Joseph P. Cusumano

Local dynamic stability versus kinematic variability of continuous overground and treadmill walking.

This study quantified the relationships between local dynamic stabiliht and variabilitr during continuous overground and treadmill walking. Stride-to-stride standard deviations were computed from temporal and kinematic data. Marimum finite-time Lyapunov exponents were estimated to quantify local dynamic stability. Local stability of gait kinematics was shown to be achieved over multiple consecutive strides. Traditional measures of variability poorly predicted local stability. Treadmill walking was associated with significant changes in both variability and local stability. Thus, motorized treadmills may produce misleading or erroneous results in situations where changes in neuromuscular control are likely to affect the variability and/or stability of locomotion.

Nonlinear time series analysis of normal and pathological human walking.

Characterizing locomotor dynamics is essential for understanding the neuromuscular control of locomotion. In particular, quantifying dynamic stability during walking is important for assessing people who have a greater risk of falling. However, traditional biomechanical methods of defining stability have not quantified the resistance of the neuromuscular system to perturbations, suggesting that more precise definitions are required. For the present study, average maximum finite-time Lyapunov exponents were estimated to quantify the local dynamic stability of human walking kinematics. Local scaling exponents, defined as the local slopes of the correlation sum curves, were also calculated to quantify the local scaling structure of each embedded time series. Comparisons were made between overground and motorized treadmill walking in young healthy subjects and between diabetic neuropathic (NP) patients and healthy controls (CO) during overground walking. A modification of the method of surrogate data was developed to examine the stochastic nature of the fluctuations overlying the nominally periodic patterns in these data sets. Results demonstrated that having subjects walk on a motorized treadmill artificially stabilized their natural locomotor kinematics by small but statistically significant amounts. Furthermore, a paradox previously present in the biomechanical literature that resulted from mistakenly equating variability with dynamic stability was resolved. By slowing their self-selected walking speeds, NP patients adopted more locally stable gait patterns, even though they simultaneously exhibited greater kinematic variability than CO subjects. Additionally, the loss of peripheral sensation in NP patients was associated with statistically significant differences in the local scaling structure of their walking kinematics at those length scales where it was anticipated that sensory feedback would play the greatest role. Lastly, stride-to-stride fluctuations in the walking patterns of all three subject groups were clearly distinguishable from linearly autocorrelated Gaussian noise. As a collateral benefit of the methodological approach taken in this study, some of the first steps at characterizing the underlying structure of human locomotor dynamics have been taken. Implications for understanding the neuromuscular control of locomotion are discussed. (c) 2000 American Institute of Physics.

Slower speeds in patients with diabetic neuropathy lead to improved local dynamic stability of continuous overground walking

Patients with diabetic peripheral neuropathy are significantly more likely to fall while walking than subjects with intact sensation. While it has been suggested that these patients walk slower to improve locomotor stability, slower speeds are also associated with increased locomotor variability, and increased variability has traditionally been equated with loss of stability. If the latter were true, this would suggest that slowing down, as a locomotor control strategy, should be completely antithetical to the goal of maintaining stability. The present study resolves these seemingly paradoxical findings by using methods from nonlinear time series analysis to directly quantify the sensitivity of the locomotor system to local perturbations that are manifested as natural kinematic variability. Fourteen patients with severe peripheral neuropathy and 12 gender-, age-, height-, and weight-matched non-diabetic controls participated. Sagittal plane angles of the right hip, knee, and ankle joints and tri-axial accelerations of the trunk were measured during 10 min of continuous overground walking at self-selected speeds. Maximum finite-time Lyapunov exponents were computed for each time series to quantify the local dynamic stability of these movements. Neuropathic patients exhibited slower walking speeds and better local dynamic stability of upper body movements in the horizontal plane than did control subjects. The differences in local dynamic stability were significantly predicted by differences in walking speed, but not by differences in sensory status. These results support the hypothesis that reductions in walking speed are a compensatory strategy used by neuropathic patients to maintain dynamic stability of the upper body during level walking.

