Confused with EKF Localization

Question

I have been trying to understand EKF localization from Probabilistic Robotics by Thrun Burgard and Fox.

There the covariance prediction is given by $$\overline{\Sigma }_t=G_t\Sigma_{t-1}G^T_t+V_tM_{t-1}V^T$$

Where $G_t$ is the Jacobian matrix of the motion model $g(x_{t-1},u_t)$ with respect to $x_{t-1}$ and evaluated at $(\mu_{t-1},u_t)$ and $V_t$ is the Jacobian matrix of the motion model $g(x_{t-1},u_t)$ with respect to $u_{t}$ and evaluated at $(\mu_{t-1},u_t)$ and $M_t$ is the covariance motion noise which is a function of the actions.

I have been trying to relate that covariance prediction equation with the original equation derived earlier in the book $$\overline{\Sigma }_t=G_t\Sigma_{t-1}G^T_t+R_t$$ with no success.

The authors mention that we used the first formulation because we had to map the noise from the control space to the state space but I don't see how it follows from the original EKF derived earlier.

What further confuses me is that there are other EKF formulations which assume that $g$ could be nonlinear in the noise and replace $V$ with the Jacobian matrix of $g(x_{t-1},u_t)$ with respect to the noise (and not the actions as in here.)

I'm really confused. Any help is appreciated.

Answer 1

Like any good engineering formula you can use derive formulations that are appropriate to many different situations. Let's take a rock simple example that doesn't need an EKF at all but is sufficient to illustrate the point.

Take the simplest model possible

$x = 100 * u$

Let's say you know the variance of u and it is .1, what is the variance of x? Well, we have

$100 (.01) 100^T$ = 1000.

If instead I'd observed my overall system, I'd have observed the variance of x and see that it would be 1000 and I'd use the formula with R.

Put another way if I just blindly used the variance of u for x I'd be off by 3 orders of magnitude, not great. So that is the mapping between control and state space. The state space formulation above also handles spaces of different dimensions u and x don't need to be the same and states that depend on complex ways on the input.