Minimum Time Maneuvering of Ship with Wind Disturbances

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K1 S

338

Minimum Time Maneuvering of

K. Ohtsu

, K.Shoji

and T.Okazaki

Tokyo Unzverity of Mercantile Marne "IHI Heavy Industry

Tokyo University of Mercani1e Marine

Abstract. Ship's rrainimuxn tizne maneuvering prob a nornlinear two-points boundary value one in the conjugate gradient restoration method proposed by Key Words. Ship's Minimum Time Maneuvering gate Gradient Restoration Method.

1. Introduction

It is very important work for a shiprnaster to

build a shiphandling plan before approaching a

berth, leaving it or altering her head and so on.

One possible way to find the satisfactory plan be-forehand is to use a shiphandling simulater and

choose the best way from various trials. However,

there are individual differences among the ways chosen by those trials. Unlike this, a mathemat-ical method using some optimal theory is more reliable, if the mathematical model representing a ship's maneuvering motion is accurate. How-ever, it must be noticed that the model becomes

highly nonlinear especially in low speed and large

maneuvering motion such as berthing.

In order to take enough account of the nonlin-earity, the authors formulated the problems as

one of the nonlinear two-points boundary-value

ones in the calculus of variations. And the

for-mulated problem was solved, using the numerical

method developed by Miele and his associates over

the past several years called the conjugate gradi-ent restoration (OCR) method (Wu, 1980; Miel,

1970).

Unfortunately, though the solutions does not yield

on-line control laws, it can be considered that the information or diagrams gained from them

are useful for building the maneuvering plan

be-fore the actual shiphandling. Enumerated the problems which have been already solved by this

method, it is listed up as follows (Shoji, 1992;

Ohtsu, 1994):

The minimum time course alternating prob.

km,

The minirnimi time stopping problem,

Ship with Wind Disturbances

Deift University of Technology

Ship Hydromechanics Laboratory

Library

Mekelweg 2, 2628 CD Deift

The Nether'ands

Phone: +31 15 2786873 - Fax: +31 15 2781836

lemi with wind djstu_rbances are formulated as calculus of variations. And it is solved by the Miele et al.

Two Points-Boundary Value Problem,

Conju-The minimum time parallel deviating

prob-lem.

However, all in these have been solved under the conditions with no disturbances.

The problems treated in this paperare typical two

patterns of shiphandling methods with wind dis-turbances. The ship chosen as object of the study is T.S.Shioji Maru(425 gross tonnages), which is installed with a bow and stern thrusters besides a rudder and a controllable pitch propeller.

2. Minimum Time Maneuvering Problems and its Formulation

2.1. Minimum Tim e Meneur e ring Problems

Let the minimum time maneuvering problems

treated here be defined as follows:

AssumecI that a ship travels at a certain speed to-ward the giren direction at the initial approaching

point. A shipmaster must make her shift io the

destination point. Her bearing end speed at the destination point are free or given. How should he steer her and use her engine and thrusters in

order to accomplish the work in rn.inirnum time?.

2.2. The Formulation as a Two Point Boundarp Point Problem

The problem stated in the above might be

forinu-lated as two-points boundary value problem in the calculus of variations as follows:

Let z be defined as the n(= 4) dimensional state vector, whose elements are composed of forward

(2)

speed u, sideways speed y and rate of turn r and u

as the rn(= 2or4) dimensional control one, whose elements, of rudder angle S, CPP blade angle 8,

powers of bow and stern thrusters ofT6 and T1, re-spectively. Furthermore, it is assumed that the

in-dependent variable is represented by 9 but a time

normalization is used for the simplicity of

compu-tations. Thus, the actual time 8 is replaced by

the normalized time t = 9/i-, which is defined in

such a way that the initial time is t0 and the

flr.tal time is t=1. Since i- is free in the minimum time maneuvering problem, it is regarded as the parameter to be optimized.

With the above notation, this type of minimum time maneuvering problem can be formulated as

follows (Shoi, 1992):

Minimize the functional

1

jf(zur,t)

(1)

with respect to the state z, the control u and the î- which satisfy 1) the differential constraints,

0<t<1

(2) where denotes a non linear hydrodynasuic model

for representing ship's motions and 2) the

bound-ary conditions;

The initial ship's state,

z(o) = given. (3)

The final state of the ship is specified by the

function

[(z,r)J1 = o.

