Chinese Journal of Electronics
Vol.25, No.3, May 2016
Electromagnetic Force Balanced Single-Wheel
Robot∗
ZHU Xiaoqing1,2 , RUAN Xiaogang1 , CHEN Zhigang1 , WEI Ruoyan1 and XIAO Yao1
(1. College of Electronic Information and Control Engineering, Beijing University of Technology. Beijing 100124, China)
(2. Center for Intelligent Machines, McGill University, Montreal H3A 2K6, Canada)
Abstract — It is a challenging task to carry on the research concerning the lateral stabilization of Single wheel
robot (SWR). The lateral dynamics of earlier flywheel stabilizing SWR is derived to explain the shortcomings of
such a mechanism, and the recovery torque proportional
to the acceleration and moment of inertia of the flywheel
is proved. But as the motor being a speed server system,
it is difficult to control its acceleration to actuate the flywheel to provide recovery torque for SWR; and the moment of inertia of the flywheel must be large enough to
produce adequate recovery torque, which makes such a system rather cumbersome. We proposed a new mechanism
to solve the problem by introducing an electromagnetic
force. The proposed mechanism is described briefly, and
the dynamic analyses of such a new SWR is given, then
the stability analysis is also done. Three-dimensional (3D)
multi-body simulation, numerical simulation and physical
invented pendulum prototype experiments were conducted
to verify the proposed mechanism. The experiment results
verified that the proposed mechanism is feasible and have
some advantages over the flywheel balanced SWR.
Key words — Single-wheel robot (SWR); Electromagnetic force; Lateral stabilization mechanism; Dynamic
analyses; Stability analysis.
I. Introduction
Robot is a popular topic for researchers[1−2], with the
advantage of compact structure and excellent maneuverability, especially the ability to go through narrow spaces;
SWR (also called unicycle robot) has aroused the attention of researchers for the past two decades[3−7] . SWR
presents a novel example in the study of multi-body dynamics and control[8] in theory, and practically it plays
an important role in inspection tasks[9] for space exploration. For our daily life, the SWR can be used under special circumstance, where human beings are not proper to
perform task with danger or space limit. Moreover, SWR
can be used as an entertainment robot, bringing the children a lot of fun and knowledge as well. On the other
hand unicycle robot has only one wheel in contact with
the ground, its design and control is also a great challenge,
especially because lateral stability is a key research problem. Schoonwinkel[10] proposed a SWR inspired by human
balancing monocycle, a turntable was mounted on top of
the SWR chassis, which could be rotated with respect to
the vertical axis. MIT performed a similar research[9], but
the formulation of dynamics was different from which in
Ref.[8]. Subsequently, different kinds of SWR were built.
Gyrover was built by the Xu group[11,12], a spinning flywheel mounted inside the wheels was its unique feature,
and it explored a gyroscopic effect to keep its lateral balance. A SWR which used air for lateral balance was built
by Lee[13] et al. The Murata girl is another famous SWR
with a height of 50cm and is the smallest unicycle robot
among those reported. Its lateral-balancing mechanism is
an inertia wheel, but the formulation of its dynamics was
not done. The Ruan group proposed an inertia flywheel
SWR, which was similar to the mechanism of the Murata girl. Formulation of its dynamics was done by EulerLagrange equation[14] , and a control method was proposed
in Ref.[15]. Guo[16] et al. proposed a mass block sliding
control method for SWR by changing the position of the
mass block to realize lateral balance. The latest SWR was
built by the Buratowski group[17,18] , a weighted lever was
placed on the upper part of the wheel frame, which was
tilted to either side within the body of the wheel to provide counter balance. From all of the proposed mechanisms, lateral stabilization problem were solved by different ways. However there still exist some disadvantages,
for example using the gyroscopic effect will consume a lot
∗ Manuscript Received Sept. 11, 2014; Accepted Dec. 14, 2015. This work is supported by the National Natural Science Foundation of
China (No.61075110, No.61375086), National Basic Research Program of China (973 Program) (No.2012CB720000) and China Scholarship
Council Program (No.201306540008).
