MATHEMATICAL TREATMENT

OF OPTIMAL OCEAN SHIP ROUTEING

PROEFSCHRIFT

TER VERKRUGING VAN DE GRAAD VAN DOCTOR IN DE

TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE

HOGESCHOOL DELFT OP GEZAG VAN DE RECTORMAGNI FICUS DR. IR. C.J.D.M. VERHAGEN,

HOOGLERAAR IN DE AFDELING DER TECHNISCHE

NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT

TE VERDEDIGEN OP WOENSDAG 20 NOVEMBER 1968

TE 14 UUR

DOOR

CORN EUS DE WIT

DOCTORANDUS WIS- EN NATUURKUNDEGEBOREN TE ROYFERDAM

1968

"BRONDER-OFFSET" ROTTERDAM

Dit proefschrift is goedgekeurd door de promotorPROF. DR. R. TIMMAN.

CONTE NTS

page

1. INTRODUCTION 9

1. 1 The stage preceding ship routeing 9

1.2 The K. N.M. I. ship routeing department 10

1. 3 Description of the dynamical system 11

2. COORDINATE SYSTEM 14

3. ANALYSIS OF THE CONSTRUCTION OF AN OPTIMAL TRACK 19

3. 1 Controlled dynamical systems 19

3. 2 Restricted coordinates 20

3. 3 The unrestricted problem 20

3.4 Extremals, optimal controls, timefronts 24

3.5 PontryaginTs maximum principle 28

3. 6 Behaviour of the tangent and the normal to a timefrontalong an extremal 31

3. 7 Construction of the solution. Examples 39

3. 8 Remarks on the occurrence of more than one solution 463. 9 Modifications in case of coordinate restrictions 50

4. PRACTICAL DATA 57

4. 1 The wave prediction problem 57

4. 2 Some basic elements of ocean wave theory 57

4. 3 The Sverdrup-Munk wave forecasting method 59

4. 4 The Pierson-Neumann theory 62

4. 5 The performance of a ship in a given wave field 70

5.

page

4. 5. 1 Practical approach 70

4. 5. 2 Scientific approach 71

EVALUATION OF THE LEAST TIME TRACK 75

5. 1 Oceanographic and meteorological data 75

5. 2 Evaluating the tirnefronts 76

5. 3 Revision of the timefronts 78

5. 4 Determination of the trajectory and the time gain 79

5. 5 The Algol program to compute the least time track 80

5. 6 Considerations regarding data incertainties andpractical use 87

MATHEMATICAL TREATMENT OF OPTIMALOCEAN SHIP ROUTE ING

1. INTRODUCTION

This chapter contains a description of the practical data, that formed the basisof ocean navigation for merchant ships in the past.

The foundation of a Ship Routeing Department at the Royal Dutch Weather

Institute (K. N.M. I.) in 1960 can be seen as an end of this period for two reasons.In the first place the issue to a ship of navigational advices by a shore stationwas an important renewal in navigation history, while secondly this was a seriousattempt to furnish ships with information regarding the sea conditions to beexpected, which was never but occasionally done before.This chapter concludes with the description of a suitable mathematical modelof the ship's movement with respect to the earth's surface, suitable in thesense that it can be used for solving the problem of constructing the shortest -i.e. least time - track between two fixed points.

1.1. The stage preceding ship routeing.

The entirely autonomous navigation of merchant ships across the North AtlanticOcean in either East or West direction was committed on the following basis:

Statistical information regarding the occurrence of bad weather andrough sea furnished by books, called "Ocean Pilots" and by Pilot Charts,issued by the U. S. Hydrographic Office.Information regarding areas to be avoided on account of iceberg danger.

e. Determination of a great circle track and computation of the intersectionso oof this track with meridians of 10 W, 20 W etc.

d. Meteorological information from a weather forecast for a period of oneto five days ahead.

This meteo information contained little or nothing regarding the sea conditionsto be expected. The ship's master was compelled to draw his conclusions fromthese weather forecasts all by himself, led by his experience and by his commonsense. Therefore, the deviations from the predetermined track seldomly were

9

more than occasional and for low powered ships, this procedure not rarelyresulted in a considerable delay of the time of arrival.

1.2. The K. N.M. I. Ship Routeing Department.

Seeking to avoid these delays and the high costs they implied, two Dutch shipping

companies studied the possibility of furnishing their ships with more detailedinformation on the sea to be expected, such as height and direction of sea waves,direction and period of swell.For this purpose, the K. N. M. I. established a special Ship Routeing Department"

with the assignment of predicting sea conditions for and giving navigational ad-

vices to ships "under treatment'.This routeing office was thus confronted with various problems of which nosystematical study had been made so far.To start with, a reliable picture of the weather situation was needed, not onlyfor the present and the past, but also for the next (at most) eight days to come.In the case of an obviously stable weather situation, this stability was extra-polated over the period the crossing was expected to take and a least time track

was constructed in accordance with it.In the more frequently occurring case of a rapidly changing weather picture,no assumptions were made at all for more than two days ahead. Here therouteing department confined itself to pilot the ship through the dangers of thenext two days to come. Working this way, disappointments still occur, althoughwith considerably less frequency than in the preceding period with no shore basednavigating control.The second difficulty was the prediction of the sea conditions as a result of agiven weather situation. Although many aspects of this problem are still subjectto serious studies, the routeing department has by now attained a satisfyingdegree of reliability regarding these forecasts.Thirdly the ship's response to the sea and swell waves had to be considered.As this response depended upon various characteristic properties of the routedship, such as its draught, shape, length, longships and thwartships stability,the routeing was started on a strictly voluntary basis with two ships of theHolland America Line. After some experience, obtained with these trial models,the service was extended to other ships of the same general cargo type, whilenowadays also tankers and container carriers are frequently accompanied bythe K. N.M. I. routeing office on their way across the North Atlantic.

lo

1.3. Description of the dynamical system.

It is the aim of this thesis to indicate a construction of the least time track ona mathematical basis, mainly formed by Pontryagin's theory of optimallycontrolled processes.To start with, the weather and sea situation will be assumed to be known in theentire navigating area and for a time last, that covers the average trip's duration.From a mathematical point of view it seems perhaps most appropriate todescribe the ship's movement in a two-dimensional manyfold - the earth's sur-face - by means of complete equations of motion. However, practical navigationmethods as well as some numerical facts regarding the motion of an ocean vesselnecessitate some simplifying assumptions.In the first place ocean navigation is carried out with the aid of a conformalocean chart of a small scale with such a slight scale alteration, that a straightline between two points on this chart can be taken as a good enough approximation

of the shortest distance between these points. A chart with these properties woildbe Kahn's oblique cilindrical projection. This chart will be treated more closelyin the second chapter.For the time being, it will be assumed that the ship's position and movementcan be accurately enough described with the aid of an orthonormal 2-dimensionalcoordinate system.

The dynamical system, describing the horizontal movement of the vessel, callsfor the following notations:

x,y : Ship's position coordinates. (It is customary to neglect the ship'sdimensions in sea navigation)Components of the ship's speed in the and directions, relativeto water.Number of ship's engine rotations per time unit.Steering angle, i. e. the sharp angle between the rudder plane andthe plane of keel and stems.

(The numbers n and y could be regarded as control parameters)Longships and thwartships directions of the driving power per massi.mit, excited by propellor and/or rudder.

Resistance force components per mass unit in 1- and d-directions asa result of ship's speed and situation of sea and swell.Ship's course.

11

u, V

4n

y

l,d

wl,wd

t : Time.t , t. : Times of departure and arrival.

e', c2 : X and components of the sea current, assumed to be stationary.: Time derivatives of x,y,u,v.

E., i=1(1)m: Sea resistance parameters.

With these notations, the equations of motion are:ii = (i - w1)cos a - (d - wd)sin a

(1. 3. 1)=

(I- 1)sin (d - wd)cos a

2y=v±cThe driving force component lis a function of n and y. while d is dependent ofn, y, u and y, so(1.3.2) l=l(n,y); d=d(n,y,u,v)

The controls n and are limited: tn N lvi 4 F.The resistance components w1 and Wd depend on the longships speed (u2 + y2)2

and the rudder angle y as well as on the parameters,

which are functions

ofx, yandt.The problem to be solved would now be to find a trajectory satisfying (1. 3. 1)

and begin and end conditions like x(ta)=xa, y(ta)=y , x(tb)=xb, y(t.0)=y and to

choose n and y as measurable functions of t in such a way as to minimize the

time t. for a given value of to a

This system, however, is unfit for practical use for two reasons:On account of safety considerations regarding the ship's oscillationsthe necessity to reduce n in heavy weather frequently occurs. Thus nno longer satisfies the requirements of a freely choosable control.The practical result of (1. 3.1) always consists of long periods (12 hours)of ultimately slightly changing u and y, interrupted by short time inter-

vals of a few minutes, during which u and y change to anoTher value

as a result of a pulse type alteration of y and possibly of n.On account of the fact, that the influence of the sea parameters -such as height and direction of the sea waves direction and period ofswell - on the ship's resistance is by now far from exactly known forone thing, while on the other hand one has good reasons to expect errorsin these parameters, the values of u and y, obtained from integrating(1. 3. 1) can hardly be expected to have any practical use.

12

It is for these reasons, that another model will be adopted.Given certain sea conditions in a point (x, y) at a time t, the maximum attainablespeed relative to water can be regarded as a function of the course angle a.Denoting s = (u2 + v2)2, the equations of motion are:

1 3 3 = s(x,y,a,t)cos a + c1(x,y).) 2= s(x,y,a,t)sin a + c (x,y)The problem is now to find a trajectory satisfying (1.3. 3) and the startingcondition:

(1. 3.4) (Given ta) : (x(t), Y(ta (x,while for some, yet undetermined time t,0 the condition(1. 3. 5) (x(tb) y (» '

must hold. Furthermore a as a function of time is to be chosen so that itminimizes tb.This system is fairly in accordance with the practical case.The way s depends on a can in some occasions be very peculiar. In heavy sea,

the highest attainable speed can for certain value intervals of a be extremelysmall or even zero, meaning thatavalue of a in that interval is highly unad-visable or even forbidden.

This is the main reason, why an analogy with the air navigation problem, studiedand treated by H. M. de Jong (K. N. M. I. publ. 64) was not possible. In thatcase s was entirely independent of a, which made the problem solvable as adirect application of the classical calculus of variations.It must be noticed that the course angle a is to be considered as a "course madegood". It differs from the steering course a by the drift angle d, as shown infigure 1. 3.a.

Figure 1.3.a.

ship's velocity relativeto water

13

2. COORDINATE SYSTEM

System (1. 3. 3) supposes a plane orthonormal coordinate system. Crossingthe North Atlantic involves an area of the geoid of 4000 * 1200 square nautical

miles, that must be scanned to find the best ship's route.As it is a practical habit of navigators to work with a conformal sea chart - toavoid tedious corrections of measured or plotted angles, like courses, sightand radio bearings - it is most convenient to construct a plane conformalmapping of this area with very slight scale alteration.Considering the length/width ratio of the area, amounting to about 3. 3, it seems

most appropriate to make an oblique Mercator projection.In a Mercator projection the equator is mapped as the X-axis and the mappingof some meridian is the Y-axis. The isometric grid of parameter lines, thatis being mapped as the net of lines parallel to the X- and Y-axis, is formed byparallel circles of equal geographic latitude and meridians. It would implyquite a few computational difficulties to construct such an isometric system onan arbitrary part of the geoid.In order to avoid these difficulties, one can make a conformal mapping of theearth's surface onto a sphere by way of intermediate step. Beside being con-formal, this mapping must have a practically constant scale.Denoting the spherical mappings of A and B - the vertices of the ship's oceanroute - as A' and B', we can consider the great circle over A' and B' as anoblique equator, while the collection of great circles through the poles of thisgreat circle over A' and B' serve as skew meridians. The conformal mappingwith this basis is called a Kahn projection.

Notations:

cp , . : geographical latitude and longitude.

a : radius of the equator.

e : excentricity of an elliptic meridian.

p : curvature radius of a meridian.

N : curvature radius of the prime vertical., L latitude and longitude of the spherical mapping of a point (p , X).

R : radius of the sphere.o : index, referring to the projection centre, when mapping the (Hayford)

14

ellipsoid onto a sphere.n : longitude scalek : distance scale sphere/ellipsoid.pQ : great circle distance from P to Q.VpQ : maximal latitude of the great circle over P and Q.8 : spherical distance from a point p? of the sphere to the oblique equator.y : angle between the oblique meridians of P' and S, the intersection of the

oblique and normal equator closest to the projection centre.m : angle between P'S and the normal equator.s : scale Kahn chart/sphere.s :valueofsforô=o.

Adopting the Hayford ellipsoid as a suitable approximation of the earth's surface,we work with:

a = 6378388 meterse = 0. 08226889

a(1 - e2)2 2 3J(1 - e sin cp)

aN=I 2.2- e sin w)

The transformation of (cp , X) to ( , L) will now be described.Taking = (cp) and L = L(k), a conformal mapping requires the equality:

k Rd R cos dLpdcp Ncoswd?

dLTaking n = - constant and L X , this gives:dX o o

L=X +n(X-X ) ando o

w (1 - e2)secf dfSsecpdp=n $ 2wo 1-esinf

The apparent freedom to select the constants and n is now used to make2 °

and equal to zero for w=

The results are:w

dw

15

16

= arctg(tg

n sin p0/sin

Rri(pN0)

The coordinates (, L) of the spherical mapping of a point on the earth (p, X)follow from:

e= 1n[tg(r / + /2)(

- e sin cp3+ e sin p

= 2 arctg(exp(In(tgfrr/4 /2)) + n@ - a ))) - rr/2(p (p0

L=X +n(X-Xo o

In order to demonstrate the slight scale alteration,, k was computed, with= 46°, for cp = 150(10) 600:

Taking k=1 implies an error over a 2000 miles stretch from the edge to the

centre of less than 0. 5 mile. For navigational practice this is negligible.

Taking the great circle on the sphere as an approximation of the geodesic

on the ellipsoid introduces another error of a still smaller magnitude. The

small value of keeps this error smaller than 210' miles over a 100 milesd(p

stretch.

((p)

150

20°

(k

1. 00029868

1.00017889

25 1. 00009431

30° 1. 00004036

40° 1.00000025500 1.00000086

o55 0. 99999470

60° 0. 99997084

P,

Figure 2. a.

Concentrating now on the transformation of (, L) to (6,y), the quantities y andL are computed by:

dA,B,arccos(cos ATc08 B'

+ Sfl A?Sfl BTCOS(LA, - LB,)),R -

dA,B,y = arccos(cos A,COS B?S1fl'Aj - LB,)cosec( R

L = LA, - arcsin(cotg y tg 'A'

Then the angle m follows from m = arctg(tg cosec(L - L)).Finally, ô and can be computed:

dsp,R - arccos(cos(L - L)cos r)

dsp,ô = arcsin(sin(--) sin(m - y))

dspi= arctg(tg(--) cos(m - y))

The x- and y-components of the mapping P" of P' into the oblique Mercator pro-

jection can be computed by means of:

arcsin(sin 'VB' cosec y),

X = sR( - 'BT)

17

y = sR ln(tgT-r/4 + /2)

ss secio

One can take accoimt of the scale alteration by multiplying a line element

ds = (dx2 + dy2)a with the factor sec ö, where ô follows from y according toô = 2 arctg(exp(y/s R)) -

Another possible error source is the fact, that a straight line segmentis taken as the shortest track between two points.For a distance d between two points P and Q, measured along a straight linein the Kahn chart, the difference with the great circle distance between P and Q

is maximal if ô1) = ôQ

namely:

dsecôd - 2 arcsin(cos ô sin( 2 R

With IôI loo and working with distance steps of at most 200 miles, the errorper step never exceeds 0.06 miles, so the relative error is at most 0.03%. Ona 4000 miles stretch, this could accumulate to 1. 2 miles, which is negligible

for navigational practice.

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3. ANALYSIS OF THE CONSTRUCTION OF ANI OPTIMAL TRACK

Optimal ship routeing can be incorporated into the class of optimal controlproblems. These problems were investigated and analyzed by a group of

Russian mathematicians, directed by Pontryagin, while Halkin later publisheda more generalized theory. It was Pontryagin's merit to effectuate somethingmuch like a breakthrough from the classical variation calculus into modernapplied mathematics by announcing his maximum principle. Halkm gave theproblem statement and the adjacent theory in a somewhat different and moregeneral form. His concept of the "set of reachable events" can be seen asa continuation of Hamilton's wave front theory. It also opens pracdcal possi-bilities for an actual solution of the problem.With this basis, the third chapter has no other pretentions than to be an ex-planation of this application of the general theory.

3.1. Controlled dynamical systems.

- 12Consider a ship, starting in A:(x ,x ) at a time t , following a course , whicha a 12is a given function of the time t, and arriving in B:(xb,xb) at a time t,. This

sequence of events can be conceived as a dynamical system. The events y = (x, t)are elements of the event space, which is the Cartesian product of the twodimensional space X - with points x = (x1,x2) - and the real time axisT={ttt.Soy=(x,t)EX*T.A dynamical system is specified by the fact that the various elements (x, t) aremutually connected by a binary relation R on x * T:

yRy for all y in X * T.if y1Ry2 and y2Ry3 then y1Ry3.

(e) if y1Ry2 and y2Ry1 then y1 = y2.

(d) if y1ly2 and y1Ry3 then either y2Ry3 or y3Ry2.

