Control of yaw and roll by a rudder/fin-stabilization system

z m

CONTROL OF YAW AND ROLI1SM*,ABILIZATION SYSTEM

by Claes G. Käliström

Swedish Maritime Research Centre, SSPA ABSTRACT

I Different discrete time regulators for control of a ship's yaw and roll motions are designed using linear quadratic control theory. A simulation study to investigate the performances of the different regulators is presented. _{A ro-ro shïp equipped with two active} sta-bilizing fins sailing in irregular beam seas is used as test ship in the simulation study.

1. INTRODUCTION

Traditionally course-keeping and roll stabilization of ships are treated as two separate control problems. _{Since both yaw and roll} mo-tions are induced by deflecmo-tions of rudder as well as stabilizing fins, it is, however, important to consider the interactions when designing the regulators. This can be done in different ways. _{Regulators based} on dc-coupling of the yaw-roll motions were discussed in references

(1) and (2). Linear quadratic control theory was employed in (3) to design continuous time regulators.

Efforts to design regulators for reducing the roll by the rudder only have also been made. See references (4) , (5) , (6) , (7) and (8). The key problem using this approach is, of course, that two outputs, yaw and roll, have to be controlled by one input. _{An aütoilòt for}

course-keeping, where not only the yaw motion was considered but also the roll motion, was discussed in (9).

In this paper different discrete time regulators for _{a ro-ro ship} are designed based on linear quadratic control theory. This theory is described in several text books, for example reference _{(10). A loss}

function approximately describing the rate of fuel consumption is used as a criterion for the design of an ordinary course-keeping autopilot. This loss function is based

_{on a.}

linear combination of squared heading and rudder angles. _{The autopilot is then modified by}

sing a criterion, where not only heading and rudder angles are pena-lized but also roll motions. _{A regulator fo reduction of roll by an}

ctive fin-stabilization system is also designed without considering he yaw-roll interactions. _{Finally, an optima]. multivariable} con-roller for a rud1er/fin-stahjljzatjon system is described, where fuel onsumption as well as roll motions are minimized. The

performances

f the different regulators _{are compared in a simulation study. A} re-atively complex non-linear mathematical model of a ro-ro ship sail-ng in irregular beam seas is used in the simulations.

Ï

The paper is organized as follows. _{The test ship is described in} action 2 and the mathematical simulation model is summarized

_in

Sec-ion 3. _{The performance criteria and the design of the different} re-ulators are discussed Section 4. _{The simulation results are given in}

:Se&tion 5 and the major conclusions to be drawn can be found in Sec-!tion 6. An appendix with the detailed simulation model is also given.

2. TEST SHIP

A single screw/single rudder ro-ro ship was selected for the si-mulation study. The bow and stern draught is 11.0 in and the displace-!mentis 52 010 in3. The metacentric height H iz 0.45 in. The ship

particulars are surnma.rized in Table 1.

Table 1. Particulars of Test Ship

Propulsion and Rudder

A diesel engine of 13 000 HP delivers the propulsion power. The ship is fitted with a four-blade propeller _{with a diameter of 6.3 in.} The rudder is of "Mariner" type with _{a total area of 35.8 in2. Th}

ropeller and rudder data are given in.Table 1. ctive Fins

The test ship has no roll stabilization _{system. Hoiever, in the} irnulation model the ship is fitted with _{two active fins. See Fig. 1.} ach fin has an area of 25.0 in2 and the aspect ratio is 1.0. The ac-ive fins are placed 50 in aft of L/2.

HULL DATA Length between perpendiculars (L) Beam Drauçht, bow Draught, stern Displacement (Sl) 173.2 in 32.3 in

11.0

in 11.0 in 52 010 rn3 0.45 in Metacentric height () PROPELLER DATA Propeller diameter (D) Pitch ratio (P/D) Number of blades 6.3 in 0.715 4 RUDDER DATA Total rudder area

Aspect ratio

35.8 in2

Ptnci! ¡n ta:

Figure 1. _{Active Fins Designed for the Test Ship.} 3. MATHEMATICAL MODEL

The simulation model, which describes the surge, sway, yaw and roll motions of a ship sailing in waves, is briefly outlined in this Section. _{The complete simulation model is given in the appendix,} where parameter values for the test ship are also summarized. _The numerical values have been obtained from captive scale model tests and theoretical estimations. _{Another alternative of determining the} parameters for a specific ship is the application of system identifi-cation to full-scale experiments or free-sailing scale model _tests. See references (11), (12), (13) , (14) and (15).

