control of a 2-dof omnidirectional mobile inverted pendulum

Journal of Mechanical Science and Technology 26 (9) (2012) 2921~2928

www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-012-0710-2

Control of a 2-DOF omnidirectional mobile inverted pendulum†

Tuan Dinh Viet, Phuc Thinh Doan, Hoang Giang, Hak Kyeong Kim and Sang Bong Kim*

Department of Mechanical and Automotive Engineering, College of Engineering, Pukyong National University, Busan, 608-739, Korea

(Manuscript Received October 20, 2011; Revised February 20, 2012; Accepted May 24, 2012)

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Abstract

In this paper, stabilization of a 2-degrees-of-freedom (2-DOF) omnidirectional mobile inverted pendulum (OM-IP) is studied. The

OM-IP consists of the rod that rotates around a rotary point of a universal joint which is connected at the center of the omnidirectional

mobile platform (OMP). For ease of analysis, the OM-IP is decoupled into two subsystems: a 2-DOF inverted pendulum (IP) and an

OMP. The IP is a rod that rotates around a universal joint with 2-DOF. The OMP is a body consisting of disk and three omnidirectional

wheels that moves on plane and keeps the rod in balance. Dynamic modeling of the 2-DOF OM-IP is presented. From the dynamic equa-

tion, an adaptive backstepping control method is proposed to keep the rod in balance. Update law is presented as differential equation of

an unknown parameter when the distance from the center of gravity of the rod to the rotary point on the OMP is unknown. Stability of the

adaptive controller is proven by using Lyapunov function. Simulation and experimental results show the effectiveness of the proposed

controller.

Keywords: Omnidirectional mobile inverted pendulum (OM-IP); 2-DOF inverted pendulum (2-DOF IP); Omnidirectional mobile platform (OMP); Adap-tive backstepping controller (ABC)

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1. Introduction

The inverted pendulum is a classical problem. However, it

is a perfect benchmark for the design of a wide range of non-

linear control theories. Most inverted pendulums considered in

previous paper are: single, double or triple inverted pendulum

on a cart [1, 4, 5, 7], a rotational single-arm [8] or two-link

pendulum and an inverted pendulum on an x-y robot [2, 11].

In this paper, a new idea to develop a 2-DOF inverted pendu-

lum using an OMP is introduced. The OMP employs three

omni-wheels in a triangular configuration. Therefore, it has

the ability to move in all directions on the floor and keep the

rod, 2-DOF IP, balance consistently.

The inverted pendulum has been widely studied by many

researchers around the world. The reason for studying the

pendulum relies on the fact that many important engineering

systems can be approximately modeled as the pendulum. In

previous studies, there are a large number of systems proposed

for inverted pendulum. The most popular and simplest method

used to control the inverted pendulum is PID controller. H.

Zhang et al. developed a robust PID with disturbance, noise

and unknown parameter [9, 10]. P. Ali et al. stabilized the

inverted pendulum on the cart by using feedback linearization

method converted to fuzzy controller based on Taylor series

[1]. H. Wang et al. controlled 2-DOF inverted pendulum using

a contact-less feedback low cost CCD camera [2]. H. Wang

transformed the 2-DOF pendulum problem to a 1-DOF one.

They realized two control loops controlled by a linearization

and stabilization method based on an observer. Stabilization

control of double inverted pendulum system was proposed by

Q. Li et al. [4]. To control the IP, some kinds of controller can

be used such as fuzzy controller [1], neural network controller

[11], sliding mode controller and adaptive controller [4-7].

However, stabilization problem of 2-DOF inverted pendulum

based on OMP still remains difficult task because of the slip-

ping of omni-wheels and the limitation of the maximum ac-

celeration of OMP driving by DC motors.

This paper proposes a new idea about stabilization of a 2-

DOF IP using an OMP. It is the first time that an idea to de-

velop a 2-DOF IP using an OMP is introduced. A control

method using adaptive backstepping control technique is pro-

posed when the distance from the center of gravity of the rod

to a rotary point of a universal joint on the OMP is unknown.

