Multibody Mechanics/Euler Parameters

Euler parameters are an alternative to Euler angles as a means to describe the orientation of a body (or alternatively, a reference frame) in three dimensions.

The significant difference between Euler parameters and Euler angles is that where there are only three Euler angles, there are four Euler parameters. The Euler parameters are not independent, and a valid set of parameters must satisfy the constraint that the sum of their squares is equal to one.

The Euler parameters are defined by an axis and angle, i.e. if a body is rotated about an axis defined by the unit vector $u_1,u_2,u_2$, by some angle $\varphi$ to arrive at its current orientation, then the Euler angles are

$e_0=\cos \left(\frac{\varphi }{2}\right)$

$e_i=\sin \left(\frac{\varphi }{2}\right)u_i,i=1...3$

Note that as long as the vector $\overrightarrow{u}$ is a unit vector, then the Euler parameters will, by definition, satisfy the constraint that their squares sum to one.

See also Rodriguez parameters.