## SUMMARY

Walking humans spontaneously select different speed, frequency and step
length combinations, depending on which of these three parameters is
specified. This behavior can be explained by constrained optimization of cost
of transport (metabolic cost/distance) where cost of transport is seen as the
main component of an underlying objective function that is minimized within
the limitations of specified constraints. It is then of interest to ask
whether or not such results are specific to walking only, or indicate a more
general feature of locomotion control. The current study examines running gait
selection within the framework of constrained optimization by comparing
self-selected running gaits to the gaits predicted by constrained optimization
of a cost surface constructed from cost data available in the literature.
Normalizing speed and frequency values in the behavioral data by preferred
speed and frequency reduced inter-subject variability and made group
behavioral trends more visible. Although actual behavior did not coincide
exactly with running cost optimization, self-selected gait and predictions
from the general human cost surface did agree to within the 95% confidence
interval and the region of minimal cost+0.005 ml O_{2} kg^{-1}
m^{-1}. This was similar to the level of agreement between actual and
predicted behavior observed in walking. Thus, there seems to be substantial
evidence to suggest that (i) selection of gait parameters in running can
largely be predicted using constrained optimization, and (ii) general cost
surfaces can be constructed using metabolic data from one group that will
largely predict the behavior of other groups.

## Introduction

It has been shown that humans and animals choose to move in a way that
minimizes the cost of locomotion
(Alexander, 2000;
Alexander, 2001;
Hoyt and Taylor, 1981;
Saibene, 1990). Until
recently, it was generally assumed that the least metabolically costly gait
for any given forward speed (*v*), step frequency (*f*), or step
length (*d*) could be described by a single functional relationship
between these parameters. Thus, one should be able to generate a single
behavioral relationship representing the least costly gait in
speed-frequency-step length (*v-f-d*) space by controlling any one gait
parameter, measuring the self-selected value of either one of the two other
parameters, and calculating the value of the third using some form of the
relationship *v=fd*.

However, Bertram and Ruina
(2001) suggested in a walking
study that not one but three different `least costly' relationships are
generally obtained by following such a procedure. The behavioral relationship
obtained depends on which parameter is specified. Thus, one `least costly'
behavioral relationship was obtained by specifying *v*, another by
specifying *f*, and yet another by specifying *d*. It is
apparent from these results that optimal gait is not rigidly predetermined by
internal factors, but rather depends on the conditions presented to the
individual and emerges from interaction between factors, both internal and
external to the individual.

But how can three different curves all represent the least costly gait? To explain this apparent paradox Bertram and Ruina formulated the constrained optimization hypothesis (Bertram and Ruina, 2001). According to this hypothesis, gait parameters are selected to optimize (minimize) some objective function within the limitations of imposed constraints. In keeping with the original observation that animals and humans move in a way that minimizes cost, Bertram and Ruina (2001) proposed that cost of transport (metabolic cost/distance) serves as the objective function and that the controlled gait parameters serve as constraints. Bertram (2005) compared self-selected behavioral relationships to behavioral predictions obtained by applying constrained optimization to a metabolic cost surface and found that these were strikingly similar for walking. This suggests that metabolic cost does indeed strongly influence choice of gait parameters, and validates constrained optimization as a model for predicting gait selection.

Is this result specific to walking, or does it apply to other aspects of
human movement control? There are many features of the mechanics of walking
that differ substantially from running. Identifying a similar control strategy
in both running and walking would indicate a general feature of movement
control effective at levels beyond the mechanics of each specific gait. The
objective of the present study was to test the applicability of the
constrained optimization hypothesis to running. We did this by comparing
self-selected running behavioral data collected under *v*-constrained,
*f*-constrained and *d*-constrained conditions to predictions
obtained by performing constrained optimization on metabolic data available in
the literature. This allowed us to see whether or not constrained optimization
of metabolic cost can reliably predict gait selection in other modes of
terrestrial locomotion besides walking, and whether or not constrained
optimization of metabolic cost data from one group of subjects can predict the
gait selected by another group.

## Materials and methods

### Subjects

Five healthy subjects (one female and four males) participated in the study. Anthropometric data for each subject are given in Table 1. We obtained informed consent from all subjects prior to experimentation. All testing was done according to the guidelines set by the Florida State University Human Subjects Committee Review Board.

### Self-selected running behavior

We employed methods similar to those detailed in Bertram and Ruina
(2001) and Bertram
(2005) for human walking, but
used constraint values appropriate for running. Since steady state locomotion
can be defined by the simple relationship *v*=*fd*, we evaluated
running behavior under three different constraint conditions:
*v*-constrained, *f-*constrained and *d-* constrained. In
each case one variable was controlled (either *v, f* or *d*),
one variable was directly measured, and the third variable was calculated
using the relationship *v*=*fd*. We briefly outline the specific
procedures below.

For constrained *v*, subjects ran on a treadmill (Desmo Pro,
Woodway, Wakeshaw, WI, USA) at constant belt speed. Eleven different belt
speeds were used, ranging from 0.49 to 4.32 m s^{-1}. We presented the
belt speeds at random to reduce the potential for systematic bias and
cross-trial interference. Between trials the subjects walked at a comfortable
speed until they had fully recovered. At each *v, f* was measured by
timing the duration of two sets of 20 steps using an electronic stopwatch. The
two trial results were averaged to obtain a reliable measure of *f* for
that speed and individual. We calculated step length using
*d*=*v/f*. Measurements were made after at least 1 min of
running at each *v*.

For constrained *f*, subjects ran in time to the beat of an
electronic metronome (KDM-1, Korg Inc., Tokyo, Japan) at ten different
frequencies ranging from 2 to 3.33 steps s^{-1}. Again, step
frequencies were randomly presented and subjects were allowed to fully recover
between trials. We measured *v* by timing how long it took subjects to
travel a 10 m segment of a 30 m level runway (using a portion of an outdoor
athletic track). Accurate measurements of speed were facilitated by use of two
portable cameras (TK-S241U, JVC, Victor Co., Yokohama, Japan), mounted
perpendicular to the path of the runner on tripods placed at the starting and
ending points of the 10 m distance along the straight portion of the track. We
combined the signals from both cameras into a single viewing channel
*via* a signal inserter (SCS splitter/inserter, American Video
Equipment, Houston, TX, USA) and fed the signal into a video monitor
(Panasonic, Matsushita Electric Industrial Co., Ltd., Kadoma, Japan). This
allowed the timer a perpendicular view of the starting and ending points. We
timed each 10 m run using an electronic stopwatch. We calculated step length
using *d*=*v/f*.

Finally, for constrained *d*, subjects ran by stepping on evenly
spaced markers (2 inch roofing nails with colored plastic washers inserted
into a grass athletic field) over level ground at ten predetermined step
lengths, ranging from 0.3 m to 2 m. Some subjects were unable to reliably
maintain 2 m step lengths, so only nine step lengths were used for these
individuals. Step lengths were randomly presented and subjects were allowed to
fully recover between trials. We also gave the subjects one or more practice
trials at each step length and did not take measurements until the subject
felt fully comfortable with the step length requirements of the trial. This
was especially necessary for step lengths approaching 2 m. We measured
*f* at each speed by timing the duration of 2 sets of 20 steps within
30 markers for each given step length. We then averaged the two measured
frequencies to obtain *f.* We calculated speed using
*v*=*fd*.

