# What does the covariance of a quaternion *mean*?

If I have a set of Euler angles (representing the orientation of an object) and I find the covariance of those angles then I have some intuition that $\sigma^2$ is in units of $\text{rad}^2$ and I can visualize what a normal distribution of angles with a given mean is.

If I represent that same set of orientations with quaternions, I have no idea what the resulting variance $\sigma^2$ means. How can I visualize it or gain an intuitive sense of how to interpret it?

• This answer has more than earned the bounty, because on pp 73-76 it derives the discrete time process error covariance $Q_k$ for a Kalman Filter whose state contains a quaternion orientation and an angular velocity $\omega$. – Ben Jackson Oct 20 '14 at 4:49