Mechanical
work
basic mechanisms
in terrestrial
locomotion:
two
for minimizing
energy expenditure
GIOVANNI
A. CAVAGNA,
NORMAN C. HEGLUND,
AND C. RICHARD TAYLOR
Istituto di Fisiologia Umana dell’Universitci
di Milano, Centro di Studio per la Fisiologia
de1 Lavoro Muscolare de1 CNR, 20133 Milano, Italy; and Museum of Comparative
Zoology,
Harvard University, Cambridge,
Massachusetts 02138
elasticity;
ciency
walking;
running;
hopping;
trotting;
galloping;
efi-
ON FIRST CONSIDERATION,
the methods which vertebrates have utilized to move along the earth’s surface
seem both diverse and complex. Some use two legs
while others use four. They walk, amble, trot, pace,
canter, gallop, and hop (19). Biologists have concentrated on describing
differences
between locomotory
types, and detailed descriptions exist for the foot-fall
patterns and the anatomic features associated with the
different modes of locomotion (16, 17, 19).
In this paper we have tried to find mechanisms which
are common to these different
modes of terrestrial
locomotion.
We selected animals to represent
what
appear to be three very difftirent types of locomotion:
bipedal birds which walk and run (turkey and rhea);
quadrupedal
mammals
which walk, trot, and gallop
(dog, monkey, and ram); and bipedal mammals which
hop (kangaroo and springhare).
We measured the forces applied by the animals to
the ground as they moved at different speeds and used
different gaits. These force measurements
were used to
calculate the mechanical
energy which must be provided by the locomotory system to move the animal’s
center of mass forward relative to the ground. We
considered only the situation after an animal had accelerated and reached a constant average speed. Locomotion at a constant average speed consists of a series of
cycles (steps or strides), during which both the gravitational potential energy and the kinetic energy of the
center of mass oscillate between maximum
and minimum values as the center of mass rises, falls, accelerates, and decelerates. Forces must be applied to the
ground to raise and reaccelerate the center of mass (12,
15), and mechanical work is performed (work = force x
displacement).
This has been called the external work
output Wezt of locomotion
(7), and the rate at which
this work is performed is then external power output
( Wed. west is only part of the total work done by the
animal during locomotion.
The muscles also perform
work to change the kinetic energy of the limbs relative
to the center of mass. This has been called the internal
work output Wint of locomotion;
we did not measure
Wint in this study, but it has been measured for man
(5), quail (lo), and kangaroos (2) by other investigators.
Contracting
muscles use chemical energy to supply
some, but not necessarily all, or even most, of Wert.
Almost no additional
energy would be required
to
maintain
a constant forward speed of the center of
mass, if the decrements of the gravitational
potential
energy (as the center of mass decreases in height during
each stride) and the decrements of kinetic energy (as
the animal slows during each stride) could be stored
and/or used to reaccelerate and raise the center of mass
during another part of the stride.
There are two mechanisms
for alternately
storing
and recovering energy within each stride: 1) an exchange between gravitational
potential energy and kinetic energy, as occurs in a swinging pendulum; and 2)
an exchange between mechanical energy stored in muscle’s elastic elements and recovered as both kinetic and
gravitational
energy, as in a bouncing ball. Both of
these storage-recovery
mechanisms have been found to
be important for minimizing
the chemical energy input
required for Wexl in human locomotion: the pendulum
in the walk and elastic storage in the run (6-8, 13, 20).
We wanted to know if these mechanisms are utilized in
R243
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
CAVAGNA,
GIOVANNI
A., NORMAN
C. HEGLUND,
AND C.
RICHARD TAYLOR. Mechanical
work in terrestriaz locomotion:
two basic mechanisms
for minimizing
energy expenditure.
Am. J. Physiol. 233(5): R243-R261,
1977 or Am. J. Physiol.:
Regulatory
Integrative
Comp.
Physiol.
2(3): R243-R261,
1977. -The work done during each step to lift and to reaccelerate (in the forward
direction)
and center of mass has been
measured
during
locomotion
in bipeds (rhea and turkey),
quadrupeds
(dogs, stump-tailed
macaques,
and ram), and
hoppers (kangaroo
and springhare).
Walking,
in all animals
(as in man), involves an alternate
transfer
between gravitational-potential
energy and kinetic energy within each stride
(as takes place in a pendulum).
This transfer is greatest at
intermediate
walking
speeds and can account for up to 70% of
the total energy changes taking place within a stride, leaving
only 30% to be supplied by muscles. No kinetic-gravitational
energy transfer
takes place during
running,
hopping,
and
trotting,
but energy is conserved by another mechanism:
an
elastic “bounce”
of the body. Galloping
animals
utilize
a
combination
of these two energy-conserving
mechanisms.
During
running,
trotting,
hopping,
and galloping,
1) the
power per unit weight required to maintain
the forward speed
of the center of mass is almost the same in all the species
studied; 2) the power per unit weight required
to lift the
center of mass is almost independent
of speed; and 3) the sum
of these two powers is almost a linear function of speed.
R244
some regular
so, how.
MATERIALS
CAVAGNA,
way by all terrestrial
AND
vertebrates,
and if
METHODS
Animals
Procedures
The horizontal and vertical components of the resultant force applied by the animal to the ground were
measured by means of a force platform.
The lateral
movements
were disregarded.
The animals were not
restrained in any way when they ran across the force
platform. We called, coaxed, or chased them (more or
less vehemently)
to achieve a range of speeds. The
force platform was inserted with its surface at the same
level as the floor, about 30 m from the beginning of a
corridor. Thus the animals had plenty of space to reach
a constant speed over the platform, even during a fast
run. Since we wanted to study locomotion at a constant
average speed, we accepted only those trials where the
accelerations taking place during one or more complete
strides differed from the decelerations
by less than
25%. The force platform was 4 m long and 0.5 m wide
and this length allowed us to record the force exerted
by the feet against the ground over several strides.
AND
TAYLOR
The horizontal force and the vertical force minus the
body weight were integrated electronically to determine
the instantaneous
velocity of the center of mass of the
body to yield the instantaneous
kinetic energy. The
change in potential energy was calculated by integrating vertical velocity as a function
of time to yield
vertical displacement,
and multiplying
this by body
weight. The total mechanical energy as a function of
time was obtained by adding the instantaneous
kinetic
and potential energies. The positive external mechanical work was obtained by adding the increments
in
total mechanical
energy over an integral number of
strides.
Both the mechanical
details of our force platform
and the procedures involved in using force platforms to
measure external
mechanical
work have been thoroughly described in a recent article by Cavagna (3).
For the convenience of the reader, we have included a
list of the symbols which we use throughout
the paper
with their definitions in Table 1.
RESULTS
Walking
We found a remarkable
similarity
in the force and
velocity records from all of our experimental
animals
when they walked across our force plates (Fig. 1). In
fact it was often not possible to identify which record
came from which animal without knowing the absolute
values of the forces involved. A great deal of information (e.g., the patterns of thrust from individual
feet)
can be obtained from these records, but we have omitted
this detailed information
in this paper. Instead, we
have concentrated
on trying to understand
general
mechanisms which animals utilize to keep their center
of mass moving forward at a constant average speed.
Walking in all of our experimental
animals involved
an alternate exchange between the kinetic energy and
the gravitational
potential energy of the center of mass
within each stride. In this respect, walking is similar
to a swinging pendulum or an egg rolling end-over-end
(7). The mech anical energy which the muscles have to
provide to keep the center of mass moving forward is
decreased by the amount of this exchange which depends on three factors: I) the phase relationship
between the changes in kinetic energy and gravitational
potential energy within the stride; 2) the relative magnitude of the two; and 3) the degree of symmetry
between the two (i.e., how closely they approximate
mirror images of each other).
If the kinetic and gravitational
potential
energy
changes are 180” out of phase, if their magnitudes
are
equal, and if the changes are symmetrical,
then the
muscles will not need to provide any additional energy
to keep the center of mass moving, provided of course
friction and wind resistance are neglected. The amount
of muscular work required to keep the center of mass of
a walking
animal moving at a constant speed will
depend on the deviation of the measured kinetic and
potential energy changes from this optimal condition.
Phase relationship
between kinetic and gravitational
potential energy. The phase relationship
between the
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We tried to achieve as great a diversity of locomotory
modes as possible in our selection of animals. We chose
a large bird, the rhea (Rhea americana,
one animal
weighing 22.5 kg) and a smaller bird, the wild turkey
(Meleagris
gallopavo,
two animals each weighing
7
kg). Locomotion in these two bipeds could be compared
with that of man, who had already been carefully
studied with the same apparatus (9). Bipedal locomotion
in birds looks quite different
from that of humans,
since the ankle of the bird occupies a similar position
relative to the ground as the knee of humans. Thus the
leg of a bird appears to bend in the opposite direction
from a leg of a human.
For our quadrupedal
animals we selected one which
was a highly specialized and efficient runner and appeared to move gracefully and effortlessly at all speeds:
the dog (Canis familiaris,
a small terrier weighing 5
kg and a large mongrel weighing 17.5 kg). We compared
the dog with another quadruped
which was highly
specialized to use its legs for nonlocomotory
tasks, and
which appeared to move awkwardly
and inefficiently
at all speeds (i.e., with its legs flailing and swinging in
all directions): a monkey, the stump-tailed
macaque
(Macaca speciosa, two males, each weighing 3.6 kg).
