In this lecture, we will discuss non-linear approaches to the Kalman filter. A standard linear model has some limitations that require different methods to model motion uncertainty. Non-linear motion updates break the Gaussian properties of the state distribution. We will now discuss the extended common filter and the unscented common filter. With a non-linear dynamical system, the state transition can be a function of the state, and thus it's harder to predict where the model should be going. A differentiable function helps by allowing for approximation through linearization at the current time step. The Jacobian matrix represents the differentiation of this matrix motion function, but it does not capture higher order dynamics. If there is not a large deviation in the time step, then this can be a valid approach to track the state distribution through time. The linear form, shown above, is rewritten to project the state using the non-linear function instead of the state transition matrix. Effectively, the Jacobian replaces the state transition matrix in the calculation of the predicted covariance. Similarly, the Kalman gain is rewritten with the Jacobian of the observation function. The Jacobian is evaluated at the point x of t. The overall update to track the state over time is very similar to the linear system. The extended Kalman filter is a straightforward method to retain the gassing concepts given a differentiable motion and observation model. The next approach to dealing with non-linearities utilizes a small set of sample points. This filter is called the unscented Kalman filter or UKF. The UKF continually re-estimates the distribution statistics of the mean and covariance, by transforming characteristic points through the non-linear dynamical system. First, we model the distribution based on a set of sigma points. These sigma points typically characterize the covariance of the matrix around about a standard deviation away from the mean and including the mean as one of the sigma points. The statistics of the sigma points will have the same mean and covariance of the underlying Gaussian distribution. The new distribution after the motion model is applied will not be Gaussian, however, we can recalculate the mean and covariance of the transformed characteristic signal points in order to approximate the new distribution as Gaussian. With the idea of tracking statistics over time we first look at tracking the expected state over time. The expected state is the average of the transformed sigma points. Similarly the predicted covariance is the covariance of the sigma points having been run through the dynamical system. Each sigma point will have an associated expectation for the observation. We can model the distribution of the expected observation then by calculating the statistics again. The Kalman gain is slightly modified from the linear system. To calculate the Kalman gain, we disregard the observation model and utilize the covariances of the observation points where signal points run through the motion model. The final update is just as the linear filter. However, the covariance update will be slightly different. Please look at the attached notes for good resources on further details. Finally, in a preview to week four, we can take the unscented Kalman filter to the limit and use many points to characterize the distribution. This modified distribution of points then will not limit the underlying model to be Gaussian. We will explore this particle filter in week four.
