A skewed approach to filtering

The dynamics of many physical systems are nonlinear and non-symmetric. The motion of a missile, for example, is strongly determined by aerodynamic drag whose magnitude is a function of the square of speed. Conversely, nonlinearity can arise from the coordinate system used, such as spherical coordinates for position.If a filter is applied these types of system, the distribution of its state estimate will be non-symmetric. The most widely used filtering algorithm, the Kalman filter, only utilises mean and covariance and does not maintain or exploit the symmetry properties of the distribution. Although the Kalman filter has been successfully applied in many highly nonlinear and non-symmetric systems, this has been achieved at the cost of neglecting a potentially rich source of information. In this paper we explore methods for maintaining and utilising information over and above that provided by means and covariances. Specifically, we extend the Kalman filter paradigm to include the skew (third order moments) and examine the utility of maintaing this information. We develop a tractable, convenient algorithm which can be used to predict the first three moments of a distribution. This is achieved by extending the sigma point selection scheme of the unscented transformation to capture the mean, covariance and skew. The utility of maintaining the skew and using nonlinear update rules is assessed by examining the performance of the new filter against a conventional Kalman filter in a realistic tracking scenario.