SWEDISH MARITIME RESEARCH CENTRE SSPA GÖTÉBORG
PUBLICATION NO 93
1982IDENTIFICATION AND ADAPTIVE CONTROL
APPLIED TO SHIP STEERING
by
CLAES G KÄLLSTRÖM
Thesis for the degree of "Teknologie Doktor" in Automatic ContrOl
Reprint of a thesis for the degree of
"Teknologie Doktor in Automatic Control,
Lund Institute of Technology,
P 0 Box 725, S-22O 07 LUN 7, Sweden
CODEN: LUTFD2/(TFRT-10l8)/1192/(1979) To Margareta Distributed by: Liber Distribution s-162 89 STOCKHOLM Sweden ISBN 91-38-06815-X ISSN 0373-4714
n o n 0 ut ut C o -I I-o I-z w o o 4 Dokrn.entuflgi,ere
Lund Institute of Technology
Hendlággate Dept of Automatic Control
Karl Johan Aström
Förtatte,o
Claes G Käliström
Dokumeflttltel och undertitel
-tdentification and Adaptive Control Applied to Ship Steering
Referat (tammandrea)
System identificatipn methods are applied to determine ship steering dynamics
from full scalé e,ctiments. The techniques uséd include output error and more general prediction error methods The experiments were performed both under open loop and closed loop conditions. Both linear and nonlinear modéls are considered.
Two adaptive autopilots for ships are designed. The autopilots are basedon velocity scheduling, a self-tuning regulator for steady state coursekeeping,
a high gain turning regulator with variable structure, and a Kalman filter Méthods for design of the autopilots are discussed. Rêsuits fromsimulations and- full scale experimènts are presented.
Referat sKuivet Bu
Author
Fönleg till ytterligare nyck.lOrd
Adaptive control; adaptive systems; computer-aided design; computer control; marine systems; optimal control; parameter estimation; PID control; self-adjusting
systems; stochastic systems. KleUltikOtiont3ytUm OCR -kleie(er)
lndexterrner (ange kölle)
Adaptive systems; automatic control; automatic pilots; computerized simulation; experimentation; shipboard computers; ship maneuvering; ships; time series analysis.
(Thesaurus of. Eñgineering and Scientific Terms, -Engineers Joint Council, NY, USA)
Omfiag Ovriga bibliogrefiska uppglf ter
192 pages Sprgk
English
Sakreteáuppgifter ISSN ISBN
Dokurnöntet ken erhillas frgn Motiegarens uppgifter
Department of Automatic Control Lund Institute of Technology Box 725, S-220 07 Lund 7, Sweden
Prit Blankett LU 11:25 1976-07 Dokun,.ntnnn Dokum.nlb.teCknlng REPORT LUTFD2/(TFRT-].018)/1-192/(1979) UtgiunlngudatUm Arendeb.t.cknlng May 1979
Table of contents
INTRODUCTION
SHIP STEERING DYNAMICS il
2.1 Equations of Motion 11
2.2 Hydrodynamic Forces and Moments 13
2.3 Nonlinear Models 14
2.4 Linear Models 16
2.5 Linear Models with Added Nonlinearities 23
2.6 Determination of Hydrodynamic Derivatives 27
2.7 Reférences 28 3. DISTURBANCES 31 3.1 Current 31 3.2 Wind 33 3.3 Waves 37 3.4 Surrimary 45 3.5 References 49 4. SYSTEM IDENTIFICATION 51
4.1 LISPID Model Structures 53
4.2 IDPAC Model Structure 61
4.3 The SecL Scûut 63
4.4 The
Sea Swt
804.5 The
Sea St/Latu4
904.6 The
Compa4s I4and
1044.7 Summary 116
4.8 References 118
5. ADAPTIVE SHIP STEERING 123
5.1 Simulation Model 124
5.2 Velocity Scheduling 127
53 Kalman Filtering 128
5.4 Steady State Course Keeping 141
5.5 Turning 153
5.6 Summary 159
5.7 References 161
6. EXPERIMENTS 165
6.1 The
Sea
Scout Experiments 1656.2 The Sea. SwL Experiments 166
6.3 The
Sea. St'ia-to
Experiments 1686.4 Economic Benefits 170
6.5 References 171
7. CONCLUSIONS 173
8. ACKNOWLEDGEMENTS 177
APPENDIX A - SHIP CHARACTERISTICS 179
APPENDIX B - EXPERIMENTS FOR SYSTEM IDENTIFICATION 181
APPENDIX C - SIMNON PROGRAMS FOR SIMULATION MODEL 183
1. INTRODUCTION
This work summarizes results obtained from several research projects supported by the Swedish Board for Technical Devel-opInent. The ship dynamics projects at the Department of Automatic Control started in 1973, although preliminary
studies were initiated iñ 1969-. System identification tech-niques were app-lied to determine s-hip steering dynamics from full scale experiments. This was done in collaboration with the Swedish State Shipbuilding Experimental Tank (SSPA) in Gothenburg. Adaptive autopilots for large ships were designed in a joint projec.t with Kockums Mekaniska Verkstads AB and Kockums Automation AB in Malmö. Thé adaptive autopilots were tested on three oil tankers built by Kockums for the Salén Shipping Companies in Stockholm. Experiments designed for system identification within the SSPA projects were also performed with the \Salén tankers.
Ship steering dynamics and disturbances
Mathematical models of ship steering dynamics and disturb-ances are useful for, computer simulations, for prognosis of manoeuvring qualities, and for autopilot and predictor design. The parameter values of such models are normally estimated from theoretical calculations or from scale model tests.
The dynamics of ship steering is reviewed in Chapter 2. The influence of wind, waves, and currents on the ship motion is discussed in Chapter 3. These chapters summarize known infor-mation. Their purposes are to obtain a hierarchy of models of different complexity which are suitable for control sys -tems design. The main result is a mathematical model wh-ich
is suitable for computer simulation of the steeriñg of large ships. The standard. notation in naval hydrodynamics has been adopted.
System identification
System idéntification offèrs a älternative approach of
determining ship steering dynamics. It requires data of ho a ship reacts to changes ïn ruddèr angle. Applications of
thistechnique to data from 14 full scale experiments are suthmarized in Chaptér 4. The tést- ships âre threè oil tankers and a cargo ship. Experiments designed for system idéntifiôätion were performed ;with the tankers. Three zig-zag tests, whidh. are common for ship testïng, performed with the cargb ship are also analysed. These expériménts were fourd in the literature.
Different identification techniques such as output error and prediction error methods are explored. Both linear and nOn-.
linear models are fitted to the experimental data. Input--output mòels in terms of both difference and differential equations are considered. The experiences and results o-tàined f tom applyïng identification methods to determine ship steering dynamics arè summarized.
Adaptive ship steering
The requirements on ship steering are increasing for reasons of safety and economics. The autopilots corranonly used today aré based on simple PID-control algorithms. An autopilot
must be properly adjutèd to give a Ood performance..
Adjust-ments are required to compénsate for wind, waves, currents,
speed, trim, draught, an4 water depth. The adjUstment are
tedious äñd time cohsumiñg. Fixed settings arethereföre often used. It is a common experience that the autopilots do
not work well in bad weather or when the speed is changed. It has also been observed that autopilots may have difficul-ties when the ship is making large manoeuvres. The reason is partly that the autopilot is not properly tuned and partly that the PID-algorithzn is too simple.
Some of the disadvantages of conventional autopilots can be avoided by using an adaptive autopilot. Such a pilot can
adjust its parameters in order to compensate for changes in the environment. Because the parameters of the controller are tuned automatically it is also possible to use control
lgorithms which are more complex than the ordinary PID-algo-rithrn. The design of adaptive autopilots for large ships is discussed in Chapter 5. Examples of computer simulations are given. The basic autopilot functions such as velocity sched-uling, Kaimari filtering, steady state course keeping, and
turning are described. A self-tuning regulator is used for steady state course keeping, änd a high gain regulator with variable structure is used for turning.
The adaptive autopilots were tested on three oil tankers. Over 130 experiments were recorded. A summary of the experi-mental results is given in Chapter 6. The economic benefits of improved steering are also estimated.
The major conclusions of the work are summarized in Chapter 7. Suggestions for further work are also given.
2. SHIP STEERING DYNAMICS
A review of the dynamics of ship steering is given in this chapter. Different nonlinear and linear mathematical models for the ship motion in deep and unrestricted waters are discussed. The models are confined to calm water and to motions in the horizontal plane only. Parameter values of different ship models found in the literature are given.
