model structure, remain.

2.2.2 The adjustment of the moue model

The moue model is based on the Newtonian force equation [moue et al.,1981:2] which yields in the earth fixed coordinate system:

mk(t)

= X0(t); (2.2 3. a)

m30(t) = (2.2.3.b)

m{û(t)-v(t)r(t)} =

_{X(t) =} XH(t) + XR(t) + Xp(t); (2.2.4.a)

m{'(t)+u(t)r(t)} =

Y(t) =

YH(t) + YR(t); (2.2.4.b)

I(t)

= N(t) _{NH(t) + NR(t),} (2.2.4.c)

20 Modelling the ship dynamics

with m the mass of the ship;

I the moment of inertia around the z-axis;

X0(t) and Y0(t) the forces acting on the ship; and

N(t) the total moment acting on the ship.

If Eq.(2.2.3) is transformed to the ship fixed coordinate system it gives:

with Y11 the hydrodynamic force on the hull along the y-axis; the rudder force acting in the y-axis direction;

NH the hydrodynamic moment acting on the hull around the z-axis; and

NR the moment caused by the rudder around the z-axis.

The forces XH(t), YH(t), NH(t), XR(t), YR(t), NR(t) and X(t) are described in a

series of publications [Hirano,1980; Hirano and Takashina,1980; moue et al.,1981; Ogawa and Kasai,1978]. In this thesis only the adjustments of the model and the manoeuvring

model finally used are discussed. For a description of the original model readers are

referred to the original publication [moue et al.,1981:2J. The manoeuvring model of moue has been changed in two respects to make it suitable for use in a navigation system. First the model of the propulsion force is changed and next the hydrodynamic coefficients are redefined to simplify the identification of the hydrodynamic coefficients.

moue describes the propulsion force as a function of the number of revolutions n(t). If a

propeller with controllable pitch is used that kind of description is no longer useful,

because the number of propeller revolutions is constant; the propulsion force is varied by

the pitch. In this paragraph it is shown how it is possible to estimate the propulsion

force in the case of both a fixed and a controllable pitch propeller. Here two models for the propulsion model are given; the first model can estimate the propulsion force for a

propeller with a fixed pitch, when the number of revolutions is measured. The second

model estimates the propulsion force for both fixed and controllable pitch propeller when the shaft torque and the propeller pitch are measured.

x(t) = (1-t0)Pn(t)2DKt{J(t)}

u(t ) { 1W (t) }

where J(t)

n(t)D

p

t the thrust deduction factor

Po

the propeller diameter; wf(t) the wake fraction;

Kt the thrust coefficient; and

p the density of the water.

The thrust deduction factor compensates the propulsion force for the resistance increase

caused by the propeller, and the wake fraction describes the difference between the

inflow velocity of the water by the propeller and the velocity of the ship. The propeller characteristics Kt{J(t)} can be found for a known propeller in, for example, {Lammeren

.8

(2.2.5)

2.2 The ship manoeuvring model 21

The first propulsion force model is based on the model which is used by moue, who

describes the propulsion force with

.3 1.2 .4 .6 .8 1 1.2 1.4

J Propeller pitch (Pp/Dp)

Figure 2.5: The characteristics of a propeller, values based on a B j-55 screw series /Lammeren et al.,19691, in the left hand diagram (P/D=l.2):

Kt the thrust coefficient; 10 Kq the torque coefficient;

the propeller efficiency; and

in the right hand diagram: Ckt the ratio between X('t) and Q(t) as a function of the dimensionless propeller pitch (P/D).

22 Modelling the ship dynamics

et al.,1969J, an example of the propeller characteristics is given in Fig.2.5. In Nomoto [1978] a method is given to model K{J(t)} for different kinds of situations; for a forward sailing ship it yields

K{J(t)} = c1 + c2 J(t),

with n(t)>O and u(t)>O, (2.2.6)

and with c1 and c2 arbitrary constants depending on the type of propeller. A comparable

model is given for a forward sailing ship with the engines in reverse mode. If the propeller type is unknown it is not possible to determine c1 and c2 by means of the

propeller characteristics. To solve that problem Eqs(2.2.5) and (2.2.6) are combined to

X(t) = (l-t 0)pn(t)2D{c1 + e2 J(t)}.

When J(t) is eliminated it leads finally to propulsion model I

X(t) = C1n(t)2 + ë2 u(t)n(t) {1-wt)},

(2.2.7)

with ë1 = (l-t )pDc1; and

Po

= (l-t

)pDc2.

