AMER. ZOOL., 28:237-245 (1988)
Why Mammals Gallop1
R. MCNEILL ALEXANDER
Department of Pure and Applied Biology,
University of Leeds, Leeds LS2 9JT, England
SYNOPSIS. Most mammals use symmetrical gaits (such as the trot) at moderate speeds but
change to asymmetrical gaits (gallops) at high speeds. A mathematical model of quadrupedal gaits failed to show any advantage in this change: it seemed to show that, even at
high speeds, there was always a symmetrical gait that was at least as economical as galloping.
That model treated the back as rigid, but another model seemed to show that back
movements such as occur in galloping could only increase the energy cost. However,
metabolic measurements on horses showed that galloping is more economical than trotting
at high speeds. The explanation seems to be that kinetic energy fluctuations, due to
backward and forward swinging of the legs, become very large at high speeds. Galloping
makes it possible for kinetic energy associated with leg movements to be stored briefly as
strain energy in elastic structures in the back, and returned in an elastic recoil. The most
important of the strain energy stores in the back, that have been discovered so far, is the
aponeurosis of the longissimus muscle.
INTRODUCTION
My colleagues and I think of ourselves
as practitioners of animal mechanics, a subject whose scope is indicated by my book
(Alexander, 1983). We use the methods of
engineering mechanics to elucidate the
structure and movements of animals. Until
1970, I worked mainly on fish (references
in Alexander, 1974). Since then we have
worked almost exclusively on the movements of tetrapods (including people) and
on their skeletons, muscles and tendons.
Our interests in the optimization of gaits,
and in the roles of elastic mechanisms in
locomotion, are illustrated by this paper,
but are reviewed more comprehensively by
Alexander (19846). Another interest, in the
consequences of size differences for animal
structure and movements, led to our
dynamic similarity hypothesis for gaits
(Alexander and Jayes, 1983). Yet another
main interest, in the mechanical stresses
imposed by strenuous activities, led to a
theory of skeletal safety factors (Alexander, 1984c).
Our approach is always quantitative. We
make rough calculations to check the plausibility of explanations, in undergraduate
1
From the Symposium on Questions, Explanations,
Models and Tests in Morphology: The Interaction Between
Hypotheses and Empirical Observations presented at the
Annual Meeting of the American Society of Zoologists, 27-30 December 1985, at Baltimore, Maryland.
textbooks (Alexander, 1979, 1981) as well
as in research. We do not think a functional
explanation complete until we can show
that a structure or movement is optimal
(by some plausible criterion) for the proposed function (Alexander, 1982). We
make much use of mathematical models
which we try to keep as simple and general
as possible. These attitudes are illustrated
by the example of our work that I have
chosen to present.
THE PROBLEM
Most of the larger mammals walk to
travel slowly, trot to go faster and gallop
at their highest speeds. Cats, dogs, antelopes, horses and a great many other mammals change gaits like this, as they increase
speed.
Walking, trotting and some other gaits
(including the pace of camels and the amble
of elephants) are described as symmetrical
because the left and right feet of each pair
move approximately half a cycle out of
phase. In contrast, the various types of gallop are asymmetrical: the phase difference
between the left and right feet of each pair
is distinctly different from half a cycle (Hildebrand, 1977). Mammals of different sizes
change gaits at different speeds but Alexander and Jayes (1983) presented a general
rule using a Froude number (a parameter
of a type that is often useful in analysing
the mechanics of systems influenced by
237
238
R. MCNEILL ALEXANDER
0.5
pace
trot
amble
0.5
left/right
phase
0.25 -
canter
difference
pronk
|
0.25
0.25
gallop
boynd
0.5
0.75
0.2 5
0.5
0.7 5
fore/hind phase difference
FIG. 1. (a) A diagram showing the phase differences between the feet, in quadrupedal running gaits, (b) A
similar diagram with contours showing relative metabolic power requirements of gaits in fast locomotion,
calculated for an idealized quadruped with a rigid back and massless legs. Darker tones indicate larger power
requirements. Possible savings, by storage of elastic strain energy, are ignored. Modified from Alexander et
al. (1980).
gravity). This Froude number was u2/gh,
where u is the speed of locomotion, g is
the acceleration of free fall (9.8 m sec"2)
and h is leg length (defined as the height
of the hip joint from the ground). Mammals generally use symmetrical gaits at
speeds corresponding to Froude numbers
less than two, and asymmetrical gaits at
Froude numbers greater than three. They
change from a symmetrical to an asymmetrical gait at some Froude number
between two and three.