This relation R is caused by a system of differential equations(3.1.1) = f(x,ci (t), t) almost everywhere on T.As n(t) is a given function of t, this equation can be written as

k = f*(x,t).

However, by varying n(t), the solution of (3.1.1) will in general vary as well.

19

System (3. 1. 1) can therefore be seen as a controlled dynamical system - or"control system" for short - and the, yet to be chosen, function c(t) is calleda control or steering function for the obvious reason that it can vary thesolution of (3. 1. 1) to a certain extent.

The optimal control problem now consists of choosing a (t) so that the arrivaltime at B is not greater than any other , generated by some

3.2. Restricted coordinates.

Before studying this control system more precisely, the fact that the coordinatesx1 and x2 of the ship's position are limited, cannot remain unmentioned, becauseit might play a part when determining the least time track.In practice, the only restrictions of importance are the 10, 20 or 30 fathomsdepth lines - dependent of the ship's draught - and the boundaries of icebergdanger areas.These restrictions can be expressed mathematically by the requirement, thatthe coordinate vector x is an element of G, a bounded subset of X, determinedby the condition G = [ x p(x) 0) , where cp(x) is a scalar function of x.As the coordinate restrictions appear to be working as slight modifications ofthe general maximum principle, this problem will be taken into considerationlater in this chapter.

3.3. The unrestricted problem.

Consider the following data: 12The initial point A : x = (x x ), from where the ship starts at a time ta a'a a

The time halfiine T : [t ta t

The point of destination B : Xb = (x,x).

A collection C of available courses a. What this collection consists of, maydepend on the ship's position (x) as well as on the time (t) the ship is at that

position, so C = C(x,t).In a field of low waves C is the entire interval [ O,2i-r).However, if the wave heights exceed 4 meters, some course intervals maybecome prohibited.For a wave direction , C(x, t) can thus be the union of the intervals

20

Figure 3.3.a.

Thinking of a certain trajectory with points x(t) = (x1(t),x2(t)), the valuesof a, chosen in these points, must be elements of C(x(t), t). Following

some track, a can thus be seen as a function of t.For practical reasons, we confine ourselves to control functions a (t) thatare piecewise differentiable, piecewise continuous with a finite number of

discontinuous jumps.The actual motor of the control system is the vector valued function

112 212f(x,a,t) (f (x ,x ,a,t),f (x ,x ,a,t))

f1 and--r- - with i, j = 1,2 - are assumed to be bounded, continuously diffe-

rentiable with respect to x , x and t and differentiable with respect to a.ÒX

1 2

A special study will be made of functions a(t), as mentioned in (5), with theproperty that the differential system(3. 3.1) * = f(x,a (t), t) with the initial condition

(3. 3. 2) x(t) = X and the end condition

(3. 3. 3) x(tb) = Xb for some t.0 3t has a unique and continuous

solution.

The arrival time t,0 obviously depends on the choice of the function (t).

The problem can now be formulated as to find a function a(t), defined in (7) sothat the arrival time t.0, determined by a(t), is not greater than any othergenerated by some

21

Some additional remarks have to be made concerning the components of f(x,a ,t).According to (1. 3. 3) we havei12 12 . . il 2(3.3.4) (x ,x ,a,t) =s(x ,x ,a,t) sin(i-ri/2 -a) c (x ,x ).12 . 1Considering a fixed point P with coordinates (x ,x ) at a time t, the values of f

2 . 1 2.and f can be plotted out into the X - and X -direction respectively for allpossible values of a in C(x, t), using arbitrary, but equal units for and

The curve, that is thus obtained, will be called the "original velocity indicatrix".In a field of low to moderate waves, a can vary continuously from O to 2rr.The scalai function s(x,a,t), indicating the maximum attainable speed for atrue course a, is in that case differentiably dependent of a . The indicatrix isnow a closed curve with a continuous tangent, but not necessarily convex, asit may be necessary to reduce speed in certain courses to avoid too heavy rollingor pitching.

22

Figure 3.3. b.

As the scalar value of the sea current c never exceeds 1. 5 miles per hour,while s is not less than 8 miles per hour, the centre P, from where thevectors f were plotted out, always lies well inside the original velocity in-dicatrix. It is clear that in this case, the ship is able to make headway fromP into all directions.Considering the case, that there are prohibited sectors for a, the originalindicatrix consists of separate arcs (AB and CD in fig. 3. 3. c).

At first hand the possibilities for a ship to proceed from P seem to belimited to the sectors APB and CPD. However, it is still possible to makeheadway into a direction, that lies within one of tue gaps BPC or DPA.To explain this, let us replace the velocity indicatrix by a position indicatrix,meaning that from P, the values of f1 (x, , t) St and f2(x, a , t) ô t are plotted out

Figure 3.3.c.

into the X1- and X2-directions, with It> O and using the adopted units for ¿and x2.As f1 and f2 are assumed to be partially differentiable with respect to x1, x2and t, we have for some fixed a:

t+5tx'(t+ It) =x'(t) + J f1(x,cY,T)dT

t

= x'(t) + f1(x, a, t) It + o(lt),

- o.where o(t) has the property: umItO

The end points of the vectors

x(t) + f(x,ty,t)ôt

can thus be considered as approximations of the furthest attainable pointsfrom P at time t after a small time interval 5t. The errors in the coordinatesof these points can be made arbitrarily small by taking ôt small enough.If one thinks of figure 3. 3. c to be constructed this way, it is clear that allpoints of the straight line segment BC are now attainable by taking linearcombinations of the courses ab and ac during the It-time interval. The pointK for instance is attainable by taking course ab during a time XIt and course

23

during the interval (t+Xôt,t+ôt), where X follows from KC X* BC.

Points of AD can be reached by tacking between the courses OEa and OEd in a

proper time ratio.The objection to this idea could be made, that it is practically impossible fora ship to tack on two onsidrably different courses during a small timeinterval. However, practical data reveal, that in wave fields on the NorthAtlantic circumstances are changing very slightly, so that one hour is asmall enough choice of lt to obtain a practically negligible error in the sensementioned above.In 3. 5 it will be shown, that it is the (smallest possible) convex envelope ofthe original indicatrix, that plays an important part in the constructionalprocedure of the least time track. This envelope will be called the "effectivevelocity indicatrix".As a result of practical experience it can be asserted, that for the merchantships that were routed on the North Atlantic, every point P lies inside its owneffective indicatrix, so that the foregoing considerations justify the conclusion,that it is always and everywhere possible to make headway from P into allhorizontal directions from O to 2îr.

3.4. Extremals, optimal controls, timefronts.

If a trajectory from A to B, satisfying (3. 3. 1) for some control function (t),

generates an arrival time t,0 not greater than any other tb caused by some

(t), the control a(t) is called "optimal" and the corresponding trajectory iscalled an "extremal".

Given the initial event (Xta) and the system (3.3.1), the set of reachablepoints H1(t;x ,t) can be defined as the collection of points x(t) that can bereached at the instant t, by starting from A at a time ta and then followinga trajectory, governed by (3.3.1) for all possible control functions a(t).So H(t;x,t) = [x(t)Ix(t) = X, * = f(x,OE(T),T),TE (t ,t]This point collection, shortly denoted as H+(t), is everywhere dense, closed

and bounded.1 2The boimdedness is an evident consequence of the fact that f and f are

bounded. The density of H(t) means to express that, given a trajectory

x x(-;o(T)), we can always find a control function (T), different from a(T),with the property that, given an arbitrary c > O, every point x*(r) of the

24

trajectory, produced by this control function (T), has a distance to x(T) less

than e. (t<T t).

This property, proved extensively by Halkin'), is guaranteed by the boundedness1 2 1 2

of f and f and of their partial derivatives to x and x . It is, for instance,possible to choose (T) arbitrarily different from a(T) during the interval

(t, ta + ôt), while taking (T) = a(T) for ta 5t î t. By taking it sufficientlysmall, we can always find x*(T) in an c-vicinity of X(T).A consequence of these two properties is the possibility of covering H(t) by afinite number of these c-circles, which means that H+(t) is compact and - by

virtue of this - closed.I now wish to consider the boundary of H(t), called the timefront S(t).

+S (t)

Figure 3.4.a.

In what follows I assume, as is plausible from topological considerations,that S(t) is a continuous closed curve with an almost everywhere continuoustangent. As this boundary of H(t) can be seen as the collection of ultimatelyreachable points at a time t, every trajectory from A to a point of S(t) is anextremal.An important property of these boundaries follows from the considerations andpractical data regarding f(x,ci,t), mentioned in 3.3.Consider the points of S(t) for some t ta As all these points lie inside theirown effective indicatrix, a ship is able to move away from such a point intoall directions. This implies that for every ôt > O, the boundary S(t + it) ofH(t + it) lies wholly "outside" S(t), i.e. two S-timefronts never have pointsin common.

1)Journal d'Analysc Mathììariquc. Vol. XII, J5rusalem Acid. Press.

j' extremals

25

Another useful concept is the set of initial points H(t;x,0, tb). This set can bedefined as the collection of points, from where, starting at a time t, B can bereached at a given time t,0, following (3. 3.1) for some proper ci(r). So we can

write

H (t;x.O,tb) = [x(t)lx(tb) = Xb, t. t,0, k = TE(tt)

This set, shortly denoted as H(t), has the same properties as H(t), whileits boundary S(t) has the same properties as those, that were mentioned forS(t).I now wish to state and prove a fundamental property of extremals:Lemma 3. 4. y: If a trajectory from A to B is an extremal and P is an arbitrarypoint of this trajectory, then the arcs AP and PB of this extremal are "extre-mais" as well.The extremal arc AP is an extremal in the sense that, starting from A at atime t and following the extremal AB, one arrives at P at a time t t, where

at' is the arrival time at P for an arbitrary trajectory from A to P and governedby k = f(x,*(T),T).The extremal arc PB is to be considered an extremal in such a way, that everyother trajectory from P to B with arrival time t,0, following (3. 3. 1), demandsa starting time t t.

C

26

S (t*)S(t)

Figure 3.4.b.

To prove this lemma, let us assume that - fig. 3.4.b - the fully drawn curveAB is an extremal from A (at ta) to B. (Whenever a part of this curve, sayfrom C to P, is considered, it will be indicated as CP ).Taking an arbitrary point of this curve, let us first suppose that PBe is notan extremal. This would mean, that a moving point M', starting from A at

time ta following ACPe arriving at P at a time t, could then reach B in ashorter last of time than t,-t by following some other track - the dotted linefor instance - and would thus arrive in B at a time t < t. This would contradictthe assumption that t,0 was optimal, so PBe must be an extremal.As for the proof, that AP is an extrema!, suppose it is not and assume, thatthe dashed arc ADP leads from A (at time ta) to P at a time t < t. Considernow two points M and Mt, moving from A to B.M takes track ACPB , arrives at P at time t and is in B at time t , not later

e bthan any other point, going from A to B.Mt starts from A at the same moment t, but follows the dashed curve ADP,which brings it in P at an earlier moment t". Then Mt goes to B along trackPB

For the arrival time t of Mt there are now two possibilities, merely thinkingof the fact that Mt leaves P earlier than M

t, < t,0. This is in contradiction with the optimality assumption regardingtb and therefore impossible.Mt is being overhauled by M somewhere on the way between P and B, sayin Q. Let this overhauling take place at a time t'. Q may even coincidewith B, so for t' the inequality t < t' tb holds.

From that moment on, the positions of M and Mt coincide permanently untilt.0, so they arrive in B at the same moment.Consider now the sets H(tt; X,0, t,0) and H(t; xb t,0) with S(tt) and S(t) astheir respective boundaries.As t* was assumed to be smaller than t, the boundary S(tt) must be whollyoutside S(t) and these two curves have no common points. Now, as M, startingin P at time t, follows an extrema! to B, this starting point P is situated onS(t), the boundary of H(t). This implies that P is an interior point of H(t*).Now the conclusion is justified, that PB is not an extremal for Mt to reach B.In other words, Mt could reach B at an instant t < t, if some other track wouldbe taken from P to B. This however contradicts the optimality assumption oft,0 once more.

So the supposition, that P could be reached at an earlier time than t, is erro-neous. This completes the proof of lemma 3.4.n.The main reason for giving this lemma some more attention is, that it indicatesa possibility of constructing the least time track from A to B by means of theS -timefronts.

Construction of these fronts for various values of t leads to the final timefront

27

S(tb), on which B is located. The other points x(t) of the extremal from

A to B are subsequently located on the S(t)-fronts. Working backwards fromB, all these points can be traced and thus the track can be determined.The way, this working backwards's as well as finding S(tb) takes place, willbe indicated after having treated some properties regarding the relation betweentimefronts and extremals. These properties will be discussed in the following

paragraphs.

3.5. Pontryagin's Maximum Principle.

Reconsidering the result of the preceding paragraph, it can be stated that a(total) extremal from A to B is an extremal all the way through. In other words:Every line element of an extremal between the times t and t + e S t - with e andSt positive - is optimal when it comes to crossing the lane between the twoconsecutive timefronts S(t) and S(t + St).

28

Figure 3.5.a.

This optimal crossing characterizes the extremal's direction.Let F be an extremal, intersecting the fronts S(t) and S(t + e St) in P and Qrespectively. The width PR of the lane between these two fronts, measured

+ o(e)perpendicular to S (t) in P, is proportional to e St + o(e), with hrn - O.

As the vector IQ is equal toe-0

t+ 5tx(t + e St) - x(t) = f(X(T),(r),T) dT =

f(x(t),cr(t),t)eSt + o(e)

we see that, with the extremal's velocity f(x(t),a (t),t) at P, this width iscovered faster than with any other velocity f(x(t) , a , t).

Neglecting terms of o(e), it turns out that the projection of this velocity vectoronto the vector drawn in P perpendicular to S+(t), has to be greater thanor at least equal to the projection of any other vector f(x(t), a t) ontoMaximum principle. Analytically, this means that in every point of an extrema!the control parameter a must be so chosen, that it maximizes the inner product

f(x,a, t)) = H(a;x,*,t)

for given values of the vectors x and and the parameter t.So for all t ta the control function ae(t) that belongs to an extrema!, mustsatisfy the requirement

( (t), f(x(t) , a e(t) , t)) ( (t), f(x(t) , a, t))

for all possible values of oThis requirement is the formulation of Pontryagin's maximum principle inthis case.Transversality. The special crossing of the time fronts by the extremals is called

"transversal intersection".In this case it may be useful to point out the possibility of a discontinuous jumpof the velocity x along an extrema! as a result of a continuous change of i).This can occur, whenever the maximum value of H(a, x, , t), denoted asM(i,x,t) is attained for two different values of a, saya1 and a2, with a1 <a2,while for all ' between these values, H is smaller. In such a case a jumpsinstantaneously from a1 to a2 or reversely. This means that the extremaltrajectory makes an abrupt bend.

Figure 3.5.b.

_...polar curve of f(x,a,t)

29

These bending points are allowed for in Pontryagin's theory, as (3. 3. 1) wasto hold almost everywhere on T, which practically means, that a finite numberof discontinuous jumps of a - and therewith of x - is permitted. Let us oncemore consider the original velocity indicatrix, as described in 3. 3.

30

Figure 3.5.c.

The true course a, for which the inner product2

(,f) = (,c) + . s sin(r 1/2 - a)11

is maximized, can be graphically determined with the aid of the original indi-

catrix:Draw a line perpendicular to the given -direction, denoted by cp, so that

at least one point of the indicatrix lies on this line, while all other pointsare located on the same side of it as the centre P.

Letting p vary from O to 2i-r, we find the optimal course as a function of cp.

This function a(ç) turns out to be piecewise continuously differentiable.Discontinuous jumps can be expected in two cases:

1. is perpendicular to a tangent of the indicatrix with two or more touching

points.

2. is perpendicular to a straight line, that covers a gap in the originalindicatrix.

As for the determination of e it is now clear that the original indicatrixcan be replaced by its convex closure, for which the name "effective indicatrix"seems suitable.Figure 3.5.d shows the graph of e() following from the indicatrix of figure3.5. c.

2-r

cp

Figure 3.5.d.

3.6. Behaviour of the tangent and the normal to a timefrontalong an extremal.

Let [FI be the collection of all extremals from A to a point of S(T), with

T > ta Stated more precisely, [FI is the collection of solutions of = f(x,a,t)with initial value x(t) Xa and x(-1-) on S(-T-).

3rr/2

31

3 i-r/2

1-T

ae

Figure 3.6.a.

A subcollection C of ÇF1 is formed by a field of extremals - i.e. a family of

extremals with no mutual intersection points but A - with the additional property

that the timefronts S+(t) are differentiable curves as long as they are drawn in

relation with elements of C.The extremals of C, covering a subregion G of can be characterized by

attaching a real number s to each specimen.Thus the coordinates of a point of G are functions of t and s:

X' = x'(t,$) with i = 1,2, or in vector notation x = x(t,$).In view of the assumptions regarding C, these functions are partially differen-tiable with respect to s everywhere in G, except of course in the edge points,where there is only left or right differentiability.Let Ç be an extremal of C with s = s, generated by the control function

Take t and t so that t < t <t T.1 2 a 12 +

F intersects the timefronts S (t1) and S (t2) in P1 and P2 respectively.

