Optimal speed in Thoroughbred horse racing
aa r X i v : . [ phy s i c s . pop - ph ] J un Pacing strategy in horse racing
Quentin Mercier , Amandine Aftalion , Ecole des Hautes ´Etudes en Sciences Sociales, Centre d’Analyse et de Math´ematiqueSociales, CNRS UMR-8557, 54 boulevard Raspail, Paris, France.* [email protected]
Abstract
Thanks to velocity data on races in Chantilly (France), we set a mathematical model whichprovides the optimal pacing strategy for horses on a fixed distance. It relies on mechanics,energetics (both aerobic and anaerobic) and motor control. We identify the parametersuseful for the model from the data. Then it allows to understand the velocity, the oxygenuptake evolution in a race, as well as the energy or the propulsive force and predict thechanges in pacing according to the properties (altitude and bending) of the track.
Introduction
Very little is known about the optimal strategy for a horse to run and win a race. Inparticular, due to limitations in the measurement of the mean oxygen uptake ( ˙
V O
2) for ahorse at high exercise, no information is available on the full ˙
V O
V O max ) much quicker than humans, nevertheless no preciseestimate of the time needed to reach a steady state ˙ V O
V O
V O
V O Materials and methods
Data.
The data consist of two dimensional position and speed sampled at 10 Hz for horsesracing in Chantilly (France), at the end of the 2019 thoroughbred horse racing season. Theyare provided by France Galop and are from roughly ten races. The tracking system isdeveloped by Mc Lloyd. It is a miniaturized device which does not bring any discomfortto the horse or the jockey. Reliable data is obtained thanks to a patented positioningtechnology and robust mobile network data transmission, even during crowded events. Thedevice mean accuracy is 25cm or 2 hundredth of a second. The accuracy of the system wasvalidated on horses by comparing 1ms-accurate photo finish data on more than one hundredraces with gap data on the finish line obtained after processing latitude and longitude data.The tracking system provides the latitude and longitude data sampled at 10Hz, as well as thevelocity. The latitude and longitude data given by the tracker are projected over a referencetrack leading to the position of horses in the race.
Model.
Once we have these raw data, we have to smooth the speed using a third orderSavitzkyGolay filter. Therefore, the raw data provide two curves for each horse sampled at10Hz: • the curve of time vs distance from start, projected on the reference track, so thateach horse runs the same distance, • the curve of velocity vs distance from start, projected on the reference track.We use the model developed by [10, 11, 12] for the optimal strategy in running and adapt itfor horses. For a fixed distance, this model predicts the final time, the velocity curve and theeffort developed by the horse to produce the optimal strategy. It depends on the geometryof the track, the ramps and slopes and the physiology of the horse. This physiology is takeninto account through a number of parameters that have to be identified numerically for eachhorse thanks to the data: • the maximal propulsive force per unit of mass f M , • the global friction coefficient τ which encompasses all kinds of friction, both fromjoint and track. In total, f M τ is the maximal velocity, • the maximal decrease rate and increase rate of the propulsive force which is relatedto the motor control of the horse: a horse, like a human being, cannot stop or startits effort instantaneously, but needs some time or distance to do it. This is what ourcontrol parameters u − and u + will provide, • the total anaerobic energy or maximal accumulated oxygen deficit e , • the ˙ V O d at which the max-imum of ˙ V O d at which ˙ V O σ ( s ) where s is therelative distance from start, but in fact, in the model, it is a curve σ ( e ( s )) where e ( s )is the remaining anaerobic energy. The profile of σ is to be identified from the data.For fixed values of these parameters, the model of [10, 11, 12] adapted to horses yields anoptimal control problem based on a system of coupled ordinary differential equations forthe instantaneous velocity v ( s ), the propulsive force per unit of mass f ( s ), the anaerobicenergy e ( s ), where s is the distance from start. The system relies on Newton’s second lawof motion taking into account the positive slope or negative ramp α ( s ), the energy balance(between the aerobic contribution ˙ V O σ ( e ), the anaerobic contribution e ( s ) and the power developed by the propulsive force) and finally the control on the variations of thepropulsive force. A crucial piece of information to be taken care of is the centrifugal forcein the bends. This force does not act as such in the equation of movement but limits thepropulsive force through a constraint which yields a decrease in the effective propulsive forcein the bends.For the ease of completeness, we provide below the full optimal control problem though theresults of the paper do not require to understand it and the reader can skip this paragraphas a first reading. Instead of writing the equations of motion in the time variable, we writethem using the distance from start s . This amounts to dividing by v the derivatives in timein order to get the derivatives in space. We also write the equations per unit of mass. Let d be the length of the race and g = 9 .
