163
1 INTRODUCTION
Despite the fact that the total number of navigational
accidents (collisions, allisions, groundings) has
decreased in the last decade, emergency cases
involving large-tonnage vessels are quite frequent. As
for today, modern container fleet keeps growing in
size and capacity. For example, the container ship
OOCL Hong Kong with a length of 400 meters, a
width of 59 meters and a draft of 16 meters, with a
capacity of 21,413 TEU was launched in 2017. At the
same time, according to insurers’ assessments (Allianz
2018) the loss of a container ship with a cargo capacity
of 20,000 TEU could cost as much as 1 billion US
dollars. Obviously, with the increase in the size of
ships, the problem of ensuring their navigation safety
in narrow waters becomes even more critical.
Mathematical modelling and simulation are necessary
processes involved into design and operation of ships
and port facilities. At the same time, physical
modelling using scaled models is time consuming and
expensive, which, if necessary, is performed at the
final design stage. Proper mathematical modelling
helps to find out limitations and possible problems or
look for optimal solutions at early design stage as
well as in the subsequent design process.
This research is directed to the ULCS class
container ship mathematical model adjustment on the
basis of existing sea trial data.
2 PREVIOUS RESEARCH ANALYSIS
Extensive research dedicated to the vessel
maneuvering modeling was carried out in the past
and published by numerous authors. Generally
speaking, we can divide existing models in two
groups: linear models, which include course control
with constant speed, which are widely used for
autopilot design (Pipchenko, Shevchenko 2018) and non-
linear models, which include vessel dynamic
calculation in wide motion parameters range.
Features of an Ultra-large Container Ship Mathematical
Model Adjustment Based on the Results of Sea Trials
O. Pipchenko, M. Tsymbal & V. Shevchenko
National University “Odessa Maritime Academy”, Odessa, Ukraine
ABSTRACT: The research addresses the problem of an ultra-large container ship mathematical model
adjustment based on sea trials. In order to verify the model’s adequacy, simulated data had to be compared to
the trial report data, which was obtained in ballast condition with significant trim. In such circumstances,
model coefficients cannot be calculated by known methods and have to be corrected as per trial data. It is
proposed to determine translational motion coefficients first. To get optimal results, it was also proposed to
divide the objective function into kinematic and dynamic components, with each component being assigned a
weighting factor. A separate objective function component was assigned to the zig-zag maneuver, which
includes the first and second overshoot angles.
http://www.transnav.eu
the
International Journal
on M
arine Navigation
and Safety of Sea Transportation
Volume 14
Number 1
March 2020
DOI:
10.12716/1001.14.01.20
164
The most common are 3 DoF (degrees of freedom)
maneuvering models of two main types. In the first
case it’s a system of equations for longitudinal and
transverse speed and rate of turn in relation to a
vertical axis as shown by Fossen (2002), Kijima et al.
(1993), Perez & Blanke (2003), Yasukawa et al. (2015),
Yoshimura et al. (2012); in the second case it’s a
system of equations for forward movement speed,
drift angle and rate of turn in relation to a vertical axis
as shown by Gofman (1988) and Pershitz (1983).
From a mathematical modeling perspective, when
forces of different nature such as wind and wave
forces, currents, tugs, thrusters are considered,
especially in case of maneuvering calculation at near
zero speeds, it is more convenient to build a model
with the motions separated by dedicated axis.
Forces and moments acting on a ship can be
calculated using equations from various sources such
as Kijima et al. (1993), Perez & Blanke (2003),
Yasukawa et al. (2015), Yoshimura et al. (2012), ITTC
(2005), ABS (2006) and others.
Model coefficients may be found by formulas,
generalized for a number of ship types, which in
return usually leads to calculation errors, still too big
for navigational safety evaluation.
The other approach is to apply both parametric
and functional approximation using neural networks
as suggested by Pipchenko and Zhukov (2007), but
later requires a substantial amount of experimental
data, which is not always cost-effective.
Therefore, if we choose the approach way, after
preliminary model coefficients calculation, it should
be adjusted according to available experimental data.
3 EQUATIONS OF MOTION
The system of equations which describes vessel
motion on the horizontal plane can be presented as:
( )( )
( )( )
()
xy
G GG
yx
G GG
zz zz
GG
mmu mmvr X
mmv mmur Y
I J r N xY
+ −+ =
+ ++ =
+=−
(1)
where m – vessel displacement; m
x, my – added
masses, I
kk, Izz – moments of inertia, Jkk, Jzz – added
moments of inertia, u
G, vG, pG, rG– longitudinal and
transverse speed and rate of turn with respect to
horizontal and vertical axes related to the vessel
center of gravity; X, Y, K, N – hydrodynamic forces
and moments acting on ship.
