International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 1
Number 3
September 2007
305
Method of Synthesis of Flexible Strategies for
Preventing Collisions
M. Tsymbal
Odessa National Maritime Academy, Odessa, Ukraine
ABSTRACT: Flexible strategy is the method, which allows forming a strategy of the operating vessel for
avoiding collision, when navigating in congested waters and the risk of collision with more than one vessel
exists. This strategy does not conflict with the ColReg requirements and take into account multi-variant ap-
proach of conduct of targets and possibility of worsening of situation in time. The methodological base for the
solution of this problem is the theory of the dynamic n-guided systems.
For analytical description of the process of manoeu-
vering in congested waters for preventing collision,
the methods of theory of the dynamic n-guided
systems were used. This theory is the logical
evaluation of differential game theory, which is
applied by many authors in the synthesis of models
for the search of optimum manoeuvers for collision
avoidance.
Now we will show that the process of manoeu-
vering of several ships in confined waters for
collision avoidance is adequately described by the
methods of the theory of dynamic n-guided systems.
The dynamic system defines any object or process
for which the initial state is determined as aggregate
of any variables and constants values and a function
law, which describes the change of the initial state in
time and can be used for the prognosis of the future
state of the dynamic system. In the case when the
state of the dynamic system depends on input control
influences of a few participants, i.e. input control
influence is distributed between the participants, and
to every participant the non-empty set of strategies
of control is put in accordance, such dynamic system
is called as n-guided. Depending on the aims of
participants’ control, cooperation between them can
carry the co-operative, antagonistic, coalition or
other type of interaction.
For the process of manoeuvering of a few vessels
in confined waters, assume that in some area of
control S
c
there are n vessels. This area of control S
c
is fixed with a definite ship, which we will name as
an operating vessel. The local proximity of ships
belonging to the control area allows to consider their
aggregate, as some dynamic system which must be
described from positions of their safe relative
motion. Such dynamic system Σ is controlled by all
n vessels.
The state of system Σ is described by the
coordinates ξ
i
(t) and η
i
(t) of each ship and their
parameters of movement V
i
(t) and K
i
(t) in the two
dimensional rectangular system of coordinates,
related to the area of control S
c
and oriented in N-S
direction. Therefore the state x=x(t), as element of
set of X, is a 4n – measured vector. The change of
the state x of the system Σ is described by differential
equations of the ship movement, taking into account
their inertia descriptions.
306
......
)(
)(
cos
sin
)(
)(
)(
)(
,
......
......
......
......
......
3
1
ik
iiv
ii
ii
i
i
i
i
n
i
f
Vf
KV
KV
K
V
tx
tx
tx
tx
β
β
η
ξ
==
=
The dynamic system Σ is characterized by two
additional factors. At first, each of the ships of the
system Σ is proceeding by planned route, i.e.
expedient activities of each of the ships of the
system. And, secondly, strategies for collision
avoidance D
i
of ships depend on each other and on
current position. If dependence exists, it is necessary
to get its formal description. The second factor
determines the type of interaction, arising between
ships at dangerous approaching, expressed in the
applied strategies for collision avoidance.
This factor (type of interaction between ships) is
the most essential for providing safe passing, and
from the point of formalization is the most
indefinite. In all cases, a basic normative document
regulating the conduct of ships when the risk of
collision exists is the ColReg, which foresee
coordination of only binary interactions of the pair of
ships. When operating in congested waters,
manoeuvering can be limited by existing dangers for
navigation and more than two vessels can be
involved in the risk of collision. Even in the
situation of the meeting of two vessels in the open
sea the ColReg generate the row of ambiguousness.
So, each of the meeting vessels shall not only define
the presence of dangerous situation, but also define
the range where she “may take action” (Rule
17(a)(ii)), and where she “shall take action” (Rule
17(b)). Each of the vessels involved as estimation
can get different ranges of mutual duties at the same
beginning position. Therefore, observing the ColReg
requirements, the vessels are forced to make
decision of involving risk of collision and choice of
proper strategy for safe passing in the conditions of
considerable vagueness.
