Optimizing the breakaway position in cycle races using mathematical modelling
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Abstract
In longdistance competitive cycling, efforts to mitigate the effects of air resistance can significantly reduce the energy expended by the cyclist. A common method to achieve such reductions is for the riders to cycle in one large group, known as the peloton. However, to win a race a cyclist must break away from the peloton, losing the advantage of drag reduction and riding solo to cross the finish line ahead of the other riders. If the rider breaks away too soon then fatigue effects due to the extra pedal force required to overcome the additional drag will result in them being caught by the peloton. On the other hand, if the rider breaks away too late then they will not maximize their time advantage over the main field. In this paper, we derive a mathematical model for the motion of the peloton and breakaway rider and use asymptotic analysis techniques to derive analytical solutions for their behaviour. The results are used to predict the optimum time for a rider to break away that maximizes the finish time ahead of the peloton for a given course profile and rider statistics.
Keywords
Mathematical model Air resistance Asymptotic analysis Optimization1 Introduction
Cycling science is a lucrative and competitive industry, in which small advantages are often the difference between winning and losing. For example, the 2018 Tour de France was won by a margin of less than one minute for a total race time of more than 86 h [10]. Improvements in performance typically require the combined expertise of a wide range of specialists, including sports scientists, engineers, and dieticians. In addition, mathematics can provide a fundamental underpinning for the race dynamics and strategy, to complement the results of more sophisticated analyses such as computational fluid dynamics simulations.
Longdistance cycle races, such as a stage of the Tour de France, typically follow a prescribed pattern: riders cycle together as a main group, or peloton, for the majority of the race before a solo rider or small group of riders makes a break from the peloton, usually relatively close to the finish line. The main reason for this behaviour is that cycling in a group reduces the air resistance that is experienced by a cyclist. With energy savings of up to around a third when cycling in the peloton compared with riding solo [8], it is energetically favourable to stick with the main field for the majority of the race. However, if a cyclist wishes to win a race or a Tour stage then they must decide on when to make a break. In doing so, the rider must provide an additional pedal force to offset the effects of air resistance that would otherwise be mitigated by riding in the peloton. However, the cyclist will not be able to sustain this extra force indefinitely, with fatigue effects coming into play. As a result, a conflict emerges: if the cyclist breaks away too soon then they risk fatigue effects kicking in before the finish line and being caught by the peloton. On the other hand, if the cyclist breaks too late then they reduce their chance of a large winning margin.
The mathematics of drag and air resistance in the context of cycling science is well known and has led to many studies considering strategies for minimizing this drag reduction, ranging from cycling behind a rider (see, for example, [3] for a summary) to the foot positioning on downhill sections [6]. Strategies for shortdistance races have been examined (see, for example, [9]) in which minute changes in tactics can be the difference between winning and losing. In longdistance races, many more tactics come into play, such as pacing [5, 18]. In such races, while drag reduction plays a much more important role, the combination of the knowledge of drag with strategies for winning a longdistance race are much less common, at least in the public literature.
In this paper we consider, for a given course profile and rider statistics, the optimum time to break away, which maximizes the time difference between the rider finishing and the peloton crossing the finish line. To answer this question we derive a mathematical model for cycling that captures the advantage of riding in the peloton to reduce aerodynamic drag and the physical limitations (due to fatigue) on the force that can be provided by the leg muscles.
We begin in Sect. 2 by forming a mathematical model for the motion of a single rider, which is extended in Sect. 3 to describe a group of cyclists and a breakaway rider, as seen in professional cycle races. The model for a single rider is derived by considering the forces acting on a cyclist by appealing to Newton’s Second Law. To make analytical progress, we use an asymptotic expansion that exploits the fact that variations from a mean course gradient are typically small. We then examine the validity of these solutions by comparing them to numerical calculations of the full mathematical model that do not rely on the assumption of small undulations. The asymptotic solutions also provide a method to draw direct relationships between the values of physical parameters and the time taken to cover a set distance.
The physiological factors that limit the force that muscles are able to provide are then explored in Sect. 4. We assume that the concentration of potassium ions in the muscle cells is a strong factor in the fatigue of the muscles after a period of exertion, and we form equations to model the evolution of force output over time. This is applied to a breakaway situation to understand how the muscles respond after a rider exerts a force above their sustainable level.