The stability and the convergence properties of OM-IP are

guaranteed by Lyapunov function. First, the OM-IP is sepa-

rated into two subsystems, the OMP and the 2-DOF inverted

pendulum. The IP is a rod that rotates around a universal joint

structure with 2-DOF. The OMP consists of platform and

three omnidirectional wheels that are moving on x y− plane

and keeping the rod in balance. Then, dynamic modeling of

the OMP and the 2-DOF IP are presented. In order to reduce

*Corresponding author. Tel.: +82 51 629 6158, Fax.: +82 51 621 1411

E-mail address: [email protected] † Recommended by Associate Editor Yang Shi

© KSME & Springer 2012

2922 T. D. Viet et al. / Journal of Mechanical Science and Technology 26 (9) (2012) 2921~2928

the control of the 2-DOF inverted pendulum into the controls

of two subsystems of 1-DOF inverted pendulum, the motions

of 2-DOF inverted pendulum and the OMP are also separated

into two independent motions, motion in x z− plane and

motion in y z− plane. Simulation and experimental results

show the effectiveness of the proposed controller.

2. System modeling and analysis

This section presents dynamic modeling of the OM-IP sys-

tem. For a convenient modeling process, the modeling of the

system is decomposed into two parts: OMP modeling and 2-

DOF inverted pendulum modeling.

2.1 2-DOF inverted pendulum modeling

Fig. 1 shows the rod of the 2-DOF inverted pendulum rotat-

ing around the rotary point of the universal joint, .O As

shown in Fig. 1, the 2-DOF inverted pendulum can be de-

coupled into 1-DOF of x z− plane and y z− plane. R is

the center of gravity of the rod, ,l is the distance from R to

the rotary point, O ; x

R and y

R are the projection of R

on x z− and y z− planes and xzl and

yzl are the dis-

tances from x

R and y

R to the rotary point O , respectively;

θ is the angle between the rod and z axis; α and β are the projection angles of the rod on x z− and y z− planes

from z axis, respectively.

The 2-DOF inverted pendulum is modeled under the fol-

lowing assumptions:

(1) The angle between the rod and z axis is small;

, 1α β ≪ .

(2) Mass is concentrated on the center of gravity (COG).

(3) The friction forces and inertial moment of the rod about

COG are zero.

According to geometry in Fig. 1, the followings are ob-

tained:

2 2 2tan tan tanθ α β= + (1)

2

2

2

cos 1

cos cos1 sin

cos

xzl l l

θα α

ββ

= =

+

(2)

2

2

2

cos 1.

cos cos1 sin

cos

yzl l l

θβ β

αα

= =

+

(3)

Fig. 2 shows the projection of the 2-DOF inverted pendu-

lum on x z− plane. With the assumption (1) of , 1α β ≪ ,

the following is obtained:

.xz yzl l l≈ ≈ (4)

Referring to Fig. 2, by applying Newton’s 2nd law at the

center of gravity of the pendulum along the horizontal and

vertical components by using Eq. (4), the followings are ob-

tained:

( ) ( )2 2

2 2cos cos

xz

d dV mg m l m l

dt dtα α− = ≈ (5)

( ) ( )2 2

2 2sin sin

x xz

d dH m x l m x l

dt dtα α= + ≈ + (6)

where m is the mass of the rod; g is the gravitational ac-

celeration; and x is the horizontal displacement of the OMP.

Taking moments about the COG yields the torque equation

as:

sin cos

sin cos

x xz x xz

x

I c Vl H l

Vl H l

α α α α

α α

+ = −

≈ −

ɺɺ ɺ (7)

( ) ( )2 2

2 2sin sin

x xz

d dH m x l m x l

dt dtα α= + ≈ + (8)

where xc is the pendulum viscous friction coefficient in x

axis; I is the moment of inertia of the rod about the COG;

V and x

H are the vertical and horizontal reaction forces on

the rod.

Applying Newton’s 2nd law for the OMP yields:

x x xF H Mx k x− = +ɺɺ ɺ (9)

where M is the mass of the OMP; xk is the viscous friction

coefficient of OMP in x axis; and x

F is the horizontal con-

trol force on the OMP in x direction.

Combining Eqs. (4)-(9), the modeling equations of the 1-

DOF inverted pendulum in x z− plane are given as follows:

yx

zR

l

βα

θ

O

xzl

yzl

yR

xR

COG

Fig. 1. 2-DOF OM-IP.

α

x

xzl

z

xH

V

x

α

x

xF

mg

M

m

xc α&

xF

xMg

xH

xk x&

O

O

O

Fig. 2. Projection of the 2-DOF inverted pendulum on x z− plane.

