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