realization of an unmanned bicycle robot with balancer

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2010 Doctoral Dissertation

Realization of an Unmanned Bicycle Robot

with Balancer

Lychek Keo

No. 07D51214

Tokyo Institute of Technology

Graduate School of Science and Engineering

Department of Mechanical and Control Engineering

Supervisor

Prof. Masaki Yamakita


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Copyright c

Lychek Keo

All Rights Reserved 2010


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ABSTRACT

Recently, robotic systems are becoming increasingly common, as are robotic systems

integrated with smart environments. However, some robots are used to rescue or search

activity in the disaster area where human cannot do the same jobs due to a possibility of

additional disasters. Therefore, contribution of an autonomous robotic system is highly

expected in this condition. Such as of the kind specified, an unmanned bicycle robot is

proposed since it has high mobility and does not require wide contact space to the ground.

An unmanned bicycle robot can be used to collect the information or transportation in the

disaster areas and / or in the mountains.

In this dissertation, we discuss about realizing an unmanned bicycle robot with bal-

ancer, which can work in general environment. Hence, the stabilization of the bicyclerobot, perform an acrobatic turn via a wheelie motion and path tracking any trajectories

in the ground plane should be studied in this research. For realizing autonomous bicycle

robot with balancer, first we develop a simplified nonlinear dynamic model that it is de-

rived from Lagrange’s equations and nonholonomic constraints with some assumptions

on bicycle system. Then we can use this model for developing bicycle simulator, balanc-

ing controller design, trajectory controller design for moving in the ground plane. Finally,

we come up with the realization of an acrobatic turn via a wheelie motion.

As modeled here, the bicycle with the balancer is an under-actuated system, subject to

nonholonomic contact constraints associated with the rolling constraints on the front andrear wheels. The bicycle and the balancer is considered as an inverted two-link system

where the first link is the bicycle body frame with steering frame and the second link is

the balancer. The bicycle with balancer joints can be fixed by using holonomic constraints

for the joints. Therefore, we are able to use the balancer and / or the steering handlebar to

stabilize the bicycle.

In this research, stabilizing control and trajectory tracking control of an autonomous

bicycle are derived independently based on the simplified model. The balancing control

designs based on an output zeroing controller and we extend the controller designed by

improving the stability performance using several techniques. First we derive a bicycleinternal roll angle with capturing the steering eff ect and adding into the system state for

suppressing the steering eff ect. This technique can eliminate the interferences from the

turning handlebar or from the front wheel disturbances to the system. Next we propose a


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balancing controller using both a steering and a balancer to stabilize the bicycle. In order

to add the steering torque into the controller designed, we need to modify the angular

momentum of the two-link system by adding the steering eff ect into the first link, then

the steering torque will appear in the control law. The cooperation between steering and

balancer can be enlarging the region of the stability. The last technique we propose a

new balancer configuration which is attached with the bicycle. This new balancer hastwo motors, one is a rotating motor which is used to rotate the balancer. Another motor

is a linear motor which is used to shift the center gravity of the balancer. With these

configurations, we can use the balancer as a flywheel mode or a balancer mode. The

balancer is configured as a flywheel, when disturbances to the system are large, and it

will switch to the balancer mode when the position of the center of the gravity should

be shifted. The advantage of the flywheel is to stabilize the bicycle with large region of

stability. The advantages of the balancer are to shift the bicycle body from obstacle and

to control the bicycle to follow the trajectory. The last two techniques are used when

the disturbances to the system are large or the system is at startup mode. The trajectorytracking control is derived using an input-output linearization approach to track the path

in the ground plane. The steering and the back wheel are used to control the system. The

connection of the steering input signal between the balancing control and the trajectory

control are coupling and scaling by weighting gain.

In order to make the bicycle robot more useful in the narrow place, we are developing

one of the unmanned bicycle robot systems with a two D.O.F balancer and it can do a

wheelie and move to track the path in the ground plane. Using a two D.O.F balancer,

it is expected that the acrobatic motion can be realized and the body can be turned by

swinging the balancer in two directions. In principle, the motion of the system can be

decomposed into two planes (frontal plane and sagittal plane). The model of the system

in each plane is diff erent and the previous balancing control method cannot be used for

control the bicycle in the sagittal plane. Thus, we modify and extend the output-zeroing

controller to control the bicycle in the sagittal plane by using a generalized momentum.

Using the both methods, a wheelie on a bicycle with quick turn can be realized.

The validity of these proposed control methods are validated by numerical simula-

tions and experimental setup. The balancing and trajectory control are implemented with

the real bicycle by using Matlab

XPC-Target

. An autonomous bicycle robot can becontrolled remotely via a host PC.


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ACKNOWLEDGEMENTS

First and foremost, I would like to thank my advisor Assoc. Prof. Masaki Yamakta,

who has been a good adviser with continuous encouragement and guidance in completing

the research. Without his valuable advice and comment, this dissertation would never becompleted.

Special thanks also go to the member or Yamakita Laboratory for the advices and sup-

port throughout the whole research and the staff of Mechanical and Control Engineering

Department, Tokyo Institute of Technology.

Many thanks go to Japan International Cooperation Agency (JICA) who has supported

me during my study in Japan and Japan International Cooperation Center (JICE) for their

kindness and help.

I wish to give my very special thanks to both Dr. Om Romny and Mr. Phol Norith,

rector and vice rector of Institute of Technology of Cambodia (ITC), who give me the

best opportunities to do this research through the AUN / SEED-Net and JICA project.

An immeasurable debt of gratitude also goes to my colleagues for their advices and

helping hands in the difficult moment and I would like to thank to all the people who have

contributed towards the development of this project.

Finally, I am incredibly grateful to my family. My parents, my wife, my son, and my

sisters, gave me their unconditional love and support during all these years. While I never

had the opportunity to see my family during my three years of studies at Tokyo Institute

of Technology, my heart has always been with them. They never stopped believing in me

and did everything they could to help me achieve my goals.


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vii

Contents

Abstract iii

Acknowledgements v

List of Figures xi

List of Tables xvii

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 A Brief History of the Bicycle . . . . . . . . . . . . . . . . . . . 2

1.1.2 Dynamics and Control . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Contributions of this Dissertation . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Dynamic Model of a Bicycle with a Balancer 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Nonlinear Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Generalized Coordinates . . . . . . . . . . . . . . . . . . . . . . 14

2.2.3 Generalized Velocities . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.4 Inputs and Generalized Forces . . . . . . . . . . . . . . . . . . . 18

2.2.5 Kinetic and Potential Energy for a Bicycle with a Balancer . . . . 18

2.2.6 The Euler-Lagrange Equation . . . . . . . . . . . . . . . . . . . 212.2.7 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.8 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Linearized Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . 24


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2.3.1 Case 1: A Bicycle with Only a Balancer . . . . . . . . . . . . . . 25

2.3.2 Case 2: The bicycle with Steering and Back Wheel . . . . . . . . 27

2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.1 Bicycle Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.2 Linear model of Bicycle with a Balancer . . . . . . . . . . . . . 31

2.4.3 Validation Linear Model Using Zero Input-Responses . . . . . . 322.4.4 Self-Stabilization of a Bicycle . . . . . . . . . . . . . . . . . . . 34

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Stabilization of a Bicycle with a Balancer 37

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Bicycle Dynamics and Its Internal Equilibrium . . . . . . . . . . . . . . 38

3.2.1 Bicycle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.2 Internal Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.3 Internal Equilibrium Angle . . . . . . . . . . . . . . . . . . . . . 393.3 Balancing Control Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.1 Model of a Two-Link System . . . . . . . . . . . . . . . . . . . 40

3.3.2 Output-Zeroing Controller . . . . . . . . . . . . . . . . . . . . . 41

3.3.3 Actuated Control . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.4 Closed-Loop Diagram . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.1 Balancing Bicycle without Captured Steering Eff ect . . . . . . . 43

3.4.2 Balancing Bicycle with Captured Steering Eff ect . . . . . . . . . 49

3.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.5.1 Balancing at Zero Velocity . . . . . . . . . . . . . . . . . . . . . 54

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Control of a Bicycle with Balancer and Steering 61