PO

The coefficients and ë2 are unknown and have to be estimated by the sea trials.

The second method to describe the propulsion force is used when the propeller torque Q(t) is measured instead of the number of revolutions n(t). That method is also useful

when the ship is fitted with a controllable pitch propeller, but in that case the propeller

pitch P(t) has to be measured. To start with, it is assumed that a function exists which

describes the relation between the propulsion force, the torque of the propeller shaft and the propeller pitch, in the formula

X(t) = f{Q(t) P(t)},

(2.2.8)

where P(t) is constant for a propeller with a fixed pitch. To model the function

f{Q(t)P(t)) a typical propeller characteristic is studied (Fig.2.5). It can be seen that

the propeller efficiency i increases almost linearly with J(t) in the working range of the propeller, which is for this type of propeller J(t)<1. In the formula

J( t )Kt{J(t)}

(t) =

- c

*

2xK {J(t)}

q

with c an arbitrary constant; and

Q(t)

Kq{J(t)} pn (t)2D5 p

X(t) = 2ir/D

c Q (t) = Ckt Q(t)

p p (2.2.9. a) (2.2.9.b)

which is the torque coefficient. The combination of Eq.(2.2.5) and Eq.(2.2.9) leads to (2.2.10)

with ck an arbitrary constant. Eq.(2.2.10) gives the relation between the torque and the

propulsion force in the case of a propeller with a fixed pitch. The next step in the

derivation of the function f{Q(t)P(t)}

is the extension of Eq.(2.2.10) for a

controllable pitch propeller. For this purpose the value of Ckt is calculated for different values of the pitch of the same type of propeller (Fig.2.5). Now it is assumed, based on Fig.2.5, that ck can be modelled as a quadratic function of the propeller pitch, with

Ckt = cl +

c2P(t) + c3P(t)2.

(2.2.11) The combination of Eqs(2.2.10) and (2.2.11) leads to propulsion model II

X(t)= {c1 + c2P(t) + c3P (t)2} Q(t),

(2.2.12) with c1, c2 and c3 arbitrary constants depending on the type of propeller. From this the function f{Q(t)P(t)} of Eq.(2.2.8) is defined, and the propulsion force is modelled.

The second adjustment of the moue model is a change in the definition of the

hydrodynamic coefficients to simplify the system identification techniques. The method

which is used to change the definition of the coefficients is illustrated by an example;

moue [1981:2] describes the hull force XH(t) as follows

XH(t) = -mû(t) + cmmyv(t)r(t) + R{u(t)}, (2.2.13)

u(t) ₌ (2.2. 17.a) (t) ₌

f{0(t),t),T(t),8(t),Dp(t),1t)}

(2 .2 . 17 .b)

t(t)

_{= fr{u(t),v(t),r(t),}

S(t), '(t)P(t)n(t)};

( 2. 2. 17. c)

= u(t)cos{(t)} - v(t)sin{(t)};

(2. 2. 17. d)

u(t)sin{(t)} + v(t)cos{(t)};

(2.2.1 7e)

1(t) = r(t),

(2.2.1 7f)

24 Modelling the ship dynamics

with m m the added mass terms

_{X y}

cm an empiricalcoefficient O.S<cm<O.75; and R{u(t)} the ship's resistance.

If Eq.(2.2.13) is implemented in Eq.(2.2.4.a) it yields for the first few terms:

(m+m) û(t) = (m+c m ) v(t)r(t) +

. (2.2.14)

my

It can be seen that terms which depend on the same variables are combined. When

Eq.(2.2.14) is divided by the mass term m+m it follows:

(m+c m )

u(t) m ' v(t)r(t) + , (2.2.15)

(m+m)

or u(t) = c v(t)r(t) + .... , (2.2.16) ul

(m+c m)

withc_ul

m y

(m+mX)

By this method the number of unknown coefficients is reduced and terms of the equation

which depend on the same variable are eliminated. This results in a simpler model

structure, which in turn simplifies the system identification procedure. The procedure described above finally leads to the following model:

