INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 14, No. 10, pp. 1775-1782 OCTOBER 2013 / 1775

© KSPE and Springer 2013

A Study of the Tracking Control of an Transfer Craneusing Nonholonomic Constraint

Ji-Hyun Jeong1, Young-Bok Kim2, and Dong-Won Jung1,#

1 Department of Mechanical Engineering, Jeju National University, 102 Jejudaehakno, Jeju-si, Jeju-do, South Korea, 690-7562 Department of Mechanical System Engineering, Pukyong National University, San 100, Yongdang-dong, Nam-Gu, Busan, South Korea, 608-739

# Corresponding Author / E-mail: jdwcheju@jejunu.ac.kr, TEL: +82-64-754-3625, FAX: +82-64-756-3886

KEYWORDS: Rubber-tired gantry crane, Nonholonomic system, Sliding mode, LMI

The automation of container cranes is one of the significant factors of global competitiveness, by improving the celerity of distribution

processing in container terminals. Most container terminals operate with Rubber-Tired Gantry Crane (RTGC) type container cranes.

However, the RTGC displays various uncertainties, including slips and scarcities of air pressure of tires during tracking mode, and

position error during suspension, and therefore it is difficult to secure reliability for the automation level, and systematic research

developments remain in inadequate status. First, the problem of tracking control that tracks the route determined at the yard with high

speed and accuracy must be solved, to achieve automation of the RTGC. Since it is virtually impossible to use the steering unit of the

RTGC, there exist constraints of having to perform tracking control through the speed control of both drive units. Thus, the RTGC is

a system with nonholonomic constraint, as the typical nonlinear system driven by tires. In this study, the nonholonomic system is applied

to conduct the overall system modeling, through kinematic modeling and dynamic modeling of the RTGC. Furthermore, the sliding

mode control technique is applied as one of the discontinuous time invariant control techniques, for the purpose of simultaneously

accomplishing system stabilization, and overcoming model uncertainties. The gains of sliding planes will be analytically found by using

the LMI technique, and performance of the proposed control unit will be confirmed through the experiment.

Manuscript received: March 4, 2013 / Accepted: July 16, 2013

1. Introduction

Recently, container terminals have striven to secure international

competitiveness, by efficiently dealing with mega-sized container

ships. Korea has also considered measures with various efforts in

various fields, including new port constructions, to secure its

competitiveness from the threatening challenges of China. The most

basic requirement of a port with international competitiveness can be

described as the speedy processing of distribution. Therefore, the

automation of container cranes in container terminals will improve the

speed of distribution processes, to serve as one of the significant factors

in fulfilling global competitiveness. Most of the cranes operated in

container terminals across the globe are the Rubber-Tired Gantry Crane

(RTGC) type Transfer Cranes (TC). With the exclusion of new

developments, the domestic container terminals, like the foreign

container terminals, are operating the RTGC type TCs in most

situations, including Busan North Port, Gwangyang Port, and Incheon

Port. Furthermore, such operations have caused a great burden of

operating expenses. Due to such reasons, there is a need to understand

the distinct characteristics of RTGC that occupy most of the existing

port cranes, and to automate the system. In comparison with the Rail

Mounted Gantry Crane (RMGC), the RTGC secures autonomous

movements; however, it displays various uncertainties of self-twist

angle and position error during suspension mode. In particular, the

usage of the steering unit is virtually impossible, causing difficulties for

automation, with the constraint of having to control tracking through

the speed control of drive units on both sides. Tracking systems can be

subdivided into two types, the holonomic System, capable of

simultaneously operating the steering unit, to instantaneously change

the course of direction in any way, and the nonholonomic System,

incapable of instantaneously changing the course of direction, to

change the course of direction while in motion. With the mere

consideration for kinematics of the controlled system, the position

control unit assumes perfect velocity tracking. However, it is difficult

to conduct perfect velocity tracking in actuality, and therefore in recent

times, studies are being conducted with simultaneous consideration of

both kinematics and dynamics. The nonholonomic System has the

constraint that disables the system from instantaneously changing

DOI: 10.1007/s12541-013-0237-1

1776 / OCTOBER 2013 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 14, No. 10

course of direction in motion, and this becomes a significant problem,

when designing the control unit of the nonholonomic System.