The necessity of formalization of the interaction
of ships in the conditions of existing risk of collision
has defined the features for analytical description of
the dynamic n-guided system Σ, containing vessels.
For the purpose of analytical formalization of the
presence of collision situation in the system Σ we
input the concept of situational disturbance
.
Situational disturbance in relation to an operating
vessel arises when a ship cannot continue realization
of programmed trajectory of motion due to existing
risk of collision with one or a few ships of the
system Σ. The possibility of the situational distur-
bance appearance is determined as a result of
prognosis of the state of the system
Σ
for any time
period. If the forecast trajectory in the space of states
of the system
Σ is safe, situational disturbance is
absent. Otherwise if there is situational disturbance,
the necessity of its compensation appears.
The space of positions M(P
n
), the description of
which are distances between the ships of the system
Σ, consists of a few subsets M(P
nk
), each of which is
characterized by some levels of risk of collision. For
every pair of ships four subsets of their relative
position are determined: the subset of safe positions
P
n0
and three subsets of positions P
n1
,P
n2
,P
n3
with a
different degree of danger situation of collision (in
accordance with the number of ranges of mutual
duties according to Rule 17 of ColReg). Thus in
each of the subsets the proper type of conduct is
prescribed to the pair of interactive ships. Indicated
representation of the space of positions M(P
n
) allows
to formalize the concept of situational disturbance.
In general, situational disturbance between the pair
of dangerous ships is offered to be characterized by
initial ω
ijn
and maximal ω
ijmx
intensity, which can
take on a whole number values depending on the
range of mutual duties in the initial moment and at
the moment of time of the CPA (closest point of
approach). Consequently, initial intensity ω
ijn
can
take on values from 0 to 2, and maximum – from 1
to 3. When the dynamic system Σ consists of more
than a pair of ships, the situational disturbance is
described by the square matrix D
bn
of n dimension,
the element of which d
ij
is initial ω
ijn
and maximum
ω
ijmx
intensity.
0...
...0......
...0
...0
2211
222121
1212
,,
,,
1
,
1
,
mxnnnmxnnn
nmxnnmxn
nmxnnmxn
bn
D
ωωωω
ωωωω
ωωωω
=
In the case when situational disturbance is
produced from an operating vessel, that disturbance
is the vector, got from the line of matrix, which
corresponds to the operating vessel.
nmxnnmxnmxn
bn
D
1
,
1
,,
...
13131212
ωωωωωω
=
Thus components of vector of situational
disturbance, relating to the ships, with which
operating vessel is passing clearly at a safe distance
is equal to zero. Otherwise they contain the values of
ω
ijn
and ω
ijmx
.
The nature of the situational disturbance is in the
forecast of finding the targets in PAD (predicted area
of danger). As a matter of fact, estimations of the
pair of situational disturbances (ω
ijn
, ω
ijmx
) and (ω
jin
,
ω
jimx
) are not always symmetric for both ships. It is
provided by subjective individual authentication of
307
initial range of mutual duties of each ship, as a result
of which different areas S
nd1
and S
nd2
can be got.
Therefore the presence of the situational disturbance
may be possible for one of the ships, and at the same
time may not exist for another.
Situational disturbance exposes the advent of
dangerous position in advance, according to the
prognosis of the change of relative position of ships.
Therefore it has conditional character, because
possible actions of ships and the method of
prognosis the change in the dynamic system state
influences on the truth of its realization.
The appearance of situational disturbance
generates interaction between ships, and there is the
task of the disturbance compensation by the choice
of the proper strategy for preventing collision. The
structure of interaction between the ships of the
dynamic system Σ is uniquely determined by the
structure of the situational disturbance. The type of
interaction between ships determines different
system states of the dynamic system Σ. As a result of
the researches made three system states of the
system Σ were found, each of which is characterized
by separate system property.
