Gait Transition in Swimming

Gait transition in swimming

Remi Carmigniani∗ Ecole des Ponts ParisTech [email protected] Ludovic Seifert & Didier Chollet CETAPS EA3832, Faculty of Sports Sciences, University of Rouen Normandy Christophe Clanet LadhyX, Ecole Polytechnique June 18, 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 efficient swimming technique as it is the only one used the artwork of ancien Egypt over 2,000 BC, mod- for long distances (over 200 m) and the fastest one ern competitive swimming started in the early 19th- (used in freestyle sprint)[2]. It is characterized by century England [22]. The search for speed in swim- alternated arm propulsion phases and arm recovery ming led to changes of the technique from the natu- out of the water. ral quadrupeds dog fashion technique to the breast- The skill to swim fast is a combination between stroke, then side-stroke and Trudgen-stroke, all the generating high thrust and minimizing drag due to way to the modern front-crawl. The front-crawl was aquatic resistance on the body. The first study in- pioneered in competition by the Australian Richard vestigating drag during human locomotion in water Cavill at the beginning of the 20th century. He was can be traced back all the way to the early beginning largely inspired by natives surfers from the Solomon of the 20th century [14]. Karpovich [13] pioneered Islands [22]. The technique was refined over time as the quantification of human body drag using a tow- arXiv:1906.06518v1 [physics.bio-ph] 15 Jun 2019 the average speed of swimmers has continued to in- ing protocol (called passive drag, Dp,b). He found crease over the century (see figure 1). Front-crawl is that the passive drag when the swimmer is fully ex- now used on a large range of distances in swimming tended in a so called streamline position near the 2 pool and open-water races. It appears to be the most surface was about Dp,b = kp,bv , where v denotes the towing velocity and k 31 (24) kg/m for men p,b ≈ ∗Corresponding author (women, respectively). Then numerous research ex-

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till et al. [6] emphasize the importance of stroke tech- 2.2 nique on the performance and defined a stroke index 2.0 SI = vLs to evaluate the swimming economy. 1.8 Focusing furthermore on the swimming technique, Chollet et al. [5] investigated the arm stroke phase 1.6 organization during a stroke cycle and defined the 1.4 index of coordination (IdC). This non-dimensional Speed (m/s) 1.2 number characterizes the temporal motor organization of propulsion phases. The two main patterns 1.0 Men Women of coordination can be simplified to the sketches of 0.8 1900 1950 2000 figure 2. The solid lines represent a simplified hand Year elevation compared to the mean water level (dotted lines)1. When the solid curve is below this level, a Figure 1: Evolution of the mean velocity over time of the propulsive phase occurs. This is further outlined by 100 m long course freestyle. The circles de- the gray blocks at the bottom. The arms are iden- note the world records evolution. The squares tified by the index i L, R , for left and right, denote the year best performance from 2001 respectively. This index∈ { enables} to track the suc- to today. cessions of propulsive phases. As an example, we consider the nth cycle of the right arm. It begins at R tstart,n and ends when this arm starts its next propul- amining passive drag have emerged as shown in the R sive phase tstart,n+1. The cycles repeat periodically review of Scurati et al. [26]. The mean drag experi- with a period T = tR tR . The propulsion enced during swimming is still not fully understood start,n − start,n phase of one arm lasts t = tR tR and the and continue being investigated[10, 18, 33, 24]. A p end,n − start,n non-propulsive phase t = tR tR . The simple way to reduce the drag is to swim in the wake np start,n+1 end,n coordination time is then defined by: − of another swimmer [37, 32, 4]. This is called drafting and the effects of drafting on the swimmer technique t = tR tL , (1) and race strategy are still to explore. c end,n − start,n The swimming performance is solely evaluated on and the index of coordination corresponds to the non- the time to reach a certain distance. To under- dimensional time of coordination compared to the cy- stand the link between the achieved performance and cle period: the swimming technique, researchers have first fo- cused their attention on the arm stroke frequency IdC = tc/T. (2) (also called stroke rate) fR, and the mean velocity of the swimmer v [9, 8]. To link these two quanti- In the case of figure 2-a), catch-up mode, the index ties, they defined the distance per stroke (or stroke of coordination is negative as the propulsive phase of length) Ls = v/fR. Craig & Pendergast [8] collected the latter arm starts after the end of the propulsive data on expert swimmers asking them to swim at a phase of the former. This technique is exhibited by given velocity using the minimum stroke frequency long distance swimmers who used glide within the they could achieve. They observed that swimmers cycle. During this glide they adopt a streamline arm did not use this minimum stroke rate technique for position as illustrated in the picture of figure 2-a). On long distance races (over 200 m). They commented the contrary, in figure 2-b), superposition mode, the that even though these swimmers could achieve the index of coordination is positive. There is a time 2 tc | | same velocity with a lower frequency (and hence a 1Note that when the solid curve overlaps the dotted one, longer stroke length), they used a higher frequency this intends to mean that the hand has entered the water but and a lower force per stroke to reduce fatigue. Cos- is not yet active in the propulsion.

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catch-up mode superposition mode T T

arm R arm R

arm L arm L

tp tnp tp tnp

a) b)

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. present the field observations of expert swimmers co- A third pattern of coordination can be defined at the ordination and discussed a simple way to compare transition between the former two and is referred to the swimmers among them. The swimmers used only opposition in the literature. It corresponds to the their arms to generate thrust. As previously noticed, case where one arm starts its propulsion phase when for low velocity, hence long distance races pace, the the other finishes. There is no time lag between the swimmers prefer a catch-up mode of swimming. Sec- two propulsion phases (IdC = 0). These three dis- ond, we propose a physical model to understand this tinctive patterns of coordination were first described choice of coordination. The model is compared to by Costill et al. [7] and then quantified by Chollet et our field observations and a linearized expression is al. [5]. They observed the choice of coordination of derived. different level swimmers. Expert swimmers were able to reach higher swimming velocity thanks to higher Field investigation and first positive index of coordination than non-expert swimmers both on incremental tests [5, 28] and 100-m coordination model races [27]. The effect of fatigue on the coordination Raw observations was also investigated by Alberty et al. [1]. They observed a general increase of the index of coordination Following the work of Chollet et al. [5], we consider with fatigue. A physical model discussing the mo- the motor coordination of national level French swim- tor coordination is proposed in our current study to mers. To simplify the discussion, the motor coordi- understand the transition from catch-up to superpo- nation of the swimmers is averaged between the two sition mode and the optimal choice of coordination arms and the legs motions are ignored. That is to depending on the targeted velocity of swimming. say that the swimmers are considered symmetrical Our study is organized in two steps. First, we and only the arms coordination are discussed.

