Abstract A new mechanism to balance an autonomous

unicycle is explored which makes use of a simple pendulum.

Mounted laterally on the unicycle chassis, the pendulum

provides a means of controlling the unicycle balance in the

lateral (left-right) direction. Longitudinal (forward-backward)

balance is achieved by controlling the unicycle wheel, a

mechanism exactly the same as that of wheeled inverted

pendulum. In this paper, the pendulum-balancing concept is

explained and the dynamics model of an autonomous unicycle

balanced by such mechanism is derived by Lagrange-Euler

formulation. The behavior is analyzed by dynamic simulation in

MATLAB. Dynamics comparison with wheeled inverted

pendulum and Acrobot is also performed.

I. INTRODUCTION

NICYCLE has been the subject of research for at least two

decades. One of the first researches about unicycle was

done in Stanford University in 1987 [1]. In [1], the lateral

balancing mechanism was the weight mounted on top of the

unicycle chassis, which could be rotated with respect to the

vertical axis. The dynamics modeling was performed using

both Newton-Euler and Lagrange-Euler formulations.

Another research was performed in MIT and was reported in

1990 [13]. In [13], the lateral-balancing mechanism is exactly

the same as that in [1], but the dynamics formulation was

done using Kane formulation. Other different

lateral-balancing mechanisms were proposed in [2], [6], [7],

[8] and [10]. In [2] and [8], Yangsheng Xu group and

Abdullah Al Mamun group built unicycles with gyroscopes

mounted inside the wheels. The balancing mechanism made

use of the gyroscope angular momentum. In [6],

Yuta group built a unicycle robot with a head as its balancing

mechanism. It has the wheel with the shape of a rugby ball to

ensure lateral stability and steering was performed by leaning

the robot head to the left or right direction. In [7], Wang

Qiyuan group proposed an inertia wheel as the

lateral-balancing mechanism. The motion of inertia wheel

provides lateral recovery torque to ensure lateral balance. In

[10], a turntable with the shape of three-plate propeller was

Manuscript received April 19, 2011.

Jian-Xin Xu is Professor in the Electrical and Computer Engineering

Department, National University of Singapore, Singapore. (corresponding author phone: +65-6516-2566; fax: +65-6799-1103; e-mail:

elexujx@nus.edu.sg).

Abdullah Al Mamun is Associate Professor in the Electrical and Computer Engineering Department, National University of Singapore,

Singapore (e-mail: eleaam@nus.edu.sg).

Yohanes Daud is Graduate Student with the Electrical and Computer Engineering Department, National University of Singapore, Singapore

(e-mail: g0800557@nus.edu.sg).

proposed as the lateral-balancing mechanism by Kazuo

Yamafuji group.

From all of the proposed mechanisms, an inertia wheel in

[7] seems to be the simplest. It is capable of providing

recovery torque without shifting the centre of mass in the

lateral direction. However, the possible application of such

mechanism is limited only for goods transportation. Once a

human rider is considered, there must be some shifting in the

centre of mass, as a human rider is very likely to move his or

her body to the left and right sides. The gyroscope mechanism

in [2] and [8] can provide good steering and balancing

capabilities, but the rotating gyroscope consumes a lot of

energy. Some more, since the shape of the robots is a simple

plain wheel, their use is limited.

Besides the researches mentioned above, there are a lot of

other researches about unicycle, which in fact investigate

wheeled inverted pendulum [4], [9], [11]. Wheeled inverted

pendulum has the same longitudinal behavior as that of

unicycle, but its lateral behavior is completely different.

Because of the two wheels, wheeled inverted pendulum is

safe from lateral instability, unlike unicycle. Probably, this is

the main reason that a lot of unicycle researches start from

wheeled inverted pendulum.

In this paper, we propose a simple lateral-balancing

mechanism. A simple pendulum is mounted on the unicycle

chassis and it can be rotated to the left and right sides of the

unicycle. A simple illustration is shown in Fig. 1. This

mechanism similar to the mechanism proposed in [6], except

that we do not consider rugby-ball-shaped wheel since most

unicycles currently available do not have such wheel. Some

more, rugby-ball-shaped wheel has wide shape which

imposes limitations on the ability to pass narrow path and the

ability to maneuver at sharp angle. Without the

rugby-ball-shaped wheel, it is expected that our control task

will be more difficult than that in [6]. Our proposed

mechanism can become the first step towards the

development of manipulator arm mounted on a single wheel.

This comes from the realization that if one link after another

is added to the current model, the pendulum itself will

resemble the manipulator arm. We believe that such futuristic

robot will be of use for a lot of varieties of tasks besides a

means of transportation.