This study will consider the RTGC as a nonholonomic System, to

construct a control system to solve the most basic problem in the

automation of RTGC, which is to accurately move to a given position,

according to a determined route.1-3 In other words, the study will

consider the various problems that may occur due to properties of tire

transfer cranes, to model the overall system through kinematic and

dynamic modeling, as well as review the controller design problem that

is enabled to track the determined routes with high accuracy, while

being tough against disturbances. To solve the control problem of

RTGC with kinematic constraints, the sliding mode control technique

is applied as one of the design techniques to enable a nonlinear

controller to simultaneously handle the system stabilization and model

uncertainties. In this study, a matrix formed of vectors generating null

spaces is introduced, to reflect the constraints of nonholonomic

Systems in the kinematic equation. Next, the sliding mode controller is

designed, which asymptotically converges all state variable errors of

the system to 0. Here, the gain of sliding surface is analytically found

using the LMI technique, and by referring to the research results of

Jeong et al.,10,11 where each parameter is used to verify the performance

of the proposed controller through experimentation.

2. Modeling of Nonholonomic RTGC

2.1 Kinematic modeling

As shown in Fig. 1, this study considers the RTGC with the same

isocenter as the drive shaft, and composed of 2 drive units. When the

position of RTGC is defined as xc and yc, and the angle formed by the

RTGC axis on the X-axis is defined as θ on the rectangular coordinate

axis, the crane can be expressed as the generalized coordinates

composed in the following 5 conditions.

(1)

where, q indicates the generalized coordinate vector, and θr, θl indicates

the rotation angle of the left and right drive-wheels.

The following constraints are established when assuming that the

crane is incapable of vertical movement on the symmetry axis of

forward direction, and the drive-wheels rotate without slip.

(2)

(3)

(4)

Here, l1 indicates the distance between drive-wheels and the

barycentric coordinates of the crane, and r indicates the radius of the

drive-wheel.

Eqs. (2) and (3) can be summarized, to indicate the following.

(5)

(6)

The two equations can be summarized as a matrix, to indicate the

following.

(7)

Here, υ = [v w]T is the velocity vector, which is mainly composed

of the tangential flow velocity v, and the angular velocity w in

barycentric coordinates of the crane, while is the vector

mainly composed of the angular velocity from the left and right drive

units of the crane.

Since Eqs. (2)-(4) are incapable of gaining clear relational

expressions between variables, due to integration, they become the

nonholonomic constraint, and can be indicated as the following, when

expressed in the form of vectors.

(8)

where,

(9)

The kinematic equation of the crane from nonholonomic constraint

expression (8) can be expressed as follows.4

(10)

Here, Sn(q) is defined in the following as the matrix in the null space

of matrix A, to remove the Lagrange multipliers.

(11)

(12)

q xc yc θ θr θl, , , ,[ ]=

x·c θsin y·c θcos– 0=

x·c θcos y·c θsin l1θ·

+ + rθ·r=

x·c θcos y·c θsin l1θ·

–+ rθ·l=

w θ· r

2l1

------- θ·r θ

·l–( )= =

v x·c θcos y·c θsin+r

2--- θ

·r θ

·l+( )= =

υ v

w

r

2---

r

2---

r

2l1

-------r

2l1

-------–

θ·r

θ·l

Jη= = =

η θ·r θ

·l[ ]T

=

A q( )q· 0=

A q( )θsin θcos– 0 0 0

θcos– θsin– l1

– r 0

θcos– θsin– l1

0 r

=

q· Sn q( )η=

A q( )Sn q( ) 0=

Sn q( )

r

2--- θcos

r

2--- θcos

r

2--- θsin

r

2--- θsin

r

2l1

-------r

2l1

-------–

1 0

0 1

=

Fig. 1 A Schematic diagram for analysing system dynamics

INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 14, No. 10 OCTOBER 2013 / 1777

2.2 Dynamic modeling

The dynamic motion model of cranes can be gained from Fig. 1 by

using the Lagrange equation, provided that the cranes move on

horizontal planes, and the potential energy is ignored.

