Vessel Dynamic Positioning System Mathematical Model

ANDRII SIMANENKOV1, HALYNA DOSHCHENKO1, VALENTYN CHYMSHYR2,

ANDRII KONONENKO3, HANNA TERZI4, IRYNA SMYRNOVA2

1Department of Operation of Ship Electrical Equipment and Аutomatic Аppliances,

Faculty of Naval Power Engineering,

Kherson State Maritime Academy,

20, Ushakova str., Kherson, 73000,

UKRAINE

2Danube Institute of the National University “Odessa Maritime Academy”,

9, Fanagoriyskaya str., Izmail, 68601,

UKRAINE

3Danube Vocational College of the National University “Odessa Maritime Academy”,

9, Fanagoriyskaya str., Izmail, 68601,

UKRAINE

4Izmail State University of Humanities,

12, Riepina str., Izmail, 68601,

UKRAINE

Abstract: - This study aims to examine two potential approaches for addressing the challenge of synthesizing

control laws in the Dynamic Positioning (DP) system. Both approaches pertain to the same ship model but are

rooted in distinct ideologies concerning how to account for the influence of external disturbances on a closed

system. The findings of this investigation could offer valuable insights for enhancing DP systems and

formulating more efficient management strategies in maritime conditions. The research delves into the structure

and principles of the DP system as a sophisticated control complex, identifying associated challenges in its

application. Despite numerous implemented projects and considerable developer efforts, sustaining a ship in a

specified position during rough seas remains a formidable task, partly due to the ship's lack of energy

armament. Exploration in the realm of regulatory system development has yet to yield the anticipated results.

The present study constructs a mathematical model depicting the dynamics of a ship during positioning,

considering two versions of automatic control laws aimed at stabilizing the ship's position. The second model

demonstrates superior efficiency in the control system, surpassing the first by at least 14%. A comparative

analysis of two control system options with filtering properties in dynamic positioning mode for the vessel was

conducted. For better results, it is recommended to implement filtering on the relevant data source before this

procedure on a specific data consumer. This preliminary testing helps to remove duplicate and inaccurate data,

reducing the load on the data link. Final filtering should be performed on high-performance systems. In

summary, the originality and novelty of this article stem from its comparative analysis of control laws,

exploration of DP system dynamics, acknowledgment of existing challenges, and practical recommendations

for data filtering in dynamic positioning. The study brings a tangible contribution to the field, paving the way

for advancements in the development of DP systems management and control methods.

Key-Words: - dynamic positioning system, offshore vessel, automatic control, modeling, power system,

Kalman filter, DPS operation control, thrusters.

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1 Introduction

In today's context, maritime transport plays a crucial

role in the global transportation system, and the

offshore fleet is undergoing rapid development.

Depending on their type and specialization, offshore

vessels are equipped with specific equipment not

generally found on conventional merchant vessels.

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The nature of work on offshore vessels differs

significantly from that on merchant ships. The

offshore fleet includes special-purpose vessels

designed to perform specialized tasks, unlike the

merchant fleet, which primarily focused on

transporting cargo between destinations.

Specialized offshore vessels are required for

various offshore activities, including offshore oil

and gas exploration, drilling oil wells, developing

subsea infrastructure, installing fixed platforms,

deploying wind turbines, laying transcontinental

cables and pipelines, and providing comprehensive

maintenance of the above infrastructure. These

complex processes include low-level construction

and installation activities in the high seas.

Offshore operations require vessels with a

specific design and specialized equipment. Anchor

handling vessels tow drilling rigs and set/moor

anchors in the correct position. Cable layers are

specially designed for laying subsea cable networks.

Diving support vessels serve as a floating base

for deepwater operations. Two types of vessels are

involved in monitoring and maintaining pipelines

and drilling rigs: pipe-laying vessels that lay

pipelines and drilling vessels that specialize in

drilling exploration wells. These vessels are

equipped with systems for securing the vessel to the

wellhead, drill pipe storage racks, and drilling fluid

tanks, [1].

