265
1 INTRODUCTION
One of the most priority issues is the problem of
ensuring the safety of navigation in narrow waters, the
relevance of which is determined by the increase in the
intensity of traffic on waterways, the increase in the
size of vessels and the difficult navigation conditions
of navigation. Solving this problem contributes to
improving the protection of human life at sea and
reducing damage to the environment, which requires
improving methods for determining the parameters of
a safe separation strategy.
2 A BRIEF REVIEW OF PUBLICATIONS ON THE
TOPIC
The principle and methods of external control of the
process of diverging vessels are set out in work [1],
which considers the main methods of external control
of the process of diverging vessels, which consist in
using the areas of dangerous courses of approaching
vessels and their dangerous speeds to assess the level
of danger of the approach situation and, if necessary,
select a diverging maneuver. The method of diverging
vessels at sea by shifting to a line parallel to the path at
one or another angle to the program course line is
proposed in the monograph [2]. As it is noted in it,
increasing the effectiveness of preventing vessel
collisions can be achieved by creating new divergence
algorithms and intelligent systems. The principles of
external and locally independent control of the process
of diverging vessels that are dangerously approaching
are set out in work [3], which also analyzes the
methods of their application in the event of a threat of
collision of vessels. In the situation of dangerous
convergence of a vessel with two targets, a method for
forming regions of unacceptable values of the vessel's
motion parameters with respect to each of them is
proposed in the work [4]. A procedure for assessing the
level of danger of the emerging convergence situation
for each target is developed. The possibility of
choosing a divergence maneuver by general evasion
from both targets is shown, three situations of
convergence of a vessel with two targets are
considered.
Dependence of the Maximum Approach Distance
on the Shape of the Ship's Safe Domain
I. Burmaka & D.S. Zhukov
National University “Odessa Maritime Academy”, Odessa, Ukraine
ABSTRACT: In work analytical expressions are resulted for a calculation minimum - possible distance of
rapprochement in the case of application of domains of elliptic and difficult form. Shown graphic dependence
minimum - possible distance of rapprochement from foreshortening of ships which are drawn together, for the
domains of both types.
It is shown that the domains of elliptic and difficult form have a similar character of change minimum - possible
distance of rapprochement depending on foreshortening of ships.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 19
Number 1
March 2025
DOI: 10.12716/1001.19.01.31
266
A description of the process of vessel separation in
terms of a differential antagonistic game is proposed in
[5], and in [6] an emergency strategy of separation in a
situation of excessive convergence of vessels is
considered. In [7] the issue of choosing the optimal
standard maneuver of separation of two vessels is
considered. In [8] the distribution algorithms for
forming a strategy of several vessels for situations of
their dangerous convergence are considered. In [9]
analytical expressions for calculating the boundaries of
the areas of dangerous courses and dangerous speeds
in the case of external control of the process of vessel
separation are given. In the work, formulas for
calculating the boundaries of the area of unacceptable
values of the courses of one vessel and the speeds of
another vessel are obtained and the procedure for
graphically displaying the area is considered. The
theoretical justification of an autonomous ship collision
avoidance system is considered in [10], which provides
an algorithm for collision avoidance. Generalizing
approaches to research on ship control automation,
which can be based on the application of mathematical
models and algorithms, or on the use of artificial
intelligence.
3 PURPOSE
The purpose of this publication is to study the
dependence of the maximum approach distance on the
shape of the ship's safe domain. Materials and methods
as shown in [3], the condition for safe separation of
ships is determined by the equality of the distances of
the shortest approach and the maximum allowable
approach, i.e., provided that the ships are approaching.
Traditionally, the maximum allowable approach
distance does not depend on the relative position of the
approaching ships and is taken to be constant in
magnitude, which means that the safe domain of the
ship has the shape of a circle and is given in the space
of relative motion. However, more than ten forms of
the safe domain other than a circle have been proposed
recently. Therefore, the maximum allowable approach
distance when using such domains is not constant, but
depends on the angle of view of the approaching ships.
The value of the maximum permissible distance of
convergence of vessels, taking into account the shape
of the safe domain, depends on the side of the relative
deviation and is determined by the maximum relative
courses and. Therefore, we denote and - the maximum
permissible distances of convergence with the relative
deviation of the vessel to the right and left,
respectively, as shown in Fig. 1. From the same figure,
we can find expressions for determining the maximum
permissible distances of convergence:
p
s s p
d ot ot
d
D D sin(K α) . D D sin(K α)= − = −
where D and - respectively, distance and bearing to
the target.
