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Abstract— This research is aimed to design and develop a
two-wheeled mobile robot with an extendable link for applications
in a confined area. The proposed system mimics a double inverted
pendulum, where the first link (Link1) is made near to the upright
position and the second link (Link2) can be at any interest angular
position within [-90,90] range with the extendable effect. This type
of configuration is very complex, highly nonlinear and less studied
by the researchers in this field. Therefore this paper focuses on
modeling of a multi degree of freedom of two-wheeled mobile
robot. The mathematical model of the robot has been derived from
its free body diagram involving the wheels, Link1 and Link2 with
extendable effect. The equations have been linearized and
represented in the form of state space model. The model was then
tested using linear quadratic regulator (LQR), which was used to
control the angular position of Link1. The results show that the
model derived works very well with its linear operating region.
The modeling stage was extended using virtual prototyping that
represent its actual system. Simulation results also shown that
with suitable controllers, the 4D model developed was able to
perform at any interest angular position.
I. INTRODUCTION
wo-wheeled mobile robot is known to have advantages on
performing better manuever in a confined space. It is a
typical mobile robot that is very complex and highly nonlinear
and comes with a double link inverted pendulum configuration.
There are many laboratory-scale inverted pendulum systems
that have been developed to study the dynamics of the system [1,
2, 3]. In fact, the idea of double inverted pendulum-like system
has been extended for different applications such as
two-wheeled wheelchair, [4]. It is also a well-known classical
subject but very exciting that is explored by researchers in the
field, most of the time used to prove different control algorithms
developed.
There are many applications of the inverted pendulum
scenario that covers the laboratory scale to complex inverted
pendulum-like systems for examples pendulum on cart, people
standing on movable two-wheeled and two-wheeled
wheelchair.
This work was supported by the research funds by the International Islamic
University Malaysia.
M.T. A. Rahman (taqqa_019@yahoo.com), S. Ahmad
(salmiah@iium.edu.my), and R. Akmeliawati (rakmelia@iium.edu.my) are with
the Intelligent Mechatronics Systems Research Unit, Department of
Mechatronics Engineering, Faculty of Engineering, International Islamic
University Malaysia, Jalan Gombak, 53100 Kuala Lumpur, Malaysia.
There are number of studies done on inverted pendulum on
two-wheeled. This type of application has emerged from the
integration of an inverted pendulum-cart and a rotational
pendulum. For example, inverted pendulum on two-wheeled
for educational purposes [5-7], human transportation [8], and
assistance robot [9]. From those studies, it is observed that most
of the researchers have stressed on the single-link inverted
pendulum on two wheels which are quite simple yet limited
flexibility.
In contrast, this research introduces a new double inverted
pendulum on two wheels with extendable link configuration.
The proposed system will give more flexibility and advantages
for many purposes. For example, a two-wheeled wheelchair
that has been modeled based on the idea of a double inverted
pendulum might help the disabled to be independent in the
sense that the user may be able to reach a higher level of height
when on the two-wheeled configuration, with the capability of
maneuvering in confined space, at home or library, without any
helper.
The challenge will be on the modeling and control of the
system. Therefore, this paper will first focus on the integrated
modeling of the system, which involves both mathematical
modeling and virtual 4D modeling technique. Both techniques
complement each other in producing a complete analysis of the
system dynamics before any hardware implementation is
conducted.
This paper is organized as follows: Section II presents the
derivation of the mathematical modeling of the double-link
two-wheeled mobile robot and the 4D virtual prototyping.
Section III presents the simulation based performance
evaluation of the system for both model. Finally, Section IV
concludes the findings.
II. MODELING OF DOUBLE-LINK TWO-WHEELED MOBILE ROBOT
A. Mathematical Modeling
In the proposed two-wheeled mobile robot, as shown in
Figure 1, there are three independent motors used, one for each
wheel, represented by 2R and 2L; and one between Link1 and
Link2, represented by 2 . In addition, a linear actuator, 2F is
introduced at Link2 that extends the link to reach higher level of
height. The system outputs are the angular position of Link1, /1,
Integrated Modeling and Analysis of an Extendable Double-Link
Two-Wheeled Mobile Robot
Muhammad Taqiuddin Abdul Rahman, Salmiah Ahmad and Rini Akmeliawati
T
2013 IEEE/ASME International Conference onAdvanced Intelligent Mechatronics (AIM)Wollongong, Australia, July 9-12, 2013
978-1-4673-5320-5/13/$31.00 ©2013 IEEE 1798
the angular position of Link2, /2, the extension, d and the linear
position, x .