Finally, in Sect. 5, we seek an optimal breakaway strategy using the framework derived. We model a race situation with the main field of riders benefiting from a drag reduction and a breakaway rider applying a higher pedal force which decays with time and find the optimal position along a course to break away from the peloton, using the asymptotic solutions for both a constant breakaway force and accounting for fatigue effects.
2 Mathematical model for a single rider
2.1 Governing equation for the rider motion
We begin by forming a mathematical model for the onedimensional motion of a single rider. We characterize the rider’s motion by the distance travelled along the course, \(\hat{x}\), at time \(\hat{t}\), and the course undulations by the angle of incline, \({\theta }={\theta }(\hat{x})\) (see Fig. 1). Note we use hats to denote dimensional quantities and, for future reference, variables without hats will be dimensionless.

The force due to gravity, \(\hat{m}\hat{g}\sin {\theta }(\hat{x})\), where \(\hat{g}\) denotes acceleration due to gravity.

The frictional force, \(\hat{F}_{\text {f}}\), capturing the resistance of the tyres with the road and the sum of all mechanical resistances including from the chain and wheel bearings. Here we assume that \(\hat{F}_{\text {f}}\) is constant, as is also assumed by Martin [13] and Pivit [15].

The phenomenological drag force \(\hat{F}_{\text {D}}=\frac{1}{2}\hat{\rho }c_{\text {D}} \hat{A} (\text {d}\hat{x}/\text {d}\hat{t})^2\), where \(\hat{\rho }\) is the density of air, \(c_{\text {D}}\) is a drag coefficient, and \(\hat{A}\) is the frontal area of the bike and rider. This form of the drag law assumes that the air is stationary, but readily generalizes to allow for wind by replacing \((\text {d}\hat{x}/\text {d}\hat{t})^2\) with \((\text {d}\hat{x}/\text {d}\hat{t}\hat{v}_w)^2\), where \(\hat{v}_w\) is the air velocity in the positive \(\hat{x}\) direction (\(\hat{v}_w>0\) corresponds to a tailwind and \(\hat{v}_w<0\) corresponds to a headwind). In what follows we will assume that \(\hat{v}_{w} \equiv 0\). While the effects of crosswinds can be important, with riders forming echelons to shield each other from crosswinds, we will neglect this effect here as this plays a small role on the forward motion of riders

The pedalling force, \(\hat{F}_{\text {p}}\), which could depend in general on \(\hat{x}\), \(\text {d}\hat{x}/\text {d}\hat{t}\) and \(\hat{t}\) to reflect a varying power output. We will begin by assuming \(\hat{F}_{\text {p}}\) to be constant, which provides a good approximation for relatively constant speeds. In this case the power output, \(\hat{F}_{\text {p}}\text {d}\hat{x}/\text {d}\hat{t}\), is constant when travelling at constant speed and increases linearly with speed. We relax the assumption of constant pedal force in Sect. 4.
2.2 Nondimensionalization
2.3 Solution for a flat course
2.4 Approximation for a weakly undulating course
2.5 Results and comparison to numerical solutions
course  \(\theta _1\)  \(\theta _2\)  \(\theta _3\)  \(\theta _4\) 

\(\phantom {\Big [}\alpha\)  0.0123  \(0.0123\)  0  \(0.0037\) 
\(\varepsilon\)  0.0123  0.0123  0.02  0.0237 
f(x)  \(1 + 2.44x/(x+3)\)  \(1  2.44x/(x+3)\)  \(\sin (30x)\)  \(0.1560.844\sin (30x)\) 
The function ode45 in MATLAB is used to solve the full system (5) numerically without making any approximations on the course gradient. This can be used to check that the asymptotic solution derived in Sect. 2.4 is close to the actual solution by plotting solutions for the choices of \(\theta\) given above. The asymptotic solutions are compared with the numerical solutions to the full system and shown in Fig. 2b, c.
For smooth inclines and declines (\(\theta _1\) and \(\theta _2\)) the asymptotic approximation captures the behaviour well. The rapidly oscillating function \(\theta _3\) in (13) corresponds to a road that has many small hills and the asymptotic prediction also provides a good approximation, even though \(\theta\) is never close to being constant. This suggests that the asymptotic solution should be a good approximation if there are small undulations in the course. We do, however, see that issues can arise when the variations become too large, as seen for \(\theta _4\), and cumulative errors can arise. However, as discussed earlier, we are able to accommodate courses like this by separating them into sections of almost constant slope and approximating with different \(\alpha\) for each segment.
