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Abstract

The skill to swim fast results from the interplay between generating high thrust while minimizing drag. In front crawl, swimmers achieve this goal by adapting their inter-arm coordination according to the race pace. A transition has been observed from a catch-up pattern of coordination (i.e. lag time between the propulsion of the two arms) to a superposition pattern of coordination as the velocity increases. Expert swimmers choose a catch-up coordination pattern at low velocities with a constant relative lag time of glide during the cycle and switch to a maximum propulsion force strategy at higher velocities. This transition is explained using a burst-and-coast model. At low velocities, the choice of coordination can be understood through two parameters: the time of propulsion and the gliding effectiveness. These parameters can characterize a swimmer and help to optimize their technique.
Gait transition in swimming
Remi Carmigniani
Ecole des Ponts ParisTech
remi.carmigniani@enpc.fr
Ludovic Seifert & Didier Chollet
CETAPS EA3832, Faculty of Sports Sciences, University of Rouen Normandy
Christophe Clanet
LadhyX, Ecole Polytechnique
June 15, 2019
Abstract
The skill to swim fast results from the interplay between generating high thrust while minimizing drag. In
front crawl, swimmers achieve this goal by adapting their inter-arm coordination according to the race pace.
A transition has been observed from a catch-up pattern of coordination (i.e. lag time between the propulsion
of the two arms) to a superposition pattern of coordination as the velocity increases. Expert swimmers
choose a catch-up coordination pattern at low velocities with a constant relative lag time of glide during the
cycle and switch to a maximum propulsion force strategy at higher velocities. This transition is explained
using a burst-and-coast model. At low velocities, the choice of coordination can be understood through
two parameters: the time of propulsion and the gliding effectiveness. These parameters can characterize a
swimmer and help to optimize their technique.
Although we can find evidences of swimming in
the artwork of ancien Egypt over 2,000 BC, mod-
ern competitive swimming started in the early 19th-
century England [22]. The search for speed in swim-
ming led to changes of the technique from the natu-
ral quadrupeds dog fashion technique to the breast-
stroke, then side-stroke and Trudgen-stroke, all the
way to the modern front-crawl. The front-crawl was
pioneered in competition by the Australian Richard
Cavill at the beginning of the 20th century. He was
largely inspired by natives surfers from the Solomon
Islands [22]. The technique was refined over time as
the average speed of swimmers has continued to in-
crease over the century (see figure 1). Front-crawl is
now used on a large range of distances in swimming
pool and open-water races. It appears to be the most
Corresponding author
efficient swimming technique as it is the only one used
for long distances (over 200 m) and the fastest one
(used in freestyle sprint)[2]. It is characterized by
alternated arm propulsion phases and arm recovery
out of the water.
The skill to swim fast is a combination between
generating high thrust and minimizing drag due to
aquatic resistance on the body. The first study in-
vestigating drag during human locomotion in water
can be traced back all the way to the early beginning
of the 20th century [14]. Karpovich [13] pioneered
the quantification of human body drag using a tow-
ing protocol (called passive drag, Dp,b). He found
that the passive drag when the swimmer is fully ex-
tended in a so called streamline position near the
surface was about Dp,b =kp,bv2, where vdenotes
the towing velocity and kp,b 31 (24) kg/m for men
(women, respectively). Then numerous research ex-
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Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
1900 1950 2000
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Year
Speed (m/s)
Men
Women
Figure 1: Evolution of the mean velocity over time of the
100 m long course freestyle. The circles de-
note the world records evolution. The squares
denote the year best performance from 2001
to today.
amining passive drag have emerged as shown in the
review of Scurati et al. [26]. The mean drag experi-
enced during swimming is still not fully understood
and continue being investigated[10, 18, 33, 24]. A
simple way to reduce the drag is to swim in the wake
of another swimmer [37, 32, 4]. This is called drafting
and the effects of drafting on the swimmer technique
and race strategy are still to explore.
The swimming performance is solely evaluated on
the time to reach a certain distance. To under-
stand the link between the achieved performance and
the swimming technique, researchers have first fo-
cused their attention on the arm stroke frequency
(also called stroke rate) fR, and the mean velocity
of the swimmer v[9, 8]. To link these two quanti-
ties, they defined the distance per stroke (or stroke
length) Ls=v/fR. Craig & Pendergast [8] collected
data on expert swimmers asking them to swim at a
given velocity using the minimum stroke frequency
they could achieve. They observed that swimmers
did not use this minimum stroke rate technique for
long distance races (over 200 m). They commented
that even though these swimmers could achieve the
same velocity with a lower frequency (and hence a
longer stroke length), they used a higher frequency
and a lower force per stroke to reduce fatigue. Cos-
till et al. [6] emphasize the importance of stroke tech-
nique on the performance and defined a stroke index
SI =vLsto evaluate the swimming economy.
Focusing furthermore on the swimming technique,
Chollet et al. [5] investigated the arm stroke phase
organization during a stroke cycle and defined the
index of coordination (IdC). This non-dimensional
number characterizes the temporal motor organiza-
tion of propulsion phases. The two main patterns
of coordination can be simplified to the sketches of
figure 2. The solid lines represent a simplified hand
elevation compared to the mean water level (dotted
lines)1. When the solid curve is below this level, a
propulsive phase occurs. This is further outlined by
the gray blocks at the bottom. The arms are iden-
tified by the index i∈ {L, R}, for left and right,
respectively. This index enables to track the suc-
cessions of propulsive phases. As an example, we
consider the nth cycle of the right arm. It begins at
tR
start,n and ends when this arm starts its next propul-
sive phase tR
start,n+1. The cycles repeat periodically
with a period T=tR
start,n tR
start,n. The propulsion
phase of one arm lasts tp=tR
end,n tR
start,n and the
non-propulsive phase tnp =tR
start,n+1 tR
end,n. The
coordination time is then defined by:
tc=tR
end,n tL
start,n,(1)
and the index of coordination corresponds to the non-
dimensional time of coordination compared to the cy-
cle period:
IdC = tc/T. (2)
In the case of figure 2-a), catch-up mode, the index
of coordination is negative as the propulsive phase of
the latter arm starts after the end of the propulsive
phase of the former. This technique is exhibited by
long distance swimmers who used glide within the
cycle. During this glide they adopt a streamline arm
position as illustrated in the picture of figure 2-a). On
the contrary, in figure 2-b), superposition mode, the
index of coordination is positive. There is a time 2 |tc|
1Note that when the solid curve overlaps the dotted one,
this intends to mean that the hand has entered the water but
is not yet active in the propulsion.