Fig. 1. Schematic diagram of Link1, Link2, Wheel and Upright position of the
proposed two-wheeled mobile robot with extendable link.
Equation of Motion of Wheels
Theoretically, the equation of motion for wheels in Figure 1
can be represented as follows, [10];
� � � �> @)(2 2 LRLR
WW
HHrJrM
rx ���
� WW��
(1) Where r is radius of wheels, Mw is mass of the wheel, Jw is
moment of inertia of wheel, HR and HL are vertical and
horizontal reaction force between the wheel and Link1 for each
right and left wheel.
Equation of Motion of Link1
From the schematic diagram of Link1 in Figure 1, by using
Newton’s law of motion, the sum of forces in the horizontal
direction,
xMNlMlMHH
xMF
LR
x
����
��
1111
2
11
1
sin)(
,
����
¦TTT
(2)
Where M1 represents the mass of Link1, �1 represent the
rotation angle of Link1.
The sum of the force perpendicular to the first link is;
111111
1111
11
coscossinsin
cos)(sin)(
,cos
TTTT
TTT
T
xMNPgM
lMHHPP
xMF
LRLR
��
��
��
���
����
¦ A
(3)
Where g is the gravitational acceleration, P and N are
action-reaction force between Link1 and Link2, and l represents
the distance between the center of gravity of Link1 (half of
length of Link1).
The sum of Moment around center of mass of Link1 can be
written as;
1111
11
)(cossin
cos)(sin)(
,
TWWWTT
TT
D
��INlPl
lHHlPP
IomentM
LR
LRLR
�����
���
¦ (4)
� Multiplying equation (3) with (- l ) and subtracting equation
(4) yields;
111111
1
2
111
cossin
cos2sin2
TWWWTT
TTT
lxMIglM
lMNlPl
LR����
��
���� �
��� (5)
Both equation (2) and (5) are nonlinear, which will be
linearized by assuming that the system operates over a small
angle. Thus, let �1=�+/1, where /1 is a very small angle,
therefore cos �1 = -1, sin �1 = - /1, d�1/dt = 0. Applying this
small angle approximation in equation (2) and (5) yields the
following equations:
»¼
º�
�
���
��
��
««¬
ª¸̧¹
·¨̈©
§
�
�
¸̧¹
·¨̈©
§
����
NlMI
lM
rlMI
NlM
lMI
PlMglM
lMI
lMM
r
JM
x
LRLR
WW
2
11
21
2
11
2
1
12
11
2
1
22
1
2
11
2
112
)()(2
2
22
1
WWWWW
G��
(6)
]]D
]WWW
D]WW
G]
G
NlNlM
r
lMPlglM LRLR
2
)()()2(
1
11
11
��
���
��
� ��
(7)
2222 GG dMLMxMN s������ ���
dMgMP s��� 2
where d is elongation of extendable-link, � is constant value in
terms of Mw, Jw, M1, l, r and I1, d�� is acceleration of
extendable-link,1G�� is angular acceleration of Link1, I1 is
moment of inertia of Link1, I2 is moment of inertia of Link2,
2G�� is angular acceleration of Link2, L represents the distance
between the center of gravity of Link2 (half of length of Link2),
Ms is mass of extendable link.
Equation of Motion of Link2
By using similar procedure, the final horizontal acceleration
of wheel, angular acceleration of Link1 and Link2 can be found
as follows;
615
24231212
2
211
AA
AdAdgMAdgMA ss
��
����
G
GGGGGG�� (8)
625
1423122
2
212
BB
BdBdgMBdgMB ss
��
����
G
GGGGGG�� (9)
615
2423122
2
21
CC
CdCdgMCdgMCx ss
��
����
G
GGGGG�� (10)
Where Ai, Bi, and Ci (for i= 1, 2, 3, 4, 5) are constant and given
as in terms of system’s parameters.
1799
On the other hand, the parameters A6, B6, C6 represent
complex variables as a function of input torque terms (2R, 2L, 2
and 2F).
),1/()( 11556 ebbeA �� � �1156 1/ ebeB �
4616 aBaC � and
r
LMe LR
���
� VH
WWHW )(2
5,
ra LR
��
V
WW4
Equation of Motion of Extendable Link
Fig. 2. Schematic Diagram of the extendable link.
