CC BY
33
DOI: 10.24412/2413-2527-2022-432-78-84
Mathematical Models for Software Development
of Control Systems for Unmanned Vessels
Grand PhD S. S. Sokolov, G. V. Danilin, PhD T. P. Knysh Admiral Makarov State University of Maritime and Inland Shipping Saint Petersburg, Russia [email protected], [email protected], [email protected]
Abstract. Unmanned navigation is a direction in the field of water transport, which is based on the idea of operating unmanned vessels. Despite the fact that the world's first model of an unmanned vehicle was the model of a remotely operated vessel shown by Nikola Tesla in 1899, for a long time humanity's interest in unmanned vessels was much lower than in unmanned aerial vehicles, since they found use in the military industry faster, and cars that would potentially have greater consumer demand, and accordingly, projects that would be commercially successful and quickly recouped. However, in recent decades, in the field of water transport, a tendency to reduce the number of ship crews has emerged and continues to develop to this day, based on the desire to reduce the influence of the «human factor» on the accident rate of navigation. In addition to improving the safety of navigation, it is expected that with the transition to unmanned navigation, the useful capacity of ships will increase and operating costs will decrease. In this regard, interest in unmanned vessels has increased and it has become necessary to develop software for their control systems, and for this, in turn, appropriate mathematical and algorithmic support is required. In this work, at the first stage, the concept of an unmanned vessel will be studied, the classification of the degree of autonomy of vessels will be considered, and the structural diagrams of control systems of unmanned vessels with remote control and autonomous unmanned vessels will be familiarized. At the second stage, the features characteristic of most mathematical models of ship movement described in the literature today will be studied first, and then systems of equations describing the movement of a ship in deep water and in shallow water will be considered.
Keywords: water transport, automation, control system, unmanned vessel, mathematical modeling, block diagram.
Introduction
Unmanned navigation is a direction in the field of water transport, which is based on the idea of operating unmanned vessels [1]. Despite the fact that the world's first model of an unmanned vehicle was the model of a remotely operated vessel shown by Nikola Tesla in 1899, for a long time humanity's interest in unmanned vessels was much lower than in unmanned aerial vehicles, since they found use in the military industry faster, and cars that would potentially have greater consumer demand, and accordingly, projects that would be commercially successful and quickly recouped. However, in recent decades, in the field of water transport, a tendency to reduce the number of ship crews has emerged and continues to develop to this day, based on the desire to reduce the influence of the «human factor» on the accident rate of navigation. In addition to improving the safety of navigation, it is expected that with the transition to unmanned navigation, the useful capacity of ships will increase and operating costs will decrease. In this regard, there has been an increased interest in unmanned vessels.
Types of unmanned vessels
The International Maritime Organization (IMO) has assigned the abbreviation MASS to unmanned vessels (from the English Maritime Autonomous Surface Ships — marine autonomous surface vessels). According to the definition given by IMO in May 2018, an unmanned vessel (MASS) is a vessel that, to one degree or another, can function independently of human involvement. At the same time, a classification of the degree of autonomy of courts was proposed:
• a vessel with automated processes and decision support, on board of which there is a crew to activate and control on-board systems and functions;
• remotely operated vessel with crew;
• remotely operated vessel without crew;
• a fully autonomous vessel, the onboard control system of which is capable of independently making decisions and determining the order of actions.
Currently, the last two classes of vessels are of the greatest interest to researchers and designers [2-4]. Since today there is no single approach to the design of unmanned vessels, the creation of projects of vessels of the same and second class has been allocated in two parallel directions.
The most significant difference between vessels from different classes is the construction of their control systems. Not a small number of domestic and foreign scientists have been engaged in the development of ship control systems [3, 5-8], however, there is still no unified approach to the creation of control systems for an unmanned vessel. At the moment, most of the developments in this area belong to private organizations seeking to preserve trade secrets and developing their systems closed, and the control systems themselves are created for specific technical characteristics of specific samples. The vision of how the structure of the control system of a remote-controlled unmanned vessel (Figure 1) and the structure of the control system of a fully autonomous unmanned vessel (Figure 2) should look like is presented in several scientific publications [2, 4].
It is likely that vessels of both classes will be equally widespread in the future for a certain period of time, but they will be focused on performing different tasks: fully autonomous vessels will make long-distance sea crossings, and their wiring through straits, channels and rivers, as well as assistance in performing maneuvers in the port, will probably be carried out using remotely controlled unmanned vessels. However, gradually fully autonomous vessels are likely to prevail. This should be facilitated by the development of mathematical and algorithmic support.
Fig. 1. Structure of the control system of an unmanned vessel with remote control
Fig. 2. Structure of the control system of an autonomous unmanned vessel
It can be seen from the figures that the structure of the control system of an autonomous unmanned vessel is more complex and contains a larger number of subsystems. Appropriate mathematical and algorithmic support is critically important for the creation of an autonomous vessel control system.