2.2 The ship m000cuvring model 25

with

f {_u } =c_ul

v(t)r(t) + c112u(t) + C3u(t) u(t) + c 4u(t)3 +

{c5 + c6P(t) + c

_{u7 p}

P (t)2) Q(t) +

c V {tn(t)Pp(t)}2Sin{ar(t)} sin{(t)}; u8 r

Ç{ } =-

(m+m)I(m+m)Ciu(t)r(t) + civ(t)V(t) + c2r(t)V(t) +

c

v(t)Iv(t)l + c

_{v(t)jr(t) + c5r(t)lr(t)J +}

v3 v4 c_v6V {t,n(t),Pp(t)}2Sin{ar(t)} cos{8(t)};_r = Criv(t)V(t) + c2r(t)V(t) + C 3v(t)v(t)r(t)/V(t) + Cr4 v(t)r(t)r(t)/V(t) + cr5r(t) I r(t) I + c6(t)V(t)2 + Cr7 (t) v(t)V(t) + Cr8 I

(t) r(t)V(t)

+ crgVr{t,n(t)Pp(t)}2sin{ar(t)} cos{«t)};

Vr{tPp(t)rì(t)}

the effective rudder inflow speed [moue et al.,1981:2J;

ar{tPp(t)n(t)}

the effective rudder inflow angle; and

V(t) the ship speed through the water; V(t) = {u(t)2 + v(t)2}.

The model coefficients can be directed to their origin:

the coefficients influenced by the hull forces: c_ul..c

_{u4' vi}

c ..v _{Crl Cr8}

y

the coefficients influenced by the rudder forces: c c , c and

u8 vG r9'

the coefficients influenced by the propeller forces: c_u5.c

_ui

2.2.3 The estimation of the hydrodynamic coefficients

In the previous section the manoeuvring model of which the coefficients are unknown has been derived. The hydrodynamic coefficients represent the behaviour of a particular ship;

every ship has her own set of coefficients. So, it is necessary to estimate the set of

hydrodynamic coefficients for each ship separately. In the previous section it was decided that the estimation of the coefficient set would be done by means of sea trials which will be discussed in this section. Three aspects of the identification procedure are discussed:

the choice of the type of sea trials, the method used to process the data, and finally the

26 Modelling The ship d,namics

It was decided to use the standard manoeuvring trials to estimate the hydrodynamic coefficients. The reason for this was that the manoeuvring

trials are known by the

nautical personnel of the ship, the trials can be performed easily in a minimum of water

area, and the track of the ship is already roughly known. The alternative, the use of a pseudo random signal, has the disadvantage that the track to

be sailed can not be

predicted,

which can lead to problems with traffic

or waterway restrictions. If

pseudo-random sequences are used it will require a good procedure to time the rudder

orders and propeller actions which are given manually, compared with the standard

manoeuvring trials which require a few standard orders.

In [NMI,1977; Glas and Schutte,1976J an overview is given of a series of standard manoeuvring trials which can be used for the identification of the hydrodynamic

coefficients of the manoeuvring model. The manoeuvríng trials are:

-

The natural stop; the ship sails on a straight line and at a

certain moment the

engines are stopped or the propeller is put onwindmilling, which causes the ship to

slow down by its own resistance. In terms of control theory; the natural stop is the

step response of the ship to an engine order. The engine order for this trial is defined as from full ahead to neutral position of the telegraph.

- The forced stop; the ship's velocity is decreased by the use of the propeller, which is

also a step response of the ship on an engine order. The forced stop can be performed

in two ways: a full stop which means that the engine is changed from full ahead to

full astern; and the gradual stop where the engine power is decreased in two or three stages.

- An acceleration test, which is the opposite of the stop manoeuvre.

- A turning circle; the ship sails a straight line and at a certain moment a constant

rudder angle is given which causes the ship to sail in a circle. In control theoretic terms: the ship reacts as a result of a step input on the rudder.

A zigzag manoeuvre; the ship sails a sinus shaped curve as result of a series of rudder actions.

A pull out test; the ship sails in a circle, and at a

certain moment the rudder is

turned in the neutral position. The pull out test can be used as end of a turning

circle, and is the opposite of a turning circle.

Spiral test; the rudder angle is increased by a few degrees at intermediate intervals; before a new rudder action is given the ship has to turn steady (it is in an equilibrium

motion). With this manoeuvre the relation betweenthe rudder angle and the rate of

turn r(t) is estimated.

- Williamson turn; a manoeuvre to pick up a man overboard, with a few rudder and

During the natural stop the position and the heading of the ship are recorded. The velocity and the acceleration of the ship are derived from the position and heading

2.2 The ship manoeuvrzng model 27

the speed decreases and the ship's speed will be low at the end of the manoeuvre,

which is at the place where the manoeuvre was initiated.

