## SUMMARY

Notably absent from the existing literature is an explicit biomechanical
model linking limb design to the energy cost of locomotion, COL. Here, I
present a simple model that predicts the rate of force production necessary to
support the body and swing the limb during walking and running as a function
of speed, limb length, limb proportion, excursion angle and stride frequency.
The estimated rate of force production is then used to predict COL
*via* this model following previous studies that have linked COL to
force production. To test this model, oxygen consumption and kinematics were
measured in nine human subjects while walking and running on a treadmill at
range of speeds. Following the model, limb length, speed, excursion angle and
stride frequency were used to predict the rate of force production both to
support the body's center of mass and to swing the limb. Model-predicted COL
was significantly correlated with observed COL, performing as well or better
than contact time and Froude number as a predictor of COL for running and
walking, respectively. Furthermore, the model presented here predicts
relationships between COL, kinematic variables and body size that are
supported by published reduced-gravity experiments and scaling studies.
Results suggest the model is useful for predicting COL from anatomical and
kinematic variables, and may be useful in intra- and inter-specific studies of
locomotor anatomy and performance.

## Introduction

As our understanding of locomotor biomechanics has improved over the past half-century, a number of theoretical models have been proposed for predicting kinematic parameters, ground reaction forces and, less often, locomotor cost from anatomical variables. However, while limb length and proportion have been linked to energy cost of locomotion (e.g., Alexander and Jayes, 1983; Hildebrand, 1985), no current model incorporates these variables explicitly as determinants of locomotor cost. This study presents a mathematical model, based on the force production hypothesis (Kram and Taylor, 1990; Taylor 1994), that predicts the energy cost of locomotion for walking and running gaits as a function of limb length and proportion, and tests this model in a sample of humans.

Physiological studies have demonstrated empirically that the metabolic cost of locomotion primarily derives from the muscular force required to accelerate the body's center of mass as it oscillates through the stride cycle. Notably, the rate of force production, as proposed by Kram and Taylor (1990), predicts the rate of oxygen consumption during locomotion more accurately than other parameters, including the work done in moving the center of mass and limbs through a stride cycle (Heglund et al., 1982; Cavagna and Kaneko, 1977). Added-mass studies of quadrupeds and bipeds (Taylor et al., 1980; Kram, 1991; Wickler et al., 2001; Griffin et al., 2003) and gravity-manipulation studies of running humans (Farley and McMahon, 1992) have shown that the muscular force needed to generate the vertical component of ground reaction force, acting in opposition to gravitational acceleration, accounts for the majority of locomotor cost (see Taylor, 1994). In addition, the muscular force generated during braking and propulsion, associated with the horizontal component of ground force generation, also contributes as much as one-third of locomotor cost (Chang and Kram, 1999; Gottschall and Kram, 2003). Comparisons of bipeds and quadrupeds (Roberts et al., 1998a,b) suggest these determinants of locomotor cost work similarly for both, although differences in muscle fiber length and effective mechanical advantage of the limb joints may lead to higher costs for bipeds.

The relationship between limb length and locomotor energy cost is less
clear. Alexander and Jayes
(1983) initially proposed that
various gait parameters, including locomotor cost, are dynamically similar and
would scale by Froude number, a dimensionless constant that corrects for size
between pendular systems. Thus, the cost of locomotion for a given animal at a
given speed could be predicted by calculating the Froude number,
*U*^{2}(*L g*)

^{–1}, where

*U*is travel speed and

*L*is limb length. While there is some support for this proposal from studies of human walking (Alexander, 1984; Minetti et al., 1994), Froude numbers do not predict the scaling of cost or kinematic parameters during running (Minetti et al., 1994; Donelan and Kram, 2000). Furthermore, recent studies that manipulate gravity during walking have shown that stride length (Donelan and Kram, 1997) and energy cost (Farley and McMahon, 1992) are not constant at a given Froude number, suggesting dynamic similarity as proposed by Alexander and Jayes (1983) may not adequately describe walking mechanics.

An inverse relationship between locomotor cost and limb length has also
been proposed for running gaits, which are typically considered to act as
mass-spring systems rather than pendular systems. Kram and Taylor
(1990) suggested that `larger
animals with longer limbs and step lengths will have lower transport costs,'
as the magnitude of the vertical impulse decreases with longer stance periods.
However, while stance phase duration, or contact time, *t*_{c},
has been shown to correlate with limb length
(Hoyt et al., 2000), numerous
within- and between-species studies have found no relationship between limb
length and the cost of locomotion (walking humans,
Censi et al., 1998; running
humans, Ferretti et al., 1991; Cavanaugh
and Kram, 1989; Brisswalter et
al., 1996; interspecific studies,
Steudel and Beattie, 1995). In
the best study to date comparing short- and long-legged humans (mean limb
length 79 cm and 95 cm, short- and long-legged groups, respectively), Minetti
et al. (1994) found locomotor
cost was lower for long-legged individuals during walking but higher during
running. This difference between walking and running gaits, and the lack of a
clear relationship between limb length and locomotor cost, suggests a simple
relationship between limb length and locomotor cost is unlikely.

