Wave feed forward as a means to improve dynamic positioning

V.-

t JUtI 1978

ARCHE

OTC 3057

WAVE FEED-FORWARD AS A MEANS TO

INPRÖVE DYNAMIC POSITIONING

by J.A. Pinkster, Netherlands Ship Model Basin

° Copyright 1978. Oftshore Technology Conference

This paper was presented at the 10th Annual OTC n Houston. Ten., May 8-11, 1978 The material is sublect to correction by the author. Permissionto copy is restricted so an abstract of riot mete than 300 wDrds

ABSTRACT

Existing dynamic positioning systems for ships depend mainly on a feed-back control using the posi-tion error as input, supplemented by a feed-forward control signal based on the instantaneous wind speed and direction. This paper explores the possibility of creating a feed-forward control signal which can reduce the horizontal motions of the vessel which are caused by the low frequency second order wave drifting forces in irregular waves. Results of model tests are presented which indicate that the relative wave height measured directly alongside and from the vessel can be used to create a wave-feed-forward

control signal which reduces significantly the low frequency wave induced surge, sway and yaw motions of a vessel moored in irregular waves.

INTRODUCTION

The last decade has seen a steady increase in the number of vessels which are stationed by means of dynamic positioning or stationing systems. Up to now, most dynamic positioning systems were used for pcsitioning drill ships in deep water. Nowadays, dynamic positioning is also being used for diving

support vessels, supply vessels, fire fighting ves-sels and naintainance and survey vesves-sels. This in-creasing interest for dynamic positioning systems stems from the need for a mea:ls of maintaining the vessel's position in the horizontal plane which is quick in implementation, does not interfere with systems already lying on the sea-floor and by means of which a reasonably accurate position may be main-tained for longer periods of tine.

In order to maintain a given average position in the horizontal plane by means of dynamic positio-ning, the mean and low frequency parts of the envi-ronmental forces due to waves, wind and current are counteracted by propulsion units. These propulsion units are either fixed tunnel mounted c.p. propeller, azimuthing right angle drive units with propellers externally fixed to the vessel, or in some cases, vertical axis propeller. The magnitude and direction of the thrust of such units are governed by a con-trol system which has as input the position error References and illustrations at end of paper.

lab.

y. Schepsbouwk'n:k

Technische Hogesch.

Deift

of the vessel relative to the required position, (feed-back part) and, in most cases the instantaneous vinci speed is used as a measure for the instantaneous value of the wind force to be counteracted by the propulsion units (wind-feed-forward part).

As has been shown in (1), including a wind-feed-forward signal in the control system can enhance the position keeping performance

considerably,espe-cially in gusty weather conditions.

This paper is concerned with the capability to generate what might be termed a wave-feed-forward signal which serves as a measure for the instanta-neous second, order low frequency horizontal wave drifting force. This force, has been shown to be capa-ble of generating large amplitude, low frequency horizontal notions of moored vessels in irregular

waves (2), (3) and (li), and is therefore also of

im-portance for vessels with DP. systems.

The method dealt with here, is a result of theoretical studies and model tests carried out during the last few years at the N.S.M.B. into the low frequency second order wave forces acting on stationary vessels in irregular waves.

SECOND ORDER WAVE FORCES ON FLOATING VESSELS

Assuming that the fluid motions may be de-scribed by a velocity potential , the hydrodynanic forces acting on a vessel are found by integration of the fluid pressures over the wetted part of the

hull:

=

_{- 5f p}

n dS

S

in which p is according to Bernoulli's

equation:

2

p = -pg x3

- - p

and is the total velocity potential including all orders:

(1) 2 (2)

rr=gr

+g

+

We are concerned here with the second order wave forces. The second order force referenced to a fixed

system of co-ordinate axes which coincides

with

the

body axes in the mean position of the body follows

fron:

(2)

₌

_-pg

f

(1)2 _{(i) (i)}

ndl+R

.m.x

g Wi -

ff(hIl)I2

(2)

- pt

so

(i)

__(i))}

_d.S

-p(x

The above expression is deduced in ref. (5)

A similar expression is deduced for the second

order moment: (2) =

-pgf

r(1)2 x ) dl R(1),I.7

-(i)

(1))}

_{(x) dS}

5

The expressions shown here differ from earlier expressions for the second order force and moment in so far, that they have to be evaluated on the surface of the body and give insight in the mechanism by which the waves and vessel interact to produce the second order force. We will see this upon closer inspection of the expression for the second order wave force and restrict ourselves to the surge force in irregular head waves.