, (i = 1, ..., 5) (13)

where, : kinetic energy, D : loss function, λ : Lagrange multiplier, τi :

torque of drive unit, and qi : generalized coordinates. The kinetic

energy and loss function are as follows.

(14)

(15)

where, Mc : total mass of crane including the container, J1 : rotary

inertia moment at the crane’s center of mass, Jw : rotary inertia moment

of drive unit, D1 : damping coefficient of rotary motion at crane’s

center of mass, and Dcx, Dcy : damping coefficient of crane’s translation.

At this time, the generalized coordinate qi can be expressed as qi =

[xc, yc, θ, θr, θl]T, to indicate the equation of motion, as displayed in the

following.

(16)

(17)

(18)

(19)

(20)

Here, τr, τi is the torque applied to the left and right drive-wheels.

The expression stated above can be expressed in the form of a

matrix, as displayed in the following.

(21)

where,

(22)

, (23)

, (24)

Both sides of Eq. (21) can be multiplied with ST, to consider

and , to produce the following results.

(25)

When Eq. (10) and the differentiation of Eq. (10) are substituted

into the above equation, it will produce the following results.

(26)

where,

(27)

is:

(28)

Therefore, if it is defined as stated above, Eq. (26) can be

summarized as displayed in the following.

(29)

3. Sliding Mode Controller Design using LMI

As the controlled system from kinematic and dynamic expressions

described in Chapter 2, the RTGC can be expressed by the following

state equation.

(30)

where, u = τ = [τr τl]T, and . If the input

voltage of each drive motor is described as V1, V2, and the torque

constant of motor is described as Km1, Km2, the driving power of left

and right drive motors can be described as τ = [Km1V1 Km2V2].

To solve the tracking problem of cranes, the target velocity vr and

target angular velocity wr are placed to display the equation of motion

for the target value, as follows.

(31)

where, xd and yd are the target coordinates where the crane’s center of

gravity must be positioned, and θd is the target angle where the crane’s

center of gravity for the standard coordinates must be accomplished.

d

dt----

∂ℑℑq· i--------⎝ ⎠⎛ ⎞ ∂ℑ

∂qi

-------–∂D

∂q·-------+ τi Aij

Tλ j

j=1

3

∑–=

ℑ

ℑ 1

2---Mc x·c

2y·c2

+( ) 1

2---J

1θ· 2 1

2---Jw θ

·r θ

·l+( )+ +=

D1

2---Dcxx

·c

2 1

2---Dcyy

·c

2 1

2---D

1θ· 2

+ +=

Mcx··c Dcxx

·c+ λ

1θsin– λ

2λ3

+( ) θcos+=

Mcy··c Dcyy

·c+ λ

1θcos λ

2λ3

+( ) θsin+=

J1θ··

D1θ·

+ l1λ3

λ2

–( )–=

Jwθ··r τr λ

2r–=

Jwθ··l τ l λ

3r–=

Mq·· Vq·+ Eτ ATλ–=

M

Mc 0 0 0 0

0 Mc 0 0 0

0 0 J1

0 0

0 0 0 Jw 0

0 0 0 0 Jw

=

V

Dcx 0 0 0 0

0 Dcy 0 0 0

0 0 D1

0 0

0 0 0 0 0

0 0 0 0 0

= E

0 0

0 0

0 0

1 0

0 1

=

λ

λ1

λ2

λ3

= ττr

τl

=

Sn

Tq( )AT

q( ) 0= Sn

Tq( )E q( ) I

2 2×=

Sn

Tq( )M q( )q·· Sn

Tq( )V q( )q·+ τ=

Sn

Tq( )M q( )Sn q( )η· Sn

Tq( )M q( )S·n q( ) Sn

Tq( )V q( )Sn+( )η+ τ=

Sn

TMS

·n

0 0

0 0=

M Sn

TMSn

r2

4l1

2------- Mcl1

2J1

+( ) Jw+r2

4l1

2------- Mcl1

2J1

–( )