c 2016 Chinese Institute of Electronics. DOI:10.1049/cje.2016.05.008
Chinese Journal of Electronics
442
of energy; the initial wheel or turntable can’t provide adequate torque and the provided torque are difficult to control; exploiting air reaction required precise control of air;
the changing position of mass block or lever has problem
of tilting angle limitation. To build a single-wheel robot
that avoids the above-mentioned problems, we propose a
novel SWR by applying an electromagnetic force. The rest
of the paper is organized as follows: In Section II, the lateral dynamic of flywheel-stabilizing SWR is derived and
analyzed; The design of the mechanism of our SWR is explained in Section III; Section IV derives the new dynamic
model of proposed SWR; Stability analysis of our SWR
is given in Section V; 3D simulation experiment are performed in ADAMS integrated with MATLAB in Section
VI, A physical experiment is also included in this Section
to verify the feasibility of the electromagnetic balanced
mechanism; Discussion is given in Section VII; and Section VIII concludes the paper.
II. Flywheel-Stabilizing SWR
1. Design of the mechanism
The flywheel-stabilizing SWR is mainly composed of
a wheel, a body, a flywheel, two DC motors, etc., and its
lateral stabilization is realized by the rotation of the flywheel, as shown in Fig.1. The details of such a prototype
can be found in Ref.[19].
2016
denotes the length between the mass center of the SWR
and the bottom of the wheel. The total mass of SWR is
denoted by M , and its moment of inertia is JM . Because
we mainly focus on the lateral dynamics of the SWR, the
wheel motor is not taken into consideration, the rotor of
the flywheel motor is rotated relative to its stator at the
angular velocity Ω , and ω = χΩ , where χ is the reduction
coefficient. The mass of the flywheel is denoted by m, and
its moment of inertia about XF axis is Jm . JD denotes
the moment of inertia of the flywheel-driven motor about
its rotational axis. τm and τf denote the torque of the
motor and the torque caused by friction respectively.
3. Lagrange equation dynamic
Since it is rather complex to calculate direct recovery
torque produced by the inertia of the flywheel at the first
instance, the Lagrange equation is employed. In Ref.[20]
the dynamic of a pendulum with a fixed suspension point
was analyzed, similarly, the SWR can be regard as an
inverted pendulum, so with reference to Ref.[20], the lateral dynamic of flywheel balanced SWR is derived. We
first choose φ and α as two generalized coordinates. The
kinetic energy of SWR is given by
T = 0.5[(Jm + ml2 )φ̇2 + Jm (φ̇ + α̇)2 + JD (φ̇ + Ω )2 ] (1)
The potential energy of SWR is
V = (ml + M b)g cos φ
(2)
The two generalized forces are
Qφ = −τf ,
Qα = τm /χ
(3)
The virtual work of nonconservative forces is
δW = τm δα /χ − τf δφ
(4)
The Lagrange function is described by
L=T −V
(5)
According to the Lagrange equation
Fig. 1. Basic configuration of the
flywheel-stabilizing SWR
Fig. 2. Definition of system variables for the robot model
2. Coordinate frames and definition of variables
Refering to Fig.2, let Σ O {X, Y, Z} and ΣW {XW ,
YW , ZW } be the inertial frame whose x-y plane is anchored to the flat surface, and the body coordinate frame
whose origin is located at the center of the wheel, respectively. ΣO {X, Y, Z} is also the world coordinate. Let
ΣW {XW , YW , ZW } be the Cartesian coordinates of the
center of mass of the flywheel with respect to the inertial frame ΣO . φ is the rolling angle of SWR; the flywheel
is rotated about the XF axis by rotational angle α and
its angular velocity is ω. l denotes the length from the
mass center of flywheel to the bottom of the wheel and b