### Data analysis

#### Self-selected running behavior

All self-selected running behavioral data were pooled to evaluate general
gait selection trends. Before pooling the data, we normalized *v* data
by apparent preferred *v* (*v*_{p}), *f* data by
apparent preferred *f* (*f*_{p}), and *d* data by
*d*_{p}*=v*_{p}/*f*_{p} for each
subject. We considered normalizing the data by speed and frequency of variants
of the Froude number, but rejected this normalization since it did not improve
the fit of the linear regressions. We determined *v*_{p} and
*f*_{p} by estimating the location of the point of intersection
of the *v*-constrained, *f*-constrained, and
*d*-constrained *v*-*f* relationships
(Fig. 1). The point of
intersection should indicate the absolute minimum cost of transport for each
individual and, therefore, should also correspond to the freely chosen
*v* and *f* selected by an individual during unconstrained
running, as it does for walking (Bertram
and Ruina, 2001; Bertram,
2005).

The pooled self-selected running behavioral data were fit with
least-squares linear regressions (SigmaStat, SPSS, Chicago, IL, USA). Since
the constrained variable (independent variable) was different for each
constraint condition, regression analyses were performed with the data plotted
on different axes for each constraint. Frequency-constrained data were plotted
with *f* on the *x*-axis and *v* on the *y*-axis;
*v*-constrained data with *v* on the *x-*axis and
*f* on the *y*-axis; and *d* constrained data with
*d* on the *x*-axis and *f* on the *y*-axis.
However, for the sake of consistency and ease of comparison, each linear
regression equation was converted into *v*(*f*) form and
replotted in *v*-*f* space
(Fig. 2), as per Bertram
(2005).

The slope of the *v*-*f* relationship for each of the three
constraint conditions was obtained from the linear regressions and the
standard error for each slope was computed. A one-way analysis of variance
(ANOVA) was used to determine whether or not the three slopes were
significantly different from one another. Once statistical significance was
determined, a Tukey *post hoc* comparison (also in SigmaStat) was used
to identify where the significant differences lay. We defined statistical
significance as *P*≤0.05.

#### Cost surface

We compiled and evaluated cost data from several sources available in the literature (Cavanagh, 1982; Knuttgen, 1961; Liefeldt, 1992) as well as from an undergraduate student honors project done in our laboratory at Florida State University (Rouviere, 2002). See Appendix for an outline of the methods used in this thesis. Information on these data is displayed in Table 2. We used data from the single-subject studies directly and average values from multiple-subject data.

Although the above data represented a reasonable assemblage of running
metabolic cost data, all data sets did not agree well. Two sets of data,
*v*-constrained data from Knuttgen
(1961) and from Liefeldt
(1992), differed substantially
from the other data available. The cost values reported by Liefeldt
(1992) were unusually low
(0.1611-0.1769 ml O_{2} kg^{-1} m^{-1}). These levels
are approximately 70% of those reported by the remainder of the studies
(0.1930-0.2992 ml O_{2} kg^{-1} m^{-1}). Knuttgen
(1961) reported
*v*-constrained data in which *f* remained virtually constant
over a wide range of speeds. This is in contrast to observations from our
study as well as *v*-*f* data from Minetti et al. (1998),
indicating that subjects increase *f* as *v* increases (at least
under the speeds considered here). The differences between these two sets of
data and the other sets of data may be due to differences in method,
differences in equipment, or peculiarities of the subjects. In any case, it
seems safe to assume that these two sets of data do not represent standard
responses, so these two data sets were not included in the metabolic cost
profile evaluated in the current study.

Gross (no baseline correction) metabolic cost measurements (ml
O_{2} kg^{-1} min^{-1}) were converted to cost of
transport (ml O_{2} kg^{-1} m^{-1}), *C*, and
the data points plotted in *C-v-f* space. Since the data set was
composed of only 28 points once data points had been averaged for multiple
subject studies and outliers rejected, the resolution of the raw data was
inadequate to reliably predict behavior. Therefore, we used the `griddata'
function in MATLAB (MATLAB 5.3, The MathWorks Inc., Natick, MA, USA), a
triangle-based cubic interpolation algorithm, to construct a continuous cost
surface between the data points (Fig.
3) that would facilitate appropriate mathematical analysis of
optimization (see below).

Next, we calculated the partial derivatives,∂
*C*(*f,v*)/∂*f* and∂
*C*(*f,v*)/∂*v*, for the cost surface,
*C*(*f,v*), to generate gait predictions for the applied
constraints. According to the principle of constrained optimization,
individuals should choose *v-f* combinations that correspond to points
where one of the partial derivatives is zero in order to minimize the cost of
transport. This is equivalent to finding points where a constraint curve is
tangent to a cost contour (Bertram and
Ruina, 2001; Bertram,
2005) (Fig. 4). For
*v-*constrained conditions (*v* held constant), running cost is
minimized when *f* is chosen such that∂
*C*(*f,v*)/∂*f*=0. Likewise, for
*f*-constrained conditions (*f* held constant), running cost is
minimized when *v* is chosen such that∂
*C*(*f,v*)/∂*v*=0. Therefore, we predicted
self-selected *v-f* relationships under *v-* and
*f-*constrained conditions by plotting regions where both∂
*C*(*f,v*)/∂*f*=0 and∂
*C*(*f,v*)/∂*v*=0. We also plotted regions
that contain points <0.001 ml O_{2} kg^{-1} m^{-1}
and <0.005 ml O_{2} kg^{-1} m^{-1} from minimal
cost (*C*_{min}) for each constraint to show how sensitive cost
of transport is to changes in *v-f*
(Fig. 5A,B). A narrow region
indicates high sensitivity to changes in *v* and *f*, whereas a
wide region indicates relative insensitivity to differences in these values.
For *d-*constrained conditions, we replotted cost of transport data in
*d*-*f* space and the data were fit to a new cost surface. We
then calculated new partial derivatives∂
*C*(*f,d*)/∂*f* and∂
*C*(*f,d*)/∂*d* and plotted as∂
*C*(*f,d*)/∂*f*=0 to show the predicted
*v-f* relationship. We also plotted regions containing points <0.001
ml O_{2} kg^{-1} m^{-1} and <0.005 ml O_{2}
kg^{-1} m^{-1} from minimal cost for *d*-constrained
conditions (Fig. 5C). We did
not plot solutions to ∂*C*(*f,d*)/∂*d*=0 since
they duplicate the ∂*C*(*f,v*)/∂*v*=0
curve.