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Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
tR
start,n tR
end,n tL
start,n tL
end,n tR
start,n+1
arm Rarm L
tptnp
|tc| |tc|
T
arm R
arm L
Propulsion
tR
start,n tR
end,n
tL
start,n
tL
end,n1tR
start,n+1
arm Rarm L
tptnp
|tc||tc|
T
arm R
arm L
Propulsion
a) b)
catch-up mode superposition mode
Figure 2: Main differences between long (left) and short (right) distance swimmers’ coordination patterns. Photos
are extracted from races at the Olympic Games with the permission of The Olympic Multimedia Library.
during which both arms performed their propulsion.
A third pattern of coordination can be defined at the
transition between the former two and is referred to
opposition in the literature. It corresponds to the
case where one arm starts its propulsion phase when
the other finishes. There is no time lag between the
two propulsion phases (IdC = 0). These three dis-
tinctive patterns of coordination were first described
by Costill et al. [7] and then quantified by Chollet et
al. [5]. They observed the choice of coordination of
different level swimmers. Expert swimmers were able
to reach higher swimming velocity thanks to higher
positive index of coordination than non-expert swim-
mers both on incremental tests [5, 28] and 100-m
races [27]. The effect of fatigue on the coordination
was also investigated by Alberty et al. [1]. They ob-
served a general increase of the index of coordination
with fatigue. A physical model discussing the mo-
tor coordination is proposed in our current study to
understand the transition from catch-up to superpo-
sition mode and the optimal choice of coordination
depending on the targeted velocity of swimming.
Our study is organized in two steps. First, we
present the field observations of expert swimmers co-
ordination and discussed a simple way to compare
the swimmers among them. The swimmers used only
their arms to generate thrust. As previously noticed,
for low velocity, hence long distance races pace, the
swimmers prefer a catch-up mode of swimming. Sec-
ond, we propose a physical model to understand this
choice of coordination. The model is compared to
our field observations and a linearized expression is
derived.
Field investigation and first
coordination model
Raw observations
Following the work of Chollet et al. [5], we consider
the motor coordination of national level French swim-
mers. To simplify the discussion, the motor coordi-
nation of the swimmers is averaged between the two
arms and the legs motions are ignored. That is to
say that the swimmers are considered symmetrical
and only the arms coordination are discussed.
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Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
In the current study, we consider the data col-
lected on 16 French male swimmers in 2007 for whom
the mean ±standard deviation (min, max) of age,
body mass, height, arm span and arm length were:
21.2±4.4 (19, 31) years, 78.8±8.5 (66.3,90.5) kg,
1.84±0.03 (1.70, 1.93) m, 1.91±0.08 (1.70, 2.14) m,
0.65±0.05 (0.60, 0.75) m, respectively. At the time
of the experiment, they were practicing a minimum
of 10 hours a week and had been swimming com-
petitively for 12.1±3.5 years, confirming their expert
level [11]. An extra swimmer was also tested. He was
highly specialised in sprint race (50 m race). He had
performed similar coordination test in the past and
showed behaviour similar to the one described in the
paper. For this test, he surprisingly performed dras-
tically differently than before. We decided to remove
him from the data set due to this change of coor-
dination, which was probably due to specific sprint
work.
The personal best record of the 16 expert swim-
mers was in average 54.2 ±1.8 (50.33, 57.8) s at the
100-m freestyle in the long pool. All swimmers com-
peted at national level. They were all tested on two
graded speed tests in a randomized order using only
the arms in front crawl. Their legs were tied and they
were equipped with a pull buoy for buoyancy. They
all volunteered for this study and gave their written
consent to participate.
One test consists of simulated racing techniques
where these expert swimmers were asked to swim at
8 different velocities corresponding to different race
paces (from 3000 m to 50 m + maximal speed) on
a single 25 m lap. During this test, the swimmers
were video recorded by two synchronized underwater
video cameras at 50 fps (Sony compact FCB-EX10L),
in order to get a front and side view, from which the
different stroke phases and the arm coordination have
been computed. The protocol is similar to the one
described in [5, 29, 30].
Figure 3 shows examples of the evolution of the in-
dex of coordination with the mean velocity vfor three
swimmers. Overall, it is observed that the swimmers
tend to increase their coordination index as they in-
crease their velocity. The maximum mean velocities
of the swimmers M1 and M2 are close to 1.5 m/s
but yet their coordination patterns are different, re-
spectively -0.5% and -5%. On the other hand, it can
be seen that the swimmers M3 and M1 are close to
the opposition mode (IdC = 0) for drastically differ-
ent velocities. At lower velocities, these three swim-
mers choose similar coordination patterns. This out-
lines the difficulty to compare swimmers technique
and to provide good advice for training and perfor-
mance in competition. Figure 4 shows the evolution
of the propulsion time for these three swimmers with
the velocity. Their propulsion time decreases as the
velocity increases. Their minimum propulsion time
ranges from 0.6 to 0.4 s. For M1 and M3, the propul-
sion time seems to plateau to a lower bound as the
velocity is increased as outlined by the vertical dashed
lines. Note that knowing the IdC and the propulsion
time tp, we can get the stroke rate.
0.6 0.8 1 1.2 1.4 1.6 1.8 2
30
20
10
0
10
v(m/s)
IdC (%)
M1
M2
M3
Figure 3: Arm coordination with the mean velocity for
3 swimmers. The dashed lines correspond to
the eq.7 with ev=v.
A second test consists of graded speed test on the
so-called MAD-system [33] and enables to Measure
the Active Drag of the swimmers. In this test, the
swimmers push off from fixed pads spaced 1.35 m and
0.8 m below the water surface with each stroke. The
system enables to estimate the drag force assuming
constant mean swimming velocity [34, 33, 30]. All
the swimmers were tested on 10 different speeds on
the MAD-system.
Figure 5 shows the obtain results for the 3 selected
swimmers. The drag force is fitted to Db=kMAD
bv2
to estimate the body drag coefficient. For the current
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Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.2
0.4
0.6
0.8
1
v(m/s)
tp(s)
M1
M2
M3
Figure 4: Propulsion time with the mean velocity for the
3 swimmers presented in figure 3. The verti-
cal dashed lines show the minimum propulsion
time achieved by the swimmers.
three swimmers, the value range from kMAD
b= 24.8
to 38.0 kg/m. The mean value on the 16 swimmers
was 30 kg/m.
0 0.511.5 2
0
50
100
150
v(m/s)
Db(N)
M1
M2
M3
Figure 5: Body drag estimated with the MAD-system
with the mean velocity for three swimmers of
the data set. The dashed lines correspond to
the fitted curve Db=kMAD
bv2.
To sum up, for all the 16 swimmers, we have col-
lected information on their swimming technique (arm
coordination, propulsion time) and also body char-
acteristics (body mass, height, body drag coefficient
during swimming). The goal is now to identify non-
dimensional numbers enabling a fair comparison of
these expert swimmers and provide a physical dis-
cussion to predict the optimal coordination. To this
end, we first propose a simple model of a swimmer.
