943
1 INTRODUCTION
Roughly speaking, there are in general distinguished
two types of natural perturbations in marine
navigation which influence a resultant ship track and
are taken into account in chart work and navigational
calculations, e.g. in the Electronic Chart Display and
Information System (ECDIS). Namely, these are a
leeway caused by wind and a drift due to a flow of
water current or stream (For the sake of simplicity, the
wave-induced motion is considered as included in the
action of wind, and not distinguished as a separate
type of perturbation). The basic data of the related
motions, i.e. direction and speed (set and rate) are of
navigational interest, however their actual effect on a
specific ship track is even more essential. This is due
to the fact that the same perturbation may have
different impact on the vessels making way through
the water and having different characteristics, for
instance, draught, dimensions, displacement, shape,
speed. As a consequence, the resultant velocity with
respect to the ground and/or water may be subjected
to substantial deviations. For more details and
investigations concerning the leeway drift, see, for
example, [2, 3, 5, 6, 8].
In this note, the main attention is paid to the way
how the perturbation vector is included in the
equations of motion of a navigating ship, taking into
consideration the issue that the lateral effect of
perturbation with respect to a true course line of own
ship may differ measurably, depending on the type
and force of natural phenomena represented by wind
or water (sea, river) current as well as ship heading. In
particular, the transverse impact can be taken in full,
causing the maximum drift angle in given conditions.
On the other edge, it can be compensated completely
in some circumstances, e.g. by heeling a ship’s hull,
drag (fluid resistance). As a standard, the
measurements of the real wind data are available on
the vessel’s bridge as well as the ship true velocity
( )
U
and the position over ground, making use of the
global navigation satellite navigation sts (GNSS),
gyrocompass, speed logs. Comparing the input data
(wind, water stream/current) resulting in a
perturbation
( )
W
and in consequence, the
A Note on Model-as-Usual for Leeway and Drift Track
in Marine Navigation
N. Aldea
1
& P. Kopacz
2
1
Transilvania University of Braşov, Braşov, Romania
2
Gdynia Maritime University, Gdynia, Poland
ABSTRACT: In this note, we briefly discuss and compare the simplified models of the water current and wind
effects on ship’s resultant track in marine navigation, which are represented by drift and leeway. Our focus is
on a compensation of the lateral component of perturbation with respect to the ship heading line, which is
included in the equations of resultant motion. The variable magnitude of the compensation affects ship
navigation, in particular, total drift angle, speed and course over ground. The corresponding unit time fronts,
speed changes and drift angle in relation to the various ranges of the cross compensation and own ship true
course are also compared by some illustrative examples.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 18
Number 4
December 2024
DOI: 10.12716/1001.18.04.22
944
perturbation-induced force, as well as the output data
referring to the resultant ship velocity
( )
V
, one can
observe that the common vector addition including
the entire
W
-action, i.e.
=+V U W
may not hold
explicitly, while deriving the total (angular) drift or
the resultant (linear) deviation from the true course
line in the presence of the perturbing natural forces
which have a direct impact on ship (craft) navigation.
In what follows, we aim at emphasizing that the entire
perturbation-induced force (motion) that pushes a
ship sideways is not necessarily distributed so that
direction of its actual impact on the ship motion is
collinear with
W
. This implies that its components
i.e. orthogonal and collinear with respect to the
heading line can be subjected to reduction (scaling),
which then affects the ship navigation.
2 DRIFT VERSUS LEEWAY
For simplicity, but without loss of generality, we
apply the simplified model based on the Euclidean
plane with the Cartesian coordinate system x0y and
the kinematics of a material point, where axis y points
downwards. Therefore, the unperturbed sailing in a
calm sea reads
( ) ( )
cos , sin
==x t y t
, where
)
0, 2
stands for a ship heading (true course) that
is measured clockwise from the axis x, and the dot
indicates derivative with respect to time t, i.e.
=
dx
x
dt
and
=
dy
y
dt
. Moreover, the true speed is normalized,
i.e.
1 =U
in what follows. In order to have a full
control of navigation we assume that
WU
, so the
ship is able to reach any waypoint (point of
destination) in the presence of perturbation.
Furthermore, we consider a passive navigation, where
the action of perturbation is not counteracted by a
ship in order to keep the fixed (preplanned) track over
ground. Namely, the heading is arbitral but fixed, and
we let the ship to be drifted due to action of
W
in
the exposition presented.