We selected the ram (Ovis musimon,
two animals
weighing
60 and 85 kg) because it was the largest
quadruped
we could conveniently
find that was still
within the capacity of our force plates.
We selected, two hoppers which were very far apart
phylogenetically:
the kangaroo
( MegaZeia rufa, one
female weighing 20 kg and one male weighing 21 kg),
which is a marsupial,
and the springhare
(Pedetes
cafer, one animal weighing 2.5 kg), which is a rodent.
This suggests that their saltatory modes of locomotion
have evolved independently.
HEGLUND,
MECHANISMS
--iif
Ee
Ekf
Eku
EP
E tot
OF
Ff
F,
K’
L
Lc
L dc
Lc
p”
s*
t act
tc
tdec
47
Vf
W
K
l
ext
Wf
WV
+ Ekv
step or stride
frequency
horizontal
component
of the resultant
force exerted
by the
feet against
the ground
vertical
component
of the resultant
force exerted
by the feet
ainst t&e ground
acceleration
of gravity
slope of a linear
relationship,
starting
from
the origin,
calculated
for the function
-tif = f(Vf);
i.e., K = -aflQf
=
V-f
Knt
~ K?
’ rir,,
(t,-Jt,)
length
of the step or stride,
L = 6,-T
forward
displacement
of the center
of mass taking
place
during
each step or stride
while the body is in contact
with
the ground,
L, = vf* tc
forward
displacement
of the center
of mass taking
place at
each step of walking
when both feet contact
the ground,
L&
~ WV
=
I
vf
l
t&
forward
displacement
of the center
of mass taking
place at
each step of walking
when one foot only is in contact
with
the ground,
Lx = vf*t,
body mass
body weight
sum of the upward
displacements
of the center
of mass
taking
place during
a cycle of movement
T; S, = W,/P
fraction
of the period
T during
which
the forward
speed Vf
increases
fraction
of the period
T during
which
the feet contact
the
ground
fraction
of the period
7 during
which
the forward
speed Vf
decreases
fraction
of the period
T during
which
the body is off the
ground
rate of oxygen
consumption,
above the rest rate, which
we
changes in kinetic and gravitational
potential energy
can be easily seen from the records of instantaneous
kinetic and potential energy of the center of mass over
a series of strides in Fig. 2 (computed from the force
and velocity data presented in Fig. 1). The top line of
each experiment in Fig. 2 represents the kinetic energy
due to the velocity of the center of mass in the horizontal
direction (E&, the two middle lines (which almost
completely overlap) are the gravitational
potential energy (E,) and the sum of the gravitational
potential
energy plus the kinetic energy due to the velocity of
the center of mass in the vertical direction (E, + E,J.
The kinetic energy in the vertical direction (Ekv) is so
small compared to E, that two separate lines representing E, and E, + Ekv can only be distinguished
in a few
of the records (e.g., the rhea walking at 3.7 km* h-l).
The records of the turkey and the rhea clearly show
that at slow walking speeds E, is at its maximum
when E, is at its minimum,
i.e., they are almost
completely out of phase; as the animals walk faster, E,
and E, become more and more in phase with one
another (compare the experiments
where the turkey
walks at 3.4 km. h-l vs. 4.6 km h-l or where the rhea
l
Wf
R?lt
wmetab
WI
w
have converted
to metabolic
power input using the energetic
equivalent
of 4.8 Cal/ml O2
instantaneous
speed of forward
motion
of the center
of mass
average
speed of locomotion,
vf = L/T
positive
work done at each step to increase
the mechanical
energy
of the center
of mass; Wed is the sum of the increments of Etot during
7; this is called “external”
work
positive
work
done at each step to increase
the forward
speed of the center
of mass; Wf is the sum of the increments
of Ekf during
7
positive
work done at each step to lift the center
of mass; W,
is the sum of the increments
of E, during
7
the positive
work done during
each step to accelerate
parts
of the body relative
to the center
of mass. This
is called
“internal”
work
rate at which Ee is delivered
during
T; We = E,/T
= W,,JT:
this mechanical
power
is smaller,
about
half or
less, than
the average
power
developed
by the muscles
during
contraction;
in fact not only positive
work but also
negative
work is done during
7; in addition
7 may include
a
“flight”
period t,
= Wf/T; other indications
as for West
= W&;
other indications
as for West
= W&
rate
at which
the muscles
consume
chemical
energy
for
locomotion;
Wmetd = (Wmetd during
locomotion)
- (Wmtti
at
rest)
work done per unit distance
and per unit of body mass;
4
angle between
the vertical
and a line connecting
the hip
joint with the point of contact
on the ground
(Fig. 3)
period
of a repeating
change
in forward
velocity
and height
of the center
of mass; in walking,
running,
and trotting
7 is
the period
of the step; in galloping
7 is the period
of the
stride
overall
efficiency
of positive
work production,
i.e., the ratio
between
the total mechanical
power
output
(We,
+ W&
and the net energy
expenditure
per unit time (Wmet,);
the
latter
includes
the cost of negative
work
Y’
=
wextlrir,,,;
Y’ < Y
walks at 2.2 km* h-l vs. 5.1 km* h-l). The bottom line
in Fig. 2 (E,,,) is the instantaneous
sum of Ekf and (E,
+ E&. When the two curves are MO0 out of phase, the
oscillations of Etot are less than those of E, or (E, +
Ekv) individually
by the amount of the exchange between the two (see Fig. 2, man walking at 3.9 km h-l).
When the Ekf and the (E, -+ Ekv) curves are almost
completely in phase, little exchange is possible, and
the increments of Etot approach the sum of the increments of the two curves (see Fig. 2, ram walking at 4.6
km h-l). Other experiments in Fig. 2 show a range of
phase relationships between these two extremes.
Magnitude
of kinetic and gravitational
potential energy changes. In a simple pendulum
the change in
gravitational
potential
energy (AE,) will equal the
changes in kinetic energy in the forward (Al&J and
vertical directions (AE,J at each instant in time
l
l
Al3 P
=
hEkf
+
mkv
(1)
In the simplest inverted pendulum system which has
been used to characterize
walking - the stiff-legged
walk of Alexander ( l)-Ek, would be lost from the system
in each step when the front foot hits the ground (Fig.
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i
K
R245
LOCOMOTION
average
deceleration
forward
of the center
of mass taking
place at each step as a consequence
of the impact
with the
ground,
-if
= -bvf/tdec where
- AVf is a decrement
of Vf
which this decrement
takes place
and tee is the time during
elastic
energy
stored
during
the negative
work
phase
of
each step and delivered
during
the positive
work phase
kinetic
energy
of forward
motion
of the center
of mass
kinetic
energy
of vertical
motion
of the center
of mass
gravitational
potential
energy
of the center
of mass
total mechanical
energy
of the center
of mass, Etot = Ekf +
E,
f
TERRESTRIAL
R246
CAVAGNA,
HEGLUND,
AND
TAYLOR
WALK
0.5
VF(m/scc)
const.
u
-~
I
,’
-0.5 E
4.
0 -40 E
F,(h)
V, ( m/see
)
..'
0.5
const.
_’ : ‘.
-
04
1
-0.5 E
F,(kg)
i
/LA./-II,. . c.--E b',--j
.f
E *c?upt/-
cE
,-
-1 t
-0.4
0
20 '
40
0
2ooL
0
_
--__
a5 c. F
-0.5L
4
0 -.~/" ,, '.. ,." , ~,' (_,," ;, ".\ _
-4
E
E
FF(kd
v,(ll+cc)
Q8
const.
~I~_ "
..;' ,,._,
'.-,,,,...- .-
F,(kg)
_.'
0.5
(., - ~/- /' \ ;'. \_ I \
-0.5 E
-
__
c.-..
\,
- 0.5
-10
-------’
40
10
r
-
,-\ ,,/‘#,’
1 .” /? ~,.‘i . .._“-._ .._.._
-0.4
4
8
_- \_ 1_ ._I'
c. -
," /?
-8
i
-1
0.4
const.
.'",,, .A" /i+ ,,,.,I‘ ,/f/A
'G
J
1
i
-
4
)
0 ,,
-10
-4
V, ( m/set
0.2
c.'
F
-0.2'
8
,,/
o @,'.,..,,, '.," ,/x. r.",-,/ir
-
10
,
-0.8
F,(kg)
3.7 k-i?hr
8
_
E
E
E
km~~~
~-o.5
we-.---E
E
0E- .._I
:,.,<.//'
- 0E..._/* FJ"ih/"
;,,'-'r-'I,/,,'.(,_.0E- I (/iI,.",~",1'
-j
E
E
E
0 _---._
0- 0 ‘E
E
0.4
const. I
-0.4
VF(m/sec)
RHEA
4-L--'~-~~
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
100 c
-40
a8
”
1
(.. _
:-
\.’
%t"*
,,.
-
c.
i
/
-
-Q8 '
-1
'd2-d
,‘I
,'
20
40
x'
w
100
c
200 i
FIG. 1. Experimental
records
obtained
during
walking.
Each set
of records
indicates
from bottom
to top: vertical
force exerted
by the
feet against
the ground
(F,), oscillations
of vertical
component
of
velocity
of the center
of gravity
(V,.), forward
component
of force
exerted
on the ground
(F,) and oscillations
of forward
component
of
velocity
of the center
of gravrty
(V,). Positrve
values
of F, mdicate
a
backward
push of the foot against
the ground
during
this push the
1
2 set
forward
velocity
V, increases;
the opposite
occurs
when F, is negative. Velocity
tracings
begin and end when the animal
crossed
two
photocells
over the platform.
