A mathematical model for simulating cycling: applied to track cycling

Abstract

A review of existing mathematical models for velodrome cycling suggests that cyclists and cycling coaches could benefit from an improved simulation tool. A continuous mathematical model for cycling has been developed that includes calculated slip and steering angles and, therefore, allows for resulting variation in rolling resistance. The model focuses on aspects that are particular, but not unique, to velodrome cycling but could be used for any cycling event. Validation of the model is provided by power meter, wheel speed and timing data obtained from two different studies and eight different athletes. The model is shown to predict the lap by lap performance of six elite female athletes to an average accuracy of 0.36% and the finishing times of two elite athletes competing in a 3-km individual pursuit track cycling event to an average accuracy of 0.20%. Possible reasons for these errors are presented. The impact of speed on steering input is discussed as an example application of the model.

Full text

Vol.:(0123456789)
Sports Engineering
https://doi.org/10.1007/s12283-018-0283-0
ORIGINAL ARTICLE
A mathematical model forsimulating cycling: applied totrack cycling
BillyFitton1 · DigbySymons1,2
© The Author(s) 2018
Abstract
A review of existing mathematical models for velodrome cycling suggests that cyclists and cycling coaches could benefit
from an improved simulation tool. A continuous mathematical model for cycling has been developed that includes calculated
slip and steering angles and, therefore, allows for resulting variation in rolling resistance. The model focuses on aspects
that are particular, but not unique, to velodrome cycling but could be used for any cycling event. Validation of the model
is provided by power meter, wheel speed and timing data obtained from two different studies and eight different athletes.
The model is shown to predict the lap by lap performance of six elite female athletes to an average accuracy of 0.36% and
the finishing times of two elite athletes competing in a 3-km individual pursuit track cycling event to an average accuracy
of 0.20%. Possible reasons for these errors are presented. The impact of speed on steering input is discussed as an example
application of the model.
Keywords Cycling· Simulation· Velodrome· Dynamics
1 Introduction
1.1 Motivation
Cycling is a sport which lends itself to performance analy-
sis. The relative ease of data collection means that com-
petitive teams carry out much analysis and spend significant
resources determining optimal choices of equipment, athlete
or strategy. As power meters and other measurements have
become more accurate, the desire for accurate mathematical
models has grown. A review of existing predictive cycling
models has revealed scope for improvement in the modelling
of tyre forces in track cycling in particular.
1.2 Literature review
Since the release of Schoberer Rad Meßtechnik’s (Jülich,
Germany) first power meter in the early 1990s, mathematical
models of cycling have become increasingly sophisticated.
Olds etal. [1] presented a comprehensive equation for power
demand accounting for aerodynamic drag (including head
winds), rolling resistance and equipment specifications.
Olds etal. [2] improved this model, including the impact
of drafting, crosswinds and the kinetic energy of limbs and
wheels. This revised model was validated by testing 41 ath-
letes over a 26-km time trial. Using measured power as an
input the model predicted finishing time to an accuracy of
5%. Martin etal. [3] developed a model more complicated in
its calculation of aerodynamic drag, with wheel drag varying
with velocity, but otherwise less sophisticated. This model
had an accuracy of 3% when validated for six test subjects.
Basset etal. [4] derived a model to compare the several dif-
ferent world hour record attempts in the 1990s. This model
included some limiting assumptions: i.e. equal groundspeed
and air speed, and frontal surface area, a constant fraction
of body surface area.
It was not until Martin etal. [5] and Lukes etal. [6] who
presented models for velodrome cycling that cornering on
a banked track was addressed. Martin etal. [5] considered
the track as circular with constant radius and banking angle.
In contrast, Lukes etal. [6] modelled the velodrome as two
straights and two corners, albeit with no transition between
the two. A similar approach was taken by Caddy etal. [7]
in an investigation into the impact of cyclist posture on
event performance. All only approximate the true shape of
* Billy Fitton
bf264@cam.ac.uk
1 Department ofEngineering, University ofCambridge,
Trumpington St, CambridgeCB21PZ, UK
2 Mechanical Engineering Department, University
ofCanterbury, Christchurch, NewZealand

