375
1 INTRODUCTION
The desired route of a given ship is usually expressed
using waypoints (Fossen, 2011). This description
method is extremely attractive, since the route can be
easily stored in the onboard computer's memory.
Waypoints can be designated and programmed
before or during the cruise, taking into account such
factors as weather conditions, avoiding obstacles, and
mission planning (Śmierzchalski & Łebkowski, 2002;
Lazarowska, 2016; Lisowski, 2016). Each waypoint,
defined in Cartesian coordinates (x
i, yi), is used for
creating the desired route as a set of straight line
segments connecting pairs of successive waypoints.
When the ship reaches the designated acceptance
circle surrounding the waypoint i, the path is
switched to the next line segment connecting
waypoints i and i+1. In a more advanced solution,
arcs of circles connecting line segments of the route
are defined around each waypoint, and are then used
to determine the desired turning at this point (Kula &
Tomera, 2017). Once the waypoints are established, it
is usually desirable for the sea unit to track the
waypoints as closely as possible, even in the presence
of unknown environmental disturbances.
Analyzing the operation of ship control systems
for surface ships moving along the desired route
began in the 1980s. In principle, it is easy to design
a system to control the ship course along the set
trajectory passed from a conventional autopilot by
using the information obtained from the positioning
system (Amerongen & Nauta Lemke, 1986). However,
better quality is obtained when considering the
system as a whole, including the ship, the ship-acting
environment, and the regulator, as in this case all
relevant state variables can be used in control
synthesis. The entire system can be analyzed through
the use of techniques known as analytical control
strategies, such as self-tuning control (Kallstrom,
1982), LQG (Holzhutter, 1990; Bertin, 1998; Morawski
& Pomirski, 1998), adaptive control (Chocianowicz &
Waypoint Path Controller for Ships
M. Tomera & Ł. Alfuth
Gdynia Maritime University, Gdynia, Poland
ABSTRACT: The paper presents and discusses tests of a waypoint controller used to sail a ship along the
desired route. The planned desired route for the moving ship is given as a set of waypoints connected with
straight lines. The ship's control is based on the rudder blade deflection angle as a commanded parameter. The
task of the controller is to determine the rudder angle which will allow the ship to sail along the desired route
segment. The controller algorithm consists of two parts, the first of which is used for controlling the ship
motion along linear segments of the desired route, while the second part is used when changing to the next
route segment. A switching mechanism is designed to choose the relevant part of the control algorithm. The
quality of operation of the ship motion control algorithm was tested on the training ship Blue Lady, at the Ship
Handling Research and Training Centre located on the lake Silm at Kamionka near Iława, Poland.
http://www.transnav.eu
the
International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 14
Number 2
June 2020
DOI:
10.12716/1001.14.02.14
376
Pejaś, 1992), H
∞
(Messer & Grimble, 1993) and LMI
(Miller & Rybczak, 2015).
A common feature of all the above analytical
control strategies is their dependence on the reliability
of the mathematical model describing the
maneuvering dynamics of the ship. In addition, it is
often necessary to linearize the ship's model before
applying the above analytical control strategies.
In order to avoid the above difficulties associated
with the accuracy of the applied mathematical model
of ship dynamics, other control strategies, making use
of the fuzzy set theory (Vukic et al., 1998; Velagic et
al., 2003; Gierusz et al., 2007; Ahmed & Hasegawa,
2016; Yu & Xiang, 2017), artificial neural networks
(Zhang et al., 1996; Kula, 2015; Zhuo & Guo) and
nonlinear control (Pettersen & Lefeber, 2001; Do
& Pan, 2003; Fredriksen & Pettersen, 2006; Baker et al.
2013; Witkowska & Śmierzchalski, 2018) have also
been developed.
Conventional ships are usually equipped with one
or two main propellers for controlling the surge
velocity and fins for controlling the course. Even if
additional transverse thrusters are installed, they do
not provide a significant extra force at transit speeds.