d ∂L
∂L
(
)−
= Qj
dt ∂ q̇j
∂qj
(j = 1, 2, · · · , n)
(6)
We have to calculate the following expressions
∂L
= (JM + ml2 )φ̇φ̈ + JD φ̇φ̈ + JD α̇φ̈/χ
∂φ
+Jm φ̇φ̈ + Jm α̇φ̈ + (ml + M b)g sin φ
(7)
d ∂L
) = (JM + ml2 + JD + Jm )φ̈ + (JD /χ + Jm )α̈ (8)
(
dt ∂ φ̇
∂L
= JD φ̇α̈/χ + JD α̇α̈/χ2 + Jm φ̇α̈ + Jm α̇α̈
∂α
d ∂L
(
) = (JD + Jm )φ̈ + (JD /χ2 + Jm )α̈
dt ∂ α̇
(9)
(10)
Electromagnetic Force Balanced Single-Wheel Robot
Then we neglect a small quantity of high order and
the obtained equation is
Jχφ̈ + (JD + Jm χ)α̈ = (ml + M b)gχ sin φ − χ × τf (11)
(JD + Jm χ)χφ̈ + (JD + Jm χ2 )α̈ = χ × τm
(12)
where J = JM + ml2 + Jm + JD .
Eq.(11) and (12) imply that the recovery torque by inertia of the flywheel is proportional to the angular acceleration of the flywheel, and the direction of the recovery
torque is opposite to its angular acceleration. It is common knowledge that the DC motor is an angular velocity
server system, but from Eq.(11) and (12), one has to control the motor’s acceleration. Besides, owing to the limit
of the speed of the motor, the acceleration cannot be done
at all times. Thus the recovery torque cannot be provided
by the flywheel at all times either. Moreover, Eq.(11) and
(12) imply that the recovery torque by the inertia of the
flywheel is proportional to the moment of inertia of the
flywheel. So, to produce large recovery torque to stabilize such SWR, the momentum of the flywheel must be
large, and then the momentum of SWR increases and thus
requires larger recovery torque. It becomes an unending
circle. The above-mentioned disadvantages have been verified by physical experimental results carried out on our
SWR, but owing to the limitation of the paper length we
omit describing them, and mainly focus on analyses of the
reason why such mechanism is not ideal for the SWR to
balance itself laterally, while proposing a new mechanism
by introducing an electromagnetic torque.
III. New Design of the Mechanism of the
Proposed SWR
To overcome the disadvantages of flywheel-stabilizing
SWR, we propose a new mechanism for lateral stabilization of SWR. Inspired by Amper force and the principle
of generator[21], we design a new prototype of SWR.
Our SWR mainly consists of a wheel driven a hub
motor, a coil winding frame to fix a set of coil winding
around it, two holders with permanent inserted driven by
two symmetrical DC motors respectively, three platforms
to place necessary units such as lithium battery, gyro, and
DSP control panel as shown in Fig.3.
In order to make the SWR self-stable, the whole system is designed symmetrically, especially the driven motor
to actuate the driving wheel is a hub motor rather than
a DC motor. The holders with permanent act as “rotor”,
spinning at high speed to provide the magnetic field. The
left parts together called the main body of the single wheel
robot, acting as “stator”. The two DC motors are placed
at the center of the circularly symmetric “rotor”, so as to
drive the “rotor” to spin around the “coil winding frame”.
443
The parameter details of the system are illustrated in Section VI. The recovery torque is produced by the electromagnetic induction between the “stator” and the “rotor”.
According to the formula of Lorentz force, to produce the
maximum force, the direction of current I should be perpendicular to the direction of magnetic field B. In order
to show the direction of current of the “stator” and the
magnetic field, a left hand is included in Fig.3, taking the
SWR lean to left side situation as an example: the direction of the magnetic field is depicted by the middle finger,
the direction of the current is pointing as the index finger, then the direction of the force is along the thumb.