B.Fitton, D.Symons
a velodrome. Lukes etal. [8] refined their approximation by
splitting the velodrome geometry into eight sections rather
than four. This improved model also included tyre scrubbing
effects. It could predict finishing times with an accuracy of
2%. In a separate study investigating the aerodynamics of
track cycling, Underwood [9, 10] created the most accu-
rate track cycling model to date. Using measured power and
field-derived values for drag area, CdA, the model predicted
elite individual pursuit finishing times with an average error
of 0.42%.
1.3 Paper overview
Although four of the above models [4, 5, 8–10] concern vel-
odrome cycling, they all neglect some aerodynamic aspects
and model the bends simplistically. To address these limi-
tations, this paper describes a more complete mathemati-
cal model for all cycling events. It proposes a method of
predicting the slip and steer angles necessary to navigate
turns (allowing for banking), as well as allowing for rotat-
ing bodies in the system. The model is derived, a method of
implementation is explained and then the results of two dif-
ferent validation studies are presented. The impact of speed
on steering input is discussed as an example application of
the model.
2 Mathematical model
2.1 Model principles
The focus of this model is velodrome cycling; however, the
equations derived are generic and could be applied to other
cycle sports. The model is a significant extension of the
velodrome-specific model presented in a previous study [7].
The model is a quasi-steady-state analysis that assumes
instantaneous equilibrium of the cyclist at each time step
but allows for changes in speed and configuration between
time steps. At each time step, the cyclist is assumed to be
following a path of known local curvature at constant speed,
i.e. with all accelerations other than centripetal neglected.
The rate of work done against dissipative forces is calcu-
lated based on this instantaneous equilibrium. Any differ-
ence between the cyclist’s input mechanical work and that
dissipated is attributed to changes of gravitational potential
and/or kinetic energy (any change in the latter implying
acceleration).
Forces are resolved both tangential and perpendicular to
the cyclist’s direction of motion at every time step. The lean
angle, tyre slip angles and steering angle are calculated. The
model assumes that the heading angle of the cyclist, χ, and
the steering input, δ, are small and will, therefore, only be
considered in determining the tyre slip angles (which are of
a similar magnitude).
2.2 Derived terms
2.2.1 Governing equation
The governing equation for this model is an energy balance:
Over a time period, δt, the available mechanical work
is the product of the cyclist’s input power, Pin, and the effi-
ciency, η, of the bicycle transmission. Drag forces dissipate
much of this energy, Ediss. The remaining power results
in changes in the total kinetic, ΔT, and/or potential, ΔV,
energies.
Figure1 shows forces on the cyclist viewed in a direction
parallel to the ground surface and perpendicular to the direc-
tion of motion (see also Fig.2, a view along the direction
of motion). β is the banking angle of the ground surface;
λ is the angle of vertical inclination of the cyclist’s direc-
tion of motion and Fλ is the consequent component of the
(1)
𝜂Pin𝛿t=ΔT+ΔV+Ediss .
+
Fig. 1 Summary of forces acting on a cyclist, view is in a plane per-
pendicular to ground surface (β to the vertical)
,
1
3
2
Fig. 2 Forces acting on cyclist in a plane perpendicular to their direc-
tion of motion (λ to the vertical)

