On the algebraic definition of total rotation in RSA

Abstract: Total rotation is a quantity that has been used for years in RSA. However, its definition has no mathematical sense, since the Euler angles do not form a vector space, since angles cannot define a multiplication group. With this work I tried to give a mathematical definition of the total rotation connecting the Euler description of the rotations with the helical axis. The approximation for small angles was used to connect Euler's angles and helical angle. With this approximation Euler angles acquire the properties of a vector space and it is possible to justify the meaning of this parameter. Validation test showed that total rotation has an approximation error between 5\% and 7\% for angles in the range $\left[-\frac{\pi}{6}, \frac{\pi}{6} \right]$. Since usually RSA uses smaller angle ranges, the approximation is perfectly suitable for use in RSA.