The exnression for the surge force becomes:

F1(2)

_-pgf

r(1)2 n1dl +

x5.m.g3(1)

-

ff

{(i)2

-(i)

(i))}

_{n1dS 6}

The above expression contains three components; The first part is a line integral around the static water-line of the vessel of the square of the

relative wave height. The relative wave height is the wave height as measured from the vessel. This contri-bution arises from the fact that near the surface the pressure in the waves can be approximated by the static pressure. The static pressure increase at the mean water-line on the hull is -proportional to the relative wave height and the additional area on which the pressure acts is also proportional to the rela-tive wave height. This results in an inwardly directed force proportional to the square of the relative wave height which has the form of the first part of the above equation.

The second contribution to the second order wave drifting force is a consequence of the choice of the axes to which the second order force is referenced. The exoressions 14 and 5. apply to force components parallel to and about a fixed system of axes with X3 axis vertically upwards. The surge force given in expression C. is along a horizontal X1 axis and not along the moving longitudinal xi body axis. As a result of this, first order vertical hydro-dynamic and hydrostatic forces acting along the vertical body axis G-x3 will give a longitudinal second order force component according to:

F3 (see Figure 1) 7

where: F = first order vertical force

= first order pitch angle.

Taking into account that the vertical force F3(1) is (i)

F3

=in.

this contribution

m.x

the total fluid force which equals:

The third contribution to the second order wave drifting force stems from the integration of all second order fluid pressures over the mean submerged part of the hull. These pressure contributions follow from the second order part of the Bernoulli

equation 2.

If the velocity potential of the fluid mqtion is defined relative to a fixed system of co-ordinates and the body is carrying out small axplitude motions about the mean position, it can be shown that the second order pressure in a point on the submerged hull becomes:

(2) (2)

ij(1)j2

(_(1) (1))

in which: = first order approximation for the total velocity potential

= (i)

(i)

(i)

w d m

(i)

(i)

(i)

w ' d ' n =

first order

ve-locity potentials associated with respectively the undis-turbed incoming waves, the

diffraction potential and the

body motion potential.

(2)

second order approximation for the total velocity potential consisting of contributions from the incoming waves,

diffraction and body motions

(6).

THE RELATIVE MAGNITUDE OF COMPONENTS OF THE SECOND ORDER WAVE DRIFTING FORCE

8

Q

In the previous section the components of the second order wave force were introduced and discussed briefly. Im (5) equations . and 5. were evaluated

for the case of a rectangular barge moored in regular waves using a computer progran based on three

dimen-sional first order potential theory. Good agreement was found between results of measurements and

calcu-lations. From the results of calculations it was found that the mean second order force is dominated by the contribution from the relative wave height

.

This is in agreement with previously found experi-mental results on the measurement of the mean

longitudinal wave drifting forces on the bow, mid-body and stern sections of a barge moored in regular head waves (3).

In the same study it was found that the low frequency surge notions of the barge could be reasonably well predicted for different sea states, using only the

Ir

The estimate of the wave drifting forces and moment depend on the scuare of the relative height in

irregular waves. The relative wave height at a point along the water-line is an oscillating quantity with zero mean and may be represented by an expression of the following form:

ç (t) = E ç . sin(w.t + e.) 16.

r . ri i i

i=1

Squaring this expression results in

M M

ç 2(t) = E E ç .ç . sin(w.t + e.)

r

rirj

i i

sin(w.t

+ e.)

17.

which, by using a well knom trigonometrical relationship results in:

M M ç 2(t) = E E ç .ç . r 1=13=]. rir.J i J i 3 M M - E E ç .ç . cos(w.+w..)t+(c.+ei} i=lj=l

rirj

i J i .3

i8.

From this it is seen that the square of the relative wave height contains both low frequencies and high

frequencies corresponding to the difference and sum frequencies of the components of the relative wave heights in irregular waves.