2.1. Equations of Motion
A ship,which is considered as a rigid body, has six degrees of freedom corresponding to translation along three axes and rotation about these three axes. The equations of motion are conveniently expressed using a co-ordinate system fixed to the ship (see Fig. 2.1). The hydrodynamic forces and. moments acting on the ship are easy to describe in such a co-ordinate system because the synirrietry of the hull can be exploited.
Fig. 2.1 - Definition of co-ordinate system fixed to the ship. Translatiofls along the co-ordinate axes are called surge, sway and heave and rotations about these axes are called roll, pitch and yaw,
The equations describing the motion in the horizontal plane can be simplified, if the couplings between this rnotion and the heave, roll and pitch motions are neglected. This approxi-mation is commonly accepted (Abkowitz, 1964; Eda and Crane,
1965; Norrbin, 1970; Clarke, 1971'and 1976), and at least for tankers and similar ships the coupling, effects aré negligibly small. Since only tankers and other large ships
are considered in the sequel, the coupling effects of heave, roll and pitch motions into the horizontal motion are assumed
to be negligible.
The variables used to describe the horizontal motion of a ship are explained in Fig. 2.2. The projections of the total ship speed V on the x- and y-axes are called the surge
velocity u and the swây veloöity V. The turning rate is
denoted r and the heading and rudder angles are denoted i
and S. ,
Fig. 2.2 - Variables used to describe tie motion in the horizontal plane. The origin is placed half-way between the perpendiculars andin the centre-line of the ship.
The equations for the horizontal motion are obtained from Newton's laws expressing conservation of linear and angular momentum (see, for example, Abkowitz, 1964):
m(û-vr_xGr2) = X
m(r+ur+xG) =
(2.1)Ir+mxG(v+ur) = N
where X and Y are the components of the hydrodynamic forces on the x-axis and y-axis, N is the z-component of the
hydro-dynamic moments, m is the mass of the ship, I is the moment
of inertia about the z-axis, and XG is the x co-ordinate of the centre of gravity. Notice that the y co-ordinate of the centre of gravity is equal to zero, because the co-ordinate
origin has been set in the centre-line of the ship.
2.2. Hydrodynamic Forces and Moments
The hydrodynamic forces X and Y and the hydrodynamic moment N of (2.1) are complicated functions of the ship motion. Much theoretical and experimental work has been devoted to deter-mine the functions.
Abkowitz (1964) suggested
X = X(u,v,r,ô,û,',i)
Y = (2.2)
N =
and approximated the functions with Taylor expansions about
the steady state condition u = u0 and v=r=='7=.=O,
where terms of order 4 or higher were dropped. By considering the lateral symmetry of a ship several terms of lower order
can also be omitted. Discussions of the importance of the different terms of the Taylor expansions are found in Eda
and Crane (1965), StrØm-Tejsen (1965), and Mandel (1967).
A drawback with the Taylor expansions of (2.2) is that the high-order terms have no physical interpretations. Norrbin (1970) has developed other expressions for the hydrodynamic
forces and moments, where the high-order terms have such
interpretations.
2.3. Nonlinear Models
It is customary to normalize the nonlinear models (2.1) and
(2.2). This can be done in several differént ways.. In the
'prime' system (SNANE, 1950), which is most common, the length unit is the length of the ship L, usually specified
as the distance between the perpendiculars the time unit
is L/V, where V is the total ship speed, and the mass unit is
where p is the mass density of water. T1-ie mass unit
can alternatively be defined as pL2T (Norrbin, 1960), where
T is the draught of the ship, but this normalizatioñ will
not be used. Another normalization, the 'bis' system, was
proposed by Norrbin (1970). In this system the length unit
is L, the time unit is
/i7',
where g is the acceleration ofgravity, and the mass unit is m.
The simulations described in Chapter 5 are based on the
following nonlinear model which was developed by Norrbin
(1970) (cf. also Dyne and Trägrdh, l975 Norrbin and
(1-
x'i:i
=x"
u2 + _L
J.
x» u4 + L \ u) L 2 uu gL2 24 uuuu + (i + xi' vr +x"
uivlv2 + vr) uivivv +XyR4.YU
'5 u2'5+Y(T/rn'5]'5 + CT/rn) (l_tv)f
\.
f
,,\.
i
f
I i - Y. iv + Li
X" -
Y. 1r = - Y" uy + I Y" - 1 Jur +\ vj G rj L uy ur
+
ly"
u2r+Y"11 vivi+
/i L 2 uuv 2 uur
+ L
1
" rin + Y" Ivir + Y" viri +2 rin Ivir viri
+
i:
y" u2 + Y(T/m)6 + kTY(T/m)L 2 uu'5
(x_N;)
+ L((kZ)2_N)i =
N" +(N» _x)ur
+1 1 2 1 1 2
11
+
N uv+--N
ur+--N
vIvI+V?L 2
uuv 2 uurL2 vivi
+ L N" rin + N" ivir + N" viri +
2 rin Ivir viri
+ 1 !
N" u2'5 + N(T/rn)'5 + kTN(T/m)L 2 uu'5
where dimensionless hydrodynamic.derivatives normalized in the 'bis' system have been introduced. Notice that t, is the thEust deduction factor and that (kz)2=I/(rnL2). The pro-peller thrust T is given by
T = pn2D4(J/JI)2
K(J')
(2.4)where n is the propeller rate of revolution, D is the propeller diameter, and J is the propeller advance coeffi-cient. The following relations are also used:
+1
x"
r2 + G 2 rrjJ
= u(l-w) / (nD) JI =J/I1*J
(2.5)
where w is the wake fraction. The thrust coéfficient is
given by the approximate relation
= k1(J')2 + k2J' + k3. (2.6)
Parameter values of the model (2.3) for a tanker are given in Table 2. 1.. The ballast condition corresponds to a dis-placement V = 172 400 m3 and a draught at bow TB = 9 m and
at stern T = 12 m. The values of Table 2.1 are used in the
simulations described in Chapter 5. Sirmilations of manoeu-vring trials with the same model are given in Källström
(1976). Parameter values of models that aresimilar to (2.3) for other tankers and a large container ship can be found in van Berlekom and Goddard (1972), van Berlekom, Trägrdh, and Dellhag (1974)., and Norrbin and co-workers (1978).
2.4. Linear Models
The surge equation is nonlinear principally, while linear terms are present in the sway and yaw equations. Therefore it is assumed in this section that the surge velocity u is constant and equal to u0. Also it is assumed that the ship is perfectly syïrimetrical with respect to its xz-plane, i.e. asymmetric forces and moments due to a single screw action1 for example, are neglected.
The steady state solutions to the equations of motion (2.1) and (2.2) are given by
f(V,r) = Y (u0, y, r, 0, 0, 0, 0) - mu0r = O
(2.7)
Table 2.1 - Dimensionless parameter values of the môdel (2.3) for a
356 000 tdw oil tanker of Kockuxns' design adopted from Dyne and Trägrdh (1975). The parameter values have been estimated from theoretical òalculations and scale model tests with similar tankers. They are normalized by use of the 'bis' sytexn (L=350 m). The full load condition
V=389 100 m' and TB=TS 22.3 rn implies that k=0.24
and x = 0.0267. The ballast condition V = 172. 400 m3,
TB=9 m, and Tsi2 m implies that k= 0.24 and x=
= 0.0197. The nominal válue n0 =87.6 rpm gives uQ=l5.8
knots in full load condition and u0 =17.25 knots in ballast.
The parameter values of equations (2.4), (2.5), and (2.6)
are: p =1025 kg/rn3, D=9..i rn,
w0.42, k1=-0.33,
k2=-0.38, and k3 =0.35. The ship characteristics are
sürnmarized in Appendix A.
X-equation Y-equation N-equation
Ballast Ballast Ballast
1-x 1.070 1.060
1-Y
1.820 1.750 xe-N O OX -0.03444 -0.068 x-Y'. O O (k)2-N 0.103 0.103
l.nrn o o Y -0.700 -0.630 -0.350 -0.170
x+X'!
