Modelling road cycling as motion on a curve

Abstract

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.