A mathematical model forsimulating cycling: applied totrack cycling
cyclist’s weight acting in their direction of motion. Fd is the
aerodynamic drag force and ζ is the angle from the direc-
tion of motion through which it acts. FR and FN are rolling
resistance and normal contact forces, respectively, where
subscripts 1 or 2 refer to the front or rear tyre. FF is the
propulsive force acting at the contact patch of the rear tyre.
Fx is the horizontal component of the centripetal, Fc, and
drag, Fd, forces (Fig.2). Fy is the sum of the forces acting
on the cyclist at an angle λ to the vertical and perpendicular
to the direction of motion (Fig.2). Both Fx and Fy have
components acting in the plane of Fig.1 (as shown). Ω is
the angular velocity of the cyclist and bicycle.
2.2.2 Cyclist dynamics
Figure2 shows the forces acting on a cyclist that determine
the angle of lean, θ. This view is of a plane normal to the
direction of motion, i.e. at angle λ to the vertical. Within this
plane, the weight of the cyclist and bike has component Fw.
As the wheels navigate a bend with instantaneous radius of
curvature Rw, the cyclist’s centre of gravity moves on a path
with instantaneous radius RCG. The corresponding centrip-
etal force, Fc, acts in a direction perpendicular to both the
direction of motion and the axis of rotation and thus at an
angle, κ, to the horizontal. Taking moments about the wheel
contact point, and using
an iterative formula for θ can be derived:
where m is the total mass of the cyclist and the bicycle,
g is the gravitational acceleration, vCG is the velocity of
the cyclist/bicycle centre of gravity and hCG the distance
between the centre of gravity and the wheel/ground contact
line.
The roll angle of the cyclist, φ, is given by
(2)
Fc
=mv
CG
2/
R
CG,
(3)
Fx=Fccos
𝜅
+Fdsin
𝜁
cos
𝛽
,
(4)
Fw
=mg cos 𝜆,
(5)
Fy=Fw−Fcsin 𝜅−Fdsin 𝜁sin 𝛽,
(6)
F𝜆=mg sin 𝜆,
(7)
𝜃
=tan−1
(
Fx
F
y),
(8)
RCG =Rw−hCG sin (𝜃−𝜅),
(9)
𝜃
n+1=tan−1
⎛
⎜
⎜
⎜
⎝
mvCG2
(Rw−hCG sin (𝜃n−𝜅)) cos 𝜅+Fdsin 𝜁cos 𝛽
mg cos (𝜆)−mvCG2
(Rw−hCG sin (𝜃n−𝜅)) sin 𝜅−Fdsin 𝜁sin 𝛽
⎞
⎟
⎟
⎟
⎠
,
2.2.3 Aerodynamic drag
Aerodynamic drag can exceed 90% of a cyclist’s resistance to
motion [11, 12]. In this model, the aerodynamic drag force is
calculated via
using the drag area, CdA, air density, ρ, velocity of the cen-
tre of drag, vd, and local air velocity, vair. vd/air is the velocity
of the centre of drag relative to the air and is found using
The drag force acts in the same direction as vd/air, at an
angle ζ to the direction of motion and parallel to the ground
surface (it is assumed that vair is in a plane parallel to the
ground surface).
2.2.4 Slip, camber andsteer angles
To determine the necessary steering input, the tyre loading and
slip and camber angles must be calculated (Fig.3a). The tyre
loading depends on the overall equilibrium of the cyclist (see
Figs.1, 2, 3b). The resultant reaction force, P, acting through
the tyres and its components can be derived by
where FN and FS are the total normal contact force and side
force, respectively. FN and FS can be separated into forces
acting through front and rear tyres by taking moments about
the front tyre contact patch, resolving in two directions and
using the lengths a and b (see Fig.3b).
Thus it can be found that
Having determined the side force required at each tyre,
it is possible to calculate the slip and camber angles by the
use of
(10)
𝜑=𝜃−𝛽
(11)
F
d=1
∕2𝜌CdA
|
|
|
𝐯𝐝∕𝐚𝐢𝐫
|
|
|
2
(12)
𝐯
𝐝
∕
𝐚𝐢𝐫
=𝐯
𝐝
−𝐯
𝐚𝐢𝐫
.
(13)
P
=
√
Fx
2+Fy
2=
√
FN
2+FS
2
,
(14)
FN=Pcos 𝜑=Fxsin 𝛽+Fycos 𝛽,
(15)
FS=Psin
𝜑
=Fxcos
𝛽
−Fysin
𝛽
,
(16)
F
N2 =
FNa+
(
Fdcos 𝜁+F𝜆
)
hCG cos 𝜑
a+b,
(17)
FN1 =FN−FN2,
(18)
F
S2 =
FSa+
(
Fdcos 𝜁+F𝜆
)
hCG sin 𝜑
a+b,
(19)
FS1 =FS−FS2.

