The Higher Order Unscented Transformation (HOUF)

As mentioned in the previous text the random variables are practically described by a finite set of moments. It is obvious that a more precise description of the moments yields a more accurate nonlinear filter. The CoDE laboratory is currently involved in the development of higher order unscented filters [5].

To capture higher order moments, the sigma-set is symmetrically expanded, which introduces additional design parameters. The augmented sigma-set is shown in the following figure:

The augmented sigma-set is capable of matching higher order moments, where the sigma points and the weights are the design variables.  Depending on the number m of augmented sigma points, a variable order of moments can be approximated. The two-sigma-set, for example, is capable of matching up to the eighth order moment. In [5] we have shown the solution of the 5 unknowns for the two-sigma-set, which are repeated here:

These equations can be further simplified by substituting the sigma-points into the equations of the weights.

The expansion to the 3-sigma-set enables the higher order unscented filter to match up to the twelfth order moment.

1. Simon Julier, Jeffrey Uhlmann and  Hugh Durrant-Whyte. New Approach for Filtering nonlinear Systems. In Proceedings of the American Control Conference, Vol. 3, pages 1628-1632, 1995.
2. Simon Julier. Scaled Unscented Transformation. In American Control Conference, May 8-10, Anchorage, Alaska, 2002.
3. Simon Julier. Minimum Skew Unscented Transformation. In American Control Conference, May 8-10, Anchorage, Alaska, 2002.
4. Simon Julier. Spherical Unscented Transformation. In American Control Conference, June 4-6, Denver, Colorado 2003.
5. Dirk Tenne and Tarunraj Singh. The Higher Order Unscented Filter. In American Control Conference, June 4-6, Denver, Colorado 2003.