T. D. Viet et al. / Journal of Mechanical Science and Technology 26 (9) (2012) 2921~2928 2923

( ) ( )2cos sinx x

M m x lm F k xα α α α+ + − = −ɺɺ ɺɺɺ ɺ (10)

( )2 ( sin cos ) .x

I l m lm g x cα α α α+ − − = −ɺɺ ɺɺɺ (11)

With the assumption (3), the Eqs. (10) and (11) become:

( ) ( )2cos sinx

M m x lm Fα α α α+ + − =ɺɺ ɺɺɺ (12)

sin cos 0 .l g xα α α− + =ɺɺ ɺɺ (13)

Similarly, the modeling equations of the 1-DOF inverted

pendulum in y z− plane are given as follows:

( ) ( )2cos siny

M m y lm Fβ β β β+ + − =ɺɺ ɺɺɺ (14)

sin cos 0l g yβ β β− + =ɺɺ ɺɺ (15)

where ,x α and x

F of Eqs. (12) and (13) are replaced by

,y β and y

F , respectively. y is the vertical displacement

of the OMP; β is the projection angle of the rod on y z−

plane and y

F is the vertical control force on the OMP in y

direction.

2.2 OMP modeling

Fig. 3 shows the configuration of OMP. The OMP consists

of three omnidirectional wheels equally spaced at 0120 from

one another. The three wheels have the same radius denoted

by r and are driven by DC motors. L is the distance from

wheel’s center to the center of OMP’s geometry, O .

The OMP uses omni-wheels, so it can move in any direc-

tion of x y− plane. Therefore, it can keep the balance of the

rod.

Fig. 3 shows the schematic of the OMP on x y− plane.

The forces applied to the OMP consists of the forces gener-

ated by DC motors on the omni-wheels, ( 1,2,3)i

F i = , the

reaction forces on x axis and y axis by the rod, x

H and

yH . y axis is assumed to be a line connected from the cen-

ter of OMP’s geometry, O , to the center of wheel 1. In Fig. 3,

the direction of ( 1,2,3)i

F i = is assumed is positive direc-

tion.

In this section, to separate the motion of OM-IP into two in-

dependent motions, motion in x z− plane and motion in

y z− plane, the motion of OMP is also separated into motions

of the direction of x axis and the direction of y axis.

The motion of OMP in the direction of x axis means that

the OMP moves only in direction of x axis without rotating

and the total of external forces generated on three wheels only

has the component on x axis, and the component on y

axis is zero. From this statement, the following is obtained:

1 2 3

1 2 3

2 3

0

cos cos3 3

sin sin 0 .3 3

x

F L F L F L

F F F F

F F

π π

π π

+ + =− + + =

− =

(16)

( 1,2,3)i x

F i = is denoted as the force applied to the thi

wheel of the OMP to move in direction of x axis. 1 2,

x xF F

and 3x

F are the solutions of Eq. (16).

From Eq. (16), the followings are obtained:

1

2

3

2

3

1

3

1.

3

x x

x x

x x

F F

F F

F F

= −

==

(17)

Fig. 4 shows the OMP moving in the direction of x

axis. 1xF is negative because its direction is opposite with

1F in Fig. 3.

Similarly, to make the OMP move in the direction of y

axis, the following equations have to be satisfied.

1 2 3

2 3

1 2 3

0

sin sin3 3

cos cos 03 3

y

FL F L F L

F F F

F F F

π π

π π

+ + =

− =− + + =

(18)

( 1,2,3)i y

F i = is denoted as the force applied to the thi

wheel of the OMP to move in direction of y axis. 1 2,

y yF F

Ox

y

1

23

L

1F

3F

2FxH

yH

3π3

π

23π

Fig. 3. Schematic of the OMP on x y− plane.

Ox

y

1

23

L

1xF

3xF

2xFxH

yH

xF

3π3

π

23π

Fig. 4. OMP moving in the direction of x axis.


and 3 y

F are the solutions of Eq. (18). From Eq. (18), the

following equations are obtained:

1

2

3

0 ;

1;

3

1.

3

y

y y

y y

F

F F

F F

=

=

= −

(19)

Fig. 5 shows the OMP moving in the direction of y

axis.

Because the motion in the direction of x axis is independ-

ent on the motion in the direction of y axis, the total force on

each wheel is given as follows:

1 1 1

2 2 2

3 3 3

2

3

1 1

3 3

1 1.

