International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 3
Number 1
March 2009
81
Method of Safe Returning of the Vessel to
Planned Route After Deviation from Collision
M. Tsymbal & I. Urbansky
Odessa National Maritime Academy, Odessa, Ukraine
1 INTRODUCTION
Flexible strategies for collision avoidance, take into
account the presence of dangerous and obstacle ves-
sels, hazards for navigation and the Colreg require-
ments. The strategy of collision avoidance depends
on the realized range of mutual duties, relative posi-
tion of vessel and target and, correlation of their
speeds. In general strategy of deviation foresees
transfer the current position of the vessel from sub-
set of dangerous positions to subset of safe positions,
calculation the deviation course and then returning
to the planned route by a course tangent to the circle
of the assigned CPA (Tsymbal 2006, 2007, 2008).
During modeling the flexible strategies on com-
puter, it was determined, that on short distances the
risk of collision can arise again when the vessel re-
turning to the planned route after deviation from col-
lision.
The paper presents the method for calculating the
parameters for ship manoeuvring, when returning to
the planned route after deviation from collision.
2 MATHEMATICAL MODEL
2.1 Three types of returning trajectories
The detailed analysis, shows that for safe returning
of the vessel on the planned route in default of co-
ordination between the ship and target it is necessary
the initial situation G to identify with one of three
subsets Mn1, Mn2 or Mn3, each of which deter-
mines the type of trajectory for returning the ship
and mathematical model for calculation the parame-
ters of manoeuvre.
The first subset M
n1
includes safe situations G,
when are assured increase the distance L
t
between
the vessels and target, i.e. dL
t
/dt>0. Second subset
M
n2
includes situations, when distance between the
ship and target reduces, i.e. dL
t
/dt<0. And, finally,
the third subset of situations M
n3
includes those situ-
ations, for which it is possible increasing or reducing
the distance L
t
.
For identification the type of initial situation G it
is necessary to calculate the initial relative course
K
oto
and rate of its change ω
otb
.
ABSTRACT: Flexible strategies for collision avoidance, presented at TransNav 2007, were examined using
computer program for its correctness in different situations of ships interaction. It was determined, that on
short distance the risk of collision can arise again when the vessel returning to the planned route after devia-
tion from collision. For controlling ship’s safe returning, the mathematical model was developed. This model
describes the analytical dependence of the rate of changing relative course with respect to rates of turning of
the vessels and its initial relative position. This method can be used in automatic systems for controlling the
safe returning of the vessel to the planned route.
82
2.2 Calculation of relative course and rate of its
change
The initial relative course K
oto
means the relative
course of the ship’s deviation. Its calculation is pro-
duced on the parameters of motion of the vessel and
target by expression:
]
sinsin
arcsin[
oto
cocvov
oto
V
K - VKV
K =
where K
vo,
V
v
, K
co,
and
V
c
, = the values of initial
course and speed of the vessel and target according-
ly; V
oto
= initial relative speed:
2122
]cos2[
/
covocvcvoto
) - K(K V V - V VV +=
For calculation the value of relative angular speed
ω
otb
, it is necessary to know the values of rate of turn
of the vessel ω
vb
and the target ω
c
.
Calculation of the value relative angular speed
ω
otb
for a situation, when the vessel and target
change course simultaneously is produced by the
following analytical expression:
−
+
−
=
)cos()cos(
ω)cos(ω)cos(
ω
ctcvtv
cctcvbvtv
otb
KVKV
KVKV
2
)]cos()cos([
ω)ωsin()]sin()sin([
otctcvtv
bbycvctcvtv
VKVKV
tKVVKVKV
+
∆∆+∆−
where K
vt
= (K
vy
+
ω
yb
t); K
ct
= (K
vy
+
ω
yb
t);
Δω
b
= ω
yb
- ω
c
; ΔK = K
vy
– K
cy
; K
vy
= vessel’s
course deviation; K
cy
= target’s course deviation. K
vy
and K
cy
determined previously by using the method
of flexible strategies for collision avoidance.