We chose to fit a new surface to the data points once we had replotted the
data in *C-d-f* space to make the numerical calculation of∂
*C*(*f,d*)/∂*f* simpler. It is relatively
straightforward to numerically calculate partial derivatives parallel to the
axes of the plot, whereas more involved calculations are required to determine
partial derivatives along other directions. This is because the `griddata'
interpolation algorithm generates points on the surface in a rectangular grid
aligned with the plot axes. However, one downfall of replotting the data is
that the two interpolated surfaces are not identical. Still, we do not feel
that the two surfaces differ enough to substantially affect the behavioral
predictions. This is supported by Fig. 6, which shows a comparison of the
curves generated by plotting points satisfying (i)∂
*C*(*f,v*)/∂*v*=0 and (ii)∂
*C*(*f,d*)/∂*d*=0.

Most of the metabolic data were taken under a single applied constraint
condition, so we did not have enough information to obtain
*v*_{p}*, f*_{p} and minimum cost of transport
for each data set. Therefore, we normalized predicted *v-f*
relationships from the metabolic data by *v*_{p} and
*f*_{p} for the pooled data in order to compare the predictions
made using the metabolic data to the self-selected behavioral data. We
determined *v*_{p} and *f*_{p} for the pooled
data by finding the coordinates of the absolute minimum metabolic cost.

## Results

### Self-selected running behavior

The slopes of the three different *v-f* relationships were
significantly different (Fig.
2). The *v-f* relationships for *v*-constrained and
*f*-constrained conditions were particularly well defined
(Table 2). The overall
*P*-values from the ANOVA and the pairwise *P*-values from the
Tukey *post hoc* analysis were all <0.001. This strongly suggests
that the observed differences between the slopes are caused by subjects
choosing gait parameters in specific response to the imposed constraints.

### Shape of metabolic cost surface

The metabolic cost surface has an ovoid bowl shape when plotted in
*C-v-f* space (Fig. 3).
The long axis of the bowl lies along the line of the *v-* constrained
behavioral curve. The bowl has relatively little curvature along the long axis
(contour lines are widely spaced), and thus along the *v-*constrained
behavioral curve, and higher curvature perpendicular to it (contour lines
closely spaced) (Figs 4 and
5).

### Self-selected vs predicted behavior

#### Speed constrained

Predicted and self-selected running behavior data agreed within the region
of minimal cost (*C*_{min})+0.005 ml O_{2}
kg^{-1} m^{-1} and 95% confidence interval for
*v-*constrained conditions (Fig.
5A). In the area where metabolic data were available, 21 out of 24
data points fell within the region of *C*_{min}+0.001 ml
O_{2} kg^{-1} m^{-1}, and two of the remaining points
fell within the region of *C*_{min}+0.005 ml O_{2}
kg^{-1} m^{-1}, while only one point fell outside these
regions. There was more scatter in the data for speeds roughly in the middle
third (∼0.8-1.4*v/v*_{p}) and for the lowest speeds
(∼0.14*v/v*_{p}).

#### Frequency constrained

Predicted and self-selected running behavior data also agreed within the
region of *C*_{min}+0.005 ml O_{2} kg^{-1}
m^{-1} and 95% confidence interval for *f*-constrained
conditions (Fig. 5B). In the
area where metabolic data were available, 17 out of 35 data points fell within
the region of *C*_{min}+0.001 ml O_{2} kg^{-1}
m^{-1}, 13 of the remaining points fell within the region of
*C*_{min}+0.005 ml O_{2} kg^{-1}
m^{-1}, and five points fell outside these regions. There was somewhat
more scatter in the data for higher speeds:∼
1.05-1.2*v/v*_{p}.

#### Step length constrained

Likewise, predicted and self-selected running behavior agreed within the
region of *C*_{min}+0.005 ml O_{2} kg^{-1}
m^{-1}and 95% confidence interval for *d*-constrained
conditions (Fig. 5C). However,
unlike *v-* and *f*-constrained conditions, only 7 out of 22
behavioral data points fell within the region of
*C*_{min}+0.001 ml O_{2} kg^{-1} m^{-1}
in the area where metabolic data were available, and only five more fell
within the region of *C*_{min}+0.005 ml O_{2}
kg^{-1} m^{-1}, whereas ten fell outside these regions. This
reflects a relatively high degree of scatter over all step lengths.

For all three constraint conditions, the data points that fall outside the
region of *C*_{min}+0.005 ml O_{2} kg^{-1}
m^{-1} came from a variety of individuals. This indicates that scatter
in the behavioral data was not due to the peculiar behavior of any one
individual.

## Discussion

### Shape of metabolic cost surface

It is well known that cost of transport (metabolic cost/distance) for
running is relatively constant when measured under *v-*constrained
conditions at commonly used speeds, and that cost of transport increases more
dramatically under *f-, d-* or fully constrained conditions
(Cavanagh, 1982; Diedrich and
Warren, 1995; Hreljac, 1993;
Knuttgen, 1961;
Kram and Taylor, 1990).
However, there is evidence that cost of transport does increase under
*v*-constrained conditions at extremely high and low speeds - i.e. at
speeds much higher or lower than *v*_{p}
(Hreljac, 2002). The metabolic
cost surface we created reflects these observations, since there is relatively
low curvature along the *v-*constrained behavioral curve and higher
curvature perpendicular to it.

### Normalization

Normalizing *v-f* values of the metabolic data after pooling can be
thought of as normalizing by an average *v*_{p} and
*f*_{p}. This normalization method did not reduce inter-subject
variability. The sole purpose of using this method was to facilitate
comparison between normalized self-selected behavior and predicted behavior.
However, if self-selected behavior does indeed reflect the shape of the
metabolic cost surface, then successfully collapsing the behavioral data into
generalized behavioral trends *via* normalization (i.e. scaling to
reduce inter-subject variability) implies that one should be able to
successfully generate any subject's cost surface by scaling a generalized cost
surface by the *v*_{p}, *f*_{p}, and minimum
cost of transport of that subject. Table
3 and Fig. 7 show
that inter-subject variability in behavioral trends was indeed reduced by the
normalization method used in this study.

### Actual behavior vs predicted behavior

#### Speed constrained

The narrow region of *C*_{min}+0.005 ml O_{2}
kg^{-1} m^{-1} slope indicates that the cost of transport
increases quite rapidly for *f* values to either side of the optimal
gait along horizontal lines representing *v-*constraints. Therefore,
there is a stiff energetic penalty associated with deviating from the optimal
gait under *v*-constrained conditions. So, we would expect to see
little scatter in the behavioral data. And, indeed, the *f*-constrained
data has the highest *R*^{2} value
(*R*^{2}=0.88) (Table
3).

#### Frequency constrained

As with *v*-constrained running, a narrow region of
*C*_{min}+0.005 ml O_{2} kg^{-1} m^{-1}
indicates that cost of transport increases quite rapidly for *v* values
to either side of the optimal gait along vertical lines representing
*f-*constraints. So again there should be a reasonably stiff energetic
penalty associated with deviating from the optimal gait under
*f*-constrained conditions and, consequently, little scatter in the
data. And again, prediction matches the observed behavior fairly well since
the *f*-constrained data has the second highest *R*^{2}
value (*R*^{2}=0.78) (Table
3).

