International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 1
Number 4
December 2007
365
Availability of the Certain Value of Position
Error in the Navigational Systems – Model
Application
C. Specht
Polish Naval Academy of Gdynia, Poland
ABSTRACT: The distributions and parameters of the random variable (density functions, distribution
functions, expected values, variances, moments etc.), which in navigation is represented by position error, are
well defined. This approach is similar like in geodesy, where passing time hasn’t any impact on statistical
solution. Presented paper evaluates alternative approach to position error analyses, where the random values
are: working and failures times of the system. Arbitrary accepted certain value of position error was assumed
as a decision criterion of the system for working state definition. The general mathematical model of the
availability of certain value of position error is presented and the special case – exponential distributions of
lifetimes and failure times model – was also analysed. The model was adopted for positioning based on
EGNOS system measurements and some results are present.
1 INTRODUCTION
The parametric assessment of navigation systems
during the last decade has been the most common
way of their classification with regard to their
quality. Within the scope of this evaluation critical
space is provided. This space is very closely related
to the navigation requirements set for its various
forms. The comparable criteria of navigation
systems were often discussed in world literature as
well as in Polish publications. The analysis of the
criteria allows distinguishing three main groups,
which are identical with particular phases of
positioning systems development over the years and
they are as follows: positioning, reliability and the
safety of exploitation criteria.
1 Positioning criteria - system characteristics in
quality of position fixing. They have in their
scope 3 types of accuracy (predictable, repeatable,
relative) as well as fix rate and dimension,
system: capacity, ambiguity, and coverage.
2 Reliability criteria – they form a separate group of
indicators with reference to characteristics of
systems exploitation. Reliability, availability and
continuity are among them.
3 The safety of exploitation criteria – their function
is to give the user current information about the
quality (status) of operating system allowing for
the proper level of their utility. So far, integrity,
the only criterion belonging to this group, has
been characterised by a wide range of variables
such as: time to alarm, the probability of false
alarm etc. [Ober P. B., 1999].
So far perceiving of those two groups
(positioning and reliability) has differentiated them
due to the methods of statistical and probabilistic
inference. The classic approach to analogous
estimation of a system is characterized by the
following assumptions: the determining error is a
random variable, it does not respect playtimes of the
system work and also the lack of estimation of the
reliability of the system. In this case the new
366
probability characteristic, which joins the accuracy
and one of the reliability criterions – availability of
the certain value of position error could be
introduced. Let’s define the availability of certain
value of position error – as a probability that in any
moment of time
( )
t
the position error of determining
coordinates
( )
n
δ
is lower or equal then the arbitrary
acceptable value
( )
U
, which mean than
U
n
≤
δ
,
,...2,1=n
. The suggested approach treats the lifetimes
and the times of failure as the random variables
being in relation in the fixed value of the position
error and also it introduces the measures which
making the reliability estimation possible.
2 WORKING PROCESS OF THE POSITIONING
SYSTEM
Let the position error of any navigational system, in
the function of time, be a variable taking its values
from the given interval of errors
),0 ∞
. Assume that
the process of determining the coordinates is
alternating with the renew in general reliability
theory sense. Then we can recognize two states: the
working one – the state where the error
for
,...2,1=n
and the state of failure where
U
n
>
δ
. Let
,...,
21
XX
be the working times while
,...,
21
YY
are the
times of failures. Hence the moments:
nnn
XYYXYXZ ++++++=
−12211
'
...
,
,...,2,1=n
are the moments of failures and
nnn
YZZ +=
'''
, are the
moments of renewal. Assume also that the random
variables
ii
YX ,
,
,...2,1=i
are independent and the
working failure times have the same distributions.
Let’s define the analytical form of the distributions
of the variables
n
X
and
n
Y
as
( ) ( )
yFxXP
i
=≤
, (1)
( ) ( )
yGyYP
i
=≤
for
,...,2,1=i
(2)
where:
( )
xF
,
( )
yG
means the distribution functions
of
n
X
and
n
Y
.
Introduce also the notations of the expected value
and the variance as follows
( ) ( )
XEXE
i
=
,
( ) ( )
YEYE
i
=
(3)
and
( )
2
1
σ
=
i
XV
,
( )
2
2
σ
=
i
YV
,
,....2,1=i
, (4)
where:
( )
i
XE
,
( )
i
YE
- the expected values of the
working and failure times. We have also to admit
that
0
2
2
2
1
>+
σσ
.
3 GENERAL MODEL
Let’s define the reliability process in which the
relation between the single measurement error
n
δ
and the parameter
U
decide about its state (work
or failure). Let
( )
t
α
be the binary interpretation of
the reliability state of the process as:
( )
<≤
<≤
=
++
+
''
1
'
1
'
1
''
,0
,1
nn
nn
ZtZ
ZtZ
t
α
for
,...1,0=n
(5)
The state
1)( =t
α
means that in the moment
t
the error of the single measurement is less or equal
than
U
. In the opposite case for
U
n
>
δ
, the system
is in the state of failure. The availability of a certain
value of position error will be denoted as
[ ]
UtPtD ≤= )()(
δ
. (6)
Let us consider the following sequence of
events such that
{ }
'
1
''
+
<≤=
nnn
ZtZV
,
...2,1,0=n
. (7)
It means that at
t
moment
Ut ≤)(
δ
and up to
t
moment exactly
n
renewals took place. As the
events
n
VVV ,...,,
10
are pairwise mutually exclusive
then:
[ ]
∑
∞
=
∞
=
=
=≤
0
0
)()(
n
n
n
n
VPVPUtP
δ
. (8)
To define the value of
∑
∞
=0
)(
n
n
VP
we introduce the
additional notations:
nn
X
XXS +++= ...