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In the current study, we consider the data col- spectively -0.5% and -5%. On the other hand, it can lected on 16 French male swimmers in 2007 for whom be seen that the swimmers M3 and M1 are close to the mean standard deviation (min, max) of age, the opposition mode (IdC = 0) for drastically differ- body mass,± height, arm span and arm length were: ent velocities. At lower velocities, these three swim- 21.2 4.4 (19, 31) years, 78.8 8.5 (66.3,90.5) kg, mers choose similar coordination patterns. This out- 1.84±0.03 (1.70, 1.93) m, 1.91±0.08 (1.70, 2.14) m, lines the difficulty to compare swimmers technique 0.65±0.05 (0.60, 0.75) m, respectively.± At the time and to provide good advice for training and perfor- of the± experiment, they were practicing a minimum mance in competition. Figure 4 shows the evolution of 10 hours a week and had been swimming com- of the propulsion time for these three swimmers with petitively for 12.1 3.5 years, confirming their expert the velocity. Their propulsion time decreases as the level [11]. An extra± swimmer was also tested. He was velocity increases. Their minimum propulsion time highly specialised in sprint race (50 m race). He had ranges from 0.6 to 0.4 s. For M1 and M3, the propul- performed similar coordination test in the past and sion time seems to plateau to a lower bound as the showed behaviour similar to the one described in the velocity is increased as outlined by the vertical dashed paper. For this test, he surprisingly performed dras- lines. Note that knowing the IdC and the propulsion tically differently than before. We decided to remove time tp, we can get the stroke rate. him from the data set due to this change of coordination, which was probably due to specific sprint 10 work. The personal best record of the 16 expert swim- 0 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- 10 peted at national level. They were all tested on two − graded speed tests in a randomized order using only IdC (%) the arms in front crawl. Their legs were tied and they 20 M1 − were equipped with a pull buoy for buoyancy. They M2 all volunteered for this study and gave their written M3 30 consent to participate. − 0.6 0.8 1 1.2 1.4 1.6 1.8 2 One test consists of simulated racing techniques v (m/s) where these expert swimmers were asked to swim at 8 different velocities corresponding to different race Figure 3: Arm coordination with the mean velocity for paces (from 3000 m to 50 m + maximal speed) on 3 swimmers. The dashed lines correspond to ∗ a single 25 m lap. During this test, the swimmers the eq.7 with ve = v . were video recorded by two synchronized underwater video cameras at 50 fps (Sony compact FCB-EX10L), A second test consists of graded speed test on the in order to get a front and side view, from which the so-called MAD-system [33] and enables to Measure different stroke phases and the arm coordination have the Active Drag of the swimmers. In this test, the been computed. The protocol is similar to the one swimmers push off from fixed pads spaced 1.35 m and described in [5, 29, 30]. 0.8 m below the water surface with each stroke. The Figure 3 shows examples of the evolution of the in- system enables to estimate the drag force assuming dex of coordination with the mean velocity v for three constant mean swimming velocity [34, 33, 30]. All swimmers. Overall, it is observed that the swimmers the swimmers were tested on 10 different speeds on tend to increase their coordination index as they in- the MAD-system. crease their velocity. The maximum mean velocities Figure 5 shows the obtain results for the 3 selected MAD 2 of the swimmers M1 and M2 are close to 1.5 m/s swimmers. The drag force is fitted to Db = kb v but yet their coordination patterns are different, re- to estimate the body drag coefficient. For the current

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1 dimensional numbers enabling a fair comparison of M1 these expert swimmers and provide a physical dis- M2 cussion to predict the optimal coordination. To this M3 0.8 end, we ﬁrst propose a simple model of a swimmer.

(s) 0.6 p

t Maximum force model Writing Newton’s second law on the swimmer system 0.4 in the direction of the race, we get :

dv 0.2 (m + m ) = T D , (3) 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 a dt b − b v (m/s) where m0 is the mass of the swimmer, ma denotes the 2 Figure 4: Propulsion time with the mean velocity for the added masses due to the acceleration of the water , 3 swimmers presented in figure 3. The verti- v is the instantaneous velocity, Tb the total instanta- cal dashed lines show the minimum propulsion neous thrust generated by the swimmer and Db the time achieved by the swimmers. body drag. Averaging on a stroke cycle and assuming a periodic regime is reached, it comes: three swimmers, the value range from kMAD = 24.8 2 b 0 = T b kbv , (4) to 38.0 kg/m. The mean value on the 16 swimmers − was 30 kg/m. 2 where we assumed Db = kbv and the overline denotes the average on a cycle. For instance for a quan- 150 R T tity a: a = 1/T 0 a (t) dt. We can further separate i the thrust of each arm and define Ta (t) as the instantaneous thrust of the arm i at t. In this simplified 100 symmetrical model, on a cycle, the two arms will produce the same mean thrust and thus we can define: (N) b

D 1 Z tp 50 Tea = Ta (t) dt, (5) M1 tp 0 M2 M3 where Tea denotes the mean thrust generated by one 0 0 0.5 1 1.5 2 arm during the propulsion. It is reasonable to assume v (m/s) that this thrust can be controlled by the swimmer and is bounded, Tea [0,T ∗]. T ∗ corresponds to the ∈ a a Figure 5: Body drag estimated with the MAD-system maximum thrust they can generate. Injecting this in with the mean velocity for three swimmers of eq.4, we get: the data set. The dashed lines correspond to MAD 2 the fitted curve Db = k v . 2tp 2 b 0 = Tea kbv . (6) T − To sum up, for all the 16 swimmers, we have col- Using the fact that T = 2t 2t , we get 2t /T = lected information on their swimming technique (arm p c p 1 + 2IdC. We then find a relationship− between the coordination, propulsion time) and also body char- coordination index and the velocity, which depends acteristics (body mass, height, body drag coefficient 2 during swimming). The goal is now to identify non- In all the applications ma = 0 as we did not evaluate it.

