International Journal of Engineering Trends and Technology- Volume2Issue2- 2011
GPS User Position Using
Extended Linearization Technique
B.Hari Kumar #1, Dr. Dixit #2, N.Namassivaya #3, G.V.Chalapathi Rao #4
Professor & Head, ECE Department, M.V.S.R.Engineering College, Hyderabad, A.P., India.
Professor, ECE Department, Allahabad University, Allahabad, U.P., India
Associate Professor, ECE Department, M.V.S.R.Engineering College, Hyderabad, A.P., India
Assistant Professor, ECE Department, M.V.S.R.Engineering College, Hyderabad, A.P., India
Abstract - The position of a GPS receiver can be determined by
are observed. GPS data of Chitrakut station in RINEX format has
obtaining pseudoranges from a minimum of four different GPS
been used for this purpose.
satellites. The measured ranges do not represent the true ranges as
the signal coming from a GPS satellite will be contaminated by
Keywords - Earth Centered Earth Fixed (ECEF), Global
various errors like Ephemeris error; Propagation error in the form
Positioning
of Ionospheric and Tropospheric delays; Satellite and Receiver
Independent Exchange (RINEX).
System
(GPS),
GPS
Time
(GPST),
Receiver
clock biases with respect to GPS Time (GPST); Multipath error
etc. Most of these errors can be estimated accurately and can be
I. INTRODUCTION
accounted for. After making necessary corrections to the observed
The Global Positioning System is all weather, space based
pseudoranges the receiver position in ECEF coordinates (xu, yu, zu)
navigation system. It is a constellation of a minimum of 24
and the receiver clock bias with GPS Time (GPST) tu can be
determined either by using Linearization Technique [1] or Method
of least squares using Bancroft Algorithm [2]. As it is easy to
implement, most of the GPS receivers employ former method for
satellites in near circular orbits, positioned at an approximate
height of 20,150 km. above the earth. The satellites travel with a
velocity of 3.9 km/sec with an orbital period of 11 hours 58
fixing the user position when a great degree of accuracy is not
minutes. The satellites transmit C/A (Coarse/Acquisition) and
required. The latter method is found to be more accurate when
P (Precision) codes, Navigation message, Clock parameters etc.
pseudoranges from more than four satellites are considered [3]. So
at
far only Bancroft method has been suggested for an accurate
L2 (1227.60 MHz). These signals can be received and processed
estimation of a GPS receiver position when more than four
to obtain the user position in 3-D (latitude, longitude and
satellites are observed. In this paper we are proposing a new
altitude), velocity and time accurately.
technique namely, Extended Linearization Technique (ELT) for
the estimation of an user position. Results show that the accuracy
obtained using ELT is better compared to the Linearization
Technique and is on par with the Bancroft algorithm for over
specified cases when pseudoranges from more than four satellites
ISSN: 2231-5381
two
frequencies
namely L1
(1575.42 MHz)
and
The user estimates an apparent or pseudorange to each SV
(Satellite Vehicle) by measuring the transit time of the signal.
The measured ranges do not represent the true ranges as the
signal coming from a GPS satellite will be contaminated by
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International Journal of Engineering Trends and Technology- Volume2Issue2- 2011
various errors. Most of these errors can be estimated accurately
technique makes use of only four observables at a given epoch
and can be accounted for. After making necessary corrections to
time, this proposed technique makes use of all the observations
the observed pseudoranges the receiver position in ECEF
thereby accuracy in determining the position of a GPS user is
coordinates (xu, yu, zu) and the receiver clock bias with GPS
improved. To determine the user position in three dimensions
Time (GPST) tu can then be determined. If the unknown
(xu, yu, zu) and the receiver clock offset tu, pseudoranges are to
coordinates of the user position are represented by xu, yu and
be obtained from a minimum of four satellites.
zu and the known positions of Satellite Vehicles are with xj, yj,
zj, ( where j = 1,2,3,4) in ECEF coordinate system, the user
ρj 
x
 x u   y j  y u   z j  z u   ct u ;
2
j
2
j = 1,2,3……,m.
position (in 3-D) and time offset ‘tu‘ are obtained by
simultaneously solving the nonlinear equations given below.
ρj 
x
 x u   y j  y u   z j  z u   ct u ;
2
j
2
j = 1,2,3,4.
2
(1)
Where ‘c’ is the free space velocity of electromagnetic wave in
m/s.
2
(2)
Where m is the number of satellites observed.
The resulting equations can be written as a function of user
coordinates and clock offset as
 j  f  xu , y u , z u , t u 
(3)
Using an approximate position location ( x̂ u , ŷ u , ẑ u ) and time
The measured ranges do not represent true ranges as the signal
bias estimate t̂ u , an approximate pseudorange can be calculated
coming from a satellite is affected by various errors like
ˆ j 
x
j
 xˆ u

