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What do we expect by setting a too low measurement error in a Kalman Filter model, much lower than the noise existing in the measurements? And why?

Let's say we simulate some measurements and we add some noise to them with variance $0.1$, but we set $Q$ to $0.001$. I have run some toy experiments with these values already, but the results are not really that clear and I thought I would require a more theoretical answer before drawing any conclusions.

My guess is that since we will not account for the measurements' noise through $Q$, the model will be overconfident about the measurements and the estimates can diverge from the real state values as more noise will be used to update the estimate's parameters.

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If you carefully inspect the Kalman filtering equations, you come to realize that the actual value of the covariance matrix, process noise, and measurement noise aren't what affect the state updates. It's the ratios between them.

So assuming that your process noise matrix reflects reality, then your intuition is correct -- using a measurement noise matrix much lower than the actual measurement noise means that the filter puts entirely too much credence on the measured values, and allows measurement noise to jerk the estimates around.

(Note that if you want a decent estimation of the state estimation's covariance, you need to start with realistic state, process, and measurement covariance matrices -- but up to the point where you care about that, the filter's dependence on $\mathbf P_0$, $\mathbf R$ and $\mathbf Q$ will be immune to scaling them all by constant values).

For the purposes of developing intuition about this, I found it very helpful to make up low-order filters and just play with this. You can even make 1-state toy filters -- you'll find that a 1-state Kalman filter ends up being a simple low-pass filter, whose bandwidth settles out in the long run to be dependent on the ratio of $\mathbf R / \mathbf Q$.

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