3 3

x y x

x y x y

x y x y

F F F F

F F F F F

F F F F F

= + = −

= + = +

= + = −

(20)

3. Controller design

Because the dynamic equations in x z− plane and y z−

plane are similar, the controllers applied for them are designed

in the same way. In this section, we only write the controller

for x z− plane.

From Eqs. (12) and (13), the dynamic equation of the in-

verted pendulum in x z− plane is as follows:

( ) ( )2cos sinx

M m x lm Fα α α α+ + − =ɺɺ ɺɺɺ (21)

sin cos 0 .l g xα α α− + =ɺɺ ɺɺ (22)

From Eq. (22), the following is obtained:

sin.

cos

g lx

α αα−

=ɺɺ

ɺɺ (23)

Substituting Eq. (23) into Eq. (21) yields:

( ) ( )2sincos sin

cosx

g lF M m lm

α αα α α α

α−

= + + −ɺɺ

ɺɺ ɺ

( )( ) ( )

2

2 2

cos sin sin cos

sin sin

xF M m g m

l M m M m

α α α α αα

α α

− +⇔ = +

− + − +

ɺɺɺ

u

lα ϕ⇔ = +ɺɺ (24)

where

( )( )2

cos sin

sin

xF M m gu

l l M m

α α

α

− +=

− + (25)

( )2

2

sin cos

sin

m

M m

α α αϕ

α=− +

ɺ (26)

where u is part of the control law to be designed.

By defining 1x α= ,

2x α= ɺ , Eq. (23) is rewritten in state

space form as:

1 2x x=ɺ (27)

2.

ux

lϕ= +ɺ (28)

A feedback control law is chosen as following:

( )( )2

cos sin

sin

xF M m gu

l l M m

α α

α

− +=

− +

( )( )2sin

tan .cos

x

u M mF M m g

αα

α

+⇔ = + − (29)

In this section, the backstepping algorithm is used to design

a controller to keep the rod in balance. The angle of the rod

converges to its desired value from a wide set of initial condi-

tions. The controller has not only to guarantee the stability and

the regulation of the tracking error, but also to converge the

estimated value of the unknown parameters towards its true

constant values.

Step 1) A tracking error variable is defined as:

1e rx x x= − (30)

where rx is the reference angle of .α

In applying the backstepping technique, a backstepping er-

ror be is defined as:

2be x δ= − (31)

where 2x is considered as a virtual control input and a stabil-

ity function δ for 2x is defined as:

1 e rK x xδ = − + ɺ (32)

Fig. 5. OMP moving in the direction of y axis.


where 1

0K > is a design parameter.

From Eqs. (27) and (30)-(32), differentiating ex with re-

spect to time yields:

1.

e b ex e K x= −ɺ (33)

The first Lyapunov function candidate associated with the

tracking error is chosen as:

2

1

1V .

2ex= (34)

From Eq. (33), the time derivative of the first Lyapunov

function candidate is obtained as:

2 2

1 1 1

1V ( ) .

2e e e e b e e e bx x x x e K x K x x e= = = − = − +ɺ ɺ ɺ (35)

Eq. (35) cannot guarantee 1

V 0≤ɺ when 0ex ≠ and

0be ≠ . Thus, the second Lyapunov function candidate must

be considered.

Step 2) From Eqs. (31) and (32), differentiating be with re-

spect to time becomes:

2 2 1.

b e re x x K x xδ= − = + −ɺɺ ɺ ɺ ɺ ɺɺ (36)

Eqs. (28), (33) and (36) can be rewritten in the ( ),e bx e

space as:

1e b ex e K x= −ɺ (37)

2

1 1.

b e b r

ue K x K e x

lϕ= − + − + +ɺ ɺɺ (38)

Depending on whether l is known or not, two cases are

considered.

Case 1) If l is known, the second Lyapunov function candi-

date is chosen as:

2

2 1

1V V .

