Constrained optimization in human walking: cost minimization and gait plasticity
Department of Nutrition, Food and Exercise Sciences, Florida State University, Tallahassee, FL 32306, USA
(e-mail: jbertram{at}ucalgary.ca)
Accepted 12 January 2005
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Summary |
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Key words: gait, locomotion, metabolic cost, control, human
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Introduction |
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To account for this observation we have proposed the constrained
optimization hypothesis (Bertram and Ruina,
2001; Fig. 2).
Walking parameters (such as speed, step frequency or step length) will be
selected to minimize an underlying objective function for the constraints
imposed on the system. The combination of gait parameters that provide the
minimization will depend on the criteria that constrain the function, i.e.
what feature of walking is the end objective. This implies that a variety of
combinations of speed, step length or frequency can be selected, depending
only on the conditions externally imposed on the individual. Thus, the human
walking system is expected to display substantial behavioral plasticity,
determined by the circumstances under which it is operating and largely
unconstrained by higher order control features built into the coordination
system (Holt, 1996
).
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The above implies an integration between neuromuscular function and the physical circumstances that ultimately determine the most appropriate behavioral features of the gait. The constrained optimization hypothesis, if validated, would provide a new tool for investigating this integration. It provides a theoretical framework with specific predictive expectations that can be tested against hypotheses designed to evaluate and identify the limits and precision of the integration, and provides a means to explore the underlying mechanisms responsible. Assuming that human walking responds to other environmental constraints in a manner similar to that of the fundamental gait-determining parameters such as speed, step length and frequency, full understanding of constrained optimization might also provide a more precise means of predicting locomotion behavior under a range of normal and abnormal circumstances.
Behavioral data from a range of subjects fit well with the expectations of
the constrained optimization hypothesis
(Bertram and Ruina, 2001). In
fact, the response is so predictable that this experiment can be replicated as
an undergraduate laboratory exercise
(Bertram, 2002
). However, two
important questions remain: what is the nature of the objective function that
determines the optimization, and what are the limits of its influence?
Before the second question can be approached, the first must be determined.
It has been suggested that the most obvious candidate for the objective
function is the metabolic cost of locomotion, and there is some circumstantial
evidence supporting this view (Anderson and
Pandy, 2001; Bertram and Ruina,
2001
; Kuo, 2001
).
It is the purpose of this study to evaluate the metabolic cost of locomotion
as the objective function of constrained optimization in human walking. This
is accomplished in three steps: (1) generate a map of the metabolic cost
function for all combinations of walking parameters for a group of healthy
subjects and plot this in speed-step frequency space (a metabolic cost
surface), (2) use this empirical function to predict the optimum walking gait
parameters under specific imposed walking conditions using the constrained
optimization hypothesis as a predictive model and (3) compare the predicted
behavior to the behavior independently selected by the subject group under the
specified constraint conditions.
The above experiment constitutes a direct test of the constrained optimization hypothesis using metabolic cost of travel as the objective function. Association of the observed and predicted behaviors will indicate that metabolic cost is a substantial consideration in the automatic selection of basic parameters in human walking, even though the selected parameters may not match those routinely used in walking.
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Methods |
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All procedures performed were approved by the Florida State University Human Subjects Institutional Review Board and approved informed consent forms were acquired from each subject prior to participation.
Metabolic cost measurement
The metabolic cost of walking was determined for each subject by measuring
O2 consumption and CO2 elimination rates when walking on
a treadmill at a set speed while the subject matched their step frequency to
the audible tone of an electronic metronome. Note that constraint of both
speed and step frequency also implies constraint of step length, because only
a single step length will provide the appropriate speed for a given frequency.
For each subject, 49 metabolic measurements were taken over a range of speeds
and frequencies that covered the majority of the range physically possible.
Table 2 illustrates the
treadmill speeds, step frequencies, the sequence of speed-frequency
combinations used and mean energy expenditure (J kg-1
m-1) for the subject group.