(4)

where the function P is a q dimensional

vec-tor (0 < q < n)

In order to increase reality, non differential con-straints:

S(u,-r,t)=o,

O<t<1.

(5)

may be added, by whose constraints it is

possi-ble to set the maximum liniits of rudder angle, propeller blade angle, powers of bow and stern

thrusters to be applied.

Remarks: In many cases, the constraints to con-trol variables are given by inequality equations. For example, since rudder angle of S must be re-stricted to the hard over angle ofSmr

Sma < S < (6)

must be held, in order to obtain the equality given

by the form (5), introduced a new independent

variable of S, (6) is transformed to

S = sin(Si)

Mathematical Maneuvering Model 3.1. Baue Equation of Iviotion

Table 1 shows T.S.Shioji Maru's principal dimen-sions. Length Breadth Tonnage Propeller Bow Thruster Stern Thruster 49.93 in 10.00 in 425.0 GT CPP 2.4 tons 1.8 tons

Table 1 Principal Dimensions of T.S.Shioji Mar11

This ship is installed with a bow and stem

thrusters for low speed maneuvering besides single

rudder and single propeller. The propeller

revolu-tion are regulated by the change of propeller pitch

angle.

Fìg.1 depicts the ship-fixed coordinate system. By

reference to it, the mathematical model is written

by

(7m +

r) -

= XH+XR+XP+XW

(m± + TfltL7'

= YH±YR+YT+YW

(I

+ J);

NH + NR + Nr ±

where m and I are mass and turning moment of

inertia, m, m, and J, added masses along n and

y axes and added moment of inertia around z axis

and u, y and r, ship's speed along x and y axes and the rate of turit around z respectively. The

subscripts H, P, R, T and W denote the

hydro-dynamic forces induced by hull, propeller, rudder ,thrusters and wind disturbances, respectively.

X

Fig.1 Coordinate System

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340

3.2. Hydrodyn.amic Force:

The concrete hydrodynamic forces are written by

polynomial representations as follows:

= _CtuIuI+X,Vv+Xvr

+ X,Vr+X,,v+X,r2

= Y,Vv±Y,,vIv+Y,Vr

± Yi!r + Y,?vIrJ

NH = N,Vv +N,,vvI +N,Vr

+ N,.?rjrI+Y,yvIrI

where X,, for example, means 8X/t9v etc and

V,ship's ordinary speed.

The thrust force of the propeller at pitch angles

as follows:

(i - t)pn2D(Co + C39p ± C2Jp C30PJP + ClOy,

c54 + C66'J ± C78pJ

8Up + L9.JpP' 73 (7) where and wp denote thrust deduction fraction

and wake fraction and n, Dp and J, propeller

revolutions, its diameter and advance coefficient. C1 C9, empirical coefficients, respectively. The rudder forces at a rudder angle S are repre-sented by

where tR,aH and Zif denote empirical coefficients

due to hull-propeller interactions and ZR, rudder position and the rudder normal force FN is

sim-pliiled by

FN = PARfa(U sinS ± 7R(V + 1RT)UR cosS)

where AR and fa denote rudder projected area and its normal force coefficient and 7R,1R,

elflpir-ical coefficients representing the fairing effects of

the stream behind the hull. Effective rudder

in-flow t1R

= (

- k)(l - wp)u + k(0.7Dpn)tan8p

where e denotes ratio of axial velocity at propeller

position to rudder position. k denotes propeller acceleration fraction simplified by

= kro(

±

using a sigrnoid function for representing the

dis-continuity of Ier, where k, means k at e > 0.

The last two variables UR and kr were simplified, in order to make easy partial differentiation by the

state or control variables. The actuator's dynam-ics are also considered in the model. For the

de-tails on the model, see the reference (Shoji, 1992).

3.3. Wind Di,tvrba.nce,

Wind loads on the ship's super structure can be

represented as follows:

= PaAojUìC1 (8)

Yw =

1paAosUirCy (9)

Nw =

paAo,LppUvCN (lo)

where C1,Cy and CN denote the experimental

coefficient of wind force and moment acting on the ship's super structure, p is the density of air and Aof and Ao: are lateral and transverse pro-jected area of super structure,respectively. Since

the coefficients C1,Cy and CN are function of

the relative wind direction to the ship, they can

be approximated by (Fossen, 1994):

J,:

4. Optimization Technique 4.1. Optimal Condition.,

The above problem can be solved by the theory of

calculus of variations. It is referred as one of the Boiza type of problem and it can be recasted as that of minimizing the augmented functional

i (f + Ar'(± - ) + pTs)dt

fu

Ç

±SaTp

= o,0 t 1

À +rTA

= 0,0

t <1

C1 = cos a (11) Cy

= Ôy sin a

(12)

CN = CN sin 2a.