2be= + (39)

A control law is chosen as:

( ) ( )21 2 11 .

b e ru l K K e K x x ϕ = − + + − − +

ɺɺ (40)

It would render the time derivative of the second Lyapunov

function candidate negative as follows:

2 1

2 2

1 1 1

2 2

1 2

V V

0 .

b b

e e b b e b r

e b

e e

uK x x e e K x K e x

l

K x K e

ϕ

= +

= − + + − + − + +

= − − ≤

ɺ ɺ ɺ

ɺɺ (41)

Case 2) If l is unknown, the second Lyapunov function can-

didate is chosen as:

2

3 2

1V V

2l

lλ= + ɶ (42)

where l l l= −⌢

ɶ is the estimating error of l ; l⌢ is the estima-

tion value for l ; 0λ > is an adaptation gain.

To design the control law such that 0ex → and 0

be →

as t→∞ , the following control law is proposed as:

( ) ( )21 2 11 .

b e ru l K K e K x x ϕ = − + + − − +

⌢ɺɺ (43)

The time derivative of the second Lyapunov function can-

didate negative as follows:

( )( )

2 2

3 2 1 2

1 2

2

1

V V

1.

1

e b

b

b

e r

l lK x K e

l

K K ell e

l K x x

λ

λ ϕ

= + = − −

+ + − + + − − +

ɺɶɶɺ ɺ

ɶ ⌢ɺ

ɺɺ

(44)

So update law for l⌢ is chosen for

3Vɺ to be negative as

follows:

( ) ( )21 2 11 .

b b e rl e K K e K x xλ ϕ = + + − − + ⌢ɺ

ɺɺ (45)

The block diagram for the proposed control algorithm of the

OM-IP is shown in Fig. 6.

4. Prototype of the experimental OM-IP

A prototype of the experimental OM-IP is shown in Fig. 7.

The OM-IP is considered as two subsystems of a 2-DOF IP

and an OMP. The 2-DOF IP rotates around a universal joint.

The potentiometer sensor is connected at the universal joint

Fig. 6. Block diagram of the proposed controller.


for measuring its angle. The mobile platform has three omni-

directional wheels that are driven by 3 DC motors.

A potentiometer sensor is used to measure the angles of the

rod as shown in Fig. 8. The sensor has two potentiometers

corresponding to its position in x and y directions, respec-

tively.

A control system is developed based on microcontrollers

PIC18F452’s which are operated with the clock frequency

40MHz. A hardware configuration of the proposed control

system using seven PIC18F452 is shown in Fig. 9.

The projection angle α of the rod on x z− planes from

z axis can be measured by a potentiometer sensor as shown

in Fig. 10.

V (V V ) / 2

/ 3(OV ,OV ) (V V )[ ]

(V V )

0 min max

0 x x 0

max min

radπ

α

= +

= ∠ = × −−

�� (46)

where Vmin and V

max are the minimum and maximum volt-

age which are returned from the potentiometer sensor, respec-

tively. V0 is average voltage of V

min and V

max. The V

x is

the real voltage value on x z− planes which is returned by

the sensor in the real time.

Similarly, the projection angle β of the rod on y z−

planes from z axis can be measured as follows:

V (V V ) / 2

/3(OV ,OV ) (V V )[ ]

(V V )

0 min max

0 y y 0

max min

radπ

β

= +

= ∠ = × −−

�� (47)

where x

V and α of Eq. (47) are replaced by Vy and β ,

respectively.

5. Simulation and experimental result

To verify the effectiveness of the proposed controllers, si-

mulations and experiments have been done for the OM-IP.

In the simulation, the numerical parameter values and the

initial values are given in Tables 1 and 2.

The tracking error be in x z− and y z− planes in the

full time of 20 seconds in simulation are shown in Fig. 11.

They go to zero from 4 seconds. Figs. 12 and 13 show the

projection angles in x z− plane and y z− , respectively.

The continuous line is simulation result and the dot line is

experimental result. The angle errors converge to zero after 4

seconds. The experiment result is bounded around simulation

with 0.025 rad± . Fig. 14 shows the control forces applied

three omni-wheels in simulation and Fig. 15 shows the control

forces applied to three wheels in experiment. In simulation

result, the control forces become zero after the angle errors go

to zero. In experimental results, the control forces are bounded

around zero with 0.04 N m± ⋅ . Fig. 16 shows the estimated

Fig. 9. Control architecture of hardware system of the OM-IP.

Fig. 10. Configuration of measuring the α angle.