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Each subject participated in four metabolic cost measurement sessions, each running on a different day but all occurring within one week. The first session was used to familiarize the subject with the protocol of the study, the gas analysis equipment (including mouthpiece, nose clip and supporting head gear) and to practice walking on a treadmill in time with a metronome beat. Each of the following three sessions were used to evaluate metabolic cost of walking, with the 49 measurements distributed between the three measurement sessions. All metabolic measurements were taken at least 2 h postprandial. A baseline metabolic rate was taken at the beginning and end of each measurement session, determined as the mean of the average over the last 2 min of a 10 min period of quiet standing. This served two roles. Consumption rate before and after the session was checked for equivalence. Metabolic rate at the end of each measurement session was within 3% of the original (and was usually indistinguishable from the original value), indicating that the procedure did not overtax the subjects. The mean of the beginning and ending daily baseline was also used to adjust for occasional small metabolic rate differences on different measurement days.
During each of the three walking cost measurement sessions, O2 consumption and CO2 elimination were monitored from 16-17 5 min activity bouts in which speed and step frequency were controlled. Rest periods of at least 2 min were allowed between activity bouts to reduce the effects of fatigue. The subjects were encouraged to take more time if they felt short of breath or fatigued. The O2 consumption and CO2 elimination rates were taken as the mean of the last 2 min of the 5 min walking trial. The exercise duration was purposely kept as brief as possible so that more measurement points could be collected in each session. It is recognized that the 5 min collection period, with the initial 3 min constituting the transition to steady consumption rate, is marginal. Oxygen consumption and CO2 elimination rates were visually monitored following the 3 min point and the measurement period was increased by one or more minutes if a convincing steady state had not been achieved by the beginning of the fourth minute. Note that almost all of the walking trials used in this analysis were not aerobically challenging. The subjects were not rigorously trained athletes but all were active individuals with good fitness levels and many subjects did not require data collection extensions for any of their speed-frequency combinations. Oxygen consumption and CO2 elimination rates (STP) were determined using a commercially available metabolic analysis system (TrueMax 2400, ParvoMedics, Salt Lake City, UT, USA).
The sequence of speed-frequency combinations followed a psuedo-randomized order. For each subject the complete grid of speed-frequency combinations was pre-determined and divided into rows and columns. The rows and columns were numbered in alternate sequence. The three sessions in which measurements were made were divided into 16 or 17 speed-frequency combinations each (for a total of 49 measurements over three sessions). The order of speeds and frequencies for a given subject was determined by randomly selecting the number for the row or column. Once a row or column was selected, metabolic measurements were made for the speed and frequency of every second cell. Rows and columns were randomly selected until all cells were measured. This reduced the chance of an order effect in the measurement of the metabolic data. The order for one subject is listed in Table 2 by the numbers in the lower right corner of each cell (1 through 49).
Oxygen and CO2 exchange rates were converted to caloric
equivalents (Lusk, 1924),
according to the assumption of low-to-moderate intensity work load and a
respiratory exchange ratio (RER) below the blood CO2 buffer level
(Romijn et al., 1992
).
Metabolic cost measurements were considered acceptable if the average RER
value over the final 2 min of each trial was 0.92 or less. Caloric cost rate
was converted to Joules s-1 and normalized for the mass of the
individual. This metabolic cost rate was then normalized as a metabolic cost
of travel by converting the consumption rate to energy per distance traveled,
as determined by the treadmill belt speed. Belt speed was directly measured
for each trial by hand timing belt marker progression while the subject was
moving on the belt.
Self-selected walking behavior
Walking behavior under each of the three walking constraints was determined
for each subject on a day distinct from those in which metabolic measurements
were made. Three constraints were applied: walking at a set speed (on a
treadmill), walking at a set frequency (level walking to a metronome beat) and
walking using a set step length (level walking matching step length to evenly
spaced floor markers). The subject's response was determined by measuring the
selection of the remaining two variables in the relation
v=df, where v=forward speed, d=step length
and f=step frequency. In each condition the subjects were told simply
to walk in a manner that felt comfortable under the imposed conditions. This
part of the study matched the procedures described in Bertram and Ruina
(2001), which are briefly
reiterated below.
