Optimal pacing strategy for a race of two competing cyclists
Abstract
Background: For optimal pacing strategies in the case of two or more competing or cooperating cyclists only few approaches take slipstreaming into account. However, by incorporating the slipstream effect in the model of a race of two runners on a flat course, it has been shown, how the trailing runner can position himself at striking distance behind the other and when he should start the final sprint. (Pitcher, 2009: Optimal strategies for a two-runner model of middle-distance running. SIAM Journal on Applied Mathematics, 70(4), 1032–1046).Purpose: We transfer this approach to cycling on a track of fixed length with real-world height data. In particular on descents, high speed is involved and increases the significance of the slipstreaming strategy.
Methods: We adopt the standard mechanical bicycling model that accounts for pedaling power, gravity, friction, inertia, and aerial drag. The nominal aerial drag force is multiplied by a slipstream factor that has its minimum when the cyclist is located closely behind his opponent (Pitcher, 2009). Our physiological model defines the remaining anaerobic capacity that de-/increases non-linearly when the pedaling power exceeds/falls below critical power (Gordon, 2005: Optimizing distribution of power during a cycling time trial. Sports Engineering, 8(2), 81–90). The mechanical and physical parameters may be different for the two cyclists. We use a state-of-the-art optimal control method for the numerical computations (Patterson et al., 2013: GPOPS-II: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive gaussian quadrature collocation methods and sparse nonlinear programming. ACM Transactions on Mathematical Software, 39(3).).
Results: The optimal pacing strategy for two cyclists on a competition between Saint-Gildas-des-Bois to Redon (stage 3 of the Tour de France 2013), France, is shown in Figure 1.
Discussion and Conclusion: Slipstreaming has a significant impact on pacing strategies in cycling. Incorporated into the mechanical bicycling model, optimal control algorithms can be used to compute the optimal tactic for the final sprint. Future work should take into account that the cyclists will often cooperate to stay ahead of the peloton before they compete in the final sprint phase.
Downloads
Published
How to Cite
Issue
Section
Copyright (c) 2014 Journal of Science and Cycling
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Authors contributing to Journal of Science and Cycling agree to publish their articles under a Creative Commons CC BY-NC-ND license, allowing third parties to copy and redistribute the material in any medium or format, and to remix, transform, and build upon the material, for any purpose, even commercially, under the condition that appropriate credit is given, that a link to the license is provided, and that you indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
Authors retain copyright of their work, with first publication rights granted to Cycling Research Center.