With brushes, the directions of armature current can be
changed as desired. The upward and bottom part of the
coil winding are useful for recovery, but the side parts are
useless, owing to the direction of the current at the two
opposite sides of the coil, the Lorentz forces acting on the
two sides counter each other. The “rotor” forms a spinning
magnetic field, and when there is current passing through
the “stator”, an electromagnetic force occurs between the
“rotor” and the “stator”. Since the “rotor” also driven by
torque provided by DC motors, which will produce extra torque to compensate such “disturbance”, thus there
is little influence to its spinning. But for the “stator”, it
will rotate around the roll axis, and that torque is exactly
what we need to control the lateral balance of our SWR.
The electromagnetic force is approximately proportional
to the “stator” current, so it is rather simple for controlling. By controlling the direction and intensity of “stator”
current, the SWR can stabilize itself laterally. The driving wheel is driven by a hub motor, by controlling the
rotation of the driving wheel, the longitudinal dynamic
stability can maintain. For the yaw control, we control
the single wheel robot to lean a specific angle to one side
to produce a centripetal force, such centripetal force will
let our robot to yaw. By the control of the roll angle and
the rotation velocity of our single wheel robot, it would
turn as desired.
IV. Dynamic Analyses
Here we derive the new lateral dynamic of the proposed SWR, as shown in Figs.4–6, because the proposed recovery torque can be rather easily calculated and
Newton-Euler equation is employed. In this section C denotes the mass centre point of SWR, its roll angle is denoted by φ, its pitch angle is denoted by θ, and the yaw
angle is denoted by ψ. The rolling angle of the driving
wheel is β. B denotes the magnetic field intensity, the
“stator” current intensity is denoted by I; L1 , L2 , L3 denotes the length from the upward part of the coil winding,
the downward part of the coil winding, the mass centre
to the contact point between the wheel and the ground,
respectively. The radius of the driving wheel is denoted
Chinese Journal of Electronics
444
by R; m1 , m2 , m3 denotes the mass of the driving wheel,
the mass of the “rotor”, the mass of the left parts except
for the driving wheel and the “rotor”, respectively. The
details of the parameters are given in Table 1. Owing to
the interference of the wheel, the “rotor” could not rotate
freely as a whole, we use two armatures symmetrically.
Half of the effective length between the holders and the
coil winding is denoted by L4 . F denotes the electromagnetic force and generally F = ILB, where L generally
denotes the effective length of a conductor in magnetic
field.
Table 1. Details of the designed SWR parameters
Symbol
L1
L2
L3
L4 m1 m2
m3
R
Parameter 50cm 10cm 30cm 10cm 1kg 0.5kg 4.5kg 4cm
2016
the anti-torque from the driving wheel. Similarly we obtain the longitudinal dynamic equation as follows:
τβ R − (m2 + m3 )g sin θL3 = Jθ θ̈
(14)
Where Jθ is the inertial moment of the SWR about the
pitch axis, τβ is the actuate torque from the hub motor.
For the yaw dynamic, even though there is no direct
yaw torque, we can still make our SWR turn by leaning to
one side, according to the centripetal formula we obtain
Eq.(15)
(m1 + m2 + m3 )gL3 tan φ = Jψ ψ̈
(15)
Where Jψ is the inertial moment of the SWR about the
yaw axis.
For the driving wheel dynamic, there are two torques
applied on it, that are the actuate torque from the hub
motor τβ , and the friction torque from the ground τf g.
Then its dynamic equation comes:
τβ − τf g = JW β̈
(16)
Where JW is the inertial moment of the driving wheel
about its rolling axis.