Walking at constant v was controlled using a treadmill (Woodway, Desmo Pro, Waukeshaw, WI, USA) set at constant belt speed. Ten belt speeds were used, covering a range both above and below the subject's preferred walking speed (from 0.258 to 2.34 m s-1). Not all subjects were able to walk at 2.34 m s-1, so these subjects were measured at nine speeds. Step frequency at each v was measured by timing the duration of two sets of 20 steps after at least 1 min of walking at that v. Times for the two sets were averaged. Since v was set as treadmill belt speed and f was directly measured, step length could be calculated as d=v/f.
Constant f was imposed by asking the subject to walk to an electronic metronome beat. Frequencies were both above and below the subject's preferred step frequency. Ten frequencies were selected at intervals between 0.80 and 2.93 steps s-1. The subject walked on a level corridor and was timed over a measured distance twice during the circuit. The two measured v values were then averaged. Timing was accomplished using a stop watch, but this was facilitated by displaying video images of the beginning and end points of the measured distance for the stationary timer as the subject passed. In this case f was set, v was measured and step length again calculated as d=v/f.
Constant d was imposed by asking subjects to step in registry with tape markers arranged at even distances on a level hallway over a 30 m length. Ten marker spacings were used, ranging from 0.236 to 1.01 m (normal step length for an average individual is approximately 0.56 m). For each d, presented to each subject in a different random order, the duration of 20 steps was timed. This measurement was made for each of two runs per individual and the average of the two was taken. In this case d was set, f was measured and speed was calculated as v=fd.
Analysis
The most challenging aspect of the analysis is the characterization of the
metabolic cost contours, from which predictions of minimum cost walking could
be derived. Metabolic cost data were pooled within the subject group and the
cost surface was estimated from means at the 49 measured speed-frequency
combinations.
The wide range of speed and step frequency combinations for human walking mean that even with 49 metabolic cost data points this represents only a sparse characterization of the cost surface (Fig. 3A). Such a characterization does not provide the resolution necessary to reasonably predict the selection of walking parameters directly from the raw measurements (Fig. 3B). In order to estimate the metabolic cost surface from the data, it was necessary to generate a continuous cost surface between the measured data points (interpolate). To accomplish this we use a custom-designed plate-fitting routine written in Matlab® (The Mathworks Inc., Natick, MA, USA).
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The analysis routine is an approximate mathematical realization of a
physical idea that may be described as follows. Say we have a few measured
points representing a function of two independent variables. Plotting the
independent variables along x and y directions, and the
measured function value along the z direction, we get a set of points
in three dimensional space. These points lie on, or (allowing for measurement
error) close to, some curved surface. Our aim is to reconstruct that surface.
The physical idea is to imagine a large, thin, nearhorizontal plate with
finite bending stiffness. Imagine attaching each measured (x, y, z)
data point to the corresponding (x, y) point on the plate with a very
short, stiff spring. Each spring tries to move its attachment point on the
plate close to its corresponding measured data point (where the data point is
considered fixed in parameter space) but the properties of the plate resist
bending. The net result is that the plate takes on a smoothly deformed (bent)
shape as it passes somewhat close to each anchored data point. If we use
stiffer springs, then the fit at the data points is better; however, softer
springs give a smoother overall surface. The stiffness of the spring thus
provides a user-adjustable parameter for the surface-fitting process. This
data-fitting approach, and variants thereof, are widely used. For example, the
well known Bezier surfaces employed in many computer-assisted drawing
applications use this basic idea (Foley et
al., 1990). The approximations in the realization involve
constraining the plate at its edges, which are taken rather far from any of
the data points; and also in describing the deflected shape itself using a
truncated double Fourier sine series. Such functions are routinely employed in
a number of surface-fitting applications
(Brown and Churchill, 2001
).
Examples include applied mathematics (Yee,
1981
), climate modeling (Cheong,
2000a
,b
)
and material analysis (Winterbottom and
Gjostein, 1966
, Saylor et al.,
2000
).