81 the gravity. Let c ( s ) denote the curvature at distance s from the start, which is provided for each track and α ( s ) be the slope coefficient. Let v bethe initial velocity. Let e , f M , τ , u − , u + and the function σ be given. They are identified fora specific race and horse. The optimal control problem (where minimizing the final time isequivalent to minimizing the integral of the inverse of the velocity) coming from the Newtonlaw of motion and the energy conservation ismin v,f,e,u Z d v ( s ) ds, where v ′ ( s ) = 1 v ( s ) (cid:18) − v ( s ) τ + f ( s ) − gα ( s ) (cid:19) , v (0) = v ,e ′ ( s ) = σ ( e ( s )) − f ( s ) v ( s ) v ( s ) , e (0) = e , e ( s ) ≥ , e ( d ) = 0 ,f ′ ( s ) = u ( s ) v ( s ) , u − ≤ u ( s ) ≤ u + , and under the state constraint f ( s ) + v ( s ) · c ( s ) ≤ f M for s ∈ [0 , d ].The optimal control problem for horse performance is solved using Bocop, an open licencesoftware developed by Inria-Saclay France [13]. This yields the optimal strategy dependingon the length of the race, including the velocity profile and the ˙ V O f M , τ , u + , u − , e , σ ( e ) from the availabledata. Identification process.
The identification process is made through a bi-level optimizationprocedure looking for minimizing errors between the response of a Bocop simulation and thedata through the following objectivemin p | t simu ( d ) − t data ( d ) | + 1 N N X i =1 ( v simu ( s i ) − v data ( s i )) ! , where the subscript simu (resp. data ) refers to the variable extracted from the simulation(resp. data). The distance s = 0 is the beginning of the race and s N = d is the length of therace, while s i are intermediate distances. The parameter p refers to the vector containingall the variables to identify: p = ( τ, e , f M , d , d , σ M , σ f , u − , u + ) , where σ M is the maximal value of σ and σ f its final value. The objective is made up of twoparts: first the difference in final time at the end of the race d , and then the mean squareerror over the speed measured at N points. For our identification process, N is taken equalto one thousand and the points are evenly distributed between s and s N .The algorithm used here is a particle swarm optimization method [14] available in thepyswarm library in Python 3, part of the family of the heuristic optimization methods[15]. The main advantage of such a method is its good ability to explore the design spaceand its ease of use and implementation. A swarm of designs { p i } is tested, that is theoptimal control problem is solved for these parameters using Bocop, and its objective valueis calculated. The performance influences the speed and the direction of the particles insidethe design space for the next iteration. At the end of the process, the best score particle iskept as the result. The stopping criterion of the algorithm is set such that the algorithmis unable to find new particles for which the objective is at least 10 − better than the bestscore observed until then. For all the examples treated in this paper, the swarm size is set to50 particles and a maximum of 150 iterations. Each identification process has reached thestopping criterion described. This method is well suited to our problem since the objectivespace has a lot of local minimizers so that gradient based methods can get stuck in localminimizers. It insures overall robustness in the results as the design space is always wellexplored before converging toward a particular area of the design space. Topology of the tracks.
We have studied three types of races: a 1300 meters, a 1900meters and a 2100 meters in Chantilly. The GPS track is shown in Fig 1. The 1300m starts m m m finish linestart m start m start m Fig 1.