For the holders dynamic, the actuate torque from the
DC motor and the electromagnetic torque applied on it,
so its dynamic equation is as follows:
Fig. 3. Proposed electromagnetic balanced SWR
Fig. 4. Front view of dynamics
analyses
2τm − F (L1 − L3 ) = Jh ϕ̈
(17)
Where Jh is the inertial moment of the holder, ϕ is the angle of the holder. The degree of freedom of the SWR is 5,
and we choose independent coordinate q = [φ θ ψ β ϕ]T ,
then the whole system dynamic equation can be written
as follows:
M (q)q̈ + C(q, q̇)q̇ + N (q) = τ
(18)
Where M (q) ∈ R5×5 , C(q, q̇) ∈ R5×5 , N (q) ∈ R5×1 denote the generalized inertial matrix, convective inertial
term, nonlinear term, and control torque, respectively.
The control law is chosen as:
Fig. 5. Side view of dynamics
analyses
Fig. 6. Isometric view of SWR
τ = Kp e + Kd ė + N̂ (q)
As for the main body of SWR, two external forces are
applied on it, which are gravity and electromagnetic force,
and the SWR rotates about its rolling axle. According to
the angular theory of momentum, we obtain Eq.(15) on
the assumption that there is no lateral slippage between
the wheel and the ground.
F L1 + F L2 − ((m2 + m3 )g sin φ)L3 = Jφ φ̈
(19)
Where e is the tracking error, e = qd − q, qd is the desired
output, N̂ (q) is the estimate value of the gravitational
moment satisfies
M (q)q̈d + C(q, q̇)q̇d + N̂ (q) = 0
(20)
V. Stability Analysis
(13)
Where Jφ is the inertial moment of the SWR about the
roll axis.
For the longitudinal dynamic, it is obvious that two
torque affect the pitch angle θ, that are the gravity and
The dynamic equation is rewritten by misusing
Eq.(20) from Eq.(18) and substituting Eq.(19) into the
result:
M (q)(q̈d − q̈) + C(q, q̇)(q̇d q̇)+
(21)
Kp e + Kd ė + N̂ (q) − N (q) = 0
Electromagnetic Force Balanced Single-Wheel Robot
When the measurement devices can provide the numeric
results of the status of SWR, the estimate nonlinear term
N̂ (q) → N (q), then Eq.(21) comes:
M (q)ë + C(q, q̇)ė + Kp e = −Kd ė
(22)
The Lyapunov function is chosen by
VL = 0.5ėT M (q)ė + 0.5ėT Kp e
(23)
Owing to the positive definiteness of M (q) and Kp , VL is
global positive definite, then we evaluate its time derivative.
V̇L = ėT M (q)ë + 0.5ėT Ṁ (q) ė + ėT Kp e
(24)
Since Ṁ (q) − 2C(q, q̇) is skew symmetric, we have
ėT Ṁ (q) ė = 2ėT C(q, q̇)ė
(25)
then,
V̇L = ėT Ṁ (q) ë + ėT C(q, q̇)ė + ėT Kp e
= ėT (M (q)ë + C(q, q̇)ė + Kp e) ≤ 0
(26)
It is obvious that V̇L is semi-negative definite while Kd
is positive definite. So when V̇L ≡ 0, we have ė ≡ 0 and
ë ≡ 0. Substitute the above result into Eq.(26), we have
Kp e = 0, since Kp is not singular, then e = 0. According to LaSalle theorem, we know that (e, ė) = (0, 0) is
the global asymptotic stable equilibrium. In other words,
given arbitrary initial condition (q0 , qd ), we can always
have q → qd , q̇ → 0, which implies that the system
asymptotic stable.
VI. Experiments
As shown in Fig.7, the control system consists four
main parts, the control plant is our SWR, the sensors
such as gyro and accelerometer measuring the attitude of
SWR and feedback to the controller, and then the controller gives out the control signal to actuator to make the
plant to reach desired goal.
Fig. 7. Schematic of control system
1. Lateral control
Firstly we focus on the lateral balance by the proposed electromagnetic force. The gyro acts as a sensor to
measure the roll angle and its velocity and sends measuring results to the controller. The controller calculates the
445
control output current through the “stator”, the electromagnetic force received from the “rotor” applied on the
“stator”, and make it to reach vertical position. We import our designed unicycle to the ADAMS environment.