The mathematical approximations of the metabolic cost surface used in the current analysis may be viewed as arbitrary but useful simplifications that do not affect the quality of the surface fit (as is borne out by our graphical results and the modest error measures generated). We believe, moreover, that any other reasonable fitting process would give essentially the same results, since it is simply an issue of creating a continuous surface guided by the location of the measured data, which are relatively numerous and evenly distributed over the entire region of concern.
Once a smooth surface is fit to the collected data (Fig. 3C), contours can be generated (Fig. 3D). Fit is evaluated by comparing the original data to the surface estimate (Fig. 3E). Speed vs frequency relations for the three constraint conditions can be predicted from the cost contours as described in Fig. 2 (Fig. 3F).
The fit in the analysis used here has two control characteristics. First,
the stiffness matrix, which determines the resistance of the plate to diverge
from the data points (the stiffness of the spring analogy described above).
The second control parameter sets the number of Fourier nodes, which
determines the complexity of the Fourier models that generate the surface
function. The fewer the nodes the more basic the shape of the surface.
Although the interpolation procedure is non-standard, it suited the purposes
of this study as the evaluation strategy was developed. Currently other
strategies are available that generate similar surface predictions; nonuniform
rational B-splines (Chaturvedi and Piegl,
1996) accomplish the same task and are gaining in popularity.
As indicated by the raw contours derived from simple prismatic connection of the data shown in Fig. 3B, some smoothing and interpolation of the cost surface between measured points was necessary to generate usable gait predictions. Not having a previous convention to work from, the strategy in this study was to employ a modest degree of smoothing and modify the data as little as possible. The parameters set in the analysis used in this study are shown in Fig. 4D-F where 142 (196) nodes are used with a stiffness matrix of 105. The smoothing function with these control parameters can be compared to that generated by equivalent nodes and a more pliant stiffness matrix of 103 (Fig. 4A-C) or to 42 (16) nodes with an equivalent stiffness matrix of 105 (Fig. 4G-I). The plate-fitting function control parameters used in the current analysis generate modest error values (note in Fig. 4F that this error landscape is an order of magnitude less than the other examples).
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The gait parameter combinations predicted by constrained optimization from the metabolic cost contours are quite specific. The smoothing strategy left as much information content as possible in the surface. However, this leaves the specific prediction vulnerable to minor irregularities of the surface, and the `optimum' thus determined may jump in unlikely ways (see for instance the step length constrained prediction, green dotted line, in Fig. 3F). If walk parameter selection is guided not so much by absolute optima but instead by limits, i.e. a range beyond which it is disadvantageous to operate, then it is best to evaluate the predictability of the cost function from its shape in the region of the optimum. The optimum is determined as the tangent to the cost contour for each parameter (Fig. 2). The curvature of the cost contour in the region of the tangent determines the specificity of the optimization. That is, a highly curved contour will require the selection of gait parameters that closely satisfy the optimization requirement. Under these circumstances deviation from the tangent would incur substantial extra costs. However, in the region of a straight or gradually curving contour, the selection of gait parameters that differ slightly from the absolute optimization will not have such large cost implications. In Fig. 5 the optima predicted by the constrained optimization criteria are indicated by ranges around the optima. This is done to provide a sense of the influence of variation in the contour slope at the tangent point. A series of ranges are shown as shaded regions where the deviations represent ±1% (red), ±5% (orange), ±10% (yellow), ±15% (gray) differences from the slope at the tangent point (Fig. 5).
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Data display and regression analysis
Self-selected walking behavior and optimization predictions are most
conveniently compared between different constraint conditions when plotted on
identical axes. However, this means that the independent variable may not be
represented on the abscissa as least-squares regression technique demands.
When necessary for plot comparisons, the regression equation was determined
using the appropriate independent variable, but the variables were then
mathematically replaced to represent the arrangement on the comparison plot.
For example, the data of Fig.
