||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
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
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