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 Control Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2.1 Balancing Control . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2.2 Trajectory Control . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.3 Closed-Loop Control Diagram . . . . . . . . . . . . . . . . . . . 684.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3.1 Balancing at Zero Linear Velocity . . . . . . . . . . . . . . . . . 70

4.3.2 Path Tracking with Balance . . . . . . . . . . . . . . . . . . . . 77


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4.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.4.1 Balancing at Zero Linear Velocity . . . . . . . . . . . . . . . . . 86

4.4.2 Path Tracking with Balance . . . . . . . . . . . . . . . . . . . . 87

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5 Control of a Bicycle with a Flywheel Balancer 935.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2 Bicycle with Flywheel Balancer Dynamics . . . . . . . . . . . . . . . . . 95

5.3 Control Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3.1 Model of Two-Link System . . . . . . . . . . . . . . . . . . . . 96

5.3.2 Output-Zeroing Controller . . . . . . . . . . . . . . . . . . . . . 97

5.3.3 Switching Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.4.1 Stabilizing Bicycle with a Balancer . . . . . . . . . . . . . . . . 98

5.4.2 Stabilizing Bicycle with a Flywheel . . . . . . . . . . . . . . . . 1005.4.3 Stabilizing Bicycle with Flywheel Balancer . . . . . . . . . . . . 103

5.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.5.1 Balancing control of the bicycle robot with balancer . . . . . . . 110

5.5.2 Balancing control of the bicycle robot with a flywheel . . . . . . 111

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6 Acrobatic Turn via Wheelie Motion 121

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.2 Bicycle Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.3 Control Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.3.1 Frontal Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.3.2 Sagittal Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.4.1 Balancing Control near the Upright Position . . . . . . . . . . . . 133

6.4.2 Both Wheels Contact with the Ground . . . . . . . . . . . . . . . 135

6.4.3 Perform a Turn with Wheelie by a Bicycle . . . . . . . . . . . . . 136

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7 Conclusions 139

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140


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Bibliography 141

List of Publications 149


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List of Figures

1.1 Riding a bicycle in disaster area. . . . . . . . . . . . . . . . . . . . . . . 1

1.2 First Bicycle (“Draisienne”) made by Baron Karl von Drais (1817). . . . 2

1.3 Velocipede by Pierre Michaux et Cie of Paris, France circa (1865). . . . . 3

1.4 The ordinary, or high-wheeler, or penny-farthing. . . . . . . . . . . . . . 4

2.1 Coordinate system of the bicycle with the balancer. . . . . . . . . . . . . 13

2.2 Geometric of a bicycle with a balancer parts. . . . . . . . . . . . . . . . . 142.3 Coordinate system of a bicycle looking from backward. . . . . . . . . . . 15

2.4 Coordinate system of the bicycle with the balancer in a plane. . . . . . . . 16

2.5 Coordinate system of the top-view bicycle. . . . . . . . . . . . . . . . . . 16

2.6 Top view of the bicycle showing body velocities. . . . . . . . . . . . . . 17

2.7 Bicycle looking from backward. . . . . . . . . . . . . . . . . . . . . . . 19

2.8 User interface for a bicycle with a balancer simulator. . . . . . . . . . . . 29

2.9 Virtual reality of a bicycle with a balancer. . . . . . . . . . . . . . . . . . 31

2.10 Initial position setup for zero-input response. . . . . . . . . . . . . . . . 32

2.11 Roll angle of zero-input response. . . . . . . . . . . . . . . . . . . . . . 332.12 Balancer angle of zero-input response. . . . . . . . . . . . . . . . . . . . 33

2.13 Response for self-stabilization of a bicycle. . . . . . . . . . . . . . . . . 34

3.1 Two-link system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Closed-loop diagram for stabilization of a bicycle. . . . . . . . . . . . . . 43

3.3 Simulation results of the bicycle stabilization with the balancer. . . . . . . 45

3.4 Dynamic output function y. . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5 Simulation results of the bicycle stabilization with adding noises into the

system state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.6 Simulation results of the bicycle stabilization using LQR controller. . . . 48

3.7 Simulation results of the bicycle stabilization without adding steering eff ect. 50

3.8 Simulation results of the bicycle stabilization with adding steering eff ect. . 51


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3.9 Bicycle with a balancer hardware. . . . . . . . . . . . . . . . . . . . . . 52

3.10 Inertial measurement unit. . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.11 Control panel box. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.12 Bicycle with a balancer hardware scheme. . . . . . . . . . . . . . . . . . 55

3.13 Experimental results of the bicycle stabilization with a balancer. . . . . . 56

3.14 Experimental results of the bicycle stabilization without adding steeringeff ect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.15 Experimental results of the bicycle stabilization with adding steering eff ect. 58

4.1 Bicycle robot with balancer moving on a path. . . . . . . . . . . . . . . . 62

4.2 Top-view of a bicycle model with a reference point. . . . . . . . . . . . . 66

4.3 Bicycle closed-loop control diagram. . . . . . . . . . . . . . . . . . . . . 68

4.4 Simulation results of the bicycle stabilization with only a balancer: angle

responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.5 Simulation results of the bicycle stabilization with only a balancer: angu-lar velocity responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.6 Simulation results of the bicycle stabilization with only a balancer: con-

trol inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.7 Power dissipate by balancer motor. . . . . . . . . . . . . . . . . . . . . . 72

4.8 Simulation results of the bicycle stabilization with only a steering: angle

responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.9 Simulation results of the bicycle stabilization with only a steering: angu-

lar velocity responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.10 Simulation results of the bicycle stabilization with only a steering: controlinputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.11 Power dissipate by steering motor. . . . . . . . . . . . . . . . . . . . . . 75

4.12 Simulation results of the bicycle stabilization with both balancer and

steering: angle responses. . . . . . . . . . . . . . . . . . . . . . . . . . . 75


steering: angular velocity responses. . . . . . . . . . . . . . . . . . . . . 76


steering: control inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.15 Power dissipate by balancer and steering motors. . . . . . . . . . . . . . 774.16 Simulation results of the straight line tracking with variable speed: angle

responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78


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4.17 Simulation result of the straight line tracking with variable speed: rear

wheel velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.18 Simulation result of the bicycle tracking a straight line in the plane. . . . . 79

4.19 Simulation result of the bicycle tracking error. . . . . . . . . . . . . . . . 80

4.20 Simulation results of the sinusoidal path tracking: angle responses. . . . . 81

4.21 Simulation result of the sinusoidal path tracking: rear wheel velocity. . . . 814.22 Simulation result of the bicycle tracking a sinusoidal path. . . . . . . . . 82

4.23 Simulation results of the circle path tracking: angle responses. . . . . . . 83

4.24 Simulation result of the circle path tracking: rear wheel velocity. . . . . . 83

4.25 Simulation result of the bicycle tracking a circle path. . . . . . . . . . . . 84

4.26 Simulation result of the figure eight path tracking: angle responses. . . . . 84

4.27 Simulation result of the figure eight path tracking: rear wheel velocity. . . 85

4.28 Simulation result of the bicycle tracking a figure eight path. . . . . . . . . 85

4.29 Experimental results of the bicycle stabilization with only a balancer and

the scale gain γ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.30 Experimental results of the bicycle stabilization with balancer and steer-

ing and the scale gain γ = 2. . . . . . . . . . . . . . . . . . . . . . . . . 87

4.31 Experimental results of the bicycle stabilization with balancer and steer-

ing and the scale gain γ = 1. . . . . . . . . . . . . . . . . . . . . . . . . 88

4.32 Bicycle robot running on the walkway in front of the classroom. . . . . . 89

4.33 Experimental results of the bicycle tracking a straight line: angle responses. 90

4.34 Experimental results of the bicycle tracking the straight line: rear wheel

linear velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.35 Experimental result of the bicycle tracking a straight line in the plane. . . 91

4.36 Experimental results of the bicycle tracking a sinusoidal path: angle re-

sponses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.37 Experimental result of the bicycle tracking a sinusoidal path. . . . . . . . 92

