Engineering, 2010, 2, 212-219
doi:10.4236/eng.2010.24031 Published Online April 2010 (http://www.SciRP.org/journal/eng)
The Development of Self-Balancing Controller for
One-Wheeled Vehicles
Chung-Neng Huang
Graduate Institute of Mechatronic System Engineering, National University of Tainan, Taiwan, China
Received November 30, 2009; revised February 16, 2010; accepted February 19, 2010
Abstract
The purpose of this study is to develop a self-balancing controller (SBC) for one-wheeled vehicles (OWVs).
The composition of the OWV system includes: a DSP motion card, a wheel motor, and its driver. In addition,
a tilt and a gyro, for sensing the angle and angular velocity of the body slope, are used to realize
self-balancing controls. OWV, a kind of unicycle robot, can be dealt with as a mobile-inverted-pendulum
system for its instability. However, for its possible applications in mobile carriers or robots, it is worth being
further developed. In this study, first, the OWV system model will be derived. Next, through the simulations
based on the mathematical model, the analysis of system stability and controllability can be evaluated. Last,
a concise and realizable method, through system pole-placement and linear quadratic regulator (LQR), will
be proposed to design the SBC. The effectiveness, reliability, and feasibility of the proposal will be con-
firmed through simulation studies and experimenting on a physical OWV.
Keywords: Self-balancing Controller, One-wheeled Vehicle, Mobile-inverted-pendulum, Pole-placement,
Linear Quadratic Regulator (LQR)
1. Introduction
In recent years, because of the surging consciousness of
global pollution and energy-shortage crises, automobiles
and motorcycles are no longer the best for transportation.
In order to fit the daily required and improve above
problems, exploring new energy or developing lighter
and innovative mobile carriers are beginning to be
known as new trends. The earliest two-wheeled balanc-
ing robot was published in 1987 by Prof. Yamafuji [1].
From then on, the concerning researches with this topic
have been increasing [2-6] and have even been a com-
mercialized product. For example, the Human Trans-
porter, was developed by Segway Co., U.S.A., which is a
very famous two-wheeled balancing vehicle [7,8]. In
addition, NASA’s Robonaut, Segway platform puts ro-
bots in motion, is now aim in the military projects [9].
However, the balancing mechanism of a two-wheeled
system is rather complicated. Whereas, for the problems,
the two-wheeled synchronization and body balancing
should be considered simultaneously, making a lot of
sensors a requirement. It does not only complicate the
system, but also increases the cost. For the reasons above,
how to simplify the two-wheeled system has become one
of the studying motives of OWV. By this motive, not
only can it maintain the advantages of a two-wheeled
system, but it can also simplify the system mechanism
further, decreasing the cost.
Nowadays, although a lot of references engaging in
the studies of two-wheeled balancing carriers can be
found [2-6], the ones for OWV studies are still insuffi-
cient [10,11]. Trevor Blackwell [12], an American engi-
neer, has proposed an OWV design on his personal web-
site, and the concept OWV, EMBRIO, is published by
BRP [13], a motorcycle manufacturer in Canada. Besides,
[10] is the newest one based on fuzzy controls for bal-
ancing.
According to the above studies, one can find that the
proposed OWV is rather new and is a worthy topic in
studies on modern robots and carriers. However, most of
the studies on robot-balancing subjects are often adopt-
ing the fuzzy control theory to handle those balancing
control problems [4,14,15], but carrying out system
modeling and analysis, and confirming the effectiveness
and feasibility through experiment studies [14,15]. For
fuzzy control, using linguistic information, can model
complex systems without employing precise quantitative
analyses, particularly for the controlled plant with in-
Identify applicable sponsor/s here. (sponsors)

