Identification of ankle joint stiffness during passive movements--a subspace linear parameter varying approach

This paper describes a novel method for the identification of time-varying ankle joint dynamic stiffness during large passive movements. The method estimates a linear parameter varying parallel-cascade (LPV-PC) model of joint stiffness consisting of two pathways: (a) an LPV impulse response function (IRF) for intrinsic mechanics and (b) an LPV Hammerstein cascade with time-varying static nonlinearity and a time-invariant linear dynamics for the reflex pathway. A subspace identification technique is used to estimate a statespace representation of the reflex stiffness dynamics. Then, an orthogonal projection decouples intrinsic from reflex response and subsequently identifies an LPV-IRF model of intrinsic stiffness. Finally, an LPV model of the reflex static nonlinearity is estimated using an iterative, separable least squares method. The LPV method was validated using experimental data from two healthy subjects where the ankle was moved passively by an actuator through its range of motion first without and then with perturbations. The identification results demonstrated that (a) the dynamic response of the intrinsic pathway changes systematically with joint position; and (b) the static nonlinearity of the reflex pathway resembles a half-wave rectifier whose threshold decreases and gain increases as ankle is moved to dorsiflexed position.