5A weregenerated using speed as the controlled (independent)
variable. However, for comparison to the other constraint conditions they are
plotted with frequency on the abscissa. The regression line depicted was
generated using a least-squares power function from a plot with speed as the
abscissa, then the equation so determined was plotted against the transposed
axes in Fig. 5A. This
regression differed substantially from that which would have been generated
using the variables as plotted, i.e. with frequency treated as the independent
variable for all constraints. Such manipulation was done whether the
independent variable is represented on the plot but not as the abscissa (as in
Fig. 5A), or whether the
independent variable was simply inferred by the plot but was not actually
represented (as in Fig. 5C).
Inspection indicates that the regressions provide a reasonable representation
of the mean behavior of the subject group. Note that the use of a power
function to fit some of these data is not intended to imply that a power
function determines the relationship between these variables. Rather, it is a
curvilinear relationship that appears to fit the data and is also convenient
to manipulate, as described above.
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Results |
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First, observe that the constraint manipulations that make up this experiment are evident from these data. In the speed constrained condition the data points lie at discrete speed levels (Fig. 5A), as expected for the pre-selected treadmill speeds used as the control manipulation in this case. Inter-subject variability at each predetermined speed is expressed as variability in step frequency (and therefore also step length). Likewise, in the frequency constrained data the variability within each discrete frequency is expressed as a range of selected speeds (Fig. 5B). Slightly less obvious are the discrete step lengths in the step length constrained plot. Since v=df, on this plot a constant step length is indicated as a slope emanating from the origin, where greater slopes indicate longer step lengths and lesser slopes shorter steps (Fig. 5C). In this latter case inter-subject variability is expressed as a combination of speed and frequency, but that variability is limited to a constant slope. Attempts to reduce the data variability within any of the constraint conditions using standard normalization techniques based on conversion to non-dimensional terms (i.e. based on speed and frequency variants of the Froude number) did not substantially reduce the inter-subject variability. The data are presented in the absolute form as the most direct illustration of the experiment and its results.
The subject group was purposely selected to represent a wide range of body forms and sizes and thus is expected to indicate a general test of the constrained optimization hypothesis. However, this diversity resulted in substantial scatter of the behavioral data. The inherent variability presents a challenge when evaluating the association between the self-selected walking parameters and those predicted by constrained optimization of walking cost. In order to represent appropriately the behavior selected for each speed, frequency and step length constraint, the mean and standard deviation for the group are plotted (Fig. 5). The speed and step length constrained walking behavior data have been fit with a power-function regression. Some of the frequency constrained walking behavior data are not well represented by a power function. These data are fit with quadratic regressions, using either the entire data set (solid line), the subset that chose the higher speed option (long dashes) and the lower speed subset (short dashes).
For speed constraint conditions (Fig. 5A) the data correspond well with the ±1% optimality. The ±5% optimality zone includes all mean values and a majority of the standard deviations and clearly represents the behavior data. Greater scatter of the data is noted at extremely slow speeds.
For frequency constraint conditions the range of all optimality zones are highly restricted up to frequencies of 1.8 Hz. Below this frequency the means of the behavior data follow the trend indicated by the optimum, but are displaced toward higher speeds than predicted (by approximately 1 standard deviation). At higher step frequencies, cost optimization predicts a wide range of cost-effective speeds, including multiple optima for each frequency. The behavioral data display a substantial degree of variability in this area as well. At higher frequencies the subject population displayed two behavioral strategies: one that appeared to fit the higher speed optimization, and one that matched the lower speed optimization. Although substantial scatter exists, speeds between the two optima were not selected.
For step length constraint conditions, ±1% and ±5% optimality winds back and forth through the major distribution of the behavior data. The ±5% optimality zone includes six of the ten means. Three of the means outside the prediction range lie within an area of the cost optimization zone bounded on both sides by more optimum areas. The regions in which these means lie do not vary much from the optimization criteria. Many of the standard deviations reach beyond the optimization zone. This is particularly true for the right boundary where many selected speeds and step frequencies exceed those predicted.
