Indirect-Inversion Algorithm via Precise Integration for Ill-Conditioned Matrix in Ambiguity Resolution

doi:10.1007/s12204-020-2200-5

Abstract

Abstract: Global navigation satellite system (GNSS) positioning depends on the correct integer ambiguity resolution (AR). If the double difference equation for solving the float solution remains ill-conditioned, often happening due to the environment complexity and the equipment mobility, the correct AR is difficult to achieve. Concerningthe ill-conditioned problem, methods of modifying the equation coefficient matrix are widely applied, whose effects are heavily dependent on modifying parameters. Besides, the direct-inversion of the ill-conditioned coefficient matrix can lead to a reduction in the accuracy and stability of the float solution. To solve the problem of ill-conditioned matrix inversion and further improve the accuracy, the present study for the first time proves the positive definite symmetry of the coefficient matrix in AR model and employs precise integration method to the indirect inverse of coefficient matrix. AR model for the GNSS positioning and the general resolving strategies introduction are briefly introduced. An indirect-inversion algorithm via precise integration for ill-conditioned coefficient matrix is proposed. According to the simulations and comparisons, the proposed strategy has higher precision and stability on float solution, and less dependence on modifying parameters.

Key words: ill-condition| indirect-inversion| ambiguity resolution| precise integration| global navigation satellite
system positioning

摘要： Global navigation satellite system (GNSS) positioning depends on the correct integer ambiguity resolution (AR). If the double difference equation for solving the float solution remains ill-conditioned, often happening due to the environment complexity and the equipment mobility, the correct AR is difficult to achieve. Concerningthe ill-conditioned problem, methods of modifying the equation coefficient matrix are widely applied, whose effects are heavily dependent on modifying parameters. Besides, the direct-inversion of the ill-conditioned coefficient matrix can lead to a reduction in the accuracy and stability of the float solution. To solve the problem of ill-conditioned matrix inversion and further improve the accuracy, the present study for the first time proves the positive definite symmetry of the coefficient matrix in AR model and employs precise integration method to the indirect inverse of coefficient matrix. AR model for the GNSS positioning and the general resolving strategies introduction are briefly introduced. An indirect-inversion algorithm via precise integration for ill-conditioned coefficient matrix is proposed. According to the simulations and comparisons, the proposed strategy has higher precision and stability on float solution, and less dependence on modifying parameters.

关键词: ill-condition| indirect-inversion| ambiguity resolution| precise integration| global navigation satellite
system positioning

CLC Number:

P 228.1

ZHAO Fang, SUN Jin, ZHAO Jianjun, YANG Libin . Indirect-Inversion Algorithm via Precise Integration for Ill-Conditioned Matrix in Ambiguity Resolution[J]. J Shanghai Jiaotong Univ Sci, 2020, 25(6): 762-768.

References 22

[1]	DENG J, WANG S L, CHEN R J. Resolving strategies for ill-posed equation in ambiguity resolution of GNSS network real-time kinematics [J]. Journal of Chinese Inertial Technology, 2016, 24(1): 14-19 (in Chinese).
[2]	COUNSELMAN C C, GOUREVITCH S A. Miniature interferometer terminals for earth surveying: ambiguity and multipath with global positioning system [J].IEEE Transactions on Geoscience and Remote Sensing,1981, GE-19(4): 244-252.
[3]	HATCH R. Instantaneous ambiguity resolution [M]//Kinematic systems in geodesy, surveying, and remote sensing. NewYork. Springer, 1991: 299-308.
[4]	DING J, GAO F, WANG F, et al. An improved FARA algorithm for GPS integer ambiguity resolution[J]. GNSS World of China, 2013, 38(3): 47-50(in Chinese).
[5]	CHEN D, LACHAPELLE G. A comparison of the FASF and least-squares search algorithms for on-thefly ambiguity resolution [J]. Navigation: Journal of The Institute of Navigation, 1995, 42(2): 371-390.
[6]	TEUNISSEN P J G. Least-squares estimation of the integer GPS ambiguities [C]//General Meeting of the International Association of Geodesy. Beijing, China:IAG, 1993: 1-16.
[7]	TEUNISSEN P J G. The least-squares ambiguity decorrelation adjustment: A method for fast GPS integer ambiguity estimation [J]. Journal of Geodesy, 1995,70: 65-82.
[8]	OU J K, WANG Z. An improved regularization method to resolve integer ambiguity in rapid positioning using single frequency GPS receivers [J]. Chinese Science Bulletin, 2004, 49: 196-200.
[9]	CAI J Q, HU C W, GRAFAREND E W. The optimal regularization method and its application in GNSS rapid static positioning [C]//20th International Technical Meeting of the Satellite Division. Fort Worth,TX, USA: Institute of Geodesy, 2007: 299-305.
[10]	CAI J Q, GRAFAREND E W, HU C W. The uniform Tykhonov-Phillips regularization (α-weighted ShomBLE) and its application in GPS rapid static positioning [M]//VI hotine-marussi symposium on theoretical and computational geodesy. Berlin, Heidelberg:Springer, 2008: 216-224.
[11]	LI B F, SHEN Y Z, FENG Y M. Fast GNSS ambiguity resolution as an ill-posed problem [J]. Journal of Geodesy, 2010, 84(11): 683-698.
[12]	LUO X W, OU J K. A new approach for fast integer ambiguity resolution suitable for medium-long baseline in GPS network RTK [J]. Geomatics and Information Science of Wuhan University, 2007, 32(2): 156-159 (in Chinese).
[13]	LUO X W, OU J K, YUAN Y B, et al. An improved regularization method for estimating near realtime systematic errors suitable for medium-long GPS baseline solutions [J]. Earth, Planets and Space, 2008,60(8): 793-800.
[14]	WANG S L, WANG Q, NIE W F, et al. Analysis and solution of ill-conditioned problems in BeiDou system high-precision high-orbit satellite positioning model[J]. Journal of Navigation and Positioning, 2013, 1(3):31-35 (in Chinese).
[15]	GUI Q M, GUO J F. A new ambiguity resolution based on partial ridge estimator [J]. Journal of Information Engineering University, 2004, 5(2): 137-139 (in Chinese).
[16]	LIU Y Y. Research on biased estimation theory and its application in GPS data processing [D]. Zhengzhou,China: The PLA Information Engineering University,2005 (in Chinese).
[17]	GUI Q M, HAN S H, WU B R, et al. Double-k-type ridge estimation and its applications in GPS rapid positioning[J]. Journal of Zhengzhou Institute of Surveying and Mapping, 2006, 23(1): 8-10 (in Chinese).
[18]	GUI Q M, HUANG W B, ZHANG J J. Robust universal ridge estimation [J]. Acta Geodaetica et Cartographica Sinica, 1998, 27(3): 211-216 (in Chinese).
[19]	LIU G Y. Some key issues relating to high precision GPS positioning and crustal deformation analysis[D]. Wuhan, China: Institute of Geodesy and Geophysics,Chinese Academy of Sciences, 2004: 60-61 (in Chinese).
[20]	CUI X Z, YU Z C, TAO B Z, et al. General surveying adjustment [M]. 2nd ed. Wuhan, China: Wuhan University Press, 2009 (in Chinese).
[21]	ZHONG W X. On precise time-integration method for structural dynamics [J]. Journal of Dalian University of Technology, 1994, 34(2): 131-136 (in Chinese).
[22]	ZHONG W X, WILLIAMS F W. A precise time step integration method [J]. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 1994, 208(6): 427-430.