In this study, we successfully designed a sliding mode
controller for a complicated operation of an overhead
crane: simultaneously combining control of cargo lifting,
trolley moving, and cargo swing vanishing. From the
simulation and experiment results, all system responses
are asymptotically stable: cargo swing completely
vanished and trolley motion and cargo lifting/lowering
accurately reached the reference values. Furthermore, the
proposed controller stabilized the crane system even if
the overhead crane is an under-actuated system with a
wide range of varying uncertainties. For the next research,
we will enhance this sliding mode control problem for 3D
overhead cranes.

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Design of Sliding Mode Controller for the 2D
Motion of an Overhead Crane with Varying
Cable Length
Le Anh Tuan
Department of Mechanical Engineering, Vietnam Maritime University, Vietnam
Email: tuanla.ck@vimaru.edu.vn
Abstract—Overhead cranes are under-actuated systems.
They have three outputs that need to be controlled,
consisting of cargo swing angle, trolley displacement, and
suspended cable length, but only two actuators: cargo lifting
and trolley driving forces. The main objective of this study
is to design a robust controller for an overhead crane that
transfers cargo from point to point as fast as possible and, at
the same time, keeps the cargo swing angle small during the
transfer process and make it completely vanish at the
desired cargo destination. The proposed controller must
simultaneously carry out three duties: minimize cargo swing,
track trolley to the desired destination, and lift/lower cargo
to the reference length of cable. The controller is designed
based on a sliding mode control technique. To validate the
proposed control quality, a stability analysis of the system is
discussed and the response analysis is executed with both
MATLAB simulation and experimental research. The
simulation and experiment results show that the crane
system is stable and has the desired behavior.
Index Terms—Lyapunov function, overhead cranes, sliding
mode control, switching suface, under-actuated systems
I. INTRODUCTION
Overhead cranes are widely used in many different
industrial fields such as shipyards, automotive factories,
and other industrial factories. To increase productivity,
many types of cranes are required in fast operation. This
means the time cycle of cargo transport must be short.
The fast operation of overhead cranes without control
leads to cargo swing on wire rope - the faster the cargo
transport, the larger the cargo swing angle. This results in
a dangerous situation during the operation process; it is
possible to damage the factory, the crane, and other
equipment. More seriously, it may cause accidents if the
cargo swing angle is too large.
Papers on the control of overhead crane could be
divided into two groups: control of two-dimensional (2D)
overhead crane and three-dimensional (3D) overhead
crane. Many crane control techniques are also available.
Manuscript received August 24, 2014; accepted June 3, 2015.
This research is funded by the Vietnam National Foundation for
Science and Technology Development (NAFOSTED) under grant
number 107.01-2013.04.
Some authors concentrate on crane nonlinear control [1]-
[3]. Many researchers recently focused on intelligent
control approaches of cranes such as fuzzy logic [4], [5],
neural network control [6], and so on.
The robust control of cranes, especially structure
variable control (SVC) technique, has been studied by
many researchers. Lee [7] suggested a sliding mode anti-
swing control for overhead cranes designed based on
Lyapunov stability theorem. Shyu [8], [9] presented a
sliding mode control (SMC) to minimize swing angle and
maximize trolley speed. Almutairi [10] dealt with SMC
for 3D overhead crane using a fully dynamic model
including five nonlinear second order differential
equations. His study proposed an observer to estimate
immeasurable states of 3D crane system. Liu [11]
considered an adaptive sliding mode fuzzy control
approach for overhead cranes in case of combination of
trolley moving and bridge traveling. However, the cargo
suspended cable is viewed as a constant length element.
The works [8]-[10] only achieve the simulation results
without experiment. Motivated by [8]-[11], we propose
the sliding mode controller for overhead crane in which
the variation of cargo lifted cable is taken into account.
And, both theoretical and experimental results are shown
in our study.
The SMC approach is classified as a variable structure
control technique with many advantages. It is known to
be robust, easy to implement, and insensitive to
uncertainties and disturbance [12]. Its robustness is due to
a natural capability to deal with uncertain objects. It is
especially suitable for under-actuated systems with
uncertainties. 2D overhead cranes are under-actuated
systems in which cargo mass is considered uncertain.