Why do mammals gallop instead of trotting, at their highest speeds? Our investigations of this question have involved film
analysis and mathematical modelling, but
the most promising solution has been
reached by morphology and mechanical
testing (Alexander et al., 1985). This paper
describes the principal stages of our investigations.
masses of the legs. The pattern of force
exerted by each foot on the ground was
represented by equations designed to imitate the forces that had been recorded when
animals had been made to run over force
platforms.
A muscle that shortens while exerting
tension does (positive) work. Alternatively,
a muscle may lengthen while exerting force,
in which case it exerts a braking action,
doing negative work. Running at constant
speed on level ground requires almost equal
quantities of positive and negative work
(not quite equal, because frictional losses
occur). Metabolic energy is required for
both. The mathematical model added up
separately the positive and negative work
required of each leg in the course of a stride.
Calculations were performed only for
gaits in which the phase difference between
the movements of the two forefeet was the
same as the phase difference between the
A RIGID MODEL
two hindfeet. This saved a great deal of
Alexander et al. (1980) adopted the plau- effort, and seemed unlikely to lead to serisible hypothesis that mammals choose their ously misleading results. For symmetrical
gaits in such a way as to minimize the energy gaits, this left-right phase difference is 0.5,
cost of locomotion. This implies that gaits and for asymmetrical gaits it is less than
which are not used, at any particular speed, 0.5. In each of the diagrams in Figure 1,
would be more expensive of energy than the vertical axis shows left-right phase difthose that are. To try to discover whether ferences and the horizontal axis shows forethis was true, Alexander et al. (1980) for- hind phase differences. Every possible
mulated a mathematical model capable of combination of phase differences is reppredicting the energy costs of gaits involv- resented by a point on the diagram. Figure
ing all conceivable footfall patterns. We la shows the phase relationships for named
kept the model as simple as seemed rea- gaits. Contours in Figure lb show the calsonable, treating the head and body as a culated energy cost for running with each
single rigid unit and initially ignoring the combination of phase differences, at a par-
WHY MAMMALS GALLOP
ticular Froude number (corresponding to
a fast galloping speed). They indicate that
galloping is as economical as any other gait,
but is no more economical than the amble
and only slightly more economical than
trotting and pacing.
Figure lb ignores elastic effects, assuming that all work done by the legs is achieved
by length changes of muscles, at metabolic
cost. However, each foot exerts a braking
force when first set down and an accelerating force later in the step: the leg does
negative work followed by positive work.
Similarly, springs do negative work when
they are loaded, storing strain energy which
is returned as positive work when they are
allowed to recoil. Muscles and particularly
tendons serve as springs, reducing the metabolic energy cost of running (Alexander,
1984a). Further calculations taking account
of possible elastic savings failed to reveal
any clear advantage in the gallop. Galloping was found to be about as economical
as ambling, or more expensive than other
gaits, depending on the precise assumptions made. The conclusions were not
altered by taking account of the masses of
the legs.
239
FIG. 2. Diagrams of successive stages of a galloping
stride, with schematic graphs showing energy fluctuations in the course of the stride.
est speeds, but it cannot explain why they
break into a canter or gallop at quite modest speeds. For example, most horse races
are won at speeds of 16-17 m/sec (as the
sporting pages of newspapers record) and
speeds exceeding 13 m/sec are achieved
in trotting races, but when horses are
allowed to choose their gait, they change
from trotting to cantering at only 5 m/sec
(Heglund et al, 1974).
