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Gait Posture. Author manuscript; available in PMC 2007 February 1.

Published in final edited form as:

Published online 2006 April 18. doi: 10.1016/j.gaitpost.2006.03.003

PMCID: PMC1785331

NIHMSID: NIHMS12864

Musculoskeletal Biomechanics Laboratories, Department of Engineering Science & Mechanics, School of Biomedical Engineering & Science, Virginia Polytechnic Institute & State University, 219 Norris Hall (0219), Blacksburg, VA 24061, United States

The publisher's final edited version of this article is available at Gait Posture

See other articles in PMC that cite the published article.

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.

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 λ_{1} > λ_{2}
> > λ_{n}. To avoid confusion,
λ_{1} 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.

Kinematic data were recorded from 19 healthy adult subjects including 6
males and 13 females; mean age (±S.D.) 22.5 ± 2.8 years;
mean height 1.7 ± 0.1 m and mean weight 65.7 ± 12.7 kg.
Lower-body kinematic data were recorded from 21 reflective markers using a
6-camera, 3D, video motion analysis system at a data sampling rate of 240 Hz
(Vicon, Oxford Metrics). Markers were placed on the sacrum, anterior superior
iliac spine, posterior superior iliac spine, anterior thigh, lateral epicondyle
of the femur, anterior shin, lateral malleolus of the fibula, dorsum of the
foot, 5th metatarsal, calcaneous and hallux. Subjects walked barefoot on a
treadmill at 20%, 40%, 60% and 80% of their Froude velocity, V_{F}.
Walking velocity was expressed in terms of Froude velocity to appropriately
scale the walking speed to leg length and pendulum dynamics [24]. Each subject's Froude velocity (square
root of Froude number) was calculated based on the equation:

$${V}_{\mathrm{F}}=\sqrt[]{\mathit{Rg}}$$

(1)

where *R* is the distance between the greater
trochanter and lateral malleolus of the fibula and *g* is the
acceleration due to gravity. Comfortable walking speed is typically 0.42
V_{F} and running is initiated at 0.70 V_{F} [2,15]. Therefore, stability was recorded at walking velocities near the
comfortable walking speed, slow walking, fast walking, and at speeds in excess
of the natural walk-run transition.

Four repeated collections of 30 walking strides per velocity condition were recorded for each subject. Ankle, knee and hip angles were calculated from the 3D locations of the marker set using standard techniques (MATLAB, Mathworks, Inc., Natick, MA). Analyses were limited to the plantarflexion/dorsiflexion dimension of the ankle, knee flexion angle and hip flexion angle. Previous studies recommend against filtering the data before Lyapunov analyses so as to retain spatio-temporal fluctuations and nonlinearities [8]. However, we believe that kinematic signals at frequencies greater than 10 Hz are unlikely related to the musculoskeletal motion and therefore filtered the data with a 10 Hz, low-pass second-order Butterworth filter. Regardless of this difference in opinion regarding filtering the results were similar between these studies.

Since cadence changes with walking velocity but data sampling frequency
remained fixed, a dilemma arises with respect to the proper way to compare data
collected at different walking velocities. Specifically, since stride-duration
decreases with walking velocity, data collected at 80% V_{F} would
likely have less than half as many data points per stride, on average, as data
collected at 20% V_{F}. Data were time-normalized in two separate
manners and results compared. First, every stride was time-normalized to 100
data points per stride. This provides an equal number of data points per stride
regardless of velocity but it removes stride-to-stride temporal variations that
are an important component of Lyapunov stability analyses. Second, data sets of
30 contiguous strides were re-sampled to be 3000 data points long, i.e.
approximately 100 data points per stride on average but any individual stride
could be greater than or less than 100 data points. This permits
stride-to-stride temporal variation while normalizing the data such that the
average number of data points per stride were similar for each velocity
condition. In an attempt to understand the influence of re-sampling frequency on
the Lyapunov analysis, the same data were also analyzed with data re-sampled to
1500 data points. Independent stability analyses were performed on each
time-dependent joint angle, *x*_{j}(*t*), where *j* = 1:6 was the joint number
and *t* was the re-sampled time interval.

