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1 INTRODUCTION
Some of the results presented in this paper were
obtained by evaluating the accuracy and resistance of
radio navigation systems to interference. In [1] we
modeled the motion of five flying objects. The results
of this modeling were used to obtain information
about the geocentric position of users of the aviation
communication network. The aim of the research was
to evaluate the accuracy of determining the position of
flying objects (LO) using the telemetry method. The
first part of our research is devoted to a general model
of the motion of a flying object. We determined the
initial LO coordinates from the real flight situation
from flightradar24. We have created a general model
of the trajectory of the aircraft's motion, which
consists of a straight flight and two turns. The
advantage of the model created in this way is that it is
flexible and we can modify it. It can be used to model
the motions of any flying object located on Earth,
whose initial coordinates are in the geodetic
coordinate system connected with WGS-84. We
subsequently used this model in the creation of LO
trajectories and evaluation of accuracy and resistance
of radionavigation systems to interference. The
disadvantage of this model was the use of multiple
coordinate systems, which causes complications in
performing the simulation. Modeling of the motion of
a flying object using several coordinate systems is
given in [2]. The authors of this paper used such a
model to evaluate the accuracy of a navigation system.
In the work [6] the methodology of creating a model
of the motion of a flying object is presented. Even in
this case, the authors of the model use several
coordinate systems. The advantage of this model is its
flexibility and the ability to model different types of
flight trajectories. The disadvantage of this model is
its complexity and relatively large computational
time. In work [3] is presented a recursive least squares
(RLS) algorithm to extract and predict the position of
a flying object in a 3D environment. The authors of the
article state that the recursive nature of the
computations lets use this model for real-time
applications although it is relatively complex. In the
Model of the Motion of a Navigation Object in a
Geocentric Coordinate System
M. Džunda
Technical University of Kosice, Kosice, Slovakia
ABSTRACT: In this paper we describe the creation of a model of the motion of a flying object in a geocentric
coordinate system (ECEF - Earth-Centered, Earth-Fixed). Such a model can be used to investigate the accuracy
and resistance of radio navigation systems to interference. The essence of the design of the model lies in the
mathematical description of the motion of a flying object in a geocentric coordinate system. The flight trajectory
of a flying object consists of one straight section and two turns. When creating a model, we assume a flight at a
constant altitude. In this paper, we present one of the possible procedures for modelling the motion of a flying
object in a geocentric coordinate system. We chose the initial coordinates of the flying object according to
flightradar 24. We used the Matlab software for computer simulation.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 15
Number 4
December 2021
DOI: 10.12716/1001.15.04.10
792
work [4] the authors state the algorithm for tracking
moving objects is based on the extension of simple
online and real-time tracking. This algorithm it was
developed by integrating a deep learning-based
association metric approach with simple online and
real-time tracking, which uses a hypothesis tracking
methodology with Kalman filtering and a deep
learning-based association metric. The results of the
experiment confirmed that target detection algorithm
performed very well. The work [7] contains
algorithms for satellite motion control. The proposed
method shows considerable intelligence and certain
universality, and has a strong application potential for
future intelligent control of satellites performing
complex space tasks. Such models are not very
suitable for modeling the flight trajectory of aircraft
because they are relatively complex. In work [8] is a
description of the model of the flying object is given.
In this model, the flying object consists of four objects:
the delegates that encapsulates the implementation,
the message handler that interprets the messages, the
event handler that provides adaptation strategies and
the context object that holds a state beyond the
adaptation. Such a model is complex and not suitable
for evaluating the accuracy of navigation systems. In
our paper, we present one of the possibilities of
modeling the flight trajectory of a flying object in a
geocentric coordinate system based on a mathematical
description of spatial curves. When creating the
model, we used some of the findings presented in [1,
5]. The advantage of the presented solution is its
simplicity, flexibility and high speed of simulation.
Such a model is suitable for evaluating the accuracy of
navigation devices. Also when designing a
communication network of flying objects and
performing relative navigation in this network.
Simple models of the motion of flying objects are
suitable for solving the mentioned tasks, which must
be sufficiently accurate and correspond to the physical
meaning of the solved task.