II. CONCEPTUAL DESIGN

Fig. 1 shows the conceptual design of our

Pendulum-Balanced Autonomous Unicycle: Conceptual Design and

Dynamics Model

Jian-Xin Xu, Senior Member, IEEE, Abdullah Al Mamun, Senior Member, IEEE and Yohanes Daud,

Student Member, IEEE

U

51978-1-61284-251-6/11/$26.00 c© 2011 IEEE

pendulum-balanced autonomous unicycle. It has three main

parts: wheel, chassis and pendulum. There are only two

actuators: one brushless DC motor at the wheel hub and one

brushless DC motor at the pendulum pivot.

A. Moving Forward and Backward

Longitudinal motion in the forward and backward

directions is achieved through actuation of brushless DC

motor at the wheel hub. The wheel is rigidly coupled to the

stator is rigidly attached to the chassis. When the brushless

DC motor is activated, the wheel will move with speed and

direction depending on the control signal applied to the

motor. The chassis is also affected by the reaction torque from

. The torque at wheel and the

reaction torque at the chassis have the same magnitude, but

they are in the opposite directions [5]. If the pendulum angle

and the lateral lean angle are set to be zero, this

longitudinal motion is exactly the same as the motion of

wheeled inverted pendulum.

B. Maintaining Lateral Stability

Lateral motion in the left and right directions is controlled

by the brushless DC motor at the pendulum pivot. The

rotor, while the brushless

attached to the chassis. Hence, when the brushless DC motor

is activated, the pendulum will move in one direction due to

the action torque, while the chassis and wheel will move in

the other opposite direction due to the reaction torque. If the

longitudinal lean angle is set to be zero and there is

enough friction at the wheel-ground contact, the lateral

motion is very similar to the motion of Acrobot [12].

C. Turning

Turning of pendulum-balanced autonomous unicycle is

achieved by controlling the wheel and pendulum actuators at

the same time. The main concept in turning motion is

gyroscopic precession. If a disturbance torque is applied to a

rolling wheel, instead of falling down immediately, the wheel

will turn (precess) to the direction where it falls. This natural

behavior is due to the conservation of angular momentum [2,

8]. Thus, for pendulum-balanced autonomous unicycle to

perform turning motion, wheel actuator must first generate

torque to accelerate the wheel. Once constant speed has been

achieved, the pendulum actuator can start being activated to

give lateral torque for the wheel in order for it to precess.

III. DYNAMICS MODEL

The dynamics model of the unicycle is derived by

Lagrange-Euler formulation following the procedure in [3]

and [8]. Five frames are needed for coordinate representation.

The assignment of these frames is shown in Fig. 2. The

assumptions in the dynamics model formulation are listed

below.

1. All unicycle components are perfectly rigid bodies.

2. There is no slip between the wheel and the ground.

3. The wheel is always in contact with the ground.

4. The ground is perfectly flat.

No-slip condition imposes motion constraints on the

unicycle. It is governed by the equations:

(1)

(2).

Lagrange-Euler formulation has six steps: 1.

Determination of position vectors, 2. Determination of linear

velocity vectors, 3. Determination of angular velocity vectors,

4. Calculation of kinetic and potential energies 5. Calculation

of Lagrangian and 6. Formulation of dynamics equations by

Lagrange equation.

chassis and pendulum are given as followed, where sine and

cosine are shortened as s and c respectively and lc + lcp is

shortened as l .

(3)

Fig. 1. Conceptual Design of Pendulum-Balanced Autonomous

Unicycle

Wheel

Pendulu

Chassis

Front Side View

Fig. 2. Assignment of Coordinate Frames for Dynamics Modeling of

Pendulum-Balanced Autonomous Unicycle

zw1 zw2

yw2

xc

yc

zc

zp yp

xp

yground

xground

zground xw1, xw2 yw1

zc

xw1, xw2

xc

yc zw1, zw2

yw1, yw2

zground

xground

yground

yp

xp

zp

52 2011 IEEE 5th International Conference on Robotics, Automation and Mechatronics (RAM)

(4)

(5)

For step 2, the linear velocities are computed as the

derivatives of the position vectors with respect to time.

(6)

(7)

(8)

For step 3, the angular velocity vectors are given as

followed.

(9)

(10)

(11)

The angular velocities are given with respect to the local

frames. The advantage of this is that the computation of

kinetic energy due to the angular velocities is simpler because

the inertia matrices are diagonal matrices.

For step 4, the kinetic and potential energies for the wheel,

chassis and pendulum are computed as:

(12)

(13)

(14)

(15)

(16)

(17).