Meaning
of integration
constant
is
described
by Cavagna
(3). For turkey
and rhea,
records
are given
for different
speeds of walking
to show modifications
of the tracings
with
speed. Note similarity
of records
between
different
animals
and man.
3), and the transfer of kinetic energy into gravitational
potential energy could only take place between E,, and
E,. Alexander has concluded that most of the energy
used during this stiff-legged
walking would go into
replacing the AEk,. lost during each step. It would
therefore be advantageous
to keep AE,,, small. One
way to do this is to keep the amplitude
of the swing
small. For example, in Fig. 3 (top) AE,, = AE,,, when v,
is 45” and LU?,~ > AE,,. when C,C~45”. It can be seen
clearly in Fig. 2 that walking animals keep AEk,. small,
in fact it never exceeds 6% of AEkf.
Not only is AE,,. small during walking, but it appears
also to be converted directly into AE,, by the application
of a force normal to the direction of the velocity of the
center of mass as it falls forward. The extension of the
back foot which occurs during walking (9) may apply
0
MECHANISMS
OF
TERRESTRIAL
R247
LOCOMOTION
WALK
MONKEV
MAN
3.9 km/hr
RHEA
26 km/b
2.2 km/b
T
Q2cal
3.4 km/hr
RHEA
3.7 km/hr
TURKEV
4.6 km/M
RHEA
5.1 km/hr
RAM
TIME
FIG. 2. Experimental
records
of mechanical
energy
changes
of
center
of mass of body during
walking.
Curves
were calculated
from
records
of Fig. 1; they were drawn
directly
using the plotter
output
of a computer.
In each set of tracings,
the upper
curve
refers
to
kinetic
energy
of forward
motion
&
= 4 m Vt; the middle
curve
to the sum of gravitational
potential
energy
E, and kinetic
energy
just such a normal force (Fig. 3, bottom).
Also the
center of mass continues to fall afier the front foot hits
the ground during part of the period of double support.
If the two feet act to apply a resultant force normal to
the direction of the velocity of the center of mass during
this time, then there could also be a transfer ofEkv into
E,, Our records of the potential energy changes of the
center of mass before and during the period of double
contact show that E, decreases gradually as the center
of mass approaches its lowest point (as in Fig. 3,
bottom); this is consistent with a transfer ofEk, into Ekp
How do the relative magnitudes
of AE, and AZ&
compare during walking in bipedal birds and quadrupedal mammals?
Summing
the increments
of (E, +
Eke) over a stride gives the positive work done against
gravity (WV) and dividing by the stride period gives the
average power (I&,>. Summing
the increments of Ekf
gives the positive work done to accelerate the body
F--l
\
SCCF
of vertical
motion
Ekv = 4 m
E tot = Ekf + E, + Eke. Curves
line),
but
Etol curves,
average
pendulum.
4.6 km/hr
V,Z; the bottom curve to total energy
E, and (E, + Ekf) are also given (thin
be distinguished
from
(E, + Ekv) and
often they cannot
since Ekv is very small in walking.
Note that at low and
speeds E, and Ekf change
in opposition
of phase
as in a
forward (W,> and dividing by the stride period gives
the average forward power (W&. Mass specific power,
l8$,lrn and Wflm, is plotted as a function of walking
speed in Fig. 4. Using these graphs it is easy to compare
the relative magnitudes
of the positive energy changes
in E, and E,,-, and to see how they vary as a function of
walking speed. In both birds (rhea and turkey) W, =
Wf at the slowest walking speeds, and Wf > W, at the
highest walking speeds as has been found in man (9).
There is a similar trend in the ram, but at its highest
walking speeds W, = WV. In the dogs Wf = W, over the
entire range of walking speeds and in the monkey W,
> Wf over the entire range of speeds. The point which
emerges from the measurements
of W, and W, is that
their relative
magnitudes
are often similar during
walking: Therefore, from the viewpoint of relative magnitude alone, it is possible to have a significant transfer
of energy between kinetic and gravitational
potential
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
TURKEV
CAVAGNA,
HEGLUND,
AND
TAYLOR
the amount of power actually expended (W&.
W,,,
can be calculated from EtOl (Fig. 2) by summing all the
increments in + over a stride and dividing by stride
period, just as W, and W, were calculated from E,, and
(E, + Ekv). This was done and W,,, is plotted as a
function of walking speed in the second (from the top)
set of graphs in Fig. 4. The magnitude
of the exchange
between gravitational
potential energy and kinetic energy is then equal to the total power, which would be
required
if there were no exchange (1Wfi + 1WUI) minus
.
Wext. This difference can be expressed as a percentage
of the total power required with no exchange
P&l + I%( - Kxt x 1oo
(2)
I %I + I Kq I
If the exchange were complete, then W,,, would be
zero, and the recovery would be 100%.
Percentage recovery is plotted as a function of walking speed in the third (from the top) set of graphs in
Fig. 4. The figure shows clearly that there was a large
exchange between gravitational
potential
energy and
kinetic energy during walking in all of our animals.
This exchange reached a maximum
at intermediate
walking
speeds in bipeds: 70% in the rhea at 3-4
km* h-l, and 70% in turkeys at 3-4 km. h-l. The exchange declined at slower and faster walking speeds.
This is identical to the situation which Cavagna et al.
(9) found in man, and their curve has been plotted as a
dotted line in the graph of percent recovery for the
turkey for comparison. Percent recovery in the quadrupedal ram also reached a maximum
(35%) at intermediate walking
speeds (3-4 km h-l) and decreased at
slower and faster speeds. In dogs and monkeys, percent
recovery reached values of 50%, but it did not change
in a regular way with speed.
The external work required to move one kilometer
has been calculated by dividing We,, by speed and is
plotted in the bottom set of graphs in Fig. 4. It appears
that there is an optimal walking speed for both birds
and the ram, where West to move a given distance is
minimal and the exchange between gravitational
potential energy and kinetic energy is maximal. This is also
the case for man (9) as can be seen from the dotted line
in the turkey graph.
% recovery
energy within the stride in all of the animals.
How important
is the exchange between kinetic and
gravitational
potential
energy within each stride? As
mentioned
at the beginning
of this discussion, phase
angle and the relative amplitude
of kinetic and potential energy changes are only two of the factors which
determine
the completeness of the exchange between
kinetic and gravitational
potential energy; symmetry
is also important.
The two changes plotted as a’function
of time must be mirror images of one another if the
exchange is to be complete. If there is an increase in
potential
energy without
a simultaneous
and equal
decrease in kinetic energy, then contracting
muscles
(or stored elastic energy) must provide the missing
energy. Likewise,
if there is a decrease in potential
energy without
a corresponding
increase in kinetic
energy, the energy must ether
be lost as heat, or
stored in elastic elements.
Perhaps the simplest way to quantify the net effect
of all three parameters is to compare the magnitude
of
the power output required to maintain a constant walking steed if there were no exchange (I Wtl + I W,,l) with
l
Running,
Trotting,
and Hopping
Just as we had found a remarkable
similarity
in the
force and velocity records obtained during walking (Fig.
I), we also found a striking similarity in the force and
velocity records of humans, birds, dogs, monkeys, rams,
kangaroos,
and springhares
as they ran, trotted,
or
hopped across the force plates (Fig. 5). Running,
trotting, and hopping animals all used their muscles to
push upward and forward simultaneously
during one
part of the stride and to break their fall and slow their
forward speed simultaneously
in another part of the
stride. There was usually, but not necessarily, an aerial
phase. The relative proportion of the time spent in the
air was largest in the hoppers (up to 75% of the step
period), intermediate
in runners (up to 30% of the step
period), and smallest in the trotters (O-15% of the step
period). It is easy to identify this aerial phase on the
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
FIG. 3. Schematic
representation
of inverted
pendulum
system
in walking.
Top diagram
represents
stiff-legged
walk
of Alexander
(1) where
only one foot makes
contact
with
the ground
at a time.
Step length,
therefore,
equals
forward
displacement
of the center
of
mass while
one foot is in contact
with the ground
(L,).
EkF is lost
during
each
stride
when
the foot first
makes
contact
with
the
ground,
while Ekf is transferred
into E,. Ekv can be kept small by
keeping
small
the angle cp between
a line connecting
the hip joint
with the ground
and a line drawn
perpendicular
to the ground.
A
closer approximation
to pendulum
system
used by walking
animals
is given
in bottom
diagram.
The rear foot pushes
upward
before
it
leaves
the ground’,
exerting
a force normal
to the velocity
of the
center
of mass. This results
in a conversion
of Ekv directly
into EkP
This normal
force continues
during
the period
when both feet make
contact
with the ground
as a resultant
of the forces exerted
by both
feet. The center
of mass moves forward
during
the period
of double
contact
by an amount
Ldc, and the step length L equals Ldc + L,.
=
MECHANISMS
OF
TERRESTRIAL
Fu49
LOCOMOTION
QUADRUPEDS
BIPEDS
--RAMS
--MONKEYS
25
15
10
5 I
I
I
I
I
I
I
i
I
I
0
50
8
0
l
0
i
I
I
1
1
f
1
t
WALR*QALL()p
e--
---
/
t
1
wm
l
-t
25
I
0
I
I
2
3
4
51
I
1
I
I
234561
I
I
I
1
I
I
I
I
--
+
l
0 00
1
I
l
N /.... A’
,,
-....
w -‘-...*
,_.....