GPS track of the 1300m, 1900m and 2100m in Chantilly, France.with a straight, then there is a bend before the final straight. The 1900m starts earlier withan almost straight and follows the 1300m. The 2100m starts in the final straight of the otherraces, makes a closed loop before reaching the same final straight and finish line.The elevation and curvature profiles are provided by France Galop and illustrated in Fig2. The tracks are made up of straights (zero curvature), arcs of circles (constant curvature)and clothoids (curvature increasing linearly with distance). A clothoid is the usual way tomatch smoothly a straight and an arc of a circle since the curvature increases linearly. It isused for train tracks and roads as well. It allows smooth variations of velocities which aremore comfortable for horses.The specificity of the track is that there is a bend of 500 meters before the final straight,where the track is going down in the first quarter (about 1 . . . . . . A l t i t u d e ( m ) (a) . . . . . A l t i t u d e ( m ) (b) . . . . . A l t i t u d e ( m ) (c) Distance ( m ) . . . C u r v a t u r e ( m − ) Distance ( m ) . . . C u r v a t u r e ( m − ) Distance ( m ) . . . C u r v a t u r e ( m − ) Fig 2.
Altitude in meters (first line) and curvature (second line) vs distancefor different tracks: (a) 1300m (first column), (b) 1900m (second column)and (c) 2100m (third column). The last 1300m are always the same.start, which is first going up and then down. We will see that curvature and altitude havea strong effect on the pacing.Let us point out that the track is banked but, because data correspond to horses close tothe inner part of the track, the banking is not meaningful for the data and will not be takeninto account here.
Results
We have chosen three significant races of 1300m, 1900m and 2100m. For each one, wehave taken the data of a horse which seems to have run an optimal race. We describe belowthe results of our simulations.1300 meters.
The parameters identified for this race are in Table 1. The velocity data τ e f M d d σ M σ f u − u + Table 1.
Identified parameters for the 1300m(raw and smoothed) and the velocity computed with our model are plotted in Fig 3 for the1300m. We observe a very good match between the curves: there is a strong start with themaximal velocity being reached in 200 meters. Then the velocity decreases, and in particularin the bend, between 300 and 600 meters from start. Though the track is going down, thecentrifugal force reduces the propulsive force as we see in Fig 4b (the black curve shows thelimitation due to the centrifugal force). It is only when reaching the clothoid, before the finalstraight, after 600 meters, that the horse can speed up again. The end of the race is uphilland the velocity decreases though the horse reaches the straight. Nevertheless, a decrease invelocity at the end of such a race takes place even on a flat track.The ˙
V O σ in J/s/kg but we want to plot the results in terms of ml/mn/kg knowing that
Distance ( m ) . . . . . . . . . V e l o c i t y ( m · s − ) Raw dataSmoothed dataIdentification
Fig 3.
Velocity data (raw data and smoothed data, t f = 76 . t f = 76 . kJ (see [16] and also [17]), so we have to multiplyour data for σ by 60 /
21. For a maximal value of σ equal to 47, this yields a ˙ V O max of133.6 ml/mn/kg . We see that the ˙ V O
V O
Distance ( m ) . . . . . . V O ( m l · m n − · k g − ) (a) Distance ( m ) . . . . . . . . . F o r c e ( N · m − ) (b) f M f tot f ( s ) Fig 4. ˙ V O f ( s ) in the direction of movement, black is theeffective propulsive force f tot = p f + c v taking into account thecentrifugal force, where c is the curvature.1900 meters. The parameters identified for this race are in Table 2. The velocity data (rawand smoothed) and the velocity computed with the model are plotted in Fig 5 for the 1900meters. We observe that there is a strong start with the maximal velocity being reached in300 meters. Then the velocity decreases. Between 900 and 1400m, we see the effect of the τ e f M d σ M d σ f u − u + Table 2.
Identification parameters for the 1900 metersbend: at the beginning of the bend, the track is going down and the horse slightly speedsup; then the centrifugal force reduces the velocity but the velocity increases again at the endof the bend. The end of the race is with a strong acceleration before the final slight slowdown. The ˙
V O
Distance ( m ) . . . . . . . . . V e l o c i t y ( m · s − ) Raw dataSmoothed dataIdentification
Fig 5.
Velocity data (raw and smoothed, t f = 116 . t f = 116 . Distance ( m ) . . . . . . V O ( m l · m n − · k g − ) (a) Distance ( m ) . . . . . . . F o r c e ( N · m − ) (b) f M f tot f ( s ) Fig 6. ˙ V O f ( s ) in the direction of movement, black is theeffective propulsive force f tot = p f + c v taking into account thecentrifugal force, where c is the curvature.We see that ˙ V O
V O the ˙
V O
Distance ( m ) . . . . V e l o c i t y ( m · s − ) No topology t f = . sslope only t f = . sCurvature only t f = . sAll topology t f = . s Fig 7.
