International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 5
Number 4
December 2011
A number of models for ship-ship collision probabil-
ity estimation can be found in the literature. They
can be divided into two major groups namely: static
and dynamic models. The static models are simpler
and less time consuming for computation, however
their accuracy can be questioned. The dynamic
models are more complex, in principle they need
more input data than static models, but their results
are comprehensive. In this section the short over-
view of existing models will be made, and our con-
tribution to the existing knowledge will be put for-
1.1 Static models
The most known approaches were introduced by
(Fujii et al. 1970) and (MacDuff 1974). Models of
this kind have been commonly used in recent dec-
ades and won the popularity among researches main-
ly due to simplicity and robustness. However they
have also some drawbacks, lack of ship dynamics or
assumption regarding a collision between two ships.
A collision is defined as a meeting of two ships in a
distance named the ”collision diameter”, which
means almost the physical contact. Such an assump-
tion may lead to an understanding that in any ship-
ship encounter at a distance greater than the ”colli-
sion diameter” these ships are able to avoid a colli-
sion, which in most cases is not true. Despite the
drawbacks the model was adopted by (Pedersen
1995), and with minor modifications was used to de-
termine the safety of navigation in many European
waters: (Otto et al. 2002), (Sfartsstyrelsen 2008).
Hence in Europe it is mostly known as Pedersen
model. Another method for the frequency of colli-
sion estimation, making an assumption regarding
uncorrelated traffic, was outlined by (Fowler and
Sorgrad 2000). A critical situation is assumed to oc-
cur when ships come to close quarters to a distance
of 0.5 Nm of each other, which is constant regard-
less of a meeting scenario. A model for encounter
probability estimation proposed by (Kaneko 2002)
defines a critical area of an optional form of a closed
boundary, around a ship which violation means col-
lision. Kaneko in his model recognizes two shapes
of the critical area: rectangular and circular, but
again the size of the area is fixed. A series of papers
utilizing the ship domain approach to ship-fixed ob-
ject collision assessment was published also by
(Gluver and Olsen 1998) and (Pedersen 2002).
However, none of the model listed above takes
ship dynamics into consideration.
1.2 Dynamic models
Another group of models utilize marine traffic simu-
lations. A group of researches led by Merrick pro-
posed a risk analysis methodology for maritime traf-
fic in coastal areas based on system simulation
A New Definition of a Collision Zone For a
Geometrical Model For Ship-Ship Collision
Probability Estimation
J. Montewka ‡, F. Goerlandt & P. Kujala
‡Aalto University, School of Engineering, Finland; Maritime University of Szczecin, Poland
Aalto University, School of Engineering, Finland
ABSTRACT: In this paper, a study on a newly developed geometrical model for ship-ship collisions probabil-
ity estimation is conducted. Most of the models that are used for ship-ship collision consider a collision be-
tween two ships a physical contact between them. The model discussed in this paper defines the collision cri-
terion in a novel way. A critical distance between two meeting ships at which such meeting situation can be
considered a collision is calculated with the use of a ship motion model. This critical distance is named the
minimum distance to collision (MDTC). Numerous factors affect the MDTC value: a ship type, an angle of
intersection of ships’ courses, a relative bearing between encountering ships and a maneuvering pattern. They
are discussed in the paper.
(Merrick et al. 2002), (Merrick et al. 2003). Mari-
time traffic is simulated in the time domain based on
routes obtained from expert opinion and vessel arri-
val records. Finally, these were combined with the
simulation output in order to carry out a risk analysis
(van Dorp and Merrick 2009).
Another probabilistic model for the assessment of
navigational accidents in an open sea area was out-
lined by (Gucma and Przywarty 2007). The method
makes use of a simplified model of maritime traffic,
which is simulated in the time domain. A recent
model, introduced by (Goerlandt and Kujala 2011) is
based on an extensive time-domain simulation of
maritime traffic in a given area. Vessel movements
are modelled based on data obtained from a detailed
study of route-dependent vessel statistics. The colli-
sion candidates are detected by a collision detection
algorithm which assesses the spatio-temporal propa-
gation of the simulated vessels in the studied area.
Markov, semi-Markov and Random Field theory
based models for maritime traffic safety estimation
were introduced recently (Smalko and Smolarek
2009), (Smolarek and Guze 2009), (Smolarek 2010),
(Guze and Smolarek 2010). However the main as-
sumption of the models proposed is that traffic flow
is stationary, which is not applicable to areas with
scheduled traffic. Recently a geometrical model for
estimation the probability of ship collisions while
overtaking were introduced by (Lizakowski 2010),
his model considers human factor and the fairway
and ship dimensions. However all these models are
advanced mathematically they do not take into ac-
count ship dynamics nor human factors.
