The influence of gait speed on local dynamic stability of walking

Abstract

The focus of this study was to examine the role of walking velocity in stability during normal gait. Local dynamic stability was quantified through the use of maximum finite-time Lyapunov exponents, λ_Max. These quantify the rate of attenuation of kinematic variability of joint angle data recorded as subjects walked on a motorized treadmill at 20%, 40%, 60%, and 80% of the Froude velocity. A monotonic trend between λ_Max and walking velocity was observed with smaller λ_Max at slower walking velocities. Smaller λ_Max indicates more stable walking dynamics. This trend was evident whether stride duration variability remained or was removed by time normalizing the data. This suggests that slower walking velocities lead to increases in stability. These results may reveal more detailed information on the behavior of the neuro-controller than variability-based analyses alone.

Introduction

Stability is a critical component of walking [10], [14]. It can be defined as the ability to maintain functional locomotion despite the presence of small kinematic disturbances or control errors. Stability of standing static postures is often recorded from kinematic variability associated with the center-of-pressure under the equilibrium base of support. However, walking is a dynamic condition wherein the joint control torques change with time and posture. Therefore, stability of walking requires analyses that account for both time and movement [16]. Kinematics of walking and associated variability are influenced by walking velocity thereby indicating potential velocity effects on stability [17], [19]. Some studies suggest that one possible motivation for slower walking speed in the elderly and in individuals with joint disease and neuropathology is to improve stability [6]. This assumes that stability of walking is improved at slower velocities. The purpose of this study was to test this assumption.

There is a difference between kinematic variability and stability. Studies have measured the magnitude of kinematic variability as an estimate of stability [11], [25]. It is often assumed that increased variability corresponds to decreased stability. However, measurements of kinematic variability are subtly different than stability. It is reasonable to assume that every walking stride could be similar to every other stride. Natural kinematic variance observed in empirical data is therefore attributed to mechanical disturbances or control errors. These disturbances are attenuated in time by the neuro-controller and musculoskeletal system in order to maintain a stable walking pattern. Thus, stability must be estimated from the time-dependent expansion or attenuation of kinematic variability [10], [14].

Stability of human walking can be estimated from temporal analyses of multi-dimensional variability [4]. Disturbance to the walking trajectory is an ongoing process so the attenuation of kinematic variability is continually manifest. Poincare maps quantify the attenuation of kinematic variability between consecutive strides [14]. This method has the advantage of measuring stability in a multi-degree-of-freedom system. However, it provides limited insight regarding intra-stride effects and often ignores expansion in temporal variability, e.g. stride-duration variance. Effects from a kinematic disturbance can be observed over a time scale that influences both intra-stride and inter-stride movement [11]. Dynamic analyses can be used to track the time-history of individual disturbances recorded from the time-dependent kinematics [16]. The time-dependent rate of kinematic expansion is measured by the Lyapunov exponent, λ. One Lyapunov exponent exists for every movement dimension of the analyzed kinematic trajectory. These can be arranged, in order of most rapidly diverging to most rapidly converging, as λ₁ > λ₂ > ⋯ > λ_n. To avoid confusion, λ₁ may be referred to as λ_Max to represent the largest Lyapunov exponent. Rosenstein et al. concluded that when using the full Lyapunov spectrum, a system is stable when the sum of these Lyapunov exponents is negative, i.e. the rate of convergence is greater than the rate of divergence [22]. Calculation of the full Lyapunov spectrum from experimental data, however, is exceedingly difficult. These calculations may be simplified greatly by realizing that two randomly selected initial trajectories should diverge, on average, at a rate determined by the largest Lyapunov exponent, λ_Max. Calculation of λ_Max is relatively easy and can be used to evaluate the influence of walking velocity on dynamic stability of walking.

The goal of this study was to (1) implement Lyapunov analyses to characterize stability of dynamic steady-state walking, and (2) test whether walking velocity influences stability of walking. This is the first in a series of studies planned to quantify the stability of gait in normal-developing subjects and patients with developmental neuro-impairment.