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Determination of GPS receiver position using Multivariate Newton-Raphson Technique for over specified cases.

Introduction

The user estimates an apparent or pseudorange to each SV (Satellite Vehicle) by measuring the transit time of the signal. Using the pseudo ranges, user position in 3-D (latitude, longitude and height) and the time offset between the transmitter and receiver clock can be estimated. If the unknown coordinates of the user position are represented by [x.sub.u], [y.sub.u] and [z.sub.u] and the known positions of Satellite Vehicles are with [x.sub.j], [y.sub.j], [z.sub.j], (where j = 1,2,3,4) in ECEF ECEF Earth Centered Earth Fixed
ECEF Earliest Completing Edge First (algorithm) 
 coordinate system, the user position (in 3-D) and time offset '[t.sub.u]' are obtained by simultaneously solving the nonlinear equations given below.

[[rho].sub.j] = [square root of [([x.sub.j] - [x.sub.u]).sup.2] + [([y.sub.j] - [y.sub.u]).sup.2] + [([z.sub.j] - [z.sub.u]).sup.2] + [ct.sub.u]; j = 1,2,3,etc. (1)

Where 'c' is the free space velocity of electromagnetic signals in m/s.

The measured ranges do not represent true ranges as the signal coming from a satellite is contaminated by various errors like ephemeris ephemeris (ĭfĕm`ərĭs) (pl., ephemerides), table listing the position of one or more celestial bodies for each day of the year.  error, propagation error in the form of ionospheric and tropospheric delays, satellite and receiver clock biases with respect to GPST GPST GPS Time
GPST Global Positioning System Tester
GPST Granite Peak Ski Team (Wausau, WI)
GPST German Poop Shelf Toilet
, multipath error etc. In order to determine the receiver position accurately, all these errors have to be estimated and compensated for. In this paper, the ionospheric delay is estimated using Klobuchar model [3]. Hopfield model has been used for the estimation of tropospheric delay [4]. Satellite clock bias and the relativistic effects also have been estimated and accounted for. Finally the user position is estimated using the Linearization In mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential  technique, Method of least squares Noun 1. method of least squares - a method of fitting a curve to data points so as to minimize the sum of the squares of the distances of the points from the curve
least squares
 using Bancroft algorithm and also by the proposed Mutivariate Newton--Raphson Technique. The results show that the accuracy of MNRT MNRT Ministry of Natural Resources and Tourism (Tanzania)
MNRT Minister of State for Research and Technology
 is better than the linearization method and is comparable to Bancroft algorithm.

Multivariate Newton--Raphson Technique

To determine the user position in three dimensions ([x.sub.u], [y.sub.u], [z.sub.u]) and the receiver clock offset [t.sub.u], pseudorange measurements are to be made to four or more number of satellites (Eq. 1). The resulting equations can be written as a function of user coordinates and clock offset

[[rho].sub.j] = [f.sub.j] ([x.sub.u], [y.sub.u], [z.sub.u], [t.sub.u)], j = 1,2,3 ---, m (2)

where 'm' is the number of observations made.

The above set of nonlinear equations can be written as

[f.sub.j](x) = 0, j 1,2 ----m

Where the vector 'x' is given by x = [[x.sub.u], [y.sub.u], [z.sub.u], [t.sub.u] = [[x.sub.1], [x.sub.2], [x.sub.3], [x.sub.4]] (3)

The derivatives of the above functions can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ]

Alternatively the above equation can be written as

[df.sub.j] = [4.summation over (i=1)] ([partial derivative][f.sub.j]/[partial derivative][x.sub.i]] [dx.sub.i], ; j = 1, 2, ---, m and i = 1,2,3,4 (4)

We can discretize this equation as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Where 'j' is the index over functions, 'i' is the index over variables and the superscript in parentheses stands for the iteration.

The next iteration shall take us to the root, so we assume that f([{x}.sup.(2)]) = 0

This system can be written in matrix form as:

[J.sup.(k)] [delta][x.sup.(k)] = -[R.sup.(k)] (6)

where [R.sup.(k)] is called the residual vector at the [k.sup.th] iteration and is defined as [R.sup.(k)] = f ([{x}.sup.(k)] (7)

and [J.sup.(k)] is called the Jacobian matrix at the [k.sup.th] iteration and is defined as

[(J.sup.(k)]).sub.f,i] = ([partial derivative][f.sup.(k).sub.j]/[partial derivative][x.sup.(k).sub.i]) (8)

and [delta][x.sup.(k) = [x.sup.(k+1)] - [x.sup.(k)] (9)

The new guess for 'x' is

[x.sup.(k+1) = [x.sup.(k) + [delta][x.sup.(k)] (10)

If 'm' number of satellites are observed at a given time, Jacobian matrix J will be mx4 matrix, R will be mx1 matrix and [delta]x would be 4x1 matrix. As J is not a square matrix its inverse can be obtained as

Inv (J) = ( Inv ( J' * J ) * J' ), where J' is the transpose trans┬Ěpose
v.
To transfer one tissue, organ, or part to the place of another.
 of J.

The above procedure is to be repeated till sufficient accuracy is obtained.