The structure of the basic mathematical model was given by Norr-bin (16) and further developed _{by Norrhin ('17)}

. The mathematical

rodel, excluding

the roll

_{equation, has been used in many simulation} tudies, for example in (14) _{and (18).}

quations of Motion

The equations of motion are conveniently expressed using _a co-rdinate system fixed to the ship. _{The origin is placed in the free} iater surface plane half-way between the perpendiculars and in the

entre-Une of the ship.

The variables used to describe the _{surge, sway, yaw and roll} mo--ions are explained in _{Fig. 2.}

The projections of the total ship peed V on the x- and y-axes arc the surge velocity u and the sway

veiocity y.

The yaw rate is denoted r and the heading and rudder

angles are denoted 4 and S.

The roll motion is characterized by the

roll angle

and the roll rate p. The angle a of the active fins is

defined positive when a negative roll moment is induced.

Figure 2.

Variables Used to Describe the Surge,

Sway, Yaw and Roll Motions of a Ship.

The equations of motion are obtained from Newtonts laws

express-ing conservation of linear and angular momentum:

m(û - vr

_{- xGr}

.+ zGLp) 1.

pn

X + T(1-t )

+ X + X

i-x

F w

p

R

m(r+ur+xG_zGf)

= Y+Y +Y i-Y

_{R F w}

I

_ZZ

?-i ,+mx (+ur)

_ZX

_{= N+N +N +N}

G R R w

I

p - I

i

- mz (r + ur) + mgsin

_{K + KR + K}

+ K

I

xx

zx

G F w

(3.1)

where Xand Y are the components of the hydrodynarnic forces

on the

x-axis and y-axis, N and K are the z-cornponent and

_{x-cornponent of the}

hydrodynarnic moments, m is the mass of the ship,

_{Izz and I}

_are

mo-ments of inertia,

is a product of inertia, and

xG and ZG are x

and z co-ordinates of the centre of gravity.

_{Note that the y}

co-ordinate of the centre of gravity is equal

to zero, because the

=

(-

5 ₊

&

=

(-ct+ct)/TF

ordinate origin has been set in the centre-line of the ship.

The

pro-peller thrust is denoted T and the thrust deduction factor tp.

_XR,

XF and X

are forces in the x-direction from rudder, active fins and

waves, respectively.

The added forces and moments of the other

equa-tions of (3.1) are defined analogously.

;Propeller Thrust

The propeller thrust is computed as

where p is the water density, D is the propeller diameter and n is the

propeller rate of revolution.

_{The propeller is characterized by}

_a

polynomial expression for KT, which is given in the

appendix.

A

con-stant propellr rate of revolution n equal

_{to 1.93 1/s (116 rpm) is}

assumed in the simulations.

_{This propeller rate corresponds to}

_a

nominal ship speed of 7.7 rn/s (15 knots).

Rudder and Active Fin Forces

The lift forces from the rudder and

_{-one of the active fins are}

obtained as:

=

_{(k1u2 + k2T) (1}

+ s2) 5

(3.3)

=

k3u2(1 + s2ct2)a

Note that

F =

2-sin y-F, where y is the angle between

_{the fin axis}

and the free water surface plane (cf. Fig.

1).

The details of (3.3)

and the other forces and moments from rudder

_{and active fins are given}

in the appendix.

First-order models describing the rudder

_{and active fin dynamics}

are included in the simulation model:

where

and

_{c are the commanded angles and TR and TF are the time}

constants.

_{Both rudder and fin rates of (3.4) are limited as well as}

the maximum rudder and fin angles.

iave Forces

The forces and moments from

_{beam seas acting on the ship}

_are

pproxirrì'ated by:

(3.4)

= pDkn2

_(3.2)

'Yw = In . 5(t)

K = mL a S(t)

w

whére (t) is the z co-ordinate of the sea level at origin and 5(t) is the wave slope.