B.Fitton, D.Symons
which follows from Eqs.(14)–(18). The ratio of side to nor-
mal forces is also given by
where
𝛼i
and
𝛾i
are slip and camber angles of the wheel in
question,
C𝛼i
the cornering stiffness (/rad) and
C𝛾i
the camber
stiffness (/rad) of the tyre.
Camber and slip angles for front and rear tyres are defined
by [13]
where ε is the steering rake angle.
un
and
ut
are the nor-
mal and tangential components of the bicycle velocity with
respect to the bicycle frame, see Fig.3a and
where χ is the heading angle. The relationship between
ground steer angle, δ′ and steer angle, δ, is given by [13]
(20)
tan
𝜑=
F
S
F
N
=
F
S1
F
N1
=
F
S2
F
N2
,
(21)
F
Si
FNi
=𝛼iC𝛼i+𝛾iC𝛾i
,
(22)
sin 𝛾1=sin 𝜑+𝛿sin 𝜀cos 𝜑,
(23)
𝛾2=𝜑,
(24)
𝛼
1=𝛿�−
u
n
+aΩcos 𝛽
u
t
,
(25)
𝛼
2=
bΩcos 𝛽−u
n
u
t
,
(26)
un
=vwsin 𝜒,
(27)
ut=vwcos 𝜒,
Equations (20), (21) and (23) enable the rear tyre slip
angle to be given as a function of the known roll angle φ
and the rear tyre stiffness coefficients:
Since α2 is now known Eqs.(26) and (27) can be sub-
stituted into Eqs.(25) and (26) to give
Using Eqs.(20)–(29), it is now possible to obtain a
function of the bicycle and track geometry, tyre coeffi-
cients and roll and heading angles that can be solved itera-
tively, by substituting Eq.(28) for the steer angle, δ:
2.2.5 Rolling resistance
The total rolling resistance is given by
where Crr1 and Crr2 are coefficients of rolling resistance for
front and rear tyre, respectively. These coefficients depend
(28)
tan (
𝛿�
)
=
𝛿cos 𝜀
cos 𝜑−𝛿sin 𝜑sin 𝜀
.
(29)
𝛼
2=
tan 𝜑−𝜑C
𝛾2
C𝛼2
.
(30)
𝜒
=sin−1
bcos 𝛽∕R
w
√
𝛼
2
2+1
−tan−1𝛼2
.
(31)
𝛿
�=tan 𝜒
(
1−
C𝛼2
C𝛼1
)
+cos 𝛽
Rwcos 𝜒
(
a+b
C𝛼2
C𝛼1
)
+
C𝛾2
C
𝛼1
𝜑−
C𝛾1
C
𝛼1
sin−1(sin 𝜑+𝛿sin 𝜀cos 𝜑)
.
(32)
FR=FR1+FR2=FN1Crr1+FN2Crr2,
Fig. 3 Top view in a plane
parallel to the ground of a the
kinematics of, and b the forces,
on the bicycle and cyclist
Ωcos
̇
1
2′
-
Kinematics
1
2
21
cos
−
sin
,cos

A mathematical model forsimulating cycling: applied totrack cycling
on the characteristics of the particular tyre and the instanta-
neous slip and camber angles. Measurement of the rolling
resistance characteristics of a tyre requires careful experi-
mentation, see, e.g. Fitton and Symons [14].
2.2.6 Potential energy
Work done against gravity is determined by changes in
potential energy, V, which is equal to
where the overall height of the centre of gravity, z, depends
on both the varying height of the path of the wheels, hw, and
the lean angle of the cyclist, θ.
2.2.7 Kinetic energy
The kinetic energy of the system is given by
where mw and Iw are the mass and moment of inertia of one
bicycle wheel, vcw and ωw are the translational and angular
velocities of the wheel, respectively, mc and Ic are the mass
and moment of inertia of the cyclist/bicycle, and vc and ωc
are the translational and angular velocities of the cyclist/
bicycle. Rotational kinetic energy of the limbs, pedals and
cranks is neglected; Olds [2] showed them to account for
only 0.07% of total kinetic energy, compared to 2% for the
wheels. To make use of Eq.(34), the different velocities
must be determined from the geometry of the cyclist and
bicycle (Fig.4).
By assuming no longitudinal slip for the tyre/ground con-
tact, the angular velocity of the wheel relative to the cyclist,
ωw/c, can be found by
(33)
V
=mgz=mg
(
h
w
+h
CG
cos 𝜃
),
(34)
T
=2
(1
2
mw𝐯𝐜𝐰
2
)
+
1
2
mc𝐯𝐜
2+2
(1
2
𝐈𝐰𝛚𝐰
2
)
+
1
2
𝐈𝐜𝛚𝐜
2
,
where r is the outer radius of the tyre and the components
are in the first, second and third directions (Fig.2).
The cyclist and bicycle frames rotate as a rigid body,
with an angular velocity of
The components of the wheels’ and cyclist’s angular
velocities in the first, second and third directions are given
by
Due to the quasi-steady-state approach adopted for this
model, the component of angular velocity in the third
direction (i.e. dθ/dt, the rate of change of lean angle) is
assumed to be zero at each instant.
Moments of inertia about the centre of gravity in the
first, second and third directions (Fig.2) are assumed to
be principal moments of inertia for both wheel and cyclist;
thus:
With all terms in Eq.(34) determined, the total kinetic
energy, T, can be defined in terms of vCG by
where
and
2.3 Numerical solution andimplementation
A straightforward implementation is forward integra-
tion of the acceleration, aCG, over fixed time increments,
δt. Forces and configuration are calculated assuming
(35)
𝛚
𝐰
∕
𝐜=
[
−
v
w
r
00
],
(36)
Ω=
v
c
Rc
=
v
CG
RCG
=
v
cw
Rcw
=
v
w
Rw
.
(37)
𝛚𝐜=𝛀=[Ωsin(
𝜃
−
𝜅
)Ωsin(
𝜃
−
𝜅
)0],
(38)
𝛚
𝐰
=𝛚
𝐰
∕
𝐜
+𝛚
𝐜
,
(39)
𝛚
𝐰=
[
Ωsin(𝜃−𝜅)−
v
w
r
Ωcos(𝜃−𝜅)0
].
(40)
𝐈
𝐰=
⎡
⎢
⎢
⎣
Iw100
0Iw20
00Iw
3
⎤
⎥
⎥
⎦
𝐈𝐜=
⎡
⎢
⎢
⎣
Ic100
0Ic20
00Ic
3
⎤
⎥
⎥
⎦
.
(41)
T=KvCG
2,
(42)
T
=