Local dynamic stability was determined based on the maximum finite-time
Lyapunov exponent, λ_{Max}. These were used to quantify the
exponential attenuation of variability between neighboring kinematic
trajectories. The approach assumes that every stride could be identical to every
other stride. Stride-to-stride differences in kinematic measurements are
attributed to small perturbations. Therefore, kinematic variability can be used
to evaluate the stability of the system by tracking the progression of a
perturbed gait cycle back to the mean. Since the recorded time-series data,
*x*_{j}(*t*), are one dimensional column vectors of joint angles
it was necessary to reconstruct an *n*-dimensional state-space
out of the kinematic data in order to accurately determine dynamic perturbations
to the ideal gait cycle. One typical method of creating an
*n*-dimensional state-space from scalar data is by method of
delays. Using this method a joint angle in *n*-dimensional space
would appear as

$${Y}_{j}\left(t\right)=[{x}_{j}\left(t\right),{x}_{j}(t+{T}_{\mathrm{d}}),{x}_{j}(t+2{T}_{\mathrm{d}})\cdots {x}_{j}(t+(n-1){T}_{\mathrm{d}})]$$

(2)

where *x*_{j}(*t*) is the original scalar data of joint angle and
*T*_{d} is a constant time delay. A reconstructed state-space *Y*_{j}(*t*) with an embedding dimension of *n* = 3
can be seen in Fig. 1A.

(A) Reconstructed state-space kinematics of knee angle with three embedded
dimensions. Fifteen contiguous strides are illustrated. (B) Divergence of
nearest neighbors with temporal variability permitted. **...**

The success of state-space reconstruction by the method of delays is
sensitive to the time delay, *T*_{d} [23]. Several methods exist for
calculation of *T*_{d}. There is no consensus on which method provides optimal results. Three
methods have been popularized including: (1) time delays estimated from the
average mutual information function [6,9], (2) time delays
estimated from the time it takes for the autocorrelation function to drop to a
pre-specified fraction of its initial value [20], and (3) time delays estimated using geometric approaches based on
maximizing some component of the reconstructed state-space [3,23]. Using the average mutual information function and the
autocorrelation approach, we observed time-delay estimates ranging from 9 to 40
samples. To assure that all of the trials were analyzed similarly, a constant
*T*_{d} of 10 samples (10% of the length of the gait cycle) was used for all
reconstructed state-space.

The number of state-space dimensions, *n* in Eq. (1), is selected based on a
global false-nearest-neighbor analysis [22]. This method incrementally increases *n* until the
number of false-nearest-neighbors approaches zero. False nearest neighbors are
defined as sets of points that are very close to each other at dimension
*n* = *k* but not at *n* =
*k* + 1. For example, a plot of
*x*(*t*) = sin(2π^{t}) +
cos(π*t*) with *n* = 3 embedding
dimensions in Fig. 2A illustrates that the
curve does not intersect itself. When this same data is viewed in
two-dimensions, *n* = 2, it artificially appears to cross-over
itself, i.e. a false nearest neighbor would occur at the crossover point as in
Fig. 2B. A global
false-nearest-neighbor analysis suggested that an embedding dimension of
*n* = 5 was appropriate for the analyzed data.

Plot of *x*(*t*) =
sin(2π*t*) + cos(π*t*)
with embedding dimension *n* = 3 (A) and *n* =
2 (B). Note the false-nearest-neighbor illustrated by the intersection at
**...**

