New Algorithm for GNSS Positioning Using System of Linear Equations

B. Oszczak

Abstract:	The GNSS standard positioning solution determines the coordinates of the GPS receiver and the receiver clock offset from measurements of at least four pseudo-ranges. The new algorithm, given in the paper, allows to calculate the coordinates of the GNSS receiver and the receiver clock error and does not require both linearisation of observation equations and use of the least squares method. As it is known, the method of point positioning with the code ranges (Hofmann-Wellenhof, 2008) consists in determining the coordinates of the GNSS receiver antenna based on the known coordinates of minimum of four satellites and the measured pseudoranges corrected for ionospheric and tropospheric refraction using algorithms given e.g. by Klobuchar (Klobuchar J., 1986) and Hopfield (Hopfield HS, 1969). In this standard method, the subject of determination are the Cartesian coordinates of GNSS antenna and the receiver clock error. After introducing necessary corrections to the observed pseudoranges the antenna position in ECEF coordinates and the receiver clock bias, can be determined either by using linearization technique (Strang, G. and Borre, K.,1997) and/or method of least squres using Bancroft Algorithm (Bancroft, 1985). Many authors discussed the different concepts of finding solution of positioning algorithm. Bancroft algorithm consist of a 4x4 matrix inversion and the solution of of scalar equations of the second degree. The algorithm has been widely discussed and further analysed by Abel and Chafee (Abel and Chafee, 1991) and Chafee and Abel (Chafee and Abel, 1991). Another solution given by Graffarend and Chan (Graffarend and Chan, 1996) based on quadratic form of observation equations for algebraic reduction of numbers of observation equations. Kleusberg (Kleusberg, 1994) also discussed subsequent options of possible solution for determination of unique solution among many other solutions as well as discussed geometrical conditions of various cases. In the presented paper it will be shown that position determination for five or more observation equations is possible with no need of using both linearisation technique and least squares method basing on introduced definition of index of reference point relative to the point to be determined (Oszczak B., 2012). In the proposed algorithm, between the device sending a radio signal (satellite), and the device receiving the sent signal (GNSS), we can take into account the corrected values of pseudoranges to calculate the index of each satellite in regard to the GNSS receiver. We assume, similarly as in the GPS single point positioning algorithm (Hofmann-Wellenhof, 2003), to determine the unknown coordinates and clock bias it is necessary a priori elimination from the measured pseudoranges the errors resulting, inter alia, from the ionospheric and tropospheric refraction and the satellite clock error. We derive a direct solution for any number of pseudo-ranges equations without linearisation of the observation equations and application of least squares method. The article presents a basic principles of the new method to solve the problem, gives the formulas, the method of their derivation and explains the importance of new introduced unknown. Introducing the unknown in the system of equations makes it possible to calculate the correct coordinates of the point to be determined without regard to the magnitude of this unknown, which represents a systematic error of measured distance (e.g. the receiver clock error in GNSS positioning). The numerical example given for 5 satellites confirms correct performance of the proposed algorithm. The generalised algorithms for n-number of observed satellites using n-index definition is developed and discussed by the author (Oszczak B., 2012). The method given in the presented paper can be also extended to the system of n-observational equations. In three dimensional space the index for n-reference points in regards to point to be determined can be defined as the sum of all indexes of n-reference points relative to point to be determined. References Abel, J.S. and Chaffee, J.W. Existence and Uniqueness of GPS Solutions. IEEE Transactions on Aerospace and Electronic Systems. 1991, Vol. 27, No. 6, pp. 952-956. Bancroft, S., "An algebraic solution of the GPS equations" IEEE Transactions on Aerospace and Electronic Systems. 1985, Vol. 21, pp:56-59. Chaffee, J. and Abel, J. On the Exact Solutions of Pseudorange Equations. IEEE Transactions on Aerospace and Electronic Systems, 1991, Vol. 30, No.4, pp.1021-1030. Grafarend, E.W. and Chan, J. A Closed-form Solution of the Nonlinear Pseudo-Ranging Equations (GPS), Artificial Satellites Planetary Geodesy, 1996, No. 28, pp. 133-147. Hofmann-Wellenhof B., Lichtenegger H., Wasle E., Navigation, Principles of Positioning and Guidance, Legat, Wieser, 2003, Springer-Verlag Wien. Kleusberg, A. Die direkte Losung des raumlichen Hyperbelschnitts. Zeitschrift fur Vermessungswesen, 1994, Vol. 119, No. 4, pp.188-192. 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, 1986, November 4-7, pp:280-286. Hofmann-Wellenhof B., Lichtenegger H., Wasle E., GNSS Global Navigation Satellite Systems, GPS, Glonass, Galileo&more, 2008, SpringerWienNewYork. Hopfield HS, "Two-quartic tropospheric refractivity profile for correcting satellite data", Journal of Geophysical Research, 1969, Vol. 74, No. 18, p:4487-4499. Oszczak B., Zeszyty Naukowe Wyzszej Szkoly Oficerskiej w Deblinie, Algorytm wyznaczania wspolrzednych punktu i dodatkowej niewiadomej z zastosowaniem ukladow rownan liniowych (Algorithm for determination of point coordinates and additional unknown using the system of linear equations), nr 2(19)/2012 ZN Deblin, Strang, G. and Borre, K. "Linear Algebra, Geodesy and GPS" Wellesley-Cambridge, Wellesley, MA, 1997.
Published in:	Proceedings of the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2013) September 16 - 20, 2013 Nashville Convention Center, Nashville, Tennessee Nashville, TN
Pages:	3560 - 3563
Cite this article:	Oszczak, B., "New Algorithm for GNSS Positioning Using System of Linear Equations," Proceedings of the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2013), Nashville, TN, September 2013, pp. 3560-3563.
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