So x =x(t.,s ) with ,,j=l,2.Pi i o

Analogously Ql and Q2 are points of another extremal re of C, generated by

(t), so that x = x'(t., s +e). This situation is exposed in figure 3. 6.be J O +

The components of the tangent vector ôx(t), touching S (t) in a point of are

defined by:

(3.6.1)

so that

òx(t,s )ôx'(t)

= òs°

, (i = 1,2)

(3.6.2) x1(t,s + e) = x'(t,s ) + e x'(t) + o(e)o o

o(e)where o(e) has again the property: hrn = 0.e-.o

For what follows a lemma is needed regarding the optimal steering functions(t) and (t). I first want to prove:

Lemma 3. 6. a: If f(x) is a Lebesque-integrable function, defined on Ca, bi with

the property f(t)dt = O for x E Ea,b] then f(x) = O almost everhere on [a,bi.a

To prove this, I introduce the collections B, B1 and B2:

B = [xl f(x) O , XE Ca,bl j , B1 = [xj f(x) >0, x ECa,b]j

B2 = [xl f(x) < O , X E [a,b] j

Then B = B1 + B2, B1 and B2 are disjoint, so (B) = i(B1) + (B2). Now B1

and B2 can be sequentially covered by a finite or countably infinite number of

measurable sets:

B. C E B.. so (B.) = mf E (B..)i (j) 1J (i) '

The sets B.. are so defined, that, with a.. = um inf(x j x E B..) and1) 1) 1)

b.. = um sup (x j x .E B..), it follows from a.. x b.. that x E B.., in otheri) 'J 'J ii liwords: B.. is a closed interval.

ii

From the assumption that $ f(t)dt = O it follows that

b. b. a..

$1(t) dt

= $f(t) dt

i)1(t) dt = O

ajj a a

As f(t) > Olor i = 1 and 1(t) < O for i = 2, it is clear, that j.(B..) 0.

Now from the choice of B.. we see that B.. and B. are disjoint for j k, soij ij ikthat

.L(B.)=

i(B..) = O while p.(B) = i (B1) ±

so j.(B) = 0, which means that f(x) = O almost everywhere on Ca,bl.We are now in a position to state and prove:

33

Lemma 3. 6. : um f(x(t, s0 ± e), a (t), t) = f(x(t, s ) , a (t), t)O O

almost everywhere on [t,T.As we are dealing with a field of extremals, it is obvious, that for t E [t,Twe have:

(3. 6. 3) lim x(t, s + e) x(t, s0)

(If F is a boundary curve of G, then this 'e -. O" should be replaced by 'e i O"

or "e î O")

Comparing now:

tx(t,s + e) = x + $ {f(x(,s +

o ata

34

x(t,so

and

= X f(x(,$),a(),) d , it is now clear thatta

$ [f(x(,s+ e),a(),) -1(x(,$),a(), d = O

ta

for t E [ta1] , or written componentwise:

t

Ç um ?(x(,s +e),a (),) -f'(x(,s ),a (),)J d = O.o o o

ta

According to lemma 3. 6. a we may now conclude:

um f'(x(t, s ± e ),a (t), t) = f(x(t, s ),a (t), t)o e o o

almost everywhere on [t,T]

Corollary:In regard of the assumption (6) in paragraph (3. 3), that f1 and f2 were

continuously differentiable with respect to x1 and x2 and as a result of thislast lemma we may state, that

almost everywhere on t,T]

ume-.0

f(x(t,s + e),cy (t),t)o

xJ

Figure 3.6.b.

Seeking now a relation between 5x(t1) and ôx(t2), we subtract:

a -2(3. 6.5 ) x(t2, s + e) = x(t1, + ) + f(x(t, + e), a (t), t)dt and

ti

(3. 6.5b) x(t2, s) =x(t1,$) + f(x(t, s), cy(t), t) dt and find,

ti

in view of (3.6.2):t2

(3.6.6) e âx(t2) = e ôx(t1) + [f(x(t,s + e), (t), t) -f (x(t,s ), a (t), t)) dto o oti

where o2(e) denotes a vector in R2 with components of the type o(e)We can split up (3. 6. 6) in two ways:

(3. 6. 7) 1: e 5x(t2) = e ô x(t) + I + J + 02(e) with

t2

(3.6. 7a)J

=$ [f(x(t, s + e),a (t),t) - f(x(t, s + e),a (t), t)) dto e o otit2

(3. 6.7b) J

= Ç [f(x(t, s + e), a (t), t) - f(x(t, s ), a (t), t)) dtt:,

O O O O

f1(x(t, s),(t), t)

òx3(i,j = 1,2)

and

35

(3.6. 8) 2: âx(t2) = 6x(t1) + 1* + + 02(e) with

(3. 6. 8 1* = [f(x(t, s), (t), t) - f(x(t, s), (t), t)) dt

(3.6. 8b) J*= $ [f(x(t,s+ c), (t),t) - f(x(t,s ), (t),t)) dt.ti

The expressions for J and J* can be written respectively as:

2 f(x(t, s (t), t)(x(t, s+ e) - (t, s)) + 02(e)) dtJ =

ij=iti

r 2 àf(x(t, s e), (t), t)(x(t, s0+ c)-x(t,s0)) + o2()} dtJ*=

ij=iti

In view of the corollary to lemma (3. 6. ) we may conclude now, that

2 f(x(t,s ), (t),t)(3. 6. 9) hm J/e = hm j*/6

= $ E0 ô x3(t)dt

e_0 e-.0 t1=1 xJ

Concentrating on I and 1*, we can remark two things:

From (3. 6. 7) and (3. 6. 8) it follows that

(3.6. 7C) ôx(t ) = ôx(t ) + hm I/e + hm J/e and2 i e-0 e-0

(3. 6. 8c) ôx(t2) = ô x(t1) + lim 1*/e + hm j*/e-.0

Considering (3. 6.9) it can be asserted now that

um i/ = him i*/ = Le-0

f(x(t, s ), (t),t) - f(x(t,s ),a (t),t)C O OWriting L as him

° dte-0 C

ti

we may conclude that L is almost everywhere differentiable to its upper bound,

36

and

dLso that-- exists almost everywhere on it1,T1

From (3. 6. 7 ) we see that

dl(3.6.10) -a- = f(x(t,s 4- e),a (t),t) - f(x(t,s + ),a (t),t)t o e o o

Remembering the maximum principle (-. 3.5) and the assumption that a (t)

was an optimal control function - as is an extremal - it is clear that

(, ,f(x(t,s + e),o' (t),t)) (t f(x(t,s + e),a (t),t))o e e' o o

where is a normal vector to S4-(t) in the point x(t,s + e)

So according to (3. 6. 10) we have:

(3. 6. 11) dt - and obviously

(3.6.12 & 13) for e >0 : (dI/) O and for

Regarding 1*, we find from (3.6. 81):

dl*(3.6.14) -j- = f(x(t, s ),o? (t), t) - f(x(t, s ),a (t),t)

o e o o

As ci(t) is an optimal control function, generating the extremal F0, the maximumprinciple reveals:

(,f(x(t,$),(t),t))

so from (3. 6. 14) it follows that for e > O

(3. 6. 15 & 16)

analogous:

dL <(3.6.13) gives = O a.e. on [t ,T anda

Letting e descend to zero, tends toe

From (3. 6. 12) it follows now, that ($dL O

while (3.6.15) leads to ( ) O a.o' dtdL(,--) = O almost everywhere on [ta

dI/ee<0 : dt ) O

dJ*10 and for e < 0: dt O

on every timefront S4-(t), t E [t,rlO almost everywhere on [taT

e. on [taT] So for e O we see that

,T . For e i O the arguments are

37

(3.6.16) gives O a.e. on [t,T1

So the conclusion is that

(3.6.17) = O a.e. on

Differentiating (3. 6 7') to t2 and omitting the index (t2:=t) we find in view of

(3. 6.9):

d(ôx) 2 f(x(t,$),(t),t)ôx(t) +--- a.e. on [t ,T1(3.6.18) dt j=1 dt a

Investigating the behaviour of a vector (t) along an extremal, for which the

relation ( (t), x(t)) = O holds for all t E [tat t,j , we see that it has to satisfythe requirement

((t), ôx(t)) = O.

In view of (3. 6. 18) we see that

2 i 2 f1(x(t),a(t),t) dL1E +. E

1=1 j=1âx3 + = O

Because of (3. 6. 17) and arranging this relation somewhat differently, thecomponents and

2of the covariant vector are found to vary with t

according to:

38

(3.6.19) = - òf3(x(t),a(t),t)j=1 J

A vector with this property and the condition, that

(i (t ), 5x(t )) = Oo o

for some to E [tatb] is called an adjoint vector to the trajectory x(t;(t))

(i = 1,2) a.e. on [t,t)

3.7. Construction of the solution. Examples.

Summarizing the foregoing paragraphs, a solution x = x(t) was sought of thecontrolled dynamical system * = f(x, a, t) with the initial condition X(ta) = X and

for some - yet undetermined - value tb the requirement x(tb) = xb. Among allthe real valued functions (t), almost everywhere continuous, that generate asolution of this problem, an optimal control function a (t) was sought, generatinga minimal arrival time tbThe trajectory that satisfies all these requirements, could be traced by carry-ing out the following procedure:Select an arbitrary vector = Select a(ta)sothat it maximizes theinner product H(0) = (°f(xa,(ta),ta))With t (t) O let the components of ) (t) vary according to (3.6.18), whilex(t) changes as indicated by (3. 3. 1) and for all t ta let a (t) maximize theinner product

H(t) = ( (t),f(x(t),'(t),t)

Considering the maximum principle and the fact, that (3. 6. 18) is homogeneousin 2' it can be seen, that the initial choices ° and p° - for somereal p > O - generate the same trajectory, so can be taken with unit length,say

o = (coscp sincp )o' o

The coordinates of points of the timefront S(t) are consequently functions oftandç, sox1x'(t,cp)If analytical expressions for these functions can be deduced, tb and can be

solved from the equations

(3. 7. 1)

1 1x (t,cp) = Xb12 2

X (t,p)=x

lt should be remarked here, that these equations may very well have morethan one set of solutions (t,cp*). With the object of finding the least time trackit is obvious, that of all solutions for t,0, we are only interested in the smallestvalue of it, providing it is greater than taIf q is the solution of cp, corresponding to this minimal arrival time t, then

39

the extremal trajectory from A to B can be constructed by solving (3. 3. 1) and(3.6.18), using the maximum principle for the determination of and takingx'(t) = X' and (t) = (cos cp*,sinp*) as initial conditions.

Remark 1:Considering the effective indicatrix of fig. 3. 7. a, drawn with respect to thestarting point A, it appears that all initial values with

m < rn"o - T0 - o

generate the same initial courseo

40

Figure 3.7.a.

From this We can conclude, that the initial arcs of extremals, issuing from A,may partly coincide. This happens when C(Xata) the collection of possiblecourses in A, does not consist of the entire interval [0, 2,-r] , so that theeffective indicatrix has points where the tangent is not continuous.Let, in figure 3. 7. b, F be such an arc of partly coinciding extremals.Iii a point K of that curve, the value of is not quite determined, because itdepends on which neighbouring extrema! F is compared with.Let correspond with a bending point like Q in figure 3. 7.a and let ' and "

be the vectors perpendicular to the left and right tangents to the indicatrix ofK in such a bending point.

If w' and cp" are the directions of ' arid iii" respectively, it is clear that thetangent to the timefront S(t) in K bends abruptly from p' + i-t/2 to p" + rr/2

Remark 2:Along every extremal the maximum value of H(a,x,J,t,t), equal to M(,x,t)satisfies the relation

(3.7.2)

To prove this, consider the three possibilities for a in a point of an extremaltrajectory:a. a is continuously variable and fis partially differentiable with respect to

a. We then know, that for H = M:

fi=a i=1 i a

dM 2 d. f dx +dt i=1 dt j,j=ii dt t

With (3. 3. 1) and (3. 6. 17) we find that

Figure 3.7.b.

(_L;. f1 + f3) +dt i=1 'j i j

òxso

dM M-j =--- a.e. on [ta=)

and so:

dMdt òt

41

If a is constant - compare remark i - then there simply is no change in aas a result of a change oft, so the same relation holds.If a jumps discontinuously from a1 to a2, we have the same maximum of Hfor a1 and a2, so in this case there is no change in H as a result of acontinuous change in and x.

For an autonomous system: k = f(x,a) it now follows that the maximum value

of H is constant along an extremal.

Remark 3:The Hamilton function H enables us to write (3. 3. 1) and (3. 6. 2) in a canonical

form:

(3. 7.3)

(3. 7. 1. 1)

42

f

.i Hx=(i = 1,2)

I now wish to enlighten this procedure by means of some elementary examples:

Example 1:On the X-axis a moving point has at a time t = to a position x0 with respect tothe origin and a velocity y along the X-axis. The movement of this point along

othe X-axis can be controlled by:

ic=cy with the limitations -1 a 1.

The assignment is to choose a as a function of t, so that the moving point arrivesat the origin with velocity zero in the least possible time.

Introducing the velocity as a new variable y = k, the equations of motion are:

Application of (3. 3. 1) and (3. 6. 2) gives

H is maximized by taking o = signTaking the arrival time t1 = O and putting (0) = (cos , sin (p), the solutionof (3. 7. 1.2) becomes

(3. 7. 1. 3)= cos (

{o

tCOS(Do

(tO)

Concentrating on the S-timefronts, various possibilities are considered:O 1V2

Fort<Oweseethat2=sifl -tcos(p> O, soo(t)=+1. Forthis(p-sector, the S-timefront consists of merely one point:(t2/2,t), solution of (3.7.1.1) for o i-r/2 < (p < i-r.

Now2

= sin - t cos is positive for tg < t O and negative fort < tg p - So, proceeding in time, we have:o(t) = -1 for t < tg (p and (t) = +1 for tg (p0 < t O

Solving (3. 7.1.1), we find:x = t2/2, y = t for t E [tg (p , O],

x = -tg2(p0+2ttg(p0 - t2/2, = 2 tg - t for t E [t0,tgp0]2Eliminating (p, we find the equation for S(t) : 4 x (y - t) = 2 t

At the time t = tg (p, the switching from o -i to o = + i takes place.

= sin cp - t cos (Ç is negative for t <O. So o(t) = -1 maximizes H(cy,x,).The trajectory is (-t2/2, -t) and the S-timefront is just this one point.

d! 3rr/2< (p0<211.As 2(t) = sin - t cos (p is negative for tg < t O and 2(t) > O fort < tg (ç0, we find:

x(t) = tg2(ç0 - 2 t tg (p + t2/2 , y(t) = - 2 tg (p + t for to t < tg

The switching point is (- tg2(p , - tg (p) , the switching time is t = tg (p

and a jumps from +1 to -1.For t E [tg0,O] we have x = - , y = - t.The S-timefront arc has the equation 4 x - + t)2 = 2 t2

Figure 3.7.c shows the various possibilities. Taking (x,y0) = (2,2) , we

43

= O

(3.7.1.2)2-1,whjle H=1y+2o

44

Figure 3.7.c.

Example 2:In a 2-dimensional plane with an orthonormal coordinate system, the velocity

field is defined as follows:In a direction with respect to the X-direction the velocity is vj sin(y> O). The problem is to find the least time track from a given point A to

another given point B.

The velocity components in the and Y+_directions are:

5c=v sinjcossino

The adjoint system is awfully simple:=

= O with solutions = cos p

= sin , where is a constant angle.In this case we have to maximize

H = y sin a cos a cos cp + V Sifl a J sin a Sin =

=v0Isinacos0).For various values of we find:

find to = - 6 , tg p0 = - 2 , and the optimal track is:

= - - 4 t - 4

y=-t-4,

- 2 t O:y=t

w O : li = y for a = or 7,-r/4o max o

O<p <r:H v(1+sinp)for a=.îr/4+p.c.cp rr:H =v for a=3rr/4 or5rr/4.

o max od. ri<cp <2i-r :Hmax=v(1 -sincp0)for a = 3i-/4+ cp

Taking the origin in A, the S (t)-timefronts can be described by:cp0 O: x= v0t , - vtyvtO <ir : X = - Vt COS , y vt(1 + sin cp)

C. w :x_vt, -v0t yvt.d.i-r<<2ir:x=vtcoscp,y=vt(sincp-1).

Figure 3.7.d.

Figure 3. 7. d shows the extremal tracks for various cases. Denoting thedirection of AB as , we find:1. If 7ir/4 <2ir or O r-r/4 , then the extremal exists of a number of

line segments with direction r-r/4 and 7-V4. The tracks AP1B1 and AQ1 B1

45

give the same arrival time at B1 and so does any track along the peripheryor inside the rectangle AP1B1Q1, as long as it leads from A to B1 in courses

of 45 and 315 alternatively.If rr/4 < < 3rr/4 , a = and the track is one straight line.

For 3î/4 5/4, the extremal track consists of line segments withdirections 135° and 225°. The number of 90°-left or -right turns can befinite (one at least) or even countably infinite. This has the practical sensethat this number of turns can be as large as one could possibly like.

If 5îr/4 < < 7î/4 then a = . The extremal track is again a straight line.

This example can be applied with slight modifications to a head - to - windsailing ship. It shows that a windward position can sometimes better be reachedby tacking up against the wind than by steering a constant course.

3.8. Remarks on the occurrence of more than one solution.

In regions of small values of l the speed decrease causes a decrease of the

distance between two consecutive timefronts S+(t) and S(t+ôt). As these regions,in the case of ship routeing, very frequently have a more or less elliptic struc-ture, they also cause a convergence of the extremals. The result of these twophenomena often consists of a sector of ambiguity, i. e. more than one solutionof the system of equations (3. 3.1) and (3. 6. 17) combined with the maximumprinciple, with A as starting point and a point in this sector as point of arrival.This phenomenon will now be explained at the hand of an elementary example.

