Modelling the helmsman of a super tanker

Author:

Title:

Rap or t Date:

Sumarv:

DELFI UNIVERSITY OF TECHNOLOGY

LABORATORY FOR

MEASUREMENT AND CONTROL

Ir. W. Veldhuyzen

Modelling the helmsman of a super-tanker.

N 93

March 1973.

A supertanker can he considered as a non-linear system which

responds very slowly to changes in the rudder posision. _Moreover, this type of ships is often unstable in loaded condition, i.e. it has a tendency to start

turning

either to the left or to the right.

These properties make the supertanker very hard to handle.

A number of tests is performed using a ship maneuvering simulator. The trained subjects had to steer a 200.000 tons tanker along a varying course. The records of this tests were used _{to define a} mathematical model, describing the behavior of a helmsman. The mathematical model obtained in this way is fairly good. A discus-sion of the further research is given.

The research mentioned in this report was sponsored by _{the Deift} University Foundation (Delfts Hogeschool Fonds).

MODELLING THE HELMSMAN OF A SUPERTANKER.

Introduction

Most of the investigations concerned with the behavior of the human operator as a controller have been executed with _reference to the pilot of an aircraft or spacecraft. Some work in this field has been done on the control of submarines. The human _{operator as} a controller of surface ships, however, did not get very much atten-tion until recently.

In the Netherlands about two years ago the Institute TNO _for Mechanical Construction (TNO-IWECO) at Deift built a ship

maneu-vering simulator [i] in order:

To study ship maneuverability.

o To obtain data for ship and harbor design. To establish criteria for ships and harbors. To execute traffic control studies.

s To study nautical instrument design and automatic control. To train officers and pilots.

The simulator design was based on data and experiences _obtained with a previously built experimental simulator, designed _by TNO-IWECO in cooperation with the TNO Institute for Perception (TNO-IZF) at Soesterberg; another partner in the _{simulator project} was the Shipbuilding Laboratory of the Delft University of Techno-logy.

Earlier Stuurman [2] executed a number of trials _{on the} experimen-tal simulator in which he showed that for small ships the _control behavior of the helmsman could very well be approximated _{by means} of a describing function model. He also found evidence _{that for} larger ships a nonlinear model probably would give _{a more realistic} description of the helmsman's behavior.

in consult with TNO-IWECO it was decided to continue _{the work of} Stuurman as a joint activity of the Shipbuilding _{Laboratory and} the Man-Machine Systems Group. Special emphasis _{was to be laid on} modelling the helmsman of a supertanker with the _{following goals}

2

in mind:

o To provide data on which the maneuvers of this type of ship under human control can be predicted in a number of situations. o To enable an evaluation of the employment of a human pilot

versus the use of an autopilot.

o To investigate in more detail 'the advantages of predictive dis-plays in supertanker control.

Some introductory experiments have been executed using the simu-lator as a supertanker moving at constant speed on an almost straight course. A first attempt has been made to describe the behavior of the helmsman by means, of a nonlinear model using four

trainees of the School of Navigation at Rotterdam as subjects.

2 Ship dynamics

The dynamics of a ship depend not only on the properties of the ship itself like shape, dimension, mass and engine power, but also on the topology of the surrounding water. The motions of a ship in the horizontal plane can be described by a set of nonlinear differ-ential equations. These equations describe the translations of the ship in a direction corresponding to the longitudinal axis of the ship and in a direction perpendicular to this axis as well as the rotation about a vertical axis through the center of gravity.

Fig. 1 gives an indication of the variables concerned.

