Revealing Time-Varying Joint Impedance With Kernel-Based Regression and Nonparametric Decomposition

During movements, humans continuously regulate their joint impedance to minimize control effort and optimize performance. Joint impedance describes the relationship between a joint’s position and torque acting around the joint. Joint impedance varies with joint angle and muscle activation and differs from trial-to-trial due to inherent variability in the human control system. In this paper, a dedicated time-varying system identification (SI) framework is developed involving a parametric, kernel-based regression, and nonparametric, “skirt decomposition,” SI method to monitor the time-varying joint impedance during a force task. Identification was performed on single trials and the estimators included little a priori assumptions regarding the underlying time-varying joint mechanics. During the experiments, six (human) participants used flexion of the wrist to apply a slow sinusoidal torque to the handle of a robotic manipulator, while receiving small position perturbations. Both methods revealed that the sinusoidal change in joint torque by activation of the wrist flexor muscles resulted in a sinusoidal time-varying joint stiffness and resonance frequency. A third-order differential equation allowed the parametric kernel-based estimator to explain on average 76% of the variance (range 52%–90%). The nonparametric skirt decomposition method could explain on average 84% of the variance (range 66%–91%). This paper presents a novel framework for identification of time-varying joint impedance by making use of linear time-varying models based on a single trial of data.