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5
th
Asia-Pacific Congress on Sports Technology (APCST)
Optimizing pacing strategies on a hilly track in cross-country
skiing
David Sundström
*
, Peter Carlsson, Mats Tinnsten
Department of Enginnering and Sustainable Development, Mid Sweden University, Akademigatan 1, Östersund SE-83125, Sweden
Received 18 March 2011; revised 9 May 2011; accepted 10 May 2011
Abstract
During events involving locomotive exercise, such as cross-country skiing, it is believed that pacing strategies (i.e.
power distribution) have a significant impact on performance. Therefore, a program was developed for the numerical
simulation and optimization of cross-country ski racing, one that numerically computes the optimal pacing strategy
for a continuous track. The track is modelled by a set of cubic splines in two dimensions and can be used to simulate
a closed loop track or one with the start and finish at different locations. For an arbitrary point on the two dimensional
track, equations of motion are formulated parallel and normal to the track, considering the actual slope and curvature
of the track. Forces considered at the studied point are the gravitational force, the normal force between snow and
skis, the drag force from the wind, the frictional force between snow and ski and the propulsive force from the skier,
where the latter is expressed as the available power divided by the actual speed. The differential equations of motion
are solved from start to finish using the Runge-Kutta-Fehlberg method. The optimization of the ski race is carried out
with the Method of Moving Asymptotes (MMA) which minimizes the racing time by choosing the optimum
distribution of available power. Constraints for minimum, maximum and average power are decided by conditions of
scaling by body size. Results from a simulated ski competition with optimized power distribution on a real track are
presented.
© 2011 Published by Elsevier Ltd. Selection and peer-review under responsibility of RMIT University
Keywords: Numerical simulation; optimization; cross-country skiing; splines
* Corresponding author. Tel.: +46-63-165-994.
E-mail address: david.sundstrom@miun.se.
1877–7058 © 2011 Published by Elsevier Ltd.
doi:10.1016/j.proeng.2011.05.044
Procedia Engineering 13 (2011) 10–16
David Sundström et al. / Procedia Engineering 13 (2011) 10–16
11
1. Introduction
Cross-country skiing is a winter sport where the athlete’s (skier’s) ability to cover the course distance
in the shortest time possible is of decisive importance to his/her performance. But since the athlete’s
amount of power is limited, the efforts must be distributed in a rational way. High power should be used
where it is most advantageous, and low power on sections of the track where it is possible or perhaps
necessary to rest and recover. Mathematical modelling of a ski race, combined with efficient nonlinear
optimization routines, provides the potential to analyze how the skiers’ efforts should be distributed in the
most advantageous way. Individual differences, such as size, mass, the available power and ability to
recover, etc., can be considered, as well as the influence of different course profiles on different skiers.
The aim of this investigation was to create a model that can approximate the optimal pacing strategy for a
typical world class cross-country sprint skier on a real course profile. And in the long run, the athlete can
apply this optimal pacing strategy to improve performance.
Nomenclature
t, n local coordinate t in the course direction and n normal to the course, aimed at the centre of
curvature
x, y global coordinates horizontally and vertically
f(x) equation for the course section, i.e. y = f(x)
α
course incline at studied point
R course curvature at studied point
N normal force from snow
F
g
gravity force, F
g
= mg
F
D
drag force (air resistance) also taking into account any tail/headwind w in addition to skiers
speed
t
F
μ
frictional force between ski and snow (glide)
F
S
skier's forward propulsive force
w tail or headwind speed on current course section
P available (propulsive) power
2. Description of the Numerical Simulation Model
An arbitrary two-dimensional course profile can be created as a connected chain of cubic splines. As
the entire course is expressed as a connected chain of third degree polynomials, there is no problem in
obtaining the actual gradient and curvature radius for an arbitrary point on the course. The model only
takes account of the effective propulsive force generated by the athlete in the direction of the track. This
12 David Sundström et al. / Procedia Engineering 13 (2011) 10–16
in contrast to Moxnes and Hausken [1] or Holmberg and Lund [2] where certain parts of body movements
in skiing were studied. Moxnes and Hausken show analytical results for the relationships between glide
length, friction and kicking force during a diagonal stride, and in Holmberg and Lund a simulation of the
biomechanics of double-poling was performed and the load distribution between different muscles was
analysed.
