Optimal statistical operators for 3-dimensional rotational data: geometric interpretations and application to prosthesis kinematics

Rotational data in the form of measured three-dimensional rotations or orientations arise naturally in many fields of science, including biomechanics, orthopaedics and robotics. The cyclic topology of rotation spaces calls for special care and considerations when performing statistical analysis of rotational data. Relevant theory has been developed during the last three decades, and has become a standard tool in some areas. In relation to the study of human kinematics and motion however, these concepts have hardly been put to use. This paper gives an introduction to the intricacies of three-dimensional rotations, and provides a thorough geometric interpretation of several approaches to averaging rotational data. A set of novel, simple operators is presented. Simulations and a prosthetics-related real-world example involving wrist kinematics illuminate important aspects of the results. Finally generalizations and related subjects for further research are suggested.