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

Abstract

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 thirdorder 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 timevarying joint impedance by making use of linear time-varying models based on a single trial of data.

Separating neural influences from peripheral mechanics: the speed-curvature relation in mechanically constrained actions

“…Interestingly, smoothness can also account for the relation between hand path curvature and tangential speed. For simple curves (e.g., ellipses), this is the so-called "two-thirds power law," as hand speed decreases at regions with higher curvature in a robust power law (Lacquaniti et al 1983;Richardson and Flash 2002;Schaal and Sternad 2001;Sternad and Schaal 1999;Viviani and Flash 1995). More complex curves display more complex relations, a "spectrum of power laws" (Huh and Sejnowski 2015).…”

“…In principle, knowledge of mechanical impedance combined with simultaneous measurement of force and motion during object manipulation would allow us to "subtract off" or "peel back" peripheral biomechanics to uncover underlying neural influences. In practice, mechanical impedance is nonlinear and time-varying, and measuring it during movement, although possible, is challenging (Bennett et al 1992;Guarín and Kearney 2017;Lacquaniti et al 1993;Lee et al 2016;Lee and Hogan 2015;Rouse et al 2013Rouse et al , 2014van de Ruit et al 2020). Moreover, measurement inevitably introduces perturbations that may alter behavior.…”

Unifying system identification and biomechanical formulations for the estimation of muscle, tendon and joint stiffness during human movement

“…In equation ( 1), the system dynamics are modeled by means of the TV FRF, while in the SDS the TV IRF is used instead. Since the TV IRF is the time-domain equivalent of the TV FRF [39], the same considerations hold.…”

“…Equation with quasi-param. time variation van de Ruit 2020 [39] Lumped joint impedance modeled by a second order IBK difference equation from angle to torque. The statistical properties of the parameters in time are expressed by a kernel, which assumes smooth behavior.…”

Locally Periodic Kernel-Based Regression to Identify Time-Varying Ankle Impedance During Locomotion: a Simulation Study

“…The KBR method has been previously applied to the identification of joint impedance in [6]. In the current study, the method is modified to be applicable to ankle joint impedance identification during (quasi)periodic data, like locomotion.…”