mwRcw
RCG 
2
+1
2mcRc
RCG 
2
+Iw1sin (𝜓)
RCG
−
Rw
RCGr
2
+1
2Ic1

sin (𝜓)
RCG

2
+

Iw2+1
2Ic2

cos (𝜓)
RCG

2

vCG
2
(43)
𝜓=𝜃−𝜅.
(−)
Fig. 4 Showing velocities of the cyclist and bicycle in a plane parallel
to the bicycle frame

B.Fitton, D.Symons
instantaneous steady-state cornering using the method
above; both are assumed constant throughout the time step.
aCG is calculated, and also assumed constant throughout
the time step. This approach is a computationally efficient
approximation that should be sufficiently accurate if δt
remains small (i.e. less than 2s).
The rate at which energy is lost to dissipative forces is
the product of the force magnitudes and the corresponding
velocities and can, therefore, be calculated by
Differentiating Eq.(33) means that the rate of change of
potential energy can be determined by
Note that the quasi-steady-state approximation assumes
that the angular velocity of roll, dθ/dt, is approximately zero
at each time step; therefore, in the simplest implementation
dθ/dt must be estimated via linear extrapolation from previ-
ous time steps.
If the input power
Pin
of the cyclist is known, the power
PT associated with a change in kinetic energy can be calcu-
lated by modifying Eq.(1) to
If K is assumed to be approximately constant over a time
step then PT can be calculated by
The acceleration of the cyclist’s centre of gravity is given
by
The velocity, displacement, configuration and forces act-
ing on the cyclist at the beginning of the next time step can
then be determined.
To avoid the discontinuity that arises from vCG equal-
ling zero at the start of the initial time step, Eq.(48) can be
modified to
where G is the gear ratio and Q the starting torque.
(44)
dE
diss
dt
=Fdcos 𝜁vd+FRvw
.
(45)
dV
dt
=mg
(
vwsin 𝜆−hCG
d𝜃
dt
sin 𝜃
).
(46)
P
T=
dT
dt
=𝜂Pin −
dV
dt
−
dE
diss
dt
.
(47)
P
T≈K
d
dt(
vCG
2
)
=2KvCGaCG
.
(48)
a
CG =
P
T
2KvCG
.
(49)
a
CG =Q
R
cw
KR
CG
rG
,
3 Validation
3.1 Method, assumptions andfixed terms
3.1.1 Method
The model was implemented in Matlab (Mathworks, Cam-
bridge, UK) and validated by two different methods. The
investigation has been approved by the Cambridge Univer-
sity Engineering Department Research Ethics Committee.
First, the tool was used to predict the lap times of six dif-
ferent elite female athletes cycling at approximately constant
speed. These athletes each took part in three sub maximal
efforts at the Manchester velodrome on three separate days.
Throughout each effort, the athletes were asked to maintain
a specified speed and to follow the 250-m datum line as
closely as possible. The specified speed was varied for each
session and athlete. In total, the six athletes completed 174
laps at speeds between 42 and 51km/h. The recorded power
data and measured athlete characteristics were then used to
predict the athlete’s performance throughout the 174 laps.
All of the participants gave informed consent for their data
to be used in this investigation.
Second, the tool was used to predict the finishing time
of two elite female cyclists competing in the 3km Individ-
ual Pursuit (3KIP) event at the 2017 UEC European Track
Championships (ETC2017) in Berlin from the input power
recorded for each cyclist during the same event.
3.1.2 Athlete power
Input power of the athletes was recorded using a power
meter (Schoberer Rad Meßtechnik, Germany) which had
been calibrated according to the manufacturer’s instructions.
3.1.3 Atmospheric conditions
Air density, ρ, was calculated from local atmospheric con-
ditions at the time using Teten’s formulation [15]. Gravi-
tational acceleration, g, was determined for the velodrome
locations [16].
3.1.4 Track geometry andcyclist trajectory
Track geometry (banking angles, radii, inclinations) was
required for two different velodromes. The Manchester (UK)
velodrome geometry was found from a survey of the 250-m
datum line using a TC403L total station (Leica Geosystems,
Heerbrugg, Switzerland). The geometry of the Berlin velo-
drome was determined from a combination of expert knowl-
edge and information given by the track designers: Schuer-
mann Architects (Muenster, Germany). It was assumed that