(13) ±

(T)

=

(f - A

+ pS

-± (ATz + pTp) - (ATz) (14)

subject to equation (2) - (5).

where A, p are

variable Lagrange multipliers and _j. is a constant Lagrange multiplier. The second equation arises

after the customary integration by parts is

per-formed. The functions z(t),u(t),i- and the multi-pliers A(t),p(t), must satisfy equations (2) - (5) and the following optimality conditions (Euler's

equations) =

-(l-tR)FNsinS

= (1+aH)FNcosS

(4)

L

(f' -

')dt + (Jz)i = o

4.2. SeqiLential Gmdien. Re:oretion Method Since the differential systems (2)-(5) has nonlin-ear properties, it is difficult to get the analytical solution. Thus, approximate and iterative

numer-ical methods are employed to find it. The nu-merical method used in this paper, is the conju-gate gradient-restoration one developed by Miele et al.(Wu, 1980; Miel, 1970). In this method, the constraint error P

pi pi

P = jN( -

0 )dt

+ j

0 N(S) ± N(b), (18)

and the error in the optimality conditions Q

L' N(A

-

+

+ L' N(f

T) ₊

+

N[j(f -

_{)d +}

± N(A+l,&TiL),

(19)

are defined, where N() denotes the norm squared of a vector z', i.e.

N(z') = ₍₂₀₎

For the exact optimal solution,

P=0,

Q=0

(21)

But as approximation to the optimal solution, the numerical method aims at

(22)

where e and e2 are small, prescribed number.

More details about the CGS technique are de-scribed in, for examples, Wu and Miele (Wu, 1980)

and Miele and Iyer (Miel, 1970)

5. Computational Time

The calculations described in the below will be implemented under the convergent conditions of

e, < 0.1_10 and e1 < 0.1. In this case, if it is

desired the solution under the above high accurate

conditions, the computational time takes about 2

or 3 hours when using a work station with a speed

of 80 mips. However LI a rough but enough prac-tical solution is permitted, about 3 or 4 minutes until gained.

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6.1. Wimd Effeci on the Ship

Fig.2 shows the effects of wind pressures on the

ship to the fore-and-aft and athwart directions and its turning moment in Shioji Meru. The

small circles in each figure denote the empirical

results and the solid lines, the first

approxima-tions of them in Eq(11)-Eq.(13). As understood from these, the bow f11c off the wind, when the

ship receives the forward winds of the beam, while

it turns away from the wind when it receives the aft winds of the beam. These characteristics on the ship's behaviour are important effects that a ship master must take enough account of in the shiphandling with wind disturbances.

', 0.25 (i o o 0. 50 0. 00 0. 08 0.12 0.06 0.00 -0. 12

6. Minimum Time Deviations with Wind

Disturbances

0. 0 45.0

Rcl3tive .,r,d

135:0

Fig.2 Wind Effects,C1,Cy and CN

6.2. The Problemi end The Optimal Solionj

The first examples are minimum time deviation problems with wind disturbances. The ship must deviate 500 m away from the initial approaching line in minimum maneuvering time with the

rud-der. The winds blow from the bow quarter and the

stern one at relative wind velocities of 20 rn/sec

O o o

coco

O 180. IDeg 341 o o o o o 0.80 _o _o o o o ' 0.40 o o 900 1350 R&ative nd 0.0 450

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342

and 3Orn/sec, respectively. The initial

approach-ing ship's speed is 12 laiots and the sideways speed

must vanish and the head must direct again to the original course after ending the deviation. Fig.3 shows the optimal paths and ship's heads in each

case, solved by the CGR method. Fig.4 shows

the corresponding time histories of the rudder an-gles and the heading anan-gles. It is noticed that the time histories of the heading angles have almost same patterns, whereas those of the rudder angles different ones in each case. Thus, it is

gener-ally concluded to be sufficient that a ship master pay his attention to only maintain the ship's head along the minimum time solution with no wind to accomplish the minimum time deviation

maneu-vering. o th - o o Heading Angle

Absolute wind velocity is zero.

