# Determination of GPS receiver position using

## Determination of GPS receiver position using

Determination of GPS receiver position using

Multivariate Newton-Raphson Technique for over specified case

B.Hari Kumar, Dr. K.Chennakesava Reddy*

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

*ECE Department, JNTU college of Engineering, Hyderabad, A.P.

E-mail: hari_kumarin@yahoo.com

Abstract:

GPS user can fix his position by obtaining pseudoranges from a minimum of four different GPS satellites. The user position in ECEF coordinates (xu, yu, zu) and the receiver clock bias with GPST tu can be obtained from the measured pseudoranges after making appropriate corrections. The receiver position can then be determined normally either by using linearization technique or method of least squares using Bancroft Algorithm. As it is easy to implement, most of the GPS receivers employ former method when a great degree of accuracy is not required. The latter method is found to be more accurate when pseudoranges from more than four satellites are considered. So far only Bancroft method has been suggested for the estimation of a GPS receiver position for the over specified case when more than four satellites are observed. Here we have used a new method namely, Multivariate Newton - Raphson Technique (MNRT) for the estimation of a user position. The accuracy is found to be better compared to linearization technique and is on par with the method of least squares using Bancroft algorithm for over specified case when pseudoranges from more than four satellites are considered. GPS data of Chitrakut station in RINEX (Receiver Independent Exchange) format has been used for this purpose.

Key words: Global Positioning System (GPS), Receiver Independent Exchange (RINEX) format, Satellite Vehicle (SV)

I. Introduction

The user estimates an apparent or pseudo range 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. Let the user be at xu, yu and zu in ECEF (Earth Centered Earth Fixed) coordinate system and the SVs be at xj, yj, zj, ( where j = 1,2,3,4) in the same coordinate system as the user. Assume that tu is the time offset between the user and SV. User’s position (in 3-D) and time offset are obtained by simultaneously solving the nonlinear equations.

; j =1, 2, 3, 4 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 position of the receiver accurately, all these errors have to be estimated and compensated for. Most of these errors can be estimated accurately and can be accounted 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, Bancroft algorithm and Mutivariate Newton – Raphson Technique. The results have shown that the accuracy of MNRT is better than the linearization method and is comparable to Bancroft algorithm.

II. Multivariate Newton - Raphson Technique

To determine the user position in three dimensions (xu, yu, zu) and the receiver clock offset tu, pseudorange measurements are made to four or more number of satellites (see Eq. 1). The resulting equations can be written as a function of user coordinates and clock offset

ρj = fj (xu, yu, zu, tu) , j =1,2,3 ---, M ….. (2)

where M is the number of observations made.

The above set of nonlinear equations can be written as

Where the vector ‘x’ is given by ……. (3)

The derivatives of the above functions can be written as

; j =1, 2, 3, 4

Alternatively the above equation can be written as

……. (4)

We can discretize this as

…. (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

This system can be written in matrix form as:

……. (6)

where is called the residual vector at the kth iteration and is defined as

……. (7)

Where is called the Jacobian matrix at the kth iteration and is defined as

……. (

and ……. (9)

and the new guess for x is

……. (10)

If M is the number of observations made at a given time,

J will be Mx4 matrix. R will be Mx1 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 the required accuracy is obtained.

III. Results and Discussion

For the determination of user position, data from Chitrakut station in RINEX format is considered [6]. Programs have been written in MATLAB to sort the ephemeris data into matrix format, to find the satellite position in ECEF coordinates and for the determination of receiver position in ECEF coordinates for both the methods. 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 accounted for. The results are as follows:

Sv no Az

(deg) El

(deg) Sv clock+ relativistic(m) Iono delay(m) Tropo

Delay

(m)

3 89.75 46.29 19048.06 1.9858 3.31

13 315 53.02 9807.00 1.8118 2.996

16 45 21.03 6064.55 3.1902 6.632

19 135 39.19 -7308.55 2.2292 3.785

20 180 25.42 -10893.93 2.9077 5.555

23 75.96 83.30 46843.87 1.5062 2.412

27 296.5 23.79 8954.52 3.0088 5.909

Table1. Estimation of GPS errors

Exact position of the user obtained by a high precision

GPS receiver is Xu = 918074.1038m,

Yu= 5703773.5389 and Zu =2693918.9285m.

User position by linearization technique:

Xu= 918050.65m;

Yu= 5703751.91m; and Zu = 2693899.70m.

User position by Bancroft algorithm:

Xu = 918075.38m;

Yu = 5703776.40m; and Zu = 2693918.73m.

User position by MNRT

Xu = 918075.35 m;

Yu = 5703776.43 m; and Zu = 2.693918.74 m

IV. Conclusion

It can be seen from the above results that the Multivariate Newton – Raphson Technique is more accurate compared to linearization technique and is comparable to Bancroft algorithm for computation of user position when data from additional satellites is 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.

6. http://home.iitk.ac.in/~ramesh/gps/gpsdata/gpsdata.htm

B. Hari Kumar was born in July 1962 at Ongole, Andhra Pradesh. He obtained B.E. from Andhra University in 1983 and M.E. from Osmania University in 1986 with Microwave and Radar Engineering as Specialization. Presently he is working as Associate Professor in MVSR Engineering College, Hyderabad. He has presented about 15 papers in several International and National conferences. He has 22 years of teaching experience. His present areas of interests include LAAS, GPS and Pseudolites, He is a member of IEEE and a Life Member of ISTE.

Dr..K.Chennakesava Reddy is working as a professor in ECE Department of JNTU College of Engineering, Hyderabad. He obtained B.E. from Osmania University, M.Tech. from REC, Warangal and Ph.D from JNTU. He is specialized in Power Electronics. He has 23 years of teaching experience

Multivariate Newton-Raphson Technique for over specified case

B.Hari Kumar, Dr. K.Chennakesava Reddy*

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

*ECE Department, JNTU college of Engineering, Hyderabad, A.P.

