Kinematic Relative GPS Positioning UsingState-Space Models: Computational Aspects

Xiao-Wen Chang, Mengjun Huang

Abstract:	"For kinematic relative GPS positioning the often used approach is to establish a discrete linearized statespace model, and then apply a standard Kalman filtering technique to estimate the positions. However, there are a few shortcomings with this approach. Since the integer ambiguities are constant (suppose there are no cycle slips), the state-space model for positioning is special. In order to apply a standard Kalman filtering technique, this approach enlarges the timevariant state vectors (e.g., position-velocity vectors) to include the integer ambiguities. This not only makes the estimation problem largerÑleading to higher computational cost, but also results in singular covariance matrices for the process noise vectors in the process equations since the covariance matrix corresponding to the integer ambiguity vector is a zero matrix. Then some standard Kalman filtering techniques, such as the square root information filter cannot be applied. The typical method people have used to handle this singularity problem is to artificially assign a small covariance matrix to the ambiguity vector. But this is an approximation and does not look elegant in theory. In order to avoid the above problems, for short baseline kinematic relative positioning, we present a computationally efficient and numerically reliable approach. Our approach can be regarded as an extension of the standard information square root filtering technique. The basic idea is to write all available measurement equations (we use both code and carrier phase measurements based on L1 signals) and process equations together to form a large linearized model, which has special structures. Then based on this model, we develop a recursive algorithm to compute the least squares estimates of positions and velocities. We obtain not only the estimate of the current position (this is called filtering), but also the estimates of previous positions (this is called smoothing). Our algorithm is computationally efficient and numerically reliable. One thing which makes the estimation problem much more complicated and also more interesting is that the integer ambiguity vector may change due to cycle slips, loss of signals, and satellite rising/setting. In this paper we show how to handle this problem in our algorithm. Unlike the typical literature in GNSS, we give details about the algorithm so that people can implement it without difficulties."
Published in:	Proceedings of the 61st Annual Meeting of The Institute of Navigation (2005) June 27 - 29, 2005 Royal Sonesta Hotel Cambridge, MA
Pages:	937 - 948
Cite this article:	Chang, Xiao-Wen, Huang, Mengjun, "Kinematic Relative GPS Positioning UsingState-Space Models: Computational Aspects," Proceedings of the 61st Annual Meeting of The Institute of Navigation (2005), Cambridge, MA, June 2005, pp. 937-948.
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