0.02670.0197 Y-1
-0.820-0.820 N-x
-0.163 -0.1551+X"
vr 1.492 1.450 2 Y"uuv -0.910 -0.820 2 N"uuv -0.450 -0.220
-300 -300 0.234 0.234 -0.177 -0.168 -0.42 -0.42 Y" -2.150 -l.9Ö0 0.064 0.060
ltp
0.80 0.80 IrI O O N;iri -0.129 -0.120 ïvIr 0.500 0.250 .N'(1 -0.180 -0.150 IrI -0.500 -0.250 N1r1 0.180 0.150 .. 0.176 0.252 N" -0.083 -0.118 Y - 1.200 1.200 IÇ -0.564 -0.564 lç 0.040 0.040 k, -0.020 -0.020where it is assumed that the rudder is kept constant at the centre positiön. A stationary solution to (2.7) is given by
y = 0 and r = O, since the force Y and the moment N will
vanish for these values. Depending on the proportieS of Y
and N there may, however, also be other solutions. Two typical cases are shown in Fïg. 2.3, where the functions f and.g are plotted for two différent ships. The case of one stationary
point P only, as indicated in Fig. 23(a), is most common.
The point P is
a
stable node which corresponds to a stablestraight line motion. For special classes of ships, for example large tankers, the case shown in Fig. 2.3(b) cañ,
however, occur. In such a case the point Q i a saddle point,
which corresponds to an unstable straight line motion, while
the points R1 and R2 are stable nodes. Thesé orrespond to
stable turns. A ship with these properties cän not be kept on a straight cour'se with zero rudder angle, since a small
disturbance will make the ship to enter a spiral turn and
the motion will end up in a stationary port or starboard yaw.
o. -2 Fig. r Ldeg/s] - r rdeg/s] 2 £knotsl -2 2 Eknots] 0.4 g(v,r)=O f) y. r) O
2.3 - The determination of the stationary motions as
the intersections of the curves f(v,r) = O and g(v,r) = O is illustrated. In case a, which is
a Man.Lne class vessel, the curves intersect at
the origin only, but in case b, which is a fully
loaded 356 000 tdw oil tanker (cf. Table 2.1),.
m' -:
VL(m'x_Y)
m'x-N
L(I_N)
- Y'Lv
Isi ULv
v(N' -m'x' G y G rxe-N;
L((kZ)2_N
LLNU
6 6 (2.8)where the hydrodynamic derivatives are normalized using the 'prime' system. Linearization of the model (2.3) gives
Linearizations about the stationary solution y= r =6 = O of
nonlinear models will now be discussed. The standard linear model is given by
!I',.
Li., uy
L. 1
2 uv) ur-1
+3L
..
ur) yU L N y2 1 L 2 + ! N"
N")
r/L
2 uuvj uut5vN» -xU
ur+_
GvL
+ y2 i L 2N6
(2.9)where the correction terms for the rudder efficiency
Y6(T/m)6 and N6(T/m)6 have been neglected and where it
has been assumed that u= u0 =V. The relations between the
parameters of the models (2.8) and (2.9) are given by V r V + r y r
=m'Y
m' Y' = m' (Y»L.
.y"
V uv uuv = m: ur) 6 - m 2 uii6 N: = m' N V V N: = m' N'.' r r N' = m' (N» +--- - N" yuv
/L2
uuvN' =m' (N"+---N"
rur V2 uur
N' = m' 6 2uuc5 = I' G G I =where m' = m/(PL3) = 2V/L3.
The criterion for asymptotic stability of the linear - equations of motion (2.8) is (see, for example, Abkowitz,
1964):
y
(N;_m'x)
- (Y.._m) N > 0 (2.11)The linear equations of motion (2.8) or (2.9) are easily converted to standard state space notatiòn by solving for
the derivatives r and :
(2.10)
V L
11 +
L2 21
where the 'prime' notation has been retained.
ô (2.12) all V aj V , V
L22'
L 22 V rThe input-output relation between the rudder angle ô and the yaw rate r can be represented by the transfer function
Gô(s) =
G(s)
= y2 y3- b's+ b'
L2 i L3 22
ajs+3L a K1 (s + =(s+-)(s-)
(1 +ST3)
1 + sT1)(1+ ST2)
- c s + c K ( +sT3)
52V
- as+a'
y2(l+sT1)(1+sT2)
L2 2 (2.14) K T3where K1y V . The parameters of the transfer functions
T1T2
(2.13) and (2.14) are related to the parameters of the state
space model (2.12) aj = aj1 -=
a11 a2
b'=b'
1 21 ba1 b1
cjb1
ca2 b1
through- aj2 a1
- a1 b1
- a2 b1
(2.15) Kl(+)
(s _)(s + (2.13) KT3 where K1= T1T2 . If the sway velocity y is taken instead as
the output, then the f011owing transfer function is
Table 2.2 - Examples of parameter values of the models (2.8), (2.12), (2.13), and (2.14) for different ships.
Ship Cargo ship, series 60, block-coef f. Cargo ship, Maft.&iVL class Oil tanker, 190 000 tdw Oil/ore carrier, 282 000 tdw Oil tanker,356 000 tdw
Full load Full load Ballast FuU load Ballast
0.70
Length L (Lpp) (a) 160.0 160.93 304.8 321.56 350
Breadth B (a] 22.9 23.17 47.17 54.56 60
Draught at bow T [a] 9.14 6.86 18.46 21.67 12.25 22.3 9.0
Draught at stern T5 (ii') 9.14 8.08 18.66 21.67 14.91 223 12.0
Displacement V (a3] 23 400 16 622 220 000 312 200 188 900 389 100 172 400
Nominal speed u0 (-y) (knots] 15.18 15.00 16 16.00 17.09 15.80 17.25
Nominal propeller rate n0 (Ppm] 74.4 69.0 80 100.8 98.4 87.6 87.6
a' _Y,, m'x - Yj 0.0229 0.00039 0.01566 -0.00009 0.03139 0 0.03705 0.00152 0.01899 0.00057 0.03303 0 0.01407 0 m'x-N 0.00048 -0.00023 0 0.00115 0.00048 0 0 0.00i2 0.00083 0.00192 0.ò0205 0.00112 0.00187 0.00083 Y -0.0222 -0.01160 -0.01873 -0.01767 -0.00982 -0.01500 -0.00607 -0.0076 -0.00499 -0.01169 -0.01284 _-O.00779 -0.01429 -0.00631
Ç
-0.00567 -0.00264 -0.00701 -0.00930 -O00352 -0.00749 -0,00164--0.0034 -0.00166 -0.00359 -0.00368 -0.00209 -0.00340 -0.00145 0.00211 O.O027 0.00323 0.00314 0.00227 0.00319 0.00203 -0.00105 -0.00139 -0.00152 -0.00145. -0.00106 -0.00151 0.0O095 il -0.895 -0.770 -0.597 -0.298 -0.428 -0.454 -0.431 a12 -0.286 -0.335 -0.372 -0.279 -0.359 -0.433 -0.448 2l -4.367 -3.394 -3.651 -4.-370 -2.959 -4.005 -1.976 a2 -2.719 -2.093 -1.870 -1.638 -1.712 -1.818 1.747 b1 0.108 0.170 0.103 0.1.16 0.150 0.097 0.144 b1 -O918 -1.627 -0.792 -0.773 -1.011 -0.807 -1.145 K' -1.092 -3.855 3.511 1.008 2.660 0.831 5.881 Ç -0.918 -1.628 -0.793 -0.771 -1.010 -0.807 -1.145 Ç 0.470 1.898 -2.015 -0.555 -1.881 -0.579 -5779 K'1v 0.108 0.170 0.103 0.116 0.150 0.097 0.144 T'1 2.743 5.660 -10.593 -3.091 -6.932 -2.882 46.910 T 0.308 0.372 0.390 0.443 0.438 0.382 0.447 T 0.710 0.889 0.933 1.048 1.153 1.069 1.472 T'3v 0.194 0.189 0.211 0.286 0.242 -0.184 0J88 K (1/s] -0.053 -0.185 0.095 0.026 - 0.073 0.019 0.149 K1 -. [1/si] -0.00219 -0.00374 -0.00058 -0.00051 -0.00076 -0.00044 O00074 tm/sI 3.7 14.6 -16.6 -4.6 -16.5 - -4.7 -51.3 K1 (m/2] 0.041 0.063 0.023 0.024 0.036 0.018 0.032 T1 - - (sI 56.2 . 118.0 -392.3 -120.8 253.4 -124.1 -666.9 T2 (s] 6.3 7.8 14.4 17.3 16.0 16.4 17.6 T3 (s] 14.5 18.5 34.5 40.9 42.1 46.0 58.1 T3 [s] 4.0 3.9 7.8 11.2 :8.8 -7.9 7.4 Reference Zuidweg Chislett van van Bérlekom, - Dyne and Träg&rdh
-(1970) and Berlekom TrägArdhandDellhag (1975) Stróm- and (1974)
-Tejsen Coddard (1965) (1972)
Resarks - Origin - - Cf. Table 2.1 and
- at the- Appendix A.
centre of gravity.