One complication in predicting the effect of limb length on locomotor efficiency is the cost of accelerating the limb during swing phase. While initial studies suggested swing cost is negligible (Taylor et al., 1974, 1980; Mochon and McMahon, 1980), more-recent studies have demonstrated that the muscular force required to swing the limb can constitute a significant portion of total locomotor cost. Several studies have measured the increase in energy cost when mass is added to the limb and have shown that energy costs increase directly with increased moment of inertia (Martin, 1985; Myers and Steudel, 1985; Steudel, 1990). More recently, Marsh et al. (2004) in a study of guinea fowl, measured energy consumption using blood flow in the limb muscles of guinea fowl and found limb swing contributed over 20% of total locomotor cost over a range of speeds. Thus, it appears a tradeoff may exist between the force required to support the body and the force required to swing the limbs. As limb length increases contact time increases and a lower rate of force production is necessary to support the body, but more force is required to swing the longer limb.

The model presented here predicts both the force required to support bodyweight and the force required to swing the limb as functions of limb length and proportion. Following Kram and Taylor (1990), the predicted rate of force production (i.e. the mean muscular tension required per step multiplied by step frequency) is then used to predict the rate of oxygen consumption. Predicted oxygen consumption is then tested against observed oxygen consumption in a sample of human recreational runners over a range of running and walking speeds. These results show that the model provides a useful framework for understanding the link between limb length and the cost of locomotion for both walking and running, incorporating both the cost of supporting the bodyweight and swinging the limb.

## Materials and methods

### The LiMb model

#### Justification and assumptions

The model presented here predicts the mass-specific rate of energy
expenditure (*V*_{O2} kg^{–1}
s^{–1}), hereafter cost of locomotion, COL, from the muscular
force generated to support the body's center of mass (COM) and swing the
limbs. To facilitate discussion and comparison with other models, the present
model is termed the LiMb model, as it incorporates the force generated to
swing the limb and support body mass (*M*_{b}) in predicting
COL.

The primary assumption of the LiMb model follows directly from the force
production hypothesis (Kram and Taylor,
1990; Taylor,
1994): the mass-specific rate of energy consumption is a linear
function of the rate of muscular force production. This assumption requires
that the relative shortening velocities of muscles and the force exerted on
the ground per unit of active muscle are independent of body size and speed.
Empirical studies suggest these criteria are met. *In vivo* studies
have shown that most muscles supporting bodyweight during locomotion contract
isometrically during steady walking and running, and therefore at a constant
relative shortening velocity. This has been demonstrated in turkeys
(Roberts et al., 1997),
walking humans (Fukunaga et al.,
2001), and wallabies (Biewener
et al., 1998), although other studies suggest shortening
contractions may be employed as well
(Gillis and Biewener, 2001;
Daley and Biewener, 2003).
Nevertheless, as long as the relative amount of muscle shortening work does
not change with speed or size, this assumption still holds. Additionally,
interspecific comparisons over a range of animal size have demonstrated that
the effective mechanical advantage, EMA, of extensor muscles scales inversely
with muscle fiber length (EMAα
*M*_{b}^{–0.26},
Biewener, 1989; fiber lengthα
*M*_{b}^{0.26},
Alexander et al., 1981)
suggesting that a given volume of muscle should exert the same force on the
ground independent of body size. Indeed, one important result of Kram and
Taylor (1990) was that the
ratio of metabolic energy expenditure to the rate of muscular force production
was constant across running speed, body mass, and species. Thus the available
evidence suggests that energy expenditure during locomotion may be predicted
by the rate of muscular force production without including major complexities
of muscle physiology.

Total predicted force production for the LiMb model is considered to depend on three components: vertical force, horizontal force and limb swing. Estimated force production for each component is derived separately.

#### Vertical forces

Vertical force production for both walking and running is estimated from
the change in vertical momentum of the body's COM. While `passive' mechanisms,
such as energy storage and release *via* tendons or the exchange of
potential and kinetic energy, may reduce the mechanical work done by the
muscles thereby improving energy economy
(Roberts et al., 1997), such
mechanisms still require muscular force to prevent the limb from collapsing
(i.e. to support bodyweight) and, to the extent the muscles perform true
mechanical work, to lift the COM. Therefore, in the LiMb model, positive
(upward) accelerations are viewed as a product of muscular force production
while negative (downward) accelerations of the COM are a product of gravity.
`Passive' mechanisms to reduce mechanical work, not considered explicitly in
the LiMb model, can be viewed as the efficiency with which oxygen consumption
is translated into force production. Accelerations of the COM require an
equivalent muscular force; passive mechanisms mediate the cost of producing
this force. Because the COM experiences free-fall during running but not
during walking, vertical accelerations are linked to kinematic and anatomical
variables differently for each gait and are, therefore, derived separately for
walking and running.

### 1. Running

Vertical force production during running is derived as follows (see
Fig. 1): during steady running
on level surface, the positive (upward) vertical acceleration,
*a*_{y}, produced by muscular force must be equal in magnitude
to that of gravity, * g*. Furthermore, average positive vertical
acceleration during contact time,

*t*

_{c}, must equal that of gravity during step period,

*T*

_{step}, or

*t*

_{c}

*a*

_{y}=

*T*

_{step}

*. For a simple two-dimensional model in which the limbs are treated as simple cylinders with no feet, the knees are modeled as telescoping (prismatic) joints, and protraction of the hind limb is equal to retraction, contact time during one step is a function of hind-limb length*

