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Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018
Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.1683
AN ENHANCED COUPLING NONLINEAR TRACKING CONTROLLER
FOR UNDERACTUATED 3D OVERHEAD CRANE SYSTEMS
Menghua Zhang, Xin Ma,
Xuewen Rong, Rui Song, Xincheng Tian, and Yibin Li
ABSTRACT
An enhanced coupling nonlinear tracking control method for an underactuated 3D overhead crane systems is set forth in
the present paper. The proposed tracking controller guarantees a smooth start for the trolley and solves the problem of the
payload swing angle amplitude increasing as the transferring distance gets longer for the regulation control methods. Different
from existing tracking control methods, the presented control approach has an improved transient performance. More
specifically, by taking the operation experience, mathematical analysis of the overhead crane system, physical constraints,
and operational efficiency into consideration, we first select two desired trajectories for the trolley. Then, a new storage
function is constructed by the introduction of two new composite signals, which increases the coupling behaviour between
the trolley movement and payload swing. Next, a novel tracking control strategy is designed according to the derivation form
of the aforementioned storage function. Lyapunov techniques and Barbalat’s Lemma are used to demonstrate the stability of
the closed-loop system without any approximation manipulations to the original nonlinear dynamics. Finally, some simulation
and experiments are used to demonstrate the superior transient performance and strong robustness with respect to
different cable lengths, payload masses, destinations, and external disturbances of the enhanced coupling nonlinear tracking
control scheme.
Key Words: Tracking control, Overhead crane systems, Underactuated systems, Lyapunov techniques, Barbalat’s lemma,
Coupling behavior.
I. INTRODUCTION
Due to high payload capacity, less energy consumption, and wide flexibility of operation, overhead cranes have
been widely applied in construction sites, harbors,
workshops, and so on. Overhead crane systems are
underactuated systems, which have fewer independent
control inputs than the number of to-be-dominated degrees
of freedoms (DOFs) [1–6]. Underactuated systems have
some advantages, such as simple structure, low power
consumption, low cost, and low weight due to the omission
of partial actuators. However, unexpected payload swings
occur for reasons like inertia or external disturbances during
the transportation process, which makes it challenging to
design the crane controllers due to the underactuated nature
Manuscript received March 3, 2016; revised June 23, 2017; accepted August 19,
2017.
The authors are with School of Control Science and Engineering, Shandong
University, Jinan 250061, China.
Menghua Zhang, Xin Ma (corresponding author, e-mail: maxin@sdu.edu.ch),
Xuewen Rong, Rui Song, Xincheng Tian, and Yibin Li are with the School of Control
Science and Engineering, Shandong University, Jinan 250061, China.
This work is supported by Shandong Province Science & Technology Development
Foundation, Shandong Province Independent Innovation & Achievement
Transformation Special Fund, the Fundamental Research Funds of Shandong
University, China, and the National High-tech Research and Development
(863 Program) of China.
of cranes. At present, payload swing suppression and elimination are usually achieved by skilled workers through reducing crane speeds, leading to a loss of transportation
efficiency. Moreover, accidents may occur due to workers’
fatigue and negligence. Therefore, it is of great practical
significance and engineering value to design crane
controllers.
In order to improve the reliability and transportation
efficiency of overhead cranes, a considerable number of
studies have been undertaken. Fang et al. [7,8] proposed
several nonlinear coupling regulation control laws for
underactuated overhead crane systems by analyzing system
energy, and demonstrated that the more enhanced coupling
of the controllers the better transient responses performance
for crane systems. Based on this, Sun et al. [9,10] designed
some enhanced coupling regulation controllers for 2D overhead crane systems. However, for the above coupling-based
regulation control method, the payload swing amplitude
increases as the transferring distance gets longer. Furthermore, when analyzing state convergence, either approximation operations are required or specific terms are neglected.
In [11], an adaptive controller based on dissipation theory
was designed, achieving rapid stabilization control. In
[12,13], a hierarchical sliding-mode controller and an adaptive sliding-mode controller were proposed, respectively,
© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018
realizing precise positioning of the trolley and effective
swing suppression control. An input shaping technique
proved to be a simple and effective anti-swing control
method through a series of experiments [14–19]. In [20–23],
reference trajectories subject to trolley displacement,
velocity, and acceleration limitation were defined, and then
combined with anti-swing components, guaranteeing precise trolley positioning and anti-swing control. Radac et al.
[24] presented an optimal behavior prediction mechanism
for multi-input–multi-output control systems in a hierarchical control system structure, using previously learned
solutions to simple tasks called primitives. Zhang et al.
[25] achieve simultaneous motion regulation and payload
swing suppression and elimination within a finite time.
Hashim et al. [26] proposed an adaptive synchronization
control method with prescribed performance to control
transient as well as steady state behavior. In addition to the
aforementioned model-based control methods, intelligent
control methods such as fuzzy logic control method
[27,28], which was introduced into the sliding-mode control
method in order to eliminate the swing angle induced by the
sliding-mode control method, has been applied in crane
systems.
The aforementioned studies were presented for 2D
overhead crane systems. Because 3D overhead crane systems have more states, stronger state coupling and stronger
non-linearity, it is more challenging to propose appropriate
control methods for them. Passivity and energy-based
frameworks [29], second-order sliding-modes controller
[30], fuzzy controller [31], and motion planning control
methods [32] were employed to drive the trolley to the desired positions and suppress the payload swing effectively.
Sun et al. [33] proposed an energy coupling output feedback (OFB) controller of a 4-DOF underactuated crane with
inputs constraints, achieving both accurate trolley positioning and efficient payload swing elimination. Zhang et al.
[34] proposed an offline minimum-time trajectory planning
approach for underactuated overhead cranes. Khatamianfar
and Savkin [35] presented a novel tracking control method
based on model predictive control, leading to high control
performance with small payload swing.