5.1 Bicycle with flywheel balancer. . . . . . . . . . . . . . . . . . . . . . . . 94

5.2 Flywheel balancer configurations. . . . . . . . . . . . . . . . . . . . . . 94

5.3 Coordinate system of the bicycle with the balancer. . . . . . . . . . . . . 95

5.4 Bicycle with flywheel balancer consider as a two-link system. . . . . . . 96

5.5 Switching between flywheel and balancer. . . . . . . . . . . . . . . . . . 985.6 Simulation results of the bicycle stabilization with the balancer: Roll an-

gle α, Roll angular velocity α̇. . . . . . . . . . . . . . . . . . . . . . . . 99


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5.7 Simulation results of the bicycle stabilization with the balancer: Balancer

angle β, Balancer angular velocity ˙ β. . . . . . . . . . . . . . . . . . . . . 99

5.8 Simulation results of the bicycle stabilization with the balancer: Balancer

control input τb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.9 Simulation results of the bicycle body tracking desired value: Roll angle

α, Roll angular velocity α̇. . . . . . . . . . . . . . . . . . . . . . . . . . 1015.10 Simulation results of the bicycle body tracking desired value: Balancer

angle β, Balancer angular velocity ˙ β. . . . . . . . . . . . . . . . . . . . . 101

5.11 Simulation results of the bicycle stabilization with the flywheel and a1 =

0: Roll angle α, Roll angular velocity α̇. . . . . . . . . . . . . . . . . . . 102


0: Balancer angle β, Balancer angular velocity ˙ β. . . . . . . . . . . . . . 102


0: Flywheel control input τb. . . . . . . . . . . . . . . . . . . . . . . . . 103


40: Roll angle α, Roll angular velocity α̇. . . . . . . . . . . . . . . . . . 104


40: Balancer angle β, Balancer angular velocity ˙ β. . . . . . . . . . . . . . 104

5.16 Length of the balancer to COG hb. . . . . . . . . . . . . . . . . . . . . . 105

5.17 Simulation results of the bicycle stabilization with the flywheel balancer:

Roll angle α, Roll angular velocity α̇. . . . . . . . . . . . . . . . . . . . 106

5.18 Simulation results of the bicycle stabilization with the flywheel balancer:

Balancer angle β, Balancer angular velocity ˙ β. . . . . . . . . . . . . . . . 106

5.19 Bicycle with flywheel balancer hardware. . . . . . . . . . . . . . . . . . 107

5.20 Flywheel balancer configurations. . . . . . . . . . . . . . . . . . . . . . 108

5.21 Bicycle hardware system. . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.22 Control flow chart for bicycle robot. . . . . . . . . . . . . . . . . . . . . 109

5.23 Balancer fix at half length. . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.24 Experimental result of the bicycle stabilization with a balancer mode in

full length: Roll Angle α. . . . . . . . . . . . . . . . . . . . . . . . . . . 111


full length: Balancer angle β. . . . . . . . . . . . . . . . . . . . . . . . . 1125.26 Experimental result of the bicycle stabilization with a balancer mode in

full length: Balancer torque τb. . . . . . . . . . . . . . . . . . . . . . . . 112


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half length: Roll Angle α. . . . . . . . . . . . . . . . . . . . . . . . . . . 113


half length: Balancer angle β. . . . . . . . . . . . . . . . . . . . . . . . . 113

5.29 Experimental result of the bicycle stabilization with a flywheel mode and

a = 0.3: Roll Angle α. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.30 Experimental result of the bicycle stabilization with a flywheel mode and

a = 0.3: Roll angular velocity α̇. . . . . . . . . . . . . . . . . . . . . . . 115


a = 0.3: Balancer angle β. . . . . . . . . . . . . . . . . . . . . . . . . . 115


a = 0.3: Balancer angular velocity ˙ β. . . . . . . . . . . . . . . . . . . . . 116


a = 0.3: Flywheel torque τb. . . . . . . . . . . . . . . . . . . . . . . . . 116


a = 10: Roll Angle α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117


a = 10: Roll angular velocity α̇. . . . . . . . . . . . . . . . . . . . . . . 117


a = 10: Balancer Angle β. . . . . . . . . . . . . . . . . . . . . . . . . . 118


a = 10: Balancer angular velocity ˙ β. . . . . . . . . . . . . . . . . . . . . 118


a = 10: Flywheel torque τb. . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.1 Perform a wheelie on a bicycle with a balancer. . . . . . . . . . . . . . . 122

6.2 Coordinate system of the bicycle robot. . . . . . . . . . . . . . . . . . . 123

6.3 The back view of the bicycle robot. . . . . . . . . . . . . . . . . . . . . . 125

6.4 The side view of bicycle robot. . . . . . . . . . . . . . . . . . . . . . . . 128

6.5 Simulation results start from near the upright position. . . . . . . . . . . 134

6.6 Simulation results when both wheels always contact with the ground. . . 135

6.7 Simulation results of acrobatic turn via wheelie for a bicycle with a balancer.136

6.8 Simulation snaps of going straight, wheelie, turning, and landing usingScherzo simulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137


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xvii

List of Tables

2.1 The bicycle with balancer parameters. . . . . . . . . . . . . . . . . . . . 30

3.1 The control parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 New control parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Yamaha electricbicycle “Passol” technical specifications. . . . . . . . . . 52

3.4 Balancer motor RKS-25-6018s technical specifications. . . . . . . . . . . 53

3.5 Steering motor Maxon RE36 technical specifications. . . . . . . . . . . . 53

4.1 The balancing and trajectory control parameters. . . . . . . . . . . . . . . 69

5.1 The bicycle with flywheel balancer parameters. . . . . . . . . . . . . . . 98

6.1 Bicycle with a 2 D.O.F balancer parameters. . . . . . . . . . . . . . . . . 133

6.2 Control parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133


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xviii


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1

Chapter 1

Introduction

In recent years, the researches on robotic systems are increasing very fast around the

world, especially the robotic systems integrated with smart environments. However, some

robots are used to rescue or search activity in the disaster area where human cannot do

the same jobs due to a possibility of additional disasters (Figure 1.1). Such as of the kind

specified, unmanned bicycle robot is proposed since it has high mobility and does not

require wide contact space to the ground. An unmanned bicycle robot can be used to

collect the information or transportation in the disaster areas and / or in the mountains.

Figure 1.1: Riding a bicycle in disaster area.

In order to make the bicycle become an autonomous robot, the bicycle must be self-


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2 CHAPTER 1. INTRODUCTION

stabilization, it can do acrobatic turn via a wheelie motion and it also moves to track any

trajectories in the ground plane. As in [2], unmanned bicycle robot with a flywheel was

constructed and the still and running motion were realized.

1.1 BackgroundResearch on the stabilization of the bicycles has been gained momentum over the last

decade in a number of robotic laboratories around the world. Modeling and control of

the bicycles became a popular topic for researchers in the latter half of the last century.

During the early 20th century several authors studied the problems of self-stabilizing bi-

cycles, balancing, and steering the bicycle. The bicycle literatures in this dissertation are

comprehensively reviewed from a brief history of the bicycle, the modeling and ending

with the control theory for balancing and trajectory tracking of the bicycle.

1.1.1 A Brief History of the Bicycle

The purpose of this section is to give an overview of the state of knowledge on the steering

behavior of single-track vehicles up to date. Vehicles for human transport that have two

wheels and require balancing by the rider date back to the early 19 th century. The first

such development came in 1817 when the German inventor Baron Karl von Drais (Figure

1.2), inspired by the idea of skating without ice, invented the running machine or draisine

[1]. After the fundamentals of bicycle design had been conceived by the end of the 1860s,

Figure 1.2: First Bicycle (“Draisienne”) made by Baron Karl von Drais (1817).


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1.1. BACKGROUND 3

a multitude of subsequent improvements were suggested and tried. The next appearance

of a two-wheeled riding machine was in 1865, when pedals were applied directly to the

front wheel. This machine was known as the velocipede (“fast foot”) (Figure 1.3), but

Figure 1.3: Velocipede by Pierre Michaux et Cie of Paris, France circa (1865).

was popularly known as the bone shaker, since it was also made entirely of wood, then

later with metal tires, and the combination of these with the cobblestone roads of the day

made for an extremely uncomfortable ride.

