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SIMULATION ANALYSIS OF RIDER

EFFECT ON BICYCLE

STABILIZATION

THESIS

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS OF

BITS C421T/ 422T THESIS

BY

UMASHANKAR N

ID NO. 2002A4TS901

UNDER THE SUPERVISION OF

MR. HIMANSHU DUTT SHARMA

SCIENTIST, DIGITAL SYSTEMS GROUP

CENTRAL ELECTRONICS ENGINEERING RESEARCH INSTITUTE (CEERI)

PILANI (RAJASTHAN)

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE (BITS)

PILANI (RAJASTHAN)

28th NOVEMBER, 2006

i

ACKNOWLEDGEMENT

I am very grateful to my supervisor, Mr. Himanshu Dutt Sharma, Scientist, Central Electronics

Engineering Research Institute (CEERI), Pilani, for introducing me to the world of robotics research

and help me build all skills to pursue a career in research; I owe a lot to him for all the support he has

given me in the successful completion of my thesis. If it had not been for him, this thesis would have

been impossible.

I wish to thank Dr. Ravi Prakash, Dean, Research and Consultancy Division, Birla Institute of

Technology and Science (BITS), Pilani for helping me pursue my thesis in CEERI, Pilani. I also wish

to thank him for consenting to be the examiner for my final thesis viva.

My thanks are due to Dr. L. K. Maheshwari, Vice Chancellor, BITS, Pilani and Dr. Chandrasekhar,

Director, CEERI, Pilani for facilitating co-operative use of research facilities between the two

institutes.

I am highly indebted to the engineers at MathWorks Inc. for developing the MATLAB package

without which all my research work would have remained incomplete.

ii

LIST OF SYMBOLS & ABBREVIATIONS USED

ROBI - Robot Bicycle

χ - Lean Angle

ψ - Steer Angle

θ& - Angular Velocity

τ - Torque

L - Lagrangian function

K - Kinetic energy

P - Potential energy

GUI - Graphic User Interface

iii

--------------------------------------------------------------------------------------------

Thesis Title : Simulation Analysis of Rider Effect on Bicycle Stabilization

Supervisor : Mr. Himanshu Dutt Sharma, Scientist, CEERI, Pilani

Semester : First Session : 2006 - 07

Name of the Student : Umashankar N ID No. : 2002A4TS901

--------------------------------------------------------------------------------------------

ABSTRACT

This thesis work involves the application of robotics based approach in formulating the generalized

dynamic equations of motion of an autonomous bicycle system running under lab conditions. The

bicycle system is considered to be composed of three rigid bodies connected by revolute joints. The

mathematical model facilitates study of various parameters of motion namely- lean, steer, angular

speed, lean-rate, steer-rate, lean-torque, steer-torque, speed-torque, inertial forces, coriolis and

centrifugal forces, gravity forces, kinetic energy, potential energy and total energy of various rigid

body parts forming the system. The equations are programmed as a MATLAB model, which also

provides a GUI to experiment with it. In this work, simulation experiments have been conducted and

the model has been verified for its behavior, with reference to system’s stability at various initial

speeds when running without a rider. Also, a simplified set of model equations is also proposed to

capture the basic lean and steer characteristics. Finally, the effect of rigid rider on the autonomous

bicycle system is studied. Simulation experiments have been conducted on the same platform and

conclusions on drawn based on the results.

TABLE OF CONTENTS

ACKNOWLEDGEMENT i

LIST OF SYMBOLS & ABBREVIATIONS USED ii

ABSTRACT iii

1. INTRODUCTION 1

2. BICYCLE MODEL 3

2.1. COMPONENT TABLE 4

2.2. FRAME AND JOINT CONSIDERATIONS 4

2.3. GRAPHIC USER INTERFACE 6

3. BICYCLE DYNAMICS MODEL USING LAGRANGE – EULER (L-E) FORMULATION 7

3.1. MOTION EQUATIONS OF THE BICYCLE 9

4. MODEL VALIDATION FOR AUTONOMOUS BICYCLE 11

4.1. SIMULATION EXPERIMENTS 11

4.2. ANALYSIS OF SIMULATION RESULTS 13

5. MODEL VALIDATION FOR AUTONOMOUS BICYCLE WITH RIGID RIDER 16

5.1. SIMULATION EXPERIMENTS 16

5.2. ANALYSIS OF SIMULATION RESULTS 18

6. CONCLUSIONS 19

APPENDIX 20

A. TERMS IN THE EQUATION OF MOTION 20

B. SIMULATION RESULTS 21

REFERENCES 39

1

1. INTRODUCTION

Bicycle stability problem has remained difficult to analyze for decades. The main reason for this is the

non-availability of a mathematical model that closely represents a practical bicycle’s dynamics.

