Assessing the stability of human locomotion: a review of current measures

Abstract

Falling poses a major threat to the steadily growing population of the elderly in modern-day society. A major challenge in the prevention of falls is the identification of individuals who are at risk of falling owing to an unstable gait. At present, several methods are available for estimating gait stability, each with its own advantages and disadvantages. In this paper, we review the currently available measures: the maximum Lyapunov exponent (λS and λL), the maximum Floquet multiplier, variability measures, long-range correlations, extrapolated centre of mass, stabilizing and destabilizing forces, foot placement estimator, gait sensitivity norm and maximum allowable perturbation. We explain what these measures represent and how they are calculated, and we assess their validity, divided up into construct validity, predictive validity in simple models, convergent validity in experimental studies, and predictive validity in observational studies. We conclude that (i) the validity of variability measures and λS is best supported across all levels, (ii) the maximum Floquet multiplier and λL have good construct validity, but negative predictive validity in models, negative convergent validity and (for λL) negative predictive validity in observational studies, (iii) long-range correlations lack construct validity and predictive validity in models and have negative convergent validity, and (iv) measures derived from perturbation experiments have good construct validity, but data are lacking on convergent validity in experimental studies and predictive validity in observational studies. In closing, directions for future research on dynamic gait stability are discussed.

Figure 1.

Calculation of the maximum Lyapunov exponent. (a) A three-dimensional attractor (state space reconstruction of q). (b) Close-up view of part of the attractor; for each point on the attractor, the nearest neighbour was calculated, and divergence of these points was calculated as dist_j(t) (expanded view of part of (a)). (c) Average logarithmic rate of divergence, from which maximum Lyapunov exponents, λ_S and λ_L, can be calculated as the slope of the curve at 0–0.5 strides and at 4–10 strides, respectively (black line: divergence curve; grey line, λ_S: 2.2156; dotted line, λ_L: 0.0712).

Figure 2.

Calculation of the maximum Floquet multiplier. (a) A three-dimensional attractor, with a schematic of the Poincaré section, which is perpendicular to the direction of flow (state space reconstruction of q). (b) Close-up of the Poincaré section (expanded view of part of (a)). The larger point in the middle (S*) represents the limit cycle, that is, the average of all data points in the Poincaré section. The Jacobian maps the relative position (with respect to S*) of all points S_k to the relative position (with respect to S*) of all points S_k+1. The largest Floquet multiplier (the largest eigenvalue of the Jacobian) thus indicates whether the distance from S* grows or shrinks from one cycle to the next.

Figure 3.

Calculation of the scaling exponent α. (a) Data series of stride times, sampled at 300 samples per second. (b) Part of the data series of (a) integrated, with linear fits for n = 4. (c) The average residual around fits as shown in (b) (f(n)) is then plotted against n, and the slope of this line is the scaling exponent α (logf(n) versus log(n); black line, data; grey line, fit: α = 0.71871).

Figure 4.

An inverted pendulum model. m, mass; g, gravitational constant (−9.81); l, pendulum length; CoM, centre of mass; BoS, base of support; CoP, centre of pressure.

Figure 5.

Reconstructing a state space using embedding delays. (a) The original recorded velocity signal, with point q(t) and points q(t+τ) and q(t+2τ) (time-normalized first derivative of the time series q). In (b), the original time series (q) from t onwards is plotted on the x-axis, with corresponding values of q(t+τ) on the y-axis, and values of q(t+2τ) on the z-axis (state space reconstruction of q). In matrix form, this state space is