In order to allow tracking of individual responses to the imposed constraints, the data for each subject are provided in Fig. 6. Female and male subjects are separated, as are specific constraint conditions, but each individual is labeled independently within the gender. These are the same data plotted as in Fig. 5.
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Discussion |
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What are the potential explanations of this result? One possibility is that the observed responses that differ from normal, speed constrained walking simply represent neurologically determined artifacts, where the observed relationships indicate unnatural expressions of walking in the peculiar circumstances forced upon the subject. Such neurological factors could be psychological in origin, where such things as attention to foot placement to match floor markings is the source of interference with normal gait, or more functional interference with the neural control system itself, such as central pattern generator (CPG) coordination. In either case, the suggestion is that imposed constraints disrupt the normal gait pattern by interfering with the natural control system. This is most easily imagined with the step length constraint, where the overall speed-frequency relationship (Fig. 5C) differs markedly from what we consider `normal' (i.e. speed constrained walking, Fig. 5A). This may suggest that something unnatural is indeed occurring in the system. In normal walking, both step length and frequency increase to achieve increases in speed. In contrast, when walking is constrained to set step lengths, the tendency is for step frequencies to remain constant or decrease slightly as step lengths and speeds increase (Fig. 5C).
Rather than indicating an artifact, however, the relationship found for
step length constraint is consistent with both the constrained optimization of
cost (the prediction from the measured cost contours,
Fig. 2) and is to be expected
from simple mechanical considerations of walking dynamics for this constraint.
Consider the objective of step length constrained walking. In its purest
sense, the goal would be to move the foot (and as a consequence, the center of
mass of the body) from marker to marker for the least metabolic cost. In this
case the objective of the task becomes dominated by the mechanics of the swing
limb because it is placement of the foot near the marker that is the main
objective of the step, with no predetermined requirement for the details of
progress achieved. It is well recognized that the swing limb, though complex,
ultimately has pendular characteristics (Mochon and McMahon,
1980a,b
;
McGeer, 1990
;
Garcia et al., 1999
). One
fundamental feature of a pendulum is that the period is approximately constant
regardless of swing excursion. Thus, one might assume that if the body uses
the natural pendular frequency characteristics of the swing limb as one
component of a cost minimization strategy, then under constrained step length
a constant `natural' frequency would be employed. Certainly from
Fig. 5C, it can be seen that a
single step frequency (2 steps s-1) is used by one or more subjects
at all step lengths. In spite of the wide range of morphology in the subject
group (body mass, height and leg length) and the extreme range of step lengths
(0.24-1 m), a remarkably small range of frequencies were employed. This small
range of frequencies bracketed the frequency used at the preferred walking
speed, suggesting the natural frequency of the swing leg was a major
determinant of this aspect of the step under constrained step lengths. In
spite of the limited frequencies used, a range was observed. This simply
indicates that the motion of the swing leg is not the only determinant of
walking cost, and other rate-related costs also need to be considered. Such
factors as negative work and collision losses in the contact limb
(Donelan et al., 2002
;
Ruina et al., 2005
) would
contribute to locomotion cost in step length constrained walking, just as they
do for speed or frequency constraints. The main difference in step length
constrained walking is that, within the assigned task, motion of the swing
limb becomes a more determining factor of the overall cost function. Thus,
even the apparently odd relationship between speed and frequency in step
length constrained walking can, on closer inspection, be seen as a logical
(natural) response to the conditions imposed.
Interpreting the results
If it can be assumed that the walking parameters selected by the subjects
under each of these constraint conditions are not an artifact, how well does
the constrained optimization hypothesis, using metabolic cost per distance
traveled as the objective function, explain the selection of gait
parameters?
