423
1 INTRODUCTION
Airplane position is detected by air navigation
systems, which consist of a combination of different
types of air navigation devices. The position of the
aircraft onthetrackisdefined as the intersection of
two or more position lines. The first indication of
addressing the positioning problem was
heavenly
navigation. She used the knowledge of the mutual
geometric arrangement of the stars and the
measurementoftheanglesbetweenthem.Inabilityto
measure distance to the stars has led to automatic
positioningofflyingobjects(FO)usingradiosignals.
The principle of radionavigation is based on
measuringthe
timeoftransmissionofthesignalfrom
thetransmittertothereceiver.Attheknownvelocity
of propagation of electromagnetic waves (light
velocity), it is possible to calculate the distance
betweenthetransmitterandthereceiver.Inthiscase,
theemphasisisplacedontheprecisionofthesignal
propagation time
measurement [Dzunda, M. &
Kotianova,N.2016,Dzunda,M.&Kotianová,N.2015,
Dzunda,M.KotianovaN.&Holota,K.&EtAl.2015,
Dzunda,M.&Hrban,A.1998],becauseatimeerror
of1μscausesa300merroratthemeasureddistance.
For precise positioning, we require
that the receiver
beabletomeasurethepropagationtimeofthesignal
to1nsaccuracy.ecosystememitscontinuousradiation
wavesintothesurroundingspace.
2 POSITIONGPOSITIONSWITHSATELITE
NAVIGATIONSYSTEMS
Air Traffic Services of the SR are using
communication, radio navigation and radar systems
to control air traffic. Communication
systems are
operatingatfrequenciesof100‐150MHz.Theoutput
powerofthesesystemsis5.0to20.0W.Theprimary
surveillance radar is operating at frequencies of 2‐4
GHz. Transmitted power of these systems is 14‐25
kW. The power of older types of radar is several
hundredkW.Secondaryradarsoperateatfrequencies
of1030MHzand1090MHz.Transmissionpowerofa
deviceisabout2kW.
Determination of Flying Objects Position
M.Dzunda,P.Dzurovcin&L.Melniková
TechnicalUniversityofKosice,Kosice,Slovakia
ABSTRACT: This paper describes various methods of flying object positioning with the emphasis on their
accuracy.Basedontheaccomplishedanalysis,we will select the most appropriatemethodto determine the
positionofaflying object for relative navigation purposes.Theprimary criterionforchoosing
apositioning
method is the accuracy of distance measurement within users working in the aviation communications
network. The results presented in the paper have been based on mathematical modelling and computer
simulationperformedintheMatlabprogrammingenvironment.Theresultsobtainedcanbeusedtonavigate
flyingornon‐flying
objectsthatworkintheaircommunicationsnetwork.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 13
Number 2
June 2019
DOI:10.12716/1001.13.02.21
424
NDB navigation systems and ADF work on the
frequency kHz 200‐525. Transmission power of a
deviceis25‐50W.VORoperatesonafrequencyMHz
108‐112. Transmission power has a 25‐100 W. DME
measures the distance and works on the frequency
960‐1215MHz.Transmissionpower
ofadeviceis100
W.
TheILSprecisionapproach:
Localizer LLZ (device frequencies: 108‐112 MHz,
transmissionpower2W)
GPGlidepathbeacon(devicefrequencies:328.6to
335.4MHz,transmissionpower2W)
VHF marker beacons (frequency 75 MHz,
transmission power 3W) Using satellite navigation
systems, we can
determine the location in two basic
ways:
1 Pointpositioning‐absolutepositioning
2 Relativepositioning‐relativepositioning
In absolute positioning, it is the positioning of
individual points to the used positional system that
formsanetworkontheEarthʹssurface.Theabsolute
positioning method is the basic GPS assignment. In
addition to determining the instantaneous position,
these methods also serve to determine the speed of
the receiverʹs movement and to navigate it on the
groundorinthespace.Withabsolutepositioning,we
can determine the position in general with an
accuracywithin±10mto±30m(SA
mode)andwithin
±5mto±10m,(withoutSAmode).We canincrease
theaccuracyofgeocentriccoordinatesdetermination
by applying differential corrections of measured
pseudo‐distances. Accuracy in coordinate
determinationisincreasedandrangesfrom±1mto±
5m.