A multicomponent model for an inland ship safe-
ty estimation was presented by (Galor 2010). How-
ever each of the proposed model’s component is es-
sential, the model itself constitutes rather an
introduction to the further quantitative analysis of
the problem.
For the first time the idea of ship manoeuvrability
implementation into a collision assessment model
was presented by (Curtis 1986). However, this mod-
el was limited to one ship type, which was a very
large crude carrier (VLCC), and only overtaking and
head-on situations were considered.
1.3 Authors’ contribution
A new criterion for ship-ship collision probability
estimation and a new model have been introduced
by (Montewka et al. 2010). The model considers
ship maneuverability and traffic parameters; the new
collision criterion is named the Minimum Distance
To Collision (MDTC). MDTC is a critical distance
between two ships being on collision courses, at
which they must perform collision evasive actions,
in order to pass safely. The MDTC is estimated by
means of ship motion model and series of experi-
ment for various ship meeting scenarios.
This paper is a continuation of our previous re-
search, it consists of the detailed analysis of the
MDTC values for a wide ranges of input variables
(they are defined in the following Chapters) and two
patterns of performing collision evasive action. The
maneuvering patter one means that own ship is per-
forming a collision evasive action and the other ship
is not acting, in the maneuvering patter two both
ships are involved in avoiding a collision. Perfor-
mance of turning circle is considered a collision eva-
sive action.
The MDTC model introduced in a previous work of
(Montewka et al. 2010) and developed further in this
paper, is based on an initial assumption, that two
ships collide if the distance between them becomes
less than a certain value, named a MDTC. This
MDTC value is not a fixed number, but it is calcu-
alted dynamically for each type of vessel and en-
counter individually. Thus it changes with the situa-
tion. The main factors affecting the MDTC value
are: the vessels maneuverability, the angle of inter-
section labelled α in Figure1a, the relative bearing
from one vessel to the other labelled β in Figure1b
and a pattern of evasive maneuvers (one vessel
swinging or both). In the previous study, a simpli-
fied methodology was applied, which assumed that
two vessels met at a constant relative bearing while
proceeding with their service speeds. Presented
study considers a wide range of relative bearings,
varying from 10 do 80 degrees (counting from the
own ship’s bow) and takes into account two differ-
ent engine settings for each ship type, therefore
providing more detailed results.
Figure 1: A definition of MDTC and major factors affecting it
Source: (Montewka et al. 2010)
The theory of the model and preliminary research
aiming to define the ”collision zone” were presented
by (Montewka et al. 2010). In this paper the results
of studies with respect to different ship types and
ship speeds and varying meeting angles are shown.
However only planar motion of a ship is taken into
account and assumption regarding ship navigating
through deep water is made. We also assume, that
the prevailing weather conditions do not deteriorate
significantly the maneuverability of ships sailing in
the analyzed part of the Gulf of Finland. In order to
validate it, we simulated a maneuver of turning cir-
cle to starboard side, for the chosen ship type, which
was a RoPax (for ship particulars see Table 1), for
two different wave conditions (no wave, and an av-
erage wave height for the Gulf of Finland). Accord-
ing to (Pettersson et al. 2010) and (Raamet et al.
2010) the average monthly weave height recorded in
the analyzed area (sea between Helsinki and Tallin)
does not exceed 2 meters, and as a such was adopted
for the simulation. For this purpose the Laidyn ship
motion model was adopted (Matusiak 2007).
The results allowed us to keep our assumptions,
as a difference between the trajectories of a ship in
two different heights of a wave seems to be negligi-
ble for the purposes of our research (Figure 2).
Figure 2: Turning circles of RoPax performed for two different
wave heights
3.1 Ships considered
In the course of our analysis we are considering four
major ship types: a passenger ship, a containers car-
rier, a RoPax and a tanker. In each scenario, ships
are assumed to proceed with two different engine
settings (except for a passenger vessel which is as-
sumed to sail always at a maximum speed) which re-
sult in forty two encountering scenarios, as depicted
in Figure 4. The following abbreviations are used:
’FA’ is full ahead and ’HA’ means half ahead. The
’FA’ abbreviation corresponds to a mean speed of a
ship of given type as obtained from recorded AIS
data. The abbreviation ’HA’ does not correspond to
an actual engine setting, it rather reflects a spread of
recorded speed values for a given class of ships in
the analyzed area. The value of ’HA’ for given ship
type was calculated by subtracting the standard de-
viation from the mean value for a given type of ship.
The main particulars of the analyzed vessels are
listed in Table 1.
Table 1: Ships particulars.
