629

1 INTRODUCTION

Analysis of the voyage cycle of sea vessels shows that

they spend about 20% of the sailing time in confined

waters, but it accounts for about 80% of all emergency

events.

Turning a ship in the open ocean, when sailing

away from navigational hazards, is not difficult, and

when using maps 1: 50,000 and smaller, its details are

not expressed on the map at all. In confined waters,

the water area for maneuvering is commensurate with

the geometrical dimensions of the vessel and planning

the trajectory of movement, and especially its

curvilinear section, becomes of paramount

importance.

The most dangerous section of the track, which

determines the safety of the vessel's movement, is the

curved one. The reasons for this are the speed of the

process of moving the vessel when turning, the

absence of the planned coordinates of the curved path

Algorithms and Calculation Scheme for Planning the

ay of Movement of Trajectory Point During

aneuvering for Anchoring

. Surinov

National

University “Odessa Maritime Academy”, Odessa, Ukraine

ABSTRACT: The analysis of the historical aspect of the development of the maneuvering during anchoring

shows that during the movement there is no time to control the position by technical means, due to the speed of

the process of changing the parameters of movement. The main purpose of this research is to modify the

methodological basis for the preparation of flight planning during anchoring with increased accuracy to

determine the coordinates along which the ship will move. The methodology of this work is based on the

calculation of trajectory points of the path in combination with the method of segments on the map using the

characteristics of the maneuverability of the vessel. The method is based on determining the coordinate matrices

of rectilinear and curvilinear sections through which the vessel passes during maneuvering for anchoring. The

calculation program is developed in Excel, and allows without the help of a micro calculator that takes into

account the maneuverability of the vessel in the automatic system. Thus, the control of the center of gravity of

the vessel relative to a given path line is performed.

A high-precision system for automatic determination of planned coordinates by trajectory points on track and

traffic control during anchoring has been developed, which is based on recently developed algorithms,

calculation schemes and methods at the Maritime University, which are based on the latest meaningful models

of high-precision planning movement on them. This approach automates the process of controlling safe traffic,

including the use of decision support systems, including stranding prevention and collisions with other vessels.

The results of the research can be used on a ship for automated planning of coordinates on waypoints and

control of traffic on them for safe maneuvering, as well as for training navigators on specialized simulators to

perform trajectory planning, including limited conditions.

http://www.transnav.eu

the

International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 15

Number 3

September 2021

DOI: 10.12716/1001.15.03.18

630

and the necessary data on the turnability, which are

required for their planning.

To eliminate such reasons, it is necessary: to carry

out preliminary planning of the coordinates of curved

track sections or to automate the process of their

calculation, which can be performed while sailing;

perform automated control of lateral displacement,

relative to the planned coordinates, select a maneuver

for divergence and use the maneuver in a timely

manner to correct movement; use high-precision

methods of determining the position of the vessel,

including geodetic ones. Therefore, the purpose of

this work is a method for planning the specified

coordinates of trajectory by waypoints (WP), based on

determining the matrixes of coordinates of trajectory

points (TP) of rectilinear and curved track sections

through which the ship passes when maneuvering for

anchoring. Another aspect of the problem is the lack

of the necessary data on the maneuvering properties

of braking and turnability of the vessel for all modes

of using control actions. Therefore, such tasks are

highly relevant.

2 ANALYSIS OF RECENT STUDIES AND

PUBLICATIONS, WHICH BEGAN TO SOLVE

THIS PROBLEM AND HIGHLIGHT THE

PREVIOUSLY UNSOLVED PARTS OF THE

GENERAL PROBLEM

There are three ways to plan a turn - manual, semi-

automatic and automatic.

When planning a turn manually, the required

initial data [5] is determined by the Master according

to the turnability characteristics before the start of the

turn. They include the coordinates of the points of the

beginning and end of the turn, bearings and distances

to noticeable landmarks at these moments. A sailor

controls the steering wheel during the turn. Turn

control consists in determining the position of the

vessel at the moments of its beginning and end.

Movement on a curved path is not controlled due to

the fact that location determination takes a long time,

and such information is delayed by the time a control

decision is made. In order to control the position

when turning, use the accelerated methods of

determining the place.

With semi-automatic, part of the turn planning

functions [2, 9] is performed manually, then part of

the motion control functions are performed

automatically.

Currently, there are navigation devices [3, 5],

which make it possible to quickly determine the

position of the pivot point (PP) and the width of the

maneuvering lane during movement, which

significantly increases the speed of the process of

obtaining parameter values and making decisions on

control.

Using the method of control of the abscissa of the

position of the centre of gravity (CG) [8], PP,

recalculation of the coordinates of the satellite dish to

the CG and the method of planning a given path using

coordinate matrices [4] of trajectory points (TP) allow

automating the process of high-precision planning

and control along the trajectories including curved

sections. Methods for solving these problems,

developed at the National University "Odessa

Maritime Academy" recently [1–5, 8, 9] , make it

possible to propose algorithms and calculation

schemes and develop a navigation system for

automatic planning of coordinates of the path of

movement and control along it.

