Ehsan Sobhani Tehrani

A Subspace Approach to the Structural Decomposition and Identification of Ankle Joint Dynamic Stiffness

Objective: The purpose of this paper is to present a structural decomposition subspace (SDSS) method for decomposition of the joint torque to intrinsic, reflexive, and voluntary torques and identification of joint dynamic stiffness. Methods: First, it formulates a novel state-space representation for the joint dynamic stiffness modeled by a parallel-cascade structure with a concise parameter set that provides a direct link between the state-space representation matrices and the parallel-cascade parameters. Second, it presents a subspace method for the identification of the new state-space model that involves two steps: 1) the decomposition of the intrinsic and reflex pathways and 2) the identification of an impulse response model of the intrinsic pathway and a Hammerstein model of the reflex pathway. Results: Extensive simulation studies demonstrate that SDSS has significant performance advantages over some other methods. Thus, SDSS was more robust under high noise conditions, converging where others failed; it was more accurate, giving estimates with lower bias and random errors. The method also worked well in practice and yielded high-quality estimates of intrinsic and reflex stiffnesses when applied to experimental data at three muscle activation levels. Conclusion: The simulation and experimental results demonstrate that SDSS accurately decomposes the intrinsic and reflex torques and provides accurate estimates of physiologically meaningful parameters. Significance: SDSS will be a valuable tool for studying joint stiffness under functionally important conditions. It has important clinical implications for the diagnosis, assessment, objective quantification, and monitoring of neuromuscular diseases that change the muscle tone.

Linear parameter varying identification of ankle joint intrinsic stiffness during imposed walking movements

This paper describes a novel model structure and identification method for the time-varying, intrinsic stiffness of human ankle joint during imposed walking (IW) movements. The model structure is based on the superposition of a large signal, linear, time-invariant (LTI) model and a small signal linear-parameter varying (LPV) model. The methodology is based on a two-step algorithm; the LTI model is first estimated using data from an unperturbed IW trial. Then, the LPV model is identified using data from a perturbed IW trial with the output predictions of the LTI model removed from the measured torque. Experimental results demonstrate that the method accurately tracks the continuous-time variation of normal ankle intrinsic stiffness when the joint position changes during the IW movement. Intrinsic stiffness gain decreases from full plantarflexion to near the mid-point of plantarflexion and then increases substantially as the ankle is dosriflexed.

Ankle Joint Intrinsic Dynamics is More Complex than a Mass-Spring-Damper Model

This paper describes a new small signal parametric model of ankle joint intrinsic mechanics in normal subjects. We found that intrinsic ankle mechanics is a third-order system and the second-order mass-spring-damper model, referred to as IBK, used by many researchers in the literature cannot adequately represent ankle dynamics at all frequencies in a number of important tasks. This was demonstrated using experimental data from five healthy subjects with no voluntary muscle contraction and at seven ankle positions covering the range of motion. We showed that the difference between the new third-order model and the conventional IBK model increased from dorsi to plantarflexed position. The new model was obtained using a multi-step identification procedure applied to experimental input/output data of the ankle joint. The procedure first identifies a non-parametric model of intrinsic joint stiffness where ankle position is the input and torque is the output. Then, in several steps, the model is converted into a continuous-time transfer function of ankle compliance, which is the inverse of stiffness. Finally, we showed that the third-order model is indeed structurally consistent with agonist–antagonist musculoskeletal structure of human ankle, which is not the case for the IBK model.

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.

A novel algorithm for linear parameter varying identification of Hammerstein systems with time-varying nonlinearities

This paper describes a novel method for the identification of Hammerstein systems with time-varying (TV) static nonlinearities and time invariant (TI) linear elements. This paper develops a linear parameter varying (LPV) state-space representation for such systems and presents a subspace identification technique that gives individual estimates of the Hammerstein components. The identification method is validated using simulated data of a TV model of ankle joint reflex stiffness where the threshold and gain of the model change as nonlinear functions of an exogenous signal. Pilot experiment of TV reflex EMG response identification in normal ankle joint during an imposed walking task demonstrate systematic changes in the reflex nonlinearity with the trajectory of joint position.

