Available online at www.sciencedirect.com
Procedia Engineering 60 (2013) 118 – 123
6th Asia-Pacific Congress on Sports Technology (APCST)
On Optimization of Pacing Strategy in Road Cycling
David Sundströma*, Peter Carlssona, Mats Tinnstena
a
Department of engineering and sustainable development, Mid Sweden University, Akademigatan 1, Östersund 83125, Sweden
Received 15 April 2013; revised 6 May 2013; accepted 9 May 2013
Abstract
The wide-spread use of powerve for optimizing
the distribution of power output i.e. the pacing strategy. Therefore, the aim of this study was to examine the effects of
course profile on the optimal pacing strategy in road cycling. For that reason, three course profiles, built up by
cubical splines, were simulated with a numerical program for a fictional cyclist. The numerical program solves the
equations of motion of the athlete and bicycle while an optimal design algorithm is connected to the simulation,
aiming to minimize the time between start and finish. The optimization is constrained by a power-endurance concept
named the critical power model for intermittent exercise. Three course profiles with the same total elevation but
different number of hills were studied. The time gains of an optimized pacing strategy were
%,
%, and
%
and the speed variances at the optimized pacing strategy were
%,
%, and
% for the single plateau,
double hill, and quadruple hill courses respectively. Hence, the course profile has great effect on the optimal pacing
strategy. In addition, the results show that the potential improvement of adopting an optimized pacing strategy is
substantial at the highest level of competition.
© 2013 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license.
©
2013 Published
by Elsevier
Ltd. Selection
andSchool
peer-review
under Mechanical
responsibility
RMIT University
Selection
and peer-review
under responsibility
of the
of Aerospace,
andof
Manufacturing
Engineering,
RMIT University
Keywords: Numerical optimization, simulation, pacing strategy, road cycling.
1. Introduction
Pacing strategy may be defined as the way of varying speed along the course in locomotive exercise.
Sometimes the term power distribution is used interchangeably although it refers to the corresponding
alteration of work-rate. Recently, optimization of pacing strategies has been introduced to investigate how
to best distribute the power output in road cycling [1-4]. Gordon [1] used analytical optimization on
simple linear course profiles without considering the inertia of the athlete-bicycle system, while he
optimized the power distribution in road cycling. Cangley et al. [2] used a more sophisticated model for
simulating road cycling and optimized the pacing strategy with optimal control theory, first introduced to
1877-7058 © 2013 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license.
Selection and peer-review under responsibility of the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University
doi:10.1016/j.proeng.2013.07.062
David Sundström et al. / Procedia Engineering 60 (2013) 118 – 123
this field by Maronski [4]. However, the model used to constrain the power output was not as intuitive as
Gordon´s. Dahmen et al. [3] used a more sophisticated constraint while also using optimal control theory
to optimize the pacing strategy in road cycling. The aim of this study was to investigate the influence of
course profile on pacing strategy and power output distribution in road cycling. This study examines the
importance of correct power distribution for a shorter part of the whole course near the end of the race
where the athlete has made a solo breakaway.
2. Methods
A numerical model was developed for simulation of road cycling and optimization of the athlete´s
pacing strategy. Equations of motion were programmed into the MATLAB software and solved
numerically with the Runge-Kutta-Fehlberg method [5]. The optimal design routine called Method of
Moving Asymptotes (MMA) [6, 7] was connected to the simulation to minimize the time from start to
finish.
kg was considered in this study. The
A world-class male road cyclist with a body mass of
. The total mass
total mass of the athlete, bicycle (as used in mass start races) and clothing was
moment of inertia of the two wheels was . The total inertia of the system in the direction of travel was
, where was the wheel radius.
This study investigates the optimal pacing strategy for three different course profiles that are named
after their appearance. They are termed Single Plateau (SP), Double Hill (DH), and Quadruple Hill (QH).
m and a total elevation of
m. The course profiles are
Every course has a horizontal length of
cubical splines each. The splines only extend in two dimensions ( and ) to simplify the
built up by
model. No environmental wind is considered on the courses. The course profiles are presented in Fig. 2.
In previous work [8], we have transformed motion equations for cross country skiing so that the
independent variable was horizontal distance instead of time. This has the advantage of ending up at the
exact finish line when solving the equations of motion, instead of some distance longer. Thus, in this
study we incorporated a slightly adjusted model of Martin et al. [9] and transformed it so the independent
variable was horizontal distance. All forces acting on the athlete and bicycle can be seen in Fig. 1.
Fig. 1. Arbitrary course section with local and global coordinates, as well as the forces acting on the athlete.