21
'
,
( )
( )
xFxSP
nn
=≤
'
(9)
and
nn
YYYS +++= ...
21
''
,
( )
( )
yGySP
nn
=≤
''
. (10)
The variables
'
n
S
and
''
n
S
corresponds to
cumulative times of work and failure of the process
of determining the coordinates of position.
367
The distribution functions
( )
xF
n
and
( )
yG
n
could
be found by the n-times convolution operation. For
any
n
we get the final form [Barlow R. E., Proshan
F. 1975] as
( ) ( ) ( )
∫
−=
−
t
nn
xdFxtFtF
0
1
(11)
( ) ( ) ( )
∫
−=
−
t
nn
ydGytGtG
0
1
,
...3,2=n
, (12)
As
'''''
nnn
SSZ
+=
so
( )
( )
( ) ( )
∫
−=≤=Φ
t
nnnn
udGutFtZPt
0
''
,
,...2,1=n
,(13)
where
( )
t
n
Φ
is the distribution function of
''
n
Z
.
To determine
)(tD
let’s compute the probabilities
( )
0
VP
and
( )
∑
∞
=1n
n
VP
separately. As for
0≥t
{ }
{ }
1
'
1
''
00
0 XtZtZV
n
<≤=<≤=
+
(14)
hence
( ) ( ) ( ) ( )
tFtXPtXPVP −=≤−=>= 11
110
, (15)
however
( )
n
VP
,
...2,1=n
we find from the formula
of total probability
( )
( ) ( )
[
)
[ ]
( )
∫
−>⋅+∈=
=+<≤=<≤=
++
t
n
nnnnnn
xtXPdxxxZP
XZtZPZtZPVP
0
11
,
''
''''''
. (16)
Then
( ) ( ) ( )
[ ]
( )
[ ]
( )
∫∫
Φ−−=−−Φ=
t
n
t
nn
xdxtFxtFxdVP
00
11
. (17)
The availability is calculated as a sum
probabilities of the mutually exclusive events
[ ]
( )
[ ]
( ) ( ) ( )
∑∑
∞
=
∞
=
+==
===≤
=
1
0
0
1)()(
n
n
n
n
VPVPVP
tP
UtPtD
αδ
. (18)
Substituting (15) and (17) to (18) we obtain final
form for availability of the certain value of position
error as follows
( ) ( ) ( )
[ ]
( )
∫
Φ
−−+−=
t
xdHxtFtFtD
0
11
, (19)
where
( ) ( )
∑
∞
=
Φ
Φ=
1n
n
xxH
(20)
is a renewal function of stream made of the renewal
moments.
4 EXPONENTIAL MODEL
Typical realizations of the operating time in
navigational systems are characterized by the
exponential distributions of the lifetime and the time
of failures due to the property called the
”memoryless” property. Let define the exponential
process where the distribution functions as
( )
≤
>−
=
−
0for0
0for1
t
te
tF
t
λ
, (21)
( )
≤
>−
=
−
0for0
0for1
t
te
tG
t
µ
, (22)
where
λ
,
µ
are failure and renewal rates. When
substituting (21) and (22) to (19) we obtain
( ) ( ) ( )
[ ]
( )
( )
( )
[ ]
( )
, 11
11
0
0
exp
∫
∫
Φ
−−−
Φ
−−+=
=−−+−=
t
xtt
t
xdHee
xdHxtFtFtD
λλ
(23)
where
( )
tD
exp
denotes the availability of the certain
value of position error in the navigational system in
the case of the exponential life and failure times
distributions. After few simple transformations
[Specht C., 2003] finally form could be find as
( )
( )
t
etD
µλ
µλ
λ
µλ
µ
+−
+
+
+
=
exp
(24)
5 EXPERIMENT
The measurement campaign focuses on methodology
verification. Two-weeks registration was done based
on EGNOS augmentation system. More than 2
368
millions fixes were used for availability of the
certain value of position error calculations (fig. 1).
Fig. 1. Scatter plot for EGNOS system (500.000 fixes, values in
meters)
For establishing availability functions, three
decision limits were fixed:
m1<
δ
,
m2<
δ
,
m3<
δ
.
Fig. 2. Availability of the certain value of position error
functions for:
m
1<
δ
(red),
m
2<
δ
(blue),
m
3<
δ
(brown)
with limited values
6 CONCLUSIONS
The article presents the mathematical model of
availability of the certain value of position error
calculation. The classic approach to analogous
estimation is characterized by the following
assumptions: the determining error is a random
variable, it does not respect playtimes of the system
work and new one suggested approach treats the
lifetimes and the times of failure as the random
variables being in relation with the fixed value of the
position error. General and also exponential
statistical models were presented. The model were
also verified based on EGNOS system measurements.
ACKNOWLEDGEMENTS
Polish State Committee for Scientific Reseach has
supported this work. Grant No 4 T12C 064 27.
REFERENCES
Barlow R. E., Proshan F. 1975. Statistical Theory of Reliability
and Life Testing, Holt, Rinehart and Winston, inc., USA.
Global Positioning System Standard Positioning Service,
Performance Standard, Assistant Secretary of Defense,
2001.
Hofmann-Wellenhof B., Legat K., Wieser M. 2003.
Navigation: Principles of Positioning and Guidance,
Sprinter-Verlag, Vien.
Ober P.B. 1999. Designing Integrity into Position Estimation,
Proceedings of the 3
rd
European Symposium on Global
Navigation Satellite Systems – GNSS’99, Genova, Italy.
Specht C. 2003. Availability, Reliability nad Contiunuity
Model of Differential GPS Transmission, Polish Academy
of Sciences, Annual of Navigation no 5.