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# Modelling road cycling as motion on a curve

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We present a mathematical model of road cycling on arbitrary routes using the Frenet–Serret frame. The route is embedded in the coupled governing equations. We describe the mathematical model and numerical implementation. The dynamics are governed by a balance of forces of gravity, drag, and friction, along with pedalling or braking. We analyse steady-state speed and power against gradient and curvature. The centripetal acceleration is used as a control to determine transitions between pedalling and braking. In our model, the rider looks ahead at the curvature of the road by a distance dependent on the current speed. We determine such a distance (1–3 s at current speed) for safe riding and compare with the mean power. The results are based on a number of routes including flat and downhill, with variations in maximum curvature, and differing number of bends. We find the braking required to minimise centripetal acceleration occurs before the point of maximum curvature, thereby allowing acceleration by pedalling out of a bend.
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Vol.:(0123456789)
Sports Engineering (2022) 25:12
https://doi.org/10.1007/s12283-022-00376-3
ORIGINAL ARTICLE
P.J.Nee1,2· J.G.Herterich1,3
Accepted: 15 June 2022
Abstract
We present a mathematical model of road cycling on arbitrary routes using the Frenet–Serret frame. The route is embedded
in the coupled governing equations. We describe the mathematical model and numerical implementation. The dynamics are
governed by a balance of forces of gravity, drag, and friction, along with pedalling or braking. We analyse steady-state speed
and power against gradient and curvature. The centripetal acceleration is used as a control to determine transitions between
pedalling and braking. In our model, the rider looks ahead at the curvature of the road by a distance dependent on the current
speed. We determine such a distance (1–3s at current speed) for safe riding and compare with the mean power. The results
are based on a number of routes including ﬂat and downhill, with variations in maximum curvature, and diﬀering number
of bends. We ﬁnd the braking required to minimise centripetal acceleration occurs before the point of maximum curvature,
thereby allowing acceleration by pedalling out of a bend.
Keywords Cycling· Mathematical modelling· Dynamics· Aerodynamics
1 Introduction
Cycling concerns human power transmission on a bicycle to
overcome forces to produce motion. Pedalling drives motion,
while braking, friction, and drag act against motion. Gravity
assists motion on descents and opposes on ascents (Fig.1).
Studies in cycling analyse these forces on each bicycle com-
ponent individually and as a whole [1].
Aerodynamics in cycling has been studied extensively.
Drag forces while drafting vary by seating position [2] on
ﬂat and ascents [3]. Bicycle design is continuously develop-
ing to improve aerodynamics [4, 5]. Cornering inﬂuences
aerodynamics as a change in direction increases the magni-
tude of the airﬂow and pressure [6]. When taking a corner,
cyclists tend to use an anticipatory steering strategy. The
depend on current speed and position [7]. Studies of vehicle
drivers show that the tangent point on the inside of a bend,
is an observational focus [8]. In cycling, faster riders over
a set course brake later at corners and use the full width of
Mathematical models for cycling dynamics use decom-
posed forces parallel and perpendicular to the direction of
motion. Simple models accounting for friction and aerody-
namics may accurately predict output power [10]. On a track
(velodrome), forces incorporate curvature and banking, lean,
and tyre slip angles [11, 12], improving the calculation of
rolling resistance [11]. A Brachistochrone problem in Carte-
sian coordinates solves the optimal dynamics in a velodrome
[13]. For road cycling, weakly undulating courses in one
dimension allow for analysis of riding dynamics, incorporat-
ing physiology for energy expenditure [14]. A further model
considers the time evolution of the energy balance between
kinetic and potential energy for external forces of mechani-
cal power, drag, and rolling resistance [15]. This approach
allows for comparison of pre-deﬁned strategies for racing
on diﬀerent routes, but does not include braking at turns.
On a three-dimensional route, pacing and cornering strate-
gies may be modelled based on diﬀerential geometry relating
and velocity. Pedalling rates to maintain constant power output
* J. G. Herterich
james.herterich@ucd.ie
1 School ofMathematics andStatistics, University College
Dublin, Belﬁeld, Dublin 4, D04V1W8, Ireland
2 Department ofApplied Mathematics andTheoretical
CB30WACambridge, UK
3 UCD Earth Institute, University College Dublin, Dublin,
Ireland
P.J.Nee, J.G.Herterich 12 Page 2 of 14
may be adjusted via gears to account for varying elevation [15,
16]. Optimal control techniques are used to minimise time,
steering, and power [17]. Furthermore, combining mechanical
and physiological parameters, optimal control models suggest
pacing strategies for endurance races [18]. A comparison of
optimal pacing and constant power strategies for road cycling
shows a small, but consistent time saving for optimal pacing
[19]. The gradient and drag terms are by far the most sensi-
tive parameters in many cycling models, followed by rolling
resistance [20, 21].
We model the motion of a rider along a three-dimensional
road, given as a continuous curve. Motion is governed by a bal-
ance of forces: drag, gravity, friction, pedalling, and braking.
Linear acceleration solely applies when travelling in a straight
line. However, centripetal acceleration is also experienced
along a curved path. By describing motion in the Frenet–Serret
frame following the road, rather than in Cartesian directions,
we simplify the governing equations. We consider a number of
prescribed routes including bends and downhill sections. We
set limits for centripetal acceleration to ensure safe passage
determine when to slow down or speed up based on centrip-
etal acceleration restrictions. The routes chosen, and riding
parameters used, allow for successive periods of pedalling and
braking without prolonged periods of pedalling.
