3D Kinematics: Using Quaternions for Modeling Orientation and Rotations in Biomechanics

Abstract

Analysis of human movement requires mathematical formulation of rotations in three-dimensional space. The need for appropriate description of rotations occurs both in the analysis of experimental data and in the modeling of human movement. To capture the characteristics human body motion using experimental data, one often needs to initially assume joints with three rotational degrees of freedom (spherical joints). Then, using measured data, one can determine the number of rotational degrees of freedom of a joint under consideration. In complex whole-body movements, such as in various athletic disciplines, the athlete may undergo a rotation in three-dimensional rotation that requires also adequate mathematical formulation. In this chapter we introduce a mathematical representation of orientation and rotation that is based on the notion of quaternions. The quaternion-based representation of rotations has a physical interpretation that is meaningful in biomechanical context. It is computationally efficient since it has small number of parameters with the least number of constraints without mathematical singularities that lead to false results and numerical instability. Another important feature of the quaternion formulation is that successive rotations that often occur in human movements are easily composed within the representation. Finally, efficient identification and interpolation procedures of data obtained experimentally based on the quaternion representation are presented.