Then we set the gravity direction opposites of y direction
in the world coordinate, and the initial lean angle is 20◦ .
To make comparison, step one we do not apply a control
torque (setting the current zero). Then the robot just falls
down freely, and the change of roll angle is shown in Fig.8.
From Fig.8, we can see that the robot’s roll angle changes
from 20◦ to 340◦. The result is consistent with physical
experience. But the desired goal of our control is to stabilize the robot to the vertical position, that is to set the
roll angle and its velocity to zero. So step two we add the
control current to produce electromagnetic force, and see
what happens. In order to fulfill our goal, the proper control input should be calculated. Then we using ADMAS
integrated with MATLAB simulation as our testing tool.
The control input is calculated by MATLAB, and the 3D
simulation is performed by ADAMS. In MATLAB, a PD
controller is designed, and by tuning the selected parameters we finally stabilize the robot, as shown in Fig.9.
Fig. 8. Roll angle without control input
Fig. 9. Roll angle with control
input
As shown in Fig.9, the roll angle changed from 20◦
to 0◦ smoothly, then it is verified that the proposed
electromagnetic balanced mechanism works for lateral
stabilization. During the stabilization process the control
signal is shown in Fig.10, by control the stator current,
the desired recovery torque can be obtained.
In order to make the proposed electromagnetic balanced mechanism more convincing, a physical experiment
has been carried out on an electromagnetic balanced invented pendulum. The pendulum leaned to right side at
angle of 10◦ as shown in Fig.11(a), and then the “rotor”
permanent magnet rotated as shown in Fig.11(b) and (c),
finally the pendulum stabilized at its vertical position as
shown in Fig.11(d). The changing process of the roll angel
of pendulum is shown in Fig.12.
2. Longitudinal control
Secondly, we focus on the longitudinal stabilization
and the velocity control of the driving wheel. The initial pitch angle is also 20◦ , and the longitudinal stabilization process started after the lateral stabilization finished.
Chinese Journal of Electronics
446
The result of pitch angle stabilization process is shown in
Fig.13. From Fig.13, we see that the pitch angle changed
from 20◦ to 0◦ smoothly, and reached the stabilization at
0.7s. Then we set the driving wheel velocity to be 10◦ /s,
its result is also depicted in Fig.13. From Fig.13, it is
shown that the driving wheel move backward to realize
the longitudinal stabilization, then speeded up forward to
track the setting velocity.
Fig. 10. Control signal in experiment
Fig. 11. Stabilizing process of
inverted pendulum
2016
To verify the proposed electromagnetic force balanced
SWR is better than flywheel balanced one, we conduct
a comparison experiment. The initial roll angle was set
to be 20◦ , and the lateral balance results are shown in
Fig.17. It reveals in Fig.17 that both methods can realize the lateral balance, but the proposed electromagnetic
force balanced SWR stabilized itself more quickly, with
less overshot and shorter setting time. Then we tried to
find out the maximum initial roll angle of the two balanced method, the results showed that when the initial
roll angle is greater than 25◦ , the flywheel balanced SWR
failed to stabilized itself, while the electromagnetic force
balanced SWR can still realize lateral balance when the
roll angle is 50◦ . This is because both the flywheel’s moment of inertia and the speed of motor are limited. For
the electromagnetic force balanced SWR there is no such
problem, since the SWR can have greater recovery torque
by increasing the intensity of the current. Moreover we
can also provide greater recovery torque by adding more
circles of wires while the mass increased only a little. To
put it more clearly, Table 2 summarize the performances
comparison between the two balanced methods.