Although the existing models give a very good insight to the understanding of bicycle system yet

these have lots of approximations and assumptions, which drive them away from the practical one.

The works of Whipple [1], Klein and Sommerfeld [2] had small-angle approximations. Jones [3] and

Le Hénaff [4] concentrated on geometrical considerations that neglected dynamical forces on the

steering system. Lowell & Mckell [5] and Frank et. al [6] derived equations of motion of a rigid-rider

bicycle system using Newtonian mechanics. Papadopoulos [7] and Hand [8] developed the linearized

equations of motion for general bicycle geometry and obtained ranges of stable motion depending on

the velocity and parameters of the bicycle. Most of these models have assumed the velocity of bicycle

to remain a “constant” while modeling the system, consequently, the linear acceleration of the system

is also eliminated from the model. The bicycle design too is largely simplified in existing models so

the equations further get approximated; the simulation results indicate the basic bicycle behavior but

sophisticated analysis of the motion can not be done using them in an accurate manner.

The Newtonian and Lagrangian mechanics based models [5, 8] were implemented in the earlier work

of ours [9, 10, 11] and it was realized that a more comprehensive model is needed to explore the

system in detail. This thesis presented here, eliminates most of the limitations mentioned above and a

generalized bicycle model is proposed that very closely represents practical bicycle dynamics. For the

ease of forming equations of motion, techniques from robotics are used and the bicycle is modeled

analogous to a robotic arm. Using the standard robotics approach, optimal number of coordinate

frames is assigned to the bicycle system, and later Lagrangian-Euler method is used to find equations

representing the dynamics of bicycle motion [12]. This approach leads to a generalized set of

equations representing the dynamics of a bicycle system with lean, steer and velocity as three

variables. This non-linear model is a step ahead of exiting models wherein velocity has been

approximated to a constant, and it is also capable of providing most of the motion parameters

accurately.

The model is developed in MATLAB wherein coupled differentials are solved, which generate

solutions with time for various parameters i.e. lean, steer, angular velocity, lean rate, steer rate, lean

torque, steer torque, velocity torque, KE & PE of all links and forces on these links namely- inertial,

gravity, coriolis and centrifugal. The model also has a GUI, in which the bicycle parameters can be

changed for analysis and experiments. Also, a set of model equations is found out by curve fitting to

represent a simplified but sufficiently accurate model for the autonomous bicycle system. Finally, the

2

effect of the presence of rigid rider on the bicycle is analyzed through the similar set of simulation

experiments are conducted and compared with that of the autonomous bicycle without the rigid rider.

The report is organized as follows- section 2 deals with physical model of bicycle, section 3 develops

Lagrangian-Euler dynamics, section 4 describes results of simulation experiments on autonomous

bicycle and their analysis, section 5 describes results of simulation experiments on autonomous

bicycle with rigid rider and their analysis and conclusions are presented in section 6. Appendix B

contains the detailed simulation results.

3

2. BICYCLE MODEL

The bicycle has been designed to closely resemble a practical bicycle. The entire bicycle is

constructed using rigid rods and blocks. The three-dimensional model, designed using Pro-Engineer is

shown in Fig. 1. The design of the bicycle model considers the following-

• All rods have been designed as hollow cylinders with constant wall thickness.

• The wheels are also designed as hollow cylinders.

• The mass of the spokes are accounted for in that of the wheel centre, which is a solid

cylinder.

• The mass of all parts have been calculated based on their density and volume values.

• The material used for all parts, except wheels and seat, is mild steel.

• Wheels and seat are considered to be made of synthetic rubber.

• Both the front and rear wheels are identical.

• The pedaling set-up has been reduced to a circular disk, with the chains neglected.