Maximum force model
Writing Newton’s second law on the swimmer system
in the direction of the race, we get :
(m0+ma)dv
dt=TbDb,(3)
where m0is the mass of the swimmer, madenotes the
added masses due to the acceleration of the water2,
vis the instantaneous velocity, Tbthe total instanta-
neous thrust generated by the swimmer and Dbthe
body drag. Averaging on a stroke cycle and assuming
a periodic regime is reached, it comes:
0 = Tbkbv2,(4)
where we assumed Db=kbv2and the overline de-
notes the average on a cycle. For instance for a quan-
tity a:a= 1/T RT
0a(t) dt. We can further separate
the thrust of each arm and define Ti
a(t) as the instan-
taneous thrust of the arm iat t. In this simplified
symmetrical model, on a cycle, the two arms will pro-
duce the same mean thrust and thus we can define:
e
Ta=1
tpZtp
0
Ta(t) dt, (5)
where e
Tadenotes the mean thrust generated by one
arm during the propulsion. It is reasonable to assume
that this thrust can be controlled by the swimmer
and is bounded, e
Ta[0, T
a]. T
acorresponds to the
maximum thrust they can generate. Injecting this in
eq.4, we get:
0 = 2tp
Te
Takbv2.(6)
Using the fact that T= 2tp2tc, we get 2tp/T =
1 + 2IdC. We then find a relationship between the
coordination index and the velocity, which depends
2In all the applications ma= 0 as we did not evaluate it.
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Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
Figure 6: Arm coordination with the mean velocity for
3 swimmers of the data set. The dashed lines
correspond to the eq.7. The solid red line is
a guide for the eyes outlining the plateau the
swimmers seem to converge toward.
on the mean thrust generated by one arm during its
propulsive phase and the body drag:
v=ev(1 + 2IdC)1/2,(7)
where ev=qe
Ta/kb. This basic discussion enables to
define a characteristic velocity vwhich depends on
the swimmers mean maximum thrust, T
aand their
body drag coefficient kb. It also appears in this simple
model that to swim faster the swimmers can play on
their coordination once they maximize their thrust,
assuming constant body drag coefficient kb. We can
use this model to characterize the swimmer veloc-
ity. It can be assumed that at the maximum velocity
of the previous coordination test (see figure 3) the
expert swimmers used their maximum thrust to pro-
duce their highest speed. It comes:
v=vmax
1 + 2IdCmax
,(8)
where the index max denotes the test with maximum
velocity for the swimmer. The dashed lines in figure 3
correspond to the eq.7 with ev=vand is referred to
as the maximum force model. The swimmers follow
nicely the model of maximum force when their ve-
locity increases. It is observed that as they simulate
longer races (lower velocities) they tend to diverge
from this simple maximum force model and seem to
use less thrust. This is in agreement with the ob-
servations of Craig & Pendergast [8]. This method of
characterization of the velocity is applied to the three
swimmers presented previously. The results are dis-
played in figure 6. It appears that the swimmers use
similar coordination at similar non-dimensional ve-
locities. We observe two swimming strategies: one
following the red dashed line and corresponds to the
maximum force model and one with a rather constant
index of coordination (solid red line).
Non-dimensional velocity and coordination
We apply this analysis on the 16 expert swimmers
tested in 2007. In all the cases, we use their maxi-
mum velocity to define their characteristic velocity v
(see eq.8). We group the swimmers in pools of similar
v/vwith steps of 0.05 and averaged the data. Each
pool contains at least 4 points and 4 different swim-
mers. In average, there are 14 observations per pool
with 10 different swimmers. The results are displayed
in figure 7. This figure is one of the main results of
the present paper. The data are also provided in SI
appendix 1 for the separated swimmers.
In this data set, we observe that these expert swim-
mers follow nicely the maximum force model pre-
sented in the previous section for non-dimensional
velocity higher than 0.8. Below this value, the index
of coordination is almost constant and near a value
of -15% – -20%.
For sprint races, the expert swimmers do not have
to worry about their energy consumption and should
maximize their velocity. This is achieved by using
the maximum thrust and the highest reachable index
of coordination. On the contrary, for mid and long
distance races, the expert swimmers need to man-
age their energy consumption and adopt the index
of coordination that enables them to maximize the
distance of the race they can swim maintaining this
velocity. The race distance they can reach depends
on physiological data such as their maximum rate of
energy supplied by the oxygen and their anaerobic re-
serve [15, 16, 3]. In other words, an expert swimmer
should adopt the arm coordination that minimizes
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Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
Figure 7: Arm coordination with the mean non-
dimensional velocity for the 16 swimmers of
the data set. The dashed line corresponds to
the maximum force model. The solid lines
show the optimal coordination with τ0= 0.335
and = 0 and = 0.035 in black and gray, re-
spectively.
the energy cost at a given velocity for mid and long
distance races [10, 2, 19]. In the next section, we
propose a simple model to understand the observed
coordination at low velocities (hence simulating long
races) and predict the index of coordination plateau
for a given swimmer depending on their physical char-
acteristics.
Burst-and-coast in catch-up mode
Physical model
Burst-and-coast swimming behavior is quite common
in nature. It consists of cyclic burst of swimming
movements followed by gliding phase in which the
body is not producing thrust. This surprising strat-
egy of propulsion is observed in fishes such as cod
and saithe and was shown to be actually energetically
cheaper than steady swimming at the same average
velocity [35]. Mathematical model helped understood
this non-intuitive behavior [36] where non-continuous
propulsion could be cheaper. In the present paper,
the model proposed by Weihs [36] and Videler and
Weihs [35] is adapted to human swimming with the
arms only.
The main assumption in the model is that the re-
sistance is not the same during active and passive
swimming. In the catch-up mode of coordination,
the swimmers alternate between phases with active
propulsion and phases of gliding. During active swim-
ming the water resistance will be assumed to have the
form Db=kbv2and the swimmer produces a thrust
Tb, which will be considered constant (in order to
keep the model simple). For the gliding phase, the
swimmers have one arm forward fully extended (sim-
ilarly to the figure 2-a). The drag should be reduced
and will be modeled by Db= (1)kbv2, where 0.
