Some aspects of automatic steering of ships

Introduction

The automatic steering system of-a ship will be equipped for two main purposes. One of them is to navigate a ship between two ports at the least expenditure. As a vessel does not necessarily run in a straight course but may wind its way a little, it is necessary in this case to search for the method of appraisement to distinguish the best system of steering.

The other purpose is to navigate a ship correctly along the designated route in the inland sea or in the bay. _{Flow accurately the ship can trace the} appointed course and how quickly it can turn are the important problems, and for this _{purpose the} matter of fuel consumption can be neglected. There are two ways in this case, one of which is to control the ship completely automatically along the

pre-scribed route and the other is to help the

helms-man with the improvement of the apparent maneu-verability by employing subsidiarily the automatic control of a minor loop.

Works carried out by the authors regarding these problems are mainly introduced in the following:

Equation of Motion

When the co-ordinate systems _{are taken as:}

o - ,

Fixed in the space

Gx,y: Moving together with

the ship, the

center of gravity of it is taken as the origin of the co-ordinate system, which are shown in Fig. I, and the nomenclatures

are designated as:

U: Velocity of ship

U, V: _{x-cornponent and y-component of the}

velocity of ship

O: Angle of yaw of ship July 1968

Lab.

_{y. ScFeepsbouwkunde}

Technische Hogeschool

Deift

o Automatic Steering o Ships

By Seizo Motora, Professor Tokeo Koyama Assist. Professor

University of Tokyo

o

Fhg. i Co.ordinate System

¿

5

r:

Angular velocity of ship

¡9: Drift angle Rudder angle

x-component of the external force act-ing on ship

y-component of the external force

act-ing on ship

N: Moment acting on ship

M: Mass of ship

Mx: Added mass in the direction of x axis My Added mass in the direction of y axis

Iz:

Moment of inertia of ship Jz: Added moment of inertia of ship, we get the equations of motion as follows:

(M+Mx)Û=(M+My)rV+X

(M'+My')ß'=Y/3'.ß'+( (M'+Mx')

+ Yr' )r' + Y8 '.8' + Y' (Iz' + Jz')i'= Nf3 ' _{f3' + Nr'. r' + N8 '.6' + N'}

t

ARCHEF

some socs

where, the symbols with ' stand for the values non-dimensionalized by the length of the ship L, the

draught d and the velocity U, and Y or N with

the subscript indicates the derivative of them

dif-ferentiated with respect to the subscript.

Optimum Automatic Steering at Seafl

It has been considered up to the present that the purpose of the auto-pilot of ships at sea is to make the deviation from the expected course as small as possible. This may be because the more winding course is regarded to bring the greater total length

of path than the rectilinear

course. However, it will be understood that this consideration

is not

correct, when we ask if any excessive rudder angle is forgiven in order to get less deviation.

Then, what is the true purpose of the auto-pilot? According to the most general definition, it is to

navigate the ship from the port of departure

to

that of arrival at the least expenditure as

men-tioned. But this over-comprehensive definition makes

the solution of the problem impossible, as many complicated phenomena are concerned. Therefore,

we now treat the problem of the auto-pilot based on the relatively simple definition as follows:

"The purpose of the auto-pilot is to get the least loss (due to disturbances) to go forward, when the ship is running along the prescribed course

over-coming surrounding disturbances."

According to such a definition, the aim will be further explicit, as steering is of course restricted.

In this chapter, we at first consider the way to appraise the auto-pilot and what disturbances need to be taken into account. Then we investigate into

the auto-pilot most suitable for such a purpose. Method to Appraise Auto-Pilots

Let us consider what are the losses of a ship to go forward, that is, increase of resistance in a wide sense, by the aforementioned definition of the ap-praisement of the auto-pilot. It is important that we must take the losses concerning steering only, for the right appraisement of the auto-pilot. For instance, however large the increase of resistance due to heaving or pitching may be, it has nothing to do with the loss of the ship under discussion.

Then, it is enough to regard X in Eq. (1) as:

X= T(l -t)-(RhUll +R8) (4)

where, T is thrust of propeller, t is thrust deduction coefficient, is resistance of ship and R is drag of rudder. Thus, Eq. (1) will be:

(M+Mx)Ú=T(l t)(M+My)Ußr(Rs0i+R6).