This means that independent control is possible with
only two degrees of freedom (DOF): surge and yaw.
In this article, the problem of control is defined in
such a way that the ship follows straight line
segments between the route waypoints at constant
speed, and the ship motion control is performed using
the rudder blade as a commanded parameter.
2 PROBLEM FORMULATION
The movement of a ship sailing on the water surface
is described in three degrees of freedom. Two
coordinate systems are used for its description
(Figure 1). The first of them is the Earth-fixed
coordinate system (X
N, YN), related to the water map,
in which the X
N-axis points north and the YN-axis
points east. The other coordinate system (X
B, YB) is
associated with a moving ship, and its origin is on the
waterline, at the point consistent with the position of
the center of gravity of the ship. The state variables x
describing the ship movement are collected in two
vectors
η
= [x, y,
ψ
]
T
and
ν
= [u, v, r]
T
(Fossen, 2011).
The components of the vector
η
are defined in the
Earth-fixed coordinate system (X
N, YN), while those of
the vector
ν
in the body-fixed coordinate system
(X
B, YB). The resultant vector of the ship’s movement
status has the form
x = [
η
T
ν
T
]
T
= [x, y,
ψ
, u, v, r]
T
(1)
The velocity vector
η
defined in the Earth-fixed
coordinate system is associated with the velocity
vector
ν
determined in the body-fixed coordinate
system using the following kinematic relationship
νRη )(
ψ
=
(2)
(North)
u
v
x
{Inertial frame}
{Body frame}
U
(surge)
(sway)
y
r
β
ψ
(yaw)
X
N
Y
N
X
B
Y
B
c
ψ
δ
Figure 1. Quantities describing ship's movement in the
horizontal plane, (X
N, YN) – Earth-fixed reference system,
(X
B, YB) – body-fixed reference system, (x, y) – position
coordinates,
ψ
– ship's course, u – surge speed, v – sway
speed, r – yaw rate, U – total speed,
δ
– rudder angle,
β
- sideslip angle.
where R(
ψ
) is the rotation matrix determined from the
formula
−
=
100
0cossin
0sin
cos
)(
ψψ
ψψ
ψ
R
(3)
The ideal desired route of the ship consists of
a number of linear segments N (Figure 2). The ship is
assumed to move along straight line segments
between the waypoints. The desired speed u
dk is
assumed constant over each individual route
segment. The course
ψ
k resulting from the current
route segment is the right-handed angle, relative to
the X
N axis
ψ
k = atan2(yk+1 − yk, xk+1 − xk ) (4)
During the turning maneuver at a waypoint, the
ship moves along a circular arc connecting the
adjoining linear segments at this point. In order to be
able to perform such a maneuver, it is necessary to
start the turning maneuver at a distance L
k ahead of
the waypoint, the length of which depends on the
course difference between two consecutive linear
route segments
)()(
1 kkkk
ffL
ψψψ
−=∆=
+
(5)
and can be determined experimentally.
To facilitate determining the deviation of ship's
position in relation to the implemented segment of
377
ψ
Y
N
X
N
Y
B
(x
k
,y
k
)
x
y
(x
k+1
,y
k+1
)
(x
k−1
,y
k−1
)
ψ
k
Y
R
X
R
x
r
y
r
X
B
R
k+1
(East)
(North)
Γ
ψ
r
ψ
k−1
ψ
k+1
L
k
Figure 2. Concept of surface ship track-keeping along
desired route
the desired route, the third coordinate system (XR, YR)
is introduced. The origin of this system is at the
starting point of the executed line segment of the
desired route (x
k, yk), while the XR-axis points towards
the waypoint (x
k+1, yk+1).
The control task consists in finding such an
algorithm that will allow the ship to follow the ideal
route of the passage (Figure 2). The control signal is
assumed to be a two-element vector having the form
S
c(t) = [
δ
c(t) ngc(t)]
T
(6)
where
δ
c(t) it is the commanded rudder blade
deflection angle, whereas n
gc(t) is the commanded
rotational speed of the propeller.