A Pierson-Moskowitz wave spectrum given by (see ref. (19))

A

_eBh'

-is simulated by feeding a white no-ise signal into a filter. A signi-ficant wave height of 7 m and a mean wave period of 9.4 s are chosen in the simulations. _{This corresponds to A = 0.78 and B = 0.063 in}

(3.6). A rational spectrum

b2

22

i3) =

+ (a12-2a2)w + (a22_2a1a3)uj2 + a32

with ai = 0.5, a = 0.33, a3 = 0.07 and b2 = 0.415 is used as an ap-proximation of (3.6) for the chosen sea state. _{The spectra (3.6) and}

(3.7) are illustrated in Fig. 3.

lo (m2s) 8 6 4 2 Pc;ci! in

!!i1;t'r

(3.6 (3.7) 0 0.4 0.8 1.2 1.6

co(rad/s)

igure 3. _{The Rational Spectrum (3.71}

(continuous line) and the

Pierson-Iosko.jt Spectrum (3.6) (dashed line) for a Signi-ficant Wave Height of _{7 in and a Mean Wave Period of 9.4 s.}

;Gs(s) =

where c = -The two tions by the

rA stochastic process

output from the filter

G(s)

=

when the input the wave slope

= --w)

IS

given analogously by s3 + a1s2 + a2s + a3 c b2 S s3 + a1s2 + a2s + _a3

0.065 for the chosen sea stafe.

filters (3.8) and (3.9) are represented in the simula-differential equation system:

b2s

I-with spectral density (3.7)

is white noise (see ref. (20)). An approximation spec trum ?erc n :ç r,..r is obtained as (3.8) of (3.10)

where e is a discrete time white noise signal with zero mean and standard deviation 2.22. _{A time step of 0.5 s was used.} Realiza-tions of the wave level _{and the wave slope S are shown in Fig. 4.} These realizations are used in all the simulations.

(3.9) *1 X 2 X3 -j

s

-a1 i O -a2 O i -a3 0 0

[1

o o [_aic: c o

I

r

xl X2 X3 X

_i

X2 X₃ + O b2 O e

to.

-Ia.

-0.4

o.

0.4

t!AVE LEVEL (ti) VERSUS TIME Cs)

WAVE SLOPE C-) VERSUS TillE CS)

0.

Figure 4. Realizations of wave level and wave slope S. DESIGN OF REGULATORS

The regulator design is based on linear quadratic control theory (see ref. (10)). _{The design is carried out efficiently by use of the} nteractive computer program SYNPAC, ref. (21).

A preliminary investigation showed that a sampling interval of 1-5 s was appropriate. _{It was decided to use the interval 2 s in all}

he simulations. inearjzed Models

The following linearized state space model is th basis for the esign: -'i Pcnr.i1 ¡: l!cr ' 0. 100. 200.

300.

_400.

i

tab.

20b.

303.

«b.

-0.019 -3.1 ir -0.0012 -0.087 = 0.00066 -0.14 o o o 0 0 1 0 0 + p-0.044 0.015 -0.0018 -0.00026 -0.0017 -0.0040 0 O O -0.018 -0.022 1 0 Performance Criteria

A simple criterion for steady state course-keeping has _been pro-posed by Koyama (22) and Norrbin (23):

co f(p2 ₊

_{62) dt}

(4.4) ô 1

--

O

r

_KIT + i i O O C P C + [ -0.0040 O a C 11) + (4.1)

where the equations of surge, rudder and fin motions have been exclud-ed and the linearization is performexclud-ed around u = 7.7 rn/s,

iv-r-p- i

-4)-

iS

- a= 0.

If the roll motion is neglected in equation (4.1), an analysis shows that the time constants of the y and r equations are Ti = 8 s,

T2 -50 s and T3 = 22 s. _{Thus, the steering equations have a very} unstable mode. Nomoto's steering model:

(4.2)

where T = T1 + T2 - T3 = -64 s and K = 0.040 l/s, is used for the de-sign of a simple yaw regulator.

If the couplings of y and r into the roll equations of (4.1) are neglected, the ship's natural period of roll is then calculated to 42 s and the damping coefficient to 0.061. The disign of a simple roll controller is based on the model:

(4.3)

icI in îj3

n'.);

0.069 0 0.031 V

0.00055 0 -0.000031 r

It was verified in reference (24) by full-scale experiments that the criterion (4.4) with an appropriate value of A gave a good descrip-tion of the rate of fuèl consumpdescrip-tion in weather condidescrip-tions varying be-tween light and fresh breeze. In hard weather the criterion (4.4) has not been verified. Based on formulas in reference (23) the value of

was calculated to 0.17 for the test ship. A corresponding value

for the active fin deflections was also calculated: A = 0.24.