It should be pointed out that most of the above control
methods are aimed at stabilization control. From the perspective of practical application, workers prefer to operate
the trolley along a positioning reference trajectory to improve transportation efficiency and offset adverse effects,
such as the large initial control moment brought by stabilization control. For this reason, Fang et al. [36] planned an Sshaped smooth trajectory for the trolley based on operation
experience and mathematical analysis, and then designed
an adaptive control law to make the trolley track the planned
trajectory. By taking system friction and air resistance into
account, an adaptive trajectory tracking controller was
designed for a 3D overhead crane systems in [37]. In [38],
by introducing the potential function, a nonlinear tracking
controller for 2D overhead crane systems guarantees the
tracking error always within a priori set bounds. However,
the existing tracking control schemes can only guarantee
the trolley displacement and the payload swing angels converge to the desired value (trajectory) and zero, but cannot
guarantee the transient performance of the cranes.
In response to the above-mentioned practical issues, a
nonlinear coupling tracking control scheme for 3D overhead
crane systems is proposed in this paper. Specifically, we take
operation experience, the mathematical analysis of the
system, physical constraints, and operational efficiency into
consideration. Two desired trajectories are selected for the
trolley. Then, inspired by [7–10], a novel positive definite
function is constructed by introducing two new composite
signals integrating the trolley velocity and payload swing,
increasing the transient performance of the controller.
Lyapunov techniques and Barbalat’s Lemma are used to
demonstrate the asymptotic stability of the closed-loop
system. Simulation and experimental results are included
to prove the superior robustness and transient performance
of the nonlinear coupling tracking control method set forth
in the present paper.
The merits of the proposed nonlinear tracking
controller are as follows.
1. As validated by numerical results, the proposed controller admits strong robustness against different cable
lengths, payload masses, destinations, and external
disturbances.
2. By comparing the proposed controller with PD controller and OFB controller, an increased transient performance of the proposed controller is achieved.
Furthermore, the proposed control method guarantees
“soft trolley start”, addressing the problem of existing
crane regulation control methods.
3. A strict mathematical analysis of the control method
without any approximation to the original nonlinear
dynamic equations is presented in this paper, providing theoretical support for the superior performance
of the controller.
4. As will be seen from simulation and hardware experiment results, the transient performance of the controller is improved.
The rest of the paper is organized as follows: In
Section II, the model of 3D overhead crane systems is
illustrated. Section III provides the nonlinear coupling
tracking controller. In Section IV, we exhibit some numerical simulation and hardware experiment results to demonstrate the superior performance of the proposed controller.
Section V summarizes the paper.
© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
M. Zhang et al.: Enhanced coupling nonlinear tracking controller
Z
II. 3D OVERHEAD CRANE SYSTEMS
trolley
In this paper, the control problem for a 3D overhead
crane system is addressed. The dynamic equations of 3D
overhead cranes are described as follows [30–38]:
M ðqÞ€
q þ C ðq; q_ Þq_ þ G ðqÞ ¼ F
(1)
where q ∈ R represents the state vector, M(q) ∈ R
denote the inertia matrix,
C ðq; ; q_ Þ∈R44
is the
Centripetal-Coriolis matrix, G(q) ∈ R4 denote the gravity
vector, F ∈ R4 is the resultant force imposed on the trolley.
The vectors/ matrixes are detailed defined as
T
q ¼ x y θx θy
4
2
4×4
mp þ M x
0
6
0
mp þ M y
6
M ðq Þ ¼ 6
6 mp lC x C y
0
4
mp lS x S y mp lC y
2
mp lC x C y
0
mp lS x C y θ_ x mp lC x S y θ_ y
0 0
0
mp l2 S y C y θ_ y
mp l 2 S y C y θ_ x
G ðqÞ ¼ 0 0 mp glS x C y
mp lC y
mp l 2 C 2y
0 0
6
60 0
C ðq; q_ Þ ¼ 6
6
40 0
mp lS x S y
0
3
7
7
7;
7
5
mp l 2
0
3
mp lC x S y θ_ x mp lS x C y θ_ y
7
_
7
mp lS y θy
7;
7
2
_
mp l S y C y θ x
5
0
mp glC x S y
h
F ¼ F x f rx F y f ry d θx C 2y θ_ x
T
d θy θ_ y
Y
x
y
payload
Fig. 1. Illustration for 3D overhead crane. [Color figure can be
viewed at wileyonlinelibrary.com]
1 _
M ðqÞ C ðq; ; q_ Þ δ ¼ 0; ∀δ∈R4
(4)
2
For ease of the forthcoming controller design and
stability analysis, 1 can be rewritten as
δT
M x þ mp €x þ mp lC x C y θ€ x mp lS x S y θ€y mp lS x C y θ_ 2x
2mp lC x S y θ_ x θ_ y mp lS x C y θ_ 2y ¼ F x f rx
M y þ mp €y þ mp lC y θ€y mp lS y θ_ 2y ¼ F y f ry
mp lC x C y €x þ mp l 2 C 2y θ€x 2mp l 2 S y C y θ_ x θ_ y
(5)
(6)
(7)
þmp glS x C y ¼ d θx C 2y θ_ x
iT
mp lS x S y €x mp lC y €y mp l 2 θ€y mp l 2 S y C y θ_ 2x
where mp, Mx and My denote the payload mass, the trolley
mass, the sum mass of the trolley and the bridge, respectively, l is the cable length, g represents the gravitational
constant, Sx, Cx, Sy and Cy are the abbreviations of sinθx,
cosθx, sinθy and cosθy, respectively, C1 and C2 are two
auxiliary variables, Fx and Fy represent the control inputs
along the X and Y axis, respectively, frx and fry denote the
frictions in direction X and Y, respectively, dθx and dθy are
air resistance coefficients, x(t) and y(t) are the trolley
displacements along the X and Y axis, respectively, θx(t)
and θy(t) are projected swing angles (as shown in Fig. 1).
According to the nature of friction, the following
friction models are selected [3,9,33]:
f rx ¼ f r0x tanhðx_ =ηx Þ þ k rx jx_ j_x
f ry ¼ f r0y tanh y_ =ηy þ k ry jy_ j_y
bridge
X
(2)
(3)
where fr0i, ηi∈R+, kri∈R1, i = x, y, are friction-related
parameters.
The Centripetal-Coriolis matrix C ðq; q_ Þ and the inertia
matrix M(q) have the following relationship:
(8)
mp glC x S y ¼ d θy θ_ y
In the process of crane transportation, for safety’s
sake, the cable length remains the same and the payload
swing angles are within permitted ranges. Based on this,
we made the following assumptions.