As the electrical equipment and control systems

on these vessels become increasingly complex, the

competence of personnel, especially in the support

fleet, is crucial. The reliable operation of shipboard

electrical equipment relies on obtaining and

processing accurate measurement data, making

research in this area relevant.

Dynamic Positioning (DP) systems are

extensively used to meet safety requirements for

performing various special tasks at sea and to ensure

proper control of the course and positioning of

offshore vessels. From an economic standpoint, the

use of DP systems on offshore vessels is preferred,

as it eliminates additional costs associated with

ensuring effective control of the vessel's location

and course during operations.

The application of DP systems extends beyond

offshore activities, encompassing support for diving

operations, subsea pipe and cable laying,

transportation, research tasks, and more. This

technology addresses crucial challenges in modern

shipping, especially with the growing interest in

exploring natural resources in the world's oceans.

Given their specific function, offshore vessels

are equipped with a substantial amount of electrical

and electronic apparatus overseen by an electrical-

technical officer, [2]. Reliable data from measuring

instruments and their timely and correct

interpretation are crucial for ensuring the reliable

operation of electrical equipment on ships, which

ensures the relevance of this study.

1.1 Problem Statement

Modern ship control systems and power plants are

highly complex automated technical complexes

designed to efficiently perform operations

determined by the purpose and specifics of the ship's

operation in different conditions. The main method

of studying automatic systems is their mathematical

modeling. Mathematical models of real systems

must reflect their characteristics with the required

accuracy, which leads to complex nonlinear

dependencies.

1.2 Analyses of Recent Sources

The wealth of published papers in the field of ship

modeling is vast. To illustrate, [3], presents ship

dynamics models showcasing the ship's response to

rudder shifting and fixed pitch propeller (FP) speed,

while a nonlinear model with 6 degrees of freedom

is displayed, [4]. Numerous publications, including,

[5] and [6], provide an overview of ship models and

experimental methods for identifying ship

dynamics. Addressing the Dynamic Positioning

(DP) control problem, a publication details a

nonlinear multivariable simulation model for a

floating production, storage, and offloading (FPSO)

vessel, as analyzed in [7], where the system's

cascade models are also presented, [8].

Dynamic positioning (DP) has two main

categories: absolute and relative. Absolute

positioning records the position of a vessel at a

specific point, while relative positioning records the

vessel to a moving object, such as another vessel.

As explained above, a vessel can also be

strategically positioned to take advantage of wind,

waves, or currents - a principle called weathervane,

[9].

2 Theoretical Basis

DP systems autonomously manage the vessel's

position and heading through the continuous

operation of thrusters, effectively balancing

environmental forces such as wind, waves, and

current. These environmental forces naturally

attempt to displace the vessel from its intended

position. However, the automatically controlled

thrust strategically counteracts these forces,

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ensuring the vessel remains steadfast in the desired

position.

Dynamic positioning systems can be described

as the integration of several shipboard systems to

achieve precision maneuverability (Figure 1).

Support and maintenance vessels in the offshore

industry, for example, need the ability to hold

positions within a centimeter as they may have to

move dangerously close to drilling rigs, [9].

The DP system allows vessels to:

- follow a predefined trajectory for precise

maneuvering or staying on course within the

pipeline laying area;

- move at a predetermined speed;

- stay at a predetermined drilling point.

- The DP system is used by:

- drilling vessels and semi-submersible drilling

rigs to stay on point while drilling in deep

water;

- offshore platform supply vessels to maintain a

position in the open sea, unloading supplies or

loading extracted resources or waste

materials;

- pipe-laying and offshore construction vessels

to maintain position.

This is becoming increasingly important in the

development of offshore fields. As oil and gas

exploration moves into deeper waters, the demand

for DP systems increases, reducing the risk of

emergencies during exploration and production.

The main components of any DP system are:

- a positioning system, usually GPS;

- DP computer;

- engines/thrusters.

The positioning system monitors the vessel's

position, the DP computer calculates the required

thrust, and the thrusters apply the calculated thrust

to maintain the vessel's position.

