Using a Flywheel to Stabilize a Self-Balancing Bicycle
I. I. SILLER-ALCALÁ, J. U. LICEAGA-CASTRO, R. A. ALCÁNTARA-RAMÍREZ
S. CALZADILLA-AYALA
Departamento de Electrónica, Área de Control de Procesos,
Universidad Autónoma Metropolitana,
Av. San Pablo No. 180, Col. Reynosa Tamaulipas, Del. Azcapotzalco, C. P.0200, México D.F.,
MÉXICO
Abstract: - The designs of two linear control systems approach to stabilize the balance of an unmanned bicycle
system are presented. Both approaches are based on the use of a reaction wheel or flywheel to balance the
bicycle. The two linear control approaches, based on the linearization of a nonlinear model obtained using
Lagrange formalism, are the classic linear controllers, PID and State Feedback control. The performance of
both controllers is verified by digital simulation and real-time experimental results.
Key-Words: - Reaction Wheel, Balancing control, State feedback controller, Bicycle robot, Classic Control,
PID.
Received: March 21, 2023. Revised: January 20, 2024. Accepted: March 9, 2024. Published: April 16, 2024.
1 Introduction
In the last two decades, scientists have focused on
achieving the goal of balancing a two-wheeled
bicycle. The problem of unmanned balancing the
bicycle when it is moving at a certain speed or zero
speed is very attractive to systems control
researchers, [1], [2], [3], because it presents three
interesting problems for this community: the system
is unstable, a zero at the origin, and the presence of
disturbances. To solve this problem, authors usually
use a robust control algorithm and mechanical
devices, such as flywheels or gyroscopes, to add
them to the bicycle to stabilize it at zero speed
wheel, [4], [5], [6], [7], [8], [9], [10], [11], [12],
[13], [14], [15], [16]. Although, in this paper, the
flywheel could be assumed as a basic gyroscope
working only on an axis of rotation, it may be a first
step in the use of full gyroscopes, not only as rigid
body position sensors but for the stabilization of
mechanical vehicles. For instance, gyroscopes could
be used as satellite position controllers. However, it
requires first designing, controllers for the
gyroscopes themselves. Unfortunately, this is not a
simple task because gyroscopes are highly nonlinear
multivariable systems, [17] and [18].
In this work, the prototype of an autonomous
bicycle is stabilized at zero speed, that is, when the
bicycle is not moving; so, the bicycle is treated as an
inverted pendulum, neglecting the induced torques
generated by maneuvering the bicycle handlebar.
Nonetheless, these torques could be dumped into the
bicycle perturbations. The prototype, the basis of
this article, is one of three mechatronic systems that
can be assembled using a kit called “Arduino
Engineering Kit Rev2” [19], developed by
Mathworks and Arduino. The two control strategies
selected to stabilize the bicycle are PID and state
feedback control. These controllers were chosen due
to their easy implementation and because they are
well suited for an engineering context. Additionally,
the State Feedback controller is a natural option
because the prototype allows access to all process
states. In this context, the article is divided into the
following sections: In Section 2, the dynamic model
of the self-balancing bicycle based on the Lagrange
formalism, which also includes the actuator electric
DC motor, is presented. In Section 3 the design of
the PID controller and the State Feedback controller
are shown. In Section 4, the performance of the
designed controllers is verified through simulations
and the implementation of the controller. Finally, in
Section 5, the conclusion of the research is
presented.
2 Mathematical Model of a Self-
balancing Bicycle
In this section, the model of the unmanned bicycle
system is presented. The development is based on
the principle of the inertial wheel pendulum to
obtain simplified Lagrange dynamic equations.
Subsequently, the nonlinear differential equations
are linearized at an equilibrium point. These linear
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DOI: 10.37394/23203.2024.19.8
I. I. Siller-Alcalá, J. U. Liceaga-Castro,
R. A. Alcántara-Ramírez, S. Calzadilla-Ayala
E-ISSN: 2224-2856
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differential equations are the basis for carrying out
the design of the two linear controller approaches.
The Bicycle to be stabilized is shown in Figure 1.