My colleagues and I wondered whether
back movements could save energy in locomotion, but we concluded initially that they
could not (Alexander et al., 1980). We
developed the model shown in Figure 3a
A FLEXIBLE BACK
but did not publish our analysis of it,
Mammals bend and extend their backs because it failed to explain why mammals
as they gallop (Fig. 2). Cheetahs (Acinonyx) gallop. This model has legs that swing
and greyhounds make very marked back backwards and forwards, but do not bend.
movements but horses and many other The legs of each pair move in synchrony.
ungulates bend much less. Notice that the The trunk consists of two rigid segments,
back is extended when the forefeet are set hinged together. The anterior and postedown (Fig. 2a) but is bent before they leave rior ends of this mathematical model are
the ground (Fig. 2b). This lengthens the identical, but the flexible back makes it
step (the distance travelled while a foot is more realistic (in that respect) than the
on the ground). A similar effect is achieved rigid-backed model of Alexander et al.
for the hind feet by setting them down (1980).
while the back is bent (Fig. 2d) and
We wanted to know how the movements
straightening the back before they are lifted of the back affected the energy cost of loco(Fig. 2e).
motion. In other words, we wanted to comHildebrand (1959) argued that galloping pare the energy cost in galloping of the
enables mammals to increase their speed model shown in Figure 3a with that of a
by lengthening the stride: he calculated that similar rigid-backed model (Fig. 3b). For
back movements added 5% to the stride this it was convenient to distinguish
length of a galloping cheetah. Also, it between several components of the energy
enables the back muscles to contribute to of a moving animal.
the power required for locomotion. This
Consider a rigid body moving in twomight seem a convincing explanation of dimensional Cartesian coordinates. (This
why mammals gallop to travel at their high- passage will be simplified by limiting motion
240
R. MCNEILL ALEXANDER
FIG. 3. Diagrams of idealized quadrupeds used as
models of galloping. Model (a) has a flexible back but
model (b) has a rigid one.
to two dimensions.) It has mass M and its
moment of inertia (about the centre of
mass) is /. At time / its centre of mass has
coordinates X, Fand components of velocity U, V, and the body has angular velocity
ft. The gravitational potential energy of
the body (the energy due to its height) is
MgY. Its kinetic energy (the energy due to
its motion) has components VzM(JJ2 + V2)
due to its linear motion, and V2IQ2 due to
its rotation.
Our model is not rigid, but has parts that
move relative to each other. For this reason, a more complex analysis of kinetic
energy is required. A standard approach
in textbooks of mechanics is to regard the
body as consisting of small particles. The
first particle has mass mx and components
of velocity w,, u,; the second mass m2 and
velocities u2, va, and so on. The total kinetic
energy is regarded as consisting of external
kinetic energy (EKE) due to motion of the
centre of mass and internal kinetic energy
(IKE) due to motion of parts of the body,
relative to the centre of mass. The EKE is
V2M(U2 + V2) and the IKE is '/•>>«,[(«, U)2 + (u, - V)2] + Vim2[(u2 - U)2 + (v2 -
the moment of inertia about the centre of
mass (which may change as the system
moves) and ft is an angular velocity defined
by specifying that the net angular momentum of the system is /fi. Thus this energy
can be thought of as being associated with
the net angular momentum of the system.
IKE(b) is energy due to motion of the
individual particles, relative to a system of
coordinates moving with components of
velocity U, V and rotating with angular
velocity 0.
The model shown in Figure 3b has IKE(b)
because its legs move relative to its trunk.
That of Figure 3a has IKE(b) for this reason, and also because the two halves of the
trunk move relative to each other.
Let the models shown in Figure 3 differ
only in the presence or absence of the joint
in the back. Let them move at the same
speed, using the same gait and exerting
identical patterns of force on the ground.
Gravitational potential energy (PE) and
EKE fluctuate in precisely the same way in
the two models. Net angular momentum
fluctuates in the same way in the two models
and so also does IKE(a), if the fluctuations
of moment of inertia are small enough to
be ignored. IKE(b) is the only component
of the kinetic energy that fluctuates in
markedly different fashion in the two
models. The flexible-backed model (Fig.
3a) moves its hip and shoulder joints
through smaller angles than the rigid model
(Fig. 3b), because back movements contribute to the forward and backward
swinging of the legs. Nevertheless, mathematical analysis showed that fluctuations
of IKE(b) would be larger in the flexible
model than in the rigid one, increasing the
energy cost of galloping.
