299

1 INTRODUCTION

Air transport is one of the regular modes of transport

and it is therefore essential that it be as safe as

possible. The intensive development of aviation,

which occurred after the Second World War, led to an

increase in capacity and a reduction in air traffic. At

the same time, this caused an increase in more

frequent collisions of aircraft in the air. For this

reason, it was necessary to develop a system that

would increase air safety and help prevent such

collisions. Several types of anti-collision systems have

been developed in the history of aviation. Significant

progress in their development did not come until

1981. At this time, the currently best-known active on-

board anti-collision system TCAS (Traffic Alert and

Collision Avoidance System) was improved. The

development of this system continues today even

today due to the ever-increasing demands on air

safety. TCAS means additional safety insurance in air

traffic. There are also financially and technically less

demanding anti-collision systems, such as the PCAS

system, the so-called mobile anti-collision system.

Thanks to its specific properties, this passive system is

also suitable for its installation in smaller aircraft and,

due to its financial availability, it is intended mainly

for the needs of general aviation. Anti-collision

systems are currently a key element in increasing the

level of air safety. In our research we have devoted the

alternative possibility of determining the position of

aircraft without the use of satellite navigation systems,

based on the principles of relative navigation

(RelNav). In order for this method of navigation to be

applicable, it is necessary to analyze the accuracy of

the system and its application within the concept of

the future communication and navigation

infrastructure. The main goal of the paper is to

evaluate the accuracy of position determination by the

RelNav system depending on the position of users of

the communication network. Work related to relative

navigation is focused on research in formation flights,

in-flight refueling and initial navigation, which is

directly dependent on GNSS. Currently, relative

navigation is most important for mapping and

navigating UAV resources in a GNSS-free

environment. We published the results of the

evaluation of the accuracy of the relative navigation

system in [2–4]. The results of the simulation of the

Influence of Mutual Position of Communication

Network Users on Accuracy of Positioning by Telemetry

Method

M. Džunda & P. Dzurovčin

Technical University of Kosice, Kosice, Slovakia

ABSTRACT: In this paper we solve the problem of the influence of the mutual position of the users of the

communication network on the accuracy of the telemetric navigation system. We present the principle of

operation of a telemetry navigation system and examine the accuracy of determining the position of users of the

communication network depending on their mutual position. The telemetric method of determining the

position of users of a communication network can be used in shipping or air transport. The simulation of the

telemetry system will be performed in the Matlab software environment.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 15

Number 2

June 2021

DOI: 10.12716/1001.15.02.04

300

RelNav system presented in these works confirm that

the accuracy of this system deteriorates as the distance

between the users of the communication network

increases. We have also found that the accuracy of the

RelNav system depends on the relative position of the

users of the communication network. The [6] work

presents a method improve DME distance measuring

accuracy by using a new DME pulse shape. The

proposed pulse shape was developed by using

Genetic Algorithms and is less susceptible to

multipath effects so that the ranging error reduces by

36.0–77.3% when compared to the Gaussian and

Smoothed Concave Polygon DME pulses, depending

on noise environment. The [5] paper introduces

optimal ground station augmentation algorithms that

help to efficiently transform the current DME ground

network to enable a DME/DME positioning accuracy

of up to 0.3 nm or 92.6 m with a minimal number of

new ground DME sites. The positioning performance

and augmented ground network using the proposed

Stretched-Front-Leg (SFOL) pulse-based DME are

evaluated in two regions which have distinct terrain

conditions. The [10] Distance Measuring Equipment

(DME) is seeing renewed interest for its ability to

support future aviation navigation and surveillance

needs. It is one of the major technologies being

examined by the FAA Alternative Position Navigation

and Timing (APNT) program to support needs to

provide an alternative to GNSS in a NextGen airspace.

