Joint impedance is a common way of representing human joint dynamics. Since ankle joint impedance varies within the gait cycle, time-varying system identification techniques can be used to estimate it. Commonly, time-varying system identification techniques assume repeatably of joint impedance over cyclic motions, without taking into consideration the inherent variability of human behavior. In this paper, a method that assumes smooth, cyclic joint impedance, yet allows for cycle-to-cycle variability, is proposed. The method was tested on isometric, cyclic experimental data from the ankle under conditions with a time variation comparable to the expected one during the gait cycle. The estimated model could describe the data with high accuracy (VAF of 94.96%) and retrieve realistic inertia, damping and stiffness parameters. The results provide motivation to further apply the method on experiments under dynamic conditions and to employ the proposed method as a tool for investigating the human joint dynamics during cyclic movements.

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Accurate and swift tuning of joint impedance is crucial to perform movement and interaction with our environment. Time-varying system identification enables quantification of joint impedance during movement. Many methods have been developed over the years, each with their own mathematical approach and underlying assumptions. Yet, for the identification of joint impedance, a systematic comparison revealing each method's unique strengths and weaknesses, is lacking. Here, we propose a quantitative framework to compare these methods. The framework is used to review five time-varying system identification methods using both simulated data and experimental data. These methods included three time-domain methods: ensemble, short data segment, and basis impulse response function; and two frequency-domain methods: ensemble spectral, and kernel-based regression. In the simulation study, joint stiffness – the static component of impedance – was simulated as a square wave to mimic the most extreme case of time-varying behavior. The identification results were compared based on the (1) variance accounted for (VAF), (2) bias, (3) random, and (4) total estimation error with respect to the simulated joint stiffness; and (5) rise time between two stiffness levels. In the experimental study, human ankle joint impedance was identified. Identification performance was compared using the variability in estimating joint stiffness – representative of the random error – and VAF. The performance metrics revealed distinct identification properties for each method. Therefore, researchers must make a well-justified decision which method is most appropriate for their application. The combination of simulation and experimental work with extensive performance quantification creates a framework for quantitative assessment of newly developed time-varying system identification methods.

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In vivo joint stiffness estimation during time-varying conditions remains an open challenge. Multiple communities, e.g. system identification and biomechanics, have tackled the problem from different perspectives and using different methods, each of which entailing advantages and limitations, often complementary. System identification formulations provide data-driven estimates of stiffness at the joint level, while biomechanics often relies on musculoskeletal models to estimate stiffness at multiple levels, i.e. joint, muscle, and tendon. Collaboration across these two scientific communities seems to be a logical step toward a reliable multi-level understanding of joint stiffness. However, differences at the theoretical, computational, and experimental levels have limited inter-community interaction. In this article we present a roadmap to achieve a unified framework for the estimation of time-varying stiffness in the composite human neuromusculoskeletal system during movement. We present our perspective on future developments to obtain data-driven system identification and musculoskeletal models that are compatible at the theoretical, computational, and experimental levels. Moreover, we propose a novel combined closed-loop paradigm, in which reference estimates of joint stiffness via system identification are decomposed into underlying muscle and tendon contribution via high-density-electromyography-driven musculoskeletal modeling. We highlight the need for aligning experimental requirements to be able to compare both joint stiffness formulations. Unifying both biomechanics' and system identification's formulations is a necessary step for truly generalizing stiffness estimation across individuals, movement conditions, training and impairment levels. From an application point of view, this is central for enabling patient-specific neurorehabilitation therapies, as well as biomimetic control of assistive robotic technologies. The roadmap we propose could serve as an inspiration for future collaborations across broadly different scientific communities to truly understand joint stiffness bio- and neuromechanics.