**g***L*, excursion angleϕ , and running speed

*U*, such that: (1) Average acceleration during stance phase,

*a*

_{y}, must then equal: (2) Because

*F*=

*Ma*, mean positive vertical acceleration,

*α*

_{y}, is equivalent to the mean mass-specific force produced during contact time (), or the average force of the vertical GRF impulse produced during stance phase. Treating the vertical impulse produced during stance phase as a single (virtual) muscle contraction with mean tension , the rate of muscular force production is equivalent to Eq. 2 multiplied by step frequency,

*f*

_{step}. Because

*f*

_{step}=

*T*

_{step}

^{–1}, the rate of muscular force production given by Eq. 2 is: (3) As outlined above, this model assumes the rate of oxygen consumption to be proportional to the rate of force production. As force is related to oxygen by some constant

*k*, the LiMb model predicts the mass-specific cost of locomotion (

*V*

_{O2}kg

^{–1}s

^{–1}) for running, COL

_{run}, based on vertical ground force production: (4)

Eq. 4 is similar to the prediction for locomotor cost proposed by Kram and
Taylor (1990; their equation
1: COL_{run}=*k gt*

_{c}

^{–1}), with the exception that

*t*

_{c}

^{–1}has been replaced with the equivalent expression (

*U*[2

*L*sin(ϕ/2)]

^{–1}). However, by incorporating hind-limb length, running speed and excursion angle as independent variables, this model has greater utility for comparative studies investigating the different effects of these variables on locomotor costs.

### 2. Walking

Vertical force production during walking is predicted as follows (see
Fig. 2). In a walking gait the
COM follows a sinusoidal trajectory, alternately accelerating upward
*via* muscular force production and downward *via* gravity. The
rate of muscular force production necessary to achieve this change in momentum
(i.e. to prevent the limb from collapsing) is a function of the vertical
change in position of the COM through stance phase (i.e. the amplitude of
oscillation) and the duration of each step. In a simple `stick-figure' model,
the amplitude equals *L*[1–cos(ϕ/2)], and step duration
equals *U*^{–1}[2*L*sin(ϕ/2)]. Given equal
periods of upward and downward acceleration of duration
*U*^{–1}[*L*sin(ϕ/2)], the average vertical
velocity of the COM during the first half of stance phase equals
+*L*[1–cos(ϕ/2)](*U*[*L*sin(ϕ/2)])^{–1},
while the average vertical velocity during the second half equals–
*L*[1–cos(ϕ/2)](*U*[*L*sin(ϕ/2)])^{–1}.
Note that in this case the vertical velocity of the COM is a function of
walking speed, which determines both the horizontal and vertical velocity of
the COM as is traverses its sinusoidal trajectory with gravity acting as a
restoring force. The movement of the COM in this case is analogous to that of
a ball rolling along a sinusoidal track; the forward speed of the ball also
determines its vertical velocity.

Assuming maximum vertical velocity is equal to twice the average vertical
velocity (the mathematically simplest case), the change in velocity between
steps (i.e. during the trough of the COM trajectory) must equal
4*L*[1–cos(ϕ/2)]. This change in velocity occurs over a
period of time equivalent to
*U*^{–1}[*L*sin(ϕ/2)], and therefore average
vertical acceleration for one step is given by:
(5)
Two such periods of upward acceleration occur per each stride. Treating one
period of acceleration as the product of one virtual muscular contraction with
average force
(as for
running), the mass-specific rate of force production for walking is found by
multiplying Eq. 5 by 2*f*, where *f* is stride frequency. Doing
this gives the predicted COL for walking based on vertical ground force
production:
(6)
As in running, *k* is a constant relating force generation to oxygen
consumption.

In addition to the force needed to perform this change in momentum (Eq. 6),
there is the constant acceleration of gravity, * g*. Here, I make
the simplifying assumption that the force required to resist gravity and
maintain an upright posture is equivalent to the metabolic cost of standing
quietly, although these costs probably differ. As the cost of standing is
subtracted from the `net' cost of locomotion, it is not included in the model
prediction of walking cost.

Note that the form of Eq. 6 is similar to a Froude number in that the rate
of energy expenditure scales with *U*^{2} and
*L*^{–1}. However, Eq. 6 differs in that gravitational
acceleration is not included: vertical force production is a function of
inertia, not weight. Furthermore, stride frequency and excursion angle are
included. The implications of these differences are discussed below.

#### Horizontal forces

To estimate horizontal forces, I make the simplifying assumption that the
combined vertical and horizontal ground reaction forces, GRF, produce a
resultant vector that passes through the COM throughout stance phase.
Empirical studies of ground reaction forces suggest this assumption is valid
(Chang et al., 2000;
Lee et al., 2004), and that
horizontal GRF covaries with vertical GRF
(Breit and Wahlen, 1997;
Chang et al., 2000).
Furthermore, it is assumed that horizontal deceleration during the first
portion of stance phase and horizontal acceleration during the second portion
are generated *via* muscular contraction.