The relative positioning method determines the
coordinatesofthe
newpointsrelativetotheposition
of the reference point whose geocentric coordinates
are known. Relative methods are of primary
importance for geodetic applications, as they allow
measurementswhoseresultsleadtocoordinateswith
accuracyinmm.Itproceedsfromthemeasurementof
thephaseofthecarrierwave,while
themathematical
modeldoesnotdirectlyusethephasemeasurements
in the processing of the measurement results, but
differentiates them in the appropriate way‐simple,
double and triple. Both methods are suitable for
determining the position of the moving and
immovable object. Several methods are used to
determine location by satellites.
According to the
measuredparameter,theyaredividedinto:
anglemeasuringmethod,
longdistancemethod,
Dopplermethod,
interferometricmethod,
combinedmethod.
The common feature of all methods is that we
need to know the location of the satellites to
determine the position. The different methods
differ
from one another in what the electromagnetic wave
(propagation signal) parameter is measured. These
methods can be used to create new navigation
systems,whethersatelliteoralternate
2.1 Anglemeasuringmethod
Theanglemeasuringmethodisoneoftheoldestand
at least the least accurate methods. The principle of
positioning lies in the measurement of the elevation
angletothesatellite.Thegeometricpointofthepoints
withtheconstantelevationangletothesatelliteisthe
cone with the top in the space of the satellite, as
shown in Fig. 1 [Kotianova, N. 2017]. To determine
the elevation
angle accurately, it is necessary to use
directional antennas with a narrow beam in the
direction of the maximum radiating characteristic.
This method did not spread any more, because it
required very large antenna systems. If we make a
measurementofthesamesatellite(atanothertime)or
anothersatellite(at
thesametime),wewilldetermine
the second cone. Due to the rapid movement of the
satellites and thus the rapid change of the azimuth
andthealtitudeofthesatelliteitispossibletorepeat
the measurement after about 2 minutes. The cross
member of the two cones with
the Earthʹs surface,
respectively.withtheheightatwhichthepositionof
thepointislocatedintersectwiththemeasuredpoint.
In the case of an angle method with increasing
distance from the reference point, the measurement
errorisincreasing.Thepositioningaccuracyisgiven
bytheaccuracyofthe
measurementof the elevation
angulardirectionalantennas,andthisisnottoohigh.
Figure1. The principle of the angel measuring method of
positioning[Kotianova,N.2017].
2.2 Dopplermethod
The basis of this method is the Doppler effect when
thetransmitterandreceivermoveatdifferentspeeds.
In this method, the satellites must not move on a
geostationary path. If their angular velocity is the
sameastheangularvelocityofapointontheEarthʹs
surface,itwouldnotmoverelativetotheobserveron
Earth,andthuswouldnotshowaDopplershift.
A satellite that moves over a non‐geostationary
orbit sends a signal with a stable f
TX frequency. The
signaltransmitsinasuitablemannerthetimestamps
transmittedattimepointst
i,ti+1,ti+2withaconstant
timeintervalΔT=t
i+1‐ti.Thefrequencyofthesignal
receivedbytheuseratthepositionmeasurementsite
isduetotheDopplereffectequaltothef
rxvaluethat
differsfromf
TX.Thereceivedsignaliscoupledtothe
oscillatorsignalwiththefrequencyfedtothemixer.
The output signal is a signal with the frequency
difference fo‐f
TX. Periods of this signal read the
counter triggered and blocked by consecutively
received time stamps. If the distance between the
425
satelliteandtheuserwould not change,thenumber
ofperiodswouldbeequalto:
.
ioTX
NTff (1)
However, the distance between the satellite and
theuserchangesbetweentwotimestampsfromdito
d
i+1(Figure2[Kotianova,N.2017]).Inthiscase,the
timestampisreceivedbytheuseratthetimeti+Δi,
whereΔi = di / c represents the time required for
signaltransmissiontodistancedibetweenthesatellite
and the user at the velocity propagation velocity c.
The
frequency difference meter period counter
essentially measures the phase change of the signal
betweentworeceivedtimestamps.