Figure 1 shows the vessel’s dynamic positioning

system schematic diagram.

Advantages of the DP system include:

- efficient & easy vessel positioning and

maneuverability without the need for

moorings, tugs, or labor-intensive anchor

operations;

- capability to operate in ultra-deep waters

where establishing mooring lines is

challenging;

- flexibility to change location or direction

swiftly to avoid adverse weather effects;

- quick disconnection and the ability to sail

away if necessary;

- safety, when working on congested seabeds

with numerous pipelines, buried munitions,

mooring ends, or underwater structures.

The disadvantages of a dynamic positioning

(DP) system include:

- Designing and installing DP requires

significant capital investment.

- Dynamic positioning systems are associated

with increased fuel consumption and

maintenance costs, increasing overall

operating expenses.

- In shallow water, DP systems may be less

cost-effective than traditional mooring

systems.

- There is a risk of severe consequences in the

event of equipment failure, especially during

critical operations such as pipe-laying or

near fixed offshore platforms.

Given the above factors, when using DPs for

fixed positioning of a vessel, it is essential to

consider the vessel's course and all the opposing

forces acting on it. One method often used to assess

the impact of these risks is mathematical modeling.

Fig. 1: Schematic diagram of the dynamic positioning system, [9]

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The DP task involves keeping a ship at a

specific point in the horizontal plane with a given

heading angle. Controlling forces and torque

generated by onboard actuators are employed for

this purpose.

Using various hardware and software, the

Dynamic Positioning System (DPS) automatically

maintains a vessel or offshore structure in a fixed

position, with or without propeller or thruster thrust,

[11].

The ship's current heading position is

determined by an estimation process based on the

ship's model, position, and previous heading

measurements, as well as consideration of the forces

at work. The engines are provided with an

appropriate compensating load to counteract the

external forces and moments, while the thrusters

generate the necessary forces to maintain the desired

position.

The evaluation module processes all signals.

This process includes tests to detect high scatter,

peaks, and signal drift. False signals are ignored and

compensated for by considering the ship's tilt. The

estimation module's main goal is to provide accurate

position, heading, and speed data. The estimation

system filters out fast, purely oscillatory movements

caused by pitching.

The Dynamic Positioning System (DPS) uses

information from the sensor system or position

reference system in the ship model to estimate the

ship's position. A typical control strategy is the

proportional-integral-derivative (PID) controller,

which uses the position and heading estimates. The

integral action is necessary to compensate for static

environmental disturbances, and the controller

feedback includes reference and forward feedback.

The thruster distribution unit displays the

parameters at the controller outputs at a given point.

The hydrodynamic and derivative coefficients,

critical components of the equations of motion, are

determined by experimental testing on a physical

model.

The ship's behavior during dynamic positioning

is nonlinear, and accurate prediction for ship control

is achievable primarily with the use of a nonlinear

mathematical model. System identification methods

are increasingly used to determine ship dynamics,

identifying various input signals.

2.1 The Purpose of the Study

The article analyses the structure and principles of

operating a dynamic positioning system (DPS) as a

complex control system. It also discusses the

problems associated with these systems. Despite the

considerable efforts of developers and many

implemented projects, it is impossible to achieve

long-term maintenance of a ship in a given position

undersea heave. This is because of the lack of

energy armament of the vessels in use. Numerous

studies in the field of regulation systems

development have not yet yielded the expected

result.

This paper aims to analyze two possible

approaches to solving the problem of synthesizing

control laws in the DP system. Both methods can be

applied to the same ship model, but they are based

on different ideologies of accounting for the effect

of external disturbances on a closed system.

3 Methods

A ship model comprises a set of motion equations

employed for forecasting the movement of a vessel

under the influence of specified forces and

moments. For optimal Dynamic Positioning System

performance, it is crucial to maximize the level of

detail in the model. Validation of model parameters

is essential through sea trials. Nevertheless, it's

important to note that the model offers only an

approximation of certain aspects of the ship's

behavior and is not flawless. The ship model

implements a mathematical model of flat, parallel

ship motion that takes into account the

hydrodynamic characteristics of the ship's hull,

pitching and wind drift forces, and moments, which

allows to simulation of the movement of a given

design ship under external environment influence,

[12].