An approximation to the transfer function (2.13) has been proposed by Nomoto and co-workers (1957):
K K1
= r
l+sT
(2.16)where the effective time constant T is equal to T1+T2-T3. This first-order approximation is only valid for low frequen-cies. There are several procedures proposed to determine the parameters of (2.16) from experimental data, and these
parameters have also been determined for many ships (see Nomoto and co-workers, 1957; Nomoto, 1960).
Examples of parameter values of linear models for different ships are shown in Table 2.2. The two cargo ships are stable
(cf. (2.11)), while the three tankers are unstable in both
full load and ballast conditions. The parameter values of the state space model (2.12) are remarkably similar for the different ships of Table 2.2, while the dimensionless gains
K' and K and the dimensionless time constant Tj vary
signi-ficantly. This depends on the fact that the value of a =
= aj1 a2 - a2 a1
is always close to zero, and a small relative change of aj, aj2 -a1, or a2 may give a large relative change of a. - Notice, however, that the variationsof K, Kj, T, T, and T
are small for all ships of Table2.2.
2.5. Linear Models with Added, Nonlinearities
Since the nonlinear models discussed in Sections 2.1-2.3 are complex, a class of simpler nonlinear models has been deve-loped. They are obtained by adding nonlinear terms to the linear models described in Section 2.4.
Linear models have been used extensively in different applica-tions. The validity of these models is, however, limited to
moderate déviations from the stationary point y = r = = 0.
It was shown in strom, Källström, Norrbin, and Byström' (1975)
that a 20°/20° zig-zag test with a container ship is domi-nated by nonlinear effects, while a linèar model hàs at least some relevance when it is applied to a 100/100 zig-zag test.
The range of validity of a linear model can be estimated from a spiral test, where the steady state relation between rudder angles and turning rates isinvestigated. It is concluded from Fig. 2.4, where two typical graphs are shown, that the validity of a linear model is limited to a very small region for an unstable tanker (case b), while the region is somewhat larger for a stable cargó ship (case a). Notice that the unstable, linear model of the tanker must not be interpreted in such a way that a starboard rudder will give a port turh,
since the dashed line of Fig. 2.4(b) represents
urt4tab!
stationary points. A sufficiently large starboard rudder angle will thus always give a starboard turn independent of' the i-nitïal yaw rate, éven if an unstable, linear model is considered.
In order to extend the range óf validity several authors
have suggested the addition of a nonlinear term to the linear
model: '
T1T2i + (T1+T2) + KH(r) = KÔ+KT3. (2.17)
Notice that the transfer function (2.13) is obtained if
H(r) = r/K. It follows from (2.17) that the relation =H(r)
is obtained in the steady state, which means that H'(r) can be determined from a spiral test (cf. Fig. 2.4).
An extension of the linear equations of motion (2.8) or (2.9) to be used for system id'entification (cf. Chapter 4) has been süggested by Norrbin (1976). See also Norrbin, Byström,
Aström, and Källström (1977). The following modified version of (2.8) is then obtained:
Stbd Ê Port & [deg] Stbd -20 a. Fig. 2.4 -m'x' -N: G V
- Steady state relation between rudder angle and
rate of turn r for a MaitLnv class vessel (case a), adopted from Chislett and Strøm-Tejsen (1965), and
for a fully loaded 356 000 tdw oil tanker (case adopted from Källström (1976). Notice that the Mcitinjr. class vessel is stable, while the tanker is unstable. The linearized models are represented by the dashed lines. The graphs are not perfectly symmetrical, because asymmetric forces due to a single screw action are present.
L(m1x -L(I' -N:
\Z
r V r / / / r Edeg/sJ 0.5,
/
/
/
/
/
Port 6 [degl-
Lv
Y'v(Y'-m')
- N'v(N _m'x)
V V r + L y2 L -0.5 b. f(v,r) (2.18) fN(v,r)where the effective cross-flow drag coefficient C is a dimensionless parameter,
fi
lfv'T1v
- < ---
1v
1f(v,r)= TfIri{_
_<-<
The mean draught TM is defined by
= (TB
/2
(2.21)and
the trim coefficient t byt =TSTB/TM.
(2.22)The following simplifiéd versions of (2.19) and (2.20) are obtained when r = 0:
f(v,r) = TM vivi IL2
(2.23)
fN(v,r) = TM vivi IL2
The nonlinear functions fand
N have been derived by
integrating the local cross-flow resistance over the hull (see Norrbin, 1976).
TMrjri{_+T[+.(!)2J}
<_i!<_i
_.<_!<.
.<_!!<
(2.20)fi
i(v'?tivl
11v
(2.19) and fN(v,r) =2.6. Determination of Hydrodynamic Derivatives
The different models descrIbed in Sections 2.3, 2.4, and 2.5 are only useful, if there are procedures available by which the values of the hydrodynamic derivatives can be determined for a specific ship in a specific load condition. There is no complete theoretical method known by which the hydrodynainic derivatives can be calculated from a knowledge of the hull geometry. Many different approaches to solve this problem have been made, and it is common to introduce empirical
corrections to the theoretical results to improve the methods. A review of different approaches can be found in, for example,
Norrbin (1960, 1970) or Motora (1972). Semi-empirical
relations for the basic stability derivatives have been derived by linear regression in Norrbin (1970).
Scale model tests are commonly, used instead of, or as a
complement to, theoretical calculations of the hydrodynamic derivatives. In such experiments the scale model is forced into a specific motion and the forces acting on the hull are measured. The hydrodynamic derivatives are then estimated by polynomial fitting or by similar methods. See, for example, Strøm-Tejsen and Chislett (1966). The main drawback of this
method is, of course, that it is difficult to eliminate the
effects of scaling. The mathematical models obtained from scale model experiments are sometimes improved by comparisons' with trial manoeuvres performed with the corresponding full scale ship.
Another method to estimate the parameters of different ship steering models is the application of system identification techniques to data from full scale experiments. The procedure is described in Chapter 4 where many examples are also
2.7. Reférences
Abkowitz M A (1964):
Lectures on ship hydrodynamics, steering and maneuver-ability. Report No. Hy-5, Hydrodynamics Department, Hydro- änd Aercdynamics Labóratory, Lyngby, Denmark.
Astrom, K J, C G Kallstrom, N H Norrbin, and L Bystrom (1975) The identification of linear ship steering dynamics using,
maximum likelihood parameter estimation Publ No 75,
Swedish State Shipbuilding Experimental. Tank, Göthenburg, Sweden.
van Berlekom, W B, and T A Goddard (1972):
Maneuvering of large tänkers.
T'Lan6. SoeLe.
o NaiiaAhect and MaìLe. En ¿nav
80, New York. àn Bêrlèkom, W B, P Trägrdh, and A Deilhag (1974):Large tankers - wind coefficients and speed lÒss due to
wind and sea.
T'tana. Royas In.tULon o
MauaA'hLte.c.
117, 41, London, 1975.
Chislett, M S, and J StrØin-Tejsen (1965):
Planar motion mechanism tests and full-scale steering
and manoeuvring predictions for a
Ma'ine'
class vesselReport No. Hy-6, Hydrodynamics Department, Hydro- and Aerodynamics Laboratory, Lyngby, Denmark.
Clarke, D (1971):
A new non-linear equation for ship manoeuvring
Invtnaonae. Sh.pbuÁiding P'ogneo
18, No. 201, 181.Clarke,D (1976):
Sorne Ape.c
o.he VynamJc6 o
ShLp S-te.vtng.
Thesis, Faculty of Engineeriñg, University of London, London.Dyne, G, and P Trägrdh (1975):
Simuleringmode11 för 350 000 tdw tanker i fullast- och
bãllastkonditioner p djupt vattèn. Report 2075-1,
Swedish State Shipbuilding Experimental Tank, Gothenburq, Sweden (in Swedish).
Eda, H, and C L Crane, Jr (1965):
Steering characteristics of ships in calm water and waves.
T'an. SocLe
06 Navas Mch-te.ct4 and Maine. Engnéen
73, New York. Käliström, C G (1976)
Simulation of ship yawing. Department of Automatic Cbntrol, Lúnd Institute of Technology, Lund, Sweden.