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10 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- 0 servations of Craig & Pendergast [8]. This method of characterization of the velocity is applied to the three 10 − swimmers presented previously. The results are dis- IdC (%) played in ﬁgure 6. It appears that the swimmers use

20 M1 similar coordination at similar non-dimensional ve- − M2 locities. We observe two swimming strategies: one M3 following the red dashed line and corresponds to the 30 − 0.4 0.6 0.8 1 1.2 maximum force model and one with a rather constant v/v∗ index of coordination (solid red line).

Figure 6: Arm coordination with the mean velocity for Non-dimensional velocity and coordination 3 swimmers of the data set. The dashed lines correspond to the eq.7. The solid red line is We apply this analysis on the 16 expert swimmers a guide for the eyes outlining the plateau the tested in 2007. In all the cases, we use their maxi- swimmers seem to converge toward. 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 on the mean thrust generated by one arm during its ∗ pool contains at least 4 points and 4 different swim- propulsive phase and the body drag: mers. In average, there are 14 observations per pool 1/2 with 10 different swimmers. The results are displayed v = v (1 + 2IdC) , (7) e in figure 7. This figure is one of the main results of q the present paper. The data are also provided in SI where ve = Tea/kb. This basic discussion enables to appendix 1 for the separated swimmers. define a characteristic velocity v∗ which depends on In this data set, we observe that these expert swim- the swimmers mean maximum thrust, Ta∗ and their mers follow nicely the maximum force model pre- body drag coefficient kb. It also appears in this simple sented in the previous section for non-dimensional model that to swim faster the swimmers can play on velocity higher than 0.8. Below this value, the index their coordination once they maximize their thrust, of coordination is almost constant and near a value assuming constant body drag coefficient kb. We can of -15% – -20%. use this model to characterize the swimmer veloc- For sprint races, the expert swimmers do not have ity. It can be assumed that at the maximum velocity to worry about their energy consumption and should of the previous coordination test (see figure 3) the maximize their velocity. This is achieved by using expert swimmers used their maximum thrust to pro- the maximum thrust and the highest reachable index duce their highest speed. It comes: of coordination. On the contrary, for mid and long

vmax distance races, the expert swimmers need to man- v∗ = , (8) age their energy consumption and adopt the index √1 + 2IdCmax of coordination that enables them to maximize the where the index max denotes the test with maximum distance of the race they can swim maintaining this velocity for the swimmer. The dashed lines in ﬁgure 3 velocity. The race distance they can reach depends correspond to the eq.7 with ve = v∗ and is referred to on physiological data such as their maximum rate of as the maximum force model. The swimmers follow energy supplied by the oxygen and their anaerobic re- nicely the model of maximum force when their ve- serve [15, 16, 3]. In other words, an expert swimmer locity increases. It is observed that as they simulate should adopt the arm coordination that minimizes

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The main assumption in the model is that the re- M 10 τ0 = 0.335 & = 0.035 sistance is not the same during active and passive τ0 = 0.335 & = 0 swimming. In the catch-up mode of coordination, 0 the swimmers alternate between phases with active propulsion and phases of gliding. During active swim- 10 ming the water resistance will be assumed to have the IdC (%) − 2 form Db = kbv and the swimmer produces a thrust 20 Tb, which will be considered constant (in order to − keep the model simple). For the gliding phase, the 30 swimmers have one arm forward fully extended (sim- − 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 ilarly to the ﬁgure 2-a). The drag should be reduced v/v∗ 2 and will be modeled by Db = (1 )kbv , where 0. Note that if < 0 then clearly,− the swimmers should≥ Figure 7: Arm coordination with the mean non- never try to glide as their resistance is greater dur- dimensional velocity for the 16 swimmers of ing this phase (this could be the case for non expert the data set. The dashed line corresponds to swimmers). The parameter denotes the gliding ef- the maximum force model. The solid lines fectiveness of the swimmer and is part of swimming show the optimal coordination with τ0 = 0.335 and = 0 and = 0.035 in black and gray, re- technique. As the stroke is supposed periodic and spectively. 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 the energy cost at a given velocity for mid and long extreme values denoted v and v+. The equations to distance races [10, 2, 19]. In the next section, we solve can be written as: − propose a simple model to understand the observed dv 2 coordination at low velocities (hence simulating long (m0 + ma) = Ta kbv , 0 t tp, (9) races) and predict the index of coordination plateau dt − ≤ ≤ dv 2 for a given swimmer depending on their physical char- (m0 + ma) = (1 )kbv , tp t T/2, acteristics. dt − − ≤ ≤ (10)

Burst-and-coast in catch-up mode with the boundary conditions:

Physical model v(0) = v(T/2) = v , (11) − Burst-and-coast swimming behavior is quite common v(tp) = v+. (12) in nature. It consists of cyclic burst of swimming Similarly to the previous section, we will consider movements followed by gliding phase in which the that the swimmers can control their thrust and that body is not producing thrust. This surprising strat- it is bounded T [0,T ]. An extension to superpo- egy of propulsion is observed in fishes such as cod a a∗ sition is discussed∈ in SI appendix 2. and saithe and was shown to be actually energetically cheaper than steady swimming at the same average To non-dimensionalize this set of equations, we de- velocity [35]. Mathematical model helped understood fine τ = t/τ ∗, u = v/v∗ and ϕ = Ta/Ta∗, where τ ∗, v∗ this non-intuitive behavior [36] where non-continuous are the characteristic time and velocity defined by: propulsion could be cheaper. In the present paper, m0 + ma the model proposed by Weihs [36] and Videler and τ ∗ = , (13) kbv∗ Weihs [35] is adapted to human swimming with the p arms only. v∗ = Ta∗/kb. (14)

