Dynamic joint stiffness defines the dynamic relationship between the position of a joint and the torque acting about it and can be separated into intrinsic and reflex components. Under stationary conditions, these can be identified using a nonlinear parallel-cascade algorithm that models intrinsic stiffness-a linear dynamic response to position-and reflex stiffness-a nonlinear dynamic response to velocity-as parallel pathways. Experiments using this method show that both intrinsic and reflex stiffness depend strongly on the operating point, defined by position and torque, likely because of some underlying nonlinear behavior not modeled by the parallel-cascade structure. Consequently, both intrinsic and reflex stiffness will appear to be time-varying whenever the operating point changes rapidly, as during movement. This paper describes and validates an extension of the parallel-cascade algorithm to time-varying conditions. It describes the ensemble method used to estimate time-varying intrinsic and reflex stiffness. Simulation results demonstrate that the algorithm can track rapid changes in joint stiffness accurately. Finally, the performance of the algorithm in the presence of noise is tested. We conclude that the new algorithm is a powerful new tool for the study of joint stiffness during functional tasks.

Previously, we described a time-varying, parallel-cascade system identification algorithm that estimates intrinsic and reflex stiffness dynamics. It uses an iterative technique, in conjunction with established, time-varying, identification methods, to estimate the two pathways from ensembles of input and output realizations having the same time-varying behavior. This paper presents the results of a study that systematically evaluated the performance of the algorithm. Simulations were used to determine the algorithm's ability to track rapid changes in dynamic stiffness, and quantify its performance limits. There was close agreement between the simulated and estimated joint stiffness demonstrating that the algorithm estimates stiffness correctly even when it changes rapidly. However, the algorithm's ability to identify the reflex pathway was shown to depend on the relative contributions of the intrinsic and reflex pathways to the overall torque. As the intrinsic contribution to the output grew it became increasingly difficult to identify the reflex pathway accurately. The quality of the reflex identification greatly improved as the number of realizations in the data ensembles increased. More realizations were needed as the signal-to-noise ratio decreased and the relative contribution of the reflex pathway decreased. For good results, under typical time-varying experimental conditions, between 500 and 800 realizations are required.