RESEARCH ARTICLE
Optimizing running a race on a curved track
Amandine AftalionID1*, Pierre Martinon2
1 Ecole des Hautes Etudes en Sciences Sociales, PSL Research University, Centre d’Analyse et de
Mathe´matique Sociales, Paris, France, 2 Inria Paris and LJLL Sorbonne Universite´, Paris, France
* amandine.aftalion@ehess.fr
Abstract
In order to determine the optimal strategy to run a race on a curved track according to the
lane number, we introduce a model based on differential equations for the velocity, the pro-
pulsive force and the anaerobic energy which takes into account the centrifugal force. This
allows us to analyze numerically the different strategies according to the types of track since
different designs of tracks lead to straights of different lengths. In particular, we find that the
tracks with shorter straights lead to better performances, while the double bend track with
the longest straight leads to the worst performances and the biggest difference between
lanes. Then for a race with two runners, we introduce a psychological interaction: there is an
attraction to follow someone just ahead, but after being overtaken, there is a delay before
any benefit from this interaction occurs. We provide numerical simulations in different
cases. Overall, the results agree with the IAAF rules for lane draws in competition, where
the highest ranked athletes get the center lanes, the next ones the outside lanes, while the
lowest ranked athletes get the inside lanes.
Introduction
In athletics, inside lanes are considered a disadvantage due to curvature, while in outside lanes,
there is no one to chase. The aim of this paper is to understand from a physical and mathemat-
ical point of view the effect of the curved part of a track and of the lane number on the running
performance both for a single runner and for a two-runner race.
To our knowledge, no optimal control problem including these effects has been studied.
There is a huge literature on the way of running on a curved track, see for instance [1–8]. Nev-
ertheless, it is never coupled with the psychological effect to have a neighbor on the next lane,
which is mentioned as important. Furthermore, though the IAAF regulations [9] do not
impose a fixed shape of track, but allow the straights to vary between 80m and 100m, we are
not aware of any study discussing the effect of the the lane and the track coupled with the psy-
chological effect.
In this paper, we will build on a model introduced by Keller [10] and extended by [11, 12],
to investigate how the shape of the track and the centrifugal force change the optimal strategy
in a race: this leads to longer race times for higher curvatures, and therefore favors the outer
lanes. Estimating the performance of champions based on the modeling of Keller [10] has
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OPEN ACCESS
Citation: Aftalion A, Martinon P (2019) Optimizing
running a race on a curved track. PLoS ONE 14(9):
e0221572. https://doi.org/10.1371/journal.
pone.0221572
Editor: Gordon Fisher, University of Alabama at
Birmingham, UNITED STATES
Received: December 13, 2018
Accepted: August 10, 2019
Published: September 5, 2019
Copyright: © 2019 Aftalion, Martinon. This is an
open access article distributed under the terms of
the Creative Commons Attribution License, which
permits unrestricted use, distribution, and
reproduction in any medium, provided the original
author and source are credited.
Data Availability Statement: All relevant data are
within the paper.
Funding: The authors received no specific funding
for this work.
Competing interests: The authors have declared
that no competing interests exist.
been developed by various authors [6, 13–16], but never taking into account so many parame-
ters as in this paper. We will also introduce a model taking into account the psychological
effect between two runners. This is made up of two effects: on the one hand, the attraction by a
runner close ahead, and on the other hand, the delay before any benefit from the interaction
occurs again after being overtaken. This delay model is inspired by a paper on walking [17].
Let us point out that the mathematical problem encompassing delay in the equations is quite
involved. We model the attraction by a runner close ahead as a decreased friction, since
the focus on chasing someone ahead improves the runner’s economy. Due to the staggered
start positions in the curved part of the track, this “rabbit effect” is less favorable on the outer
lanes.
After introducing the model, we perform simulations using the optimal control
toolbox BOCOP [18]. Since the IAAF regulations do not impose a single shape of track, we ana-
lyze the effect of the shape of the track on the optimal velocity profile, as well as the influence
of the various parameters of the runner for a single runner. Then we perform simulations for
two runners and our results show that the combination of the centrifugal force and the two
runners interaction brings a numerical justification to the fact that the central lanes are the
most favorable to win a race.