A suitable criterion for the design of roll regulators is:

=

5(Ap2

+

Regulator 2

A yaw and roll controller, where the rudder only is used, is de-signed based on the criterion:

J

_{¡(Ap2 +}

₊

_Ac2

_{+ A} 2) _dt

(4.7)

ôc

+ 6a2)

dt (4.5)

where A0 and A _{should be chosen in such away that a good roll}

ré-ductionis obthned.

A few preliminary simulations resulted in the following values: Ap = 55 s2 and Xc,

0.5.

The values can pro5ably be improved by a mo5e comprehensive simulation study.

Regulator 1

A simple yaw controller is designed.based on Nomoto's model (4.2) and the criterion (4.4):

and a combination of the models (4.1) and (4.2):

=

_{- 27r - 5.O.p - 2.34 - 1.44}

Regulator 3

This regulator consists of one rudder/heading loop and one fin/roll loop. _{The yaw-roll interactions are not considered. The} yaw controller is the same as (4.6) and the roll controller _is de-signed using the model (4.3) and the criterion (4.5):

(4.8) Pcrtni! i rçe n'..:;::

o

Note that the roil rate gain is the _{important factor and that the roll} gain is almost negligible.

= - 112r - 2.3 _(4.6)

a _{= - 13.p + 0.10}

L!

ERegulator 4

A true rnultivariable controller is designed based

on a

combina-:ti0n1 of the models (4.1) and (4.2), and the criterion:

'The following controller was obtained:

= - 26r

_-

2.8.p

- 2.4q, - 0.87.q

=

5.O.r - l3p + 0.0204

- 0.21.

:The reason for excluding the

_{sway velocity V is simply that it is}

usually difficult to obtain reliable

_{measurements of y.}

.5. RESULTS OF SIMULATIONS

The simulations were carried

_{out efficiently by use of the}

inter-active simulation

_{program SIÎ4NON, reference (25).}

_{The results of the}

simulations are shown in Figs. 5-8.

_{A summary of root-mean-square}

(rrns) values and maximum

_{values are given in Table}

_2,

_{where the}

steer-ing loss function J

(eqn.

(4.4)) is also tabulated.

_{Note that the}

maximum rudder and fin angles

_{were limited to 200. in all the}

simula-fions.

Table 2.

_{Simulation Results}

J =

f(Xp2 + *2 + A2

+

_óc

2

+ A a

2)

dt

1/ ;U(I¡fl fl1T))I _L':_______

(4.10)

(4.11)

Regulator

rrns (6)

deg

rms (a)

deg

rms

(ip)

deg

max

deg

rms ()

deg

max

deg

(deg)2

'1

8.84

-

0.50

1.29

5.32 16.38 13.53 2

7.87

-

_1.14

_2.71

_4.86

14.93

11.81

3

8.78

_12.12

_0.48

_1.32

_4.86

14.58

13.34

4

4.22

_12.28

_0.87

_1.93

4.70

13.37

3.79

-2

RJDDR ANCLE (DEG) VERSUS TillE CS)

HEADfl'G ANCLE COEG) VRtJ IThE CS)

tob.

ROLL ANtLE (DEG) VESUS TIIE _CS)

20b.

Figure 5.

_{Regulatcr i}

_{- Simulation Results.}

RUDDER ANGLE (DEG) VERSUS TIIE CS)

AD.INC tGLE CDD) VERSUS TI CS)

ROLL AWSLE CD) VERSUS TINE CS)

Figure 6. Regulator 2 - Simulation Results.

0.

RUDDER ANGLE (PEG) VERSUS TINE (S)

FIN ANGLE (PEG) VERSUS TINE (S)

L-EADING AWGLE CDE.) VERSUS TIRE CS)

-2.

0. tOO. 2Db.

ROLL ANGLE (LEG) VIEZstJS TINE (S)

20.

:.;.

sob.

4Db.

-20.

O. 100. 200. GOD.

«'o.

20.

0.

RU!)DER ANGL! CDEG) VERSUS TINE CS)

FIN ANGLE CP3) VERSUS TI!1 _(S)

PS

Pci)'It in p.:j iUar

2.0.

IEADIN AtLE (DZG) VEISUS TIEE CS)

2.1

i

r

too.

200.