Since thrust commands may not contain high frequencies from the point of view of wear and tear of the propulsion units etc., this part must be re-moved by filtering, leaving only the mean and low frequency thrust commands. Care must be taken to select a filter which, while removing the sum frequen-cies present in the signals, does not introduce sig-nificant phase lag in the low frequency components. This is a well known problem in conventional dynamic positioning systems where the input, i.e., the posi-tien error signal, contains wave frequency components and a mean and low frequency component. This input signal is filtered to eliminate the wave frequency components. With wave-feed-forward, however, the high frequencies are in the order of twice the freauencies of the waves. This means that the demands placed on the filter are less stringent than is the case with the normal feed-back system.

After filtering the high frequency component off the signals generated by equations 13., 111. and

15.,

the

thrust control or command signals are then determined such that the longitudinal thrust, lateral thrust and yawing moment are equal and opposite to the estimated instantaneous second order wave drifting forces and moment.

EXPERThNTAL VERIFICATION 0F THE APPLICABILITY OF WAVE-FEED-FORWARD

In order to demonstrate the afore-goin, model tests were carried out in irregular lonG crested waves with a

_1:82.5

scale model of a loaded 200 KDWT tanker in the wave and current laboratory of the Netherlands Ship Model Basin. This basin measures

60m X 140m X 1m.

The main particulars of the vessel are as

follows:

Length 310.0 rs

Breadth 147.2 n

the contribution of the relative wave height. This zuggested that, at least for ship type floating otructures, both the mean and the low frequency part of the second order wave drifting forces were dorsi-mated by the relative wave height contribution.

In this paper we will show that practical use can be made of this result for improving the posi-tioning accuracy of dynamically positioned vessels.

APPLICATION 0F THE SECOND ORDER WAVE DRIFTING FORCE EQUATION FOR GENERATION OF A WAVE-FEED-FORWARD THRUST CONTROL SIGNAL

On the basis of previous studies touched upon in the afore-going section it could be concluded that the horizontal second order wave drifting force is approximated by the following term:

(2)

Cf

ç2

ai 11.

and the yaw drifting moment by

ç2

x dl 12.

in which ç is the instantaneous value of the relative rwave heigth in a point along the water-line of the vessel. F(2) and M(2) are the resultant instantaneous second order wave drifting force and moment and CF and CM are constants. The problem to evaluate the instantaneous force and moment and then to generate a wave-feed-forward

sig-nal reduces to the problem to evaluate continuously equations 11. and 12., while the vessel is lying in waves. Ideally, this means that the relative wave height or wave elevation against the hull has to be known at all points around the water-line of the vessel. This would mean using an infinite number of wave probes fixed to the vessel. However, if the

amplitude of the relative wave height does not vary too rapidly with position around the water-line, the above equations can be evaluated by measuring the relative wave height ata discrete, finite, number

of points.

The integrals can then be evaluated by standard methods used in evaluating line integrals of which

the argument is known at a discrete number of pointS, for instance by means of Simpson's rule. In this study use was made of a simple summation of the form

shown below.

The procedure is as follows: The instantaneous wave elevation or relative wave height ç(t) is measured at n = 1....N points and squared to produce N signals corresponding to ç2(t). These signals are then multiplied by constants appropriate to the position of the wave probes around the water-line and summed to produce the longitudinal and lateral second order forces and the yawing moment.

N F '2'(t) C E ç 2(t) n - dl 13. X X rn in n N F

2(t)

C E ç 2(t) n . Al

i.

y rn 2n n N 11

2(t)

C E ç 2(t) (x n - x n ).Al M rn

2nln

in2n

n n i 15.

r,

L'raft

Displacement

Transverse metacentric height Natural period of roll

Instead of using propulsion units, the horizon-tal position and heading angle of the vessel vere governed by three servo units

which

could apply a horizontal longitudinal force and two horizontal transverse forces. This set-up is shown in Figure 2. The light rods connecting the vessel to the servo units incorporated axial force transducers. The servo units applied forces on the vessel in response to two control signals i.e. the feed-back control signal and the wave-feed-forward control signal.

