Unscented Kalman Filter Equations for Constant Turn Rate and Velocity Process Model

Question

I am learning about Unscented Kalman Filters in Udacity's Self-Driving Car Nanodegree. The UKF is designed to track an object moving under the assumptions of constant turn rate $\ddot\psi$ and velocity (speed) $v$, the so-called CTRV model. The process noise is assumed to come from a longitudinal acceleration component $\nu_a = {N(0,\sigma_a^2)}$ and a yaw acceleration component $\nu_{\ddot\psi} = N(0,\sigma_{\ddot\psi}^2)$.

The mean state vector $$x = \begin{bmatrix} p_x \\ p_y \\ v \\ \psi \\ \dot\psi \\ \end{bmatrix}$$ tracks the mean values of x and y coordinates, speed, turn angle and the turn rate. The matrix $P$ stores the covariances of these variables.

The screenshots describing the model are below.

My understanding is that since the noise components are zero-mean, they should only affect the covariances not the means. However in the lecture noise is being added to the means too. For example $e=\Delta t \cdot \nu_{\ddot\psi}$ is the effect of of the yaw acceleration noise on the yaw rate. Also the notation seems odd. What does it mean to multiply by $\Delta t$ a stochastic variable $\nu_{\ddot\psi}$ with a mean and variance components?

Answer 1

From a statistical point of view, the noise parameters are zero mean gaussian distribution and that does not mean that at all times the value of noise would be zero. All it says is that if you were to randomly sample the noise (assuming the number of samples taken are statistically significant, i.e >=30), the mean would tend to zero.

In this specific case, the process model is to estimate the state of the tracked car at x_k+1 , given the state of the car at x_k. Now, the noise component at this specific instant wouldn't have to be necessarily zero. You add the noise component because you only know about the noise distribution and not the noise at that specific instant. It can be any number or matrix in that noise distribution.

Answer 2

First, the state vector isn't the mean, it is the state vector.
In the Kalman Filter model, where we assume the model is linear and noise is AWGN is matches the mean of the process.

The Kalman filter in its non linear form is no longer guaranteed to hold this assumption.

Now, the dynamic model is a discretized form of a continuous model.
In the 1st order discretization we use Zero Hold filters, so if there is noise add at time $ {t}_{0} $ it will effect the model within the time interval. So in the piece wise constant model it means its effect is the value multiplied by the the time interval of the model.