2 Mathematical model
2.1 Frenet–Serret frame
In diﬀerential geometry, the Frenet–Serret frame is a com-
mon description of the kinematic properties along a continu-
ous curve in
3
. For a curve
r(s)
, parameterised with respect
to arc-length s, the frame is formed by three orthonormal unit
vector-valued functions
T(s)
,
N(s)
, and
B(s)
, deﬁned as
(1a)
T
=
dr
ds
,
(1b)
N
=dT
ds
/||||||||
dT
ds
||||||||
,
where
||
||
is the vector magnitude. Physically,
is the
unit vector tangent, and points along the curve;
N
is a unit
normal vector, normal to the curve; and
B
is the binomial
unit vector, a second normal vector. The vectors are related
to each other via two scalar functions,
𝜅(s)
and
𝜏(s)
, such that
where
𝜅(s)
is the curvature and
𝜏(s)
the torsion [22]. We take
curvature always to be positive.
The functions
𝜅(s)
and
𝜏(s)
completely determine the
shape of a regular curve [23], providing important environ-
mental properties for a road. The vectors
,
N
, and
B
deter-
mine the orientation of the curve. Most of the forces and
accelerations acting on a rider naturally arise in the direc-
tions of these vectors.
Consider a rider on a road
r(s)
with Frenet–Serret frame
(T,N,B,𝜅,𝜏)
in terms of the arc-length s. The route is
parameterised by s, but the rider’s position is parameterised
by time t. These two are connected by the velocity
v
of the
rider along the route
so that the speed
u
is
in the
direction. Applying an additional time derivative,
we express acceleration
a
in the Frenet–Serret frame,
(1c)
B=T×N,
(2a)
dT
ds
=𝜅N
,
(2b)
dN
ds
=−𝜅T+𝜏B
,
(2c)
d
B
ds
=−𝜏N
,
(3)
v
=
d
r
(
s
)
dt
=ds
dt
T
,
(4)
u
=
ds
dt
,
(5)
a
=
dv
dt
=
du
dt
T+𝜅u2N
.
Fig. 1 Sketch of forces acting
on a rider. All forces but grav-
ity align with the direction of
motion along the route
Modelling road cycling asmotion onacurve Page 3 of 14 12
Acceleration has components in two directions, linear accel-
eration in
and centripetal acceleration in
N
.
2.2 Forces onarider
We model a rider at speed
u
(4) in a direction
determined
by a road. Forces acting on the rider depend on position,
orientation, and speed: drag is due to motion through air;
the gravity component depends on orientation; friction may
depend on both motion and orientation. The rider pedals to
generate motion and applies brakes to slow down. Applica-
tion of pedalling and braking is a choice of the rider, with
safety concerns on a curved route and other factors such as
fatigue.
The hydrodynamic drag force
Fd
acts against the direc-
tion of motion in the
direction, given by
where
𝜌
is the density of air,
Cd
the drag coeﬃcient, and
A
the projected surface area in the direction of motion. We
assume that no wind is present, though this may be added
as a relative velocity.
Gravity
Fg
acts in the negative
z
-direction (Cartesian
frame). The component experienced by the rider in the
direction is
where
m
is the mass of rider and bicycle,
g
the acceleration
due to gravity, and
Tz
the component of
in the vertical
direction. On an ascent (
Tz
>
0
), gravity opposes motion.
Conversely, on a descent (
Tz<0
), gravity assists motion.
The friction force
Ff
is due to the bicycle tyres in con-
tact with the road surface. We model Coulomb friction [24],
neglecting corrections for speed and heat generated on tyres,
though a simpler constant friction force is also widely used
[14]. As
T2
x
+T
2
y
+T
2
z
=
1
, the component experienced by
the rider in the
direction is proportional to the normal
force due to gravity, given by
where
𝜇R
is the rolling resistance and
Tx
and
Ty
are the two
horizontal components of
T
. The friction force opposes
motion and increases with a ﬂatter surface, as expected. The
coeﬃcient of rolling resistance is much less than that of both
sliding and static friction [1, 25].
A rider generates force via pedalling
Fp
. A linear
force–velocity relationship is determined experimentally
[26]. The pedalling force is related [13] to the maximum
(6)
F
d=−
1
2
𝜌CdAu
2T
,
(7)
F
g
=mg [T
(0, 0, 1)] T
=−
mg T
z
T
,
(8)
F
f=−𝜇Rmg(T
2
x
+T
2
y
)
12
T
,
torque M, maximum pedalling frequency rotation
Ω
, devel-
opment D, and linear velocity
u
, so that
The maximum torque is that which occurs at zero pedalling
frequency, and maximum pedalling frequency is that which
occurs at zero torque [13]. This form(9) is a simpliﬁcation
of the force and torque proﬁles during a pedal stroke [27].
Heuristic ﬁts for power-to-mass ratios may be employed
using rider-speciﬁc data [15]. The development is the dis-
tance travelled in one rotation of the pedals, the product of
the wheel perimeter and gear ratio [13].
Pedalling force(9) decreases with speed, though we do
not consider fatigue as an element. However, we require
u
<
DΩ
2𝜋
for a positive force. We impose this limit as a speed
above which the rider is freewheeling (neither pedalling nor
braking). We write the force as
where
H[
]
is the Heaviside function. Freewheeling typi-
cally occurs on descents.
A key quantity to evaluate and compare work done by a
rider is power
P
, the rate at which work is done,
using velocity
v
(3) and pedal force
Fp
(10) [13]. Hence,
power has a quadratic relationship with velocity [26], and
in our case is maximised when
that is, half the speed before freewheeling. Power is based
on instantaneous force and velocity, and is zero when the
rider is not pedalling.
The rider may slow down by applying the brakes. This
generates a force
Fb
,
where b is a braking factor. The b factor depends on whether
the front, rear, or both brakes are applied, as well as the
coeﬃcient of friction between the brake pad and wheel [1].
When braking we take
Fp
=0
, and when pedalling
Fb=0
.