We begin with a simpler scenario, where the
perturbation
W
is caused only by a horizontal
motion of water, e.g. tidal stream, surface sea current,
river (laminar) current. In this case, the set and rate
are usually taken into a velocity triangle “as they are”,
clearly defining the drift track. Consequently, the
resultant equation of motion reads
=+V U W
(1)
Assuming that
( ) ( ) ( )
, , , , , , ,
=
xy
t x y W t x y W t x yW
points in the x-direction, the last relation in the model
under consideration yields (Actually, Wy=0, since
W
is always pointing in the positive direction of axis x in
the adopted model, and such set-up will be kept in the
whole exposition)
( )
cos=+
x
x t W
,
( )
sin=+
y
y t W
(2)
The above means that the impact of the
perturbation described by its velocity vector on the
related drift effect of the ship is added entirely, i.e.
with the maximum longitudinal and cross
components of
V
. The analogous situation is also
described by a standard wind triangle in air
navigation of the aircrafts, where wind plays the role
of water current in marine navigation; see, for
example, [7] in this regards. Obviously, the velocity
vectors
U
and
V
are not collinear in general. For
the sake of clarity, this is illustrated in Figure 1, where
0
V
determines the resultant destination, i.e. the
estimated position EP (or the observed position Fix 1)
as well as
U
indicates the dead reckoning position
(DR).
Figure 1. The effects of the standard drift-like perturbation
( )
0
V
on the resultant ship’s motion over the ground
including the maximum cross component of
W
and the
leeway-like perturbation
( )
1
V
in the case, where the cross
component is entirely compensated. The ship true velocity
vector is denoted by
U
and the heading
runs clockwise
from the axis x.
In turn, a different situation may arise in the case
of leeway being the effect of wind impact on an
unperturbed ship’s track along the heading line. This
time the perturbation can in fact be compensated in
practice, since a part of it can result in, for instance,
heeling a ship (rolling), namely, it is not allocated
entirely to altering the initial course, unlike the above-
mentioned water drift. More precisely, the cross
component with respect to the heading direction is
subjected to compensation (reduction of magnitude)
due to some physical phenomena, e.g. transverse
drag, fluid resistance, while the longitudinal one is
usually taken in full in the usual models. By analogy,
the latter corresponds to a tailwind or a headwind in
aviation, which increases or decreases the forward
(a.k.a. effective) speed of a craft, respectively.
Therefore, by contrast to the previous scenario with a
water current, the resultant velocity in the case of the
maximum compensation of the sideways movement
(off the
U
-track deviation) leads to
=+proj
U
V U W
(3)
where by
proj
U
W
we denote the perpendicular
projection of
W
onto the true velocity
U
. Recalling
the local coordinate system applied, the last relation
yields
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( )
( )
1 cos cos=+xt W
,
( )
( )
1 cos sin. =+yt W
(4)
In such edge case there is no lateral drift (or it is
treated as negligible in practice), namely, the cross
track distance (a.k.a crosswind leeway component)
due to action of the perturbation is kept (almost)
zeroed. For the sake of clarity, this is shown in Figure
1, where the velocity
1
V
determines the sea position
SP2 (or the observed position Fix 2). In consequence,
the resultant position varies in time of sailing along
the true course line, which coincides with the water
track in this case. In other words, the leeway angle is
vanished and therefore
V
and
U
point in the same
direction, where in general
VU
. The effective
ship speed is increased (as presented in Figure 1) or
decreased, depending on direction of
W
with
respect to the heading
. For more details and
advanced study on evaluation of resistance increase
and speed loss of a vessel in wind and waves, see, e.g.
[4].
There are some direct analogies to similar
scenarios in other models, e.g. iceboat sailing under
wind, driving a craft, skiing or running on a hill slope
under gravity ([1]), where the requested direction of
motion (over ground) is collinear with the control
velocity (heading), although some natural forces
(modelled as the perturbing vector fields) endeavour
to push the moving craft or runner sideways off the
initial track. However, due to the ability of applied
wheel (tyre) or skid to hold the ground without
sliding, i.e. traction, it is possible to resist the pushing
side force and to continue the passage without any
angular deviation.
Both above-mentioned settings yield different
impacts on the resultant ship speed. The speed change
for a given perturbation is expressed by the ratio
( )
1
0
, , ,
=k t x y
V
V
, which leads to
( )
2
1 cos
, , , .
1 2 cos
+
=
++
k t x y
W
WW
(5)
We mention that if
W
is of arbitrary (strong)
force in (9), then the absolute value of the numerator
should be applied instead. It is easily seen that the
usual drift-like setting results in higher ground speed
than the above leeway-like model, i.e.
01
VV
for
all directions
)
0, 2
; see also Figures 4 and 8 in
this regards. Remark, however, that for any fixed
(requested) course over ground, i.e. when
0
V
and
1
V
are collinear, the inequality holds in the opposite
direction and both velocities correspond in general to
different headings, i.e.
01
.