3 FORMATION OF THE GOAL OF THE WORK

AND THE FORMULATION OF RESEARCH

OBJECTIVES

Algorithms and computational schemes determine the

procedure for increasing the accuracy of planning the

route of movement to anchorage area, taking into

account the water area for maneuvering,

characteristics of braking and turnability and the use

of methods for planning the route of movement by

trajectory points, including curved sections. To

achieve this goal, we will perform the inverse route

planning when anchored m / v "MSC Canaberra" on

the roadstead of the Chernomorsk Port, as shown in

Fig.1.

Figure 1. Planning of the TP maneuvering route when the

vessel is proceeding to anchored area No.351 on the

roadstead of the port of Chernomorsk

4 PRESENTATION OF THE MAIN RESEARCH

MATERIAL

To perform route planning, the following vessel data

are required: permanent vessel data; two tables of

characteristics of braking and turning ability [2, 9]; a

map with an area for maneuvering and recommended

paths of movement, determined by geodetic methods.

The further algorithm of the meaningful model

will be as follows:

1. The coordinates of the anchoring point are plotted,

which are agreed with the VTS.

2. The total vector of the wind and current direction is

determined and the approach line is drawn in its

direction for releasing the anchor until it intersects

the recommended path line on the map or the

water area of the beginning of maneuvering, free

from hazards and suitable in depth, and determine

the coordinates of the penultimate maneuvering

631

point, as shown in Fig. 1.

For this, the navigator must set the initial data in

the computer, namely, the coordinates of the

waypoints and the maneuvering characteristics of

the vessel, which must be available on the vessel in

electronic form in the form of tables. After that, the

trajectory point’s matrices will be automatically

given to the skipper, along which the vessel will

follow every 2 cables.

3. Make a table of waypoints and calculate the TC, the

distance from the previous WP to the next one, the

angle of rotation at each waypoint and the

required rudder shift angle as follows.

The procedure for calculating these points is as

follows: using formula (1) from a spherical triangle,

taking latitude and longitude as the basis, we

determine the true course that the ship will follow.

( )

21 1

arctan sin tan sin sin cosTC DLon DLON

φφ φ

= ÷−

(1)

where TC - true heading from the previous point to

the next; DLon - the difference between the longitudes

of the end and start points; - latitude of the starting

point; - latitude of the end point.

These calculations were performed in Excel. The

results of calculating the initial data and calculating

the course to follow to each of the previous to the next

point are shown in table. 1. In this case, calculations

are performed in radians. Therefore, it is necessary to

convert the coordinates of points from degrees to

radians in advance.

Table 1. Calculation of true courses to follow between points

_______________________________________________

WP LAT LON Course

_______________________________________________

0 46⁰18,0' N 30⁰53,5' E

1 46⁰17,2' N 30⁰52,0' E 233

2 46⁰18,5' N 30⁰46,5' E 289

3 46⁰18,0' N 30⁰44,0' E 265,7

4 46⁰15,5' N 30⁰43,0' E 204,1

5 46⁰14,0' N 30⁰44,0' E 164

_______________________________________________

After the TC value is determined, you can proceed

to the automatic calculation of the distance between

the specified WPs using the following formulas:

1ii

DLon

λλ

−

= −

(2)

1ii

DLat

φφ

−

= −

(3)

The value of the meridional parts in equatorial

minutes for the Krasovsky’s ellipsoid can be

calculated by the formula:

1 sin

3437.7468 ln tan 45

2 1 sin

φφ





−⋅





=⋅+





+⋅







(4)

where MP - is the meridional part of the parallel φ, eq.

min .; φ - geographic latitude; e - the eccentricity of

the terrestrial ellipsoid.

The name of the MP coincides with the name of the

latitude. To obtain the difference in the meridional

parts (DMP), it is necessary to calculate the MP

i, then

the MP

i-1, and after that the algebraic difference:

1i ii

DMP MP MP

−

= −

(5)

The name DMP

i is indicated from MPi to MPi-1, that

is, it corresponds to the DLat name. The direction of

TCi is calculated by the formula:

tan

DMP

λλ

−

(6)

The direction is obtained in a quarter count and for

recalculation in a circular one it is necessary to take

into account that the first letter of the quarter

corresponds to the name of the DLat, and the second

to the DLon.

The distance traveled between the WPs is

determined by the formula:

sec

S DLat TC= ⋅

(7)

The calculation results are summarized in table 2.

Table 2. Calculation of distances to follow between points

_______________________________________________

WP LAT LON Distance S, cables

_______________________________________________

0 46⁰18,0' N 30⁰53,5' E

1 46⁰17,2' N 30⁰52,0' E 13,2

2 46⁰18,5' N 30⁰46,5' E 40,0

3 46⁰18,0' N 30⁰44,0' E 18,5

4 46⁰15,5' N 30⁰43,0' E 15,8

5 46⁰14,0' N 30⁰44,0' E 16,9

_______________________________________________

After receiving the TC, it is necessary to determine

the angle of rotation θ. To do this, it is necessary to

subtract the course that was before the commence of

the turn from the course that the ship will follow after

the turn, as shown in formula (8).

1ii

TC TC

−

= −

(8)

Rotation angle data between points are given in

table. 3. All calculations are performed in degrees.