I
I
I
I
l
H
0
I
I
I
2
3
4
1
I
1
I
1
2
3
4
5
1
1
I
I
I
2
3
4
5
6
AVERAGE SPEED OF WALKING (km/hr)
4. Top set of graphs
gives
power
required
to lift center
of
mass W, and to reaccelerate
center
of mass in the forward
direction
W1 during
each step of walking
at different
speeds.
Second
(from the
top) set of graphs
gives
total external
power
Wert as a function
of
walking
speed. Third
set of graphs
gives percent
recovery
of mechanical energy
(Eq. 2) resulting
from an exchange
between
gravitational
potential
energy
E, and kinetic
energy
of forward
motion EkP
Bottom
set of graphs
gives external
work
per unit distance
(W,,,/
km), which
reaches
a minimum
when recovery
of mechanical
energy
is maximum;
the same
is true
for man
(dotted
Lines).
Vertically
crossed
symbols
refer
to a walk-run
transition,
obliquely
crossed
symbols
to an unidentified
gait. The two turkeys
performed
different
W, but the same W,; this is true
also for the two monkeys;
the
monkey
doing more work against
gravity
in walking
always
shiRed
to a trot, the other to a gallop.
force and velocity records of Fig. 5, since both F, and F,
are zero when the animal is not touching the plate.
Also V, remains constant and V, changes at a rate of
9.8 me sS2 due to the acceleration of gravity.
Phase relationship
and exchange between kinetic and
gravitational
potential energy within each stride. Energy changes due to the motion in the vertical (E, +
Eke) and forward (E,,) directions were always almost
completely in phase during running, trotting,
and hopping (Fig. 6); therefore there was little possibility for
an exchange between the two. Wett = 1Wfi + 1IV,1 and
percent recovery calculated
using Eq. 2 was nearly
zero (Fig. 7).
E,, becomes quite large in parts of the stride during
running (much greater than during walking). It reaches
maximum values twice: at the instant the animal leaves
the ground (easily seen in Fig. 6 as the point where (E,
+ Ekv) becomes constant) and at the instant the animal
lands (seen in Fig. 6 as the point where (E, + Ek,,)
starts to decrease again). After the animal leaves the
ground, Ekv decreases as the center of mass rises, until
it is completely converted into E, when the center of
mass reaches its maximum
height. At this instant the
vertical velocity is zero and the E, and (E, + Ek,,)
curves in Fig. 6 coincide. Then the animal begins to
fall as E, is converted into E,,; this then decreases as
FIG.
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
20
R250
CAVAGNA,
HEGLUND,
AND
TAYLOR
RUN = HOP = TROT
VF(m/sec)
Q5r
const. t
0.4r'
-
const.
- 0.5L
FF(kg)
\
E
-40
2
'
E
-20
2,
I,
-
const. -2 E
-2 E
0.8
\
\
\
const.
-a*
E
- _ -
-
-‘__
-
t
FIG. 5. Experimental
force and velocity records
obtained
during
running
(man,
turkey,
and
rhea),
hopping
(kangaroo
and spring
hare),
and trotting (monkey,
dog, and rami.
Two different
speeds are given for the kangaroo to show increase
of force,
particularly
of the forward
component,
with
speed. Right-hand
ordinates
give vertical acceleration
of center
of mass as
a multiple
of gravity;
note that this is
greatest
in hopping
(particularly
in
springhare,
2.5 kg body wt), intermediate in running,
and minimal
in trotting. Other
indications
as in Fig. 1.
P lb
Ii
0E."L,?.
,i,.
i- 0E---7.---L
:‘, “r
.‘ -()'
E
*
*
o-^ L..__."_(
1
20
4
8r
a
(_
-20
---i
-8 -1
0.5
const.
* :
.‘?,
\,
:
'
.I(
*’ / -
i
-
p '
L
I
20
40.
the animal breaks its fall until it reaches zero when
the center of mass is at its lowest point. Since E,,. is
zero when (E, + E,,.) is both at its maximum and at its
minimum values, the increments in (E, + IX,,,) per step
equal the increments in E,.
Magnitude
of kinetic and gravitational
potential energy changes. Perhaps the most unexpected and intriguing finding of this study was that the mass specific
power output for reaccelerating
the center of mass (in
the forward direction) was nearly the same for bipedal
running, quadrupedal trotting, and hopping when plot-
*
"
.:g
3
$2
ted as a function of speed ( WJrn in Fig. 8). In all of the
animals wf/rn seemed to approach zero at zero running
speed and increased more and more steeply with increasing speed. The functions relating wflrn and speed
for birds, springhares, kangaroos, dogs, monkeys, and
rams are almost completely superimposable.
Even the
data which Cavagna et al. (9) obtained from humans in
running (dotted line in kangaroo graph in Fig. 8) are
hardly distinguishable
from the data from our diverse
assortment of animals.
The mass specific power output for raising the center
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
”
- 0.2
-4
r
“,’
const.-
F&kg)
, x -%,A I:
or
,
~_
const.
-
,
-
- 0.4 t
20
40
1
const.
-0.4L
0
Vv(m/sec
t
-
MECHANISMS
OF
TERRESTRIAL
R251
LOCOMOTION
RUN = HOP = TROT
MAN
14.3
km/hr
KANGAROO
10.2
km/hr
MONKEY
5.7
km/hr
rnEKF
EP
T
2Scal
ti
r
25cal
E TOT
’
g-r\
EP*
13.5
km/hr
KANGAROO
24.4
DOG
km/hr
10.4
km/hr
l/mAn
T
2cal
1
RAM
SP.
RHEA
16.9
HARE
15.5
6.0
km/hr
FIG. 6. Experimental
records
of mechanical
energy
changes
of center
of
mass of the body while
running,
hopping, and trotting.
Curves
were calculated
from
tracings
of Fig.
5. Their
meaning
is described
in Fig. 2. Note
that gravitational
potential
energy
E,
and kinetic
energy
of forward
motion
Ekf change in phase in running
as in
hopping
and trotting.
Other
explanations are given in text.
km/hr
km/hr
T
Q-7
5cal
Y
‘4’
$7-j7j
TIME
1
-1
set----+
of mass within each stride (l&/m
in the top’ set of
graphs in Fig. 8) was almost independent
of running
speed in all of the animals. Since the vertical displacement of the center of mass S, generally decreased with
increasing speed (Fig. 9) and
w Jm = S,*g
l
f = const
(3)
(where g is the acceleration of gravity) an increase in
stride frequency f must compensate for the decrease of
S,. l&,/m was greater in hopping than in running or
trotting as a result of the greater vertical displacement
of their center of mass. S, does not seem to depend on
body mass alone, nor on the overall dimensions of the
(2.5 kg), S, is greater
body; e.g., in the springhare
than in the ram (85 kg). It seems more likely that S,
depends on the peculiar characteristics
of the elastic
system on which the body bounces at each step. For
example when the kangaroo decelerates downward and
forward, its feet are flexed so that the Achilles tendons
are stretched storing elastic energy. In order to release
this elastic energy an extension of the feet is necessary,
but this implies an appreciable upward displacement
of the body of the order of magnitude
of the dimensions
of the foot of the kangaroo (Fig. 9). In other words, to
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
TURKEY
EKF
Ft252
CAVAGNA,
c
60
I -
I
I
I
I
I
I
I
I
RAMS
HEGLUND,
I
AND
I
TAYLOR
I
SPRING HARE
lo0
40
RHEA
i
0
z
g
40
0
20
09
0
0
t
60
f-1
0
TURKEYS
0
i
DOGS
15
20
AVERAGE
25
30
SPEED
0
5
10
OF LOCOMOTION
(km/hr)
20
15
25
of mechanical
energy
(Eq. 2) in walking
(open
symbols),
running
(closed
symbols)
and galloping
(half-open
symbols) as a function
of speed. “Recovery”
indicates
extent
of mechanical energy
reutilization
through
the shift
between
potential
and
kinetic
account).
gallop,
utilize the elastic energy stored by his elastic system,
the kangaroo must ‘(jump over his feet.”
In running,
trotting,
and hopping the mass specific
power for external work (We,Jm) is
pare the mechanical power output of the animal with
its chemical power input and define the efficiency (y)
with which an animal converts stored chemical energy
into positive mechanical work as
FIG.
7. Recovery
Westlm
= IWjmI+ (WJmI
energy
as in a pendulum
(elastic
energy
is not taken
This
recovery
is maximal
in walking,
intermediate
and minimal
in running
(run = hop = trot).
30
(4)
since there is no exchange between E, and Ekf. l&Jrn
increased linearly in all the animals with increasing
speed as in man (Fig. 8). The equations for these lines,
calculated using the method of least squares, are given
in Table 2. The high correlation coefficients for a linear
relationship
indicate that the curvilinear
relationship
for l&/m must be compensated
for by a curvilinear
relationship
for l&,/m as has been described for man
(9). Thus WJ m is probably not independent
of speed
but our data are too scattered to define this relationship
precisely.