In 1957, Nornoto [3] showed that if it is assumed thai the ship is sailing at a constant speed, then the relation between the

rudder angle and the rate of turn r can be described by means

of a second order linear differential equation. For most of the

maller ships this equation gives an adequate description of the ship's behavior in a number of standard maneuvers. For a

super-tanker, however, it was found that the behavior was essentially

nonlinear. Based on full scale trials }3ech [;4] _{proposed to extend}

Nomoto's equation with a nonlinear term. This leads _{to the} follow-ing relation [6]

V Yo V, centor of grevity-ships trajectory u -X0 FIGURE 2:

The quantities invoLved in the desc?'iption of the

ship's maneuvers.

where r(t) d(t)/dt is the rate of turn, t) is the heading

angle, t(t) is the rudder angle, and where the quantities a1, a2, T, T2, T3 and K are constants. Here, it should be noted that all constants in this equation are dependent on the hydrodynamic

proper-ties of the ship, which are related to, among other things, its speed, ìts load condition and the possible restrictions in the surrounding water.

In this study a particular ship viz, a 220.000 tons deadweight tanker in loaded condition has been chosen. Table 1 shows the principdl data of the ship. The constants in

q.

C i) for this

ship have been determined by Clansdorp [5,6] in a series of full

scale trials; they are given in Table 2. 1f a stationary situ-ation is considered, that is, i(t) = 0, '(t) 0, t) 0, then

Eq. ( 1) changes into:

TABLE 2:

Principal data of the 2003000 tdw tanker i _fully loaded condition.

TABLE 2:

Constants in the equation describing the relation _between

the rudder angle and the rate

of

_{turn for the supertanker3}

in deep and still water; the speed considered _{is 7.72 rn/sec.} Length Breadth Depth Draft D i s p 1 a c e m e n t Froude number 310.00 rn 47.16 rn 24.50 m 18.90 F) 238.000 n 0. 14 4

Fig. _{2 represents Eq. ( 2); this static characteristic}

is given

for the ship in fully loaded and _{bailasted condition., Tine ship is}

course unstable in loaded condition; it has _{a natural tendency to} deviate from the straight course and to start turning either in one direction or the other.

3 _{The maneuvering simulator}

The simulator consists of _{a wheelhouse which has the same appearance}

as that of a real sea-going vessel. The fore part of the ship, the

sea and a coastline are displayed on a _{screen in front of the}

wheel-constants dimension

numerical values fully baden ballasted 1 2 2 -1 a2 sec /rad 80,000 16,200 T sec _{250 80} T sec 10 3 T3 sec 20 6 K

sec1

_{-0.0434 -0. 0471}

starboard -16 -2 --03 - 0.4 rOte of Q turn r [o ort 8 12 rudcJr

I

5 starboa rd - -12 -03 01 -OEt -Q2 - 03 rateof turn r fa ¡sec port 0 4 8 12 -+-s-rudder crrgle 6[] FIGURE 2:

Re lation be tween the rudder angle and the rate of turn r in the stationary state for the ship considered in this

investigation as found by Glansdorp [5] (a: loaded condition;

b: ballasted condition).

house. The total angle of vision of the helmsman is120°. The image of the fore part of the ship is static; it is produced by two slide projectors which have a fixed position. The coast line is generated by means of a point light source arid a movable model with three degrees of freedom simulating two translations and one

rotation in the horizontal plane. Fig. 3 shows the simulator

during a simulated approach of a harbor. A blockdiagram of the system, including the helmsman, is given in Fig. LI..

On an analog computer the dynamics of the ship to be simulated, including the characteristics of thrust engine and rudder engine, can he programmed. The computer yields the signals which control the environmental display system and also the instruments such as

compass, rudder position indicator, log, etc. The helmsman has the same controls at his disposal for maneuvering as on a real ship viz, the wheel, which gives the input to the rudder engine, and

prcj

LJ

FIGURE 3:

The Ship Research and Maneuvering 3imulator

of

the

Instiute TNO for Mechanical- Constructions _{at Del-ft.}

L

::::1i

.--- i'---J desir.d t s ate II C

ii

FIGURE 4:

Blcckdiagrarn

of

the TNO simulator.

d stu r b a r e s 6

r

---i

L.