Let the starting point A have coordinates (- 3 , 0) and let the velocity compo-

nents be given by:2 2

(3.8.1) x(2-exp(-x -y ))cosa=(2-exp(-x2-y2))sina , a E[O,2rí.

This implies that in every point of the X-Y-plane the velocity indicatrix is acircle with radius r(x,y) = 2 - exp( - x2 - -y2). This radius has a minimal

length in the origin,Min r(x,y) = r(0,0) = 1

(x, y)

and r is obviously constant on a circle with (0,0) as centre.Applying the maximum principle, the form

46

= ( cos a+2

sin a) r(x,y) (3.8.2)

has to be minimized for given values of, 2' x and y.

The adjoint variables and2

change with time according to:

(3.8.3){l=-x(2-r)(lcos+2sina)

Writing = m sin(îri/2 - p) with i = 1,2, it follows that H is maximal for a = p.Thus the optimal dynamical system is described by:

k = (2 - exp(- - y2»cos p

= (2 - exp(- - y2))sin(3.8.5) 2 2= exp(- x - y )(x sin - y cos p)

th= -mexp(-x2 -y2)(xcoscp +ysincp)

with starting conditions x(0) = - 3, y(0) = 0, cp(0) = p and m(0) = 1/(2 - exp(- 4f))The fact that H = (2 - exp(- - y2))m is constant in this stationary velocityfield, implies that

H(0)(3.8.6) m=2 2 2 22 - exp(- x - y ) 2 - exp(- x -

By taking various initial values, like p: = 0(0. 01)1.20 the equations (3. 8.4)can be integrated numerically by means of a Runge -Kutta method with it = 0. 1as a sufficiently small step width. This was carried out with the aid of a TR-4computer and for t = 1(1)12 the values of x, y, p and m were printed out. Theresult is exposed in figure 3. 8. a.As for a qualitative explanation, it can be remarked that m = is constantlyproportional to grad S, where S(x, y, t) = O is the equation of a timefront S(t).According to (3.8.6), m has a maximum value i for (x,y) = (0,0). This meansthat the distance between two subsequent timefronts S(t) and S(t + it) isminimal in the origin. This explains that the timefronts become less convexnear the X-axis as t increases.The overlapping of the timefronts can be explained by thinking of a timefrontS(t + 5t) to originate from S(t) as the envelope of the It-position-indicatrices,drawn for all points of S(t).Building up the subsequent timefronts this way and reminding that the radius of

47

48

Figure 3.8.b.

Figure

the indicatrix, r(x,y)ot for a fixed value of x is minimal for y = O, there is afair possibility that the timefront through O has a zero curvature in O or evena concave part for - y0 y y. In that case the next timefront has a deeperdent, which is symmetric with respect to the X-axis.Let us consider a small line element P*** of S+(t) near the X-axis, with Pon the X-axis and the distance = P** small enough to be able to assumer(x,y) equal in P, P* and **

Now if the curvature radius of S(t) in P is equal to r(x,y)5t, the trajectoriesthrough P, P* and ** coincide in Q, the curvature centre of S+(t) for P.

Q is known to be called the conjugate point to A for the central extremal, with= O

Figure 3. 8. c.

The solutions of (3. 8. 4) with fixed x, y and variable as starting values forx(0), Y(0) and cp(0) are functions of t and In Q the partial derivatives of x andy with respect to are zero, while for all points on the line through Q parallelto the Y-axis the derivative -- is positive.

poTurning back to the approximative construction, it is now obvious that the twoparts of S+(t + it) on each side of the X-axis are transformed, by a next timeincrease it, to the intersecting fronts R*S* and R**S**. The frontal arc be-tween R* and R** is formed by the "t + 2 It-points' of extremals that inter-sected the timefront through P between P* and P'.If the curvature radius of S(t + it) in Q is equal to r(xQ,yQ)ôt, then R* andR** are the conjugate points to A for the extremals through * and ** res-

pectively.

'I'he further propagation of the timefronts is analogous. For t"> t + it theyapp. reni ly have a double point on the central "extremal" through Q and turning

49

points on the two dashed curves through Q, which form the locus of conjugatepoints to A for extremals with various starting valuesThe practical importance of this phenomenon is, that for every point in thesector T**QT* we can find three solutions of (3. 8. 4) that lead to that point.

Of these solutions only one is the absolute extrema!, except when the pointis located on the locus of double points of the timefronts, which is the continua-tion of the (3. 8.4)-solution through Q. As these circular fields of low velocityfrequently occur in ocean navigation, we can conclude from this example, thatsolving the problem of finding the least time track by means of some sort of atrial-and-error method has to be rejected, because such a procedure onlyleads to a trajectory, that is locally optimal. In order to find the real extrernalfrom A to B, it is necessary to construct the timefronts as far as possible,i.e. between the boundaries of the region G, mentioned in (3. 2).

3.9. Modifications in case of coordinate restrictions.

The region G, within which the extrema! from A to B has to be located, can inthe case of a North Atlantic crossing between the English Channel and the U.S.East Coast be described by the requirements:

Assume the data, necessary to construct the complete figure of extremals andtimefronts, to be known in a somewhat larger region Ô, so that it is alwayspossible to extend an extremal arc for a short stretch outside of G.1f extremals, starting from A on the basis of an initially chosen value ofintersect the boundary of G at a too early time t - meaning that the correspondingregion H+(t) does not contain the destination B - then these trajectories areimmaterial and have to be cancelled. If such an extrema! F touches the

gboundary - meaning that one point R of Fg lies on the boundary, while all otherpoints of tg in a not too large neighbourhood of R lie inside G, the arc of Fgfor t> tR is an unwanted border of the field of extremals, with which the regionG is being scanned until B is hit by H(tb) for some t, > t. For example, infigure 3. 9a the part of G between the arc RP and the boundary part PQSTR

50

0x17-2y 3 for 0x< 9

-1y 3 for 9x<1O-1.5y 3 for 10x< 15

-1y 3 for 15x17

K

Figure 3.9. a.

would remain uncovered by extremals. In such a case, it is necessary to coverthis shadow region with a collection of supplementary extremals, all startingfrom R.Let the adjoint vector with respect to Fg have the value for t = tRIf is left unchanged, then the procedure, described in (3. 7) generates thetrajectory FgThe solution seems to be to turn over some angle, varying for example

from O to -'n/2, before the integrating procedure is continued. However, thisnormal procedure would generate extremals, that lead to points outside of G.The construction of extremal arcs from R, meant to cover the region "below"the arc RI' clearly has to deviate from the "free" procedure, described in(3.7).This deviation practically consists of covering an arbitrary part RK of theboundary RT with a velocity, given by x = f(x,n,t), where the ships's courseng is determined by the direction of RT, the local value of the stream vectorand the indicatrix at the point x (shown in figure 3.9. b).

Having reached K, the direction of , belonging to the boundary course g atthat point, can be deduced from the maximum principle. Continuation of thetrajectory as a free extremal then generates a curve within the yet unscannedpart of G.When the boundary is covered as far as the corner point T and the trajectoryis continued in the way, just mentioned, the remaining part of G can be coveredwith free extremals, starting at T and constructed on the basis of properly

Q

51

chosen initial values for the adjoint vector.

r

Figure 3.9.b.

The coincidence of an extremal arc with a part of the boundary will now be

briefly investigated.With x'(t ) = X'(3.9.1)

-[

0

i,a,t) and p(x) O

suppose there is a class A of functions nit) so that a solution x'(t) of (3.9. 1)attains the value x for some time t1.Among these control functions there is at least one, say (t), that generates aminimal arrival time Ç.Suppose (t) determines the solutions 1(t) of (3.9.1), where cp ('(t)) = O for

to t t1 meaning that the extremal coincides with the boundary. If (t) is

given a small variation ôa(t), while (t) + la(t) is still a function of class A,this varied control will generate the solutions x' = '(t) + ôx1(t) of (3.9. 1) and

an arrival time Ç + ot1 with ôt1 O

Substituting: x' = X , X1 = X0 , =

(3. 9.2)X = y , X0 = y0 , X1 = y1

2 2 2t =y , t=y0 , t1y11/fl = G2 ,f2/f1 = G1

52

dxdt

(i = 1,2)

we have:

(3.9.3)

(3.9.4)

- =G',a,x)

y'(x)= y' , y'(x1) = y , (y and y are fixed, y is variable)

p(x,y ) = O, p(x,y) ='p(x,y1) = O

Denoting the minimal solution by y = '(x) and introducing the abbreviative

notations

=G'k andkòG'(,,x) =G'a

the variations ly' and ôa have to meet the requirement:xl

(3.9.5) j (G2. ôy'+G25)dx O.y'

X0

The x-derivatives of ôy' are found to be

(3.9.6) d(ôy') G1 ôy +G'5

Introducing a slack variable z, the inequality (3.9.4) can be replaced by:

(3.9.7) (x,y1) + z = O, where z O and z = O for y1 ='(x)

The variations in '(x) imply slack variations ôz(x), so

(3.9.8) = O

Differentiating this with respect to x and using (3. 9. 6) gives:

(3.9.9) ( 18y' +p 1G'6y1 +p 1G1.5y' +p1G' ôa + 5(-) = O.xy yy

1Putting p = p 1G and o = l/pya=

1 + i 1G' + G'1 , x2 =J- xy yy y y

(Using the Einstein summation

convention)

53

equation (3.9.9) becomes:

(3. 9.10) xôy

+ p Sa 5(dz) = 0

Considering the minimal time condition (3. 9.5), we find:xl

(3.9.11) $ ((G2.y'X0

I now introduce the adjoint vector p = (p1,p2) by defining:

(3.9.12) - a G2xdp

+ p(G3 - G OEXai dx j i a

Before further development, consider

d dp.d(Sy')-(P Sy) 5y1

+ i dx

dp.=

(- + p G) 5y1 + G Sa

dp.dz

= ( + p Ga.) 51 +p G' (-a ôy -adx

With N = p. G' a, this becomes:i a

dp.dz(3. 9. 13) ( 8y1) = ( + p G - X x) Sy' - X s ()dx

Now (3.9. 11) can be changed to:

xl(3.9.14) $ ((pây') + (N -a G2) ô()) dx O or

X0

Xl Xl

ôy]+

S() dx O with L = X - a G2o

Remembering that ôy'(x) = O and 5y1(x1) = O , this leads to:

54

- a G2X) Sy' -a G25(4) dx O

dzor Sa = - a ây - a 5 ()

xl- r

d 5zdxO.(3.9.15) p2 8y2(x1) + .L ÔZ

X0

The requirements 5y2(x1) = 6t1 O and Sz O now mean that

p2O, and

The last conclusion means that the 'boundary extremal", for which S.L is positive,tends to a free extremal.Summarizing the result, the variables y1, a, and p. have to be solved fromthe following equations:

= G'(y3,a,x)

dp.p G + + G2.dx y y'

1p(x,y ) = O

G2+p G' - p.G' = O.a y1 i a

Transforming this set into a system of equations with the time as independentvariable and putting p1

= 2and = - ± p2 - 1)/f1 , I find:

dx=

fJ.+ fJ +

dt - J x1 xi x1 xxJ(3. 9. 16)

c(x) = O

- P) = O

The last expression determines .L . This can be geometrically explained bymeans of figure 3.9.c.In a point P of the boundary cp (x') = O the stream vector c is drawn. The timeand the coordinates of P determine the dimensions of the ship's velocity

55

indicatrix. The gradient vector vcp is pointed to the outside of G.The adjoint vector has a direction, so that bare application of the maximumprinciple would produce a velocity vector f, that has a positive inner product

with vcp. This would mean a trespassing into the prohibited region Gc. Sub-

traction of vp from - with > O - has to deliver a velocity f perpendicular

to vp . This number can be found as follows:Determine the ship's velocity s so that e + s touches the boundary.In the endpoint of s draw the tangent to the indicatrix.

The vectors and vcp now must have equal projections on this tangent.

This determines .. , as long as it is positive. As soon as becomes zero,the extremal leaves the boundary and leads to points with cp(x') < O

56

Figure 3.10. C.

4. PRACTICAL DATA

This chapter contains a summary of a few methods of wave prediction frommeteorological data, followed by a curse treatment of the possibilities of aship's response to a given wave pattern.

4.1. The wave prediction problem

Let us assume the weather and sea conditions in a region G to be known exactlyat the present time to, while a weather forecast for a time t1 can be consideredsufficiently reliable to serve as a basis for a wave forecast, to be made up forthis future time.The first serious approach to solve this problem was performed by H. U. Sverdrupand W.H. Munk. As a result of their investigations, a pack of practical direc-tives was compiled by Bretschneider, thus enabling sea navigators to make afair estimation of the sea conditions to be expected.Although these forecasts were averagely in fair accordance with practice, thebasic mathematical model was a bit incomplete. An important improvement ofthis model was introduced by W. J. Pierson and a practical method of deter-mination - or rather estimation - of sea conditions was issued as a result, i)

The following three paragraphs contain a brief summary of these theories.

4.2. Some basic elements of ocean wave theory.

Consider a wave train with very long crests in water of constant depth h. Selectthe X, Y-plane perpendicular to the crest lines, the X+_direction horizontalto,the right, the Y-direction upward.lithe fluid is irrotational and incompressible, the velocity components u and yof a fluid particle satisfy the relations:

(4.2.1) Uy - y = O (irrotationality),

(4.2.2) + Vy = O (incompressibility).

This justifies the introduction of a velocity potential p(x,y,t) with the properties:

1) 1955: Practical methods for observing and forecasting ocean waves. U.S. Navy H. 0. 603.

57

(4.2.3) , p=v , w = 0

Let the wave profile be represented by y = T(x,t). The free surface boundarycondition, with the assumption of a small steepness to permit neglection of

higher order terms, means that for y = 1 we must have

(4.2.4) = = lit.

The Bernoulli equation, with constant pressure and neglection of 2nd order

terms, leads to the relation between w and li:

(4. 2. 5) + gfl = 0 for y = li

At the sea bottom the velocity is purely horizontal so we have the relation:

(4.2.6) V=CPy=0 for y=-h.

Solving (4.2. 3) and taking account of (4.2.4, 5, 6), we find:

cosh(k(y + h)) sin(k x - a t) and(4.2.7) =ca sinh(kh)

caa cosh(kçfl ± h))(4. 2. 8) li = - cos(k x - a t) sinh(kh)g

where c = phase velocity,a = a constant with the dimension of length,k = wave number, (wave length L = 2i-r/k)

a = circular frequency, with c = a/k.As li 0. 02*h we replace li + h by h, finding:

(4.2.9) e2 = (g/k)tanh(kh)

For deep water - h> L - this leads to the approximations

2e =g/k and

li = a cos(kx - at)

Introducing the period T = 2rr/c, we find:

(4.2.12) e = gT/2rr

Expressing c in knots and T in seconds, it appears that

(4.2.12a) c=3T

The energy of an elementary sinewave per unit surface area can be found to be:

(4.2.13) E=p ga2

in which p is the density of sea water.The deep water assumption leads to the approximations for the group wave speedVg and the speed of energy propagation Ve:

(4. 2. 14) V =V =cg e

4.3. The Sverdrup - Munk wave forecasting method.

In 1943 Sverdrup and Munk introduced a wave prediction method, composed

after an assignment by the U.S. Navy Hydrographie Office. The practical resultsof this method can still fairly well match those of later investigations, especiallywhere the growth of waves is concerned.

With the information, cursely mentioned in the preceding paragraph, one cancompute the average rates, at which energy is transmitted from wind to waves bynormal pressure and by tangential stress. These rates are denoted by R and

R. Their values were adopted as:

R1 222

= sp (U - e) k a e sign(U - e)

2212 2R y u p leU

with s = sheltering coefficient, p1 = air density

= resistance coefficient, 5 = wave steepness = 2a/L.

59

As for the growing of waves, the 'significant wave" was introduced. Consideringa sufficiently large number (n) of wave heights, the mean height of the n/3 heighestwaves is called the "i-significant wave height", denoted by 111/3. The entirecomposite sea is now (supposed to be) represented by just one classical wave

with height 1/3 This idea meets some objections:For an elementary wave in deep water the quantities c, a and L can all bewritten as functions of k. From c2 = g/k and a2 = gk

total rate of increase of L with time is:

L+ C-dt - òt

Neglecting higher order terms, the wave crest velocity increase over oneòc ÒL ÒL -L=0.wave length is L , so + ct òx

Putting =d(a/k) ÒL - J da L c L

x dL òx - d(2/k) x 2i-î(c -) s- = -

(4. 3. 1) we get: òt 2a =0.

This relation implies, that a steady state, meaning = 0, cannot occur ifthere is a change of L with distance. Yet it is well known, that a sea in steadystate, generated by an offshore wind, has waves with lengths, that increasewith the wind fetch.On the other hand, experience has shown, that ocean waves, being generatedin a sufficiently large area, grow independant of their position, while (4. 3. 1)

ÒL ÒLwould suggest, that = O if = O

These difficulties, arising from the assumption to have the sea pattern repre-sented by one "significant wave", were attempted to be overcome by stating,that significant waves are not conservative in a storm area.

