Figure 1.
A schematic of the walking model.
A point mass is restrained by rigid massless legs. The trailing ankle is actuated as a pre-loaded spring released at the beginning of double stance. The hip joint and the leading ankle do not exert any torque.
Figure 2.
One stride of the walking model.
The end and beginning of a step is the moment when the leading foot collides with ground (Frame 1, 5 and 9). During double stance the model moves as four linked bars (Frame 2, 3, 6 and 7). During single stance the model moves as an inverted pendulum (Frame 4 and 8).
Table 1.
Parameter values for the model.
Figure 3.
One stride of the model in the limit of zero θ0.
L, F, vi and Ti are the radius of the disk, the stochastic force with zero mean applied to the center of mass while the disk is rolling from 0 to π radian, the velocity of the disk at the end of ith cycle, duration of ith cycle, respectively. The velocity vi+1 is determined right after the disk rolls by π radian because the stochastic force is no longer applied to the disk hence momentum is conserved.
Figure 4.
Scaling exponents α and β for 500 strides of walking with two different values of θ0.
The leading leg angle θ0 defines the Floquet multiplier as cos22θ0, and determines how fast the perturbed dynamics converges to the nominal limit cycle. Small θ0 makes the Floquet multiplier close to unity yielding slow convergence, and large θ0 makes the Floquet multiplier close to zero yielding fast convergence. Both α and β are slightly different from those of the shuffled data with no correlation when θ0 is π/6 and the model yields relatively strong orbital stability (Floquet multiplier = 0.25). The difference in the scaling exponents between the original time series and their shuffled counterparts becomes much more prominent when θ0 is π/12 and the attraction to the limit cycle becomes relatively weak (Floquet multiplier = 0.75); α and β become similar to those of human walking with long-range correlations with clear difference from those of the shuffled data. The bottom panels show time series of the normalized stride intervals (stride intervals divided by their mean value). The structure of the time series changed due to shuffling particularly when θ0 is π/12.
Figure 5.
Distribution of scaling exponents α and β when the model walked 500 strides.
In A, the scaling exponents α and β were evaluated for each of 20 simulations in four different cases. In addition to cases with two different values of θ0 (π/12 and π/6), two extreme cases were investigated for comparison. When the stride intervals are artificially generated as a random variable from a normal distribution, they become white noise. The stride intervals with θ0 = π/6 show a slight indication of long-range correlations. When θ0 = π/12, a similar leg angle to that of normal human walking, the stride intervals present evident long-range correlations with similar α and β to those observed in human walking. When θ0 approaches zero and the model becomes a rolling disk, the stride intervals approach Brownian noise. In B, the distributions of α and β for θ0 = π/12 and θ0 = π/6 are compared with those of their shuffled counterparts. The scaling exponents of the original time series are significantly different from those of the shuffled data which approximate α and β of white noise. This confirms that temporal structure, rather than a specific distribution of variability, gives rise to the long-range correlation in stride intervals of the walking model.
Figure 6.
Distribution of scaling exponents α and β when the model walked 3,000 strides.
In A, the scaling exponents α and β were evaluated for each of 20 simulations. When θ0 = π/6, α and β are no longer different from those of uncorrelated noise; the large stride number has attenuated the long-range correlation. In contrast, when θ0 = π/12, the stride intervals still present evident long-range correlations with similar α and β to those observed in human walking. When θ0 approaches zero and the model has marginal orbital stability, the scaling exponents remain close to those of Brownian noise regardless of the large stride number. In B, the distributions of α and β for θ0 = π/12 are compared with those of the shuffled time series. (Note the change of plot scale.) As in 500 stride walking, the scaling exponents of the original time series are significantly different from those of the shuffled data which are statistically indistinguishable from α and β of white noise.
Figure 7.
Scaling exponents α and β for 3,000 strides walking with θ0 = π/12.
Exponents α and β are still similar to those of human walking with long-range correlations, and clearly different from those of the shuffled data. Also in the right panels, it is visible that shuffling changed the structure of the time series of normalized stride intervals (stride intervals divided by the mean value).
Figure 8.
Scaling exponents α and β for 100,000 stride walking with θ0 = π/12.
The scaling exponents are significantly different from those of the uncorrelated noise, but the evidence of long-range correlations is less prominent. In A, α is close to that of white noise though β is similar to that of human walking with long-range correlations. In B, the exponents are evaluated with window sizes larger than 1,000 strides (1,000≤n≤100,000/2, and 2/100,000≤f≤0.01); the exponents are not significantly different from those of white noise. C shows the representative data. On the whole, the local slope of the curve relating log F(n) and log n decreases as window size n increases.
Figure 9.
Distribution of coefficient of variance (COV).
In both cases of θ0 = π/6 and θ0 = π/12, the distribution of COV obtained from each set of 500 or 3,000 strides is not significantly different from 3%, the COV of stride intervals observed in normal human walking. The circle and error bar indicate the mean and the standard deviation interval, respectively.