Hydrodynamic forces and moments can be
presented as:
HRP
HRP
HRP
XX X X
YY Y Y
NN N N
= ++
= ++
= ++
(2)
where H - hull; R – rudder; P – propeller; W – wind;
BT – bow thruster.
Forces and moments acting on a ship hull can be
derived on the basis of the model offered by
Yoshimura (2012). Water resistance forces and
moment (X
H, YH, NH) together with forces and
moment of inertia can be given as follows:
2
2
''2' ''''''2' 4
0
2
2
' ' ''' 3' 2'' '2''3
2
2
X m v r LdU
ygg
H
X X X m r X xm r X
y rr y
rG
Y m u r LdU
xgg
H
Y Y mrY Y rY r Y r
r x rrr
r rr
N L dU
H
ρ
ββ β
ββ β ββββ
ρ
β ββ β
β βββ ββ β
ρ
+= ×
+ + − +− +
−= ×
+− + + + +
=
2
' ''' 3' 2'' '2''3
N NrN N rN r N r
r rrr
r rr
β ββ β
β βββ ββ β
×
++ + + +
(4)
Figure 1. Coordinate system for ship movement modelling
where
ρ
– water density,
β
- drift angle, positive to
port side; X
0, X
ββ
, X
ββββ
, Xrr, X
β
r, Y
β
, Y
βββ
, Yr, Yrrr, Y
ββ
r,
Y
β
rr, N
β
, N
βββ
, Nr, Nrrr, N
ββ
r, N
β
rr – resistance forces
coefficients.
Thrust force created by propeller can be calculated
as:
(1 ) ;
PP
X tT=−⋅
(4)
(
)
24
;
PPTP
T nDKJ
ρ
=⋅⋅ ⋅
2
01 2
() ;
TP P P
KJ k kJ kJ= +⋅ +⋅
(5)
( )
1
,
P
P
PP
uw
J
nD
−
=
165
where T – propeller thrust, t
P – thrust deduction
coefficient, n
P – propeller revolutions, DP – propeller
diameter, K
T – thrust coefficient, JP – propeller slip, wP
– hull influence coefficient.
Forces and moment created by rudder can be
defined by formulas:
( )
( )
( )
1 sin
1 cos
cos
R RN
R HN
R R HH N
X tF
Y aF
N x ax F
δ
δ
δ
=−−
=−+
=−+
(6)
where F
N – normal force created on the rudder: tR, aH,
xH – coefficients, which reflect hydrodynamic
interaction between hull, propeller and rudder; x
R –
distance from midship section to rudder stock.
2
1
sin ,
2
a
RR R
FN A U f a
ρ
=
where A
R – rudder area, UR – water flow speed on the
rudder, f
a – lifting factor, aR – effective inflow angle on
the rudder.
Coefficients of equations (3), (4) and (6) can be
defined according to methods given in Kijima et al.
(1993), Perez & Blanke (2003), Yasukawa et al. (2015),
ITTC (2005), ABS (2006) and others or can be taken
from databases for ship with proportional dimensions
(Yoshimura et al. 2012).
4 LOGITUDINAL MOTION MODEL
ADJUSTMENT
It is reasonable to start the mathematical model
coefficients adjustment from ship forward motion
equation as it can be separately allocated from
common system of equations (Pipchenko et al. 2017).
During further adjustment ship forward motion
equation coefficients will not be changed.
Corresponding scripts for ship motion calculation
and further adjustment were written in MATLAB
R2016b.
Typical trial maneuvers, which involve
longitudinal motion, are acceleration, crash stop and
inertial stopping. In this case data was taken from sea
trials report of 10000 TEU, 2015 year-built container
ship Maersk Sirac. Main parameters of this vessel are
given in Table 1.