_sob.

ROLL qLE CD) VERSUS TI

CS)

100.

í'

[\_

fl. V1

!\i

Figure 8. Regulator 4 - Simulation Results.

-ob.

sob.

400.

20.

-

R;ci! n

tmbr

By comparing the simple course-keeping regulator 1 with regula-tor 2 (yaw/roll-control by rudder), it is concluded that the rms of roll is reduced by 8.6 % and the maximum roll angle by 8.9 %. Thus, roll stabilization by rudder is a reasonable concept for the test ship. Also note that the J is decreased for regulator 2. This re-sult indicates that the roll motion should be considered in all auto-pilot designs for ships (cf. ref. (9)).

Traditional yaw/roll-control by rudder/fins (regulator 3), with-Out considering the yaw/roll-interactions, irnpizpved the roll perfor-mance approximately as much as regulator 2 did. However, it should be noted that the value of J _{is larger for regulator 3 than for} re-gulator 2 because of the induced resistance of the active fins.

Regulator 4, the true multivariable controller, reduced the _rms of roll by 11.7 % _{and the maximum roll by 18.4% compared to} regula-tor 1. _{The improvements compared to regulator 3 were 3.3 % and 8.3 %,} respectively. _{Note the extremely low value of J obtained for} regu-lator 4.

6. CONCLUSIONS

A relatively complex non-linear mathematical model for a ship moving in irregular seas has been presentd. _{An approach to simulate}

continuous wave spectra by feeding white noise into a filter was given.

Different discrete time yaw/roll-regulators were designed using linear quadratic control theory. _{These were investigated by} simula-tions, where aro-ro ship was sailing in irregular beam seas of a significant wave height of 7 m. It was concluded:

o _{A true multivariable rudder/fin-controller improved both roll}

and steering performances significantly.

o Some, not insignificant, improvements o _{both roll and steering}

behaviour were obtained with a system for yaw/roll-control by the rudder.

However, not many simulations of this type have been carried out up to now, so the simulation results must be considered as preliminary.

Future work should include _{a feasibility study of applying} ad--vanced filtering techniques, for example 1alman filtering, and adapt-ive control techniques for the combined yaw/roll-control problem. adaptive autopilots for _{yaw-control only and Kalinan filters}

were

de-signed in references (14) and (2G) . Kaimari and adaptive filtérs have

also been designed for wave filtering in connection with dynamic po-sitioning systems (references _{(27) and (28))}

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I;ì ri i!i;1r

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Automatica 15(1979), _{pp. 255-268.}

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and

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p.

. I

C TI

i'L.j.r

L. Byström, C. G. Kllström, "System identification of

linear and non-linear ship steering dynamics", 5th Ship

Control Systems Symposium, Annapolis, Maryland, USA

1978, Proceedings Vol. 3, 21

_p.

C. G. Kllström, "Identification and adaptive control

applied to ship steering", Department of Automatic

Control, Lund Institute of Technology, Lund,

Sweden,

CODEN: LTJTFD2/(TFRT-1O18)/1-192/(1979)

C. G. Käliströrn, K. J. Aström, "Experiences

_{of system}

identification applied to ship steering",

Automatica

17(1981):1,

_{p. 187-198.}

N. H. Norrbin, "Theory and observations

_{on the use of a}

mathematical model for ship Inanoeuvring

_{in deep and}

con-fined waters", SSPA, Göteborg, Sweden, Publication No 68,

_{1971, 117 p.}

N. H. Norrbin, "A method for the prediction

_{of the}

lnanoeuvring lane of a ship in a channel of varying

_width",

Symposium on Aspects of Navigability

_{of Constraint}

Water--

ways, Including Harbour Entrances, Deift, the

Nether-lands 1978, Proceedings Vol. 3, Paper 22, 16 p.

N. H. Norrbin, S. Göransson, R. J.

_{Risberg, and D. H.}

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Panama canal bend", 5th Ship Control

_{Systems Symposium,}

Annapolis, Maryland, USA 1978, Vol.