One interesting feature of the predicted behavior for
*f*-constrained conditions is that multiple optima appear to exist for
each frequency at lower frequencies. This is similar to the predictions for
constrained walking (Bertram,
2005) in which multiple optima were predicted at higher
frequencies under *f*-constrained conditions. However, in walking,
observed behavior within the subject population was distributed between the
optima, whereas all observed behavior in the present study was concentrated at
the lowest speed optimum. This concentration could be due to the low number of
subjects recruited for the behavioral part of the study. It is possible that
if more subjects were included, some may have chosen the higher speed optimum.
It is also possible, however, that the lowest speed optimum was chosen because
it corresponds to a slightly lower metabolic cost than the higher speed
optimum (the method we used to locate optima does not distinguish between
local and global optima). This hypothesis is supported by the slope of the
metabolic cost contour lines. Contour lines at lower frequencies have roughly
positive slopes. Thus, for any given frequency, a lower speed should, in
general, have a lower metabolic cost. Since the lowest speed optimum is near
the edge of the cost surface, more low speed metabolic data would be needed to
conclusively confirm this hypothesis. A third possibility is that the lower
speed optimum would provide an adequate cost/distance solution and lower cost
rate (cost/time). Possibly this appealed to the subjects involved in this
study because this study involved more rigorous activity than the previous
walking study. At this time the interaction between factors potentially
influencing the objective function are not established
(Bertram, 2005).

#### Step length constrained

As under *v*- and *f*-constrained conditions, the fairly
narrow region of *C*_{min}+0.005 ml O_{2}
kg^{-1} m^{-1} indicates that cost increases quite rapidly for
*v-f* combinations along diagonal lines representing
*d*-constraints. This indicates that a rather large energetic penalty
should be associated with deviating from the optimal gait under
*d*-constrained conditions. However, this does not agree with the
measured behavioral response of the subjects studied. The
*d*-constrained behavioral data is the most scattered of all three
constraint conditions. One possible explanation for this discrepancy is that
the shape of the cost surface was distorted because we were not able to
normalize speed, frequency and cost values for each individual before pooling
the metabolic data. Another explanation is that other types of metabolic cost
such as cost per time may modify the shape of the cost surface under
*d*-constrained conditions (Bertram,
2005). Cost per step would not alter the results for
*d*-constrained conditions because cost per step and cost per distance
differ by only a constant, *d*, so both cost per step and cost per
distance surfaces would have the same minima along lines of constant
*d*. (Note that similar logic holds for cost per distance and cost per
time surfaces under *v*-constrained conditions.)

#### Implications

Although the predicted and actual behaviors do not coincide exactly, they
do agree quite well considering the confounding factors with this study, e.g.
small number of metabolic data points available, inability to normalize
metabolic data prior to pooling, etc. This indicates that minimizing cost per
distance can largely account for the complex behavior observed in human
running. This is especially important because walking and running employ
fundamentally different mechanics (Cavagna
et al., 1977). Walking is generally modeled using an inverted
pendulum to emphasize exchange of kinetic and potential energy, whereas
running is generally modeled using spring mass system to emphasize storage and
release of elastic strain energy. The fact that self-selected gait correlates
well with gait predicted *via* constrained optimization of metabolic
cost for both walking and running indicates that the control of these two
gaits might be quite similar. This, in turn, suggests that constrained
optimization might even be capable of predicting gait parameters for forms of
motion with even more radically different mechanics.

However, there is substantial evidence that constrained optimization of metabolic cost would not successfully predict the self-selected behavior for cycling or other human-machine forms of locomotion. It has been shown that experienced cyclists train themselves to pedal at a cadence that is significantly higher than that which minimizes metabolic cost per distance cycled for a given speed. Interestingly, less experienced cyclists spontaneously choose cadences that are closer to (although still higher than) the energetic optimum (Marsh and Martin, 1997). One possible explanation for this is that the body might judge optimality in a way that is inappropriate for locomotion when a machine intervenes. For example, the body might optimize whole body muscle work to minimize cardio-pulmonary metabolic cost per distance as appears appropriate for walking and running, when localized muscle fatigue is a more important limitation for performance in cycling (Foss and Hallén, 2005). Therefore, experienced cyclists train themselves to override their instincts in order to optimize race performance in the artificial human-machine integration of cycling.

Although some discrepancies exist between the behavior predicted by constrained optimization of cost per distance and the observed self-selected behavior, the basic form of the predicted and observed behavioral curves agreed. This was similar to the level of agreement demonstrated for walking (Bertram, 2005). Optimization of alternative objective functions such as metabolic cost per time and cost per step did not predict running gait as well as metabolic cost per distance (Fig. 8). However, it is likely that these other types of cost might still help shape the objective function and influence features of gait parameter selection (Bertram, 2005). Also, other factors not directly related to metabolic cost, such as local muscle fatigue and body temperature, might play a role in running, which places specific demands on the locomotory system due to the vigor of the activity. Finding a way to measure the extent to which these elements contribute to the objective function, and under which circumstances, would be a worthy and challenging goal for future studies.

## Appendix

### Summary of methods used in Rouviere's thesis

The purpose of C. Rouviere's honors thesis (Rouviere, 2002) was to examine a few hypotheses regarding signals that might trigger the transition between running and walking. Testing one of these hypotheses involved determining the shape of the cost surfaces for walking and running near their intersection.

To build these cost surfaces, Rouviere measured the metabolic cost of walking and running near the gait transition for 5 subjects; 3 male, 2 female (age=25±1.87 years, mass=79.1±10.8 kg, height=179.4±13.2 cm). Subjects came to the laboratory twice and performed 12 walking or running trials on the treadmill over a complete range of constrained speed, frequency and step length conditions. The order of frequencies and gaits were originally randomly assigned, but all subjects used the same random sequence. A recovery period of at least 5 min was provided between trials to reduce the effects of fatigue and ensure valid metabolic measurements.

Oxygen consumption and carbon dioxide release rates were obtained using
standard metabolic analysis techniques (TrueMax 2400, Parvo Medics, Salt Lake
City, UT, USA). Subjects were tested at least 2 h post-prandial. A 7 min
baseline consumption level was determined prior to each test session. This was
used to normalize the metabolic rates of the two testing days. Values for the
metabolic data points were calculated by averaging metabolic data from the
last 3 min of each 5 min trial. None of the running trials were particularly
strenuous as the highest running speed was in the range of 3 m s^{-1},
only slightly faster than the natural gait transition speed. However, to
ensure that all metabolic data were obtained during steady state exercise,
only trials where the rate of oxygen consumption had reached a steady value by
the third minute were used. Also, RER was monitored and values for all trials
were 0.92 or below.

## ACKNOWLEDGEMENTS

The concept of constrained optimization in locomotion was initially formulated by Andy Ruina, Theoretical and Applied Mechanics, Cornell University. We wish to thank the individuals who volunteered to participate in this study. We also wish to thank Sharon Bullimore for commenting on the original manuscript. This paper is based on work supported under a National Science Foundation Graduate Research Fellowship awarded to A.K.G. Any opinions, findings, conclusions or recommendations expressed in this paper are solely those of the authors and do not necessarily reflect the views of the National Science Foundation.