Mandel, p (1967):
Ship maneuvering and control. In J P Comstock (Ed.):
Pkinaip.ee. o
Ncuci Ahectk.
Society of Naval Architects and Marine Engineers, New York.Motora, S (1972):
Maneuverability, state ofthe art. International Jubilee Meeting on the Occasion of the 40th Anniversary of the Netherlands Ship Model Basin, Wageningen, the Netherlands.
Nornoto, K, T Taguchi, K Honda, and S Hirano (1957):
On the steering qualities of ships. In
atLonai
Skip-bdng Pko'te.o
4, Ño. 35, 354. Nomoto, K (1960).:Analysis of Kempf's standard maneuver test and proposed steering quality indices. Proc. ist Symposium on Ship Maneuverability, Washington (Report 1461, David Taylor Model Basin, Washington).
Norrbin, N H (1960):
A study of course keeping and manoeuvring performance.
Proc. 1st Symposium on Ship Maneuverability, Washington
(Report 1461, David Taylor Model Basin, Washington), and Publ. No. 45, Swedish State Shipbuilding Experimental Tank, Gothenburg, Sweden.
Norrbin, N H (1970):
Theory and observations on the use of a mathematical model for ship manoeuvr-ing in deep and confined waters. Proc. 8th Symposium on Naval Hydrodynamics, Pasadena, California, USA, and Pubi. No. 68, Swedish State Ship-building Experimental Tank, Gothenburg, Sweden.
Norrbin, N H (1976):
0m en kvasi-stationär integral f ör icke-linjär dämpning i fartygs girmanöver. PM Bl87, Swedish State Shipbuilding Experimental Tank, Gothenburg, Sweden (in Swedish).
Norrbin, N H, L Byström, K J Aström, and C G Käliström (1977): Further studies of parameter identification of linear and nonlinear ship steering dynamics. Report 1920-6, Swedish State ShIpbuilding Experimental Tank, Gothenburg, Sweden.
Norrbin, N H, S Göransson, R J Risberg, and D H George (1978): A study of the safety of two-way traffic in a Panama Canal bend. Proc. 5th Ship Control Systems Symposium, Vol. 3, paper Kl 3-1, Annapolis, Maryland, USA.
SNAME (1950):
Nomenclature for treating thé motion of a submerged body through a fluid. Technical and Research Bulletin No. l-5, Society of Naval Architects and Marine Engineers, New York.
Str$rn-Tejsen, Y (1965):
A digital computer technique for prediction of standard maneúvers of surface ships. Report 2130, David Taylor Model Basin, Washington.
StrØm-Tejsen, J, and M S Chislett (1966):
A model testing technique and method of analysis for the prediction of steering and manoeuvring qualities of surface vessels. Report No. Hy-7, Hydrodynamics Depart-ment, Hydro- and Aerodynamics Laboratory, Lyngby,
Denmark.
Zuidweg, J K (1970):
AutomatL
GuLdanco, ShLp4
a Conto
Pxobm.
Thesis,3. DISTURBANCES
All disturbances were neglected in the mathematical models
of Chapter 2. A ship is, however, influenced by wind, waves
and currents. They appear as disturbances in the equations of motion. There is a substantial literature on disturbances. Simple models for disturbances, describing the orders of magnitude involved, are selected. They are summarized in this
chapter. The purpose is to obtain models which are. suitable
for simulation of the steering of large tankers. As in Chapter
2 the models are confined to motions in the horizontal plane
only. It is also assumed that the couplings of heave, roll,
and pitch motions are negligible. Another important
assuxnp-tion is that the disturbances are modelled by forces and
moments which are added to the calm sea models outlined in
Chapter 2. This is an approximation because the superposition
principle does not necessarily hold for large manoeuvres.
3.1. Current
It is assumed that the current is constant and homogeneous.
The inertial velocity of the ship is thus influenced, but not
the hydrodynamic forces and moments.
When a current is present the equations of motion become
(cf. (2.l)):
m [û- (v+vC)r_xG r2] = X
m [<7+ (u+uC)r+xG = (3.1)
Fig. 3.1 - Definition of the velocity VC and the direction of the current.
whre u
and v are the, velocity components of the current on the x-axis and y-axis. The following relations are obtàined from Fig. 3.1:= V cos(y--)
(3.2)
v = V
where the velocity of the current is denoted V and the
direòtion is dénoted
.
The influence of currents on the ship motion is thus given by
the fòrces and moment (cf. (3.1))
Foot-note: The derivatives and of the velocity components of the current are assumed zero in (3.1), which is an approximation. By using
the correct values of and obtained from (3.2), it can be shown that the forces and moment from the current are zero (cf. eqn (3.3)).
X Current = m v r Y Current = - m u r N Current - m XG uC r
which are added to the equations of motion (2.1).
3.2. Wind
The wind is regarded as the sum of two components, the mean wind and the turbulence. The mean wind is a constant, homo-geneous air-current and the turbulence is considered as stochastic fluctuations.
The forces induced by a mean wind have been investigated by
wind tunne'l tests on different. ships (see, for example,
Wagner, 1967). Large tankers with different superstructures were investigated by van Berlekom, Trägrdh, and Deilhag (1974)
They used the following expressions for the forces and moment:
1 2
Xid =
a VR Cx(YR 1 2 2 '1wind = a VR CY(yR)L 1 2 3Nid
a VR CN(YR)Lwhere X and Y are the components of the wind induced
wind wind
forces on the x-axis and y-axis, Nwjd is the z-component of
the wind induced moments,
a is the density of air a = 1.23
kg/rn3), and L is the ship length. The wind speed VR and the
wind direction 1R relative to the ship are explained by Fig.
3.2. They are related to the speed VT and the direction 1T of
the true wind through the following identities:
(3.4) (3.3)
L
Fig. 3.2 - Definition of the speed VT and the direction of the true wind as well as the wind speed V
and the wind direction R relative to the ship.
VR =
v'i4+v
= arctg(v/u) +r
if O (3.5)= arctg(v/u) (+2w if VR> O)
if UR < O where = VT COS ir) - u VR = VT sin(yT__ir) y-are the components of VR on the x-axis and y-axis.
The wind coefficients C, C, and CN have been determined by van Berlekom, Trägrdh, and Delihag (1974) for different tankers. The following relations describe the principal forms:
Cx(YR) x cos
= (3.7)
CN(YR) = CN sin 2R
These expressions are good approximations for a ship which is not too asymmetrical with respect to the yz-plane. The approximation of CN is poor for a tanker which has a large aft superstructure. Ari -improvement is given by
CN1
°<R<-:'
<y<27r
CN-' ir
CN2 if
The parameters for the influence of a mean wind on a 356 000 tdw oil tanker of Kockums' design are given in Table 3.1. They have been estimated by the method described in van Berlekorn, Trägrdh,and Dellhag (1974). The parameter values are used in the simulations of Chapter 5.
Much work has been done to describe wind turbulence. It has been found that, at least at high altitudes, the wind fluctua-tions can be described as homogeneous isotropic turbulence
(Press, Meadows, and Hadlock, l956 Etkin, 1961). The nature
of the turbulence close to the ground differs from that at high altitudes. A number of measurements have been carried out (see, for example, Panofsky and McCormick, 1959). They
indicate that the scale of turbulence Lt is approximately proportional to the altitude h. The formula
(3.-8)
Tablé 3.1 - Parameter values for the influence of a mean wind on a 356 000 tdw oil tanker of KockunLS'
désign (L=350 m). Cf. (3.4), (3.7), and (3.8).
The ship characteristics are summarized in
Appendix A.
has been proposed by Etkin (1961) as an approximation valid up to h = 300 m.
Since only large vessels are considered, the ship lengths are typically of the order of 200-300 ni. The effective time constants -are 50 s or larger, and the speeds are about 5-lo rn/s. Both the ratio of the ship length to the turbulence
sqale and: the ratio of the distanbe travelled in a time
coñstant to the -turbulence scale are much-larger than one. It is -thus reasonable to assume thät the influence of the wind turbulence on the ship motion can be regarded as a
rea-lization of white- noise (cf. Aström and Käliströrn, 1976).
