Comparison of time-varying system identification methods to assess joint impedance

Abstract

Joint impedance describes the dynamic resistance of a joint in response to position perturbations. Joint impedance is known to vary nonlinearly during movement caused by varying joint angle and muscle activation. The nonlinear behaviour can be described using linear time-varying models under certain assumption. Recently a number of time-varying system identification algorithms to estimate joint impedance have been developed. System identification algorithms are typically validated in simulation. These algorithms have not yet been compared using the same simulated data or validated using real experimental data in this application. In this study the algorithms are assessed on their ability to estimate joint stiffness using three data sets. The first data set is from a simulation model representing joint dynamics with time-varying stiffness and damping. The second data set is from a mechanical variable stiffness device. A mapping of the true stiffness of the device was extracted by interpolating the estimated stiffness of time-invariant trials. The last data set is real experimental data from a human ankle with varying contraction levels to which small position perturbations were applied. The simulation study and the experimental mechanical study suggest that when estimating stiffness the linear parameter varying (LPV) method has a bias, the kernel based regression (KBR) method overall has the highest error, the ensemble impulse response function (eIRF) method needs many repetitions, the basis impulse response function (bIRF) method is able to achieve the lowest error, the short data segment (SDS) method is the most robust to different perturbation signals, and the ensemble spectral methods (ESM and mESM) are able to achieve reasonable results. The results of the experimental human study show that the estimated stiffness by the ensemble and short data segment methods have a trend similar to that of the EMG signal, albeit with different offsets. The bIRF, SDS, ESM and mESM make a reasonable compromise between smoothness and required repetitions.