6

lt is known that, even if yawing or swaying of the ship were considerable, flhls in the above equation would hardly increase. The second term

on the right hand side is the forward component of the centrifugal force acting on the ship and is

determined by the resultant motion caused by

steering against external disturbances. When r and $ are both cyclic functions of time with phase lag

between each other, the loss corresponding to the mean value of this term will be:

l/2(M + My)U7 cos ( 5 )

where, ? and stand for each amplitude. We shall

call this "loss due to centrifugal force".

Next, let us consider the loss due to steering. However, only the direct effect of steering on re-sistance is taken into consideration in this case, so that the loss generated by the resultant motion of

steering is not comprised in this loss. Although

only R6 has been noted as such a loss, not only R, the increment of thrust deduction coefficient t and vtriotis influences upon all propulsive factors also slicu "I be deliberated on.

Now, we must call to our minds the increase of the total length of path mentioned in the first place in addition to these two losses. In the following sections, let us explain these three losses in details. Increase of Length of Path

Due to Serpentine Motion

When the ship does the serpentine motion as shown in Fig. 2, the ratio of the length ot path to the distance on the straight course will be given as follows, if the amplitude of serpentine motion Is

represented by :

Sinusoidal

Broken Lure

Fig. 2 Serpentine Motion

6

4oQ

2

cos O

O

Fig. 3 Etongatinn of Distance Rin due to

Serpentine Motion

Sinusoidal motion (a)

V'I+ U2 cos2

d -

(6)

Broken line motion (b)

sec G (7)

Fig. 3 shows the increased portion of them, which makes clear that the increase of length of path is little as long as o is not too much.

1f the deviation of ship from the course is Ee, the

following relation of B, E and $ is given:

11=_ro

-I

then, is not equal to strictly. However, since

it is in the low frequency range, _{the serpentine}

motion will grow considerably, $ can be regarded so small in the present discussion that _{may be}

nearly the same as E0.

Losses due to Centrifugal Force Loss Caused by Waves

lt is necessary to kno' _{the accurate data about}

r and $ and their pha

angles for waves with various directions in orthr to get the loss due to

waves. _{But, we have no conclusive methods to}

estimate exciting forces _{or coefficients of terms of}

the equation of motion, which are functions of freq uenc:v of wave.

l'he authors think thst Dr. Eda's method2 is the most piacticible ami the most reasonable of

Fig. 4 Increase of Resistance due to Centrifugal Force Caused by Waves

many proposed prediction methods. Fig. 4 shows tise loss calculated by this method for the ship of Series 60 form with Cb = 0.60, when,

Length of ship: 150m

Speed: 18 kn

Amplitude of wave:

lin

at every 30 degrees of the angle of encounter, which

is taker as zero in following sea.

We can see great negative increases of resistance remarkably in following sea. This fact will not be

a strange phenomenon, however, their absolute values

may be, because the increase of resistance given by Eq. (5) becomes, of course, negative when,

90°<<270°

The curves in Fig. 4 are not experimental or

empirical data but computed results, and so they fairly lack reliability, particularly in the estimation of phase lag . However, L may be predicted that

the loss due to centrifugal force caused by waves will be not small.

Can we check these losses by stering? Though this loss does not have anything to do with steering

for it is _{caused by yawing or swaying, the}

fre-quencies of these motions are too high to be followed by the rudder.

il

90

I

150V 03 0.5 w (rad/sec) 0,7 0,9 30 July 1968 7 20 I O

w (rad/sec)

FIg. 5 Increase of Resistance due to Centrifuga! Force

Caused by Gust

Accordingly, even if steering were hard-to-hard

in good timing, there would scarcely be effects. Thus, it will be concluded that the loss caused by individual wave, however large

it may be, must

not be considered in appraisement of

the

auto-pilot.

Loss Caused by Gust

The velocity of wind is generally composed of steady component and fluctuating blow. As we are

now discussing the problem of auto-pilots, the rudder

angle indispensable to overcome the side force (or swaying force) and yawing moment is

out of the

present question and we need merely examine steer-ing against fluctuatsteer-ing blow.