The following forces and moment are thus obtained from a
mean wind and turbulence (cf. (3.4)):
3.
wind=x.
wind+x
Vwind
=Y.
wind=N.
wind wind
(3.10)
Full load Ballast
TB=TS=22.3m
TB=9rn,TS=l2m
-0.007 -0.013
-0.016 -0.048
CN1 O -0.0015
where
5,
, and are realizations of white noise. The intensities are given approximately by= a VT I Cx(yR) L2 G
= a VT I CY(YR) L2 G (3.11)
= a VT ICN(YR) L3 o
where o is the intensity, of turbulence. The approximation
G = 0.2 VT (3.12)
is used when simulating tankers in Chapter 5.
3.3. Waves
A proper description of the forces and moments generated by waves is difficult to obtain. Since the only reason for having such a characterization is to obtain reasonable simu-lations, a simplified model which describes the character of the forces is used. The model, largely based on material available in the literature, is explained briefly in this section. The steering of ships in waves has been discussed by Rydill (1958), Eda and Crane (1965), Lewis (1967), Zuidweg
(1970), Eda (1972), Price and Bishop (1974), and Reid and Williams (1978). The work of Zuidweg is particularly appro-priate for our purposes.
Empirical observatioñs
When being on a large tanker in the open sea one has the distinct impression that the motions generated by waves have
a strong periodïc component. This is partly due to the
periodic nature of the waves and partly due to the fact that the tanker acts like a low pass filter. Since the waves
usually are generated by the wind, the steady state ampli-tude and frequency of the waves depend on the mean value of the wind velocity. There will, of course, also be
flutua-tions in the wave amplitude and frequency, but the dominating effect on a tanker is definitely that of a sinusoidal disturb-ance. When attempting to model the waves it is therefore
useful to look for models having this property. It is thus natural to start with the standard gravity wave model for a
regùlar sea.
-Regular waves
A simple two-dimensional wave train progressing over an infinite watei' surfabe with an infinite depth is considered. This situation is comonly refer-red to as a regular sea. If it is assumed that the wave amplitude is small compared with the wavelength and the water is incompressible, then the followïng expression for the wave profile may be obtained
(see, for example, Lamb, 1932)
(x0,t) = (h/2) cos(kx0-wt) (3.13)
where (x0,t) is the z0 co-ordinate of the sea level at X0
at the time t, and h i-s the wave height (see Fig.. 3.3). The
wave number k is defined by k = 2rr/X, where X is the
wave-length. The wave frequency is equal to 21r/T, where T is
the wave period. The theory of gravity waves gives.the
re 1-at ion
2
k 4ir g
(3.14)
where,g is the acceleration of gravity. The wave slope s(x0,t) is easily obtained from (3.13):
s(x0,-t) = = -(kh/2) sin(kx0-wt). - (3.15) dx0
z0
A
h ,. X0
Fig. 3.3 - A sinusoidal wave with wavelength X and wave height h.
A block-shaped ship with length L,. breadth B, and draught T moving in a regular sea at the speed V will now be considered
(see Fig. 3.4). The hull of a large tanker does not differ very much from a rectangular parallelepiped. It follows from Fig. 3.4 that
x =
-
(3.16)where
T is the directioñ of the true wave propagation (cf.
also Fig. 3.2) and x is the direction of the wave
propaga-tion relative to the ship.
By introducing the frequency of encounter
'e defined by
= w - ku cos x + kv sin (3.17)
expressions for the wave profile, the wave slope and the pressure p in the moving co-ordinate system xyz can be derived (seePrice and Bishop, 1974):
= (h/2) cos(kx cos X-ky sn Xet)
.
s(x,y,t) = -(kh/2) sin(kx cos x-ky sin
Xwet)
(3.18)-kz . .
p(x,y,z,t) = pg(z-e (x,y,t)]
Direction
of wove
propagation
Fig. 3.4 - Block-shaped ship in a regular sea.
To derive expressions for the forces and moments induced by waves the following two assumptions are made (see Rydill, 1958; Zuidweg, 1970; Price and Bishop, 1974):
- the forces and moments only result from pressure
- the wave field is not disturbed by the presence of the ship.
Zuidweg (1970) also proposed the following assumption to obtain expressions which are not too complicated:
- the influence of the waves is accounted for by assuming a fluctuating pressure distribution below the water surface, whereas the surface itself is assumed to be undisturbed.
By using all three assumptions the following forcés and moment induced by a regular see on a 6lock-shaped ship can be derived (cf. Zuidweg, 1970):
Xwaves = 2a B sin b
S]fl C
s(t)C
sin b sin c
waves = -2a L b s(t)
(3.19)
Nwaves = a kLB2 sin b
COS CS1fl C
-
L2 bCosb
b] (t) where -kT 2 a = pg(1-e )/k b = kL/2 cos x c = kB/2 ' sin x s(t) = s(0,O,t) = (kh/2) sin(wt) (t) = (O,O,t) = (h/2) cos(uet) Cf. (3.18).nother expression for a block-shaped ship can be obtained if the, wave surface across the hull is approximated as a plane
surface.: Xwaves = pg BLT cos x s(t) 1waves = -pg BLT sin s(t) Nwaves = pg BL(L2 - B2' --24 sin 2x s2(t) Nwaves = pg BL(L2_B2) sin 2 Tk2 . (t) (3.20) (3.21)
Notice that Nwaves of (3.20), contrary to (3.21), always
turns the ship in the same direction for a certain angle x.
Thes equations will hold only if the ship is small compared
to ti&e wavelength. Otherwise the wave can not be regarded as being plane. Notice that Xwaveg and 1waves of (3.20) are the same as those obtained by assumingkL, kB, and kT small in
This is a consequence of the assumption of a plane water surface across the hull.
The wavelengths are usually Of the order of 50-150 in, which méans that the formulas (3.20) only can be.used for small
ships. Since large tankers typically have a length of 200 -300 m, the simplifid expressioñs can not be used.
Empirical equations for wave height and period as functions of wind speed
Visual observations of waves in different weather situations have been compiled by several authors. A review can be found in Price and Bishop (1974). An approximate relationship between wind speed ànd significant wave height has been pro-posed (cf. Price and Bishop, 1974, pp.161-162). The following polynomial describes approximately the relationship:
h(VT) 0.015 V + 1.5 (3.22)
where h [mJ is the wave height and VT [m/s) is the wind speed. Figure 3.5 shows the relation (3.22). By using observations of wave däta compiled by Hogben and Lumb (1967), reprinted in Price and Bishop (1974, p.151), and the relation (3.22), the
following approximate relationship between wind speed V and
wave period T [s] can be derived:
Tw(VT) = -0,0014 V + 0.042 V + 5.6 (3.23)
This relation is shown in Fig. 3.6. The equations (3.22) and
(3.23) are approximately valid for O VT 20 rn/s. Hogben
and Lumb (1967) pointed out that visual observations of wave period often were unreliable, while it was possible to
estimate the wave height accurately. This implies that the relation (3.23) must be used with caution. Graphs similar to those in Fïgs. 3.5 and 3.6 have also been presènted by Roll
(1958). They are reprinted in Lewis (1967). Roll's graphs and Figs. 3.5 and 3.6 differ tosome extent.A possible explanation
Lm/sJ
C m/s1
Fig. 3.5 - Wave height h versus wind velocity VT as described by the relation (3.22).
Fig. 3.. 6 - Wave period T versus wind velocïty VT as described by the relatión (3.23).
for this is that Roll's observations were made in the North Atlantic Ocean, while Hogben and Lumb made a world wide compilation of wave data.
By combIning (3.14), (3.16), (3.17, (3.19), (3.22), and
(3.23) the forces and moment indüced by a regular sea on a
block-shaped ship are obtained for a true wind of speed VT
and direction
T
It is then assumed that the regular sea isgenerated by a constant wind which has been blowing for a long time. It is also assumed that the directïon of the wave propagation and the direction of the true wind are the same.
The perforinañce of ships in regular waves has also been investigated by scale model tests. Such investigations have been published by, for example, Eda and Crane (1965) and Burcher (1971)
Irregular waves
The characteriation of the real sea surface as a train of regular waves is an approximation. There are mainly two different ways to describe the irregular sea. The first method uses superposition of a large number of regular waves
of different amplitudes, f requences, phase angles, and directions to obtain a resulting water surface which is similar to that of an irregular sea. The previously developed wave induced forces and mòments can then be used for each regular wave component.
Another, approach is to regard the sea level, and the wave
such stochastic processes have been determined both
theoret-Based on the empirical observation that the wave induced motions of a large tanker have a strong periodic
slope, as stochastic processes. The spectral densities of
ically and empirically under many différent conditions (Lewis, 196.7; Price and Bishop, 1974).
component, however, it was decided to include only regular waves in the simulation model.