3 MATHEMATICAL MODEL OF SHIP’S
DYNAMICS AND ACTUATORS
The control plant is a 1:24 scale physical model of the
tanker Blue Lady. The most important parameters of
this model are summarized in Table 1.
Table 1. Main particulars of the training ship Blue Lady
_______________________________________________
Parameter Value
_______________________________________________
Total length LOA = 13.78 (m)
Breadth B = 2.38 (m)
Draft (full load) T
d = 0.86 (m)
Displacement (full load)
∆
= 22.83 (m
3
)
Position of center of gravity x
G = 0.00 (m)
_______________________________________________
A complex mathematical model for this tanker was
developed by Gierusz (2001). The model includes all
actuators installed on the ship and allows to analyze
its movement in the entire speed range.
In a general form, the mathematical model of
ship's dynamics is given as
τν
ν
Dν
νCνM =
++ )
()(
(7)
The matrix M contains the parameters of inertia of
the rigid body, its dimensions, weight, mass
distribution, and volume, as well as the added weight
coefficients
−−
−−
−
=
rzv
G
rGv
u
NINmx
YmxYm
Xm
0
0
00
M
(8)
The centripetal and Coriolis force matrix C contains
hydrodynamic coefficients associated with the liquid
in which the ship moves
−−
+
−
+
−
=
0
)(
)
(
)
(0
0
)(0
0
)
(
uX
m
vr
xm
uXm
vr
xm
u
G
u
G
ν
C
(9)
The damping matrix D is associated with
hydrodynamic damping forces and makes it possible
to determine these forces for high velocities.
−−
−−
−
=
)()
(0
)(
)(0
0
0)(
)(
3332
23
22
11
νν
ν
ν
ν
νD
dd
dd
d
(10)
where
|
|)
(
|
|
11
uXd
u
u
=v
,
|||
|
||
)(
|||
||
|
22
rYv
Y
uY
d
v
rvv
vu
+
+=
v
,
( )
|||||
|
||
||||23
rYvYuY
d
rrrvru
++=v
,
( )
|||
|||
||||||
32
rNvNu
Nd
vrv
vvu
++
=v
,
( )
||||||
||||||33
rNvNuNd
rrrvru
++=v
.
Table 2 presents all parameters related to the
mathematical model of Blue Lady dynamics given by
Eq. 7. The vector of forces acting on the ship's hull is
composed of forces generated by the propeller and
rudder blade and those generated by interacting
external environmental disturbances.
[ ]
w
thNYX
τττ +==
T
,,
τττ
(11)
Table 2. Parameters of Blue Lady dynamics
_______________________________________________
No Variable Value No Variable Value
_______________________________________________
1 m 22 934.4 11
vr
Y
||
−29 634.8
2 I
z 436 830.2 12
ru
Y
||
7 841.9
3
u
X
−730.5 13
rv
Y
||
18 521.8
4
v
Y
−18 961.8 14
rr
Y
||
12 502.0
5
r
Y
0.0 15
vu
N
||
−9 984.6
6
v
N
0.0 16
vv
N
||
−9 260.9
7
r
N
−183 519.1 17
vr
N
||
−40 007.0
8
uu
X
||
−193.9 18
ru
N
||
−55 614.0
9
vu
Y
||
−2 350.9 19
rv
N
|
|
−12 502.0
10
vv
Y
||
−6 859.9 20
rr
N
||
−843 900.0
_______________________________________________
3.1 Mathematical model of propeller and rudder blade
For a propeller with a fixed blade angle, the generated
thrust force is more or less proportional to the square
of the shaft speed n
g. The propeller/rudder model can
378
be divided into two parts. The first part describes the
nominal pressure (at rudder angle
δ
= 0).
≤
≥
=
0
0
2
gggTn
ggTp
nnnk
nnk
T
(12)
The second part concerns additional forces: drag
and lift, produced by the rudder blade associated
with the propeller.