When target keep her course, ω
c
= 0 and, only the
vessel change her course
2
)]cos()cos([
ω)ωsin()]sin()sin([
otcycvtv
ybybycvcycvtv
VKVKV
tKVVKVKV
+
+∆−
−
The sign of the vessel’s rate of turn depends on
the side of turn Δ
y,
so, that sign(ω
yb
) =-sign(Δ
y
);
Δ
y
=1 when deviation to starboard.
2.3 Identification of initial situation
Belonging of the initial situation G to the subset M
n1
is analytically expressed as follows:
1
MnG ∈
, if {
],2/[
παπα
++∈
oooto
K
and
otb
ω
>0},
or {
]2/,[
παπα
−+∈
oooto
K
and
otb
ω
<0};
)()ω(
yVb
signsign ∆−=
where α
0
= initial bearing from the vessel to the tar-
get.
Condition which describes belonging of initial
situation G to the subset M
n2
, expressed by a next
correlation of relative initial course K
oto
and sign of
rate of turn ω
otb
:
2
MnG ∈
, if {
],2/[
oooto
K
απα
−∈
and
otb
ω
>0}, or
{
]2/,[
παα
+∈
oooto
K
and
otb
ω
<0};
)()ω(
yVb
signsign ∆−=
Belonging of initial situation G to subset M
n3
, is
determined by analytical expressions:
3
MnG ∈
, if {
],2/[
oooto
K
απα
−∈
and
otb
ω
<0}, or
{
]2/,[
παα
+∈
oooto
K
and
otb
ω
>0}, or
{
],2/[
παπα
++∈
oooto
K
and
otb
ω
<0}, or
{
]2/,[
παπα
−+∈
oooto
K
and
otb
ω
>0};
)()ω(
yVb
signsign ∆−=
The type of returning trajectory depends on the
subset to which the initial situation belongs.
2.4 First type of returning trajectory
If G belonging to Mn1, the most preferable is the
first type of vessel’s returning trajectory to the
planned route. This type of trajectory requires mini-
mum time. The first type trajectory is shown on Fig-
ure 1.
Figure 1. First type of returning trajectory in true motion
This manoeuvre includes the turn of vessel from
the course of deviation K
y
to the returning course K
b
,
and, when approaching to the planned route the ves-
sel shall turn from the course K
b
on a programmatic
course K
0
, as shown on a Fig. 1 in true motion. The
value of K
b
depends on K
0
and ΔK
y
. We proposed
ΔK
y
=40°.
The parameters of this manoeuvre are the values
of: returning course K
b
; the moments of beginning
the turn t
bn
and ending the first turn t
bk
; the moments
t
kn
and t
kk
– which determine the beginning end end-
ing of the second turn. These parameters calculated
by next equations:
)cos1(
yck
KRL ∆−=
83
where R
c
= radius of circulation of the vessel.
Vybn
Kt ω/2∆=∆
bnbnbk
ttt ∆+=
y
ky
b
K
LL
L
∆
−
=
sin
)(
yv
ky
bkkn
KV
LL
tt
∆
−
+=
sin
)(
Vyknkk
Ktt ω/∆+=
2.5 Second type of returning trajectory
In case when G belonging to M
n2
,
the vessel use se-
cond type of returning trajectory which consist with
the first turn to the same side as deviation from col-
lision with angular speed ω
yb
=-sign(Δ
y
)ω
ymax
and,
then returning to the planned route by course K
b
.
The second type of trajectory is shown on Figure 2
in true motion.
Figure 2. Second type of returning trajectory in true motion
On the first step it is necessary to determine the
coordinates X
vo
, Y
vo
X
vk
Y
vk
X
vp
Y
vp
:
o
v
vo
KX S cos−=
o
v
vo
KSY sin=
)](cos[cos
byycvk
KKKX R −−
=
])( sinsin[
ybycvk
KKKRY −−
=
)
](
22
sin(cos
sin)cos[cos
oo
ovovkovoo
vp
KK
KKK
YYX
X
−
−+
=
oovovpovovp
KKK
XXYY sincos)(sin
/][
−
+=
Then we calculate the distance L
b
:
22
)()(
vk
Y
vpvkvpb
YXL
X
−− +=
The parameters of manoeuvre calculated by next
equations:
Vbybnbk
KKtt ω/)](2[ −−+=
π
yv
kb
bkkn
KV
LL
tt
∆
−
+=
sin
)(
Vobknkk
KKtt ω/)( −+=
2.6 Third type of returning trajectory
If G belonging to M
n3
,
the vessel shall continue mo-
tion with relative course of deviation K
oty
till the
moment, when the returning to relative course K
otb
guarantee that CPA will not less then L
d
. This third
type of returning trajectory shown on Figure 3 in
relative motion.