• The bicycle model is supposed to be moving on a running belt, as shown in Fig. 1, whose

speed is equal to bicycle’s linear speed and friction is sufficient to produce motion without

slipping. This avoids need of considering its translatory motion with reference to the world

/Base coordinate frame.

Figure 1: The 3-dimensional Bicycle Model.

4

2.1. COMPONENT TABLE

The bicycle is assumed to consist of three rigid bodies, namely, rear part, front part and front wheel.

All these three rigid bodies are in turn made up of various components. The components that

constitute the three rigid bodies of bicycle are given in Table 1.

Table 1: Bicycle Components

Rear Part Front Part Front Wheel

Wheel Seat Handle bar

Wheel centre Seat support Handle holds Wheel

Carrier Middle bar Steer rod Wheel centre

Carrier supports Slant bars Wheel hold centre

Connectors Pedal disk Wheel holds

2.2. FRAME AND JOINT CONSIDERATIONS

The bicycle has been modeled as a robotic arm (Fig. 2) consisting of a sequence of three rigid bodies

(links) connected by revolute joints. The first link corresponds to the rear part, which includes the rear

wheel. Here, the bicycle is considered to be autonomous and rider-free and hence the mass of the rider

is neglected in this model. The second and third links correspond to the front part and the front wheel

respectively. To describe the translational and rotational relationships between adjacent links, we have

used the Denavit – Hartenberg algorithm [12] for establishing a coordinate system to each link of the

articulated chain. The base frame has been fixed at the contact point of the rear wheel to the ground

and it is virtually fixed to that position. All coordinate frames have been determined and established

on the basis of three rules:

● The zi-1 axis lies along the axis of motion of the ith joint.

● The xi axis is normal to the zi-1 axis and pointing away from it.

● The yi axis completes the right-handed coordinate system as required.

The joint angles, namely, lean, steer and angle of rotation of the front wheel (and hence the angular

and linear velocities) correspond to the three degrees of freedom of the bicycle model.

5

Figure 2: Bicycle Model with Frames attached.

The mass, inertia and centre of mass calculations for different links, with the corresponding frames,

have been tabulated in Table 2. The values correspond to the default bicycle parameters that have been

used for experimentation.

Table 2: Mass, Inertia and Centre of Mass calculations

Link Mass

(in kg )

Inertia Terms

(in kg m2 )

Coordinates of

Centre of Mass

(in m)

Rear Part

22.9029

Ixx 29.7644 x -0.005

Iyy 17.7778

Izz 12.0837 y 0.6902

Ixy -0.0042

Iyz 13.9992 z 0.8605

Izx -0.0056

Front Part

2.5480

Ixx 0.6936 x 0.0201

Iyy 0.0348

Izz 0.6722 y 0.4292

Ixy 0.0379

Iyz 0 z 0

Izx 0

Front Wheel

2.5411

Ixx 0.1268 x 0

Iyy 0.1268

Izz 0.2527 y 0

Ixy 0

Iyz 0 z 0

Izx 0

6

2.3. GRAPHIC USER INTERFACE

A graphic user interface (Fig. 3) has been created using MATLAB – GUIDE that can be used to

change the dimensions of various components of the bicycle. It also allows the user to change the

initial conditions and input functions, which are used for solving the dynamic equations (developed in

Section 3) of the bicycle model. The model has been named as ROBI (RObot BIcycle), as we are

modeling the bicycle as a robot arm.

Figure 3: Graphic User Interface.

7

3. BICYCLE DYNAMICS MODEL USING LAGRANGE – EULER (L-E)

FORMULATION

The derivation of the dynamic model based on the L-E formulation is simple and systematic. The

dynamic equations for the three-joint Robot Bicycle (ROBI) are highly nonlinear and consist of

inertia forces, coupling reaction forces between joints (coriolis and centrifugal), and gravity effects.

Furthermore, these torques/forces depend on the bicycle’s physical parameters, instantaneous joint

configuration, joint velocity and acceleration. The L-E equations of motion provide explicit state

equations for ROBI’s dynamics and can be utilized to analyze and design advanced joint-variable

space control strategies.

