How to compute the angular and linear velocities Quaternions? I am new to this area and although I have studied the algebra I am unable to understand how to compute the velocities.
The best way is of course to get yourself familiar with geometric algebra first.
Lounesto, Pertti. Clifford algebras and spinors. Vol. 286. Cambridge university press, 2001.
After that, everything can be derived by yourself. But I guess most people here only like quick and dirty approaches. Use this book instead:
Kuipers, Jack B. Quaternions and rotation sequences. Vol. 66. Princeton: Princeton university press, 1999.
straightly answers your question.
If you want to have more geometric sense of the algebra manipulations, John McCarthy's couldn't have done it better.
McCarthy, J. Michael. Introduction to theoretical kinematics. MIT press, 1990. McCarthy, J. Michael, and Gim Song Soh. Geometric design of linkages. Vol. 11. Springer, 2010.
When I studied the relationship between the rotation matrix and quaternion, I have found a literature, here is the link. I think it will instruct you how to compute the velocity derivatives using quaternions.