Race model
Model for a single runner race
Single runner on a straight track.
When a runner is running on a straight, as used by
Keller [10], according to Newton’s second law, the acceleration is equal to the sum of forces.
We can list two forces, the propulsive force f(t) in the direction of motion, and the friction
force, that we assume to be linear in velocity. This leads to the first equation of motion for the
velocity v(t) written by unit of mass:
_vðtÞ ¼ f ðtÞ considered in this paper, we assume a linear function σ and note σf the final value, thus
sðeÞ ¼ sf
e0 centrifugal force, which, by unit of mass, is fc = v2/R where v is the velocity of the runner and R
the curvature radius. In order to produce a mathematical model for the dynamics in the curved
part, we have to take into account the centrifugal force in Newton’s law of motion and project
this equation on the 3 directions of motion.
Even on straights, there is an equation to be written in the z direction: the reaction of the
ground, N, is equal to the weight. By the principle of action/reaction, the reaction of the
ground is equal to the runner’s propulsive force in the z direction. Note that the runner does
not have his feet on the ground all the time in the stride: he rather pushes (propulsive force)
only for some time in a stride [19]. Some remarks in [26] can be found related to this issue.
We point out that there is an interesting explanation of the effect of arms to counterbalance
the torque, and that since there are two legs, the reaction on each leg is not exactly the same
[1]. In this paper, we do not include these effects as we believe them to be of minor importance.
The specificity of our work is that although we consider a mean force and mean velocity in a
stride, our model allows us to compute an instantaneous force and speed along the race.
On a curve, the runner makes an angle α with the vertical axis to balance the centrifugal
force. The runner is subject to gravity g, to the reaction of the ground N along the angle α, and
to the centrifugal force fc = v2/R (see Fig 1). One has to consider the equations of motion in the
centrifugal direction and the z direction, which lead to
v2
R ¼ N sin a; g ¼ N cosa
ð9Þ
which provides the angle according to the velocity and the value of N:
tana ¼ v2
Rg ; N2 ¼ g2 þ v4
R2 :
ð10Þ
By the principle of action/reaction, the propulsive force in the transverse direction is the
opposite of the reaction of the ground in the horizontal direction, hence is equal to N sin α.
Moreover, the propulsive force in the vertical direction is N cos α. The total propulsive force F
is therefore such that F2 = f2 + N2 where we recall that f is in the direction of movement. From
(10), we find
F2 ¼ f 2 þ N2 ¼ f 2 þ g2 þ v4
R2 :
ð11Þ
Since F has to be bounded and g is constant, this leads to the new constraint
f 2 þ v4
R2  f 2
M:
ð12Þ
We point out that eventually the effect of the centrifugal force is taken into account in the
force constraint. It cannot have an energy effect directly since the centrifugal force does not
produce any work.
Study of the track shape.
It is important to know the exact shape of the track since it
influences the runner’s optimal pacing strategy and performance. However, there is no fixed
regulation to build an athletic track. Actually, as indicated in the IAAF manual [9], the length
of the straight part can vary between 80 and 100m, while the curved part can be a half circle
(‘standard’ track) or two different circular sections (‘double bend’ tracks). We choose to study
a standard track with an 84.39m straight part, and then two double bend tracks with straight
parts of 79.996m and 98.52m respectively. The shapes and dimensions of theses tracks are
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detailed in Fig 2 and Table 1. Note that for races longer than 100m, runners start the race in
the curved part. The starting positions are therefore adjusted in order to have the same total
distance for all lanes (‘staggered start’).
Note that each runner is assumed to run at a distance of 30cm from the inner limit of the
lane. This is how the radius for the circular parts is set in order to obtain a 400m distance for
lane 1. Then the width of each lane is 1.22m. This leads to different radii of curvature Rk(s)
depending on the lane k and the distance s run on the lane since the start. On the straight part,
1/Rk(s) = 0. For more details on the value of Rk(s) according to the track, we refer to the
Appendix “Track Shape Details”.
