New Formulae to Approximate an Infinitesimal Rotation Followed or Preceded by a Large one

Rotations of the Cartesian reference frames are used extensively in many fields of research such as navigation and guidance systems, and motion tracking systems. Of particular interest is the case when an infinitesimal rotation is followed or preceded by a large rotation. If the order of rotations is required to be changed, there is no formula to describe a good approximation. Instead, researchers usually use dot product of the unit vectors of the coordinate systems before and after the reversal. In this work, we provide a direct formula for such approximation. Further, we show that the proposed formulae can be used to construct the inverse of any transformation comprising two successive rotations about different axes by taking many successive infinitesimal rotations approximated properly. Moreover, we present a case study for which the proposed formulae can be used to find the solution in a direct way compared to other techniques.