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Note that v∗ is the same as the one defined in the where λp (λc) is the non-dimensional distance trav- previous simple model. Using these definitions, we elled during the propulsive (gliding) phase (respec- rewrite the eq.9–10 in the form: tively). It is important to outline that τc is the non-dimensional coordination time and is negative in du 2 = ϕ u , 0 τ τp, (15) catch-up mode. The mean velocity can be evaluated dτ − ≤ ≤ as: du 2 = (1 )u , τp τ /2, (16) dτ λp + λc − − ≤ ≤ T u = . (23) with boundary conditions: τp τc − u(0) = u( /2) = u , (17) Then the objective of expert swimmers is to swim a T − given distance in the minimum time. To achieve this u(τp) = u+. (18) goal for long distance races, they have to manage Figure 8 shows half a cycle with the different nota- their energy consumption. At a given mean veloc- tions. ity, they should select the coordination that enables /2 them to minimize their propulsion cost. This will en- T able them to swim the longest distance at this mean velocity. The energy consumed during catch-up mode u+ is the energy expended solely during the propulsion phase:

Z τ p cu = ϕudτ, E 0 u − = ϕλ . (24) Push Glide p time For this amount of energy, the swimmers travel a non- dimensional distance λp + λc. To travel the same τp τc − Distance distance in opposition mode (steady swimming), they would use: λp λc /2 Z T op,cu = ϕ udτ, Figure 8: Burst-and-coast model for swimmers intra- E 0 cycle velocity variations and notations. = ϕ (λp + λc) , (25)

One of the main motivations behind this simple where ϕ is the thrust required to swim at the constant model is that it can be solved analytically. It is rather velocity u and using eq.15 it comes: simple to show that: ϕ = u2. (26) 1 u u+ = √ϕ tanh τp√ϕ + tanh− − , (19) The economy can therefore be deﬁned as: √ϕ

1 u+/u cu ϕ λp τc = − − , (20) cu = E = 2 . (27) (1 )u R op,cu u λp + λc + E −s ! 1 u2 /ϕ Most of the development are similar to the one in the λ = log − − , (21) p 1 u2 /ϕ paper of Videler and Weihs [35]. We further add the − + assumption that the propulsion phase duration τp is 1 u+ λc = log , (22) limited by the arms dynamic. Note that in super- 1 u position mode, the economy is always larger than 1. − −

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Therefore, for u < 1 the swimmer should at worst The propulsion time increases with the arm length prefer the opposition mode. Above u > 1, superpo- (larger distance to travel) and the hand size (αh in- sition is the only possible mode of coordination. An creases). Note that it does not depend on the swim- extension of the present model to the superposition mer’s force. Using the dimensional parameters, we mode of coordination is discussed in SI appendix 2. get τ0 2La√khkb/(m0 + ma), with La = 0.6 m the arm length,≈ m = 80 kg, k 13 kg/m the drag co- 0 h ≈ Propulsion time assumption eﬃcient of the hand [20] and kb = 30 kg/m . This gives τ0 0.30. In human swimming, the propulsion phase duration ≈ corresponds to the time the hand needs to travel 0.8 from the fully extended forward position to the re- M lease from the water near the hips. To estimate this τ0 = 0.335 & = 0.035 τ = 0.335 & = 0 time of propulsion, we will simplify the arm+hand to 0.6 0 a simple paddle, which travels twice the arm length on a straight line at constant velocity (we neglect the p τ 0.4 acceleration phases). The thrust generated by the swimmer solely comes from this paddle. The propulsion time is then: 0.2

2λa τp , (28) 0 ≈ uh/b 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 v/v∗ where λa is the non-dimensional arm length and u , the non-dimensional hand velocity in the body h/b Figure 9: Non-dimensional Propulsion time of expert frame. As the hand travels much faster than the body swimmers with the non-dimensional mean ve- through the water, we neglect the contribution of the locity. The dashed red lines show the evalua- body velocity. We then expect the propulsion time tion of the τ0 based on the observation. The to be of the order of: solid lines show the optimal propulsion time

2λa with τ0 = 0.335 and = 0 and = 0.035 in τp , (29) black and gray, respectively. ≈ uh/w where uh/w is the hand velocity with respect to the To verify this assumption, we look at the propul- water. This velocity depends on the resistance coef- sion time of our swimmers. We can evaluate the ficient of the hand αh = kh/kb and the force used by propulsion time as: the swimmer to move it through the water3: tp 2 τp = . (33) αhuh/w = ϕ, (30) τ ∗ and therefore: Figure 9 shows the obtained mean value of the non-dimensional propulsion phases with the non- τ0 τp , (31) dimensional velocity. We see that the propulsion time ≈ √ϕ plateaus to τ = 0.31 0.03 when the points are on 0 ± where τ0 is a a characteristic time of propulsion. It the maximum force model in figure 7: depends on the arm length λa and the αh coefficient: v 1 u = = (1 + 2IdC) /2 . (34) τ 2λ √α . (32) v 0 ≈ a h ∗ 3we discuss the impact of the arm speed on the achievable This corresponds to ϕ = 1 and = 0. It is in good force in SI appendix 3 agreement with the previous estimations.

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Comparison to the data set relative velocity variation with the mean velocity is consistent with the observations of Matsuda et al. To summaries, the swimmers are characterized by [21]. two measurable parameters: the propulsion time parameter τ0 and the gliding eﬀectiveness . Both parameters could be measured on swimmers. To min- Linear approximation and two regimes imize their propulsion cost, they can play on their It is interesting to note that in eq.19–22, we can de- force used to produce the thrust ϕ [0, 1]. We can ∈ ﬁne X = u /√ϕ and write all the parameters as a write this problem in the form: − function of X only. Therefore, is a function of X R min cu (ϕ) . (35) only. ϕ, s.t. u=u0 R

This problem can be solved and would yield the op- IdCc (%) timal coordination strategy for our model swimmer for a given mean velocity u = u0. 40 30 20 10 0 20 To compare the present burst-and-coast model to 0.6 − 10 15 − − 25 − 30 our data set, we used a Powell’s conjugate direction − − − − − method [25] with Golden-section search [17] on and 0.5 τ0 to minimize the error on the model prediction on 35 the index of coordination and propulsion time. The 0.4 −