- © The Company of Biologists Limited 2006

## References

## SUMMARY

Walking humans spontaneously select different speed, frequency and step
length combinations, depending on which of these three parameters is
specified. This behavior can be explained by constrained optimization of cost
of transport (metabolic cost/distance) where cost of transport is seen as the
main component of an underlying objective function that is minimized within
the limitations of specified constraints. It is then of interest to ask
whether or not such results are specific to walking only, or indicate a more
general feature of locomotion control. The current study examines running gait
selection within the framework of constrained optimization by comparing
self-selected running gaits to the gaits predicted by constrained optimization
of a cost surface constructed from cost data available in the literature.
Normalizing speed and frequency values in the behavioral data by preferred
speed and frequency reduced inter-subject variability and made group
behavioral trends more visible. Although actual behavior did not coincide
exactly with running cost optimization, self-selected gait and predictions
from the general human cost surface did agree to within the 95% confidence
interval and the region of minimal cost+0.005 ml O_{2} kg^{-1}
m^{-1}. This was similar to the level of agreement between actual and
predicted behavior observed in walking. Thus, there seems to be substantial
evidence to suggest that (i) selection of gait parameters in running can
largely be predicted using constrained optimization, and (ii) general cost
surfaces can be constructed using metabolic data from one group that will
largely predict the behavior of other groups.

- gait
- locomotion
- cost
- control
- human

## Introduction

It has been shown that humans and animals choose to move in a way that
minimizes the cost of locomotion
(Alexander, 2000;
Alexander, 2001;
Hoyt and Taylor, 1981;
Saibene, 1990). Until
recently, it was generally assumed that the least metabolically costly gait
for any given forward speed (*v*), step frequency (*f*), or step
length (*d*) could be described by a single functional relationship
between these parameters. Thus, one should be able to generate a single
behavioral relationship representing the least costly gait in
speed-frequency-step length (*v-f-d*) space by controlling any one gait
parameter, measuring the self-selected value of either one of the two other
parameters, and calculating the value of the third using some form of the
relationship *v=fd*.

However, Bertram and Ruina
(2001) suggested in a walking
study that not one but three different `least costly' relationships are
generally obtained by following such a procedure. The behavioral relationship
obtained depends on which parameter is specified. Thus, one `least costly'
behavioral relationship was obtained by specifying *v*, another by
specifying *f*, and yet another by specifying *d*. It is
apparent from these results that optimal gait is not rigidly predetermined by
internal factors, but rather depends on the conditions presented to the
individual and emerges from interaction between factors, both internal and
external to the individual.

But how can three different curves all represent the least costly gait? To explain this apparent paradox Bertram and Ruina formulated the constrained optimization hypothesis (Bertram and Ruina, 2001). According to this hypothesis, gait parameters are selected to optimize (minimize) some objective function within the limitations of imposed constraints. In keeping with the original observation that animals and humans move in a way that minimizes cost, Bertram and Ruina (2001) proposed that cost of transport (metabolic cost/distance) serves as the objective function and that the controlled gait parameters serve as constraints. Bertram (2005) compared self-selected behavioral relationships to behavioral predictions obtained by applying constrained optimization to a metabolic cost surface and found that these were strikingly similar for walking. This suggests that metabolic cost does indeed strongly influence choice of gait parameters, and validates constrained optimization as a model for predicting gait selection.

Is this result specific to walking, or does it apply to other aspects of
human movement control? There are many features of the mechanics of walking
that differ substantially from running. Identifying a similar control strategy
in both running and walking would indicate a general feature of movement
control effective at levels beyond the mechanics of each specific gait. The
objective of the present study was to test the applicability of the
constrained optimization hypothesis to running. We did this by comparing
self-selected running behavioral data collected under *v*-constrained,
*f*-constrained and *d*-constrained conditions to predictions
obtained by performing constrained optimization on metabolic data available in
the literature. This allowed us to see whether or not constrained optimization
of metabolic cost can reliably predict gait selection in other modes of
terrestrial locomotion besides walking, and whether or not constrained
optimization of metabolic cost data from one group of subjects can predict the
gait selected by another group.

## Materials and methods

### Subjects

Five healthy subjects (one female and four males) participated in the study. Anthropometric data for each subject are given in Table 1. We obtained informed consent from all subjects prior to experimentation. All testing was done according to the guidelines set by the Florida State University Human Subjects Committee Review Board.

### Self-selected running behavior

We employed methods similar to those detailed in Bertram and Ruina
(2001) and Bertram
(2005) for human walking, but
used constraint values appropriate for running. Since steady state locomotion
can be defined by the simple relationship *v*=*fd*, we evaluated
running behavior under three different constraint conditions:
*v*-constrained, *f-*constrained and *d-* constrained. In
each case one variable was controlled (either *v, f* or *d*),
one variable was directly measured, and the third variable was calculated
using the relationship *v*=*fd*. We briefly outline the specific
procedures below.

For constrained *v*, subjects ran on a treadmill (Desmo Pro,
Woodway, Wakeshaw, WI, USA) at constant belt speed. Eleven different belt
speeds were used, ranging from 0.49 to 4.32 m s^{-1}. We presented the
belt speeds at random to reduce the potential for systematic bias and
cross-trial interference. Between trials the subjects walked at a comfortable
speed until they had fully recovered. At each *v, f* was measured by
timing the duration of two sets of 20 steps using an electronic stopwatch. The
two trial results were averaged to obtain a reliable measure of *f* for
that speed and individual. We calculated step length using
*d*=*v/f*. Measurements were made after at least 1 min of
running at each *v*.

For constrained *f*, subjects ran in time to the beat of an
electronic metronome (KDM-1, Korg Inc., Tokyo, Japan) at ten different
frequencies ranging from 2 to 3.33 steps s^{-1}. Again, step
frequencies were randomly presented and subjects were allowed to fully recover
between trials. We measured *v* by timing how long it took subjects to
travel a 10 m segment of a 30 m level runway (using a portion of an outdoor
athletic track). Accurate measurements of speed were facilitated by use of two
portable cameras (TK-S241U, JVC, Victor Co., Yokohama, Japan), mounted
perpendicular to the path of the runner on tripods placed at the starting and
ending points of the 10 m distance along the straight portion of the track. We
combined the signals from both cameras into a single viewing channel
*via* a signal inserter (SCS splitter/inserter, American Video
Equipment, Houston, TX, USA) and fed the signal into a video monitor
(Panasonic, Matsushita Electric Industrial Co., Ltd., Kadoma, Japan). This
allowed the timer a perpendicular view of the starting and ending points. We
timed each 10 m run using an electronic stopwatch. We calculated step length
using *d*=*v/f*.

Finally, for constrained *d*, subjects ran by stepping on evenly
spaced markers (2 inch roofing nails with colored plastic washers inserted
into a grass athletic field) over level ground at ten predetermined step
lengths, ranging from 0.3 m to 2 m. Some subjects were unable to reliably
maintain 2 m step lengths, so only nine step lengths were used for these
individuals. Step lengths were randomly presented and subjects were allowed to
fully recover between trials. We also gave the subjects one or more practice
trials at each step length and did not take measurements until the subject
felt fully comfortable with the step length requirements of the trial. This
was especially necessary for step lengths approaching 2 m. We measured
*f* at each speed by timing the duration of 2 sets of 20 steps within
30 markers for each given step length. We then averaged the two measured
frequencies to obtain *f.* We calculated speed using
*v*=*fd*.