2
  y j  yˆ u

2
 z j  zˆ u

2
 c tˆu ;
ephemeris error, propagation error in the form of ionospheric
j =1, 2, 3, -------, m
and tropospheric delays, satellite and receiver clock biases with
respect to GPST, multipath error etc. In order to determine the
(4)
= f ( xˆ u , yˆ u , zˆ u , tˆu )
receiver position accurately, all these errors have to be estimated
The unknown user position and receiver clock offset are
and compensated for. In this paper, the ionospheric delay is
considered to consist of an approximate component and an
estimated using Klobuchar model [4]. Hopfield model has been
incremental component as stated below.
used
for the estimation of tropospheric delay [5]. Satellite
x u  xˆ u  x u
clock bias and the relativistic effects also have been estimated
y u  yˆ u  y u
and accounted for. Finally the user position is estimated using
the Linearization technique, Method of least squares using
Bancroft algorithm and also by the proposed Linearization
Technique. The results show that ELT is more accurate in
determining the user position than the linearization method and
is comparable to Bancroft algorithm.
II. EXTENDED LINEARIZATION TECHNIQUE (ELT)
z u  zˆ u  z u
t u  tˆu  t u
This allows writing Eq. (3) as
f xu , yu , zu ,tu   f xˆu  xu , yˆu  yu , zˆu  zu ,tˆu  tu 
This latter function can be expanded about the approximate
point using a Taylor series. It can be shown that
If more than four GPS satellites are observed at a given epoch
time, using this proposed Extended Linearization Technique a
(5)
ˆ j   j 
x j  xˆ u
rˆj
xu 
y j  yˆ u
rˆj
yu 
z j  zˆu
rˆj
zu  ct u
(6)
better accuracy can be obtained. Where as the Linearization
ISSN: 2231-5381
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International Journal of Engineering Trends and Technology- Volume2Issue2- 2011
x
Where rˆj 
This procedure is to be repeated for about 50 times until the
 xˆu    y j  yˆ u   z j  zˆu 
2
j
2
2
required accuracy is obtained.
 j  ˆ j   j
a xj 
x j  xˆ u
rˆj
; a yj 
(7)
y j  yˆ u
rˆj
; a zj 
III. RESULTS AND CONCLUSIONS
RINEX data from Chitrakut station (Near IIT Kanpur) is used
z j  zˆ u
(8)
rˆj
for this purpose [7]. The observation data of 3rd January 2006 at
Where axj, ayj and azj terms denote the direction cosines of the
(0 hrs.
unit vector pointing from the approximate user position to the jth
(SV PRN. Nos. 3 13 16 19 20 23 27) are observed at the
satellite.
epoch time. Algorithms have been implemented to sort out the
Rewriting Equation (6) results into
ephemeris data into matrix format and for the determination of
 j  a xj x u  a yj y u  a zj z u  ct u
(9)
0 min. 30 sec.) have been used. Seven satellites
satellites’ position at the epoch time [6]. By using clock
correction parameters which are available as part of the
When pseudorange measurements are made to ‘m’ satellites
(j = m), Equation (9) can be represented in matrix form as
 1 
  
 2
   3  will be m x 1 matrix;


....... 
  
 m
 x u 


y u 
x = 
 z u 


 ct u 
Navigation message, the satellite clock bias and error due to
relativistic effect have been obtained. The Ionospheric delay has
been estimated using Kloubachar model. All the eight
coefficients for the implementation of Kloubachar model are
available as part of Navigation message. The Tropospheric
delay has been estimated using Hopfield method. The estimated
errors and the corrected ranges have been represented in
Table 1. The receiver position is then determined using the
Linearization technique, Bancroft algorithm and also by the
will be 4 x 1 matrix
and
proposed ELT. All the calculations have been carried out by
writing programs in MATLAB. The results are summarized
below. The (x, y, z) positions in meters of the seven observed
a x1 a y1 a z1 1 


a x 2 a y 2 a z 2 1 
H  a x 3 a y 3 a z 3 1  will be m x 4 matrix.