With the above cases in mind we note that the
perpendicular component of leeway track with
respect to the velocity
U
can actually be
compensated in arbitrary range (partially), since the
effect is in particular subjected to speed and
characteristics of ship, relative direction and speed of
perturbation. Such approach represents more accurate
model of a real perturbation effect, where the
magnitude of compensation is defined more generally
by a monotonic (increasing) function
( )
0,1 fa
of a
parameter
0,1 a
(say, a cross perturbation
coefficient or leeway-like coefficient). In practice the
formula for
( )
fa
(e.g. experimental, theoretical)
depends on a specific ship characteristics and
hydrometeorological conditions. We thus get
( ) ( )
( )
1= + + −
a
f a proj f a
U
V U W W
. However, for our
purposes and simplicity, this can be represented by an
identity function, i.e.
( )
=f a a
. Then we get the
improved relation including the perturbation effect,
namely
( )
1= + + −
a
a proj a
U
V U W W
(6)
which is also illustrated in Figure 2. Recalling the
adopted coordinate system again, it follows that
( )
( )
( )
( )
( )
1 cos cos 1
1 cos sin
= + + −
=+
x t a a
y t a
WW
W
(7)
Figure 2. The unified model of the perturbation velocity
vector (
a
V
, red) including the arbitrary compensation of the
lateral component which is determined by a parameter
0,1
a
. In particular, if a=0, then this yields the standard
full drift model
( )
0
V
and if a=1, then this leads to the
leeway-like model with the cross component vanished
( )
1
V
. The total drift angle is denoted by
and measured
from
U
to
a
V
(positive if clockwise and negative if
counter clockwise)
The last outcome can be applied to modelling both
drift-like and leeway-like effects with a given (partial)
compensation of the sideways movement caused by a
resultant perturbation (total drift), i.e. the cross
component. In the extreme cases, if a=0, then (6)
implies (2) and for a=1 we are led from (6) to (4).
Scaling the component of
W
results in the changes
of sea (estimated) position, which is then translated
from SP1 (a=0) to SP2 (a=1) in the case of leeway as
shown in Figure 1. The scaling factor
( )
fa
is specific
for the given conditions and vessel, taking into
account, in particular, the ship (hydro)dynamics in
the advanced model.
The below formula shows the explicit dependence
of speed over ground on the magnitude of
compensation that is determined by the parameter
0,1 a
. From
( ) ( )
22
=+
a
x t y tV
we obtain
( ) ( )
( )
( )
( )
2
2
2
2
, , 1 cos 1 sin
= + + −
a
x t y t aV W W
(8)
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Furthermore, the angular deviation
a
(a-drift
angle) from the true course line due to the a-
dependent effect of perturbation
W
is given as
follows
( ) ( )
( )
( )
( )
1
1
1
2
2
2
2
, , sgn(sin )cos
1 cos
sgn(sin )cos .
1 cos 1 sin
−
−
= − =
+
=−
+ + −
a
x t y t
a
a
V
V
W
WW
(9)
Remark that if one would like to consider arbitrary
force of perturbation in (9), then the modulus of the
numerator should be instead. Being in line with the
standard definitions of leeway and drift angles in
marine navigation,
a stands for the angular difference
between a ship heading and resultant direction of
motion, which is measured from
U
to
a
V
(positive
if clockwise and negative if counter clockwise). This
includes the notions of usual drift and leeway
individually as well as it can be applied to express the
total drift, referring to the resultant (combined)
perturbation. If the perturbation is weak, i.e.
,WU
then in general
90
a
; see also Figure 5
in this regards.
3 UNIT TIME FRONTS BY EXAMPLES
In this section, we aim at presenting the influence of a-
compensation on unit time front (UTF) that is a set of
positions P at which a ship can arrive in equal (unit)
time under the action of perturbation, commencing
from a certain initial point O in all directions. More
formally, we can write
( )
2
: , 1= =TF P O P
,
where
denotes a time distance. In the case of unit
time, the front is generated by the rotating velocity
vector
a
V
centred at O according to (6) – (8).
We first let the perturbation be constant, i.e.
, 0= c
c
W
, where
0.25, 0.5, 0.75c
, and the initial
position is located at the origin. The outcome
including a partial compensation, where a=0.5,
together with the scenarios considering the extreme
values of the parameter a is presented in Figure 3.
Note that the time fronts create the Pascal’s Snails
(a>0) under the action of constant perturbation as well
as the common Euclidean circle for a full drift case
(a=0). Clearly, the stronger the wind force
c
W
, the
more the fronts differ from each other.
Figure 3. The unit time fronts created by the resultant
velocities
a
V
for the full range of headings
)
0, 360
in
the case of constant perturbation of force
( ) ( ) ( )
0.25 ,0.5 , 0.75 dashdotted solid dashed
c
W
for the
standard drift-like perturbation (blue, a=0) and leeway-like
perturbation with the cross component vanished (green,
a=1). The unit time front generated by the true velocity of
own ship is indicated in dashed white and the initial
position is located at (0, 0). The effect of the partially
compensated perturbation is presented in red, where a=0.5
and
0.5=
c
W
As mentioned in the preceding section, the a-
compensation also affects the resultant speed changes.