Facility data-base resolution and fidelity, and

precision of DME ground system location surveys are

a function of today's 0.2 nm system error limit. These

errors become noticeable when viewing flight data

that is processed to the tighter tolerances required by

APNT. Their contribution to system error will likely

require reduction [1]. The navigation subsystem in

most platforms is based on an inertial navigation

system (INS). Regardless of the INS grade, its

navigation solution drifts in time. To avoid such a

drift, the INS is fused with external sensor

measurements such as a global navigation satellite

system (GNSS). In situations when maneuvering helps

to improve state observability, virtual lever-arm

(VLA) measurements manage to gain additional

improvement in accuracy. These results are supported

by simulation and field experiments with a vehicle

mounted with a GNSS and an INS [8]. Positioning

accuracy of area navigation (RNAV) is directly related

to geometric dilution of precision (GDOP). For the

distribution of three DME beacons, this paper

establishes the GDOP algorithm model based on

DME/DME RNAV, the GDOP calculation is deduced,

and the relation between GDOP and DME ground

station distribution is given. The affection of the

geometric configuration and the baseline length of

DME stations, and of the aircraft height on GDOP

values are studied and simulated by MATLAB. In [7]

the optimal methods of DME beacon distribution for

DME/DME RNAV are proposed and all these results

will provide theoretical guidance for implementation

of DME/DME RNAV in China. Alternative

positioning navigation by DME service network is

used in many cases on-board of aircraft. This problem

is topical in case of bad accuracy of GNNS, spatially

during takeoff or landing phases of aircraft flight. In

the paper, an accurate approach of positioning by data

from DME with support of all available DMEs signals

in particular point of airspace has been discussed. An

availability area of Ukrainian DME service network

for altitude of FL320 has been evaluated. Dilution of

precision coefficient for geometrical factor estimation

in accuracy calculation of positioning by DME

network for Ukrainian airspace was used [13]. A

typical type of a DME pulse being used in practice is a

Gaussian pulse, and the achievable DME ranging

accuracy is primarily determined by the pulse shape.

[Kim, E. 2013] presents an alternative DME pulse

waveform that is able to provide much higher range

accuracy than the conventional Gaussian pulse. The

alternative pulse waveform is compliant with the

pulse shape requirements in the current DME

specifications to maintain the compatibility with

existing DME ground transponders and avionics. In

addition, the paper discuss the potential constraints in

implementing the alternative pulse in the existing

transponders and avionics. The Multi-constellation

Global Navigation Satellite System (Multi-GNSS) has

become the standard implementation of high accuracy

positioning and navigation applications. It is well

known that the noise of code and phase

measurements depend on GNSS constellation. Then,

Helmert variance component estimation (HVCE) is

usually used to adjust the contributions of different

GNSS constellations by determining their individual

variances of unit weight. However, HVCE requires a

heavy computation load [12]. [Zhao, J. et. Aall 2018]

presents a new method to improve the accuracy in the

heading angle estimate provided by low-cost

magnetometers on board of small Unmanned Aerial

Vehicles (UAVs). This task can be achieved by

estimating the systematic error produced by the

magnetic fields generated by onboard electric

equipment. The magnetic biases’ determination

problem can be formulated as a system of non-linear

equations by exploiting the acquired visual and GNSS

data [19]. The Chinese Area Positioning System

(CAPS) is a new positioning system developed by the

Chinese Academy of Sciences based on the

communication satellites in geosynchronous orbit.

The CAPS has been regarded as a pilot system to test

the new technology for the design, construction and

update of the BeiDou Navigation Satellite System

(BDS). BDS-3 type is equipped with two major

functions, namely navigation and positioning, as well

as data communication. BDS can provide seven types

of services. The results indicate a potential application

of CAPS for highly accurate positioning and speed

estimation and the availability of a new navigation

mode based on communication satellites [16]. At the

same time, many applications do not require precise

absolute Earth coordinates, but instead, inferring the

geometric configuration information of the

constituent nodes in the system by relative

positioning. The Real-Time Kinematic (RTK)