### Data analysis

#### Self-selected running behavior

All self-selected running behavioral data were pooled to evaluate general
gait selection trends. Before pooling the data, we normalized *v* data
by apparent preferred *v* (*v*_{p}), *f* data by
apparent preferred *f* (*f*_{p}), and *d* data by
*d*_{p}*=v*_{p}/*f*_{p} for each
subject. We considered normalizing the data by speed and frequency of variants
of the Froude number, but rejected this normalization since it did not improve
the fit of the linear regressions. We determined *v*_{p} and
*f*_{p} by estimating the location of the point of intersection
of the *v*-constrained, *f*-constrained, and
*d*-constrained *v*-*f* relationships
(Fig. 1). The point of
intersection should indicate the absolute minimum cost of transport for each
individual and, therefore, should also correspond to the freely chosen
*v* and *f* selected by an individual during unconstrained
running, as it does for walking (Bertram
and Ruina, 2001; Bertram,
2005).

The pooled self-selected running behavioral data were fit with
least-squares linear regressions (SigmaStat, SPSS, Chicago, IL, USA). Since
the constrained variable (independent variable) was different for each
constraint condition, regression analyses were performed with the data plotted
on different axes for each constraint. Frequency-constrained data were plotted
with *f* on the *x*-axis and *v* on the *y*-axis;
*v*-constrained data with *v* on the *x-*axis and
*f* on the *y*-axis; and *d* constrained data with
*d* on the *x*-axis and *f* on the *y*-axis.
However, for the sake of consistency and ease of comparison, each linear
regression equation was converted into *v*(*f*) form and
replotted in *v*-*f* space
(Fig. 2), as per Bertram
(2005).

The slope of the *v*-*f* relationship for each of the three
constraint conditions was obtained from the linear regressions and the
standard error for each slope was computed. A one-way analysis of variance
(ANOVA) was used to determine whether or not the three slopes were
significantly different from one another. Once statistical significance was
determined, a Tukey *post hoc* comparison (also in SigmaStat) was used
to identify where the significant differences lay. We defined statistical
significance as *P*≤0.05.

#### Cost surface

We compiled and evaluated cost data from several sources available in the literature (Cavanagh, 1982; Knuttgen, 1961; Liefeldt, 1992) as well as from an undergraduate student honors project done in our laboratory at Florida State University (Rouviere, 2002). See Appendix for an outline of the methods used in this thesis. Information on these data is displayed in Table 2. We used data from the single-subject studies directly and average values from multiple-subject data.

Although the above data represented a reasonable assemblage of running
metabolic cost data, all data sets did not agree well. Two sets of data,
*v*-constrained data from Knuttgen
(1961) and from Liefeldt
(1992), differed substantially
from the other data available. The cost values reported by Liefeldt
(1992) were unusually low
(0.1611-0.1769 ml O_{2} kg^{-1} m^{-1}). These levels
are approximately 70% of those reported by the remainder of the studies
(0.1930-0.2992 ml O_{2} kg^{-1} m^{-1}). Knuttgen
(1961) reported
*v*-constrained data in which *f* remained virtually constant
over a wide range of speeds. This is in contrast to observations from our
study as well as *v*-*f* data from Minetti et al. (1998),
indicating that subjects increase *f* as *v* increases (at least
under the speeds considered here). The differences between these two sets of
data and the other sets of data may be due to differences in method,
differences in equipment, or peculiarities of the subjects. In any case, it
seems safe to assume that these two sets of data do not represent standard
responses, so these two data sets were not included in the metabolic cost
profile evaluated in the current study.

Gross (no baseline correction) metabolic cost measurements (ml
O_{2} kg^{-1} min^{-1}) were converted to cost of
transport (ml O_{2} kg^{-1} m^{-1}), *C*, and
the data points plotted in *C-v-f* space. Since the data set was
composed of only 28 points once data points had been averaged for multiple
subject studies and outliers rejected, the resolution of the raw data was
inadequate to reliably predict behavior. Therefore, we used the `griddata'
function in MATLAB (MATLAB 5.3, The MathWorks Inc., Natick, MA, USA), a
triangle-based cubic interpolation algorithm, to construct a continuous cost
surface between the data points (Fig.
3) that would facilitate appropriate mathematical analysis of
optimization (see below).

Next, we calculated the partial derivatives,∂
*C*(*f,v*)/∂*f* and∂
*C*(*f,v*)/∂*v*, for the cost surface,
*C*(*f,v*), to generate gait predictions for the applied
constraints. According to the principle of constrained optimization,
individuals should choose *v-f* combinations that correspond to points
where one of the partial derivatives is zero in order to minimize the cost of
transport. This is equivalent to finding points where a constraint curve is
tangent to a cost contour (Bertram and
Ruina, 2001; Bertram,
2005) (Fig. 4). For
*v-*constrained conditions (*v* held constant), running cost is
minimized when *f* is chosen such that∂
*C*(*f,v*)/∂*f*=0. Likewise, for
*f*-constrained conditions (*f* held constant), running cost is
minimized when *v* is chosen such that∂
*C*(*f,v*)/∂*v*=0. Therefore, we predicted
self-selected *v-f* relationships under *v-* and
*f-*constrained conditions by plotting regions where both∂
*C*(*f,v*)/∂*f*=0 and∂
*C*(*f,v*)/∂*v*=0. We also plotted regions
that contain points <0.001 ml O_{2} kg^{-1} m^{-1}
and <0.005 ml O_{2} kg^{-1} m^{-1} from minimal
cost (*C*_{min}) for each constraint to show how sensitive cost
of transport is to changes in *v-f*
(Fig. 5A,B). A narrow region
indicates high sensitivity to changes in *v* and *f*, whereas a
wide region indicates relative insensitivity to differences in these values.
For *d-*constrained conditions, we replotted cost of transport data in
*d*-*f* space and the data were fit to a new cost surface. We
then calculated new partial derivatives∂
*C*(*f,d*)/∂*f* and∂
*C*(*f,d*)/∂*d* and plotted as∂
*C*(*f,d*)/∂*f*=0 to show the predicted
*v-f* relationship. We also plotted regions containing points <0.001
ml O_{2} kg^{-1} m^{-1} and <0.005 ml O_{2}
kg^{-1} m^{-1} from minimal cost for *d*-constrained
conditions (Fig. 5C). We did
not plot solutions to ∂*C*(*f,d*)/∂*d*=0 since
they duplicate the ∂*C*(*f,v*)/∂*v*=0
curve.