Lagrange’s equation is commonly applied in the dynamic
system that has complex coordinates system [11]. Thus it is
used for deriving the equation of motion of extendable link.
Figure 2 shows the schematic diagram of the extendable link
where a1, a2, and a3 are coriolis, sliding and gravity
acceleration, respectively. Using Lagrange’s equation method
[12], the force applied to extend the Link2 is given by the
following equation;
)sincos( 22
222
2 gMdMMdF SSSd TTIT ��� ���� (11)
Where Fd = Torque of motor, 2F / Radius of motor, rm.
After linearizing and rearranging equation (6), the extendable
link configuration of motion is described as follows;
2)/( GW dgMMrd ssF �� �� (12)
The linearized model of the system is obtained by using the
Jacobian method and represented by the state-space model as
follows;
uxx BA � � (13)
»»»»
¼
º
««««
¬
ª
»»»»»»»»»»
¼
º
««««««««««
¬
ª
�
»»»»»»»»»»
¼
º
««««««««««
¬
ª
»»»»»»»»»»
¼
º
««««««««««
¬
ª
»»»»»»»»»»»
¼
º
«««««««««««
¬
ª
F
L
R
dcba
dcba
dcba
dcba
d
x
d
x
YT
WUV
SQR
NLM
d
x
d
x
WWWW
GG
GG
GG
GG
4444
3333
2222
1111
0000
0000
0000
0000
000000
00000
00000
00000
10000000
01000000
00100000
00010000
2
1
2
1
2
1
2
1
�
�
�
�
��
��
��
��
�
�
�
�
Where M, L, N, R, Q, S, V, U, W, T , Y, a1, a2, a3, a4, b1, b2, b3,
b4, c1, c2, c3, c4, d1, d2, d3 and d4 are the constant values in
term of system’s parameters, the state vector, xT
],,,,,,,[ 2121 dxdx ���� GGGG and the output is given as;
xCy (14)
Where
»»»»
¼
º
««««
¬
ª
00001000
00000100
00000010
00000001
C�
Table 1 shows the designed for the two-wheeled mobile robot.
TABLE I
SYSTEM’S PARAMETERS
Parameters Value Unit
G 9.81 ms-1
Mw 0.700 Kg
M1 0.900 Kg
M2 0.075 Kg
Ms 0.075 Kg
Jw 0.225 kg.m2
I1 0.04 kg.m2
I2 0.0225 kg.m2
L 0.2 M
L 0.15 M
R 0.065 M
B. Virtual 4D Prototype Modeling
The model of double link two-wheeled mobile robot by using
four-dimension (4D) software called MSC Visual Nastran 4D
(VN). This software takes into account the effect of the gravity
(as the fourth dimension) and has the ability to visualize the
motion of the model. This environment also considers the
friction between bodies thus mimics the actual physical system.
It also allows the integration with any control algorithm
developed in Matlab-Simulink. Figure 3 shows the virtual
model of the proposed double-link two-wheeled mobile robot
with extendable link with the details.
1800
Fig. 3. Virtual Prototype with details of material specification
III. SIMULATION BASED PERFORMANCE EVALUATION
A. Open Loop poles and Response
The first thing that we should inspect is the open poles of
system, which are [0, 0, 125.28, -125.28, 10.91, 0.0000+2.15j,
0.0000-2.15j, -10.91]. The position of the poles verifies that the
system is not stable in open loop as there are two poles lie in the
right-half plane (10.91 and 125.28). The open loop response in
Figure 4 confirms that the system is not stable.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
2
4
6
8
10
12
14x 10
25
Time (s)
ou
tpu
ts
Delta1(Degree)
Delta2(Degree)
x-displacement(m)
d-displacement(m)
Fig. 4. Open loop step response of the system.
B. Linear Quadratic Regulator (LQR)
To stabilize the system, we use the linear quadratic regulator
(LQR). Figure 5 shows the block diagram of the state space
model with LQR implementation.
Fig. 5. Block Diagram of LQR Realization.