1. The cable connecting the trolley and the payload
should be mass-less and rigid.
2. During the whole transportation, the swing angles
should be kept within the following bounds:
π
π
< θx ; θy <
2
2
III. NONLINEAR COUPLING TRACKING
CONTROLLER DESIGN
In this section, we propose a nonlinear coupling tracking controller for 3D overhead crane systems and present
the stability analysis of the closed-loop system.
© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018
3.1 Selection of target trajectory
It is well known that to keep the trolley running
smoothly, the target trajectory should be an S-curve or a series of shaped impulses [36]. Therefore, we use the S-curve
trajectories in [36] as the trolley positioning reference trajectories, which are given by
"
#
pdx
k 2vx
coshð2k ax t=k vx εx Þ
þ
ln
xd ¼
2
4k ax
cosh 2k ax t=k vx εx 2pdx k ax =k 2vx
(9)
3
2
k 2vy
pdy
cosh 2k ay t=k vy εy
4
5
þ
ln
yd ¼
2
2
4k ay
cosh 2k ay t=k vy εy 2pdy k ay =k vy
(10)
where kvx and kax represent the maximum velocity and the
maximum acceleration of the trolley in direction X, respectively. The maximum velocity and the maximum acceleration of the trolley along the Y axis are denoted by kvy and
kay, respectively. Pdx and pdy are the destinations in direction
X and Y. The purpose of introducing εx and εy is to regulate
the initial accelerations of the trolley.
The desired trajectories of 9 and 10 have the following
properties [36]:
3.2 Nonlinear coupling tracking control law
development
To facilitate the design of the controller, the tracking
errors are defined as
T
e ¼ x xd y yd θx θy
T
¼ ex ey θx θy
(16)
where ex and ey denote the tracking errors of the trolley
along the X and Y axis, respectively.
Based on the energy form of the crane systems, a positive definite function is constructed as
1
(17)
E ðt Þ ¼ e_ T M ðqÞ_e þ mp gl 1 C x C y
2
By taking the derivative of 17 with respect to
time, along the trajectories of 4, we are led to the following result:
1_
e_ þ M€e þ mp glS x C y θ_ x þ mp glC x S y θ_ y
E_ ðt Þ ¼ e_ T M
2
¼ e_ x F x f rx mp þ M x €x d
h
i
þ_ey F y f ry mp þ M y €y d
þmp l S x S y θ_ y C x C y θ_ x €x d mp lC y θ_ y €y d
d θx C 2y θ_ 2x d θy θ_ 2y
(18)
lim xd ðt Þ ¼ pdx ; lim yd ðt Þ ¼ pdy ; lim x_ d ðt Þ
t→∞
t→∞
t→∞
¼ 0; lim y_ d ðt Þ ¼ 0; lim €x d ðt Þ
t→∞
¼ 0; lim €y d ðt Þ ¼ 0
t→∞
t→∞
(11)
0≤_xd ðt Þ≤k vx ; 0≤_yd ðt Þ≤k vy
(12)
j€x d ðt Þj≤k ax ; j€y d ðt Þj≤k ay
(13)
x_ d ∈L2 ; €x d ∈L2 ; y_ d ∈L2 ; €y d ∈L2
(14)
During the transportation process, we cannot control
the payload swing angles directly due to the underactuated
nature of the crane systems. The only way for achieving
swing damping is to utilize the coupling relationship between the trolley motion and the payload swing. Hence,
we can’t plan the trajectory of payload swing. In this paper,
the desired swing angles are selected as
ξ y ¼ e_ y þ γΨ θy
Then, the desired state variable of a 3D overhead crane
system is written as
(15)
(20)
where λ, γ∈R+ are positive control parameters, g(θx)denotes one θx-related function to be determined, f (θy) and
Ψ(θy) represent θy-related functions to be determined. By
taking the time derivative of 19 and 20, it can be
obtained that
0
0
ξ_ x ¼ €e x þ λg ðθx Þf θy θ_ x þ λgðθx Þf θy θ_ y
(21)
0
ξ_ y ¼ €e y þ γΨ θy θ_ y
θ x ¼ θy ¼ 0
qd ¼ ½xd yd 0 0T
We notice that, for the first two terms of E_ ðt Þ related
to control inputs, no terms are associated with the payload
swing motion θx or θ_ x andθy or θ_ y . To solve this problem, two generalized signals are introduced as
ξ x ¼ e_ x þ λg ðθx Þf θy
(19)
(22)
where g0 (θx) denotes the derivative of g(θx) with respect to
θx, f 0 (θy) represents the derivative of f (θy) with respect to
θy , and Ψ0 (θy)is the derivative of Ψ(θy) with respect to θy.
© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
M. Zhang et al.: Enhanced coupling nonlinear tracking controller
0
λmp lC x C y g ðθx Þf θy θ_ 2x
0
þλmp lC x C y gðθx Þf θy θ_ x θ_ y
0
λmp lS x S y g ðθx Þf θy θ_ x θ_ y
0
λmp lS x S y gðθx Þf θy θ_ 2y
2
¼ λmp l C x C y θ_ x S x S y θ_ y ⇒gðθx Þ
¼ sin θx ; f θy ¼ cos θy
By integrating 19 and 20 with respect to time, one has
t
t
∫0 ξ x dt ¼ ∫0 e_ x þ λgðθx Þf θy dt
t
(23)
¼ ex þ λ∫0 g ðθx Þf θy dt
t
t
t ∫0 ξ y dt ¼ ∫0 e_ y þ γΨ θy dt ¼ ey þ γ∫0 Ψ θy dt
(24)
Then, the new state vector is accordingly constructed as
T
ξ ¼ ξ x ξ y θ_ x θ_ y
T
¼ e_ x þ λg ðθx Þf θy e_ y þ γΨ θy θ_ x θ_ y
Further, according to the structure of 27 and the
conclusions of 29 and 30, the controller is designed as
(25)
t
F x ¼ k d1 ξ x k p1 ∫0 ξ x dt þ f rx
þ mp þ M x €x d þ λ mp þ M x C x C y θ_ x
λ mp þ M x S x S y θ_ y
Inspired by the form of 17, we define a new positive definite function as
1
E L ðt Þ ¼ ξ T M ðqÞξ þ mp gl 1 C x C y
2
the time derivative of which is calculated as
(26)
0
C y Ψ θy ≤0
(28)
Considering the practical conditions where the payload
swing is always beneath the trolley, that is, π2 < θy < π2 ⇒
Cy > 0. Therefore, one natural and convenient choice for
Ψ (θy) is obtained as
0
Ψ θy ¼ 1⇒Ψ θy ¼ θy
(29)
(31)
t
F y ¼ k d2 ξ y k p2 ∫0 ξ y dt þ f ry
þ mp þ M y €y d þ λ mp þ M y θ_ y
h
0
i
0
E_ L ðt Þ ¼ ξ x F x f rx mp þ M x €x d þ λ mp þ M x g ðθx Þf θy θ_ x þ gðθx Þf θy θ_ y
h
0 i
þ ξ y F y f ry mp þ M y €y d þ γ mp þ M y Ψ θy θ_ y
þ mp l S x S y θ_ y C x C y θ_ x €x d mp lC y θ_ y €y d d θx C 2y θ_ 2x d θy θ_ 2y
0
0
þ λmp lC x C y g ðθx Þf θy θ_ 2x þ λmp lC x C y gðθx Þf θy θ_ x θ_ y
0
0
0
λmp lS x S y g ðθx Þf θy θ_ x θ_ y λmp lS x S y g ðθx Þf θy θ_ 2y þ γmp lC y Ψ θy θ_ 2y
Hence, to guarantee the last term of 27 non-negative, the
following formula should be satisfied:
(30)
(32)
(27)
where kp1, kp2, kd1, and kp2 ∈ R+ represent positive
control gains.
3.3 Stability analyses of the closed-loop system
Theorem 1. The nonlinear coupling tracking controller
31 and 32 can guarantee the displacement, velocity, and acceleration of the trolley converges to the desired trajectories
and the payload swing angles, angular velocity, and angular
acceleration converge to zero, in the sense that
T
lim x y x_ y_ θx θy θ_ x θ_ y ¼ ½xd yd x_ d y_ d 0 0 0 0 T
t→∞
Further, to guarantee the following terms of 27 nonnegative, that is,
(33)
0
0
λmp lC x C y g ðθx Þf θy θ_ 2x þ λmp lC x C y gðθx Þf θy θ_ x θ_ y
;
0
0
λmp lS x S y g ðθx Þf θy θ_ x θ_ y λmp lS x S y gðθx Þf θy θ_ 2y ≤0
Proof. The following positive definite function is chosen
as the Lyapunov function candidate
based on complete square formula, one choice for g(θx)
and f (θy) are obtained as
(34)
1
V ðt Þ ¼ ξ T Mξ þ mp gl 1 C x C y
2
2 1 t
2
1 t
þ k p1 ∫0 ξ x dt þ k p2 ∫0 ξ y dt
2
2
© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018
We can easily conclude from 36 that
By taking the time derivative of 34, and substituting the
conclusions of 27, 31, and 32, it can be derived that
t
t
t 3d θ
t 3d θ
k d1 ∫0 ξ 2x dτ þ k d2 ∫0 ξ 2y dτ þ ∫0 x C 2y θ_ 2x dτ þ ∫0 y θ_ 2y dτ
4
4
2
t
_
_
≤V ð0Þ V ðt Þ λmp l∫0 C x C y θx S x S y θy dτ
2
2 2
x d mp lC y €y d
t mp l 2
t mp lS x S y €
€x dτ þ ∫0
þ∫0
dτ∈L∞
d θx d
d θy
2
V_ ðt Þ ¼ k d1 ξ 2x k d2 ξ 2y λmp l C x C y θ_ x S x S y θ_ y
þmp l S x S y θ_ y C x C y θ_ x €x d mp lC y θ_ y €y d
3d θx 2 _ 2
C θ
d θx C 2y θ_ 2x d θy θ_ 2y ≤ 4 y x
2
mp lS x S y €x d mp lC y €y d
m2 l 2 2 3d θy _ 2
€x θ þ
þ
d θx d
4 y
d θy
k d1 ξ 2x
k d2 ξ 2y
λmp l C x C y θ_ x S x S y θ_ y
(40)
It follows from 41 that
ξ x ; ξ y ; θ_ x ; θ_ y ∈L2
2
(35)
The following formula can be obtained from 37–39,
41 and the Barbalat’s Lemma [1] as
lim ξ x ¼ 0; lim ξ y ¼ 0; lim θ_ x ¼ 0; lim θ_ y ¼ 0
Integrating 35 with respect to time and performing
some arrangements yields
2 2
3d θx 2 _ 2
t mp l
€x 2 dτ
C y θx dτ þ ∫0
4
d θx d
2
x d mp lC y €y d
t 3d θy 2
t mp lS x S y €
_
∫0
θ dτ þ ∫0
dτ
4 y
d θy
t
V ðt Þ≤V ð0Þ ∫0
(36)
It follows from 13 and 14 that
t
t
V ðt Þ∈L∞ ⇒ξ x ; ξ y ; θ_ x ; θ_ y ; ∫0 ξ x dt; ∫0 ξ y dt∈L∞
t→∞
(37)
(38)
t→∞
t→∞
M y þ mp €y þ mp lC y θ€y mp lS y θ_ 2y
t
¼ k d2 ξ y k p2 ∫0 ξ y dt þ mp þ M y €y d
þ λ mp þ M y θ_ y
€x ¼ g 1 þ g2
(39)
(42)
(44)
After performing some mathematical operations like
