Identification of physiological systems: estimation of linear time-varying dynamics with non-white inputs and noisy outputs

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.

Estimation of the Two Degrees-of-Freedom Time-Varying Impedance of the Human Ankle

An understanding of the time-varying mechanical impedance of the ankle during walking is fundamental in the design of active ankle-foot prostheses and lower extremity rehabilitation devices. This paper describes the estimation of the time-varying mechanical impedance of the human ankle in both dorsiflexion–plantarflexion (DP) and inversion–eversion (IE) during walking in a straight line. The impedance was estimated using a two degrees-of-freedom (DOF) vibrating platform and instrumented walkway. The perturbations were applied at eight different axes of rotation combining different amounts of DP and IE rotations of four male subjects. The observed stiffness and damping were low at heel strike, increased during the mid-stance, and decreases at push-off. At heel strike, it was observed that both the damping and stiffness were larger in IE than in DP. The maximum average ankle stiffness was 5.43 N·m/rad/kg at 31% of the stance length (SL) when combining plantarflexion and inversion and the minimum average was 1.14 N·m/rad/kg at 7% of the SL when combining dorsiflexion and eversion. The maximum average ankle damping was 0.080 Nms/rad/kg at 38% of the SL when combining plantarflexion and inversion, and the minimum average was 0.016 Nms/rad/kg at 7% of the SL when combining plantarflexion and eversion. From 23% to 93% of the SL, the largest ankle stiffness and damping occurred during the combination of plantarflexion and inversion or dorsiflexion and eversion. These rotations are the resulting motion of the ankle's subtalar joint, suggesting that the role of this joint and the muscles involved in the ankle rotation are significant in the impedance modulation in both DP and IE during gait.

Summary of Human Ankle Mechanical Impedance During Walking

The human ankle joint plays a critical role during walking and understanding the biomechanical factors that govern ankle behavior and provides fundamental insight into normal and pathologically altered gait. Previous researchers have comprehensively studied ankle joint kinetics and kinematics during many biomechanical tasks, including locomotion; however, only recently have researchers been able to quantify how the mechanical impedance of the ankle varies during walking. The mechanical impedance describes the dynamic relationship between the joint position and the joint torque during perturbation, and is often represented in terms of stiffness, damping, and inertia. The purpose of this short communication is to unify the results of the first two studies measuring ankle mechanical impedance in the sagittal plane during walking, where each study investigated differing regions of the gait cycle. Rouse et al. measured ankle impedance from late loading response to terminal stance, where Lee et al. quantified ankle impedance from pre-swing to early loading response. While stiffness component of impedance increases significantly as the stance phase of walking progressed, the change in damping during the gait cycle is much less than the changes observed in stiffness. In addition, both stiffness and damping remained low during the swing phase of walking. Future work will focus on quantifying impedance during the “push off” region of stance phase, as well as measurement of these properties in the coronal plane.

Mechanical Impedance of the Non-loaded Lower Leg with Relaxed Muscles in the Transverse Plane

This paper describes the protocols and results of the experiments for the estimation of the mechanical impedance of the humans' lower leg in the External-Internal direction in the transverse plane under non-load bearing condition and with relaxed muscles. The objectives of the estimation of the lower leg's mechanical impedance are to facilitate the design of passive and active prostheses with mechanical characteristics similar to the humans' lower leg, and to define a reference that can be compared to the values from the patients suffering from spasticity. The experiments were performed with 10 unimpaired male subjects using a lower extremity rehabilitation robot (Anklebot, Interactive Motion Technologies, Inc.) capable of applying torque perturbations to the foot. The subjects were in a seated position, and the Anklebot recorded the applied torques and the resulting angular movement of the lower leg. In this configuration, the recorded dynamics are due mainly to the rotations of the ankle's talocrural and the subtalar joints, and any contribution of the tibiofibular joints and knee joint. The dynamic mechanical impedance of the lower leg was estimated in the frequency domain with an average coherence of 0.92 within the frequency range of 0-30 Hz, showing a linear correlation between the displacement and the torques within this frequency range under the conditions of the experiment. The mean magnitude of the stiffness of the lower leg (the impedance magnitude averaged in the range of 0-1 Hz) was determined as 4.9 ± 0.74 Nm/rad. The direct estimation of the quasi-static stiffness of the lower leg results in the mean value of 5.8 ± 0.81 Nm/rad. An analysis of variance shows that the estimated values for the stiffness from the two experiments are not statistically different.

Dynamic primitives in the control of locomotion

Humans achieve locomotor dexterity that far exceeds the capability of modern robots, yet this is achieved despite slower actuators, imprecise sensors, and vastly slower communication. We propose that this spectacular performance arises from encoding motor commands in terms of dynamic primitives. We propose three primitives as a foundation for a comprehensive theoretical framework that can embrace a wide range of upper- and lower-limb behaviors. Building on previous work that suggested discrete and rhythmic movements as elementary dynamic behaviors, we define submovements and oscillations: as discrete movements cannot be combined with sufficient flexibility, we argue that suitably-defined submovements are primitives. As the term “rhythmic” may be ambiguous, we define oscillations as the corresponding class of primitives. We further propose mechanical impedances as a third class of dynamic primitives, necessary for interaction with the physical environment. Combination of these three classes of primitive requires care. One approach is through a generalized equivalent network: a virtual trajectory composed of simultaneous and/or sequential submovements and/or oscillations that interacts with mechanical impedances to produce observable forces and motions. Reliable experimental identification of these dynamic primitives presents challenges: identification of mechanical impedances is exquisitely sensitive to assumptions about their dynamic structure; identification of submovements and oscillations is sensitive to their assumed form and to details of the algorithm used to extract them. Some methods to address these challenges are presented. Some implications of this theoretical framework for locomotor rehabilitation are considered.