2.1. Forces and scaling
External forces to take into account during a race are the effect of gravity, the frictional forces between
the skis and snow and the effect of air resistance in which any tailwind or headwind can be included. The
skier’s propulsive force ܨ
௦
in the direction of the course is used in the equations of motion. The propulsive
force is the available power, generated by the skier, divided by the current speed (ܨ
௦
ൌܲ ݒ
Τ
). When
scaling between skiers of different size (mass), it has been assumed that the athletes have uniform
physiques. Scaling the terms of ܲ
ത
and P
max
in Equations 7-9 for the individual skier was performed using
the exponential relationship;
భ
మ
ൌቀ
భ
మ
ቁ
Ǥଽସ
(1)
where ܲ
ത
is the allowable maximum value of the mean power and P
max
is the maximum power for the
optimization variables. Index represents skiers 1 and 2 with masses m
1
and m
2
etc., see also Bergh [3]. In
the expression used for air resistance, ܨ
ൌ
ଵ
ଶ
ܥ
ܣߩݒ
ଶ
ൌ݇ݒ
ଶ
the air resistance coefficient k
ሺ݇ ൌ
ଵ
ଶ
ܥ
ܣߩሻ
in uniform scaling is only proportional to the projected frontal area A and is therefore scaled according to
Equation 2, as the projected frontal areas behave like masses raised to 2/3.
భ
మ
ൌቀ
భ
మ
ቁ
Ǥ
(2)
2.2. Derivation of equations of motion for an arbitrary course section
The equations of motion are drawn up for a skier in horizontal and vertical directions for an arbitrary
point on the course, see Figure 1. Speed and acceleration relationships in Equation 3 and geometric
relationships from the course profile equation in Equation 4:
Fig. 1. Arbitrary section of the course with forces on the skier and chosen coordinates.
David Sundström et al. / Procedia Engineering 13 (2011) 10–16
13
ݐ
ሶ
ൌ
ඥ
ݔሶ
ଶ
ݕሶ
ଶ
ݔሷ
ݕሷ
൨ൌቂ
ܿݏߙ ݏ݅݊ߙ
ݏ݅݊ߙ െܿݏߙ
ቃήቂ
ݐ
ሷ
݊ሷ
ቃ (3)
Į = arctan(y')
ଵ
ோ
ൌ
௬ᇱᇱ
ሾ
ଵା
ሺ
௬ᇱ
ሻ
మ
ሿ
యమ
Τ
(4)
During a ski race a skier is obliged to follow the given course profile. As a first step, it is thus useful
to describe the equations of motion in the natural directions of the movement, i.e. in the direction normal
to the course and tangentially [4]. After transformation to x- and y-coordinates and inserting the
expressions of the acting forces, the following equations of motion are obtained:
݉ݔሷ ൌ
௧
ሶ
ߙ െ݇ሺݐ
ሶ
ݓሻ
ଶ
ߙ െ݉ቀ݃ߙ െ
௧
ሶ
మ
ோ
ቁ
ሺ
ߤߙߙ
ሻ
,
݉ݕሷ ൌ
௧
ሶ
ݏ݅݊ߙ െ݇ሺݐ
ሶ
ݓሻ
ଶ
ݏ݅݊ߙ െ݉ቀ݃ܿݏߙ െ
௧
ሶ
మ
ோ
ቁ
ሺ
ߤݏ݅݊ߙܿݏߙ
ሻ
െ݉݃. (5)
where the frictional force is ܨ
ఓ
ൌߤ݉ሺ݃ߙെ
௧
ሶ
మ
ோ
ሻ, the propulsive force is ܨ
௦
ൌ
௧
ሶ
and the drag force is
ܨ
ൌ
ଵ
ଶ
ܥ
ܣߩሺݐ
ሶ
ݓሻ
ଶ
ൌ݇ሺݐ
ሶ
ݓሻ
ଶ
.