We want to point out that at the junction between the circular and straight parts, the runner
will experience a discontinuity in the centrifugal force. This force is 0 on the straight part and
Fig 1. Illustration of the forces on the runner.
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can reach a value of the order of 2.5N per kilo on the circular part (since v * 10m/s and
R * 40m), which is about one quarter of the weight.
We will see on the numerical simulations that this may lead to an acceleration of the runner
when reaching the straight part of the track. One could think that it would be better to build a
track where the curvature goes smoothly from 0 to the value of the matched circle so that the
runner experiences a continuous variation of his centrifugal force. This type of curve, known
as a clothoid, is used for instance for railways and roads. The simulations in Section “Numeri-
cal simulations for a single runner” indicate that the final time is actually larger on a clothoid,
because the smooth transition leads to a smaller radius for the circular part, therefore a larger
centrifugal force.
One of the main results of our simulations is that the tracks with shorter straights lead to
better performances (see Section “Effect of different track shapes”).
Final model for a single runner race.
The optimal problem is to minimize T = y(d) with
y(s), f(s), e(s) solving (6) and (7), σ being given by (3), with the bounded control
df
ds

  0:015
ð13Þ
and the force constraint coming from (12)
f 2ðsÞ þ
1
_y4ðsÞR2
kðsÞ  f 2
M
ð14Þ
where the curvature radius Rk(s) is prescribed according to the lane k and the track shape, see
the Appendix “Track shape Details”. We use the convention Rk(s) = + 1 on a straight.
Table 1. Track parameters.
Track
Straight
Circle
Standard
84.39m
(36.50m, 180˚)
Track
Straight
Circle 1
Circle 2
Double Bend 1
79.996m
(34.00m, 2 × 70˚)
(51.543m, 40˚)
Double Bend 2
98.52m
(24.00m, 2 × 60˚)
(48.00m, 60˚)
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Fig 2. Shape for standard track (left) and and double bend 2 track (right).
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Finally, introducing a state variable for the inverse of speed z(s) = 1/v(s), the optimal control
problem for a single runner is
ðOCPÞ1
min yðdÞ;
_yðsÞ ¼ zðsÞ;
s 2 ½0; d;
yð0Þ ¼ 0;
_zðsÞ ¼ z2ðsÞ=t basis for runners and the focus on chasing produces better running economy [29–31]. This
psychological effect is indeed acknowledged by runners (sometimes called “rabbit effect”) and
can allegedly have an effect as high as 1 second per 400m lap [31].
The differential equations for y1 and y2 are therefore
€y1ðsÞ ¼ multiply the interaction term F by an characteristic function Iη defined by
IZðsÞ ¼
0
if an overtaking has occured on ½s Final model for a 2-runner race.
We define Ti = yi(d) and F from (18). The optimal con-
trol problem becomes:
ðOCPÞ2
minð minðT1; T2Þ þ kwðT1 þ T2ÞÞ;
_yiðsÞ ¼ ziðsÞ;
s 2 ½0; d;
yið0Þ ¼ 0; zð0Þ ¼ _yið0Þ ¼ 1=v0
i ;
i ¼ 1; 2;
_eiðsÞ ¼ sðeiðsÞÞziðsÞ Single runner on a straight track
We start with a simple straight 200m race to illustrate the effect of parameters fM, τ, and e0. We
take as reference athlete A1 of Table 2. The corresponding speed and force profiles are shown
with black lines in Fig 4. The velocity increases to its peak value vm * fM τ and then decreases.
The runner does not have enough energy to run the whole duration of the race at maximal
force.
The propulsive force starts at its maximal value fM, then decreases at the constant rate
|df/ds|max. The time at which the force begins to decrease depends on the values of fM and
e0. Indeed, increasing e0 does not change the beginning of the race but allows to run longer at
f = fM. On the other hand, increasing fM increases the peak velocity but does not change much
the second part of the race. Finally, increasing τ has a more uniform effect and increases the
velocity for the whole duration of the race.