Instantaneous horizontal force (see Fig.
3) during stance phase must, therefore, equal
*F*_{i}tanθ_{i}, where *F*_{i} is
the instantaneous vertical GRF and θ_{i} is the instantaneous
protraction or retraction angle of the limb. The summed horizontal force for
one stance phase, *F*_{x}, is therefore:
(7)

The time course of both vertical ground force production and excursion
angle are, therefore, necessary to compute *F*_{x}. To
approximate this value, I assume the average force for both braking and
propulsive impulses, , is
equivalent to the product of the mean value for θ_{i} and the
mean vertical mass-specific GRF during stance phase,
. The mean value
for θ_{i} is equivalent to 0.5(ϕ/2),
for the braking
or propulsive GRF is estimated as
,
and the combined horizontal force production (braking + propulsion) for one
step is estimated as:
(8)
It follows that mass-specific horizontal force production can be calculated
using the mass-specific mean vertical GRF,
:
(9)
and the rate of horizontal force production is therefore estimated as:
(10)
Note that
is the value estimated for running (Eq. 3) as the mass-specific rate of
vertical force production. Combining Eq. 3 and 10, therefore, gives combined
(vertical plus horizontal) force production for running:
(11)
Similarly, combined (vertical plus horizontal) force production for walking is
estimated as:
(12)
The LiMb model therefore predicts COL for walking and running, based on
vertical and horizontal ground force production, as:
(13)
(14)

### Limb swing

The work done to swing the limb can be calculated using the equation for
work done on a pendulum (Hildebrand,
1985):
(15)
where *D* is the radius of gyration, *M*_{L} is the
mass, *T* is the driven period, and *T*_{0} is the
natural, or resonant, period of the limb. Work is in Nm×rad, orτϕ
, and τ=*Fr*, where *r* is the effective lever arm
of the muscle. The rate of force production,
, is found by
multiplying both sides of the equation by the stride frequency *f* and
dividing by excursion angle ϕ, and *r*, which produces:
(16)

Dividing both sides of this equation by body mass *M*_{b}
produces the mass-specific rate of force production necessary to swing the
limb. This rate of force production is the predicted cost of limb swing,
*C*_{limb}:
(17)
where *b* relates force to oxygen. Note that *b* subsumes
*r*, as the oxygen/force ratio will be a function of the mechanical
advantage of the muscles (Roberts et al.,
1998a,b).
Also, because limb swing involves different muscle groups than those used to
support the body, and requires non-isometric contractions, it is not expected
that *b*=*k*, the term relating force production to oxygen
consumption for vertical and horizontal ground forces (Eq. 4, 6, 13, 14). When
Eq. 17 is used to predict limb swing cost without *b* (i.e. when
*b*=1) the estimated value is less than 1% that of stance phase cost
(Eq. 13, 14), more than an order of magnitude less than found for guinea fowl
(Marsh et al., 2004), and less
than is suggested by weighted-limb studies
(Martin, 1985;
Myers and Steudel, 1985).
Since the relationship between muscular architecture, contraction rate, force
production and oxygen consumption are not currently known well enough to
predict *b a priori*, a reasonable estimate must be made. Here, I use
*b*=30, as this produces values of *C*_{limb} in line
with Marsh et al. (2004). The
sensitivity of the model predictions to different estimates of *b* will
be discussed below.

Total mass-specific force production (vertical + horizontal + limb swing)
for walking and running is, therefore, predicted *via* the LiMb model
as:
(18)
(19)

### Testing the LiMb model

To test the LiMb model, nine human subjects (five male, four female; body
mass range: 53.3–94.3 kg) volunteered to perform a set of walking and
running trials at a range of speeds on a custom-built treadmill (tread
dimensions: 2×0.6 m) at the Concord Field Station in Bedford, MA, USA.
Subjects were recruited to maximize variation in limb length (range:
79–112 cm); all were healthy, fit, recreational runners (self reported
miles/week running: 6–25, median *N*=15) ages 20–35 with no
history of running-related injury or illness. Subjects wore their personal
running shoes for all trials. IRB (Human Subjects Committee) approval was
obtained from Harvard University prior to the study, and written informed
consent was obtained from each subject prior to participation. Subjects were
paid for their participation in accordance with Harvard University IRB
guidelines.

Limb length was measured as the vertical distance from the greater
trochanter, determined by palpation, to the floor while shod. Subjects walked
at four speeds, ranging from 1.0–2.5 m s^{–1}, or their
fastest sustainable walking speed, and ran at three speeds ranging from
1.75–3.5 m s^{–1}. The range of running speeds was
tailored to each subject such that the slowest running speed was slower than
the subject's volitional walk–run transition speed. Subjects performed
6–10 min trials at each speed while wearing a loose-fitting mask. Air
was pulled through the mask at 200–300 l min^{–1}; this
air was sampled and oxygen concentration monitored at 5 Hz using a
paramagnetic analyzer (Sable Systems PA-1B; Las Vegas, NV, USA). The system
was checked for leaks for each trial by bleeding N_{2} into the mask
at a known rate and plotting (N_{2} rate/mass-flow rate) against the
observed decrease in O_{2} content of the sub-sampled air; this
relationship was consistent across trials (*N*=9,
*r*^{2}=0.98, *P*<0.001, second-order polynomial
regression, no outliers).

Oxygen consumption was monitored during the trial in real time to ensure
that steady-state aerobic metabolism was achieved, and the rate of oxygen
consumption *V*_{O2} (s^{–1}) was measured
following Fedak et al. (1981)
using data from the last minute of each trial. The resting rate of oxygen
consumption, measured while standing for 6 min prior to the start of locomotor
trials, was subtracted from the rate of oxygen consumption for each trial, and
the difference divided by body mass to calculate the COL
(*V*_{O2} kg^{–1} s^{–1}) for each
trial, where *V*_{O2} is in ml.