<
≥
=
00
0,sin5.0
g
gR
n
nF
D
δ
(13)
<
≥
=
00
0
,cos
g
g
R
n
n
F
L
δ
(14)
where F
R is the operating force of the rudder blade,
expressed as:
<+⋅
≥+⋅
=
0)sin(
0)sin(
2
2
uuk
uuk
F
RFn
RFp
R
βδ
βδ
δ
δ
(15)
The local rudder blade drift angle
β
R is determined
as:
),(2atan
δδ
β
uv
R
−=
(16)
where u
δ
is the effective inflow of the water jet to the
rudder blade in the longitudinal direction
≤
>+++
=
0
0)(
2
5
2
4
2
321
Tu
TukTkukukk
u
δ
(17)
and v
δ
is the effective inflow of the water jet to the
rudder blade in the transverse direction, determined
from:
2
rL
vv −=
δ
(18)
For a system that includes a propeller and the
associated rudder blade, the following force and
torque vector is applied to the ship hull
+
+
−
=
xRnLnT
yL
yT
N
Y
X
LL
kTk
LkTk
D
T
)
(
τ
τ
τ
(19)
Table 3. Parameters of the propeller/rudder control system
_______________________________________________
No Variable Value No Variable Value
_______________________________________________
1 kTp 4.5658 8 kFp 272.1
2 k
Tn 3.2903 9 kFn 204.1
3 k
yT −0.1333 10 k1 0.3850
4 k
nT −0.2024 11 k2 0.3000
5 k
yL 1.1760 12 k3 0.4900
6 k
nL −0.5493 13 k4 0.0217
7 L
xR 5.7800 14 k5 0.1150
_______________________________________________
4 STRUCTURE OF CONTROL SYSTEM
The above defined control was implemented in the
system shown in Figure 3. The input signal to this
system is the desired route given by the path planning
system as the safe path of ship movement. The
desired route has the form of a broken line, defined
by the coordinates of subsequent waypoints (x
k, yk).
The ship motion on the water surface is described by
the vector x consisting of six state variables (1), where
(x, y) are the ship position coordinates measured by
the DGPS system,
ψ
is the ship's heading measured
by the gyrocompass, (u, v) are the linear body-fixed
velocity components (surge, sway), and r is the yaw
rate. Usually, the velocity components are not
measured. The measured coordinates of the ship
motion state are collected in the vector
η
= [x, y,
ψ
]
T
.
Way-point
path
controller
Ship
DGPS
Gyrocompass
Environmental
disturbances
η
δ
c
n
gc
Desired route
x
Figure 3. Block diagram of ship's motion control system
along desired route
Based on the information received from the path
planning system in the form of the desired route and
the measured ship position coordinates and course
collected in vector
η
= [x, y,
ψ
]
T
, the waypoint path
controller determines the commanded rudder angles
δ
c. The second commanded value, which is the main
propeller revolutions n
gc, is constant and not
regulated. The commanded rudder angle
δ
c, is
determined using a set of two component controllers,
as shown in Figure 4.
Two modes of waypoint path controller operation
are considered. The first mode, called track-keeping,
consists in controlling the ship's movement along a
straight line segment of the route, while the second
mode is used during the maneuver of changing to the
next straight line route segment and is called the
turning maneuver. Conditions for switching between
these two operation modes are shown in Figure 5. In
Mode 1 (
σ
= 1), both component controllers are
involved in determining the commanded value of
rudder blade deflection
δ
c. The PD component
379
controller minimizes the course error e
ψ
, while the PI
controller minimizes the ship cross-track error e
y. The
path controller switches to the turning maneuver
when the ship arrives at a distance L
s from a
waypoint, which is smaller than the distance L
k for
this waypoint (L
s < Lk).