addition, subtraction, multiplication, and division on 5–6
and 43–44, one has
It is clear from 13, 21–22, and 37–38 that
ξ_ x ; ξ_ y ∈L∞
t→∞
Substituting 31 and 32 into 5 and 6, respectively, 5
and 6 are rewritten as
M x þ mp €x þ mp lC x C y θ€x mp lS x S y θ€ y
mp lS x C y θ_ 2x 2mp lC x S y θ_ x θ_ y mp lS x C y θ_ 2y
t
¼ k d1 ξ x k p1 ∫0 ξ x dt þ mp þ M x €x d
þ λ mp þ M x C x C y θ_ x
λ mp þ M x S x S y θ_ y ;
(43)
Then, according to 5–6, 19–20, 31–32, and 36
we have
F x ; F y ; €x ; €y ; θ€ x ; θ€y ; x_ ; y_ ∈L∞
(41)
(45)
where the expressions of g1 and g2 are
1
mp lM y C 2y S x C y θ_ 2x mp lM y S x C y θ_ 2y þ M y þ mp S 2y k d2 ξ y mp þ M y €y d λ mp þ M y θ_ y
@
A
_
_
þmp C y S x S y mp þ M x €x d þ λ mp þ M x C x C y θx λ mp þ M x S x S y θy k d1 ξ x
0
g1 ¼
g2 ¼
M x M y þ M x mp S 2y þ M y mp C 2y
t
t
mp gM y C 2y S x C x M y þ mp S 2y mp k p1 ∫0 ξ x dtC y S x S y k p2 ∫0 ξ y dt M y þ mp S 2y
M x M y þ M x mp S 2y þ M y mp C 2y
© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
M. Zhang et al.: Enhanced coupling nonlinear tracking controller
For 54, owing to lim ω1 ¼ 0; lim ω_ 2 ∈L∞ ; lim ξ y ¼ 0, we
It follows from 37–39, and 43 that
t→∞
lim g 1 ¼ 0; g_ 2 ∈L∞
(46)
t→∞
It can be obtained from 21 that
ξ_ x ¼ €x €x d λC x C y θ_ x þ λS x S y θ_ y
(47)
Substituting 47 into 42, the following result
is obtained
ξ_ x ¼ g1 €x d λC x C y θ_ x þ λS x S y θ_ y þ g 2
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |{z}
(48)
ϑ2
ϑ1
t→∞
lim θ€y ¼ 0; lim S y ¼ 0⇒θy ¼ 0
t→∞
t→∞
t→∞
t→∞
(49)
By invoking the extended Barbalat’s Lemma [1],
we have
lim ξ_ x ¼ 0⇒ lim €x €x d λC x C y θ_ x þ λS x S y θ_ y
t→∞
t→∞
¼ 0⇒ lim €x ¼ 0
t→∞
(50)
With some similar analysis, it can be obtained from
8 and 44 that
€y ¼ β1 þ β2
(51)
where
k d2 ξ y þ λ mp þ M y θ_ y mp S x S y C y €x þ mp þ M y €y d
β1 ¼
;
M y þ mp S 2y
β2 ¼
mp lS y C 2y θ_ 2x
t
k p2 ∫0 ξ y dt
(57)
t→∞
It can be obtained from 7 that,
t→∞
lim ϑ 1 ¼ 0; ϑ_ 2 ∈L∞ ; lim ξ x ¼ 0
t→∞
mp lS x S y €x mp lC y €y mp l 2 S y C y θ_ 2x þ d θy θ_ y gC x S y
þ
θ€ y ¼
l
mp l 2
(56)
In a similar way, it follows from 9, 37, 42, 50, and
55 and the extended Barbalat’s Lemma [1] that
From the conclusions of lim €x d ¼ 0; lim θ_ x ¼
0; lim θ_ y ¼ 0;and 46, 42, it is concluded that
t→∞
can utilize again extended Barbalat’s Lemma [1] to obtain
lim ξ_ y ¼ 0⇒ lim €y €y d γθ_ y ¼ 0⇒ lim €y ¼ 0 (55)
t→∞
t→∞
t→∞
The following formula can be obtained from 8,
dθ
þ mp lS y θ_ 2y þ mp gC x S y C 2y y C y θ_ y
l
:
M y þ mp S 2y
θ€ x ¼
mp lC x €x þ 2mp l 2 S y C y θ_ x θ_ y d θx C 2y θ_ x
mp l 2 C y
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
α1
gS x
;
lC y
|{z}
α2
where
lim α1 ¼ 0; α_ 2 ∈ L∞
(59)
t→∞
Then, by using 42 and the extended Barbalat’s
Lemma [1], the following conclusions can be derived
lim θ€x ¼ 0; lim α2 ¼ 0⇒θx ¼ 0
t→∞
t→∞
lim x_ ¼ lim x_ d ¼ 0
t→∞
(61)
t→∞
By substituting 11, 42, and 57 into 20, one has
lim y_ ¼ lim y_ d ¼ 0
t→∞
(62)
t→∞
It can be obtained from 11, 42, 50, 60, and 43 that
t
lim ∫0 ξ x dt ¼ 0
(63)
It holds from 38–39, and 42 that
Rearranging 23 results in the conclusion of
lim β1 ¼ 0; β_ 2 ∈L∞
ex ¼ ∫0 ξ x dt þ λ∫0 S x C y dt :
|fflffl{zfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl}
t
t→∞
t
ℑ1
(64)
ℑ2
From 37, 61, and 63, it is derived that
It can be derived from 22 that
ξ_ y ¼ €y €y d γθ_ y
(60)
Substituting 11, 42, and 60 into 20 yields
t→∞
(52)
(58)
(53)
lim ℑ 1 ¼ 0; lim ℑ_ 2 ∈L∞ ; lim e_ x ¼ 0
t→∞
t→∞
t→∞
(65)
By invoking Barbalat’s Lemma [1], we have
By substituting 53 into 51, one has
ξ_ y ¼ β1 €y d γθ_ y þ β2 :
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |{z}
ω1
ω2
lim ex ¼ 0⇒ lim x ¼ lim xd
(54)
t→∞
t→∞
t→∞
We can conclude from 11, 42, 55, 57, and 44 that
© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
(66)
Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018
t
lim ∫0 ξ y dt ¼ 0
(67)
t→∞
It can be obtained from 24 that
t
t
ey ¼ ∫0 ξ y dt þ γ∫0 θy dt :
|fflffl{zfflffl} |fflfflffl{zfflfflffl}
(68)
Then, by using 42, 62, and 67, the following conclusions can be derived
lim σ 1 ¼ 0; lim σ_ 2 ¼ 0; lim e_ y ¼ 0
t→∞
t→∞
(69)
t→∞
We can utilize the extended Barbalat’s Lemma [1]
again to derive
lim ey ¼ 0⇒ lim y ¼ lim yd
t→∞
t→∞
κy = μy + kd2υy, μ_ y ¼ k d2 μy þ k d2 υy ,where μx and
μy denote auxiliary functions.