Many investigators utilize the Denavit – Hartenberg matrix representation to describe the spatial

displacement between the neighboring link coordinate frames to obtain the link kinematic information,

and they employ the lagrangian dynamics technique to derive the dynamic equations. The direct

application of the lagrangian dynamics formulation, together with the Denavit – Hartenberg link

coordinate representation, results in a convenient and compact algorithmic description of the bicycle

equations of motion. The algorithm is expressed by matrix operations and facilitates both analysis and

computer implementation. The derivation of the dynamic equations of bicycle [12] is based on the

understanding of:

1. The 4 X 4 homogeneous coordinate transformation matrix, i-1Ai, which describes the spatial

relationship between the ith and the (i - 1)th link coordinate frames. It relates a point fixed in

link i expressed in homogeneous coordinates with respect to the ith coordinate system to the

(i – 1)th coordinate system.

2. The Lagrange – Euler equation

.3,2,1==∂∂

−

∂∂

iq

L

q

L

dt

di

ii

τ&

where

L = lagrangian function = kinetic energy K – potential energy P

K = total kinetic energy of the bicycle

P = total potential energy of the bicycle

qi = generalized coordinates of the bicycle

iq& = first time derivative of the generalized coordinate, qi

τi = generalized force (or torque) applied to the system at joint i to drive link i

Generalized coordinates are used as a convenient set of coordinates which completely describe the

8

location (position and orientation) of a system with respect to a reference coordinate frame. Since the

angular positions of the joints are readily available because they can be measured by potentiometers

or encoders or other sensing devices, they provide a natural correspondence with the generalized

coordinates. Thus, in the case of a rotary joint, qi ≡ Өi, the joint angle span of the joint.

In the case of ROBI, there are three generalized coordinates, namely, χ (lean), ψ (steer) and Ө (angle

of rotation of the front wheel). The Homogeneous Coordinate Transformation Matrices [12] are given

below:

−

−

−−

=

1000

sincos0

0sincossinsincos

0coscoscossinsin

1

10

aA

ααχαχαχχαχαχ

−

=

1000

010

0sin0cos

0cos0sin

2

21

aA

ψψψψ

−

=

1000

0100

00cossin

00sincos

32 θθ

θθ

A

21

10

20 AAA =

32

20

30 AAA =

32

21

31 AAA =

where α = slant angle of the steering axis with the vertical,

a1 = distance between two wheel-ground contact points + (Rwheelout * tan α)

a2 = Rwheelout / cos α

The Kinetic Energy of the bicycle system is given by,

( )[ ]rp

T

iriip

i

r

i

pi

qqUJUTrK &&∑∑∑===

=11

3

1

21

where

χ&& =1q (leanrate)

9

ψ&& =2q (steerrate)

ω=3q& (angular velocity of the front wheel)

The formulae and matrix representations of ijU , iQ and

iJ are given in Appendix A.

The Potential Energy of the bicycle system is given by,

( )i

i

iii

rAgmP 03

1−∑=

=

where

[ ] 2/81.9,000 smggg =−=

[ ]Tiiii

i zyxr 1=

mi = mass of link i

3.1. MOTION EQUATIONS OF THE BICYCLE

The Lagrangian function: L = K – P is given by

( )[ ] ( )i

i

iii

rp

T

iriip

i

r

i

pi

rAgmqqUJUTrL 03

111

3

1

21

====

∑+= ∑∑∑ &&

Applying the Lagrange – Euler formulation, the necessary generalized torque τi for joint i is given by,

ii

iq

L

q

L

dt

d

∂∂

−

∂∂

=&

τ

The expression for generalized torque can be simplified (procedure shown in Appendix - A) in matrix

notation as:

3,2,13

1

3

1

3

1

=++= ∑∑∑===

icqqhqD imkikm

mk

kik

k

i&&&&τ

or in matrix form as

))(())(),(()())(()( tqctqtqhtqtqDt ++= &&&τ

where

)(tτ = 3 X 1 generalized torque vector applied at joints i=1, 2, 3. i.e.