10 15 0

0 20 obtained best-fit parameters are = 0.035 and τ0 = τ − − 25 − 0.335. Note that τ0 is in the error bar of the previous 0.3 − 30 estimation. Figures 7 and 9 show the obtained best − 35 − fit compared to the data set. We further added the 40 0.2 − optimal choice for the limit case = 0. For this case, the model predicts the intuitive choice of the 0 0.05 0.1 0.15 0.2 0.25 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. ≤ Figure 10: Colormap of the optimal coordination in the The model shows that there exists a single op- Burst-And-Coast model with and τ0. timal coordination (here IdC 17%) for non- c ≈ − dimensional velocity lower than uc = 0.81. The For our expert swimmers, we already observed that swimmers would save 1% energy. It is indeed pos- ≈ the maximum force model yields rather good pre- sible to be more efficient in catch-up mode at cer- dictions for the coordination and velocity (see red tain speed with this model, even-though this can be dashed line in figure 7). We expect the coefficient counter-intuitive [12]. Above this critical velocity, the to be close to 0 for our swimmers. We also found swimmers use their maximum force and hence follow that τ0 was reasonably small. To linearize the equa- the maximum force model presented before. tions developed in the two previous subsections, we The present model also provides information on will assume τ0 1 and 1. We further assume the intra-cycle velocity variation (IVV) of the swim- that τ0. Keeping only the smallest order terms mers. With the best-fit parameters, our swimmers in and τ0, it comes: have a relative velocity variation of 13% during the IdC plateau and it decreases once the swimmers reach τ 2 1 + X4 (X) 1 0 1 + X2 1 . the maximum force model. In this simplified model, Rcu ≈ − 2 − 2X2 − the relative velocity variation goes to zero as the IdC (36) goes to 0 due to constant force approximation during the propulsion phase. Yet the reduction of the This approximated function has a single minimum in

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X [0, 1]: swimmer characteristic velocity v∗. This will affect ∈ the non-dimensional propulsion time τ0. If we sup- 1 pose that the legs do not affect the gliding effective- X0 = , (37) 2 1/4 (1 + 4/τ0 ) ness , the coordination is then affected accordingly by an increase of the IdCc. We predict an increase and its value is: of the IdCc from -17% to -12% with this assumption τ 2 q and observe it on a group of similar level swimmers. min = 1 + 0 1 + 4/τ 2 1 . (38) Rcu − 2 0 −

Injecting this expression in the evaluation of the Conclusion and applications index of coordination and keeping only the first order term, it comes that the optimal index of coordination In the current study, we tried to understand how the is: arm coordination patterns of 16 expert front crawl swimmers vary according to the active drag per- ! 1 1 forming two tests: an incremental coordination test, IdCc 1 p . (39) ≈ −2 − 1 + 4/τ 2 where swimmers were requested to simulate their rac- 0 ing techniques at 8 different speeds and a drag mea- This will be the optimal choice of coordination as surement test to measure their active drag using the 1/2 MAD-system. In both tests, the swimmer legs where long as u < (1 + 2IdCc) . Above the critical speed: tied and they were equipped with a pull-buoy to avoid 1 that their legs sank. To compare the evolution of uc 1 , (40) ≈ (1 + 4/τ 2) /4 the different swimmer coordination (IdC) with their 0 mean velocity (v), we defined a characteristic veloc- the swimmers will then switch to the maximum force ity v∗ = vmax/√1 + 2IdCmax using their maximum regime. This is in good agreement with the observa- velocity test. We observed that swimmers used simi- tions of Craig & Pendergast [8]. Figure 10 shows a lar coordination patterns for a given non-dimensional colormap of the numerically found optimal index of velocity u = v/v∗. At low u, the swimmers seem to coordination with τ0 and . select a constant negative index of coordination and From eq.39, we observe that the optimal index above a critical non-dimensional velocity of about of coordination decreases (increases) when (τ0) in- 0.8 their coordination increases with their velocity. creases. It is not surprising to expect that the swim- To further understand these two regimes, we pro- mer will tend to glide more when increases. Note pose a physical model of burst-and-coast [36] adapted that if = 0, then the expression yields that the opti- to swimming in front-crawl. The main idea of this mal coordination is the opposition mode. The effect model is that the swimmers experienced a reduced of varying τ0 is maybe less obvious. Increasing τ0 drag while gliding with one arm extended forward. (keeping all the other parameters constant) can be We compare this drag to the active swimming drag compared to swimming with paddles. This will in- during the underwater stroke by defining a gliding crease the parameter αh in eq.32. From the present effectiveness coefficient > 0. One additional key model, we then expect the swimmers to change their assumption in our model is the arm propulsion time. coordination pattern toward the opposition mode We proposed that this time depends on the force used (IdC closer to zero) with paddles. This prediction to propel the body through the water and is bounded is observed by Sidney et al.[31]. by a lower value τ0 which depends on the swimmer In all the results, presented so far the legs were tied characteristics only. We then showed that this model and could not be used by the swimmers. We discuss predicts that there exist two swimming regimes sim- their effects in SI appendix 4. It is rather clear that ilarly to the observations. allowing the legs to kick will lead to an increase of the 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 References propulsion force to minimize their propulsion cost. A ”high velocity” regime, where the swimmers increase [1] M. Alberty, M. Sidney, P. Pelayo, and H. Tous- their index of coordination to push at maximum force saint. Stroking characteristics during time to ex- and to gain more speed. It is in this latter regime that haustion tests. Medicine and Science in Sports the transition from catch-up to superposition can oc- Exercise, 41(3):637, 2009. cur. The optimal index of coordination in the low ve- [2] T. M. Barbosa, J. A. Bragada, V. M. Reis, D. A. p 2 Marinho, C. Carvalho, and A. J. Silva. Ener- locity regime is IdCc (1 1/ 1 + 4/τ0 )/2 and the transition from the≈ − ”low− velocity” to ”high ve- getics and biomechanics as determining factors 1/4 of swimming performance: Updating the state locity” regime will occur at u 1/ 1 + 4/τ 2 in c 0 of the art. Journal of Science and Medicine in the limit τ 1 and τ . This≈ transition can be 0 0 Sport, 13(2):262 –269, 2010. linked to swimming distances through an energetic equation. This transition is similar to the one ob- [3] H. Behncke and B. Brosowski. Optimization served by Craig & Pendergast [8] in between the 200 models for the force and energy in competitive m and 400 m races (see SI appendix 5). sports. Mathematical Methods in the Applied Using this model, it is possible to advice on indi- Sciences, 9(1):298–311, 1987. vidual optimal arm coordination for each swimmer based on his/her physical characteristics. To predict [4] J.-C. Chatard and B. Wilson. Drafting distance their optimal coordination, we need to evaluate the in swimming. Medicine and Science in Sports two parameters and τ0. It is possible to use a gliding and Exercise, 35(7):1176–1181, 2003. test to estimate the value of by varying the arm position of the swimmer at the surface. For the propul- [5] D. Chollet, S. Chalies, and J. Chatard. A new sion time parameter, one could evaluate the time to index of coordination for the crawl: description perform a single arm pull with the maximum possi- and usefulness. International Journal of Sports ble thrust on a 25m sprint with index of coordination Medicine, 21(01):54–59, 2000. measurements. Then τ0 = t0kbv∗/m, where kb would be the drag measured from the previous gliding test [6] D. Costill, J. Kovaleski, D. Porter, J. Kirwan, with the arms along the body. These tests could be R. Fielding, and D. King. Energy expenditure also done on swimmers with disabilities and advice during front crawl swimming: predicting success them on individual optimal arm coordination based in middle-distance events. International Journal on their physical characteristics and type of impair- of Sports Medicine, 6(05):266–270, 1985. ment. [7] D. L. Costill, E. W. Maglischo, and A. B. Richardson. Swimming. 1992.