(i) Speed constraint
The behavioral data generated in speed constrained conditions matches that
observed in numerous previous studies under these same circumstances
(Minetti et al., 1994;
Nilsson et al., 1985
) and fits
well documented cost data (Atzler and
Herbst, 1927
; Margaria,
1938
; Ralston,
1958
). Subjects walking on a treadmill are expected to select this
speed-frequency relation, and this is attributed to the step frequency-step
length combinations providing cost minimization for each speed
(Kuo, 2001
). As such, these
data provide no surprises other than their relationship to the other
constraints employed in the present study. If walking parameters are selected
in normal, speed constrained conditions in the same manner as in notable
unusual circumstances like frequency and step length constraints, then the
walking relations generated for the alternative constraints may be viewed
simply as normal walking responses under those constraints. The functional
plasticity displayed in these other constraints suit the imposed
circumstances, just as speed constrained walking does under its constraint
condition (the required speed).
(ii) Frequency constraint
The data generated for frequency constrained walking are simultaneously the
most complex, the least closely related to the prediction and potentially the
most telling regarding the constrained optimization hypothesis. At the lower
step frequencies (below 1.5 steps s-1) the mean speed selected for
each frequency was systematically above the optimization (by approximately 1
S.D.). This is in a region of the cost surface where the
contours predict a sharp optimum (i.e. the contours are highly curved,
indicating a substantial cost for deviating from the optimum). At these lower
frequencies the metabolic cost rate is relatively low, even if the
cost/distance is higher. Since the subjects were not required to walk for
extended periods, it is possible that they selected higher walking speeds
because they were not familiar with walking at such low speeds, i.e. basically
out of impatience with the experimental protocol, and the deviation was
allowed by the low cost rate even though walking at those speeds was
relatively inefficient. It is also possible that at such low speeds features
of the swing and stance period were dissociated, giving a different
optimization solution. Changes in relative stance and swing period were not
measured in this study but may be useful to consider in future studies. The
predicted cost minimization should hold, however, unless differences existed
between frequency constrained over-ground walking (where gait selection was
measured) and the speed-frequency-step length constrained treadmill walking in
which walking cost was assessed. Alternatively, the deviation from the model's
expectation may indicate the influence of optimization criteria not considered
in the current model. For instance, if cost minimization at these or other
speeds was influenced in some part by a penalty for going slowly or by such
considerations as cost/step (rather than the cost/distance considered here),
then the observed behavior might fit an alternative model. Evaluation of such
alternatives will likely require a sophisticated set of experiments involving
isolation of specific variables influencing the subject's perception of the
task at hand. Such information is not available from the current study. The
systematic deviation from the model closely follows the slope of the
optimization in this region of the curve, which may suggest some influence on
its determination by the cost profile. Full explanation of this aspect of the
model and these differences between prediction and selected behavior will
require further evaluation, possibly with substantially longer duration
walking tasks and/or alternative constraint regimes.
At a constrained step frequency of 2.1 steps s-1 all subjects
but one walked at a speed exceeding preferred walking speed (1.1 m
s-1, Ralston, 1958;
Sun et al., 1996
); the
exception consistently walked at substantially lower speeds than any other
subject (Fig. 6D). At this step
frequency the cost contours indicate a single optimum speed that lies very
close to the mean selected by the subject group
(Fig. 5B). Beyond this
frequency, however, two distinct optima are indicated by the cost contours,
one at increasing speeds and another centered slightly below preferred walking
speed. For all frequencies above 2.1 steps s-1 the subject pool
spontaneously divided into two groups, one composed of four subjects that
appeared to choose the higher speed optimum and a second composed of the
remaining subjects that walked at decreasing speeds as frequency increased.
The mean behavior of each of the groups matched well with one or the other
optimization opportunity available in this region, as indicated by regression
of each subgroup (Fig. 5B).
This is compelling evidence that the selection of walking parameters is indeed
strongly influenced by constrained optimization of the metabolic cost. The
match between the selected behavior and the multiple optimizations in this
region is even more striking when it is observed that, despite the high
variability in these data, no speeds intermediate between the two optima were
selected, i.e. the scatter in these data is centered on each of the two optima
rather than evenly distributed over the entire range.
