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Advanced Kalman Filtering, Least-Squares and Modeling: A Practical Handbook by Bruce P. Gibbs

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CHAPTER 10

FACTORED (SQUARE-ROOT) FILTERING

From the first application of Kalman filtering—midcourse navigation and guidance for the Apollo moon mission (see McGee and Schmidt 1985)—engineers were concerned about the numerical accuracy of the Kalman filter covariance measurement update

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since it differences two positive definite arrays. This was a potential problem for the Apollo mission because the filter was implemented on a 16-bit fixed-point arithmetic flight computer. This project provided the incentive for Potter to develop the first square-root implementation of the Kalman filter (see Battin 1964), although this version could not handle process noise. Within the next few years other square-root variants or extensions were developed by Bellantoni and Dodge (1967), Andrews (1968), Dyer and McReynolds (1969), Schmidt (1970), Carlson (1973), Bierman (1974, 1977b), and others. In the same time period investigators compared the numerical accuracy and execution speed of the various algorithms with the standard and Joseph forms of the Kalman equations. Results of some studies were published (Gura and Bierman 1971; Kaminski et al. 1971; Bierman and Thornton 1977; Verhaegen and Van Dooren 1986) while other studies remained as internal white papers. To this author’s knowledge the results were fairly consistent: the covariance forms of the Kalman filter measurement update—equations ...

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