So, if the number n
b
of the interactive ships is
equal to zero, i.e. n
b
=0, there is no situational
disturbance in the dynamic system, interaction of
ships is absent, and the elements of the dynamic
system (ships) are unrelated. They execute presence
of a special purpose functions, realizing the
programmatic trajectories (planned route) of motion.
The dynamic system Σ is in the initial (first) system
state, which is characterized by independent
differential equalizations, describing the controlled
motion of ships along the planned route.
In this case the dynamic system Σ is characterized
by the zeroing matrix of situational disturbance,
which contains zeroing elements only.
0...00
...0......
0...00
0...00
1
=
bn
D
For the system Σ the absence of interaction
between ships is principally important. The structure
of the system Σ is characterized by the absence of
interaction between elements, which determines its
first system property.
For the second system state of the dynamic
system Σ the matrix of situational disturbances D
bn
contained the «isolated» elements d
ij,
which does not
equal zero, thus no more than one on the line of
matrix D
bn
.
0...00
...0......
0...0
0...0
2121
1212
,
,
2
mxn
mxn
bn
D
ωω
ωω
=
It means that there is only independent conduct of
pair of ships in the dynamic system Σ.
In this case situational disturbance converted the
dynamic system Σ into a new system state, because a
new (second) system property appears. A new
system property characterized by relative position
and interaction appears in the pair of ships, the
parameters of distance appear between ships and
bearing from a ship to the ship, which did not exist
for a separate ship. Note should be taken on that
appearance of the indicated system property which is
conditioned by the change of the system Σ structure -
origin of interaction between elements. Thus the
structure of relations has the following feature: the
separate ship of the dynamic system Σ can be
conducted only with one ship of the system.
In this case there is only binary interaction, in
which strategy of compensation gets out depending
on intensity of situational disturbance, and the
parameters of the manoeuvre for collision avoidance
are counted. Thus differential equalizations
describing the motion of the pair of interactive ships
are linked between themselves and, producing the
common management of position, ships, joined by
some rule, compensate situational disturbance,
changing the strategy of expedient motion to the
strategy of situational disturbance compensation.
The dynamic system Σ appears in the third system
state, if one of ships in the system will interact with
more than one partner. As it was specified before,
the system state of the dynamic system Σ was
determined from the analysis of the formed matrix of
situational disturbance D
bn
. A matrix D
bn
in the
beginning is checked up consequently on lines, here
the non-zeroing elements of matrix are fixed in
every line, if such are present. If even one line of
matrix contains two or more of such elements, Σ is
in the third system state.
If each line of matrix contains no more than one
non-zeroing element, it is required to check up all
columns of matrix for the presence of non-zeroing
elements d
ij
. In the case when even one column
308
contains more than one nonzeroing element of
matrix, the dynamic system Σ is in the third system
state.
System Σ gains new system characteristic in the
aspect of controllability and indignation, as well as
strategy for preventing collision, becomes’
complicated, new objects called - coordinating
frameworks appear which are the consequently-
parallel structures of surrounding ships necessary to
put in order by complex strategy of situational
disturbance compensation. Relations uniting a few
elements with each other appear in the structure of
the dynamic system Σ.
The matrix of situational disturbance D
bn
becomes the source of forming situation frameworks
linking the structure of dangerous and obstacle ships
for a definite vessel. In this case the groups of
dependent equalizations of interactive ships are
selected from the independent initial system of
differential equalizations, for which it is required to
find the concerted strategies (interdependent)
providing compensation of situational disturbance.
It should be noted that in the third system state
the dynamic system Σ
ns
is characterized by
situational disturbances having a matrix form.
Therefore for compensation of situational disturban-
ces in this system state the method of external
control, which would allow to prang all interactions
of situational disturbance simultaneously and in
complex, would be the most effective, converting
the dynamic system Σ
ns
into the first system state.
