In this study, we develop methods to model and simulate road cycling on real-world courses, to analyze the performance of individual athletes and to identify and quantify potential performance improvement. The target is to instruct the athlete where and how to optimize his pacing strategy during a time trial.
We review the state-of-the-art mechanical model for road cycling power that defines the relationship between pedaling power and cycling speed. It accounts for the power demand to overcome the resistance due to inertia, rolling friction, road gradient, friction in bearings and aerial drag.
For several model parameters the measurement proves to be difficult. Thus, we estimate four compound parameters from a fit of the dynamic model to varying real-world power and speed measurements. The approach guarantees precise estimation even on courses with moderately varying slope as long as that slope is known with sufficient precision. An experimental evaluation shows that our calibration improves the model speed estimation significantly both on the calibration course and on other courses with the same type of road surface. A sensitivity analysis allows to compute the change in speed for small parameter perturbations proving in detail that the influences of the coefficients for aerial drag and rolling friction dominate.
We designed a simulator based on a Cyclus2 ergometer. The simulation includes real height profiles, virtual gears, a video playback that was synchronized with the cyclist's current virtual position on the course and online visualization of course and performance parameters. The ergometer brake is controlled so that it imitates the resistance predicted by the outdoor road cycling model. The software can partly compensate the physical limitations of the eddy current brake.
The road cycling model and thus the simulator resistance depend sensitively on an accurate estimation of the slope of the road. Commercial gps enabled bicycle computers do not provide a sufficient precision since the differentiation of the height data in order to compute the slope amplifies high frequency noise. A differential gps device provides height data of sufficient quality but only in case the satellite signals are not hidden by obstacles such as houses, trees, or mountains, which is often a serious limiting factor. For this purpose, we also present a method that combines model-based slope estimations with noisy measurements from multiple GPS signals of different quality.
We validated both the model and the simulator with field data obtained on mountain courses. The model described the performance parameters accurately with correlation coefficients of 0.96–0.99 and signal-to-noise ratios of 19.7–23.9 dB. We obtained similar quality measures for a comparison between model estimation and our simulator. Thus the model prediction errors can be attributed to measurement errors in differential gps altitude and model parameters but not to the ergometer control.
The athlete represents the motor of the system. Power supply models quantify his ability to sustain time-variable power demand. We briefly review the Morton-Margaria model that illustrates the interplay between the aerobic and anaerobic metabolism as a hydraulic system. Due to the complexity of human physiology and the inability to measure the required quantities, the model needs coarse simplification before it is usable quantitatively in practice. We present three physiological power supply models:
1. The 3-parameter critical power model extends the classical critical power model with the two parameters critical power and anaerobic work capacity by introducing a maximum power constraint and has an exertion rate that depends linearly on the pedaling load.
2. Gordon's modification, denoted by exertion model, suggests an alternative non-linear exertion rate that, in addition, defines an implicit maximum power constraint.
3. Our own 4-parameter model introduces an additional steering parameter for the nonlinearity and adopts the power constraint from the 3-parameter critical power model, thus combining – as we believe – some of the favorable properties of both models.
Having the power demand and different versions of supply models at hand, we compute minimum-time pacing strategies for both synthetic and real-world cycling courses as numerical solutions of optimal control problems using the Matlab package GPOPS-II.
In order to verify and discuss the numerical solutions, we derive candidate solutions for each problem. It turns out that for the 3-parameter critical power model, we deal with a singular control problem and, remarkably, the optimality criterion is that on sections, where the slope varies only moderately, the speed is perfectly constant.
Direct transcription methods as they are used in GPOPS-II often have severe numerical difficulties with singular optimal control problems. However, we found that if our problem is parametrized using kinetic energy instead of speed, significantly more detailed optimal strategies may be obtained on courses with real complex slope data and the computing time decreases.
We plot and discuss minimum-time pacing strategies for three real uphill courses in Switzerland, for which we have accurate height profile data, combined with the three physiological models. For Gordon's model we conducted an experiment, where an athlete was instructed on our simulator to follow the optimal strategy and finished the course in less time than when pacing himself based on his experience.
Finally, we give a numerical example how a weaker athlete rides in the slipstream of a stronger leading competitor and overtakes just in the right moment towards the end of the race in order to win the competition.