(iii) Step length constraint
In step length constrained walking the optimization prediction itself was
not well determined due to the congruency of cost contours and the slope of
the step length constraint. This feature can be seen by comparing the slope of
the speed-frequency data for each step length (where speed and step frequency
covary at a slope determined by the step length) and the slope of the cost
contour in the region of the data. The cost contours and step length
determined slopes are nearly the same throughout the range of step lengths
used (Fig. 5C). As a result of
this congruence there is little or no cost consequence for a wide range of
speed-frequency combinations and the ±1% and ±5% optimization
solutions meander back and forth across the data. These drifts in optimum
position likely indicate subtle variations in the cost profile that remain
because a minimal level of smoothing was used to generate the cost surface
from the original discrete point data, or might result from subtle artifacts
produced by the complex Fourier surface modeling routine. It is likely that
such minor fluctuations in cost are not meaningful, particularly for pooled
data such as these. This remains to be determined by future studies. The
±10% and ±15% ranges describe a broad swath of options near the
optimum that include a large proportion of those selected by the subjects.
This shows a general insensitivity of the speed-cost relation at any given
step length, though lower overall costs are associated with step lengths in
the normal range. Perhaps this cost insensitivity is indicative of a useful
design feature allowing for flexibility in step length selection and the
negotiation of obstacles that would routinely obstruct the path of a walker in
natural habitats. As discussed above, some aspects of the cost congruence and
behavior selection likely derive from the pendular action of the swing limb,
but a complete understanding of these features will depend on a full
exploration of the relation between walking dynamics and metabolic cost.
Implications
Of interest in the context of the general applicability of the constrained
optimization hypothesis is the indication from the current data that the same
explanation, constrained optimization of locomotion cost, can predict general
features of the self-selected response to both commonly (speed) and uncommonly
(frequency and step length) imposed walking conditions. In this case, the
control of a single feature of the walking gait (either speed, step frequency
or step length) appears adequate to elicit the myriad of other coordination
factors (nerve firing, muscle activation and force stimulation, feedback
integration, etc.) that ultimately result in coordinated (and effective)
walking. Such responsiveness suggests the coordinating system has both
substantial plasticity and is able to spontaneously respond to the integrated
physical and physiological conditions with which it is confronted. At this
point not enough information is available to determine how this is
accomplished in human walking, but the results of experiments like those
described here indicate that new opportunities are available to explore the
details of the process and the mechanisms responsible.
Although a reasonable correspondence occurs between the observed behavior
under each of the three walking constraints and the predicted behavior using
metabolic cost per distance (J kg-1 m-1), there are
several instances where the predictions are not as close as one might expect
if cost/distance is indeed the determining objective function. Notable
differences between cost minimization strategy and gait parameter selection
suggest that factors other than global cost of travel also influence the
control of human walking. Such consistent differences parallel previously
identified cost anomalies in gait such as the preference to change gait
between walking and running at speeds in which apparent cost is not minimized
(Hreljac, 1993;
Thorestensson and Roberthson,
1987
; Tseh et al.,
2002
). It is likely that the observed differences indicate that
cost/distance is not the only factor responsible for the behavior selected.
The complete objective function may well prove to be more complex, involving
partial contributions from a number of considerations. Such a function
(F) could take the form:
![]() |
where C0-Ci are proportional constants indicating the influence of each contributing function. Such contributing functions could range from the currently considered cost/distance through cost/time, cost/step, velocity (v) and other currently unidentified functions (Fi).
From the present study it is apparent that the cost/distance function has a substantial influence on the cost surface that makes up the landscape of the objective function, but each additional component could contribute a modifying influence. By providing a means to experimentally explore the process by which the human locomotion system selects the general features of gait, and by implication also the underlying specifics, constrained optimization provides a strategy for evaluating the factors that determine the control of walking in humans. The current study provides only a superficial examination of the potential for this line of examination. Constrained optimization implies that gait selection parameters should be predictable on an individual basis, rather than the general group characteristics considered here. The present work has demonstrated that taking the analysis to this level will require consideration of numerous potentially confounding factors such as individual motivation and experience with the task. The constrained optimization perspective, however, provides a new strategy for exploring this aspect of walking control and determining the role of these fundamental factors.
![]() |
Acknowledgments |
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Footnotes |
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References |
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