60

dawe see that = e. The

Making up the account of energy and neglectingd(EL)stated that the entire rate of energy gain dt

waveblock equals the total input by wind, (RT +

of wave power over one wave length. This

(4 3 2) d(EL) = (R + R +-(cE/2))Ldt T 'J ô

higher order terms, it can beof a crest-to-crest movingR)L plus the first order gaingives the relation

dL c c ldLAs = L, so that while the deep water phase speed answers

c2 gL/ 2dc ldL= 21r, so - - = L

the expression (4. 3. 2) can be converted into:c dt

(4. 3. 3)

(4. 3. 5)

In a free ocean the energy of the significant wave is taken independent of x.

Thus follows the duration equation":

(4. 3.4) dt cdt T

This equation is used to compute H113, if the wind fetch is unlimited. Todeduce an analogous equation for a steady wave with limited fetch, consider

a water sheet of height h + 1 , unit width and length Ox, fixed in space. In thissheet, potential energy is lost at the amount of ò(cE/2)0 while the gain is

ÒEagain (R + R)ôx . As the local change of energy is - ox, we have

(cE/2)Ox ôx+(R +R)Ox.T V

For a wave in steady state = O, so the fetch equation" becomes:

dE Edc cE =R +RT V

+ dc2dx 2=R ±R

T V

Before solving these equations, the dimensionless parameters a H/L (wave

steepness) and a c/U (wave age) are brought up.Practical experience has affirmed a univalent functional relation between Oand .

As a result of this, the equations for duration and fetch can subsequently bewritten in the form

dt ' and = g(dO

The difficulty of determining O as a function of was overcome in both cases

by assuming a constant partition of energy, R + R in total, over the energiesTdE ' cdEto contribute to the increase of wave height (-s-- and -,---) and to the increase

Edc Edcof phase speed (- - and - -)As there is little sense in reconstructing all the details, I shall confine myself

61

with the exhibition of the result, given by figure 4. 3. a. This figure shows graphs

of the dimensionless quantities gH/U2 and c/U for a given value of gt/U

(t = duration), in the case of an unlimited fetch and a limited duration, and for2a given value of gF/U (F = fetch length) for the case of a limited wind fetch.

Obviously, for a given set of values for U, t and F, one should take thesmallest of the two values, both for gH/U2 and for c/U.

62

5

2

'2.

10 - 2 5 iO2 2 5 10 2 5 1O 2 5 10 2

- gt/U and gF/U2

Figure 4.5. a.

4.4. The Pierson - Neumann theory.

It was W.J. Pierson's idea to approach the problem of ocean wind waves bymeans of stochastics. Therefore this summary of Pierson's theory is precededby a curse treatment of a few statistical theorems.

A stochastic process xt(w) can be defined as an indexed set of stochastic varia-bles. The set of values of the process is (part of) the real line or the complexplane in most cases. The index set T = [t can be the real line, the set of allintegers or any other point collection. The points w are elements of a space .

A probability measure is defined on a a-algebra of subsets in Û. For a fixedpoint w, xt(w) is a function of t, called a sample function or realization of theprocess.Consider now a complex valued stochastic process. The mean value function

gH/U25

C

c/IJ'3

and

(4. 4. 1) mt = E(xt) = Sxt(w) d(w) is assumed zero.

The covariance function, created to give quantitative information on the mutualdependence of the process values at different points of T, is:

(4.4.2) r = E(x ) =$x(w) xt(w) d(w) st tsso r =rst st1h

If T = [t.) with i = 1(1)n, then the matrix [rt } is Hermitian and non negativeiidefinite.

A process, for which EI x(t) - x(t)l 2_O if t-. to, is called "continuous in

the mean". This property is equivalent to the continuity of r5t for s = t.Let T be a cx -field BA on a finite dimensional space A.

Suppose a cx-finite measure is defined on this field. A stochastic set functionz(s) with s E BA is now defined by:

(4.4.3) z(s1 U = z(s1) + z(s2 n s)

E(z(s)) O

E(z(s)z(s')) .(s n s')

This definition implies, that the process has orthogonal increments. If f(X) isdefined on A so that 5J1(X)j 2d (X) < a stochastic process can be defined in

terms of integrals with respect to this orthogonal set function z(s). This canbe done by selecting a sequence of step functions f(), so that

rlf(X) - f (X)nA

2d = O

Defining J =$f(X) dz(X), it can be proved, that J converges in the mean to a

stochastic variable J, where

E 1j12 = f(X)12 d

so that J c in b" represented by the expression f(X) dz(X) . These variables havethe property A

63

E(JT)=

$f 5) d(X)A

(4. 4.4) The stochastic variable= $

f(t, ) dz(X) now has the covariance

function A

(4.4.5) r = E(xi) = $f(sX) TU7) d(Ñst stA

Inversely, it is possible to deduce the representation (4. 4. 4) from the covariancerepresentation (4. 4. 5).

The stochastic process x - where t is an rn-vector - is called strictly stationary

if the variables have the same joint probability distribution asi n

x+h'

,x+h'

for any rn-vector h.i n

For a strictly stationary process the covariance function rt has the property:

r5h,t+h=rS,t=rSt,O=r(s_t).

Taking the real line for T, the covariance function r(t) of a stationary processenables one to construct:

Xi r _-it1_,

- $n' dT(

From this it follows that hm F) = O, whileX-,

r(t) = $ e1tf(1l)d1, so that r(0) = 5f(11)d = lirn F(X).

Furthermore F(X) is a bounded, non decreasing function. It is called the"spectral distribution function" of the process.The covariance function can be found by

r(t)= $

e1tXdF(X)

64

But this means that the covariance representation

r = r(s-t) = I e1_t»dF(X)st

can be seen as (4.4. 5) with f(t, X) = e1tX and (X) = F(X) . So now there is anorthogonal set function z(s) defined on the Borel sets of the real line, so that

x(t) = $eltXdz(. With F(X) as the probability measure, we have

E(z(s1)T) = $ dF(X) or E(dz(X)J2) dF(X).S 1fl

The stochastic variable x(t), just mentioned, is therefore denoted abbreviouslyby

x(t) = $e1tJ(dF().

To describe the wave profile of a wind generated sea, Pierson adopted thestochastic model:

2(4.4.6) (x1,x2,t)

= $cos( (x1 cos8 + x2 sine) -at +

(a, e)=(O, -rr)

In this expression, e(a,e) represents a random phase shift, uniformly dis-tributed over (-n,i-r) for each element thde.z(a, e) is an orthogonal set function, with

(4.4.7) E(dz2(c, )) = A2(a, )thd

Figure 4.4. a shows a graph of the spectral function A2(cy, e) on the (a, e) -plane.

As dz(a, 8) has the dimension of length, the integral

-'- eo+e$ da { $ A2(a,B)dB } has the dimension i2 and

a0-5a e-e

65

66

Figure 4.4.a.

Is thus proportional to the energy of waves within the frequency band of width

Aa around « coming in from directions between 8 - .A8 and 8 +

Apart from the question if this model is a good aid to determine as many aspossible relations between winds and sea waves - practical experience has sofar shown little need for refinement - the kernel of Pierson's problem is tofind a proper energy spectrum.The Neumann spectrum, designed in 1952, has to be considered as a first guess

of A2(a,8).

Taking the mean wind direction ed O, the dependence of A2 on e was assumed

to be proportional to cos28 for -i/2 < e <ir /2, and A2 was taken zero else-

where, i. e. in the leeward region. This assumption was based on observationsby Arthur (1949), who concluded that the wave height varied approximately

proportional to the cosine of the bearing relative toThe adopted expression for the energy spectrum became:

2 -6 2-2-2 2A (a,e) =Ca exp(-2g a U )cos e for -i-r/2< 8<rr/2 and

A2(a,e) = O for <e -ir/2 or ir/2 O

The frequency interval may be limited by fetch (F) or duration (t). At a given

wind speed U, the lower bound of the frequency intrvaI, over which thespectral function A2 has to be integrated to get the total energy Qf the wavepattern, is the maximum of two quantities:

af a function of U and F,

a function of U and t.The total energy now follows from

rî/2E $ $ A2(a,e)dade, where a =Max(af,at)

The -significant wave height corresponds with this total energy amountaccording toH1/3 = 2.83JE, workingwith c.g.s. units. Figure 4.4.b exhibitsthe results of these formulas.

.03 .04 .05 .06 .080.1 0.150 2frequency in sec1

Figure 4.4.b.

67

As an example, take a 28 knots Northerly wind, generating a sea pattern at200 miles south of Nova Scotia. The diagram shows, that this wind has to beblowing for about 19 hours to generate a fully developed sea. Before thattime, the lower frequency bound is determined by the duration, while the 200

miles fetch is a barrier for further development of the waves for a longerduration than 19 hours.Various attempts were made to improve the Neumann representation of a

wind generated sea. Kitaigorodskii paid an important tribute in this respect by

introducing his similarity theory.Writing A2(a,8) = S(a)cos28, he assumed that S(o) was a function of a, g, U and

F in case of a limited fetch.This function can be transformed to a dimensionless spectrum by putting:

S S(a,g,U,F)g3U5

= l.Jag

F = gFU2.

According to Kitaigorodskii, this function S only depends on the dimensionless

quantities and . In addition, the spectral function for a fully developed sea

has to have a couple of properties, like

H1/3 u2

a Ug' = constantmax

S(a,g,U2) S(a,g,U1) for U2> U1 and allG

An attempt to construct a spectral function S with these properties was made

by L. Moskowitz in 19631).

His proposal not only met the above mentioned requirements, but it was also

found to be in satisfactory accordance with data, collected from practical

measurements.Denoting w = 21-Ta and w = gU , the proposed spectral function was:

S(w)2 -5 4 -4

=csg w exp(-w0w ), where

= 8. , 0. 745 for c. g. s. units.

Pierson, Moskowitz. - Proposed spectra] forni for fully developed wind seas, N.Y. University.Geophysical Sciences Laboratory. Report 63-12.

68

As for the propagation of windwaves outside a storm area, consider an areawith fully developed waves of length L and width W.

+Y

F

gt gt

2(d + F) 2d

Figure 4.4.c.

Assume the windwave pattern in this area to be a linear composition of wavesof all sorts of frequencies and directions. In that case, the waves that leavethe storm area, will all propagate with their own group velocity.In a point P(x0,y) at a time to, the sea will only contain part of all the fre-quencies and part of all the directions that the sea in the source area consistedof. These frequency and direction intervals may have widths &7 and A.The bounds of these frequency and direction bands are

and arctan((y - W/2)/x)< e + i-r< arctan((y + W/2)/x0)

These bands determine, what part of the entire spectrum is observed in P attime t

o

gFtApproximating and by = 2d (d F)

and e = W/d00

it is clear that these bands become smaller as d and t increase, while d ItO 0 00

is constant. This is a remarkable affirmation of practical experience: Thefurther one gets from a storm area, the more the waves look like elementarywaves with small frequency band an parallel wave crests.

69

yo

x+X

o

4.5. The performance of a ship in a given wave field.

4. 5. 1. Practical approach

For a given i-significant wave height H and a mean wave direction ed, expe-rience has shown that the maximum ship's speed r for a course a is a function

of H and of °d = a - ed . The problem of finding this function r(ad,H) is at

this moment far from being completely solved. Attempts to determine r for

ad = 0, n/2, on a strictly experimental basis have been made - among severalothers - by G. Verploegh of the K. N. M. I. -routeing office and by R. W. Jamesof the U.S. Naval Hydrographic Office. Mainly based on ship's log data, theycomposed graphs of r as a function of H for head, beam and following seas. One

of these graphs is shown in figure 4. 5. a.

70

15o

Q)Q)o,

10o,

4 8 12 16 20 24 28

p H(feet)

Figure 4.5.a.

In addition to this, the K. N. M.I. discovered a curious policy of ships in the

case of quarterly incoming waves, i. e. u/2 < ad <IT.Practically all ship's masters avoided too heavy rolling in these courses byeither reducing speed or altering course. This means that H may determinea prohibited sector (g1,g2) for ad so that the choice of ad is limited by the

constraints

O E ad g1 or g2 ad Ir , with g1 g2

With the experimentally found values for r(O,H), r(rr/2,H), r(i-r,H), g1 and g2at hand, I assumed the effective indicatrix of that ship to be an ellipse foradmissible values of ad and a straight line segment for the prohibited sector.This is shown in figure 4. 5. b.

Figure 4.5. b.

4. 5. 2. Scientific approach.

In this country, a scientific research of various aspects of a ship's behaviourin sea waves is being coordinated by the Netherlands Ship Research CentreT. N. O. This institution has issued reports of investigations, carried out atthe Shipbuilding Laboratory of this technological university and at theNetherlands Ship Model Basin (N. S. P.) in Wageningen.A brief summary of the results of this work will now be given.

If a ship is moving through a wave field at an average speed s with respectto the water, the sea waves work as an input for several oscillatory shipmovements, the most important of them being:

Heaving: Vertical oscillation of the ship's gravity centre G.Swaying: Horizontal thwartships oscillation of G.

Pitching: Angular oscillation about a thwartships axis through G.Rolling: Angular oscillation about a longships axis through G.

Yawing: Angular oscillation about an axis, perpendicular to the axes of

71

'

O, G

z,ya a

' a

W' Wa

2 2A , A etc.z 1

H, H etc.z

etc.

m , m etc.z y

The quantities a and a * are related by

72

Wave height and amplitude,

True and apparent wave circular frequency,

Heaving and swaying amplitudes,

Pitching angle and amplitude,

Rolling angle and amplitude,

Yawing angle and amplitude,

Spectral functions of the corresponding quantities,

Frequency responses of z, etc.,

Arguments of the complex numbers etc.,

Mean squares of z, y etc.

rolling and pitching. Yawing can also be identified with an oscillatory

alteration of the ship's compass course.In an attempt to analyse these movements the principle of linear superposition

was adopted:The sea wave pattern is assumed to be a linear combination of sinewaveswith frequencies varying from zero to infinity, each with its own amplitude

and phase.The Pierson-Moskowitz model was selected as the most appropriate one to

work with.The ship's movement as a response to this composite sea can be regarded

as a linear combination of sinussoidal responses to the various sine com-

ponents of the wave pattern, each response with its own amplitude gain and

fase lag with respect to the sinussoidal input oscillation.

Although these assumptions may make the impression of being somewhathazardous, the results have so far proved to be rather useful. The ship'sresponses can be characterized by complex numbers with a module equal tothe amplitude ratio (output/input) and with an argument equalling the fase lagof the output oscillation relative to the wave input.In an attempt to enlighten the use of these complex numbers by means of a fewexamples, let us adopt the following notations:

A =HA , A =HThA , A =z z li y y

A =HThA ksinecp cpl

z=

HHA. kcose and

Regarding the last two expressions it can be remarked that the input amplitudesfor pitching and rolling oscillations are k fl cos e and k a sin e respectively.As for the yawing response it can be noted that this complex number isdependent of both a and 8, probably like

H (*, e) = (a*)sjn 2e

For the rest, little or nothing is known about this quantity numerically.For the heaving we can now give the stochastic representation

SA cos(o*t + £ + ) (do*de*)z z

where c is a stochastic variable with a uniform distribution over (O, 2v).

The mean square follows from m = 5 5A2(a*, 8*)da*de*

-îV2OThe probability for an amplitude to exceed a given value can be found fromRayleigh's formula

2P B1 = i r x

a m j x exp(- 2m ) = exp(- 2m

For the other quantities we have analogous expressions. Concentrating now onthe case 8d = O (ship's head to wind), we can evaluate the chance that the shipmakes water on deck and hatches. To do this, we have to find an expression

73

(4. 5. 2. 1) c*=o+c2scos(e -8)/.

(Without loss of generality the ship's course is taken zero)

Furthermore, put e* = e - edLet A2 be known as a function of o and e. Then use (4. 5. 2. 1) to transformA to a function of o* and e*.

The other spectral functions can now be found from

74

for the total water height at the ship's stem. Denoting this quantity by S, wehave S = ReTb - z + , where L is the ship's length and

= a exp(i(a*t+kL cos e)).

With z/ = exp(ia*t), /Çflk cos 8) = H exp(iY*t) and

ba = exp(i(kL cos 8 + a*t)) = exp(io*t) we find:

S = Re['fl (H - H + -kL cos e H)exp(io*t)}

Denoting this by S = Re{T H exp(ia*tfl, we can find the mean square of this

value by taking

rr/2

= J$IH2A2th*des s

-rr/2 O

If the ship makes water on deck, the amplitude Sa S exceeds the freeboard'sheight at the stem. The probability that this occurs, can now be evaluated byusing Rayleigh's distribution formula. On the other hand the frequency responses

of and z enables one to calculate the speed S, at which the probability ofmaking water does not exceed a desired upper bound.The avoidance of slamming - the ship's hull rises above the sea and then slamsthe sea surface, causing a short, but rather frightening vibration of the ship'sstructure - can be treated in an analogous way.Stated very briefly and generally, a captain's set of wishes with regard to theavoidance of various undesired phenomena and the knowledge of the ship'sresponses can thus lead to the evaluation of a best possible speed for variouscourses in a given wave field. The knowledge of the ship's responses can alsoserve to compute the ship's resistance increase in waves. This resistance in-

crease has been found to be determined by the wave amplitude, the ship'sdimensions and the values of the heaving and pitching responses. Thus one canget a fair idea about the ship's speed in a given wave pattern, if the engines

are working at maximal power capacity.

5. EVALUATION OF THE LEAST TIME TRACK

5.1. Oceangraphic and meteorological data.

In the KaIm projection of the concerned area - I suggest to concentrate on theNorth Atlantic from now on - the X-axis is taken to be the mapping of thecentral great circle from Bishop Rock, starting point of ocean navigation forWest bound merchant craft, to Providence Channel near the Bahama Islands.

o oThe origin is the mapping of a point on the earth with coordinates 49 N, 7 W.