In the drag force expression ݇ൌ
ଵ
ଶ
ܥ
ܣߩ, C
d
is the drag coefficient,
A is projected frontal area,
ρ
is air density and w is the tailwind or headwind speed. The second order
differential equations in Equation 5 can be transformed into a system of four connected first order
equations. Subsequently the system may be solved by any standard numerical solver, see Carlsson et al.
[5].
3. Optimization
Numerical optimization is performed on the previously described simulation program. The natural
choice of objective function in the optimization is to minimize the total race time (Equation 6). The
chosen optimization variables are available power on carefully selected points along the course (P
j
).
Power levels between these points are available from linear interpolation between the variables. Since the
maximum power a skier can generate over a longer time is limited, certain restrictions must be connected
to the objective function. The chosen constraints are maximum and minimum levels of power (Equation
7), combined with the global maximum mean power ܲ
ത
during the whole race (Equation 8). In order to
prevent the skier from using excessively high power on a longer part of the course, local constraints are
added to those previously described (Equation 9). These constraints prevent the skier from developing
maximum power on more than two consecutive optimization variables and allow a period of recovery
before the next maximum outtake. In mathematical terms, the optimization problem is formulated as;
Minimize ܶൌ
σ
οݐ
ୀଵ
(6)
Such that ܲ
ܲ
ܲ
௫
݆ൌͳǡʹǡǥǡܰ (7)
ଵ
்
ܲ
ሺ
ݐ
ሻ
݀ݐ ܲ
ത
்
(8)
൫ܲ
ܲ
ାଵ
ܲ
ାଶ
൯ʹܲ
௫
ܥήܲ
ത
݆ൌͳǡʹǡǥǡሺܰെʹሻ (9)
where T is the total race time,
Δ
t
i
is the time segment during iteration i, K is the total number of time
segments during the actual simulation, P
j
is the j:th optimization variable (i.e. power level at j), P
min
and
P
max
are the minimum and maximum power levels for the optimization variables, ܲ
ത
is a reference value of
the maximum mean power for the simulated skier, C is a constant (0 ≤ C ≤ 1) and N is the number of
optimization variables. To ensure that a good speed is maintained at the very end of the race, it might
14 David Sundström et al. / Procedia Engineering 13 (2011) 10–16
sometimes be necessary to complete the constraints with restrictions to certain speeds or power levels for
the last variables.
4. Numerical Results
The transformed system of connected first order differential equations of movement has been
implemented in a MATLAB program for solutions with the Runge-Kutta-Fehlberg method. In this paper,
the gradient-based Method of Moving Asymptotes (MMA [6]) has been used for the optimization. The
routine has performed excellently in a number of optimization applications, including structural dynamics
[7].
Experience shows that it is important to keep the time segments small when solving the differential
equations, otherwise the optimization variables (i.e. the power variables) will occur on slightly different
places on the track with different levels of power. The input data for solving the differential equations are
the skier’s mass, starting speed and available power during the various parts of the race. Input data also
includes the course profile expressed as a connected chain of cubic splines, the glide friction coefficient
μ
, tailwind or headwind speed w and any different postures’ effect on air resistance on various sections of
the course. When scaling the power constraints (ܲ
ത
, P
max
) and the coefficient k in the expression of air
resistance, the reference value has been taken from values that may be appropriate for a skier with the
mass m = 70 kg. For this skier the reference value of maximum mean power ሺܲ
ത
ሻ has been set to ܲ
ത
= 325
W and the air resistance coefficient k
70
= 0.49. The reference values used for power, air resistance and
friction have, when simulating skiing 10,000 m on a flat course in conditions with no wind, given a skiing
time of just under 24 min. for a skier with a mass of 70 kg.
