Daniel Ludvig

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.

Voluntary modulation of human stretch reflexes

It has been postulated that the central nervous system (CNS) can tune the mechanical behavior of a joint by altering reflex stiffness in a task-dependant manner. However, most of the evidence supporting this hypothesis has come from the analysis of H-reflexes or electromyogram (EMG) responses. Changes in overall stiffness have been documented but, as yet, there is no direct evidence that the CNS can control reflex stiffness independently of the intrinsic stiffness. We have used a novel identification algorithm to estimate intrinsic and reflex stiffness and feed it back to subjects in real-time. Using this biofeedback, subjects could learn to control reflex stiffness independently of intrinsic stiffness. At low torque levels, subjects could vary their reflex stiffness gain by a factor of 4, while maintaining elastic stiffness and torque constant. EMG measurements confirmed that the contraction levels of the ankle muscles remained constant. Further experiments showed that subjects could change their reflexes rapidly on command. Thus, we conclude that the CNS can control reflex stiffness independently and so has great flexibility in adjusting the mechanical properties of a joint to meet functional requirements.

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.

Task-relevant adaptation of musculoskeletal impedance during posture and movement

One approach to designing robotic prostheses that interact with the environment in a naturally compliant fashion is to design them with mechanical properties that replicate the functions of an intact limb. Limb and joint mechanics can be quantified using estimates of impedance, a measure that can also be regulated in robotic systems using feedback control. Numerous studies have quantified the impedance of intact joints under static postural conditions. However, the few studies that have quantified impedance during movement have shown that it differs drastically from estimates obtained during static postural conditions. Specifically, the static component of impedance, known as stiffness, is substantially lower during movement control than during postural control. This difference has important implications for designing biomimetic prostheses and other robotic systems, though the factors contributing to the differences between posture and movement and the extent of these differences under different movement conditions are not yet known. In this paper, we systematically explore how human knee stiffness is affected by the kinematic and mechanical variables that constantly vary during movement. To do so we used a non-parametric system identification algorithm that makes few assumptions on the structure of the system or the relationship of the system to these changing kinematic and mechanical variables. We found that stiffness did not correlate with the net joint torque, as occurs during postural conditions, but rather with computed active muscle torque. Furthermore, we found that externally imposed movements during passive conditions cause a drop in joint stiffness, implying that at least some of the observed results are due to changes in intrinsic muscle or joint mechanics rather than altered neural control.

Real-Time Estimation of Intrinsic and Reflex Stiffness

Joint stiffness defines the dynamic relationship between the position of the joint and the torque acting about it. Joint stiffness is composed of two components: intrinsic and reflex stiffness. Separating the two stiffness components is difficult because they appear and change together. A number of approaches have been used to distinguish the components, but all these are inherently off-line. We have developed a novel algorithm that estimates the two components of ankle stiffness in real time. Cross-correlations between torque and position, velocity, and acceleration are used to estimate intrinsic stiffness. The reflex torque is then estimated by subtracting the estimated intrinsic components and the reflex stiffness estimated by computing the impulse response function (IRF) between the estimated reflex torque and the half-wave rectified velocity. A novel position perturbation, consisting of pseudo-random pulses of different lengths, is used to eliminate covariance between intrinsic and reflex stiffness estimates. Simulation results showed that the algorithm estimates intrinsic and reflex stiffness very accurately and responds to changes in stiffness in less than 15 s. Validation with experimental data showed that the real-time estimates were in close agreement with the estimates generated by an established off-line intrinsic and reflex stiffness identification algorithm.

Identification of intrinsic and reflexive contributions to low-back stiffness: medium-term reliability and construct validity.

This study aimed at testing the reliability and construct validity of a trunk perturbation protocol (TPP) that estimates the intrinsic and reflexive contributions to low-back stiffness. The TPP consists of a series of pseudorandom position-controlled trunk perturbations in an apparatus measuring forces and displacements at the harness surrounding the thorax. Intrinsic and reflexive contributions to low-back stiffness were estimated using a system identification procedure, leading to 12 parameters. Study 1 methods (reliability): 30 subjects performed five 75-s trials, on each of two separate days (eight weeks apart). Reliability was assessed using the generalizability theory, which allowed computing indexes of dependability (ϕ, analogous to intraclass correlation coefficient) and standard errors of measurement (SEM). Study 2 methods (validity): 20 healthy subjects performed three 75-s trials for each of five experimental conditions assumed to provide different lumbar stiffness; testing the construct validity of the TPP using four conditions with different lumbar belt designs and one control condition without. Study 1 results (reliability): Learning was seen between the first and following trials. Consequently, reliability analyses were performed without the first trial. Simulations showed that averaging the scores of three trials can lead to acceptable reliability results for some TPP parameters. Study 2 results (validity): All lumbar belt designs increased low-back intrinsic stiffness, while only some of them decreased reflex stiffness, which support the construct validity of the TPP. Overall, these findings support the use of the TPP to test the effect of rehabilitation or between-groups differences with regards to trunk stiffness.

Considering Limb Impedance in the Design and Control of Prosthetic Devices

The mechanical properties of our limbs and how those properties are regulated by the nervous system endow us with the ability to interact with our environment in numerous predictable and effective ways. While there have been many recent advances in the design and control of prosthetic limbs, none yet have the capacity to regulate their mechanical impedance over the rangeachievable by human limbs, or to replicate the functions that neuromuscular impedance control makes possible. The premise of this chapter is that designing prosthetic limbs capable of replicating the essential functions endowed by impedance control would lead to more natural and capable devices. The chapter summarizes current understanding of how human limb impedance is regulated, and attempts at replicating the functions afforded by impedance control in prosthetic limbs. It also highlights challenges and possible solutions in each of these areas.

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 techniques strong ability to predict the systems dynamics.

Estimation of Joint Stiffness with a Compliant Load

Joint stiffness defines the dynamic relationship between the position of the joint and the torque acting about it. It consists of two components: intrinsic and reflex stiffness. Many previous studies have investigated joint stiffness in an open-loop environment, because the current algorithm in use is an open-loop algorithm. This paper explores issues related to the estimation of joint stiffness when subjects interact with compliant loads. First, we show analytically how the bias in closed-loop estimates of joint stiffness depends on the properties of the load, the noise power, and length of the estimated impulse response functions (IRF). We then demonstrate with simulations that the open-loop analysis will fail completely for an elastic load but may succeed for an inertial load. We further show that the open-loop analysis can yield unbiased results with an inertial load and document IRF length, signal-to-noise ratio needed, and minimum inertia needed for the analysis to succeed. Thus, by using a load with a properly selected inertia, open-loop analysis can be used under closed-loop conditions.

Estimation of joint impedance using short data segments

Joint impedance is an important property of the human muscular system and plays a role in the control of movement and posture. Previous studies showed that joint impedance varies with the position of the joint and activation level of the surrounding muscles; however, it remains unknown how it varies during movement. Non-parametric algorithms that estimate time-varying impedance do exist; however these algorithms require hundreds of realizations of the same time-varying behavior. In this paper we develop a non-parametric algorithm that can estimate slowly time-varying impedance using multiple short data segments. Using simulated data we evaluate the desired data segment length and the number of realizations needed to yield accurate estimates.