technique shows its efficiency and accuracy in

calculating the relative position. The efficiency test

shows that the proposed method can be a real-time

method, the time that calculates one epoch of

measurement data is no more than 80 ms and is less

than 10 ms for best results. The novel method can be

used as a more robust and accurate ambiguity free

tracking approach for outdoor applications [15]. To

bridge the GPS outage for multicopters is designed

proposes a novel navigation reconstruction method

for small multicopters, which combines the vehicle

dynamic model and micro-electro-mechanical system

301

(MEMS) sensors. Firstly, an induced drag model is

introduced into the dynamic model of the vehicle, and

an efficient online parameter identification method is

designed to estimate the model parameters quickly. In

addition, the nongravitational acceleration estimated

from body velocity and radar height are utilized to

yield a more accurate attitude estimate. Some types of

Multi-Static Surveillance Radars are described in [17-

18]. Fusing the information of the attitude, body

velocity, magnetic heading, and radar height, a

navigation system based on an error-state Kalman

filter is reconstructed. Simulation results show that

the proposed navigation reconstruction algorithm

aided by the vehicle model can significantly improve

navigation accuracy during a GPS outage [14]. This

article demonstrates the superiority of a fuzzy

tracking system over the standard Kalman filter

tracking system under the conditions of uneven

accelerations and sudden change of direction of the

targets, as well as in the case of failure to observe the

target during successive scans. A cascading Kalman

filtering algorithm was used to solve the speed

ambiguity and to reduce the measurement error in

real-time radar processing. The cascade filters are

extended Kalman filters with controlled gain using

fuzzy logic for tracking targets using radar equipment

under difficult tracking conditions [11]. An automatic

landing of an unmanned aerial vehicle (UAV) is a

non-trivial task requiring a solution of a variety of

technical and computational problems. The most

important is the precise determination of altitude,

especially at the final stage of approaching to the

earth. With current altimeters, the magnitude of

measurement errors at the final phase of the descent

may be unacceptably high for constructing an

algorithm for controlling the landing manoeuvre. One

of the possible and straightforward ways is the

camera resolution change by pixels averaging in

computer part which performed in coordination with

theoretically estimated and measured OF velocity.

The article presents results of such algorithms testing

from real video sequences obtained in flights with

different approaches to the runway with simultaneous

recording of telemetry and video data [9]. Over the

past years Time-of-Flight (ToF) sensors have become a

considerable alternative to conventional distance

sensing techniques like laser scanners or image based

stereo-vision. Due to the ability to provide full-range

distance information at high frame-rates, ToF sensors

achieve a significant impact onto current research

areas like online object recognition, collision

prevention or scene and object reconstruction. The

main contribution, in this context, is a new intensity-

based calibration model that requires less input data

compared to other models and thus significantly

contributes to the reduction of calibration data.

2 POSITIONING BY TELEMETRY METHOD

The principle of the telemetric method is described in

detail in [4]. Next, in accordance with [4], we will state

algorithms for calculating the flying object position.

Telemetry is the automatic measurement and wireless

transmission of data from remote sources. The

location of the FO is determined by acquiring the

transmissions from three (or more) different locations

to triangulate the location of the device. Each user in

communication network transmits a signal which

contains information about its position and the

current time at regular intervals. These signals,

travelling at the speed of light, are intercepted by a

receiver of an unknown user, which calculates how far

each user is based and how long it takes for the signal

to arrive. The range from the other users is

determined by the time the signal is received. To

ensure the work of the RelNav system, at least three

users must be visible to the unknown receiver at all

times. The location of the unknown user is calculated

by the following solution system of three equations

[4]:

( ) ( ) ( )

2 2 2

i i i i

x x y y z z d− + − + − =

(1)

where: xi, yi, zi- geocentric coordination of the user, di

– range between an unknown receiver and each

transmitter, x, y, z- geocentric coordination of an

unknown receiver.