2 COORDINATE SYSTEMS USED IN FO MOTION
MODELLING
Furthermore, in accordance with the literature [1, 5],
we will present the coordinate systems that we will
use in creating a model of the trajectory of a flying
object. When creating the flight trajectory of a flying
object, we placed the main emphasis on the fact, to the
proposed model was not very complex and
corresponds to the physical meaning of the solved
task. Our task was to create a model of the flight
trajectory of a flying object in a geocentric coordinate
system. Because we determined the initial conditions
of the flight trajectory in accordance with the real
flight trajectories of transport aircraft from
flightradar24, we also used the geodetic coordinate
system.
2.1 ECEF Geocentric Coordinate System
It is a three-dimensional coordinate system with a
centre in the centre of the Earth. The X axis of the
system passes through the intersection of the zero
meridian and the equator, the Y axis is pointing from
west to east, the Z axis is having the north-south
direction. It is a rectangular coordinate system and is
shown in Fig. 1. [1, 2, 5]. In this coordinate system we
will be model the trajectory of the motion of a flying
object.
Figure 1. ECEF Geocentric Coordinate System [5]
Figure 2. Geodetic coordinate system [5]
Figure 3. Ellipsoidal height h and the normal height H [5]
2.2 Geodetic coordinate system
The local model of FO motion is necessary to transfer
to the geodetic coordinate system. The Geodetic
Coordinate System (LLH) is shown in Fig. 2 and used
in aviation in the processing of data in the
autonomous navigation, in radio air navigation
devices and the like. The geodetic coordinate system
determines the position of the point on the surface of
the ellipsoid. The latitude φ, the longitude λ and the
ellipsoidal height h are coordinates. The difference
between the ellipsoidal height h and the normal
height H ( Fig. 3) is the so called height anomaly for
which: ς = h-H, where: H - altitude, ς - height of geoid
or quasi geoid [1, 2, 5]. The area of a quasi-godium is
defined by models of density and topography of the
terrain, satellite ellite models of the Earth and ground
- based measurement of gravity acceleration.
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3 THE MODEL OF FLYING OBJECT MOTION
The model of FO motion is designed to verify the
accuracy and resistance of radionavigation systems
against interference. The model of FO motion serves
to gain information about their geocentric location in
space. We have abstracted from the forces acting on
FO during the flight. For the purpose of simulating
the radionavigation systems, we may consider this
fact irrelevant. To clarify the principle of the model of
FO motion, we chose the real point according to
flightradar24, in which we place the FO and simulate
its further movement [1, 2]. As initial co-ordinates
(starting point) LO, we have chosen real FO
coordinates, i.e., FO ETD 37A with coordinates:
latitude 0,8500800654 rad, longitude 0,3771307447 rad.
See Figure 4 and text below. Height above ellipsoid
10933 m and altitude 10973 m. Geoid curl 40 m. These
coordinates are considered as the reference and
represent the starting point of the XP flight trajectory
Xp [xp, yp, yp].
Figure 4. Trajectory of motion of FO ETD37A
FO coordinates in the geographical coordinate
system. Identification ETD37A, starting point:
− latitude, degree 48.706
− longtitude, degree 21.608
− height above ellipsoid, m 10933
− geoid curl, m 40,0
Identification ETD37A, end point:
− latitude, degree 49.136202
− longtitude, degree 20.333297
− height above ellipsoid, m 10933
− geoid curl, m 40,0
FO coordinates in the a geocentric coordinate
system. Identification ETD37A, starting point:
− X, m 3927434.0
− Y, m 1555616.0
− Z, m 4777290.0
End point:
− X, m 3927194.0
− Y, m 1455308.0
− Z, m 4808781.0
The coordinates of the starting point are
determined by flightradar24 in the geographical
coordinates (LLH). The model created represents the
FO flight in the a geocentric coordinate system ECEF.
In modelling FO trajectories, each part of its trajectory
is modelled in the geocentric coordinate system
(Figure 1). The FO trajectory is composed of a direct
flight and two turns. Based on this, it will be possible
to evaluate the accuracy of FO position determination
in the navigation system not only in the level flight,
but also in FO manoeuvres. Trajectory model input
parameters that can be changed according to current
requirements are:
− the initial position of the FO,
− the duration of the level flight or in turns,
− trajectory of motion,
− radius of curvature.
Therefore, we must make the corresponding
transformations of the starting point FO. The process
of transforming the co-ordinates of the resulting FO
motion with the start in the given initial coordinates
in the LLH system to the ECEF includes the
transformation of geodetic starting point coordinates
into the ECEF system. We use the Matlab function
llh2xyz to transform coordinates. Function llh2xyz
serves to transform initial co-ordinates from LLH to
ECEF. The function serves to convert the geographical
coordinates (latitude, longitude and altitude in WGS-
84) into the rectangular geocentric coordinates X, Y, Z
in meters. The latitude and longitude are given in
radians and the ellipsoidal height in meters is given.