r2

4l1

2------- Mcl1

2J1

–( ) r2

4l1

2------- Mcl1

2J1

+( ) Jw+

= =

V Sn

TVSn

r2

4---- Dcx Dcy

D1

l1

2------+ +

⎝ ⎠⎜ ⎟⎛ ⎞ r

2

4---- Dcx Dcy

D1

l1

2------–+

⎝ ⎠⎜ ⎟⎛ ⎞

r2

4---- Dcx Dcy

D1

l1

2------–+

⎝ ⎠⎜ ⎟⎛ ⎞ r

2

4---- Dcx Dcy

D1

l1

2------+ +

⎝ ⎠⎜ ⎟⎛ ⎞

= =

Mq· Vη+ τ=

η· Aη Bu+=

A M( )1–

V–= B M( )1–

=

x·d vr θdcos=

y·d vr θdsin=

θ·d wr=

1778 / OCTOBER 2013 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 14, No. 10

Also, the tracking error ev is defined as the following equation.

(32)

The above equation is differentiated, to present the following

equation.

(33)

For stabilized interpretation, the Lyapunov inequation is selected, as

displayed in the following equation,

(34)

In the tracking error Eq. (32), V0 can be differentiated from the

trajectory ev, to produce the following equation.

(35)

where, k1, k2 is the scalar parameter of the amount, and V0 is the correct

scalar function. To satisfy the V0 ≤ 0 condition for the asymptotic

stability of the system, the desired linear velocity vd and angular

velocity wd must be selected as follows.

(36)

When using Eq. (7), the desired angular velocity of the left and right

drive units θrd, θld can be indicated as follows.

(37)

In order to apply the sliding mode control on the servosystem, the

difference of target value d and output y is integrated, to find the state

variable z1, which is added to the state variable of Eq. (30), to compose

the servosystem to find the following results.

(38)

where, ,

Furthermore, Eq. (38) can be written as follows.

(39)

The following can be established for the system of Eq. (39), from

the assumptions.5,6

A1: , can be stabilized.

A2: All status information is possible for usage.

A3: rank

Also, the linear sliding surface for sliding motion is defined as

follows.7-9

(40)

Provided that S must satisfy the following properties in the m×n

matrix.

A4: is regular.

A5: Sliding mode dynamics of (n − m), restricted to the sliding

surface of Sz = 0, are asymptotically stable.

Ultimately, the designing of a sliding mode controller is similar to

finding the matrix S, which satisfies the conditions A1-A5.

The following equivalent control system can be found when

applying , the condition of occurrence of sliding motion, to Eq.

(39).

(41)

To find the gain of sliding surface S, the method of selecting the

feedback gain F, based on the optimal control theory, is used, as shown

in Eq. (42).

(42)

The necessary and sufficient conditions for the equivalent control

system of Eq. (41) to achieve asymptotic stability are defined by the

following existence of positive definite matrix P that satisfies LMI.12-14

(43)

The LMI of Eq. (43) is enabled to efficiently find the value through

the optimized algorithm, like the MATLAB LMI Control Toolbox.15

In the sliding mode control technique, the control input is generally

composed of the linear equivalent control input ueq and the control

input uN, which includes the nonlinear elements, as displayed in the

following equation.

(44)

If the condition for occurrences of sliding motion is defined as =0

in Eq. (40) to find the equivalent control input ueq, and is summarized

by substituting in Eq. (39), the equivalent control input of sliding

servosystem can be found as follows.