Single runner on a standard curved track
We simulate the same runner on the so-called standard track, i.e. 115.61m half circle of radius
36.80m followed by a 84.39m straight. Fig 4 shows the race profiles obtained for the inner and
outer lanes (respectively 1 and 8), and the straight race. The time splits for 50–100m, 100–
150m and 150–200m are indicated in the figure: we have chosen the parameters for A1 so that
they match the order of magnitude of time splits for athletes in World Championships. The
velocity profiles of the curved track are quite different from the straight track:
1. the runner starts slower because of the curvature: even though he puts his maximal propul-
sive force at the start, part of it is used to counterbalance the centrifugal force, resulting into
a lower effective force and a lower velocity
f 2
M  f 2
init þ ðv0Þ
4
R2
k
:
Fig 4. Single runner A1 on a standard track, lanes 1 and 8, and straight track. Force in N/kg vs distance on the right graph. Speed vs distance on the left graph, with
the constant speeds given by Eq (21) in dashed lines. Time splits for 50–100m, 100–150m and 150–200m are indicated.
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2. in the middle part of the race, the maximal propulsive force is reached and we can derive
from (1) and (12) the relation between f and v:
v ¼ f t and f 2
M ¼ f 2 þ f 4t4
R2
k
ð20Þ
with Rk the curvature radius on lane k. We can compare this formula with our simulations:
on the straight vs = fM τ, while from (20), the velocity in the middle of the race on lane k is
v2
s ¼ v2
k þ v4
kt2
R2
k
; that is v2
k ¼ Effect of different track shapes
Now we study the effect of different track shapes defined in Fig 2: standard with 84.39m
straight (STD), double bend 1 with 80m straight (DB1), double bend 2 with 100m straight
(DB2), and two modified standard tracks with smoothed curvature, including clothoid junc-
tions of 10m (CL1) and 30m (CL2). For the clothoid tracks, we choose a straight of 84.39m as
the standard track. As explained in the Appendix, the length of the junctions provides the
radius of the circle and the angle, which are respectively 33.32m and 164˚ for (CL1) and
29.95m and 118˚ for (CL2).
For each track shape, we simulate the race on the inner and outer lanes (1 and 8). The
results are summarized in Table 3, with the races for runner A1 (on lane 5) shown in Fig 6. Ref-
erence athlete A1 has a difference of 0.17s between the best (DB1 track, lane 8) and worst
(DB2, lane 1) case. As mentioned previously, runner A2 with a very high force fM = 13 is almost
unaffected by the curvature, with times varying only between 20.32s and 20.36s. Yet, the DB2
track is still worse than the others. Conversely, athlete A3, with a lower force fM = 6.5, is more
affected, with 1.01s between the best and worst cases. The DB2 is his worst track and his best
performance is on the standard track.
Our results show a time difference between inner and outer lanes ranging from 0.02s for
the standard track to 0.15s for the worst double bend track. This is consistent with [6] who also
finds the double bend track to be the worst, using a simplified model based on constant mean
Fig 5. Single runner A3 on a standard track, lanes 1 and 8, and straight track. Force in N/kg vs distance on the right graph. Speed vs distance on the left graph, with
the constant speeds given by Eq (21) in dashed lines. Force and velocity increase when the centrifugal force disappears.
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Table 3. Times for different runners and track shapes.
runner
shape:
STD
DB1
DB2
CL1
CL2
Straight
A1
lane 1
20.48
20.49
20.62
20.50
20.56
20.43
A1
lane 8
20.46
20.45
20.47
20.46
20.49
20.43
A2
lane 1
20.32
20.33
20.36
20.33
20.34
20.31
A2
lane 8
20.32
20.32
20.32
20.32
20.33
20.31
A3
lane 1
20.72
20.80
21.55
20.80
21.03
20.32
A3
lane 8
20.54
20.55
20.66
20.57
20.63
20.32
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velocity and curvature. Let us point out that in the next section, we will study a two-runner
race where the effect of the lane becomes more pronounced: we find a larger difference
between the best and worst mean time per lane.
Focusing on runner A1 in Fig 6, we analyze more closely the effect of the track shape and
lane:
• DB1 is the quickest track for the outside lane, though it is very close to STD.
• The standard track has the smallest difference between lanes.