The task our solution is to determine and display the
FO position in the ECEF rectangular coordinate
system with the centre at the Earth's ground. After
performing simulation, we can visualize the model of
FO motion in the ECEF system. The FO position will
be determined in each second by the coordinates x, y,
z in meters. The model of first phase of FO flight
(straight movement) we created as follows. The first
phase of the FO motion, straight and level flight, is
400 s at an altitude of 10973,0 m. Altitude of flight
does not change. We chose the initial coordinates for
local movement as follows: x = 3927434.0 m, y =
1555616.0 m, z = 4777290.0 m.
The model of first phase of FO flight we created as
follows. We started from the equation of a line in
three dimensional space. We assume that the line is
uniquely given by two points or one point and a
direction vector. Line Xk Xp can be defined as follows.
The line passes through the starting point XP [xp, yp,
yp] and endpoint Xk [xk, yk, zk]. The coordinates of
the end point are given in table no. 1 and 2. We
express the direction vector Ū in the form:
, , 1, 2, 3U Xk Xp xk xp yk yp xk xp u u u= − = − − − =
(1)
Then the parametric expression of the line Xp Xk
has the form:
1; 2; 3; 0;1x xp t u y yp t u z zp t u t= + = + = +
(2)
When creating a model, we choose the end point of
the flight trajectory arbitrarily in the geographical
coordinates. In Figure 5, the end point of the straight
flight is marked with a red asterisk. Subsequently, we
create a flight model of a flying object, which consists
of two turns. When creating this model, we assume
that the flying object flies in a constant height. We will
model the curves using circles. The radius of the circle
is denoted by the letter r. We assume that the radius r
can be arbitrary. If we have the point S [x; y] = S [0; 0],
so for all points of the circle:
2 2 2
x y r+=
(3)
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If we have a point S [x; y; z] = S [0; 0; 0], so the
implicit expression of a circle is:
2 2 2
0; 0x y r z+ − = =
(4)
We express the parametric equations of this circle
as follows:
( ) ( ) )
cos , sin , 0, 0,2x r t y r t z t
= = =
(5)
In our case, we assume that the center of the turn
of the flying object is at the point Xk [xk, yk, zk] and
zk = constant. We will express the parametric
equations of such a circle as follows:
( ) ( ) )
cos , sin , , 0,2x r t xk y r t yk z const t
= + = + =
(6)
Based on equation (6), we can simulate flight after
a circle . The parameters of the model (6) are the
radius of the circle r and the angle t.
Figure 5. Trajectory of motion of FO ETD37A
In Figure 5, the end point of the flight in a turn is
marked with a blue ringed. According to equation (5),
we modeled the flight in the second turn. In
accordance with algorithms 1 through 6, we have
simulated FO trajectory. The FO motion trajectory is
shown in Fig. 5. From figure 5 it is clear that said
algorithms allow us to simulate the flight of a flying
object, which consists of a straight section and two
turns. The advantage of this solution is that the model
is simple and does not require a long simulation time.
4 CONCLUSION
The result of modelling is a model describing FO
motion in a geocentric coordinate system. For the
simulation to be as accurate as possible, we have
performed air traffic observation over the territory of
the Slovak Republic via the Flightradar24 application.
We randomly selected FO and his geographic
coordinates and altitudes have been implemented in
our model. For the purpose of solving the problem, it
was necessary to transform his coordinates into a
geocentric coordinate system. The simulation results
have confirmed that the created model sufficiently
accurately describe FO flight in real-world conditions.
The generated simulation model can be used for
further research and development of communication,
navigation, radar systems or anti-collision system.
Also for examination the accuracy and resistance of
radio navigation systems to interference. In order to
solve this problem, we strive to create such models
that allow us to simulate the trajectory of a flying
object under conditions that are close to real.
Therefore, we have created a model of a flying object,
which is characterized by flexibility and by changing
the parameters of this model it is possible to get as
close as possible to real flight conditions. At this stage
of the research, we do not consider the turbulence of
the atmosphere and other factors that affect the flying
object. We created our model so that the flight
trajectory consists of a straight flight and two turns.
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