Table 3. Calculation of distances to follow between points

_______________________________________________

WP LAT LON Course

_______________________________________________

0 46⁰18,0' N 30⁰53,5' E

1 46⁰17,2' N 30⁰52,0' E 233 56

2 46⁰18,5' N 30⁰46,5' E 289 23,3

3 46⁰18,0' N 30⁰44,0' E 265,7 61,6

4 46⁰15,5' N 30⁰43,0' E 204,1 40,1

5 46⁰14,0' N 30⁰44,0' E 164

It should also be remembered that the circulation

parameters depend on the rudder shift. The rudder

shift, in turn, depends on the steering angle. The

formula (9) gives recommendations for choosing the

rudder shift.

(9)

632

Table 4 shows the definition of the recommended

rudder shift for each steering angle, according to

formula (9).

After the main parameters of the track lines are

obtained, you can commence calculating the points of

the beginning and end of the turn.

Table 4. Steering angle depending on the steering angle

_______________________________________________

WP LAT LON Course

_______________________________________________

0 46⁰18,0' N 30⁰53,5' E

1 46⁰17,2' N 30⁰52,0' E 233 56 10

2 46⁰18,5' N 30⁰46,5' E 289 23,3 5

3 46⁰18,0' N 30⁰44,0' E 265,7 61,6 15

4 46⁰15,5' N 30⁰43,0' E 204,1 40,1 10

5 46⁰14,0' N 30⁰44,0' E 164

_______________________________________________

When performing a turn automatically, the routine

work of calculating the necessary data - the moment

of the beginning of the turn, choosing an angle rudder

shifting, determination of the coordinates of the

current position, the onset of the moment of holding

the vessel is performed by a computer. The navigator

gives the necessary commands to the helmsman and

controls on the computer screen the actual position of

the vessel relative to the planned one and corrects its

movement.

All the calculations performed are summarized in

one table 5, which are necessary for planning the

coordinates of the planned path by trajectory points

(TP) and navigation control of control along it,

including curved sections of the path, when anchored.

With the automatic execution of the turn, the

process of movement is planned and carried out by a

computer without the participation of the navigator

and the helmsman. The navigator exercises control

over the normal operation of the control system and,

if possible, visually evaluates the position of the vessel

relative to the signs of the navigational situation.

Table 5. Summary table of turnability parameters

_______________________________________________

WP LAT LON Course Distance, S,

cables

_______________________________________________

0 46⁰18,0' N 30⁰53,5' E

1 46⁰17,2' N 30⁰52,0' E 233 13,2 56 10

2 46⁰18,5' N 30⁰46,5' E 289 40,0 23,3 5

3 46⁰18,0' N 30⁰44,0' E 265,7 18,5 61,6 15

4 46⁰15,5' N 30⁰43,0' E 204,1 15,8 40,1 10

5 46⁰14,0' N 30⁰44,0' E 164 16,9

_______________________________________________

Currently, there are two ways of planning the

planned coordinates of the trajectory points of the

beginning and end of the turn, which differ in the

method of taking into account the characteristics of

turnability when planning a given path and control of

steering. The first is based on the choice of the rudder

shift angle according to the value of the course

change, the calculation or transfer of the

corresponding section of the turnability characteristics

curve to the map with the subsequent determination

of the coordinates of the beginning and end of the

circulation. At the same time, the moment of giving

the command to shift the rudder is taken as the

beginning of the turn, as in the performance of full-

scale tests to determine the characteristics of

turnability. The turning process is controlled by a

gyrocompass.

The second method is based on the choice of the

transfer angle in accordance with the value of the

steady-state radius of curvature of the trajectory. It

involves maintaining the angular rate of rotation

constant during the entire circulation time by variable

rudder masses. Rotation control is carried out by an

angular velocity sensor and a gyrocompass. The

disadvantage of this method is that it does not allow

you to establish a correspondence between the current

and valid locations. The control is carried out

according to the steady-state value of the angular

velocity while the vessel is moving in maneuvering

mode. Variable rudder angles, which are used,

increase the width of the maneuvering lane. The latter

in hazardous areas of confined waters is of paramount

importance. Both methods do not provide for the

formalization of the process of movement and control.

When planning the coordinates of movement, the

path is usually used in the form of straight-line

segments with their intersection at waypoints. This

planning method is suitable for use on the high seas.

Turn planning is not expressed in terms of the scale of

the map used at the crossing, so it does not make

sense.

When sailing in confined waters, when the

geometrical dimensions of the vessel are

commensurate with the distance to hazards, it

becomes necessary to carefully plan safe coordinates

of movement and ensure their passage using the

vessel's movement controls.

Since the process of movement in cramped

conditions and when making turns is fast, planning,

and when making turns, and controlling the

parameters of maneuvering is not always possible.

To take into account the characteristics of

turnability when planning a curvilinear movement,

we will use the method of segments [6, 7].

When planning a curved trajectory, it is necessary

to transfer the corresponding section from the

turnability data presented in the ship's documents in

graphic form to the map. For a more convenient

performance of such work, it is recommended to

graphically plot the circulation curves, which are

available in the ship's documents, on a maneuverable

tablet, as shown in Fig. 2.