Pierre Michaux certainly produced pedaled velocipedes in increasing numbers in

1867-1869. In 1870 the first all metal machine appeared (Figure 1.4). The pedals were

still attached directly to the front wheel with no freewheeling mechanism. Solid rubbertires and the long spokes of the large front wheel provided a much smoother ride than

its predecessor. The front wheels became larger and larger as makers realized that the

larger the wheel, the farther you could travel with one rotation of the pedals. Because the

rider seat so high above the center of gravity, if the front wheel was stopped by a stone

or rut in the road, or the sudden emergence of a dog, the entire apparatus rotated forward

on its front axle, and the rider, with his legs trapped under the handlebars, was dropped

unceremoniously on his head. Thus the term “taking a header” came into being.

Improvements to the design began to be seen, many with the small wheel in the front

to eliminate the tipping-forward problem. One model was promoted by its manufacturerby being ridden down the front steps of the capitol building in Washington, DC. These

designs became known as high-wheel safety bicycles. This machine, which was the first

to be called a “bicycle,” was the world’s first single-track vehicle to employ the center-


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Figure 1.4: The ordinary, or high-wheeler, or penny-farthing.

steering head that is still in use today [3]. These bicycles enjoyed great popularity among

young men of means during their hey-day in the 1880s.

The 20th Century, Cycling steadily became more important in Europe over the first

half of the twentieth century, but it dropped off dramatically in the United States between

1900 and 1910. Automobiles became the preferred means of transportation.

The first company to produce bicycles in Japan was Miyata Industry Co. who began

bicycle production in 1890 and is still in operation. Today the bicycle is the most common

means used for running errands in and around your neighborhood and for commuting to

the nearest train station.

1.1.2 Dynamics and Control

Bicycles display interesting dynamic behavior. For example, bicycles are statically un-

stable like the inverted pendulum, but can, under certain conditions, be stable in forward

motion. As state in [4], A detailed model of a bicycle is complex because the system has

many degrees of freedom and the geometry is difficult to understand. Some important

aspects to consider are the choice of bicycle components to include in the model, the

treatment of elasticity of the bicycle parts, the modeling of tire-road interaction, and thecomplexity of the rider model. Modeling and Control of bicycles became a popular topic

for the researchers in the latter half of the last century [3]-[30].


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1.1. BACKGROUND 5

The first publication of the full nonlinear and also the linearized equations of motion

for an upright uncontrolled bicycle was by Whipple [5] in 1899. His linearization was

found to be correct except for typographical errors. Whipple’s model, which is essen-

tially the model considered in the “Basic Bicycle Model” section, consists of two frames,

the rear frame and the front frame, which are hinged together along an inclined steering

head assembly. This model is compared with numerical linearization of nonlinear modelsobtained from two multibody programs in [6]-[9], where the bicycle is described by 23

parameters and the model is validated by experimental study in [10] and [11]. These lin-

earized equations of motion for a bicycle developed as a benchmark and it is suitable for

research or application. By using this model, many researchers did analysis of the bicycle

stability and self stabilizing the bicycle ([12]-[16]). By extended the Whipple’s nonlin-

ear model, G. Franke [17] was aimed at eliminating some of the limitations mentioned

above: a general model of the moving bicycle was to be found, with as few simplifications

as possible, with any given geometry, allowing us to determine stable and unstable riding

situations, and to investigate the ability of the bicycle to stabilize itself under any given

initial conditions, the rider being able to steer with the help of small displacements of the

center of gravity.

On the other hand, the bicycle model were also attractive to the French’s researchers

since 1899s ([18]-[19]). This bicycle model is simplify by assuming that the inclination

angles of the rear and front wheels are the same, only the mass of the rear frame is taken

into account, the trail is zero, which is only true for small angles if the steer axis is vertical.

Due to advantage of the bicycle behaviors, some researchers used the bicycle models to

teach system dynamics [20], [21].

Y. Yavin ([22]-[23]) derived a kinematic model from the dynamic model of the motion

of a riderless bicycle. It is assumed that the bicycle is controlled by a pedalling torque, a

directional torque and by a rotor mounted on the crossbar that generates a tilting torque.

Then, using this kinematic model, a constraint point-to-point control ([24]) problem is

dealt with. The problem of navigation and control of the motion of a riderless bicycle by

using a simplified dynamic model is studied in [25].

Recently, N. Getz studied a simplified bicycle model, this mathematical model for abicycle has several attractive features when compared with other models [26],[27]. To de-

rive these bicycle models, he used the potential and kinetic energy by quadratic terms and

applying Lagranges equations to these expressions, but these models cannot be control


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when the linear velocity is zero. N. Getz published many papers related to balancing and

trajectory control, path and roll angle tracking control [28], internal equilibrium control

for path-tracking with balance by using steering and rear-wheel torque as inputs [29].

In order to improve the Getz’s models, J. Yi et al. [30] were modified these dy-

namic models by adding the bicycle caster angle, and captured the steering eff ect on the

vehicle tracking and balancing. The trajectory tracking control is designed using an ex-ternal / internal model decomposition approach, and the system balancing is developed by

the nonlinear control methods. The motion planning algorithms are based on the fused

global positioning systems (GPS) and on-board computer vision systems information.

W. Ham [31],[32] derived the nonlinear inverse kinematic problem in unmanned bi-

cycle system by using iterative method. This idea is similar to Piccard’s iterative method

in basic concept. The stabilizing bicycle used the lateral motion of mass, a steering angle

and driving wheel speed. For the path tracking control strategy, he used moving the center

of load mass left and right respectively based on the nonlinear compensation-like control.

L. Guo et al. [33] used the linearized dynamic model of bicycle robot to design a

linear controller for balancing bicycle robot with high speed. A linear quadratic optimal

controller was designed based on linear control theory for the linear dynamic model of bi-

cycle robot. There are many researchers have been successful to use diff erent controllers

for stabilizing the bicycle by controlling the steering torque such as feedback lineariza-

tion [34], robust and optimal control design [35]-[36], sliding mode control [37],[38],

fuzzy logic control [39]-[42], gain scheduling with LPV [43], [44], etc. Many of these

researches have been concerned with high speed range and have been conducted by bike

motion.

Speed of the bicycle is also a problem for stabilizing bicycle, it is well known that

the control of bicycle with steering at zero or slow linear velocity is very difficult. R.

F. Chidzonga [45] discussed about stabilization of a bicycle below critical speed and

he is demonstrated in his article that simple linear control can be applied to steady a

bicycle in the upright position at speed well below the critical self stability speed limit.

T. Yamaguchi [46] and D. J. N. Limebeer [47] studied the dynamics of the accelerating

bicycle under straight-running and cornering conditions. If the bicycle is cornering at

constant acceleration and roll angle, it is shown that for low values of acceleration (andbraking), it follows closely a logarithmic spiral shaped trajectory. The acceleration based

control is useful for bicycle to improve running stability in low-speed range. M. Weaver

[48] developed controllers to steer bicycles at very high speeds (70-100 mph) by using the


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on a steering handlebar and a balancer. The simulation results seem to be good with the

system performances, including the tracking performance for nonlinear time-varying un-

certainty, the pulse disturbance for simulating the wind eff ect, the variable speed motion

of bicycle.

An other researcher, Y. Liu [59] proposed a method for stabilizing an unmanned bicy-

cle by a pedalling torque, a directional torque that generated using the steer adjustment,and by a tilting torque that generated using a 2-RHR parallel mechanism mounted on the

crossbar.

1.2 Motivation

The motivation of this research is to realize an unmanned bicycle robot with a balancer

for collecting information or transportation in the disaster areas and / or in the mountains.

The research includes modeling of the bicycle with balancer for realizing the bicyclesimulator, controller design for stabilization of the bicycle, trajectory tracking of any

given path in the ground plane and acrobatic turn via a wheelie motion.

1.3 Contributions of this Dissertation

The main new results of this dissertation are:

1. A simplified dynamic model of a bicycle with a balancer.

2. A bicycle with a balancer simulator and a bicycle virtual reality work with Matlab-Simulink.

3. A balancing control that design based on output zeroing controller with suppressing

the interferences between the steering and the balancer.

4. A cooperating control for stabilizing the bicycle with both steering and balancer.

5. A new balancer configuration that works as a flywheel or a balancer for stabilizing

bicycle.

6. A trajectory tracking control for bicycle robot.

7. An acrobatic turn via a wheelie motion.

Hardware implementation is done to realize an unmanned bicycle robot.


