Jooeun Ahn

A Simple State-Determined Model Reproduces Entrainment and Phase-Locking of Human Walking

Theoretical studies and robotic experiments have shown that asymptotically stable periodic walking may emerge from nonlinear limit-cycle oscillators in the neuro-mechanical periphery. We recently reported entrainment of human gait to periodic mechanical perturbations with two essential features: 1) entrainment occurred only when the perturbation period was close to the original (preferred) walking period, and 2) entrainment was always accompanied by phase locking so that the perturbation occurred at the end of the double-stance phase. In this study, we show that a highly-simplified state-determined walking model can reproduce several salient nonlinear limit-cycle behaviors of human walking: 1) periodic gait that is 2) asymptotically stable; 3) entrainment to periodic mechanical perturbations only when the perturbation period is close to the models unperturbed period; and 4) phase-locking to locate the perturbation at the end of double stance. Importantly, this model requires neither supra-spinal control nor an intrinsic self-sustaining neural oscillator such as a rhythmic central pattern generator. Our results suggest that several prominent limit-cycle features of human walking may stem from simple afferent feedback processes without significant involvement of supra-spinal control or a self-sustaining oscillatory neural network.

Long-Range Correlations in Stride Intervals May Emerge from Non-Chaotic Walking Dynamics

Stride intervals of normal human walking exhibit long-range temporal correlations. Similar to the fractal-like behaviors observed in brain and heart activity, long-range correlations in walking have commonly been interpreted to result from chaotic dynamics and be a signature of health. Several mathematical models have reproduced this behavior by assuming a dominant role of neural central pattern generators (CPGs) and/or nonlinear biomechanics to evoke chaos. In this study, we show that a simple walking model without a CPG or biomechanics capable of chaos can reproduce long-range correlations. Stride intervals of the model revealed long-range correlations observed in human walking when the model had moderate orbital stability, which enabled the current stride to affect a future stride even after many steps. This provides a clear counterexample to the common hypothesis that a CPG and/or chaotic dynamics is required to explain the long-range correlations in healthy human walking. Instead, our results suggest that the long-range correlation may result from a combination of noise that is ubiquitous in biological systems and orbital stability that is essential in general rhythmic movements.

On the dynamics of a quadruped robot model with impedance control: Self-stabilizing high speed trot-running and period-doubling bifurcations

The MIT Cheetah demonstrated a stable 6 m/s trot gait in the sagittal plane utilizing the self-stable characteristics of locomotion. This paper presents a numerical analysis of the behavior of a quadruped robot model with the proposed controller. We first demonstrate the existence of periodic trot gaits at various speeds and examine local orbital stability of each trajectory using Poincar`e map analysis. Beyond the local stability, we additionally demonstrate the stability of the model against large initial perturbations. Stability of trot gaits at a wide range of speed enables gradual acceleration demonstrated in this paper and a real machine. This simulation study also suggests the upper limit of the command speed that ensures stable steady-state running. As we increase the command speed, we observe series of period-doubling bifurcations, which suggests presence of chaotic dynamics beyond a certain level of command speed. Extension of this simulation analysis will provide useful guidelines for searching control parameters to further improve the system performance.

Feasibility of entrainment with ankle mechanical perturbation to treat locomotor deficit of neurologically impaired patients

Entraining human gait with periodic torque from a robot may provide a novel approach to robot-aided walking therapy that is competent to exploit the natural oscillating dynamics of human walking. To test the feasibility of this strategy we applied a periodic ankle torque to neurologically impaired patients (one with stroke and one with multiple sclerosis). As observed in normal human walking, both patients adapted their gait periods to synchronize with the perturbation by phase-locking the robotic torque at terminal stance phase. In addition, their gait cadence became significantly faster due to the training with clear after effects when the perturbation ceased. These results support a new strategy for walking therapy that exploits an embedded neural oscillator interacting with peripheral mechanics and the resulting natural dynamics of walking, which are essential but hitherto neglected elements of walking therapy.