.. I d y

n.j

I _:1 Sfl*A t k;naWwr.a'-..,

-wheet'

UCAG .rnp-tr :1.1 i

- j

7

the telegraph to the engine, which

_{governs the speed of the}

pro-peller'. External disturbances simulating

_{the effects}

of

wind, waves

and currents can also be introduced into _{the model on tne analog}

computer.

4 _{The experiments [7; 8; 9]}

In the experiments described

here, the simulator

_{has been}

_{used as}

a supertanker at full sea

in fully loaded condition

_{as well as in}

ballasted condition, moving at a constant speed. The analog com-puter

was programmed according

_{to Eq.}

( 1) based on the constants

given by Glansdorp as indicated _{in Table 2. The rudder engine}

was also included in the simulation. _{Its dynamics have been}

chosen

according to the Eqs

C 3) and C LI.).

T4 (S(t) (t)

(

t)

I M,

where 6d(t) is the position of the steering wheel or the desired

rudder angle; where T4 is _{a time constant of 1 sec and where}

the

quantity M is the maximal value _{of the rotation speed of}

the rudder (0.045 rad/sec).

The subjects were four trainees _{of the School of Navigation}

at

Rotterdam. They were studying for the rank of first or second _mate

after having been at _{sea for several years. All four}

trainees were experienced in steering conventional cargo ships; only one of them

(subject A) had sailed _{on tankers up to 90,000 tons dead}

weight. Their task consisted of following a straight course for about half an hour, or of following a preprogrammed _{but unpredictable}

course which changed between +20

_and

_20 _{around a certain nominal}

course

for

_{about fourty minutes. This prescribed course acted as a test}

signal

for

_{the man-supertanker system.}

Because it is

not a

real-istic situation

to follow a

_{continuously changing}

_{course, a binary}

signal was chosen. The construction

_{of the binary test signal}

_was

based on the idea that with the experiments _{to be executed also} linear models of the helmsman's _{behavior might be investigated.}

Therefore the input signai (desired course) was constructed in such a way that about 70 percent of the input's energy was concen-trated at only four frequencies. The amplitude of the binary signal was equal to the two degrees earlier mentioned. The four sinusoidal

components were the third, fifth, eight and thirteenth harmonics of a sinusoid with a period of forty minutes; the phase at the initial point was chosen at random.

TABLE

3: S:nimary of ti

tests performed.

* With disturbances due to waves.

**- With rate of turn indicator.

--- Because of a short age of time this run could not be performed.

)ATE 1972 April 25 1972 Anni 26 1972 May 2 1972 May 3 7.00pm 8.00pm 0.00pm 10.00pm Subject A loaded cand. course chang. Subject c loaded cond. course chang. Subject A 13 ball. cond. course chang. Subject D loaded cond. course chang. Subject B 2 loaded cand. coursa keep, Subject D 8 loaded cand, course chang. Subject B loaded cand. course chang. Subject c 20 loaded cond. course chang. Subject A 3 loaded cand. course keep. 9 Subject C loaded cand, course keep. 15 Gubject A loaded cond. course keep. 21 Subject D loaded cand. course keep. Subject B loaded cand. course chang. Subject D 10 loaded cond. course keep. Subject B loaded cand. course keep. 22 Subject C loaded cond. course keep. Subject A 5 ball, cand, course keep. Subject C 11 loaded cand. course keep. Subject A 17 loaded cond. course keep. Subject 23 loaded cond. course keep. Subject 3 6 ball. cond. course keep. Subject D 12 4* loaded cand. course chang. Subject B * loaded cond. course keep. Subject C 24 * loaded cand. course keep.

g

1n Table 3 a summary of the tests is given. During the experi-ments the compass as well as the rudder angle indicator were used.