Table 2. Performances comparison
Method
Maximum angle Overshot Setting time Mass
Flywheel
25◦
80%
1s
20kg
balanced
Electromagnetic
50◦
30%
0.2s
5kg
balanced
Fig. 12. Roll angel changing
process in experiment
Fig. 13. Longitudinal control
results
3. Yaw control
As mentioned before the yaw control is fulfilled by
leaning the SWR to a proper angle and controlling the
driving wheel at certain velocity. the yaw control result
was shown in Fig.14, the SWR moving in a circle with
radius being 5m. During the yaw process, the roll angle
of SWR is 30◦ as shown in Fig.15, while the velocity of
driving wheel is 60◦ /s as shown in Fig.16.
Fig. 16. Angular velocity of
driving wheel during
yaw process
Fig. 17. Lateral balance comparison between two
different methods
VIII. Conclusions
Fig. 14. Projection of mass centre on the ground during yaw process
Fig. 15. Roll angle of SWR during yaw process
VII. Discussion
To conclude, in this paper the lateral dynamics of earlier flywheel balanced SWR is analyzed firstly, which reveals that such mechanism isn’t an ideal one. Then we
proposed a new lateral stabilization mechanism for SWR
to overcome the disadvantage of flywheel balancing. In
the proposed mechanism, we introduce electromagnetic
force to balance SWR laterally. By controlling the current intensity, we directly control the recovery torque.
In contrast to fly-wheel stabilizing SWR, whose recovery
torque is proposed to accelerate the flywheel, the proposed mechanism is easier to control and the recovery
torque is more convenient to understand and calculate,
Electromagnetic Force Balanced Single-Wheel Robot
the maximum roll angle increased to 50◦ while the mass
decreased greatly. But a problem also exists in our proposed mechanism that is the Electromagnetic interference
(EMI). Owing to the recovery torque is produced by magnetic field, then the electro component such as gyro sensor and control panel may be disturbed. So the necessary
electromagnetic shielding should be considered, and in our
physical prototype SWR electromagnetic shielding material will be equipped. Moreover, the proposed lateral stabilizing mechanism for SWR can be used in other underactuated systems. For example, the attitude adjustment
of spacecraft, satellite, missile, helicopter and so on. The
proposed mechanism provides a torque generation method
for under-actuated system. The complete physical SWR
prototype building and testing in the really word and so
on are also urgent work.
References
[1] G. Yang, R. Dong and H. Wu, “Viewpoint optimization using genetic algorithm for flying robot inspection of electricity
transmission tower equipment”, Chinese Journal of Electronics, Vol.23, No.2, pp.426–431, 2014.
[2] Z. Cao, L. Yin, Y. Fu and J. Zhang, “Visual servo stabilization of nonholonomic mobile robot based on epipolar geometry
and 1D trifocal tensor”, Chinese Journal of Electronics, Vol.22,
No.4, pp.729–734, 2013.
[3] A. Ajorlou, M.M. Asadi, A.G. Aghdam and S. Blouin, “A consensus control strategy for unicycles in the presence of disturbances”, 2013 American Control Conference, Proceedings
of the American Control Conference, Washingtion DC, USA,
pp.4039–4043, 2013.
[4] K. Amar and S. Mohamed, “Visual servo stabilization of nonholonomic mobile robot based on epipolar geometry and 1D
trifocal tensor”, Chinese Journal of Electronics, Vol.22, No.4,
pp.729–734, 2013.
[5] K.D. Do, “Bounded controllers for global path tracking control of unicycle-type mobile robots”, Robotics and Autonomous
Systems, Vol.22, No.4, pp.729–734, 2013.
[6] J. Lee, S. Han and J. Lee, “Decoupled dynamic control for pitch
and roll axes of the unicycle robot”, IEEE Transactions on Industrial Electronics, Vol.60, No.9, pp.3814–3822, 2013.
[7] J.H. Park and S. Jung, “Development and control of a singlewheel robot: Practical Mechatronics approach”, Mechatronics,
Vol.23, No.6, pp.594–606, 2013.