To explain the essence of the method of segments,

let us consider the circulation curves of the m / v

"MSC Canaberra".

Figure 2. Circulation curves of m / v "MSC Canaberra" in

ballast

633

The following procedure is recommended for

calculating rotation elements.

1. By the value of the angle of rotation, the side of the

rotation and the angle of rotation, mark the end

point of the rotation E with a pencil.

2. Apply a parallel ruler to the center of the tablet and

a point on its circumference corresponding to the

angle of rotation 120°. Move the ruler to point E

and draw a straight line from it until it intersects

with the initial radius of the tablet, get the point M

of intersection with the path line before turning.

3. On the scale of the tablet, the values of the

segments MC and ME are taken, which are plotted

accordingly from point M on the map, which gives

the points of the beginning and end of the ship's

trajectory when turning as shown in Fig. 3.

4. The bisector of the CME angle is drawn on the

tablet, the MS segment is removed and transferred

to the card. Connect points C, S and E with a

smooth curve. Get the trajectory of the ship at the

turn. The coordinates of points H and K are taken,

data is obtained for turning (bearing and distance).

5. The duration of the turn is determined according to

the table, which is usually available in the ship's

documents.

Figure 3. Plotting a turn using line segments on the map

6. The coordinates of the trajectory points are

analytically calculated for certain segments, the

known coordinates of the waypoint M and the

coordinates of the turning point, as shown in Fig.

4, for turning at the entrance to the channel of the

Yuzhny Port.

Figure 4. Construction of a curved by TP path graphically

When calculating the coordinates, the distance

should be taken in miles, calculations are made to the

fourth decimal place of a minute, and rounding to the

third.

A detailed calculation is made in the following

sequence for the selected vessel “MSC Canaberra”.

1. Using the formulas (2, 3), we calculate the

parameters of circulation in the cargo, give them in

the form of Table 6 and convert them into miles

after the corresponding calculations.

Table 6. Calculated circulation parameters

_______________________________________________

Rudder Parament Legend Laden In

Shift ballast

_______________________________________________

Advance, cables l1 3.32 3.19

Direct displacement, cables l2 2.11 2.02

Tactical diameter, cables D

T 4.73 4.53

Constant diameter, cables D

Y 4.86 4.53

_______________________________________________

We calculate the coefficients for the MC in cargo:

3.32 2.37 0.95 0.095

2.37 0.237

a l cables miles

b miles

=− =−= =

= = =

Coefficient for ME in laden

2.11 0.211a l cables miles= = =

2. We calculate the segments for the angle of rotation

θ = 122.6°, and after 10° within the limits of

rotation according to the formulas:

122.6

tan 0.095 0.237 tan 0.528

MC a b miles

= +⋅ = + ⋅ =

Table 7. Results of calculating turning segments

__________________________________________________________________________________________________

Angle of rotation 10

100

110

122,6

__________________________________________________________________________________________________

МC, cables 1,373 1,600 1,833 2,078 2,339 2,622 2,936 3,290 3,7 4,189 4,792 5,808

МE, cables 0,227 0,458 0,697 0,946 1,212 1,501 1,821 2,182 2,6 3,099 3,713 4,749

__________________________________________________________________________________________________

Table 8. Results of calculation by segments of perpendiculars

__________________________________________________________________________________________________

Angle of rotation 10

100

110

122,6

__________________________________________________________________________________________________

HP, cables 1,597 2,03 2,437 2,803 3,118 3,373 3,559 3,669 3,7 3,651 3,522 3,249

KP, cables 0,039 0,157 0,349 0,608 0,928 1,3 1,711 2,149 2,6 3,052 3,489 4,001

__________________________________________________________________________________________________

634

122.6

tan 0.211 tan 0.211 1.827 0.386

ME l miles

=+=⋅ =⋅=

The calculation results are summarized in Table 7.

We calculate the perpendiculars by the formulas

cosHP HM ME

= +

sinKP ME

and summarize

in table. 8.

3. We plot the circulation curve graphically in the

following sequence, Fig. 4:

− On the map, using navigational tools, we build

a waypoint M of intersection of the

recommended courses, and determine its

coordinates.

− From point M with segments MC and ME, from

Table 7 for a rotation angle of 122.6°, on a scale

we plot the points of the beginning and end of

the rotation Нц and Кц and determine the

coordinates of these points analytically by the

coordinates of the waypoint and the direction

of TC before and after the rotation.

− From point E we postpone the MC segment

along the line of the original path and from its

end we draw a line, at an angle of 10° at which

we set aside the segment ME10 and obtain the

trajectory point when turning by 10°. For

analytical calculation according to the

coordinates of the point Нц, the direction and

size of the segments EМ10° and М10°К10°, the

coordinates of ТP К10° are calculated according

to the data in Table 7.

− Similarly, for all values of the turning angles

from Table 7, we construct the trajectory points

(TP) and obtain the planned path at the turn. If

necessary, we connect the TP with the help of a

template and get a turn plan.

− The results of calculating trajectory points using

the “Rotate” program, given in work [7], is

shown in Fig. 5.

