Kalman Filters with Uncompensated Biases

A N UNDERLYING assumption of the Kalman filter is that the measurement and process disturbances can be accurately modeled as random white noise. Various mitigation strategies are available when this assumption is invalid. In practice, sensor errors are often modeled more accurately as the sum of a white-noise component and a strongly correlated component. The correlated components can, for example, be random constant biases. In this case, a standard technique is to augment theKalman filter state vector and estimate the random biases. In an attempt to decouple the bias estimation from the state estimation, Friedland [1] estimated the state as though the bias was not present and then added the contribution of the bias. Friedland showed that this approach is equivalent to augmenting the state vector. This technique, known as two-stage Kalman filtering or separate-bias Kalman estimation, was then extended to incorporate a walk in the bias forced by white noise [2]. To account for the bias walk, the process noise covariance was increased heuristically, and optimality conditions were derived [3,4]. In this work, the effects of the noise and biases are considered as sources of uncertainty and not as elements of the state vector. This approach is applicable in situations when the biases are not observable or when there is not enough information to discern the biases from the measurements. A common approach would be to tune the filter using process and measurement noise such that the sample covariance obtained through Monte Carlo analysis matches the filter state estimation error covariance. The technique presented here takes advantage of the structure of the biases to obtain a more precise representation of their contributions to the state estimation uncertainty. The resulting algorithms are useful in quantifying the uncertainty in a single simulation along the nominal state trajectory. This process can aid in tuning the filter as well as be employed onboard to obtain an accurate measure of the uncertainty of the state estimates. The approach taken is similar to that of the consider filter proposed by Schmidt [5]. The consider filter can be applied to the present problem and the two solution approaches, although different in form, are functionally the same. The goal of this paper is to introduce the uncompensated bias Kalman filter as presented by the authors in [6,7]. More recently Hough [8] independently derived a similar algorithm and applied it to orbit determination. This engineering note shows the relation between these two recent techniques as well as their equivalency to the Schmidt consider filter [5].