A novel method for detecting inherent spin in rigid-body rotation

This paper presents a novel method for detecting inherent spin in rigid-body rotation in space through a comparative analysis of alternative kinematic representations. The study examines elementary sequential rotations, Euler-vector-based formulations, conic coordinates, orthodromic motion, and quaternions, emphasizing how each description captures or obscures the rotational spin associated with the body motion. To achieve this, we utilize a mnemonic representation that streamlines the description of angles and apply the principles of tilde and hat matrices. The main contribution of the work is a unified framework that makes the intrinsic spin explicit and allows direct comparison between rotation descriptions that lead to the same final orientation but differ along the trajectory. The significance of the findings does not stem from innovation but rather from the methodology employed to track the phenomena and the results derived. The proposed approach is particularly useful for identifying when a representation introduces unnecessary spin or when a spin-free motion can be preserved by an appropriate choice of coordinates. Numerical examples illustrate the differences among the formulations and show how the proposed method improves the interpretation of rigid-body rotation in space.