2.6 Time to complete a course
Again we use MATLAB to obtain numerical solutions as shown in Fig. 2d, and find good agreement with the asymptotic solution (15), with slight discrepancies arising for course \(\theta _4\) in the same way as seen in Fig. 2b, c.
Now that we have confidence in our asymptotic solutions we will use these results to compare the course completion time for a breakaway rider and the main field of riders. The explicit nature of the asymptotic solutions enable us to perform efficient parameter sweeps.
3 Modelling a group of riders and a breakaway
We now use the equations found in the previous section to seek an optimal strategy for a rider to break away from the main field of riders and finish in the shortest possible time. For simplicity we restrict our attention to the case of a single breakaway rider, who loses the benefit of drag reduction from being in the peloton. However, we note that the methods we use readily generalize for a small group of breakaway riders who benefit from drag reduction themselves.
We consider a course of length X and find the time it takes the main field to complete the course, \(T_\text {end}\). We then suppose a breakaway rider travels with the peloton up to some distance along the course, \(X_{\text {b}}\), before breaking away and travelling alone until the end of the course. Using this formulation we will find the time for the breakaway rider to complete the course, \({T}_{{\text {b}},\text {end}}\). We will seek the maximum difference in completion times, \(\Delta T_\text {end}=T_\text {end}{T}_{{\text {b}},\text {end}}\), by varying the breakaway position, \(X_{\text {b}}\).
3.1 Governing equations
In this section we assume the extra pedal force to be a constant, that is we set \(F_{\text {b}}=F_{\textrm{b}0}\). We will relax this assumption in Sect. 5, but for now we will find that this allows us to model the situation analytically using the solutions found in Sect. 2 and gives an explicit understanding of the parametric dependence of the system. We will also assume for simplicity that the main field of riders do not react to the breakaway in terms of altering their pedal force.
3.2 Model solutions for a constant breakaway force
3.3 Results
4 Fatigue effects when attempting a breakaway
In the previous section we assumed the breakaway force was constant to allow us to solve the system using asymptotic methods. In this section, we seek an improved model for how the force provided by a breakaway rider might vary.
4.1 Cause of the fatigue
As seen in Sect. 3.1, for a rider to break away from the peloton the excess force they apply, \(F_{\text {b}}\), must be high enough to counter the increase in drag experienced by a solo rider (Eq. 18). As a consequence of applying this excess force, fatigue effects will come into play, resulting in the force that the rider is actually able to exert subsequently dropping back towards the sustainable force of the peloton (or possibly even below for a short period of time while the rider recovers, as we will discuss later on). We expect the fatigue to play a role both on a short timescale, due to anaerobic respiration, preventing sustained bursts of speed, and on a longer time, due to depletion of energy reserves.
The shortterm fatigue effect is commonly attributed to lactic acid buildup as a consequence of anaerobic respiration. However, another cause of fatigue is the net movement of potassium ions (K\(^+\)) out of contracting skeletal muscle [12]. There are contrasting opinions on whether lactic acid or potassium has the larger effect on force output, as noted by Hargreaves [4]: “Generally, the lactate ion does not appear to have any major negative effects on the ability of skeletal muscle to generate force, although conflicting data exist in the literature. Of greater consequence is...the movement of strong ions (e.g., K\(^+\)) across the muscle cell membrane.” Here we choose to model the movement of potassium ions, but note that a similar mathematical relationship would apply if we were considering lactic acid and so this assumption will not affect our conclusions. In the following sections we incorporate the effect of fatigue due to anaerobic respiration and depletion of energy reserves.
4.2 Potassium transport model
4.3 Stamina model
The stamina effects considered in this model lead to an initial slow decrease in pedal force before the deficit potassium in the muscle cells has a severe effect on the force output, resulting in a subsequent rapid drop (Fig. 5). As expected, the addition of the stamina limitation results in the force dropping below the sustainable force, \(F_{\text {s}}\), for a marked period of time, before the potassium level recovers towards the threshold value, \(p_1\). In reality we might expect a rider to recover after a while, meaning that if they were caught by the peloton they could break away a second time. However, we are only concerned with single breakaways, as our model predicts no advantage of breaking away twice.