(45)

ev

e1

e2

e3

θcos θsin 0

θsin– θcos 0

0 0 1

xd xc–

yd yc–

θd θ–

= =

e·1

e·2

e·3

1– e2

0 e1

–

0 1–

v

w

vr e3

cos

vr e3

sin

wr

+=

V0

1

2---e

1

2 1

2---e

2

2 1 e3

cos–

k2

--------------------+ + 0≥=

V·0 e

1e·1

e2e·2

e3

sin

k2

------------e·3

+ +=

=e1

v– vr e3

cos+( )e3

sin

k2

------------ wr w– k2vre2+( )+

vd

wd

vr e3

cos k1e1

+

wr k2vre2

e3

sin

k2

------------+ +=

θ·rd

θ·ld

J1– vd

wd

=

z1

θ·rd θ

·ld[ ]

Tθ·r θ

·l[ ]T

–( ) d

dt----

z1

z2

∫=

=0 I–

0 A

z1

z2

0

Bu I

0d+ +

z2

θ·r θ

·l[ ]T

= d θ·rd θ

·ld[ ]

T=

z· Az Bu Ed+ +=

A B

B( ) m n<=

σ z( ) Sz S1

S2

z1

z2

S1z1

S2z2

+= = =

SB

σ· 0=

z· A B SB( )1–

SA–{ }z B SB( )1–

SEd–=

F S BTP= =

PA ATP+ PB

BTP I

0<

u ueq uN+=

σ·

ueq S2B( )

1–

S1d– S

1A+( )z

2+{ }–=

Fig. 2 Servosystem with sliding mode controller

INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 14, No. 10 OCTOBER 2013 / 1779

The nonlinear control input uN should enable the system condition

to reach the sliding surface when the system condition is outside the

sliding surface, and also, be defined as displayed in the following, to

play the role of disabling deviation from the sliding surface.

(46)

is defined as the positive definite matrix. The sliding

mode base controller of RTGC induced, along with the above equations,

can be indicated as block diagram, as seen in Fig. 2. Ultimately, each

matrix calculated to construct the sliding mode servosystem, is as

follows.

(47)

(48)

(49)

(50)

(51)

(52)

(53)

4. Experimental Apparatus and Results

4.1 Plant model and measurement system

As shown in Fig. 3, the plant model was constructed for the

experimentation in this study. The properties and specifications of the

plant model are summarized in Table 1. In this study, the sliding mode

controller was designed, with the controlled system of pilot RTGC

produced as shown in Fig. 3. The experimental method set the

calculated control signals onto the two drive motors. At this point, the

uN

B1–

Kσ

σ

σ------–

0⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

=if σ t( ) 0≠

if σ t( ) 0=

Kσ

Rn n×∈

A

0 0 1.0000– 0

0 0 0 1.0000–

0 0 1.1641– 0.6133–

0 0 0.6133– 1.1641–

=

B 0 0 159.1108 16.1049–

0 0 16.1049– 159.1108

T

=

B1– 0.0063 0.0006

0.0006 0.0063=

P

0 0 0.3638– 0.0728–

0 0 0.0728– 0.3638–

0.3638– 0.0728– 0.4682– 0.3079–

0.0728– 0.3638– 0.3079– 0.4682–

=

S 0.0057– 0.0006– 0.0070– 0.0041–

0.0006– 0.0057– 0.0041– 0.0070–=

Kσ 3 3

T=

ueq1.1269– 0.0030

0.0030 1.1269–z2

0.0071 0.0032–

0.0032– 0.0071d 0.0210–

0.0210–σ( )sgn+ +=

Table 1 Specification of the plant model

Items Specification

Scale 1/24

Trolley winding speed 0.150 [m/sec]

Crane speed [max] 0.270 [m/sec]

Height of crane (h) 1.013 [m]

Width of crane (l) 1.010 [m]

Total weight of crane (Mc) 15.0 [kg]

Motor

Nominal voltage 12 [V]

Nominal torque 2.45×10-3 [N·m]

Nominal power 1.8 [W]

Fig. 3 Controlled system (Pilot RTGC)

Fig. 4 Step responses when the reference values are varied

1780 / OCTOBER 2013 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 14, No. 10

encoders were installed on the wheels of both sides, to measure the

traveling distance. Furthermore, two potentiometers were installed in

the front and rear to the progressing direction on the right side of the

drive unit, to measure the degree of breakaway from the set route. The

two potentiometers survey to measure the crane’s traveling direction,

and breakaway from the set route. For example, if the crane breaks

away from the driving route due to tire slip, a voluntary rotary angle

occurs between the crane and the guideline that is the set route. Since

the actual traveling distance of the opposite drive axis can be calculated

through the rotary angle and the encoder signal on the other side of the

drive axis, it is possible to correct the occurred errors through the

controller. In this study, the required rotary angle was measured by the

two potentiometers, and the traveling distance was measured by the

encoders.