Maximum finite-time Lyapunov exponents were calculated based on the
algorithm published by Rosenstein et al. [22]. The Euclidean distance between nearest neighbors, *d*^{i}(*t*), was computed for each data-point,
*i*, in the reconstructed state-space *Y*^{j}(*t*) for all time, *t*. Nearest neighbors
are found by selecting data points from separate cycles that are closest to each
other in reconstructed state-space as in Fig.
1B. If repeated strides were kinematically identical, then a plot of the
trajectories would illustrate each cycle on top of the others in state-space. In
this condition, the distance between nearest neighbors, *d*^{i}(*t*), would be zero for all pairs of nearest neighbors,
*i*. However, in the empirically measured data the distance
between nearest neighbors, *d*^{i}(*t*), was greater than zero as in Fig. 1A. Hence, there are clearly kinematic disturbances
observable in the data. The distance between all nearest neighbors was tracked
forward in time to record time-dependent changes in kinematic variability as in
Fig. 1B. The rate of change in the
distance between nearest neighbors is quantified by the Lyapunov exponents,
λ:

$$d\left(t\right)={D}_{0}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{\lambda t}$$

(3)

where *D*^{0} is the average displacement
between trajectories at *t* = 0. Two randomly selected initial
trajectories should diverge, on average, at a rate determined by the largest
Lyapunov exponent, λ^{Max} [22]. Therefore, the maximum Lyapunov exponent,
λ^{Max} was approximated from the experimental joint-angle
data as the slope of the linear best-fit line to the curve created by the
equation:

$$\langle \mathrm{ln}\phantom{\rule{thinmathspace}{0ex}}\{{d}_{i}(k+\Delta t)\u2215{d}_{i}\left(k\right)\}{\rangle}_{i}={\lambda}_{\mathrm{Max}}\Delta t$$

(4)

where ln *d*^{i}(*k*) represents the average logarithm of
displacement for all pairs of nearest neighbors, *i*. The maximum
finite-time Lyapunov exponent, λ^{Max}, was calculated as the
slope of the logarithm of the average divergence across the span of
0–1 strides as shown in Fig. 3.
A value of λ^{Max} was computed for each joint of every
subject at each walking velocity. λ^{Max} was interpreted as
a measure of dynamic stability.

Average logarithmic divergence vs. time. The best fit linear slope of the
logarithmic relation from 0 to 1 stride represents
λ^{Max}. In this example the stride duration was 0.74
**...**

Statistical analyses were performed to determine the effects of walking
velocity on stability. Lyapunov exponents were computed independently for the
ankle, knee and hip for each subject and each walking velocity,
*V*^{F} = 20%, 40%, 60%, 80%. Preliminary analyses
revealed no statistically significant differences in stability between the right
and left limbs. Therefore, data from the right and left limbs were pooled for
statistical analyses. Two-factor repeated measures analysis of variance (ANOVA)
tested the within-subject effects of joint and walking velocity on
λ^{Max}. Analyses were performed using commercial
software (Statistica, 4.5 Statsoft, Inc., Tulsa, OK) using a significance level
of α < 0.05.

When the full 30 stride data sets at each velocity were resampled to be 3000
data points in duration, variability in stride-duration was observed despite walking
at constant velocity. Mean and standard deviations of stride time were 1.57
± 0.06 s at 20% *V*^{F}, and 1.12 ±
0.02, 0.94 ± 0.01, and 0.79 ± 0.02 s for 40%, 60%, and 80%
*V*_{F}, respectively. This stride-time was significantly
longer (*p* < 0.001) at 20% *V*^{F}
than at 40%, 60%, and 80% *V*^{F}.

Maximum finite-time Lyapunov exponents, λ^{Max}, were
calculated for each subject and velocity condition to estimate dynamic stability of
walking. There was a significant main effect of joint on the stability value
(*p* < 0.001) illustrated in Fig. 4. Mean value of λ^{Max} was 1.08
± 0.35 mm/s for the ankle, 1.40 ± 0.37 mm/s for the knee, and
1.27 ± 0.34 mm/s for the hip. Post hoc analyses demonstrated that
λ^{Max} for the ankle was significantly (*p*
< 0.001) less than in the hip and knee. λ^{Max} for
the hip was in turn significantly (*p* < 0.01) less than
in the knee. Recall that smaller values of λ^{Max} suggest
greater stability thereby indicating greater neuromuscular stabilizing control in
the ankle joint than measured in the hip and knee.