[ ]Ttttt )(),(),()( θψχ ττττ =

10

=)(tq 3 X 1 vector of the joint variables of the bicycle and can be expressed as

[ ]Tttttq )(),(),()( θψχ=

=)(tq& 3 X 1 vector of the joint velocity of the bicycle and can be expressed as

[ ]Tttttq )(),(),()( θψχ &&&& =

=)(tq 3 X 1 vector of the acceleration of the joint variables of the bicycle and can be

expressed as

[ ]Tttttq )(),(),()( θψχ &&&&&&&& =

D(q) = 3 X 3 inertial acceleration-related symmetric matrix whose elements are

== ∑=

kiUJUTrDn

kij

T

jijjkik ,)(),max(

1, 2, 3

h(q, q& )= 3 X 1 nonlinear Coriolis and centrifugal force vector i.e.

[ ]Thhhqqh 321 ,,),( =&

where == ∑∑==

iqqhh mkikm

mk

i&&

3

1

3

1

1, 2, 3

and == ∑=

mkiUJUTrhn

mkij

T

jijjkmikm ,,)(),,max(

1, 2, 3

c(q) = 3 X 1 gravity loading force vector .i.e.

[ ]Tcccqc 321 ,,)( =

where =−∑==

irUgmc j

j

jijij

i )(3

1, 2, 3

The formulae and matrix representations of ijU , ijkU , iQ and

iJ are given in Appendix A.

11

4. MODEL VALIDATION FOR AUTONOMOUS BICYCLE

4.1. SIMULATION EXPERIMENTS

All existing bicycle models assume the velocity of bicycle to be a constant in their formulation. But,

we know that it does not happen when we ride the bicycle. Hence, it is slightly rough approximation

in the development of a practical bicycle model. The ROBI model allows time variation of bicycle’s

velocity as well. However, in order to validate the model, we perform experiments with constant

bicycle velocity, i.e. there is zero angular velocity torque applied at this joint and hence there is no

direct variation of the bicycle velocity. The results are compared with the already existing benchmarks

for validation.

In cases of zero angular velocity torque, the initial angular velocity tends to remain constant. So, cases

with different initial angular velocities, from 10 to 100 radians/second in the steps of 10

radians/second were studied. Four cases involving initial angular velocity values of 10, 20 (Table 3),

40 and 60 (Table 4) radians/second respectively are illustrated here.

Table 3: Simulation Results of Autonomous Bicycle at Lower Velocities

Plots At Initial Angular Velocity of 10 rad/s

(Linear Velocity of 3.55 m/s)

At Initial Angular Velocity of 20 rad/s


Lean

12

Steer

Angular

Velocity

Table 4: Simulation Results of Autonomous Bicycle at Higher Velocities





Lean

Steer

13

Angular

Velocity

4.2. ANALYSIS OF SIMULATION RESULTS

● At lower velocities, 10 and 20 radians per second, the system of rigid bodies connected and

moving as a bicycle is unstable and falls asymptotically i.e. lean and steer both sharply fall to one side.

Thus at lower velocities there is no tendency of the system to try to balance itself. The same is

observed in practice that bicycle would fall without oscillation of steer and lean to one side at lower

velocities. It is further noticed that the total energy [Table B.1] of the system remains constant with

slight reduction in angular velocity as it falls and at the same time, the lean and steer rates increase

[Table B.1].

● At higher velocities, 40 and 60 radians per second, the bicycle system is unstable but it shows

tendency to stabilize itself through oscillations in steer, lean and velocity, eventually the oscillating

system falls exponentially to one side. The inherent stabilization efforts are thus noticeable in the

system in the form of oscillations which delay the falling phenomenon. This represents the inherent

stability characteristic present in bicycle system.

● The frequency of aforementioned steer, lean and velocity oscillations in the system is high at

higher velocities and low at relatively lower velocities. Such a behavior is consistent with practical

observations.

● The forces generated in the system, show that at lower velocities the centrifugal and coriolis

effects [Table B.1] are smaller in magnitude as compared to gravity generated force and thus the

bicycle tends to fall quickly. In contrast the forces of coriolis and centrifugal origin have larger

magnitude than gravity force at higher velocities thus the motion is controlled by Centrifugal and

Coriolis forces [Table B.2]; therefore, falling due to gravity is delayed, showing the balancing effort

of the system.

● The total energy of the system remains constant but KE and PE oscillate as the parameters, lean

rate, steer rate and velocity change with time.