The distances to the users are derived from the

transmitted signals and they are affected by

uncertainties in clock setting, therefore they are

normally referred to as pseudo-ranges. There is a time

synchronization problem of each user´s clock. The

Pseudo-range measurements lead to a pseudo-ranging

four point problem which is the problem of

determining the four unknowns. The unknowns

comprise the three components of the receiver

position {X, Y, Z} and the stationary receiver range

bias. Four pseudo-range equations are expressed

algebraically as [4]:

( ) ( ) ( ) ( )

2 2 2 2

1 1 1 1

0x x y y z z b d− + − + − − − =

(2)

( ) ( ) ( ) ( )

2 2 2 2

0x x y y z z b d− + − + − − − =

(3)

( ) ( ) ( ) ( )

2 2 2 2

3 3 3 3

0x x y y z z b d− + − + − − − =

(4)

( ) ( ) ( ) ( )

2 2 2 2

4 4 4 4

0x x y y z z b d− + − + − − − =

(5)

where: d1, d2, d3, d4 – measured pseudo-distances

from a source of transmission to receivers, x1-4, y1-4, z1-4

– coordinates of users, x, y, z- coordinates of the

receiver, b – shift of the time basis of the receiver

converted into distance.

For subtraction (5) from (2), is valid:

14 14 14 14 41 14

f x x y y z z d b e= + + + +

(6)

For subtraction (5) from (3), is valid it holds:

24 24 24 24 42 24

f x x y y z z d b e= + + + +

(7)

For subtraction (5) from (4), is valid:

34 34 34 34 43 34

f x x y y z z d b e= + + + +

(8)

302

If variable b is considered as constant (the so-called

factor of homogenization), then, depending on

expressions (6-8), a system of three equations with

three unknown variables is obtained. We have applied

the method of resultants of the multi-polynomials to

solve the system of linear equations x=g(b), y=g(b),

z=g(b) [4].

Following is valid:

( )

1 14 41 14 14 14

f x x d b e k y y z z= + + + +

(9)

( )

2 24 42 24 24 24

f x x d b e k y y z z= + + + +

(10)

( )

3 34 43 34 34 34

f x x d b e k y y z z= + + + +

(11)

If the variable “k“ turns into the factor of

homogenization, Jacobi’s determinant of the

coordinate x by (9), (10), (11) is expressed as:

3 3 3

det

df df

dy dz

df df

J det

dy dz

df df df

dy dz dk







(12)

In compliance with (9-11), for y = g (b) is valid:

( )

4 14 41 14 14 14

f y y d b e k x x z z= + + + +

(13)

( )

5 24 42 24 24 24

f y y d b e k x x z z= + + + +

(14)

( )

6 34 43 34 34 34

f y y d b e k x x z z= + + + +

(15)

Jacobi’s determinant for the coordinate y is

expressed as:

4 4 4

5 5 5

6 6 6

df df df

dx dz dk

df df df

J det det

dx dz dk

df df df

dx dz dk







(16)

In compliance with (9-11), for z = g (b) is valid:

( )

7 14 41 14 14 14

f z z d b e k x x y y= + + + +

(17)

( )

8 24 42 24 24 24

f z z d b e k x x y y= + + + +

(18)

( )

9 34 43 34 34 34

f z z d b e k x x y y= + + + +

(19)

Jacobi´s determinant for coordinate z is expressed

as:

9 9 9

df df

dx dy

df df

J det det

dx dy

df df df

dx dy dk







(20)

From expressions (12), (16), (20) we obtain the

value of the determinants Jx, Jy, Jz. We get the variable

equations x=g(b), y=g(b), z=g(b) by applying the value

of each determinant. Substituting the expression for x,

y, z into the equation (2), the quadratic function for

the unknown variable b is obtained:

0Ab Bb C+ + =

(21)

Solutions of the quadratic equation (21) have two

roots, b+ and b-. Therefore we have to calculate the

coordinates of the FO (x+, y+, z+) and P (x-, y-, z-). To

come to the correct solution, we have to calculate the

norm (

2 2 2

norm x y z= + +

) for (x, y, z) b- and (x, y, z)

b+. If the coordinates of the receiver are fed into the

coordinate reference system, the norm of the position

vector will be close to the value of the radius of the

Earth (Re = 6372.797 km) and this solution for the FO

(x, y, z) will be regarded as a correct one [4].