Figure 5. Turn when entering the port of Yuzhny TP in

increments of 5°

In order not to perform graphic constructions

every time when it becomes necessary to determine

the point of the beginning and end of the rotation, it is

advisable to perform all the described actions on the

tablet once, for rotation angles through 5° or 10 °, and

the results are summarized in a table. When working

with a programmable micro calculator, as well as

taking into account the characteristics of turnability in

automated systems, it is advisable, instead of using a

tabular form, to program the calculations of all two

necessary values of MC and ME which we denote by

the letter d according to the general approximating

formula 10 [2, 9].

tan

d ab

= +⋅

(10)

where a and b are the turnability coefficients; θ - angle

of rotation.

The coefficients a and b included in formula (10)

can be determined by the method of least squares, or

more simply, and with sufficient accuracy for

practical purposes of navigation, by the method of

selected points. For this, the graphs of the dependence

of d on θ are plotted from the table of segments. Two

further spaced points are selected on the graph and

the corresponding values of d and θ are written out.

Substituting these values into formula (10), a

system of two linear equations with two unknowns a

and b is obtained, from which they are determined.

The values of the thus calculated segments are given

in table. 9.

For an approximate calculation, you can use the

following formulas:

MC a l b

ME a b l



=−=







= =



Taking into account all the above parameters, we

determine the points of the beginning and end of the

turn and bring this to Table 9.

Table 9. Elements of turns in the form of segments when

maneuvering for anchoring in the roadstead of the port of

Chornomorsk

_______________________________________________

WP LAT LON C

MC ME

_______________________________________________

0 46⁰18,0' N 30⁰53,5' E

1 46⁰17,2' N 30⁰52,0' E 233 56 2,7166907 1,9939104

2 46⁰18,5' N 30⁰46,5' E 289 23,3 1,8351734 1,2927493

3 46⁰18,0' N 30⁰44,0' E 265,7 61,6 1,379717 1,5677947

4 46⁰15,5' N 30⁰43,0' E 204,1 40,1 2,1030485 1,3685956

5 46⁰14,0' N 30⁰44,0' E 164

_______________________________________________

C - Course

The next task is to calculate the coordinates of the

rotation matrices every 10 degrees by the method of

segments. To do this, it is necessary, according to the

method described above, to determine for each section

the beginning, end of the turn and TP with a step of 10

degrees. These data for each turning angle when

anchored in the port of Chornomorsk are given in

Tables 10-13. It should be emphasized that the

calculation takes place for small sections of the path.

Therefore, in order to increase accuracy, it is necessary

to leave 10 decimal places.

Having received the data of the lines for each

parcel, you can determine the geographic points in

which they will be located. These calculations are best

done in tabular form. To do this, it is necessary to

initially find the difference in latitude (DLat) and the

difference in longitude (DLon) between each point,

using formulas (12) and (13).

635

Table 10. Rotation elements of angle of rotation 1 as line segments

__________________________________________________________________________________________________

Angle 10⁰ 20⁰ 30⁰ 40⁰ 50⁰ 56⁰

__________________________________________________________________________________________________

МC 1,081958282 1,408883289 1,746053028 2,099410462 2,476012182 2,71669

МE 0,328082488 0,661226178 1,004809472 1,364888378 1,748653718 1,99391

__________________________________________________________________________________________________

Table 12. Rotation elements of angle of rotation 3 as line segments

__________________________________________________________________________________________________

Angle 10⁰ 20⁰ 30⁰ 40⁰ 50⁰ 61,6⁰

__________________________________________________________________________________________________

МC 0,031844958 0,267266499 0,51006536 0,764521121 1,035715294 1,37972

МE 0,230095185 0,463739959 0,704706376 0,957241716 1,226389141 1,56779

__________________________________________________________________________________________________

Table 13. Rotation elements of angle of rotation 4 as line segments

__________________________________________________________________________________________________

Angle 10⁰ 20⁰ 30⁰ 40,1⁰

__________________________________________________________________________________________________

МC 1,081958282 1,408883289 1,746053028 2,103048453

МE 0,328082488 0,661226178 1,004809472 1,368595571

__________________________________________________________________________________________________

Table 11. Rotation elements of angle of rotation 2 as line

segments

_______________________________________________

Angle 10⁰ 23,3⁰

_______________________________________________

МC 1,121244311 1,835173414

МE 0,54855392 1,292749344

_______________________________________________

cos

DLat MC TC

= ⋅

(12)

where TC is the heading before the turn.

tan

DLon DMP TC= ⋅

(13)

The difference between the meridional parts can be

determined by the formula (14).

tan 45

3437.75 ln

tan 45

DMP











= ⋅















(14)

It must be remembered that for the calculation of

the first section, it is the coordinate of the turning

point. For each section of the turn, the data of the

difference in latitude and longitude between each

segment of the MC and ME were calculated. Having

received the segments of the difference in latitude and

the difference in longitude, using simple navigation

formulas (15) and (16), we determine the coordinates

of these points.

DLat

ϕϕ

= +

(15)

where

is the latitude of the turning point.

DLon

λλ

= +

(16)

where

is the longitude of the turning point.

