# Multibody Mechanics/Euler Parameters

The Euler parameters are defined by an axis and angle, i.e. the relative orientation of frames is described by a rotation about a common axis. If the axis defined by the vector ${\displaystyle {\vec {u}}}$ decomposed as ${\displaystyle u_{1},u_{2},u_{2}}$ in both frames (the axis is common), and the rotation by an angle ${\displaystyle \phi }$ (right-handed about the vector) to arrive at orientation of the second frame, then the Euler parameters are
${\displaystyle e_{0}=\cos \left({\frac {\phi }{2}}\right)|{\vec {u}}|}$
${\displaystyle e_{i}=\sin \left({\frac {\phi }{2}}\right)u_{i},i=1...3}$
Note that it is common for one to consider Euler parameters in normalized form where the vector ${\displaystyle {\vec {u}}}$ is a unit vector ${\displaystyle {\hat {u}}}$, in which case the Euler parameters will, by definition, satisfy the constraint that their squares sum to one.