They have three output variables that need to be
controlled (cargo swing angle, trolley displacement, and
cable length) and only two control inputs (trolley moving
and cargo lifting forces). Therefore, the SMC technique is
a proper selection in this case. Furthermore, cargo swing
is directly concerned with trolley motion and length of
cable. Thus, simultaneously combining the control of
these output variables is not easy to implement and needs
proper control actions. In this article, we propose a new
robust controller for 2D overhead cranes based on the
SMC technique. The proposed controller concurrently
Journal of Automation and Control Engineering Vol. 4, No. 3, June 2016
©2016 Journal of Automation and Control Engineering 181
doi: 10.18178/joace.4.3.181-188
executes three duties consisting of vanishing cargo swing,
tracking trolley, and lifting/lowering cargo to desired
positions. The suggested controller stabilizes the crane
system and guarantees convergence of system responses
to desired values.
Figure 1. Physical modeling of 2D overhead crane
This paper is organized as follows. In Section 2, we
establish a physical model, a fully nonlinear
mathematical model of 2D overhead crane, and its
compact form. Section 3 presents the design of a sliding
mode controller including switching surface selection and
SMC scheme design. System stability analysis is shown
in Section 4. Simulation of system responses,
experimental study, and results analysis are given in
Section 5. Finally, some concluding remarks are
presented in Section 6.
II. SYSTEM DYNAMICS
In this section, we build a dynamic model of an
overhead crane that simultaneously combines trolley and
cargo lifting motions. A physical model is given in Fig. 1.
The dynamic system has three masses composed of mt, mc,
and ml. The cargo mass mc and trolley mass mt are
considered point masses concentrated at their centers. ml
denotes equivalent mass of all rotating components of the
cargo lifting mechanism. Chosen generalized coordinates
of the system include x(t), l(t), and (t), namely, trolley
displacement, cable length, and cargo swing angle,
respectively. Furthermore, frictions of trolley moving and
cargo hoisting are respectively characterized by bt and br.
Forces of driving motors of trolley travelling and cargo
lifting ut, ul, are created so that the trolley moves and
handles cargo from the starting point to its destination as
fast as possible and at the same time, minimizes the cargo
swing.
For convenience, the following assumptions are given.
(i) The mass and elastics of wire rope are neglected. (ii)
There is no effect of disturbance caused by wind outside
the factory floor since the overhead crane usually works
indoors. (iii) The motions of all components of system
are considered in a plane.
By using virtual work principle and Lagrange’s
equation, we can derive the motion equations describing
the system dynamics as follows
2
sin cos
2 cos sin
t c c c
t
t c c
m m x m l m l
u t
b x m l m l
(1)
2
sin
cos
c c l
l
r c c
m x m m l
u t
b l m l m g
(2)
cos 2 sin 0x l l g
(3)
The system dynamics including Equations (1), (2), and
(3) can be rewritten in the matrix form
, M q q C q q q G q F
(4)
where, TM q M q is symmetric mass matrix. ,C q q
denotes damping and centrifugal matrix. G q is a matrix
of gravity. F denotes a matrix of control forces of
driving motors. These matrices are determined as follows
11 12 13
21 22
31 33
0
0
m m m
m m
m m
M q ;
11 12 13
22 23
32 33
, 0
0
c c c
c c
c c
C q q ;
0
t
l
u
u
F ; 2
3
0
g
g
G q ;
x
l
q ;
The coefficients of M q matrix are given by
11 12 13
21 22 31
2
33
; sin ; cos ;
in ; ; cos ;
;
t c c c
c c l c
c
m m m m m m m l
m m s m m m m m l
m m l
The coefficients of ,C q q matrix are determined by
11 12 13
22 23 32 33
; cos ; sin cos ;
; ; ; ;
t c c c
r c c c
c b c m c m l m l
c b c m l c m l c m ll
The nonzero coefficients of G q vector are given by
2 3cos ; sin ;c cg m g g m lg
An overhead crane is an under-actuated system. The
system has three controlled outputs but only two
actuators, ut and ul. Therefore, we separate the
mathematical model of the crane into two auxiliary
system dynamics: un-actuated and actuated mathematical
models. Similarly, three generalized coordinates need to
be separated: 1
T
x lq for actuated and for un-
actuated dynamics. To determine the un-actuated state ,
we can rewrite dynamics (3) as follows
1
cos 2 sinx l g
l
(5)
From the previous equation, we can realize that the
cargo swing angle is directly affected by properties of
the trolley motion x and the length of wire rope l.
