International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 2
Number 2
June 2008
161
Accuracy of Relative Navigation in Automated
Systems of Data Communication
M. Dzunda & V. Humenansky
Faculty of Aeronautics, TU of Košice, Slovakia
ABSTRACT: The architecture of wide-band synchronous automated systems of data communication (ASDC)
enables utilization of high accuracy measurement of the signal reception times to relative navigation in the
network. In this contribution an operation analysis of such systems in the relative navigation mode is
conducted. The accuracy and possibilities of the utilization of these systems were examined by modeling in air
navigation.
1 INTRODUCTION
Wide-band ASDC, utilizing the TDMA (Time
Division Multiple Acces) technique feature high
capacity of transferring digitalized and coded data.
Each user has a “synchronized clock” and is alocated
a necessary portion of throughput of the system in
accordance with the task under solution. In the
instances of time alocated for transmission the user
is “transmitting data”, which can be used by the rest
of the users. Digital processing of the signal enables
each user access to all kinds of information
transmitted by other users, while it is unimportant
for him to know who produced them.
Fig.1. The ASPD for relative navigation within the network of
users
Fig. 2. Clock time and time scale FO
This is achieved by filters with variable
parameters ensuring high accuracy of measuring
time of the signal when received from other users
and also make it possible to use the ASPD for
relative navigation (RELNAV) within the network
of users of the given system (Fig. 1, 2).Suppose
ASDC terminals are on board of flying objects (FO),
or non-flying objects (NFO). Adding the ASDC
board computer with a program module one can
perform RELNAV function and absolute navigation
without the need to complement the board of the
flying object by a new, extra piece of a navigation
equipment. This way, one can enhance the ASDC
capabilities and, on the basis of it, design unified
systems of data communication and navigation
[1, 2].
162
2 DEFINING THE TASK
Suppose we have a right angle coordinate system
XOY, wherein there are navigation points (NP)
marked with digits from 1 to N. For the sake of
simplification let us reduce the three-dimensional
task of determining the position of the FO to a two-
dimensional one. In that two-dimensional coordinate
system, let there be located N number of NPs with
known coordinates x
k
, y
k
, where k = 1, 2,....N and
the flying object FO with y, x coordinates is to be
measured (Figure 3).
Fig. 3. Determining position of a flying object
Difference R
i
can be obtained by expression
:
R
1
2
=(y
1
- y )
2
+ ( x
1
- x )
2
; (1)
R
2
2
= ( y
2
- y )
2
+ ( x
2
- x )
2
; (2)
R
3
2
= ( y
3
- y )
2
+ ( x
3
- x )
2
; (3)
Solving the equations (1-2) the y is expressed as
follows:
y = c.x + e, (4)
where
c = a/b; e = d/2.b; (5)
a = x
2
- x
1
; b = y
2
-
y
1
; (6)
d = - y1 + y2 - x1 + R21 - R22; (7)
Subsituting (4) into (3) we arrive at expression:
h.x
2
+ f.x + g = 0, (8)
where h = c
2
+ 1; f = 2.e.c - 2.y
3
.c - 2.x
3;
g = e
2
- 2.
y
3
.e + y
3
2
+ x
3
2
- R
3
2
; (9)
Then, by solving the equation (8), we obtain the
coordinates of the flying object (FO point) by the
following functions:
X
LO1
= (- f - ( D )
0,5
)/2.h; (10)
Y
LO1
= c.X
c1
+ e;
X
LO2
= (- f + ( D )
0,5
)/2.h (11)
Y
LO2
= c.X
c2
+ e;
where D
2
= f
2
- 4.h.g.
From expressions (10-11) it is evident that the
task of determining the coordinates of a FO is
ambiguous. As it follows from Figure 1, this
problem can be solved through proper choice of the
coordinate system and choosing the proper
coordinates of the FO (X
LOi
, Y
LOi
) on the basis of
known coordinates of NB and the distance R
i
.