The ship model encompasses representations of

the hull, propulsion system, and active control

components such as propeller columns (PCs) and

thrusters (TPs), which initiate the ship's movement

and dictate its course.

Similar to an actual vessel, the ship model is

furnished with various sensors:

- position and heading sensors are utilized to

ascertain the ship's spatial coordinates (X and

Y) and heading angle;

- motion sensors provide data on the ship's

velocity, along with projections of the speed

vector on the X and Y axes, as well as the

circular speed;

- rotational speed sensors monitor the

propellers of rudder columns (RC) and

thrusters (TP), along with the rotation angle of

RC and TP;

- accounting external influences, including

wind (which determines wind speed and

direction, denoted as u and v) and pitching

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(which determines the pitching score and

wave direction).

In motion, the ship is exposed to environmental

forces - wind, pitching, and current (Figure 2).

Fig. 2: Forces acting on the ship and its displacement, [12]

The external influences model simulates

environmental factors that disrupt the ship's

position, encompassing various disturbances such as

waves of different intensities and directions, as well

as wind exerting force on the ship with specific

strength and direction, [13]. Relevant sensors on the

ship model capture these parameters, serving as

input data for calculating the forces responsible for

the ship's longitudinal and transverse movements.

The resultant forces are then transmitted to the

calculation blocks within the ship's DPS model for

further consideration and compensation.

A sea vessel contends with forces from wind,

waves, currents, and the propulsion system. The

vessel's reactions, manifesting as position, heading,

and speed alterations, are gauged by position

reference systems, a gyrocompass, and vertical

reference sensors. Data from the vertical reference

sensors are used to correct for roll and pitch in the

reference system readings. Wind sensors measure

wind speed and direction.

The DPS control system computes the forces

required for the engines to manage the ship's motion

across three degrees of freedom in the horizontal

plane—wave, oscillation, and pitch.

In describing the dynamics of a surface vessel, a

horizontal motion-based model with variable

parameters such as wave, oscillation, and pitch is

employed. Several main forces influence the

movement of a vessel: hydrodynamic forces and

torques. The main input variables for calculating the

current course are the shaft angular velocity related

to the propeller thrust and the rudder deflection

angle. This model assumes negligible changes in

roll and pitch, allowing their omission from the

equations. Consequently, the ship is considered a

solid body moving in a plane, possessing three

degrees of freedom, as outlined in Equations 1-3,

[12].



  

 

 

(1)



 



 

 



(2)



 

 

 

 



(3)

In the given expression

t is the time index,

u, v - wind speed and direction,

r is the angular velocity,

m and Iz are the mass of the vessel and the

moment of inertia relative to the axis of normal to

the X0Y0 plane,

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xG - Cartesian coordinates of the centre of

gravity along the X0 axis,

δ is the deviation of the rudder angle,

n - shaft rotation frequency,

X(...), Y(...) and N(...) correspond to external

forces (longitudinal waves X0, oscillation axis Y0)

and moment (for X0-Y0 rotation). Accurate,

dependable, and continuous location data plays a

crucial role in dynamic positioning, with a requisite

data rate of once per second to attain high precision.

The dynamic positioning of a vessel depends on

a particular coordinate system that differs from a

conventional navigation system. In DPS mode, the

model provides automated steering from the current

position to a predetermined point, calculating the

basic parameters for the active steering mechanisms.

Built-in algorithms in the DPS model dynamically

determine the necessary adjustments through the

remote pilot unit (RPU) and power plant (PP),

compensating for counteracting forces and guiding

the vessel on a predetermined course. Within the

DPS model, various calculation modules handle

distance and heading correction, speed correction,

torque correction, compensation for forces, optimal

distribution of thrusts, conversion of thrusts to

angles, and propeller speed calculation.

