Rotation Transformation Matrix of the Joint Coordinate System with the Application to the Knee Joint

Abstract

The human musculoskeletal system is comprised of various joints which are characterized by their unique morphology and kinematics. The joint coordinate system is tailored to comply with the observed mechanical properties of joints, making their research more intuitive and in accordance with standard clinical terminology. This coordinate system allows for spatial orientation to be described by three sequence independent Euler angles (the so-called “jaw-pitch-roll” angles), which mirrors the sequence independency of the final position of human limbs. The knee joint is the most complex, both anatomically and mechanically. It can be modeled as a four link open kinematic chain to which the joint coordinate system is applied. This coordinate system is usually described in Denavit-Hartenberg parameters, and the rotation transformation matrix is obtained using standard methods of vector algebra. In this paper another approach is used to derive the rotation transformation matrix, namely, the Rodrigues’ rotation formula. The utility of this algorithm is evident: the Rodrigues’ method provides an elegant solution which is obtained in a concise and efficient manner.