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1.4. OVERVIEW OF THE THESIS 9

1.4 Overview of the Thesis

This dissertation presents about realization of an autonomous bicycle robot with balancer,

first we develop a simplified nonlinear dynamic model that it is derived from Lagrange’s

equations and nonholonomic constraints with some assumptions on bicycle system. Then

we can use this model for developing bicycle simulator, balancing controller design, trajectory controller design for moving in the ground plane. Finally, we come up with the

realization an acrobatic turn via a wheelie motion.

The dissertation is organized as follows: In chapter 2, we present dynamic model of a

bicycle with a balancer. The models derive from Lagrangian equations and nonholonomic

constraints. Chapter 3 discusses on stabilization of a bicycle with a balancer using an

output zeroing controller and adding the steering eff ect in the system state to eliminate

the interferences from the steering frame. The cooperation between a steering handlebar

and a balancer use to stabilize of the bicycle and trajectory tracking control based on aninput-output linearization are presented in chapter 4. A new balancer configuration and

a switching control algorithm for stabilizing an unmanned bicycle robot is proposed in

chapter 5. A realization of an acrobatic turn via wheelie motion is studied in chapter 6.

The main results of the dissertation are summarized and a number of ideas and problems

for future work are presented in chapter 7.


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11

Chapter 2

Dynamic Model of a Bicycle with a

Balancer

2.1 Introduction

Modeling and control of bicycles became a rich problem off ering a number of consider-

able challenges of current research interest in the area of mechanics and robot control as

described in the previous chapter. As stated in [4], there are many models of diff erent

complexities starting with simple models and ending with more realistic models derived

from multibody simulation software. Most of the researchers discussed only the mechan-

ical model of the bicycle that consists of four rigid bodies namely as: the rear frame withthe rider rigidly attached to it, the front frame being the front fork and handle bar assem-

bly and the two knife-edge wheels. These bodies are interconnected by revolute hinges at

the steering head between the rear frame and the front frame and at the two wheel hubs.

This kinds of mechanism is very difficult to control when the bicycle at low or zero linear

velocity.

However, N. Getz [27] used these kinds of properties to develop a simplified bicycle

dynamic model and this mathematical model for a bicycle has several attractive features

when compared with other models. Unfortunately, this model cannot control the bicycle

when the linear velocity is zero and J. Yi et al. [30] modified this bicycle model by addingthe bicycle caster angle, and captured the steering eff ect on the vehicle tracking and bal-

ancing then he can successful to stabilize the bicycle at zero velocity. This technique


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12 CHAPTER 2. DYNAMIC MODEL OF A BICYCLE WITH A BALANCER

cannot apply with the conventional bicycle, because the conventional bicycle has a short

trail that it make the steering angle less eff ect to the bicycle body angle.

By understanding the inconvenient of the conventional bicycle mechanism, we de-

cided to attach a balancer on the top of the bicycle and we can control this balancer for

stabilizing the bicycle. There are two kinds of bicycle dynamics model that used in this

dissertation. The first bicycle with balancer model is developed by M. Yamakita et al.[53] in 2005. The whole bicycle system derived by finding the dynamic equations for

subsystems using Lagrange’s dynamic equations, and they are combined by Lagrange’s

multiplier according to nonholonomic and holonomic constraints between subsystems.

This model use to simulate and design controller for realization of an acrobatic turn via

wheelie motion that it is in chapter 6. The second bicycle with balancer model is a simpli-

fied nonlinear model [60] that it is derived from Lagrange’s equations and nonholonomic

constraints with some assumptions on bicycle system.

As modeled here, the bicycle with the balancer is an underactuated system, subject to

nonholonomic contact constraints associated with the rolling constraints on the front and

rear wheels. We can use this model for bicycle simulator, balancing controller design by

using balancer and / or steering handlebar, trajectory controller design for moving in the

ground plane.

2.2 Nonlinear Dynamic Model

In this section, we will present a simplified model of a bicycle with a balancer that it

is derived from Lagrange’s equations and nonholonomic constraints associated with therolling constraints on the front and rear wheels. The bicycle nonlinear model is derived

based on N. Getz model [27] and [63].

2.2.1 Model Assumptions

A detailed model of a bicycle is complex since the system has many degrees of freedom.

Thus, we need to make some assumptions to simplify the model. We consider a simplified

bicycle model as shown in figure 2.1 with the following assumptions.

1. We consider the bicycle as a point mass with two wheel contacts with the ground.

2. The moments of mass of the bicycle are neglected.


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2.2. NONLINEAR DYNAMIC MODEL 13

on the wheelsRolling constraints

Steering Axis

Balancer

C 2C 1

l

h

mc

hb

mb

Wheelbase b

Figure 2.1: Coordinate system of the bicycle with the balancer.

3. The rigid frame of the bicycle is assumed to be symmetric about a plane containing

the rear wheel.

4. The bicycle is assumed to have a steering-axis fixed in the bicycle’s plane of sym-metry, and perpendicular to the flat ground when the bicycle is upright.

5. The front and rear wheel tires are thin, we do not consider the thickness of the tire.

6. The moments of mass of the front and rear wheels are neglected.

7. Only rear wheel longitudinal force has been considered and the front wheel longi-

tudinal force is neglected.

8. The bicycle is moving in a flat plane xOy , and no vertical motion is considered.

9. The balancer is considered as a point mass and it rotate with the vertical direction.

We define the parameters of a bicycle with a balancer system as below:


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• m Bicycle total mass

• h Height of the bicycle center of mass

• c Horizontal distance between rear wheel contact point and bicycle center of mass

• b Bicycle wheelbase

• l Vertical distance between ground and contact point of the balancer

• ∆ Bicycle trail

• mb Balancer mass

• hb Height of the balancer center of mass

2.2.2 Generalized CoordinatesThe coordinate system used to analyze the bicycle is defined in figures 2.1-2.5. We con-

Balancer Center of Mass

Active Joint

Bicycle Center of Mass

Figure 2.2: Geometric of a bicycle with a balancer parts.

sider the bicycle and the balancer as a two link system where the first link is the bicycle


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Bicycle

BalancerActive Joint

α

z

y

β

Figure 2.3: Coordinate system of a bicycle looking from backward.

body with steering and the second link is the balancer and it connects by active joint as

illustrate in figure 2.2. The balancer is fixed on the top of the bicycle body frame and the

balancer angle β is relative to the bicycle body frame and it is rotated in the vertical plane(Figure 2.3).

In the reference position, the global Cartesian coordinate system is located at the rear-

wheel contact point C 1 with x and y axes in the ground plane and z axis perpendicular to

the ground plane in the direction opposite to the force of gravity as shown in figure 2.4.

The intersection of the vehicle’s plane of symmetry with the ground plane forms its

contact line. The contact line is rotated about the z direction by a yaw-angle θ . The

contact line is considered directed, with its positive direction from the rear to the front of

the bicycle. The yaw angle θ is zero when the contact line is parallel to the x axis. The

angle that the bicycle’s plane of symmetry makes with the vertical direction is the roll

angle α. The steering shaft angle ψ is the rotation of the front frame with respect to the

rear frame about the steering axis. Consider the line of intersection between the plane of

the front wheel and the ground plane. Let φ be the front wheel direction angle between

this intersection and the contact-line.

Denote the radius of the trajectory of the rear-wheel contact point C 1 as R and the

trajectory curvature is defined as σ or we can call this angle as the virtual steering angle.

From figure 2.5, we can obtain the steering angle as

σ =: 1 R

= tan φ

b, (2.1)


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Balancer

ψ

b

vr

CG

α β

h

c

θ

v⊥

y

z

x

hb

η

φ

∆

C 2

C 1

l

O

Figure 2.4: Coordinate system of the bicycle with the balancer in a plane.

Back Wheel Front Wheel

x

θ

δ

φv f

v⊥

O

R

φ

O′y

Figure 2.5: Coordinate system of the top-view bicycle.


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and refer to σ as the steering variable. We will use σ rather than φ to parameterize steering

will help to give the nonholonomic rolling constraints of the bicycle a very simple form.

