Derivation of Unscented Kalman Filter (UKF)
Linearization of non-linear dynamics/observation in the EKF is a clever trick to use the KF equations. We approximate a new linear function from non-linear function around one point i.e. the mean of the Gaussian. However, there is still a better way to approximate it, say we approximate around multiple points. This is the inituition behind the powerful method called Unscented Kalman Filter (UKF). We take some points on source Gaussian and map them on target Gaussian after passing points through some non linear function and then we calculate the new mean and variance of transformed Gaussian. These special set of points are called "Sigma points". It can be very difficult to transform whole state distribution (computationally expensive) through a non linear function but it is very easy to transform some individual points. These sigma points are the representatives of whole distribution.
Unscented Transform
Consider a non-linear function
The Unscented Transform gives us a way to accurately estimate the mean and covariance of the transformed distribution through a nonlinearity. We can use the UT to modify the EKF to make it a more accurate state estimator. The resultant algorithm is called the Unscented Kalman Filter (UKF). The UT can aslo be modified in varied ways through the inclusion of mean as one of the sigma points and the scaling factor.
Consider a non-linear dynamical system with non-linear observations,
Step 1: Propagating the dynamics by one timestep
Given our current state estimate
Computing general form of Kalman gain
The observation step is also modified using the UT. We need a way to compute the Kalman gain in terms of the sigma points in the UT. Until now we have written the Kalman gain using the measurement matrix
Let's get back to our standard Kalman Filter equations (Step 2 of Part 2) to obtain a general form of the kalman gain and update equations, i.e. without the measurement matrix
Step 2: Incorporating the observation
It is clear from the above derivation that we need the estimate of the new observation to proceed further. We generate new sigma points
The mean of the transformed sigma points is nothing but the mean of the estimation of the observation
As compared to the Extended Kalman Filter, the UKF is a better approximation for nonlinear systems. Of course, if the system is linear, both EKF and UKF are equivalent to a Kalman Filter. In practice, we typically use the UKF with some tuning parameters in the Unscented Transform. An important point to remember about both the UKF and EKF is that even if they can handle nonlinear systems, they still approximate the filtering distribution
Still wondering why it is called Unscented Kalman Filter?
The guys who invented UKF thought that EKF stinks because it was a very poor idea to linearize a non linear function around a single point i.e. mean. They called it Unscented on purpose so that they can tell the world that EKF stinks!!
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