A two-runners model: optimization of running strategies...

1
A two-runners model: optimization of running strategies according to the physiological parameters Camilla Fiorini Université de Versailles Saint-Quentin-en-Yvelines [email protected] Aftalion and Bonnans’ model [2]: ˙ x ( t )= v x (0)= 0, x ( T )= D ˙ v ( t )= f ( t ) - v ( t ) τ v (0)= 0, ˙ e ( t )= σ ( e ) - f ( t ) v ( t ) e (0)= e 0 , (1) x ( t ) position at time t; v ( t ) velocity at time t; e ( t ) anaerobic energy at time t; τ constant coefficient which models the friction effects, linear in v σ = σ ( e ) oxygen uptake ˙ VO2 (Figure below). e 0 - e [J/kg] 0 (1- φ) e 0 (1 - e cr )e 0 e 0 m 2 /s 3 r f max Piecewise linear Smoothed curve Physiological constraints: e ( t ) 0 t 0; (2) f ∈F : = { f :0 f ( t ) f M t 0}. (3) AIM: solving (1)-(2)-(3) in such a way that, given a distance D, the time T to reach it is minimal. Optimal control problem: f control variable; T cost functional to be minimized; Resulting problem: min f ∈F T ( f ) s.t (1)-(2). (4) Single runner model Our model: for i = 1, 2 ˙ x 1 = v 1 x 1 (0)= 0 ˙ x D = v 2 - v 1 x D (0)= 0 ˙ v 1 = f 1 - v 1 τ 1 - cv 2 1 (1 - γ ( e -α( x D - β) 2 )) v 1 (0)= 0 ˙ v 2 = f 2 - v 2 τ 2 - cv 2 2 (1 - γ ( e -α( x D + β) 2 )) v 2 (0)= 0 ˙ e i = σ i ( e i ) - f i v i e i (0)= e 0 i , (5) the subscript i refers to the runner; x D ( t ) : = x 2 ( t ) - x 1 ( t ) distance between the runners at time t. Boundary condition: ( x 1 ( T ) - D )( x 2 ( T ) - D )= 0. (6) The physiological constraints (2) and (3) do not change, however the value f M depends on the runner. x D [m] β 1-γ 1 1-γ (e -α (x D -β ) 2 ) The term 1 - γ e -α( x D ± β) 2 , shown in the figure above, encompasses both friction and a psy- chological factor, which consists in trying to follow one’s competitor, in order to be able to overtake. It is a potential which has a minimum at distance β behind and decreases global fric- tion because it increases the will to follow. On the other hand, when the other runner is too far, there is no benefit. Two-runners models We minimize the following quantity, given a proper constant weight c w > 0: J ( f 1 , f 2 )= T + c w | x D ( T ) | . (7) The resulting problem is: min f i F i J s.t. (5)-(6)-(2), (8) where F i is the set of the admissible controls which depends on the athlete and is defined as follows: F i : = { f :0 f ( t ) f M,i , | ˙ f ( t ) |≤ K i t (0, T ) }. Optimization problem Mathematical models All the results presented in this section are obtained with the free software BOCOP [4]. [J/kg] [N/kg] normalized time 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1.5 -1 -0.5 0 0.5 1 1.5 2 x D [m] Different initial energies: e 0 1 = 1400 J / kg and e 0 2 = 1275 J / kg. x 1 ( T )= 1498.13m T = 249.43s,(-2s w.r.t best performance running alone) Overtaking at 99% of the race. [J/kg] [N/kg] normalized time 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1.5 -1 -0.5 0 0.5 1 1.5 x D [m] Different initial τ : τ 1 = 1.33s τ 2 = 1.31s x 1 ( T )= 1498.82m T = 249.536s,(-2s w.r.t best performance running alone) x[m] 0 500 1000 1500 0 1 2 3 4 5 6 7 v[m/s] Runner-1 Runner-2 x[m] 0 500 1000 1500 3 3.5 4 4.5 5 f Runner-1 Runner-2 x[m] 0 500 1000 1500 0 200 400 600 800 1000 1200 1400 e Runner-1 Runner-2 x[m] 0 500 1000 1500 5 10 15 20 25 sigma[m 2 /s 3 ] Runner-1 Runner-2 [J/kg] [N/kg] normalized time 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5 0 5 10 15 20 25 30 35 40 45 x D [m] Stronger runner starts behind: τ 1 = 1.31s τ 2 = 1.33s T = 248.726s,(-1s w.r.t best performance running alone). Overtaking at 87% of the race. Real races: Beijing 2008: overtaking at 84.6% of the race; Rome 2014: overtaking at 96.9%; Singapore 2015: overtaking at 91.8%. Numerical results new model for a two-runners problem, which takes into account psychological factors; the numerical results show how a runner can improve his personal best performance by exploiting the advantage of running behind someone else; the major application for Olympic training could be for an athlete to estimate whether he should stay behind or lead, and when is the best time to overtake; the curvature of the track and the parameter identification are the aim of upcoming papers. Conclusion [1] A. A FTALION AND C. F IORINI , A two-runners model: optimization of running strategies according to the physio- logical parameters, submitted, 2015. [2] A. A FTALION AND J.-F. B ONNANS, Optimization of running strategies based on anaerobic energy and variations of velocity, SIAM Journal on Applied Mathematics, 74(5):1615-1636, 2014. [3] A. B. P ITCHER, Optimal strategies for a two-runner model of middle-distance running, SIAM Journal on Applied Mathematics, 70(4):1032-1046, 2009. [4] F. B ONNANS , D. G IORGI , V. G RELARD , S. M AINDRAULT, AND P. M ARTINON, BOCOP - A toolbox for optimal control problems. Mathematics and Social Sciences Workshop November 16 th and 17 th , 2015 - Imperial College, London References

Transcript of A two-runners model: optimization of running strategies...