Table 1. Maersk Sirac – vessel information
_______________________________________________
Parameter Value
_______________________________________________
Overall length, m 300
Length between perpendiculars, L, m 287
Breadth of vessel, В, m 48.2
Draught (mean / maximum) at load, d, m 12.5/15.0
Forward draught at trials, m 4.02
Aft draught at trials, m 10.16
Propeller diameter, D
P, m 9.7
Block coefficient (ballast), C
b 0.6044
Wet surface area,
Ω
, m
2
11656
Midship section plane coefficient, C
M 0.9735
Rudder area, A
R, m
2
78.95
_______________________________________________
To perform calculations, ship motion equation
along X axis can be expressed in following form:
( )
24 2
0
1
(1 )
2
x
P P P TP
mm
t n D K J X LdU
u
ρρ
+
− ⋅⋅ ⋅ ⋅ + ⋅⋅⋅⋅
=
. (8)
Coefficient Х
0 = - 0.014 for this case, was estimated
from sea trials. It is important to note that absence of
Х
0 credible value increases uncertainty of other
coefficients values during adjustment. When Х
0
experimental value is absent it is useful to apply
resistance calculation methods on still water (i.e.
Holtrop, 1982).
Coefficients t
P and wP can be defined using
approximate formulas:
0.1885
0.325 ;
0.5 0.05.
P
Pb
Pb
D
tC
Bd
wC
⋅
= ⋅−
⋅
= −
Coefficients of the J
P can be approximated by
known propeller trials data. In our case this data is
absent and in first approximation relation between
ship speed and propeller revolutions was received
(figure 2).
If we have a close look on a thrust K
T and advance
ratio J
P coefficients formulas when negative
revolutions are set, the thrust coefficient can gain
incorrect value. This is because the J
P will be negative
when the speed is positive and, as follows, parts of
the equation (5) will be deducted from coefficient k
0.
To obtain realistic values for astern maneuver
equations (4) and (5) shall be presented as:
( )
4
;
PTP PP
T DKJ nn
ρ
=⋅ ⋅ ⋅⋅
(9)
( )
1
.
P
P
PP
uw
J
nD
−
=
(10)
After equation (4) coefficients adjustment using
Nelder–Mead method the calculations result is almost
matches with the experiment, with average deviation
of 0.26 knots. The objective function used in
optimization has following form:
1
N
Tn
Sn
n
UU
Z
N
=
−
=
∑
, (11)
where U
Т – sea trials measured speed, US – speed as
result of simulation.
But further calculation of crash and inertial
stopping maneuvers doesn’t give a satisfactory result.
This is because the optimization program adjusts only
coefficient k
0 while other coefficients decrease almost
to zero. This, in turn, excludes propeller advance
effect from the model. Therefore, to achieve adequate
optimization results it is necessary to include all three
maneuvers: acceleration, inertial and crash stopping
into objective function calculation.
166
Considering the above, the objective function (11)
will look like:
12
3
45
11
1
11
NN
Tn Sn CSTn CSSn
nn
N
ISTn ISSn
n
NN
CSTn CSSn ISTn ISSn
nn
ww
UU U U
Zw w
NN
UU
w
N
D D DD
NN
= =
=
= =
++
+
+
−−
=
−
−−
∑∑
∑
∑∑
, (12)
where D – track reach, w – weighting factor, index CS
– crash stop, IS – inertial stopping.
As errors in distance and speed have different
order it is necessary to normalize them using the
weighting factors. In our case w = [1 1 1 0.001 0.001].
As a result of re-optimization, the coefficients k
1, k2
and k
3 will be adjusted which gives the result with
satisfactory accuracy, shown on figures 2-4 and in
table 2.
Table 2. Speed trial adjustment results
_______________________________________________
Parameter Value
_______________________________________________
Average speed deviation during 0.26 knots
acceleration, ∆
U
Average speed deviation during crash 0.59 knots
stopping, ∆
UCS
Average speed deviation during inertial 0.50 knots
stopping, ∆
UIS
Crash stop track reach calculation relative 0.04 % / 1.0 m
error, ∆
DCS
Inertial stopping track reach calculation 1.5 % / 77.1 m
relative error, ∆
DIS
Coefficients before adjustment Coefficients after
adjustment
k
0 k1 k0 k1
0.16 -0.068 0.06104 0.8632
k
2 k3 k2 k3
0.074 0.022 -1.0901 0.067
_______________________________________________
5 MANEUVERABILITY MODEL ADJUSTMENT
Typical maneuvers for ships’ turning capacity trials
are turning circles and zig-zag 10/10°.
According to the trial report data the propeller
revolutions during turning circle will change from 83
rpm to 54 rpm.
As shown in table 1, sea trials were conducted
with ship in ballast condition with the trim of 6.14 m
and the average draught of 7.09 m. Ship’s average
operational draught is usually twice bigger and trim
is close to zero. In this regard, model coefficients
calculation using empirical formulas will lead to big
errors.