3.4. Summary.
In the presence of wind, waves, and currents the equations of motion (2.1) are changed to:
m(ü-vr-xGr2)=X+X
current+X+X
wavesm( + ur + xGr) = Y + Ycurrent+ 1wirx +
waves + I i + mx ( + ur) = + N
current+ + N + N
where the forces and moments from the disturbances are given
by (3.3), (3.4), and (3.19). The realizations of white noise
, , and describe the wind turbulence '(cf. (3.10)).
It should be emphasized that the equations of motion (3.24) are only approximations. They form, however, a suitable model for computer simulation Of the steering of large tankers. The equations of motion (3.24) are combined with Norrbin's model
(2.3) in the simulations of Chapter 5. The disturbances are
then completely characterized by the velocities V and Vr and
the directions y and 1T of the current and thé wind.
The influences of different disturbances on a 356 000 tdw tanker have been estimated. Balläst condition and a ship speed of 8 rn/s were assumed. A typical weather situation
character-ized by VT = 10 rn/s and 450 (cf. Fig. 3.2) was chosen.
The moment disturbances from currents of 1 rn/s combined with a yaw rate of 0.1 deg/s can be counteracted by a rudder
de-flection of 0.1°. Rudder angles of the 'order of 0.4° are
required to compensate for 'the mean wind and about 0.04° for the turbulence. The wave frequency of encounter We was about 1 rad/s. Since the tanker acts like a low pass filter, the
amplitude of the yaw rate generated by the waves is only 0.03 deg/s. The frequency
e can typically vary between 0.1
and 2 rad/s depending on the waves. Since the yaw rate amplitude depends on the frequency we. it can be as high as
O.3 aegis when We = 0.1 rad/s. Thus the rudder authority is not suffIcient to compensate for waves in all cases,' while
the other disturbances usually can be counteracted.
It is interesting to linearize the equatiàns of motion (3.24) to investigate the influence of the disturbances on a linear model. Normalization using the 'prime' system arid
lineariza-tionaboutu=u0,v=r=6=O,=p0gjve(cf.(2.8)):
+ y2 L 6 y2 N"Ló
o6+
L(m'x' -y' O \ G rjL(I' -Nñ
O Z rj o iv(Y_m' +)
v(N -m'xi
o y2 -L O y2LO
O + (3.25)where the heading isintroduced as a new state variable.
The parametérs V,,
V, V, N', N,
ánd aré due to wind andwave disturbances, while and N only appear when a current
is present. Notice, that it is possible to include unsyrnmetri-cal forces and moments from, for example, a single screw action
V L y2 L V + m' x -o
Lv
V(N +Ñ
oy
r
+
V-a11
Va2
- -1-aj3y2-y -, y -, y2
-a21
a22-a23
i o
6+
y2 L y2 L2 21 Oin the parameters V and Noticealso that the, parameters
V', V, V, FÇ,
, and are time varying when wave disturb-ances are present. If the sway velocity y is neglected when the forces and moments from wind and waves are considered, which is an adequate approximation for a model linearizedabout y O, then V' and i' are both zero.
The following state space model is derived from (3.25) by
solving for the derivatives r, i, and ij (cf.(2.12)):
(3.26)
where w1 and w2 are realizations of white noise. Notice that
1
= b11
= b1, and, if the v-dependence of forcesfrom wind and waves is neglected, = aj1, a21 = a1.
Furthermore = aj2 and a22 = a2 if no current is present.
The following transfer functions relating the sway velocity y,
the yaw rate r and the heading i to the rudder angle 6 are
obtained from (3.26) if the time-dependence due to wave
disturbances is neglected (cf. (2.13) and (2.14)):
O L 11 y2 o y r + t) W1 + w2 O
,
2V3
s+
G (s) = 2-
y3-,
V6 s3 +j
s2 a s-
a34
+4
3 G6 (s) s + s+4
s + a 2y
s+Y- b G6(s)3V
2V ¡
S+
=c'-a'
b'-& b'
2 - 12 21 22 11= ¡
b' - - 'b'
3 13 21 23 11Cf. (2.15). The transfer function G6 contains an integrator
when there are no wind and wave disturbances, but it follows from (3.27) that this is not necessarily the case in the presence of wind and wave disturbances.
(3. 27) where I 1 a a' 3 2 - I
-
11 =a11 a2
- 11 23 == ¡i
, 21 11 22-
a12 a1
' a' 13 21 I 11 21- a3
(3.28)
3.5. References
Aström, K J, and C G Källström (1976):
Identification of ship steering dynanics. A ornati..ca 12, 9.
van Berlekorn, W B, P TrägArdh, and A Delihag (1974):
Large tankers - wind coefficients and speed loss due to
wind and sea. Tan4. Royal I titatZon o
PJavoi A.tect4
117, 41, London, 1975.
Burcher, R K (1971):
Developments in ship manoeuvrability. T'tan4. Royal
lnst-ta-t-on o
Naval Aect
114, 1, London, 1972.Eda, H, and C L Crane, Jr (1965):
Steering characteristics of ships in calm water and waves.
Tnani. Soce.ty o Naval AJr.chL.ta-t4 and Mane Engnv
73, New York. Eda, H (1972)
Directional stability and control of ships in waves.
Joatnal o ShÁ..p Reak, September 1972, 205.
Etkin, B (1961)
Theory of the flight of airplanes in isotropic turbulence; review and extension. UTIA Report No. 72, Iñstitute of Aerophysics, University of Toronto, Toronto, Canada.
Hogben, N, and F E Lumb ('1967):
Oaean StatLotLc4. H.M.S.O., London.
Lamb, H (1932)
Hydodynami4, 6th ed. Cambridge University Press,
Cambridge, Great Britain.
Lewis, E V (1967):
The motion of ships in waves. In J P Comstock (Ed.):
PncLples o
Naval Society of Naval Archi-tects and Marine Engineers, New York.Panofsky, H A, and R A McCormick (1959):
The spectrum of vertical velocity near the surface. lAS Report No. 59-6, Institute of the Aeronautical Sciences, New York.
Press, H, M T Meadows, and I Hadlock (1956):
A reevaluation of data oñ atmospheric turbulence and air-plane gust loads for application in spectral calculations.
Report 1272, Natiönal Advisory Committee for Aeronautics, Washington.
Price, W G, and R E D Bishop (1974):
PkobabÁlitic Thoky o ShLp Vynami4. Chapman and Hall,
Reid, R E, and V E Williams (1978):
A new ship control design criterion for improving heavy weather steering. Proc. 5th Ship Control Systems
Sympo-sium, Vol. 1, paper Cl-1, Annapôlis, Maryland, USA.
Roll, H U (1958):
Height, length and steepness of seawaves in the North
Atlantic and dimensions of seáwaves as functions of wind force. Technical and Research Bulletin No. l-19, Society
of Naval Architects and Marine Engineers, New York.
Rydill, L J (1958):
A linear theory for the steered motion of ships in waves.
Tavi. In
¿tuCLon o
Navas AchL.tc.t4
101, 81,
London,1959.
Wagner, B (1967):
Windkräf te an UbérwassersChiffen. Sc.hL
u.nd Ha1n,
Heft 12 (in German).
Zuidweg, J K (1970):
Automc.tLc GaLdahce o
ShLp4 a4 a Con.titoe PkobUm.
Thesis, Technische Hogeschool, Delft, the Netherlands.t SYSTEM IDENTIFICATION
Applications of system identification techniques to data
from full scale experiments performed with four different
ships are described. The test ships are three oil tankers
and a cargo ship. The ship characteristics are given in
Appendix A. Fourteen experiments corresponding to a total time of 13 h are analysed. The data contain over 6 000
samples. The experiments are summarized in Appendix B.
In the tanker experiments coniinand signals were given to the
rudder engine. The resulting motion was observed. The expe-riments were performed using the oñ board process computer. The tests were made both under open and closed loop condi-tions. In the open loop experiments the autopilot was
dis-connected and changes in the rudder were commanded. The
inpüt signals were chosen so that they excited the ship
steering dynamics properly. In most cases signals which were
close -tO pseudo-random binary sequences (PRBS) were used.
Some care was used to make sure that the ship maintained a
reasonable heading during the tests. In the closed loop
experiments PRBS-like perturbations were introduced as changes in the heading reference to the autopilot. It was also attempted to excite the system parametrically by
chang-ing the autopilot gain while keepchang-ing the heading reference
Constant.