Is there storage and recovery of mechanical energy
in elastic elements? Measuring
the magnitude
of the
exchange between kinetic energy and gravitational
potential energy is easily done and its importance can
be quantified
as percent recovery using Eq. 2. Determining the magnitude
of an exchange between kinetic
and/or gravitational
potential energy and elastic potential energy (Ee) is much more difficult. There is a way,
however, to approach the question of elastic storage
using our force plate measurements
(8). We can com-
Y
=
wett+
wint
into
in
(5)
W* metab
where IVest is external power output, PVint is internal
power output, and VVmetabis the chemical power input
into the muscles for locomotion ( VSImetab
during locomotion, -YSlmetabat rest).
The maximum
efficiency with which muscles can
convert chemical energy into positive mechanical work
is about 0.25. If the efficiency of locomotion at steady
state exceeds 0.25, then the explanation
has to be
storage and recovery of energy using muscle’s elastic
elements. Neglecting
the cost of negative work, the
efficiency with which the contractile machinery of muscle transforms chemical energy into positive work becomes
(wead
Ym
=
+
.
~intl
-
we
W metab
where VVe is the power output delivered free of cost
during the recoil of stretched elastic elements (i.e., the
energy stored during the deceleration,
when the muscle
performs negative work, and released during the sub-
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
c)
g
-I
a
m
WV
I
1
0
WEXT
ix
WV
I
1
1 m
0
0
00
50
50 -
_
l
.
I
I
10
1
I
1
RHEA
f
t
0
RHEA 1
I
BIPEDS
I
I
I
I
I
I
I
0
1
I
I
I
t
TURKEYS--
I
10
TURKEYS 1
I
I
I
I
I
I
I
I
1
I
I
I
1
1
I
I
I
I
I
1
wud
I
20
AVERAGE
I
1
I
4-. smw
l
I
10
0
.I’
.
10
I
1
10
I
.
I
I
I
I
I
I
20
I
I
I
20
I
I
I
0
0
KANGAROOS /
I
I
I
I
I
10
I
1
I
10
SPEED OF LOCOMOTIQN
I
I
I
I
HOPPERS
SPRING HARE 1
1
I
1
20
I
20
(km/hr)
I
,
I
---I
1
1
I
DOGS
-
I
1
I
30
1
0
I
I
I
QUADRUPEDS
I I
DOGS I
10
I
I
10
1
I
I
I
I
20
I
MONKEYsl
I
20
MONKFYS
I
0
0
1
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
I
I
I
I
I
I
10
I
I
RAMS
1
10
I
RAMS
i
(middle
set
of
graphs);
and mass specific power for total
external work, w&m, are plotted as a function of running
speed. Curves through
I&/m
data were calculated using Eq.
12. Straight lines through w&m
data were calculated using leastsquares method (equations are
given in Table 2). Dotted lines
are data from man (9). Some
values of I@“/ti for rhea and
springhare
(open
circles)
have
been calculated from the equa. tion W+2 = Avf*vf*f, using
data such as those in Fig. 5. All
other values (closed
and
half
closed symbols)
were calculated
as described in (3). Below: mass
specific power data, given above,
were divided by the average
speed v’ to obtain work done per
unit distance, Wl( m l L).
wf/rn
FIG. 8. Above:
mass specific
power for raising center of mass
against gravity within each stride
WV/m
(top set of graphs);
mass
specific power for reaccelerating
center of mass in the forward
direction after the deceleration
due to impact with the ground
CAVAGNA,
HEGLUND,
I
KANGAROOS
I
AND
TAYLOR
I
1
15
20
SPRING HARE
l
l
25
l
l
l
l
l
l
0
l
l
l
0
l
0
l
0
l
l
m
0
#
0
l
l
.
.
0
.
l
l
0
0
l
0
l
10
15
20
AVERAGE
FIG.
during
bols),
9. Vertical
each step
and during
displacement
of center
of walking
(open symbols),
each stride
of galloping
of
mass,
running
(half-open
25
0
.5
10
25
SPEED OF LOCOMOTION (km/hr)
taking
place
(closed
symsymbols)
is
sequent acceleration,
when the muscle performs positive work).
A minimum
value for We can then be calculated
using the maximal value for ym of 0.25
.
W 4 = ( wezt + wint> - 0.25 wmetab
(7)
(W, is defined
30
only when We > 0)
It must be emphasized that this value for W; is only a
minimum
value since 1) muscles usually operate at
efficiencies below 0.25 (e.g., isometric contractions have
an efficiency of zero); and 2) even in the optimal
conditions of contraction,
Ym in Eq. 6 cannot be 0.25
because Wmetab includes some chemical energy which
is used by active muscles when they are stretched by
external forces and perform negative work. Therefore,
the approach of using y to study elastic storage can
tell us that elastic storage may exist if y is greater
than 0.25. It can also give us a minimum
value for
this storage, but it cannot tell us that elastic storage
does not exist when y is less than 0.25.
plotted
as
interpretation
a function
of this
of the
figure.
speed
of
locomotion.
See
text
for
We have measurements
of Werl and Wmelab for our
animals, but not of Wint. However, if we calculate the
efficiency of doing external work (7’) from our data
.
W
(8)
y'=i@=
metab
and if y’ > 0.25 this clearly indicates a large amount
of storage of energy in elastic elements, since W, must
then account for both (We, - 0.25 Wmet&) and all of
the Wint according to E+ 7.
y’ has been calculated as a function of speed for all
of the animals and is plotted in Fig. 10. It increases
from about 0.10 to 0.15 as turkeys increase running
speed from 5 to 20 km. h-l; from 0.25 to 0.27 as dogs
increase trotting speed from 5 to 15 km h-l; from about
zero to 0.23-0.25 as monkeys increase trotting or galloping speed from 5 to 25 km h-l; from 0.16 to 0.25 as
springhares increase their hopping speed from 10 to 25
km= h-l; and from 0.24 to 0.76 as kangaroos increase
their hopping speed from 10 to 30 km h-l.
l
l
l
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
‘5
0
MECHANISMS
TABLE
( W&m>
OF
TERRESTRIAL
R255
LOCOMOTION
2. External power per unit of body mass
relative to average forward speed (Yr>
Kvtlm
%sm
@L/m
=
=
=
6.05
-6.45
7.84
+ 4.00 Tf
+ 5.71 Yr
+ 4.88 v,
r = 0.903
r = 0.952
r = 0.980
Hoppers
Springhare
Kangaroo
K?rrlm
Kttlm
=
=
4.04
11.53
+ 5.63 yf
+ 6.29 Vf
r = 0.924
r = 0.973
Ksrlm
l&,,/m
=
=
Kwlm
%stlm
KTtlm
%tlm
=
=
=
=
9.84
12.30
-23.03
-18.65
-39.57
15.96
+
+
+
+
+
+
1.98
2.65
4.24
6.68
8.05
1.95
r =
r=
r =
r =
r =
r =
and
Q,
is
Quadrupeds
Dog, 5 kg, trot
Dog, 17.5 kg, trot
Dog, gallop
Monkey,
trot
Monkey,
gallop
Rams,
trot
l&,/m
h.
is measured
* From Cavagna
in
Cal/kg- min
et al. (9).
I$
t_‘,
Yr
Yr
y/
Vf
measured
in
0.945
0.988
0.966
0.987
0.936
0.622
km/
7 =
t,
+
t,!
where 7 is the period of a repeating change in forward
velocity and height of the center of mass. In both
running and trotting there was a symmetrical
change
in forward velocity and height at each step: in this
case 7 is the period of the step. When the animal
changed from a trot to a gallop there were no longer
two symmetrical steps in each stride and 7 became the
period of a stride. Values of t, and t, were measured
from the force records and are plotted in Fig. 11. The
value of t, decreased markedly with increasing speed
in all of our animals while t, either increased or remained constant. The increase in t, in Fig. 11 when
animals changed from a trot to a gallop is the result
of 7 changing from the period of a step to the period of
a stride.
The distance an animal moves forward while it is in
contact with the ground can be measured fairly accurately from t, and &
L c = t;P,
AVERAGE
SPEED
OF LOCOMOTION
(km/Iv)
External
power
output,
west, total energy
expenditure
calculated
from the oxygen
consumption,
wrnptab (data for birds from
Fedak
et al. (14), data for kangaroo
from Dawson
and Taylor
(ll),
and data for springhares
and dogs from unpublished
measurements
by Taylor)
and efficiency
of doing external
work
(7’ = werJwmelab)
are given
as a function
of speed.
Dotted
horizontal
line indicates
upper
limit
of 0.25 for efficiency
of transformation
of chemical
energy
into mechanical
work by the contractile
machinery
of muscle.
High
y’ values
attained
by some animals
suggest
a substantial
recovery
of mechanical
energy
through
muscle
elasticity.
FIG.
10.
We can reasonably conclude from the values of y’
that dogs, monkeys, kangaroos, and springhares
rely
on power recovered from elastic elements. This can be
very large; for example, when a kangaroo hops at 30
km- h-l, We amounts to a minimum
of two-thirds
of
the measured W,,, plus all the Wint.
It is interesting
to note that y’ often increases with
increasing speed, suggesting that elastic storage and
recovery become relatively
more important
at higher
speeds. This is in agreement with the findings on man
(5, 6) and on isolated muscle (4).
How does the external power absorbed and delivered
when the animal is in contact with the ground change
with speed? The muscles and tendons can absorb and
(9)
(10)
because the oscillations
in the speed around P, are
small. Two curves are plotted in Fig. 11 using Eq. 10
for two values of L, which just bracket the observed
values for t,. These two values for L, approximate
minimum
and maximum
values and show how L,
changes with speed.