Kinematic data was also collected during each trial using a high-speed
infrared camera system (Qualysis®; Qualysis Motion Capture Systems,
Gothenburg, Sweden) operating at 240 Hz. Reflective markers were adhered to
the skin overlying the greater trochanter and to the subject's shoe over the
calcaneal tuberosity and distal fifth phalange. Qualysis® data analysis
software was then used to measure: protraction angle at heelstrike
(heel–trochanter–floor), retraction angle at toe-off
(toe–trochanter–floor), contact time duration (heel strike
frame–toe-off frame) and stride frequency. Swing period was estimated as
(stride frequency)^{–1}. These variables, with speed, were used
to calculate the rate of force production for each trial using equations from
the model.

### Estimating limb swing cost

To determine limb swing costs it was necessary to estimate the radius of
gyration, mass and resonant frequency of the limb for each subject. Limb mass
was estimated as 16% of body mass, following Dempster
(1955). The radius of gyration
*D*=*hL*, where *h* is a measure of mass distribution; a
value of 0.56 was used for *h* following Plagenhoef
(1966).

The resonant period *T* is also a function of limb length *L*
and shape *h*, as
*T*_{0}=2π[*I*(*M gL*)

^{–1}]

^{0.5}, and

*I*=

*h*

^{2}

*L*

^{2}

*M*

_{L}. Thus: (20)

### Predictions

To test the utility of the LiMb model, predicted COL was plotted against observed COL. This was done using the equations for vertical force production (Eq. 4, 6), vertical plus horizontal force production (Eq. 13, 14), and total force production (Eq. 18, 19) to determine the contribution of each component in predicting COL. It was predicted that each additional component would improve the correlation between predicted and observed COL.

Next, the LiMb model (Eq. 18, 19) was compared with
*t*_{c}^{–1} and Froude number as alternative
predictors of COL_{run} and COL_{walk}, respectively. The LiMb
model was predicted to outperform contact time and Froude number in predicting
locomotor cost, as the model incorporates horizontal force production and limb
swing costs.

Finally, I tested the prediction that *k*, the constant relating
oxygen consumption to force production, was the same for walking and running
gaits. This was predicted because walking and running employ isometric
contractions in the same muscle groups over similar ranges of limb excursion.
Therefore, muscle length, effective mechanical advantage and relative
shortening velocity and, therefore, *k*, should be independent of gait.
In fact, Kram and Taylor
(1990) found *k* was
nearly constant across a large range of body sizes and limb design (e.g.
rabbit to horse); it seems likely, therefore, that *k* should be
similar between gaits within a species.

Least squares regression was employed in each of the above tests to
determine the percentage of observed variation explained by a given predictor
with each trial treated as an independent data point. To test for differences
in *k* between gaits, *k* was determined as the slope of the LSR
for predicted *versus* observed COL for walking and running, and these
slopes were compared following Zar
(1984, pp. 292).

## Results

The predicted rate of force production correlated significantly with
observed COL for both running and walking
(Table 1). For running, the
correlation between predicted and observed COL improved as each component of
force production was added (Table
1). Vertical force production (Eq. 4) predicted only 16% of the
variance in observed COL_{run} (*N*=27,
*r*^{2}=0.16, *P*<0.05), while including horizontal
force production (Eq. 13) increased explained variance to 25% (*N*=27,
*r*^{2}=0.25, *P*=0.01). Adding the cost of limb swing
(Eq. 18) had the largest effect, increasing explained variance in observed
COL_{run} to 43% (*N*=27, *r*^{2}=0.43,
*P*<0.001) (Fig. 4).
The low correlation coefficient is likely a result of between-subjects
differences in the force/oxygen constant, *k*, as will be discussed in
the following section.

For walking, vertical force production (Eq. 6) and vertical + horizontal
force production (Eq. 14) each explained 92% of the variance in observed
COL_{walk} (*N*=34, *r*^{2}=0.92,
*P*<0.001 for both conditions), while total force production (Eq.
19) explained 94% (*N*=34, *r*^{2}=0.94,
*P*<0.001) (Fig. 5).
Thus, while predicted force production explained a much higher percentage of
the variance in observed COL_{walk}, the horizontal-force and
limb-swing components did not improve the correlation between predicted and
observed COL_{walk} significantly.

The model performed as well or better than other predictors of cost. While
predicted COL_{run} (Eq. 18) explained over 40% of the variance in
observed COL_{run}, the inverse of contact time predicted less than
30% (*N*=27, *r*^{2}=0.29, *P*<0.01). For
walking, predicted COL_{walk} (Eq. 19) predicted over 90% of the
variance in observed COL_{walk}, as did Froude number (*N*=34,
*r*^{2}=0.91, *P*<0.001). While the model
outperformed (i.e. produced greater correlation coefficients) contact time and
Froude number as predictors of observed COL, comparisons of correlation
coefficients (Zar, 1984, pp.
313) revealed these differences were not significant (*P*>0.05).