Thus, having determined its coordinates for each

point of the turning section, it is possible to construct

the rotation matrices, which are given in Tables 14-17.

The next task is to determine the matrices of

straight sections. It should be noted that the matrix for

the first segment will start from the zero point and

end with the turn start point 1. The second straight

segment matrix will start from the turn end point 1

calculated above. It will end as the starting point of

the turn at WP 2. Subsequent matrices of straight

sections, except for the final one, will have the same

construction principle. For the last leg, the straight

matrix will start at turn end point 4 and end with

waypoint 5.

Table 14. Matrix of trajectory point for turning angle 1

_______________________________________________

Mt1 φc1 46⁰17,29810' N λE1 30⁰52,3879' E

E11 46⁰17,26745' N λE11 30⁰52,3532' E

E12 46⁰17,23683' N λE12 30⁰52,3528' E

E13 46⁰17,21277' N λE13 30⁰52,2773' E

E14 46⁰17,20443' N λE14 30⁰51,9039' E

E15 46⁰17,21461' N λE15 30⁰51,8827' E

E1 46⁰17,23895' N λE1 30⁰51,7195' E

_______________________________________________

Table 15. Matrix of trajectory point for turning angle 2

_______________________________________________

Mt2 φC2 46⁰18,46415' N λC2 30⁰46,8146' E

E21 46⁰18,46738' N λE21 30⁰46,5507' E

E2 46⁰18,44780' N λE2 30⁰46,313' E

_______________________________________________

Table 16. Matrix of trajectory point for turning angle 3

_______________________________________________

Mt3 φC3 46⁰18,00621' N λC3 30⁰44,0808' E

E31 46⁰17,98465' N λE31 30⁰44,0734' E

E32 46⁰17,98112' N λE32 30⁰44,0678' E

E33 46⁰17,98275' N λE33 30⁰44,0207' E

E34 46⁰17,95989' N λE34 30⁰43,7965' E

E35 46⁰17,94024' N λE35 30⁰43,7916' E

E3 46⁰17,91413' N λE3 30⁰43,7229' E

_______________________________________________

Table 17. Matrix of trajectory point for turning angle 4

_______________________________________________

Mt4 φC4 46⁰15,61518' N λC4 30⁰43,04254' E

E41 46⁰15,60160' N λE41 30⁰43,02962' E

E42 46⁰15,58431' N λE42 30⁰43,01078' E

E43 46⁰15,43542' N λE43 30⁰43,01337' E

φE4 46⁰15,42106' N λE4 30⁰43,03575' E

_______________________________________________

Well-known navigation formulas are used to

construct matrices of straight-line segments. The

formula for determining the difference in latitude will

have a different form. The section MC for rectilinear

matrices is replaced with a step , which for the first

point is 2, for the next 4, and so on. The formula for

calculating the DLat of a straight section is given

below.

cosDLat TC

= ⋅

(17)

Calculations are carried out up to the point where

the pivot starts.

