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Optimal strategy for trail running with nutrition and fatigue factors

Bogna Jaszczak, Łukasz Płociniczak

TL;DR

This paper tackles optimal strategies for long-distance trail running by extending Keller's classical model to account for varying terrain, fatigue accumulation, and in-race nutrition. The authors formulate a generalized dynamical system with gravity, drag, fatigue $Q$, energy $E$, and logistic carbohydrate oxidation $N(t)$, nondimensionalize it, and solve the resulting optimal-control problem via the Pontryagin Maximum Principle (PMP). Validation on five Golden Trail World Series routes shows finish-time predictions within a few percent of records, with characteristic optimal controls that adapt to terrain and altitude. This framework offers a practical, route-aware tool for pre-race planning of nutrition and pacing in endurance mountain running.

Abstract

This paper presents an extension of Keller's classical model to address the dynamics of long-distance trail running, a sport characterized by varying terrains, changing elevations, and the critical influence of in-race nutrition uptake. The optimization of the generalized Keller's model is achieved through rigorous application of optimal control theory, specifically the Pontryagin Maximum Principle. This theoretical framework allows us to derive optimal control strategies that enhance the runner's performance, taking into account the constraints imposed by the changing terrain, nutritional dynamics, and the evolving fatigue factor. To validate the practical applicability of the model, simulations are performed using real-world data obtained from various mountain races. The scenarios cover various trail conditions and elevation profiles. The performance of the model is systematically evaluated against these scenarios, demonstrating its ability to capture the complexities inherent in long-distance trail running and providing valuable insight into optimal race strategies. The error in the total race-time prediction is of the order of several percent, which may give the runner a reliable tool for choosing an optimal strategy before the actual race.

Optimal strategy for trail running with nutrition and fatigue factors

TL;DR

This paper tackles optimal strategies for long-distance trail running by extending Keller's classical model to account for varying terrain, fatigue accumulation, and in-race nutrition. The authors formulate a generalized dynamical system with gravity, drag, fatigue , energy , and logistic carbohydrate oxidation , nondimensionalize it, and solve the resulting optimal-control problem via the Pontryagin Maximum Principle (PMP). Validation on five Golden Trail World Series routes shows finish-time predictions within a few percent of records, with characteristic optimal controls that adapt to terrain and altitude. This framework offers a practical, route-aware tool for pre-race planning of nutrition and pacing in endurance mountain running.

Abstract

This paper presents an extension of Keller's classical model to address the dynamics of long-distance trail running, a sport characterized by varying terrains, changing elevations, and the critical influence of in-race nutrition uptake. The optimization of the generalized Keller's model is achieved through rigorous application of optimal control theory, specifically the Pontryagin Maximum Principle. This theoretical framework allows us to derive optimal control strategies that enhance the runner's performance, taking into account the constraints imposed by the changing terrain, nutritional dynamics, and the evolving fatigue factor. To validate the practical applicability of the model, simulations are performed using real-world data obtained from various mountain races. The scenarios cover various trail conditions and elevation profiles. The performance of the model is systematically evaluated against these scenarios, demonstrating its ability to capture the complexities inherent in long-distance trail running and providing valuable insight into optimal race strategies. The error in the total race-time prediction is of the order of several percent, which may give the runner a reliable tool for choosing an optimal strategy before the actual race.
Paper Structure (9 sections, 2 theorems, 39 equations, 6 figures, 7 tables)

This paper contains 9 sections, 2 theorems, 39 equations, 6 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Derivation
  4. Nondimensionalization
  5. Optimal control
  6. Real-world examples
  7. Data and parameters
  8. Results
  9. Conclusion

Key Result

Lemma 1

$\lambda_E(t)$ is positive and $\lambda_Q(t)$ is negative for $t\in [0, 1)$.

Figures (6)

  • Figure 1: Logistic curve fitted to experimental data. The value of the determination parameter $R^2$ is 0.9459.
  • Figure 2: Exemplary values of model variables for a 20 km race on a flat route.
  • Figure 3: Optimal velocity strategy for Pikes Peak Ascent race. Dashed line represents the slope expressed as $\tan \alpha$.
  • Figure 4: Optimal velocity strategy for Zegama Aizkorri Maraton race. Dashed line represents the slope expressed as $\tan \alpha$.
  • Figure 5: Values of model variables for Pikes Peak Ascent.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Lemma 1
  • Theorem 1
  • proof