The characterization of the kinematic and dynamic properties of the ankle joint for an artificial ankle joint design

The kinematic and dynamic properties of the ankle joint (talocrural region) in normal subjects provide important information for the design of rehabilitation robots, below-knee prostheses, ankle-foot orthoses, and exoskeletons. This paper presents a quantitative analysis of published experimental data, simulation studies of human gait, and a dynamic model of ankle joint intrinsic and reflex stiffness to determine design requirements for such ankle devices to operate in the sagittal plane (i.e. ankle plantarflexion/dorsiflexion). The design requirements are derived in terms of average torque, rotatum, range of motion, velocity, acceleration, system bandwidth, torque-velocity curve, and the torque probability density function.

Identification of time-varying dynamics of reflex EMG in the ankle plantarflexors during time-varying, isometric contractions.

The dynamic relationship between the joint position and reflex EMG in ankle muscles of healthy human subjects was studied for time-varying (TV) contractions. A linear parameter varying (LPV) identification algorithm was used to estimate the Hammerstein system relating ankle position to the reflex EMG response. The estimated Hammerstein system comprised a time-invariant (TI) linear element and a TV static nonlinearity that resembled a half-wave rectifier with a threshold and linear gain. The results demonstrated a systematic change in the reflex nonlinearity with the activation level. The gain of TV nonlinearity increased with activation level reaching its peak at 20-30% maximum voluntary contraction and then decreased. The threshold of the nonlinearity decreased with increasing activation level reaching it minimum at the same point where the gain was maximal. Using the LPV-Hammerstein method in this work, the underlying TV dynamics were extracted from small number of trials. Thus, this method can be used to study stretch reflexes in subjects with neuromuscular disorders.

Linear Parameter Varying Identification of Dynamic Joint Stiffness during Time-Varying Voluntary Contractions

Dynamic joint stiffness is a dynamic, nonlinear relationship between the position of a joint and the torque acting about it, which can be used to describe the biomechanics of the joint and associated limb(s). This paper models and quantifies changes in ankle dynamic stiffness and its individual elements, intrinsic and reflex stiffness, in healthy human subjects during isometric, time-varying (TV) contractions of the ankle plantarflexor muscles. A subspace, linear parameter varying, parallel-cascade (LPV-PC) algorithm was used to identify the model from measured input position perturbations and output torque data using voluntary torque as the LPV scheduling variable (SV). Monte-Carlo simulations demonstrated that the algorithm is accurate, precise, and robust to colored measurement noise. The algorithm was then used to examine stiffness changes associated with TV isometric contractions. The SV was estimated from the Soleus EMG using a Hammerstein model of EMG-torque dynamics identified from unperturbed trials. The LPV-PC algorithm identified (i) a non-parametric LPV impulse response function (LPV IRF) for intrinsic stiffness and (ii) a LPV-Hammerstein model for reflex stiffness consisting of a LPV static nonlinearity followed by a time-invariant state-space model of reflex dynamics. The results demonstrated that: (a) intrinsic stiffness, in particular ankle elasticity, increased significantly and monotonically with activation level; (b) the gain of the reflex pathway increased from rest to around 10–20% of subjects MVC and then declined; and (c) the reflex dynamics were second order. These findings suggest that in healthy human ankle, reflex stiffness contributes most at low muscle contraction levels, whereas, intrinsic contributions monotonically increase with activation level.

Design of a hydraulic ankle-foot orthosis

This paper presents the design and simulation of an ankle-foot orthosis (AFO) to assist human walking. Design requirements were established based on a quantitative study of published data, simulations of human walking, and a model of intrinsic and reflex ankle joint stiffness. The design of an AFO that meets these requirements is then presented; it comprises a small linear, hydraulic actuator, a servo-valve, a hydraulic power supply, and an accumulator. Two methods of selecting the kinematic parameters of the AFO are introduced. One is based on force minimization and the other on compactness maximization. The performance expected of the AFO is demonstrated in a simulation study.