As speed increases, either t, must decrease and/or
L, must increase. For a given W&step,
the power
must increase as t, decreases. However,
the power
could be kept constant with increasing
speed, if L,
increases proportionately
to P, Some animals increase
L, appreciably as they move faster; however t, decreases
in all animals, approaching a minimum
value of about
0.1
s (also observed in man by Cavagna et al. (9)).
The animal decelerates and falls (i.e., performs negative work) during the first part of the time of contact
( tdec), and it accelerates and lifts its center of mass
during the second part (t,,,)
tc =
tdec
+
t,cc
(11)
Since t, decreases and W,,, increases with increasing
speed, the external power which must be absorbed and
delivered by the muscles and tendons must increase.
Both the external power which has to be dissipated
and/or stored while the animal decelerates (W&/t&c)
and the external power which has to be recovered and/
or supplied by the muscles while it reaccelerates ( W,,,I
t,,,) have been calculated
(Table 3). In the turkeys
is
more
than
60%
greater
than Wezt/tacc. This
Wextlt
dec
would be expected from the force-velocity relationship
of muscle: larger forces are developed when the muscles
are stretched
(during deceleration)
than when they
shorten (during acceleration).
Thus muscles will require less time to decelerate the body than to reacceler-
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
Bipeds
Rhea
Turkeys
Humans*
deliver external power (thanks to external forces) only
when an animal is in contact with the ground. During
each step (or stride) there is a time when the animal
is in contact with the ground (t,) and a time when it
is in the air ( tv)
Fu56
CAVAGNA,
04.
I
’ RiEA
I
)
’
’
TbRKiYS
L, = Q5m
\\ ‘\,0.7m
\\ \
\\ 0‘1,
\
‘y* ‘\‘\
\
\
\
\
i __KPINGA~~~SI
;t
&*-...
c
0
-0
0 0
--\
ooo*-.
9
00
t
OV
0
10
20
0
10
17.5kg-DOG
m
\
\
\
\\ 2Sm
\\
\\
\’ ’
\\
\\
\
tc
\A
A
+4
m
:,; :0.6m
0 t\ .. ::
\.\ \\
\ \
‘,
0
l
\
\
\
8
\
\
\
L,
\\ GALLOP
‘\\ .
‘. --.
\
a’.
\‘k,T
\\.Ii&T
..\
tV
A
A
’
A
A
\,
A
A
30
AVERAGE
l
LC
=0.3 m
‘\, 0.4 m
\ 0 ‘\
? 2,
\t,
l
‘.
\
‘8,\ TROT
A
A
20
A\
A
A
A
A
AA
Ooo
Oo
O-LOO
0
’
L OI 1 1 I
10
MONKEY TROT
A’,
CC
\
0
\
A
‘a
\
\
A’
“\
\
5kg- DOG
Lc=0.6m
\
\\1.1m
\
‘\1,4
‘\,,
\\
a “\ k
\
=0,2m ’ \ \\ ’
Y’\
y io.3; \\\
\\0 ‘*
‘=e
\\
\\\ GALLOP
:, 3
\
: &
‘\\,&&,
%1ds
~CQ”
’1*AAA
Iahi
I I
A
A
0
“-iO
A
20
0
‘k\
\
\\
0
0
0
00
k
\
\
l -. -.
I
I
all others
ho
10
0
I
t,=O 1
0
10
SPEED OF LOCOMOTION (km/hr)
FIG. 11. Time
in which
the feet contact
the ground
(t,, closed
symbols’)
and in which
the body is off the ground
(t,, open symbols)
during
each step of running
(circZes and squares)
and each stride
of
galloping
(triangles)
are given
as a function
of speed.
Interrupted
lines were constructed
assuming
that forward
the body is in contact
with the ground
(L, =
speed and equal to the indicated
value.
ate it. However, if the energy were absorbed in and
delivered by passive springs (e.g. tendons) then about
the same amount of time would be required for deceleration and acceleration,
so that W,,lt,,
= Wertltacc.
This is what we found in the kangaroos (Table 3).
It is therefore reasonable to conclude that the contractile
component
of muscle is more important
in
delivering positive work in the turkey than it is in the
kangaroo and that passive elastic elements play a more
important
role in the kangaroo than in the rhea. This
is consistent with the larger efficiency values (y’) which
we found for the kangaroo compared with the other
animals (Fig. lo), indicating
independently
a larger
storage and recovery of elastic energy per unit time,
We. The difference between Wdec and IV,,, shows that
another hopper, the springhare,
does not use passive
elastic elements to an unusual extent; this is in agreement with the lower efficiency values y’ that we found
for this animal (Fig. 10).
Why do animals use nearly the same mass specific
power to reaccelerate in the forward
direction
at a
given speed? Cavagna et al. (9) derived an equation
for the mass specific power output due to the forward
speed changes in man
.
vt
(12)
w/ fm=K’
tv
l+-Tf
L
where K’ = K
and K is the slope of a linear
relationship
found between the average deceleration
forward of the center of mass during each step (A/>
and the average speed of locomotion (VJ.
l
tdec/tc,
t,
l
displacement
when
vf) is independent
of
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
0
\
t
0
’
l
20
0
’
l .
Off
000
0.4
I-~ARE1
smk
Lc= ‘0.5m
\\
\\
‘\\\0.85 m
\
\
8
0
LCF1;4
’
TAYLOR
\
l
Y,
LlJ
r
’
AND
lt
.‘00 .
\fl
0
1
HEGLUND,
MECHANISMS
OF
TERRESTRIAL
R257
LOCOMOTION
3. Mass specific external power stored andlor
dissipated after animal lands while it decelerates
(We&&
and mass specific external power recovered
and/or supplied by muscles while animal reaccelerates
before taking off (W,,,lt,,)
TABLE
Galloping
Animal and
Rody Mass
Bipeds
Turkey,
7 kg
Rhea, 22.5 kg
20.5 kg
Springhare,
2.5 kg
12.5
17.5
245.4
410.4
10
15
19
144.0
221.4
286.2
10
409.2
925.2
,
149.4
249.6
64%
114.0
175.2
226.8
26%
379.2
8%
20
28
1443.6
857.4
1338.0
10
15
20
352.2
659.4
1110.0
288.0
539.4
908.4
22%
5
10
13
58.8
101.4
131.4
46.2
79.8
103.2
27%
4
9
39.7
32.1
57.6
24%
71.4
Like the two other modes of locomotion, galloping
also involved a similar mechanism
in all animals,
although this might not be obvious when one first
looks at the complex force and velocity tracings of
galloping animals (Fig. 13). The force tracings are
particularly
difficult to interpret because of the “ringing’ of the force plate. The artifacts due to these
vibrations were excluded in our computations of energy
changes (3).
TABLE 4. Constants for equation
I&lm and vf (Eq. 12 in text)
relating
QU&drUpdS
Dog, 17.5 kg
Dqt, 5 kg
Bipeds
Turkeys
m-l) =
0.656- 0.043v’ (m -s-l)
0.378 2 0.043
0.442 2 0.030
n = 25
0.469 2 0.030
n = 78
0.240
0.133 f 0.033
n = 24
0.119 2 0.035
0.251
0.116 + 0.031
0.481 + 0.022
0.539
0.474 + 0.087*
0.656
0.576 2 0.084
0.329
n = 24
To see whether Eq. 12 applies generally to running,
trotting,
and hopping, we calculated values for the
constant K’ and t,/L, for all our animals.
K’ was calculated as follows: -& (measured from
the force and velocity records) was plotted as a function
of &, and a line for the best fit of a linear relationship
between -& and Vf (with its origin at zero) was calculated using the method of least squares. The slope of
this line (K) was multiplied
by an average value for
tdec/tc (taken directly from the force records and included
in Table 4) to give K’.
In all the animals except the kangaroo, tJL, was
independent
of speed, and we were able to calculate
average values for this constant (Table 4). In the
kangaroo tJL, increased linearly with Vf, and we used
the equation for this linear relationship
4J
-(sLc
0.871
(13)
instead of a constant.
The relationships
between I&/m and Vf, calculated
using Eq. 12, are plotted as solid lines in the second
from the top set of graphs on Fig. 8. The calculated
lines are in good agreement
with the experimental
data, in spite of the many simplifications
which we
made when calculating the constants.
Both K and t,lL, vary a great deal from animal to
animal (Table 4), but they change in such a way that
the relationship
between I&/m and vf remains nearly
the same in all our animals. Animals which have the
largest K (like the hoppers) also have the largest t,/L,;
thus the effect of a large K on increasing WJrn is
nearly exactly compensated for by a larger t,/L,. This
Rhea
0.544
Human
0.536
Hoppers
Kangaroos
1.120
n=
1.458
Springhare
n = 25
n = 78
46
n=
0.450 * 0.040
n=60
Quadrupeds
Monkey, trot
1.150
n=60
0.427 + 0.054
0.491
0.162 + 0.036
0.262
0.054 + 0.038
n = 45
0.039 2 0.039
n = 14
0.034 2 0.018
n = 35
0.001~ 0.004
n = 18
n=8
Dog, 5 kg, trot
0.587
Dog, 17.5 kg, trot
0.479
n=8
0.447 2 0.027
n = 45
0.441 + 0.022
0.211
n = 14
0.331
Dog, gallop
0.425 + 0.045
0.141
n=35
0.442
Ram, trot
46
0.508 + 0.024
0.225
n = 18
tdcclte and tJLc values are means + SD.