There is a further frictional force
Fs
that prevents lateral
sliding when passing through a bend in the road. This force
is normal to the direction of motion,
(9)
F
p=
2𝜋M
D(
1
2𝜋u
DΩ)
T
.
(10)
F
p=
2𝜋M
D(
1
2𝜋u
DΩ)
H
[
1
2𝜋u
DΩ]
T
,
(11)
P=v
Fp,
(12)
u
=
DΩ
4𝜋
,
(13)
Fb=−b mg T,
(14)
Fs
=−𝜇mgN,
P.J.Nee, J.G.Herterich 12 Page 4 of 14
where
𝜇
is a minimum required coeﬃcient of friction, related
to the lean angle [1, 28].
We neglect a number of other eﬀects in our model such as
wheel bearing friction, slope resistance (ascending) and assis-
tance (descending), frictional loss in the drive chain, and the
inertial force [1, 10, 18], as well as environmental and physi-
ological factors [14, 29]. For simplicity, we neglect changes in
aerodynamic eﬀects when cornering [6]. The model is appli-
cable to cycling on a known smooth three-dimensional route.
Deviation from the route is not possible, but centripetal accel-
eration may be controlled by braking. An additional feature to
use for control that we do not consider is the eﬀect of jerk when
accelerating or decelerating [30], where evasive action such as
powerful braking leads to large jerk forces [31].
2.3 Equations ofmotion
Given a road route, we determine the Frenet–Serret frame
(T,N,B,𝜅,𝜏)
in terms of arc-length s. The forces of the
system depend on the position along any particular route.
We develop a coupled system of diﬀerential equations for
motion on an embedded route.
The position on the route, given by arc-length s, allows
us to determine the forces acting on the rider. The arc length
is given by
with
s0=0
at the beginning of the route.
The speed is determined by a balance of forces in the
direction. The representation depends on whether the rider is
pedalling or braking, typically not applied at the same time.
Newton’s second law gives
where
ue
is the velocity at an event at time
te
, for continu-
ity. This velocity may be the initial at
te=0
or at a time of
transition between pedalling and braking. The forces, given
by Eqs.(6), (7), (8), (10), and (13), incorporate the route by
their dependence on s.
The work done
W
is the total power (11), given by
(15a)
ds
dt
=u
,
(15b)
s(0)=s0,
(16a)
m
du
dt
=
{(
Fp+Fd+Ff+Fg
)
T
(pedalling),
(
F
b
+F
d
+F
f
+F
g)
T(braking),
(16b)
u(te)=ue,
(17a)
dW
dt
=vFp
,
(17b)
W(0)=0.
When braking or freewheeling,
Fp=0
work is done.
The equations for position(15), velocity (16), work
done(17), and the transition between pedalling and brak-
ing(20) comprise the coupled and nonlinear governing equa-
tions of the system. The forces, given by(6), (7), (8), (10), and
(13), depend on position s, speed
u
, and an external factor, the
road geometry. When a transition occurs, at
t=te
, we impose
continuity in the dynamics (15)–(17). We solve the governing
system until the rider reaches the end of the route at arc-length
sl
at some
tend
, i.e.,
s(tend)=sl
.
Finally, we discuss the transition between braking and ped-
alling. The main concern in this paper is when taking a bend.
Deviation from the line of the road may lead to an accident
due to an impact, sliding, or skidding. We consider the force
balance in the normal
N
direction where the centripetal accel-
eration(5) should be less than the sliding force
Fs
(14). To
ensure the rider does not exceed such a threshold, we set a
lower bound with a parameter
Γ
such that
with
0<Γ<1
. When entering a bend in the road, the rider
is typically pedalling. As the curvature
𝜅
increases, the
threshold on centripetal force
m𝜅u2
(18) may be exceeded.
In that case, pedalling is stopped and braking is applied. We
take
Γ=Γ
1
with, for illustrative purposes,
Γ1=0.8
. Brak-
ing is applied until it is safe to start pedalling again, when
Eq. (18) is satisﬁed with
Γ=Γ
2<Γ1
(otherwise there is an
Γ2=0.2
to reﬂect a
decrease in velocity or curvature, or both, before pedalling
again. Hence, we have the following switch conditions:
There is an additional consideration to model. A rider on
speed. The conditions in (19) are instantaneous at the arc-
length position s of the rider’s current location. We introduce
a new parameter,
s>0
, so that the rider looks ahead by this
distance to judge the centripetal acceleration. This parameter
is only applied to the road curvature
𝜅
, while we take the
speed at the rider’s current location, s. Hence, the applied
There is choice in
s
. Absolute or bespoke values do not
scale up on generic routes where the magnitudes of speed
(18)
m𝜅u2<Γ𝜇mg,
(19a)
𝜅u2>Γ1𝜇g, pedalling
braking,
(19b)
𝜅u2<Γ2𝜇g, braking
pedalling.
(20a)
𝜅(
s
+
s
)
u
(
s
)2
>
Γ1
𝜇g, pedalling
braking,
(20b)
𝜅
(s+s
)u(s)
2
<Γ
2
𝜇g, braking
pedalling.
Modelling road cycling asmotion onacurve Page 5 of 14 12
and curvature may vary considerably. We choose
s
to
depend on the current velocity,
so that the rider looks ahead to the position in
t
seconds at
the current velocity.
The dynamics of pedalling and braking may be under-
stood by centripetal acceleration. By looking ahead a dis-
tance
s
(21), the rider observes upcoming increases in
curvature in the route. Note that even while braking, the
centripetal acceleration may increase if deceleration does
not compensate enough for increases in curvature as motion
continues. This potential increase in centripetal acceleration
is a key reason for always looking ahead by some distance
s
.
