Dynamic Stability of Passive Dynamic Walking on an Irregular Surface

[+] Author and Article Information
Jimmy Li-Shin Su

Department of Biomedical Engineering, University of Texas, 1 University Station, D3700, Austin, TX 78712

Jonathan B. Dingwell1

Department of Kinesiology and Health Education, University of Texas, 1 University Station, D3700, Austin, TX 78712jdingwell@mail.utexas.edu



Corresponding author.

J Biomech Eng 129(6), 802-810 (May 10, 2007) (9 pages) doi:10.1115/1.2800760 History: Received November 25, 2005; Revised May 10, 2007

Falls that occur during walking are a significant health problem. One of the greatest impediments to solve this problem is that there is no single obviously “correct” way to quantify walking stability. While many people use variability as a proxy for stability, measures of variability do not quantify how the locomotor system responds to perturbations. The purpose of this study was to determine how changes in walking surface variability affect changes in both locomotor variability and stability. We modified an irreducibly simple model of walking to apply random perturbations that simulated walking over an irregular surface. Because the model’s global basin of attraction remained fixed, increasing the amplitude of the applied perturbations directly increased the risk of falling in the model. We generated ten simulations of 300 consecutive strides of walking at each of six perturbation amplitudes ranging from zero (i.e., a smooth continuous surface) up to the maximum level the model could tolerate without falling over. Orbital stability defines how a system responds to small (i.e., “local”) perturbations from one cycle to the next and was quantified by calculating the maximum Floquet multipliers for the model. Local stability defines how a system responds to similar perturbations in real time and was quantified by calculating short-term and long-term local exponential rates of divergence for the model. As perturbation amplitudes increased, no changes were seen in orbital stability (r2=2.43%; p=0.280) or long-term local instability (r2=1.0%; p=0.441). These measures essentially reflected the fact that the model never actually “fell” during any of our simulations. Conversely, the variability of the walker’s kinematics increased exponentially (r299.6%; p<0.001) and short-term local instability increased linearly (r2=88.1%; p<0.001). These measures thus predicted the increased risk of falling exhibited by the model. For all simulated conditions, the walker remained orbitally stable, while exhibiting substantial local instability. This was because very small initial perturbations diverged away from the limit cycle, while larger initial perturbations converged toward the limit cycle. These results provide insight into how these different proposed measures of walking stability are related to each other and to risk of falling.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

(a) Schematic representation of the PDW walking down a “bumpy” slope. Random perturbations (δ) were applied to the relative leg angle at which foot strike occurred (Eq. 3). These perturbations directly corresponded to either bumps or dips in the walking surface, Δh=−2Lsin(ϕ∕2)sin(δ∕2), as shown. (b) Complete sequence of leg angle configurations used to define one complete continuous stride.

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Figure 2

Schematic representation of orbital and local dynamic stability analyses. (a) Time series data, plotted in a four-dimensional state space: x(t)=[θ(t),θ̇(t),ϕ(t),ϕ̇(t)]. (b) Representation of a Poincaré section transecting the state space perpendicular to the system trajectory. The system state, xk, at stride k evolves to xk+1 one stride later. The Floquet Multipliers quantify whether the distances between these states and the system fixed point, x*, grow or decay on average across consecutive cycles (Eq. 8). (c) Expanded view of a local region of the attractor shown in (a). An initial naturally occurring local perturbation, dj(0), diverges across i time steps as measured by dj(i). Local divergence exponents (λ*) are calculated from the averaged instantaneous slopes (Eq. 12) of the average logarithmic divergence, ⟨ln[df(i)]⟩, of all pairs of initially neighboring trajectories versus time.

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Figure 3

Phase portraits of the two configuration variables, θ and ϕ, for increasing amplitudes of ε. Note that trajectories spread out across larger regions of the state space as the average perturbation amplitudes increase.

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Figure 4

Mean standard deviations (Eq. 5) for each of the four state variables (θ,θ̇,ϕ,ϕ̇) as a function of the mean perturbation amplitude ε. Each dot represents one of the ten simulations conducted at each ε magnitude. The amplitude of the variability of the PDW output walking kinematics increased exponentially as the amplitude of the variability of the applied perturbations increased. All exponential curve fits were highly statistically significant (all r2>99.6%; p<0.001).

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Figure 5

(a) Average maximum FM values over all trials as a function of perturbation amplitude (ε) and percent gait cycle. (b) Average FM versus percent gait cycle for three perturbation amplitudes (ε∊{0.02,0.06,0.10}rad). Dashed lines indicate between-trial standard deviations for the ten trials simulated for each ε value. Small increases in maximum FM values were exhibited at and immediately after each foot strike, where the perturbations were applied. (c) Linear regression between ε and the maximum FM computed from the Poincaré section taken at 50% of the gait cycle (FM50%). Each dot represents one trial. For all ε>0.00, the maximum FM changed little with increases in the amplitude of the applied perturbations.

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Figure 6

(a) MLD, ⟨ln[dj(i)]⟩, as a function of normalized time for the six different mean perturbation amplitudes ε. Each curve shown is the average of the ten curves obtained from the ten trials simulated for each ε. (b) Instantaneous slopes (Eq. 12) of the MLD curves shown in (a). Insets show close-up views of these curves. (c) Linear regressions between the short-term (λ*S) and long-term (λ*L) divergence exponents and ε. Each dot represents one trial. The λ*S exponents exhibited a strong linear relationship with ε; however, the λ*L exponents did not.

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Figure 7

(a) Schematic representation of the time evolution of two state space trajectories passing through a Poincaré plane. The limit cycle trajectory passes through the Poincaré plane at the fixed point (x*). The two trajectories are initially very close to each other: i.e., d(0) is very small. After one stride, these two trajectories have diverged away from each other: i.e., d(T)⪢d(0) reflecting the presence of local instability. However, they have also both converged toward the central limit cycle trajectory (i.e., they are orbitally stable) at the same time. (b) Magnitude of the perturbation response (yk+1=xk+1−x*) versus initial perturbation magnitude (yk=xk−x*) and % gait cycle for simulated walking data with ε=0.10rad. Very small perturbations grow larger (above the horizontal plane), while large perturbations are damped out (below the horizontal plane). Variations along the gait cycle are also observed.




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