From the equation above, we take the time derivative and obtain

φ̇ = b

1 + tan2

φ

σ̇ = b

1 + b2

σ2 σ̇. (2.2)

We note that the front wheel direction angle φ is not the angle of rotation of the steering

shaft of the bicycle. Call the steering shaft angle ψ. From the geometry of the front

wheel steering mechanism (Figure 2.4), we can find the following relationship among the

steering shaft angle ψ, front wheel direction angle φ, bicycle roll angle α, and head angle

η,

tan φ cos α = tan ψ sin η. (2.3)

If we assume that the roll angle is in the upright position α = 0 and the steering axis

is on the z axis (head angle η = π

2 ), then the steering shaft angle ψ is equal to the frontwheel direction angle φ. Finally, we get the complete set of the generalized coordinates

for the bicycle with balancer system as x, y,θ ,α, β , and σ.

2.2.3 Generalized Velocities

We consider the generalized velocities ˙ x, ẏ, θ̇ , α̇, ˙ β, and σ̇ such that it will be convenient

when we introduce the constraints. Let vr be the component of the velocity of the rear

wheel contact along the contact line as measured from the ground frame, and let v⊥ be the

component of the velocity of the rear wheel contact perpendicular to the contact line andparallel to the ground plane as measured from the ground frame. From figure 2.6, we can

˙ x

vr v⊥ẏ

C 1

x

C 2

φy

θ

v f

Figure 2.6: Top view of the bicycle showing body velocities.


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calculate the relationship between vr , v⊥, ˙ x, ẏ and yaw angle θ as.

˙ xẏ

= cos θ − sin θ sin θ cos θ

x

y

. (2.4)

2.2.4 Inputs and Generalized Forces

Let τr be the reaction force that the ground exerts on the bicycle at the rear wheel contact

point and τb be the reaction force that generate by the balancer motor. Another torque

generator is associated with the steering variable σ that it is corresponding generalized

torque τσ. Thus our bicycle is riderless and under automatic control, driven by three

torque generators.

2.2.5 Kinetic and Potential Energy for a Bicycle with a Balancer

In this subsection, we find the kinetic and potential energy for a bicycle and a balancerseparately.

• Kinetic energy for bicycle’s body with steering part

From figure 2.4 , we can obtain the kinetic energy for the bicycle’s body with the

steering mechanism as follow:

K 1 = 1

2

J s ψ̇2 +

m

2vr + hθ̇ sin α

2+ v⊥ − hα̇ cos α + cθ̇

2+ (hα̇ sin α)2 , (2.5)

where J s is the constant moment of inertia of the steering mechanism and we can

use σ instead of φ. From equation (2.2), we get the moment term for the steering

mechanism as

J s ψ̇2 = J s(φ̇ sin η)

2 = J sb2

(1 + b2σ2)2 sin2 ησ̇2 = J s(σ)σ

2. (2.6)

Substitute equation (2.6) to (2.5), we get

K 1 = 12 J s(σ)σ̇2+

m

2

vr + hθ̇ sin α2+v⊥ − hα̇ cos α + cθ̇

2+ (hα̇ sin α)2

. (2.7)

• Potential energy for bicycle’s body with steering part


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In order to capture the stabilization mechanism of the bicycle at a low moving

velocity, we can modify the potential energy by considering the mass center gravity

change due to steering. The change of the mass center position due to the steering

action can be calculated by the estimate of the bicycle frame rotation angle δ in the

horizontal plane xOy.

From figure 2.5, δ can be estimated approximately as

δ = ∆ sin η

b φ. (2.8)

From figure 2.5 and [14], we can get the reduction of the mass center height as

hCG∆ = δc sin α. (2.9)

Substitute 2.8 into 2.9, we obtain

hCG∆ = c∆ sin η

bφ sin α ≈ c∆σ sin η sin α. (2.10)

From figure 2.7, the potential energy for the bicycle’s body with the steering mech-

CG

z

β

αmg

mbg

lh

hb

y

Figure 2.7: Bicycle looking from backward.

anism with the reduction of the mass center as follow:

P1 = mg (h cos α − c∆σ sin η sin α) . (2.11)


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• Kinetic energy for a balancer part

Let vc be the linear velocity of the center of gravity of the balancer, giving by

vc = J vc

α̇˙ β +

vr + (l sin α + hb sin(α + β))θ̇

v⊥

0

, (2.12)

where the Jacobian matric is define as

J vc =

0 0

−l cos α − hb cos(α + β) −hb cos(α + β)

−l sin α − hb sin(α + β) −hb sin(α + β)

.

The angular velocity of the balancer is given by

ω = cos θ sin θ 0

T α̇ + ˙ β

+ 0 0 1

T θ̇. (2.13)

We assume that I b is a diagonal matrix representing the moment inertia of the bal-

ancer, where I bx = I by = I b and I bzθ̇ 2 is a small value.

Thus, the kinetic energy for the balancer is

K 2 = 12mbv2c +

12ω

T I bω, (2.14)

or

K 2 =mb

2

vr + (l sin α + hb sin(α + β))θ̇

2+v⊥ − (l cos α + hb cos(α + β))α̇ − hb cos(α + β) ˙ β

2+(l sin α + hb sin(α + β))α̇ + hb sin(α + β) ˙ β

2+

I b

2

α̇ + ˙ β

2.

(2.15)

• Potential energy for a balancer part


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From figure 2.7, the potential energy for a balancer part can be computed as

P2 = mbg (l1 cos α + hb cos(α + β)) . (2.16)

2.2.6 The Euler-Lagrange Equation

The Euler-Lagrange equations for such a system can be derived as follows.

L = (K 1 + K 2) − (P1 + P2),

and it is become

L =12

J (σ)σ̇2 + m

2

vr + hθ̇ sin α

2+v⊥ − hα̇ cos α + cθ̇

2+ (hα̇ sin α)2

+mb

2 vr + (l sin α + hb sin(α + β))θ̇ 2+ (v⊥ − (l cos α + hb cos(α + β))α̇

− hb cos(α + β) ˙ β2+(l sin α + hb sin(α + β))α̇ + hb sin(α + β) ˙ β

2

+ I b

2 (α̇ + ˙ β)2 − mgh cos α + cmg∆σ sin η sin α − mbg (l cos α + hb cos(α + β)) .

(2.17)

2.2.7 Constraints

As in assumptions in Subsection 2.2.1 the front and rear wheels are assumed to roll with

neither lateral nor longitudinal slip. Front and rear wheel contacts are constrained to

have velocities parallel to the lines of intersection of their respective wheel planes and the

ground plane, but free to turn about an axis through the wheel / ground contact and parallelto the z axis.

The generalized velocities of the bicycle are partitioned as ṙ = [α̇, vr , ˙ β, σ̇]T and

ṡ = [θ̇, v⊥]T . In these velocity coordinates the nonholonomic constraints associated

with the front and rear wheels, assumed to roll without slipping, are expressed simply by

ṡ + A(r , s)ṙ = 0 (2.18)

θ̇ v⊥ + 0 −σ 0 0

0 0 0 0

α̇

vr

˙ βσ̇

= 0.

The first equation show that the yaw rate θ̇ of the bicycle is the product of the steering


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variable σ := 1b tan φ and the rear wheel velocity vr ,

θ̇ = σvr . (2.19)

The second equation tell us that v⊥, the componentof the velocity of the rear wheel contact

point perpendicular to the plane of that wheel, is zero.

v⊥ = 0. (2.20)

The Constrained Bicycle Lagrangian’s Equation

Substitute equations (2.19) and (2.20) into (2.17), obtaining the constrained Lagrangian

Lc.

Lc =1

2

J (σ)σ̇2 + m

2(vr + hvr σ sin α)2 + (−hα̇ cos α + cvr σ)2 + (hα̇ sin α)2

+mb

2

(vr + (l sin α + hb sin(α + β))vr σ)

2+ (−(l cos α + hb cos(α + β))α̇

− hb cos(α + β) ˙ β2+(l sin α + hb sin(α + β))α̇ + hb sin(α + β) ˙ β

2

+ I b

2 (α̇ + ˙ β)2 − mg(h cos α − c∆σ sin η sin α) − mbg (l cos α + hb cos(α + β)) .

