A Model-Based Adaptive Control of Turning Maneuver for Catamaran
Autonomous Surface Vessel
IGOR ASTROV, IRINA ASTROVA
Department of Software Science,
Tallinn University of Technology,
Akadeemia tee 15a, 12618, Tallinn,
ESTONIA
Abstract: - The effective computer control of airborne, waterborne, and ground autonomous vehicles has
become one of the highest priorities in the area of cyber-physical systems, Industry 4.0 in particular, and the
world economy in general. Despite extensive research on Unmanned Aerial Vehicles (UAVs), the study of
Autonomous Surface Vessels (ASVs) has been more than ten times less intense. As an attempt to fill a gap in
that field, this article discusses the control mathematics of a realistic ASV nonlinear model of a real
autonomous electric catamaran “Nymo”, which was designed at Tallinn University of Technology (TalTech).
More technically, the article offers a novel adaptive control system that is based on knowledge of the main
parameters of ASV and is specially designed for a Simulink/MATLAB environment. The article also enables
adjusting variables like transition time and heading angle overshoot value. The control of the desired tracking is
represented in such a maneuver as turning the catamaran at different angles. The designed control system has
shown good quality in terms of accuracy in tracking the desired heading angles.
Key-Words: - adaptive control, computational modeling, computer simulation, dynamics, kinematics, marine
vehicles, mathematical model, nonlinear systems, numerical methods, system analysis and
design.
Received: April 19, 2023. Revised: February 15, 2024. Accepted: March 23, 2024. Published: May 9, 2024.
1 Introduction
Unmanned autonomous vehicles are becoming
increasingly important in both cyber-physical
systems, Industry 4.0, and the whole global
economy, [1]. These vehicles cannot be developed
and operated without reliable mathematical models
to replace the mental models existing in the heads of
vehicles’ human drivers, [2].
At Tallinn University of Technology (TalTech),
one of the highest priority areas of study has been
the research on catamaran sailboats. The control
structures for autonomous sailboat navigation by
using conventional or adaptive PID controllers were
presented in [3]. Sail control was used to reduce
heel and roll angles. The sail angle was optimized,
whereas the rudder angle was controlled depending
on the specified wind direction and the desired
torque. The main goal was to ensure the long-term
autonomous operation of sailing catamarans.
Recently, a new activity at TalTech has become
the development of an electric catamaran called
Nymo, an autonomous surface vehicle (ASV) for
unmanned cargo transportation and environment
monitoring purposes. Figure 1 shows the catamaran
“Nymo” with a weight of 200 kg, a load capacity of
up to 100 kg, and a possible distance of over 50 km.
Fig. 1: Catamaran “Nymo”
In this article, we model and simulate an ASV
that corresponds to the catamaran “Nymo” in Figure
1. It is expected that the research results will be
suitable for further development of real ASV and
improvement of its control in harsh environmental
conditions.
Here we offer and simulate a specially designed
control system that effectively takes into account and
adjusts the desired parameters like transition times,
heading angle overshoot values, and the ratio of
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Igor Astrov, Irina Astrova
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coordinate-to-acceleration transition times for
excellent performance at different heading angles
stabilization tasks for ASV.
2 Model of Catamaran
Let us look at the catamaran model in Figure 2. This
catamaran type of ASV has two separate hulls with
separate propellers, where and are the port
(left) and starboard (right) forces, respectively,
which are provided by different thrusts; and is the
side hull separation. Figure 2 shows the definitions
of the control system variables, where the heading
angle represents the orientation of the fixed frame
of the ASV’s body relative to the northeast-down
frame. For convenience, the instantaneous ASV’s
heading angle is measured in an anticlockwise
direction from the global -direction. Then the
angle is related to the heading angle by the next
relation
.
From Figure it can be seen that the control of the
catamaran is based on the difference in the thrusts of
the left and right propellers. If , then the
catamaran will move in a straight line. If ≠ , it
will cause a heading change in the catamaran
because the difference in thrust forces between the
two stern motors provides the turning moment.
