Okey!
When using log, do you have any good method/example of how to do this?
I did not have this problem you are describing, but I can see it happening. What also can be done is to change the beta and tau update equations to include absolute value.
When it comes to omega, a bias is of course needed! I wasn't thinking about that.
Edit: Got exponential mapping to work.
However, I am having problems with estimating the bias correctly. Looking into it.
Edit2: The bias does not seem to be observable... Looking into it.
Edit3: The bias is not observable without a fixed reference. To get the bias as well, attitude estimation must be coupled with this estimation, or if some other filter does bias estimation.