(2.21)

2.2.8 Equation of Motion

The equation of motion based upon the constrained lagrangian are derive

d

dt

∂ Lc

∂ṙ α −

∂ Lc

∂r α + Aaα

∂ Lc

∂sa = −

∂ L

∂ṡb Bbαβṙ

β (2.22)

where Bbαβ

denote the components of the curvature of the connection A(r , s),

Bbαβ =

∂ Abα

∂r β −

∂ Ab β

∂r α + Aaα

∂ Ab β

∂sa − Aa β

∂ Abα

∂sa

(2.23)

By using the generalized forces in Subsection 2.2.4, thus the equation is modified to be

d dt

∂ Lc∂ṙ α

− ∂ Lc∂r α

+ Aaα ∂ Lc∂sa = − ∂ L

∂ṡb Bbαβṙ

β + τi (2.24)

The elements of the matrix A(r , s) from equation (2.18) are A11 = 0, A12 = −σ, A

13 = 0,

A14 = 0, A21 = 0, A

22 = 0, A

23 = 0 and A

24 = 0.


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Thus, the curvature of the connection Bbαβ

is computed using equation (2.23) to be

B124 = − B142 = −1 (2.25)

with the remaining Bbαβ

are zero. Solving equation (2.24), we can obtain the bicycle

dynamics as M ̈q = K + Bu (2.26)

or, we can rewrite as

M 11 M 12 M 13 0

M 21 M 22 0 0

M 31 0 M 33 0

0 0 0 M 44

α̈

v̇r ¨ β

σ̈

=

K 1

K 2

K 3

K 4

+

0 0 0

1 0 0

0 1 0

0 0 1

F r

τb

τσ

where

M 11 = I b + mh2+ mb(h

2b + l

2) + 2hblmb cos β,

M 12 = M 21 = −chmσ cos α,

M 13 = M 31 = I b + hbmb(hb + l cos β),

M 22 = m(c2σ2 + (1 + hσ sin α)2) + mb(1 + lσ sin α + hbσ sin(α + β))

2,

M 33 = I b + h2bmb,

M 44 = J sb

2

sin2 η(1 + b2σ2)2,

K 1 = cmg∆σ sin η cos α + hm(g sin α + vr (σvr (1 + hσ sin α) + cσ̇)cos α)

+mb(g(l sin α + hb sin(α + β)) + σv2r (l(1 + lσ sin α)cos α

+hb(1 + hbσ sin(α + β))cos(α + β) + hblσ sin(2α + β)) + hbl ˙ β(2α̇ + ˙ β)sin β),

K 2 = −m(h(cσα̇2 + vr σ̇(1 + hσ sin α))sin α + σvr (2hα̇(1 + hσ sin α)cos α + c

2σ̇))

−mb(2hbσvr (( ˙ β + α̇)(1 + hbσ sin(α + β)) + lσ ˙ β sin α)cos(α + β)

+σvr σ̇((l sin α + hb sin(α + β))2) + hbvr σ̇ sin(α + β) + lvr σ̇ sin α + 2lσvr α̇((1

+lσ sin α)cos α + hbσ sin(α + β))),

K 3 = hbmb(−lα̇2

sin β + g sin(α + β) + v2r σ cos(α + β)(1 + σ(l sin α + hb sin(α + β))),

K 4 = c∆gm sin η sin α + 2b4 J sσσ̇3

sin2 η(1 + b2σ2)3.


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Finally, we obtained the dynamic model of a bicycle with a balancer that it contained of

the steering eff ect. In order to fix the steering joint or the balancer joint, we introduce a

holonomic constraint for the joints

J q̈ = − ˙ J q̇ (2.27)

where J is a constrained Jacobi matrix which is switched according to the joint situations.

The equation of motion becomes

M J T

J 0

q̈

λ

= Bu + K

− ˙ J q̇

. (2.28)

where λ is the corresponding Lagrange multiplier.

2.3 Linearized Dynamic Model

In order to simplified the nonlinear model for linearizing, we assume that the bicycle

linear velocity is constant (vr = V r ). Thus, the nonlinear equation (2.26) becomes

M 11 M 13 0

M 31 M 33 0

0 0 M 44

α̈

¨ β

σ̈

=

K 1

K 3

K 4

+

0 0

1 0

0 1

τb

τσ

(2.29)

where

M 11 = mh2 + (h2b + l

2)mb + I b + 2hblmb cos β,

M 13 = M 31 = hbmb(hb + l cos β) + I b,

M 33 = h2bmb + I b,

M 44 = J sb2

(1 + b2σ2)2 sin2 η,

K 1 = cmg∆σ sin η cos α + hm(g sin α + V r (σV r (1 + hσ sin α) + cσ̇)cos α)

+mb(g(l sin α + hb sin(α + β)) + σV 2r (l(1 + lσ sin α)cos α

+hb(1 + hbσ sin(α + β))cos(α + β) + hblσ sin(2α + β)) + hbl ˙ β(2α̇ + ˙ β)sin β),K 3 = hbmb(g sin(α + β) − lα̇

2 sin β + V 2r σ cos(α + β)(1 + σ(l sin α + hb sin(α + β))),

K 4 = c∆gm sin η sin α + 2b4 J sσσ̇3

sin2 η(1 + b2σ2)3.


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2.3. LINEARIZED DYNAMIC MODEL 25

Because of the complexity of this nonlinear model, we divide the nonlinear equation

(2.29) into two cases:

1. Case 1: The bicycle with only a balancer

2. Case 2: The bicycle with steering and back wheel

2.3.1 Case 1: A Bicycle with Only a Balancer

In this case, we consider the steering is fixed at zero position (σ = 0) and the bicycle is

in the still motion (vr = 0). Thus, the dynamic system for a bicycle with only a balancer

becomes M b11 M b12 M b21 M b22

α̈¨ β

= K b1

K b2

+ 01

τb (2.30)where

M b11 = mh2 + (h2b + l

2)mb + I b + 2hblmb cos β,

M b12 = M b21 = hbmb(hb + l cos β) + I b,

M b22 = h2bmb + I b,

K b1 = hmg sin α + mb(g(l sin α + hb sin(α + β)) + hbl ˙ β(2α̇ + ˙ β)sin β),

K b2 = hbmb(g sin(α + β) − lα̇2 sin β),

This system dynamic model is based on an inverted pendulum, there are two equilibriumpoints, one is at the upright position and the other is downward position.

Firstly, we linearized the model around upward position. Since the control objective

is to keep the bicycle and the balancer vertical, we can linearize the models around the

operating point α∗ ≈ 0◦ and β∗ ≈ 0◦. Assume that cos α ≈ 1, sin α ≈ α, sin(α+ β) ≈ α+ β,

α̇2 ≈ 0, ˙ β2 ≈ 0 and α̇ ˙ β ≈ 0. Let X = [α β α̇ ˙ β]T be a state variable and u = τb be the

input. The linearized model of the nonlinear equations in (2.30) can be determined as

E ˙ X = FX + Gu, or (2.31)˙ X = E −1FX + E −1Gu,


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where,

E =

I 0

mh2 + (hb + l)2mb + I b hbmb(hb + l) + I b0

hbmb(hb + l) + I b (h2bmb + I b)

,

F =

0 I

g(hm + mbl + mbhb) hbmbg

0

hbmbg hbmbg

,

G = 0 0 0 1

T .

Next, we will find the linearized model using the equilibrium points where α∗ ≈ 180◦ and

β∗ ≈ 0◦. Assume that cos β ≈ 1, sin α ≈ π − α, sin(α + β) ≈ π − α − β, α̇2 ≈ 0, ˙ β2 ≈ 0

and α̇ ˙ β ≈ 0. Let X = [α − π β α̇ ˙ β]T be a state variable and u = τb be the input. The

linearized model of the nonlinear equations in (2.30) becomes

M ˙ X = N X + Pu

˙ X = M −1 NX + M −1Pu (2.32)

where,

M =

I 0

mh2 + (hb + l)2mb + I b hbmb(hb + l) + I b0

hbmb(hb + l) + I b (h2bmb + I b)

,

N =

0 I

−g(hm + mbl + mbhb) −hbmbg

0

−hbmbg −hbmbg

,

P = 0 0 0 1 T

.