Needless to say, it

is of no use to get detailed

information of wind force in case of such a problem, then we have tried a most rough estimation. When we supposed that the lateral area of above-water form of the same ship as in the previous section was 1,500 in2, and that the amplitude of wind pressure was that the velocity of it is 10 m/sec, we have obtained the computed results as shown in Fig. 5, the loss is remarkable in

the range of very low

frequencies. But, the anguiar velocity, which brings about a gTeat loss due to centrifugal force when the frequency of wind is

such a low one, naturally

gives rise to rather excessive deviation of the course.

In fact, when we compare the loss due to

centri-fugal force caused by gust with that due to the

increase of length of path caused by deviation of the course, we can see that the former is less than

8

the latter in almost the whole range of frequencies, though both have tile same order remaining in the narrow range.

Loss Caused by Steering

This loss is considerable as pointed out in

Ref.3> For instance, when the ship of Series 60

form is running in

still water at 18 kn, the loss caused by sinusoidal steering with the amplitude of 0.1 radian will be as shown in Fig. 6 obtained from

some calculations.

Although this loss attains a very large value as in this figure, it does not arise while steering is

enough to cancel needless motions of tile ship

produced by disturbances because the centrifugal force

exerted on the

ship in

a seaway can be

attributed to the mutual action between disturbances

and steering as mentioned above.

This loss may be regarded trivial in the range of lower frequency than the frequency of the broken point, when usual auto-pilots are able to cancel the additional motions of the ship generated by dis-turbances, and also small in the range of high

fre-quency, when the motion of the ship is

scarcely changeable even if there is tile fairly large steering.

There is a question around the frequency of

the broken point when a great loss may be caused

for a bad control system4>. However, if we intend to take account of the stability of control systems

in appraisement of auto-pilots, it makes tile problem

merely more complicated. Accordingly, we will assume as a major premise that the control systems treated in this article are all stable.

Thus, it is unnecessary to take into account the

loss due to centrifugal force caused by steering under the condition of the good stability of control systems.

Japan Shipbuilding & Marine Engineering

0.01 01

10.0 _{0.01 0.1}

w (rad/sec)

1.0

Fig. 6 Increase of Resistance due to Centrifugal Force Caused by Sinusoidal Steering

1.0

July 1968

1 .0

20 30

e

Fig. 8 Increment of Thrust

testing results for the model ship of mariner type, we may suppose a similar quantity for other high speed cargoliner. But this loss may be pretty less for ships of different forms such as oil tankers be-cause the resistance of rudder occupies only a

negligible portion in total resistance for such full

ships.

Method to Appraise Auto-Pilots

To put together aforementioned facts, only the two losses due to increase of length of path and steering have been left to be adopted in

appraise-ment of auto-pilots, for other losses can be neglected

as compared with these two.

When the yawing motion of a ship and steering are both sinusoidal, each loss is proportional to the square of amplitude and then the total loss becomes

as follows:

Total loss =-2 +22

When a ship is navigating in random disturbances,

neither yawing motion nor rudder angle are always sinusoidal, therefore, though it is not strict, the

authors would like to propose the following formula to assess auto-pilots in a seaway:

J=2+À2

(8)

02: _{Mean square of deviation of path in a seaway}

Mean square of rudder angle

À: Weighting constant of 2 and 2 _{(about 8}

for a cargohiner of mariner type)

where, it should be remarked that this performance

index is

based on the major premise

that the

stability of control system is satisfactory.

!/

/4:

/

30

i.:

8=20' 10 1.1 12 1.3 V,, (m/)

Fig. 7 Increase of Thrust

Loss due to Steering

As previously mentioned, it is dissatisfied that RÒ

in Eq. (4) is only considered as this loss. Then, the authors made a new attempt to measure the increase

of thrust due to steering for

a model ship self-propelled straightly with a constant rudder angle.

The test is similar to the usual self-propulsion test in the process of which the friction correction is applied, so that the influence of steering on

pro-pulsive factors can be known correctly.

The tested model ship is a model of mariner

type cargoliner which is 2.5 m long. The increases

of thrust due to steering are shown in Fig. 7 and their rate against various rudder angles in Fig. 8, supposing the ship speed is 18 kn.