1
Δ
.
ii
i
t
LO RX
t
Ni f f dt
=
1
Δ. . Δ.
o
oi i TX
f
Tf d d Tf
c
(2)
The number of periods of the signal emitted
betweenthetwoadjacenttimestampsisthesameas
thenumberofperiodsofthesignalreceivedbetween
theadjacentmarks,becausethedopplereffectwillof
coursealsooccurinthetimedomain:
1
..Δ
TX i i TX
f
ttfT
=
11
.Δ Δ
RX i i i i
f
tttt
(3)
IfwedenoteF=f
o‐fRXandthecoordinatesofthe
satellite at the moment t as an ordered triple (x
i, yi,
d
ay),respectively.atthemomentti+1asthetriple(xi+1,
y
i+1,zi+1),andtheuserʹscoordinatesas(xu,yu, zu),
weobtaintheequation:
222
111
LO
iiii
f
NTF xx yy zz
c
222
iii
x
xyyzz
(4)
Figure2.Dopplerfrequencyshift[5].
If we perform a measurement of at least three
periods between the four time stamps, we obtain
threetimesofthedifferencesignalperiodaftertheN
i,
N
i+1, Ni+2 mixer. If we know the coordinates of the
satellitesinthemomentsoft
i,ti+1,ti+2,wecansolvea
systemofthreeequationsofthreeunknowns, which
arethecoordinatesoftheuseratthelocationlocation
(x
u,yu,zu).Thepositionofthesatelliteisdetermined
from the current Keplerian parameters of its path,
whichwillbetransmittedbythesatelliteintheform
ofanavigationmessagesothatthepositioningerror
inthetimepoints t
i,ti+1,ti+2isassmall as possible.
The Doppler method has been applied to some
navigationsystemsastheprimarymethod,theParus
/CycladicSystemandtheUSNavyTransit.Today,it
isratherusedasasupportmethodforthetelemetric
method.
2.3 Anglemeasuringmethod
Thelong‐distance method
isone of the most widely
used positioning methods. It is used by current
systems, GPS, GLONASS and the future European
GALILEO system. In the distance method, the
position of the object is determined by its distance
fromthebroadcast source(satellite).The distance di
between the receiver and the satellite
can be
calculatedfromthetime of
di
,whichelapsesfrom
thetransmissionofthesignalfromthesatellitetoits
receptionatthereceiverinrelationto:
.
idi
dc
(5)
wherec‐thespeedof light. If the coordinatesof the
satellite are known, it is possible to calculate the
distancediinrelationto:
222
iiii
x
xyyzzd
, (6)
where:
x
i,yi,zi,‐thecoordinatesofthetransmitter(satellites),
d
i‐the distance between the receiver and the
transmitter, x, y, z‐the coordinates of the receiver.
The time
di
according to equation (5) can be
precisely determined only if the perfect time
synchronization of the pair of satellite‐receiver is
ensured. With an electromagnetic wave propagation
velocityc=ofabout3x10
8
m∙s
‐1
,1mdistanceinthe
free space corresponds to a 3.3 ns time delay of the
signal. Ensuring such synchronization of remote
independent clock generators in two independent
systems(on‐boardsatelliteanduserreceiver)isavery
challenging task, leading in particular to the higher
complexityofthewhole
system.
For this reason, the additional b [Awnage, J.L. &
Grafarend, E.W. 2004] is introduced into the
calculation, which represents the time differenceΔt
calculatedover the distance.To calculate positionin
three‐dimensional space, it is necessary to process a
signalfromatleastfoursatellites.
Considering(6)the
coordinatesofthereceiverwe
obtainbysolvingfourequationsintheform:
2222
1111
0xx yy zz bd
, (7)
2222
2222
0xx yy zz bd
,(8)
426
2222
3333
0xx yy zz bd
, (9)
2222
4444
0xx yy zz bd
, (10)
wherex,y,z‐coordinatesofthereceiver,x
1‐4,
y
1‐4, z1‐4‐ satellite coordinates, d1‐4‐ pseudo‐distance
distancebetweenthereceiverandthesatellite,b‐time
delaycalculatedoverdistance.