At this stage of model adjustment, it is important
to define which parameters reflect the accuracy of the
obtained results and a corresponding form of
objective function.
In this case, it is useful to divide an objective
function Z into dynamic Z
D and kinematic ZK parts.
From sea trials data on a turning circle maneuver we
can get the following parameters: ship’s speed,
heading, coordinates, advance and tactical diameter.
Consequently:
12
11
NN
Tn Tn
Sn Sn
nn
D
UU rr
Zw w
NN
= =
+
−−
=
∑∑
; (7)
( )
( )
( )
( )
( )
( )
( )
3
4
1
max max
max
max max
max
max
T
S
T
K
T
S
T
N
n
T
XX
X
Zw
YY
Y
D
w
NX
=
−
+
= +
−
∆
⋅
∑
(8)
( ) (
)
22
TT
SS
D X X YY∆= − + −
,
where ∆D – position error; w
i – weighting factor;
index T – trial data; index S – calculated data; max
(X
T) – position, indicating turning circle tactical
diameter; max(Y
T) – position, indicating advance.
Let’s pick first and second overshoot angles errors
as zig-zag maneuver objective function component:
11 22
5
12
TS TS
Z
TT
Zw
ψψ ψψ
ψψ
∆ −∆ ∆ −∆
= +
∆∆
(9)
As errors in distance and speed have different
order lets normalize them using the weighting factors.
In our case w = [1; 180⋅60/π; 2; 1; 0.2].
Consequently, objective function will be defined
as:
DKZ
ZZ Z Z=++
(10)
Further it is useful to define coefficients which
have to be adjusted. In our case the algorithm will
vary 19 coefficients included in hull resistance and
rudder forces equation:
' ' ''
' '' ' ' '
' '' ' ' '
, ,, ,
,, , , , ,
, , , , , ,, ,
rr
r
r rrr
r rr
r rrr
R
r rr
h
X X XX
YYY Y Y Y
NNN N N N a
ββ β ββββ
β βββ ββ β
β βββ ββ β
εγ
167
Figure 2. Relationship between the ship’s speed and propeller revolutions
Figure 3. Crash stopping curves
Figure 4. Inertial stopping curves
168
Ship movement model coefficients adjustment
algorithm block-diagram is shown on figure 5. As
described above, the first stage is ship longitudinal
motion model adjustment. Model initial coefficients
are chosen from appropriate database. Then the
model calculation and trials data comparison is
conducted. If the model accuracy doesn’t satisfy
chosen criteria, adjustment by the Nelder–Mead
method is conducted. As a result, refined coefficients
will be recorded to database.
The modelling results at the first step and after
coefficients adjustment are given on figures 6-8 and in
table 3. As seen, adjustment procedure allows to
decrease modelling errors sufficiently.
Figure 5. Ship motion mathematical model adjustment
algorithm block-diagram
Figure 6. Starboard side turning circle trajectory
Left – before adjustment; right – after adjustment; o – trials data, * - calculation data.
Figure 7. Starboard turning circle ship motion parameters.
Left – before adjustment; right – after adjustment; o – trials data, * - calculation data.
169
Figure 8. Course-keeping abilities parameters: zig-zag 10/10.
Left – before adjustment; right – after adjustment; dashed line – trials data; solid line – calculation data
Table 3. Mathematical model adjustment results
__________________________________________________________________________________________________
Parameter Trials Before adjustment After adjustment
__________________________________________________________________________________________________
Advance, m 830.9 1255 836
Turning circle tactical diameter, m 1164.3 1630 1148.3
1-st overshoot angle, ° 2.5 3.8 3.0
2-nd overshoot angle, ° 3.2 - 3.3
RMSD position - 589.2 49.9
RMSD course - 28.5 10
__________________________________________________________________________________________________
Coefficients before and after adjustment / difference %
__________________________________________________________________________________________________
'
X
ββ
'
r
X
β
'
rr
X
'
X
ββββ
'
Y
β
'
r
Y
'
Y
βββ
'
r
Y
ββ
'
rr
Y
β
'
rrr
Y
__________________________________________________________________________________________________
-0.0626 -0.1149 -0.00068 0.4182 0.3099 0.1207 1.5816 0.6323 0.7173 0.0088
-0.2617 -0.1531 -0.00069 0.47811 0.1044 0.1795 2.7160 0.9423 1.3620 0.001
318 33 2 14 66 49 72 49 90 89
__________________________________________________________________________________________________
'
N
β
'
r
N
'
N
βββ
'
r
N
ββ
'
rr
N
β
'
rrr
N
ε
R
γ
h
a
__________________________________________________________________________________________________
0.0179 -0.03025 0.2407 -0.6018 0.077 -0.03 0.902 0.350 0.3674
0.0087 -0.03 0.2259 -0.6445 0.109 -0.055 1.344 0.312 0.3422
52 4 6 7 41 79 49 11 7
__________________________________________________________________________________________________
6 CONCLUSIONS
In order to verify the model’s adequacy, simulated
data had to be compared to the trial report data,
which was obtained in ballast condition with
significant trim. In such circumstances, model
coefficients cannot be calculated by known methods
and have to be corrected as per trial data.