Our model is based on the kinetics of a moving particle. This means that the acceleration of the
athlete-bicycle system is determined by the net force acting on the body. The net force is the sum of the
propulsive force and all the resistive forces acting to restrict motion. The propulsive force is expressed as:
119
120
David Sundström et al. / Procedia Engineering 60 (2013) 118 – 123
(1)
is the mechanical transmission efficiency and is
where is the power output at the crank spindle,
the ground speed. The acceleration of the athlete-bicycle system in the global directions can thus be
expressed as:
(2)
and the
where and is the local coordinates of motion (Fig. 1), the rolling resistance is
[10].
is the rolling resistance coefficient and is the
wheel bearing friction is
normal force. The coefficients and are derived from the study of Dahn et al. [10]. Expressions for
the remaining forces are presented in a previous study [8]. The course inclination is . The dependent and
independent variables are switched in accordance with the previous study [8] and the following equation
is derived:
(3)
where
is the drag coefficient, is the projected frontal area,
is the incremental drag area associated
with wheel spoke rotation, is the air density, is the course curvature radius and is the acceleration
of gravity. The prime denotes differentiation by the -coordinate. Finally, we introduced a new variable
to create a system of first order ordinary differential equations that can be solved with the Runge-KuttaFehlberg method.
In order to optimize the pacing strategy, an optimization routine called MMA was connected to the
simulation model. It was set to minimize the time ( ) between start and finish while altering a set of
design variables ( ) that determined the power output along the distance. The variables were 81 equally
spaced discrete values of power output and in between these variables; the power output was determined
by linear interpolation. The optimization was constrained by the minimum and maximum power output
limits as well as the critical power model for intermittent exercise [11]. This model has two fixed
parameters. One aerobic parameter that is named Critical Power ( ) and one anaerobic parameter named
). Along with these parameters, we introduced a dependent parameter
Anaerobic Work Capacity (
), which was the remaining part of
at the studied point.
named Available Anaerobic Work (
must be evaluated at each optimization variable and every time ( ) the linear interpolation of
The
the variables equals . The available anaerobic work was formulated as:
(4)
is the time step between
and , is the number of variables and is the number of
where
intersections between
and the linear interpolation of the power variables. Available anaerobic work
at
and after total exertion,
.
cannot exceed
started with
-1
according to the model. Hill and Smith [12]
in male
121
David Sundström et al. / Procedia Engineering 60 (2013) 118 – 123
athletes. However, concerning that the simulations are meant to mimic near end conditions of the race, a
was used. The optimization problem was expressed as:
halved
Minimize
(5)
subject to
(6)
(7)
where is the finishing time, is the time segment during distance step , is the total number of
and
is the lower and upper limits of the optimization variables respectively. These
distance steps,
power output limits are demanded by the MMA but also makes sense concerning the limitations of the
athlete s physiology.
The input data for the simulations were chosen in order to mimic a world class road cyclist. With this
-1
min-1. For our
kg cyclist this
in mind, he was given a maximum oxygen consumption of
-1
gives a
of
and a maximal rate of aerobic energy expenditure of about
W.
% (at
rpm) [13] and a degree of capacity
Further considering a gross efficiency of
of %, resulted in a
W. The
was calculated to
J while the
utilization at
W and
W respectively.
minimum and maximum power output was set to
Considering a brake hoods position the combined athlete-bicycle frontal area and drag coefficient was
m2 [14] and
[15] respectively, using the scaling laws of Heil. The drag
calculated to
coefficient was considered constant for differences in speed, in agreement with the minor variations
kg m-3 considering dry air of °C
reported by Chowdhury et al. [16]. The air density was set to
kPa. The acceleration of gravity was set to
m s-2.
and standard atmospheric pressure of
, using the relationship
The rolling resistance coefficient of the tires was calculated to
kPa. The coefficients in the expression for wheel
reported by Grappe et al. [17] and a tire pressure of
and
considering greased cartridge bearings [10].
bearing resistance were set to
2
and
The moment of inertia and the incremental drag area of the wheels were set to
2
m respectively [9] and the wheel radius was measured to
m. The total mass of
kg and the total inertia was
kg. To mimic a flying
athlete-bicycle system was
-1
. All design variables were given the initial value
start, the initial value of speed was set to
W.
of
3. Numerical Results and Discussion
The optimization routine completed 30 iterations for each course. The finishing times were
s,
s, and
s and the remaining
s at the finish line were 0.041%,
0.41%, and 0.22% of
for the SP, DH, and HQ courses respectively. The optimized pacing strategies
and power output distributions can be seen in Fig. 2. As one might expect, the optimization strives to
level the speed by adjusting the power output distribution to follow the course inclination.
, thus only making modest use
The optimization for course SP yielded nearly no recovery of the
model. We can see on all three courses that the point where
reaches zero, without
of the recovering, occurs at the top of the last hill. Therefore, this distance of even power output became longer
for the SP than for DH and longer for DH than for QH. This greater distance of unused anaerobic
%,
%, and
% for the
resources caused greater speed variance. The variances of speed were
SP, DH, and QH courses respectively. Ultimately, this variation is responsible for the differences in
finishing times betw
The starting strategy was similar for all courses when it comes to the shape of the power output curve.