Average λ^{Max} increased with walking velocity. This
trend was observed when data were re-sampled to 3000 samples per 30 strides
(A) and similarly when temporal variability was removed by re-sampling **...**

Stability was significantly (*p* < 0.001)
influenced by the main effect of walking velocity. This effect was similarly
observed whether stride-duration variability was maintained by time-normalizing at
3000 points per 30 strides, or whether stride-duration variability was eliminated by
time-normalizing to exactly 100 points per stride. λ^{Max} at
each velocity was significantly different than at every other velocity and
monotonically increased with increasing velocity. Regression analyses were performed
to analyze for trends related to velocity. A best-fit trendline with linear and
quadratic terms was developed for each set of data for comparison with similar
methods by others [8]. Results suggest the
λ^{Max} values for the ankle were quadratically related to
walking velocity, but the stability of the knee dynamics were significantly
correlated with a linear behavior of walking velocity as in Table 1. Data re-sampled to 3000 data points per 30 strides were
compared to data re-sampled to 1500 data points per 30. Halving the effective
sampling frequency caused a significantly (*p* < 0.001)
reduced λ^{Max}.

It has been suggested that individuals with impaired neuromuscular control may walk with reduced velocity in order to improve their stability [5]. Existing evidence reveals that kinematic and spatio-temporal variability are influenced by walking velocity [19]. However, stability might be poorly represented by the magnitude of variability. Instead, assessment of stability requires examination of how the neuro-control system handles kinematic variability, i.e. the active and passive control of disturbances. Therefore, the goal of the current study was to determine the relationship between walking velocity and the Lyapunov exponent that represents the time-dependent change in joint angle variability. Dingwell and Marin [8] recently reported results from a similar study to investigate the influence of walking velocity on stability. They observed that kinematic variability demonstrated a quadratic behavior, i.e. variability was least near the comfortable walking velocity but it was increased at both slow and fast walking velocities. Conversely, their analyses suggest that stability increased linearly with walking velocity. Our analyses agree with this monotonic trend.

Lyapunov analyses revealed that stability was significantly influenced by
walking velocity. When the data from every stride were time-normalized to 100 data
points per stride, a linear relationship was observed between
λ^{Max} and velocity. Smaller λ^{Max}
represents a more stable system. Therefore, every joint was significantly more
stable at lower velocities. These results agree with the conclusions of Dingwell and
Marin [8] despite differences in measurement
techniques and processing. However, this method of time-normalizing the data
artificially removes stride-to-stride temporal variations. A kinematic disturbance
can influence not only the movement pattern but it can also influence the time
duration of the movement trajectory. To accommodate stride-duration variability
separate analyses were performed wherein the data from 30 strides were
time-normalized to 3000 data points in total. This process causes a mean
stride-duration of 100 points per cycle but all strides were not necessarily of
equal length. In fact, results demonstrate mean stride-to-stride variation of 2.3%.
Using this time-normalizing procedure walking velocity continued to significantly
influence the Lyapunov exponent, λ^{Max}, i.e. monotonic trend of
lower stability at faster walking velocities. Unlike the analyses with exactly 100
points per stride, the temporal variation introduces a more quadratic trend.

Fast walking velocity may influence dynamic stability by a combination of several mechanisms. Walking velocity affects kinematics, double-support time, step width and other clinical correlates of stable walking [1]. Modified support mechanics may influence the ability to control movement disturbances. Analyses of bipedal walking demonstrate the existence of a biomechanical resonance associated the pendulum-like behavior of the skeletal structure and muscle stiffness [12]. These may contribute to stability at the preferred walking velocity [18]. Therefore, walking at velocities that are faster or slower than this resonant frequency require greater active neuromuscular control to maintain stable periodic movement [21]. In other words, faster walking velocities increase the segmental momentum thereby requiring greater effort from the neuro-controller to attenuate kinematic disturbances. Short stride durations limit the allowable time for neuromuscular corrections to compensate for mechanical disturbances or controller errors. The slow walking velocity requires active control that is out-of-phase with movement in order to slow the natural dynamics of the passive system. Hence, most of the variability observed by the Lyapunov exponents during the slow walking velocity appeared to be attributable to fluctuations in stride duration whereas during fast walking the variability was primarily associated with kinematic disturbances. This suggests that these subjects may be temporally less stable at the slow walking velocity than at fast walking velocities but spatially more stable at the slow velocity. Nonetheless, results suggest that the neuro-control system more effectively controlled kinematic disturbances at the slow velocity than during fast walking.