A two-runners model: optimization of running strategiesaccording to the physiological parameters

Camilla FioriniUniversité de Versailles Saint-Quentin-en-Yvelines

[email protected]

Aftalion and Bonnans’ model [2]:x(t) = v x(0) = 0, x(T) = D

v(t) = f (t)− v(t)τ v(0) = 0,

e(t) = σ(e)− f (t)v(t) e(0) = e0,

(1)

• x(t) position at time t; v(t) velocity at time t; e(t) anaerobic energy at time t;

• τ constant coefficient which models the friction effects, linear in v

• σ = σ(e) oxygen uptake VO2 (Figure below).

e0 - e [J/kg]

0 (1- φ) e0

(1 - ecr

)e0

e0

m2/s

3

r

f

max Piecewise linearSmoothed curve

Physiological constraints:e(t) ≥ 0 ∀t ≥ 0; (2)

f ∈ F := { f : 0 ≤ f (t) ≤ fM ∀t ≥ 0}. (3)AIM: solving (1)-(2)-(3) in such a way that, given a distance D, the time T to reach it is minimal.Optimal control problem:

• f control variable;

• T cost functional to be minimized;

Resulting problem:minf∈F

T( f ) s.t (1)-(2). (4)

Single runner model

Our model: for i = 1, 2

x1 = v1 x1(0) = 0xD = v2− v1 xD(0) = 0v1 = f1− v1

τ1− cv2

1(1− γ(e−α(xD−β)2)) v1(0) = 0

v2 = f2− v2τ2− cv2

2(1− γ(e−α(xD+β)2)) v2(0) = 0

ei = σi(ei)− fivi ei(0) = e0i ,

(5)

• the subscript i refers to the runner;

• xD(t) := x2(t)− x1(t) distance between the runners at time t.

Boundary condition:(x1(T)− D)(x2(T)− D) = 0. (6)

The physiological constraints (2) and (3) do not change, however the value fM depends on therunner.

xD

[m]β

1-γ

1

1-γ(e-α(x

D-β)

2

)

The term 1 − γe−α(xD±β)2, shown in the figure above, encompasses both friction and a psy-

chological factor, which consists in trying to follow one’s competitor, in order to be able toovertake. It is a potential which has a minimum at distance β behind and decreases global fric-tion because it increases the will to follow. On the other hand, when the other runner is too far,there is no benefit.

Two-runners models

We minimize the following quantity, given a proper constant weight cw > 0:

J( f1, f2) = T + cw|xD(T)|. (7)

The resulting problem is:

minfi∈Fi

J s.t. (5)-(6)-(2), (8)

where Fi is the set of the admissible controls which depends on the athlete and is defined asfollows:

Fi := { f : 0 ≤ f (t) ≤ fM,i, | f (t)| ≤ Ki ∀t ∈ (0, T)}.

Optimization problem

Mathematical models

All the results presented in this section are obtained with the free software BOCOP [4].

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[J/kg]

[N/kg]

normalized time

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

2

xD

[m]

•Different initial energies: e01 = 1400J/kg and e0

2 = 1275J/kg.

• x1(T) = 1498.13m

• T = 249.43s, (−2s w.r.t best performance running alone)

•Overtaking at 99% of the race.

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[J/kg]

[N/kg]

normalized time

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

xD

[m]

•Different initial τ: τ1 = 1.33s τ2 = 1.31s

• x1(T) = 1498.82m

• T = 249.536s, (−2s w.r.t best performance running alone)

x[m]0 500 1000 15000

1

2

3

4

5

6

7v[m/s]

Runner-1Runner-2

x[m]0 500 1000 15003

3.5

4

4.5

5f

Runner-1Runner-2

x[m]0 500 1000 15000

200

400

600

800

1000

1200

1400e

Runner-1Runner-2

x[m]0 500 1000 15005

10

15

20

25sigma[m2/s3]

Runner-1Runner-2

[J/kg]

[N/kg]

normalized time

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-5

0

5

10

15

20

25

30

35

40

45

xD

[m]

• Stronger runner starts behind: τ1 = 1.31s τ2 = 1.33s

• T = 248.726s, (−1s w.r.t best performance running alone).

•Overtaking at 87% of the race.

Real races:

• Beijing 2008: overtaking at 84.6% of the race;

• Rome 2014: overtaking at 96.9%;

• Singapore 2015: overtaking at 91.8%.

Numerical results

•new model for a two-runners problem, which takes into accountpsychological factors;

• the numerical results show how a runner can improve his personalbest performance by exploiting the advantage of running behindsomeone else;

• the major application for Olympic training could be for an athleteto estimate whether he should stay behind or lead, and when isthe best time to overtake;

• the curvature of the track and the parameter identification are theaim of upcoming papers.

Conclusion

[1] A. AFTALION AND C. FIORINI, A two-runners model: optimization of running strategies according to the physio-logical parameters, submitted, 2015.

[2] A. AFTALION AND J.-F. BONNANS, Optimization of running strategies based on anaerobic energy and variationsof velocity, SIAM Journal on Applied Mathematics, 74(5):1615-1636, 2014.

[3] A. B. PITCHER, Optimal strategies for a two-runner model of middle-distance running, SIAM Journal on AppliedMathematics, 70(4):1032-1046, 2009.

[4] F. BONNANS, D. GIORGI, V. GRELARD, S. MAINDRAULT, AND P. MARTINON, BOCOP - A toolbox for optimalcontrol problems.

Mathematics and Social Sciences WorkshopNovember 16th and 17th, 2015 - Imperial College, London

References