Thus the rigid model seemed to show
that galloping was at least as expensive of
energy as the most economical symmetrical gaits. The flexible model seemed to
show that back movements made galloping
even more expensive. The hypothesis, that
mammals gallop to minimize energy costs,
seemed to have been destroyed.
V)2]+ . . . (see Chorlton, 1967). In this
analysis, any energy due to rotation of the
system is included in the IKE.
For our analysis, we wanted to subdivide
the IKE. We formulated and proved a
theorem that we have not found in textbooks of mechanics. The kinetic energy of
a system moving in two dimensions is the
sum of EKE, IKE(a) and IKE(b).
T h e EKE, as already explained, is
ViMfU* + V2). It can be thought of as energy
associated with the net linear momentum
METABOLIC DATA
of the system, which has components MU
and MV.
The hypothesis was saved by the work
IKE(a) is energy due to rotation of the of Hoyt and Taylor (1981). They trained
system as a whole, and is V2IQ2. Here / is small horses to run on a treadmill and to
WHY MAMMALS GALLOP
241
FIG. 4. Diagrams showing muscles that could bend
or extend the back. The muscles are indicated by
broken lines, and their effects by arrows.
walk, trot or gallop on command. The
speed of the treadmill controlled the speed
of locomotion so it was possible to induce
the horses to trot at speeds at which they
would normally walk or gallop, and to gallop at speeds at which they would normally
trot. They used face masks to collect the
expired air from the horses, and analysed
it to determine their rates of oxygen consumption. They found that each gait was
more economical than the others, in the
range of speeds at which it is normally used.
A mechanical explanation had already
been found, for walking being more economical at low speeds and running at higher
speeds (Alexander, 1980). It remained a
mystery why, among possible running gaits,
trotting should be more economical than
galloping at moderate speeds, and galloping should be the more economical at high
speeds.
BACK MUSCLES
Alan Jayes and I were convinced that, to
understand galloping, we needed to understand the role of the back. We reasoned
that the most promising animal for study
would be one with a long, flexible back and
short legs. We chose the ferret (Mustela
putorius) and spent a great deal of time
making and analysing films and force-platform records of it galloping. It was reasonably easy to make a ferret gallop over
a force platform, by dragging a dead mouse
in front of it on a string. The ferret was
nevertheless a most unfortunate choice of
animal, because its movements were highly
irregular. Successive strides were apt to differ markedly, making it hard to believe
that any optimization principle was being
applied. It would probably have been bet-
FIG. 5. Diagrams of the longissimus muscle and its
aponeurosis; (a) is a dorsal view, (b) a transverse section and (c) a lateral view.
ter to have studied the back movements of
greyhounds, which make very regular
strides, each almost identical to the one
before (Jayes and Alexander, 1982).
The work on ferret back movements was
unprofitable, but we were also investigating the structure of the muscles that move
the back (Jayes and Alexander, 1981). We
wanted to assess the contribution each
could make to bending or extension. Longitudinal muscles dorsal to the vertebral
column tend to extend the back (Fig. 4a).
Longitudinal muscles ventral to the vertebral column (the psoas muscles) and in
the abdominal wall (rectus abdominis) tend
to flex the back (Fig. 4b). Circular muscles
in the abdominal wall would tend to extend
the back by increasing the pressure in the
abdominal cavity (Fig. 4c), but none of the
muscles are strictly circular. Oblique muscles in the abdominal wall tend to flex or
extend the back, depending on their angle.
In our analysis, each muscle was represented by a simple geometrical model. It
was assumed that the fibers of every muscle
could exert the same stress (force per unit
cross-sectional area), and the bending
moment each muscle could exert on the
back was calculated. It emerged that the
longissimus muscle, with the associated
iliocostalis, was much the most important
extensor of the back in all the species studied.
The longissimus originates on the transverse processes of the vertebrae of the
trunk and on the ribs (Fig. 5). It inserts by
an aponeurosis which is attached to the
ilium, the sacrum and the lumbar neural
242
R. MCNEILL ALEXANDER
spines. The muscle fibers take curving
paths, approximating to arcs of helices. The
fibers of the aponeurosis converge posteriorly.
