A nonlinear control model-based scheme is designed to achieve the three objectives: i drive the crane system to the desired positions, ii suppresses the vibrations of the payload, and ii
Trang 1Nonlinear Adaptive Control of a 3D Overhead Crane
Quoc Toan Truong, Anh Huy Vo and Quoc Chi Nguyen
Ho Chi Minh City University of Technology e-Mail: tqtoan@hcmut.edu.vn ; vahuy@yahoo.com; nqchi@hcmut.edu.vn
Tóm tắt
Trong bài báo này, vấn đề phát triển một bộ điều khiển phi tuyến của hệ thống cầu trục trong không gian 3D được trình bày Mô hình phi tuyến của cầu trục được xây dựng với giả thuyết hệ thống có khối lượng tập trung (lumped mass model) Giải pháp điều khiển phi tuyến sẽ nhằm đạt được ba mục tiêu: (i) điều khiển hệ thống crane đến vị trí mong muốn, (ii) triệt tiêu dao động lắc của tải trong khi di chuyển, (iii) điều khiển bám cho vận tốc nâng hạ tải Trong bộ điều khiển phi tuyến, luật ước lượng các thông số chưa xác định rõ của hệ thống (lực ma sát và khối lượng tải) được sử dụng Với luật điều khiển được thiết kế, ổn định tiệm cận của hệ thống cầu trục được chứng minh bằng phương pháp Lyapunov Hiệu quả của luật điều khiển được kiểm chứng thông qua mô phỏng số
Abstract:
In this paper, a nonlinear adaptive control of a 3D overhead crane is investigated A dynamic model of the overhead crane is developed, where the crane system is assumed as a lumped mass model Under the mutual effects of the sway motions of the payload and the hoisting motion, the nonlinear behavior of the crane system
is considered A nonlinear control model-based scheme is designed to achieve the three objectives: (i) drive the crane system to the desired positions, (ii) suppresses the vibrations of the payload, and (iii) velocity tracking of hoisting motion The nonlinear control scheme employs adaptation laws that estimate unknown system parameters, friction forces and the mass of the payload The estimated values are used to compute control forces applied to the trolley of the crane The asymptotic stability of the crane system is investigated by using the Lyapunov method The effectiveness of the proposed control scheme is verified by numerical simulation results
1 Introduction
Overhead crane systems are widely used to move
heavy cargo from one place to another place in
factories and harbors A crane is naturally an
underactuated mechanical system, in which the
number of actuators is less than the degree of
freedom of the system For an overhead crane, the
degree of freedom is five (i.e., trolley and rail
positions, rope length, and two sway angles), but
the number of actuators is three (i.e., trolley, rails,
and hoisting motors) To improve the transferring
efficiency, the trolley and rails should travel as fast
as possible However, fast trolley/rail movement
will cause sway of the load, which is dangerous
for the operator and the crane system Therefore,
the research field of crane control (i.e., sway
vibration control, trolley motion control, and
hoisting motion control) is focused by many
researchers
3D models of overhead crane systems were
developed in a number of researches The
development of modeling and control method for a
3D crane has been reported in Lee (1998) Nonlinear dynamic modeling and analysis of a 3D overhead gantry crane system with system parameters variation was introduced Ismail et al (2009) These researches developed 3D models of overhead cranes with four degrees of freedom (DOF), i.e., trolley and rail positions and two sway angles However, the two factors, hoisting motion and effects of friction were not considered In practice, it should be noted that the variation of the length of cable affects significantly to the crane dynamic Moreover, it is well known that the consideration of the friction forces is very important in many mechanical systems, especially,