[8] D.W. Vos A.H. Flotow, “Dynamics and nonlinear adaptive control of an autonomous unicycle theory and experiment”, Proceedings of IEEE CDC’29, Honolulu, pp.182–187, 1990.
[9] E.D. Ferreira, S.J. Tsai and C.J. Paredis, “Control of the Gyrover: A single-wheel gyro-scopically stabilized robot”, Adv
Robotics, Vol.14, No.6, pp.459–475, 2000.
[10] A. Schoonwinkel, “Design and test of a computer stabilized unicycle”, Ph.D. Thesis, Stanford University, California, 1987.
[11] Y.S. Xu, S.K. Au, G.C. Nandy and H.B. Brown, “Analysis of
actuation and dynamic balancing for a single-wheel robot”, Pro-
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
447
ceedings of IEEE-RSJ ICIRS, Victoria, Canada, pp.1789–1794,
2000.
Y.S. Xu and S.K. Au, “Stabilization and path following of a single wheel robot”, IEEE/ASME Transactions on Mechatronics,
Vol.9, No.2, pp.407–419, 2004.
J.H. Lee, H.J. Shin and S.J. Lee, “Novel air blowing control for
balancing a unicycle robot”, Proceedings of IEEE-RJS ICIRS,
Taipei, pp.2529–2530, 2010.
X.G. Ruan, J.M. Hu and Q.Y. Wang, “Modeling with eulerlagrange equation and cybernetical analysis for a unicycle
robot”, Proceedings of ICICTA’2, Changsha, China, pp.108–
111, 2009.
X.G. Ruan, Q.Y. Wang and N.G. Yu, “Dual-loop adaptive decoupling control for single wheeled robot based on neural PID
controller”, Proceedings of ICARCV’11, Singapore, pp.207–212,
2010.
L. Guo, Q.C. Liao and S.M. Wei, “Dynamical modeling of unicycle robot and nonlinear control”, Journal of System Simulation,
Vol.21, No.9, pp.2730–2736,2009.
P. Cieslak, T. Buratowski, T. Uhl, et al., “The mono-wheel
robot with dynamic stabilization”, Robotics and Autonomous
Systems, Vol.59, No.9, pp.611–619, 2011.
T. Buratowski, P. Cieslak, M. GIERGIEL, et al., “A selfstabilising multipurpose single-wheel robot”, Journal of Theoretical and Applied Mechanics, Vol.50, No.1, pp.99–118, 2012.
X.G. Ruan, Q.Y. Wang and J.M. Hu, “A single wheel robot system and its control method”, US Patent, US 12774578, 201005-05.
A.V. Beznos, A.A. Grishin, A.V. Lenskii, D.E. Ohotsimskii, et
al., “A flywheel use-based control for a pendulum with a fixed
suspension point”, Proceedings of IEEE-RSJ ICIRSThe Journal of Computer and System Sciences International, Vol.43,
No.1, pp.27–38, 2004.
X. Zhu, X. Ruan, R. Sun, R. Wei and X. Wang, “Study on a device of torque generation based on electromagnetic effect”, Acta
Electronica Sinica, Vol.42, No.2, pp.335–340, 2014. (in Chinese)
ZHU Xiaoqing was born in Jiangsu
Province, China. In 2010, he received
the B.E. degree in automation from Nanjing University of Information Science and
Technology. He is now a Ph.D. candidate in the College of Electronic Information and Control Engineering at Beijing University of Technology, he is also
a joint Ph.D. candidate in the Center for
Intelligent Machines at McGill University.
His research interests include robotic design and control. (Email:
[email protected];[email protected])
RUAN Xiaogang
was born in
Sichun Province, China. He received Ph.D.
degree from Zhejiang University in 1992,
Hangzhou, China. Now he is a professor of
the Beijing University of Technology, and
he is also as a director of IAIR (Institute
of Artificial Intelligent and Robots). His
research interests include Automatic Control, Artificial Intelligence, and Intelligent
Robot. (Email: [email protected])