The percentage of estimated total force production contributed from
vertical, horizontal, and limb-swing components was similar for walking and
running. For running, vertical force production accounted for 63.7%
(s.d. ±7%) of estimated force production, while
horizontal forces accounted for 19.8% (±1%) and limb-swing 16.5%
(±6%). For walking, vertical forces accounted for 49.7% (±11%),
horizontal forces for 21.3% (±5%), and limb-swing for 29% (±15%)
of total estimated forces. However, limb-swing estimates are considerably
lower when only normal walking speed (1.5 m s^{–1}) is
considered; vertical forces at this speed, rated as the most `comfortable'
speed by subjects, account for 59.8% (±5%), horizontal for 25.0%
(±3%), and limb-swing for 15.2% (±8%) of total estimated force
production. The contribution of each force component changes markedly with
speed, particularly for walking, as shown for a representative subject in
Fig. 6. Clearly, the proportion
of total estimated force production was not correlated with the proportion of
variance in COL explained by a given component.

The force/oxygen constant, *k*, was not significantly different
between gaits. The slope of the LSR for predicted *versus* observed
COL_{run} was 0.0044, which was not significantly different than the
slope for walking (slope=0.0045; *P*>0.05). Similarly, the
*y*-intercept of the LSR equations for walking (0.021) and running
(0.012) were not significantly different (*P*>0.05).

Values for *k* from this dataset are lower than that reported by
Kram and Taylor (1990) (mean
*k*=0.0092, s.d.=0.0022). This is probably a
result of estimated forces being greater when incorporating vertical,
horizontal and limb swing costs as in the LiMb model, rather than only
considering vertical forces as in Kram and Taylor
(1990). When only vertical
force production is used to predict COL (Eq. 4, 6), the value for *k*
given by LSR is 0.0061, which is near the 95% confidence interval calculated
from Kram and Taylor (1990)
(95% CI=0.0063–0.0119).

## Discussion

*Force production and* COL

The rate of force production predicted by the LiMb model for both walking
and running explained a significant amount of the variance in observed
COL_{walk} and COL_{run}. The agreement between predicted
force production and observed locomotor cost further supports the proposal
that the metabolic cost of locomotion is a function of muscular force
production (Kram and Taylor,
1990; Taylor,
1994; Griffin et al.,
2003). The success of the model suggests force production, and
therefore COL, can be predicted reliably from anatomical variables (limb
length and proportion) and basic kinematic parameters (speed, stride frequency
and excursion angle). As results here show, such an approach can be more
successful than other indices of locomotor cost, at least during running.

Vertical force production accounted for 50% or more of total estimated
force production across speeds for both walking and running gaits, more than
horizontal force and limb-swing combined. However, while vertical force
production alone is a good predictor of COL for walking, it is a poor
predictor in running in this dataset, with horizontal force and limb-swing
contributing markedly to the predictive power of the LiMb model. Thus, the
proportion of total force accounted for by a given component does not
necessarily reflect the power of that component in predicting COL. This might
be expected, as the regularity with which a given force component increases
with total cost, and therefore the predictive power of that component, need
not necessarily correspond with the magnitude of the force. Furthermore, force
components that are highly correlated will not improve the fit of the model
when combined. For example, estimated vertical and horizontal forces are
highly correlated for walking (*N*=34, *r*^{2}=0.99,
*P*<0.01) but less so for running (*N*=27,
*r*^{2}=0.48, *P*<0.01); consequently, combining
horizontal and vertical force production improves the fit of the LiMb model
for running, but has no effect for walking
(Table 1). This distinction
between the magnitude of a given force component and its reliability as an
index of COL may be relevant to studies seeking to identify discrete
components of locomotor cost.

If muscular force production is the primary determinant of energy
expenditure during locomotion, predicted rate of force production should
relate to observed COL similarly across gaits in which similar muscle groups
and shortening velocities are employed. Indeed, this appears to be the case;
the relationship between predicted force production and observed energy
expenditure, as determined by LSR, is similar for walking and running
(Fig. 7). Thus, while no single
model may successfully describe the mechanics of mass-spring running gaits and
pendular walking gaits (Donelan and Kram,
2000), results of this study suggest models using a common
paradigm of force production to predict energy cost may be successful across
different gaits and activities. If so, it may be possible to compare directly
force production and cost across widely different activities (e.g. walking
*versus* climbing), providing a new means of comparing anatomical
specialization and locomotor performance.

### Walking

The LiMb model was particularly effective in predicting COL_{walk}.
Predictions of the model fit observed COL_{walk}
(*r*^{2}=0.94) as well or better than other proposed models for
walking cost, including Froude number (*r*^{2}=0.91) and the
collision model proposed by Donelan et al.
(2002,
fig. 6;
*r*^{2}=0.89). The success of the model for walking suggests
the cost of walking is primarily a function of the change in momentum inherent
in the sinusoidal trajectory of the COM through a stride: the upward
acceleration of the COM through the troughs of this trajectory requires
muscular force. In addition, limb swing contributes to cost, particularly at
higher walking speeds in which swing periods are far shorter than the natural
period of the lower limb. The `determinants of gait' described by Saunders et
al. (1953) and others serve to
minimize walking cost by lowering the amplitude of the COM trajectory and by
increasing the duration (thereby decreasing the magnitude) of upward
acceleration.