in crane systems, the effects of friction forces are considerable under heavy payload
In this paper, we introduce a 3D model of an overhead crane system having five DOF (trolley and rail positions, rope length, and two sway angles) under two friction forces (appearing between trolley and horizontal rails and between horizontal rails and vertical rails)
In crane control, there are two approaches, semi-automated and semi-automated approaches In the first approach, operator is kept in the loop and
Trang 2dynamics of the load are modified to make his job
easier A number of researchers developed
damping controllers employing feedback signals
of the load sway angles and their rate (Henry et al.,
2001; Masoud et al., 2002) Robinett et al (1999)
developed a filter to remove noise of the input to
avoid exciting the load near its natural frequency
Balachandran et al., 1999 used a mechanical
absorber to suppress the vibration of the payload
considerable energy consumption, which is not
cost effective
In the second approach, the operator is removed
from the loop and the operation is completely
automated Many various techniques can be
applied for this The first technique is based on
generating trajectories to transfer the load to its
destination with minimum swing Sakawa and
Shindo (1982), in which optimal-velocity
profiles of the trolley that minimize the sway
angle and its derivative were proposed In their
work, the trolley motion was split into five
different sections Another important method of
generating trajectories is input shaping, which
consists of a sequence of acceleration and
deceleration pulses These sequences are generated
such that there is no residual swing at the end of
the transfer operation (Karnopp et al., 1992; Teo et
al., 1998; Singhose et al., 1997; Singhose, Porter,
Kenison, & Kriikku, 2000; Hong, Huh, & Hong,
2003; Sorensen, Singhose, & Dickerson, 2006;
Ngo & Hong, 2009; Kim & Singhose, 2010)
Unfortunately, it often meets great difficulty when
trying to obtain these system coefficients which
vary with the changes of the payload or the rope
length, and coefficient friction As mentioned
before, the objective of the crane control is to
move load from point to point and at the same
time minimize the load swing Two tasks will be
done by designing two feedback controllers:
controller for prompt sway suppression by a
proper feedback of the swing angle and its rate and
controller designed to make trolley follow a
reference trajectory with trolley position and
velocity are used for tracking feedback The
tracking controller can be either a classical
Proportional-Derivative (PD) controller (Henry,
1999; Masoud 2000) or a Fuzzy Logic Controller
(FLC) (Yang et al., 1996; Nalley and Trabia, 1994;
Lee et al., 1997; Ito et al., 1994; Al-Moussa,
2000) Similarly, the anti-swing controller is
designed by different methods Henry (1999) and
Masoud (2002) used delayed-position feedback,
whereas Nalley and Trabia (1994), Yang et al
(1996), and Al-Moussa (2000) used FLC
Generally, the cable length is considered in the design of the anti-swing controller Lee, Liang, & Segura (2006) proposed a sliding-mode anti-swing control for overhead cranes to realize an anti-swing trajectory control with high-speed load hoisting A feedback linearization control of container cranes with varying rope length proposed by Park, Chwa, & Hong (2007); these studies were used for 2-D modeling overhead crane