Fig. 1: Complete robotic bicycle
Fig. 2: Front view of the robotic bicycle and
reference coordinate system
To obtain the equations describing the dynamics
of the system, a behavior like an inverted pendulum
with a flywheel is assumed.
The variables and coordinate system that were
used are shown in Figure 2, from the front view of
the robotic bicycle, where:
θ: It is the angle of inclination of the robotic
bicycle concerning the vertical axis.
ω: It is the angular velocity of the flywheel or
reaction wheel.
A: It is the axis of rotation of the inverted
pendulum, in this case, the robotic bicycle.
B: It is the center of mass of the robotic bicycle.
Table 1. Dynamic Parameters
First, the model of the electric DC motor actuator
is developed. Using the basic circuit that represents
a DC motor in Figure 3, the equation below is
obtained:
(1)
Fig. 3: Diagram of DC motor
Where, is the motor supply voltage, and
are the armature coil resistance and inductance,
respectively; is the armature current, and is
the counter-electromotive force, given by the
following equation:
(2)
Where is the rotor speed, and
therefore, the flywheel angular velocity and, is
the electromotive constant. In DC motors, it is
generally assumed that the generated torque is
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proportional to the current provided, this
relationship is clarified in the following equation:
(3)
where is the torque generated by the motor
and is the torque constant of the motor, the
following equation is obtained by substituting (1)
into (3).
(4)
It is considered that the term corresponding to
the inductance can be neglected, since its value is
much lower than that of the resistance ,
considering this and substituting (2) into (4), the
following equation is obtained:
(5)
By using the Lagrange formalism, the following
equations that describe the mechanical dynamics of
the system are given.
The torque about a given axis of rotation is the
sum of all the torques that act in the system on this
axis, and is defined as:
(6)
where,
: It is the net torque applied to the axis of
rotation.
: These are the torques applied on the axis of
rotation.
: Moment of inertia of the system.
: Angular acceleration of the system
If no external torque acts on the robotic bicycle,
other than that due to gravitational acceleration, we
have two torques that act on the bicycle:
: Torque due to gravitational acceleration.
: Torque due to the flywheel.
Therefore, the net torque on the bicycle with
respect to the axis of rotation A, Figure 2, results in:
(7)
Where the bicycle’s moment of inertia is . The
torque, provided by the flywheel which in turn
is generated by the DC motor, must be of equal
magnitude but in the opposite direction to the torque
, in such a way that the angular momentum of
the robotic bicycle is conserved. Expanding the
terms of equation (7) we obtain:
(8)
The torque which is provided by the flywheel
can be described by the following equation:
(9)
Subsequently, adding equations (5) to (8) and (9),
we obtain:
(10)
(11)
Defining states = ω, the non-
linear state space representation is given by:
(12)
Linearizing the nonlinear equations (12), [20], at
the equilibrium point
; , the state space
representation, where:
(13)
with is given by:
(14)
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g
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1 2 3
2 2 2 2
1 2 3
3
333
1 2 3
0
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; 1 0 0 ;;
u
X
fff f
x x x u
f f f f
A B u C
x x x u
f
fff
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x x x
uV
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Substituting the values of the parameters
provided by the prototype maker, Table 1, we obtain
the linear state space model:
(15)
3 Control Design
In this section, the design of two linear controllers is
presented. The PID and state feedback controllers
were chosen. Both controllers were chosen mainly
because of their well-proven effectiveness, for the
PID controller, and because there is complete access
to the state vector, in the case of State Feedback.
Also, both controllers are easy to implement and are
well suited for engineering contexts.
3.1 PID Control Design
The Transfer Function associated with the linear
state space model of equation (15) is as follows:
(16)
Rewriting the above equation:
(17)
The system is unstable, with a pole at 9.818, two
stable poles {-9.838, -0.6412}, and a zero at the
origin. The root locus of is presented in Figure
4.
Fig. 4: Root locus of G(s)
One of the closed-loop poles lies on the right half
of the S plane, making the system unstable. In
addition, there is zero at the origin, which becomes
an uncomfortable issue because it represents a
derivative behavior in the system and can lead to
stabilizing the system using a PID controller.