200îRThe mapping of a stretch of 200 nautical miles, i.e. 180*60 , is taken as a unit

of length, where R is the radius of the sphere on which the ellipsoid wasconformally mapped. The points with integer coordinates are now taken as datagrid points. This implies that the sea current and sea wave data are introducedas an array, like sxr, syr, kir, tzr[0:18, -3:5The array sxr, syr gets the values of the ocean current components for timestretches of 12 hours. The elements hzr and tzr are the significant wave heightsand the mean directions these waves come from. The 200*200 m2 grid issufficiently dense to serve as a basis for computing the current components andthe wave direction for arbitrary coordinates by means of linear interpolation.As for the wave heights, the grid is occasionally liable to neglect small areasof exceptionally high waves. To meet this imperfection, the partial derivativesof the wave heights with respect to x and y were also taken into account. Thewave data can be taken from wave prediction charts, constructed by the K. N.M. I.routeing office. These charts are still made up manually, taking account of windvelocity, duration and fetch. Automation of this procedure is being studied, but

results fit for practical use have not yet been attained.A,s the meteorological situation seldom or never can be taken stationary, it isnecessary to calculate the wave heights and directions and to have them read inby the computer at the beginning of each step of integration, which is a 12 hours

time stretch.Weather forecasts and weather maps have until now never been produced for

more than five days ahead. This means that for the future time, that exceedsthis period, - a North Atlantic crossing takes a merchant ship some 7 to 11days - an estimation of the weather situation has to be made up. This couldbe done, for instance, by assuming that the periodicity with regard to wind

75

velocity and direction, concluded from the 3-to-5-days -ahead weather forecasts,is being extrapolated over the rest of the crossing time. As for the wave heightsto be deduced from these rough weather estimations, it seems reasonable toassume a fully aroused sea, thus simplifying the calculation of wave heights forthose areas, where one guess is just as good as any other one.The uncertain area, implied by the limitation in forecasting period, has theadvantage of being further away than the area, the ship is going to cross firstand it reduces to zero as soon as the ship is so close to its destination, thatthe reliable weather forecasting period is greater than the rest of the trip's

duration.The data for the ocean current components can be taken from statistical issueslike the U.S. Hydrographic Office Pilot Charts. Because of their small magni-tude they can be taken constant for a trip's duration.

5.2. Evaluating the timefronts.

suppose the coordinates of a number of points of a timefront S - for a time= to + 12h - are known. Denote these coordinates by x i, O and y[i, Oj

with i = ik(1)im.The timefront gradient's direction (a) can now be approximated:aa: = arctan((xLi+1, O - xLi, O)/(yi, O] - y[i+1, O])) and

bb: = arctan((xLi, O] - xi-1, O)/(y[i-1, Oi - y[i, O]))If i = ik then a:=aa, if ik< i< im then a:=(aa+bb)/2 andif i = im then a:=bb. This process can be executed by calling a previouslydeclared procedure "normaal".The first (Euler-) approximation of points of S1 is now computed by callingthe procedure "optistep", which will now be briefly described. The wave heightand direction are determined as indicated in 5. 1. The ship's response to thesewaves is characterized by five parameters d1, d2, d3, g1, g2, shown in figure5.2. a.Given the timefront gradient's direction relative to the mean wind direction:

= a - tz

the problem of determining the ship's 12h_displacement vector that has amaximal inner product with the vector (cos , sin ) can now be solved. As for

h ...............the curvilinear parts of the 12 -position indicatrix this amounts to determiningthe polar line of the point with homogeneous coordinates (- sin , cos , O)

76

Figure 5.2.a.

with respect to the ellipse. Denoting this "conjugate gradient direction" by p,the angle cç + tz can be taken as the ship's course, provided p or

w g2.

If g1 <p < g2, the maximum principle would prescribe to take either+ tz or ± g2 + tz as the ship's course, depending on which of the corre-

sponding velocity vectors has the greatest inner product with (cos , sin ).

However, on account of the 12h_step integrating procedure, this policy mightcause an unpractical accumulation and even a coincidence of points of the nexttimefront. Therefore, the ship's displacement (relative to water) is taken inthe "course" p + tz, from the centre of the indicatrix to a point on the rectilinearpart. This point can be reached by steering courses + g1 + tz and ± g2 + tz in

a proper time ratio.In both cases - free and bounded courses - the vector of the ship's indicatrix,obtained in the just described way, gives the ship's displacement for 12 hours,

relative to the water. This vector (dx, dy) is now, together with the sea currentcomponents, added to the ship's starting position (x[i3O,y[i3O), which givesthe Euler approximation of the point of S1 with index i.A Heim approximation of these points is now obtained by carrying out the"optistep"-procedure once more for this first approximation (xi, 1],y[i, 1]),using wave heights expected at a time to + (+l)*l2h. This results in anotherdisplacement (dx,dy), including sea current.

77

The point of S. with index i is now taken:

x[i, 21 :(x[i, 0] +dx+x[i, 11)12 and y[i, 2] :=(y[i, 0] +dy±y[i, 1])/2

The Euler approximations of points of the first timefront are obtained by taking:

a:= i-r'(25 - i)/180 with i = 0(1)40

oThis way the area is scanned by a sector of 40 width, which produces a time-front of some 1600 miles length after 7 days. Selecting the initial values of awith lo intervals, the mutual distance between consecutive points of a timefrontis liable to become too large. To prevent this, a refining procedure is declared,adding half-way-between points to the timefront points, as soon as they are morethan 40 miles apart.Special precautions must be taken in the event that the regiones boundary preventsfree extremals from covering part of the area. This is actually the case afterpassing the New-Foundland bank. With Northerly winds off the New-Englandcoast, the free extremals are even inclined to bend Southward, so that an areawith favourable navigating conditions because of its position close to a windwardcoast may remain uncovered. This can be avoided by the addition of a sector of"subextremals", as described in the last paragraph of the 3rd chapter.

5.3. Revision of the timefronts

When passing regions of high waves or regions with considerable vorticity inthe field of wind vectors - these phenomena frequently coincide - the field ofextremals is liable to be split up into three families (as described in 3.8), oneof which is irrelevant for the solution of the problem. Therefore the trajectorieswith ultimate points for a time t, that are not located at the very border of Ht'have to be eliminated. Because of the various possibilities of configurations, thiseliminating process is carried out by two procedures, called "ontlussen" and!schoonmaak.

20

78

17 e

Figure 5.3. a.

In the event of overlapping timefronts, like shown in figure 5. 3. a., the pointswith indices 14(1)18 are deleted. This kind of configuration is traced by seeingif a timefront segment between points with indices i and i+1 has an intersectionwith any of the following segments. Another possible configuration is shown infigure 5. 3. b.

Figure 5.3. b.

This situation is checked for by seeing if the extension of the segment of atrajectory from S.1 to S. has an intersection with any of the segments betweentwo consecutive points of timefront S..A procedure "rand" cancels points of a timefront, that lie outside of G, thenavigating area.Data regarding stream and sea in gridpoints just outside of G are not changed.Assumption of extremely high wave heights in those points might work as aprevention of coming outside of G, but this trick has the disadvantage thattrajectories with arcs close to a coastline are thus being eliminated, whilethese routes can be favourable, especially in case of an off shore wind.

5.4. Determination of the trajectory and the time gain.

As soon as the destination is estimated to be less than 48 hours away, a searchis started to see, if the destination B is located between S. and S. . This is

) j-1executed by starting with tracing the index "i0" of the point of

-1 closest to B.Then B is checked on lying in one of the quadrangles, formed by the points withindices [i3OI1,[i,2,[i-1,2 and [i-1,O or

[i3O,[i,2],[i+1,2] and [i+1,O

As soon as this is found to be true, the arrival time and the initial value ofthe timefront gradient can be evaluated. The least time track can then be con-structed.

A fair idea about the time gain of the optimal track can be obtained by comparingit with the time, needed for the same ship to cross the same composition of

79

80

Figure 5.4.a.

obstacles when steering a great circle's course. Replacing this great circleby a straight line in the Kahn projection - which is allowable on account of thevery slight scale alteration in this chart - the positions on this line at the endof each 12 hours interval can be computed by calling a procedure named stap.

5.5. The Algol program to compute the least time track.

This paragraph mainly contains a program in Algol code, to compute the leasttime track between two arbitrary points. The program was used to executethis for a ship, starting from a point near Bishop Rock on March ist '67, 00h

G.m.t. and bound for a point near Nantucket Shoals, from where it cari navigateto New York without further routeing support.The result of the computation, carried out on an I. C. T. -1905 computer, ope-rating at the mathematical department of this technical university, is exhibitedon the North Atlantic chart, folded in at the inside back cover. The, yet un-revised, timefronts S0 S1 etc. and the extremal trajectory were copied froma Calcomp plotter, which is attached to the I.C.T. -computer.To illustrate the - time dependent - wave field, arrows are drawn near thetimefronts, indicating the mean wave direction at the corresponding time andposition. The adjoint numbers give the significant wave heights in meters.Fully drawn arrows indicate wind driven sea waves, while dashed arrows re-

present swell.The dotted line from (0, 0) to (13, -1) is the great circle track. The small crosseson this line are the positions on this track after a half day, one day etc.

rj -i 2-o

GENEAAL LISTING (XQLP)

1 FjXALE#0lSC1*2 G0XALE201*3 'SENtDTO' (E0,SEMICOMPFILE, .PR0G9A? W012)4 PROGRAM'(W012)5 jNP1JT'ØCF0

6 OUTPUT'O=LPO7 CREATE I5MT0(PL0TTAPE(t.0))8 '1 G!N''INTEGEP'IJ,H,K.rK.1M.F,G,ro,!1,Io,Jl,IK1.IM1; CWO120009 'PROcFDURE'PLOTsTART;ALGOL' C1Ol200llo 'PRGCFDU9EPL0T(I,A,B);INTFGFPI;'RELA,8;'XTRNAL; CW01201#

il 'PP0CEDUPEPLOTFAMEjxMtNfl..TM1Nn.YMAj0,MAxP,YMAXP.SCX,SCY); CWCl2OIB12 IREAL'XMINr),XMAXD.YMIND.YMAAD,SCA.SCY; CWOI2OIC13 INTEGEPXMAXP,YMAXP;ALGOL'; CW01201014 E0OLEAN'ALFA.BETA; 'RFALA,R,PI,OXDy,wO,yO,xB.yB.GR,L,M,N.0,P,Q,R,S.CWO120O215 GK,SGK,jG.YGK,C0SG<,S1NK. CW01202A

T,X!1,YjI.XHI,VHI.x'.1H,V4,0ET,DM1M,PP,OQ.RR,SS.XC.YK; CW01200317 .INTEGER.'AAv'NU,Mul1:2O,1:j7],IB,1oCl :17]; CW01200418 AkRAy.sxp,syR.HzR,HRx,Hpy,T7Co:I4,-2:1,x,yco:so,o:2],u,vL0:6o,O: 17], CW01200519 D,cospslocl:3,o:3o:,r)E,cosGrl :3]jG9,Dy(P,SING1 :2]; cw 1200620 PR0CEDU'ENORMAA1 (1,Y,1I,JJ,AA);'ARPAV'x,Y;'TNTEÇ,FPII,JJ; REALAA; CW012001

21 'GI4REAL'rn4,EDT,ET.BB;11I<lM'THN22 .BEGIN.DH:=jCII+1,JJJ-xClI,JJJ;EH:=YcII,JJJYc!J+1,JJJ;23 1FBSIDH)<A5S(E'TE:=ARCTANH/EH)E1SE24 AA:=pI*SIGN(r»4)*sIGN(Eu)/2_ascTAN(Eu,DH)25 END';IFII>IkTHEN'

26 BEG1NpT:=xcII,JJJ_xCT1-1JJ];FT:yCI1_1,JJ]_YEl1,JJ]; CW01201327 '1F'ABS(DT)<A5S(FT)'THEN'B8:ARCT4N(DT/ET) 0W01201428 ELSE'Be:=PI*SIGN(0T*SIGNTI,2_APCTANET/DT CW01201529 'END';' IF'II>IV'AND I1<lM'TkENAA:(AA+BA)/2 CW01201630 IF'II=IMTHFNAA:=BE cwOl2011

31 ENONOkM#4L; CW01201832 'PROCEDuREoPTISTEp(xS,YS,j);REAL'xç,ys,g; Cw01201933 'BEGIN' 'EALFP,FPC,FO,FûC.T2,2,TZ0,TZl ,TZOO,TZO1 ,TZIO, CW01202034 TZ11.HZ1H?7.AI.BlEI.00PSISIPSI,Dxl,Dv1,c0STz.SINTZ; CW01202135 :=ENTIEp(xs);K:NT1Ep(s);Fp:S_I.;rpç:_Fp; CW012022

36 Fo:=ys-K;Fo( fli-Fo;37 0X:=FDC*(FpC*SxR[H,]+Fp*SxRC4+l,<J)38 +FQ*(FpC*SAR[H,Kl]+Fp*SXR[9+j,K+j]);39 DY:=FQC*Fpc*SYRLH.KJ+EP*SyCH+1,K])40

41 HZIFPC*(I.47RCH,KJ+FP*HPACI.1,K))+FP*(MZPCH+l,K] Cw01202842 -Fpc*NBxcH+1,<J); CW01202943 Hz2:=Fpc*(kzRcH.K+1)+Fp.HRxtH,,1J) CW01203044 +Ep*($.ZR+1+j3-Fpc*HRck+j,(+j]); CW01203145 HZ:=FQC*(HZl+FQ*(FPC*HRY[u,K]+rp*HycH+l,K7)) CW012032

+ FO * C HZ 2-FQC*( FPC* HEY CH, K+1J+ Fp* HR VC H+l , (+17CCHZ:=2aC47; 'IF'HZ'GE'30'THFN 'GOTr'DELFT;î zoo : T z ECH. K]; Izol: =1 ZR Cl,

Tzlo:=Tzp[H+l,K:;TzIl:=TzpcH+j,K+17;IFTZOO-TZI0>PI 'THEN' 'BFGIN'T710=TZIO+2*P1 'GOTOAI 'END'

IF '12 10-TZOO>PI THEN' BEG IN' 1700: =TZoo+2*PT; ' OnTO' 42' END''IF'TZOl-TZII>PI'THEN' 'BFGIN'1711:TZII+2*pU 'GOTO'A3END;'lF'TZII-TZOl>Pl'THFN 'BFGIN'T701:*TZO1+2*PI; 'GOTD'A4'END';TZo:=Fpc*Tzoo+Fp*TZ1o;Tzl:Fpc*Tzoj+Fp*T7I1;'IF'TZO-TZI>PITHEN''BEGIN'TZ1:*TZl+2*PI; 'GOTO'A5'END'

81

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CEOl 203eCEO 12039CEO 12040CEO 12041CEO 12042

5657585960

IFTZI-TZ0>PIIHEN BEGINTZQ:=TZ0.2*PI; GOTOA6ENDTI:=Foc*Tzo+Fo*Tzl;cosTz:=cos(Tz1;s1NTz:sINcTz);cops1:cos(A-Tz);sIps1:=sIN(A_Tz1;H: =FNT IERI HZ I ; FP: 1z_u;Fpc:1Rp;F0RK:=1,2,3.D0.0EcKJ:=oCKH3*pcDcK.H+1)*Fp;

76 END'OPTISTEP;77 PROCEDUREONTLUSSEN;78 BEGIM 1:r1;H:=I+2;79 LII: xI1:=xE1+I,2]_xc1,2);yII:=yn+1,2:)_yI,2;80 LL2: xH1:=1cI4+I,2J_xc12J;yHI:=ycH+l,n_ycr,2;

CWO1 2043CWO 12044CWOI 2045CWO 12046CWOI 2047

CWO 12065CWO 12066CWO 12067CWO 12068CWO 12069

81 xHH:=xcH+1,2J_xcH2J;*4:=yH+l,2J_y[H.2J;oET:=x1I*yHH_v1I*xI4H;cwo12o7o82 L:=XHI*YHH_YHI*1HHI/DET;M:=xI!*YHI_v11*xHn,DFT; CW01207183 1F'L>0ANDLLEIANDM>0N0LE1T.1EN CW01207284 BEGIN4.FoP'K:I+1.sTEp.l.uNT1L.IM_H+1.D0. CW01207385 BEGINXCK.01:xCK+H_1,0J;yCK,n-I:=YCK+u._I.o); CW012074

86 xcK,2J:=XcK+H_I.2;ycK.2o:=vK+H_I,2D CW01207587 ED;IM:=1M-H+1;1:I+1;H:1+2;IFH<1M.THEw.GoT0LL1 CW01207688 ELSE"GOIOLO CW01207789 END;=H+1;1FHLE1.15ANoH<iTHENGoToLL2; CW01207890 I:rI+l;k:=I+2;1FH<IplTHENGrToLL1; CW012079

91 LoENcoNTLussEN; CW01208092 PROCEDURESCHÙONMAAK; Cw01208193 BEGINFOR.I:=IK.STEP1uN11L.11.YDO. CW 1208394 LI: BEß1NX11XC1,2]-XCj,0Jy1jyC12J--yC1,0) CW01208495 FoRH:IKSTEp1'UNTIL1MDo CW012085