Do humans optimally exploit redundancy to control step variability in walking

It is widely accepted that humans and animals minimize energetic cost while walking. While such principles predict average behavior, they do not explain the variability observed in walking. For robust performance, walking movements must adapt at each step, not just on average. Here, we propose an analytical framework that reconciles issues of optimality, redundancy, and stochasticity. For human treadmill walking, we defined a goal function to formulate a precise mathematical definition of one possible control strategy: maintain constant speed at each stride. We recorded stride times and stride lengths from healthy subjects walking at five speeds. The specified goal function yielded a decomposition of stride-to-stride variations into new gait variables explicitly related to achieving the hypothesized strategy. Subjects exhibited greatly decreased variability for goal-relevant gait fluctuations directly related to achieving this strategy, but far greater variability for goal-irrelevant fluctuations. More importantly, humans immediately corrected goal-relevant deviations at each successive stride, while allowing goal-irrelevant deviations to persist across multiple strides. To demonstrate that this was not the only strategy people could have used to successfully accomplish the task, we created three surrogate data sets. Each tested a specific alternative hypothesis that subjects used a different strategy that made no reference to the hypothesized goal function. Humans did not adopt any of these viable alternative strategies. Finally, we developed a sequence of stochastic control models of stride-to-stride variability for walking, based on the Minimum Intervention Principle. We demonstrate that healthy humans are not precisely “optimal,” but instead consistently slightly over-correct small deviations in walking speed at each stride. Our results reveal a new governing principle for regulating stride-to-stride fluctuations in human walking that acts independently of, but in parallel with, minimizing energetic cost. Thus, humans exploit task redundancies to achieve robust control while minimizing effort and allowing potentially beneficial motor variability.

A Dynamical Systems Approach to Failure Prognosis

In this paper, a previously published damage tracking method is extended to provide failure prognosis, and applied experimentally to an electromechanical system with a failing supply battery. The method is based on a dynamical systems approach to the problem of damage evolution. In this approach, damage processes are viewed as occurring in a hierarchical dynamical system consisting of a “fast”, directly observable subsystem coupled to a “slow”, hidden subsystem describing damage evolution. Damage tracking is achieved using a two-time-scale modeling strategy based on phase space reconstruction. Using the reconstructed phase space of the reference (undamaged) system, short-time predictive models are constructed. Fast-time data from later stages of damage evolution of a given system are collected and used to estimate

A Dynamical Systems Approach to Damage Evolution Tracking, Part 1: Description and Experimental Application

In this two-part paper we present a novel method for tracking a slowly evolving hidden damage process responsible for nonstationarity in a fast dynamical system. The development of the method and its application to an electromechanical experiment is the core of Part 1. In Part 2, a mathematical model of the experimental system is developed and used to validate the experimental results. In addition, an analytical connection is established between the tracking method and the physics of the system based on the idea of averaging and the slow flow equations for the hidden process. The tracking method developed in this study uses a nonlinear, two-time-scale modeling strategy based on the delay reconstruction of a system’s phase space. The method treats damage-induced nonstationarity as evolving in a hierarchical dynamical system containing a fast, directly observable subsystem coupled to a slow, hidden subsystem. The utility of the method is demonstrated by tracking battery discharge in a vibrating beam system with a battery-powered electromagnetic restoring force. Applications to systems with evolving material damage are also discussed. @DOI: 10.1115/1.1456908#

Re-interpreting detrended fluctuation analyses of stride-to-stride variability in human walking

Detrended fluctuation analyses (DFA) have been widely used to quantify stride-to-stride temporal correlations in human walking. However, significant questions remain about how to properly interpret these statistical properties physiologically. Here, we propose a simpler and more parsimonious interpretation than previously suggested. Seventeen young healthy adults walked on a motorized treadmill at each of 5 speeds. Time series of consecutive stride lengths (SL) and stride times (ST) were recorded. Time series of stride speeds were computed as SS=SL/ST. SL and ST exhibited strong statistical persistence (α≫0.5). However, SS consistently exhibited slightly anti-persistent (α<0.5) dynamics. We created three surrogate data sets to directly test specific hypotheses about possible control processes that might have generated these time series. Subjects did not choose consecutive SL and ST according to either independently uncorrelated or statistically independent auto-regressive moving-average (ARMA) processes. However, cross-correlated surrogates, which preserved both the auto-correlation and cross-correlation properties of the original SL and ST time series successfully replicated the means, standard deviations, and (within computational limits) DFA α exponents of all relevant gait variables. These results suggested that subjects controlled their movements according to a two-dimensional ARMA process that specifically sought to minimize stride-to-stride variations in walking speed (SS). This interpretation fully agrees with experimental findings and also with the basic definitions of statistical persistence and anti-persistence. Our findings emphasize the necessity of interpreting DFA α exponents within the context of the control processes involved and the inherent biomechanical and neuro-motor redundancies available.

