TABLE OF CONTENTS The Kalman filter algorithm is used extensively in integrated navigation systems. With the application of this algorithm, independent redundant sources of navigation information are combined with a reference navigation solution to obtain an optimal estimate of navigation states-position, velocity, and attitude-and other variables that contribute to navigation solution error.
In this chapter, the Kalman filter algorithm is derived in incremental stages and extended for applications. This development and its extensions proceed as follows:
recursive weighted least squares: constant systems.
recursive weighted least squares: dynamic systems.
discrete linear minimum variance estimator.
U- D factored form.
summed measurements.
combined estimate from two Kalman filters.
Following the approach of Sage, [2] the least-squares estimation algorithm is developed. Initially, the algorithm developed is for a constant system with no dynamics. Proceeding from the weighted least-squares (WLS) estimation algorithm, a recursive weighted least-squares (RWLS) estimation algorithm is developed. Time-varying system dynamics are incorporated to the algorithm's development, using the RWLS approach. This final least-squares step results in a linear discrete estimation algorithm form.
The algorithm is derived again by establishing a minimum variance linear estimator. The linear estimator form, obtained from the least-squares approach, is assumed, however, in this derivation, it is for a fully dynamic system model. The resulting estimator is the linear discrete Kalman filter algorithm. The two approaches, least-squares and minimum variance estimators, result in the same algorithm form.
The standard Kalman filter algorithm has been shown to exhibit poor numerical accuracy for ill-conditioned measurements. This problem is...