Note that if  < 0 then clearly, the swimmers should
never try to glide as their resistance is greater dur-
ing this phase (this could be the case for non expert
swimmers). The parameter denotes the gliding ef-
fectiveness of the swimmer and is part of swimming
technique. As the stroke is supposed periodic and
the two arms symmetrical, we limit the study to half
a stroke cycle. In other words, we focus on a single
arm. The swimmer velocity oscillates between two
extreme values denoted vand v+. The equations to
solve can be written as:
(m0+ma)dv
dt=Takbv2,0ttp,(9)
(m0+ma)dv
dt=(1 )kbv2, tptT/2,
(10)
with the boundary conditions:
v(0) = v(T/2) = v,(11)
v(tp) = v+.(12)
Similarly to the previous section, we will consider
that the swimmers can control their thrust and that
it is bounded Ta[0, T
a]. An extension to superpo-
sition is discussed in SI appendix 2.
To non-dimensionalize this set of equations, we de-
fine τ=t/τ,u=v/vand ϕ=Ta/T
a, where τ,v
are the characteristic time and velocity defined by:
τ=m0+ma
kbv,(13)
v=pT
a/kb.(14)
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Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
Note that vis the same as the one defined in the
previous simple model. Using these definitions, we
rewrite the eq.9–10 in the form:
du
dτ=ϕu2,0ττp,(15)
du
dτ=(1 )u2, τpτ≤ T /2,(16)
with boundary conditions:
u(0) = u(T/2) = u,(17)
u(τp) = u+.(18)
Figure 8 shows half a cycle with the different nota-
tions.
time
Distance
λpλc
τpτc
Push Glide
u
u+
T/2
Figure 8: Burst-and-coast model for swimmers intra-
cycle velocity variations and notations.
One of the main motivations behind this simple
model is that it can be solved analytically. It is rather
simple to show that:
u+=ϕtanh τpϕ+ tanh1u
ϕ,(19)
τc=1u+/u
(1 )u+
,(20)
λp= log s1u2
1u2
+!,(21)
λc=1
1log u+
u,(22)
where λp(λc) is the non-dimensional distance trav-
elled during the propulsive (gliding) phase (respec-
tively). It is important to outline that τcis the
non-dimensional coordination time and is negative in
catch-up mode. The mean velocity can be evaluated
as:
u=λp+λc
τpτc
.(23)
Then the objective of expert swimmers is to swim a
given distance in the minimum time. To achieve this
goal for long distance races, they have to manage
their energy consumption. At a given mean veloc-
ity, they should select the coordination that enables
them to minimize their propulsion cost. This will en-
able them to swim the longest distance at this mean
velocity. The energy consumed during catch-up mode
is the energy expended solely during the propulsion
phase:
Ecu =Zτp
0
ϕudτ,
=ϕλp.(24)
For this amount of energy, the swimmers travel a non-
dimensional distance λp+λc. To travel the same
distance in opposition mode (steady swimming), they
would use:
Eop,cu =ZT/2
0
ϕ udτ,
=ϕ(λp+λc),(25)
where ϕis the thrust required to swim at the constant
velocity uand using eq.15 it comes:
ϕ=u2.(26)
The economy can therefore be defined as:
Rcu =Ecu
Eop,cu
=ϕ
u2
λp
λp+λc
.(27)
Most of the development are similar to the one in the
paper of Videler and Weihs [35]. We further add the
assumption that the propulsion phase duration τpis
limited by the arms dynamic. Note that in super-
position mode, the economy is always larger than 1.
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Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
Therefore, for u < 1 the swimmer should at worst
prefer the opposition mode. Above u > 1, superpo-
sition is the only possible mode of coordination. An
extension of the present model to the superposition
mode of coordination is discussed in SI appendix 2.
Propulsion time assumption
In human swimming, the propulsion phase duration
corresponds to the time the hand needs to travel
from the fully extended forward position to the re-
lease from the water near the hips. To estimate this
time of propulsion, we will simplify the arm+hand to
a simple paddle, which travels twice the arm length
on a straight line at constant velocity (we neglect the
acceleration phases). The thrust generated by the
swimmer solely comes from this paddle. The propul-
sion time is then:
τp2λa
uh/b
,(28)
where λais the non-dimensional arm length and
uh/b, the non-dimensional hand velocity in the body
frame. As the hand travels much faster than the body
through the water, we neglect the contribution of the
body velocity. We then expect the propulsion time
to be of the order of:
τp2λa
uh/w
,(29)
where uh/w is the hand velocity with respect to the
water. This velocity depends on the resistance coef-
ficient of the hand αh=kh/kband the force used by
the swimmer to move it through the water3:
αhu2
h/w =ϕ, (30)
and therefore:
τpτ0
ϕ,(31)
where τ0is a a characteristic time of propulsion. It
depends on the arm length λaand the αhcoefficient:
τ02λaαh.(32)
3we discuss the impact of the arm speed on the achievable
force in SI appendix 3
The propulsion time increases with the arm length
(larger distance to travel) and the hand size (αhin-
creases). Note that it does not depend on the swim-
mer’s force. Using the dimensional parameters, we
get τ02Lakhkb/(m0+ma), with La= 0.6 m the
arm length, m0= 80 kg, kh13 kg/m the drag co-
efficient of the hand [20] and kb= 30 kg/m . This
gives τ00.30.
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
0
0.2
0.4
0.6
0.8
v/v
τp
M
τ0= 0.335 & = 0.035
τ0= 0.335 & = 0
Figure 9: Non-dimensional Propulsion time of expert
swimmers with the non-dimensional mean ve-
locity. The dashed red lines show the evalua-
tion of the τ0based on the observation. The
solid lines show the optimal propulsion time
with τ0= 0.335 and = 0 and = 0.035 in
black and gray, respectively.
To verify this assumption, we look at the propul-
sion time of our swimmers. We can evaluate the
propulsion time as:
τp=tp
τ.(33)
Figure 9 shows the obtained mean value of the
non-dimensional propulsion phases with the non-
dimensional velocity. We see that the propulsion time
plateaus to τ0= 0.31 ±0.03 when the points are on
the maximum force model in figure 7:
u=v
v= (1 + 2IdC)1/2.(34)
This corresponds to ϕ= 1 and = 0. It is in good
agreement with the previous estimations.
9
Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
Comparison to the data set
To summaries, the swimmers are characterized by
two measurable parameters: the propulsion time pa-
rameter τ0and the gliding effectiveness . Both pa-
rameters could be measured on swimmers. To min-
imize their propulsion cost, they can play on their
force used to produce the thrust ϕ[0,1]. We can
write this problem in the form:
min
ϕ, s.t. u=u0Rcu (ϕ).(35)
This problem can be solved and would yield the op-
timal coordination strategy for our model swimmer
for a given mean velocity u=u0.