The ahead distance L
k at which the turning
maneuver should be started depends on the course
difference between two consecutive segments of the
desired path L
k = f(∆
ψ
k). This distance was determined
experimentally in the here reported tests. After
switching to the turning maneuver, the integral in the
PI controller is reset to zero using the signal Reset and
the switching signal
σ
stops passing the side
deviation e
y to the PI controller input (
σ
= 2). During
the turning maneuver, the specified deflection of the
rudder blade
δ
c is determined with the assistance of
the PD component controller. The turning maneuver
is terminated when both the course error e
ψ
and its
derivative
ψ
e
are smaller than their limits. In the
present case, these limits were assumed as: e
ψ
< 5 and
ψ
e
< 0.5.
δ
c
e
y
σ
ψ
d
Reset
PD
PI
0
u
PD
u
PI
ψ
e
ψ
Figure 4. Internal structure of the waypoint path controller
Track-
keeping
Turning
maneuver
L
s
< L
k
5.0&5 <<
ψψ
ee
Figure 5. Directed graph illustrating conditions for
switching between path controller operation modes.
5 SYNTHESIS OF CONTROL ALGORITHM
The tested path controller is composed of two
components connected in parallel. The first
component is the PD course controller, used to
minimize the course error, while the second is the PI
controller, used to minimize the cross-track error from
the desired path segment.
For the purpose of path controller synthesis, the
dynamics of the tested ship, given by Eq. (7), has been
simplified, assuming the constant surge velocity of
the ship, u = u
0 ≈ constant, and low values of velocities
v and r. This allowed linearizing the nonlinear matrix
D given by Eq. (9) to the following form
−−
−−
−
=≈
rv
rv
u
L
NN
YY
X
0
0
00
DD
(20)
After the linearization of Eq. (7), the longitudinal
ship dynamics was decomposed assuming its
longitudinal symmetry. The longitudinal force, which
depends on the rotational speed of the main propeller
screw n
g, was linearized to the form
τ
X = Xnng. The
forces acting on the ship's hull are usually linearly
dependent on the rudder deflection
δ
, according to
the relations
τ
Y = −Y
δ
δ
and
τ
N = −N
δ
δ
. As a result, the
finally obtained maneuvering model consists of the
excluded longitudinal ship dynamics
gnGuu
nXrmxmvruXuXm =−−−−
2
)(
(21)
and the angular-positive dynamics, which is the
Davidson and Schiff model (1946) obtained from
linearization of Eq. (7)
δ
BνNνM =+
101
1
)(u
(22)
where
ν
1 =[v, r]
T
is the state vector, and
δ
is the
rudder deflection. The matrices M
1, N(u0) and B in Eq.
(13) are defined as follows (Davidson & Schiff, 1946)
−−
−
−
=
rzvG
r
Gv
NINmx
YmxYm
1
M
(23)
+
−+−
−+−−
=
00
0
0
)(
)(
u
mxNuXN
uXmYY
u
G
ruv
urv
N
(24)
−
−
=
δ
δ
N
Y
B
(25)
Table 4. Parameter values of the simplified mathematical
model of Blue Lady (
0
u
= 1.1 m/s)
_______________________________________________
No Variable Value No Variable Value
_______________________________________________
1
u
X
−217.0 5
r
N
−68103.0
2
v
Y
−2 972.0 6
δ
Y
−549.7
3
r
Y
9 238.0 7
δ
N
1 487.7
4
v
N
−11 622.0 8 Xn 33.6
_______________________________________________
The obtained linearized Davidson and Schiff
model (1946) of ship dynamics, given by Eq. (22), has
a sway velocity v which in the proposed control
algorithm (10) was not planned to be stabilized.