• DB2 is the slowest track, from 0.01s on the outside lane to 0.14s on the inner lane. When on
the outer radius of curvature 24m, the velocity significantly decreases.
• CL1 is quite close to DB1 and STD, though a little slower. CL2 is slower than DB2 on the
outer lanes, although not as bad in terms of difference between lanes.
It may seem surprising that the tracks with smoothed curvature do not perform better than
the ones with a discontinuous curvature. This comes from the fact that the clothoid junction
actually results in a smaller radius for the circular part, and thus a greater curvature. The lon-
ger the clothoid junction, the more pronounced the effect, and the slower the times.
To conclude the single runner races, it appears that the track with the shortest straight is the
quickest track for strong athletes in outer lanes. The standard track shape is the one with the
best race times overall, and also the smallest time gap between the inner and outer lanes. On
the opposite, the double bend with the long 100m straight (DB2) yields the worst times overall,
and the highest gap between the inner and outer lanes. These conclusions seem consistent
with runners’ feelings though there is no study yet of what the ideal shape of track would be
for a specific runner.
Numerical simulations for two runners
We move to the simulations for two-runner races, combining the interaction effect with the
curvature effect previously studied for the single runner case. Firstly, we study races with two
runners competing in adjacent lanes, to see the effect of the interaction. Then we compute the
mean times corresponding to all possible races of a runner versus himself and find that the
best lanes are indeed the center ones.
Races on different lanes and illustration of the interaction effect
We perform simulations for the optimal control problem (OCP2) for two runners, combining
the interaction effect with the centrifugal force. We first set A1 to be the runner on each lane 1
and 2.
Fig 6. Effect of the track shape on the race time for Runner A1 vs lane number (left graph). Speed profile for lane 5 (right graph)
with zoom. The tracks are standard (STD), double bend 1 (DB1) with short straights, double bend2 with long straights (DB2), and
curves with short and long clothoid junctions (CL1 and CL2).
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We recall that if A1 runs alone, his time on lane 1 is 20.485s and on lane 2 20.480s, so of
course because of the centrifugal effect, lane 2 is quicker. Due to staggered starts, as soon as we
set the interaction, the runner on lane 1 benefits from the interaction at the beginning of the
race. First, we set the interaction term γ = 0.04 but with no inhibition η = 0. The results are
illustrated in Fig 7, with the velocity profile in lane 1 on the left and the interaction for each
runner and relative distance on the right. When the relative distance is negative, the runner in
lane 1 is behind. So in this case, the runner in lane 2 wins the race and they overtake each
other twice: lane 1 starts behind because of the staggered starts, benefits from interaction and
overtakes at 50m; then lane 2 benefits from interaction right away and is able to overtake again
at 150m. Then they are on the straight, very close to each other, lane 1 benefits from interaction
and is ready to overtake again but loses in the end by 0.04s.
Then in Fig 8, the interaction term is set at γ = 0.04, and the inhibition frame is η = 20m.
This means that the positive interaction is disabled when a runner is overtaken in the previous
20m of race. Fig 8 shows the speed profile (left graph) and interaction / inhibition terms (right
Fig 7. Race A1 vs himself at lanes 1-2, with interaction γ = 0.04 and a frame η = 0 that is no inhibition. Left graph: speed profile
and time splits of the runner at lane 1. Right graph: distance gap and interaction term for both runners. The sign change of the
distance gap corresponds to the overtaking. Lane 2 wins by 0.04s.
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Fig 8. Race A1 vs himself at lanes 1-2, with interaction γ = 0.04 and a frame η = 20m for the inhibition. Left graph: speed profile
and time splits of the runner at lane 1. Right graph: distance gap and interaction term for both runners. The sign change of the
distance gap corresponds to the overtaking. Lane 1 wins by 0.27s.
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graph). Compared to the race without inhibition in Fig 9, we observe a different behaviour
with only one overtaking and the runner on the inside winning by 0.27s. Note that since we
optimize the whole race, there is no reason for the race with inhibition to coincide with the
race without inhibition, even before any overtaking occurs. We observe that the inhibition (on
the right graph) correctly detects the overtaking and suppresses the interaction accordingly.