A STRAIN ENERGY STORE
A few years later, my colleagues and I
were investigating the elastic properties of
leg tendons of deer (Dimery et al., 1986).
Our results indicated that, in fast galloping, most of the PE and EKE lost at stages
(a) and (d) of the stride (Fig. 2) was stored
as elastic strain energy in leg tendons. It
was returned by elastic recoil at stages (b)
and (e). Thus the energy cost of galloping
was apparently greatly reduced. The same
would also happen in fast trotting.
While dissecting a deer carcass for these
experiments, we were struck by the size of
the longissimus aponeurosis. Its total mass
was approximately equal to the mass of the
tendons we were investigating in the forelegs. We wondered whether the aponeurosis, like the leg tendons, could be an
important strain energy store (Alexander
etal, 1985).
The possibility that elastic elements could
have an important energy-saving role, in
balancing fluctuations of IKE, had previously been considered and rejected (Alexander et al., 1980). The apparent problem
was that, in fast gaits, legs swing backward
fast (relative to the centre of mass of the
body), and swing forward more slowly. The
IKE associated with the forward swing is
much less than for the backward swing (see
also Fedak et al., 1982). Therefore energy
must be discarded at the end of the backward swing and replaced (at metabolic cost)
at the beginning of the next backward
swing. Only a small fraction of the IKE
associated with a backward swing can be
stored as strain energy and re-used.
The objection holds only if the legs are
mounted on separate springs with no possibility of transferring energy from one to
another. Our new hypothesis seemed to
overcome the objection. IKE associated
with the movement of one leg might be
stored as strain energy in the aponeurosis,
and reappear in the elastic recoil as IKE
associated with another leg. At stages (a)
and (b) of a galloping stride (Fig. 2) the
forelegs are swinging backwards with large
associated IKE and the hindlegs are swinging forwards with small IKE. At stages (d)
and (e) the forelegs have small associated
IKE and the hindlegs have large IKE. The
total IKE is probably about the same in
both cases, so if the IKE lost at stages (c)
and (f) were stored as strain energy, it could
be re-used following the elastic recoil. The
aponeurosis is suggested as a major strain
energy store at stage (c) (when the longissimus muscle must be taut, to halt and
reverse the bending of the back). Other
elastic elements may store strain energy at
stage (f) but their roles have not yet been
assessed.
To show that the hypothesis was plausible, it was necessary to show that the aponeurosis could store a useful quantity of
strain energy. A rough calculation showed
that a 50 kg deer (Dama dama), galloping
at its likely maximum speed of about 13
m/sec, would lose and regain about 80 J
IKE at stage (c) of the stride. Could the
aponeurosis store a useful fraction of this?
The effective cross-sectional area of a
muscle or tendon can be calculated by
dividing its volume by the length of its
fibers. The volume can be calculated from
the mass and density. Thus it was shown
that the total cross-sectional area of the
longissimus muscle fibers of one side of the
50 kg deer was 95 cm2, and the cross-sectional area of the aponeurosis was 1.11 cm2.
Major leg muscles of mammals tend to exert
peak stresses of about 0.3 MPa in fast locomotion (Jayes and Alexander, 1982). If the
longissimus muscle of the deer exerted this
stress, the stress in the aponeurosis would
be 25 MPa. We wanted to know the strain
energy corresponding to this stress.
Strips from the aponeurosis were subjected to tensile tests, in a dynamic testing
machine. The actuator of the machine was
made to move up and down sinusoidally,
stretching the strip and allowing it to recoil
at a frequency close to the stride frequencies used in galloping. Outputs from the
machine give extension and force, which
are plotted against each other, for a single
cycle of a test, in Figure 6a. The record is
a loop rather than a single line because
some energy is lost by hysteresis effects: it
243
WHY MAMMALS GALLOP
0.2
40 i-
1.0
force,kN
stress,
MPa
strain
energy,J/g
0.5
20
2
4
extension,mm
0.02
strain
20
stress,MPa
40
0.04
FIG. 6. (a) A record of tensile test on a sample of the longissimus aponeurosis of a deer (Dama dama). Further
explanation is given in the text, (b) A graph of strain energy against peak stress, in tensile tests on samples
of the aponeurosis. Modified from Alexander el al. (1985).
would be a narrower loop if the effects of
clamping the ends of the specimen could
be avoided (Ker, 1981). Areas, in graphs
of force against length, represent work.