Substituting (5) into (1) and combining with (2), we
obtain the following actuated mathematical model
Journal of Automation and Control Engineering Vol. 4, No. 3, June 2016
©2016 Journal of Automation and Control Engineering 182
2cos sin
sin sin cos
t c c c
t
t c c
m m m x m l
u t
b x m l m g
(6)
in
cos
c c l r
l
c c
m s x m m l b l
u t
m l m g
(7)
The previous actuated dynamics can be rewritten in
matrix equation form
1 1 1, M q q Bq C q q +G q U
(8)
where,
2sin sin
;
sin
t c c
c c l
m m m
m m m
M q
0
;
0
t
r
b
b
B
sin
, ;c
c
m l
m l
C q q
sin cos
;
cos
c
c
m g
m g
G q
1 ;
t
l
u
u
U
Equation (8) can be represented into reduced order
dynamics
1 1 M q q Bq +G q U
(9)
where, 1 , U U C q q
mathematical model governed by Equation (9) is used to
section.
III. DESIGN OF SLIDING MODE CONTROLLER
In this section, we propose a sliding mode controller
that moves the trolley from an initial position to its
destination as fast as possible. Simultaneously, cargo
vibration must completely vanish when the trolley arrives
at the desired destination. Assume that all state variables
are measurable. The design of the sliding mode controller
is composed of two phases. First, we design a sliding
surface in which the state trajectories restricted to that
surface has the desired system behavior. Second, we
design a control scheme in which the system is stable on
the sliding surface. For this system, the switching surface
is proposed that the actuated states 1
T
x lq
must
come to desired constant values 1
T
d d dx lq and the
cargo swing angle vanishes; this means un-actuated
parameter approaches 0.dθ Let us define tracking
error vectors
1 d dx x l l e ; 2 de
Next, let us define a sliding surface as a linear
combination of position and velocity errors
T
1 2 1 1 2s s e s = e e
(12)
where,
and are the design parameters determined
by 1 2diag , and 1 0 .
T
Differentiating the sliding surface s with respect to
time leads to
1 1 s = q q
(13)
After designing a sliding surface, we construct a
feedback controller. Matrix M q is positive definite for
every 0l and / 2. Equation (9) can be rewritten
as
11 1 q M q U Bq G q
(14)
Substituting Equation (14) to Equation (13) and setting
,s 0 we obtain
1 U B M q q M q G q
(15)
The matrix Equation (4) does not completely describe
system behavior; it is just an approximation. Therefore,
an approximated control law where s 0 can be
presented as
1ˆ U B M q q M q G q
(16)
Furthermore, to maintain the state trajectory of the
system on the sliding surface, we must introduce the
switching action as
sw U Ksign s
(17)
Therefore, the overall sliding mode control law
composed of approximated control and switching action
can be written as
sw
1
ˆ
U U + U
B M q q M q G q Ksign s
(18)
where, is a 2x2 constant matrix, is a 2x1 constant
matrix, and 1 2diag , .K KK The design parameters
, , and K are chosen so that s approaches zero as
fast as possible. Uˆ is used for low-frequency control
action. Conversely,
swU corresponds to high-frequency
control. sign s is a sign function whose i-th component
has form
1 if 0
0 if 0
1 if 0
i
ii
i
s
s
s
sign s
(19)
However, a switching control usually causes chattering
of state trajectory around the switching surface. To
reduce chattering, we replace the sign(s) function by a
saturation function as follows
sw U Ksat s
(20)
where,
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©2016 Journal of Automation and Control Engineering 183
(10)
The matrix differential Equation (9) describes
reduced-order dynamics of the overhead crane. The
design the sliding mode control scheme in the next
(11)
1 if 1
if 1 1
1 if 1
i
i
i i
i
s
s s
s
sat s
(21)
And is a constant denoted thickness of boundary
layer.
IV. SYSTEM STABILITY
In this section, we find constraint conditions to make
the system stable. The design of a sliding mode
controller is composed of determining a sliding surface
and designing a switched control. The sliding surface is
chosen so that the state trajectories are attracted to this
surface and the switched control action (17) must
guarantee the stability of system states on the sliding
surface. In other words, the sliding mode control scheme
(18) guarantees that all state trajectories reach the sliding
surface (reaching condition) and slide into the desired
values on this surface. The reaching condition [13] is
determined by considering the Lyapunov function
0.5 TV s s such that
TV s s s (22)
where, 1 1diag , is a positive matrix. The
switched gain 1 2diag ,K KK of the sliding mode
control (18) is chosen so that the reaching condition is
satisfied. Substituting (13), (14), and (18) into (22) and
simplifying leads to
1 1TV s M q Ksign s M q K s
(23)
M q
is
positive definite for every 0l and / 2.