Fig. 8 proves that increment of thrust is pro-pol'tional to the square of rudder angle and that it

is nearly given by the following equation:

in radian.

According to _{this conclusion, the}

_{loss due to}

steering will be 22 for sinusoidal steering with the

amplitude . As this value is obtained from the

400

300

20

Full-Sca'e Measurements of Yawing

On M.S. florida Maru

We have been able to determine the assessment formula in the previous section, but we have no knowledge of disturbances acted on ships.

The authors had a chance to conduct full-scale measurements of yawing on the cargoliner M.S. Florida Maru by the cooperation and goodwill of Kawasaki Kisen Co., Ltd. and Kawasaki Heavy Industry Co., Ltd. The following is

part of the

results.

Principal particulars of M.S. Florida Maru: 145.00m

BM 19.40m

DM l2.20m

d 8.72m

Cb 0.68

The measurement was carried out at regular times everyday. The yawing angle and the rudder angle were continuously recorded in automatic steering

condition.

The auto-pilot equipped on board

is

Sperry-Rate Pilot produced by Tokyo Keiki Co.,

Ltd.

The data about yawing of ship due only to ex-ternal disturbances, that is motion of ship with no

1 oO.01 10.0 0.1 0.00 0.1 0.00 1 10.00.01 w (rad/sec)

Fig. 10 Beam Sea V, 15K Wave 80 Left Sea 6 Swell 7 Wind 11 0 Left 1 5m/s Rudder Adj. 2 Weather Adj. 3 &5X 0.01

::

1 V, 16K Wave 20 Left Sea 5 Swell 5 Wind 80' Left 6m/s Rudder Adj.2 Weather Adj. 3 S5 X 0.01

.-7A

V, 14k Wave 120rn Left Sea 5 Swell 5 Wind 150' Left 15m/s Rudder Adj.2 Weather Adj. 3 S5 X 0.01 3rd

10 Japan Shipbuilding ¿r Marine Engineering

Fig. 9 FoI!owing Sea Fig. 11 Heading Sea

w Q 0.0 bu Q w (rad/sec) 01 10 w (rad/sec) 01 10 Q 0.01 Q 0.1 0.01 Q 0.001

These facts

auto-pilots.

July 1968

steeling, is the most required but cannot be directly obtained. Therefore, we got it by subtracting the component of steering from the measured yawing angle.

Some examples of results are given in Figs. 9,

IO, and 11.

The nomenclature in those figures is as follows:

Sr : Spectrum of angular velocity of yawing motion of sin1) in automatic steering con-elition

Srl: Spectrum of angular velocity of yawing motion of ship due to external disturbances only

So : Spectrum of measured rudder angle.

The measuring conditions are described in each figure. Among them, the values of sea and swell are those described in the log book, the wind velocities are relative to the ship, and the values of rudder adjust and weather adjust show the read values of the scale of the auto-pilot.

We can find in these figures distinct differences between two sea conditions of following and head-ing. While there exist very sharp peaks in the

case of following sea, they are not so much striking in heading sea. This fact may be predicted because

the abscissas stand for frequencies of encounter to waves.

We can see Sr coincident with Srd about this peak, which means that steering is incapable of controlling motions of a ship due to external dis-turbances of high frequencies caused by waves.

Design of Optinium Auto-Pilot Proportional Control

At first, we will study the proportional control system, one of the simplest. The block diagram will be as shown in Fig. 12. In the present problem,

WC can SUOSC 0 = O without lack of generality, and then the deviation of path from the appointed course is given by 0e.

The relation between r and O in the block

dia-gram will be as follows:

00 s

1+k K(T3S+l) I (T1S±l)(T25+l)

in the region of extreme low frequency, that is, as o in this equation:

(9)

Fig. 12 Block Diagram of Automatic Steering System

2

9o_l

S k

raS'

raS

Therefore, in the low frequency range, the great-er k is, the less 00/rd becomes, while the angle of rudder retains a constant value. On the other hand,

it

is of no use to take the large k value, for the

rudder angle increases in proportion to k but 00/rd

is constant.

It will be deduced from the above discussion that there exists the optimum k value. We will detect it employing Eq. (8). Now, let be 8.