U=20m/s, a=45deg.

® U..=30rn/s, a=45deg.

® U. = 20 rn/s. a=-l35deg.

l2kt U30flhIS, a=-I35deg.

Fìg.3 The Calculated Paths of Minimum Deviatings with Winds

Fig.4 The Calculated Time Histories of Rudder angles and Heading Angles

6.3. The Ac1ucJ Sea. Te,

Li order to evaluate the reliability on the formu-lations and its calcuformu-lations, the following actual deviating tests were carried out at the sea using Shoji Mcru. In these trials, the distances between

the first approaching course and the final onewere

set up to distances of 200m and 300m. The

con-trol law following up the calculated steering order was constituted by the following simple forms:

= Sg ± K1(0 -

) + K2(ro - r) (23)

where the 8o, 'l'o and ro denote the optimal solu-tions of the steering order, the heading angle and the rate of turn.

Fig.5 shows the ship's actual paths measured

through a Doppler type of speed log (solid Line)

and the calculated ones (dotted line). The last ones were calculated under the conditions with no

wind. g 300m Deviation Wind 12.4m/a iI Wind Velocity - Absolute _{- +} + + 41 200m 300m E

t

ges

t

o es a-2Ò 0 Sò sb 100 Tiene (a) Time Histories - Measured Value Optimal Solution

rig.6 The Time Histories of Heading Angies,Rates

of Turn,Rudder Angles,Speeds and CPP Angles Fig.5 The Paths of Actual Automatic Deviation

Tests

o loo 200 300 400 500 Y(m)

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It is of interest that despite the intermediate paths are slightly different between the calculations and the actual tests, there are not large differences

at the final positions in each case. Fig.6 shows

the comparisons between the measured and cal-culated heading angles, rates of turn, the rudder angles and the forward speeds. The last two

fig-arcs clearly demonstrate that the actual rudder

angles and OFF blade angles faithfully follow up

to the signals of the steering and pitch angle of

the OPP ordered from the computer and thus the

actual heading angles and forward speed also

co-incide with the intended ones, respectively.

7. Minimum Time Inward Stopping Problems with Wind Disturbances

7.1. The Sethmg op of the Problemi

As the second examples of the minimum time

ma-neuvering with wind disturbances, the minimum

time stopping problems arc considered. The prob-lemns treated here axe set up as follows:

At the initial time, the ship is traveling at the

position locating at the position of 12 times of

her length (600m) far from the final stopping point, whose bearing from her head is a de-grees to the starboard side(Fig.7). Her speed is normal sailing speed, namely 12 knots.

The winds blow on or blow off the final

stop-ping point with a relative wind velocity of 5

or 10 rn/sec.

The ship's head at the final stopping point must direct again to the original coarse, in the attitude of the so called 'inward stopping'.

__4Q or, 5,V30c

bIo'.,a or, I8,,'5,c

btoag arr 58'c blOWin5 arr l8m'ec

Fig.7 Minimum Stopping Problem with Winds

7.2. Minimum Time Inwerd 5iop ping with Wind

Blowing ou

Fig.8 shows the maneuvering time until stopping at the given point in the minimum inward

stop-ping with the wind blowing on. It is noticed that the time differences between the minimum stop-pings with wind velocity 5 rn/sec and 10 m/sec

are longer than those between the minimum

stop-pings with no wind and 5 rn/sec. In order to

im-derstand the reason for them, let the paths and

other related maneuvering elements be examined in detail. [Seal 220. D 210.0 150.0 15.0 70.0 45.0 60.0 75.0

Lni1ial approaching angle

Fig.8 The Minimum Inward Stopping Times

with the Wind Blowing on

NON 5.0CM/SS IO. OC M/S) 80. [Deg So .0' .0 .0 2.0 3.0 .0 5.0 6.o 7.0 8.0 .0 3.0 CL) L=49.SM

Fig.9 Paths of The Minimum Time Inward

S toppings with Wind Blowing On

Fig.9 shows the ship's paths. From this figure, it is found that the paths when the wind velocity is

at 10 m/sec, swell up toward the outside

direc-tions. Fig.10 and Fig.11 show the time histories of the relative wind directions to her head and the yaw moments when the initial bearing to the final point is 15 degrees. It is clearly recognized that when the wind velocity is 10 in/sec, the relative wind to her head changes to the aftward from the abeam direction after 100 sec due to her swelling up path and as the result, the yaw moment in the last approaching stage to the final point changes