So far, we have built a model for motion on a three-
dimensional route. In the remaining sections we analyse that
model, speciﬁcally with the objective to ﬁnd a generic esti-
mate of
t
to minimise centripetal acceleration along routes.
This is a safety-ﬁrst approach to using pedalling and braking
at bends, rather than minimising total time by braking as
late as possible [9]. One potential issue is that for a region
of large curvature, the rider may be unable to see beyond the
bend, limiting
t
. We neglect this consideration, assuming
the rider is familiar with the route, and allowing for greater
scope in terms of application to GPS- and map-led devices.
3 Methods andimplementation
3.1 Interpolation
We consider a three-dimensional route
r
presented in
a discrete three-tuple set of
n+1
position coordinates
{(xi,yi,zi)}n
i=0
. The x and y coordinates represent the
East–North plane and z the vertical height. Standard trans-
formations exist to transform GPS coordinates into this form
[32], allowing for distance and gradient along the route to be
calculated and used in the governing dynamics [33]. Interpo-
lation techniques and curve ﬁtting provide for smoothness in
recreating a route from discrete data points [33, 34].
We require a continuous representation of the route as a
curve to call properties of the curve, such as curvature, at
any point. We parameterise the curve in terms of a computed
arc length, s, via
with
𝛿xi=xi+1xi
,
𝛿yi=yi+1yi
, and
𝛿zi=zi+1zi
, and
(21)
s=tu,
(22)
𝛿
si=
(𝛿xi)2+(𝛿yi)2+(𝛿zi)2
,
(23)
s
i=
i1
j=0
𝛿sj
,
and
sl=sn
. We interpolate each coordinate of
r
as a func-
tion of s, producing
r(s)
. Each interpolation is composed of
piecewise cubic Hermite interpolating polynomials (PCHIP)
[35, 36]. This interpolation scheme suppresses oscillations
for non-smooth data. Note that we ﬁnd no practical diﬀer-
ence in using other local interpolating functions, such as
spline functions [37].
We use the cubic interpolant
r(s)
to evaluate the
Frenet–Serret frame via Eqs. (1) and(2). For the govern-
ing dynamics(15)–(17), we require only the tangent vector
T(s)
and curvature
𝜅(s)
. However, these may be computed
explicitly using only
r(s)
. For any parameterisation
r(𝜙)
,
we evaluate the geometric properties as
where
×
is the vector product and
[
,
,
]
the scalar triple
product. Using cubic interpolating functions, the expressions
in (24) are well deﬁned.
3.2 Numerical integration
Given the route properties, we solve the governing sys-
tem(15)–(17) using an explicit Runge–Kutta scheme
(MATLAB’s ode45). For convergence, when transition-
ing between pedalling and braking at time
te
, we impose
a continuous transition in the forces rather than an on/oﬀ
switch. We ramp up or down either force using modulating
functions of the form
These functions tend to 1 and 0, respectively, as t increases
from an event at
te
. A ramp-down is imposed quicker than
a ramp-up to impose continuity, while also reducing the
signiﬁcant time during which the rider is simultaneously
pedalling and braking.
The conditions (20) for transitions between pedalling
and braking are identiﬁed with MATLAB’s ode event
location option [38].
Typically, we solve the dynamics until the rider reaches
a certain point, at a distance
sl
. However, we note that in
(24a)
T
(𝜙)=
r(𝜙)
||
r
(𝜙)|| ,
(24b)
𝜅
(𝜙)=
||
r
(𝜙r
��
(𝜙)
||
||
r
(𝜙)||
3
,
(24c)
𝜏
(𝜙)= [r
(𝜙),r
��
(𝜙),r
���
(𝜙)]
||
r
(𝜙
r��
(𝜙)||
2
,
(25a)
tanh[tte]
ramp up force,
(25b)
1tanh[10(tte)]
ramp down force.
P.J.Nee, J.G.Herterich 12 Page 6 of 14
distance
s
(20). For simplicity, we use a route that is
longer than
sl+max(s)
with a continuous straight sec-
tion beyond
sl
.
3.3 Parameter values
In Table1 we list the parameter values used for physical
constants. Many parameters are based on a racing bike with
rider in a touring position, with
CdA=0.3
[1]. The coeﬃ-
cient of friction is 0.4 for a wet road and 0.6 for a dry road.
Rolling resistance depends on friction and tyre deforma-
tion (mass of bicycle and rider, tyre width, and tyre pres-
sure), and is usually in the range
𝜇R
(0.002–0.008) [1]. We
choose an intermediate value [33]. The development may
be calculated by the gear ratio and perimeter of the wheel.
Parameter values of maximum torque and pedalling frequency
are based on world-class performance [39, 40]. For example, the
maximum values of torque (260Nm) and pedalling rotation
1
) are based on a velodrome where speeds are higher
than on the road [13]. Our range of D takes the upper limit of 10
[13], with lower values to represent lower gears.
The braking factor b is also subjective. An upper limit of
b=0.5
is required for safety (exceeding this may result in
going over the handlebars [1]). If only the rear wheel brake
is applied, then we may have
b=0.26
, resulting in a stop-
ping distance twice that of the front brake [1]. We choose
this for a smoother ride, i.e., no sudden braking force.
ofthemodel
In this section, we analyse steady-state motion and calibrate
the model for appropriate parameters.
We consider steady motion in our model, via (16a) when
pedalling or freewheeling. We examine the balance of forces
such that
u
, requir-
ing positive solutions.
When pedalling, the steady speed is
where
T=(Tx,Ty,Tz)
. Eq.(27) is analogous to the equilib-
rium velocity (5.2) in [13], however for an arbitrary route.