It is well known that

2 and 2 _{are given as}

follows:

-

Coo ₉ 2 92=\

_{- Srdtha}

J_oo ra Coo

62

Srddw J_oo ra, k

Fig. 13 Optimum Value of h

91

8

rakK

_R

are very significant in design of and, in the high frequency range,

(9), rj Disturbance) (10)

as S°co in Eq.

I'I oi K jr0 s+1) 00

(T,S+fl(7S+1)L.

2 3 4

12 10.0 1.0 0.1 0.01 S, and Sa Without Rate Contro! (rad/sec) 0.1 / S and Sa With Rate Contro!

S, With and Witho Ratn Control

Fig. 14 Effect of Rate Control

1.0

S, With

Rate Control

The computation of the performance index for various values of k with the aid of these two rela-tions and Srd in Figs. 9 and 11 turns out as shown in Fig. 13. This figure proves that the large value of k does not so much harm in heading sea for the motions generated by waves are small, and that it gets much worse as k grows larger in following sea.

We know k value of the usual auto-pilot equipped today can be varied from 0.8 to 4.0, and that it is usually set around l.5-2.5. This ordinary value seems to be pretty greater than the optimum one in Fig. 13.

Rate Control

Nowadays, the method of rate control is applied to auto-pilots so as to keep the stability of the whole control system together with proportional control.

It must be noted that

rate control increases the angle of rudder at high frequency.

Fig. 14 shows the spectra of Oa for Srd in Fig. 9

with and without rate control. Tile stability of the whole control system is improved by rate control in

the oblique region in

the figure, but tile mesh region points out the increase of rudder angle on

the contrary. Tile evaluation will be given by the difference of these two areas, so that it will be evident which is better considering logarithmic scales of titis figure, if we ignore the stability of

control system.

It is concluded that the quantity of rate control

should be kept as small as possible if only it satisfies

tile requirements from point of view of stability, and that tile employment of filter needs to be cared around tile frequency of wave.

10.0

1,0

0.1

0.01

Weather Adjust

The feed back loop of present auto-pilots possesses

back lash to prevent too much sensitive response of the steering system to individual encountered wave. But it was pointed out that back lash injured tile

stability of control system5).

Tile authors think the present idea of design of auto-pilots is confused about this point. To imagine

the history of automatic control, back lash was

introduced for avoiding steering for each wave, which

macle control systems unsteady, but the unexpected defect was thought removable by the use of rate control without any reflection of back lash. There-fore, we can easily deduce that the effect of back

lash will be cancelled by tite increase of rudder angle owing to rate control. This will be proved by the comparison of tile results computed as for with and without back lash and rate control.

Fig. 15 exhibits one example of such comparison. Conclusion

The above mentioned remarks come to the

fol-lowing conclusion:

I) _{lt goes without saying that auto-pilots should} be worked to reduce the deviation of path from tile appointed course, and as a matter of more importance, it is necessary to avoid too much steering.

2) The elimination of vain steering against in-clividual waves is the most significant for _titis purpose. Namely, the ratio

of rudder angle

should be taken the appropriate value smaller

Fig. 15 Effect of Weather Adjust

1.0

Rudder Area

\\rger

Course Stability Better

(+-)

Fig. 16 Relation between Turning Ability and

Course Stability

than today's usual value, and the quantity of rate control must be restrained as little as pos-sible, and yet filtered

out in high frequency

range.

l) Today's method of weather adjust does harm

only.

Improvement of Maneuverability

By Automatic Steering6

Course stability and turning abilities, the two main properties of maneuverability, are

contradic-tory characteristics, whence they cannot be improved

simultaneously by any way other than the enlarge-ment of rudder area, and scarcely possible even if we alter the hull form of a ship. This is illustrated

in Fig. 16.

This figure represents the relationship of the index of course stability l/T' and the reciprocal of tite index of _{turning ability 1/K' according} t

Nomoto's definition71.

When the rudder area

is

constant, the variance of maneuverability due to the alteration of 1juli form always appears as the ratio K'/T' is hardly changed, namely, moving on a

straight line through the origin in Fig. 16.

There-fore, when we deform the hull form of a ship of inferior course stability such as B Ship intending to make better course stability, to say practically to append a dead wood at the after part of hull, the turning ability of such a ship is inevitably injured and then it becomes similar to that of A Ship.