The shaded area in Figure 6a represents
the strain energy stored in this particular
test.
The greater the stress on the aponeurosis, the more strain energy is stored. Figure 6b shows that, at a stress of 25 MPa,
each gram of aponeurosis would store about
0.3 J. The 50 kg deer had 48 g aponeurosis
on each side of its body, so the two sides
together could store 30 J.
Strain energy would also be stored in the
longissimus muscle fibers: it was estimated
that this would amount to 5 J. Additional
strain energy would be stored in the vertebral column, which would be compressed
by the tension in the muscle. Compressive
tests on vertebrae and intervertebral discs,
in the dynamic testing machine, indicated
that about 9 J would be stored in this way.
Thus the aponeurosis, muscle and vertebral column could store a total of about 44
J, more than half the IKE fluctuation. Further strain energy (not calculated) would
be stored in leg muscles and their tendons.
It seems reasonable to suggest that most of
the IKE lost and regained at stage (c) (Fig.
2) is stored temporarily as elastic strain
energy.
In a further check of the plausibility of
the hypothesis, we considered how much
the aponeurosis would stretch. Experiments with the carcass showed that if
the longissimus muscle remained taut
throughout the galloping stride, its muscle
fibers and the aponeurosis together would
have to lengthen and shorten by 35 mm.
If, as seems more appropriate, the muscle
is taut only during stage (c) (Fig. 2), the
lengthening and shortening would be considerably less. The tensile tests showed that
the calculated stress of 25 MPa would produce a strain of 0.03 in the aponeurosis.
The longest fibers in the aponeurosis were
about 360 mm long and would stretch by
11 mm. This seemed consistent with the
hypothesis.
CONCLUSION
In galloping, bending of the back helps
to swing the forelegs back and the hindlegs
forward. Extension of the back has the
reverse effect. The mechanism just proposed, which exchanges IKE with strain
energy in the longissimus aponeurosis,
244
R. MCNEILL ALEXANDER
depends on this. It would not work in symmetrical gaits, in which the left leg of a
pair swings forward while the right leg
swings back. This seems to be the advantage of galloping.
IKE fluctuations become relatively more
important as speed increases. Heglund et
al. (1982) found that energy fluctuations
associated with the centre of mass (PE plus
EKE) were linearly related to speed in
mammals and birds. Fluctuations of IKE
were proportional to (speed)1-53. IKE fluctuations were predicted to exceed PE plus
EKE fluctuations, for mammals and birds
in general, at all speeds above 2.2 m/sec.
Changing to a gallop at high speeds
makes IKE fluctuations even larger than if
a symmetrical gait were used, because of
the movements of the back. However, it
enables IKE fluctuations to be balanced, at
least in part, by strain energy storage. Fluctuations of kinetic energy may be increased,
but metabolic energy requirements may be
reduced, making galloping advantageous.
As speed increases and the IKE fluctuations associated with leg movements
become larger, it becomes possible to save
energy by galloping.
My colleagues and I ask questions about
the structure and movement of animals,
seeking to discover selective criteria that
have been important in their evolution. We
do not try to measure fitness but we try to
identify quantities correlated with fitness,
that have been optimized (Alexander,
1982, p. 94). In this particular investigation we have sought to explain why mammals change from symmetrical gaits to gallops at high speeds. Our hypothesis
throughout has been that galloping minimizes the energy cost of fast locomotion.
Initial attempts to test the hypothesis, by
mathematical modelling, produced results
that seemed to refute it. However, metabolic data encouraged us to persist with the
hypothesis. Insight came unexpectedly,
when we realized that the aponeurosis and
vertebral column could serve as useful
strain energy stores. We proposed a mechanism that seems to show how galloping
may minimize energy costs. Anatomical
measurements and mechanical tests provided quantitative data that seem to con-
firm that the mechanism is plausible. It
seems desirable now to try to devise a new
mathematical model, in which IKE is interconverted with strain energy in a spring in
the back. The model should show that galloping becomes more economical than
symmetrical gaits, above a critical speed.
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