Implementations of modern DPS control

systems often require velocity estimation from

position and direction measurements and filtering

out the oscillatory components of motion due to

waves. This type of filtering, known as wave

filtering, is a key aspect in the motion control of

marine surface vessels.

It is recommended that the ship model be

improved by considering the oscillatory wave

motion model to estimate the wave velocity and

filtering. The states of the various components of

the model can be accurately estimated using a

Kalman filter. This methodology integrates ship

velocity estimation with effective wave motion

filtering, resulting in significantly improved

accuracy and reliability of dynamic positioning

systems.

This model is directly used in one of the

approaches to construct the control law and is also

used in simulation modeling in numerous

experiments.

A first-order Markov process is used to describe

the slowly changing forces acting on the ship, so we

write:

󰇗   

(4)

In Equation 4, b



R3 is a vector of slowly

varying forces and moments, n



R3 is white noise,



R3+3 is a diagonally positive definite matrix, and



R3+3 is a matrix that scales the disturbance by

components.

The sea disturbance model will be:

   

󰇗,

 .

(5)

Here, ξ



R6 is the model state vector, w



R3 is

white noise, and ηw



R3 is the component of the

measurement signal that arises due to the wave.

The matrices in the equation are as follows:

(6)

The parameters ω01 (і = 1..3) are the central

frequencies of the disturbance, ζ01 (і = 1..3) are the

attenuation coefficients, and σ01 (і = 1..3) are the

intensity of the disturbance for each component.

As noted above, the measured signal is affected

by measurement errors. These errors are represented

as a sum:

    

(7)

where,  – ship’s position; v – white noise.

Data obtained from a positioning system

introduces undesirable negative white noise, a factor

influenced by the sensor type and the measurement

method employed to gauge the ship's position. The

ensuing challenge revolves around accurately

estimating the vessel's position when confronted

with imprecise knowledge of its dynamics and

measurements affected by noise. The solution to this

quandary lies in applying Kalman filtering, [13]. In

the context of a dynamic positioning program, the

Kalman filter estimates the vessel's state, leverages a

previously established dynamic model, and relies on

noise-influenced measurements from the reference

system and sensors.

The Kalman filter is a mathematical algorithm

for solving the problem of linear optimal filtering of

discrete random nonstationary processes. It has

proven itself very well in solving digital signal

processing problems. No GPS navigator can do

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without a software implementation of the Kalman

algorithm, and the algorithm is successfully used in

sensor readings processors to implement control

systems.

The relevance of using mathematical filters in

signal processing is caused by the invariably present

error in the readings of various sensors and devices

caused by the finite accuracy of the device itself and

the influence of random influences. The situation is

aggravated by the inability to directly measure the

parameters of processes inside certain devices

without disrupting the operation of these devices in

a large number of cases.

The use of filtering methods, in particular the

Kalman method, minimizes the error in observations

and sensor readings. To a first approximation, we

can say that the task of the filter is to find a good

approximation for the true data, knowing the

incorrect sensor readings (sensor, meter, or just

observations) and knowing the mathematical model

of the process under study.

The Kalman filter employs a feedback control

mechanism to estimate the process; it gauges the

state of the process at a specific time and then

incorporates feedback through (noisy)

measurements.

The Kalman filter equations consist of time

update equations and measurement update

equations. Time update equations project the current

state and error variation estimates forward in time,

generating a priori estimates for the next time step.

Measurement update equations incorporate new

measurements into the a priori estimate to produce

an improved a posteriori estimate.