Experimental Measurements of Dimensionality and Spatial Coherence in the Dynamics of a Flexible-Beam Impact Oscillator

Experiments on a flexible-beam impact oscillator are described in which the spatial structure of typical motions is explored. The beam is held in a fixed mount with clamped-free boundary conditions, and the beam is driven by impacts between its free end and a sinusoidally driven impactor. Bifurcation diagrams using impactor frequency and offset as the bifurcation parameter are obtained using a computer-driven data acquisition system. The dimensionality of the system is studied by analysis of delay-reconstructed time series of experimental data. Valid delay reconstructions are obtained using mutual information and false nearest neighbour algorithms, and the correlation dimensions is estimated for the resulting experimental attractors. The relation of these topological characterizations of the system to the spatial structure of the vibrations is studied using two-point spatial correlation measurements and the proper orthogonal decomposition. It is shown that over 90% of the mean square response amplitude is captured by the first proper orthogonal mode for the cases examined and that the spatial coherence measurements can be used to distinguish between responses with similar dimensionality.

Stability and the time-dependent structure of gait variability in walking and running

Participants were asked to walk and run continuously (5 min trials) at speeds associated with preferred gait transition speeds. During slow running the local dynamic stability of the head was decreased compared with fast walking, with the reverse being true for the local dynamic stability of the ankle. The standard deviation of relative phase of the knee and ankle also was greater during slow running than fast walking. These findings for stability were mirrored in the detrended fluctuation analysis of the peak to peak interval of the head and ankle. Taken collectively these results support the proposition that larger long range correlations in the stride interval are associated with decreases in measures of stability.

A Dynamical Systems Approach to Damage Evolution Tracking, Part 2: Model-Based Validation and Physical Interpretation

In this paper, the hidden variable damage tracking method developed in Part 1 is analyzed using a physics-based mathematical model of the experimental system: a mechanical oscillator with a nonstationary two-well potential. Numerical experiments conducted using the model are in good agreement with the experimental study presented in Part 1, and explicitly show how the tracking metric is related to the slow hidden variable evolution responsible for drift in the fast system parameters. Using the idea of averaging, the slow flow equation governing the hidden variable evolution is obtained. It is shown that the solution to the slow flow equation corresponds to the hidden variable trajectory obtained with the experimental tracking method. Thus we establish in principle the relationship of our algorithm to any underlying physical process. Based on this result, we discuss the application of the tracking method to systems with evolving material damage using the results of some preliminary experiments.@DOI: 10.1115/1.1456907# In Part 1 of this paper, motivated by the need to track damage evolution in machinery, we have developed a nonlinear method for tracking slowly evolving hidden variables. From this perspective, damage is a hidden process causing nonstationarity in a fast, directly observable dynamical system. The method uses a phase space formulation of the damage tracking problem, and uses a tracking metric developed using the short-time reference model prediction error. The method was successfully applied to an electromechanical experimental system consisting of a vibrating beam with a nonlinear potential perturbed by a battery powered electromagnet. The connection between the tracking metric developed in Part 1 and the hidden drift state variable was demonstrated empirically. It was shown that, as expected from the theoretical derivation of the method, the tracking metric is in a one-to-one relationship with the local time average of the measured voltage signal. In this, Part 2, of our paper, a physics based mathematical model of the experimental system is used to study analytically the direct connection between the tracking metric and the hidden drift process. Numerical experiments performed with the model are used to validate the experimental method. The idea of averaging is then used to show that the output of the tracking method is in fact following the solution to the slow flow equation for the drifting process. This provides a physical interpretation for the output of the tracking algorithm, and shows how, in principle, the experimental method can be related to the physics of the damage process. Based on this physical interpretation, we return to the experimental application of the tracking method, and discuss some preliminary results for a system with a crack growing to failure. In the next section, we develop the mathematical model of the battery discharge experiment using a lumped parameter, Lagrangian formulation of the electromechanical system. In Section 3, the tracking method developed in Part 1 is applied to the mathematical model in numerical experiments. Using the output from numerical simulations of the mathematical model, we are able to validate the tracking algorithm. In Section 4, we discuss the method of averaging as it relates to our problem. Finally, in Section 5, we finish with concluding remarks.