_{.3, 34 p.}

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_{Theory of}

Ship Dynamics", Chapman and Hall,

_{London 1974, 318 p.}

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Academic Press, New York and London,

_{1970, 299 p.}

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_{- User's guide",}

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_{of Technology,}

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_{/1-1301 (1980).}

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-,

¡

_/

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APPENDIX - SIMULATION MODEL

The simulations are based on the following mathematical model 'describing the surge, sway, _{yaw and roll motions of a ship moving in}

waves: -V x"-N

_Gv

L

(z+K'.')

y" L uv N" L

uv

K" L Uy.

rL3

JC- f(v,r) L IC

L

'K"

L 2

--(mL) L (x-Y') L( (k) 2-Ni)

-L(I" +K)

_zx r Y" -1 ur -N" x" ur G K" +z" ur G sinq +

The model is _{normalized using the 'bis'}

system (Norrbin (16)). The propeller thrust is computed as:

'T/m ₌

_KTDn/

where

C1 + cJ

_{+ c3(P/D) + c4J2 + c5(P/D)6}

-L(z+Y)

-L(I" +N) zx p L((k) 2-Kr) Y', up N" up K" up i _À12 À21 '22 r p uy ur up + i(l-X)ú u =

L2 uu

X" u2 + 2 X"uvvuy2 + L(x"+X" )r2 + (l+X")vr_{G zr}

Lzpr + (T/m)(l-t)

_{+ XRÔ(Y /m)S} R

X"'2.(F/m)a + X/m

Fa + YR/rn

2F/m

o _{(-6+6 )/T}

C

R'

a = (_a+a)/TF

and the propeller advance coeff. ient u(l-w)

nD

The nonlinear functions fy(v,r) _{and fN(v,r) have been}

derived by Norrbin (17). They are reprinted in Käliström (14).

The lift forces from the rudder and one of the _{stabilizing fins} are computed as:

YR/rn =

_{L2 uu6}

y" u2 + _{Y0(T/rn))(1+s162)d}

F/rn =

F"u2j

_(l+s2a2)a

The rudder ani fin angles are governed by first order differen-tial equations:

6.

1 im . s a Lun , O

um

uìrn

where O _{and a are the commanded angles.}

-C.

-The heading angle _{and roll angle.q}

are obtained as:

In beam seas the wave forces can be _{approximated by:}

X/m

Y/m

= a1.S(t).

7 (mL) _(t)

K/(mL) = a3.S(t)

ihere (t) _{is the z}

co-ordinate of the sea level at origin and _{S(t) is}

he wave slope.

The

follo'7i1g

parameter values for the test ship are used in the ;irnulatjon study:

= length between perpendiculars = 173.2 m V displacement = 52 010 in3

in mass = 53.3.106 kg

'g

= acceleration of gravity = 9.81 rn/s2

= normalized x co-ordinate of centre of gravity = 0.018 = normalized z co-ordinate of centre of gravity = -0.015 k" _{= normalized radius of gyration around xaxis = 0.074}

= normalized radius of gyration around z-axis _{= 0.25} = normalized product of inertia = O

= metacentrjc height = 0.45 _in D _{= propeller diameter = 6.3 in} P/D _{= propeller pitch ratio = 0.715} n _{propeller rate = 1.93 1/s} X = -0.060 . X -0.032 .1 _{= 0.00040} 2 uvv X"

=0

rr = 0.15

thrust deduction factor = 0.19 -0.40 X = -0.67 Y.& = -0.73

=0

uy Y't = .010 N'J _{= -0.040} r

=0

Pt.ii;U ¡u íief ,22

N'.' p

=0.

N" uy = -0.50 ur = -0.19 up 0.0014 = 0.017 p = -0.0012 K" uy = 0.018 = -0.0051 = -0.0027 = 0.0011 = -0.78 = 0.22

= 12 = 0.39 121 = -0.48 122 = 131 = -0.031 32 = -0.12 = -0.020 = -0.37 3 0.52 4 = -0.063 = -0.0016 = wake fraction = 0.27 uó = 0.14 'II = 0.95 T6 i = -0.45 F = 0.057 = 1.6 2

R = time Constant of rudder = 0.5 s

= rudder rate limit = 0.077 rad/s (= 4.4 deg/s) lim

= rudder angle limit = 0.35 rad (= 20 deg) = time Constant of fin = 0.4 s

um

= fin rate limit = 0.14 rad/s (= 8.0 deg/s) 11m fin angle limit 0.35 rad (= 20 deg)

i = 27 in/s2 Assumptions:

2 = -0.017 1/s2 u 7.7 rn/s (15 knots) ,

3 = 0.23 in/s2 J beam sea on port side