● The system shows two distinct regions in its operation. First region is at low velocities i.e. upto

14

20 radians per second, where it is non-oscillatory and sharply unstable and the second region starts

after 20 radians per second, which is oscillatory and there is a delayed falling out of equilibrium of the

bicycle, thus, it can be said to be relatively stable. But in no case the system shows exact stable

behavior in any velocity range as long as it is run as autonomous system. So it can be said that bicycle

is by nature unstable at all velocities but at higher velocity the system tries to balance itself through

inherent lean-steer feedback relationship, which at max can delay the fall.

● The data generated is at several velocities to confirm the system behavior viz. 10, 20,30,40,50, 60,

70, 80, 90 and 100 radians per seconds, though the presented data is for selected velocity for the

reason of space constraint. The interesting nature of the curves motivated us to write down simplified

model equations by curve fitting method to represent motion of a bicycle system,

Simplified Lean Equation

)1(*)()*)((*)(),( *)(0000

0 −+= tvbevatvSinvAtv ωχ

where,

00 0365.01719.00 *001828.0*00445.0)( vv

eevA−− +=

8403.0*5270.0)( 00 −= vvω

00 2865.00331.050 *002616.0*10*928.4)( vv

eeva−− +−=

00 0348.02206.00 *8408.0*594.4)( vv

eevb−− +=

Simplified Steer Equation

)1(*)()*)((*)(),( *)(0000

0 −+= tvbevatvSinvAtv ωψ

where,

202020 )7097.3

6402.14()

4723.13

8951.23()

8804.0

5779.9(

0 *01106.0*007863.0*074.0)(−

−−

−−

−++=

vvv

eeevA

52.2*5769.0)( 00 −= vvω

00 3730.00321.00 *3815.0*002464.0)( vv

eeva−+−=

00 0214.01612.00 *1096.0*218.5)( vv

eevb += −

One could use these equations for simulating bicycle’s kinematics accurately.

RMSE plots for the simplified lean and steer curve fit equations are given below:

15

Figure 16: RMSE plot for the Simplified Lean Equation.

Figure 17: RMSE plot for the Simplified Steer Equation.

16

5. MODEL VALIDATION FOR AUTONOMOUS BICYCLE WITH RIGID RIDER

5.1. SIMULATION EXPERIMENTS

The only addition made to ROBI model in this case is the incorporation of rigid rider in the rear part

(Link3) of the autonomous bicycle. The rigid rider is approximated to a cuboid made of material with

the density of human. The weight of the rider taken in this case is 50.5 kg. This value can be changed

using the GUI.

In cases of zero angular velocity torque, the initial angular velocity tends to remain constant. So, cases

with different initial angular velocities, from 10 to 120 radians/second in the steps of 10

radians/second were studied. Four cases involving initial angular velocity values of 20, 40 (Table 5),

100 and 120 (Table 6) radians/second respectively are illustrated here.

Table 5: Simulation Results of Autonomous Bicycle with Rigid Rider at Lower Velocities





Lean

Steer

17

Angular

Velocity

Table 6: Simulation Results of Autonomous Bicycle with Rigid Rider at Higher Velocities





Lean

Steer

Angular

Velocity

18

5.2. ANALYSIS OF SIMULATION RESULTS

● At lower velocities, 20 and 40 radians per second, the system of rigid bodies connected and

moving as a bicycle is unstable and falls asymptotically i.e. lean and steer both sharply fall to one side.

Thus at lower velocities there is no tendency of the system to try to balance itself. The same is

observed in practice that bicycle would fall without oscillation of steer and lean to one side at lower

velocities. It is further noticed that the total energy [Table B.3] of the system remains constant with

slight reduction in angular velocity as it falls and at the same time, the lean and steer rates increase

[Table B.3].

● At higher velocities, 100 and 120 radians per second, the bicycle system is unstable but it shows

tendency to stabilize itself through oscillations in steer, lean and velocity, eventually the oscillating

system falls exponentially to one side. The inherent stabilization efforts are thus noticeable in the

system in the form of oscillations which delay the falling phenomenon. This represents the inherent

stability characteristic present in bicycle system.

● The forces generated in the system, show that at lower velocities the centrifugal and coriolis

effects [Table B.3] are smaller in magnitude as compared to gravity generated force and thus the

bicycle tends to fall quickly. In contrast the forces of coriolis and centrifugal origin have larger

magnitude than gravity force at higher velocities thus the motion is controlled by Centrifugal and

Coriolis forces [Table B.4]; therefore, falling due to gravity is delayed, showing the balancing effort

of the system.