The related comparative example in the presence of
c
W
is analysed in Figure 4.
Figure 4. Left: The resultant speed decrease (percent) in
relation to the ship’s heading for the cases a=1 (green) and
a=0.5 (red) in comparison to the standard full drift scenario
(blue, 100%, a=0), i.e.
0
/
a
VV
in the presence of constant
perturbation
, 0 ,=
c
c
W
where
( ) ( ) ( )
0.25 ,0.5 , 0.75c dashdotted solid dashed
. Right: The
resultant speed as the function of the ship’s heading
)
0, 2
and the compensating parameter
0,1
a
,
where c=0.5. Remark that the obtained surface is not
symmetric with respect to a, i.e. it is a-dependent along any
fixed heading
, although the graphical outcome looks very
much like independent from a
Moreover, Figure 5 shows the impact of a-
compensation on the drift angle
. The maximum
angular deviation of ca.14.5° (up to sign) from the true
course track holds for the headings 104.5° (to port
side) and 255.5° (to starboard side), while the
perturbation acts along the axis x, i.e. 000.0°. On the
other end, it is obvious that the a-drift is zero for any c
if sailing exactly against or with the perturbation, i.e.
0,
.
Figure 5. The modulus of the drift angle
as a function of
and a in the case of constant perturbation, where
0.25=
c
W
;
1=U
. The maximum value
14.5
o
(red
points) is reached for the standard drift-like perturbation
(a=0), where the heading ( is approximately equal to 104.5°
(<0) or 255.5° (>0)
For the sake of completeness, we mention that the
time front for a=0.5 remains convex if
1
c
W
. More
generally, the fronts are convex for arbitrary
0,1 a
if
0.5
c
W
[1].
Next, we take a look at the scenario with variable
perturbation that depends on position. Namely, we
have
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( )
2
exp 0.9 ,0
=−
by
g
W
, where
( )
0,1 b
(10)
The corresponding time fronts are compared in
Figures 6 and 7, however for different initial positions,
i.e. (0, 0) and (0, 0.5), respectively. As above, the
graphical outcomes are generated for various wind
(current) forces
,
g
W
i.e.
0.25, 0.5, 0.75
.
Figure 6. The unit time fronts of the ship’s resultant motion
commenced at (0, 0) for the edge cases of the cross
compensation, i.e. a=0 (blue) and a=1 (green) in the presence
of the position-dependent perturbation given by (10), where
( ) ( ) ( )
0.25 ,0.5 , 0.75b dashdotted solid dashed
. The
outcome of the partially compensated lateral component of
the perturbation (a=0.5, red) and the profile curve of
g
W
are shown only for b=0.5. The unit time front in the absence
of perturbation is indicated in dashed white
Figure 7. As in Figure 6, with the initial point at (0, 0.5) for
all time fronts; t=1
Furthermore, the resultant speeds in relation to the
ship heading for various compensations (
0, 0.5,1 a
)
and
g
W
-forces
( )
0.25, 0.5, 0.75b
are juxtaposed in
Figure 8.
Figure 8. The resultant speeds with respect to the heading (
in the presence of the variable perturbation given by (10),
corresponding to the scenario presented in Figure 7, where
( ) ( ) ( )
0.25 ,0.5 , 0.75b dashdotted solid dashed
, while a=0
(blue, full drift) and a=1 (green, drift with the entirely
compensated cross component). The initial point (t=0) is
located at (0, 0.5) in each case. The ship true speed
U
is
indicated by a white horizontal line. The change of speed
a
V
in the case of the partial compensation is presented for
a=b=0.5 (red)
4 CONCLUSIONS
The magnitude of sideways movement of a ship due
to action of perturbation in the sense of wind and
water current varies between specific conditions
determined by the ship characteristics and
hydrometeorological conditions, which include,
among others, speed, draught, dimensions,
displacement, stability and type of a ship, as well as
the relative direction and speed of wind and/or water
current. Due to interaction between the transverse
water resistance (drag) and ship’s hull in reality it is
possible that the forward (backward) motion is
influenced by the action of perturbation, i.e.
increasing (decreasing) the effective ship speed
through the water along the heading line, while the
lateral drift track is in fact partially reduced and
results instead in, for example, an angle of heel. In the
brief analysis presented, the longitudinal component
of perturbation with respect to true course line of a
ship (craft) was included entirely in the equations of
resultant motion, however it may also be subjected to
compensation in the improved model by analogy to
the cross component. Therefore, it is reasonable to
take into consideration scaling both components of
the perturbation at the same time and independently
in the future investigation, since they have the
substantial impact on ship track so that modelling of
drift and leeway effects will involve a more accurate
description of forward (backward) motion of a ship
making way through the water as well.
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