# Covariance of small angle quaternion

1. May 15, 2012

### pvm

i have a quaternion representing a small angle rotation (i.e. the scalar part (q4 say) is approximated to 1). This quaternion is estimated in a Kalman Filter, so i also have the associated covariance matrix P, giving me the covariance of the error in the other thee quaternion components (q1, q2, and q3), and thus the expected error in my rotation estimate.

I need to know the expected error rotation around a given vector, h, given the above. Can anyone tell me how to calculate this? I am looking for a scalar that is the variance of the angle phi representing the likely degree of rotation around this vector h.

Hope i explained that clearly! So just like i could take a 2D position error covariance matrix in x-y for example, and work out the expected error variance in any given direction, i want to do the same thing for the rotation and quaternion. Probably a simple answer, but i want to make sure i am doing the right thing...