We chose to fit a new surface to the data points once we had replotted the
data in *C-d-f* space to make the numerical calculation of∂
*C*(*f,d*)/∂*f* simpler. It is relatively
straightforward to numerically calculate partial derivatives parallel to the
axes of the plot, whereas more involved calculations are required to determine
partial derivatives along other directions. This is because the `griddata'
interpolation algorithm generates points on the surface in a rectangular grid
aligned with the plot axes. However, one downfall of replotting the data is
that the two interpolated surfaces are not identical. Still, we do not feel
that the two surfaces differ enough to substantially affect the behavioral
predictions. This is supported by Fig. 6, which shows a comparison of the
curves generated by plotting points satisfying (i)∂
*C*(*f,v*)/∂*v*=0 and (ii)∂
*C*(*f,d*)/∂*d*=0.

Most of the metabolic data were taken under a single applied constraint
condition, so we did not have enough information to obtain
*v*_{p}*, f*_{p} and minimum cost of transport
for each data set. Therefore, we normalized predicted *v-f*
relationships from the metabolic data by *v*_{p} and
*f*_{p} for the pooled data in order to compare the predictions
made using the metabolic data to the self-selected behavioral data. We
determined *v*_{p} and *f*_{p} for the pooled
data by finding the coordinates of the absolute minimum metabolic cost.

## Results

### Self-selected running behavior

The slopes of the three different *v-f* relationships were
significantly different (Fig.
2). The *v-f* relationships for *v*-constrained and
*f*-constrained conditions were particularly well defined
(Table 2). The overall
*P*-values from the ANOVA and the pairwise *P*-values from the
Tukey *post hoc* analysis were all <0.001. This strongly suggests
that the observed differences between the slopes are caused by subjects
choosing gait parameters in specific response to the imposed constraints.

### Shape of metabolic cost surface

The metabolic cost surface has an ovoid bowl shape when plotted in
*C-v-f* space (Fig. 3).
The long axis of the bowl lies along the line of the *v-* constrained
behavioral curve. The bowl has relatively little curvature along the long axis
(contour lines are widely spaced), and thus along the *v-*constrained
behavioral curve, and higher curvature perpendicular to it (contour lines
closely spaced) (Figs 4 and
5).

### Self-selected vs predicted behavior

#### Speed constrained

Predicted and self-selected running behavior data agreed within the region
of minimal cost (*C*_{min})+0.005 ml O_{2}
kg^{-1} m^{-1} and 95% confidence interval for
*v-*constrained conditions (Fig.
5A). In the area where metabolic data were available, 21 out of 24
data points fell within the region of *C*_{min}+0.001 ml
O_{2} kg^{-1} m^{-1}, and two of the remaining points
fell within the region of *C*_{min}+0.005 ml O_{2}
kg^{-1} m^{-1}, while only one point fell outside these
regions. There was more scatter in the data for speeds roughly in the middle
third (∼0.8-1.4*v/v*_{p}) and for the lowest speeds
(∼0.14*v/v*_{p}).

#### Frequency constrained

Predicted and self-selected running behavior data also agreed within the
region of *C*_{min}+0.005 ml O_{2} kg^{-1}
m^{-1} and 95% confidence interval for *f*-constrained
conditions (Fig. 5B). In the
area where metabolic data were available, 17 out of 35 data points fell within
the region of *C*_{min}+0.001 ml O_{2} kg^{-1}
m^{-1}, 13 of the remaining points fell within the region of
*C*_{min}+0.005 ml O_{2} kg^{-1}
m^{-1}, and five points fell outside these regions. There was somewhat
more scatter in the data for higher speeds:∼
1.05-1.2*v/v*_{p}.

#### Step length constrained

Likewise, predicted and self-selected running behavior agreed within the
region of *C*_{min}+0.005 ml O_{2} kg^{-1}
m^{-1}and 95% confidence interval for *d*-constrained
conditions (Fig. 5C). However,
unlike *v-* and *f*-constrained conditions, only 7 out of 22
behavioral data points fell within the region of
*C*_{min}+0.001 ml O_{2} kg^{-1} m^{-1}
in the area where metabolic data were available, and only five more fell
within the region of *C*_{min}+0.005 ml O_{2}
kg^{-1} m^{-1}, whereas ten fell outside these regions. This
reflects a relatively high degree of scatter over all step lengths.

For all three constraint conditions, the data points that fall outside the
region of *C*_{min}+0.005 ml O_{2} kg^{-1}
m^{-1} came from a variety of individuals. This indicates that scatter
in the behavioral data was not due to the peculiar behavior of any one
individual.

## Discussion

### Shape of metabolic cost surface

It is well known that cost of transport (metabolic cost/distance) for
running is relatively constant when measured under *v-*constrained
conditions at commonly used speeds, and that cost of transport increases more
dramatically under *f-, d-* or fully constrained conditions
(Cavanagh, 1982; Diedrich and
Warren, 1995; Hreljac, 1993;
Knuttgen, 1961;
Kram and Taylor, 1990).
However, there is evidence that cost of transport does increase under
*v*-constrained conditions at extremely high and low speeds - i.e. at
speeds much higher or lower than *v*_{p}
(Hreljac, 2002). The metabolic
cost surface we created reflects these observations, since there is relatively
low curvature along the *v-*constrained behavioral curve and higher
curvature perpendicular to it.

### Normalization

Normalizing *v-f* values of the metabolic data after pooling can be
thought of as normalizing by an average *v*_{p} and
*f*_{p}. This normalization method did not reduce inter-subject
variability. The sole purpose of using this method was to facilitate
comparison between normalized self-selected behavior and predicted behavior.
However, if self-selected behavior does indeed reflect the shape of the
metabolic cost surface, then successfully collapsing the behavioral data into
generalized behavioral trends *via* normalization (i.e. scaling to
reduce inter-subject variability) implies that one should be able to
successfully generate any subject's cost surface by scaling a generalized cost
surface by the *v*_{p}, *f*_{p}, and minimum
cost of transport of that subject. Table
3 and Fig. 7 show
that inter-subject variability in behavioral trends was indeed reduced by the
normalization method used in this study.

### Actual behavior vs predicted behavior

#### Speed constrained

The narrow region of *C*_{min}+0.005 ml O_{2}
kg^{-1} m^{-1} slope indicates that the cost of transport
increases quite rapidly for *f* values to either side of the optimal
gait along horizontal lines representing *v-*constraints. Therefore,
there is a stiff energetic penalty associated with deviating from the optimal
gait under *v*-constrained conditions. So, we would expect to see
little scatter in the behavioral data. And, indeed, the *f*-constrained
data has the highest *R*^{2} value
(*R*^{2}=0.88) (Table
3).

#### Frequency constrained

As with *v*-constrained running, a narrow region of
*C*_{min}+0.005 ml O_{2} kg^{-1} m^{-1}
indicates that cost of transport increases quite rapidly for *v* values
to either side of the optimal gait along vertical lines representing
*f-*constraints. So again there should be a reasonably stiff energetic
penalty associated with deviating from the optimal gait under
*f*-constrained conditions and, consequently, little scatter in the
data. And again, prediction matches the observed behavior fairly well since
the *f*-constrained data has the second highest *R*^{2}
value (*R*^{2}=0.78) (Table
3).