636

Table 18. Straight leg matrix between waypoint 0 and turn

start point 1

_______________________________________________

M01 φ0 46⁰18,0' N λ0 30⁰53,5' E

01 46⁰17,92778' N λ01 30⁰53,3811' E

02 46⁰17,85556' N λ02 30⁰53,2622' E

03 46⁰17,78335' N λ03 30⁰53,1433' E

04 46⁰17,71113' N λ04 30⁰53,0244' E

05 46⁰17,63891' N λ05 30⁰52,9055' E

06 46⁰17,56669' N λ06 30⁰52,7867' E

07 46⁰17,49448' N λ07 30⁰52,6678' E

08 46⁰17,42226' N λ08 30⁰52,5489' E

09 46⁰17,35004' N λ09 30⁰52,43' E

C1 46⁰17,29810' N λC1 30⁰52,3879' E

_______________________________________________

Table 19. Matrix of the straight segment between the end of

turn 1 point and the start point of turn 2

_______________________________________________

M12 φE1 46⁰17,23895' N λE1 30⁰51,7195' E

11 46⁰17,27802' N λ11 30⁰51,5631' E

12 46⁰17,31709' N λ12 30⁰51,4068' E

13 46⁰17,35616' N λ13 30⁰51,2504' E

14 46⁰17,39522' N λ14 30⁰51,094' E

15 46⁰17,43429' N λ15 30⁰50,9376' E

16 46⁰17,47336' N λ16 30⁰50,7812' E

17 46⁰17,51243' N λ17 30⁰50,6249' E

18 46⁰17,55150' N λ18 30⁰50,4685' E

19 46⁰17,59056' N λ19 30⁰50,3121' E

110 46⁰17,52963' N λ110 30⁰50,1557' E

111 46⁰17,66870' N λ111 30⁰49,9993' E

φ112 46⁰17,70777' N λ112 30⁰49,8429' E

113 46⁰17,74684' N λ113 30⁰49,6865' E

114 46⁰17,78591' N λ114 30⁰49,5301' E

115 46⁰17,82497' N λ115 30⁰49,3737' E

116 46⁰17,86404' N λ116 30⁰49,2173' E

117 46⁰17,90311' N λ117 30⁰49,0609' E

118 46⁰17,94218' N λ118 30⁰48,9045' E

119 46⁰17,98125' N λ119 30⁰48,7481' E

120 46⁰18,02031' N λ120 30⁰48,5917' E

121 46⁰18,05938' N λ121 30⁰48,4353' E

122 46⁰18,09845' N λ122 30⁰48,2789' E

123 46⁰18,13752' N λ123 30⁰48,1225' E

124 46⁰18,17659' N λ124 30⁰47,9661' E

125 46⁰18,21566' N λ125 30⁰47,8096' E

126 46⁰18,25472' N λ126 30⁰47,6532' E

127 46⁰18,29379' N λ127 30⁰47,4968' E

128 46⁰18,33286' N λ128 30⁰47,3404' E

129 46⁰18,37193' N λ129 30⁰47,1839' E

130 46⁰18,41100' N λ130 30⁰47,0275' E

131 46⁰18,45006' N λ131 30⁰46,8711' E

C2 46⁰18,46415' N λC2 30⁰46,8147' E

_______________________________________________

Table 20. Matrix of the straight segment between the end of

turn 2 point and the start point of turn 3

_______________________________________________

M23 φE2 46⁰18,44780' N λE2 30⁰46,313' E

21 46⁰18,43880' N λ21 30⁰46,2675' E

22 46⁰18,42980' N λ22 30⁰46,2221' E

23 46⁰18,42080' N λ23 30⁰46,1766' E

24 46⁰18,41181' N λ24 30⁰46,1311' E

φ25 46⁰18,40281' N λ25 30⁰46,0856' E

26 46⁰18,39381' N λ26 30⁰46,0401' E

27 46⁰18,38481' N λ27 30⁰45,9946' E

28 46⁰18,37582' N λ28 30⁰45,9491' E

29 46⁰18,36682' N λ29 30⁰45,9037' E

210 46⁰18,35782' N λ210 30⁰45,8582' E

211 46⁰18,34882' N λ211 30⁰45,8127' E

212 46⁰18,33983' N λ212 30⁰45,7672' E

213 46⁰18,33083' N λ213 30⁰45,7217' E

214 46⁰18,32183' N λ214 30⁰45,6762' E

215 46⁰18,31283' N λ215 30⁰45,6308' E

216 46⁰18,30384' N λ216 30⁰45,5853' E

217 46⁰18,29484' N λ217 30⁰45,5398' E

218 46⁰18,28584' N λ218 30⁰45,4943' E

219 46⁰18,26785' N λ219 30⁰45,4033' E

φ220 46⁰18,25885' N λ220 30⁰45,3579' E

221 46⁰18,24985' N λ221 30⁰45,3124' E

222 46⁰18,24085' N λ222 30⁰45,2669' E

223 46⁰18,23186' N λ223 30⁰45,2214' E

224 46⁰18,22286' N λ224 30⁰45,1759' E

225 46⁰18,21386' N λ225 30⁰45,1304' E

226 46⁰18,20486' N λ226 30⁰45,0850' E

227 46⁰18,19587' N λ227 30⁰45,0395' E

228 46⁰18,18687' N λ228 30⁰44,994' E

229 46⁰18,17787' N λ229 30⁰44,9485' E

230 46⁰18,16887' N λ230 30⁰44,9030' E

231 46⁰18,15988' N λ231 30⁰44,8575' E

φ232 46⁰18,15088' N λ232 30⁰44,8121' E

233 46⁰18,14188' N λ233 30⁰44,7666' E

234 46⁰18,13289' N λ234 30⁰44,7211' E

235 46⁰18,12389' N λ235 30⁰44,6756' E

236 46⁰18,11489' N λ236 30⁰44,6301' E

237 46⁰18,10589' N λ237 30⁰44,5847' E

238 46⁰18,09690'N λ238 30⁰44,5392' E

239 46⁰18,08790' N λ239 30⁰44,4937' E

240 46⁰18,07890' N λ240 30⁰44,4482' E

241 46⁰18,06990' N λ241 30⁰44,4027' E

242 46⁰18,06091' N λ242 30⁰44,3573'E

243 46⁰18,05191' N λ243 30⁰44,3118' E

244 46⁰18,04291' N λ244 30⁰44,2663' E

245 46⁰18,03391' N λ245 30⁰44,2208' E

246 46⁰18,02492' N λ246 30⁰44,1753' E

247 46⁰18,01592' N λ247 30⁰44,1299' E

C3 46⁰18,00621' N λC3 30⁰44,0808' E

_______________________________________________

Table 21. Matrix of the straight segment between the end of

turn 3 point and the start point of turn 4

_______________________________________________

M34 φE3 46⁰17,91413' N λE3 30⁰43,7229' E

31 46⁰17,80459' N λ31 30⁰43,6905' E

32 46⁰17,69505' N λ32 30⁰43,6581' E

33 46⁰17,58551' N λ33 30⁰43,6256' E

34 46⁰17,47597' N λ34 30⁰43,5932' E

35 46⁰17,36643' N λ35 30⁰43,5608' E

36 46⁰17,25689' N λ36 30⁰43,5283' E

37 46⁰17,14735' N λ37 30⁰43,4959' E

φ38 46⁰17,03781' N λ38 30⁰43,4635' E

39 46⁰16,92827' N λ39 30⁰43,4311' E

310 46⁰16,81873' N λ310 30⁰43,3986' E

311 46⁰16,70919' N λ311 30⁰43,3662' E

312 46⁰16,59965' N λ312 30⁰43,3338' E

313 46⁰16,49011' N λ313 30⁰43,3014' E

314 46⁰16,38057' N λ314 30⁰43,2690' E

315 46⁰16,27103' N λ315 30⁰43,2365' E

