# 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 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 error, propagation error in the form of ionospheric and tropospheric delays, satellite and receiver clock biases with respect to GPST, 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 technique, Method of least squares using Bancroft algorithm and also by the proposed Mutivariate Newton--Raphson Technique. The results show that the accuracy of MNRT 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]

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 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 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 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, 74(18): 4487-4499.

[5] Strang, G. and Borre, K., "Linear Algebra, Geodesy, 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 College of Engineering, JNT University, Jagityala, Karimnagar District, Andhra Pradesh

N. Namassivaya

Associate Professor, ECE Department, MVSR Engineering College, Osmania University, Hyderabad, Andhra Pradesh
```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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