2.1 Evaluation of the influence of users location on the

accuracy of the RelNav system

In accordance with algorithms 1 to 21, we performed

evaluation of the influence of users location on the

accuracy of the RelNav system. Using simulation, we

investigate the effect of the geometric distribution of

flying objects on the accuracy of determining their

position. The test cases are designed to examine the

sensitivity of the relative navigation system to this

factor.The simulation results are shown in figure 1 to

In our simulation, we assume that LOT5MF is a

flying object whose position we want to determine.

We know the coordinates FO RJA39K, FHM6112,

LOT653 and WZZ3007. It is clear from the theory of

navigation that the accuracy of determining the

position of the FO rangefinder method depends on

the accuracy of measuring the distance from own

rangefinders to the rangefinders of the others systems

and on the geometry of the rangefinder systems.

Therefore, we checked how the accuracy of the

RelNav system will depend on the FO geometries

working in the aviation communication network. We

performed further simulations and investigated the

accuracy of FO positioning when changing the

geometry of communication network users. The

simulations were performed for a straight flight

lasting 1375 s. When selecting the geometry of users of

the aviation communication network, we proceeded

as follows. We will consider FO LOT5MF as an

unknown FO. The initial positions of the LO in the

simulation for a distance of 5 km are shown in Figure

no. 1 and in table no. 1 and 2. When selecting the

initial coordinates of the network users, we placed the

individual users with respect to the geographical

course of the LOT5MF flight. The geographical course

303

of the LOT5MF flight will be 345ᵒ for all simulations.

The geographic course of the initial position of the

user of the RJA39K communication network will be

48ᵒ. The geographic course of the initial position of the

user of the communication network FHM6112 will be

19ᵒ. The geographic rate of the initial position of the

user of the LOT653 communication network will be

318ᵒ. The geographic course of the initial position of

the user of the communication network WZZ3007 will

be 286ᵒ. In the individual simulations, we gradually

changed the distances between LOT5MF and users

RJA39K, FHM6112, LOT653 and WZZ3007 in the

range of 5 to 50 km. The initial positions of the LO in

the simulation for a distance of 5 km are shown in

Figure no. 1 and in table no. 1 and 2.

Table 1. Initial LO coordinates in the WGS-84 system and in

the coordinate system ECEF, distance 5 km

_______________________________________________

FO identification latitude (°) longitude (°)

_______________________________________________

1. RJA39K 48.8022 21.1971

2. FHM612 48.8143 21.1657

3. LOT653 48.8080 21.1097

4. WZZ3007 48.7856 21.0838

5. LOT5MF 48.771 21.148

_______________________________________________

latitude (rad) longitude (rad) height above sea level

_______________________________________________

1. 0.851759 0.369959 11262

2. 0.851970 0.369411 10683

3. 0.851860 0.368433 7003

4. 0.851469 0.367982 10363

5. 0.851215 0.369102 3784

_______________________________________________

X (km) Y (km) Z (km)

_______________________________________________

1. 3931.154 1524.565 4784.573

2. 3930.688 1521.907 4785.025

3. 3930.405 1517.382 4781.793

4. 3934.907 1517.076 4782.678

5. 3930.301 1520.361 4776.659

_______________________________________________

Picture no. 1 shows the distribution of FO1-4 users

with respect to FO5 for a distance of 5 km. The initial

coordinates of the FO are marked with a black (+)

sign, which indicates the identification mark of the

individual FO.

Figure 1. Flight trajectories of five FO, straight flight, shift 5

km, flight length 1375 s.

The results of the FO positioning simulation are

shown in FIG. 2 to 5. In FIG. no. 2 shows the errors of

measuring the distance ∆d1 to ∆d4 between LOT5MF

and other network users as a function of time. From

picture no. 2 shows that the mean values of the

distance measurement errors ∆d1 to ∆d4 range from -

0.03 m to 0.02 m and the dispersions of the distance

measurement errors ∆d1 to ∆d4 range from 0.98 m

1.01 m

Figure 2. Errors in measuring the distance ∆d1 to ∆d4

between LOT5MF and other network users.

In FIG. no. 3 shows the error of determining the

coordinates ∆X, ∆Y, ∆Z and the error of determining

the position FO5 ∆P as a function of time. From

picture no. 3 shows that the mean values of the

coordinate errors ∆X, ∆Y, ∆Z range from - 0.03 m to

0.06 m and the dispersion of coordinate errors ∆X, ∆Y,

∆Z ranges from 2.81 m2 to 9.80 m2. The mean value of

the positioning error FO 5 m∆P is equal to 3.84 m and

the dispersion of the positioning error FO 5 2∆P is

equal to 6.04 m2.

Figure 3. Coordinate determination errors ∆X, ∆Y, ∆Z and

position error ∆P FO5, rectilinear flight 1375 s

In picture no. 4 is a histogram of the positioning

error LO 5. It can be seen from the figure that the

maximum number of positioning errors is in the range

of 0.8 to 7.64 m.

In order to verify the sensitivity of the derived

algorithm to the accuracy of FO 5 positioning, which

works in the aviation communication network, we

performed additional simulations. We maintained the

initial coordinates of FO RJA39K, FHM6112, LOT653

and WZZ3007 in the above courses and changed their

304

distances from LOT5MF in the range of 10 to 50 km.

The results of the simulation when changing the

distances of FO 5 from other FO 1 to 4 from 5 to 50 km

are shown in Figure no. 5. From picture no. 5 shows

that the mean value of the positioning error FO 5 m∆P

was in the range of 3.5 to 10.0 m and the dispersion of

the positioning error FO 5 

∆P varied in the range of

7.0 to 72.79 m

. From picture no. 5 shows that as the

distances FO RJA39K, FHM6112, LOT653 and

WZZ3007 from LOT5MF increase, the positioning

errors increase. Based on this, we can conclude that

the accuracy of determining the position of a flying

object operating in the aviation communication

network depends on the mutual position of network

users. Our proposed RelNav system is accurate. The

subject of further research will be the dependence of

the accuracy of this system on the mutual position of

the users of the communication network. vv

Figure 4. FO5 positioning error histogram

Figure 5. LO 5 positioning accuracy for straight flight and

distances from 5 to 50 km.

3 CONCLUSION

Using derived algorithms and a model of the motion

of flying objects, we performed a computer

simulation, which is used to determine the position of

the unknown LO 5 and evaluate the accuracy of the

proposed system RelNav. The computer simulation

was performed in the Matlab programming

environment and explains the principle of operation

of the RelNav system. The result of modeling is a total

of five models that simulate the motion of LO in a

geocentric coordinate system. A description of the

models is given in [4]. LO motion models represent

input data for computer simulation. In order for the

simulation to correspond to reality as much as

possible, we performed observations of air traffic over

the territory of the Slovak Republic via the

Flightradar24 application. We randomly selected five

FOs, whose geographical coordinates and heights we

implemented in our model. For the needs of solving

the problem, it was necessary to transform the

coordinates into a geocentric coordinate system. The

result of the simulation is the evaluation of the

accuracy of determining the position of FO 5 for

different model situations. The simulation results

confirmed that the accuracy of FO position

determination in the RelNav system depends on the

geometry of the communication network users. The

principles of relative navigation can also be applied to

maritime 2D navigation. The basic condition for the

correct operation of such a system in the navy is to

create a communication network of users and to

ensure time synchronization of data transmission by

individual users. We assume that for the accuracy of

determining the location of the user of this network

will be the same as for the aviation communication

network. In further research, it would be appropriate

to verify the navigation possibilities of flying and

floating objects (ships) in the relative navigation

system.

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