From the observations made on the graph of the
root locus of the system, the following transfer
function was designed for the bicycle's PID
controller:
(18)
The pole at the origin in C(s) has the purpose of
removing the origin zero in G(s), while the complex
conjugate zeros must be responsible for determining
the trajectory of the root locus so that, with an
appropriate gain, all the poles will lie on the left-
hand plane. The transfer function of the open loop
system is given by:
(19)
The roots of the closed-loop system can be
analyzed by using the root locus.
Fig. 5: Root locus of
As seen in the graph in Figure 5, when adding the
controller, the roots locus changes according to
plan, with a gain greater than 11.7, the closed-loop
poles will lie in the left half-plane, so the system
becomes closed-loop stable. However, the stable
closed-loop pole close to the origin, which becomes
the dominant pole, may affect the performance of
the control system due to its very large steady-state
time. Therefore, redesigning the controller to reduce
the steady-state time of the dominant poles results
in:
(20)
With this new controller design, the root locus is
shown in the following Figure 6.
11.024
() 32
( 0.6608 96.5795 61.9366)
s
G s C sI A B s s s
1.024
() 9.818 9.838 0.6412
s
Gs s s s
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Fig. 6: Root locus with the new controller
In this graph, it is observed that the pole
introduced by the controller at -1000 causes one of
the poles closest to the origin to move away from
the origin and, finally, around -500 together with the
high-frequency pole, both poles separate from the
real axis symmetrically, resulting in a pair of non-
dominant complex poles.
Figure 7 shows a close-up of the trajectories of
the dominant poles that are close to the origin.
Fig. 7: Root locus close-up with new controller
Finally, a gain of 28 has been chosen so that the
poles lie on the left half-plane, thereby ensuring the
closed-loop stability of the system.
The Nyquist plot and Bode diagrams of
, shown in Figure 8 and Figure 9, show
that robustness is also guaranteed as the stability
margins result in and
Fig. 8: Bode Diagrams of
Fig. 9: Nyquist plot of
To assess the performance of the control system
based on the controller of equation (20), the result of
a digital simulation based on the nonlinear model of
equations (10) and (11), assuming an obvious
reference signal of and constant disturbances of
with a frequency of is presented in
Figure 10. This figure shows that controller (20)
stabilizes the system with excellent disturbance
rejection, achieving the objective.
Fig. 10: Output signal corresponding to the
inclination angle .
( ) ( )C s G s
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Unfortunately, it was not possible to carry out a
real-time implementation using the controller of
equation (20) as the control system became
unstable. This is due to the required pole/zero
cancellation at the origin, a cancellation that cannot
be guaranteed in practice. Furthermore, as shown in
Figure 4, if the controller does not cancel zero at the
origin using an integrator or a pole at the origin, an
unstable controller would be necessary to break the
direct path between the unstable pole at 9.818 and
zero at the origin. A possible alternative to an
unstable controller is to combine a State Feedback
controller with a PI controller, as depicted in Figure
11. The State Feedback controller will stabilize the
system, while the PI controller will ensure
performance.
Fig. 11: PI and State Feedback Control System
3.2 Pole Placement State Feedback Control
Design
It is well-known that the stability and control
performance of a closed-loop system depends on its
pole locations. In this section, the pole placement
method will be used to place the poles of the closed-
loop system in the desired positions by state
feedback. To achieve this, the sufficient and
necessary condition for the existence of the state
feedback controller is that the system must be
controllable.
To know if this system, represented by the state
equations (14), is controllable, it is necessary to
check that the controllability matrix is full range.
For this system, the matrix is defined as:
(21)
The rank of the controllability matrix is equal to the
number of linearly independent rows or columns,
therefore, the matrix is full rank, if its
determinant is different from zero.
The controllability matrix of this system
= (22)
The determinant is given by:
(23)
Since the determinant of is not zero its rank is
equal to 3, equal to the order of the system, so the
system is controllable and, therefore, state feedback
control exists.
The system dynamics given by (15) are used for
the design of the linear controllers as follows.
Let the control given by:
(24)
Where is the control signal, is the reference
signal, and is the state feedback gain vector.
The closed loop system is as follows:
(25)
Rewriting
(26)
where , and the input is the reference
The state feedback gain vector should be chosen
in such a way that eigenvalues are placed on the
desired closed-loop poles.