E-mail: hari_kumarin@yahoo.com

Abstract:

GPS user can fix his position by obtaining pseudoranges from a minimum of four different GPS satellites. The user position in ECEF coordinates (xu, yu, zu) and the receiver clock bias with GPST tu can be obtained from the measured pseudoranges after making appropriate corrections. The receiver position can then be determined normally either by using linearization technique or method of least squares using Bancroft Algorithm. As it is easy to implement, most of the GPS receivers employ former method when a great degree of accuracy is not required. The latter method is found to be more accurate when pseudoranges from more than four satellites are considered. So far only Bancroft method has been suggested for the estimation of a GPS receiver position for the over specified case when more than four satellites are observed. Here we have used a new method namely, Multivariate Newton - Raphson Technique (MNRT) for the estimation of a user position. The accuracy is found to be better compared to linearization technique and is on par with the method of least squares using Bancroft algorithm for over specified case when pseudoranges from more than four satellites are considered. GPS data of Chitrakut station in RINEX (Receiver Independent Exchange) format has been used for this purpose.

Key words: Global Positioning System (GPS), Receiver Independent Exchange (RINEX) format, Satellite Vehicle (SV)

I. Introduction

The user estimates an apparent or pseudo range 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. Let the user be at xu, yu and zu in ECEF (Earth Centered Earth Fixed) coordinate system and the SVs be at xj, yj, zj, ( where j = 1,2,3,4) in the same coordinate system as the user. Assume that tu is the time offset between the user and SV. User’s position (in 3-D) and time offset are obtained by simultaneously solving the nonlinear equations.

; j =1, 2, 3, 4 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 position of the receiver accurately, all these errors have to be estimated and compensated for. Most of these errors can be estimated accurately and can be accounted 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, Bancroft algorithm and Mutivariate Newton – Raphson Technique. The results have shown that the accuracy of MNRT is better than the linearization method and is comparable to Bancroft algorithm.

II. Multivariate Newton - Raphson Technique

To determine the user position in three dimensions (xu, yu, zu) and the receiver clock offset tu, pseudorange measurements are made to four or more number of satellites (see Eq. 1). The resulting equations can be written as a function of user coordinates and clock offset

ρj = fj (xu, yu, zu, tu) , j =1,2,3 ---, M ….. (2)

where M is the number of observations made.

The above set of nonlinear equations can be written as

Where the vector ‘x’ is given by ……. (3)

The derivatives of the above functions can be written as

; j =1, 2, 3, 4

Alternatively the above equation can be written as

……. (4)

We can discretize this as

…. (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

This system can be written in matrix form as:

……. (6)

where is called the residual vector at the kth iteration and is defined as

……. (7)

Where is called the Jacobian matrix at the kth iteration and is defined as

……. (

and ……. (9)

and the new guess for x is

……. (10)

If M is the number of observations made at a given time,

J will be Mx4 matrix. R will be Mx1 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 the required accuracy is obtained.

III. Results and Discussion

For the determination of user position, data from Chitrakut station in RINEX format is considered [6]. Programs have been written in MATLAB to sort the ephemeris data into matrix format, to find the satellite position in ECEF coordinates and for the determination of receiver position in ECEF coordinates for both the methods. 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 accounted for. The results are as follows:

Sv no Az

(deg) El

(deg) Sv clock+ relativistic(m) Iono delay(m) Tropo

Delay

(m)

3 89.75 46.29 19048.06 1.9858 3.31

13 315 53.02 9807.00 1.8118 2.996

16 45 21.03 6064.55 3.1902 6.632

19 135 39.19 -7308.55 2.2292 3.785

20 180 25.42 -10893.93 2.9077 5.555

23 75.96 83.30 46843.87 1.5062 2.412

27 296.5 23.79 8954.52 3.0088 5.909

Table1. Estimation of GPS errors

Exact position of the user obtained by a high precision

GPS receiver is Xu = 918074.1038m,

Yu= 5703773.5389 and Zu =2693918.9285m.

User position by linearization technique:

Xu= 918050.65m;

Yu= 5703751.91m; and Zu = 2693899.70m.

User position by Bancroft algorithm:

Xu = 918075.38m;

Yu = 5703776.40m; and Zu = 2693918.73m.

User position by MNRT

Xu = 918075.35 m;

Yu = 5703776.43 m; and Zu = 2.693918.74 m

IV. Conclusion

It can be seen from the above results that the Multivariate Newton – Raphson Technique is more accurate compared to linearization technique and is comparable to Bancroft algorithm for computation of user position when data from additional satellites is 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.

6. http://home.iitk.ac.in/~ramesh/gps/gpsdata/gpsdata.htm

B. Hari Kumar was born in July 1962 at Ongole, Andhra Pradesh. He obtained B.E. from Andhra University in 1983 and M.E. from Osmania University in 1986 with Microwave and Radar Engineering as Specialization. Presently he is working as Associate Professor in MVSR Engineering College, Hyderabad. He has presented about 15 papers in several International and National conferences. He has 22 years of teaching experience. His present areas of interests include LAAS, GPS and Pseudolites, He is a member of IEEE and a Life Member of ISTE.

Dr..K.Chennakesava Reddy is working as a professor in ECE Department of JNTU College of Engineering, Hyderabad. He obtained B.E. from Osmania University, M.Tech. from REC, Warangal and Ph.D from JNTU. He is specialized in Power Electronics. He has 23 years of teaching experience

**MEM**- عدد الرسائل : 70

تاريخ التسجيل : 11/04/2008

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