Therefore, 1 0V M q K s
for every
positive definite
.K The sliding surface s approaches zero as t tends to
infinity for every 1 0K
and 2 0.K Comparing
between expressions (22) and (23), K
can be determined
as
1
0
2
00
00
t
c l
m
m m
K = M q
which implies that
1 1 tK m ; 2 2 c lK m m (24)
Hence, reaching condition (24) guarantees the stability
of the sliding surface. More precisely, if the switching
gain K is chosen according to Equation (24), the control
forces (18) drive the state trajectories 1
T
q q to the
sliding surface. However, the sliding mode control
scheme (18) does not ensure that these states approach
the desired values on the sliding surface. Therefore, we
prove that the crane system is stabilized on the surface
under given conditions by analyzing the un-actuated
dynamics (5) and the switching manifold (12). Thus,
Equation (5) can be rewritten as
1 1 1C 1 1A q B q
(25)
where,
1 1 1cos / 0 ; 0 2 / ; C sin /l l g l A B
Substituting (14) to (25) leads to
1 1 1 1 1C 1M q A U Bq G q B q
(26)
By substituting (15) to (26), we obtain
1 1 1 1C 1A B q A
(27)
From the sliding surface Equation (12) s = 0 we obtain
1 1 1d s = q q q 0
which is equivalent to
1 1 1d q q q
(28)
Let us define
1 2 3 1 1
TT
dz z z z q q
Note that 4.Rz Equations (27) and (28) become
2 1 1 3 1 2 1Cz z A B z A
(29)
3 1 3z z z
(30)
Substituting (30) to (29) leads to
1 1 1 1 2
2
1 1 3 1C
z z
z h
A B A
z
A B z
(31)
Combining Equations (31) and (30) with
1 2z z
can
be represented in matrix form
2
1
1 1 1 1 2
2
1 1 3 1
3
1 3
C
z
z
z z
z
z
A B A
A B z
z
z
(32)
Linearization (32) about the equilibrium position
z 0
(or dq q ) can be rewritten into a linearized form
1 2
1 1
2 2
1 2 3
3 3
2 1
0 1
z z
h h h
z z
z z
z 0 z 0 z 0
0
z z z
z
z z
0
Or
z Az (33)
where,
1 11 1
1
d
d
h
z l
q q
z 0
z
A B
Journal of Automation and Control Engineering Vol. 4, No. 3, June 2016
©2016 Journal of Automation and Control Engineering 184
1
1
2
d
d
h
z l
q q
z 0
z
A
2
1
1 1
3
0
d
d
h
l
q q
z 0
z
A B
z
thus
2
1 1 1 1
1 1
2
0 1 0 0
0
0 0
0 0 0
d d dl l l
A
(34)
The system dynamics (33) is stable if and only if all
eigenvalues of A lie in the right-half s-plane. We find the
conditions for system stability by Routh’s criterion. The
linearized system (33) is stable if and only if all
coefficients of the characteristic polynomial of A are
positive, and all terms in the first column of Routh’s table
have positive signs. From these requirements and after
some calculations, we obtain
1 1
1 2 1 20; 0
d dl l
(35)
In summary, the sliding mode control controller (18)
stabilizes the crane system described by Equation (4) if
sufficient conditions (35) are guaranteed. The selection of
parameters of the sliding mode controller must satisfy the
given conditions (35).
V. SIMULATION AND EXPERIMENT
To obtain system responses, we simulate the system
dynamics (4) taken by the sliding mode control forces (18)
based on a MATLAB environment. The overhead crane
system is simulated in two cases as
Case 1: mc = 0.85 kg; mt = 5 kg; ml = 2 kg; bt = 20
N.m/s; br = 50 N.m/s; g = 9.81 m/s
2
; 1 = 0.9; 2 = 1.2; α1
= 2.5; = 0.05. The chosen design parameters must
satisfy the conditions (35).
Case 2: An overhead crane is an under-actuated
system with uncertain components. For this system, mc
and b = [bt br] are considered uncertain components. The
variations of uncertainties depend on each particular
operation, working condition, and environment. To verify
the robustness of the proposed controller, we simulate
this system in case of value varying of uncertain
components: mc =+400%, b=[20% 20%] and the
remaining the design parameters as Case 1.
We also select diag 40,35 for Case 1 and
diag 40,18.5 for Case 2. The different selection of
is to retain switched gain diag 200,100K for both
cases.