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Human joint impedance is a fundamental property of the neuromuscular system and describes the mechanical behavior of a joint. The identification of the lower limbs' joints impedance during locomotion is a key element to improve the design and control of active prostheses, orthoses, and exoskeletons. Joint impedance changes during locomotion and can be described by a linear time-varying (LTV) model. Several system identification techniques have been developed to retrieve LTV joint impedance, but these methods often require joint impedance to be consistent over multiple gait cycles. Given the inherent variability of neuromuscular control actions, this requirement is not realistic for the identification of human data. Here we propose the kernel-based regression (KBR) method with a locally periodic kernel for the identification of LTV ankle joint impedance. The proposed method considers joint impedance to be periodic yet allows for variability over the gait cycles. The method is evaluated on a simulation of joint impedance during locomotion. The simulation lasts for 10 gait cycles of 1.4 s each and has an output SNR of 15 dB. Two conditions were simulated: one in which the profile of joint impedance is periodic, and one in which the amplitude and the shape of the profile slightly vary over the periods. A Monte Carlo analysis is performed and, for both conditions, the proposed method can reconstruct the noiseless simulation output signal and the profiles of the time-varying joint impedance parameters with high accuracy (mean VAF ~ 99.9% and mean normalized RMSE of the parameters 1.33-4.06%).The proposed KBR method with a locally periodic kernel allows for the identification of periodic time-varying joint impedance with cycle-to-cycle variability.

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During movements, humans continuously regulate their joint impedance to minimize control effort and optimize performance. Joint impedance describes the relationship between a joint's position and torque acting around the joint. Joint impedance varies with joint angle and muscle activation and differs from trial-to-trial due to inherent variability in the human control system. In this paper, a dedicated time-varying system identification (SI) framework is developed involving a parametric, kernel-based regression, and nonparametric, “skirt decomposition,” SI method to monitor the time-varying joint impedance during a force task. Identification was performed on single trials and the estimators included little a priori assumptions regarding the underlying time-varying joint mechanics. During the experiments, six (human) participants used flexion of the wrist to apply a slow sinusoidal torque to the handle of a robotic manipulator, while receiving small position perturbations. Both methods revealed that the sinusoidal change in joint torque by activation of the wrist flexor muscles resulted in a sinusoidal time-varying joint stiffness and resonance frequency. A third-order differential equation allowed the parametric kernel-based estimator to explain on average 76% of the variance (range 52%-90%). The nonparametric skirt decomposition method could explain on average 84% of the variance (range 66%-91%). This paper presents a novel framework for identification of time-varying joint impedance by making use of linear time-varying models based on a single trial of data.

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Joint manipulation elicits a response from the sensors in the periphery which, via the spinal cord, arrives in the cortex. The average evoked cortical response recorded using electroencephalography was shown to be highly nonlinear; a linear model can only explain 10% of the variance of the evoked response, and over 80% of the response is generated by nonlinear behavior. The goal of this study is to obtain a nonparametric nonlinear dynamic model, which can consistently explain the recorded cortical response requiring little a priori assumptions about model structure. Wrist joint manipulation was applied in ten healthy participants during which their cortical activity was recorded and modeled using a truncated Volterra series. The obtained models could explain 46% of the variance of the evoked cortical response, thereby demonstrating the relevance of nonlinear modeling. The high similarity of the obtained models across participants indicates that the models reveal common characteristics of the underlying system. The models show predominantly high-pass behavior, which suggests that velocity-related information originating from the muscle spindles governs the cortical response. In conclusion, the nonlinear modeling approach using a truncated Volterra series with regularization, provides a quantitative way of investigating the sensorimotor system, offering insight into the underlying physiology.@en

During movement, our central nervous system (CNS) takes into account the dynamics of our environment to optimally adapt our joint dynamics. In this study we explored the adaptation of shoulder joint dynamics when a participant interacted with a time-varying virtual environment created by a haptic manipulator. Participants performed a position task, i.e., minimizing position deviations, in face of continuous mechanical force perturbations. During a trial the environmental damping, mimicked by the manipulator, was either increased (0 to 200 N s/m) or decreased (200 to 0 N s/m) in 1 s or 8 s. A system identification technique, kernel-based regression, was used to reveal time-varying shoulder joint dynamics using the frequency response function (FRF). The FRFs revealed that the rate at which shoulder joint dynamics is adapted depends on the rate and direction of change in environmental damping. Adaptation is slow, but starts immediately, after the environmental damping increases, whereas adaptation is fast but delayed when environmental damping decreases. The results obtained in our participants comply with the framework of optimal feedback control, i.e., adaptation of joint dynamics only takes place when motor performance is at risk or when this is energetically advantageous.

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