Time update equations can be considered

forecasting equations that project forward state

estimates and error variations. The measurement

equations can be thought of as correction equations

that integrate new measurements to improve the a

priori estimate. The final estimate produced by the

algorithm resembles a predictor-corrector algorithm

commonly used to solve numerical problems. The

equations for updating time and measurements are

as follows, [12]:

Time update, Eqs. 8-9:





󰇗 

 

(8)



 

(9)

Update measurements, formulas 10-12:

 

󰇛

󰇜

(10)



  



󰇛



󰇜

(11)

󰇛 󰇜



(12)

After each pair of time and measurement

updates, the process repeats and compares the

current iteration with the previous one. The a

posteriori estimates obtained from the measurement

updates are then confidently used to formulate new

a priori estimates during the time update, creating a

robust recursive loop. The iterative nature of the

Kalman filter allows it to continuously refine and

improve its estimates as new measurements become

available, increasing the accuracy and reliability of

the overall estimation process with each iteration.

This recursive nature stands out as a highly

appealing characteristic of the Kalman filter,

making practical implementations more manageable

than, for instance, implementing a Wiener filter.

Unlike the Wiener filter, which processes all data

directly for each estimate, the Kalman filter

conditions the current estimate recursively on all

past measurements.

The main task of the positioning control system

is to maintain a given position and compass

heading, regardless of external disturbances. This

system effectively counteracts external forces and

factors, ensuring accurate and stable positioning of

the vessel. The challenge lies in mitigating these

disturbances by applying appropriate counteracting

forces.

Computer and simulation modeling are

important ways to study control systems. The

approaches based on them, allow conducting

experiments using computer systems, resorting to

the construction of physical models and without

conducting expensive field tests. Such experiments

are designed to reveal the properties of dynamic

systems, which helps to choose control laws.

For computer modeling of dynamic objects,

[14], the MATLAB environment with the Simulink

software subsystem is used, [15].

This tool allows create computer models of

dynamic systems in a visual mode using a set of

standard elements. Simulink runs under the

MATLAB application package and has access to a

wide range of features, such as efficient numerical

methods, powerful data processing tools, and

scientific visualization. The computer model can be

used to simulate dynamic processes. The system

gives the opportunity to display the data obtained

during the simulation in special blocks or transfer

them to the MATLAB working environment for

further processing.

The structures of analyzed control systems

computer models are shown in Figure 3 and Figure

4. They consist of five main blocks: External

disturbances, Measurement noise, Vessel model,

Observer, and Controller.

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Fig. 3: Simulink model of the first control law (author's development)

Fig. 4: Simulink model of the second control law (author's development)

The computer model of external disturbances is

represented by the External disturbances block,

which consists of slowly changing components that

arise under the influence of wind, currents, and

harmonic oscillations caused by sea waves.

The vessel model represented by equations (1-3)

is implemented using the Vessel model block. The

model contains two input signals: sea disturbance

and control, which originate from the External

disturbances and Controller blocks, respectively.

The output of this model is the components of the

ship’s state vector. To obtain a model of the

measured signal y, it is necessary, according to

equation (7), to add the output η of the ship model

with a component that realizes measurement errors.

This component is the output v of the Measurement

noise block. The noise range is determined by the

accuracy of the measuring instruments. The

arrangement of the External disturbances,

Measurement noise, and Vessel blocks are

completely the same in both computer models.

The speed and position of the vessel, as well as

the control signal, are transmitted to the Scope

visualization blocks. This data can be conveniently

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used in MATLAB to compare the performance of

the systems under consideration.

Figure 3 shows the Simulink model of the first

control law. It shows that the output signals of the

Observer block are position and velocity estimates,

as well as an estimate of external disturbances. The

structure of the second computer model (Figure 4)

differs only in the output signals of the asymptotic

observer. There are no estimates of the external

disturbance vector.

Let us consider the results of the computer

models presented above. Figure 5 shows the

combined transients of the control systems, with the

blue color indicating the model of the first control

law (Figure 3) and the green color indicating the

second computer control model (Figure 4). Since

they have the same basic controller, it takes some

time to reach the neighborhood of the equilibrium

point.

Figure 6 shows the joint graphs of the systems

at the equilibrium point under the influence of

external perturbations to the ship. It can be seen

from the figures that the control systems keep the

ship in the vicinity of the set position. The ship has

similar behavior in both cases.

From the results presented in Figure 6, slightly

smaller deviations can be seen, given by the second

system in the value of the ship’s heading angle.

