Transformation relationships for two commonly utilized Euler angle representations

Finding the transformation between different rotation representations in the development of high-level software systems that must be transported to different robots utilizing different rotation representation schemes is addressed. Euler angle representations, which are widely used in commercial robots and research laboratory robots, are considered. Typically, there are three different Euler angle representation systems, and each system describes a particular orientation of a rigid body in a reference coordinate frame by specifying three angles. Mathematical derivations of the transformation relationships underlying the parameters from two most commonly utilized representation schemes are presented. These relationships are derived in two different ways. One is based on the solution of the inverse transform for Euler angles and the other is based on the Napier´s rules associated with spherical trigonometry