A spherical rotation coordinate system for the description of three-dimensional joint rotations

Three-dimensional joint rotations in human movement analysis have been mainly described by Euler/Cardan angles. Due to sequence dependence, each combination of three Euler/Cardan angles defines a single pattern of joint rotation. When the rotation pattern is unknown, it needs to be considered using a particular sequence of Euler/Cardan angles to represent joint rotations. In this paper a spherical rotation coordinate system is developed for describing three-dimensional joint rotations using a method of rotation involving two steps: a long axis rotation and a pure axial rotation. Two angles of the classical spherical coordinate system--longitude and latitude--are used to describe long axis rotations in this newly proposed coordinate system. The spherical rotation coordinate system uses a radial rotation angle to describe pure axial rotation of a limb segment whereas the classical spherical coordinate system uses a radial displacement to describe motion of a point. An application of the spherical rotation coordinate system is given to define three-dimensional rotations of the glenohumeral joint. A mathematical proof shows that the long axis rotation and axial rotation are sequence independent. Two numerical examples are investigated which demonstrate that the spherical rotation angles can be uniquely determined in both forward and inverse kinematics without considering sequences rotations.