We can get the inclination of the line of K'/T'

smaller by enlargement of rudder _{area, whence it is}

thuis possible to improve course stability without harming the turning ability, but this way has, of course, such limitations that it makes great better-nient impossible.

Tite uniform ratio of K to T nearly means that

tite angular acceleration is _{constant immediately}

July 1968

Fig. 17 Comparison of Response between

Ships A and B

after steering. Accordingly, angular accelerations of

A Ship and B Ship varies as shown in Fig. 17 on

stepwise steering. It is seen from this figure that

the good turning ability of B Ship (large K8) causes its had course stability (large T8), while it is

con-trary about A Ship's.

When we employ an automatic control apparatus for B Ship with good turning characteristics in order to make actual angle of rudder smaller than

ap-pointed one, we can get B Ship with nearly the

same course stability and turning ability as A Ship. But, as original B Ship possesses good ability of turning without an automatic control, it is enough if only we cut off the automatic control system for an urgent turning. Thus, we can produce a ship

having the saine turning ability as B Ship and

similar course stability to A Ship, stich as C Ship in Fig. 16.

Our question is whether we can convert a ship with one arbitrary steering ability into the ship with other arbitrary one. This problem will be

dis-cussed in the following sections. Conditions of Automatic Steering

The equation of motion with a rudder is given by Eqs. (2) and (3). We should like to change each coefficient of the terms of equations by means of

automatic control.

The automatic control discussed here does not mean to turn the head of a ship toward a certain direction but allows to choose an arbitrary direction. Therefore, the rudder angle will be controlled as a function of angular velocity r' and angular accelera-tion i>. When the rudder angle i is given manu-ally,

the controlled angle of rudder can be

rep-resented by:

8'= (12)

13

1.0

0.5

II.

Stability Index

D E

0.5

Fig. 20 Responses with Automatic Contro!

coefficients remain, but we have only three para-meters to be determined arbitrarily, therefore, it is not enough to satisfy all the conditions completely. To examine the error due to this incompletion, it proves to be very small.

Nuinerica Examples

Let the ship of Series 60, Cb = 0.60 be the parent

ship and we regard its course stability satisfactory. Suppose other four ships B, C, D, E. which all have better turning ability than A Ship but worse course stability.

These ships can be realized by

altering the coefficient N from that of A Ship, that

is corresponding to moving the center of lateral area

further forward than A Ship.

The course stability indexes of these ships are given as follows, and D, E Ships are seen unstable:

Values of o, O2 and O3 to give them

maneuvera-bility equivalent of A Ship by appropriate controls will be calculated as Fig. 18.

It should be noticeable that particular changes of controlled quantities cannot be found as for the unstable ships of D and E in comparison with B Ship.

The responses of each ship of stepwise steering

are examined to check how the maneuverabihities of

other ships approach to that of A Ship.

Fig. 19 shows the variation of angular velocity of each ship without the automatic control. The

most stable Ship A, of course, turns the most slowly.

The response of each ship controlled automatically as designated in Fig. 18 is represented in Fig. 20, which makes clear that each ship gets almost equi-valent to A Ship. lt is a matter of course that the

A: 1.03

B: 0.51

C: O

D:

0.51

E:

0.77

14 Japan Shipbuilding & Marine Engineering

Fig. 18 Vahes of 2 and Ø3 to Give an Equiva!ent Maneuverabffity as A Ship

o 20

Fig. 19 Responses without Automatic Contro!

Substituting Eq. (12) into Eqs. (2) and (3), it

follows that, since there are three parameters 0i, 02

and (f3 to be chosen arbitrarily while those equations have three variable coefficients, the steering qualities

of two different ships cannot be made almost equal by one rudder. So, two rudders steering will fill the above requirement, which should be set as far away as possible so that the ratios of Yò' to N&' of the two ships can be differed from each other. Con-clusively, we need combine stern rudder with a

bow rudder or a bow thruster. However, we think practically we need not necessarily make steering abilities of A and B Ships strictly equal, namely, if qualities of change of yawing angle, for instance, are identifiable, quantities of drift will be allowed to differ somewhat between both ships. When we

responses of Fig. 19 will revive if the automatic control is released. These results llave been verified

by some experiments.

