Measurement of Dynamic Joint Stiffness from Multiple Short Data Segments

Virtual Stiffness: A Novel Biomechanical Approach to Estimate Limb Stiffness of a Multi-Muscle and Multi-Joint System

In recent years, different groups have developed algorithms to control the stiffness of a robotic device through the electromyographic activity collected from a human operator. However, the approaches proposed so far require an initial calibration, have a complex subject-specific muscle model, or consider the activity of only a few pairs of antagonist muscles. This study described and tested an approach based on a biomechanical model to estimate the limb stiffness of a multi-joint, multi-muscle system from muscle activations. The "virtual stiffness" method approximates the generated stiffness as the stiffness due to the component of the muscle-activation vector that does not generate any endpoint force. Such a component is calculated by projecting the vector of muscle activations, estimated from the electromyographic signals, onto the null space of the linear mapping of muscle activations onto the endpoint force. The proposed method was tested by using an upper-limb model made of two joints and six Hill-type muscles and data collected during an isometric force-generation task performed with the upper limb. The null-space projection of the muscle-activation vector approximated the major axis of the stiffness ellipse or ellipsoid. The model provides a good approximation of the voluntary stiffening performed by participants that could be directly implemented in wearable myoelectric controlled devices that estimate, in real-time, the endpoint forces, or endpoint movement, from the mapping between muscle activation and force, without any additional calibrations.

A Clustering-Based Approach to Identify Joint Impedance During Walking

Mechanical impedance, which changes with posture and muscle activations, characterizes how the central nervous system regulates the interaction with the environment. Traditional approaches to impedance estimation, based on averaging of movement kinetics, requires a large number of trials and may introduce bias to the estimation due to the high variability in a repeated or periodic movement. Here, we introduce a data-driven modeling technique to estimate joint impedance considering the large gait variability. The proposed method can be used to estimate impedance in both the stance and swing phases of walking. A 2-pass clustering approach is used to extract groups of unperturbed gait data and estimate candidate baselines. Then patterns of perturbed data are matched with the most similar unperturbed baseline. The kinematic and torque deviations from the baselines are regressed locally to compute joint impedance at different gait phases. Simulations using the trajectory data of a subject's gait at different speeds demonstrate a more accurate estimation of ankle stiffness and damping with the proposed clustering-based method when compared with two methods: i) using average unperturbed baselines, and ii) matching shifted and scaled average unperturbed velocity baselines. Furthermore, the proposed method requires fewer trials than methods based on average unperturbed baselines. The experimental results on human hip impedance estimation show the feasibility of clustering-based technique and verifies that it reduces the estimation variability.

Unbiased Estimation of Human Joint Intrinsic Mechanical Properties During Movement

The overall mechanical properties of a joint are generated by a combination intrinsic (mechanical) and reflex (neural) mechanisms. Nevertheless, many methods for estimating joint mechanical properties have used a linear dynamic model whose parameters are commonly related to the joint inertial and visco-elastic properties. Such mechanical models cannot account for torques due to reflex mechanisms and consequently fitting them to data containing reflex torques can give biased results. This paper addresses this issue in two ways. First, using simulation studies, it demonstrates that fitting linear dynamic models in the presence of reflex torques will indeed provide biased estimates of intrinsic joint properties; the bias is significant for small reflex torques and increases proportionally with reflex torque magnitude. Second, it develops and validates a novel approach to accurately estimate the time-varying, intrinsic mechanical properties of a joint in the presence of reflex torques. The approach involves applying small position perturbations to the joint trajectory and then applying novel mathematical models and system identification techniques to analytically separate the measured total joint torque into its intrinsic and reflex components. The method first estimates a non-parametric, reflex electromyography-torque model, and uses it to predict the reflex torques which is subtracted from the total torque. Then, it estimates a non-parametric, linear, and time-varying model of the intrinsic mechanical properties from the residuals. Simulation results demonstrate that the new approach accurately tracks time-varying joint intrinsic mechanical properties during movement independently of the reflex torque magnitude. The new algorithm will be a useful tool in the study of motor control, as it supports the unbiased estimation of joint intrinsic mechanical properties during movement in the presence of reflex torques.

Estimation of Time-Varying, Intrinsic and Reflex Dynamic Joint Stiffness during Movement. Application to the Ankle Joint

Dynamic joint stiffness determines the relation between joint position and torque, and plays a vital role in the control of posture and movement. Dynamic joint stiffness can be quantified during quasi-stationary conditions using disturbance experiments, where small position perturbations are applied to the joint and the torque response is recorded. Dynamic joint stiffness is composed of intrinsic and reflex mechanisms that act and change together, so that nonlinear, mathematical models and specialized system identification techniques are necessary to estimate their relative contributions to overall joint stiffness. Quasi-stationary experiments have demonstrated that dynamic joint stiffness is heavily modulated by joint position and voluntary torque. Consequently, during movement, when joint position and torque change rapidly, dynamic joint stiffness will be Time-Varying (TV). This paper introduces a new method to quantify the TV intrinsic and reflex components of dynamic joint stiffness during movement. The algorithm combines ensemble and deterministic approaches for estimation of TV systems; and uses a TV, parallel-cascade, nonlinear system identification technique to separate overall dynamic joint stiffness into intrinsic and reflex components from position and torque records. Simulation studies of a stiffness model, whose parameters varied with time as is expected during walking, demonstrated that the new algorithm accurately tracked the changes in dynamic joint stiffness using as little as 40 gait cycles. The method was also used to estimate the intrinsic and reflex dynamic ankle stiffness from an experiment with a healthy subject during which ankle movements were imposed while the subject maintained a constant muscle contraction. The method identified TV stiffness model parameters that predicted the measured torque very well, accounting for more than 95% of its variance. Moreover, both intrinsic and reflex dynamic stiffness were heavily modulated through the movement in a manner that could not be predicted from quasi-stationary experiments. The new method provides the tool needed to explore the role of dynamic stiffness in the control of movement.

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 subject's 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.