..... ...... ...... ..... 


1
a xm a ym a zm
satellites with PRN nos. of 3 13 16 19 20 23 27 at 0 hours, 0
minutes and 30 seconds of 3rd January 2006 are found to be
1.0e+007 *
[-1.14435581932368 2.18537228998174 0.92840515634504
0.88498653721608 1.52115049991917 1.98379922835602
We can obtain error matrix x from the following equation
x  H 1 
(10)
As H is not a square matrix its inverse can be obtained using
Inv (H) = ( Inv ( H  * H ) * H  )
(11)
-1.28799462471086 0.84279115293681 2.17295977908060
-0.62238562333828 2.55024173739922 -0.38396284978272
1.04260459627803 2.18281560737286 -1.10756652807472
0.15114376130666 2.36981504953570 1.16498729017268
1.98311575365209 0.65606228041700 1.72062794938024]
where H  is the transpose of H.
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International Journal of Engineering Trends and Technology- Volume2Issue2- 2011
TABLE 1
ESTIMATION OF GPS ERRORS AND CORRECTION OF PSEUDORANGES
Sv.
no
Azimu
th
(deg)
Elevati
on
(deg)
Observed
Pseudoranges
(m)
Sv. clock+
relativistic
(m)
Iono
delay
(m)
Tropo
Delay
(m)
Corrected
Pseudoranges
(m)
3
89.75
46.29
21345372.96948
19048.06
1.9858
3.31
21364414.7719640
13
315
53.02
21123433.31848
9807.00
1.8118
2.996
21133235.1936572
16
45
21.03
23647148.85446
6064.55
3.1902
6.632
23653202.7278275
19
135
39.19
22030908.95548
-7308.55
2.2292
3.785
22023593.4330683
20
180
25.42
23234206.55447
-10893.93
2.9077
5.555
23223303.5998171
23
75.96
83.30
20047262.99349
46843.87
1.5062
2.412
20094101.9563987
27
296.5
23.79
23831204.72647
8954.52
3.0088
5.909
23840149.6080318
Using the corrected pseudoranges user position is determined and
[2] Bancroft. S., “An algebraic solution of the GPS equations”,
the results are shown below:
IEEE Transactions on Aerospace and Electronic Systems
Exact User Position as per the observation data:
Vol. 21 (1985) pp: 56–59.
Xu = 918074.1038m, Yu= 5703773.539 and Zu =2693918.9285m.
[3] B.Hari Kumar and K.Chennakesava Reddy, “Determination
User position by Linearization Technique:
Of GPS Receiver Position Using Linearization Technique
Xu= 918050.65m, Yu= 5703751.91m and Zu = 2693899.70m.
And
Bancroft
Algorithm”,
IETECH
User position by Bancroft Algorithm:
Communication
Techniques,
Volume-2,
Xu = 918075.38m, Yu = 5703776.40m and Zu = 2693918.73m.
pp-14-16, 2008
User position using Extended Linearization Technique
Journal
of
Number-1,
[4] Klobuchar J, “ Design and characteristics of the GPS
Xu = 918075.72 m, Yu = 5703777.12 m and Zu = 2693918.91 m
ionospheric time – delay algorithm for single frequency
users”, Proceedings of PLANS’86 – Position Location and
Results show that the proposed method namely the Extended
Navigation Symposium, Las Vegas, Nevada, November
Linearization
4-7, pp: 280-286.
Technique
is
more
accurate
compared
to
Linearization Technique and is comparable to Bancroft algorithm
[5] Hopfield HS, “Two – quartic tropospheric refractivity
in determination of the user position when data from more than
profile for correcting satellite data”, Journal of Geophysical
four satellites are taken into account.
research, Vol. 74, No. 18, pp: 4487-4499, 1969.
[6] Strang, G. and Borre, K., “Linear Algebra, Geodesy, and
REFERENCES
[1] Kaplan, E. D., "Understanding GPS: Principles and
GPS”,Wellesley-Cambridge, Wellesley, MA, 1997.
[7] http://home.iitk.ac.in/~ramesh/gps/gpsdata/gpsdata.html
Applications", Artech House, 1996.
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