This paper considers the problem of estimating an unknown input
(bias) by means of the augmented-state Kalman (AKF) filter. To reduce
the computational complexity of the AKF, [12]
recently developed an optimal two-stage Kalman filter (TS-AKF) that separates the bias
estimation from the state estimation, and shows that his new
two-stage estimator is equivalent to the standard AKF, but
requires less computations per iteration. This paper focuses on
the derivation of the optimal two-stage
estimator for the square-root covariance implementation of the Kalman
filter (TS-SRCKF), which is known to be numerically more robust than the standard covariance implementation.
The new TS-SRCKF also estimates the state and the bias
separately while at the same time it remains equivalent to the standard augmented-state
SRCKF. It is experimentally shown in the paper that the new
TS-SRCKF may require less flops per iteration for some problems than the Hsieh's
TS-AKF [12]. Furthermore a second, even faster (single-stage)
algorithm has been derived in the paper by exploiting the
structure of the least-squares problem and the square-root
covariance formulation of the AKF. The computational complexities
of the two proposed methods have been analyzed and compared the
those of other existing implementations of the AKF.