Unscented Kalman filter
An unscented Kalman filter is a version of the Kalman filter for systems that are not linear. Instead of approximating the system by a linear one, it represents the current estimate and its uncertainty by a few carefully chosen sample points, passes each point through the system’s model, and computes the new estimate and uncertainty from the results.
Consider a discrete-time system with state $x_k \in \reals^n$, input $u_k$, and measured output $y_k$, given by
\[x_{k+1} = f(x_k, u_k) + w_k, \qquad y_k = h(x_k) + v_k,\]where the noises $w_k$ and $v_k$ are independent, zero-mean, and white, with covariances $Q$ and $R$. The unscented Kalman filter for this system is the filter that alternates prediction and correction steps like the Kalman filter, but computes each mean and covariance by the unscented transform. To pass a mean $\hat{x}$ and covariance $P$ through a function $g$, the unscented transform chooses $2 n + 1$ points, called sigma points, whose weighted mean is $\hat{x}$ and whose weighted covariance is $P$, applies $g$ to each point, and returns the weighted mean and covariance of the results.
Properties
- The unscented Kalman filter does not need the Jacobians of $f$ and $h$, unlike the extended Kalman filter.
- The unscented transform computes the mean and covariance of $g(x)$ correctly to second order in the Taylor expansion of $g$, whereas the linearization in the extended Kalman filter is correct only to first order.
- If $f$ and $h$ are linear, then the unscented Kalman filter is the Kalman filter.
Notes
The sigma points are not random samples, unlike the particles of a particle filter.