Parametrization and Range of Motion of the Ball-and-Socket Joint

The ball-and-socket joint model is used to represent articulations with three rotational degrees of freedom (DOF), such as the human shoulder and the hip. The goal of this paper is to discuss two related problems: the parametrization and the deÞnition of realistic joint boundaries for ball-andsocket joints. Doing this accurately is difÞcult, yet important for motion generators (such as inverse kinematics and dynamics engines) and for motion manipulators (such as motion retargeting), since the resulting motions should satisfy the anatomic constraints. The difÞculty mainly comes from the complex nature of 3D orientations and of human articulations. The underlying question of parametrization must be addressed before realistic and meaningful boundaries can be deÞned over the set of 3D orientations. In this paper, we review and compare several known methods, and advocate the use of the swing-and-twist parametrization, that partitions an arbitrary orientation into two meaningful components. The related problem of induced twist is discussed. Finally, we review some joint boundaries representations based on this decomposition, and show an example.