System identification of physiological systems using short data segments

System identification of physiological systems poses unique challenges, especially when the structure of the system under study is uncertain. Nonparametric techniques can be useful for identifying system structure, but these typically assume stationarity and require large amounts of data. Both of these requirements are often not easily obtained in the study of physiological systems. Ensemble methods for time-varying nonparametric estimation have been developed to address the issue of stationarity, but these require an amount of data that can be prohibitive for many experimental systems. To address this issue, we developed a novel algorithm that uses multiple short data segments. Using simulation studies, we showed that this algorithm produces system estimates with lower variability than previous methods when limited data are present. Furthermore, we showed that the new algorithm generates time-varying system estimates with lower total error than an ensemble method. Thus, this algorithm is well suited for the identification of physiological systems that vary with time or from which only short segments of stationary data can be collected.

Identification of time-varying intrinsic and reflex joint stiffness

Dynamic joint stiffness defines the dynamic relationship between the position of a joint and the torque acting about it and can be separated into intrinsic and reflex components. Under stationary conditions, these can be identified using a nonlinear parallel-cascade algorithm that models intrinsic stiffness-a linear dynamic response to position-and reflex stiffness-a nonlinear dynamic response to velocity-as parallel pathways. Experiments using this method show that both intrinsic and reflex stiffness depend strongly on the operating point, defined by position and torque, likely because of some underlying nonlinear behavior not modeled by the parallel-cascade structure. Consequently, both intrinsic and reflex stiffness will appear to be time-varying whenever the operating point changes rapidly, as during movement. This paper describes and validates an extension of the parallel-cascade algorithm to time-varying conditions. It describes the ensemble method used to estimate time-varying intrinsic and reflex stiffness. Simulation results demonstrate that the algorithm can track rapid changes in joint stiffness accurately. Finally, the performance of the algorithm in the presence of noise is tested. We conclude that the new algorithm is a powerful new tool for the study of joint stiffness during functional tasks.

Performance evaluation of an algorithm for the identification of time-varying joint stiffness

Previously, we described a time-varying, parallel-cascade system identification algorithm that estimates intrinsic and reflex stiffness dynamics. It uses an iterative technique, in conjunction with established, time-varying, identification methods, to estimate the two pathways from ensembles of input and output realizations having the same time-varying behavior. This paper presents the results of a study that systematically evaluated the performance of the algorithm. Simulations were used to determine the algorithm's ability to track rapid changes in dynamic stiffness, and quantify its performance limits. There was close agreement between the simulated and estimated joint stiffness demonstrating that the algorithm estimates stiffness correctly even when it changes rapidly. However, the algorithm's ability to identify the reflex pathway was shown to depend on the relative contributions of the intrinsic and reflex pathways to the overall torque. As the intrinsic contribution to the output grew it became increasingly difficult to identify the reflex pathway accurately. The quality of the reflex identification greatly improved as the number of realizations in the data ensembles increased. More realizations were needed as the signal-to-noise ratio decreased and the relative contribution of the reflex pathway decreased. For good results, under typical time-varying experimental conditions, between 500 and 800 realizations are required.

Identification of time-varying intrinsic and reflex joint stiffness

We have developed a time-varying, parallel- cascade system identification algorithm to separate joint stiffness into intrinsic and reflex components at each point in time throughout rapid movements. The components are identified using an iterative algorithm in which intrinsic and reflex dynamics are identified using separate time-varying (TV) techniques based on ensemble methods. An ensemble of input-output records having the same TV behavior is acquired and used to identify the system dynamics as impulse response functions at time increments corresponding to the sampling interval. Simulation studies showed that the time-varying, parallel-cascade algorithm performed well under realistic conditions with 99.9% VAF between simulated and predicted torque. To evaluate the performance of the algorithm under realistic conditions we applied it to an ensemble of experimental data acquired under stationary conditions. Results demonstrated that the TV estimates converged to those of the established time-invariant algorithm and allowed us to determine how variance of the TV estimates varied with the number of realizations in the ensemble.

Time-varying parallel-cascade system identification of ankle stiffness from ensemble data

Measurement of joint dynamic stiffness during time-varying conditions is crucial to understand the role of joint mechanics during movement. Stiffness can be separated into intrinsic and reflex components, and are modeled as linear dynamic and Hammerstein systems, respectively. Time-varying identification methods using ensemble data have been developed previously for both pathways and were tested separately on simulated data. In this study, these algorithms were integrated into the time-varying, parallel-cascade identification method. Ankle dynamics were modeled during a ramp input and simulated impulse response functions (IRFs) were generated. Gaussian white noise was low-pass filtered and was convolved with the simulated systems over 500 realizations. The ensemble data was used to evaluate the new identification technique. The mean variances accounted for (VAFs) between the true and identified IRFs for the intrinsic and reflex pathways were 99.9% and 97.7%, respectively, demonstrating the technique's strong ability to predict the system's dynamics.