Minimum-time optimisation has been used extensively in motorsports, such as Formula One racing. Using minimum-time optimisation, the ideal racing line as well as control strategies such as braking strategies can be found. This is interesting, as Reijne et al have shown that cyclists apply diverse strategies, specifically during a descent [1]. With minimum-time optimisation, it is possible to compare a riders performance to the theoretically optimal performance. The results from such optimisations can be used to help with training, as well as improve and test equipment design.

This work describes a free-trajectory steady motion control optimisation for the descent of elite cyclists. The prediction of the individual descent performance was formulated as an optimal control problem and solved with a direct approach to finding optimal cornering and braking strategies that yield the shortest descent time. While the state equations were kept simple (3 variables only), more elaborated performance limits were represented by g-g diagrams. Such diagrams represent the longitudinal, lateral, and combined acceleration limits for cyclists. A method to numerically derive g-g diagrams for cyclists driving on 3D tracks was designed. In this method, a tire model, power limit, and steady motion equations for a cyclist are used to determine the control space. The bicycle and cyclist are modeled as a single rigid body, the tire friction model is simplified as a friction circle, and the wind speed is considered to be zero at all times. As for the 3D road geometry effects, all possible effects are considered in the method, except lateral road curvature. The resulting method
provides g-g diagrams as a function of 8 local geometry and state variables.

The optimisation model was tested against the velocity and trajectory output data measured on Team Sunweb professional cyclists at the L218 descent in Germany. The resulting trajectory was similar to the trajectory ridden by elite cyclists. The velocity profile showed large differences, which are a result of a combination of inaccurate track data, differences in friction coefficient estimation, and safety margins applied by the cyclists. The results show that descent performance can be improved, as even when adhering to safety margins harder braking is possible. Overall, the model responds as expected to changes in track, environment, and bicycle/rider parameters.

Steps can be made towards better implementation of the g-g diagrams in the minimum-time optimisation. Furthermore, a more accurate tire model and power model can improve the model and extend its applications. The presented model can be used for qualitative descent analyses, and facilitate the training of elite cyclists.

[1] A.L. Schwab, M.M. Reijne, D.J.J. Bregman, Measuring and comparing descend in elite race cycling with a perspective on real-time feedback for improving individual performance. In Multidisciplinary Digital Publishing Institute Proceedings, volume 2, page 262, 2018.