The friction between the trolley and the rails, the varying payload weight (from 12 Ton to 72 Ton) are uncertainties in crane dynamics Adaptive controller for 2D modeling overhead crane with the friction force model (Aschemann, 2000) was proposed by Ma, Fang & Zhang (2008) An adaptive tracking control of 3-D overhead crane systems was investigated by Yang et al (2006) but they didn’t consider influence of hoisting mechanism to the system We extend this research
by developing a nonlinear adaptive control for 3D overhead crane under friction forces, where the hosting motion is considered
In this paper, a modeling of 3D overhead crane is derived based on using Lagrange-Euler equations, where a five degrees of freedom-crane (motions of trolley on rail, motion of rail, motion rolling the rod by hosting mechanism, two sways of payload) system is considered Effects of friction forces are also included A nonlinear adaptive controller is proposed to drive the crane to its desired position and to suppress the sway motion of payload Adaptation laws are used to estimate the unknown parameters, i.e., coefficients friction and the payload mass Lyapunov method is employed to investigate stability of the crane system under the proposed controller The effectiveness of the proposed control method is illustrated by numerical simulation results
2 Dynamics of a 3D overhead crane
This section provides detail description on the modeling of the 3D overhead crane system, as a basis of a simulation environment for the study
on the effects of parameters variation to the system The Euler-Lagrange formulation is considered in characterizing the dynamic behavior of the crane system incorporate payload and length of the rope Fig 1 shows the swing motion of the load caused by trolley movement in XYZ is the inertial coordinate
system
Trang 3Table 1 Parameters of 3D overhead crane
system
x(t) Trolley position in X-direction
y(t) Trolley position in X-direction
l(t) rope length
c
r
p
( )t
coordinate Angle g( )t has two
components q( )t and f( )t , which are
1 0 3
2 0 3
A A A ,
respectively
c
components F X and F Y applied to
the trolley in the X- and Y-directions,
respectively
l
cable
and
f f Friction forces in the X- and
Y-direction, respectively
Fig 1 Three-dimension overhead crane
The following assumptions are made:
(i) The payload and the trolley are connected by
a massless, rigid link
(ii) The position of the trolley and the swing
angle of the payload are measureable
(iii) The trolley mass is unknown
(iv) The unknown friction forces, which occur in
the contact surfaces between the wheels of the
trolley and the rails Y, the wheels of the rail
Y and the rails X are unknown (v) The time-varying length of the rope
As shown in Fig 1, the rail, the trolley, and the payload position vectors are given as follows:
Trang 4[ , 0, 0],
[ , , 0],
[ sin cos , sin , cos cos ],
r
c
p
x
x y
r
r
r
(1)
where x and y are the trolley position in X- and Y-
directions, respectively
5-DOF-crane model yielding the generalized
( )t R
( )t x t( ) ( ) ( ) ( ) ( )y t l t q t f t
The forces applied to the system are given by
[(F x f cx) (F y f cy) 0 0].F l
The friction forces in the X- and Y-directions
respective are given as follows
( ) ( ),
( ) ( ),
f t c x t
f t c y t
(4)
where c x and c y are the viscous friction
coefficients in X- and Y-directions, respectively
The total kinetic energy K and the potential energy
P of the crane system are derived as
, ,
trolley rail payload
payload
where
1
2
car c c c
K mr r ,
1
2
rail r r r
K mr r ,
1
2
payload p p p
K m r r ,
(1 cos cos )
payload p
The equations of motion using the Lagrange Euler