Fig. 5: Control system transients (author's

development)

Fig. 6: System dynamics at the equilibrium point

η=(40m, 40m, 45°) (author's development)

Values of both control signals are presented in

Figure 7. Control systems of both types produce

similar signals, which can be seen from the figures.

It should be noted that the control system of the

second computer model has a special feature, i.e., its

ability to switch between different correctors easily.

Thus, with a small amplitude of disturbance, a

simple version of the corrector can be used, and if

necessary, switch to a more modern version.

Fig. 7: Control signals of the control systems

(author's development)

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Such an operation is not provided in the first

system since the harmonic component of the ship’s

position is included in the state vector of the

asymptotic observer.

Figure 8 demonstrates this switching. For the

first 300 seconds, the simple version of the corrector

is running, and the rest of the time, the complex

version of the control law is running. This figure

also demonstrates the advantages of control laws

that have a filtering property using the Kalman

filter.

As the comparative analysis shows, the

analyzed approaches give similar results, despite

significant differences in the ideology of building

control laws.

The control law with dynamic correction gives

good results. Its efficiency is higher because it

requires solving a system of differential equations of

a lower order. Although the procedure for setting up

the dynamic corrector is more complicated for this

approach than for the first option, there is a

possibility of further improving the dynamic

corrector to obtain new properties of the control

system, for example, setting up additional filtering

frequencies.

Fig. 8: Switching between correctors in the

control system of the second computer model

(author's development)

One of the main advantages of the control law

with correction is the flexibility of its use. The

control law makes it possible to change the dynamic

corrector during operation, which allows to

selection the control mode by changing conditions.

For example, in the absence of any disturbances,

control low can be limited by basic part, and if

necessary, a switch-over to more advanced options

is available.

The advancement in computing power has

opened up possibilities for implementing more

sophisticated control algorithms. This has led to

commercialising demanding control strategies like

predictive model control and online numerical

optimization methods.

The literature explores numerous controllers,

[16], [17], [18], [19], [20], with some successfully

integrated into various commercial Dynamic

Positioning (DP) systems.

Many DP systems leverage multivariate

Proportional-Integral-Derivative (PID) controller

algorithms in conjunction with an observer, [21].

The PID generates thrust that is proportional to the

three-dimensional position of the vessel and the

deviation vector from the desired setpoint

(proportional component), the speed deviation

vector (differential component), and the

accumulated deviation vector (integral component).

All of these vectors are time-dependent. The system

determines the required motor force vector as the

sum of the three components responsible for

proportional, differential, and integral actions.

4 Results

The approach to designing new systems marks a

significant advancement in dynamic vessel

positioning system technology, comparable in

importance to the integration of Kalman filtering

into optimal control schemes. The evident

advantages of the latter led to the widespread

adoption and practical application of the Kalman

filtering system.

Upon conducting a thorough analysis of

navigation safety challenges in various sea surface

conditions, it becomes apparent that employing a

mathematical model facilitates numerous

computational studies on ship control modes with

Dynamic Positioning Systems (DPS). These studies

contribute to the formulation and refinement of

algorithms governing ship movement during

positioning, with due consideration for the optimal

utilization of electric power system resources. This

approach simplifies the complexity associated with

the development and adjustment of control

algorithms.

A mathematical model of the ship's dynamics in

the process of its dynamic positioning is built in the

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study and two versions of the automatic control laws

that stabilize the ship’s position are considered,

while the second model shows a greater efficiency

of the control system, at least by 14% compared to

the first.

Considering the variable values of external

disturbances that affect the quality of object

management, the accuracy of forecasts and the

speed of model updating play a significant role in

the management of the DP system. The simulation

results allow us to describe the offshore vessel as a

rigid body that moves in a plane, considering three

degrees of freedom, which is used to analyze the

course control of the DP system.

A comparative assessment was conducted on

two control system variants with filtering properties

applied to a vessel under dynamic positioning mode.