C. N. HUANG213
complete knowledge [4,14]. However, it has not been
viewed as rigorous due to lack of formal synthesis tech-
nologies, which guarantee the basic requirements for
control systems such as global stability [15]. That is, it
is not only difficult to precisely know the mutual influ-
ence between each system state but also might be diffi-
cult and expensive to design the controller for its com-
plexity highly depending on the number of fuzzy rules
[16].
Consequently, finishing system modeling and analy-
sis for proposed SBC is set as the first step of this study.
Next, by linearizing the system model, pole-placement
and state-feedback controls can be found and designed.
Finally, through human-computer interaction, a DSP
card featured with motion control is adopted to handle
the real-time computation of digital signals.
However, from above studies found that only using
the pole-placement and state-feedback controls, the
feedback gains can not be adjusted easily. It is difficult
to lead the system in a better stability. Here, LQR is
used to improve the above problems. For comparison,
both the numerical and experiment studies on SBC,
basing on pole-placement and LQR controls, respectively,
are made. The studies have proved that the proposed SBC
is able to perform higher controllability and stability.
2. Modeling for OWV
The concept of the OWV prototype is simply constructed
by a wheel motor and two sensors as a tilt and gyro. Here,
the wheel motor is adopted as the driving wheel of the
OWV that uses the feedback signals from sensors to
balance the system body. Figure 1 shows the outward
appearance of OWV prototype where the box on the
upper side of wheel motor can be treated as the balancing
weight of inverted-pendulum, and in which the motor
driver, sensors are set. Figure 2 shows the free body
diagram of OWV in Figure 1.
According to the free body diagram in Figure 2, the
kinetic energy E and potential energy U can be expressed
as follows, respectively.
2
2
2
2
2
)(
2
1
2
1
)sin(
2
1
)cos(
2
1
2
1
R
X
JJlm
lXmXME
M
wm
ww
w







(1)

cosmglmglU  (2)
Basing on Lagrange’s equation as well as the coeffi-
cient transformation for deriving mathematic model of
OWV, the Lagrange equation can be written as:
R
T
X
L
X
L
dt
dw
w
w















Figure 1. OWV prototype.
Figure 2. Free body diagram of OWV.
0
















LL
dt
d
(3)
where UEL


From (3), the accelerations of body movement and in-
cluded angle between vertical axis and body can be
found as follows.
m
w
w
m
w
w
Jml
lm
R
J
mM
Jml
glm
ml
R
T
X







2
222
2
2
22 2
cos
cossin
sin



(4)
2
222
2
2
2
cos
sin
)sin(cos
R
J
mM
lm
Jml
mgl
R
J
mM
ml
R
T
ml
w
w
m
w
w
w











 (5)
Since (4) and (5) are nonlinear equations, for simpli-
fying and modeling, let them be represented on the bal-
anced point 0


and given 2
R
J
mMM w
we , then

C. N. HUANG 214
x
JS
1
S
1w
X
w X

wX

S
1
S
1

J
1

K
w
T





＋
＋
＋
＋
x
K
Figure 3. OWV control (linear).
the equations can be rewritten as
RlmJmlRM
TJml
lmJmlM
glm
X
me
mm
me
w222
2
222
22
)(
)(
)( 







(6)
RlmJmlRM
mlT
lmJmlM
mglM
me
m
me
e
222222 )()( 






 (7)
By using (6) and (7), the block diagram for OWV
control can be illustrated as shown in Figure 3. Where,
])([ 222
2
lmJmlMR
Jml
J
me
m
x

，
222
22
)( lmJmlM
glm
K
me
x


RlmJmlRM
ml
J
me
222 )( 



， gRMK e

From Figure 3, it can be found that OWV is a sin-
gle input and multiple output system (SIMO). Besides,
OWV is similar to the inverted-pendulum vehicle,
belonging to the non-minimum phase system [3,17].
3. Analysis and Design for State Variables
3.1. Performance of System Instability
Since a system can be represented in state space by fol-
lowing equations:
BuAXX 

(8)
y
CX Du (9)
for and initial conditions with respect to the
equilibrium point, , where
0
tt
)0,0,0,0()( 0tX
X
: state vector

X
: derivative of the state vector with respect to time
y: output vector
u: input or control vector
A: system matrix
B: input matrix
C: output matrix
D: feed forward matrix
(8) is called the state equation, and the vector X, the
state vector, contains the state variables. It can be solved
for the state variables. Besides, (9) is called the output
equation. This equation is used to calculate any other
system variables.
For the linear, time-invariant, second-order system as
OWV, its system dynamics can be transformed and ex-
pressed by state equations; the state space of OWV can
be taken on the following form by (6) and (7).




