Three tests have been executed in which also the rate or turn indicator was used. With the exception of two trials no external disturbances simulating ship motions originating from wind, waves and currents were introduced; these external disturbances were generated by means of a digital computer and consisted of the sum

of 23 sinusoids simulating the motions of the fully loaded tanker in a following sea with long crested waves. 'These waves were consid-ered to originate from a wind with a force of eight to nine

Beaufort. The spectral density function of these motions have been calculated in reference [io']. During the tests the following sig-nals were recorded on magnetic tape:

e The desired course

The course of the ship t).

The desired rudder angle

s The rudder angle (t).

The rate of turn r(t)

6.5 Modelling the helmsman's behavior

In the Figs 5 through 7 some examples are given of the time

histories of the desired course Yd(t); the course of the ship '4.t(t);

the desired rudder angle d(t); the rudder angle 5(t), and finally

the rate of turn r(t) as recorded during the tests. In all cases the records show that the helmsman generates a rudder angle t)

as output which consists of discrete steps. Hence the records indicate that a linear model to describe the helmsman's behavior will not fit the data very well. To check this presumption the ratio between the energy not located at the four frequencies

men-tioned in paragraph 14 ard the total energy of the signal

consid-ered has been calculated. The table 14 shows the results for the

four subjects, each having performed two runs of forty minutes. From the table it can be concluded that the remnant energy, in particular of the desired rudder angle 5d(t), is that high in most of the runs observed, that it does not seem appropriate to focus the attention on linear descriptions of the helmsman's behavior

TABLE

4:

Survey of the remnant energy in the signais observed

for eight tests with four' subjects: Loaded condition;

without rate of turn indicator; without disturbances.

any longer. It suggests that the helmsman bases his dicisions on when to move and how much to move the wheel on some criterion which

is, for instance, a function of the heading angle (t); the rate of turn r(t); the position of the rudder t), and, of course, the desired angle

By the compass the course t) is displayed to the helmsman; while in the case that no rate of turn indicator is used,it is assumed that the helmsman estimates the rate of turn r(t) directly from the time history of the course (t). Based on the records obtained by preliminary tests as well as on the records shown in the Figs

5, 6 and 7 a model for the helmsman's behavior is proposed which states that if the absolute value of a quantity s(t) which is defined as:

(t) [(t) _{- d(t)l} + c1q'(t) ( 5)

exceeds a certain threshold value d1,

the

helmsman moves the

steering wheel into a position Sd (t) according to the Eq. C 6): *

(t)

a(t) + bF(t)

±

c[(t)]3

d sign _e(t) ( 5)

In this equation the quantity Je(t) (t) _4Jd(t) and the

quan-tities tp(t), d(t) and 'ut) are the values of these variables at the moment that the threshold value is exceeded. As a result of

this action, the quantity s(t) wil decrease; if now Is(t)

becomes less than a second threshold value d2, it is assumed that

10 Subject A B C f) Run 1 2 1 2 1 2 1 2

Yd(t)

28.0

28.6

23.3

28.0 28.0 28.6 28.3 28.8

t)

10.3

21.6

7.8

27.7 50.1 19.5 1.7

26.6

td(t)

88.7

88.0

90.9

9'4.3 78.2

85.9

78.1 87.9

6(t)

86.2

86.0

88.8

91.3

77.6

Bb.7

77.1

86.5

r(t)

48.3

47.9

49.5

52.9

70.0

53.2

35.0

56.1

rI

[rad/ sec2 o

-10--20 L -10 -20 0.02 o - 0.02 1f! [degr.] _-2 10 o { degr. j 10 o [dgr.] -10 0.1 r=1/J [ radi sec] _-0.1 i 0.02 o [ rad/sec2] -0.02 o s 10 15 E'IGUJ?E 5: t [mia]

Example

of

the time histories of the desired course (t); the

course of the ship _{(t); the desired rudder angle 6(); the rudder}

anale _{(t); the rate of turn r(t), and the derivatiie of the rate}

of

turn with respect to the time '(t): subject A; without _{rate of} turn indicator; without disturbances; loaded condition.

