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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-

1

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,n−1tR

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=Tb−Db,(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 = Tb−kbv2,(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

Ta−kbv2.(6)

Using the fact that T= 2tp−2tc, 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

0.4 0.6 0.811.2

−30

−20

−10

0

10

v/v∗

IdC (%)

M1

M2

M3

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 v∗which 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=v∗and 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/v∗with 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

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

−30

−20

−10

0

10

v/v∗

IdC (%)

M

τ0= 0.335 & = 0.035

τ0= 0.335 & = 0

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 v−and v+. The equations to

solve can be written as:

(m0+ma)dv

dt=Ta−kbv2,0≤t≤tp,(9)

(m0+ma)dv

dt=−(1 −)kbv2, tp≤t≤T/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/v∗and 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 v∗is 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=ϕ−u2,0≤τ≤τp,(15)

du

dτ=−(1 −)u2, τp≤τ≤ 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

τp−τc

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 τp√ϕ+ tanh−1u−

√ϕ,(19)

τc=1−u+/u−

(1 −)u+

,(20)

λp= log s1−u2

−/ϕ

1−u2

+/ϕ !,(21)

λc=1

1−log 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

τp−τc

.(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:

τp≈2λa

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:

τp≈2λa

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:

τp≈τ0

√ϕ,(31)

where τ0is a a characteristic time of propulsion. It

depends on the arm length λaand the αhcoeﬃcient:

τ0≈2λa√αh.(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 τ0≈2La√khkb/(m0+ma), with La= 0.6 m the

arm length, m0= 80 kg, kh≈13 kg/m the drag co-

eﬃcient of the hand [20] and kb= 30 kg/m . This

gives τ0≈0.30.

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)≈1−τ02

21−1 + X4

2X2+X2−1.

(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

0−1.(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 1−1

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:

uc≈1

(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 uc≈1/1+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.

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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= 2ϕ−u2,0≤τ≤τc,(41)

du

dτ=ϕ−u2, τc≤τ≤ T /2,(42)

with boundary conditions:

u(0) = u(T/2) = u−,(43)

u(τc) = u+,(44)

where u=v/v∗is the non-dimensional velocity, τ=

t/τ∗the non-dimensional time, ϕ=Ta/T ∗

athe non-

dimensional thrust and:

τ∗=m0+ma

kbv∗,(45)

v∗=pT∗

a/kb.(46)

Note that v∗is 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 = τp−τc.(47)

Figure 12 summarizes the notations. This system can

also be solved analytically:

u+=p2ϕtanh τcp2ϕ+ tanh−1u−

√2ϕ,

(48)

τp−2τc=1

√ϕcoth−1u−

√ϕ−coth−1u+

√ϕ,

(49)

λc= log s1−u2

−/2ϕ

1−u2

+/2ϕ!,(50)

λp−2λc= log su2

+/ϕ −1

u2

−/ϕ −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

λcλp−2λc

τcτp−2τc

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 λp−λconly. 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(λp−λc).(55)

The economy can therefore be deﬁned as:

Rsu =Esu

Eop,su

=ϕ

u2

λp

λp−λc

.(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 + "(X−−1) (X−+ 1) X−2−2

4X−3#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 X−≈1.17. 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, Φ∗1−uh/w

u∗

h,(59)

where ϕ∈[0,1] is the control on the force, Φ∗and

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 Φ∗1−uh/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.

0.4 0.6 0.811.2

−30

−20

−10

0

10

v/v∗

IdC (%)

No leg τ0= 0.335

With legs τ0= 0.401

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 v∗increased 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 v∗and 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 =σ−1

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:

Er−E0≈σtr−1

η

tr

TZT

0

Ta(t)v(t) dt, (61)

where E0is the anaerobic reserve at the beginning of

the race and Er≥0 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:

Er−E0≈σtr−1

ηtrR(u)kbv3,(62)

where Ris the economy at v=uv∗and 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/β)2−β/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:

0−E0≈σtr−1

ηtrRop (u)kbv3,(67)

where Rop is the optimal economy at v=uv∗4.

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/β)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