How to Draw Euler Angles and Utilize Euler Parameters

This article presents a way to draw Euler angles such that the proper operation and application becomes immediately clear. Furthermore, Euler parameters, which allow a singularity-free description of rotational motion, are discussed within the framework of quaternion algebra and are applied to the kinematics and dynamics of a rigid body. 1 Euler Angles In rigid body mechanics we need to keep track of points for each body. The motion of such a body can be decomposed into a translation and a rotation. Here we focus on the rotational part. One way to describe the rotation (the change in orientation) of a rigid body is by means of Euler angles. Or more precisely: a way to parametrize the rotation matrix is to use the three Euler angles [1]. For a rotation about a fixed origin, the rotation matrix R is the orthogonal matrix which transforms the coordinates of a point r from the body fixed coordinate system to the space fixed coordinate system, as in