# Is there a form of rigid body dynamics that uses unit quaternions instead of Euler angles?

I’d like to know specifically about an elegant way of deriving a second derivative of an orientation quaternion from a torque and a moment of inertia matrix, if one is available.

The straight forward, but somewhat round-about way that I can think of is to solve for the derivative of the orientation of a rigid body in terms of Euler angles via classical methods. Then I would convert those angular displacement vectors into unit quaternions and continually compose the unit quaternions.

I have been trying to find a formulation for mapping net torques as a function of time to orientation quaternions as a function of time, but have been unsuccessful.

• Euler angles only contain the driving parameters (3 for a free body) and quaternions are a form of homogeneous coordinates for angles with 4 parameters. I do think there has to be alternative to Euler angles (successive rotations) but I doubt quaternions are the answer. Jun 13, 2016 at 19:44
• I assume you have looked at: https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation and https://en.wikipedia.org/wiki/Quaternion? Jun 23, 2016 at 14:03
• @honeste_vivere, that is correct, are you implying that there's something I missed there? Jun 25, 2016 at 5:48
• @SantiagoSoto - Perhaps it is I that missed something. I thought those articles referred to rigid body rotations and translations using quaternions. Jun 25, 2016 at 14:02