w
w
X
X
=



















0
)(
00
1000
0
)(
00
0010
222
222
22
lmJmlM
mglM
lmJmlM
glm
me
e
me


















w
w
X
X
+




















RlmJmlRM
ml
RlmJmlRM
Jml
me
me
e
222
222
2
)(
0
)(
0

w
T (8’)
































w
w
wX
X
X
0100
0001 (9’)
In order to confirm the effectiveness of the derived
model, through the simulation by mechanical design
software Solidworks, or CATIA, system analysis and
estimation of system coefficients are done.
Table 1 and Figure 4 show the estimated system co-
efficients of OWV by Solidworks’ analysis.
Based on above, system coefficients and system sta-
bility, the analysis finds that for OWV it is not a stable
system. Initially, it would fall down by an outside dis-
turbance. This phenomenon can be confirmed by the root
locus as shown in Figure 4. Where, for one of the poles
locating in the right of s-plane, it can be judged that it is
an unstable system. Besides, this result also can be con-
firmed by finding out the eigenvalues of system matrix in
(8’) as4116.4,4116.4,0,0


s.
Table 1. System coefficients of OWV.
m 4.31 kg
Jm 0.1984 kg-m2
l 0.1 m
R 0.07 m
Mw 5.1819 kg
Jw 0.0123 kg-m2

C. N. HUANG 215
3.2. Analysis for the Controllability of OWV
Accepting to discuss the system stability in state-space
analysis, the controllability of system is also an impor-
tant index. That is, before designing state-feedback con-
trollers, the analysis for controllability should be done in
advance to make sure the system is able to achieve the
desired responses through state-feedback control and
observing.
Pole-placement is a viable design technique only for
systems that are controllable. In order to be able to de-
termine controllability or, alternatively, to design state
feedback for a plant under any representation or choice
of state variables, a matrix can be derived that must have
a particular property if all state variables are to be con-
trolled by the plant input, u. For an nth-order plant whose
state equation is (8), which is completely controllable if
the matrix
Figure 4. Root-locus diagram.
S
1

X＋
＋
＋
＋
B
XC
A
y
D
K
-
u
＋
2n
c
QBABAB AB
1





 (10)
is of rank n, where is called the controllability matrix.
Now, by substituting the coefficients of Table 1 to the
system and inputting matrices in (8’) and using (10), then
c
Q

 4

c
Qrank can be found. That is, OWV is completely
controllable on the balancing point. Its closed-loop sys-
tem’s poles can be placed at desired locations on the
s-plane through using the state feedback )()( tKxtu

.
Figure 5. State-feedback block diagram.
3.3 State-Feedback Control
In order to improve the system instability, the state
feedback as well as the pole-placement design is adopted
to make the system stable and balanced. By which, the
block diagram of the system state feedback is shown as
Figure 5. Where, the feedback is given as. )(tKxu 
In Figure 5, for the system feedback



)(tKxu
, the closed-loop system matrix can be
expressed as
][ 4321 kkkk



















4321
4321
2696.22696.24625.192696.22696.2
1000
2717.12717.16987.02717.12717.1
0010
kkkk
kkkk
BKA (11)
Thus, the transfer function of the closed-loop system


det() 0sIA BK


, is
12 34
123 4
100
1.27171.27170.6987 1.27171.27170
00 1
2.26962.269619.4625 2.26962.2696
s
ks kkk
s
kk ksk









Here, assume an example for placing the system’s poles
at s= 0, 0, -4.5, -4, since the system transfer function
should satisfy the condition




24.54 0ss s
 7452.35062. 
, the
feedback gains of system state can be found as
. In order
 1600][ 4321  kkkkK
to confirm the effectiveness of pole-placement, a simulation
study on OWV by using the results (feedback gains) of the
above example is done. Figure 6 shows that through sys-
tem feedback control, all system’s poles of OWV have
located in the left of s-plane. In addition, the feed-back
system, being disturbed by a continuous impulse

C. N. HUANG 216
Figure 6. Root-locus diagram.
Figure 7. Impulse and system response.
motion, still can return to the desired balancing state as
shown in Figure 7.
3.4 LQR Control
In above section, it shows that pole-placement and
state-feedback controls used for SBC are available and
realizable. However, if the unstable pole could only be
placed in the left of s-plane by using assignments (ex-
perienced studies), then adjusting the feedback gains to a
better stability for OWV would be difficult.
In order to improve above problem, LQR, the con-
tinuous time infinite horizon linear quadratic regular with
control constraints, is adopted to design SBC. Determi-
nistic LQR theory was first introduced by Kalman, and
has been playing a central role in modern control theory
as well as various engineering practices; the reader can
be referred to the well-known book [18], as the following
problem:
Minimize quadratic performance index
 dttRututQXtXJ TT
 