4.2 Experimental results

In the previous passage, viewed through the prepared plant model,

the experiment was conducted by using the control input Eq. (53),

designed according to the sliding surface design technique. The

potentiometer is used to measure the crane’s degree of breakaway from

route xc and slope θ, while the encoder is used to measure the traveling

distance of the crane’s center of gravity yc. Fig. 4 displays the

experimental results of tracking during the ideal state, without

disturbances that generate tire slips. This displays the actual left and

right traveling distances of y1, y2 when the target value is initially set

at 0.7 m, and changed to 0 m, 60 seconds later. Furthermore, it can be

found that the traveling distances of both drive-wheels match the target

value. Figs. 5-8 display the experimental results of tracking, for cases

with breakaways from the tracking route. In other words, it considers

the cases with occurrences of rotary angle focused on set routes, due to

disturbances such as tire slips, or lack of air pressure. In Fig. 5,

Fig. 5 Measured results of potentiometers

Fig. 6 Control inputs to DC motor

Fig. 7 Measured results of potentiometers

INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 14, No. 10 OCTOBER 2013 / 1781

disturbances are placed 3.5 mm in the right side tire of the crane, at

approximately 25-40 seconds. At that moment, the rotary degree

information and the traveling distance of left and right sides x1, x2 were

calculated through the two potentiometers installed on the front and

rear sides, as displayed in Fig. 3. The cranes tend to breakaway to the

right side, when placed with disturbances similar to Fig. 5. As shown

in Fig. 6, the changes in control input occur by the sliding mode

controller, because the crane deviates from the target value (angle of

advance (θ = 90o), and left-right side traveling distance (x1 = x2 = 0 m)).

This means that the controller sends smaller control input (approx.

2.9 V) to the left-side drive unit, and greater control input (approx.

3.1 V) to the right-side drive unit, to restore the crane that has deviated

towards the right side from its set course. In Fig. 7, disturbances are

placed 4.5 mm in the left-side tire of the crane, at approximately 27-

38 seconds. When such disturbances are placed, the crane deviates

from the set route towards the left side. The sliding mode controller

causes control input changes, as shown in Fig. 8, because at this time,

the crane deviates from the target value (angle of advance (θ = 90o) and

left-right side traveling distance (x1 = x2 = 0 m)). This means that the

controller sends smaller control input (approx. 2.9 V) to the right side

drive unit, and greater control input (approx. 3.1 V) to the left side

drive unit, to restore the crane that has deviated towards the right side

from its set course. As a simple design method that doesn’t require

much information for the control system, the servosystem design

method using the sliding mode presented in this study, as suggested by

this study, enables the user to obtain sliding surface gain, by using the

values of LMI sufficient conditions. The suggested LMI sufficient

conditions can be less conservative when compared to the existing

conditions, and from the experimental results, it tracks the set routes

without any errors. Therefore, it was confirmed that the sliding surface

gain, analytically found using the LMI technique, was capable of

supplementing the existing methods.

5. Conclusions

This study designed the sliding mode controller for RTGC with

nonholonomic constraints, and verified its effectiveness through

experimentation. RTGC is the unloading equipment with outstanding

utility in actual ports, and excellent autonomy in tracking However,

uncertainties. Therefore, this study was conducted with consideration

for such problems, to design the controller by using the sliding mode

controller, with easy application on automation of RTGC, and strong

properties against external forces. Since the traveling direction cannot

be instantaneously changed, due to the nature of the actual RTGC,

another nonholonomic constraint was used to model the RTGC. First,

the RTGC was expressed in the generalized coordinates constructed in

5 conditions, and conducted the kinematic modeling, by using the

nonholonomic constraint. Furthermore, the dynamic model of RTGC

was expressed as velocity constant of drive unit on both sides, by

introducing the matrix that generated null spaces. To solve the tracking

problem, the target velocity and target angular velocity were introduced

to design the sliding mode controller, which asymptotically converges

all state variable errors of systems to 0. At that moment, the LMI was

used to analytically find the gain matrix of sliding surface for the

RTGC’s motion trajectory, to reach the sliding surface. The

experimental process evaluated the performance of the sliding mode

controller, and also confirmed that it was capable of sufficiently

achieving the position control target of the RTGC.