One interesting feature of the predicted behavior for
*f*-constrained conditions is that multiple optima appear to exist for
each frequency at lower frequencies. This is similar to the predictions for
constrained walking (Bertram,
2005) in which multiple optima were predicted at higher
frequencies under *f*-constrained conditions. However, in walking,
observed behavior within the subject population was distributed between the
optima, whereas all observed behavior in the present study was concentrated at
the lowest speed optimum. This concentration could be due to the low number of
subjects recruited for the behavioral part of the study. It is possible that
if more subjects were included, some may have chosen the higher speed optimum.
It is also possible, however, that the lowest speed optimum was chosen because
it corresponds to a slightly lower metabolic cost than the higher speed
optimum (the method we used to locate optima does not distinguish between
local and global optima). This hypothesis is supported by the slope of the
metabolic cost contour lines. Contour lines at lower frequencies have roughly
positive slopes. Thus, for any given frequency, a lower speed should, in
general, have a lower metabolic cost. Since the lowest speed optimum is near
the edge of the cost surface, more low speed metabolic data would be needed to
conclusively confirm this hypothesis. A third possibility is that the lower
speed optimum would provide an adequate cost/distance solution and lower cost
rate (cost/time). Possibly this appealed to the subjects involved in this
study because this study involved more rigorous activity than the previous
walking study. At this time the interaction between factors potentially
influencing the objective function are not established
(Bertram, 2005).

#### Step length constrained

As under *v*- and *f*-constrained conditions, the fairly
narrow region of *C*_{min}+0.005 ml O_{2}
kg^{-1} m^{-1} indicates that cost increases quite rapidly for
*v-f* combinations along diagonal lines representing
*d*-constraints. This indicates that a rather large energetic penalty
should be associated with deviating from the optimal gait under
*d*-constrained conditions. However, this does not agree with the
measured behavioral response of the subjects studied. The
*d*-constrained behavioral data is the most scattered of all three
constraint conditions. One possible explanation for this discrepancy is that
the shape of the cost surface was distorted because we were not able to
normalize speed, frequency and cost values for each individual before pooling
the metabolic data. Another explanation is that other types of metabolic cost
such as cost per time may modify the shape of the cost surface under
*d*-constrained conditions (Bertram,
2005). Cost per step would not alter the results for
*d*-constrained conditions because cost per step and cost per distance
differ by only a constant, *d*, so both cost per step and cost per
distance surfaces would have the same minima along lines of constant
*d*. (Note that similar logic holds for cost per distance and cost per
time surfaces under *v*-constrained conditions.)

#### Implications

Although the predicted and actual behaviors do not coincide exactly, they
do agree quite well considering the confounding factors with this study, e.g.
small number of metabolic data points available, inability to normalize
metabolic data prior to pooling, etc. This indicates that minimizing cost per
distance can largely account for the complex behavior observed in human
running. This is especially important because walking and running employ
fundamentally different mechanics (Cavagna
et al., 1977). Walking is generally modeled using an inverted
pendulum to emphasize exchange of kinetic and potential energy, whereas
running is generally modeled using spring mass system to emphasize storage and
release of elastic strain energy. The fact that self-selected gait correlates
well with gait predicted *via* constrained optimization of metabolic
cost for both walking and running indicates that the control of these two
gaits might be quite similar. This, in turn, suggests that constrained
optimization might even be capable of predicting gait parameters for forms of
motion with even more radically different mechanics.

However, there is substantial evidence that constrained optimization of metabolic cost would not successfully predict the self-selected behavior for cycling or other human-machine forms of locomotion. It has been shown that experienced cyclists train themselves to pedal at a cadence that is significantly higher than that which minimizes metabolic cost per distance cycled for a given speed. Interestingly, less experienced cyclists spontaneously choose cadences that are closer to (although still higher than) the energetic optimum (Marsh and Martin, 1997). One possible explanation for this is that the body might judge optimality in a way that is inappropriate for locomotion when a machine intervenes. For example, the body might optimize whole body muscle work to minimize cardio-pulmonary metabolic cost per distance as appears appropriate for walking and running, when localized muscle fatigue is a more important limitation for performance in cycling (Foss and Hallén, 2005). Therefore, experienced cyclists train themselves to override their instincts in order to optimize race performance in the artificial human-machine integration of cycling.

Although some discrepancies exist between the behavior predicted by constrained optimization of cost per distance and the observed self-selected behavior, the basic form of the predicted and observed behavioral curves agreed. This was similar to the level of agreement demonstrated for walking (Bertram, 2005). Optimization of alternative objective functions such as metabolic cost per time and cost per step did not predict running gait as well as metabolic cost per distance (Fig. 8). However, it is likely that these other types of cost might still help shape the objective function and influence features of gait parameter selection (Bertram, 2005). Also, other factors not directly related to metabolic cost, such as local muscle fatigue and body temperature, might play a role in running, which places specific demands on the locomotory system due to the vigor of the activity. Finding a way to measure the extent to which these elements contribute to the objective function, and under which circumstances, would be a worthy and challenging goal for future studies.

## Appendix

### Summary of methods used in Rouviere's thesis

The purpose of C. Rouviere's honors thesis (Rouviere, 2002) was to examine a few hypotheses regarding signals that might trigger the transition between running and walking. Testing one of these hypotheses involved determining the shape of the cost surfaces for walking and running near their intersection.

To build these cost surfaces, Rouviere measured the metabolic cost of walking and running near the gait transition for 5 subjects; 3 male, 2 female (age=25±1.87 years, mass=79.1±10.8 kg, height=179.4±13.2 cm). Subjects came to the laboratory twice and performed 12 walking or running trials on the treadmill over a complete range of constrained speed, frequency and step length conditions. The order of frequencies and gaits were originally randomly assigned, but all subjects used the same random sequence. A recovery period of at least 5 min was provided between trials to reduce the effects of fatigue and ensure valid metabolic measurements.

Oxygen consumption and carbon dioxide release rates were obtained using
standard metabolic analysis techniques (TrueMax 2400, Parvo Medics, Salt Lake
City, UT, USA). Subjects were tested at least 2 h post-prandial. A 7 min
baseline consumption level was determined prior to each test session. This was
used to normalize the metabolic rates of the two testing days. Values for the
metabolic data points were calculated by averaging metabolic data from the
last 3 min of each 5 min trial. None of the running trials were particularly
strenuous as the highest running speed was in the range of 3 m s^{-1},
only slightly faster than the natural gait transition speed. However, to
ensure that all metabolic data were obtained during steady state exercise,
only trials where the rate of oxygen consumption had reached a steady value by
the third minute were used. Also, RER was monitored and values for all trials
were 0.92 or below.

## ACKNOWLEDGEMENTS

The concept of constrained optimization in locomotion was initially formulated by Andy Ruina, Theoretical and Applied Mechanics, Cornell University. We wish to thank the individuals who volunteered to participate in this study. We also wish to thank Sharon Bullimore for commenting on the original manuscript. This paper is based on work supported under a National Science Foundation Graduate Research Fellowship awarded to A.K.G. Any opinions, findings, conclusions or recommendations expressed in this paper are solely those of the authors and do not necessarily reflect the views of the National Science Foundation.