Improved Assessment of Orbital Stability of Rhythmic Motion with Noise

Mathematical techniques have provided tools to quantify the stability of rhythmic movements of humans and machines as well as mathematical models. One archetypal example is the use of Floquet multipliers: assuming periodic motion to be a limit-cycle of a nonlinear oscillator, local stability has been assessed by evaluating the rate of convergence to the limit-cycle. However, the accuracy of the assessment in experiments is questionable: Floquet multipliers provide a measure of orbital stability for deterministic systems, but various components of biological systems and machines involve inevitable noise. In this study, we show that the conventional estimate of orbital stability, which depends on regression, has bias in the presence of noise. We quantify the bias, and devise a new method to estimate orbital stability more accurately. Compared with previous methods, our method substantially reduces the bias, providing acceptable estimates of orbital stability with an order-of-magnitude fewer cycles.

The Basin of Entrainment of Human Gait Under Mechanical Perturbation

Using Anklebot, a therapeutic robot module, we perturbed human gait by applying external torque to the human ankle at various frequencies. We observed that with a properly designed perturbation, 8 subjects out of 10 exhibited entrained gaits: their gait frequencies were adapted to the frequency of mechanical perturbation, and they synchronized their ankle actuation with the external torque supplied by the robot. This preliminary result suggests that a limit-cycle oscillator, a plausible element of the coupled system of central nervous system and musculo-skeletal periphery, plays a significant role in the neuro-motor execution of human locomotion. The entrainment of human gait by periodic torque from a robotic aid may provide a novel approach to walking therapy that is uniquely supportive of normal biological function.Copyright

Noise Induces Biased Estimation of the Correction Gain

The detection of an error in the motor output and the correction in the next movement are critical components of any form of motor learning. Accordingly, a variety of iterative learning models have assumed that a fraction of the error is adjusted in the next trial. This critical fraction, the correction gain, learning rate, or feedback gain, has been frequently estimated via least-square regression of the obtained data set. Such data contain not only the inevitable noise from motor execution, but also noise from measurement. It is generally assumed that this noise averages out with large data sets and does not affect the parameter estimation. This study demonstrates that this is not the case and that in the presence of noise the conventional estimate of the correction gain has a significant bias, even with the simplest model. Furthermore, this bias does not decrease with increasing length of the data set. This study reveals this limitation of current system identification methods and proposes a new method that overcomes this limitation. We derive an analytical form of the bias from a simple regression method (Yule-Walker) and develop an improved identification method. This bias is discussed as one of other examples for how the dynamics of noise can introduce significant distortions in data analysis.

A simple bipedal walking model reproduces entrainment of human locomotion

Robotic studies have suggested a contribution of limit-cycle oscillation of the neuro-mechanical periphery to human walking by demonstrating stable bipedal robotic gaits with minimal actuation and control. As behavioral evidence of limit-cycle oscillation in human walking, we recently reported entrainment of human gaits to mechanical perturbations. We observed synchronization of human walking with mechanical perturbation only when the perturbation period was close to the original walking period. In addition, the entrainment was always accompanied by phase locking at the end of double-stance. A highly-simplified state-determined walker reproduced these salient features: 1) entrainment to periodic perturbations with a narrow basin of entrainment and 2) phase-locking at the end of double stance. Importantly, the model required neither supra-spinal control nor an intrinsic self-sustaining neural oscillator (like a rhythmic central pattern generator), which suggests that prominent features of human walking may stem from simple afferent feedback processes that produce limit-cycle oscillation of the neuro-mechanical periphery without significant involvement of the brain or rhythmic central pattern generators. One limitation of that model was that it entrained only to perturbations faster than the unperturbed walking period. In the study reported here, we modified the model to have two independent steps per stride. The revised model reproduced entrainment to perturbations both slower and faster than the unperturbed cadence, as observed experimentally in human walking.

Is estimation of Floquet multipliers of human walking valid

The Floquet multiplier has been widely used to assess orbital stability of human walking. Floquet introduced it to quantify the stability of periodic motion of deterministic linear systems, but human walking involves highly nonlinear dynamics and inevitable noise. This preliminary study addresses the validity of Floquet multiplier estimation of a nonlinear system with stochastic noise. Noise was added to a simplified but nonlinear model of human walking so that the variability of model walking was comparable to that of human walking. Results show that linear regression yields a biased measure of the Floquet multiplier though its accuracy improves with a sufficient number of strides.