Closed-Loop Identification: Application to the Estimation of Limb Impedance in a Compliant Environment

The force and position data used to construct models of limb impedance are often obtained from closed-loop experiments. If the system is tested in a stiff environment, it is possible to treat the data as if they were obtained in open loop. However, when limb impedance is studied in a compliant environment, the presence of feedback cannot be ignored. While unbiased estimates of a system can be obtained directly using the prediction error method, the same cannot be said when linear regression or correlation analysis is used to fit nonparametric time- or frequency-domain models. We develop a prediction error minimization-based identification method for a nonparametric time-domain model augmented with a parametric noise model. The identification algorithm is tested on a dynamic mass-spring-damper system and returns consistent estimates of the system´s properties under both stiff and compliant feedback control. The algorithm is then used to estimate the impedance of a human elbow joint in both stiff and compliant environments.