343 200.0 190.3 a 180.0 170.3 150.0 ---n: 1O

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344 200. 0 o. o [Kg-20000. 10000.0 o. o -10000.0 0.0 100.0

Fig.11 The Yaw Moments

[Degj RUDDER ANCLE

40.0

1000. 0

0. 0

-1000.0

from the starboard moment to the port one- It is clear that the minimum inward stopping maneu-vers with wind speed 10 rn/sec takes account of

this wind effect.

EDe g j RELATIVE WIND DIRECTION

_--.lOa/sec

icc. o

-0.0 100.0

Fig.lO The Relative Wind Directions

BO! THRUSTER 200. 0 [Sec] 200. 0 [Sec] [Sec]

Fig.12 The Time Histories of Rudder Angles

Fig.13 The Time Histories of Bow Thruster

Forces 1000. 0 0. 0 -1000.0 NON --..-.: l0/sec o. o 100.0 30.0 40. 0 50 0 75.0

Initial approaching angle

200. 0 [Sec]

Fig.14 The Time Histories of Stern Thruster

Forces

Fig.12, Fig.13 and Fig.14 are the time histories of the rudder angles, bow and stern thruster forces, respectively. It is noticed that the ship is turned in full power at the last stage just before the

ter-minal, utilizing the starboard steering, and the starboard bow thrusting and the port stern one besides the above wind effect.

7.3. Minimicm Time Inwerd Stoppzng with Wind

Blowing off

As the last examples, the minimum time inward

stopping problems with wind blowing off the final

point.

Fig.l5 shows the maneuvering time until stopping

at the final point in mi_ni_mum time. Also Fig.16

shows its paths. It is noticed that differing from the last problem , ail of the inward stopping ma-neuverings with wind blowing off the final point, almost coincide with the paths with no wind

dis-turbances. The reason of these coincidences is be-cause the yawing moment to the port side are

nat-urally gained at the last approaching stage due to the wind effects desibed in the former section.

90.

[De

Fig.l5 The Minimum Inward Stopping Times with the Wind Blowing off

[Kg] STERN THRUSTER

0. 0 100. 0 200. 0

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CL)

Fig.16 Paths of The Minimum Time Inward

S toppings with Wind Blowing off

8. Conclusions

This paper has given the minimum time maneu-vering methods in two kinds of typical

shiphan-dung problems under the conditions with wind

disturbances for a small training ship with a

rud-der, a controllable pitch propeller, a bow and stern

thrusters. In the minimum time deviation prob-lem, it was concluded to be sufficient for a ship master to pay his attention to only maintain the ship's along the minimum solution with no wind, no matter how the winds blow.

In this problem, furthermore, the actual sea triais were implemented and as the interesting result, it was Iciown that despite the intermediate paths

arc slightly different between the optimal solutions and the actual tests, there are not large differences

at the final point in each case.

In the problems of the minimum time inward

stop-ping at the given position with the winds blowing on and blowing off, it was confirmed that from the view point of shiphandling practice, the op ti-mal solutions are reasonable maneuvering

meth-ods that utilize at the maximum the effects of winds, especially in the minimum inward stopping

problems with the winds blowing on.

9. REFERENCES

K.Shoji and K.Ohtsu,(1992). Automatic Berthing

Study by Optimal Control Theory. Proceedings

of CAMS'92,Genova.

K.Ohtsu and KShoji,(1994). Minimum Time feering of Ships. Proc. of MCM C'94. T.I.Fossen,(1994). Guidance and Control of Ocean

Vehicles. John Wiley and Sons.

A.K.Wu.and À.Miele (1980). Sequential Con.u-gate Gradient-Restoration Algorithm for op-tima.l Control Problems with Non-Differential Constraint, and General Boundary Condition,,

Fart i, Optimal Control Applications and

Method. Vol.1.

A.Miele and R.R.Iyer,(1970). General Technique for Solving Nonlinear Two-Points Boundary Problem, 'via the Method of Particular

Solu-tion,. Joui-nul of Optimization Theory and Ap-plications, Vol.5, No.5.