Extended Kalman Filter - Unscented Kalman Filters

Unscented Kalman Filters

A nonlinear Kalman filter which shows promise as an improvement over the EKF is the unscented Kalman filter (UKF). In the UKF, the probability density is approximated by a deterministic sampling of points which represent the underlying distribution as a Gaussian. The nonlinear transformation of these points are intended to be an estimation of the posterior distribution, the moments of which can then be derived from the transformed samples. The transformation is known as the unscented transform. The UKF tends to be more robust and more accurate than the EKF in its estimation of error.

"The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 35 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. Many of these difficulties arise from its use of linearization."

Julier and Uhlmann make several claims about the nature of the Unscented Kalman Filter and suggest three different Unscented Transforms at the heart of the UKF. A recent paper shows that the UKF fails to be as accurate as the Second Order Extended Kalman Filter (SOEKF), called also the augmented Kalman filter. The SOEKF predates the UKF by approximately 35 years with the moment dynamics first described by Bass et al. The difficulty in implementing any Kalman-type filters for nonlinear state transitions stems from the numerical stability issues required for precision, however the UKF does not escape this difficulty in that it uses linearization as well, namely linear regression. The stability issues for the UKF generally stem from the numerical approximation to the square root of the covariance matrix, whereas the stability issues for both the EKF and the SOEKF stem from possible issues in the Taylor Series approximation along the trajectory.