The speed(27) is positive provided that
a condition relating the torque M to physical and geomet-
ric parameters. Work done(17) increases linearly as
u
, and
hence
Fp
, are constant. As a route develops, with
T
vary-
ing, the speed adjusts to a new steady state over a transient
period. A rider decelerates to zero if suﬃcient time is spent
without satisfying the condition (28).
When freewheeling, with
Fp
=0
analogous to a terminal velocity with frictional resistance
and drag balancing gravity. The freewheeling steady speed
u
fw
is
We note that freewheeling an ascent,
Tz>0
, is not possible,
as expected. However, a descent (
Tz<0
) is insuﬃcient to
ensure a real and positive steady state. For a real-number
value
u
fw
(29), we require a descending gradient
𝜃
such that
For
𝜇R=0.0045
, this means
𝜃>0.2578
wheel motion. For a rolling resistance as high as 0.008, we
require
𝜃>0.458
. No work(17) is done as
Fp
=0
when
freewheeling.
(26)
mdu
dt
=
(
Fp+Fd+Ff+Fg
)
T=
0.
(27)
u
=1
4𝜋2M+
16𝜋4M2
D mg
𝜇R
T2
x+T2
y
12
+Tz

12
,
(28)
M
>mgD
2𝜋
𝜇R
T2
x+T2
y
12
+Tz
,
(29)
u
fw =
2mg
𝜇R
T2
x+T2
y
12
+Tz
𝜌C
d
A
.
(30)
𝜃>tan1
𝜇
R.
Table 1 Reference parameter values
Name Symbol Value Unit Reference
Drag coeﬃcient
Cd
1 [1]
Surface area
A
0.3 m
2
[1]
Mass
m
80 kg
Gravity
g
9.8 m/s
2
Density of air
𝜌
1kg/m
3
Rolling resistance
𝜇R
0.0045 [1, 33]
Maximum torque M50–250 Nm [39]
Maximum pedalling
Ω
2𝜋
8𝜋
Development D7–10 m [13]
Braking factor b0.26 [1]
Friction coeﬃcient
𝜇
0.4–0.6 [1]
Modelling road cycling asmotion onacurve Page 7 of 14 12
4.2 Calibration onﬂat straight section
On a constant route (
T
constant), the dynamics tend to a steady
state as opposing forces, especially drag(6) and pedalling(10),
balance. To test and calibrate the model, we consider three
state outputs of speed and power for diﬀering input parameters
(Table2) with each other and with values from the literature.
The route is parameterised by
r(s)=(s,0,0)
, without loss
of generality, for arc-length parameter
s∈[0, 1000]
. The tan-
gent vector is
and
𝜅=0
everywhere. The three cases take parameter values
listed in Table2. The resulting speed proﬁles in Fig.2 are
determined by integrating Eqs.(15)–(17).
All cases are initialised with
u0=
5m/s (18km/h). Cases
1 and 2 are designed to produce the same output speed and
power,
u=8.33
m/s (30km/h) and
P=115
W respectively.
Case 1 has a larger torque M and development D, but lower
pedalling rotation
Ω
than Case 2. We observe a small diﬀer-
ence in how each rider reaches the same steady state. The rider
in Case 1 accelerates faster than Case 2 (Fig.2), reaching a
steady state and ﬁnal position at 1000m earlier (Table2).
Case 3 is a rider with high torque M, pedalling rotation
Ω
, and
development D. The steady speed is
u=12
m/s (43.2km/h)
and power is
P=300
W. As expected, the ﬁnal time
tf
is less
than Cases 1 and 2.
Furthermore, these speed and power pairings correspond to
those of a road-racing bicycle with similar input parameters
[1]. The pedalling rotations and power pairings fall in typical
ranges corresponding to 17–19% eﬃciency [1]. The steady
speeds from each numerical solution match theoretical pre-
dicted values(27) and satisfy the condition in Eq.(28).
4.3 Parameter variation
We brieﬂy consider the eﬀect of a small change in parameter
u
(27). Since
u=u({qi})
with
{q}
the parameter set
(31)
T=(1, 0, 0),
where we use
|T|=1
to write
T2
x
+T
2
y
=1T
2
z
in (27). In
eﬀect, the magnitude of rolling friction and gravity depend
du
due to a
change in parameters
dqi
is given by
d
u=
i
𝜕u
𝜕q
i
dq
i
. How-
ever, some perturbations increase and others decrease the
speed. We measure an upper bound to be
where each partial derivative is evaluated at the reference
parameter values (Table1). We take
Tz=0.01
, a small uphill
gradient for the purpose of illustration. The coeﬃcients
𝜕u
𝜕
q
i
are the local sensitivities of the output, here steady speed, to
input parameters [21, 41].
In Table3, we consider the local sensitivities with respect
to the three reference cases (Table2). Physically, these num-
bers make sense: increase in air density
𝜌
, drag coeﬃcient
Cd
, surface area
A
, mass
m
, local gravity
g
, rolling resistance
(32)
{qi}={𝜌,Cd,A,D,Ω,M,m,g,𝜇R,Tz},
(33)
d
u
i
|
|
|
|
𝜕
u
𝜕q
i|
|
|
||
|
dqi
|
|,
Table 2 Summary of calibration
numbers for three cases on a
straight ﬂat route (31)
The inputs for each case are initial speed
u0
(m/s), torque M (Nm), pedalling rotation
Ω
opment D (m). The outputs are the steady speed
u
P
(W), and ﬁnal time
tf
(s) to
1000m
Case
u0
M
Ω
D
u
P
tf
m/s Nm rad/s m m/s W s
1 5 132
2𝜋
10 8.33 115 123.2
2 5 75
3𝜋
7 8.33 115 123.8
3 5 200
3𝜋
10 12 300 87.4
Fig. 2 Speed versus time for a rider on a 1000 m ﬂat and straight
route (31). The three cases are solutions of Eqs. (15)–(17) with
parameters given in Table2
P.J.Nee, J.G.Herterich 12 Page 8 of 14
𝜇R
Tz
reduces speed; increase in development
D, pedalling
Ω
, and torque M increases speed; and vice
versa. The coeﬃcients for
𝜇R
and
Tz
are large. However the
absolute value of rolling friction is small, and a small change
in gradient is well known to have a considerable eﬀect on
speed for the same power [20].