The study concluded that it is preferable to conduct

filtering initially on the data source and

subsequently on the information consumer.

Implementing pre-filtering enables the elimination

of redundant and erroneous data, consequently

lessening the burden on the data transmission

channel. Additionally, it was determined that the

ultimate filtration should be carried out on high-

performance systems to achieve the most effective

filtration results.

5 Discussion

In aligning the simulation results with prevailing

research trends in ship dynamic positioning, it

becomes evident that various researchers similarly

underscore the significance of advancing systems

for dynamic positioning on ships. This collective

emphasis signifies a progressive stride in

technological innovation. Much like the present

study, the integration of Kalman filtering into

optimal control schemes is recognized as a pivotal

step by other researchers, emphasizing its role in

determining system efficiency.

For instance, practical implementations of

Kalman filtering have been successfully employed

in real-world maritime scenarios, showcasing its

efficacy in enhancing the precision of navigation

systems during dynamic positioning maneuvers. The

integration of such filtering techniques aligns with

the broader industry push towards more robust and

efficient dynamic positioning systems for ships.

Moreover, beyond the theoretical framework of

research, the identified effectiveness of specific

automatic control laws for stabilizing ship positions

can be directly applied in the development and

optimization of shipboard control systems. This

could lead to the implementation of more reliable

and responsive control strategies, thereby improving

the overall safety and operational efficiency of

dynamic positioning systems in practical maritime

applications.

Extending beyond the realm of the research,

comprehensive analyses of navigation safety

problems in diverse sea surface conditions are found

in other studies, contributing valuable insights.

These broader investigations, although exceeding

the chosen topic's scope, serve to reinforce the

overarching importance of mathematical models for

studying and analyzing dynamic positioning system

(DPS) management modes.

In the context of processing externalities and

data filtering, the research conducted by

contemporary scholars lends support to the concept

of a staged approach to filtering at both the data

source and information consumers. For example,

recent applications in maritime data processing have

demonstrated the benefits of implementing pre-

filtering at the data source to eliminate redundant

and erroneous data, followed by a final filtration

stage on high-performance systems. These practical

examples substantiate the broader implications and

applications of the research outcomes in the context

of dynamic positioning systems for ships.

6 Conclusions

Although the research poses a significant value, it is

important to recognize its inherent limitations and

imperfections. It stands to reason that the fixation on

the implementation of Kalman filtering and

mathematical models may unintentionally distract

from other technologies that may offer alternative or

complementary solutions. It is also possible that the

research’s particular circumstances may not fully

reflect the varied and sophisticated real-world

maritime environments.

The effectiveness of control algorithms and

filtration methods depends on the environment.

Therefore, determining the specifics of their use by

different vessel types and service conditions

required further detailed studies.

The rigid body model for sea-going vessels

proposed in this study resulted in a significant

breakthrough. Nevertheless, it does not take into

account the peculiarities of the dynamic marine

environment or the structural features of more

complex vessels.

The above limitations demonstrate the

importance of a multi-vector approach to dynamic

positioning system (DPS) design. The obtained

results achieved through the use of the Kalman filter

and mathematical models may be further applied

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with due regard to the alternative technologies and

uncertain marine environment. The actual service

adaptability and reliability of DPS can be increased

by conducting a comprehensive study of AI-based

control systems, improved sensors, and other

alternative technologies.

Control algorithms and filtering techniques

might need to undergo adjustment for specific

vessels, conditions, and operational requirements for

practical implementation. The implementation and

design of DPS must be performed adaptively to

obtain more reliable and impactful outcomes in

different maritime conditions.

The research constitutes a substantial

development in dynamic vessel positioning,

although practical application requires addressing

the identified limitations. The outlined development

approaches should be refined and adapted for their

effective and varied employment in real-world

maritime environments. Researchers, industry

specialists, and technology developers should

continuously collaborate to bridge the gap between

theory and practice. This collaboration is crucial for

the further advancement of dynamic vessel

positioning technologies.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed to the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

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)

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Creative Commons Attribution License 4.0

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