L

u

JJ

L

ri

fl,i

1.] I [degr.] -2 [degr.] _-2 5d A ₂₀ [degr.] ₁₀ 0.1 0 - 01

r:1//

₄ {radi sec J o [egr. _J 20 10 o 10 15 t [mia] o

2 [ degr] -2

V'A

2 o [ deç.r -2 20 [ dgr.J 10 o -10 - 20 5 A 20 [ degr.]

i:

r1jf

0.1 O [ rad/sec ] -0.1

rrV'

A 0.02 [ raisec 1 -0.02

jrA2

[d2gr.] I -2 O [degr.] -10 0.1 O

[ ruìsecl

; = 0.02 o

21!

[rad/sec

_j

_-0.02 s 10 - t [mm] M

4ArrwT&rJ_

15 12 10 15 FIGURE

6:

-

t

[rni]

Example

of

the time histories

of

the desired course » ,(t); the

course of the ship (t); the desired rudder angie 6(); the rudder

angle (t); the rate

of

turn r(t), and the derivatie of the rate

of

turn with respect to the time '(t): Subject B; without rate

of

turn indicator; without disturbances; loaded condition. -1pJ

tIT

¡

ypiJ

I

. t

{degr.] li, [degr.] degrj I (5 [degr. ]

rr

[rad/sec]

;-:;

2 rad/sec [ degr.] [ degr.] l'IGURE

7:

Example

of

the tim

course of

the ship angle (t); the ra

of

turn with respe

turn indicatori Wi (5 [degr.] [rad/sec] 2L o -2 20 o -lo -20 0.1 o -0.1 L 0.02 o -0.02 2 10 0.02 [rad/sec2] _-0.02

i-y

A s

w

LLr

10

r.

15 »_ t miri

e histories of the desired course

p(t); the desired rudder angle 5 ();

te of turn r(t), and the deri?'atie

of

et to the time i'(t): Subject D; without thout disturbances; loaded condition.

13 ; the the rudder the rate rate

of

L

-

-r 10

r

2 L o -2

r

1 o -2 o -lo lo o -10 0.1 o -0.1 O s 10 is

the helmsman moves the rudder again into zero position.

Parameter Estimation

The parameters a, h, c, d, d1 and d2 can be estimated by

minimizing a quantity J1 defined as:

J1

+ J

Tc( t)] dt, _{C 7)}

where the signai c(t) represents the dif[erence between the human

operator output d(t) and the model output 6d*(t). The minimal

value of the quantity J1 can he found by partial differentiation of this quantity with respect to each of the unknown _parameters and by setting the result equal to zero. This yields as many equa-tions as there are unknown parameters. 9ecause there does _{not exist} a simple analytical relation between the input and the output of

the model of the helmsman it is not possible to solve the equations just-mentioned in a simple way. Furthermore, it should he noted that if a model with given parameters should be inserted into _the control loop instead of the helmsman, this would lead to another

time history for the quantities 5d(t), _{i(t) and r(t). Go, in order}

to get an unbiased estimate of the parameters in the human _operator model, a comparison should be made between the output of _the helms-man and the output of the model, where this model is also _{part of} a closed loop system with the ship's model (See Fig. _{8). The} parameters can be found by means of a direct search method mini-mizing the quantity J1 with respect to the model _{parameters. With}

the help of Eq. ( 7) the quantity E1 can be defined:

T T f[c(t)]2dt E o T 2 *

Q[dt]2dt

Id(t)l dt

o ( 8)

The quantity E1 indicates how well the model output approximates the actual output of the helmsman.

(t) he (msmon porarneters modo) of

ödit)

helmsrnejn direc t scrc h rn o t h o d t) FIGURE 8:

Application of an error criterion _{in such a way that} unbiased parameters can be obtained _{for a model of a} system in a closed loop.