From these results we can conclude that it is

possible to make maneuverability of a ship almost equal witil that of other arbitrary ship by means of the automatic control. Hence, tile ship which possesses a good turning ability but inferior course stability can be used usually in the condition that the automatic control works to reform the course stability but to get tile turning quality down and, if necessary, this ship is of great advantage to turn rapidly as soon as tile automatic control is shut off. Such an automatic control is no doubt additional to tile manual steering, so that tile helmsman of

such a ship may feel

as

if he were steering an

ordinary ship with an original good course stability.

Automatic Guidance of Ship to

Prescribed Course8

The method of landing with meters has been

already put in practice for airplanes, furthermore, the complete automatic landing is going to be used practically, because it is dangerous for landing of airplanes. Nevertheless, even the entrance of the way to tile practical use of automatic guidance of a ship is out of sight.

The recent enlargement of ships

brings poor maneuverability which is good to exceed the limit of the ability of manual control, and tile heavy rush of cargo invites tile great increase of traffic in bays or inland seas. Tilese pllenomena swell the risky probability of collision. For prevention of such risks or for effective utilization of restricted seaway, the

problem of automatic guidance of a ship to thç

prescribed course and on arrival will need to be discussed in the nearest future. In this chapter, we have an attempt to study on the problem of

auto-matic guidance basically.

It is supposed as the premise of argument that many kinds of information of ship such as situation, direction angle and so on, are known with sufficient accuracy.

Form of Automatic Control

Eliminating r or from Eqs. (2) and (3), we get

the following two equations:

T1T2+(T1 + T2)?+ r= K(T3,- + 8) T1T2+(T1 + T2)g +f=Kp(T3ß6 + 6)

In the present work, not so much capacity of an usable analog computer may oblige to adopt tile

July 1968 T?+r=Kr6, T/3+/3=Kß6. 8 =k1e+k2 taking, J Fig. 21

Fig. 22 B!ock Diagram of Contro! System

first order approximation and it is assumed that

T3 nearly equals to T3 . Therefore, Eq. (13) will

be rendered to be approximated by:

(14)

Provided that tile silip is located at tile position deviated from the prescribed course as shown in Fig. 21, we will try to _{carry out the following}

control form:

(15)

Considering the capability of intentional oblique running, o

and i

of the prescribed course are not probable to be independent on each other but have

the relation of:

Usin 9(t)dt+(0) (16)

(13) _{Accordingly, we get the block diagram of the linear}

system for small e1 which is exhibited by Fig. 22.

The transfer functions between main stages dotted

in Fig. 22 are given as:

15 k, K, TS +1 8 ß

r

s On(s) K(k1S+k2U) TS3±S2+(klKr_k2KßU)S+k2KrU 0(s) U (kiS+k2U)(KrKS) Of(s) - S TS3+S2+(kjKr_k2KßU)S+k2KrU8 rn(s) S2(TS+l+k1Kp) (19) O(s) - S TS3+S2+(kjKr_k2KjU)S+k2KrU (17) 6(s) (TS+l)(k1S+K2U)S O(s) TS3+S2+(kiKr_k2KU)S+kZKrU (20)

Specification of Contro! System

The characteristics of a control system is almost decided by the characteristic equation, which is a

cubic equation as seen in Eqs. (l7)'(20) for the

present problem of automatic guidance. Therefore, roots of the characteristic equation are composed of a pair of complex values and one real value.

The rapidity of response and damping quality of the whole control system are often recognized from

a pair of complex roots situated nearest to the

imaginary axis in the Gaussian plane in analysis of

general control systems. lt is similar for the auto-matic guidance of a ship too. However, under cer-tain circumstances, the absolute value of the real

root may be very small, then the resultant slow

damping requires us to make the specification related

to other items than oscillation properties.

Synthesizing these results, we have got the

follow-ing specification of the automatic guidance system

which receives

the input shown in Fig.

23 that

seems to be regarded as the representative one. Of course, this is not an authorized one but an authors'

rough estimate.

I) The oscillation period should be within 100 sec. The logarithmic damping of oscillation should

be over 0.25.

To cross the appointed course during the first

one period.