3 THE INFLUENCE OF THE ACCURACY OF
MEASURING R
I
ON ACCURACY OF
DETERMINING THE POSITION OF THE FO
Expressions (10) and (11) can be used in modelling
the influence of inaccuracy of measuring the distance
dR
i
onto the accuracy of determining the position of
the FO which can be expressed using the quadratic
error of determining the position of the FO(SKLO)
S
r
. Values of dR
i
and S
r
is calculated the following
way:
dR
i
= R
i
- R*
i
S
r
=(dx
2
+dy
2
)
0.5
, (12)
where R*
i
stands for the measured position of the
FO,; dx, dy are errors of determining the coordinates
of the FO as a result of errors in measuring the
distance dR
i
.
Further I am going to present examples of
modelling the influence of dR
i
onto the accuracy of
determining the position of the FO. To perform
simulation of the function of the ASDC in RELNAV
mode a new program was developed in the Delphi
program environment. We expect that determining
the position will be effected in compliance with the
derived algorithms on a selected area of the Slovak
Republic. NP as FO and their coordinates on the
map are given in pixels.
Figure 4. illustrates the dependence of S
r
upon
dRi, for the case when dR
1
= dR
2
= dR
3
and these
coordinates are (Figure 3): x
1
= 31.0 pixels; y
1
=
= 457.0 pixels; x
2
= 11.0 pixels; y
2
= 230.0 pixels; x
3
= 28.0 pixels; y
3
= 1.0 pixel; x = 615.0 pixels;
y = 231.0 pixels.
163
Fig. 4. Dependence of SKLO upon dRi, for the case when
dR
1
= dR
2
= dR
3
This is a typical example of relative navigation of
FO within the tactical system of data transfer [1, 2].
Based on the graph it is obvious that by increasing
the inaccuracy in measuring the distance dR
i
the
error in determining the position is increasing.
Dependence of S
r
on dR
i
is almost linear and S
r
is
roughly equalling to 1,15.dR
i
. Modelling has proved
the theoretical premises about the fact that a highly
accurate measuring of the position of a FO makes it
inevitable to perform highly accurate measurement
of distance R
i
.
Fig. 5. Dependence of S
r
upon dR
i
, for the case when dR
1
≠
0,
dR
2
= dR
3
= 0
Figure 5 is illustrating the dependence of SKLO
upon dRi, for the case when dR1 ≠ 0, dR
2
= dR
3
= 0.
Figure 6 is revealing the dependence of SKLO upon
dR
i
, for the case when dR
1
= dR
3
≠
0, dR
2
= 0 and
these coordinates (Figure 3): x
1
= 31.0 pixels;
y
1
= 457.0 pixels; x
2
= 11.0 pixels; y
2
= 230.0 pixels;
x
3
= 28.0 pixels; y
3
= 1.0 pixel; x = 615.0 pixels;
y = 231.0 pixels.
Fig. 6. Dependence of SKLO upon dRi, for the case when
dR
1
= dR
3
≠
0, dR
2
= 0
This example is typical for relative navigation of
FO. Even in these cases it has been confirmed that if
want to be able to exactly determine the position of
FO, then we have to measure correctly all the
distances R
i
. Accuracy in determining the position is
also influenced by system geometry.
4 CONCLUSION
The method submitted enables modelling the
influence of errors of measuring distance of FO from
the NP exerted on the accuracy of determining its
position within the ASDC run in the mode of
relative navigation. Modelling has confirmed the
theoretical assumption stating that a highly accurate
determination of the FO position within the ASDC
run in relative navigation mode necessitates highly
accurate measuring of all distances of the FO from
the NP. The accuracy of determining the position of
the FO is influenced by the system geometry, too.
REFERENCES
[1] Džunda M.: “Accuracy of the relative navigation
in the automated data communication systems,” VII-th
International scientific and technical conference on sea
traffic engineering, Poland, Szczecin, 1997.
[2] Labun J.: Chyby merania výšky leteckých rádiovýškomerov.
Medzinárodná konferencia "Zvyšovanie bezpečnosti v
civilnom letectve - 2006". Žilina, 2006, s. 71-75.