The planned course for the Swedish Sprint Championships in 2007 has been modelled as a chain of
totally 36 connected cubic splines, see Figure 2 (the sprint competitions were actually moved to another
place due to a lack of snow in the intended area). The course has a length of 1425 m. The course has been
used for the optimization of a skier’s power distribution during the race. In the simulated race a friction
coefficient of
μ
= 0.03 is used and no headwind or tailwind is assumed. Optimization is performed with N
= 35 variables. The power variables are spread over the course in such a way that they are closer to each
other where the slope of the course has distinct changes. Available power values between the variable
values have been calculated by linear interpolation. Maximum and minimum values for the power in
Equation 8 are set to ܲ
ൌͲήܲ
ത
resp. ܲ
௫
ൌͳǤʹήܲ
ത
and the constant C in Equation 9 is given the value
C = 1. In order to stabilize the numerical process, all constraints are normalized during the optimization.
4.1. Optimization of a skier on the planned course for the Swedish Sprint Championships in 2007
Considering a typical world class cross-country sprint skier with a body mass of 75 kg, we obtain ܲ
ത
=
346.8 W and k = 0.51. The maximum and minimum values for the power variables are P
min
= 0.0 W and
P
max
= 416.1 W. The constraints on the two last variables remain the same and optimization starts with
initial values of ܲ
ൌܲ
ത
= 346.8 W for all variables P
j
. This initial set of variables gives a total time of 3
min 27.9 s with all constraints fulfilled. After 15 iterations with the optimization routine, the total time
has decreased to 3 min 24.55 s, a time difference of -3.55 s. All the constraints are still fulfilled and the
divergence from the stipulated reference value of the mean power in Equation 8 is now less than 0.5 W.
The location of the power variables (P
j
) and their values after optimization are shown in Figure 2. The
mean power that the athlete is able to sustain for a longer period of time is shown as a dashed line in the
figure. Because of the formulation of the constraint in Equation 8, the area between the line for the power
distribution should be equal above and below the mean power level (provided that the constraint is
fulfilled with equality).
David Sundström et al. / Procedia Engineering 13 (2011) 10–16
15
Fig. 2. Course profile and optimized power distribution for a 75 kg skier.
5. Discussion and Conclusion
The planned course for the Swedish Sprint Championships in 2007 was modelled with a chain of cubic
splines. Skiing was simulated for a skier of 75 kg, and optimization was performed on the power
distribution along the track. The main goal of the optimization was to minimize the total race time, at the
same time as there were limitations to the available power during the race. Overall, the performed
optimization acts in a reliable way. From an evenly distributed power level, the optimization routine
automatically iterates to a more effective way of distributing effort along the course. The risk of fatigue
for the skier can be avoided by constraints which control power levels during the race. A global constraint
on the power level is necessary but not enough; even if the mean level of the power is below that
prescribed there is still a risk of connected passages with excessive power output. The local constraint in
Equation 9 for connected power outtake assures that this risk is eliminated.
Individual adaption of optimization is easy; different bodily constitutions, as well as different physical
capacities, can be modelled. If necessary, the optimization problem can be completed with more
constraints to make it work better. A skier’s ability to perform their very best can of course vary from day
to day and this performance “on the day” can have a great effect on the results. The simulation shown
here does not take such variations into account, but it is nonetheless of great interest to study how power
should be distributed in the most effective way during a ski race. Once the profile of a course is known,
such studies might serve as an essential part of the preparation for important championship races.
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[3] Bergh, U. The Influence of Body-Mass in Cross-Country Skiing. Med Sci Sport Exer 1987;19:324-31.
[4] Meriam, JL,Kraige, LG. Engineering mechanics dynamics. 6 ed: John Wiley and sons inc, 2008.
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