Effect of the slope and curvature on the velocity curve (zoom) for the1900 meters. Brown is the velocity curve of the race, red with the slope only(straight track), purple with curvature only (flat track) and green is a flat,straight track.
The parameters identified for this race are in Table 3. The velocity data and τ e f M d σ d σ M d σ f u − u + Table 3.
Identification parameters for the 2100 metersthe computed velocity are plotted in Fig 8 for the 2100 meters. The ˙
V O
V O
V O
V O [19] and is related to a turnpike phenomenon. It is very likely that the horse pacing strategyfor long races can be analyzed with this mathematical tool as well.
Distance ( m ) . . . . . . . . V e l o c i t y ( m · s − ) Raw dataSmoothed dataIdentification
Fig 8.
Velocity data (raw and smoothed t f = 130 . t f = 130 . Distance ( m ) . . . . . . . V O ( m l · m n − · k g − ) (a) Distance ( m ) . . . . . . . . F o r c e ( N · m − ) (b) f M f tot f ( s ) Fig 9. ˙ V O f ( s ) in the direction of movement, black is theeffective propulsive force f tot = p f + c v taking into account thecentrifugal force, where c is the curvature.The first bend has a strong curvature and therefore reduces drastically the velocity as wecan see in Fig 9b: the propulsive force is reduced in the first bend. In the last bend, as inthe previous race, the velocity decreases and increases again at the end of the bend. Theend of the race is similar to the 1300 meters, with a strong acceleration before the final slowdown. The horse in this race is not as good in terms of performance as the one in the 1900mand he cannot maintain his velocity similarly at the end of the race. Discussion
Results on ˙ V O . From experiments on human races [18, 20], it is expected that the ˙
V O • increasing to a maximal value and then decreasing for short exercises, • increasing and reaching the maximal value ˙ V O max , and then decreasing at the endof the race when the residual anaerobic energy is less than 30%, • reaching a peak value which is higher than the value along the race for moderatelength exercises.Our simulations and identifications yields that the behaviour is the same for horses. Theresults of our simulations even provide precise information on the ˙ V O • the maximal value of ˙ V O • the 1300 meters is a short exercise where ˙ V O • for the 1900 and 2100 meters race, the ˙ V O
V O
V O • in a 2100 meters race, the ˙ V O
V O max , the better the performanceis. Because the change of slope in ˙ V O
V O
V O
V O
V O
V O v ˙ V O max can allow a faster start velocity without increasing the O deficit.As soon as there is a slope or ramp, the ˙ V O
V O
V O Energy.
Horses have two distinct types of energy supply, aerobic and anaerobic. In ourmodel, it is e which estimates the anaerobic energy supply. For human races, recent researchsuggest that energy is derived from each of the energy-producing pathways during almostall exercise activities [22], which is what we also observe in our simulations.In Table 4, we have computed the percentage of anaerobic energy to the total energyaccording to the length and duration of the race. The horse of the 1900m race has a verystrong ˙ V O max and therefore uses a lower anaerobic energy. We point out that the valuesestimated in [1] on a treadmill seem to be under estimated with respect to ours: for anexercise of duration 130 seconds, they find an anaerobic contribution of around 30%, whichis smaller than our value. Indeed, in a race, for a similar duration of exercise, velocities aremuch higher than on a treadmill, leading to a bigger contribution of the anaerobic supply[22]. Race length ( m ) duration ( s ) ˙ V O . . . Table 4.
Percentage of anaerobic contribution in the total energy during therace
Effect of slopes, ramps and bends on a race.