The position of the closed-loop poles was chosen
according to the following equation:
(27)
That is, with a dominant overdamped closed loop
at -1. The two non-dominant poles were chosen
trying not to obtain excessive high state feedback
gains, as this may render saturation on the system
input signal. Therefore, solving equation (28):
(28)
The state feedback gain vector obtained is
given by:
(29)
In this section, the simulations are presented to
show the efficiency of the controller. The following
Simulink program simulates the state feedback
control system composed by
32
det det 16 65 50
LC
sI A sI sA Bk ss
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(30)
The following figures show the graphs obtained
by simulation using the Simulink program in Figure
12. As seen in the graphs of Figure 13, Figure 14,
and Figure 15, the simulations of the state feedback
control system show good performance. It is also
important to notice, from Figure 13 and Figure 14,
that title angle has an overdamped behavior due
to the chosen closed loop poles. Even though the
initial conditions were set far from ideal positions.
The initial conditions are set in the integrators of the
Simulink program in Figure 12. Also, Figure 13 and
Figure 14 show that the controller could take the
output to , maintaining bicycle balance and
stability, in approximately 4 sec. due to the selected
closed loop dominant pole at -1.
Fig. 12: Simulink program of the bicycle state
feedback control system
Fig. 13: System output that corresponds to the
system inclination angle
In Figure 15, it is observed that the control signal,
which represents the voltage that would be applied
to the DC motor of the flywheel, has a very large
magnitude. This is because the states begin with
values far from ideals; that is, to achieve stability
and recovery of the bicycle balance does not require
a high control effort. Additionally, and as expected,
when the bicycle recovers position , the
control signal so the flywheel velocity tends
to zero, Figure 15.
Fig. 14: system states: (orange line),
(blue line), (red line)
Fig. 15: State feedback control signal
To assess the performance of the state feedback
controller, in the presence of output angle
variations, with magnitudes that can occur in the
real model, an input signal disturbance was added to
the diagram in Figure 12 in the state , which
corresponds to the tilt angle of the system, as shown
in Figure 16.
The disturbance is a pulse with an amplitude of
0.1745, which would correspond to an inclination of
10°. Furthermore, it occurs 6 seconds after the
simulation starts and has a duration of 0.1 seconds.
0o
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Below are the graphs of the simulation carried out.
With the results of this second simulation, shown
in Figure 17, Figure 18 and Figure 19, it is
confirmed that the feedback system behaves
correctly, as it was able to reject the perturbation
maintaining bicycle verticality. The effort made by
the controller, which is observed in Figure 19, is
because the disturbance signal is square, but the
disturbances or changes in the inclination angle of
the real system are not so abrupt, so it was
considered that the feedback loop behaves correctly.
Fig. 16: Simulink diagram of the bicycle state
feedback control system with perturbations in state
Fig. 17: system output that corresponds to the
system inclination angle in the presence of
perturbation
Fig. 18: system states: x1 = θ (orange line), x2 = θ
(blue line), x3 = ω (red line) in the presence of
perturbation
Fig. 19: State feedback control signal in the
presence of perturbation
3.3 State Feedback plus PI Control Design
As mentioned above, an alternative to avoid the
need to cancel the zero at the origin of the system by
using an integrator or a pole at zero in the controller
to break the direct path between the unstable pole
and the zero, as shown in Figure 4, so that the
system is closed loop stable is the combining use of
a State Feedback controller with a PI controller.
Otherwise, an unstable controller is required.
Following the strategy depicted in Figure 11, the
results of a Simulink digital simulations of a control
system using the estate feedback vector of equation
(29) together with the PI controller of equation (31),
are shown in Figure 20, Figure 21 and Figure 22.
(31)
10 1
()
PI
Cs
ss
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To better assess the performance of this
approach, two perturbation signals were added: a
perturbation in the form of a square signal of
, equivalent to , with a frequency of
on position , and a random
signal with an amplitude of as sensor noise of
.