σ2
σ1
m þM
υy = y pdy + λySy,where λx < 0, λy ¼ mpp þM xy λx . κx and
κy are dynamically generated as follows:
κx = μx + kd1υx, μ_ x ¼ k d1 ðμx þ k d1 υx Þ,
(70)
t→∞
With the conclusions of 42, 50, 55, 57, 60–62, 66, and
70 we conclude that Theorem 1 is proved.
IV. SIMULATION AND EXPERIMENTS
In this section, some numerical simulation and experiments are performed to verify the superior performance of
the proposed nonlinear coupling tracking control method.
4.1 Simulation
To illustrate the control performance, some simulation
tests are implemented for the proposed nonlinear coupling
tracking control method. First, to verify the improved transient performance of the proposed controller, we compare
our method with the OFB controller [33] and the motion
planning-based adaptive (MPA) controller [36]. Second,
the robustness against different payloads, cable lengths, desired positions, and external disturbances is further tested,
and we demonstrate the proposed controller can guarantee
“soft trolley start” with respect to different destinations.
For literature completeness, the expressions for the OFB
controller and the MPA controller are provided as follows.
2. MPA controller
It should be pointed out that the MPA controller is designed for the 2D overhead crane system. To be fair, we apply it to the control of the 3D overhead crane system and
obtain that
^x
F x ¼ k p1 ex k d1 e_ x þ YTx ω
(73)
^y
F y ¼ k p2 ey k d2 e_ y þ YTy ω
(74)
where kp1, kp2, kd1, and kp2 ∈ R+ represent positive control
gains, ex = x xd and ey = y yd denote the trolley tracking
errors along the X and Y axis, respectively, xd and yd are
chosen the same as those in equations 9–10, respectively,
^ y are the online estimation of ωx and ωy, respec^ x and ω
ω
tively, which are obtained by the following update laws:
^_ x ¼ Τx Yx e_ x ,
ω
^_ y ¼ Τy Yy e_ y,where Τx, Τy ∈ R3 × 3 stand for diagonal,
ω
positive definite, update gain matrixes. The detailed expressions for Yx, ωx, Yy, and ωy are given as follows:
x_
tanh
Yx ¼ €x d
ηx
¼ mp þ M x f r0x
"
Yy ¼ €y d
y_
tanh
ηy
h
¼ mp þ M y
T
jx_ j_x ; ωx
T
k rx
!
f r0y
#T
jy_ j_y
;
ωy
iT
k ry
1. OFB controller:
F x ¼ k p1 tanhðυx Þ k d1 tanhðκx Þ
(71)
F y ¼ k p2 tanh υy k d2 tanh κy
(72)
where kp1, kp2, kd1, and kp2 ∈ R+ stand for positive control
gains, υx and υy have the following forms:
υx = x pdx + λxSxCy,
As can be seen from equations (80)–(83), the MPA
control laws do not take the payload swing angles into consideration. Therefore, the payload swing angles can only be
damped out by air resistance.
The numerical simulation is implemented by
Matlab/Simulink. The control gains for the proposed controller, the OFB controller, and the MPA controller are
shown in Table I. For the MPA controller, the initial online
estimates of ωx and ωy are set as zero.
© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
M. Zhang et al.: Enhanced coupling nonlinear tracking controller
Table I. Control gains for simulation 1.
Controllers
kp1
kd1
kp2
kd2
λ
γ
λx
λy
Tx
MPA controller
OFB controller
proposed controller
25
18
50
25
23
50
18
16
100
35
26
30
NA
NA
0.4
NA
NA
0.5
NA
1
NA
NA
0.54
NA
20I3
NA
NA
Ty
20 I
NA
NA
3
M x ¼ 7kg; M y ¼ 14kg; mp ¼ 1:025kg; g ¼ 9:8m=s2 ; k ax ¼ 0:3m=s2 ;
k vx ¼ 0:3m=s; k ay ¼ 0:2m=s2 ; k vy ¼ 0:2m=s; l ¼ 0:7m; d θx ¼ d θy ¼ 0:3;
f rox ¼ 4:4; k rx ¼ 0:5; ηx ¼ 0:01; f roy ¼ 8; k ry ¼ 1:2; ηy ¼ 0:01;
within 5 s, yet the payload swing of the proposed controller
is much better suppressed and eliminated. Moreover, there
is almost no residual swing as the trolley stops, and it consumes the fewest control efforts and time with the proposed
controller. These results have evidently proven the superior
transient performance of the proposed control method.
The system parameters are defined as
and the desired position is set as
pdx ¼ 0:7m; pdy ¼ 0:5m:
4.1.1 Comparative test
As a means to validate the superior performance of the
proposed controller, we compare it with the OFB controller
and the MPA controller.
The simulation results are recorded in Figs 2–4 and
the quantified results are detailed in Table II, which include
the following performance indices: 1 final position of the
trolley pfi; 2 maximum swing amplitude θimax; 3 payload
swing residual swing θires; 4 maximum actuating force
Fimax; 5 time consumption tsi, which refers to the time when
positioning error is within the range of (4 mm, 4 mm); 6
4.1.2 Robustness verification simulation
In this subsetion, four parts are included. In the first
part, we consider l = 0.7 m, 0.4 m, 2 m three cable lengths.
Next, three cases mp = 1.025 kg, 2 kg, 4 kg for payload
masses are considered. In the third part, the destinations are
set as pdx = 0.7 m, 1.2 m, 2 m, pdy = 0.5 m, 1 m, 1.7 m, respectively. In the concluding part, to emulate external disturbances, three kinds of disturbances included impulsive
disturbances, sinusoid disturbances (the period is 1 s and
the initial phase angle is 0°), and a uniformly distributed random disturbance to the load swing are added between 7 s and
8 s, 10 s and 11 s, 15 s and 16 s, respectively. All above
10
energy consumption ∫0 F 2i dt, where i = x, y.