Results and Discussion

For the determination of user position, data from Chitrakut station in RINEX RINEX Receiver Independent Exchange Format (GPS)  format is considered [6]. Observation data of 06 1 3 0 0 30.00 (January 3, 2006 at 0 hours, 0 minutes and 30 seconds) is taken. At this specified time seven satellites with PRN nos. of 3 13 16 19 20 23 27 have been observed. Ephemeredes of these satellites are obtained from the corresponding Navigation data.

Programs have been written to sort the ephemeredes data into matrix format; to find the satellites' positions in ECEF coordinates and for the estimation of various errors. Ionospheric delay is estimated using Klobuchar model. Hopfield model has been used for the estimation of tropospheric delay. Satellite clock bias and the relativistic effects also have been estimated and corrected [2]. The receiver position is then determined using the Lineaization technique, Bancroft algorithm and also by the proposed MNRT. The results are summarized below.

The positions of the seven observed satellites with PRN nos. of 3 13 16 19 20 23 27 at 0 hours, 0 minutes and 30 seconds of 3rd January 2006 respectively are
1.0e+007 *
[-1.14435581932368   2.18537228998174    0.92840515634504
  0.88498653721608   1.52115049991917    1.98379922835602
 -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]


Using the corrected pseudoranges user position is determined and the results are shown below:

User as per the observation data:

[X.sub.u] = 918074.1038m, [Y.sub.u] = 5703773.5389 and [Z.sub.u] = 2693918.9285m.

User position by linearization technique:

[X.sub.u] = 918050.65m, [Y.sub.u] = 5703751.91m and [Z.sub.u] = 2693899.70m.

User position by Bancroft algorithm:

[X.sub.u] = 918075.38m, [Y.sub.u] = 5703776.40m and [Z.sub.u] = 2693918.73m.

User position by MNRT

[X.sub.u] = 918075.35 m, [Y.sub.u] = 5703776.43m and [Z.sub.u] = 2693918.74 m

Conclusion

The results show that the proposed Multivariate Newton--Raphson Technique is more accurate compared to linearization technique and is comparable to Bancroft algorithm in determination of the user position when data from more than four satellites are taken into account.

References

[1] Bancroft. S., "An algebraic solution of the GPS equations", IEEE (Institute of Electrical and Electronics Engineers, New York, www.ieee.org) A membership organization that includes engineers, scientists and students in electronics and allied fields.  Transactions on Aerospace and Electronic Systems 21 (1985) 56-59.

[2] B. Hofmann Wellenhof, H.Lichtenegger & J.Collins, "GPS Theory and Practice", Springer-Verlag Wien, New York

[3] Klobuchar J, "Design and characteristics of the GPS ionospheric time--delay algorithm for single frequency users", Proceedings of PLANS'86--Position Location and Navigation Symposium, Las Vegas, Nevada, November 4-7, pp280-286.

[4] Hopfield HS, "Two--quartic tropospheric refractivity profile for correcting satellite data", Journal of Geophysical research Journal of Geophysical Research is a publication of the American Geophysical Union. JGR was formerly titled Terrestrial Magnetism from its founding by the AGU's president Louis A. , 74(18): 4487-4499.

[5] Strang, G. and Borre, K., "Linear Algebra, Geodesy geodesy (jēŏd`ĭsē) or geodetic surveying, theory and practice of determining the position of points on the earth's surface and the dimensions of areas so large that the curvature of the earth must be taken into , and GPS", Wellesley-Cambridge, Wellesley, MA, 1997. http://home.iitk.ac.in/~ramesh/gps/gpsdata/gpsdata.htm

B. Hari Kumar

Associate Professor, ECE Department, M.V.S.R. Engineering College, Hyderabad, A.P., India

E-mail: hari_kumarin@yahoo.com

K. Chennakesava Reddy

Principal, JNTU JNTU Jawaharlal Nehru Technological University (Hyderabad, India)  College of Engineering, JNT University, Jagityala, Karimnagar District, Andhra Pradesh

N. Namassivaya

Associate Professor, ECE Department, MVSR Engineering College, Osmania University, Hyderabad, Andhra Pradesh <includeonly> |Hyderabad, Andhra Pradesh]]

</includeonly> Coordinates:

Hyderabad pronunciation  or
 
Table 1 : Estimation of GPS errors

Sv.   Azimuth   Elevation   Observed           Sv. clock+
no    (deg)     (deg)       Pseudoranges (m)   relativistic (m)

3     89.75     46.29       21345372.96948     19048.06
13    315       53.02       21123433.31848     9807.00
16    45        21.03       23647148.85446     6064.55
19    135       39.19       22030908.95548     -7308.55
20    180       25.42       23234206.55447     -10893.93
23    75.96     83.30       20047262.99349     46843.87
27    296.5     23.79       23831204.72647     8954.52

Sv.   Iono        Tropo       Corrected
no    delay (m)   Delay (m)   Pseudoranges (m)

3     1.9858      3.31        21364414.7719640
13    1.8118      2.996       21133235.1936572
16    3.1902      6.632       23653202.7278275
19    2.2292      3.785       22023593.4330683
20    2.9077      5.555       23223303.5998171
23    1.5062      2.412       20094101.9563987
27    3.0088      5.909       23840149.6080318
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Author:Kumar, B. Hari; Reddy, K. Chennakesava; Namassivaya, N.
Publication:International Journal of Applied Engineering Research
Article Type:Report
Date:Nov 1, 2008
Words:1421
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