The correspondence between _{the time histories of the}

actual course

of the ship _{t) steered by the helmsman and those of the ship}

model _{'0(t) steered by the model of the helrnsmarYs behavior can be} expressed by the quantity E2, defined _as:

T

f[t)-t)]2dt

n E ( 2 T

f [(

t)] 2dt o 7 _Results

Based on the data of _{one test, in which the subject had}

been

instructed to steer _{a ship in loaded condition along}

a straight course for a period of 25 minutes, the parameters of a series of

simple models have been estimated [7; _{8; g] according}

to the

para-meter estimation method just-mentioned. _{The first model was a}

linear one; later models _{were based on the Eqs}

( 5) and ( _6),

s)

cri erion E(t)

JI

model of V'(t) ship

15

16

he it that nonlinear terms in the Eq. ( 6) were neglected. The

results with these simple models, however, were very poor, and did not give any indication how to modify the models used in order to

get a better result. Therefore, the task of the helmsman, which during these tests was to keep a ship in loaded condition on a straight course, was changed into the task of following _{a certain} unpredictable but well defined course. In this case, again the parameters were estimated, and now the results were much better; they even lead to a direction in which the structure of the model had to be changed. Finally, the model as given by the _{Eqs ( 5)}

and ( 6) was obtained.

The table

6.5

shows the values

of

the model parameters as well as

the values of the quantities E1 and E2 calculated from four _tests

TABLE 5:

Summary of the pararrie ters of the mode i given by

_the

Eqs (

5) and

(

6) and the quantities E1 and E2.

Subject A B C D Date 1972 May 2 1972 May 2 1912 May 3 1972 May 3 Test nr 15 1t _{20 21} a 3.5 1.9 F.2 2.0 i [sec.] 176.9

155.0

210.0 230.5 c [degr.2] _{0.0 0.1 -0.1 0.1} d [degr.] 1.7 1.0 1.0 1.5 C1 [sec.] 20.3 30.0 22.5 30.0 djdegr.] -5.0 0.2 0.3 0.2 d2[degr.] -5.0 0.1 0.2 0.0 40.1 70.9 43.4 32.8 E9[-c] 8.7 -- -- 3.1

17

executed by the four subjects A, B, C and D (See table 3). The Figs 9 and 10 show some typical time histories of the actual

signals _d(t) 1'(t) and d(t) compared with the model outputs

(t)and d*(t). Some remarks with respect to the results obtained

can be made.

Although a remarkable difference in the steering behavior of the subjects exists (See the Figs 5, 5 and 7), the difference of the parameters between the different subjects is rather small. o In paticular for the subjects A and D the model fits the test

results fairly well.

o Although the output of the model of the helmsman's behavior does notaiways resemble the actual output very weil the course of the

ship generated by the model experiment fits extremely well. o In a few cases, for instance the test with subject C, the

opti-mization of the parameters by minimizing the quantity

'1 was

difficult. The problem was to find the absolute minimum of the quantity J1 in stead of a local minimum.

S Further research

Until now, only the parametérs of the model have been estimated e

for four subjects steering a fully loaded tanker along a prescribed course changing at certain moments from +2 degrees to -2 degrees and back. Also some tests were performed with a tanker in hallasted condition and with disturbances (simulating for instance the motions originating from seawaves) acting on the ship [io], but at this

moment no parameter is established. Some of these tests, in particu-lar those with the ship sailing in open sea with rather high waves, showed that fairly dangerous situations can occur when the helmsman is not well trained. Therefore, there is a need to work out the re-sults of the other test runs very carefully.

Sorne points which certainly have to be further explored will be: A study of the influence of the test signal used (the desired course) on the structure of the model and on the model parameters. A study of the influence of the ship dynamics on the structure of the helmsman model and on the model parameters.

\tf 2 [degrees] L [degrees] _{b' 1G U R E V1d [ degrees ]}

Vt

11it) X. turn indicator.