● The total energy of the system remains constant but KE and PE oscillate as the parameters, lean

rate, steer rate and velocity change with time.

● The data generated is at several velocities to confirm the system behavior viz. 10, 20,30,40,50, 60,

70, 80, 90 and 100 radians per seconds, though the presented data is for selected velocities.

● The primary effect of the presence of rigid rider in the bicycle system is that it postpones the

stabilizing effect of the autonomous bicycle to a higher velocity. Without the rigid rider, the

autonomous bicycle had stabilizing effect initiated at the angular velocity of 30 radians/ second, while

the similar effect happened only around 60 radians/ second with the presence of rigid rider. However,

significant stabilizing effects are noticed only around 100 radians/ second. This makes the bicycle

system with rigid rider highly unstable.

19

6. CONCLUSIONS

Robotics approach in finding out dynamic equations of motion of a bicycle results into a generalized

set of equations which have been programmed in MATLAB and solutions to these coupled

differential equations are achieved using functions available therein i.e. ode23, ode45 etc. The plots

for different velocities verify that the system behavior is similar to what is observed in a practical

bicycle system. The system is unstable in lower velocity range and shows oscillatory behavior at

higher velocities that reflects its tendency to self stabilize, however, it is not stable in autonomous

mode at any velocity.

The lean and steer response graphs of the autonomous bicycle system follow a particular trend, which

upon curve fitting results into simplified model equations to represent bicycle behavior. The resulting

curve fitting error graphs i.e. RMSE curves show that model equations are very good fit at higher

velocities i.e. 10 m/s and above.

The presence of rigid rider in the bicycle system postpones the stabilizing effect of the autonomous

bicycle to a higher velocity; hence making the bicycle system highly unstable.

More experiments have been conducted on the same model wherein the system is autonomous but a

controller works to balance it, by providing suitable actuations in terms of lean torque, steer torque

and speed torque. This forms our future work.

20

APPENDIX

A. TERMS IN THE EQUATION OF MOTION

>

≤=

−−

ijfor

ijforAQAU i

j

jj

ij0

11

0

−

=

0000

0000

0001

0010

iQ

−+

+−

++−

=

iiiiiii

ii

zzyyxx

yzxz

iiyz

zzyyxx

xy

iixzxy

zzyyxx

i

mzmymxm

zmIII

II

ymIIII

I

xmIIIII

J

2

2

2

<<≥≥

≥≥= −

−−

−

−−

−−

kiorjifor

kjiforAQAQA

jkiforAQAQA

Ui

j

jj

k

kk

i

k

kk

j

jj

ijk

0

11

11

0

11

11

0

21

B. SIMULATION RESULTS

Table B.1: Simulation Results of Autonomous Bicycle at Lower Velocities





Lean Rate

Steer Rate

Gravity

Effects-

Rear Part

Gravity

Effects-

Front Part

22

Gravity

Effects-

Front

Wheel

Coriolis &

Centrifugal

Effects-

Rear Part

Coriolis &

Centrifugal

Effects-

Front Part

Coriolis &

Centrifugal

Effects-

Front

Wheel

23

Kinetic

Energy-

Rear Part

Kinetic

Energy-

Front Part

Kinetic

Energy-

Front

Wheel

Kinetic

Energy-

Total

Bicycle

24

Potential

Energy-

Rear Part

Potential

Energy-

Front Part

Potential

Energy-

Front

Wheel

Potential

Energy-

Total

Bicycle

25

Total

Energy

Table B.2: Simulation Results of Autonomous Bicycle at Higher Velocities





Lean Rate

Steer Rate

Gravity

Effects-

Rear Part

26

Gravity

Effects-

Front Part

Gravity

Effects-

Front

Wheel

Coriolis &

Centrifugal

Effects-

Rear Part

Coriolis &

Centrifugal

Effects-

Front Part

27

Coriolis &

Centrifugal

Effects-

Front

Wheel

Kinetic

Energy-

Rear Part

Kinetic

Energy-

Front Part

Kinetic

Energy-

Front

Wheel

28

Kinetic

Energy-

Total

Bicycle

Potential

Energy-

Rear Part

Potential

Energy-

Front Part

Potential

Energy-

Front

Wheel

29

Potential

Energy-

Total

Bicycle

Total

Energy

Table B.3: Simulation Results of Autonomous Bicycle with Rigid Rider at Lower Velocities





