I am learning about Kalman filters in Udacity's Self-driving Car Nanodegree. In one of the lectures on Unscented Kalman Filters (UKF) the state update equation is confusing me. The Filter in question is being designed to track a vehicle moving in 2-D space, under the constant turn-rate and velocity(CTRV) assumption.

The state of the vehicle is described by a mean state vector $$x_k = \begin{bmatrix} p_x \\ p_y \\ v \\ \psi \\ \dot\psi \\ \end{bmatrix}$$ that tracks the mean values of x and y coordinates, linear speed, turn angle and the turn rate and a covariance matrix $P$ that stores the covariances of the state variables.

The model assumes the two noise components, longitudinal acceleration $\nu_{a,k} \sim {N(0,\sigma_a^2)}$ and tangential acceleration $\nu_{\ddot\psi,k} \sim N(0,\sigma_{\ddot\psi}^2)$, both zero-mean Gaussians.

In the absence of any noise, the mean state prediction can be written as:

$$ x_{k+1} = x_k + \begin{bmatrix} {v_k \over \dot\psi_k} {sin(\psi_k + \dot\psi_k \Delta t) - sin(\psi_k) }\\ {v_k \over \dot\psi_k} {(-cos(\psi_k + \dot\psi_k \Delta t) + sin(\psi_k)) }\\ 0 \\ \dot\psi_k \Delta t\\ 0 \end{bmatrix} $$

In my understanding, even when we account for the presence of acceleration noises, this equation should remain unchanged as both the noises have zero means and the effect of the covariances of noise

$$ Q = \begin{bmatrix} \sigma_a^2 & 0 \\ 0 & \sigma_{\ddot\psi}^2 \end{bmatrix} $$ should be multiplied by $\Delta t$ and its effect on the state covariances should be added to matrix $P$.

However,the lecture writes the equation as in the screenshot below: UKF Efffect of Noise on Mean State

Here's the screenshot describing the nature of noise: UKF Process Noise Distributions

Adding noise to mean state vector doesn't make sense to me. It means that the uncertainties from acceleration noises affect the predicted mean states at the next time-step but the overall uncertainty in the value of state variables remains fixed. This seems counter-intuitive to me.

Shouldn't the mean position be predicted based on kinematics and any uncertainties be added to the covariance matrix P?

  • $\begingroup$ After reading the lecture notes I found that the root cause of the confusion is the assumption about the (Gaussian) distribution of acceleration noise. In an earlier lecture the product of time and acceleration, e.g. $a_x * \Delta t$ and $0.5 * a_x * {\Delta t}^2$ were assumed to be mean zero, while here the acceleration without being multiplied by time is itself assumed to be zero mean. The product with time may not be zero mean though. So I've asked a new question . $\endgroup$ – farhanhubble Sep 16 '17 at 17:04
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    $\begingroup$ The slides mention "process model" so I believe the $x_{k+1}$ here is not the (filtering) mean state vector, but the true state vector, but this is impossible to check, at least without a reference to the material. (In a nonadditive model it is well possible the process noise impacts the mean, but it's unclear whether this question has anything to do with it). The claim in the comment is a bit strange - presumably the time $\Delta t$ is a constant, but then $a_x\,\Delta_t$ must have zero mean if $a_x$ has zero mean. $\endgroup$ – Juho Kokkala Oct 19 '17 at 17:08

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