Extended Kalman Filter extends Kalman Filters by linearizing the non-linear models with first-order Taylor expansion.
Unscent Kalman Filter instead extends Kalman Filters with Unscent transformation.
Unscent transformation
- Compute a set of (so-called) sigma points
- Transform each sigma point through the non-linear function
- Compute Gaussian from the transformed and weighted sigma points
- Compute a Gaussian from weighted points
Sigma Points Q: How to choose the sigma points & How to set the weights? A: Choose the sigma points $X$ and the corresponding weights $W$ so that the chosen points and weights fit exactly the orignial Gaussian distribution:
\[\sum_{i} w_i x_i = 1\] \[\mu = \sum_{i} w_i x_i\] \[\Sigma = \sum_{i} w_i (x_i - \mu_i) (x_i - \mu_i)^T\](No unique solution for $X$, $W$)