On the derivation of a tensor to calculate six degree-of-freedom, musculotendon joint stiffness: Implications for stability and impedance analyses

Major joints, such as the knee, shoulder, and spine, can buckle along the translational degrees-of-freedom (DoF), causing injury to ligaments and other passive tissues. Despite this, stability and impedance analyses have focused primarily on the rotational DoF. As such, mathematical models quantifying musculotendon translational stiffnesses remain limited and, to our knowledge, there are no published works that explicitly describes the interactions between DoF. Using an energy approach, we derived a six DoF stiffness tensor and provided the necessary equations needed to quantify the musculotendon stiffness of any joint. Using a knee model, we then compared the derived stiffness tensor against two commonly used measures: one that excludes translational DoF and another that excludes interactions between DoF. We found that both of these measures had large over-estimations of stiffness, particularly for the rotational DoF, compared to our derived tensor. These findings indicate that previous analyses may have found rotational DoF to be stable when they were unstable.