However, the magnitude differed substantially and the initial power output seemed to increase with
122
David Sundström et al. / Procedia Engineering 60 (2013) 118 – 123
decreased length of the first hill on the course. One can also see that the power output in the uphill slopes
was greater for the shorter hills in DH and QH than in SP. In comparison with even distributions of power
output, the optimization of the pacing strategy gave time gains of 3.0%, 5.0%, and 2.3% for the SP, DH,
and QH courses respectively. Although, there is some variation, this is substantial time gains at the
highest level of competition. Some imperfections in the optimized set of variables may be obtained in this
kind of numerical simulation and optimization. Thus, there were some minor irregularities in the power
output distributions.
SP
DH
QH
Fig. 2. Optimized pacing strategies and power output distributions for the SP, DH and QH courses.
David Sundström et al. / Procedia Engineering 60 (2013) 118 – 123
4. Conclusions
The results of this study showed that courses with the same total elevation but with a different number
of hills need different pacing strategies. The results of this study unsurprisingly confirmed the findings of
prior studies [2, 18, 19], that substantial time gains can be achieved with a variable pacing strategy when
external conditions change along the course. Further development of power output constraints is needed
is a decent
to achieve a more comprehensive and realistic model. The two component model
kinetics.
estimation but it does not, for instance, consider the slow or fast components of
Acknowledgements
This work was supported by the European Regional Development Found of the European Union.
References
[1] Gordon, S. Optimising distribution of power during a cycling time trial. Sports eng 2005;8:81-90.
[2] Cangley, P, Passfield, L, Carter, H,Bailey, M. The effect of variable gradients on pacing in cycling time-trials. Int J Sports
Med 2011;32:132-6.
[3] Dahmen, T. Optimization of pacing strategies for cycling time trials using a smooth 6-parameter endurance model. PreOlympic Congress on Sports Science and Computer Science in Sport (IACSS2012): IACSS Press, 2012.
[4] Maronski, R. On optimal velocity during cycling. J Biomech 1994;27:205-13.
[5] Fehlberg, E. Low-order classical runge-kutta formulas with stepsize control and their application to some heat transfer
problems. NASA technical report. Marshall, Alabama: National aeronautics and space administration, 1969.
[6] Svanberg, K. The method of moving asymptotes - a new method for structural optimization. Int J Numer Meth Eng
1987;24:359-73.
[7] Svanberg, K. The method of moving asymptotes (MMA) with some extensions. Optimization of Large Structural Systems,
Vols 1 1993;231:555-66.
[8] Sundström, D, Carlsson, P, Ståhl, F,Tinnsten, M. Numerical optimization of pacing strategy in cross-country skiing. Struct
Multidisc Optim 2012:1-8.
[9] Martin, JC, Milliken, DL, Cobb, JE, McFadden, KL,Coggan, AR. Validation of a mathematical model for road cycling
power. J Appl Biomech 1998;14:276-91.
[10] Dahn, K, Mai, L, Poland, J,Jenkins, C. Frictional resistance in bicycle wheel bearings. Cycling Science 1991;3:28-32.
[11] Morton, RH,Billat, LV. The critical power model for intermittent exercise. Eur J Appl Physiol 2004;91:303-7.
[12] Hill, DW,Smith, JC. A comparison of methods of estimating anaerobic work capacity. Ergonomics 1993;36:1495-500.
[13] Lucia, A, Juan, AFS, Montilla, M, Canete, S, Santalla, A, Earnest, C,Perez, M. In professional road cyclists, low pedaling
cadences are less efficient. Med Sci Sport Exer 2004;36:1048-54.
[14] Heil, DP. Body mass scaling of frontal area in competitive cyclists not using aero-handlebars. Eur J Appl Physiol
2002;87:520-8.
[15] Heil, DP. Body mass scaling of projected frontal area in competitive cyclists. Eur J Appl Physiol 2001;85:358-66.
[16] Chowdhury, H, Alam, F,Mainwaring, D. A full scale bicycle aerodynamics testing methodology. Procedia Engineer
2011;13:94-9.
[17] Grappe, F, Candau, R, Barbier, B, Hoffman, MD, Belli, A,Rouillon, JD. Influence of tyre pressure and vertical load on
coefficient of rolling resistance and simulated cycling performance. Ergonomics 1999;42:1361-71.
[18] Atkinson, G, Peacock, O,Passfield, L. Variable versus constant power strategies during cycling time-trials: Prediction of
time savings using an up-to-date mathematical model. J Sport Sci 2007;25:1001-9.
[19] Swain, DP. A model for optimizing cycling performance by varying power on hills and in wind. Med Sci Sport Exer
1997;29:1104-8.
123