Non-Parametric Nonlinear Parameter-Varying Parallel-Cascade Identification of Dynamic Joint Stiffness

Objective: The paper presents a method to identify ankle joint dynamic stiffness during functional tasks where intrinsic and reflex stiffness change with a time-varying scheduling variable (SV), such as joint position or torque. Methods: The method models joint stiffness with two pathways: (1) A parameter-varying (PV) impulse response function (IRF) describing intrinsic stiffness; and (2) a reflex stiffness model comprising a PV static nonlinearity followed by a PV linear element. Results: Monte-Carlo simulations demonstrated that the method accurately estimated all elements of the intrinsic and reflex pathways as they changed with a SV. Experimental results with a healthy individual subjected to large, imposed ankle movements demonstrated that: (a) Intrinsic stiffness changed substantially as a function of ankle position; elasticity was lowest near the mid-position and increased with either dorsiflexion or plantarflexion. (b) Reflex gain increased and the velocity threshold for reflex excitation decreased monotonically with ankle dorsiflexion. (c) Reflex dynamics resembled a second-order, low-pass system that was invariant with ankle position. (d) The identified PV Parallel-Cascade (PC) model accurately predicted the torque response to novel trajectories of ankle movement. Conclusion: The PV-PC method can accurately and reliably estimate how intrinsic and reflex stiffness change with a time-varying SV. Significance: The method is novel with multiple advantages: (a) It provides a unified algorithm that characterizes the changes in the parameters of all joint stiffness elements needed to understand their role in postural/movement control; (b) It is efficient requiring only two trials; (c) The models identified can predict the joint stiffness response to novel movements informing orthoses and prostheses design.