Figure 1. Maximum permissible approach distances
Results and their discussion Let us consider two
types of domains, for each of which it is necessary to
find expressions for the limiting relative deviation rates
and with the subsequent development of procedures
for calculating the maximum permissible distances of
convergence and. First, we consider analytical
expressions for the limiting relative deviation rates for
an elliptical domain, which we will denote by. In this
case, the relative rate of minimum deviation is
determined by the tangent to the domain boundary,
the position of which depends on the target angle. In
work [11], expressions for calculating the limiting
relative deviation rates were obtained, which are
defined as follows:
In the last expression:
cc
ymin1,2
c 1 c
2
cc
ymin3,4
c 2 c
2
1
1 sin cos
1
2
K
2
1
1 cos sin
2
2
1 sin cos
2
2
K
2
2
1 cos sin
2
x
X b K x K
o
a
arctg
x
Y b K x K
o
a
x
X b K x K
o
a
arctg
x
Y b K x K
o
a
−+
=
− −
−+
=
− −
(1)
where
)
)(
(
222
2222
2
222
2
2
1
(
)
22
2
rca
rbca
rca
cba
rca
cba
x
+
−
+
+
+
−= −
267
)(
(
)(
222
2222
2
222
2
222
2
2
)
rca
rbca
rca
cba
rca
cba
x
+
−
−
+
−
+
−=
moreover
)
cc
cossin(
2
KXKYb
a
c
o
o
−
=
)sincos(
cc
KXKYr
oo
+=
To calculate the maximum relative deviation rates
and maximum permissible approach distances, a
computer program was developed, in which the value
of the maximum permissible approach distance is
calculated depending on the angle, as shown in Fig. 2.
By changing the bearing from 5° to 360°, the program
calculated the corresponding values of the maximum
relative deviation rate and distance. The calculation
results are presented in Fig. 3, in the right part of which
the graphical dependence of the maximum permissible
approach distance on the angle value is shown. Let us
consider the calculation of the maximum relative
deviation rates for a safe domain of a complex shape,
which is a combination of half an ellipse and half a
circle. As previously indicated, the maximum relative
deviation rate of the vessel is the tangent from the
vessel's location to the target's safe domain. In work
[11] it is shown that in the general case the minimum
relative deviation rate of the vessel is:
min
min1 min2 min3 min4
arcsin sin 0
max{ , , , } sin 0
b
otn
y
y y y y
R
K if
K
D
K K K K if
+
=
where
ymin1
K
,
ymin2
K
,
ymin3
K
and
ymin4
K
are
defined by the expression (1), - relative bearing from
target to ship.
Figure 2. Distance dependence by the angle
Figure 3. Changing the distance with an elliptical
domain shape
As in the previous case, the developed computer
program calculated the limiting relative courses and
the values of the maximum permissible convergence
distance for a domain of complex shape, the nature of
the change of which depending on the angle is shown
in Fig. 4.
Figure 4. Changing the distance of a complex-shaped domain
Analyzing Fig. 3 and Fig. 4, we note that the
domains of elliptical and complex shapes have a
similar nature of change in the maximum permissible
approach distance depending on the angle of the
vessels. Therefore, it is advisable to use only one of
them.
4 CONCLUSIONS
1. Analytical expressions are given for calculating the
maximum permissible approach distance in the
case of using domains of elliptical and complex
shapes.
2. The graphical dependence of the maximum
permissible approach distance on the angle of the
approaching vessels for domains of both types is
shown.
3. It is shown that the domains of elliptical and
complex shapes have a similar nature of change in
the maximum permissible approach distance.
268
REFERENCES
[1] Burmaka I.A. Warning of collisions of vessels by the
methods of external control of process of divergence./
Burmaka I.A., Kalynychenko G.E, Kulakov М.A. //
Science and Education a New Dimension. Natural and
Technical Sciences, V(14), Issue: 132, 2017.- С. 56 - 60.
[2] Vagushchenko L.L. Divergence with vessels by
displacement on the parallel line of way / Vagushchenko
L.L.- Odessa: Feniks.- 2013.- 180 p.
[3] Burmaka I. Management by vessels in the situation of
dangerous rapprochement / Burmaka I., Pyatakov E.,
Bulgakov A.- LAP LAMBERT Academic Publishing, -
Saarbryukken (Germany), – 2016. - 585 p.
[4] Pyatakov E.N. Method of determination of safe maneuver
of divergence of ship by the change of course in the
situation of dangerous rapprochement with two targets./
Pyatakov E.N., Pyatakov V.E., Omelchenko T.Yu. //
Science and Education a New Dimension. Natural and
Technical Sciences, V(18), Issue: 158, 2018.- С. 72 - 75.
[5] Lisowski J. Game and computational intelligence decision
making algorithms for avoiding collision at sea/ Lisowski
J. // Proc. of the IEEE Int. Conf. on Technologies for
Homeland Security and Safety. - 2005. – Gdańsk. – Р. 71 -
78.
[6] Burmaka I. Urgent strategy of divergence at excessive
rapprochement of vessels / Burmaka I., Burmaka A.,
Buzhbetskiy R. – LAP LAMBERT Academic Publishing,
2014. – 202 p.
[7] Safin I.V. Choice of optimum maneuver of divergence /
I.V. Safin // Avtomatizatsiya sudovykh tekhnichtskikh
sredstv. - 2002.- №7 . - p. 115 -120.
[8] Hirayama K. Distributed Collision Avoidance Algorithm
in an Environment where Both Course and Speed
Changes are Allowed / K.Hirayama, K.Miyake, T.Shiota,
T.Okimoto //TransNav, the International Journal on
Marine Navigation and Safety of Sea Transportation. -
2019. - 13(1). - P. 23-29.