To compare the present burst-and-coast model to
our data set, we used a Powell’s conjugate direction
method [25] with Golden-section search [17] on and
τ0to minimize the error on the model prediction on
the index of coordination and propulsion time. The
obtained best-fit parameters are = 0.035 and τ0=
0.335. Note that τ0is in the error bar of the previous
estimation. Figures 7 and 9 show the obtained best
fit compared to the data set. We further added the
optimal choice for the limit case = 0. For this
case, the model predicts the intuitive choice of the
opposition mode (IdC = 0) as the optimal choice of
coordination. Indeed, if there is no benefit to glide
(0), then the swimmer should not glide.
The model shows that there exists a single op-
timal coordination (here IdCc≈ −17%) for non-
dimensional velocity lower than uc= 0.81. The
swimmers would save 1% energy. It is indeed pos-
sible to be more efficient in catch-up mode at cer-
tain speed with this model, even-though this can be
counter-intuitive [12]. Above this critical velocity, the
swimmers use their maximum force and hence follow
the maximum force model presented before.
The present model also provides information on
the intra-cycle velocity variation (IVV) of the swim-
mers. With the best-fit parameters, our swimmers
have a relative velocity variation of 13% during the
IdC plateau and it decreases once the swimmers reach
the maximum force model. In this simplified model,
the relative velocity variation goes to zero as the IdC
goes to 0 due to constant force approximation dur-
ing the propulsion phase. Yet the reduction of the
relative velocity variation with the mean velocity is
consistent with the observations of Matsuda et al.
[21].
Linear approximation and two regimes
It is interesting to note that in eq.19–22, we can de-
fine X=u/ϕand write all the parameters as a
function of Xonly. Therefore, Ris a function of X
only.
10
10
15
15
20
20
25
25
30
30
35
35
40
0 0.05 0.1 0.15 0.2 0.25
0.2
0.3
0.4
0.5
0.6
τ0
40 30 20 10 0
IdCc(%)
0
Figure 10: Colormap of the optimal coordination in the
Burst-And-Coast model with and τ0.
For our expert swimmers, we already observed that
the maximum force model yields rather good pre-
dictions for the coordination and velocity (see red
dashed line in figure 7). We expect the coefficient
to be close to 0 for our swimmers. We also found
that τ0was reasonably small. To linearize the equa-
tions developed in the two previous subsections, we
will assume τ01 and 1. We further assume
that τ0. Keeping only the smallest order terms
in and τ0, it comes:
Rcu (X)1τ02
211 + X4
2X2+X21.
(36)
This approximated function has a single minimum in
10
Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
X[0,1]:
X0=1
(1 + 4/τ2
0)1/4,(37)
and its value is:
Rmin
cu = 1 +τ2
0
2q1+4/τ2
01.(38)
Injecting this expression in the evaluation of the
index of coordination and keeping only the first order
term, it comes that the optimal index of coordination
is:
IdCc≈ −1
2 11
p1+4/τ2
0!.(39)
This will be the optimal choice of coordination as
long as u < (1 + 2IdCc)1/2. Above the critical speed:
uc1
(1 + 4/τ2
0)1/4,(40)
the swimmers will then switch to the maximum force
regime. This is in good agreement with the observa-
tions of Craig & Pendergast [8]. Figure 10 shows a
colormap of the numerically found optimal index of
coordination with τ0and .
From eq.39, we observe that the optimal index
of coordination decreases (increases) when (τ0) in-
creases. It is not surprising to expect that the swim-
mer will tend to glide more when increases. Note
that if = 0, then the expression yields that the opti-
mal coordination is the opposition mode. The effect
of varying τ0is maybe less obvious. Increasing τ0
(keeping all the other parameters constant) can be
compared to swimming with paddles. This will in-
crease the parameter αhin eq.32. From the present
model, we then expect the swimmers to change their
coordination pattern toward the opposition mode
(IdC closer to zero) with paddles. This prediction
is observed by Sidney et al.[31].
In all the results, presented so far the legs were tied
and could not be used by the swimmers. We discuss
their effects in SI appendix 4. It is rather clear that
allowing the legs to kick will lead to an increase of the
swimmer characteristic velocity v. This will affect
the non-dimensional propulsion time τ0. If we sup-
pose that the legs do not affect the gliding effective-
ness , the coordination is then affected accordingly
by an increase of the IdCc. We predict an increase
of the IdCcfrom -17% to -12% with this assumption
and observe it on a group of similar level swimmers.
Conclusion and applications
In the current study, we tried to understand how the
arm coordination patterns of 16 expert front crawl
swimmers vary according to the active drag per-
forming two tests: an incremental coordination test,
where swimmers were requested to simulate their rac-
ing techniques at 8 different speeds and a drag mea-
surement test to measure their active drag using the
MAD-system. In both tests, the swimmer legs where
tied and they were equipped with a pull-buoy to avoid
that their legs sank. To compare the evolution of
the different swimmer coordination (IdC) with their
mean velocity (v), we defined a characteristic veloc-
ity v=vmax/1 + 2IdCmax using their maximum
velocity test. We observed that swimmers used simi-
lar coordination patterns for a given non-dimensional
velocity u=v/v. At low u, the swimmers seem to
select a constant negative index of coordination and
above a critical non-dimensional velocity of about
0.8 their coordination increases with their velocity.
To further understand these two regimes, we pro-
pose a physical model of burst-and-coast [36] adapted
to swimming in front-crawl. The main idea of this
model is that the swimmers experienced a reduced
drag while gliding with one arm extended forward.
We compare this drag to the active swimming drag
during the underwater stroke by defining a gliding
effectiveness coefficient  > 0. One additional key
assumption in our model is the arm propulsion time.
We proposed that this time depends on the force used
to propel the body through the water and is bounded
by a lower value τ0which depends on the swimmer
characteristics only. We then showed that this model
predicts that there exist two swimming regimes sim-
ilarly to the observations.
A ”low velocity” regime, where the swimmers se-
11
Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
lect a constant index of coordination and reduce their
propulsion force to minimize their propulsion cost. A
”high velocity” regime, where the swimmers increase
their index of coordination to push at maximum force
and to gain more speed. It is in this latter regime that
the transition from catch-up to superposition can oc-
cur.
The optimal index of coordination in the low ve-
locity regime is IdCc≈ −(1 1/p1+4/τ2
0)/2 and
the transition from the ”low velocity” to ”high ve-
locity” regime will occur at uc1/1+4/τ 2
01/4in
the limit τ01 and τ0. This transition can be
linked to swimming distances through an energetic
equation. This transition is similar to the one ob-
served by Craig & Pendergast [8] in between the 200
m and 400 m races (see SI appendix 5).