Fig. 2: Planar model of catamaran
The purpose is to capture the boat dynamics and
be able to control the trajectory, thereby ensuring
that the catamaran moves to the desired target angle
as fast and safely as possible.
The kinematics of this control system can be
presented as [4]:
(1)
where (x,y) denote the coordinates of the
catamaran’s center of mass in the Earth’s coordinate
system, is the ASV’s heading angle; and
are the velocities of surge, sway, and yaw,
respectively.
The system dynamics of the considered
simplified mechanical model of the catamaran can
be described using the next equations, [5]:
(2)
where is the catamaran mass; are
the hull added masses; is the moment inertia of
the catamaran in -direction are the
external force in the -direction, external force in
-direction and external moment, respectively.
The external forces and moment in (2) can be
defined as [6]:
(3)
where is the thrust deduction factor generated by
the propeller in -direction; is the propeller
influence factor to the and -directions.
Note that in (3) no propeller forces are generated
in the -direction.
The propeller’s thrusts in (3) are defined as [7]:
(4)
where and are the propeller’s thrust
coefficient functions of the left and right propellers,
respectively; and represent the propeller’s
advance speed coefficients of the left and right
propellers, respectively; is the seawater density;
and are the numbers of revolutions of the left and
right propellers, respectively; and are the
diameters of the left and right propellers,
respectively.
Next, the coordinates of the center of mass and
heading angle are obtained by the integration
(5)
where initial state
From (1)-(5) it is clear that the position
coordinates and heading angle of the ASV can be
computed based on a given set of equations, desired
target angle, and initial conditions.
x
OE
{
E
}
X
E
}
B
{
X
b
u
r
Y
b
v
y
ψ
T
r
F
r
Y
E
d
d
F
l
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3 Parameters of Catamaran
In this section, the ASV’s parameters, [8], will be
evaluated for the catamaran.
The next parameters of the control system can be
obtained by direct measurements, [9]:
(6)
where are catamaran length, width, and
draught, respectively.
The values of and from (3) can be
estimated as, 6:
The moment of inertia about the -axis is
calculated as [10]:
(7)
Hence, from (6)-(7), we find:
(8)
The hull-added masses from (2) are associated
with the parameters of (6) and (8) as [5]
,
(9)
Hence, the parameters in (9) have the following
values:
It was known that a parameter in (4) has the
following value:
Let us now evaluate the value of from (4) for
one of the propellers.
The advance speed coefficient from (4) for the
selected propeller can be expressed as [11]
(10)
where is the surge velocity; is the number of
revolutions of the propeller; is the diameter of
the propeller.
For simulation, the propeller thrust coefficient
function will be approximated by a quadratic
polynomial of as follows:
(11)
where are the constant polynomial
coefficients.
These polynomial coefficients in (11) can be
obtained from the next linear equations:
(12)
Alternatively, equations (12) can be written in
the form of a matrix so that:
(13)
where
The vector of polynomial coefficients from (13)
can be expressed as:
Using the experimental data, [12]
we find
Combining (4), (10) and (11), we obtain that the
value of propeller thrust can be estimated as:
(14)
Then from (14), it follows that the number of
revolutions can be calculated as follows:
4 Control System
The thrust of the left propeller can be considered
as a fixed constant. Hence, we have
(15)
By choosing (15), the complex control problem
now becomes a control problem using the starboard
rotor thrust as an input to control the heading
angle.
The control system configuration for regulating
the value of r is then designed to have the
structure shown in Figure 4.
(16)
where are constants to be determined.
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The variable can be thought of as a “fast”
function of time. Hence, assuming that from
(16), we find
(17)
The following coefficients in (17) are obtained
for not exceeding the value of
%5
, [13]
(18)
where is the desired transition time of.
From (1), we get (19)
From (19), (2) and (3), we obtain
(20)
From (20) and (15), we obtain
(21)
where
. (22)
Next, by combining (16) with (21), we obtain
(23)
where
(24)
Defining as in (23), we obtain
(25)
The variable
)(ta
in (25) can usually be
described using the following expression, [14]
(26)
where
. (27)
Let us now consider the behavior of the control
system (Figure 4) over the time interval in a
steady state.