Because walking cost is predicted as a function of inertia rather than
weight, the LiMb model for walking may explain deviations from dynamic
similarity reported previously (Farley and
McMahon, 1992; Donelan and
Kram, 1997). The LiMb model predicts relative stride length,
*S*_{rel} (stride length/limb length), will be a function of
excursion angle (*S*_{rel} ≈ 4sinϕ/2) independent of
gravity. Similarly, COL_{walk} is predicted to be dependent on speed
but independent of gravity, as the positive (upward) change in momentum of the
COM during walking is a function of speed, not gravity (see Eq. 5, 6). Both of
these predictions run counter to dynamic similarity, which predicts relative
stride length to be inversely proportional to * g*, and
COL

_{walk}to be a function of Froude number (

*UL*

^{–1}g

^{–1}). Results from reduced gravity experiments, in which gravitational acceleration,

*, is manipulated*

**g***via*a harness, fit LiMb model predictions better than those of dynamic similarity: stride length (Donelan and Kram, 1997) and COL

_{walk}(Farley and McMahon, 1992) were found to be largely independent of

*but not walking speed. Thus, while the LiMb model did not explain significantly more of the variance in observed COL*

**g**_{walk}than Froude number, it does appear to outperform predictions of dynamic similarity in reduced-gravity conditions.

Preferred step length, step frequency and speed relationships noted
previously (see Bertram and Ruina,
2001) for walking humans may also be explicable *via* the
LiMb model. Speed is equivalent to the product of step frequency,
0.5*f*, and step length, 2*L*sinϕ/2 and, therefore, at any
given speed a range of step frequencies and step lengths are possible.
However, COL_{walk} for a given speed is predicted (Eqn 14) to
increase more steeply with frequency than with excursion angle. Thus the LiMb
model predicts long step lengths and low frequencies to be preferred, which
may explain why humans do not minimize step length during walking as predicted
by collision models of walking mechanics
(Donelan et al., 2002).
Indeed, an interesting tradeoff may exist: high frequency and short steps
impose high costs as predicted by the present model while greater step length
increases collision costs, resulting in a U-shaped cost/step-length curve for
a given speed. If so, at any given speed, there will be one frequency/step
length combination that minimizes combined cost. Support for this hypothesis
is offered by Bertram and Ruina
(2001), who investigated
walking speeds, step frequencies and step lengths chosen when one of these
variables was constrained. The frequency/speed relationships chosen under the
three constraint conditions were consistent with an energy-minimizing
strategy.

### Running

The relationship between predicted and observed COL_{run} shows
considerably more variation (*r*^{2}=0.43) than in walking
(Fig. 4). While the LiMb model
outperforms contact time (*r*^{2}=0.29) as a predictor of cost,
the variation between predicted and observed cost begs explanation. One likely
source of increased variance in COL_{run} *versus*
COL_{walk} is between-subjects differences in the force/oxygen
constant, *k*. The LiMb model (Eqns 18, 19) predicts COL assuming that
*k* is constant across subjects, but this is unlikely. Differences in
the *k* (measured as COL/*t*_{c}) have been noted
previously (Weyand et al.,
2001) and might be expected, as differences in variables, such as
muscle-fiber type, running mechanics and limb-proportion, will likely lead to
differences in the efficiency with which oxygen consumption is translated into
force production. Because *k* determines the slope of the
predicted–observed COL regression, differences in *k* will lead
to greater variance in running trials, in which predicted force production is
greater. To examine whether differences in *k* explain the variance in
COL_{run}, *k* was determined for each subject empirically as
the slope of the LSR between predicted and observed COL for all trials
(walking and running). While the fit of the LSR for each individual was
excellent (*N*=7 trials per subject, mean *r*^{2}=0.98,
range: 0.93–0.99, *P*<0.001 for all subjects), differences
were observed in estimates of *k* (mean=0.0043, range
0.0028–0.0054, *N*=9 individuals, see
Fig. 8A). Using these estimates
of *k* to predict oxygen consumption *via* Eq. 18 and 19 reduced
the amount of unexplained variance considerably, and the fit of the LiMb model
was similar for walking (*N*=34, *r*^{2}=0.95,
*P*<0.001) and running (*N*=27, *r*^{2}=0.87,
*P*<0.001) (Fig. 8B).
This suggests between-subjects differences in *k* explain most of the
variance from predicted COL, but further work is necessary to test this
hypothesis.

As in walking, the LiMb model for running agrees with previous results from
reduced gravity experiments. COL_{run} is expected to increase
proportionally with gravity (Eq. 19), but be independent of body mass, because
COL_{run} predicted by the LiMb model is mass-specific. Farley and
McMahon (1992) reported
COL_{run} increases in direct proportion to gravity. Furthermore,
Chang et al. (2000), in an
investigation of the separate effects of gravity and inertia on running
mechanics, found vertical and horizontal force impulses (a measure of
predicted cost *via* the present model) changed in direct proportion to
gravity but were independent of body mass. Another study, examining the effect
of gravity on walk–run transition speeds
(Kram et al., 1997), found
preferred transition speeds decrease in proportion to gravity. Because
COL_{run} changes in proportion to gravity while COL_{walk} is
independent, the model predicts walk–run transition speeds, approximated
as the speed that COL_{run}=COL_{walk}, to decrease with
decreased gravity, in agreement with Kram et al.
(1997).