This prevents the overtaken runner at lane 2 to keep up (and eventually catch up) with the one
at lane 1, as we see that the distance gap increases after the overtaking. In the race without inhi-
bition, the overtaken runner was benefiting from the interaction right away, which allowed
him to catch up and take the lead back. With inhibition, the runner on lane 1 manages to win
the race, though he is on a disadvantageous lane. In the full race, of course, the runner in lane
2 has a neighbour on the other side which changes the total result.
There are cases where, though the inhibition η = 20, there are still two overtakings. We
study for instance the 5-4 race, with the speed and force profile of the runner at lane 5 shown
in Fig 9. Without interaction (γ = 0), lane 5 wins without any overtakings, with final time
22.47s. With interaction (γ = 0.04 and η = 20), lane 5 still wins after 2 overtakings, with final
time 22.23s. At the start, the runner on lane 4 benefits from the interaction due to the runner
at lane 5 being ahead (staggered start). He catches up then overtakes the outer runner, who in
turn gains the interaction, catches up and overtakes the inner runner again. At the end the
inner runner, being behind, has the interaction again and is catching up with the outer runner,
but too late.
We have also made simulations with runner A1 vs runner A2, and though runner A2 is
stronger in force, on some lanes, runner A1 can benefit from interaction to be able to win.
We point out that the interaction parameters can be runner dependent since some may be
very sensitive to this effect and others much less.
Mean time per lane
In a real race, there are eight runners, however our model is only for two. Therefore, we simu-
late a set of races with two identical runners, the first on a fixed lane, the second on each possi-
ble other lane. We define T
k1;k2
1
to be the time for the winner in the race between two identical
runners A1 in lanes k1 and k2. We want to compute the average performance at lane i as the
Fig 9. Race A1 vs himself at lanes 5-4, with interaction γ = 0.04 and inhibition η = 20m. Left graph: speed profile and time
splits of the runner at lane 5. Right graph: distance gap and interaction term for both runners. The sign changes of the distance gap
correspond to the 2 overtakings.
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mean time
T i ¼ 1
7
X
j¼1::8;j6¼i
T
k1¼i;k2¼j
1
:
First, we compute the times T
i;j
1 : the best times in j for each i are indicated in Table 4. In Fig
10, we have plotted the times for i = 1, 5, and 8. The best times are obtained for the maximal
interaction, namely with the second runner on an adjacent lane. For runner A1 on lane 5, his
Table 4. Athlete A1 at lane k running against himself at lane k − 1. Interaction γ = 0.04 with inhibition η = 20m. Race time and gain with respect to solo race time.
lane
2
3
4
5
6
7
8
solo time
20.480
20.475
20.471
20.467
20.464
20.461
20.459
2-runner time
20.300
20.292
20.283
20.276
20.270
20.264
20.259
time gain
0.180
0.183
0.1880
0.191
0.194
0.197
0.200
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Fig 10. Times for athlete A1 at lanes 1,5,8, running against himself.
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best performance is obtained with a neighbor on lane 4 rather than 6. We recall that the model
includes a lateral attenuation for the interaction, which is 0 when runners are more than 3
lanes apart. If we compare the best time for each case, it is decreasing with the lane.
We show in Fig 11 the mean times T i obtained for runner A1 against himself, with an inter-
action weight γ = 0.04 and η = 20 when he runs on each lane i. If we look at the overall perfor-
mance then lane 5 is the best, followed by lane 6, 4, 7, 3, 8, 2 and lane 1 is by far the worst. We
compare with the solo case (γ = 0) where of course the outside lane is the quickest.
The results are nicely consistent with the IAAF rules for the lane drawn. Indeed, according
to the IAAF rules [9], starting lanes are drawn in three lots:
• a first draw is made for the four best runners in the center lanes 3, 4, 5 and 6.
• a second draw is made for the next two runners between the outer lanes 7 and 8.
• a last draw is made between the runners with the lowest performance to get the inside lanes
1 and 2.