Fig. 7. 2-DOF OM-IP in experiment.

yx

zR

l

βα

θ

O

xzl

yzl

yR

xR

COG

Fig. 8. Structure of the potentiometer sensor.


distances from the COG to the rotary point. The real distance

denoted by continuous line is 0.3 m . The dot lines is estima-

tion distance in simulation where xl converges to 0.57 m

and yl converges to 0.18 m . The dashed lines denote for

estimation distance in experiment where xl is about 0.68 m

and yl is about 0.3 m .

6. Conclusions

In this paper, a new idea to develop a 2-DOF inverted pen-

dulum based on OMP is presented and stabilization of a 2-

DOF OM-IP is studied. Based on the dynamic equation, an

adaptive backstepping controller and update law for unknown

parameter are proposed to keep the rod in balance. The stabil-

ity and the convergence properties of OM-IP are guaranteed

by the closed-loop Lyapunov functions.

Simulation and experimental results show that the adaptive

backstepping controller can keep the rod in balance when the

distance from the center of gravity of the rod to the rotary

point on the OMP is unknown. The results show that with a

ideal inverted pendulum, the proposed controller is capable of

making the errors converge to zero. These results demonstrate

the effectiveness of the adaptive backstepping controller and

update law.

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Table 1. Numerical parameter values.

Parameters Values Units

M 0.5 [kg]

m 0.1 [kg]

l 0.3 [m]

L 0.2 [m]

g 9.81 [m/s2]

λ 2

1K 5

2K 5

Table 2. Initial values for simulation.

Parameters Values Units

0α 0.2 [rad]

0β 0.1 [rad]

refα 0 [rad]

refβ 0 [rad]

0x 0 [m]

0y 0 [m]

Fig. 11. Tracking error be in x z− and y z− planes in simula-

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Fig. 12. Projection angle of the rod on x z− plane.

Fig. 13. Projection angle of the rod on y z− plane.

Fig. 14. Forces of wheels in simulation.

Fig. 15. Forces of wheels in experiment.

Fig. 16. Estimated distance from COG to rotating point.


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Tuan Dinh Viet was born in Vietnam

on July 14, 1972. He received the B.S.

degree in the Faculty of Computer

Science, Hochiminh City Open Uni-

versity, Vietnam in 1997. He received

the B.S. degree in the Faculty of In-

formation Technology, College of En-

gineering, University of Danang, Viet-

nam in 2008. He is currently a Ph.D. student in the Dept. of

Mechanical Engineering, Pukyong National University,

Busan, Korea. His fields of interests are computer science,

robust control and mobile robot control.

Phuc Thinh Doan was born in Viet-

nam on January 31, 1985. He received

the B.S. degrees in Dept. of

Mechanical Engineering, Hochiminh

City University of Technology,

Vietnam in 4/2007. He then received

the M.S. degree in the Dept. of

Mechanical Engineering, Pukyong

National University, Busan, Korea in 2/2011. He is cur-

rently a Ph.D. student in the Dept. of Mechanical Engineer-

ing, Pukyong National University, Busan, Korea. His fields

of interests are robotic, power electric, motion control and

mobile robot control.

Giang Hoang was born in Vietnam on

April 24, 1984. He received the B.S.

degree in Dept. of Computer Science,

Hochiminh City University of Tech-

nology, Vietnam in 2009. He then

received the M.S degree in the Dept.

of Mechanical Engineering, Pukyong

National University, Busan, Korea in

2/2012. He is currently a Ph.D student in the Dept. of Me-

chanical Engineering, Pukyong National University, Busan,

Korea. His fields of interests are computer science, robotic

and mobile robot control.

Hak Kyeong Kim was born in Korea

on November 11, 1958. He received

the B.S. and M.S. degrees in Dept. of

Mechanical Engineering from Pusan

National University, Korea in 1983 and

1985. He received Ph.D degree at the

Dept. of Mechatronics Engineering,

Pukyong National University, Busan,

Korea in February, 2002. His fields of interest are robust

control, biomechanical control, mobile robot control, and

image processing control.

Sang Bong Kim was born in Korea on

August 6, 1955. He received the B.S.

and M.S. degrees from National Fish-

eries University of Pusan, Korea in

1978 and 1980. He received PhD.

degree in Tokyo Institute of

Technology, Japan in 1988. After then,

he is a Professor of the Dept. of

Mechanical Engineering, Pukyong National University,

Busan, Korea. His research has been on robust control,

biomechanical control, and mobile robot control.

control of a 2-dof omnidirectional mobile inverted pendulum

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