As evident from the data of [23], on atight bend, horses slow down a lot. In Chantilly, the bends have a radius of at most 100m,which is not tight, but still has a strong impact on the pacing strategy.To better illustrate this effect, we choose a race of 1900 meters and set an imaginary slopeor ramp or bend for one third of the race at the beginning, middle or end of the race (thatis roughly 630m) with the following configuration : • either a positive slope of +3% for 630m, • or a negative slope of −
3% for 630m, • or an arc of circle with a curvature of 1 / m − for 630m.Fig 10, 11 and 12 provide the optimal velocity vs distance for the 1900 meters parameters.The common feature is that a local change of elevation or bend does not only produce alocal change in velocity but changes drastically the whole velocity profile and mean value.For a slope going up, as illustrated in Fig 10, the best time is obtained when the slope isat the end of the race. Indeed, a good horse can still provide a strong effort at the end ofthe race, even if he is tired. If the slope is at the beginning, it has a strong effect on thevelocity which cannot reach its maximum value. In the middle of the race, the slope reducesthe mean velocity and therefore the final time.For a ramp going down, as illustrated in Fig 11, it is the opposite: the best time is obtainedwhen the ramp is at the beginning of the race. Indeed the horse speeds up more easily andmore quickly.For the bend, as illustrated in Fig 12, the best time is obtained when the bend is at themiddle of the race. This is where it has the smallest decrease on the velocity profile. At thebeginning, it prevents the horse from reaching its maximal speed. Of course, here the effectis exaggerated with respect to a real race because the bend is long but it yields the generalflavour. Similarly, at the end, it prevents the horse from sprinting. Distance ( m ) V e l o c i t y ( m · s − ) notopo t f = 116 . s begin t f = 118 . s middle t f = 117 . s end t f = 117 . s Fig 10.
Optimal velocity profile for a 1900m: flat straight track (blue), +3%slope for 630m at the beginning of the race (orange), +3% slope for 630m atthe middle of the race (green), +3% slope for 630m at the end of the race(red).
Distance ( m ) V e l o c i t y ( m · s − ) notopo t f = 116 . s begin t f = 114 . s middle t f = 114 . s end t f = 115 . s Fig 11.
Optimal velocity profile for a 1900m: flat straight track (blue), +3%ramp down for 630m at the beginning of the race (orange), +3% ramp for630m at the middle of the race (green), +3% slope for 630m at the end of therace (red).Therefore, we have seen that the pacing strategy has to be analyzed with respect to thechanges of slopes, ramps or bends in order to optimize the horse effort. For a given track, it isvery likely that the turnpike theory of [19] should yield more precise and detailed analyticalestimates of the increase of decrease of velocity. Distance ( m ) V e l o c i t y ( m · s − ) notopo t f = 116 . s begin t f = 118 . s middle t f = 116 . s end t f = 117 . s Fig 12.
Optimal velocity profile for a 1900m: flat straight track (blue), bendfor 630m at the beginning of the race (orange), bend for 630m at the middleof the race (green), bend for 630m at the end of the race (red).
Conclusion
Thanks to precise velocity data obtained on different races, we are able to set a mathe-matical model which provides the optimal pacing strategy for horses on a fixed distance. Itrelies on both mechanical, energetic considerations and motor control. The process consistsin identifying the physiological parameters of the horse from the data. Then the optimal con-trol problem provides information on the pacing. We see that horses have to start stronglyand reach a maximal velocity. The velocity decreases in the bends; when going out of thebend, the horse can speed up again and our model can quantify exactly how and when. Thehorse that slows down the least at the end of the race is the one that wins the race. Weunderstand from the optimal control problem that this slow down is related to the anaero-bic supply, the ˙
V O max and the ability to maintain maximal force at the end of the race.Therefore, horses that have a tendency to slow down too much at the end of the race shouldput less force at the beginning and slow down slightly through the whole race in order tohave the ability to maintain velocity at the end.From our simulations, we are also able to get information on the ˙ V O
V O
V O
V O
V O Future works will be devoted to taking additionally into account drafting and the horsepsychology [24] since an alternative strategy can be to stay behind to save energy andovertake in the last straight [25].
Acknowledgments
The authors acknowledge support from the LabEx AMIES (ANR-10-LABX-0002-01) ofUniversit Grenoble Alpes, within the program ”Investissements d’Avenir” (ANR-15-IDEX-0002) operated by the French National Research Agency (ANR).The authors also wish to thank France Galop and Mc Lloyd for providing the data and fortheir interest in this work. Finally, they are very grateful to Pierre Martinon for his adviceon Bocop.
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