Transformation relationships for two commonly utilized Euler angle representations

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Abstract
There is no standard representation scheme that is followed by different groups of robot designers to describe the orientation of a rotating rigid body. For a sensor-based robot, the particular rotation representation scheme used by the robot affects not only the specification of the positioning of the arm and end-effector, but also the specification of the placement of the sensors and the method of computing the sensory-to-world transformation. Thus the need for finding the transformation between different rotation representations is obvious for one involved in the development of high-level software systems that need to be transported to different robots utilizing different rotation representation schemes. Among those representation schemes, Euler angle representations are widely used in commercial robots as well as in research laboratory robots. 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 upon the solution of the inverse transform for Euler angles and the other is based upon the Napier's rules associated with spherical trigonometry.

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