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2.3. LINEARIZED DYNAMIC MODEL 27

2.3.2 Case 2: The bicycle with Steering and Back Wheel

In this case, we consider the balancer is fixed at zero position ( β = 0). Thus, the dynamic

system for a bicycle with steering and constant velocity (2.29) becomes

M s11 00 M s22 α̈

σ̈ = K s1K s2 +

0

1 τσ (2.33)

where

M s11 = mh2,

M s22 = J sb2

(1 + b2σ2)2 sin2 η,

K s1 = cmg∆σ sin η cos α + hm(g sin α + V r (σV r (1 + hσ sin α) + cσ̇)cos α),

K s2 = c∆gm sin η sin α + 2b4 J sσσ̇3

sin2

η(1 + b2

σ2

)3

.

The element matrix K s1 has a term cmg∆σ sin η cos α such the steering angle aff ects the

bicycle angle. Thus, we can control the bicycle using a steering handlebar even when the

linear velocity is zero.

In order to linearize this system, we assume the bicycle is in the upright position

α ≈ 0◦ and steering is on the straight line σ ≈ 0◦. From equation (2.33), we can obtain

the linearized model as

α̈ = cg∆ sin η

h2 σ +

g

h

α +V 2r

h

σ + cV r

h

σ̇ (2.34)

The design of the front fork has a major impact on bicycle dynamics. To take the front

fork into account, we define the torque applied to the handlebars, rather than the steer

angle, as the control variable. The contact forces between the tire and road exert a torque

on the front fork assembly when there is a tilt.

Under certain conditions, this torque turns the front fork toward the lean. Let F f and

N f be the horizontal and vertical force acting on the front wheel at the ground contact.

From [4], the static torque balance of the front fork becomes

τσ − (F f + N f )∆ sin η = 0 or

τσ − cmg∆ sin η

bα −

cm∆ sin ηb

V 2r sin η − bg cos η

σ = 0 (2.35)


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The torque balance for the front fork assembly (2.35) can be written as

σ = b

cm∆ sin ηV 2r sin η − bg cos η

τσ − gV 2r sin η − bg cos η

α (2.36)

Self-Stabilization

We consider that the bicycle is self-stabilization without external torque τσ = 0, then

equation (2.36) becomes

σ = − g

V 2r sin η − bg cos ηα (2.37)

The frame model (2.34) changes because of the geometry of the front fork. The steering

angle σ in third and fourth term of the right hand side is replaced by the eff ective steering

angle σ sin η. Substitute equation (2.37) into (2.34), gives

α̈ + g2(hb cos η − c∆ sin η)

h2

V 2r sin η − bg cos ηα + cgV r h V 2r sin η − bg cos η α̇ = 0 (2.38)

This system is stable if

V r > V c =

bg cos η (2.39)

and

bh > c∆ tan η (2.40)

This explains why it is easier to ride a conventional bicycle at higher speeds as compared

to lower speeds.

2.4 Simulation Results

In this section, we do numerical simulation to validate the bicycle dynamic model by

using Matlab Simulink Toolbox.

Firstly, we create some packages namely bicycle simulator user interface and bicycle

virtual reality that allow us easily to change the bicycle parameters in the simulation.

Next, we do the simulation to compare the linearized model with the nonlinear model

and we also perform the bicycle self-stabilization.


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2.4. SIMULATION RESULTS 29

2.4.1 Bicycle Simulator

For simulations to be use more easily and more efficiently it is important that a Graphical

User Interface (GUI) should be is well functional. Matlab has the power of handling large

amounts of data and performs necessary calculations and is therefore a good platform for

a GUI. We create bicycle graphic user interface as in figure 2.8The specifications for the most important content of the GUI are listed as follows:

Figure 2.8: User interface for a bicycle with a balancer simulator.

• The GUI should handle all necessary preparations before the simulation. Diff erent

body of bicycle profiles, steering and balancer are provided.

• Changes to the bicycle with balancer specific parameters, e.g. bicycle total mass m,

bicycle length to COG h, distance from ground to the balancer l, horizontal distance

from rear wheel contact point to COG c, bicycle wheelbase b, bicycle head angle η,

moment inertia of the steering mechanism J s, bicycle initial angle, bicycle initialpotion, balancer mass mb, balancer length to COG hb, balancer moment inertia I b,

balancer initial angle, ... etc should be possible.


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• Selecting modes, balancer, flywheel or flywheel balancer, fix the balancer joint,

steering handlebar and back wheel.

• Data and parameter values in specific files must be saved, The GUI should also be

able to handle a list of separated data.

The nominal parameters of the bicycle with balancer and range of adjustable values

are shown in Table 2.1. The parameters of the bicycle were identified from an experimen-

tal setup.

Table 2.1: The bicycle with balancer parameters.

Parameters Values Adjustable

Bicycle total mass m 51.2[Kg] ±30%

Bicycle length to COG h 0.42[m] ±30%Distance from ground to the balancer l 0.68[m] ±20%

Distance from rear wheel contact point to COG c 0.35[m] ±30%

Bicycle wheelbase b 1.06[m] ±20%

Bicycle head angle η 65◦ ±20%

Bicycle trail ∆ 0.12[m] ±20%

Moment inertia of the steering mechanism J s 0.32[Kgm2] ±30%

Balancer mass mb 10.39[Kg] ±30%

Balancer length to COG hb 0.22[m] ±30%

Balancer moment inertia I b 0.18[m] ±30%

Beside the bicycle user interface, we also develop a bicycle Virtual Reality as show

in figure 2.9 that it allows the user to see the animation of the bicycle like the real one.

The Virtual Reality is a solution for interacting with virtual reality models of dynamic

systems over time. It extends the capabilities of Matlab and Simulink into the world of

virtual reality graphics. We have five inputs for the bicycle with balancer virtual reality.

They are:

• Bicycle yaw angle

• Bicycle roll angle


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Figure 2.9: Virtual reality of a bicycle with a balancer.

• Rear wheel velocity

• Steering rotation angle

• Balancer rotation angle

2.4.2 Linear model of Bicycle with a Balancer

From the bicycle with balancer in Table 2.1, we can calculate the linear model as:

1. Linear model around upright position

By using the equation (2.31) and we substitute with the parameters from Table 2.1,

we obtain

˙ X =

0 0 1 0

0 0 0 1

22.2589 −4.9563 0 0

−40.0869 49.0748 0 0

X +

0

0

−0.3181

2.5067

u (2.41)

This linear model gives the eigenvalues as λ = [7.4243, − 7.4243, 4.0265, −

4.0265]. There are some positive eigenvalues and we can conclude that the open-


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loop linear system is unstable. We can use this model to design the linear controller

for stabilization the bicycle system.

2. Linear model around downward position

We do the same way with downward direction of linear model (2.32), we get

˙ X =

0 0 1 0

0 0 0 1

−22.2589 4.9563 0 0

40.0869 −49.0748 0 0

X +

0

0

−0.3181

2.5067

u (2.42)

This linear model gives the eigenvalues as λ = [−0.0+7.4243i, − 0.0 − 7.4243i, −

0.0 + 4.0265i, −0.0 − 4.0265i]. This system is marginal stable because the eigen-

values of this system has only imaginary part.

2.4.3 Validation Linear Model Using Zero Input-Responses

In order to validate the linear model with the nonlinear model, We examine the zero-input

response of the bicycle and balancer angle in downward direction (near the operating

points α∗ = 180◦ and β∗ = 0◦) with the initial angle α(0) = 170◦ and β(0) = −10◦ and

it is shown in figure 2.10. The zero-input response with initial condition is shown in

β = −10◦

α = 170◦

Figure 2.10: Initial position setup for zero-input response.

figures 2.11 and 2.12. The responses of linear model is exactly the same and similar


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0 1 2 3 4 5 6 7 8 9 10170

172

174

176

178

180

182

184

186

188

190

Time[s]

R o l l A n g l e ( α ) [ D e g r e

e ]

Nonlinear Model

Linear Model

Figure 2.11: Roll angle of zero-input response.

0 1 2 3 4 5 6 7 8 9 10−25

−20

−15

−10

−5

0

5

10

15

20

25

Time[s]

B a l a n c e r A n g l e ( β ) [ �

realization of an unmanned bicycle robot with balancer

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