Stability of bipedal stance: the contribution of cocontraction and spindle feedback

The aim of this study is to assess the contribution of cocontraction and spindle feedback to local stability during bipedal stance. To that aim, an existing nonlinear state space model of the human musculoskeletal system is linearized in a reference equilibrium state. The maximal real part of the eigenvalues of the linearized system matrix A and the low-frequency joint stiffness are used as a measure of local stability. Muscle properties, as represented in a Hill-type muscle model, are shown to improve the behavior, the improvement being larger at high cocontraction. However, even at maximal cocontraction the low-frequency joint stiffness generated by the muscle properties is insufficient to yield a locally stable system. It follows that feedback is necessary to ensure local stability. In this study, the potential contribution of spindle feedback is investigated by optimizing the feedback gains for contractile element length and velocity for each muscle. It is found that in the case of time-delayed negative feedback, it is impossible to stabilize the system on the basis of spindle feedback. When positive time-delayed feedback is allowed, a barely stable system is obtained. When the time delays are removed, the feedback gains can be chosen such that a locally stable system is obtained, indicating the limitations imposed by the presence of time delays. Finally, it is shown that for small perturbations the response of the linear system to an arbitrary perturbation is similar to that of the nonlinear system, indicating the validity of the approach used. It is concluded that the combination of muscle properties and time-delayed spindle feedback is insufficient to obtain a system with reasonable local stability.