I'm implementing a extended Kalman filter using quaternions.
I've extended this paper to deal with my custom observations.
My state space is analogous to the one in the previous paper : $ \mathbf{x}=[r_x, r_y, r_z, q_w, q_x, q_y, q_z]^T $ with $\mathbf{r}$ the angle rate vector and $\mathbf{q}$ the quaternion.
Here is my problem : in the paper from Xiaoping, the process noise matrix is null everywhere except on the 3 first elements of the diagonal where small equals values are considered.
With this approach, we are not considering any process noise on the quaternion components, only on the angle rates. I seems like an oversimplification to me.
I've tried to put some noise on the rest of the diagonal, but i expect the quaternion components to be correlated to each others due to the normalization procedure rotation quaternions suffers.
Right now i'm searching for a more realistic process noise matrix but i can't find the information anywhere. Does anyone have an idea of what this process noise matrix should look like ?
Best :) ,
Why do you think that quaternion process noise makes sense?
The equation for a quaternion update is:
$\dot{q} = \frac12 q \otimes \omega$
There's no noise in this process; if you knew $\omega$ perfectly, you could integrate $q$ perfectly.
All noise should be introduced through $\omega$.