�

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).

1801

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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