In Fig.3, we show the maximum relative change in speed
duu
as all ten parameters are changed by the same relative
amount to the reference values,
dqiqi05
%. The size of
the change is skewed by the well-known sensitivity to drag,
rolling resistance, and gradient [20, 21].
5 Dynamic motion oncurved routes
In this section, we analyse unsteady dynamics on a number
of routes. We include transitions(20) between pedalling
and braking to determine optimal ranges for
s
(21) at cur-
rent speed. The aim is to minimise the centripetal accelera-
tion(18) at a bend. We compare centripetal acceleration with
mean power.
5.1 Routes
We consider three non-trivial routes parameterised as
r(𝜙)=(x(𝜙),y(𝜙),z(𝜙))
. They comprise ﬂat and downhill
components, diﬀering scales of maximum curvature, and
diﬀering number of bends.
Route 1 is a ﬂat meandering (cubic) curve given by
with
𝜙 [−L, 1.2L]
for
L=50
(Fig.4a). The total distance
is
sl=287.72
m.
Route 2 is an approximate Gaussian ﬁt (Fig.5a) to a rider
travelling straight through a roundabout, with inner circle of
with
𝜙∈[0, L]
for
L=200
(Fig.4c). In other words, the
route is centred at L/2, with amplitude R and standard devia-
tion
R2
. The total distance is
sl=203
m.
Route 3 is a descending route that meanders sinusoidally,
given by
with parameter
𝜙∈[0, 1.2L]
for
L=1000
m (Fig. 4e).
While progressing in the x direction, the z direction linearly
decreases with x only. The y direction has three stages. The
(34)
Route 1
=
x=𝜙,
y=𝜙
10 3
,
z=0,
(35)
Route 2
=
x=𝜙,
y=1.5 +Rexp (𝜙L2)2
2R2
,
z=0,
(36)
Route 3
=
x=𝜙,
y=
625𝜋
L(𝜙100),(𝜙<100),
125 sin (5𝜋𝜙L),(100 𝜙900)
,
625𝜋
L(𝜙900),(𝜙>900),
z=L𝜙
20
,
Table 3 Changes to speady-state speed
du
(33) due to small param-
eter variations
dqi
from the reference values (Table1) and three cases
(Table2) with
Tz=0.01
Parameter
qi
Case 1
𝜕u
𝜕
qi
Case 2
𝜕u
𝜕
qi
Case 3
𝜕u
𝜕
qi
𝜌
− 0.8171 − 0.9593 − 1.6376
Cd
− 0.8171 − 0.9593 − 1.6376
A
− 2.7236 − 3.1976 − 5.4587
D0.4061 0.4629 0.5462
Ω
0.9476 0.5852 0.8557
M0.0143 0.0303 0.01301
m
− 0.0134 − 0.0164 − 0.0121
g − 0.1097 − 0.1342 − 0.0985
𝜇R
− 74.167 − 90.719 − 66.552
Tz
− 74.167 − 90.719 − 66.552
Fig. 3 Small change in speed
du
(33) due to a small change in
each parameter
dqi
. Each parameter is changed by the same relative
amount to its reference values (Table 1) for the three cases (Table2)
with
Tz=0.01
. The steady-state speeds (27) in each case are given
and a reference line (dashed) is
duu=dqiqi
Modelling road cycling asmotion onacurve Page 9 of 14 12
middle stage consists of two sinusoidal periods of amplitude
125m. The sections to either side are straight and continu-
ous up to the ﬁrst derivative. We run our simulations until
the rider reaches the point
x=L
(i.e.,
𝜙=L
), a distance of
sl=1768.26
m. The total descent along this route is 50m.
The unsigned curvatures (24b) for each route are shown
in Fig.4b,d,f. The routes have two, three, and four bends,
respectively, and their maximum magnitudes diﬀer for a
wider application of the model. As curvature is higher for
Route 2 with
R=7.5
m (Fig.4d), we simulate the dynam-
ics with slower Case 1 parameter values (Table1). For
Routes 1 and 3, we use Case 3 parameters. In all routes,
torsion(24c) is zero.
The curvature on a roundabout with a Gaussian ﬁt(35)
is maximised at
x=L2
, with
|
𝜅
(L2)|=1R
. Traversing
a larger roundabout has a smaller curvature, as expected.
u
, the maximum centripetal
acceleration(5) is
(u)2R
. Hence, a theoretical thresh-
old speed
u
without exceeding the centripetal accelera-
tion(20) is
taking
Γ=1
(18) for an upper limit. The maximum steady-
state speed scales with
R
(Fig.5b). The scaling with
𝜇 ,
illustrating the speed adjustment required for wet or dry
(37)
u=
𝜇gR
,
Fig. 4 a Route 1 (34). b Cur-
vature of Route 1 with
s=0
m
at
(x,y,z) = (−50, 125, 0)
and
s=sl=287.72
m at
(x,y,z)=(50, 125, 0)
. c Route
2 (35) with
R=7.5
m. d
Curvature of Route 2 with
s=0
m at
(x,y,z)=(0, 0, 0)
and
s=sl=203
m at
(x,y,z)=(200, 0, 0)
. e
Route 3 (36). f Curvature
of Route 3 with
s=0
m
at
(x,y,z)=(0, 200, 50)
and
s=sl=1768.26
m at
(x,y,z)=(1000, 200, 0)
. Solid
sections represent when pedal-
ling and the dashed section rep-
resents braking according to the
governing Eqs. (15)–(17) with
pedalling/braking subject to
(20) with
s
(21) taking
t=1
s.
