GPS is a satellite-based navigation system that is able to determine the exact position of objects on the Earth, sky, or space. By increasing the velocity of a moving object, the accuracy of positioning decreases; meanwhile, the calculation of the exact position in the movement by high velocities like airplane movement or very high velocities like satellite movement is so important. In this paper, seven methods for solving navigation equations in very high velocities using least squares method and its combination with the variance estimation methods for weighting observations based on their qualities are studied. Simulations on different data with different velocities from 100？m/s to 7000？m/s show that proposed method can improve the accuracy of positioning more than 50%. 1. Introduction Global positioning system (GPS) does the positioning with the help of a group of satellites that each travels in its orbit around the Earth [1]. These satellites by sending L1 and L2 signals with frequency of 1575.42？MHz and 1227.6？MHz, respectively, identify their time and place toward the Earth. A GPS receiver by receiving these signals from at least four satellites organizes the navigation equations and by solving them shows the position of the user [1–4]. At present the only way of positioning a kinematic receiver in real time with high accuracy is by differential mode that is a relative positioning of two receivers [5, 6]. To achieve a higher accuracy, many researches have been done on differential GPS [7–10]. In some other research efforts too, it has been tried to combine GPS with other navigation systems to attain a higher accuracy [11–13]. But none of these methods are suitable for positioning at velocities of up to 7,000？m/s. Calculation of the exact position in the movement by high velocities like airplane movement or very high velocities like satellite movement using GPS receivers is very important. Jumping at outputs data of GPS receivers’ situation and sudden acceleration at very high velocities leads to making too much error in determining the position of receiver. Some methods like least squares (LS) method that has been presented for solving navigation equations up to now generally have low precision and much error [14]. So we are looking for a method which can solve the navigation equations at very high velocities and significantly decrease positioning errors. The LS is a standard approach to the approximate solution of overdetermined systems, that is, sets of equations in which there are more equations than unknowns. According to this method, the
References
[1]
M. R. Mosavi and M. Shiroie, “Efficient evolutionary algorithms for GPS satellites classification,” Arabian Journal for Science and Engineering, vol. 37, no. 7, pp. 2003–2015, 2012.
[2]
H. Azami, M. Mosavi, and S. Sanei, “Classification of GPS satellites using improved back propagation training algorithms,” Wireless Personal Communications, vol. 71, no. 2, pp. 789–803, 2013.
[3]
M. R. Mosavi and I. EmamGolipour, “De-noising of GPS receivers positioning data using wavelet transform and bilateral filtering,” Journal of Wireless Personal Communications, vol. 71, no. 3, pp. 2295–2312, 2013.
[4]
M. R. Mosavi, M. S. Azad, and I. EmamGholipour, “Position estimation in single-frequency GPS receivers using kalman filter with pseudo-range and carrier phase measurements,” Wireless Personal Communications, vol. 72, no. 4, pp. 2563–2576, 2013.
[5]
Z. Wen, P. Henkel, and C. Güunther, “Reliable estimation of phase biases of GPS satellites with a local reference network,” in Proceedings of the ELMAR, pp. 321–324, Zadar, Croatia, September 2011.
[6]
Y. J. Cui and S. S. Ge, “Autonomous vehicle positioning with GPS in urban canyon environments,” in Proceedings of the IEEE International Conference on Robotics and Automation (ICRA '01), vol. 2, pp. 1105–1110, May 2001.
[7]
H. Evers and G. Kasties, “Differential GPS in a real time land vehicle environment-satellite based van carrier location system,” IEEE Aerospace and Electronic Systems Magazine, vol. 9, no. 8, pp. 26–32, 1994.
[8]
M. Pikander and P. Eskelinen, “Differential GPS dynamic location experiments at sea,” IEEE Aerospace and Electronic Systems Magazine, vol. 19, no. 4, pp. 36–39, 2004.
[9]
S. Rezaei and R. Sengupta, “Kalman filter-based integration of DGPS and vehicle sensors for localization,” IEEE Transactions on Control Systems Technology, vol. 15, no. 6, pp. 1080–1088, 2007.