Using this model, it is possible to advice on indi-
vidual optimal arm coordination for each swimmer
based on his/her physical characteristics. To predict
their optimal coordination, we need to evaluate the
two parameters and τ0. It is possible to use a gliding
test to estimate the value of by varying the arm po-
sition of the swimmer at the surface. For the propul-
sion time parameter, one could evaluate the time to
perform a single arm pull with the maximum possi-
ble thrust on a 25m sprint with index of coordination
measurements. Then τ0=t0kbv/m, where kbwould
be the drag measured from the previous gliding test
with the arms along the body. These tests could be
also done on swimmers with disabilities and advice
them on individual optimal arm coordination based
on their physical characteristics and type of impair-
ment.
Acknowledgment
The authors would like to thank all the athletes that
participated in the tests. They are also grateful to
Leo Chabert, Benoit Bideau and Vincent Bacot for
their help at different stages of the project and the
useful discussions. Last but not least, the authors
would like to thank The Olympic Multimedia Library
for granting us access to the footage of the race and
allowing us to use the images to illustrate our work.
References
[1] M. Alberty, M. Sidney, P. Pelayo, and H. Tous-
saint. Stroking characteristics during time to ex-
haustion tests. Medicine and Science in Sports
Exercise, 41(3):637, 2009.
[2] T. M. Barbosa, J. A. Bragada, V. M. Reis, D. A.
Marinho, C. Carvalho, and A. J. Silva. Ener-
getics and biomechanics as determining factors
of swimming performance: Updating the state
of the art. Journal of Science and Medicine in
Sport, 13(2):262 –269, 2010.
[3] H. Behncke and B. Brosowski. Optimization
models for the force and energy in competitive
sports. Mathematical Methods in the Applied
Sciences, 9(1):298–311, 1987.
[4] J.-C. Chatard and B. Wilson. Drafting distance
in swimming. Medicine and Science in Sports
and Exercise, 35(7):1176–1181, 2003.
[5] D. Chollet, S. Chalies, and J. Chatard. A new
index of coordination for the crawl: description
and usefulness. International Journal of Sports
Medicine, 21(01):54–59, 2000.
[6] D. Costill, J. Kovaleski, D. Porter, J. Kirwan,
R. Fielding, and D. King. Energy expenditure
during front crawl swimming: predicting success
in middle-distance events. International Journal
of Sports Medicine, 6(05):266–270, 1985.
[7] D. L. Costill, E. W. Maglischo, and A. B.
Richardson. Swimming. 1992.
[8] A. B. Craig and D. R. Pendergast. Relationships
of stroke rate, distance per stroke, and velocity
in competitive swimming. Medicine and Science
in Sports, 11(3):278–283, 1979.
[9] A. B. Craig, P. L. Skehan, J. A. Pawelczyk, and
W. L. Boomer. Velocity, stroke rate, and dis-
tance per stroke during elite swimming compe-
tition. Medicine and Science in Sports and Ex-
ercise, 17(6):625–634, 1985.
12
Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
[10] P. Di Prampero, D. Pendergast, D. Wilson, and
D. Rennie. Energetics of swimming in man.
Journal of Applied Physiology, 37(1):1–5, 1974.
[11] K. A. Ericsson and A. C. Lehmann. Expert and
exceptional performance: Evidence of maximal
adaptation to task constraints. Annual Review
of Psychology, 47(1):273–305, 1996.
[12] R. Havriluk. Do expert swimmers have expert
technique? comment on arm coordination and
performance level in the 400-m front crawl by
Schnitzler, Seifert, and Chollet (2011). Research
Quarterly for Exercise and Sport, 83(2):359–362,
2012.
[13] P. V. Karpovich. Water resistance in swimming.
Research Quarterly. American Physical Educa-
tion Association, 4(3):21–28, 1933.
[14] J. Katzenstein and R. Dubois&-; Reymond.
Arch. anat. u. phys. 1905.
[15] J. B. Keller. A theory of competitive running.
Physics today, 26(9):43, 1973.
[16] J. B. Keller. Optimal velocity in a race. The
American Mathematical Monthly, 81(5):474–
480, 1974.
[17] J. Kiefer. Sequential minimax search for a max-
imum. Proceedings of the American Mathemati-
cal Society, 4(3):502–506, 1953.
[18] S. Kolmogorov and O. Duplishcheva. Active
drag, useful mechanical power output and hy-
drodynamic force coefficient in different swim-
ming strokes at maximal velocity. Journal of
Biomechanics, 25(3):311–318, 1992.
[19] H. Leblanc, L. Seifert, C. Tourny-Chollet, and
D. Chollet. Intra-cyclic distance per stroke
phase, velocity fluctuations and acceleration
time ratio of a breaststroker’s hip: a comparison
between elite and nonelite swimmers at differ-
ent race paces. International Journal of Sports
Medicine, 28(02):140–147, 2007.
[20] R. B. Martin, R. A. Yeater, and M. K. White.
A simple analytical model for the crawl stroke.
Journal of Biomechanics, 14(8):539–548, 1981.
[21] Y. Matsuda, Y. Yamada, Y. Ikuta, T. Nomura,
and S. Oda. Intracyclic velocity variation and
arm coordination for different skilled swimmers
in the front crawl. Journal of Human Kinetics,
44(1):67–74, 2014.
[22] J. W. McVicar. A brief history of the develop-
ment of swimming. Research Quarterly. Ameri-
can Physical Education Association, 7(1):56–67,
1936.
[23] G. Millet, D. Chollet, S. Chalies, and J. Chatard.
Coordination in front crawl in elite triathletes
and elite swimmers. International Journal of
Sports Medicine, 23(02):99–104, 2002.
[24] K. Narita, M. Nakashima, and H. Takagi. Devel-
oping a methodology for estimating the drag in
front-crawl swimming at various velocities. Jour-
nal of Biomechanics, 54:123–128, 2017.
[25] M. J. Powell. An efficient method for finding
the minimum of a function of several variables
without calculating derivatives. The Computer
Journal, 7(2):155–162, 1964.
[26] R. Scurati, G. Gatta, G. Michielon, and
M. Cortesi. Techniques and considerations for
monitoring swimmers’ passive drag. Journal of
Sports Sciences, 37(10):1168–1180, 2019. PMID:
30449240.
[27] L. Seifert, D. Chollet, and J. C. Chatard.
Kinematic changes during a 100-m front
crawl: effects of performance level and gender.
Medicine and Science in Sports and Exercise,
39(10):1784–1793, 2007.
[28] L. Seifert, D. Chollet, and A. Rouard. Swim-
ming constraints and arm coordination. Human
Movement Science, 26(1):68–86, 2007.
[29] L. Seifert, D. Chollet, and A. Rouard. Swim-
ming constraints and arm coordination. Human
Movement Science, 26(1):68–86, 2007.
13
Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
[30] L. Seifert, H. Toussaint, M. Alberty, C. Schnit-
zler, and D. Chollet. Arm coordination, power,
and swim efficiency in national and regional
front crawl swimmers. Human Movement Sci-
ence, 29(3):426–439, 2010.