5 Numerical solutions under fatigue effects
We next study the effect of course contours, beginning with a course composed of a single valley. The optimal breakaway position depends strongly on the valley location as seen in Fig. 7. Figure 7b indicates that the best position to break away is generally just after the trough of the valley since the velocity of the main field of riders will be low as they travel uphill out of the valley, meaning they benefit less from the drag reduction. One exception to this is when the trough occurs very close to the end as seen in Fig. 7c, in which case breaking away earlier can result in a larger win margin to make the most of the extra force available. A second exception is when the valley occurs too far from the end of a race (for example \(x_\text {min}=6\) in Fig. 7a), where the advantage of breaking away in the valley is countered by the large subsequent time without slipstreaming, in which the breakaway rider has a velocity lower than that of the peloton. For valleys that occur very early on, the optimal breakaway position is \(x=X5.5\), which is the same as for the flat course. When the trough roughly coincides with this position (\(x_\text {min}\approx 14.5\) in Fig. 7c, as seen with \(x_\text {min}=14\) in Fig. 7a) we obtain the best possible win margin of all courses.
Finally, we consider the effect of the hill height. We fix the position of the maximum and vary the height and gradient of the course separately to understand when the hill offers a good breakaway opportunity. We emphasize that there is no restriction on the gradient of the hill, and so this analysis holds for shallow gradients or extreme mountain stages.
6 Conclusions
In this paper we have formulated a model for the motion of a cyclist in a race, to determine the optimal time to break away from the peloton. Using an asymptotic expansion for small deviations in the gradient of the course allowed us to produce an explicit expression for the time to complete a given course.
Initially we modelled the breakaway rider with a constant pedal force (higher than the peloton), which allowed us to use explicit asymptotic expressions. From this we obtained an equation for the time difference between the breakaway rider and the main field in terms of the breakaway position, allowing us to determine the optimal position. By considering a single hill and varying its characteristics we showed that breaking away when travelling uphill is optimal since rider velocities will be lower and thus the drag reduction in the peloton will be less significant. This analytical model can be used to predict the optimal position to break away in a given race for a given individual and course profile.
We then extended the analytical model by including fatigue effects in the pedal force, no longer assuming the pedal force of a breakaway rider would remain constant. We considered the movement of potassium ions out of the muscle cells, linking the potassium concentration to the force output and including a stamina limitation, which led to an equation describing the decay of the pedal force over time after a breakaway. This model must be solved numerically, but the asymptotic approximations enable us to perform parameter sweeps for the breakaway position on different course profiles to determine how the breakaway strategy may be altered depending on the steepness, length and position of hills along a course. The analysis imposed no constraints on the gradient of the hills, and so the methodology would hold for any course profile, including mountain stages.
The strategy for riders should be to break away at a position that allows a velocity higher than the peloton for as much of the course as possible. We found that the optimal way to achieve this is to break away just before the velocity is expected to be low for a sustained period of time, such as at the beginning of a long hill climb. Uphill sections allow the breakaway rider to create distance between themselves and the main field as a result of the reduced benefit from the drag reduction in the peloton. We also observed that if a hill occurs too early in the course the peloton will have time to catch up with a rider who uses the hill to break away, but an early uphill section may still be the optimal place to break away if it is sufficiently steep or long. As well as choosing a breakaway position where a velocity higher than the peloton can be sustained, it is of course also important to minimize the duration and magnitude of a breakaway rider velocity that is lower than the main field, where the gap will be decreasing. As a consequence of this, if an uphill section is followed by a downhill section, or if the uphill section is quite long, it may be optimal to break away halfway up the hill so that the extra force can be sustained up to the top of the hill.
This paper provides a framework for performing efficient parameter sweeps that determine the optimal strategy for breaking away in a cycling race. The methodology readily generalizes to more complex scenarios, for example, incorporating rider fitness and strategies (of which plentiful data will be available to a cycling team), the possibility of two or more riders breaking away from the main field together, and the peloton’s response to a breakaway. It would be prudent to investigate the influence of other physical and physiological factors, such as diet, nutrition, and training, which all play a vital role in rider performance [7]. Finally, it would be interesting to combine these ideas with the field of rider tactics, such as a rider faking a breakaway to tempt a reaction from riders in the main field. This opens up the enticing possibility that the type of modelling presented here may be combined with concepts from the field of noncooperative game theory (see, for example, [11]) to explore a wider class of winning strategies.
Notes
Acknowledgements
I.M.G. gratefully acknowledges support from the Royal Society through a University Research Fellowship.
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