A fishtail; a manoeuvre to stop the ship as soon as possible by using the rudder as well as the engine. The track of the ship is better controlled than with the full stop

manoeuvre.

Almost all the manoeuvres occur as result of the control of one of the input signals: a

rudder action or an engine order. Only the Williamson turn and the fish tail use both the

rudder and the engine during the action. Table 2.2 describes which manoeuvre uses

which input.

To identify the hydrodynamic coefficients it is assumed that the superposition law can be applied. That assumption makes it possible to identify the hydrodynamic coefficients

independently of each other. By means of the sea trials given in Table 2.2 an

identification procedure is designed to identify the hydrodynamic coefficients step by

step. In Table 2.3 the five steps which have to be executed are given, these five steps will now be discussed.

During the first step of the identification procedure, the constant rudder angle 8, the current velocity V0(t) and the current direction c(t) are detected. A ship is never exactly symmetric, which results in the need for a small rudder angle, to keep the ship on a straight line. So, to detect ¿5. the ship sails with a constant heading. The

average of the rudder angle during the trial gives the mean rudder angle

_«

If a square

rather than a straight line is sailed then the wind influence is also reduced. The square consists of four lines with the headings at right angles to each other (for example: O', 90', 180', 270'). If there is any current, the ground track will have another shape then the sailed square; the deformation of the ground track gives information about the

current velocity and direction.

The second step in the identification is the determination of the coefficients c2, c3 and c114, which is performed by a natural stop. A natural stop starts with the ship sailing at maximum speed on a straight line. The natural stop is initiated by stopping the engines,

which results in

a decrease in speed of the ship. The natural stop means that

v(t)=r(t)=f5(.)=0 (straight line), and Q(t)=0 (engines stopped). Thus Eq.(2.2.17) can

be rewritten to

ú(t) = c_u2

u(t) + c 3u(t)

I

u(t) + c4u(t)3.

(2.2.18)

Table 2.3: The manoeuvres to be executed for the identification procedure.

action of: engine

no yes r U d d e r n O

straight line natural stop

forced stop acceleration test y e s zigzag turning circle pull out spiral Williamson turn fish tail

step manoeuvres estimation of:

i 2 3 4 square natural stop (2x) gradual stop/acceleration manoeuvre (2x) or a series of straight lines at different speed

turning circle (2x)

zigzag manoeuvre (2x)

Vc(t) c(t), &

cu2, Cu3 cu4

c5, c6

(model I) c115. .c7 (model II)

Cui c8,

.c,

cri..cr9

5 Williamson turn (2x) validation of model

28 Modelling he ship dynamics

with n is the number of samples.

Eq.(2.2.19) is a set of n equations with three unknown coefficients, and can be solved for this case with a least square method. Some practical complications during the trials are

caused by the possibility that ship will turn as a result of disturbances. The turning of

the ship can be avoided by giving small rudder orders, which cause a small increase in

resistance. The second complication is the possibility that the propeller is not able to

windmill, but is fixed during the natural stop; this will also increase the ship resistance. Abkowitz [1980] estimates the increase in resistance due to a fixed propeller on 20% of

the ship's resistance, for which the coefficients c2, c3 and c4 can be corrected. The third step in the identification is the activation of the engine, or Q(t)0. For this

step, Eq.(2.2.17.a) can be rewritten for propeller model I as

ú(t)= c112u(t) + c3u(t) I u(t)

+ c4u(t)3 + c5n(t)2

+ cUOu(t)n(t){lwf(t)}, or for propeller model II as

ù(t) = c 2u(t) + c3u(t) 11(t)

_u

+ c4u(t)3 + c5Q(t)

+ c

_u6p

P (t)Q (t) + c P (t)2Q(t).

_{u7 p}

p

(2.2.20)

(2.2.21)

For propeller model I the coefficients c5 and c16 can be estimated in the same way as

the coefficients c112, c113 and c4 in Eq.(2.2.17). Because propeller model I requires two

coefficients,

coefficient c, is

not used. The manoeuvre required to estimate the

coefficients c115 and c6 is a gradual stop/acceleration manoeuvre. The manoeuvre staTts

when the ship sails at maximum speed, then the power is reduced in three steps until

equations:

u (1) (2)

u (n)

u(i) u(1)Iu(1)I u(i)

u(2) u(2)ju(2)

u(2)

u(n) u(n)H(n» u(n)

C u3 C u4 C-uo (2.2.19)

2.2 The 3hip m000euvrzmg model 29

signals. Every t second the signal u(t) is calculated, which results into the time series: