Optimal Racing Strategies

1. Optimal Racing Strategies Brian Pike, North Carolina State University Satesha Augustin, University of Virgin Islands Steven Blumsack, Florida State University

2. Abstract The problem is to find the distribution of exerted power, p, that minimizes the time to traverse a distance in a race. Energy is produced from glucose through two mechanisms: aerobic and anaerobic. Though the aerobic pathway is more efficient, it is a process limited by the respiratory system; the maximum such power is the aerobic limit. Beyond this threshold, energy must be produced anaerobically with the build-up of lactic acid a by-product. Lactic acid reduces the efficiency of the use of energy. We assume that lactic acid is produced at a rate proportional to the difference between the power generated and the aerobic limit. Moreover, we assume that any lactic acid is gradually removed if the power demand is less than the aerobic limit. An obvious strategy is to perform at the aerobic limit. We investigated three scenarios in which exceeding the aerobic limit could be advantageous: a sprint to the finish near the end of a race, cooperative drafting in a bicycle race with the leader in an anaerobic phase and others partially recovering, and a performer going uphill in an anaerobic phase and partially recovering downhill.

3. Our model

4. Optimal Sprint to the Finish We consider the optimal strategy to race over flat terrain (s=0), using both numerical and analytic approaches. We assume for simplicity that a=b=1. Given a function p(x), we can use numerical integration techniques to approximate the amount of time an athlete will take to complete a race. We can then use minimization routines to search a vector space to minimize this amount of time. To do this, we must represent the possible p(x)’s as a vector space. For this problem, we made the elements of p(x) represent the effort given at evenly spaced locations along the path of the race, using linear interpolation to find p(x) at any particular point along the path.

5. Optimal Sprint to the Finish (cont.)

6. Optimal Drafting Strategy In the following scenario a team of two to ten bikers are involved in a race. The plan is for each biker to “draft” behind each other for a particular period of time, where at the end of each time interval each rider changes position (the leader will become the last, the second rider will become the first, and so on). The riders will continue this performance for a number of sequences until the end of the race so that each rider will become the leader at one point in the race. This implies that each rider will depend on anaerobic respiration, hence causing each rider to accumulate lactic acid. However, in this process of drafting, the biker will have an opportunity to recover, although not entirely, from the augmentation of lactic acid. We sought the minimum power the athletes should exert to obtain the maximum average speed at equilibrium. In the graphs below, time represents the time interval each biker remains in the lead position, ‘a’ represents the anaerobic state of the athlete. Within the first graph we can observe that varying the time intervals become trivial in the case of the anaerobic state of the athlete. Nevertheless, it is also clear that as the number of bikers increase so does the optimal power. The second graph illustrates the average speed of the total number of bikers. Thus, the optimal drafting strategy is to lead at a constant power throughout the race, slightly exceeding the aerobic limit.

7. Optimal Drafting Strategy (cont.)

8. Optimal Pacing over Hills We investigate the ideal way in which one should ride over sine-shaped hills in a bicycle race so as to completely recover after each hill. For this, we let m=3 and s(x)=cos(x). We search for a power function of the form p(x)=A+B*cos(x+C) that will minimize the amount of time required to finish, while requiring the values of L to match at the beginning and the end of each hill, by representing p as a vector and using the same mechanism as in our investigation of the sprint to the finish.

9. Optimal Pacing over Hills (cont.) For most values of a and b, the coefficients of dL/dt, the ideal power function was constant with amplitude B=0. In these cases, the optimal strategy for an athlete is to exercise at their aerobic threshold. However, there are some cases where a nonzero amplitude is optimal (left). This is certainly related to the amplitude of the slope function: the steeper the slope, the longer it takes to climb and the shorter the amount of time available to recover. Thus for steep repetitive hills, it is optimal to exercise at one’s aerobic limit.

10. Optimal Pacing over Hills (cont.) When the optimal solution has a nonzero amplitude, it seems to be in phase with the slope, similar to this example (right). On the steepest part of the uphill slope, the most effort is expended, while on the steepest part of the downhill, the least effort is expended. Because of the reduced returns offered by additional effort while going downhill, it makes sense that this moment is the time for minimal effort and maximal recovery.