Wecandividethedistantsystemsinto:
1 Passive‐receiver (user) receives and processes
signals from the satellite. The receiver receives a
copyofthesatellitesignaltowhichthedistanceis
measured. This copy
is synchronized with the
receivesignalintheuserreceivertoobtainadelay
associated with the time base of the receiver. To
measure the signals from the four satellites, we
obtain a fourfold delay that corresponds to the
pseudo‐distance. Putting into a set of four
equations (7‐10) we
calculate the userʹs
coordinates.
2 Active‐thereceiver(theuser)sendsthesignalto
thesatellite(tothetransponder),whichsendsthe
signal with delay. The appropriate distance
between the satellite and the user is then
determinedbytherelationship:
..,
2
ci zi
idi
dc c
(11)
whered
iisthedistancebetweenthesatelliteandthe
beacon, i represents the measured total delay of the
signalfromtheusertothesatelliteandback.Then,t
represents a delay in the transponder of the i‐th
satellite. Therefore, no time synchronization is
required between system devices. Timeτ
ci is
measured only from the time base of the user. The
disadvantageofsuchasystemisthattheusermustbe
an active component. This leads to higher energy
demands of small navigation terminals. Another
problem is limited user capacity when at a certain
point in time the given
transponder channel is
available to only one user, which is economically
disadvantageous[Pavolová, H. & Tobisová, A. 2013,
Rozenberg, R. & Szabo, S. & Šebeščáková, I. 2014,
Sabo,J.&Korba,P.&Antoško,M.&Sekelová,M.&
Rozenberg,R.2017,Šebeščáková,I.&Melníková,L.&
Socha,
Ľ.2013].Thetotaleffectiveerrorvalueisequal
to the product of the standard deviation of the
distance errorσ
d and the DOP value (dilution of
precision):
.
rms d
E DOP
(12)
The DOP coefficient depends only on the
geometricconfigurationsofsatellitesandreceivers.It
is defined as the ratio of the mean error of the
coordinate,time,position,and mean errormeasured
bytheprecision:
o
DOP
, (13)
It is true that what the DOP is less, the more
accuratetheresults.DOPconsistsofahorizontaland
a vertical component. The total radial coefficient of
deterioration of PDOP accuracy, positional DOP
equals:
22
PDOP HDOP VDOP, (14)
The PDOP is calculated before the measurement,
fromthecoordinatesofsatellitesoriginatingfromthe
almanacandthecoordinatesofthereceiver,whichis
sufficienttoknowwiththeaccuracyperkilometer.
In real measurement with errorσ
d, the search
positionislocatedinacertainspace,which is given
by the intersection of spheres with a measured
pseudo‐radius. The accuracy of the measurement
corresponds to the size of this space. The solution
withagivenerrorσ
dcanbelocatedanywhereinthis
space. From a geometric point of view, it is also
advisable to select the satellites so that the space
createdisassmallaspossible.Thereportedaccuracy
foranormalcivilGPSreceiverthatusestheprinciple
ofalong‐distancemethodand
operatesonlywithaC
/Acodeis25meters(2drms,95%)inthehorizontal
planeandis43meters(95%)intheverticalplane.
Accuracyinheightisusually2‐3timesworsethan
horizontallineaccuracy.Theaccuracyatthesatellite
system level in terms of timing
synchronization is
satisfactory because the satellites have precision
atomic clocks and the entire synchronization is
providedbythecontrolsegmentthatisboundtothe
world reference time norms. In addition, any
deviation of satellite hours may be specified in the
navigationreporttogetherwithephemeris.Theclock
uncertainty error
inthe receiver is addressed by the
use of multiple satellite signals because the receiver
can not contain atomic clocks. In Table no. 1 is an
estimate of the impact of different error sources on
positioningaccuracy.
Table1. Estimation of the impact of different sources of
errorsontheaccuracyofpositioning
_______________________________________________
CauseoferrorEstimationofimpacton
positioningaccuracy[m]
_______________________________________________
Timesynchronizationerror 2
Multi‐waysignalpropagation
1
Movementoforbitsatellites
2,5
Troposphericeffects
0,5
Ionosphericeffects
5
Calculationerrors
1
_______________________________________________
3 SIMULATIONOFDETERMINATION
POSITIONSOFFLIGHTOBJECTSBYALONG
DISTANCEMETHOD
We created simulation models based on conditions
thatareasclosetorealaspossible.Wehavechosenan
airspace that is analogous to actual flightradar24
traffic[Vagner,J.Pappová,E.2014].Flightradar24isa
flight tracking
service that provides real‐time
information about thousands of aircraft worldwide.
Based on observations of the movement of aircraft
overtheterritoryoftheSlovakRepublicon28.06.2018
08:12. we chose 5 planes randomly and found their
427
coordinates at a given time. We assume LO is
working in aviation communications network.
BecausetheLOpositiondataoriginatesfromtheGPS,
it is determined by the latitude and longitude
according to the WGS‐84. Figure 3 shows a random
selection of 4 broadcast sources (aircrafts) of known
position,
andtheredcolorishighlightedbythefifth
aircraftwhosepositioniscalculatedaccordingtothe
algorithms5to10.Weherebyverifythesuitabilityof
theLOpositioningalgorithms5to10thatworkinthe
air communications network. Such a positioning
system is called a relative navigation
system. The
informationabouttheirstartingposition:
FO SAS777 with coordinates: latitude 0,85398056
rad,longitude0,31139381rad.Heightaboveellipsoid
10653mand altitude 10613 m. Geoid curl 40 m. FO
AIC131 with coordinates: latitude 0,84694871 rad,
longitude 0,32232797 rad. Height above ellipsoid
12192mand altitude 12152 m. Geoid curl
40 m. FO
LOT224 with coordinates: latitude 0,84418725rad,
longitude0,31173396rad.Heightaboveellipsoid8809
mandaltitude8769m.Geoidcurl40m.FOCLX856
with coordinates: latitude 0,84384361 rad, longitude
0,31586831 rad. Height above elipsoid 10668 m and
altitude10628m.Geoidcurl40m.FOOHY805with
coordinates:
latitude 0,84762904 rad, longitude
0,31724117 rad. Height above ellipsoid 10958 m and
altitude10918m.Geoidcurl40m.Wedesigned5LO
motion models to simulate movements of aviation
objects. We have simulated situations where users
have left each other. The distance between users is
between10kmand100
km:Di∈(10,100).Wemodeled
straight‐lineLOflightswithalengthof375seconds.
At the same time, we assume that there is an error
measuring distance from all four broadcast sources.
Fordistancemeasurementerrors:
1234
Δ0 0 0 0 ,dddd (15)
where:
1
Δd is the error of measuring the distance of an
unknownuserLO
5fromLO1,
2
Δd is the error of measuring the distance of an
unknownuserLO
5fromLO2,
3
Δd is the error of measuring the distance of an
unknownuserLO
5fromLO3,
4
Δd is the error of measuring the distance of an
unknownuserLO
5fromLO4;
Figure3.Layoutofaircraftinairspacefromflightradar24.
We assume that during the simulation time
synchronization has been compromised between all
users. Therefore, the information about the
coordinates of each user was loaded with an error.
Since we do not know how to fetch a time error
directlyintoouralgorithms,wereplaceditwiththeX,
Y,
Zforeach LO1‐4 usererror. Coordinate errorsΔx,
Δy andΔz were generated by a random number
generator with a normal distribution and these
parameters: mean value equal to zero, dispersion
equal to one. The magnitude of the generated
coordinate errors was gradually changed by
multiplyingthembytheconstanttothefollowing:
X
LOi=k.[Δx1
i
,Δx2
i
.,...,Δxn
i
];
Y
LOi=k.[Δy1
i
,Δy2
i
,...,Δyn
i
];
Z
LOi=k.[Δz1
i
,Δz2
i
,...,Δzn
i
],wherek=0,1
X
LOi=k.[Δx1
i
,Δx2
i
.,...,Δxn
i
];
Y
LOi=k.[Δy1
i
,Δy2
i
,...,Δyn
i
];
Z
LOi=k.[Δz1
i
,Δz2
i
,...,Δzn
i
],wherek=0,25
X
LOi=k.[Δx1
i
,Δx2
i
.,...,Δxn
i
];
Y
LOi=k.[Δy1
i
,Δy2
i
,...,Δyn
i
];
Z
LOi=k.[Δz1
i
,Δz2
i
,...,Δzn
i
],wherek=0,5
X
LOi=k.[Δx1
i
,Δx2
i
.,...,Δxn
i
];
Y
LOi=k.[Δy1
i
,Δy2
i
,...,Δyn
i
];
Z
LOi=k.[Δz1
i
,Δz2
i
,...,Δzn
i
],wherek=1
X
LOi=k.[Δx1
i
,Δx2
i
.,...,Δxn
i
];
Y
LOi=k.[Δy1
i
,Δy2
i
,...,Δyn
i
];
Z
LOi=k.[Δz1
i
,Δz2
i
,...,Δzn
i
],wherek=2
SomesimulationresultsareshowninFig.4andin
TableNo.2.InFig.4showserrorsofdeterminationof
thecoordinatesx,y,zunknownFO5.Thelastgraph
shows the position error of the unknown FO 5.