equations are derived as follows:
( 1, 2,3, 4,5),
i
with L K P
The dynamic equations (6) can be rewritten as:
M q qC q q q G q u (7)
where
0
M
0 0
0 0
0 0
0 0
C
0 cos cos sin cos cos sin 0
p p
m g m g
m gL
m gL
G
f fc
11 13 14 15
sin cos cos cos sin sin
p p p
m m l
q f
q f
q f
22 23 25
sin cos
p p
f f
31 32 33
sin cos sin
P P P
f
41
44
cos cos cos
P P
f
13
14
15
cos cos sin sin cos cos
sin cos cos sin
sin sin sin cos cos sin
p
p
q f
q f
23
25
cos
p
ff
2 34
35
cos
P P
fq f
2 43
44
2 45
cos
sin cos
P
P
fq
53
2 54 55
sin cos
P P P
C m ll
f
Trang 5where 5 5
( )R
M q is inertia matrix of the crane
( )
m R
( )R
G q is the gravity term Based
on the structure of M q( ) and Cm( , )q q given by (7),
it should be noted that the following
skew-symmetric relationship is satisfied:
5 1
( ) ( , ) 0,
2
T
ξ M q C q q ξ ξ , (8)
where M q( )represent the time derivative of M q( )
and M q( )can be upper and lower bounded by the
following inequality:
n ξ ξ M q ξn ξ ξ R , (9)
where n1 and n2Rare positive bounded
constants
3 Control design
For convenience, we define a generalized
coordinate as follows:
T
,
m a
q q q (10)
where
T
( ) y( ) ( )
m x t t l t
( ) ( )
a q t f t
The equations of motion of the overhead crane
(7) are partitioned in the following:
( ) ( )
, (11)
mf mcf
where
0 0
mm
25
0
ma
m
M
41
0
am
m
55
0 0
aa m m
M
13 23
0 0
0 0
mm
C C
25
0
ma
C
C
43 53
0 0
0 0
ma
C C
aa
C
( ) sin cos
cos sin
p a
p
m gl
q
m gl
x
mf y l
F F F
u ,
( ) ( ) 0
x mcf y
c x t
c y t
u
To achieve the control objective, with given desired signals q q qd, d, d (which are assumed to
be bounded), it needs to determine a control law mf
u that guarantees the asymptotically convergence of q to qd The error signals are
defined as:
T
,
d m a
e q q e e
(12) where
T
x y l
a a da q q d f f d e q e f
where x d, y d, , l d q f d, d are defined trajectories of
, , , ,
x y L q f, respectively
We define qr as following:
rx ry
rl
r r
q q q q q
q f
(13)
where
m
a
K K
K
1 2 3
m
k k k
5
0 0
a k k
and
K K are arbitrary positive definite matrices
We define a combined error:
r
(14)
Trang 6Then, using (11), the dynamics in terms of the
newly defined signals sm and sa can be derived
as:
,
(15) where
We can be expressed τm and τa as in term of a
known matrix ωm , ωa and unknown parameter
vectors, ψm and ψa
22 23 32
p
x m
y
x
m y
c c
t
(16) where
11
13 sin cos cos cos
sin sin (cos cos sin sin )
cos cos sin cos cos sin
sin sin cos sin sin cos
m rx
r r
q
q f
q f
t
22
23
32
2
cos sin
sin cos sin
r
q
f f
t
11 22
0
0
p a
p a
m m
t t
τ ω ψ
(17) where
11
2
2 22
2
cos sin cos cos sin sin cos
cos sin
r r
q f
f q
f fq
fglcos sin q f
As a majority of the adaptive controller, the following signal is defined:
2( ( ) ( ), ( ) 0
2 ( ), ( ) 0, ( ) 0 , ( ) 0, ( ) 0
Z a t b t Z t
d
(18) where d x is some small positive constant and
2
2
2
ˆ ( )
ˆ ( )
m
m
m
a t
b t
e e e
s
s
s
(19) Note that (18) is convenience to define a differential equation, where its variable
( )
x
Z t remains positive Define a positive function
( ) x
h t Z It can be shown that:
2
2
1
ˆ
( ) (20)
m
m
h
h t
e e
s
s
Next, we assume that there exists a measure zero set of time sequences t i i1 such that
( )i 0
Z t (i.e., h t( )i 0,i1, 2,3, ,), and then, the existence assumption is verified
Let the adaptive control law be designed as:
ˆ
mf 1 1 v mv m
u ω ψ τ K s ,
(21) where
2
( 1)
ˆ
m
m
h e
s
where ψˆ1 and ψˆ2 are the estimates of
and
ψ ψ , respectively The adaption laws are
given as
T
1 1 1 T
2 1 2
ˆ ˆ
m a
(22)
Trang 7Then the error dynamics can be obtained as:
0 0 ˆ
,
1 1 v
2 2 av a
which can be rewritten as
ˆ
2 2 av a
where
ˆ ˆ
Since y y1, 2 are constant parameters, we
obtain
.