Therefore, the next step was to eliminate this velocity,
after which the Nomoto model was designated
(Nomoto et al., 1957)
380
)1)(1(
)
1(
)(
)(
21
3
++
+
=
sTsT
sTK
s
s
r
δ
(26)
where K is the static gain of angular speed, T
1, T2 and
T
3 are time constants, and r is the angular ship
velocity
ψ
=r
. The transmittance parameters (26)
refer to hydrodynamic coefficients, according to the
following relations
|
|
|
|
1
2
1
N
M
=T
T
(27)
||
1221211211222211
21
N
mnmnmnmn
TT
−−+
=+
(28)
||
211121
N
bnbn
K
R
−
=
(29)
||
211121
3
N
bmbm
TK
R
−
=
(30)
R
K
K −=
(31)
where coefficients m
ij, nij and bi (i=1,2; j=1,2) are the
coefficients of matrices M
1, N and B (23)-(25), while
|M
1| and |N| are the determinants of matrices M1
and N, respectively.
The identification of the Nomoto model
parameters based on the sea maneuvering tests has
shown that the parameter values T
2 and T3 in Eq.
(26) do not differ much from each other (Fossen,
2011). This allowed further simplification of the
transfer function (26), after which the first order
Nomoto model was obtained
1
)(
)(
+
=
sT
K
s
s
r
δ
(32)
where T = T
1+T2−T3 is the effective time constant of the
angular velocity. The above model can be stored in
the time domain as follows
δ
barr
+−=
(33)
where a = −1/T, b = K/T
To determine the parameters for the PI controller,
which minimizes the lateral deviation, it is convenient
to record the kinematic equations of ship motion (2) in
the following form (Holzhüter, 1990):
ψψ
sincos vux −=
(34)
ψψ
cossin vuy +=
(35)
r=
ψ
(36)
The above equations are nonlinear and depend on
the values of states u, v and
ψ
. Nevertheless, linear
approximations of these equations can be made,
provided that the stationary coordinate system is
rotated in such a way that the given course
ψ
d
becomes equal to zero (
ψ
d = 0). This way, the ship's
control along the desired route will be carried out in
the coordinate system (X
R, YR) related to the currently
executed path segment. Hence, the ship's course
ψ
r
will have a small value during the control along the
desired route, and we can assume that
rr
ψψ
≈
sin
1cos =
r
ψ
(37)
Then, assuming that u
≈
U, the kinematic equations of
ship motion can be reduced to a set of linear
equations
x
r
dUx
+=
(38)
y
rr
dvUy ++=
ψ
(39)
r
r
=
ψ
(40)
Two additional elements (d
x, dy) are introduced in the
above equations. They describe the errors related to
linearization and the slideslip angle caused by
environmental disturbances. In Eq. (36), y
r
is the ship
cross-track error from the desired route, determined
from the formula
y
r
= ey(t) = [x(t) – xk]sin
ψ
k – [y(t) – yk]cos
ψ
k (41)
This error depends very strongly on changes in ship’s
surge speed U.
The task of the path regulator is to control the
ship's movement along the current route segment
with end coordinates (x
k, yk) and (xk+1, yk+1), while
minimizing the course
ψ
r
and the cross-track error of
ship's position from this segment, e
y=y
r
. The preset
course resulting from a given route segment is
determined using Eq. (4), and is changed after
reaching a new waypoint. In the path controller, the
integral of lateral ship deviation y
r
from the path is
introduced, through coupling, to its input. Hence, a
new state appears in the plant
rr
I
yy =
(42)
The designed waypoint path regulator will not
control the surge velocity of the ship, therefore Eq.
(38) can be omitted in further analysis. On the basis of
Eqs. (40), (33), (39) and (42), we can write the
dynamics equations of a simplified mathematical
model of the process for the waypoint path controller
design
381
y
r
I
r
r
r
I
r
r
d
b
y
y
r
U
a
y
y
r
⋅
+⋅
+
⋅
=
0
1
0
0
0
0
0
0100
000
000
0010
δ
ψψ
(43)
The controlled variables
ψ
r
and y
r
are determined
as follows
1000
0010
r
r
r
r
r
I
r
y
y
y
ψ
ψ
= ⋅
(44)
where
ψ
r
= e
ψ
=
ψ
−
ψ
k.