However possible realization of management
compensating matrix situational disturbance is by
the algorithm of joint co-operation of ships, which is
not compatible with the principle of coordination,
fixed on the ColReg basis. Consequently, while
manoeuvering for collision avoidance submitted to
operating by the ColReg, realization of effective
external management by the dynamic system Σ
ns
is
impossible.
Therefore, compensation of situational distur-
bance is produced by the forces of interactive ships,
thus a definite ship of the system selects the line of
the matrix of disturbance, to which she belongs and
forms vectorial situational disturbance being the
component of the matrix. Thus vectorial situational
disturbance includes a few targets interactiving with
an operating vessel, and for compensation of this
situational disturbance an operating vessel is requ-
ired to form a strategy of avoiding collision, which
must not conflict with the ColReg requirements.
Thus, in general situational disturbance converts
the dynamic system Σ from the first unrevolted
system state into more revolted states, and the
strategy of situational disturbance compensation
foresees convertion of the dynamic system Σ into the
initial unrevolted state by elimination in the structure
of the system of interaction relations between
elements (ships).
In the indefinite terms a partner is conduct, when
even grounding the probabilistic distributing of
choice by the partner of strategy for preventing
collision is difficult, a ship is required to use the
principle of flexible strategies application for
preventing collision (compensation of situational
disturbances). The sense of this principle consists of
the minimax approaches of the ship to the use of
possible alternative strategies for safe passing.
We will consider the principle of flexible strate-
gies application for collision avoidance in details.
For this purpose we will appeal to the second system
state of the dynamic system Σ. Compensation of
situational disturbance in this system state of the
system Σ is simply regulated by coordinator ColReg,
which coordinates strategies for preventing collision
of interactive ships. So, if the pair of interactive
ships is in the first range of mutual duties, the
concerted strategies of avoiding collision D
1
(t
y
,K
y
)
and D
2
(Tr
2
) are prescribed to the ships. The
privileged ship c
2
shall keep her course and speed,
realizing the programmed trajectory of motion Tr
2
(strategy D
2
(Tr
2
)), and compensation of situational
disturbance is produced by a ship c
1
by strategy,
which provides safe passing at distance of the
shortest approaching, equal to limited-possible
distance L
d
. For this purpose it is necessary to
produce the calculation of the values of time of the
beginning of deviation t
y
and course of deviation K
y
.
)]-sin([arcsin
)(2
1
)(1)( sotysotysy
KKρKK
-
+=
)]-sin([arcsin
)()(
2
1
2)(
soty
-
soty
KKρKK
sy
−+=
π
)]-sin([arcsin
)()(
2
1
1)(
poty
-
poty
KKρKK
py
+=
)]-sin([arcsin
)()(
2
1
2)(
poty
-
poty
KKρKK
py
−+=
π
where
2
1
V
V
ρ =
)arcsin(
)(
L
L
K
d
soty
+=
α
)arcsin(
)(
L
L
K
d
poty
−=
α
309
Here V
1
and V
2
the speed of vessels c
1
and c
2
,
respectively; L the distance and
α
is the bearing from
the ship to target; K
2
is the course of vessel c
2
.
)(sin
)(sin
),(
),(
psoty
ntonto
psotyd
yn
KKV
KαLL
t
−
−+
=
Here K
otn
and V
otn
is the initial relative course and
speed of target. In general four alterations of courses
are possible – two in the opposite direction courses
and two in the same direction courses.
If ships are in the second range of mutual duties,
unlike the previous situation, the privileged ship c
2
can undertake the manoeuvre of deviation, applying
strategy of avoiding collision D
2
(K
y
=K
extr
), which
maximizes minimum distance of the closest approach.
nyn
tt =
)1( ),(arccos
1
<±= ρρKK
y
)1( )],-cos([arcsin
2
1
1
≥++= ρ
KρK
-
y
α
π
α
)sin(max
min
α
−=
otextr
KLD
Here K
otextr
relative course applying strategy of
avoiding collision:
ρKK
otextr
arcsin
1
±+=
π
where K
1
is the course of vessel c
1
.