The feed-back control signal was generated within each of the three servo units independently. The control was of the proportional-differential type and acted on the low frequency horizontal dis-placement of each connecting rod relative to the servo unit. To this end, the horizontal displacement signals of the horizontal rods which contained both wave frequencies and low frequencies were filtered to remove as much as possìble the wave frequencies.

The overal characteristics of the feed-back control system were such that high damping was achieved in the low frequeny region only. The hori-zontal notions of the vessel in still water after an initial displacement out of the equilibrium posi-tion is given in Figure 3 and demonstrates the high damping introduced by the feed-back system.

The wave-feed-forward system consisted of 8 ship-mounted wave probes positioned as shown in Figure 2,an analog computer to carry out the calcula-tion of longitudinal and lateral drifting forces and yawing moment given by equations 13.through 15., and three filters to filter off the sum frequency compo-nents in the control signals. The estimated instan-taneous values of the low frequency second order wave drifting forces and yawing moment were then translated into thrust control signals for the three servo units. These signals vere simply added to the feed-back control signals.

During the tests which were all carried out in irregular waves, the feed-back system parameters re-mained unchanged except for the test in head seas

(1800)

where the damping for surge was reduced in order to show more clearly the effect of wave-feed-forward. Since the wave-feed-forward signal was based on an estimate of the wave drifting force using the dominant part only, some readjustment of the W.F.F. parameters was necessary for the different sea states. These changes related to the gain factors c, c and

c in equations 13. through 15. This was done on a trial and error basis.

For each sea condition a test was carried out twice, once with and once without the W.F.F. control signals. During the teste the surge, sway and yaw motions as well as the total longitudinal and lateral forces and yawing moment applied by the servo units were measured and recorded on f.m.-tape and analysed.

Sea conditions

18.9 m

2LO,T00 m3

5.8 m

sec.

A review of the sea conditions and wave direc-tions tested is given in Table I. Power spectra of the irregular waves are given in Figure . The

dura-tion of each test corresponded to 35 minutes in reali-ty. The wave directions are defined in Figure 2.

RESLrLTS 0F TESTS IN IRREGULAR WAVES

The results of the tests are presented in the form of examples of time traces of the horizontal motions (Figure 5 and Figure

6)

and in the form of spectra of the low frequency components of forces, moment and horizontal motions (Figures 7 through

9).

From the results it is seen that except for the surge notions in bow quartering waves (1350), the low frequency parts of the horizontal notions are signif-icantly reduced when applying wave-feed-forward. It appears that a reduction in the motions need not necessarily result in a corresponding increase in the thrust to be applied to the vessel.

The mean wave drifting forces are not affected by the control system used as is demonstrated from the results given in Figures 7 through

_9.

In Figure ₉

it is seen that the low frequency component of the sway force F does not change significantly even

-through the sway motion itself is considerably smaller when using wave-feed-forward. In the same figure it is seen that the spectral density of the yaw moment is increased. In terms of lateral forces applied at the end of the vessel the absolute value of the moment is snail.

In Figure 8 and Figure 9 it is seen that the surge notion is hardly affected by wave-feed-forward. Upon closer inspection it appeared that, due to the level of the surge force, friction effects within the servo unit for the surge force (see Figure 2) tended to distort the wave-feed-forward control signal.

Figure ₇ shows results which are believed to be more representative of the effectiveness of the wave-feed-forward control signal.

RESULTS 0F TESTS IN IRREGULAR WAVES AND CURRENT

Tests were carried out with and without wave-feed-forward in bow quartering (1350) irregular waves with a significant wave height of

4.9

metres and a mean period of 10.2 seconds and a stern quartering

current (450) of about i knot.

The spectra of the low frequency parts of the motions and forces are shown in Figure 10. The results

are similar to the results shown in Figure

_9.

The

irregular waves are in both cases according to the wave spectrum given in Figure .

The results given in Figure 10 show that current does not affect the control signal. This is to be expected since the wave-feed-forward signal is deter-mined from wave elevation signals which do not change appreciably for the normal values of the current speeds encountered. From the results it is also seen that the low frequency forces and motions are not appreciably different from the results given in

Fig-ure

9,

which indicates that the influence of current on the low frequency wave drift force in this case,

is not great.