Provided that
from (26)-(27), we find
(28)
where (29)
The desired transition time for acceleration
control lies in the overshoot zone with a value of
σ≈5%. Then from (28), it follows that
(30)
Therefore, using (22) and (30), and assuming that
and the ratio of coordinate-to-
acceleration transition times
, we get
(31)
We can see from (18) and (31) that the
coefficients of the designed control
system can be computed for a given equation (16) of
control structure.
5 Simulation Results
The developed control system is used to control the
approaching trajectory with a given heading angle at
a constant number of revolutions of the left
propeller .
Simulation results for the proposed block diagram
in Figure 3 (Appendix) are shown in Figure 5, Figure
6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11,
Figure 12, Figure 13, Figure 14, Figure 15, Figure
16, Figure 17, Figure 18 and Figure 19 for various
desired heading angles.
Note that different values were used for the ratio
of coordinate-to-acceleration transition times ratio
and desired transition time in (31) for
different heading angles during those maneuvers.
The block diagram of the designed control
system in Simulink is shown in Figure 4. The input
signals as numbers of revolutions of the right
propeller are shown in Figure 5, Figure 8, Figure
11, Figure 14, and Figure 17. The output signals as
heading angles are shown in Figure 6, Figure 9,
Figure 12, Figure 15 and Figure 18. The trajectories
of this ASV are shown in Figure 7, Figure 10,
Figure 13, Figure 16 and Figure 19.
The different values for the heading angles are
chosen for the simulations shown in Figure 5,
Figure 6, Figure 7, Figure 8, Figure 9, Figure 10,
Figure 11, Figure 12, Figure 13, Figure 14, Figure
15, Figure 16, Figure 17, Figure 18 and Figure 19.
They are , and ,
respectively.
Note that a very fast transition to the desired
heading angles with high accuracy of regulation for
those simulations was achieved.
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Fig. 4: Block scheme of the control system
Fig. 5: Input signal of ASV for
Fig. 6: Output signal of ASV for
Fig. 7: ASV’s trajectory for ,
and stop time
Fig. 8: Input signal of ASV for
Fig. 9: Output signal of ASV for
Fig. 10: ASV’s trajectory for ,
and stop time
Fig. 11: Input signal of ASV for
Fig. 12: Output signal of ASV for
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Fig. 13: ASV’s trajectory for ,
and stop time
Fig. 14: Input signal of ASV for
Fig. 15: Output signal of ASV for
Fig. 16: ASV’s trajectory for ,
and stop time
Fig. 17: Input signal of ASV for
Fig. 18: Output signal of ASV for
Fig. 19: ASV’s trajectory for ,
and stop time
Furthermore, the errors of regulation for the
chosen heading angles were significantly less than
the waited value of 5%. They are -4e-3%, -7e-4 %,
4e-4 %, -3e-4%, and 1e-4 %, respectively.
6 Conclusions
The modeling and simulation techniques have been
presented and analyzed for the real autonomous
electric catamaran “Nymo” in the
Simulink/MATLAB environment. These techniques
enable adjustment of the desired parameters like
transition times, heading angle overshoot values, and
the ratio of coordinate-to-acceleration transition
times for performing such maneuvers as the turn of
the catamaran at different angles.
The proposed controller design methodology can
be used to improve the ASV’s model design for
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various applications, including various reliable
positioning monitoring tasks. From the application
point of view, the methodology provides a valuable
practical opportunity to improve the motion
characteristics of the catamaran “Nymo” dedicated
to environment monitoring and cargo transporting
under difficult sea conditions. In addition, the
designed control system with relatively simple
realization can be easily applied to other types of
ASVs.
The simulation results have confirmed the
impressive quality of the offered optimal nonlinear
control approach to assure the smooth and fast
stabilization of ASV’s trajectories.
Acknowledgment:
Research for this publication was funded by the EU
Horizon2020 project 952360-MariCybERA.
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APPENDIX
Fig. 3: Block diagram of ASV
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed to the creation of
this article at all stages from problem formulation
to final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This article was funded by the EU Horizon 2020
project MariCybERA, agreement No. 952360.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.e
n_US
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