The algorithm designed for the trajectory
controller takes the form
r
I
r
PIPDz
ykykrkekuu
4321
+++=+=
ψψ
δ
(45)
where
() () ()
d
et t t
ψ
ψψ
= −
(46)
dt
tdet
r )()
(
ψψ
=
(47)
The parameters of the trajectory controller (45),
were determined using the pole placement method,
based on the linearized process described by Eq. (43)
for constant surge velocity u
0 = 1.1 (m/s). For further
calculations, the following eigenvalues of the
designed control system were adopted
p
1=−0.1834, p1,2=−0.0692 ± j0.155, p1=−0.0047 (48)
The desired values of the trajectory controller
gains (Table 5) were determined using the function
place included in the Matlab program function set
(Mathworks, 2019).
Table 5. Parameters calculated for the PDPI controller.
_______________________________________________
k1 k2 k3 k4
_______________________________________________
PDPI controller 1.6 19.92 2.125 92.1
_______________________________________________
6 RESULTS
To check the correctness of the designed control
system, both simulation and experimental tests were
carried out. The experimental tests were carried out
on the training ship Blue Lady at the Ship Handling,
Research and Training Centre on the Silm lake in
Iława/Kamionka. The ship was in full load. During
the tests, the wind speed did not exceed 4 m/s. In the
experimental tests, the rotational speed of the
propeller was constant and equal to n
g = 440 rpm.
The first carried out study aimed at experimental
determination of the ahead distance L
k for starting the
turning maneuver. For this purpose, several
maneuvers were made for different course angle
changes between two successive straight line
segments of the desired route. The obtained test
results are shown in Figure 6. The results of the
experimental tests, marked as asterisks (*), were
approximated using the following formula
0
1
2
2
3
3
4
4
5
5
6
6
aaaa
aaaL
k
kk
kkkk
+∆
+∆+∆+
+∆+∆+∆=
ψ
ψψ
ψψψ
(49)
The values of parameters a
0…a6 in Eq. (49) were
determined using the function
polyfit included in
the Matlab program function set (Mathworks, 2019).
These values are collated in Table 6.
Next, the operation of the designed control
algorithm along the desired route was tested
experimentally. The results of these experiments are
given in Figs. 7 and 8. Figure 7 shows the map of the
water basin, with the desired route marked by
5 waypoints (x
k, yk) connected with straight lines
(dotted lines in the figure).
This figure also shows the real path (solid line) of
the ship sailing along the desired route.
Table 6. Values of parameters describing the approximation
of the ahead distance for starting the turning maneuver (49)
_______________________________________________
No Variable Value No Variable Value
_______________________________________________
1 a6 5.987527e-09 5 a2 0.01235089
2 a
5 −1.561371e-06 6 a1 2.10745127
3 a
4 1.430259e-04 7 a0 −0.02348713
4 a
3 −0.004935727
_______________________________________________
Figure 6. Experimentally determined function Lk = f(∆
ψ
k) for
training ship Blue Lady
382
Figure 7. Example of ship path obtained in experimental test
performed on the lake Silm in Ilawa/Kamionka
Comparing these two paths reveals large cross-
track errors of ship position after the ship passes
consecutive waypoints.
The time-history of the cross-track error, shown in
Fig. 8, reveals some undamped oscillations.
Figure 8. Time-history of cross-track error recorded during
the experimental test shown in Figure 7.
7 REMARKS AND CONCLUSIONS
The developed waypoint controller fulfills its task,
which consists in steering a ship along the desired
route. Unfortunately, there is no good cooperation
between the two parts of the designed path controller,
as can be observed in the cross-track error time-
history revealing relatively large oscillations.
Further work on this path controller design will
aim to eliminate oscillations of the cross-track error
and to reduce its value. In particular, it will aim at
refining the conditions at which the PI part of the
algorithm is switched on, as this algorithm
component is responsible for minimizing the cross-
track error from the desired route segment.
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