Figure 3. Third type of returning trajectory in relative motion
In the beginning it is necessary to calculate the
co-ordinates of points A,B,C:
odA
LX
α
sin=
odA
LY
α
cos=
otbdB
KLX cos=
otbdB
KLY sin−=
where K
otb
= relative course of returning.
)tgtg(
otboty
C
KK
G
Y
−
=
otbB
otboty
BC
KY
KK
G
XX tg]
)tgtg(
[ −
−
+=
where
otbBotyAAB
KYKYXXG tgtg −+−=
K
oty
= relative course of deviation.
Distance L
AC
between points A and C calculated
as follows:
22
)()(
ACAC
YYXX
AC
L
−+−=
.
84
The moment of time to turn
*b
t
is determined by
the following formula:
oty
ACAC
oty
AC
V
YYXX
V
L
b
t
22
)()(
*
−+−
==
.
Amendment for ship’s dynamic
b
Δt
is calculated
on a formula:
]sin[
)}/sin)coscos(cos
)sinsin(sin{Δ
)K(KV
K(K
τ
VKKRK
KKRKt
otbotyoty
cotbcbycotb
ybcotbb
−
−+−
−−=
where
vb
yb
KKAbs
ω
τ
)( −
=
.
The parameters of beginning and ending the turn
to course K
b
are calculated by next equations:
bbbn
ttt
∆
−=
*
τ
+
=
bnbk
tt
3 EXPERIMENTS AND RESULTS
For verification of correctness theoretical results the
imitation software was designed for modeling the
manoeuvre of returning the vessel to the program-
matic trajectory of motion in different situations and
with different types of returning trajectory. This
computer program allows on the set initial situation
choose the safe manoeuvre of returning and builds
his trajectory on the screen. Information appears in
relative and true motion.
On Figure 4 a situation is shown, when the safe
manoeuvre of returning is possible by the trajectory
of the first type.
Figure 4. First type trajectory for safe returning
Left part of screen shows the relative trajectory of
the vessel in relation to the immobile target. Right
part of the screen contains the trajectories of the ves-
sel and target in true motion. As we can see, on rela-
tive motion, the distance between vessel and target is
increasing. Black square shows the initial position of
the vessel after deviation from collision.
On Figure 5 second type of safe returning trajec-
tory shown.
Figure 5. Second type trajectory for safe returning
Third type of safe returning trajectory shown on
Figure 6.
Figure 6. Third type of safe returning trajectory
More than 100 different initial situations were
generated and the parameters of manoeuvres of re-
turning on the programmatic trajectory of motion are
calculated and modelling. It appeared that 51% of
manoeuvres had the first type of trajectory of return-
ing, 37% is the second type and 12% is the third
type. All manoeuvres chosen by the program were
safe.
4 CONCLUSION
This paper presented the method which taking into
account high level of vagueness of target’s conduct,
and increase safe returning of the vessel to planned
route after deviation from collision.
Obviously, that at presence of co-ordination be-
tween a vessel and target on the stage of their return-
ing to planned route provides more high safety re-
turning.
REFERENCES
Tsymbal M., Burmaka I. & Tjupikov E. 2007, Flexible strate-
gies for preventing collisions. Monograph, Odessa,
Ukraine, 2007, ISBN 978-966-8128-96-7
Tsymbal M., 2007. Method of Synthesis of Flexible Strategies
for Preventing Collisions. In Weintrit, A. (Ed.): Advances
in Marine Navigation and Safety of Transportation. Mono-
graph, Gdynia, June 2007, ISBN 978-83-7421-018-8
Tsymbal M. & Urbansky I. 2008. Development of simulator
systems for preventing collision of ships. In Benedict K.
(Ed.): 35th Annual General Meeting and International Workshop Confer-
ence in International Marine Simulator Forum, Rostok, September
2008, ISBN 978-3-939159-55-1