Fig 11. Mean times per lane for runner A1 when in lane i vs himself in all other lanes. Without interaction (γ = 0), and with interaction γ = 0.04, η = 20. Lane
performance (sorted by mean time): T 5 < T 6 < T 4 < T 7 < T 3 < T 8 < T 2 < T 1. Gap T 1 Nevertheless, we find that the inside lanes 1,2 are a real disadvantage, the more so as if the
runners are not as strong.
In [4] the authors recall some average time data for Olympics 1996 and 2000, and World
Championship 2001: they indicate an average time gap of 0.16s between inside lanes 1 and 2
and outside lanes 7 and 8. We obtain a smaller gap of 0.047s, which may be due to the fact that
we consider identical runners in our simulations, while in actual races the athletes in the out-
side lanes were supposedly stronger than those in the inside lanes.
Conclusion
In this paper, we have studied how the geometry of the track and the psychological interaction
between runners affect performances. We have introduced an optimal control model taking
into account the centrifugal force as a limiting factor for the maximal propulsive force. We
couple this with a new model describing the positive interaction exerted by a runner close
ahead and the delay to benefit from it after being overtaken. We carry out numerical simula-
tions for several runner profiles on different track shapes. The results indicate that the track
with the shortest straights is the quickest for strong athletes in outside lanes. The so-called
standard track (two straights and half circles) yields the best performances overall. The double
bend tracks with longer straights (DB2) are significantly slower. In particular running on lane
1 on the DB2 track appears to be an overwhelming disadvantage.
Furthermore, the combination of the centrifugal and interaction effects leads to the center
lanes being the most favorable, followed by the outside lanes, with the inside lanes being the
worst. These results fit very well with the IAAF rules for lane draws, which follow this prefer-
ence order.
Appendix: Track shape details
Note that each runner is assumed to run at a distance of 30cm from the inner limit of the lane.
This is how the radius of the circular parts is set in order to obtain a 400m distance on lane 1.
Standard track
The standard track is made up of a circular half-circle of length lc = 115.61m followed by a
straight of 84.39m, for a total distance of 200m, which yields
R1 ¼ lc=p ¼ 36:80:
Since the runner is assumed to be 30cm away from the boundary of the lane, the radius of
construction is R1 − 0.3.
We denote by Rk the radius for the runner on lane k. Since the width of a lane is 1.22m, the
radius at lane k is
Rk ¼ R1 þ 1:22ðk the same total distance. This yields the starting angle
y0
k ¼ 1:22ðk straight part and lc the length on the circle, we find
lc
R þ 2φðlÞ ¼ p:
Since the total angle for one clothoid is 2φðlÞ ¼ l=R, this equation leads to lc þl ¼ Rp.
Moreover ls þ lc þ 2l ¼ d where d is the distance of the race, that is 200m in our case, thus
Rp ¼ d On a curved track, this approximation is adjusted by projecting the two runners on a
median circle, while also taking into account the staggered start on different lanes:
rðsÞ ¼ ðy2ðsÞ 14.
Mathis F. The Effect of Fatigue on Running Strategies. SIAM review. 1989; 31(2):306–309. https://doi.
org/10.1137/1031054
15.
Quinn MD. The effects of wind and altitude in the 400-m sprint. Journal of sports sciences. 2004; 22(11-
12):1073–1081. https://doi.org/10.1080/02640410410001730016
16.
Woodside W. The optimal strategy for running a race (a mathematical model for world records from 50
m to 275 km). Mathematical and computer modelling. 1991; 15(10):1–12. https://doi.org/10.1016/0895-
7177(91)90086-M
17.
Lemercier S, Jelic A, Kulpa R, Hua J, Fehrenbach J, Degond P, et al. Realistic following behaviors for
crowd simulation. In: Computer Graphics Forum. vol. 31. Wiley Online Library; 2012. p. 489–498.
18.
Bonnans F, Martinon P, Giorgi D, Grelard V, Maindrault S, Tissot O. BOCOP—A toolbox for optimal
control problems;.
19.
Morin JB, Slawinski J, Dorel S, Couturier A, Samozino P, Brughelli M, et al. Acceleration capability
in elite sprinters and ground impulse: Push more, brake less? Journal of biomechanics. 2015;
48(12):3149–3154. https://doi.org/10.1016/j.jbiomech.2015.07.009 PMID: 26209876
20.
