Multibody Mechanics/Euler Parameters

Euler parameters (also known as Euler-Rodrigues parameters or unit quaternions) are an alternative to Euler angles as a means to describe the relative orientation of reference frames 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 constant.

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.