The overall system potential and kinetic energies are the

algebraic summation of the potential and kinetic energies for

the wheel, chassis and pendulum.

(18)

(19)

For step 5, the difference between the total kinetic energy

and potential energy is called Lagrangian and is computed as:

(20).

For step 6, using the computed Lagrangian, the dynamics

equations of the system is described by Lagrange equation:

(21)

where:

L = Lagrangian

qi = generalized coordinate

Qi = generalized force

n = number of constraints (in this case, n = 2)

k = Lagrange multiplier

aki is constant determined from the assumed constraint.

In total, there are seven equations of motion. However, the

first and second equations represent the motion constraints

and they can be substituted to the other five equations. The

final dynamics model of the pendulum-balanced autonomous

unicycle is hence described by five nonlinear differential

equations:

(22)

(23)

(24)

(25)

(26)

The coefficients of the equations above are nonlinear and

dependent on state variables. Another note is that equations

(22) and (26) do not have gravity terms.

2011 IEEE 5th International Conference on Robotics, Automation and Mechatronics (RAM) 53

IV. LONGITUDINAL BEHAVIOR AND WHEELED INVERTED

PENDULUM

If the pendulum-balanced autonomous unicycle is

constrained to move only in the forward-backward direction,

the behavior must be the same as that of wheeled inverted

pendulum.

The motion constraint imposed on the unicycle to move

only in the forward-backward direction requires the following

condition to be satisfied:

(27)

(28)

(29).

The unicycle dynamics model hence becomes simplified and

using equation (1), the model can be written as shown below.

Only two equations are relevant in this case as the other three

equations have their left-hand sides degenerate to zero.

(30)

(31)

The dynamics model of wheeled inverted pendulum taken

from [9] is:

(32)

(33).

It can be seen that the unicycle motion in the longitudinal

direction is governed by the same equations for the wheeled

inverted pendulum.

V. LATERAL BEHAVIOR AND ACROBOT

If the pendulum-balanced autonomous unicycle is

constrained to move only in the left-right direction, the

behavior must be similar to that of Acrobot.

The motion constraint imposed on the unicycle to move

only in the left-right direction requires the following

condition to be satisfied:

(34)

(35)

(36).

The unicycle dynamics model hence becomes simplified as

shown below. Only two equations are relevant in this case as

the other three equations have their left-hand sides degenerate

to zero.

(37)

(38)

The dynamics model of Acrobot taken from [12] is:

(39)

(40).

It can be seen that the two models are essentially the same.

However, for the unicycle to exactly behave like Acrobot,

there must be enough friction at the wheel-ground contact. If

this condition is not satisfied, the wheel may slip in the lateral

direction.

VI. SIMULATION STUDY

Simulation study is performed using MATLAB for the

54 2011 IEEE 5th International Conference on Robotics, Automation and Mechatronics (RAM)

analysis of the natural behavior of pendulum-balanced

autonomous unicycle when both of the actuators are not

activated. Three cases of initial conditions are evaluated:

1. , others are zero

2. , others are zero

3. , others are zero.

The unicycle parameters in Table 1 are used in the simulation.

A. Initial Condition:

This set of initial conditions is meant to observe the

longitudinal behavior of the pendulum-balanced autonomous

unicycle. Some more, with linearly increasing longitudinal

lean angle, the nonlinearity of the unicycle longitudinal

motion can be observed.

The simulation results for this set of initial conditions are

shown in Fig. 3.

It is observed that with nonzero initial longitudinal lean

angle, the longitudinal lean angle keeps increasing

nonlinearly (its rate of change increases). At the same time,

the wheel angle decreases. The angular velocity of the wheel

increases at the beginning until it reaches a maximum value

then it drops until the wheel stops completely. Physically, the

chassis falls down while at the same time, the wheel rolls

backward.

As the initial chassis lean angle is increased linearly (100,

200, 30

0), the time it takes for the chassis to reach the horizon

increases in a nonlinear manner (0.54 s, 0.63 s, 0.78 s). From

the result, it is expected that a very small initial chassis lean

angle makes the chassis fall down very slowly.

B. Initial Condition:

This set of initial conditions is meant to observe the lateral

behavior of the pendulum-balanced autonomous unicycle.

Some more, with linearly increasing chassis lateral lean

angle, the nonlinearity of the unicycle lateral motion can be

observed.

The simulation results for this set of initial conditions are

shown in Fig. 4.