[10]
Q. Yang and J. Sun, “A location method for autonomous vehicle based on integrated GPS/INS,” in Proceedings of the IEEE International Conference on Vehicular Electronics and Safety (ICVES '07), pp. 1–4, Beijing, China, December 2007.
[11]
J. Yu and X. Chen, “Application of extended Kalman filter in ultra-tight GPS/INS integration based on GPS software receiver,” in Proceedings of the 1st International Conference on Vehicular Green Circuits and Systems (ICGCS '10), pp. 82–86, June 2010.
[12]
P. Defraigne, A. Harmegnies, and G. Petit, “Time and frequency transfer combining GLONASS and GPS data,” in Proceedings of the IEEE Joint Conference on Frequency Control and the European Frequency and Time Forum, pp. 1–5, San Fransisco, Calif, USA, May 2011.
[13]
X. Zhao, Y. Qian, M. Zhang, J. Niu, and Y. Kou, “An improved adaptive Kalman filtering algorithm for advanced robot navigation system based on GPS/INS,” in Proceedings of the IEEE International Conference on Mechatronics and Automation (ICMA '11), pp. 1039–1044, August 2011.
[14]
S. N. Thipparthi, Improving positional accuracy using carrier smoothing techniques in inexpensive GPS receivers [M.Sc. Thesis], New Mexico State University, 2004.
[15]
L. O. Krause, “A direct solution to GPS-type navigation equations,” IEEE Transactions on Aerospace and Electronic Systems, vol. 23, no. 2, pp. 225–232, 1987.
[16]
Y. He, R. Martin, and A. M. Bilgic, “Approximate iterative least squares algorithms for GPS positioning,” in Proceedings of the IEEE International Symposium on Signal Processing and Information Technology (ISSPIT '10), pp. 231–236, Luxor, Egypt, December 2010.
[17]
X. Du, L. Liu, and H. Li, “Experimental study on GPS non-linear least squares positioning algorithm,” in Proceedings of the International Conference on Intelligent Computation Technology and Automation (ICICTA '10), vol. 2, pp. 262–265, Changsha, China, May 2010.
[18]
A. R. Amiri-Simkooei, P. J. G. Teunissen, and C. C. J. M. Tiberius, “Application of least-squares variance component estimation to GPS observables,” Journal of Geodesy, vol. 82, no. 2, pp. 65–82, 2008.
[19]
A. Wieser and F. K. Brunner, “Short static GPS sessions: robust estimation results,” Journal of GPS Solutions, vol. 5, no. 3, pp. 70–79, 2002.
[20]
A. Wieser and F. K. Brunner, “SIGMA-F: variances of GPS observations determined by a fuzzy system,” in Proceedings of the IAG Scientific Assembly, pp. 1–6, 2001.
[21]
J. Li and M. Wu, “The improvement of positioning accuracy with weighted least square based on SNR,” in Proceedings of the 5th International Conference on Wireless Communications, Networking and Mobile Computing (WiCOM '09), pp. 1–4, Beijing, China, September 2009.
[22]
F. Zeng, W. Dai, J. Zhu, and X. Wang, “Single-point positioning with the pseudo-range of single-frequency GPS considering the stochastic model,” Pacific Science Review, vol. 10, no. 3, pp. 274–278, 2008.
[23]
J. P. Collins and R. B. Langley, Possible Weighting Schemes for GPS Carrier Phase Observations in the Presence of Multipath, Geodetic Research Laboratory, University of New Brunswick, 1999.
[24]
A. Wieser and F. K. Brunner, “An extended weight model for GPS phase observations,” Journal of Earth, Planets and Space, vol. 52, no. 10, pp. 777–782, 2000.
[25]
F. K. Brunner, H. Hartinger, and L. Troyer, “GPS signal diffraction modelling: the stochastic SIGMA- model,” Journal of Geodesy, vol. 73, no. 5, pp. 259–267, 1999.
[26]
R. B. Langley, “GPS receiver system noise,” Magazine of GPS World, vol. 8, no. 6, pp. 40–45, 1997.