Simulationparameters:k=0,1andD
=20,0km.The
simulation results show that the accuracy of the
relative navigation system that operates in the air
communicationsnetworkdependsonthegeometryof
the system. The accuracy is worsening from the
increasing distance between network users. See the
dispersionoftheLOpositioningerrorintable
2.
Table2. Comparison of accuracy of positioning LO at the
directflight
_______________________________________________
Mutual PositioningaccuracyLO[m]
distance(Averageradialdistancebetweenactualand
[km] calculatedposition)
_______________________________________________
k=0,1 k=0,25 k=0,5 k=1 k=2
10 1,52665 3,52231 7,63329 15,26657 30,53310
20 1,56875 3,76706 7,91753 15,83506 31,67014
30 4,45322 11,13307 22,26615 44,53233 89,06476
40 5,92641 14,81602 29,63205 59,26412 118,5283
50 9,82508 22,7662 49,1254 98,2508 196,5017
_______________________________________________
428
Mutual Dispersion[m2]
distance
[km]
2
2
1
1
n
i
i
σ xx
n
_______________________________________________
k=0,1 k=0,25 k=0,5 k=1 k=2
10 1,32642 5,930898 33,16072 132,6436 530,5801
20 1,32372 8,433033 33,88081 135,5228 542,0887
30 12,3276 77,04804 308,1918 1232,764 4931,039
40 22,0634 137,8965 551,5859 2206,340 8825,339
50 58,380 269,329 1459,51 5838,06 23352,1
_______________________________________________
Figure4.FO5CoordinateandPositionFindErrors.
It is clear from the simulation results that
positioningerrorsoftheunknownLO,whichispart
of the relative navigation system in the air
communicationsnetwork,dependontheaccuracyof
the LO coordinates that the unknown object uses to
determine its position. Also, the accuracy of
measuringitsdistance
fromLO,which an unknown
objectusestodetermineitslocation.
4 CONCLUSION
One of the basic requirements for positioning LO is
accuracy. In this paper, we describe various LO
positioningmethodswithemphasisontheiraccuracy.
The different methods differ from one another in
what the electromagnetic wave (propagation
signal)
parameterismeasured.Thesemethodscanbeusedto
create new navigation systems, whether satellite or
alternate.Themainfocusisonthelong‐distance
FO positioning method that works in the air
communications network. The analysis made shows
that if we want to determine the FO position that
works
intheaircommunicationsnetwork,weneedto
knowthepositionsofatleastfourotherFOsfromthat
network.Itisclearfromthesimulationresultsthatthe
distance method is suitable for determining the FO
position.It isusedtodetermineabsolutepositionas
wellasrelativepositioning.
Theexactpositioningofthismethodistomaintain
the maximum time synchronization of time basic
individualuserswhoworkintheaircommunications
network. The accuracy of the satellite‐level
synchronization is satisfactory because the satellites
have precision atomic clocks and the entire
synchronizationisensuredbyacontrolsegment
that
is bound to world reference time norms. In the
AviationCommunicationsNetwork,itisnecessaryto
ensure the synchronization of this network with the
accuracyofthenanosecondseries.Onepossibilityto
ensure synchronization of the air communications
network is to use the currently available atomic
norms.Itis
clearfromthesimulationresultsthatthe
accuracy of the relative navigation system also
dependsonthegeometryoftheairtrafficnetwork.At
greater distances between FO, the positioning error
substantiallyincreases. Modeling has confirmed that
atthedistanceofanunknownFOfromotherairline
communications network users that
is greater than
30.0 km, the precision of determining its position is
inadequate.
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