From Figs 2–4 and Table II, it can be seen that all of
the three controllers can drive the trolley to the destination
0.5
0.7
y
yd
x
xd
0
0
0
1
2
3
4
5
6
7
8
9
0
10
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
1.5
1
0
0
-1.5
0
1
2
3
4
5
6
7
8
9
10
10
-1
20
10
5
0
0
0
1
2
3
4
5
6
7
8
9
10
-10
Time(sec)
Time(sec)
Fig. 2. Comparative test: Results for the proposed controller. [Color figure can be viewed at wileyonlinelibrary.com]
© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018
0.5
0.7
y
yd
x
xd
0
0
0
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
2
2
0
0
-2
0
10
1
2
3
4
5
6
7
8
9
10
-2
0
20
10
5
0
0
-5
-10
0
0
1
2
3
4
5
6
7
8
9
10
Time(sec)
Time(sec)
Fig. 3. Comparative test: Results for the MPA controller. [Color figure can be viewed at wileyonlinelibrary.com]
1
0.7
0.7
0.5
0
0
2
1
2
3
4
5
6
7
8
9
10
0
0
1
2
3
4
5
6
7
8
9
10
5
6
7
8
9
10
5
6
7
8
9
10
1
0
0
-2
-4
0
30
-1
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
10
15
10
5
0
-5
0
1
2
3
4
15
0
-5
0
1
2
3
4
5
6
7
8
9
Time(sec)
Time(sec)
Fig. 4. Comparative test: Results for the OFB controller. [Color figure can be viewed at wileyonlinelibrary.com]
Table II. Control performance comparison for simulation 1.
Controller
pfx
pfy
θxmax
θymax
θxres
θyres
Fxmax
Fymax
tsx
tsy
unit
MPA controller
OFB controller
proposed controller
m
0.698
0.696
0.7
m
0.409
0.502
0.501
°
2.23
2.98
1.4
°
1.825
1.34
0.94
°
0.58
0.12
0.01
°
0.64
0.08
0.01
N
7.8
14.1
6.15
N
12.6
14.75
11.4
s
4.7
4.4
4.2
s
4.9
5.0
4.4
disturbances have an amplitude of 2°. The control gains for
the four parts are set as those in Table I.
As can be seen from Figs 5–8, the proposed controller
can accurately position the trolley as well as suppress and
eliminate the payload swing even in the different cable
lengths, payload masses, desired positions, external disturbances, which demonstrate strong robustness of the presented method. Furthermore, we can conclude from Fig. 7
that the payload swing amplitude stays the same as the
10
10
∫0 F 2x dt
∫0 F 2y dt
N2·s
99.5
96.1
95.3
N2·s
318.4
330.2
292.3
transferring distance gets longer, which makes up for the
shortage of regulation control methods.
4.2 Experiments
To further illustrate the practical control performance
of the proposed nonlinear coupling tracking control method,
two sets of experiments are implemented on a self-built
© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
M. Zhang et al.: Enhanced coupling nonlinear tracking controller
0.7
1
l=0.7m
l=0.4m
l=2m
xd
0.7
0
l=0.7m
l=0.4m
l=2m
yd
0.5
0
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
0
0
-5
-10
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
2
3
2
0
0
-2
0
2
4
6
8
10
12
14
16
18
20
-2
10
20
5
10
0
2
4
6
8
10
12
14
16
18
20
Time(sec)
Time(sec)
Fig. 5. Robustness verification simulation: Results for the proposed controller with respect to different cable lengths. [Color figure can be
viewed at wileyonlinelibrary.com]
1
0.7
mp=1.025kg
0.7
mp=2kg
0.5
mp=1.025kg
mp=2kg
mp=4kg
mp=4kg
xd
0
0
2
4
6
8
10
12
14
16
yd
18
20
0
2
2
0
0
-2
0
2
4
6
8
10
12
14
16
18
20
-2
10
20
5
10
0
0
-5
-10
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
Time(sec)
Time(sec)
Fig. 6. Robustness verification simulation: Results for the proposed controller with respect to different payload masses. [Color figure can be
viewed at wileyonlinelibrary.com]
2
1.7
pdx=0.7m
pdx=1.2m
pdx=2m
1.2
0.7
0
0
2
4
6
8
10
12
14
16
18
pdy=0.5m
pdy=1m
pdy=1.7m
1
0.5
20
0
0
2
4
6
8
10
12
14
16
18
20
2
4
6
8
10
12
14
16
18
20
2
4
6
8
10
12
14
16
18
20
1
1
0
0
-1
0
2
4
6
8
10
12
14
16
18
20
10
-1
0
20
10
5
0
0
0
2
4
6
8
10
12
Time(sec)
14
16
18
20
-10
0
Time(sec)
Fig. 7. Robustness verification simulation: Results for the proposed controller with respect to different destinations. [Color figure can be
viewed at wileyonlinelibrary.com]
© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018
1
0.7
0.5
0.7
x
xd
0
0
2
4
6
8
10
12
14
16
18
y
yd
0
20
0
2.5
2.5
0
0
-2.5
0
2
4
6
8
10
12
14
16
18
20
-2.5
0
10
20
5
10
0
0
-5
-10
0
0
2
4
6
8
10
12
14
16
18
20
2
4
6
8
10
12
14
16
18
20
2
4
6
8
10
12
14
16
18
20
2
4
6
8
10
12
14
16
18
20
Time(sec)
Time(sec)
Fig. 8. Robustness verification simulation: Results for the proposed controller with respect to different external disturbances. [Color figure
can be viewed at wileyonlinelibrary.com]
prototype overhead crane (as shown in Fig. 9) at the College
of Engineering, Qufu Normal University. The physical parameters for the crane testbed are set as
tuned for the two controllers until the best performance is
derived as shown in Table III. The corresponding experiment results are shown in Figs 10, 11.
M x ¼ 6:157kg; M y ¼ 15:594kg; mp ¼ 1kg; g ¼ 9:8m=s2 ; l ¼ 0:6m
k ax ¼ 0:1m=s2 ; k vx ¼ 0:1m=s; k ay ¼ 0:1m=s2 ; k vy ¼ 0:1m=s;
f rox ¼ 20:371; k rx ¼ 0:5; ηx ¼ 0:01; f roy ¼ 23:652; k ry ¼ 1:2; ηy ¼ 0:01
and the target location is
pdx ¼ 0:3m; pdy ¼ 0:4m
4.2.1 Comparison experiments
We compare the proposed control method with the
OFB controller in this subsection. The control gains are
As can be seen from Figs 10, 11, the proposed nonlinear coupling tracking control method is superior to the OFB
control method in terms of both swing amplitude and residual swing. Moreover, the control input amplitude of the proposed controller is the smaller. Therefore, we can conclude
from the direct comparisons that the transient performance
of the proposed controller is improved by the introduction
of two generalized signals.