From Figure 20, it is clear that the combination of
the State Feedback controller and the PI controller
presents an excellent response. Moreover, as shown
in Figure 21 and Figure 22, the flywheel velocity
and control signal are within physical limits. It
should be noted that the peak values in the control
signal are due to the sudden high perturbations
affecting . Also, the effects of the sensor
noise were practically eliminated.
Fig. 20: Tilt or inclination angle
Fig. 21: Tilt angle rate (blue line), ω (red line)
Fig. 22: Control signal
4 Implementation of the State
Feedback Controller
The software configuration shown in Figure 23
provided by [19], for the operation of the bicycle
has the advantage that the state feedback controller
designed in this work is easily implemented. The
software configuration was developed using
Matlab's Simulink, with a sampling time equal to
T=0.01.
Fig. 23: Implementation of the state feedback
controller
It should be noted that the state feedback control
was designed based on a continuous state space
representation of the system. However
controversial, it is possible, in many cases, to
implement digitally a controller, designed in
continuous time, provided a good sample time is
0.05
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selected, ensuring all system modes are properly
sampled, and the digitalized controller does not
differ significantly from its continuous counterpart
in a range of frequencies well above the control
system bandwidth. In this case, the sample period
was chosen to satisfy a sampling frequency of
628.32 rad/sec, well above the control system
bandwidth of 1 rad/sec.
Figure 24 shows the graphics of the results:
inclination angle , inclination angle rate , angular
velocity of the flywheel ω, and the control signal
produced by the state feedback controller.
Figure 24 shows that the State Feedback
controller can maintain bicycle verticality under
real-time conditions. That is, with initial conditions
far from ideal and sensor noises. This explains the
high-frequency components and the almost
“chattering” control signal behavior. Excessive
control effort could be reduced if dominant closed-
loop poles are placed with a longer steady state
time, although this could reduce the possibility of
reaching stability. That is, the bicycle could lose
verticality before the controller has enough time to
recover it.
Fig. 24: signals provided by the bicycle sensors
while it is balancing: (red line), (purple line),
ω (orange line), and control signal (blue line)
5 Conclusion
The PID and state feedback controllers were chosen
due to the advantages they have, which are: easy
implementation, proven robustness and
performance, and being well-suited for an
engineering context. The implementation of the PID
controller could not be carried out, due to the
cancellation of the system zero at the origin; that is,
the exact pole/zero cancellation at zero cannot be
guaranteed due to the approximation of the integral
action in the digital implementation of the PID
controller, obtaining an unstable response. It is
important to recall that stability margins are valid
provided there are no open-loop pole/zeros
cancellations. That is, Nyquist stability criteria
cannot cope with no controllable or no observable
systems, a phenomenon occurring under pole/zero
cancellation. Further analysis or a more complex
linear controller is required to avoid canceling the
process zero at zero, for instance, the combination
of a State Feedback plus PI control scheme. On the
other hand, the state feedback controller has
excellent behavior without the need for
cancellations. Although it does not include an
integral action, it achieves the desired outputs
because the signal reference is and the state
feedback assures exponential and asymptotic
stability in all the states so,
. Therefore,
the bicycle maintains verticality.
Nevertheless, and taking advantage of having
access to the entire state vector, it would be
advisable to design and implement a non-linear state
feedback control such as a "Back Stepping" control.
In this way, a more direct control could be designed
for each state. However, as shown by equations
(10), (11), and (17) this may not be a simple task as
the system degree is 3 while its relative degree is 2.
Also, Sliding Mode control with “super twist” could
be analyzed.
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WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2024.19.8
I. I. Siller-Alcalá, J. U. Liceaga-Castro,
R. A. Alcántara-Ramírez, S. Calzadilla-Ayala
E-ISSN: 2224-2856
83
Volume 19, 2024
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- I. I. Siller-Alcalá carried out the simulation and
the control design.
- J. U. Liceaga-Castro carried out the bike model
and the control design.
- R. A. Alcántara-Ramírez has organized and
executed the experiments.
- S. Calzadilla-Ayala was responsible for the
implementation of the controllers.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2024.19.8
I. I. Siller-Alcalá, J. U. Liceaga-Castro,
R. A. Alcántara-Ramírez, S. Calzadilla-Ayala
E-ISSN: 2224-2856
84
Volume 19, 2024