The tankers were equipped with similar measurement devices. The velocities were measured by a doppler sonar equipment,
type Ainetek Straza, on the Sez Seowt and the Se.z SwL. A
doppler log from Atlas was used on the Sect Sat. The same
resolution, about 0.02 knots, was given by both manufacturers. The yaw rates were measured by a rate gyro from AB ATEW, Fien, Sweden. The drift rate given by the manufacturer was 0.0008
deg/s-. However, the quality of the rate gyro signal varied
with the sea conditions and the way the gyro was mounted. The
accuracy waà crudely estimated to about 0.005 deg/s. The heading angles were measured by a Sperry gyrocompass. The
signal from the gyrocompass was converted with a resolution
of 0.09. deg on the
Se
Saoa.
The resolution was 0.02 deg onthe
Sea. SL
and theSea. Ska.t4.
Three zig-zag tests with the Cornpa.44 Lo.Lcnd, which is a cargo
ship of the M
.nvi
class, are also analysed. The experimentsàre reported in the literaturé. An. inertial navigation, system
was used on the Compa.44 I.o.&a.nd, which made it possible to
obtain measurements with high- precision. Iiowever, the data
analysed were obtained, from graphs in Morse. and Price (1961).
Naturally a lot of the precision inherent in the measurements were lost when the graphs were digitized.
The experiments are evaluated using a general purpose iden-tification program LISPID based oñ the prediction error method. The maximum likelihood (ML) method and the output error method are also included in LISPID.. The experimental data are analysed with different identification methods and
the results obtained are compared. The ML procedure is also
applied to some of the experiments using the interactive identification program IDPAC, where the parameters of a
geheral, linear difference equation model are estimated. The
parameter values obtained from LISPID and IDPAC are compared with initial estimates based on theoretical calculations and captive scale' model tests. The initiäl values for the tankers
were provided.by the Swedish State Shipbuilding Experimental
Tank (SSPA). The initial parameters for the Cornpa.44 I4a.vLd
were determined by the .Hyd-ro- and Aerodynamics Laboratory.
(HyA, now SL), Denmark. See Chislett and. St-r$m'-Tejsen (1965).
Hydrodynamic derivatives normalized in the 'prime' system
are usually. given with five decimals in -the literature. This
practice ïs followed here even if all digits are not
There is a substantial amount of literature available on system identification. See, for example, Aström and Eykhoff
(1971), Eykhoff (1974), Mehra and Lainiotis (1976), Aströin
(1977), Goodwin and Payne (1977). System identification
tech-niques have been applied to determine ship steering dynamics
from full scale experiments for a long time. Nomoto proposed
a straightforward least squares method for estimation of the
two parameters of (2.16) from zig-zag tests (Nomoto and
co-workers, 1957; Nornoto, 1960). Parameters of nonlinear
models have been estimated using equation error and output
error methods by Kaplan, Sargent, and Goodman (1972), van
Amerorigen, Haarman, and Verhage (1975), Gill (1975), Clarke
(1976), and Oltmann (1978). Blanke (1978) has applied
fre-quency response analysis to determine a nonlinear speed equation. The maximum likelihood method has been used to
determine linear ship steering mode1 by Aström and Källströrn
(1972, 1973, 1976), Aström, Källström, Norrbin, and Byström
(1975), Tiano, Piattelli, and Leccisi (1973), Tiano (1976),
Tiano and yolta (1978), Flôwer and Towhidi (1975), Ohtsu, Horigorne, and Kitagawa (1976), and Thöm (1976). The last reference also describes applications of different versions
of least squares identification methods. Parameter of a
nòn-linear model have been estimated by the ML method in Norrbin, Byströni, Aström, and Källström (1977) and Byström and Käll-ström (1978), where a few results of prediction error identi-fications also are given. A more comprehensive treatment of applications of system identification to ship dynamics is
given in Aström, Källström, Norrbin, and Byström (1975).
4.1. LISPID Model Structures
The computer program LISPID (Linear System Parameter
IDenti-fication) is designed for identification of linear
mu1tirari-able stochastic systems (Käliström, Essebo, and Aström, 1976;
allowed in LISPID. The model can be given in continuous ôr discrete time form. Different déscriptioñs of process and measurerüent noise are permitted. The model cari be time
vary-Ing. The sampling can be uniform or varying, añd different types of measurements, such as instantaneous and. integrating, are permitted. The model may be parametrized in an arbitrary manner. Different identifiCation procedures, such as predic-tion error, maximum likelihood, and output error methods, are included in LISPID.
Three different ship steering models are incorporated into LISPID.. T.hey are of continuous time form, but the parameters are estimated using instantaneous, discrete time measure-ments. The models are time invariant and the datà analysed are usually recorded with a constant sampling interval. The models are described below.
Linear model for estimation of hydrodynamic. derivatives
13 014
o
dt + dw (4.1)
The
Paramètérs of the equations following parametrization
Le
L2 l7e2
dv of motion (3.25) are was chosen: 1 LV85
V86
09 y (t) estimated. L 03 L2 04 dr 1V07
LV°s
8901 r(t) dt + dip 1 0 ip(t) + ô(tTD) dt +R1 (tk) It follows that a2 L1a O a2 o o 1/al o O O 1/a1 k = 0, 1, ..., N-1
where w is a stochastic process with incremental covariance R1dt. The matrix R1 is parametrized as
10181
v'1018
110191 sin 020 o y (t,,) r(tk) (tk) +V1e181
1019H
I09I OThe covariance of the measurement errors is
R2 = diag
(10211. IO22I
102311 ie24iJ.The initial state is given by
v(t0) 025/a2
r(t0) = a1026
a10
and the time delay TD is parametrized as
TD = T5-T5 sin 0281'
where T is the sampling interval.
015 016 017 o + e(tk) sin 020 0 O o
O=
i
V 02 = 09 = 03m'x-N
010 = e4= I-N
011 = -yt
- _Ts1 Ivi
5 y 'l2 5' ô 06 = 013=N
eET'
y 14 Owhere it has been assumed that
T = = = = 0.
In the model (4.1) all stochastic process disturbances are modelled by w. Possïble errors in the model structure are also absorbed in w. The special form of the ent-ries R1(l,2) and R1(2,l) guarantees that the matrix R1 is nonnegative
definite. The parameters 015 0 represent measurement
biases.
The inpùt signal ô of the model (4.1) can be chosen as the rudder command to the steering engine or as the rudder angle. Usually the time constant of the steering engine is small compared with the dominant time constant of the steering dynamics of large ships. Thus the time delay TD of the model
(4.1) can be regarded as an approximation of the time con-stant of the steering engine, when the rudder command is chosen as input signal. The special expression for TD
guarantees that O TD T5.
The outputs of the model (4.1) are the fore and aft sway velocities y1, V2, the yaw rate rm, and the heading angle
It is not necessary to use all output signals. The ship speed V, the ship length L, and the distances L1, L2 from the origin to the sway velocity sensors are given fixed values dependent on the experiment. c1 and cx2 are
conver-sion factors from degrees to radians and from rn/s to knots.
08
= N-m'x
Subsets of the parameters 0 - 028 can be estimated. Para-meters which are not estimated must be given a pn.oì.
The model (4.1) is transformed iñto standard, discrete time, state space form in LISPID. The unknown parameters are esti-mated by minimization of the criterion:
v=
det p(tk) ET(tk) (4.3)The number of samples are denoted N. The p-step prediction
errors are determined recursively through (see Källström,
Essebo, and Aström, 1976):
cp(tk) = y(tk) - C(tkJtk_p) - Du(tk)
(tjlItk_p) =
A(tjltkp)
+ Bu(t),i = k-p, ..., k-1
(tk_pItk_p) = (tk_pItk_p_1) +
K[y(t_)
-- C(tk_pItk_p_l) -- Du(tkp)]
k=p,...,N-1
The maximum likelihood method is obtained by minimizing the oné-step prediction errors, i.e. by using p=1 in (4.3) and
(4.4). The output error method is obtained from the ML method by assuming no process noise in (4.1), i.e. w=O. This im-plies that K=O in (4.4).
Different models obtained with ML and output error methods can be compared by using Akaikets information criterion
(Akaike, 1972):
AIC -2 log L + 2v (4.5)
where L is the maximum of the likelihood function and y is. the number of estimated parameters. According to Akaike the quantity AIC should be minimum for the correct model. The
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