4.2.2 Robustness verification experiments
The following two cases are taken into consideration.
Case 1. Different cable lengths. The cable length is set as
l = 0.4m, whereas the control gains remain the same as those
in experiment 1.
Table III. Control gains for experiment 1.
Controllers
Fig. 9. Overhead crane experimental testbed. [Color figure can be
viewed at wileyonlinelibrary.com]
kp1
kd1
kp2 kd2
OFB controller
4.5 10.018 4
Proposed controller 9 10
9
λ
γ
λx
λy
8.5 NA NA 1 0.43
8.5 0.2 1
NA NA
© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
M. Zhang et al.: Enhanced coupling nonlinear tracking controller
0.4
0.3
0.2
0
0
2
4
6
8
10
12
14
16
18
0
20
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
2
1
0
0
-2
-1
-4
0
2
4
6
8
10
12
14
16
18
20
20
20
10
10
0
0
-10
0
2
4
6
8
10
12
14
16
18
20
Time(sec)
Time(sec)
Fig. 10. Experiment 1: Results for OFB controller. [Color figure can be viewed at wileyonlinelibrary.com]
0.4
0.3
0
x
xd
0
2
4
6
8
10
12
14
16
y
yd
0.2
18
20
0
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
0.5
1
0
0
-0.5
-1
0
2
4
6
8
10
12
14
16
18
20
-1
10
4
5
2
0
0
-2
0
-5
2
4
6
8
10
12
14
16
18
20
Time(sec)
Time(sec)
Fig. 11. Experiment 1: Results for the proposed controller. [Color figure can be viewed at wileyonlinelibrary.com]
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Fig. 12. Experiment 2: Results for the proposed controller with respect to different cable lengths. [Color figure can be viewed at
wileyonlinelibrary.com]
© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018
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Fig. 13. Experiment 2: Results for the proposed controller with respect to different payload masses. [Color figure can be viewed at
wileyonlinelibrary.com]
Case 2. Different payload masses. The payload mass is
modified to mp = 2kg, whereas the control gains remain
the same as those in experiment 1.
Figs 12, 13 show the experiment results. As shown in
Figs 12, 13 the performance of the overhead crane system is
not influenced by the variations of cable length and payload
mass, which indicates strong robustness of the proposed
nonlinear coupling tracking control method and implies significant benefit for its practical use.
V. CONCLUSIONS
In this paper, we propose a novel nonlinear coupling
tracking controller for 3D overhead crane systems, which
can drive the trolley to the desired positions accurately
and eliminate the payload swing quickly. Two S-shape
trajectories with physical constraints are selected as the
desired trajectories for the trolley based on the operation
experience, mathematical analysis, and operational efficiency. Then, on the basis of strict mathematical analysis,
the controller is proposed to track the desired trajectories,
providing theoretical support for the superior performance
of the controller. By invoking Lyapunov techniques and
Barbalat’s Lemma, we conclude that the equilibrium point
of the closed-loop system is asymptotically stable. The
superior transient performance and strong robustness with
respect to different cable lengths, payload masses,
destinations, and external disturbances of the tracking
controller are demonstrated by some simulation and
hardware experiment results. In future studies, we will try
to apply the method to a 3D crane system with time-variable
cable length.
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Xuewen Rong received his bachelor’s and
master’s degrees from Shandong University of Science and Technology, China, in
1996 and 1999, respectively. He received
his Ph.D. from Shandong University,
China, in 2013. He is currently a senior
engineer at the School of Control Science
and Engineering, Shandong University, China. His research
interests include robotics, mechatronics, and hydraulic
servo driving technology.
Rui Song received his B.E. degree in industrial automation in 1998, M.S. degree
in control theory and control engineering
in 2001 from Shandong University of Science and Technology, and Ph.D. in control
theory and control engineering from
Shandong University in 2011. He is engaged in research on intelligent sensor networks, intelligent
robot technology, and intelligent control systems. He is currently an Associate Professor at the School of Control
Science and Engineering of Shandong University in Jinan,
China.
Menghua Zhang received her M.S. degree from the School of Electrical Engineering and Automation, University of
Jinan, Jinan, China, in 2014. She is currently working toward a Ph.D. with the
School of Control Science and Engineering, Shandong University, Jinan, China.
Her research interests include control of mechatronics,
underacutated overhead crane systems, and bionic eye
systems.
Xincheng Tian received his B.S. degree
in industrial automation and M.S. degree
in automation from Shandong Polytech
University (now Shandong University),
Shandong, China, in 1988 and 1993, respectively. He received his Ph.D. in aircraft
control, guidance, and simulation from
Nanjing University of Aeronautics and Astronautics,
Nanjing, China, in 2000. He is currently a Professor at
Shandong University. His current research interests include
robotics, mechatrotics, and CNC techniques.
Xin Ma received her B.S. degree in industrial automation and M.S. degree in automation
from
Shandong
Polytech
University (now Shandong University),
Shandong, China, in 1991 and 1994, respectively. She received her Ph.D in aircraft control, guidance, and simulation
from Nanjing University of Aeronautics and Astronautics,
Nanjing, China, in 1998. She is currently a Professor at
Shandong University. Her current research interests include
artificial intelligence, machine vision, human-robot interaction, and mobile robots.
Yibin Li received his B.S. degree in automation from Tianjin University, Tianjin,
China, in 1982, M.S. degree in electrical
automation from Shandong University of
Science and Technology, Shandong,
China, in 1990, and Ph.D. in automation
from Tianjin University, China, in 2008.
From 1982 to 2003, he worked with Shandong University
of Science and Technology, China. Since 2003, he has been
the Director of the Center for Robotics, Shandong University. His research interests include robotics, intelligent control theories, and computer control system.
© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
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