RC sports models & toys about mathematical models of ship movement
To date, a number of mathematical models of ship movement have been described in the literature [9], which combine the following:
1. All these models are special cases of a mathematical model of controlled motion of a rigid solid.
2. All models allow you to calculate the trajectory of the vessel under specified conditions.
3. Each state of the vessel S(t) considered in the model is a set of 3 spatial coordinates of its center of mass and 3 angles describing the kinematic position of the vessel: roll, trim and yaw.
4. A significant part of the models neglects the change in the draft of the vessel, the angles of its roll and trim, due to their slight change, and is reduced to considering the planeparallel movement of the vessel in the horizontal plane, that is, characterizes the state of the vessel S(t) with only three parameters: coordinates x0 and y0 with respect to the fixed coordinate system and the heading angle q, that is
S( t) = (x0( t), y0( t),q( t)). (1)
5. Models are continuous, the number of ship states S(t) in them is finite.
6. Models are described, as a rule, by a system of 3 differential equations of the 2nd order:
d2S ( dS \
— = F( t,C , - ,S(t),U(t),L(t),E(t)J,
where is a vector that includes the linear dimensions of the vessel, its areas and volumes, as well as the technical characteristics of its controls; U (t) are control actions, such as the
angle of the rudder shift SR (t), the rotation speed of the propeller nm(t), its pitch ratio H/D(t) and the position of the thruster regulator NTRrel (t), which sets the percentage of power relative to the maximum; ( ) are load parameters, such as the masses and coordinates of the cargo on board the vessel; E( t) — external disturbing influences — the depths at each point of the water area, the directions of winds and currents together with their velocities, the amplitude and phase spectrum of waves together with the spectrum of wave propagation directions at all frequencies for each point of the water area at any given time.
7. All mathematical models described in the literature can be divided into 2 types: the first type describes the movement of one particular vessel, expressed, most often, by a system of differential equations with fixed coefficients determined by the selection method when comparing the simulation results with the experimental results. Such models are suitable only for highly specialized research and the creation of navigation simulators for specified types of vessels. The second type of models describes the movement of ships belonging to a certain class. In models of this type, the systematization of experimental results is carried out on a sample of vessels, so they make it possible not only to calculate the movement of the vessel in certain navigation conditions, but also to calculate how its measurable parameters will affect the behavior of the vessel. Such models are called semi-empirical. They are usually expressed through a system of differential equations and a set of formulas, tables and graphs necessary to calculate the coefficients of these equations. There are no purely theoretical models to date.
8. For the vast majority of mathematical models described in the literature, empirical data were obtained during model experiments with mock-ups of ships, and not with real ships, which means that the scale coefficients used in the calculations introduce additional errors into the models.
Equation (1), suitable for describing any existing mathematical model today, can be written in more detail:
5F=«,if,dirdi■ "(t),yoU^uitmtXEmyi, ,
But a significant part of the authors use equations not with respect to coordinates and heading angle, but with respect to their derivatives — linear velocity v, angular velocity w and drift angle p.
After conducting a critical analysis of a number of mathematical models of ship movement, it can be concluded that today there is no ideal mathematical model of ship movement that would provide acceptable accuracy in calculating the kinematic parameters of all maneuvers performed by the ship, as there is no universal model suitable for describing not one specific, but any vessel. This currently imposes certain restrictions on the development of control systems for fully autonomous vessels. It turns out that when creating a new model
of a basecoaching vessel, it is necessary to select and adapt the most suitable mathematical model specifically for it, and to ensure that this vessel can operate in different navigation conditions, it may even be worthwhile to provide in its control system the possibility of performing calculations on several models, switching between which will be carried out depending on the readings of external sensors and sensors transmitting information about the environment.
For example, when a vessel is moving through deep water [10], the equations of motion of the vessel look like:
dV dz dp dT
where
m26 — mee— 1 — -=- Cm sin G —Cv sin G--Cx cos p
■ M M T^íí
m26— m66— 1 — ' Cm cos p —Cv cos p--Cx sin p
■ M M míí
V
sV
dtà dT
m267 m66-
m M y
—2 V
— píSy-2 _
Cx = cxr + -p^fVk CXa + mu> sin p -
- x(1 -i\>')2-£oe(1 -t')k2v ;
LT
— Pl_Sy-2 _
Cy = Cyr + ~pLTVk Cya + COs p
F
- x(1 - i]>')2j^kpOyk2;
PíSy—2
Cm — CmT + p^j Vk Cma x(1 ty')2 J^kpkslpGykv ;
M — m22m66 — m26; _
V is the linear (reduced) speed of the vessel, V — V/V0;
V — the current value of the linear speed of the vessel; V0 — initial value of the ship's linear velocity;
P — drift angle;
w — is the dimensionless angular velocity of the vessel, w — D.L/V0;
L — is the length of the vessel; Q — is the angular velocity of the vessel; m11, m22, m66 — are dimensionless hydrodynamic coefficients;
t — is dimensionless time, t — V0/L;
lp — dimensionless steering arm, lp — lp/L;
lp — distance of the rudder baller from the center of gravity of
the vessel;
CXr, Cyr, Cmr, CXa, Cya, Cma, ae, oy, t', ty' — complex nonlinear
functions of the desired parameters V, w, P; Xr, Yr, Mr — experimentally determined hydrodynamic forces; CXr, Cyr, Cmr — dimensionless coefficients of experimentally determined hydrodynamic forces;
CXa, Cya, Cma — dimensionless coefficients of aerodynamic forces;
Gy — dimensionless coefficient of transverse force;
Ge — dimensionless coefficient of thrust;
ty' — the coefficient of the associated flow;
t' — is the suction coefficient for curved motion;
Fp - hydraulic section of the propulsion and steering complex;
kv — is the coefficient characterizing the change in speed in
the area of the complex compared to the speed in the center of
gravity of the vessel;
k£ — is the coefficient of displacement of the point of application of the transverse force of the propulsion and steering complex during its operation behind the hull; p — is the density of water; p1 — air density;
Sy — sail area in the lateral projection of the vessel; Vk — wind speed relative to the vessel.