A mathematical model forsimulating cycling: applied totrack cycling
the wheels of the cyclists exactly followed the datum line.
In contrast to other studies [4, 5, 7, 8] the altitude, hw, of the
datum line was allowed to vary.
3.1.5 Drag area andaerodynamic drag
Bulk airflow is caused by cyclists circling a velodrome.
Throughout the validation process, this airflow was assumed
to remain constant in both magnitude and direction. The
magnitude of the airflow was the average of that meas-
ured during the validation session using an anemometer.
The direction of the airflow was assumed tangential to the
cyclist’s motion, i.e. ζ = 0.
Due to lack of access to a wind tunnel, the cyclist’s drag
area CdA was derived from field testing. Each athlete under-
went aerodynamic testing with identical equipment and
maintaining the same position as that used in each effort
but at an earlier date. The protocol outlined by Fitton etal.
[17] was used in each instance. CdA was assumed constant
throughout each effort.
Note that the centre of drag (the point through which the
aerodynamic drag force acts), is assumed the same as the
centre of gravity. The cyclists’ positions, and the fact that
their bodies typically account for 70% of their total aerody-
namic drag [11], means that the centre of drag is very close
to the centre of gravity.
3.1.6 Mass andinertia
In the first part of this validation, the mass of the cyclists
and their equipment was measured before and after each
session and the average used in the simulation. In the second
part mass was measured only once, as close to the event as
practical.
The model requires the centre of gravity location and
moment of inertia of the cyclist. These inputs were deter-
mined by measuring an average-sized elite cyclist, who did
not take part in either study but was part of the same team,
from a high-definition photo and then modelling each limb,
the torso and the head as separate ellipsoids. The same
cyclist was weighed and each ellipsoid assigned a proportion
of the cyclist’s total mass typical for an average human [18].
Using two more photos of the same athlete in their cycling
position, three-dimensional coordinates were assigned to
each limb. With the limb dimensions, estimated masses and
positions, it was possible to determine the centre of mass and
the inertia of the cyclist and bicycle (without wheels), IC.
The values for other athletes were determined by scaling for
their relative physical characteristics. Wheels were assumed
to be uniform discs to calculate inertia.
3.1.7 Efficiency ofthebicycle
Sources of inefficiency on a bicycle include drivetrain, frame
flexibility and wheel bearings. For this validation, a fixed
mechanical efficiency, η, of 98% [19] has been assumed.
3.1.8 Tyre properties
Tyre properties Cα, Cγ and Crr were determined in a previ-
ous study [14]. Cα and Cγ were found to be constant and
independent of tyre normal force, whereas Crr was found
to be a function of both the loading and orientation (α and
γ) of the tyre. Crr has, therefore, been modelled to increase
non-linearly with both α and γ, and most significantly with
the former.
4 Results
For part one of this validation, lap times were predicted with
an average accuracy of 0.36%. The maximum and minimum
errors of the model were 0.98% and 0.001%, respectively.
The standard deviation of the errors was 0.22%. The predic-
tion was less than the actual lap time in 86% of cases; a prob-
able explanation is the cyclists’ imperfect handling ability.
Analysis of the measured wheel speed data revealed that the
elite cyclists travelled 0.7% further than the track length.
In the second validation study, finishing times of the two
events were predicted with an average accuracy of 0.20%
and individual split times with an average absolute accu-
racy of 0.24% (Table1). Figure5 compares simulated and
recorded wheel speed data for Athlete A. Again the simu-
lation under predicted the finishing time of both athletes.
An additional contributing factor may be the assumption of
constant CdA throughout the event. In a 3KIP CdA may be
greater at the start, as the athlete pedals out of the saddle,
and at the end of the event, where the tiring cyclist may not
maintain their position.
The error associated with the simulation’s prediction of
finishing times and, importantly, split times is lower than
in any previous comparable study. The model described by
Lukes etal. [8] was capable of predicting finishing time in
a 4KIP to within 2% and individual split times (0–1km,
1–2km, 2–3km, 3–4km) with a slightly larger error than
that. Underwood’s [9, 10] proposed model was able to pre-
dict finishing times for elite athletes competing in the 3KIP
and 4KIP to within 0.42%. When investigating Underwood’s
model’s prediction of split times (0–1km, 1–2km, 2–3km),
however, the available data suggest higher errors of approxi-
mately 2.5%.
Significant factors contributing to the accuracy of the
model presented here include the consideration of rotational
kinetic energy and varying tyre forces and the care taken in