We take riding parameters of
Case 3 in (a, b); Case 1 in (c,
d); and Case 3 in (e, f). The
cases are listed in Table2
P.J.Nee, J.G.Herterich 12 Page 10 of 14
conditions (Table1). This limit is independent of
s
as it is
designed so that braking is not required.
5.2 Dynamics
On all three routes (34)–(36), the rider dynamics are
broadly similar. We calculate the geometric properties
of the route(24). We may then solve the governing Eqs.
(15)–(17), with the transition between pedalling and brak-
ing given by(20).
When looking ahead by a distance travelled of
t=1
s
at the current velocity, the regions of pedalling (solid)
and braking (dashed) for each of the three routes is shown
in Fig.4a, c, e. The corresponding route curvature dur-
ing pedalling and braking is shown in Fig.4b, d, f. We
observe that braking occurs while approaching and during
the region of largest curvature.
In Fig.6, we show the full dynamics for distance, speed,
and work done for Route 3 corresponding to Fig.4e, f
with
t=1
s. For simplicity, the rider begins with the ﬂat
Fig. 5 a Sketch of a Gaussian ﬁt route (35) (dashed) taken straight
traﬃc islands, are shown in solid. b Maximum steady speed
u
(37)
centripetal acceleration leading to sliding (18). The coeﬃcient of
friction
𝜇
is taken to be 0.6 for dry (solid) conditions and 0.4 for wet
(dashed) conditions (Table1). The left y-axis displays
u
in m/s and
the right y-axis in km/h
Fig. 6 Distance, speed, and
work done on Route 3(36)
according to the governing Eqs.
(15)–(17) with pedalling/brak-
ing subject to (20) with
s
(21)
given by
t=1
s. The pedalling
parameters are given by Case
3 in Table2 with
u0=12
m/s.
Solid sections correspond to
pedalling and dashed to braking
Modelling road cycling asmotion onacurve Page 11 of 14 12
u0=12
m/s, as opposed to a descend-
ing steady-state speed(27). The rider brakes for
3.5
s at
each bend. When braking no work is done, remaining con-
stant in this period, though the distance travelled increases
as the speed is positive. The bends are suﬃciently far apart
(Fig.4f) so that the dynamics of the rider through each
bend is independent: a pseudo-steady-state speed(27) is
reached. Variations in gradient with curvature in the middle
section(36) means an unsteady transient region is always
present, but the dynamics are periodic.
5.3 Centripetal acceleration
We use thresholds on centripetal acceleration(20) to alter-
nate between pedalling and braking by looking ahead by
s>0
, as discussed. The actual centripetal acceleration
experienced at position s remains below the threshold
Γ1𝜇g
(20a). In Fig.7a, c, e, we show the centripetal accel-
erations that are experienced (thick solid) and anticipated
t=1
s (solid/dashed), so a distance ahead cor-
responding to 1s at the current speed(21). The maximum
centripetal acceleration experienced is marked (
), and is
used as an overall measure for the dynamics of the simu-
lation with
s>0
. The improvement over the centripetal
acceleration anticipated is greater in Routes 1 and 2 than 3.
We vary
t
to ﬁnd an optimal value for each route that
minimises the maximum centripetal acceleration (
) expe-
rienced along that whole route (Fig.7b, d, f). For Route 1,
t
opt 2.06
s; Route 2,
t
opt 2.6
s; and for Route 3,
t
opt 2.55
s. In all cases, for
t<1
s and
t>3
s, the cen-
tripetal acceleration increases; in the former case the rider
brakes too late and in latter too early. For 1s
<t<3
s, in
Fig. 7 a, b Route 1 (34). c, d
Route 2 (35). e, f Route 3 (36).
Routes 1 and 3 are simulated
with parameters from Case 3
and Route 2 with parameters
from Case 1 (Table1). (Left)
Centripetal acceleration at the
current position s (thick solid)
and that by looking ahead by
s=s+tu(s)
(thin solid/
dashed) with
t=1
. The curve
solid and dashed to represent
pedalling or braking, respec-
tively. The maximum centrip-
etal acceleration experienced
(thick curve) is shown with a
marker (
). The horizontal lines
represent the thresholds(20) for
transitioning between pedalling
and braking:
Γ2
𝜇
g
(dashed)
<Γ1
𝜇
g
(dot-dashed)
<𝜇g
(dot-
ted). Here
Γ1=0.8, Γ2=0.2
,
and
𝜇=0.6
(dry). (Right)
Maximum centripetal accel-
eration (solid; left y-axis) and
mean power (dashed; right
y-axis) against
t
P.J.Nee, J.G.Herterich 12 Page 12 of 14
which the maximum centripetal acceleration is minimised,
it remains below the threshold
Γ1𝜇g
.
Interestingly, the maximum centripetal acceleration
may be found at diﬀerent bends for diﬀerent values of
t
.
We observe the switch between bends by a sharp kink. For
example, consider
t1
s in Fig.7b for Route 1. The maxi-
mum centripetal acceleration changes from being experi-
enced at the ﬁrst bend to the second bend (Fig.4b). The
approach of this transition may be seen in Fig.7a where
two peaks in centripetal acceleration, one at each bend, are
nearly equal. In this case, for
t>1
s, the rider brakes far
enough in advance, and hence may begin pedalling again,
to regain enough speed subsequently into the second bend
for maximal centripetal acceleration to occur there. Route
2 possesses two kinks (Fig.7d) in the optimal value as it
changes back-and-forth from the ﬁrst to the second bend.