For both cases, the cargo is handled on the cable with
an initial length l0 = 0.1 m, and the cable is initially
perpendicular to the ground ( 0
0 0 ). The control inputs
(18) must be created so that the cargo is lowered to 0.4 m,
the desired cable length; and the trolley moves 0.3 m, the
desired displacement ( 0.4, 0.3d dl x ). Lowering the
cargo and moving the trolley must be started at the same
initial time. The simulation results are presented in Fig. 3
to Fig. 11.
Figure 2. An overhead crane system for experiment
Furthermore, to verify the quality of simulation based
responses, an experimental study is carried out with the
realistic overhead crane system (Fig. 2). The crane
system consists of two DC motors that drive the trolley
and hoist the cargo. Three incremental encoders measure
the trolley displacement, cargo hoisting, and cargo swing.
The real-time crane system is controlled by hoist PC
based on the MATLAB and SIMULINK environments
with xPC Target solution. In this system, we use two
interfacing cards attached to the target PC. One is the NI
PCI 6025E multifunction card, which is used to send the
direction control signals to the motor amplifiers. The
other is the NI PCI 6602 card, used to acquire the pulse
signals from the encoders and send PWM signals to the
amplifiers. The experimental results are described in Figs.
6–8.
Fig. 3 describes sliding surfaces in two simulated cases.
The sliding surfaces reach 0 within a considerably short
time. The motion of the system states includes two phases.
First, state trajectories reach the switching surface, and
second, they slide to desired values on this surface. The
first phase is sensitive and the second phase is insensitive
to parameter variations [14]. Therefore, the less the
reaching time of the switching surface, the more robust
the system. The design parameters must be chosen so that
the reaching time of the sliding surface is as short as
possible. The sliding surface s1 is related to trolley
displacement and cargo swing angle and the sliding
surface s2 has the relationship with cargo hoisting motion.
The sliding surface s1 almost all retain its shape in both
cases. Therefore, the responses of trolley motion and
cargo swing are not varied obviously when the simulation
is changed from Case 1 to Case 2 (Figs. 67). Meanwhile,
the sliding surface s2 of Case 1 reach to zero faster than
that of Case 2. Hence, the cargo lowering of Case 1 is
faster than that of Case 2 (Figs. 8, 11).
Journal of Automation and Control Engineering Vol. 4, No. 3, June 2016
©2016 Journal of Automation and Control Engineering 185
Control signals, ut and ul, are respectively represented
in Fig. 4 and Fig. 5. Clearly, these forces approach
constant values as system outputs reach the reference
values. For example, Fig. 4 shows that the driving forces
of the trolley arrive at 0 after 4.5 seconds for both
simulation cases. In Case 1, the lifting force of cargo, ul,
presented in Fig. 5, tends to the value,
0.85 9.81 8.34l cu m g (N), when the cargo is
lowered down the 0.4 m cable length after 5.5 seconds
(see Fig. 8). The minus sign of the lifting force implies
that its direction is opposite that of the cargo weight.
Clearly, at steady state, the lifting force is equal to the
gravitational force of the cargo that the cargo remains
balanced on the cable. Similarly in Case 2, Fig. 5 shows
that the lifting force ul and the gravitational force of the
cargo are equal but opposite in direction
( 3.4 9.81 33.35l cu m g N) at steady state.
The cargo sway responses are illustrated on Fig. 6. The
simulated responses completely vanished after one
oscillation period. Meanwhile, the experimental curve
reaches to steady-state after more than two periods.
However, the setting time of simulated responses
relatively equal to that of experimental one, 4st s. The
trajectories of cargo also show that the cargo swing is
kept small during the transfer process: 0
max
4.173 for
simulation case and 0
max
2.637 for experimental one.
Fig. 7 represents the responses of trolley travelling for
both simulation and experiment. These responses
asymptotically approach to desired values with the
different setting time. For example, ts
= 4.5 s for
simulated responses and ts
= 5.5 s for experimental curve.
Similarly, the cargo lowering responses are shown on Fig.
8. It seems that both simulation and experiment responses
do not have maximum overshoot. These responses
achieve the steady-state after the same setting time, ts
=5.5 s.
The swing velocity
of the cargo, the velocity
of the
trolley, and the lowering velocity
of the cargo are
respectively expressed in Fig.
9,
Fig.
10, and Fig. 11.
Although they are not the outputs that need to be
controlled, they remain state trajectories of the system.