96 BEGIN"1FH=I-toRH=ITHEN»0T0L2; CW01208697 1H1:xcH.2J_xCI,oJ;yHI:=Yc.4,,J-y[I,oJ; CW012087

98 xHH:xEH,2J_xcH+1,2J;yHH:=yr,.,2J-ycH+1.2J; CW012088

99 DET:=xT!*YHH_yIl*XHH;I:=XHT*YHH-YHI*XHH1,DET; CW012089100 M:=Ix!I*yHI_111axHI),oET.1.L>1.ANo.M>o.Xwo.M<1THEw. CW012090

101 8E01M EQRK:=j'STEpf UNTILIM-I 00 CW012091102 CW012092103 xcK,23:xcK1.27;y1k,2j:=ycx+l,2J CW012093104 END;IM:IM-l;'GoTo'LI C4012094

105 ;CW012095

106 L2: END CW012096107 END ; IF !M-IK>20.TNEN.u:=2o.ELSF.H:=IM_IK; CI1012097

108 FOR'l:=lK+lslEp.lUNTILIKHDo CW012098109 EEGIN LS: 1PyC1,2J>YCIK,2J1HEN CW012099110 BEGI F0PK:1 STEP I UNTIL TM-I D0 CW01299A

82

61 FORK=1 .2D0C0SGCK1:=FPC*COSPSIGCK,u1+Fp*COSPSIGCK,H+1]; CW 1204862 AI:=(DE[17+DE[3J/2;E1:=DECII_n1;BI:=81*0Ec2J,SORT(ÛECIJ*0Ec371cw01204963 ;'1Fc0PS1GEc0s4C1JopcopsI.,FCOSG[2JTHEM CWOI2OSO64 .BEGI'P=lI*cflPs1;Q:BJ*S1ps1;R:ç0RT(p*p+0*oI; CW01205165 DXI A1*P/R_E1;ny1:=B1*Q/p; CW012052

66 ox:=ox+oxl*cosTz_Dyl*sIMTz; CW01205367 OY:=DY+Dxl*slNTz+ovl*coslz;.00To.DELFT CW01205468 EN0;FûK:=1,2flo CW01205569 EGINvp:=A1*cosccKJ;s1NGcxJ:siGI(sIps1)*soRTc1_cosGcKJ CW01205670 *C0SGCKJ)Q=B1*S1NG[KJ;RçRT(P*P+Q*Q); CW012057

71 D1GRCK]:=AI*P/R;nYGRrkJ:=BI*/R CW01205872 EN0 ;FP:=(cOsGCIJ_coPsI )/(COSGC1J_CO5GL21);Fpc:=1-FP; Cw01205973 ox1:=Fp*010p23+FpC.Dxc,pc1J_EI;nyl:=Fp.0509c23+Fpc*DYGRCIJ; CW01206074 Dx:Dx+Dx1*cosTz_Dy1*sINTz;ry:=Dy+DxI*sI4Tz+Dy1*cosTz; CW01206375 DFLFT s:=pIa(YS+Dy,2),54;s:=(ExpIçI+Exp(_sI ),2;Dx:=s*ox;o:=s*1Dy cw012o64

111 'pEGIN'xCK,0=xCK+0]EK,C1'=Y[K+I,O]112 xCK,23:=xCk+1,2J;YCK.21:=VCKl,2]113 END';IM:=IM_1;'GOTO'13114 'END'115 'END'; 'FOR' I :rIM-H'sTEp' I 'UNTIL' 1M-1 '00'

116 'BEGIN' L4: '1EYC1,2)<Y[8,2J'THFN'117 'BEGIN 'FoR'K:I'STFp'1 INTIL'TN-I 'DO'118 .BEGINxCK,oJ:=xcK+1,0);YCK,oJ:=YCK.I,OJ;119 XLK,2]:=ECK+I,2J;YCK,21:=YrK+1,2]120 'END';1M=IM-1;'G0TO'14

121 'END'122 END'123 'ENc'SCHD0NMAA;124 'PPCCEDUBE'RANO;125 'BEGIN'

126 57 'IF'YCIK,2]>2.5'fl.4EN' 'REGIN'IK:=I+1; 'G0TO'A7'FND';127 ' IF'XCIM,2]<5.8'TEN'128 BEGIN' 88 'IF'YCIM,2J<-1.5'THFN'12 'BEGIN' Is1:=IM-1; 'GOTO'ß8130 'END'

131 'ENO';' IF' XCIM,23'GE'8,8'AND'XCIM,7J<q.7'THEN'132 'BEGIN'133 Aq: IF' VC IM, 2D<-0. 9' THEN'134 BEGIN' IM IM-I; ' GOTi' 89135 'END';

136 'END' 'IF' ANM. 2]' GE' 97' ANO' lUIR. 2)<15'THEN'137 'BEGIN' 810: 'IF'YLIM,2]<-1.4'TREN'138 'BEGIN' Ill:=IM-i ; 'GOTO'Alo139 'END'140 'END';'IF'ECIM,2J'GE'15'THFN'

141 'BEGIN' All: 'IF'VEIM,2J<-I'THEN'142 'BEGIN' IM:=IM-1; 'GOTO'AI I143 'ENO'144 'ENO'145 'ENC'RANO;

161 AS2: IF'Tzlo-rzoo>PI'THEN' 'BEGIN'T200:TZ00+2*PI'GOTO'AS2'END'162 A53 ' IF.TZ01_TZ11>P1'THEN''BEG1N'TZ11:TZI1+2*PI 'GûT0'AS3'END'163 8S4: 'IF'Till-TZOl>PI'THEN''RFGIN'TZOI :T2O1+2*PI,''GOTo'ES4'EMD';164 Tzo:=Fpc*Tzoo+Fp*Tzt0;Tz1:=Fpc*TZCI+Fp*TZ1l;165 ASS: 'IF'TZO-TZl>PI'THEN' 'BEGIN'TZI :T7t+2*PI; 'GÙTO'ASS'END'

83

CWO 12998CR01 299CCRo I 299

CR0 I 299ECRO 1299F

CWO I 29ECR01 299kCWO 121 OC

CWO 12101CWO 1210;

CR0 12 1OCWO 121 0CR012105CRO 1210CR01 2107

tWO 1210CR0 121 O

CROl 211 CCR012111CR0121 i;

CRO 1211CRO 1211

CR0121 ISCR0121 1CRO 1211]

CR0121 IlCR0121 1CWO 121 2CCR0 12121CRO 1212;

CR01 2123CWO 12124CR0 12125CWO 12126C WO t 2127

146 'PROcEDURE'STAP; CR012500147 'BEGIN' 'REAL'FP,FPC,FO,FQC,TZ,HZ,TZO,TZl,TlOO,TZOI ,TZ10,TZ11 ,wZ1,H22,CW012501148 EC,EC,BC,ST,COFI,SIFI,DXI.OV1,COSTZ,SINTZ; CR012502149 :=ENTJER(XG)K:=ENTIER(YGK)FP=XGK_HFPC1_FPFO:YGK_KF0C CWO12503150 1_E0;ST:=(F0C*{FPC*SXRCH,K]+FP*SXUH+I,KJ)+ CW012504

151 F0*(FPC*SXRCH,K+1J+FP*SXPCH+1,I(+1JH*COSGK CR012548152 +{FOC*(FPC*SVp[H,KJ+FP*SVRCI..+1,KJ)+ CR012548153 Fo*cFpc*syRcH,K+1J+Fp*syprH+1,K+11))*sINGK; CWOI254C154 HZl:=Fpc*(HZRCH,KJ+FP*HRXUH,K)+0)+FP*(I4ZRCI4+1,KJ_FPC*HPKCH+1,KJ) CW012505155 FZ2:=FpC*IHZRCH,K+1J+Fp*HRXIH,k+lJt+FP*(HZRCH+l ,K+1]-FPC*HRXCH+1 , CR012506

156 K1J);Hz:FQc*(Hz1+FQ*(Fpc.HRyCH,1+Fp*HPyCH+1,KJ))+ CR012507157 FQ*(HZ2_FOC*(FPC*HPVCH,K+1J+FP*HRVCH+1,K+IJ) ;HZ:2*HZ; CR012508158 ' IF'NZ'GF'30'THEN' 'BEGIN'L:=o; 'GQTÇI'ZWET'END' ; CR012509159 ViOC TzREH,xJ;Tz0I:=TZRCH,K+I1;TZ1O:TZRCH+1,J;TZ11:=TZPCH+1,K+1]; CR012510160 ASI: IFTz0o_rzI0>Pj'THEN'.9EGIN'Tz1O:Tz10+2*PI;'G0T0'AS1'END'; CR 12511

CROl 2512CRO 12513CR01 2514CR0 12515CR012516

165 ASS: !FTZ1-TZO>PI'THEN BE31NT20:=T71,+2*P1;.00TOAS6FNO r CW012513167 Tz:=FOC*Tz0+Fo*TZ1;COSTZ:=COSTZ:ÇINTZ:=S1N)T2);H:=ENTIE9(HZ;FP CW012518168 =)z_kFPC:1_FP .FoRK:=1,2,3.Do.ErK3:=Fpc.orw,HJ+Fp*DCK,H+1D; Cw01251Ç169 F0p:=1,2.Do'C0SGCJ:=pPC*CospS1CK,H3+Fp*C0SPS1GCK,H+1J;AC:= CW01252C170 oFr14DEC3n/2;EC:=DEC1J-AC;BC:=c*DEr2J,sQRTcoEc1J*DEE3j) COFI CtJ012521

171 :=cos(Gs(_T2);sIFI:=sIN)GK-Tz);p:=L*Bc*coFr;o:=#C*AC*S1F1;p:EC* CW012522172 SIFI;R:=P*R;S:=P*CflFI;fl=Q*S!PT;L:=)-P*EC+AC*EC*SQRT)T-P.S)1/(T+S,CW012523173 ;Dx1:=L*COFI;oY1:=L*SIFI;.Fnp.1:=1.200. Cw012524174 sEG1Np:=Ac*Ac*cosGCfl;sINc3:=srGN)sIF1)*sQRT(1-COSGC1J*cosGctJcwot2525175 ;Q:=Bc*BC*sINGCIi;P:=SoT)P*CoSGJ+o*sTNGCIJ;DxGkCIJ:P/R;DYGP CW012526

176 C1)-0/R END r 1F0XG)C2]<DX1+ECANDDX1,FC<0XGPL1]T1EN CW012527177 L:(0XGRC1-EC)*0YGRL2J-(DXGP[2]_VC)*DYGACtJ)/ Cw012528178 N0YG9C2)_DYGPC1])*C0FI+DxGRC1-,_0XGRC2])*SIF1) CW012529179 ZWET: Dx:=)L+ST)*C0SG;DY:=)L+ST)*S1NK; CW012531180 s:=pI*(YGK+cry,2)/54;s:r(Exp(s)+FxP)-S))/2; CW012532

181 DXS*DX;DY:=S*DY;ENOSTAP; Cw012533182 PRCCEDUPEFU1; CW012128183 BEGrNREALo;oMIYJ:=too0;1o:=o;.FoR. r=IKSTP1UNT1L1M00 CW012129184 BEGIN.Dx:=xc1.oJ_xRrDy:=yc1,oJ_YE;o:=SQRTCDX*Dx+oy*OY); CW012130185 1F[PLEDMINTHEN 9EG1NDMIN:=U; 10:=1 EN0 Cwo12131

186 EN0 ; :10_t;RR:=(XB_1110,O7)*(yrlO,23-yCjO,OJ)- CW012132187 (YsA-YCjO.0J)(1C10.23-XC10.0J); CW012133188 C(ThTPOLE: pp:(XB_XCJO,0J)*(y[jj,0J_5r10,03) CW012134189 _56_YCIO.oJ*(XCI1,0J_xCIO,03; CW012135190 00:=(XB-XEIO,2J)*(YE1l,23-Yr10,2J) Cw012136

191 -(YB-YE10,2J)*(XCII,23-XrrO.21); CW012137192 SS:=(XB-AC!1,OJ)*(YCII.23-Yrrl.0J) CW012138193 - r VB-Y C I 1.03) * I XCII .23-Arrt 03) CWO 12139194 IF'PP*QQLE'O'AND R*S LE'0'TEN' GOTO'HOERA; CW012140195 ' 1F 11=10-t 'THEN' 'BEGINs 11 :1* ; 'GOTO'CONTPOLEEND' ; CW012141

196 SELECT OUTPUT(0);'GOTO'NOG NIET; CW012142197 .4OEPA: J1=J_1;ALEA:'TRUE'F:)OSELFrT OUTPUT)O) CW012144198 NEWLINE(1);WRITETEXT(('REISOUUP')');PP:rPP/(PP-OQ) CW012152199 K:=ENTIEErJI/21;o:=rJI-2*K.pp)*,2; PRINT(K.3.0) ; Cw012153200 wRrTETExT'DAGEN');PpINTrD,1);WRITETEXT)'('UREN%%ì'); CW012154

201 WR1TETEXT)('FINDINOEX%BAANKR0MME)');PRINTCIO,2,0);'GOTO'tBB; CW012155202 NOG NIET:NEWLINEH;WPITETEXT('t'PEISDUUA>')');PRINT(J/2,2,1) CW012159203 rIRITETEXT( ) 'OXEEN' I' t; CW012160204 LBB:'END'ELJIK;'PHOCEDuRE'VERFYN; CW012161205 BEGIN'l:=IK; CWO12SIB

206 20: XII:=x[I,o]-xC1+l,0JYI1 :yrI,o3_vcj+t.oJ; CIJOI2SIB

207 'IF'X)1*111+Y11*Y11>0.04'THFN' CWOI26IC208 8EG1N'F0R'H:=IM'STEPtUwTIL'1.I'00' CWOI26ID209 'BEG1N'xcH+1,o1:=xLHoJ;vcH+t,o1:=yCH.oj CWOI26IE210 'END' ;xCl+i OJ:=XC1 .OJ-11 I/2;Yrr+t,0]:=YC1,OJ-YI 1/2; CWOI2EIF

211 1M:=!M+1;I:=1+1; 'JF')<IM'THEN' aTo'A2o CW012616212 'ELSE"GOTOA21 CWOI26IH213 END'ELSE' CW012611214 BEGIN'I:=t+1;'IFI<IM'THEN'GOTO'120'ELSF"GOTOA2I CW0126t(215 'END'; CWOI26IL

216 2t:END.vEpFYN;P1:r3.t41592653590;GR:=PI/180;BETA=TRUE CWOI2SIM217 'FOR''t=O'STEP' l'tJNTIL'14'Dfl 'F0R.:=-2'STEP' I 'UNTIL3DO' CW0t2162218 BEGIN'sXpCH,I(J:=READ;SVRCH.KJ:=955o; CW012163219 END ; 'FOP'H:=O'STEP' t UNTIL' 14D0 EOR.K:=-2,_1 .0.1 .2.3D0' 0W012164220 BEGINHZRCH.KJ:=REAO;HRXCN,KJ:=READ;RYCH,KJ:=READ; CW012165

84

221 TZRCH,K):=READ;TZRCM.k]GR*TZPrH.K222 'END'; 'FOP'H=O'STEP'l 'UNTIL3O'OO .op.K:=l,2,3.00.DCK,H):READ;223 'FO H:0STEp' I'UNTIL'OO'DO 'F0R':=1,2D000SPSIGCK,H=PEAD224 X0:rREAD;Y0:=READX=RE8D5=PEAIK:=0;1M=40;225 SELECT OuTPUT(15)

226 PL0TsTART;pL0TPR8ME(O,14.-I.5,2.5.4370,1248,0,O); CW012698227 PLOT( 1,o,-1.5);PLOT(2,o,2.5);PLOT(2.,4,2.5)PLOT(2,14.-l.4); CW01269B228 PLOT(2,9.7.-1 .4);PLOT(2,9.7.-O9);PLnT(2,8.8.-O.9)PLÇT(2,8.8.-1.5) CW01269C229 PLOT(2,O,-1 .5);PLOT( 1,0.0;PLDT(2.14.c); CW012690230 Gk:ARcTAN((yB_yo),1xB_XO));UCO,0J:XK:=XO;v0,0J:=YGK=YO CW01269E

231232 J:1;'FOR'JIK'STp'1UNTILIM'DO'233 BEGIN'A:=pI*125-I)/180;OPTISTEP(X0,yo.A);234 xci, 1J:=xo+Dx;yCI, 13:yO+Dy235 'ENC'F0R'H0'STEP'1'UNTIL'14'DO''0R'K2,-1,O,1.2.3'DO'

236 .8EGiN.HzRCH,K]:=RE8D;HQXCH,K3:FEA0;RYCj,K):=PEAD;237 TZR[H,KJ:=READ;TZRC,KGR*TZprH.KJ238 END' ;STAP;UCJ,Oj:=XGK:=(XGK+UCJ_1 ,OJ+DX )/2239240 FOR'I:K'STEP'I'UNTIL'TM'DO'

241 EGIN'NORMAAL(X,Y,I.l,A)OPTISTEP(XrT,l].YCI,IJ,A)242 XCI.2J:=( XCI, 1J+xo+oX)/2;YCI,2:(yCI.1+y0+OY)/2 ;243 FND';sELEcT OUTPUT( 15);PLoT(l,xCiK,?1,YCIK,2)244 FOR'IIK'STEP'l'UNTIL'IM'DO'245 .eEGJN.pLoT(2.xcX,2J,YCI,21);XCI,0JuCI.J3:rxrI,2J;