[31] M. Sidney, S. Paillette, J. Hespel, D. Chollet,
and P. Pelayo. Effect of swim paddles on the
intra-cyclic velocity variations and on the arm
coordination of front crawl stroke. In ISBS-
Conference Proceedings Archive, volume 1, 2001.
[32] A. J. Silva, A. Rouboa, A. Moreira, V. M. Reis,
F. Alves, J. P. Vilas-Boas, and D. A. Marinho.
Analysis of drafting effects in swimming using
computational fluid dynamics. Journal of Sports
Science & Medicine, 7(1):60–66, 2008.
[33] H. Toussaint, G. De Groot, H. Savelberg, K. Ver-
voorn, A. Hollander, and G. van Ingen Schenau.
Active drag related to velocity in male and
female swimmers. Journal of Biomechanics,
21(5):435–438, 1988.
[34] M. Truijens and H. Toussaint. Biomechanical as-
pects of peak performance in human swimming.
Animal Biology, 55(1):17–40, 2005.
[35] J. Videler and D. Weihs. Energetic advantages of
burst-and-coast swimming of fish at high speeds.
Journal of Experimental Biology, 97(1):169–178,
1982.
[36] D. Weihs. Energetic advantages of burst swim-
ming of fish. Journal of Theoretical Biology,
48(1):215–229, 1974.
[37] J. Westerweel, K. Aslan, P. Pennings, and
B. Yilmaz. Advantage of a lead swimmer in
drafting, Oct 2016.
Supplementary Information
i. Index of coordination of the 16 swimmers
We provide here the raw data for all the swimmers for
the index of coordination with dimensional velocity
and after applying the evaluation of the characteristic
speed v=vmax/1 + 2IdCmax, where vmax denotes
the maximum velocity achieved by the swimmer. The
results are displayed in figure 11.
ii. Extension to superposition mode
The burst-and-coast model can be extended to the
superposition mode. In this case, we have the follow-
ing equations:
du
dτ= 2ϕu2,0ττc,(41)
du
dτ=ϕu2, τcτ≤ T /2,(42)
with boundary conditions:
u(0) = u(T/2) = u,(43)
u(τc) = u+,(44)
where u=v/vis the non-dimensional velocity, τ=
t/τthe non-dimensional time, ϕ=Ta/T
athe non-
dimensional thrust and:
τ=m0+ma
kbv,(45)
v=pT
a/kb.(46)
Note that vis the same as the one defined in the
previous appendix.
Here, the propulsion time does not appear directly
in the equations. We still have the relation:
T/2 = τpτc.(47)
Figure 12 summarizes the notations. This system can
also be solved analytically:
u+=p2ϕtanh τcp2ϕ+ tanh1u
2ϕ,
(48)
τp2τc=1
ϕcoth1u
ϕcoth1u+
ϕ,
(49)
λc= log s1u2
/2ϕ
1u2
+/2ϕ!,(50)
λp2λc= log su2
+1
u2
1!.(51)
14
Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
0.6 0.8 1 1.2 1.4 1.6 1.8 2
30
20
10
0
10
v(m/s)
IdC
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
30
20
10
0
10
v/v
IdC
τ0= 0.335 & = 0.035
τ0= 0.335 & = 0
Figure 11: Arm coordination with the velocity for the 16 swimmers separately. Each symbol represents a single
swimmer. The dashed correspond to the maximum force model while the solid lines show the result of
the model presented in the paper for the parameters listed.
time
Distance
λcλp2λc
τcτp2τc
Double Push Single Push
u
u+
T/2
Figure 12: Burst-and-coast model for swimmers intra-
cycle velocity variations and notations ex-
tension to the superposition mode.
The energy consumed during the superposition mode
is:
Esu =Zτc
0
2ϕudτ+Zτp
τc
ϕudτ(52)
=ϕλp.(53)
For this amount of energy, the swimmers travel a non-
dimensional distance λpλconly. It should be com-
pared to the energy used to travel the same distance
in opposition mode (steady swimming):
Eop,su =ZT/2
0
ϕ udτ(54)
=u2(λpλc).(55)
The economy can therefore be defined as:
Rsu =Esu
Eop,su
=ϕ
u2
λp
λpλc
.(56)
We still assume that τp=τ0/ϕas each arms have
their own control.
Defining X=u/ϕand X+=u+/ϕ, we
have:
1< X< X+<2.(57)
Then, assuming X+=X+ ∆X, it is possible to
show that in the limit τ01 and ∆X1:
Rsu 1 + "(X1) (X+ 1) X22
4X3#2
τ2
0>1.
(58)
15
Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
Therefore, as expected intuitively, this mode is al-
ways more expensive energetically than the opposi-
tion. This mode should be use solely for sprinting and
at maximum force (ϕ= 1). It is interesting to note
that Rsu is maximum for X1.17. It seems that
this could be defined as a critical speed the swimmers
cannot exceed. In all our observations, u < 1.15.
iii. Hill’s heuristic law and propulsion time
It is well known that the force decreases with the
velocity (Hill’s heuristic law). In this appendix, we
discuss briefly this problem in our discussion of the
propulsion time. We assume that the athletes can
control their force of propulsion and that they can
chose a bounded thrust Ta[0, T
a]. This thrust
comes from the arms (in the present discussion the
legs are tied), which are controlled by the swimmers
muscles. Then the force used to activate the arms
should be bounded by a parameter, which follows
a Hill’s law, and dynamic equation of the hand be-
comes:
αhu2
h/w = min ϕ, Φ1uh/w
u
h,(59)
where ϕ[0,1] is the control on the force, Φand
u
hdenote the maximum force that can be used to
activate the arm and the maximum velocity at which
this arm can be moved without resistance, respec-
tively. Note that if eq.59 is bounded by the Hill’s
law then Φ1uh/w,max/u
h= 1 at the maximum
hand velocity in the water uh/w,max, since we non-
dimensionalize the problem here. And then we go
back to what was done earlier.
Further note that u
hcan also provide information
on the air recovery time and thus the maximum in-
dex of coordination the swimmer can achieve IdCmax.
Let’s assume for now that the swimmer is pushing
with the maximum thrust at τ0and wants to swim
at the maximum index of coordination. Then they
have to bring the arm back to the front as fast as
they can. They will reach u
hin the air recovery
phase.
iv. Impact of the legs kicking on the coordi-
nation
In all the paper the legs were tied. In previous works
done by Chollet et al. [5] and Seifert et al.[29], the
swimmers were allowed to used their legs. The ve-
locities the swimmers could reach were larger by 20%
compared to the present one for similar level swim-
mers. We look in this appendix at other data taken
on similar level swimmers with the legs free to kick.
