In this paper we propose the Middle-Axis-Rotation (MAR) parametrization of human limbs that addresses the ill-conditioned cases of analytical Inverse Kinematics (IK) algorithms. The MAR parametrization is singularity-free in the reach space of the human limbs. Unlike the swivel representation, it does not rely on the projection of an additional fixed vector. In addition, we express the joint limits of each joint of the limb in terms of the redundancy of the new decomposition. In the specific case of the upper limb, we analyse the contribution of the clavicle to produce biomechanically meaningful postures. We illustrate various real-time applications of this approach.