Data processing and analyses techniques must be considered when interpreting
the results. Rosenstein et al. [22] observed
that Lyapunov analyses are sensitive to the sampling frequency and length of the
data set. If the sampling frequency is sufficient to characterize the kinematic
variance then the length of the data set is more critical than sampling frequency.
If one were to use a common sampling rate for all velocities, then the length of the
data set at 80% *V*^{F} would include approximately half the
number of data samples per stride when compared to 20%
*V*^{F}. The only other study that directly compared walking
velocity to dynamic stability compared 3 min of gait data collected between 60% and
140% of a subject's preferred walking speed [8]. This would indicate walking velocities of approximately 27%, 45%, and
63% *V*^{F} that were similar to the walking speeds in the
current study. In their study, a fast trial may potentially have more than twice as
many strides with half as many data points per stride compared to a slow walking
trial, i.e. greater sample density at slow velocities than at fast velocities. To
investigate the effect of halving the number of data samples per stride, we compared
results from data that were re-sampled at 3000 samples per 30 strides versus results
from the same walking trials that were re-sampling at 1500 samples per 30 strides.
Note that both re-sample rates retained the stride-to-stride temporal variation of
the original kinematic data. Shorter data set lengths reduced the mean value of
λ^{Max} by 17.2%. Thus, when comparing the stability of
different walking velocities it is necessary to account for the effects of data
sampling rate and differences in data set lengths. To avoid artifacts from sample
density we recommend that stability analyses should normalized the time scale so
that there are an approximately equivalent number data points at fast walking
velocity as at slow velocity.

Several experimental and analytical limitations should be addressed in
future research. First, the maximum finite-time Lyapunov exponent,
λ^{Max}, represents the greatest rate of divergence in the
kinematic data but there are other dimensions wherein errors are attenuated at a
rate represented by the remaining λ coefficients. To fully characterize
the stability of the walking process the full set of Lyapunov exponents should be
investigated. Second, in a previous study of walking dynamics the stability of
walking was determined from a sensor placed over the first thoracic vertebrae [8]. Those measurements represent the
instantaneous kinematic expansion from the coupled dynamics of individual segment
movement from the lower limbs. However, we observed significant stability
differences in individual joints thereby indicating that the risk of failure may be
related to specific joints. Further analyses require matrix formulation wherein
separate Lyapunov coefficients are determined for homologous and multi-joint
interactions to characterize dynamic stability of locomotion. Third, the results may
be influenced by the fact that data were collected while subjects walked on
treadmill. Due to the spatial constraints of the video motion capture system; a
treadmill was necessary to enable collection of kinematic data across many
sequential strides. Studies have shown that a treadmill may reduce kinematic
variability and increase dynamic stability of gait measures [7]. Finally, our results were limited to healthy adults. The
influence of neuromuscular impairment may modify not only the stability of walking
[13] but the relation between walking
velocity and stability.

In conclusion, dynamic stability of walking is influenced by walking velocity with different contributions from the ankle, knee and hip joints. Analyses of clinical outcomes often demonstrate improved walking velocity following interventions indicating possible improvements in stability [1]. We recommend that clinical assessments should be expanded where possible to investigate whether (1) patients with neuromuscular impairment have abnormal stability and (2) whether conventional treatments successfully improve the stability of walking performance. These dynamic stability analyses may provide improved insight into neuromuscular control of dynamic locomotion.

We wish to thank J. Dingwell for his insight and comments regarding the data processing and interpretation of results. This study was supported by a grant HD 99-006 from NCMRR of the National Institute of Health.

Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.gaitpost. 2006.03.003.

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