* In kangaroos t,/L, increases
as shown in Eq. 13. 1 table and in Fig. 12 an average value was used.
with
06.
n
va
E
.
05.
04
.
03
CA
x
l
U
v
02
l
< 01
CI
0
>
l
0
05.
1
15
K(sec-1)'
FIG. 12. Animals
that suffer
a large forward
deceleration
when
they land at a given
running
speed (large
K on abscissa)
increase
the time they spend in air (large
t,lL, on ordinate)
to minimize
V&/
m (Eq. 12). See text.
0,
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
Hoppers
Kangaroo,
suggests that the animals which suffer a big forward
deceleration when they land at a given speed (i.e.,
have a large K) increase the time which they spend in
the air in order to minimize &/m (Fig. 12). However,
they have to pay a price for their large t,lL, in a
greater WV/m.
R258
CAVAGNA,HEGLUND,
AND
TAYLOR
GALLOP
VF(m/sec)
. ,.
0.2
,_-
II-
17.1
kmlhr
const.
-0.2
2
l=,(b)
0
-2
a
g
V, (m/set)
0.4
const
-0.4
F,ckg)
Og
C
2
V, (m/set
1 0.4
consi
-0.1
F&kg)
ii
Y
g
t
(
FIG. 13. Force
and velocity
records
obtained
at different
speeds of galloping. Indications
as in Fig. 1.
-!
V~(m/sec
5:
) 0.1
cons
- 0.:
Fv(kg)
II
21
VF(m/sec
1 0.
cons
- 0.
FF(kg)
4
ZE
a
IX
-4
VV(m/sec)
0.
cons
- 0.
FV(kg)
10
2ou
III1
..*
_s.
:I ." " -. ._ :.:.i
I
I I _/_>a.._
L-- 1
-L
0
1
It is much easier to refer to the energy records (Fig.
14) than to the force and velocity records in order to
understand
galloping.
Motion pictures were taken simultaneously
with the force and velocity measurements
to determine the sequence of foot-fall patterns and to
correlate these with the energy records. In this paper
we will discuss the energy records from dogs to explain
the mechanism of galloping, since essentially the same
mechanism was found in the monkey and ram.
At slow speeds, galloping combines the two separate
mechanisms utilized in walking and running. Within
a stride there are times when E, is converted into E,,
E, is converted into E,, and E, increases simultaneously with E,. As the animals gallop faster and faster,
'2
',
:.
i"
.'.
-a
2 set
the exchange between gravitational
potential
energy
and kinetic energy becomes smaller and approaches
zero. At the highest galloping
speeds the animal
bounces first on its hind legs and then on its front legs.
Phase relationship and exchange between kinetic and
gravitational
potential energy during each stride. At
slow galloping speeds, E, decreases while the average
E,, increases after the two back feet contact the ground
(the numbers 1, 2, 3, and 4 above the E,, line in Fig.
14 indicate the sequence of foot-fails beginning
with
the first rear foot). The decrease of E,, simultaneous
with the increase ofE, indicates a shift of gravitational
potential energy into kinetic energy while the animal
is falling forward. When IAEkfJ > 1AE,I , an additional
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
1c
2(
MECHANISMS
OF
TERRESTRIAL
R259
LOCOMOTION
DOG GALLOP
/’
EKF
E KV
1l.b
70
at
7Scat
I d
0
21.3km/h,
12%
,
31 km/h,
8.1%
0.5
,
1
1.5
TIME (set)
forward thrust must be provided by the muscles of the
rear legs. After this, an aerial phase may take place
(it takes place in the 17.5kg dog galloping at 21.3
km- h-l whereas it does not in the 5-kg dog). The
forward fall is stopped by the contact of the first front
leg (no. 3 in Fig. 14). The dog then “pole vaults” over
the second front leg (no. 4 in Fig. 14): this is indicated
by an increase of E, simultaneous
with a decrease of
Ekf, i.e., Ekf is converted into E,. Since lAEkfl > IAEPI
some of Ekf either goes into stored elastic energy
or is lost as heat. After pole vaulting, the dog pushes
upward and forward simultaneously
off this second
front foot; both E, and Ekf increase, and the center of
mass returns to its original height. An aerial phase
usually takes place after this push-off until the first
rear foot strikes the ground again and the cycle repeats
itself. Thus there is little symmetry
between the
changes in E, and E,; sometimes during the stride
they change in opposite directions,
and other times
they change in the same direction.
The percent recovery calculated using Eq. 2 jumps
from about zero at the highest trotting speeds to about
30% at the slowest galloping speeds in dogs, monkeys,
and rams (Fig. ‘7). As a result of this exchange Wert
dropped in all the animals as they changed from a trot
to a gallop, even though the speed stayed the same or
increased (Fig. 8). The value of y ’ (calculated according
to Eq. 8) decreased (Fig. lo), indicating less storage in
elastic elements in a slow gallop than in a fast trot.
Thus changing from a trot to a gallop involves changing
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
17.2 km/h
11.3 Z Recov.
FIG. 14. Experimental
records
of
mechanical
energy
changes
of center
of mass of the body in the gallop of the
dog. After
landing
on the back legs,
the body falls forward
(decrease
of potential
energy
and increase
of kinetic
energy).
“Pole
vaulting”
and a push
directed
upward
and
forward
take
place immediately
before
take-off
from
a front
leg. Percent
recovery
(Fig.
7)
decreases
with
speed
indicating
that
at high speeds
the mechanism
of galloping
becomes
similar
to that of running.
Numbers
and symbols
above Ek/
tracings
refer to sequence
of foot falls,
beginning
with
1 as the first rear foot
strikes
the ground
during
a stride.
There
is only
one aerial
phase
per
stride
at low speeds
(following
thrust
from front feet) whereas
at high speeds
(31 and 34.5 km/h)
there are two aerial
phases
(following
thrust
from back feet
and from front feet).
R%o
intermediate
speeds, when a decrease in height of the
center of mass may occur within the stride, S, = WV/P
may indicate a vertical lifi greater than the actual one.
Magnitude
of kinetic and gravitational
potential energy changes. W&Z increases with speed during galloping in nearly the same manner as during running.
The curves calculated using Eq. 12 ( tv and tc are taken
as the sum of the times of flight and of contact within
the stride and -cZ, as the average of the forward
decelerations
taking place during the stride) fit the
experimental
data. There is little difference between
l&/m at a given speed whether a bird runs, a kangaroo
hops, or a quadruped gallops. In addition, one monkey
galloped at all speeds while the other monkey trotted
at all speeds, yet there was no difference between the
two functions relating WJm to I$.
WJm remains nearly constant with increasing speed
and has the same value both in the trot and a gallop
(Fig. 8).
Despite the fact that l&/m
= f( V’) and WJm =
f( V’) are the same in trotting and galloping, WJm is
smaller at low speeds of the gallop as the result of the
exchange between E, and Ekf which we have already
discussed. This leads to a discontinuity
in the function
W ext = f( Vf) when the dogs shift from a trot to a gallop.
DISCUSSION
HEGLUND,
AND TAYLOR
fact that legs appear to bend in opposite directions.
Quadrupeds walk as if they were simply two bipeds
walking nearly synchronously,
one behind the other.
During running, trotting, and hopping, animals store
energy in elastic elements as the center of mass decelerates simultaneously
in the vertical and forward directions, and they recover some of this stored energy
as it reaccelerates. Galloping involves a combination
of the two mechanisms.
Walking
There is an optimal
speed for walking
in man at
which the exchange between E, and E,/ is maximal
and both Wext and W metab (per kilometer) are minimal.
The pendulum
model is a good approximation
of the
mechanism of walking at this optimal speed (9). The
mechanics of walking
deviates from the pendulum
model both at lower and higher speeds because of 1)
the difference in the relative magnitude
of W, and Wf
i.e., W, = Wf at the optimal speed, W, > W, at lower
speeds and Wf > W, at higher speeds; and 2) the phase
relationship
between E, and EkP which become more
and more in phase as the walking speed increases.
Man pushes off from the foot just before it leaves the
ground and this tends to lift the center of mass. This
push becomes larger at higher speeds, and E, and Elzf
become more and more in phase until there is almost
no exchange between E, and E,, when walking speeds
exceed 13 km h-l. However, man can still walk at
speeds exceeding 18 km h-l.
We find that walking in our animals is essentially
the same as that described for man. This similarity
is
most clearly seen in the birds. Using measurements
of
vo, as a function of speed during walking in the rhea,
we found a minimum
W metabper kilometer where Wext
per kilometer
is also minimal
(unpublished
observations of C. R. Taylor). The push from the rear foot can
be seen at high speeds as a second peak in the F, and
l
l
V, records
of the rhea and turkey
(Fig.
1); this
is
almost identical to a secondary peak in the F,! and V,!
records observed in man at high speeds of walking.
