Truncated Unscented Kalman Filtering

We devise a filtering algorithm to approximate the first two moments of the posterior probability density function (PDF). The novelties of the algorithm are in the update step. If the likelihood has a bounded support, we can use a modified prior distribution that meets Bayes´ rule exactly. Applying a Kalman filter (KF) to the modified prior distribution, referred to as truncated Kalman filter (TKF), can vastly improve the performance of the conventional Kalman filter, particularly when the measurements are informative relative to the prior. The application of the TKF to practical problems in which the measurement noise PDF has unbounded support is achieved by imposing several approximating assumptions which are valid only when the measurements are informative. This implies that we adaptively choose between an approximation to the KF or the TKF according to the information provided by the measurement. The resulting algorithm based on the unscented transformation is referred to as truncated unscented KF.