It is observed that with nonzero initial lateral lean angle, the

lateral lean angle keeps increasing nonlinearly (its rate of

change increases). At the same time, the pendulum angle

decreases. The angular velocity of the pendulum decreases at

the beginning until it reaches a minimum value then it

bounces back. Physically, the chassis and wheel fall down to

one side while at the same time, the pendulum falls to another

side.

As the initial lateral lean angle is increased linearly (100,

200, 30

0), the time it takes for the chassis and wheel to reach

Fig. 3. Dynamic Responses 0, 200 and 300 and Zero

Control Signals

Fig. 4. Dynamic Responses 0, 200 and 300 and Zero

Control Signals

TABLE I UNICYCLE PARAMETERS

Symbol Quantity Value and Unit

mw wheel mass 0.5 kg

mc chassis mass 0.5 kg

mp pendulum mass 1.0 kg Iw1 wheel moment of

inertia in x direction

0.02 kg m2

Iw2 wheel moment of inertia in y direction

0.01 kg m2

Iw3 wheel moment of

inertia in z direction

0.01 kg m2

Ic1 chassis moment of

inertia in x direction

0.02 kg m2

Ic2 chassis moment of inertia in y direction

0.015 kg m2

Ic3 chassis moment of

inertia in z direction

0.01 kg m2

Ip1 pendulum moment of

inertia in x direction

0.1 kg m2

Ip2 pendulum moment of inertia in y direction

0.1kg m2

Ip3 pendulum moment of

inertia in z direction

0.2 kg m2

rw wheel radius 0.25 m

lc distance between

wheel c.m. and pendulum c.m.

0.5 m

lcp

lp

g

distance between

chassis c.m. and pendulum pivot

distance between

pendulum pivot and c.m.

gravitational

acceleration

0.25 m

0.5 m

9.81 m s-2

2011 IEEE 5th International Conference on Robotics, Automation and Mechatronics (RAM) 55

the horizon increases in a nonlinear manner (0.52 s, 0.63 s,

0.8 s). Hence, a very small initial lateral lean angle is

expected to make the chassis and wheel fall down very

slowly.

C. Initial Condition:

This set of initial conditions is meant to observe the

gyroscopic precession in pendulum-balanced autonomous

unicycle. When the longitudinal lean angle is nonzero, as

shown in the previous results, the wheel will roll backward

with nonzero varying velocity. Because the wheel lateral lean

angle is also nonzero, there is a gravitationally induced torque

acting on the wheel and this torque will make the robot

precess.

The simulation results for this set of initial conditions are

shown in Fig. 5. Fig. 6 plots the trajectories made by the

unicycle.

As observed from the results, the robot moves backward

and turns because of the gyroscopic precession. An

interesting phenomenon is that after moving backward, the

unicycle changes direction and moves forward. It is also

observed that although the initial lateral angle is increased

linearly (100, 20

0, 30

0), the state and trajectory change in a

nonlinear manner.

VII. CONCLUSION

In this paper, dynamics model of pendulum-balanced

autonomous unicycle has been derived using Lagrange-Euler

formulation. The simulation study of the dynamics model

sheds light on the natural behavior of unicycle, which has

resemblance with the behavior of wheeled inverted pendulum

and Acrobot in the longitudinal and lateral directions

respectively. The simulation also confirms that yaw motion

can be generated by non-zero wheel speed and non-zero

lateral lean angle. The derived dynamics model is useful for

the design of control system for the unicycle. This topic is our

future work.

ACKNOWLEDGMENT

Authors thank Mechatronics and Automation Laboratory

at National University of Singapore for facility support.

Authors also thank anonymous reviewers for pointing out

some errors in the manuscript.

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Minnesota.

[3] John J. Craig, Introduction to Robotics: Mechanics and Control, 3rd

Edition, New Jersey: Pearson, 2005.

[4] Felix Grasser

Electronics, Vol. 49, No. 1, 2002.

[5] Austin Hughes, Electric Motors and Drives: Fundamentals, Types and

Applications, pp. 35. [6] Ryo Nakajima

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[9] Sang Yeol Seo, Seok Hee Kim, Se-Han Lee, Sung Hyun Han and Han

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[10] Zaiquan Sheng tions

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[11] Akira Shimada Two-

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[12] Mark W. Spong[13] David W. Vos

Adaptive Control of an Autonomous Unicycle: Theory and th Conference on Decision and

Control, Honolulu, Hawaii, 1990.

Fig. 5. Dynamic Responses 0 0, 200

and 300 and Zero Control Signals

Fig. 6. Trajectories 0 0, 200 and 300

and Zero Control Signals

56 2011 IEEE 5th International Conference on Robotics, Automation and Mechatronics (RAM)