Acknowledgment [8] A. B. Craig and D. R. Pendergast. Relationships of stroke rate, distance per stroke, and velocity The authors would like to thank all the athletes that in competitive swimming. Medicine and Science participated in the tests. They are also grateful to in Sports, 11(3):278–283, 1979. Leo Chabert, Benoit Bideau and Vincent Bacot for their help at diﬀerent stages of the project and the [9] A. B. Craig, P. L. Skehan, J. A. Pawelczyk, and useful discussions. Last but not least, the authors W. L. Boomer. Velocity, stroke rate, and dis- would like to thank The Olympic Multimedia Library tance per stroke during elite swimming compe- for granting us access to the footage of the race and tition. Medicine and Science in Sports and Ex- allowing us to use the images to illustrate our work. ercise, 17(6):625–634, 1985.

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[10] P. Di Prampero, D. Pendergast, D. Wilson, and [20] R. B. Martin, R. A. Yeater, and M. K. White. D. Rennie. Energetics of swimming in man. A simple analytical model for the crawl stroke. Journal of Applied Physiology, 37(1):1–5, 1974. Journal of Biomechanics, 14(8):539–548, 1981.

[11] K. A. Ericsson and A. C. Lehmann. Expert and [21] Y. Matsuda, Y. Yamada, Y. Ikuta, T. Nomura, exceptional performance: Evidence of maximal and S. Oda. Intracyclic velocity variation and adaptation to task constraints. Annual Review arm coordination for different skilled swimmers of Psychology, 47(1):273–305, 1996. in the front crawl. Journal of Human Kinetics, 44(1):67–74, 2014. [12] R. Havriluk. Do expert swimmers have expert [22] J. W. McVicar. A brief history of the develop- technique? comment on arm coordination and ment of swimming. Research Quarterly. Ameri- performance level in the 400-m front crawl by can Physical Education Association, 7(1):56–67, Schnitzler, Seifert, and Chollet (2011). Research 1936. Quarterly for Exercise and Sport, 83(2):359–362, 2012. [23] G. Millet, D. Chollet, S. Chalies, and J. Chatard. Coordination in front crawl in elite triathletes [13] P. V. Karpovich. Water resistance in swimming. and elite swimmers. International Journal of Research Quarterly. American Physical Educa- Sports Medicine, 23(02):99–104, 2002. tion Association, 4(3):21–28, 1933. [24] K. Narita, M. Nakashima, and H. Takagi. Devel- [14] J. Katzenstein and R. Dubois&-; Reymond. oping a methodology for estimating the drag in Arch. anat. u. phys. 1905. front-crawl swimming at various velocities. Jour- nal of Biomechanics, 54:123–128, 2017. [15] J. B. Keller. A theory of competitive running. Physics today, 26(9):43, 1973. [25] M. J. Powell. An efficient method for finding the minimum of a function of several variables [16] J. B. Keller. Optimal velocity in a race. The without calculating derivatives. The Computer American Mathematical Monthly, 81(5):474– Journal, 7(2):155–162, 1964. 480, 1974. [26] R. Scurati, G. Gatta, G. Michielon, and [17] J. Kiefer. Sequential minimax search for a max- M. Cortesi. Techniques and considerations for imum. Proceedings of the American Mathemati- monitoring swimmers’ passive drag. Journal of cal Society, 4(3):502–506, 1953. Sports Sciences, 37(10):1168–1180, 2019. PMID: 30449240. [18] S. Kolmogorov and O. Duplishcheva. Active [27] L. Seifert, D. Chollet, and J. C. Chatard. drag, useful mechanical power output and hy- Kinematic changes during a 100-m front drodynamic force coefficient in different swim- crawl: effects of performance level and gender. ming strokes at maximal velocity. Journal of Medicine and Science in Sports and Exercise, Biomechanics, 25(3):311–318, 1992. 39(10):1784–1793, 2007.

[19] H. Leblanc, L. Seifert, C. Tourny-Chollet, and [28] L. Seifert, D. Chollet, and A. Rouard. Swim- D. Chollet. Intra-cyclic distance per stroke ming constraints and arm coordination. Human phase, velocity ﬂuctuations and acceleration Movement Science, 26(1):68–86, 2007. time ratio of a breaststroker’s hip: a comparison between elite and nonelite swimmers at diﬀer- [29] L. Seifert, D. Chollet, and A. Rouard. Swim- ent race paces. International Journal of Sports ming constraints and arm coordination. Human Medicine, 28(02):140–147, 2007. Movement Science, 26(1):68–86, 2007.