The optimal stabilization of the system can be performed by
varying the values of two weighting matrices, Q and R. The
advantage of the LQR is that it can control a multivariable
system. The selected of Q = diagonal(ax, by, cz, dw, 0, 0, 0, 0),
an eight by eight diagonal matrix. The parameter ax and by were
used to control Link1 and Link2 angular angle position
respectively, cz was used to control the position of the
horizontal direction of the wheel (x-displacement) and dw was
used to control direction of d-displacement. All parameters are
heuristically tuned to get the best system performance. The
value K matix that give a good controller was shown as;
»»»»»»»»»»»
¼
º
«««««««««««
¬
ª �
0.33980.0422-0.0546-0.0506-
0.2171-1.2247-1.07841.1138
0.2751-5.27712.7040-2.7469-
1.8079-3.98200.68211.0851
2.66732.14491.9510- 1.9775-
5.4368-17.8084-16.673716.5842
28.7919-272.7603116.0066-114.6307-
9347.46791.139367.14287.3
TK
By giving the initial values of ax = 1, by = 1, cz = 1 and dw =
1, the step response was obtained as shown in Figure 6. The
settling time is more than 10 seconds. After some heuristic
tuning, the best result found was shown in Figure 7.
0 1 2 3 4 5 6 7 8 9 10-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
Time (s)
Ou
tpu
ts
Delta1(Degree)
Delta2(Degree)
x-displacement(m)
d-displacement(m)
Fig. 6. Step response with LQR control (initial Q and R).
0 1 2 3 4 5 6 7 8 9 10-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
Time (s)
Ou
tpu
ts
Delta1(Degree)
Delta2(Degree)
x-displacement(m)
d-displacement(m)
Fig. 7. Step response with LQR control (tuned Q and R).
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The value ax, by, and cz used to get the response in Figure 7
were 1000 each meanwhile value dw was 10. The higher the
values, the more control efforts are needed and the smaller the
tracking error. The system response with the final tuned value
has the settling time under below 1s.
C. Virtual Prototype Simulation Performance
The two-wheeled mobile robot with extendable double-link
configuration is modeled using MSC Visual Nastran 4D for
simulation performance testing. Modular hybrid controller
combining Fuzzy-PD to control Link1 and Link2 and PID type
controller for linear motion control of the extendable link, [13].
The mobile robot is being test to balance with the extendable
link is at its datum position which is 0 degree and 10 cm
extension. From Figure 8, 9 and 10, it can be seen that the
system is able to balance itself by maintaining the upright
position Link1 and the angular position of the Link2 at 0°. The
figures show similar results using LQR method for the
linearized state space model earlier.
0 1 2 3 4 5 6 7 8 9 10-3
-2.5
-2
-1.5
-1
-0.5
0x 10
-3
time (s)
An
gle
Lin
k1
(d
eg
ree
)
Fig. 8. Angular position of Link1, G1.
0 2 4 6 8 10-7.5
-7
-6.5
-6
-5.5
-5
-4.5x 10
-3
time (s)
An
gle
Lin
k2
(d
eg
ree
)
Fig. 9. Angular position of Link2, G2.
0 1 2 3 4 5 6 7 8 9 10-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
time (s)
dis
pla
ce
me
nt
ex
ten
da
ble
lin
k (
m)
Fig. 10. Displacement of Linear actuator, m.
The results from both models developed have been validated
using the LQR and intelligent control respectively. They
produce a promising system performance within simulation
environment. The mathematical model of the double-link
two-wheeled mobile robot with extendable link configuration
was very complex as it involves multi variables. Many
assumptions have been considered to derive the mathematical
equations of the system for simplification purpose. This
situation can lead to the unawareness of the uncertainties of the
actual system’s parameters like the vibration of motor, friction,
etc. Although the final state space equation has been
successfully derived using partial derivative method, it
consumed more time to be completed. In addition, the
linearization made has limited the system to work only within a
small region at the upright position, where the angular positions
of Link1 and Link2 are approximately at 0 degree. This
drawback has been complemented by the virtual prototype
developed at latter. This model has considered the nonlinearity
part of the two-wheeled mobile robot and the results show that
the model works very well at different input reference of Link1
and Link2.
IV. CONCLUSION
This paper focused on the integrated design and modeling of
double-link two-wheeled mobile robot using mathematical and
4D modeling tools approached. The derived mathematical
model was successfully validated using the existing LQR
technique while the developed 4D virtual prototype was
successfully tested using the intelligent controller in
Matlab-Simulink. Both models developed have provided
sufficient confidence level for future hardware implementation.
ACKNOWLEDGMENT
The authors are grateful to the International Islamic
University Malaysia for the research funding on this project.
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