The first condition is the requirement for the rapidity of response and the second is the request for appropriate damping. It appears that the value of 0.25 for logarithmic decrement is too wrong in comparison with other usual servo-mechanism, but we lìave difficulties in choosing it greater than this value as illustrated by the numerical examples in the next section.

The third

is the condition to

secure the damping ability for the portion of the real root.

Numerica! Examples

We have evaluated the ranges of k1 and k2 to

agree with the previous specification by the aid of

=

Fig. 23 Typical Input

an analog computer for the full loaded cargo vessel with the following maneuverability, as an instance:

T =25sec.

Kr = 0.05/sec

= 0.35

U =7m/sec

Supposing the aforementioned time constant can be reduced by an adequate rate compensation, we have similarly dealt with other three cases that time constants are 10 sec, 15 sec and 20 sec. The transfer function as for rudder angle is different in each case while other transfer functions may be thought

the same.

The range of k1 and k2 suitable to the

specifica-tion concerning the four time constants is illustrated by Figs. 24-27.

lt

is

a matter of course that as

the time constant is smaller, the range is more extensive so that controlling is easier. When T = 25

sec is practical, it is evident that we could not satisfy the specification without rate control whatever values

of k1 and k2 would be selected.

The various forms of response with respect to

are compared in Fig. 28 for the cases that k1 and k2

are within such range or out of it when T = 15 sec,

that is, corresponding to the conditions as shown

by marks a, b, c, d in Fig. 25.

Discussed in this chapter is the about

the problem of automatic guidance up to the pre-sent that has been studied. The authors would like to conclude this writing with anticipation of the future direction of research of this subject. The final aims of tile automatic guidance will be able to be predicted numerically, especially the proper forecast of direction angle of a ship appears to be effective to improve the existing state. The authors are intending to study on the problem of the gen-eral optimum control, and on the way of controlling simultaneously many multifarious ships with

in-16 _{Japan Shipbuilding & Marine Engineering}

t-4

3

o

_p

-(liStiflCt properties so as to apply it to the central

control system of coming and going ships in harbors.

Such a research will be thought to give some

devia-k xio (rod/rn)

Fig. 24 T = lüsec

2 k >(10' (rad.Im)

tive but important knowledge, such as the curvature of path fit to the prescribed course or the level of

minimum maneuverability indispensable to all ships.

'k

k X 1O _(radim)

Fig. 26 T = 20 sec

k X iO (rad./m)

Fig. 25 T = l5sec _{Fig. 27 T = 25 sec}

Pote d o b T<100 sec p-'>025 C o July 1968 17 Pole r<ioosec 3 '> 025 2 3 * * 2

4,

-YOKOHAMA

(dl Insufficient Damping cf Pole k, = 4

k,0.26X10' rad./rn

Reference

T. Kayama: On the Optimum Automatic

Steer-ing System of Ships at Sea, J.Z.K., Vol. 122, 1967

H. Eda, C. L. Crane:

Steering Characteristics

of Ships in Calm Water and Waves, TSNAME,

e-7t'Û6L1o//3

i9'Ó

S. Motora:

On the Automatic Steering and

Yawing of Ships in Rough Seas, J.Z.K., Vol. 94,

1954

K. Nomoto, T. Motoyama: Loss of Propulsive Power Caused by Yawing with Particular Ref-erence to Automatic Steering, J.Z.K., Vol. 120, 1966

K. Nomoto: Stability of Auto-Piloting, J.Z.K.,

Vol. 104, 1959

Motora, T. Koyama: On an Improvement of

the Maneuverability of Ships by Means of Auto-matic Steering, J.Z.K., Vol. 116, 1964

K. Nomoto: 60th Anniversary Series

of The

Society of Naval Architects of Japan, Vol. 11, P. 48

Fuwa, I. Watanabe: On the Automatic

Guid-ances of Ships, Graduation Thesis, 1968, Univer-sity of Tokyo.

Fig. 28 Comparison of Response E

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(a) Adequate Control

k, = 3 k=Q75XlO tad/rn 1) 2) (b) Insufficient Danrpng = 3 k,= 118X10' tad/rn 3) 4) 5) 6)

Ic) Too Slow ¡n Response

k, 0.26X 10' tad/rn

7)

8)