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[30] L. Seifert, H. Toussaint, M. Alberty, C. Schnit- speed v∗ = vmax/√1 + 2IdCmax, where vmax denotes zler, and D. Chollet. Arm coordination, power, the maximum velocity achieved by the swimmer. The and swim efficiency in national and regional results are displayed in figure 11. front crawl swimmers. Human Movement Sci- ence, 29(3):426–439, 2010. ii. Extension to superposition mode [31] M. Sidney, S. Paillette, J. Hespel, D. Chollet, The burst-and-coast model can be extended to the and P. Pelayo. Effect of swim paddles on the superposition mode. In this case, we have the follow- intra-cyclic velocity variations and on the arm ing equations: ISBS- coordination of front crawl stroke. In du Conference Proceedings Archive, volume 1, 2001. = 2ϕ u2, 0 τ τ , (41) dτ − ≤ ≤ c [32] A. J. Silva, A. Rouboa, A. Moreira, V. M. Reis, du 2 = ϕ u , τc τ /2, (42) F. Alves, J. P. Vilas-Boas, and D. A. Marinho. dτ − ≤ ≤ T Analysis of drafting effects in swimming using with boundary conditions: computational fluid dynamics. Journal of Sports u(0) = u( /2) = u , (43) Science & Medicine, 7(1):60–66, 2008. T − u(τc) = u+, (44) [33] H. Toussaint, G. De Groot, H. Savelberg, K. Ver- where u = v/v is the non-dimensional velocity, τ = voorn, A. Hollander, and G. van Ingen Schenau. ∗ t/τ the non-dimensional time, ϕ = T /T the non- Active drag related to velocity in male and ∗ a a∗ dimensional thrust and: female swimmers. Journal of Biomechanics, m0 + ma 21(5):435–438, 1988. τ ∗ = , (45) kbv∗ [34] M. Truijens and H. Toussaint. Biomechanical as- p v∗ = Ta∗/kb. (46) pects of peak performance in human swimming. Animal Biology, 55(1):17–40, 2005. Note that v∗ is the same as the one defined in the previous appendix. [35] J. Videler and D. Weihs. Energetic advantages of Here, the propulsion time does not appear directly burst-and-coast swimming of fish at high speeds. in the equations. We still have the relation: Journal of Experimental Biology, 97(1):169–178, /2 = τ τ . (47) 1982. T p − c Figure 12 summarizes the notations. This system can [36] D. Weihs. Energetic advantages of burst swim- also be solved analytically: ming of fish. Journal of Theoretical Biology, p p 1 u 48(1):215–229, 1974. u = 2ϕ tanh τ 2ϕ + tanh− − , + c √2ϕ [37] J. Westerweel, K. Aslan, P. Pennings, and (48) B. Yilmaz. Advantage of a lead swimmer in 1 1 u 1 u+ drafting, Oct 2016. τp 2τc = coth− − coth− , − √ϕ √ϕ − √ϕ (49) Supplementary Information s ! 1 u2 /2ϕ λ = log − − , (50) i. Index of coordination of the 16 swimmers c 1 u2 /2ϕ − + s ! We provide here the raw data for all the swimmers for u2 /ϕ 1 the index of coordination with dimensional velocity λ 2λ = log + − . (51) p − c u2 /ϕ 1 and after applying the evaluation of the characteristic − −

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10 10 τ0 = 0.335 & = 0.035 τ0 = 0.335 & = 0

0 0 IdC 10 IdC 10 − −

20 20 − −

30 30 − 0.6 0.8 1 1.2 1.4 1.6 1.8 2 − 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

v (m/s) v/v∗

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.

/2 T For this amount of energy, the swimmers travel a non- dimensional distance λ λ only. It should be com- u+ p c pared to the energy used− to travel the same distance in opposition mode (steady swimming):

/2 Z T op,su = ϕ udτ (54) u E − 0 Double Push Single Push = u2 (λ λ ) . (55) time p − c The economy can therefore be deﬁned as: τc τ 2τ p − c Distance su ϕ λp su = E = 2 . (56) R op,su u λp λc λ λ 2λ E − c p − c We still assume that τp = τ0/√ϕ as each arms have Figure 12: Burst-and-coast model for swimmers intra- their own control. cycle velocity variations and notations ex- Deﬁning X = u /√ϕ and X+ = u+/√ϕ, we tension to the superposition mode. have: − −

1 < X < X+ < √2. (57) − The energy consumed during the superposition mode Then, assuming X+ = X + ∆X, it is possible to is: show that in the limit τ − 1 and ∆X 1: 0 " #2 Z τc Z τp (X 1) (X + 1) X 2 2 − − − 2 su = 2ϕudτ + ϕudτ (52) su 1 + − 3 − τ0 > 1. E 0 τc R ≈ 4X − = ϕλp. (53) (58)

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Therefore, as expected intuitively, this mode is al- iv. Impact of the legs kicking on the coordi- ways more expensive energetically than the opposi- nation tion. This mode should be use solely for sprinting and at maximum force (ϕ = 1). It is interesting to note In all the paper the legs were tied. In previous works done by Chollet et al. [5] and Seifert et al.[29], the that su is maximum for X 1.17. It seems that this couldR be defined as a critical− ≈ speed the swimmers swimmers were allowed to used their legs. The ve- cannot exceed. In all our observations, u < 1.15. locities the swimmers could reach were larger by 20% compared to the present one for similar level swimmers. We look in this appendix at other data taken on similar level swimmers with the legs free to kick. iii. Hill’s heuristic law and propulsion time We did not measure the resistance coefficient for these swimmers with the MAD-system. They only per- It is well known that the force decreases with the formed the coordination test. velocity (Hill’s heuristic law). In this appendix, we discuss briefly this problem in our discussion of the τ = 0.335 propulsion time. We assume that the athletes can 10 No leg 0 With legs τ0 = 0.401 control their force of propulsion and that they can chose a bounded thrust Ta [0,Ta∗]. This thrust 0 comes from the arms (in the∈ present discussion the legs are tied), which are controlled by the swimmers 10

