On the algebraic definition of total rotation in RSA

TL;DR

This paper provides a rigorous mathematical definition of total rotation in RSA by connecting Euler angles with the helical axis, validating its approximation for small angles commonly used in RSA.

Contribution

It introduces a mathematically sound definition of total rotation in RSA, linking Euler angles to the helical axis and validating the approximation for typical RSA angles.

Findings

Total rotation approximation error is between 5% and 7%.

The approximation is suitable for RSA's typical small angles.

Euler angles can be treated as a vector space under this approximation.

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.