Lean Rate

Steer Rate

30

Gravity

Effects-

Rear Part

Gravity

Effects-

Front Part

Gravity

Effects-

Front

Wheel

Coriolis &

Centrifugal

Effects-

Rear Part

31

Coriolis &

Centrifugal

Effects-

Front Part

Coriolis &

Centrifugal

Effects-

Front

Wheel

Kinetic

Energy-

Rear Part

Kinetic

Energy-

Front Part

32

Kinetic

Energy-

Front

Wheel

Kinetic

Energy-

Total

Bicycle

Potential

Energy-

Rear Part

Potential

Energy-

Front Part

33

Potential

Energy-

Front

Wheel

Potential

Energy-

Total

Bicycle

Total

Energy

34

Table B.4: Simulation Results of Autonomous Bicycle with Rigid Rider at Higher Velocities





Lean Rate

Steer Rate

Gravity

Effects-

Rear Part

Gravity

Effects-

Front Part

35

Gravity

Effects-

Front

Wheel

Coriolis &

Centrifugal

Effects-

Rear Part

Coriolis &

Centrifugal

Effects-

Front Part

Coriolis &

Centrifugal

Effects-

Front

Wheel

36

Kinetic

Energy-

Rear Part

Kinetic

Energy-

Front Part

Kinetic

Energy-

Front

Wheel

Kinetic

Energy-

Total

Bicycle

37

Potential

Energy-

Rear Part

Potential

Energy-

Front Part

Potential

Energy-

Front

Wheel

Potential

Energy-

Total

Bicycle

38

Total

Energy

39

REFERENCES

1. F. J. W. Whipple, ‘The Stability of the Motion of a Bicycle’, Quarterly Journal of Pure and

Applied Mathematics, Vol. 30, pp. 312-348, 1899.

2. F. Klein and A. Sommerfeld, Über die Theorie des Kreisels’, Teubner Verlag Stuttgart,1965.

3. D. H. Jones, ‘The stability of a bicycle’, Physics Today, Vol. 23-4, pp.34-40, 1970.

4. Y. Le Hénaff, ‘Dynamic Stability of the bicycle’, European Journal of Physics, Vol. 8, pp.

207-210, 1987.

5. J. Lowell and H.D. Mckell, ‘The Stability of bicycles’, American Journal of Physics, Vol. 50,

pp 1106-1108, 1982.

6. G. Franke, W. Suhr and F. Rieß, ‘An advanced model of bicycle’, European Journal of

Physics, Vol. 11, pp. 116-121, 1990.

7. J. Papadopoulos, ‘Preprint CBRP-JP-88b’, Cornell University, New York, 1988.

8. R. S. Hand, ‘Comparisons and Stability Analysis of Linearized Equations of Motion for a

Basic Bicycle Model’, Ph.D. thesis, Cornell University, New York, 1988.

9. Himanshu Dutt Sharma, Sameer M. Kale and Umashankar N., ‘Simulation Model for

Studying Inherent Stability Characteristics of Autonomous Bicycle’, in Proc. of the IEEE

International Conference on Mechatronics & Automation, Canada, Jul. 2005, pp 193-198.

10. Himanshu Dutt Sharma and Umashankar N, ‘Stabilization of Autonomous Bicycle using

Fuzzy Controller with Maximum Allowable Lean Constraint’, in Proceedings of The 3rd

International Conference on Computational Intelligence, Robotics and Autonomous Systems,

Singapore, Dec. 2005.

11. Himanshu Dutt Sharma and Umashankar N, ‘Design of Fuzzy Controller for Autonomous

Bicycle’, in Proc. of IEEE International Conference on Engineering of Intelligent Systems,

Pakistan, Apr. 2006, pp. 498-503.

12. K.S. Fu, R.C. Gonzalez, and C.S.G. Lee, Robotics: Control, Sensing, Vision, and Intelligence,

McGraw-Hill Book Co., 1987.

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