9: Typical time histories

of

the actual signals Pd(t)J

p(t) and

d(t) eompard with

the model outputs

(t) and

d*(t): Subject A; loaded condition; without rate of

a 0 5 10 15 20 25 30 35 . t [rnmn] FIGURE

20: Typica'.

time histor-tes

of

the actual signals Jd(thj

i(t) and Sd(t) compared with

the model outputs

*(t) and

(t): Subject D; loaded condition; without rate

of

turn indicator. 35 30 t [mm] 1S 10 5 20 25 ô [dtgrees ] if

in

A study of the influence of the disturbances acting on the ship on the structure of the model and on he model parameters.

G The meaning of the strong difference between the rather weak

resemblance between the output of the model of the helmsman's behavior with the actual helmsman's output on the one side and the excellent agreement of the ship's course and that of the output of the ship's model on the other.

The physiological and psychological interpretatLon of the model parameters, as well as the cybernetic aspects.

A c k n o w ledge men t s

The authors gratefully acknowledge the contribution of C.C. Glans-dorp of the Shipbuilding Laboratory of the Delft University of Tehnologv. They are alsc greatly indebted to TNO-IWECO for provi-ding the facilities todo the experiments. The special thanks go to the staff of the TNO-IWECO simulator group for their contribu-tions in the preparation as well as in the execution of the trials. Finally, the authors would like to express their gratitude to the subjects from the School of Navigation at Rotterdam for their wholehearted cooperation.

10 References

i. Brummer, G.M..A.; van Wijk, W.R.

The Ship Manoeuvring and Research Simulator of the Institute TNO for Mechanical ConstructiOn, Deift, Inst. TNO for Noch. Constr.

Report Nr. 8l33/ (1970), pp. 1-32.

2. Stuurman, A.M., Modelling the helmsman: A study to

define a mathematical model describing the behaviour of a helmsman steering a ship along a straight course, Deift, Inst. TNO for Mech. Constr.,

20

3. Nomoto, K.; Taguchi, T ; Honda, K. ; Hirano., S.,

On the steering qualities of ships, International Shipbuilding Progress,

Vol. , Nr. 35 (1957), pp. 35-t-370.

l4 Bech, M.; Wagner Smitt, L.,

Analogue Simu1ation of Ship Manoeu-vres, Lynghy (Denmark), HyA Report

Uy 1 (1969), pp. 1-2'4.

h. Clansdorp, C.C.; Buitenhek, M.,

Manoeuvring Trials with a 200,000 tons Tanker, Deift, Shipbuilding Laboratory of the Deift University of Technology, Report Nr. 28 (1969), pp. 1-31.

Glansdorp, C.C.,

Simulation of Full Scale Results of Manceuvring Trials with a 200,000 tons Tanker with a simple Mathemati-cal Model, Deift, ShiDbuildin

Laboratory of the Deift University of Technology, Report Nr. 301 (1971), pp. 1-2.

Veldhuyzen , W .,

Len eerste model van bet regelgedrag van de roerganger van cen supertanker (A preliminary model to describe the behavior of a helmsman of a

super-tanker), ir-thesis B 810, L.v.W.M.R., Delft (1971), pp. 1-70.

V1dhuyzen, W.,

Onderzoek naar de regel techn ische aspekten van bet gedrag van de roer-ganger van een supertanker (An Inves-tigation on the behavior of a helms-man steering a Supertanker),

Report N 87, L.v.W.M.R., Deift (1972), pp. l-47.

21

.9. Veldhuyzen, W.; Luntereri, A. _{van; Stassen, H.G.,}

Model1ingthe Heisman of. a

Super-tanker: Sorne preliminary Experiments,

preprint 8th Annual Conf, on Manual

Control, Ann Harbor (1972), _{pp. 1-17.}

10. Veldhuyzen, W.,

3orekendespektra van de gierbewegin-gen van een 220.000 TDW tanker

(Calculated spectra of the yawing movements of a 220,000 tdw tanker),