Problem. Assuming we have a fixed distance/variance $ \sigma^2 $, the problem I have, is to sample rotations $R \in SO\left(3\right) $ around the identity $ I $ with isotropic covariance given as $ \sigma^2 $.
What I know. I know, that the geodesic distance between two rotations is as follows: $ d\left(R_1, R_2\right) = \frac{1}{\sqrt{2}}||\log\left(R_1^TR_2\right)||_F $, where $ \log $ denotes the Logarithmic map $ \colon SO\left(3\right) \mapsto \mathfrak{so}\left(3\right)$. Furthermore I know how to compute the geodesic distance between the two quaternions. Fixing one of the rotations $ R_1 = I $, I can form a constraint for the parameters depending on my sampling variance. So far so good, but I know from generating random uniform samples for $ SO\left(3\right) $ that sampling can be tricky. Normal sampling the components of a quaternion and then normalizing, generates uniform samples. Random sampling Euler angles is more tricky, but doable when sampling from specific distributions according to an article from RE Miles in 1965. However, generating samples with these methods will cover the space of all rotations.
Idea. The first solution I see then, is that I could just sample repeatedly with these methods and then reject samples that lie outside of $ \mathbb{B}_\sigma $ by checking the geodesic distance. I then just repeat this, until enough samples are collected. But since I want a normal distribution, wouldn't that mean, that only most rotations lie inside the ball and some can be outside of it? I would have to carefully reject, to still have the normal distribution. Furthermore, the method is not very efficient.
If I were to stay in $ T_IM $ and sample in the space of skew-symmetric matrices, how would I proceed? I like to think that the problem can be solved for $ w $, with $ R = \exp \hat{w} $. If I sample the components of the angular velocity $ w $ and then map back to the group, how can I know the distribution of the rotations?
I am unsure on how to sample, so that the geodesic distances compared to the identity are following a specific distribution. I would be thankful for any help or advice!