But in the case of vessel movement, for example, in shallow water, this mathematical model cannot be used without changes: the dynamic characteristics of the vessel in shallow water differ significantly from those in deep water. The resistance to the movement of the vessel in deep water is less, and the speed of the vessel at a constant speed of rotation of the propellers is higher. The system of equations describing the effect of shallow water on the hydrodynamics of the vessel is as follows:
__F
-cxr - musin p + x(1 - J^^e^e x (1 - t)k2 = 0
__Fp
Cyr — mu cos p — x(1 — ^')2 x — kpayk2 = 0
Fp -
C-mr — x(1 — 40 X J^kpkelpGykv = 0 —kelpCyr + Cmr + kelpmucos p = 0
In addition, there are studies that have revealed the influence of the ratio of the draft of the vessel to the depth, as well as other coefficients on the movement of the vessel in shallow water. Thanks to research of such a plan, as well as research in related fields [11-15], it is quite possible that in the near future it will be possible to create mathematical models of vessel movement in shallow water, suitable for their use in control systems on river autonomous unmanned vessels.
Conclusion
In this work, at the first stage, the concept of an unmanned vessel was studied, the classification of the degree of autonomy of vessels was considered, and the structural diagrams of control systems of unmanned vessels with remote control and autonomous unmanned vessels were familiarized. At the second stage, the features characteristic of most mathematical models of ship movement described in the literature today were first studied, and then systems of equations describing the movement of a ship in deep water and in shallow water were considered. In the course of the work, it was noted that today there is no sufficiently accurate mathematical model of ship movement suitable for describing the movement of any vessel in any navigation conditions. This currently imposes certain restrictions on the development of control systems for fully autonomous vessels.
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Построение математических моделей для разработки программного обеспечения систем управления безэкипажными судами
д.т.н. С. С. Соколов, Г. В. Данилин, к.ф.-м.н. Т. П. Кныш Государственный университет морского и речного флота имени адмирала С. О. Макарова
Санкт-Петербург, Россия [email protected], [email protected], [email protected]
Аннотация. Безэкипажная навигация — это направление в области водного транспорта, которое основывается на идеи управления безэкипажными судами. Несмотря на то, что первой в мире моделью безэкипажного транспортного средства была модель дистанционно управляемого судна, показанная Николой Теслой в 1899 году, долгое время интерес человечества к безэкипажным судам был намного ниже, чем к беспилотным летательным аппаратам, поскольку они быстрее находили применение в военной промышленности, а автомобили, которые потенциально имели бы больший потребительский спрос и, соответственно, проекты, которые были бы коммерчески успешными и быстро окупались. Однако в последние десятилетия в сфере водного транспорта наметилась и продолжает развиваться по сей день тенденция к сокращению численности судовых экипажей, основанная на стремлении уменьшить влияние «человеческого фактора» на аварийность судоходства. Помимо повышения безопасности мореплавания, ожидается, что с переходом на безэкипажную навигацию увеличится полезная вместимость судов и снизятся эксплуатационные расходы. В связи с этим возрос интерес к безэкипажным судам и возникла необходимость разработки программного обеспечения для их систем управления, а для этого, в свою очередь, требуется соответствующее математическое и алгоритмическое обеспечение. В представленной работе на первом этапе будет изучена концепция безэкипажного судна, рассмотрена классификация степени автономности судов, а также приведены структурные схемы систем управления безэкипажных судов с дистанционным управлением и автономных безэкипажных судов. На втором этапе сначала будут изучены особенности, характерные для большинства математических моделей движения судна, описанных сегодня в литературе, а затем будут рассмотрены системы уравнений, описывающие движение судна на большой глубине и на мелководье.
Ключевые слова: водный транспорт, автоматизация, система управления, безэкипажное судно, математическое моделирование, структурная схема.
Литература
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