From the results of the tests in irregular waves with and without current it appears that is is pozzi-bic to reduce the low frequency part of the sway motion by about 10 and the low frequency yaw and

surge motion by about 50 through the use of wave-feed-forward. These figures will change if a different

array of wave probes is used.

CONCLUSIONS

On the basis of theoretical calculatjon and

model tests concerning the low frequency second order wave drifting force, it was concluded that the contri-bution due to the relative wave height played a dominant part. In this paper it has been demonstrated that this may be utilized to set up a wave-feed-for-ward control system for dynamically positioned ships which effectively reduces the low frequency horizon-tal motions induced by the second order wave drifting forces. The results obtained may be considered as being representative for ship shaped floating vessels which make use of dynamic positioning.

NOMENCLATURE

pressure

total wetted area of the hull mean wetted area of the hull normal vector to surface element dS positive direction outward from the hull

force vector

specific density of the fluid acceleration of gravity total velocity potential

de-scribing fluid motion a small parameter

first order approximation to ,

linear with wave height

second order approximation to ,

quadratic with wave height water-line

length element along the water-line

second order force

m mass of the vessel

1(1), (i) (i) _{first order surge, sway and heave}

motions

(1), 6d1) _{first order roll, pitch and yaw} motions

R(1) _{matrix containing first order}

angular motion, see ref. (5)

x first order linear motion vector

- (i) of a point on S0

first order linear acceleration vector of the centre of gravity

(i) of the vessel

C first order relative wave height

_r

measured along the water-line

x _{co-ordinates of points on the}

surface S0 or the water-line in a ship-bound G-x1 - x2 - X3

system of axes with positive X3 axis vertically upward

I _{Moment of inertia matrix of the}

vessel about axes through the centre of gravity G

first order acceleration vector of the angular motions

projection of length element dl along the x1-axis

n1 dS _{projection of surface element dS}

along the x-1-axis

(2) p Cr Al n n

.Al

in n Crn(t) Wj Cri w1 /3 T C

c ,c

x y M

F (2) F (2) M(2) estimates of the second order surg

X

and sway force and yawing moment frequency

random phase

amplitude of relative wave height component with frequency uj

spectral density of the undisturbec

wave s

significant wave height of the irregular waves

mean apparent period of the irregular waves

S (w), s (w),S(w) spectral density of surge, sway

X y

' and yaw notions

F , F , M surge force, sway force and

X

yaw moment in irregular waves

REFERENCES

i. Sjouke, J. and Lagers, G.: "Development of Dynamic Positioning for IHC Drill Ship",

Paper No. OTC 11498, Offshore Technology Conference, Houston,

_1971.

Remery, G.F.M. and Hermans, A.J.: "The Slow Drift oscillations of a moored object in random seas", Paper No. 1500, Offshore Technology Conference, Houston,

_1971.

Pinkster, J.A.: "Low frequency second order wave drifting forces on vessels moored at sea", 11th Symposium on Naval Hydrodynamics, London,

_1976.

14. Van Oortmerssen, G.: "The motions of a moored ship

in waves", Ph.d. Thesis, Delft University of Technology,

_1976.

Pinkster, J.A. and Van Oortmcrssen, G.: "Computa-tion of the first and second order wave forces on bodies oscillating in regular waves", Second Numerical Ship Hydrodynamics Symposium,

San Fransisco,

_1977.

Faltirisen, O.M. and Liken, A.: "Drift forces and slowly varying horizontal forces on a ship in waves", Timman Symposium on applied mathematics,

Deift,

_1978.

second order surge torce along the X1-axis

first order hydrodynamic and hydre. static force along the vertical X3-axis of the vessel

second order component of the fluid pressure

relative wave height as measured alongside the vessel. This in-cludes all orders

length of nth length element of the water-line

projection of the nth length element in the x1-direction tìme dependent relative wave height measured at the centre of the nth length elenent

adjustable gain factors

p s so n F p g r e(i) (2) WL dl 2)

TABLE I: Sea Conditions

Irregular waves

Current

Sign. height

Mean period

Direction

Speed

Direction

in m.

in sec.

in degr.

in kn.

in degr.