0)()()()(
 (12)
subject to linear dynamics (8), (9) and a constraint on the
input , for all
Utu )( ),0[

t.
Here, the state vector is locally abso-
lutely continuous, and the minimization is carried out
over all locally integrable controls .
Throughout the note, the standing assumptions are
n
IRX ),0[:
uk IR),0[:
1) Q and R, the weighting matrices of )(tX and
)(tu , are symmetric and with positive semi-definite and
positive definite, respectively.
2) The pair (A, B) is controllable. The pair (A, C) is
observable.
3) The set U is closed, convex, and Uint0.
For the unconstrained problem (8), (9), and (12), the
optimal solution of the problem is the following
state-feedback control.
)()()( tXtKtu


(13)
where, , and P, the unique symmetric
and positive definite matrix, is the solution of the Riccati
differential equation as
)()( 1tPBRtK T 
QtPBBRtPtPAAtPtP TT  

)()()()()( 1 (14)
Since there is only stable state considered for OWV
system, that is, (14) can be rewritten as
0)(

tP
QPBPBRPAPA TT  1 (15)
By substituting (15) into (13) with system coeffi-
cients, , the optimal feedback gains, can be found
as
)(tK
1234
K
kkkk






3.1623-3.9851 -32.5896-7.5687
]1[
, by giving
matrices

R1110([diagQ, and ])1

. The R
and Q are provided to have constraint of control action
satisfied.
Figure 8 shows the comparison of stability responses
by using pole-placement and LQR, respectively. Where,
SBC, basing on LQR control, let OWV have a better
system stability is confirmed.
4. Physical Demonstration of OWV
For further confirming above numerical studies and the
feasibility of OWV, the testing setup is accomplished
and pictured in Figure 9.
A motor driver and sensors (SSY0090 tilt and CRS03-
02 gyro) are set in the box and a 24-V 250-W wheel mo-
tor (SA176-6) [19] is used as the driving wheel. All the
algorithms for pole-placement, feedback-gain calculation,
and LQR etc. are developed with C language and im-
plemented in an ITRI PMC32-6000 DSP board with a
clock frequency of 40MHz (Figure 10).
Figure 11 illustrates that when the instability of body
is detected by those sensors, the angular signals, through
A/D transformation, transfer to DSP for real-time digital

C. N. HUANG 217
Figure 8. Comparison of stability responses.
Transforms
Low pass filter
body
State feedback
A/D
CPU
D/A A/D
Driver
DSP
Tilt
Gyro
Figure 9. OWV setup.
Figure 11. Control block diagram of OWV.
terrupting algorithm is executed by every 1 ms.
Figure 12 shows the voltage outputs of motor control,
angular signals of OWV’s body from gyro and tilt while
SBC bases on pole-placement and LQR controls, respec-
tively. Here, by examining angular velocity (Figure
12(b)), since it, the derivatives of body tilt angle can be
taken as the estimation of next-state body falls, the
Figure 10. DSP board.
computation and decision. Then, the decision signal from
DSP will send to motor driver via D/A transformation.
Here, the sampling period is set to 1 ms. That is, the in-

C. N. HUANG 218
(a) voltage outputs
(b) angular velocity
(c) tilt angle
Figure 12. Control signals.
voltage signal (Figure 12(a)) is approximately inverse
to angular velocity for falling-preventive control. Be-
sides, in Figure 12(c), even through some tiny oscilla-
tions within 2 degree occur when the body of OWV
is starting to swing up by the wheel-motor driving a
horizontal force to move OWV back and forth, after
3.75 seconds it can return to the desired balancing
state. Consequently, it proves that SBC is able to be
realized by both above controls. Moreover, through
the comparisons of the signals in Figure 12, finds that
the magnitude of the state signals by LQR control is
smaller than those by pole-placement control. It is not
only shows LQR with better system stability per-
formance but also corresponding to the objective of
LQR control in (12).

5. Conclusions
This study has proposed a SBC, based on concise con-
struction and control theories for OWV. Only based on
state-space modeling and real-time sensing, the SBC is
attractive for its conciseness and feasibility. Studied re-
sults show that the real-time SBC is not only can be re-
alized but also can let OWV with optimal stability by
LQR control.
By further developing OWV technologies, such as
SBC etc., the benefits not only can serve the handling
problems, existing in present electrical mobile-robots,
welfare carriers, or two-wheel entertainment vehicles as
large steering radius (angle), differential gear, or syn-
chronization control etc., but also with advantages on
cost, setup, light quantity, and saving energy, etc.
6
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Appendix A w
M
: single wheel mass
w
J
: single wheel moment of inertia
Nomenclature
w
X
: center cart trajectory
m: pendulum payload mass

: pendulum tilt angle
m
J
: pendulum moment of inertia
w
T: driving torque
l: Pendulum length
R: wheel radius