Ship type LOA B T v
[m] [m] [m] [kn]
Container carrier 150.0 27.2 8.5 20;17
RoPax 158.6 25.0 6.1 20;18
Tanker 139.0 21.0 9.0 14;11
Passenger 185.0 27.7 6.5 25
3.2 Encountering scenarios
Each of an encountering scenario is run for seven-
teen different crossing angles (α), varying from 010
to 170 degrees with 10 degrees increment. Where
010 degrees means almost overtaking (vessel B on a
course of 350deg) and 170 stands for almost head-on
meeting (Vessel B on a course of 190deg), as de-
picted in Figure 3. The situation shown there consid-
ers own ship seeing another at 45 degrees relative
bearing. In the course of the experiment, each cross-
ing angle is calculated for a range of relative bear-
ings, from 10 to 80 degrees, counting from the own
ship’s bow.
Figure 3: Relative positions of vessels, with three chosen cross-
ing angles, before they start to maneuver, (Montewka et
For each ship-ship encounter at a given intersec-
tion angle (α) and at a given relative bearing (β), one
MDTC value is obtained. As specified in a block di-
agram depicted in Figure 4, in total 5712 MDTC
values are obtained. Then for each intersection angle
(α) the maximum MDTC value among eight (as
there are eight relative bearings considered) is
drawn. Also the relative bearing which is the most
inconvenient from a collision evasive point of view,
and which requires the most space to make an action
is indicated. For further statistical analysis 714 out
of all 5712 MDTC values are selected for each ma-
neuvering pattern.
3.3 Maneuvering patterns
In case of a maneuvering pattern number one, the
own ship is performing an evasive action, by turning
circle, and another ship is following her initial
course. In case of maneuvering pattern number two,
two ships are performing turning circles in order to
avoid collisions. The following simplifications in the
presented methodology are done:
in case of evasive pattern where two vessels per-
form turning circles, they both start their maneu-
vers at the same time;
ships are turning away from each other, which
implies course alteration away from each other to
avoid collision and to shorten the time at close
quarters (such assumption meets requirements of
the COLREG, which states, that ships must avoid
altering courses towards each other if in close
the settings of ships’ engines and rudders are con-
stant during maneuvers;
the influence of weather conditions is omitted;
the hydromechanical ship-ship interactions are
Figure 4: Research model
3.4 MDTC estimation
In order to calculate the value of MDTC for a given
pair of vessels, an iterative algorithm is used, as de-
picted in Figure 6. The basic assumption is that the
two ships collided at a time instant t0. Then starting
from this time the reverse iterative algorithm is ap-
plied. It uses backward calculation method in a
space-time domain. Two trajectories of two ships are
drawn and the consecutive positions of ship’s centre
of gravity are plotted every second (dt=1s). If corre-
sponding ships’ contours following the trajectories
have at least one common point, indicating that they
both collided, the algorithm increases the initial dis-
tance between these two ships by constant value of
. The trajectories are redrawn starting
from the new initial positions of the ships. This pro-
cess is repeated until the two contours of ships have
no overlaps at any time instant for a given relative
New initial positions are defined by moving ship
B from ship A away. For a given meeting scenario (a
given angle of intersection α and a given relative
bearing β) the ships are moved away along a line of
a given relative bearing line). For the simplicity
of calculations it is assumed, that own ship holds her
initial position, while the other ship is moved away
along the β line.
In the situation where two trajectories have no
common points and the contours of the ships do not
over-lap, the initial position of vessel B is recorded
(as the initial position of own ship A was always
(0,0)), and the distance between these two positions
is calculated and stored. This distance, is named
MDTC for a given relative bearing. As each meeting
scenario is analyzed for a range of relative bearings
(from 10 to 80 degrees), the procedure presented is
repeated for all relative bearings, yielding eight val-
ues of MDTC for each angle of intersection α. Final-
ly, the maximum value of MDTC among these eight
is drawn. This maximum value is considered a
MDTC value for a given angle of intersection. This
procedure is repeated for all angle of intersection,
then for each maneuvering patterns. Thus the MDTC
charts are obtained.
In order to determine the MDTC charts, all rou-
tines are encoded in MATLAB. As a polygonal re-
gion, which could represent a ship contour an ellipse
is chosen. To determine, whether two contours of
the ships (represented as ellipses) overlap, the fol-
lowing MATLAB function is applied (MathWorks
IN = inpolygon(X,Y,xv,yv), (1)
it returns a matrix IN of the same size as X and Y.
Each element of (IN) is assigned the value 1 or 0
depending on whether the point (X(p,q),Y(p,q)) is in-
side the polygonal region whose vertices are speci-
fied by the vectors xv and yv.
For the sake of computation effectiveness each el-
lipse is transformed into discrete form and the num-
ber of points that represent the ellipse is 24. The el-
lipse’s axes are defined in the following way:
a = 0.5LOA
b = 0.5B, (2)
where a denotes a major axis, b is a minor axis, LOA
means length overall of a ship and B is a ship’s
A MDTC value for a given encounter implies a
safe passage of two vessels, which corresponds to a
situation where these two vessels approximated by
the ellipses, will always be separable and will not
touch each other at any time step of a collision eva-