IdC (%) − muscles. Then the force used to activate the arms should be bounded by a parameter, which follows 20 a Hill’s law, and dynamic equation of the hand be- − comes: 30 − 0.4 0.6 0.8 1 1.2 v/v∗ 2 uh/w αhuh/w = min ϕ, Φ∗ 1 , (59) − u∗ h Figure 13: Arm coordination with the mean non- dimensional velocity for swimmers with and where ϕ [0, 1] is the control on the force, Φ∗ and ∈ without legs. The dashed line corresponds to uh∗ denote the maximum force that can be used to the maximum force model. The solid lines activate the arm and the maximum velocity at which show the optimal coordination with τ0 = this arm can be moved without resistance, respec- 0.335 and τ0,† = 0.401 in gray and blue, re- tively. Note that if eq.59 is bounded by the Hill’s spectively. In both cases, we used = 0.035. law then Φ∗ 1 u /u∗ = 1 at the maximum − h/w,max h hand velocity in the water uh/w,max, since we non- We applied the same analysis as done in the present dimensionalize the problem here. And then we go paper to the data of velocity and coordination. The back to what was done earlier. results are displayed in ﬁgure 13 and compared to

Further note that uh∗ can also provide information the swimmers with no legs of the present paper. We on the air recovery time and thus the maximum in- observe a similar trend. The swimmers select a con- dex of coordination the swimmer can achieve IdCmax. stant index of coordination at low velocity and then Let’s assume for now that the swimmer is pushing follow the maximum force model. The characteristic with the maximum thrust at τ0 and wants to swim velocity v∗ increased from 1.5 m/s without leg to 1.8 at the maximum index of coordination. Then they m/s with legs in average. have to bring the arm back to the front as fast as At low velocity, the swimmers usually perform one they can. They will reach uh∗ in the air recovery kick per arm stroke and at higher velocity up to three phase. ≈ [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 the turns and the start for simplification. Behncke & legs can be whether a reduction of the effective drag Brosowski [3] discuss a way to take these parts of the kb or an increase of the propulsion force Ta. In all race into account. To keep the discussion simple and cases, it will lead to an increase of the characteristic analytical, we consider the case of an infinite pool velocity v∗ and the coordination is certainly affected in the present discussion. The second term in the by this. right hand side of eq.61 is the one we minimised in In this appendix, we will assume it is only the the previous sections for a given mean velocity v = thrust that is increased and therefore the drag is uv∗. Using the opposition mode as a reference eq.61 of the same magnitude (kb = 30 kg/m). Then if becomes: the physical propulsion time is the same, the non- 1 3 dimensional one should be rescaled τ0, = τ0v∗/v∗, Er E0 σtr tr (u) kbv , (62) † − ≈ − η R where the indicates the quantity with the legs† free † to kick. Using for τ0 without leg the value found pre- where is the economy at v = uv∗ and will depend viously, we get τ0, = 0.401 with legs. Our model R † on the coordination. then will predict for a similar an increase of the For short races (pure sprints, Er > 0), the swim- index of coordination from -17% to -12%. This is in mer does not have to worry about the economy and good agreement with the observations. should select the highest possible velocity they can achieve. A superposition mode is expected with the v. Coordination and swimming distances highest possible index of coordination, as this is how they can achieve the highest velocity. They can main- In the paper we describe the intra-cycle variations dy- tain this technique as long as Er > 0. We define the namic. We optimized the choice of coordination such aerobic velocity: that the swimmers minimize their energy consump- 1 tion. To have an order of magnitude of the distance ησ /3 reached by the swimmers with this technique, an en- vσ = , (63) kb ergy equation is necessary: the characteristic distance: dE 1 = σ Ta (t) v (t) , (60) 1/3 dt − η η E0 L = . (64) 0 1/3 σ2/3k where E denotes the energy reserves of the swimmer, b σ is the maximum rate at which the oxygen is sup- and: plied to the muscles (it is equivalent to the VO max) 2 v and η a conversion efficiency of the chemical energy β = σ . (65) to propulsive energy (which we will assume to be a v∗ constant). It then comes that the sprint technique will last as We integrates this equation on the total duration long as Lr < Ls, where: of the race tr = Lr/v: L0 Ls = , (66) Z T max 2 1 tr (umax/β) β/umax Er E0 σtr Ta (t) v (t) dt, (61) R − − ≈ − η T 0 where max is the economy at the maximum index R where E0 is the anaerobic reserve at the beginning of of coordination and force, and umax the maximum the race and Er 0 the left energy at the end of the non-dimensional speed. At worst, it will corresponds race. The approximation≥ comes from the number of to the maximum for the superposition mode (see ap- cycle the swimmer used tr/T on the second term pendix ii) which occurs for umax 1.17. We will of the right-hand-side and≈ the fact that we neglect 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- [16, 15, 3, 10]. Note that we choose η to be constant. erwise, it would mean that the swimmers could sprint di Pramparo et al. [10] evaluated the efficiency in as long as they want. This will give an upper bound sub-maximal exercise to be in between 2.6 and 5.2%. to the possible values of η. This value seems therefore reasonable. Note that to For longer races, Er = 0 and the swimmer should be consistent, here the efficiency cannot be larger minimize their energy consumption. They should than 13.5 % because then the Ls would not be de- then choose the coordination that enables to min- fined. The choice of the efficiency greatly influences imize the economy at the targeted mean velocity. the results. With the present values, we get L0 = 103 Eq.62 becomes: 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- 1 op 3 0 E0 σtr tr (u) kbv , (67) ited to the 50 m race and the 100 m race swimmers − ≈ − η R are expected to manage their energy on the length of the race. The swimmer will switch from a maximum where op is the optimal economy at v = uv 4. ∗ force technique to a constant index of coordination TheR link between the length of the race and the race in between the 200m and the 400m races. This mean velocity is then: is consistent with the observations of Craig & Pen- 2 dergast [8] and previous observations of change in L0 u β = op (u) . (68) coordination patterns with pace made by Chollet et L R β − u r al. [5] and Seifert et al.[29]. Eq.68 gives the relationship between the mean velocity and the length of the race depending on the swimmer characteristics (VO2 max, anaerobic velocity, gliding efficiency and propulsion time). The transition from the long distance race strategy with constant index of coordination and the shorter race with maximum force occurs at the v = ucv∗. Injecting min the results of the paper for cu and uc in eq.68, it comes: R

L0 Lc = . (69) (u /β)2 min β/u c Rcu − c 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 parameters. 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

4 ∗ for a velocity below the critical velocity vc = ucv , it is min Rcu discussed in the main text.