1
ψ
(24)
Theorem: Consider the system (7) or (23) with
the parameters systems unknown The proposed
control law (21) employing the adaption laws (22)
guarantees the asymptotic stability of the systems,
i.e., s 0, e 0, and e 0 as t
Proof: Lyapunov function candidate can be
defined as
V t s M q s ψ λ ψ ψ λ ψ Z
(25)
In the previous, due to the quadratic form of system states as well as the definition of Z t x( )and
( )
V t is always positive-definite and indeed a Lyapunov function candidate By taking the time
derivative of V t( ), we have
1 1
1
2
v
av a
hh hh
s
2
T
1 1
( 1)
ˆ
ˆ
(
m
m
m
m
m
h
h
h
e
e e
s
s
ω ψ s
s
2
2
2
1)
ˆ
ˆ
m
m
m
m
h
e
e e
s
s
s
s
ˆ
a
Trang 8The substitution of (22) into (26) yields and due
to the positive-definiteness of K, we have:
T
V t s Ks , when h t ( ) 0 (27)
The solution of Z t( )from equation (18) is
defined and continuous for all t 0, so h t( )is
continuous for all t i Because V t( )is a continuous
function of h t( )so it keep remains to be continuous
at time t i , i.e V t( )i V t(i) From the hypothesis,
( )i 0
V t and V t(i)0 then we can conclude that
( )
V t is non-increasing at time
i
t , which then readily implies that s h, L Therefore,
, m, a
e t t L directly from (14) and definitions
of t t m, a , then following (23) that sL On the
other hand, it is clear that the set of time instants
is t i i1 measure zero
thus
Therefore by invoking the Barbalat’s lemma,
we readily obtain that s 0 asymptotically
as t , therefore implies e and e 0as t
Finally, to complete the proof in theory, we
need to show that the above hypothesis that the set
of time instants t i i1is indeed measure zero
However, if is quite straightforward from (18)
simply using the fact that all signals are uniformly
bounded after the proposed control is employed
4 Simulation results
To illustrate the controller performance, we
controller of (21) employing the adaption laws
(22) in a crane with the following parameters:
5 kg, 5 kg, 0.5 kg, 9.81 m/s
The friction parameters are given as
c x 0.01, cy 0.01
The trolley moves to a desired position selected as
follows:
3 m, y 1 m, 0.5 m,
0, y 0, L 0
x
The initial state of the system is chosen as:
(0) 0 , (0) 0, y(0) 0 , (0) 0, (0) 1 m, (0) 0, (0) 0, (0) 0, (0) 0, (0) 0
y
The control law is turned until a best performance
is achieved, which yields the following control gains:
8.05 0
0 8.05
0 0 0.45
6.9 0
0 6.9
l13.9, l23.2, e1
Figs 2, 3 and 4 plot the tracking of trolley Fig 5 and 6 are the swing angles It can be seen that when the tracking of trolley positions reaches desired positions after 8 seconds and the swing angles go to zero asymptotically at seconds 10 The swing angles are about 2 degrees in the transferring process
Selection of controller parameters can affect the system performance Unfortunately, there is no systematic approach for the selection of these values They must be chosen using iterative simulation and a tradeoff between system response and control gains should be made
Fig 2 Position of the trolley in X-direction
Trang 9Fig 3 Position of the trolley in Y-direction
Fig 4 Rope length
Fig 5 Sway angle q( )t
Fig 6 Sway angle f( )t
Fig 7 Estimated parametersψˆ ( )m t
Fig 8 Estimated Parametersψˆ ( )a t
Figs 7 and 8 are the parameters estimation results The estimate values will converge to constant values, if the plant is stable As shown in these figures, the values may not get the true values However, getting true values of the parameters was not the purpose of this paper
5 Conclusion
In this paper, a 5-DOF dynamic model of the 3D overhead crane was developed under the effects of friction forces and the unknown parameters A nonlinear adaptive controller was proposed for the overhead crane to drive it to its desired point and
to suppress the swing of payload Under the proposed controller, asymptotic stability of the overhead crane system is proved by using Lyapunov method Simulation results illustrate the effectiveness of the proposed controller An experiment system is under construction at Mechatronics Lab (HCMUT) to verify the effectiveness of the controller
Trang 10References
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Control Engineering
received B.C degree in mechanical engineering at the Chi Minh City University
of Technology (HCMUT) in
2010 He has been pursuing a M.E program at the HCMUT since 2011 He has been a
Department of Mechatronics Engineering, Faculty of Mechanical Engineering (HCMUT) since 2010 His research interests include nonlinear control of dynamical systems, robotics, and industrial applications of control engineering
Anh Huy Vo received the B.Eng degree and
M.Eng degree both in mechanical engineering at the Ho Chi Minh University of Technology (Vietnam), in 1998 and 2003, respectively He has been a faculty member at the Department of
University of Technology since 1998 His research interests include control of offshore cranes,