that the covariance is underestimated. Optimality of the unscented transform Uhlmann noted that given only the mean and covariance of an otherwise unknown probability distribution, the transformation problem is ill-defined because there is an infinite number of possible underlying distributions with the same first two moments. Without any a priori information or assumptions about the characteristics of the underlying distribution, any choice of distribution used to compute the transformed mean and covariance is as reasonable as any other. In other words, there is no choice of distribution with a given mean and covariance that is superior to that provided by the set of sigma points, therefore the unscented transform is trivially optimal. This general statement of optimality is of course useless for making any quantitative statements about the performance of the UT, e.g., compared to linearization; consequently he, Julier and others have performed analyses under various assumptions about the characteristics of the distribution and/or the form of the nonlinear transformation function. For example, if the function is differentiable, which is essential for linearization, these analyses validate the expected and empirically-corroborated superiority of the unscented transform. Applications The unscented transform can be used to develop a non-linear generalization of the Kalman filter, known as the Unscented Kalman Filter (UKF). This filter has largely replaced the EKF in many nonlinear filtering and control applications, including for underwater, ground and air navigation, and spacecraft. The unscented transform has also been used as a computational framework for Riemann-Stieltjes optimal control. This computational approach is known as unscented optimal control. Unscented Kalman Filter Uhlmann and Simon Julier published several papers showing that the use of the unscented transformation in a Kalman filter, which is referred to as the unscented Kalman filter (UKF), provides significant performance improvements over the EKF in a variety of applications. Julier and Uhlmann published papers using a particular parameterized form of the unscented transform in the context of the UKF which used negative weights to capture assumed distribution information. That form of the UT is susceptible to a variety of numerical errors that the original formulations (the symmetric set originally proposed by Uhlmann) do not suffer. Julier has subsequently described parameterized forms which do not use negative weights and also are not subject to those issues. See also Kalman filter Covariance intersection Ensemble Kalman filter Extended Kalman filter Non-linear filter Unscented optimal control References Control theory Nonlinear filters Linear filters Signal