2.6 8.2 135 -

-4.9

10.2 135,180 -

X3

(i) F3 X3

H

r

G1

/(i)

(1) F3

.X5

Figure 1: Contribution of pitch angle and heave

force to the second order surge force

Xl

WAVE PROßES MEASURING

RELATIVE WAVE HEIGHT

CURRENT 45°

200 KDWT

6

BALL JOINT

u

G

FORCE

TR AN S DUCE R

SERVO

UNIT

x1 ,F

X7

WAVES 135°

Figure 2: Model test set-up for dynamic positioning tests

"i

WAVES 180°

SURGE

S WAY

(1600)

o

ioo

TIME (sec.)

6

4

0

nw1/ 2.6 m

8.2 sec.

4.9 m

10.2

sec.

y

/

I I I I ¡

/

¡

/

/

¡ i j ¡ I

i'

t-..'

\

\

\

_\

\

\

0

0.5

1.0

1.5

U

(rad.secH)

Ficlure 4: Spectra of irregular waves

E

3

(J)

-5--WITHOUT WAVE -FEED -FORWARD

WITH

WAVE-FEED -FORWARD

2

_SURGE

0

50

100

TIME

(sec.)

Figure 5: Surge motions in irregular head waves.

Significant height 4.9 m.

5

-5

WITHOUT WAVE -FEED -FORWARD

-WITH

WAVE -FEED -FORWARD

2

o

(m)

WAVE

0

50

100

TUÀE

(sec.)

Figure 6:

Sway and yaw motions in irregular

bow quartering waves. Siqnificant height 4.9 m.

SWAY

WITHOUT WAVE -FEED-FORWARD

WITH

WAVE-FEED-FORWARD

5 1X1

SURGE

SURGE FORCE

L) L) 1)

'I

C\J

F mean

- 28 ton

C\J E

,---.'(

o

r

3

>< (I)

3

t-i-X

w

O O

0

0.25

O

0.25

w (rad.sec.1)

Figure 7: Spectra of low frequency surge motion and force

Figure 8: Spectra of low frequency forces and motions in

irregular bow quartering waves.

Significant height 2.6 m.

WITHOUT WAVE - FEED - FORWARD

-WITH

WAVE-FEED-FORWARD

025

o

'-J

i''LL

0.25

o

1 x i

SURGE

SURGE FORCE

F

mean

-10 ton

/

-l2ton

I

-SWAY

/

SWAY FORCE

\

\

to n

-L

YAW

YAW MOMENT

/

\

I

Mmean 110 ton.m

I

\2onm\

lxiO

-7 'i) CL) E o

3

Cf)

o

C) Li cJ X C

o

3

w

1x104

L) C) Li (-'J o

3

LLX U)

o

o

WITHOUT WAVE-FEED-FORWARD

WITH

WAVE-FEED-FORWARD

0.25

o

w (radsec)

(Th ) R L). - ..)

1 xO5

o

3

(n

Figure 9: Spectra of low frequency forces and. motions in irregular

bow quartering waves. Significant height 4.9 m.

SURGE

SURGE FORCE

Fxrnean

38ton

-41 ton

SWAY

SWAY FORCE

Fymean l39ton

i40 ton

\N

YAW

YAW MOMENT

M

_mean

2360 tonm

f\

//

/

_\

/

/

I

/

\

/

\

/

\

/

r

\

\

5on.m

\

_\

10

X U)

o

lo

'J 0) ej E

o

5

o 'Ji ej (n

0

3

U)

o

ixiO

o u, ej C o

2

U)

o

O u,

(j

C o

3

U)

o

--3

X (J)

3

V)

Figure 10: Spectra of low frequency forces and motions in

irregular bow quartering waves and i kn. stern

quartering current. Significant height 4.9 m.

W!THOUT WAVE-FEED-FORWARD

WITH

WAVE-FEED-FORWARD

5

o

10

o

5

lx O

o

0

u a) t" C') C"

3

L2< (J)

1 xlO'

a X10 (-'J (J) 6 G) ç', E C o

3

SURGE

)_

SURGE FORCE

Fxrnean r-24 ton

//

S WAY SWAY FO R C E

9 to n

-YAW

YAW MOMENT

\

-Mmean-84Oton.m

-600ton.m

-o

0.25

o

025

w

_(rad.secT1)