ACKNOWLEDGEMENT

This research was supported by the 2013 scientific promotion

program funded by Jeju National University

REFERENCES

1. Chang, Y. C. and Chen, B. S., “Adaptive tracking control design of

nonholonomic mechanical systems,” Proc. of the 35th IEEE

Conference on Decision and Control, Vol. 4, pp. 4739-4744, 1996.

2. Chu, J. U., Youn, I., Choi, K., and Lee, Y. J., “Human-following

robot using tether steering,” Int. J. Precis. Eng. Manuf., Vol. 12, No.

5, pp. 899-906, 2011.

3. Dinh, V. T., Nguyen, H., Shin, S. M., Kim, H. K., Kim, S. B., and

Byun, G. S., “Tracking control of omnidirectional mobile platform

with disturbance using differential sliding mode controller,” Int. J.

Precis. Eng. Manuf., Vol. 13, No. 1, pp. 39-48, 2012.

4. Jeon, Y. B., Kam, B. O., Park, S. S., and Kim, S. B., “Seam tracking

and welding speed control of mobile robot for lattice type welding,”

Proc. of IEEE International Symposium on Industrial Electronics,

Vol. 2, pp. 857-862, 2001.Fig. 8 Control inputs to DC motor

1782 / OCTOBER 2013 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 14, No. 10

5. Cho, H. S., “The design of sliding mode controller with nonlinear

sliding surfaces,” Journal of The Korea Academia-Industrial

Cooperation Society, Vol. 10, No. 12, pp. 3622-3625, 2009.

6. Yoo, D. S., “Integral sliding mode control for robot manipulators,”

Journal of Institute of Control, Robotics and Systems, Vol. 14, No.

12, pp. 1266-1269, 2008.

7. Yang, J. N., Wu, J. C., and Agrawal, A. K., “Sliding mode control

for seismically excited linear structures,” Journal of Engineering

Mechanics, Vol. 121, No. 12, pp. 1386-1390, 1995.

8. Adhikari, R. and Yamaguchi, H., “Sliding mode control of buildings

with a TMD,” Earthquake Engineering & Structural Dynamics, Vol.

26, No. 4, pp. 409-422, 1997.

9. Kim, H. K. and Lee, B. H., “The low current starting simulation of a

single phase induction motor using sliding mode control,” Journal of

the Korean Institute of Illuminating and Electrical Installation

Engineers, Vol. 21, No. 8, pp. 44-53, 2007.

10. Jeong, J. H., Lee, D. S., Jang, J. S., and Kim, Y. B., “A study on

modelling and tracking control system design of RTGC(Rubber-

Tired Gantry Crane),” Journal of Navigation and Port Research, Vol.

34, No. 6, pp. 479-485, 2010.

11. Jeong, J. H., Lee, D. S., and Kim, Y. B., “A study on the tracking

control of a transfer crane with tire slip,” Journal of Institute of

Control, Robotics and Systems, Vol. 16, No. 12, pp. 1212-1219,

2010.

12. Boyd, S., Ghaoul, L. E., Feron, E., and Balakrishnan, V., “Linear

matrix inequalities in system and control theory,” Society for

Industrial and Applied Mathematics, pp. 7-27, 1994.

13. Chapra, S. C., "Applied numerical methods: With MATLAB for

engineers and scientists," McGraw-Hill, 2004.

14. Kiusalaas, J., "Numerical methods in engineering with MATLAB,"

Cambridge University Press, 2005.

15. Gahinet, P., Nemirovskii, A., Laub, A. J., and Chilali, M., "The LMI

control toolbox," Proc. of the 33rd IEEE Conference on Decision and

Control, Vol. 3, pp. 2038-2041, 1994.