Hoogkamer W, Kipp S, Frank JH, Farina EM, Luo G, Kram R. A Comparison of the Energetic Cost of
Running in Marathon Racing Shoes. Sports Medicine. 2018; 48(4):1009–1019. https://doi.org/10.1007/
s40279-017-0811-2 PMID: 29143929
21.
Hanon C, Thomas C. Effects of optimal pacing strategies for 400-, 800-, and 1500-m races on the
_
VO2
response. Journal of sports sciences. 2011; 29(9):905–912. https://doi.org/10.1080/02640414.2011.
562232 PMID: 21547833
22.
Spencer MR, Gastin PB. Energy system contribution during 200-to 1500-m running in highly trained ath-
letes. Medicine & Science in Sports & Exercise, (2001), 33(1), 157–162. https://doi.org/10.1097/
00005768-200101000-00024
23.
Gastin PB. Energy system interaction and relative contribution during maximal exercise. Sports medi-
cine, (2001), 31(10), 725–741. https://doi.org/10.2165/00007256-200131100-00003 PMID: 11547894
24.
Aftalion A, Despaigne LH, Frentz A, Gabet P, Lajouanie A, Lorthiois MA, et al. How to identify the physi-
ological parameters and run the optimal race. MathS In Action. 2016; 7:1–10. https://doi.org/10.5802/
msia.9
25.
Samozino P, Rabita G, Dorel S, Slawinski J, Peyrot N, Saez de Villarreal E, et al. A simple method for
measuring power, force, velocity properties, and mechanical effectiveness in sprint running. Scandina-
vian Journal of Medicine & Science in Sports. 2016; 26(6):648–658. https://doi.org/10.1111/sms.12490
26.
The Physics Of Running. https://wwwreal-world-physics-problemscom/physics-of-runninghtml;.
27.
Appert-Rolland C, Hilhorst HJ, Aftalion A. Nash equilibrium in a stochastic model of two competing ath-
letes. Journal of Statistical Mechanics: Theory and Experiment. 2018; 2018(5):053401. https://doi.org/
10.1088/1742-5468/aabc7e
28.
Hilhorst H, Appert-Rolland C. Mixed-strategy Nash equilibrium for a discontinuous symmetric N-player
game. Journal of Physics A: Mathematical and Theoretical. 2018; 51(9):095001. https://doi.org/10.
1088/1751-8121/aaa883
29.
Crews DJ. Psychological state and running economy. Medicine & Science in Sports & Exercise. 1992;.
https://doi.org/10.1249/00005768-199204000-00014
30.
Stones M. Running under conditions of visual input attenuation. International Journal of Sport Psychol-
ogy. 1980; 11(3):172–179.
31.
Pugh LGE. The influence of wind resistance in running and walking and the mechanical efficiency of
work against horizontal or vertical forces. The Journal of physiology, 1971, 213(2), 255–276. https://
doi.org/10.1113/jphysiol.1971.sp009381 PMID: 5574828
32.
Gollmann L, Maurer H. Theory and applications of optimal control problems with multiple time-delays.
Journal of Industrial and Management Optimization. 2014; 10(2):413–441. https://doi.org/10.3934/jimo.
2014.10.413
33.
Boccia A, Vinter RB. The Maximum Principle for Optimal Control Problems with Time Delays. IFAC-
PapersOnLine. 2016; 49(18):951–955. https://doi.org/10.1016/j.ifacol.2016.10.290
34.
Guinn T. Reduction of delayed optimal control problems to nondelayed problems. Journal of Optimiza-
tion Theory and Applications. 1976; 18(3):371–377. https://doi.org/10.1007/BF00933818
35.
Trelat, E., Private communication.
36.
Morton RH. Statistical effects of lane allocation on times in running races. Journal of the Royal Statistical
Society: Series D (The Statistician), 1997, 46(1), 101–104.
Optimizing running a race on a curved track
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Optimizing running a race on a curved track.
Aftalion, Amandine,Martinon, Pierre