Date of Award
Doctor of Philosophy (PhD)
This dissertation pushes on the boundaries of improving human performance through individualized models of fatigue and recovery, and optimization of human physical effort. To that end, it focuses on professional cycling and optimal pacing of cyclists on hilly terrains during individual and team time trial competitions. Cycling is an example of physical activity that provides nice interfaces for real-time performance measurement through inexpensive power meters, and for providing real-time feedback via a mobile displays such as smartphones. These features enable implementing a real-time control system that provides the optimal strategy to cyclists during a race.
There has been a lot of work on understanding the physiological concepts behind cyclists' performance. The concepts of Critical Power (CP) and Anaerobic Work Capacity (AWC) have been discussed often in recent cycling performance related articles. CP is a power that can be maintained by a cyclist for a long time; meaning pedaling at or below this limit, can be continued until the nutrients end in the cyclist's body. However, there is a limited source of energy for generating power above CP. This limited energy source is AWC. After burning energy from this tank, a cyclist can recover some by pedaling below CP.
In this dissertation, we utilize the concepts of CP and AWC to mathematically model muscle fatigue and recovery of cyclists. We use experimental data from six human subjects to validate and calibrate our proposed dynamic models. The recovery of burned energy from AWC has a slower rate than expending it. The proposed models capture this difference for each individual subject. In addition to the anaerobic energy dynamics, maximum power generation of cyclists is another limiting factor on their performance. We show that the maximum power is a function of both a cyclist's remaining anaerobic energy and the bicycle's speed.
These models are employed to formulate the pacing strategy of a cyclist during a time trial as an optimal control problem. Via necessary conditions of Pontryagin's Minimum Principle (PMP), we show that in a time trial, the cyclist's optimal power is limited to only four modes of maximal effort, no effort, pedaling at critical power, or pedaling at constant speed. To determine when to switch between these four modes, we resort to numerical solution via dynamic programming. One of the subjects is then simulated on four courses including that of the 2019 Duathlon National Championship in Greenville, SC. The simulation results show reduced time over experimental results of the self-paced subject who is a competitive amateur cyclist.
Moreover, we expand our optimal control formulation from an individual time trial to a Team-Time-Trial (TTT) competition. In a TTT cyclists on each team follow each other closely. This enables the trailing cyclists to benefit from a rather significant reduction in drag force. Through a case study for a two-cyclist team, we show that the distance between the cyclists does not need to be variable and can be set at the minimum safe gap. We then use this observation to formulate the problem for an n-cyclist team. We propose a Mixed-Integer Non-Linear Programming (MI-NLP) formulation to determine the optimal positioning strategy in addition to optimal power and velocity trajectories for a three-cyclist team. The results show improved travel time compared to a baseline strategy that is popular among cycling teams.
The results from this work highlight the potential of mathematical optimization of athletes’ performance both prior and during a race. The proposed models of fatigue and recovery can help athletes better understand their abilities at their limits. To prepare for a race, cyclists can practice over the simulated elevation profile of the race on a stationary bicycle with the optimal strategies we proposed. Additionally, if allowed by the rules of a specific competition, a cyclist can be provided with real-time race strategies. There is also the potential for integrating the proposed algorithms into the available commercial smart stationary bicycles for improving well-being and training of the general public.
Ashtiani, Faraz, "Optimal Pacing of Cyclists in a Time Trial Based on Experimentally Calibrated Models of Fatigue and Recovery" (2021). All Dissertations. 2858.