246247248249250

yC!,oJ:vCJ,JJ:=yCI,2J.END ;PLOT j ,xK,yGK+O,o2);PL0T(2,XGK,YGK-0.02CWO12182

TEDFRONT:;PLOT(1,XGK-0.02,YGK);PLnT(2,XGK+0.02,YGKL

u:=J+j;ALFA:.FALsE' SELECT r.UTPUT( 15);ST AP ; 10K GK+DX 50)< =5 DV

FOR I: = Ix STEP I UNTIL IM oD'

251 BEGINNJRMAAL(X.Y.I,0,A);OPTISTEP(XCX,O),VCI,0J.A);252 ICI,1J:=XCI,OJ+DXV[I,1J:YrT,OJ+DY253 'END' IGCJ-1J=iOO. IF'XCIhl,0J>9.3'ANO'BETATHEN'254 BEGIN.BETA:=.FALsE.;.90ToAlE255 ENC' ; IPA>-5*GR'AND'ICIM.OJ>03THN

256 816BEGIN.F:=15;'IF.IM>45.TkEN.F:=60_IM;IGCJ_fl:=IM;257 'FORI=IM+I'STEPI'UNTILIM+F'OO258 .BEGIN.A:=A_GR;IC1,oJ:=XCIM,0;yCT,OJ:=yCIM,OJ;259 OPTISTEP)xCI,o],YcI,OJ,A);XCI,IJ:=x[I,0J+Dx;VC1,1D:=YCI.O+DY260 'END'; IM:=IM+F

261 END.;FOR'H:=o'sTEp'I.UNTIL'I4DDOR.K:=_2,_l,0,l,3'Do'262 BEGIN'HZPCH,KJ:=RE8D;HRxCH,KJ:=pEAD;)RyCH,K:=READ;263 TZRCH,K:=READ;TZRCH,k=GR*TZDÇH,KJ264 END.;STAP;UCJ,o]:=XGK:=(UCJ-1,OJ+xGk'+oxj/2;265 VCJ,0I:=YGK(VCJ-I,OJ+'roKDy)/2;

85

CWOI 2166CWO 12167CWO 12168CWO 12169CWO I 2698

CWO12183CWO1 2187CWO 12878CWO 12188

266 FOR1:=IK'STEPI'UNTILIM'Do' CW012008267 'BFGINNORMAAL(X.Y,1,1,A);OPTISTEP(Xr1,I.YCI,1J,A); CW012201268 uCX,JJ:=xCI,2:=(xCI.I]+ICI,0JDx)/2 CW0122O2269 vc1,JJ:=yCI,2]:=)YCI,1J+YCI,0J+Dy)/2 CWOI2O2A270 'END;PLOT( 1.XrIK,2J.YCIK,27); 'FORI :IKSTEPI UNTIL' IMDO CW012203

271 PLOT(2,XC!.2,YCI,2J);IFJ'GE15'THEN'FUXK; CW012038272 SELECT OUTPUT( 15);PLoT(1,XGK.VGK+O.O7);PL0T(2,xGK,VGK-O.02) CW012038273 PLOT) l.XGI<-O.02,VGK;PLOT(2,XGK+0o2,YGK); IF8LF8T4EN' CWOI2O3C274 G0TO4FGELOPEN;FORI=IK'STEp'1'UNT1L'IMDO' CW012204275 BG!NIcI,lJ:=xcI,23;YCI,1J:=VCI,2J.END'; CW012205

CWOI 269FC WO 12 170

CWO 12171C WO 12 172

C WO 12173

CWO 12174C WO 12 175

CWO 12176CWO 12176CWO 12768

CWO 12177C WO 12 178

CWOI 2179CWO 12180CW0 12181

CWO1 2189CWO 12190C WO 12191

CWO 12192CWO 12)93

CWO 12194CWO 12195CWO 12196CWO 12968CWO1296B

CWO 12197CWO 12198CWO 12199CWO 12200CWO 12200

DUUP GPOOTCINKFLROUTE 8 DAGE 4.5 UNEN

86

276 'FOp'I:=IKSTEPI'UNTLLIW-I'DO' CWOI2 206277 BEGIN'IFYCI,2J<YCI.l,2J'THEN'GOTO'A14'END';'GOT0'A15; C WO 12207278 4j4: oNTLussEN;ScHo0NM4AK;A15:R8ND;Ix1:IK;Iu1:=IM CWO 12208279 FQp'I:=IK'STEP'l.UNTIL'IM'rlo. CWO 12209280 BEGINX[I ,01:=XEI .2J;VCI ,0J:=yLI,2J.ND' ;VERFYN; CWO 12210

28) H:=K:i:IKt;F:G:O; CWOI 2211282 A22:'IFXCH,1]XCI.2AND'XC1,2JrX[K,0J.THEN' CWO 12212283 .BEGIN'H:=Hl;x:=K+l;l:=I+l; IF'j'LE'TMI'THEN''GOTO'422'ELSE''GOT0' CW0 I 22t 3

284 823EN0' IFXCH. 1JXCI.2)AN0XC1,2J=1rI(,0THEN' CWO 12214285 .BEG1N'H:=H+1;G:G+1;Nl)CG.JJ:=I;.GoTn.A22 CIJO 12215

286 ENO;'IF'XCH,IJ#X[I,2PAN00C1.2)#XrI,0J'THEN' CW012216287 .BEGIN.H:H+1;G:=G+1;NucG.,JJ:=I;F:=F+1;MucF,J:=K;K:=+1;.GoTo'A22 CW012217288 END;'IFXCH. lJ=3C1,2JAN0XC1,23#XrK,0J'THEN' CW012218289 BEGIN.F:=F+1;MucF,JJ:=K;K:=K+l;'GoTn.122 Cw012219290 'ENO'; 423:FORH:=G+sTEPI 'UNTI) '20'00NUCH,JJ:=100; CW012220

291 'FORH:=F+ISTEPIUNTI,'20'D0'MUCH.JJ:=100; CWO 12221292 IF'J<17THEN"G0Tfl'TYDFRcNT'ELSE'GOTOEINOE; CWO 12222293 BFGELopEN:Jo:=J.J_1;IBcJJ:=10; C WO 12 223

294 ccO: 1F18CJJ>IG[J]TI.4EN'IPCJJ:=IG[JJ:F:=20; C WO 12 224295 CCI: iF' JBCJ)'GE'MU[F,JJTWFN' CWO 12225

296 'BEGiN' IBCJ]:IBCJJ-F; GOTOCC2ENO LSE CW012226297 'BEGIN'F:=F-l; 'IF'F>o'THEN'GOT0'CC1'FLSE"GoTo'cc2END; CW012227298 CC2: Q:); CC3:'IFIBCJJ'GE'NUCG,JJ.THEN' CW012228299 BEGIN IRCJ):=IBCJJ+G; GOTO'CC4'ENO FLSE CW012229300 8EG1NG:=G-I;IF'G>o'T4EN'.GoTo'cc3.FLsE''GoTo'cc4; Cw012230

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REISOUUR> 7.5 OAGF.NWEISDUUP) 8.0 DAGENPEISDULJR 8 DAGEN 1.3 UNEN EINDINDEB BAANKRrMME 26PUNTEN BAANKf0rME0.851 0.0611.726 0.1862.574 0.2603.401 0.2814.161 0.21?4.744 0.1005.302 -0. 1265.987 -0.4636.802 -0.6517.631 -0.7788.496 -0.8319.436 -0.86010.365 -1.05111.217 -1.21812.059 -1.18112.912 -1.078

5.6. Considerations regarding data incertainties andpractical use.

As noted before, the reliability of weather predictions can be taken inverselyproportional to the time last ahead, the prediction is given for. This means,that the least time track, constructed on the basis of weather and sea estima-tions for a time, from O to 8 days ahead, can only be taken as a real extremalfor the next one or two days to come. A change in the predicted data wouldmean, that the track has to be revised. 1f a ship would thus follow a day-by-day corrected extremal, the entire track will consist of parts, that areextremal arcs from day to day, but the composition of these extremals arcswill in general not be an extremal.It seems rather precarious to make a numerical estimation of the error, thatis made this way. A possibility to collect statistical data on this matter wouldbe to simulate ship's crossings in the past, based on weather and sea predictionsthat were really made at that time, and then compare this with the real extremal,that should have been taken.Considering the efficiency it can be stated, that the past seven years of exper-imental and manual' ship routeing have shown a comparatively small averagetime gain of about three hours per trip. It can be noted however, that the costdecrease is more than would be expected from this bare figure. This can bemade plausible by noting, that considerable time gains are booked in caseswhere a great circle track would have lead through regions with very highwaves. Inthose regions there is a greater probability of damage to ship orcargo. These damage peaks are very unwanted because of the high costs theyimply, while the time loss on account of necessary repairs can disturb theentire sailing schedule.Another advantage of ship routeing has appeared to be the better predictabilityof the arrival time. For cargo ships, carrying big unit loads, this is liableto become even more important. The rather short holding time in ports forthese vessels as well as the expensive loading installations necessitate aminimizing of the time, during which these terminals stand still. Therefore

the schematizing of a container terminal's working procedure has to be donewith a minimal tolerance on account of arrival time incertainties.

87

LITERATURE

Chapter 2:L. Driencourt, J. Laborde,Traité des Projections des Cartes Géographiques,Hermann et cJe, Paris, 1932.J. van Roon,Leerboek der Zeevaartkunde III,C. de Boer, Hilversum, 1949.

Chapter 3:L. S. Pontryagin c. s.,The Mathematical Theory of Optimal Processes,Pergamon Press, 1964.I.M. Gelfand, S.V. Fomin,Calculus of Variations,Prentice Hall, Englewood Cliffs, N.J. , 1963.

H. Halkin,On the necessary conditions for optimal control of nonlinear systems,Journal d'Analyse Mathématique, Jerusalem, 1964.H.M. deJong,Theoretical Aspects of Aeronavigation and its Application in AviationMeteorology,Staatsdrukkerij- en Uitgeverijbedrijf, The Hague, 1956.

Chapter 4:Blair Kinsman,Wind waves, their generation and propagation on the ocean surface,Prentice Hall, 1965.U. Grenander, M. Rosenblatt,Statistical analysis of stationary time series,Almqvist & Wiksell, Stockholm, 1956.

W.J. Pierson, G. Neumann, R.W. James,Practical Methods for Observing and Forecasting Ocean Waves,U.S. Hydrographic Office Pub. No. 603, Washington, 1955.

88

P. Groen, R. Dorrestein,Zeegolven,Staatsdrukkerij- en lJitgeverijbedrijf, The Hague, 1958.J. Gerritsma,Het gedrag van een schip in zeegang,Vakantieleergang "Het moderne koopvaardijschip", voordr. VII, Deift,1965.

J. Gerritsma, J.J. van den Bosch, W. Beukelman,Propulsion in regular and irregular waves,Shipbuilding Laboratory, Techn. Univ. Deift, Pub. No. 17, 1961.G. van Leeuwen,The lateral damping and added mass of a horizontally oscillaring ship-model,Neth. Research Centre TNO for Shipbuilding and Navigation, Delft,1964.

J. Gerritsma, W. Beukelman,Analysis of the modified strip theory for the calculation of ship motionsand wave bending moments,Neth. Ship Research Centre TNO, Delft, 1967.R. Wahab,Amidships forces and moments on a 'series 60" model in waves fromvarious directions,Neth. Ship Research Centre TNO, Delft, 1967.J.H. Vugts,The hydrodynamic coefficients for swaying, heaving and rolling cylindersin a free surface,Neth. Ship Research Centre TNO, Delft, 1968.

Chapter 5:

F.D. Faulkner,Numerical methods for determining optimum ship routes,Journal of the Institute of Navigation, Vol. 10, No. 4, U.S.A., 1964.W.E. Bleick, F.D. Faulkner,Minimal-Time Ship Routing,Journal of Applied Meteorology, U.S.A. , April 1965.F.W. Nagle,Ship Routing by Numerical Means,U.S. Navy Weather Research Facility, Report 32-0361-042,Norfolk, Virg., 1961.G.J. Haltiner, W.E. Bleick, F.D. Faulkner,Use of Long-Range Weather Forecasts in Ship Routing,U.S. Navy Weather Resarch Facility, Norfolk, Virg., .1967.

89

90

SAMEN VATTING

In dit proefschrift wordt aangegeven, op welke wijze men voor een gegevenschip de oceaanroute met de kortste reisduur tussen twee plaatsen kan bere-kenen.De, met de tijd veranderende, toestand van de zee wordt overal in het te be-varen gebied G bekend verondersteld. evenals de - stationair genomen -zeestroom. Tevens wordt aangenomen, dat het gedrag van het schip in een

zee met gegeven golfspectrum geheel bekend is.Ter vereenvoudiging van afstandsberekeningen wordt G conform en met mini-male schaalveraridering afgebeeld in een plat viak. Deze afbeelding wordt in

hoofdstulc 2 behandeld.In hoofdstuk 3 wordt het gestelde probleem besproken als toepassing vanPontryagin' s theorie betreffende optimaal geregelde proces sen. Hierbij wor -den tijdfronten geintroduceerd en wordt nagegaan, hoe de gradientvector vanzo'n front zich gedraagt bu het volgen van een baankromme. Ook wordt aan-dacht besteed aan de mogelijke structuur van zo'n front en aan de modificaties,die moeten worden aangebracht als een deel van de baankromme langs de rand

van het te bevaren gebied valt.Hoofdstuk 4 bevat een summiere behandeling en samenvatting van diversemethoden van voorspelling van golfhoogten, alsmede een kort relaas van wat

tot nu toe bekend is betreffende het gedrag van een schip in zeegang.

In hoofdstuk 5 wordt een rekenmachineprogramma behandeld, waarmee derouteberekening kan worden uitgevoerd. Als voorbeeld wordt dit programma

toegepast voor berekening van de kortste zeeweg van de Westelijke uitgangvan het Kanaal naar de kust van Nieuw Engeland.In de slotparagraaf wordt aandacht geschonken aan de gevolgen van onzekerheden

in de gegevens en worden enkele praktische voordelen van meteorologischescheepsroutering besproken.

SUMMARY

In this thesis a method is described to evaluate the least time track for a givenship to cross an ocean.The sea conditions - changing with time - and the stationary sea currents areassumed to be known everywhere in the navigated region G. Also the ship'sresistance and response in a given wave pattern are supposed to be given.In order to avoid metric difficulties, G is conformally mapped onto a plane,keeping the scale alteration as small as possible. This mapping is treated inchapter 2.In the 3rd chapter the minimal time problem is discussed as an application ofPontryagin's theory on optimally controlled processes. The concept of a time-front is then introduced and the behaviour of the timefront gradient along atrajectory is analyzed. Some attention is given to possible structures of thesefronts, while modifications of the general theory in case part of a trajectorycoincides with the boundary are being discussed.The 4th chapter contains a curse treatment of wind wave prediction methodsas well as a brief report on what is known so far about a ship's behaviour in

sea waves.The fifth chapter gives a treatment of a computer program to evaluate theoptimar trajectory. This program serves to determine a ship's shortest trackfrom the Western entrance of the English Channel to the New England coast.As a conclusion, the consequences of data incertainties and the practical useof meteorological ship routeing are briefly discussed.

91

CURRICULUM VITAE

The author of this thesis was born in Rotterdam in 1923. After finishing high-school there, he took a two years steersman's apprentice course at theNautical School, followed by a year of practical training at the RotterdamLloyd Navigation Cy.

After the war he sailed four years on Dutch merchant vessels of the HalcyonLine as an apprentice and as a third and second mate. In 1949 he was compelledto stop his active sailing career because he was rejected medically on accountof short-sightedness. He became a teacher of navigation and mathematics atthe Nautical School at Scheveningen and obtained the legal qualification to

prosecute this profession in 1952.While teaching at the nautical schools of Scheveningen ('49 - '56) and Rotterdam('56 - '62), he studied mathematics and astronomy at the university of Leiden,where he did his doctoral examination in 1962.From 1962 until this date he has been working at the Technical University ofDeift as an instructor of mathematics.

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f(x

STE

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ING

EN

I

Teneinde het m

inimale tijdprobleem

voor een gestuurd niet autonoom systeem

onder te kunnen brengen in de klasse van optimaal gestuurde processen, is

het noodzakelijk, deze klasse te definieren op de wijze van H

alkin, d. w. z.

Gegeven:

nlm

x ,u,u

,t) voor i = l(l)n,

x'(t ) = z'

voor i = 1(1)n,

oo

x'(t1) = xj voor

i = i(i)n-1,

im

in

(u.......u )EU

(x,,x,t),

U is een gesloten en begrensde deelruiinte van

Rm

.

Gevraagd:

uk(t)zo te bepalen - k =

1(l)m - dat x(t1) m

inimaal w

ordt.

(L.S. Pontryagin c.s.

Mathem

atical Theory of O

ptimal Processes;

H. H

alkin, On the necessary conditions for optim

al control of nonlinear systems)

II

Indien men van een schip bij gegeven golfspectrum

en gemiddelde golfrichting

de kosten per afgelegde miji als functie van de koers en vaart kent, dan kan

men tussen tw

ee plaatsen de batmkrom

me berekenen, w

aarvoorde totale kosten

minim

aal zijn.

III

In Pontryagin's boek over optimaal geregelde processen w

ordt in hoofdstuk 1,paragraaf 7 verzuirnd te bew

ijzen, dat de eigenschap: "Elk stuk van een optim

alebaankrom

me is zelf optim

aal" ook geldt voor niet autonome system

en.

(Zie paragraaf 3.4 van dit proefsclsr;ft)

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