Cartesian stiffness for wrist joints: analysis on the Lie group of 3D rotations and geometric approximation for experimental evaluation

This paper is concerned with the analysis and the numerical evaluation from experimental measurements of the static, Cartesian stiffness of wrist joints, in particular the human wrist. The primary aim is to extend from Euclidean spaces to so(3), the group of rigid body rotations, previous methods for assessing the end-point stiffness of the human arm, typically performed via a robotic manipulandum. As a first step, the geometric definition of Cartesian stiffness from current literature is specialised to the group so(3). Emphasis is placed on the choice of the unique, natural, affine connection on so(3) which guarantees symmetry of the stiffness matrix in presence of conservative fields for any configuration, also out of equilibrium. As the main contribution of this study, a coordinate-independent approximation based on the geometric notion of geodesics is proposed which provides a working equation for evaluating stiffness directly from experimental measurements. Finally, a graphical representation of the stiffness is discussed which extends the ellipse method often used for end-point stiffness visualisation and which is suitable to compare stiffness matrices evaluated at different configurations.