And, finally, if position of ships belongs to the
third range of mutual duties, where it is required to
take urgent measures for preventing collision, the
ColReg orders to both ships to take such action as
will best aid to avoid collision. This requirement
formalized by the application of strategy
],,)([
112,1 extryy
KKttKD
y
=+=
πα
regardless of conduct of the second ship, followed
on the first stage of manoeuvre by a course equal to
the reciprocal bearing on the second ship, and on the
second – course of deviation K
extr
to the output in
range of safe positions.
ny
tt =
,
)-cos(
221
1
2
α
KVV
LL
t
d
y
−
−
=
.
Flexibility of the considered strategies to avoid
collision in this system state Σ consists of the
following. At first, strategy of collision avoidance as
type of ship’s conduct for preventing collision
changes depending on the realized range of mutual
duties. Secondly, compensation of situational distur-
bance foresees transfer of current position of ships
from subset of dangerous positions to subset of safe
positions, passing by intermediate subset. So, if
compensation of situational disturbance began in the
third range of mutual duties, the ship shall take
urgent action as will best to avoid collision. Current
position of ship must pass the second, and then and
in the first range of mutual duties. During this
transition strategy of collision avoidance is transfor-
med into the next sequence
D
1
,
2
[K
y
=α(t)+π]→D
1
,
2
(K
y
=K
extr
)→D
1
,
2
(t
y
,K
y
).
And, thirdly, during realization of the required
type of strategy for preventing collision, taking into
account the high level of vagueness of target
conduct, there must be the prepared reserve strategy
of avoiding collision, which must be realized at the
unforecasted change of current situation.
During compensation of situational disturbance in
the third system state of the dynamic system Σ, an
operating vessel is in close water situation with a
few targets and must choose strategy for preventing
collision in general case having a few deviations of
course, each of which being carried out by separate
component strategy not conflicting with the
requirements of coordinator (ColReg).
Therefore in the case when situational disturbance
cannot be correctly compensated within the existent
requirements of the ColReg, by operating binary co-
operations of ships, principle of organization of
structure of complex interaction, which can have a
few consequences in time levels not conflicting with
the ColReg requirements, is offered. Realization of
this principle conduces to the concept of
coordinating framework and development of the
method of its forming. In other words, coordinating
framework is the instrument of transformation of
complex interaction of ships in the well-organized
structure of consequences co operations of operating
vessel with the group of targets in accordance with
the ColReg requirements.
However coordinating framework must be
transformed into the real, saving the attained
accordance of complex strategy of situational
disturbance compensation to the requirements of
Rules. Thus, forming of the real framework takes
into account manoeuvring of other obstacle ships,
dangers for navigation and inertia descriptions of
operating ships. The procedure forming the real
coordinating framework is the method of synthesis
of flexible strategy for preventing collision, which
has in component strategies of three above mentio-
ned types for each range of mutual duties of
310
operating vessel and targets of separate level of
coordinating framework.
Flexible strategy, as method of the operating
vessel conduct for compensation of complex
situational disturbance, at the choice of numeral
values of parameters is the manoeuvre to avoid
collision of ship with targets, thus in offered
approach optimization principle of choice of
manoeuvre parameters is realized, regardless of the
type of component strategies.
Conception of flexible strategies for preventing
collision of ships is examined as temporary measure
of transitional character, which in further
development must result in principle new optimum
cooperation control system when risk of collision
exists out of hard limits of the restricted binary co-
ordination of ColReg. A methodological base for the
solution of the outlined problem remains the theory
of the dynamic n–guided systems with the use of
principles of external management and the choice of
game co-operative principles of situational
disturbances compensation.
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