We did not measure the resistance coefficient for these
swimmers with the MAD-system. They only per-
formed the coordination test.
Figure 13: Arm coordination with the mean non-
dimensional velocity for swimmers with and
without legs. The dashed line corresponds to
the maximum force model. The solid lines
show the optimal coordination with τ0=
0.335 and τ0,= 0.401 in gray and blue, re-
spectively. In both cases, we used = 0.035.
We applied the same analysis as done in the present
paper to the data of velocity and coordination. The
results are displayed in figure 13 and compared to
the swimmers with no legs of the present paper. We
observe a similar trend. The swimmers select a con-
stant index of coordination at low velocity and then
follow the maximum force model. The characteristic
velocity vincreased from 1.5 m/s without leg to 1.8
m/s with legs in average.
At low velocity, the swimmers usually perform one
kick per arm stroke and at higher velocity up to three
[23]. We will assume it is linked to the force choice
16
Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
of the arms and also in phase. Then the effect of the
legs can be whether a reduction of the effective drag
kbor an increase of the propulsion force Ta. In all
cases, it will lead to an increase of the characteristic
velocity vand the coordination is certainly affected
by this.
In this appendix, we will assume it is only the
thrust that is increased and therefore the drag is
of the same magnitude (kb= 30 kg/m). Then if
the physical propulsion time is the same, the non-
dimensional one should be rescaled τ0,=τ0v
/v,
where the indicates the quantity with the legs free
to kick. Using for τ0without leg the value found pre-
viously, we get τ0,= 0.401 with legs. Our model
then will predict for a similar an increase of the
index of coordination from -17% to -12%. This is in
good agreement with the observations.
v. Coordination and swimming distances
In the paper we describe the intra-cycle variations dy-
namic. We optimized the choice of coordination such
that the swimmers minimize their energy consump-
tion. To have an order of magnitude of the distance
reached by the swimmers with this technique, an en-
ergy equation is necessary:
dE
dt =σ1
ηTa(t)v(t),(60)
where Edenotes the energy reserves of the swimmer,
σis the maximum rate at which the oxygen is sup-
plied to the muscles (it is equivalent to the VO2max)
and ηa conversion efficiency of the chemical energy
to propulsive energy (which we will assume to be a
constant).
We integrates this equation on the total duration
of the race tr=Lr/v:
ErE0σtr1
η
tr
TZT
0
Ta(t)v(t) dt, (61)
where E0is the anaerobic reserve at the beginning of
the race and Er0 the left energy at the end of the
race. The approximation comes from the number of
cycle the swimmer used tr/T on the second term
of the right-hand-side and the fact that we neglect
the turns and the start for simplification. Behncke &
Brosowski [3] discuss a way to take these parts of the
race into account. To keep the discussion simple and
analytical, we consider the case of an infinite pool
in the present discussion. The second term in the
right hand side of eq.61 is the one we minimised in
the previous sections for a given mean velocity v=
uv. Using the opposition mode as a reference eq.61
becomes:
ErE0σtr1
ηtrR(u)kbv3,(62)
where Ris the economy at v=uvand will depend
on the coordination.
For short races (pure sprints, Er>0), the swim-
mer does not have to worry about the economy and
should select the highest possible velocity they can
achieve. A superposition mode is expected with the
highest possible index of coordination, as this is how
they can achieve the highest velocity. They can main-
tain this technique as long as Er>0. We define the
aerobic velocity:
vσ=ησ
kb1/3
,(63)
the characteristic distance:
L0=η1/3E0
σ2/3k1/3
b
.(64)
and:
β=vσ
v.(65)
It then comes that the sprint technique will last as
long as Lr< Ls, where:
Ls=L0
Rmax (umax)2β/umax
,(66)
where Rmax is the economy at the maximum index
of coordination and force, and umax the maximum
non-dimensional speed. At worst, it will corresponds
to the maximum for the superposition mode (see ap-
pendix ii) which occurs for umax 1.17. We will
use this extreme in the applications. Obviously the
17
Gait Transition in Swimming June 2019 Submitted to PNAS–Preprint
numerator should be positive in this expression. Oth-
erwise, it would mean that the swimmers could sprint
as long as they want. This will give an upper bound
to the possible values of η.
For longer races, Er= 0 and the swimmer should
minimize their energy consumption. They should
then choose the coordination that enables to min-
imize the economy at the targeted mean velocity.
Eq.62 becomes:
0E0σtr1
ηtrRop (u)kbv3,(67)
where Rop is the optimal economy at v=uv4.
The link between the length of the race and the
mean velocity is then:
L0
Lr
=Rop (u)u
β2
β
u.(68)
Eq.68 gives the relationship between the mean ve-
locity and the length of the race depending on the
swimmer characteristics (VO2max, anaerobic veloc-
ity, gliding efficiency and propulsion time). The tran-
sition from the long distance race strategy with con-
stant index of coordination and the shorter race with
maximum force occurs at the v=ucv. Injecting
the results of the paper for Rmin
cu and ucin eq.68, it
comes:
Lc=L0
(uc)2Rmin
cu β/uc
.(69)
This transition will occur only if the denominator in
eq.69 is positive. Otherwise, the swimmer will stay in
the maximum force model for all the distances. Note
that we do not take into account fatigue here.
To have an order of magnitude of these different
lengths, we consider typical values of the different pa-
rameters. We will consider a swimmer with the legs
free to kick (and therefore τ0= 0.4 and = 0.035,
see appendix iv for the origin of the value of τ0). We
consider an athlete of m0= 80 kg, with a typical
drag kb= 30 kg/m, a characteristic velocity v= 1.8
m/s, E0= 152 kJ, σ= 2.08 kJ/s and η= 0.04
4for a velocity below the critical velocity vc=ucv, it is
Rmin
cu discussed in the main text.
[16, 15, 3, 10]. Note that we choose ηto be constant.
di Pramparo et al. [10] evaluated the efficiency in
sub-maximal exercise to be in between 2.6 and 5.2%.
This value seems therefore reasonable. Note that to
be consistent, here the efficiency cannot be larger
than 13.5 % because then the Lswould not be de-
fined. The choice of the efficiency greatly influences
the results. With the present values, we get L0= 103
m, vσ= 1.40 m/s and β= 0.78. It then comes that
Ls= 64.7 m and Lc= 301 m. A pure sprint is lim-
ited to the 50 m race and the 100 m race swimmers
are expected to manage their energy on the length of
the race. The swimmer will switch from a maximum
force technique to a constant index of coordination
race in between the 200m and the 400m races. This
is consistent with the observations of Craig & Pen-
dergast [8] and previous observations of change in
coordination patterns with pace made by Chollet et
al. [5] and Seifert et al.[29].
18
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