In Fig.8, the pedalling and braking dynamics for the
optimal solution for each route (Fig.8a, c, e) and the cor-
responding curvature (Fig.8b, d, f) are shown in solid/
dashed. Comparing the dynamics with the same routes in
Fig.4 with
t=1
s shows an optimal braking period while
approaching the maximum curvature. Pedalling may be
resumed before the maximum curvature point.
We consider the change in mean power over the range
of
t
. For Routes 1 and 2, the changes are
12
W, however
these routes are short so that small changes in acceleration
after braking have a larger overall eﬀect. For Route 3, the
Fig. 8 a Route 1 (34). b Cur-
vature of Route 1 with
s=0
m
at
(x,y,z) = (−50, 125, 0)
and
s=sl=287.72
m at
(x,y,z)=(50, 125, 0)
. c Route
2 (35) with
R=7.5
m. d
Curvature of Route 2 with
s=0
m at
(x,y,z)=(0, 0, 0)
and
s=sl=203
m at
(x,y,z)=(200, 0, 0)
. e
Route 3 (36). f Curvature
of Route 3 with
s=0
m
at
(x,y,z)=(0, 200, 50)
and
s=sl=1768.26
m at
(x,y,z)=(1000, 200, 0)
. Solid
sections represent when pedal-
ling and the dashed section
represents braking according to
the governing Eqs. (15)–(17)
with pedalling/braking subject
to (20) with
s
(21) given by:
Case 3 with
t
opt =2.06
s in (a,
b); Case 1 with
t
opt =2.6
s in (c,
d); and Case 3 with
t
opt =2.55
s
in (e, f). The cases are listed in
Table2
Modelling road cycling asmotion onacurve Page 13 of 14 12
route is long enough (
sl=1768.26
m) so that change in
mean power,
4
W, is small; the long sections in between
bends, most of which are covered at pseudo-steady state
(Fig.6), allow for mean values to be reached. The diﬀer-
ence in acceleration periods after braking become small
when measured against the whole route.
6 Discussion
We present a mathematical model of road cycling with
pedalling, braking, gravity, drag, and friction on arbitrary
curved routes. The route is embedded in a coupled model
via the Frenet–Serret frame in which many of the forces
point in these natural directions. Our model presents two
novel steps in modelling cycling motion by including:
(i)motion on curved routes and(ii) braking when cor-
nering. The route is fundamentally built into the model,
extending insights in the literature [13, 14]. Our model
allows us to study the second dynamic feature of pedal-
ling and braking at bends in the road, not modelled in the
literature to our knowledge. We incorporate a brand-new
feature, using curvature and speed, of viewing the road
Thresholds on centripetal acceleration are used to deter-
mine transitions between pedalling and braking with the
rider looking ahead by a distance dependent on current
speed. We ﬁx the route, and do not allow the rider to devi-
ate in order to reduce centripetal acceleration. The model
neglects a number of forces, such as wind and inertia, and
we restrict our simulations to a subset of riding parameters
as a full study is beyond the scope of the paper.
We provide a simpliﬁed, yet applicable model for those
with the ability to produce quality data, or indeed to be
used to inform on quality control of data. We welcome
and encourage readers to test the model for motion on
a three-dimensional route with real data. Data should
include motion (such as GPS, rider-collected data, inertial
measurement units) in combination with environmental
knowledge such as surface roughness for friction.
We calibrate the model with representative parameters,
with resulting steady-state speeds and powers matching
those in the literature (Table2). The steady-state speeds,
(27) and (29), depend on input parameters incorporating
pedalling strategy, physical parameters, and the route.
We determine conditional relationships,(28) and (30), on
inputs parameters.
Outputs of our model such as speed are most sensitive
to drag, rolling resistance, and gravity (Table3), agreeing
with the literature [20, 21]. For a 5% change in each of
ten parameter values, the combined speed variation is at
most 12%. In the dynamic model more care is needed by
the user in applications such as parameter ﬁtting with real
data, such as a sensitivity analysis about both the param-
eters and route where the environment may change in a
spatio-temporal manner and over a large range.
For our full dynamic model, we use routes (34)–(36).
We include ﬂat and downhill, with variations in maximum
curvature, and diﬀering number of bends. In all cases, to
minimise centripetal acceleration throughout, an optimal
value for the distance to look ahead is found to be the dis-
tance that would be travelled in 1–3 s at the current speed
(Fig.7(b,d,f)). This distance gives the rider time to react
gradually but also maintain speed for as long as possible.
Periods of braking corresponding to optimal centripetal
accelerations are found to occur before the point of maxi-
mum curvature (Fig.8), thereby allowing acceleration out
of a bend.
7 Conclusions
We simulate road cycling along three-dimensional routes
using the Frenet–Serret frame generated by treating a route
as a curve in space. Along with environmental parameters,
we include pedalling and braking as rider-controlled param-
eters. We optimise rider dynamics based on safety at bends
on three-dimensional routes by minimising centripetal
acceleration.
Our model forms the basis of more general models for
control of motion, including automated driving, along a
route. Forces and parameters may be tuned appropriately.
The model may be adapted for spatially varying environmen-
tal parameters and additional features such as physiology
(fatigue) may be incorporated.
Acknowledgements The authors would like to thank the Undergradu-
ate Summer Research Project scheme organised by UCD School of
Mathematics and Statisticsforsupport on initial stages of this research.
Funding Open Access funding provided by the IReL Consortium.
Declarations
Conflict of interest The authors declare that they have no conﬂict of
interest.
tion, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
included in the article's Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included in
the article's Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will
need to obtain permission directly from the copyright holder. To view a
copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.
P.J.Nee, J.G.Herterich 12 Page 14 of 14
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