B.Fitton, D.Symons
measuring inputs, particularly coefficients of aerodynamic
drag and rolling resistance.
5 Example application
One novel aspect of this model is the capability to predict
tyre slip angles and the necessary steering input, δ, to navi-
gate a particular trajectory. Using the geometry of the Man-
chester velodrome datum line, the impact of speed on δ at
the bend apex has been predicted for Athlete A (Fig.6).
At low speeds, the model predicts a low δ despite the low
lean angle, θ (Fig.7). As speed increases δ also increases to
a peak of ~ 1.7° at ~ 50km/h. This approximately coincides
with the speed at which roll angle, φ, equals zero. As speed
and φ further increase δ is predicted to decrease.
6 Conclusions
A mathematical model of simulating cycling has been
developed. The model includes aspects of particular rel-
evance to velodrome cycling. Via two different validation
studies, the accuracy of the model has been shown to sur-
pass previous comparable models: errors in predicted lap
times are consistently less than 0.36%. A key advantage of
the model is the calculation of steer and tyre slip angles;
this enables the rolling resistance to be predicted more
accurately. This makes it possible, for example, to com-
ment on the impact of handling ability and tyre choice on
event performance.
Table 1 Comparison of actual [20] and simulated split times for the 3KIP events at the ETC2017
Split Actual Simulated Total error (%) Split error (%)
Total time (s) Split time (s) Total time (s) Split time (s)
Athlete A
0–1000m 73.196 73.196 73.049 73.049 − 0.20 − 0.20
1000–2000m 140.467 67.271 140.265 67.215 − 0.14 − 0.08
2000–30,000m 209.328 68.861 208.788 68.523 − 0.26 − 0.49
Athlete B
0–1000m 74.313 74.313 73.992 73.992 − 0.43 − 0.43
1000–2000m 146.726 72.413 146.312 72.320 − 0.28 − 0.13
2000–30,000m 223.157 76.431 222.833 76.521 − 0.15 0.12
Fig. 5 Comparison of simulated
and recorded wheel speed for
cyclist competing in the 3KIP at
the 2017 ETC2017

A mathematical model forsimulating cycling: applied totrack cycling
Acknowledgements The authors would like to thank the study par-
ticipants and staff at British Cycling, EIS and Manchester Velodrome.
Open Access This article is distributed under the terms of the Crea-
tive Commons Attribution 4.0 International License (http://creat iveco
mmons .org/licen ses/by/4.0/), which permits unrestricted use, distribu-
tion, and reproduction in any medium, provided you give appropriate
credit to the original author(s) and the source, provide a link to the
Creative Commons license, and indicate if changes were made.
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