Prediction of pacing and cornering strategies during cycling individual time trials with optimal control

Modelling road cycling as motion on a curve

We present a mathematical model of road cycling on arbitrary routes using the Frenet–Serret frame. The route is embedded in the coupled governing equations. We describe the mathematical model and numerical implementation. The dynamics are governed by a balance of forces of gravity, drag, and friction, along with pedalling or braking. We analyse steady-state speed and power against gradient and curvature. The centripetal acceleration is used as a control to determine transitions between pedalling and braking. In our model, the rider looks ahead at the curvature of the road by a distance dependent on the current speed. We determine such a distance (1–3 s at current speed) for safe riding and compare with the mean power. The results are based on a number of routes including flat and downhill, with variations in maximum curvature, and differing number of bends. We find the braking required to minimise centripetal acceleration occurs before the point of maximum curvature, thereby allowing acceleration by pedalling out of a bend.

Assessment of bike handling during cycling individual time trials with a novel analytical technique adapted from motorcycle racing

A methodology to study bike handling of cyclists during individual time trials (ITT) is presented. Lateral and longitudinal accelerations were estimated from GPS data of professional cyclists (n=53) racing in two ITT of different length and technical content. Acceleration points were plotted on a plot (g-g diagram) and they were enclosed in an ellipse. A correlation analysis was conducted between the area of the ellipse and the final ITT ranking. It was hypothesized that a larger area was associated to a better performance. An analytical model for the bike-cyclist system dynamics was used to conduct a parametric analysis on the influence of riding position on the shape of the g-g diagram. A moderate (n=27, r=-0.40, p=0.038) and a very large (n=26, r=-0.83, p<0.0001) association were found between the area of the enclosing ellipse and the final ranking in the two ITT. Interestingly, this association was larger in the shorter race with higher technical content. The analytical model suggested that maximal decelerations are highly influenced by the cycling position, road slope and speed. This investigation, for the first time, explores a novel methodology that can provide insights into bike handling, a large unexplored area of cycling performance.

A mathematical model-based approach to optimize loading schemes of isometric resistance training sessions

Individualized resistance training is necessary to optimize training results. A model-based optimization of loading schemes could provide valuable impulses for practitioners and complement the predominant manual program design by customizing the loading schemes to the trainee and the training goals. We compile a literature overview of model-based approaches used to simulate or optimize the response to single resistance training sessions or to long-term resistance training plans in terms of strength, power, muscle mass, or local muscular endurance by varying the loading scheme. To the best of our knowledge, contributions employing a predictive model to algorithmically optimize loading schemes for different training goals are nonexistent in the literature. Thus, we propose to set up optimal control problems as follows. For the underlying dynamics, we use a phenomenological model of the time course of maximum voluntary isometric contraction force. Then, we provide mathematical formulations of key performance indicators for loading schemes identified in sport science and use those as objective functionals or constraints. We then solve those optimal control problems using previously obtained parameter estimates for the elbow flexors. We discuss our choice of training goals, analyze the structure of the computed solutions, and give evidence of their real-life feasibility. The proposed optimization methodology is independent from the underlying model and can be transferred to more elaborate physiological models once suitable ones become available.