It is proposed to determine translational motion
coefficients first. To get optimal results, it was also
proposed to divide the objective function into
kinematic and dynamic components, with each
component being assigned a weighting factor. A
separate objective function component was assigned
to the zig-zag maneuver, which takes into account the
first and second overshoot angles.
The mathematical model adjustment was
performed using the Nelder-Mead downhill simplex
method, which allowed to obtain high accuracy
results in order to fit both vessel transitional
dynamics process and output kinematic parameters
such as track reach, advance and tactical diameter.
It is important to note that obtained coefficients fit
only the specific vessel, on the other hand, the
algorithm and obtained objective functions may be
applied to a wider scope of vessels with different
shapes and dimensions.
REFERENCES
ABS Guide for Vessel Maneuverability, 2006
Allianz Global Corporate & Specialty. Safety and Shipping
Review 2018. http://www.agcs.allianz.com/
Fossen T. I. 2002 Marine Control Systems. Guidance,
Navigation and Control of Ships, Rigs and Underwater
Vehicles. Marine Cybernetics, Trondheim, Norway
Gofman A. D. 1988. Движительно-рулевой комплекс и
маневрирование судна. Справочник. Л.: Судостроение.
Holtrop J., Mennen G. 1982. An Approximate Power
Prediction Method. International Shipbuilding Progress, 29,
335
170
Ho-Young Lee, Sang-Sung Shin. 1998 The Prediction of
ship's manoeuvring performance In initial design stage.
Practical Design of Ships and Mobile Units. Elsevier Science.
IMO MSC 76/23, 2002. “Resolution MSC.137 (76), Standards
for Ship Manoeuvrability,” Report of the Maritime
Safety Committee on Its Seventy-Sixth Session-Annex 6
Kijima,K., Tanaka,S. Furukawa, Y. and Hori, T. 1993. On a
Prediction Method of Ship Manoeuvring Characteristics/
// Proc. of MARSIM-93, Vol.1.
MAERSK SIRAC 2015. Sea Trial results. Hyundai Heavy
Industries.
Pershitz R. Ya. 1983. Управляемость и управление
судном. – Л.: Судостроение
Pipchenko A. D., Zhukov D. S. 2007. Development of the
Ship Dynamics Model on the Basis of an Artificial
Neural Network. IAMU №5, Vol. 1. Tokyo. Japan, 2007
Pipchenko O., Kopanskiy S., Shevchenko V. 2017.
Refinement of an ultra large container vessel
mathematical model based on the speed trials results.
Shipping & Navigation. NU OMA Research journal (2017)
27
Pipchenko O., V. Shevchenko, 2018. Robust automatic ship
heading controller for various conditions. Marine
Intellectual Technologies - Scientific journal № 4 (42) V.4
Pipchenko О., Tsymbal M., Shevchenko V. 2018
Recommendations for Training of Crews Working on
Diesel-Electric Vessels Equipped with Azimuth
Thrusters. TransNav – the International Journal on Marine
Navigation and Safety of Sea Transportation. Vol. 12, Issue 3.
T. Perez, M. Blanke, 2003. Mathematical Ship Modeling for
Control Applications. (Technical Report) DTU Technical
University of Denmark.
The Manouvering Committee. Final Report and
Recommendations to the 24th ITTC, 2005.
Yasukawa H., Yoshimura Y. 2015. Introduction of MMG
standard method for ship maneuvering predictions.
Journal of Marine Science & Technology (2015) 20.
Yoshimura Y., Masumoto Y. 2012. Hydrodynamic database
and manoeuvring prediction method with medium
high-speed merchant ships and fishing vessels.
International MARSIM Conference. 2012.