### Limb swing

Limb swing costs as predicted by the LiMb model are consistent with
previous studies in that these costs are low at normal walking speeds but
considerably greater at fast walking and running
(Fig. 6). However, predicting
limb swing cost is complicated by the necessity of estimating an oxygen/force
constant, *b*, and by the necessary constraint that hind-limb inertial
properties are estimated. This dependence on *b* is especially critical
for running, in which estimates of COL_{run} have a greater impact on
the fit of the model. At low values of *b* (e.g. *b*<10),
estimated swing costs relative to the cost of accelerating the COM are so low
as to be negligible, and the fit of the model for running does not exceed that
for Eq. 13. Similarly, at high values for *b* (e.g. *b*>100),
predicted limb swing cost dominates predicted COL_{run}, and the fit
of the model is diminished. Using an estimate of *b* that produces
swing costs similar to those reported by Marsh et al.
(2004) produces a good fit,
but further work is necessary to determine if this value (*b*=30) is
reasonable. For example, it is clear that limb proportion and therefore
inertial properties differ between guinea fowl and humans. Similarly, while
the method used here to estimate hind-limb inertial properties produces a
reasonable fit to the data, future work needs to improve these estimates by
incorporating anatomical data from individual subjects.

*The effect of limb length on* COL

The LiMb model predicts a somewhat complicated relationship between limb
length and COL in which longer limbs decrease the cost of accelerating the COM
(Eq. 13, 14) but increase the cost of limb swing (Eq. 18, 19). Data from this
study as well as others strongly suggests longer limbs do in fact decrease the
magnitude of vertical ground forces (i.e. the change in vertical momentum of
the COM) at a given speed. In the present human sample, vertical ground force
at a given speed, estimated as *t*_{c}^{–1}
*U*^{–1}, was significantly greater for subjects with
shorter legs (Fig. 9), a result
predicted by the LiMb model (Eq. 4, 6). Similarly, Hoyt et al.
(2000) found contact time was
strongly correlated with limb length in comparisons between species. However,
while longer limbs decrease the force necessary to support bodyweight, the
increased cost of swinging longer, heavier limbs apparently eliminates a
simple univariate relationship between limb length and locomotor cost. As a
result, COL in this study was negatively correlated with limb length only at
moderately fast walking speeds (2.0 m s^{–1}:
*r*=–0.87, *P*<0.01, *N*=9) where the cost of
accelerating the COM is high but swing cost is low. The trade-off between the
force needed to accelerate the COM and that needed to swing the limb obviates
a simple relationship between limb length and COL, at least for within-species
comparisons in which swing costs explain much of the variance in COL.

This trade-off may be less salient for comparisons of COL between species,
resulting in a simple inverse relationship between limb length and COL. Kram
and Taylor (1990) and other
studies (Taylor et al., 1974,
1980;
Taylor, 1994) have suggested
the force produced to support bodyweight determines the scaling of COL with
body size between species. Swing cost, in contrast, may be less important in
between-species comparisons as decreases in stride frequency offset increases
in limb length with body size (Hildebrand,
1985; Heglund and Taylor,
1988). If the force produced to accelerate the COM does in fact
determine the scaling of COL, the LiMb model predicts COL to scale inversely
with limb length: vertical and horizontal ground forces are a product of
*L*^{–1} (Eq. 13, 14). This prediction is supported by
interspecific comparisons of COL. Because limb length scales as
*M*_{b}^{0.33}
(Alexander et al., 1979), the
LiMb model predicts COL to scale as
*M*_{b}^{–0.33}. This exponent (–0.33) is
similar to the scaling relationship reported by Taylor et al.
(1982; exponent: –0.32,
95% CI –0.29 to –0.34). This agreement between predicted and
observed scaling suggests the LiMb model may be useful for between- as well as
within-species investigations of locomotor cost. Moreover, it suggests limb
length may drive the interspecific scaling of COL, as suggested by Kram and
Taylor (1990) and others.

## Abbreviations

- ϕ
- excursion angle; the angle included by the limb through stance phase
- a
- acceleration (where
*a*_{y}is vertical acceleration,*a*_{x}is horizontal acceleration) - b
- ratio of oxygen consumption / force production while swinging the limb, in ml/N
- COL
- cost of locomotion; the mass-specific rate of energy consumption during locomotion
- COM
- center of mass of the body
- d
- horizontal distance moved by the COM during stance phase
- D
- radius of gyration of the limb
- EMA
- effective mechanical advantage
- F
- force (where
*F*_{y}is vertical force,*F*_{x}is horizontal force) - f
- frequency (
*f*_{step}is step frequency) - g
- gravitational acceleration
- GRF
- ground reaction force
- h
- index of mass distribution of the limb;
*h*=*D*/*L* - I
- moment of inertia
- k
- ratio of oxygen consumption / force production, in ml/N
- L
- limb length
- M
- mass (
*M*_{b}body mass,*M*_{L}limb mass) - q
- vertical displacement of the COM during walking
- r
- effective lever arm of the muscles that swing the leg
- S
- stride length
*S*_{rel}- relative stride length
- T
- period (
*T*_{0}is natural period) *T*_{step}- Step period, defined as the period between two consecutive (contra-lateral) heelstrikes
*t*_{c}- Contact time; the duration of stance phase
- U
- travel speed
*V*_{O2}- volume of O
_{2}consumed - v
- shortening velocity of a muscle (
*v*_{max}is maximum velocity)

## ACKNOWLEDGEMENTS

I thank J. Polk, P. Madden, H. Herr and D. Raichlen for useful discussions that proved invaluable in developing the LiMb model. D. Lieberman and A. Biewener provided useful critique for this manuscript and helpful suggestions for testing the model. This study was supported by a National Science Foundation Doctoral Dissertation Improvement Grant to the author and by the Harvard University Anthropology Dept.

- © The Company of Biologists Limited 2005