316 46⁰16,16149' N λ316 30⁰43,2041' E

317 46⁰16,05195' N λ317 30⁰43,1717' E

318 46⁰15,94241' N λ318 30⁰43,1393' E

319 46⁰15,83287' N λ319 30⁰43,1069' E

320 46⁰15,72333' N λ320 30⁰43,0745' E

K4 46⁰15,61518' N λK4 30⁰43,0425' E

_______________________________________________

Table 22. Straight leg matrix between turn end point 4 and

waypoint 5

_______________________________________________

M45 φE4 46⁰15,42106' N λE4 30⁰43,03575' E

41 46⁰15,30571' N λ41 30⁰43,1141' E

42 46⁰15,19036' N λ42 30⁰43,1924' E

43 46⁰15,07501' N λ43 30⁰43,2707' E

44 46⁰14,95966' N λ44 30⁰43,349' E

45 46⁰14,84431' N λ45 30⁰43,4273' E

46 46⁰14,72895' N λ46 30⁰43,5056' E

47 46⁰14,61360' N λ47 30⁰43,5839' E

48 46⁰14,49825' N λ48 30⁰43,6622' E

49 46⁰14,38290' N λ49 30⁰43,7405' E

410 46⁰14,26755' N λ410 30⁰43,8188' E

φ411 46⁰14,15220' N λ411 30⁰43,8971' E

412 46⁰14,03685' N λ412 30⁰43,9754' E

5 46⁰14,00000' N λ5 30⁰44,0004' E

_______________________________________________

637

Figure 5. Block diagram of the automation of calculations of a given algorithm for the operation of the control system in the

form of an algorithm for calculating the coordinates of the trajectory points of anchoring, presented in the form of

coordinate matrices of straight and curved track sections

Having received the DLat and DLon points for

each trajectory point with a step of 2 cables, it is not

difficult to construct matrices of straight sections

using the segment method, remembered that the

course remains unchanged. The matrices of these

areas for anchoring in the port of Chornomorsk are

shown in Tables 18-22.

Thus, in the presence of only the coordinates of the

waypoints taken from the electronic map, and the

maneuvering characteristics of the vessel, given in

electronic format, it is possible to determine the

matrices of the vessel's track section. This will allow

you to control the movement of the center of the

vessel along the planned track line with an accuracy

of 2 cables.

638

Having determined the matrix of the final straight-

line section of the path, it is necessary to find the

braking start point. Let us resort to the assumption

that the entire segment of the way to the approach to

anchor, the vessel was moving at an average forward

speed (the HA (half ahead) speed for this vessel is 10.3

knots). According to the maneuvering characteristics

of the vessel, the active braking distance for HA is

14.59 cables.

We start the calculation from the final fifth point.

The distance from it to the turning point will be 14.59

cables. To do this, replace the MC segment with a

number equal to the active braking segment and then

calculate using the formulas as usual. The calculation

results are shown in Table 23.

Table 23. Braking start point coordinates

_______________________________________________

MBS φBS 46⁰14,84149' N λBS 30⁰43,42921' E

_______________________________________________

The resulting calculations of the matrices of

curvilinear and straight track sections, as well as the

coordinates of the braking point can be displayed on

the navigation map in the form of trajectory points

and track lines, as shown in Fig. 1. This will increase

the control over the movement of the center of gravity

of the vessel relative to a given path.

Then the given algorithm for maneuvering control

during anchoring Мanch can be represented as a sum

of matrices of straight and curved track sections:

01 1 12 2 23 3

34 4 45

anch t t t

t BS

M MMMMM M

MMMM

= ++ ++ ++

+++ +

(18)

5 CONCLUSIONS

The results of this calculation algorithm can be

summarized in one flowchart for automating the

calculations of a given algorithm for the functioning

of the control system in the form of an algorithm for

calculating the coordinates of the trajectory points of

anchoring, presented in the form of coordinate

matrices of straight and curved track sections, which

is shown in Figure 6.

The developed navigation information and

analytical complex "Planning the path trajectory using

the waypoint matrices" contains modernized methods

and techniques for creating a given algorithm for the

operation of the ship control system and control over

the process of moving along the trajectories, including

curved and straight sections of the track when

anchored. It automates the anchorage route planning

process and controls safe maneuvering, including the

use of motion control techniques using dynamic

positioning.

The results of the developed navigation

information-analytical complex can be used on

unmanned ships, as well as on commercial and

passenger ships during practical work, in order to

accurately control the place of the vessel; in maritime

educational institutions in the preparation of senior

cadets for work on ships and in refresher courses.

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