The transient period of these system responses can be
divided into two phases: the acceleration and deceleration
phases. For example, the trolley accelerates in the first
0.8 seconds and then decelerates in the remaining 3.7
seconds (Fig. 10). The cargo is rapidly hoisted during the
first 0.3 seconds, with speed slowing down during the
remaining 5.2 seconds (Fig. 11).
Fig.
3
to
Fig.
11 in simulation case 2 show that the
surfaces approach 0 and all state trajectories
asymptotically reach the desired values after a finite time
despite widely varying uncertainties. Hence, we can
conclude
that the proposed
siding mode controller is
robust and insensitive even if the overhead crane is an un-
actuated system with wide parameter variations.
0 0.5 1 1.5 2
-0.4
-0.3
-0.2
-0.1
0
0.1
Time (s)
S
li
di
ng
s
ur
fa
ce
s
Sliding surfaces
s1 - case 1
s2 - case 1
s1 - case 2
s2 - case 2
Figure 3. Sliding surfaces
0 2 4 6 8 10
0
2
4
6
8
Time (s)
F
or
ce
(
N
)
Trolley driving force
Simulation - case 1
Simulation - case 2
Figure 4.
Trolley driving force
0 2 4 6 8 10
-40
-30
-20
-10
0
10
Time (s)
F
or
ce
(
N
)
Cargo hoisting force
Simulation - case 1
Simulation - case 2
Figure 5.
Cargo lifting force
0 2 4 6 8 10
-6
-4
-2
0
2
4
6
Time (s)
S
w
in
g
an
gl
e
(d
eg
re
e)
Cargo swing angle
Simulation - case 1
Simulation - case 2
Experiment
Figure 6.
Cargo swing angle
Journal of Automation and Control Engineering Vol. 4, No. 3, June 2016
©2016 Journal of Automation and Control Engineering 186
0 2 4 6 8 10
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Time (s)
D
is
p
la
ce
m
en
t
(m
)
Trolley displacement
Simulation - case 1
Simulation - case 2
Experiment
Figure 7. Trolley displacement
0 2 4 6 8 10
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Time (s)
L
en
gt
h
(m
)
Cable length
Simulation - case 1
Simulation - case 2
Experiment
Figure 8. Cargo lowering motion
To show the improvement of proposed controller, the
comparison of system behavior with study [3] is shown in
Table 1. Both SMC responses and feedback linearization
(FL) responses [3] converge to desired values without
steady-state errors. The settling times of SMC responses are
shorter than those of FL responses. However, FL cargo swing
angle [3] is smaller than SMC one.
0 2 4 6 8 10
-0.5
0
0.5
1
Time (s)
S
w
in
g
v
el
o
ci
ty
(
ra
d
/s
)
Swing velocity of cargo
Simulation - case 1
Simulation - case 2
Figure 9. Cargo swing velocity
0 2 4 6 8 10
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Time (s)
V
el
o
ci
ty
(
m
/s
)
Velocity of trolley
Simulation - case 1
Simulation - case 2
Figure 10. Trolley velocity
0 2 4 6 8 10
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Time (s)
V
e
lo
c
it
y
(
m
/s
)
Cargo hosting velocity
Simulation - case 1
Simulation - case 2
Figure 11. Cargo lowering velocity
VI. CONCLUSION
In this study, we successfully designed a sliding mode
controller for a complicated operation of an overhead
crane: simultaneously combining control of cargo lifting,
trolley moving, and cargo swing vanishing. From the
simulation and experiment results, all system responses
are asymptotically stable: cargo swing completely
vanished and trolley motion and cargo lifting/lowering
accurately reached the reference values. Furthermore, the
proposed controller stabilized the crane system even if
the overhead crane is an under-actuated system with a
wide range of varying uncertainties. For the next research,
we will enhance this sliding mode control problem for 3D
overhead cranes.
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Journal of Automation and Control Engineering Vol. 4, No. 3, June 2016
©2016 Journal of Automation and Control Engineering 187
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Le Anh Tuan graduated both B. Eng. and M.
Eng. in Mechanical Engineering and Marine
Machinery from Vietnam Maritime University
in 2003 and 2007, respectively. He received
the Ph.D. degree in Mechanical Engineering
from Kyung Hee University, South Korea in
2012. Currently, he is an assistant professor in
Mechanical Engineering of Vietnam Maritime
University. Dr. Tuan is also a faculty of Duy
Tan University, Da Nang, Vietnam. His
interested research composes of applied nonlinear control, dynamics
and control of industrial machines.
Journal of Automation and Control Engineering Vol. 4, No. 3, June 2016
©2016 Journal of Automation and Control Engineering 188

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