In conclusion,
a pendulum
model applies only at
intermediate
walking speeds; Ekv is always small and
can be converted into E,,; therefore the simplest pendulum models-such
as the stiff-legged walk of Alexander (l), which assumes that W,,, is used to replace
the Ekv lost at each step-do
not seem to explain
walking in our animals or in man. Wext is done at
each step of walking (see increments of Etot in Fig. 2)
to increase E, and/or E, above the values attained by
means of the described shift of kinetic into potential
energy and vice versa. In other words Wext is used to
Evolution seems to have been very conservative in
designing vertebrate
locomotory systems for moving
along the ground. What at first seemed a bewildering
array of modes of locomotionbipedal walking in humans and in birds, quadrupedal
walking in mammals,
trotting, galloping and hopping-can
all be reduced to
two general mechanisms,
a pendulum and a spring,
which have been utilized either singly or in combinacomplete the vertical lift and to give an additional
tion, to minimize the expenditure
of chemical energy
push forward, thus keeping the “egg rolling.”
by the muscles for lifting and reaccelerating the center
of mass within each stride. During walking, there is Running and Galloping
an alternate
exchange between kinetic and gravita.
tional potential energy. This exchange can be as large
W metab increases linearly with running and galloping
as 70% of the energy changes taking place as the center speed in most animals from a positivey intercept which
of mass rises and falls and accelerates and decelerates
is. about 1.5-2.0 times W metab predicted for rest (21).
within each stride. Walking involves essentially the W metab is used by muscles to provide mechanical power
same mechanism in humans and in birds despite the for locomotion.
This power consists mainly
of WpVI
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
from a mode of locomotion which relies heavily on
elastic recovery to one which combines both the spring
and the pendulum mechanisms.
As galloping
animals
increase their speed, the
changes in (E, + Ekv) and Ew become more and more
in phase until at the highest speeds almost no exchange
is possible. At these high speeds there are two bounces
per stride, first off the back feet followed by an aerial
phase, then off the front feet followed by a second
aerial phase. At the highest speeds, the two front feet
hit the ground almost simultaneously
as do the two
rear feet. The center of mass regains its initial height
after each bounce, and there are two nearly identical
changes in E, and Ew per stride which are in phase,
just as in the trot. Therefore at the highest speeds tC
and t, in Fig. 11 and S, in Fig. 9 (which is measured
as WJP) must be divided into half to calculate
the period and displacement
during each bounce. At
CAVAGNA,
MECHANISMS
OF
TERRESTRIAL
R261
LOCOMOTION
= II&l + II&l and Wint (t o accelerate the limbs relative
to. the center of mass). Both I& = f(V& (Eq. 12), and
l
l
l
A preliminary
account
of part of this
Symposium
on Biodynamics
of Animal
England,
September
1975.
Received
for publication
29 October
work
was presented
at the
Locomotion,
Cambridge,
1976.
REFERENCES
1. ALEXANDER, R. McN.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Mechanics
of bipedal
locomotion.
In:
Perspectives
in Experimental
Biology.
Zoology,
edited
by S. P.
Davies.
Oxford:
Pergamon
1976, vol. 1.
ALEXANDER,
R. McN.,
AND A. VERNON.
The mechanics
of hopping by kangaroos
(Macropodidae).
J. ZooZ., London
177: 265
303, 1975.
CAVAGNA, G. A. Force plates as ergometers.
J. AppZ. Physiol.
39: 174-179,
1975.
CAVAGNA,
G. A., B. DUSMAN,
AND R. MARCARIA.
Positive
work
done by a previously
stretched
muscle.
J. AppZ. Physiol.
24:
21-32,
1968.
CAVAGNA, G. A., AND M. KANEKO.
Mechanical
work
and efficiency
in level walking
and running.
J. Physiol.,
London,
268:
467-481,
1977.
CAVAGNA,
G. A., L. KOMAREK,
AND S. MAZZOLENI.
The mechanics of sprint
runnning.
J. Physiol.,
London,
217: 709-721,
1971.
CAVAGNA, G. A., F. P. SAIBENE, AND R. MARCARIA.
External
work in walking.
J. Appl.
Physiol.
18: 1-9, 1963.
CAVAGNA, G. A., F. P. SAIBENE, AND R. MARGARIA.
Mechanical
work in running.
J. AppZ. Physiol.
19: 249-256,
1964.
CAVAGNA,
G. A., H. THYS, AND A. ZAMBONI.
The sources
of
external
work
in level walking
and running.
J. Physiol.,
London, 262: 639-657,
1976.
CLARK, J., AND R. McN.
ALEXANDER.
Mechanics
of running
by
quail
(Coturnix).
J. Zool.,
London
176: 87-113,
1975.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
DAWSON,
T. J., AND C. R. TAYLOR. Energetic
cost of locomotion
in kangaroos.
Nature
246: 313-314,
1973.
ELFTMAN, H. The force exerted
by the ground
in walking.
Arbeitsphysiologie
10: 485-491,
1939.
ELFTMAN, H. Biomechanics
of muscle
with
particular
application to studies
of gait. J. Bone Joint
Surg.
48A: 363-377,
1966.
FEDAK, M. A., B. PINSHOW,
AND-K.
SCHMIDT-NIELSEN.
Energetic
cost of bipedal
running.
Am. J. Physiol.
227: 1038-1044,
1974.
FENN, W. 0. Work against
gravity
and work
due to velocity
changes
in running.
Am. J. Physiol.
93: 433-462,
1930.
GRAY, J. AnimaZ Locomotion.
New York:
Norton,
1968, p. 479.
HILDEBRAND,
M. Analysis
of Vertebrate
Structure.
New York:
Wiley,
1974, p. 710.
MARCARIA,
R. Sulla fisiologia
e specialmente
sul consumo
energetico della marcia
e della corsa a varie
velocita
ed inclinazioni
de1 terreno.
Atti Accad.
Naz. Lincei
Mem. Classe
Sci. Fis. Mat.
Nat. Sez. III 7: 299-368,
1938.
MUYBRIDGE,
E. Animals
in Motion.
New York:
Dover,
1957, p.
72 and 183 plates.
RALSTON,
H. J., AND L. LUKIN.
Energy
levels
of human
body
segments
during
level walking.
Ergonomics
12: 39-46,
1969.
TAYLOR,
C. R., K. SCHMIDT-NIELSEN,
AND J. RAAB. Scaling
of
energetic
cost of running
to body
size in mammals.
Am. J.
Physiol.
219: 1104-1107, 1970.
Downloaded from http://ajpregu.physiology.org/ by 10.220.32.247 on June 16, 2017
W- = f(v’),
(5), originate
from zero and increase
rnze and more steeply with increasing speed; they
therefore
cannot explain the positive y intercept
of
.
W metab = f(v,). W,, however, is about constant and
does not go to zero at zero running speed; work against
gravity has to be performed at about the same rate at
low as well as high speeds (Fig. 8). This rate of working,
required to start the mechanism of running, may help
to. explain the y intercept of the relationship
between
W metab and p,
The same argument
can be seen in terms of the
work done per unit distance,. W&L.
m) and Wf/
(I, n) increase while W,/(L . m) decreases with increasing speed (Fig. 8). Within the speed range of our experimental
data the increase in W&L am) and the
decrease in WV/( L . m) approximately
compensate for
each other and West/(L m) is approximately
independent of speed in most of the animals. However,
if we extrapolate these functions to lower speeds, Wezt/
+ (W/Y,)
would increase and tend to
L = (W,/Q
infinity when Vr approaches zero, because W, is high
also at the lowest speeds. This may help to explain
the similar increase for W,,,,,/(L
. n) which has been
observed in most animals at low speeds (21). The increase in y’, which we find in most of our animals with
increasing
speed (Fig. lo), will also increase the y
intercept in the relationship
Wmetab = f( &), i.e., it may
also be important
in explaining the increase in Wmetad
(L- m) at low speeds.
Values of y (Eq. 5), increasing linearly with P, and
similar to those of y’ (Eq. 6) for kangaroo (Fig. lo),
have been found for man by Cavagna and Kaneko (5).
Wint in kangaroo is much smaller than in man, whereas
Wext is slightly greater. It follows that the kangaroo
develops much less mechanical power than man as vr
increases above about 15 km. h-4 However Vo, increases
with speed in man (l-8), whereas it remains about
constant in the kangaroo or even decreases slightly
with vf (ll), SO that the ratios West/Wmetab = 7’ and
= y are about equal in both man
Cwest
+ wint)lwmetab
and kangaroo.
Alexander and Vernon (2) explain an increase of y
with vf in the kangaroo by a progressively
greater
recovery of elastic energy due to an increase with speed
of the forces stretching the tendons. According to the
calculations of these authors (based on tendon dimensions and compliance), elastic storage saves nearly 40%
of the energy which would otherwise be required by a
wallaby hopping at 8.6 km. h-l whereas, for the kangaroo, the saving would increase with speed from 23%
at 9.7 kmgh-l
to 40% at 22 km h-l. These figures
correspond to maximum
values of y’ much smaller
than those given in Fig. 10 and found in man; the
discrepancy was also noticed by Alexander and Vernon.
Possibly the role of elasticity
is underestimated
by
calculations taking into account only the elastic compliance of tendons and not that of the contractile component itself.
On the other hand, the evidence of elastic recovery
reached in the present paper could be invalidated
by
errors of measurement
of I) metabolic power minus
resting metabolism: the metabolic measurements
were
made at different time and on a treadmill,
whereas
the mechanical power was measured during running
on the floor; and 2) mechanical
power output: these
can derive from both the methods used (3) and from
irregularities
of the run of the animals. In addition,
the assumption
that the maximal
efficiency of the
contractile
component
is 0.25 may be incorrect; and
mechanical energy could be stored not in elastic elements but in some other unrecognized sites.