Optimal Tracking with Sway Suppression Control for a Gantry Crane System

European Journal of Scientific Research
ISSN 1450-216X Vol.33 No.4 (2009), pp.630-641
© EuroJournals Publishing, Inc. 2009
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Optimal Tracking with Sway Suppression Control
for a Gantry Crane System
M.A. Ahmad
Control and Instrumentation Research Group (COINS),
Faculty of Electrical and Electronics Engineering
Universiti Malaysia Pahang, Lebuhraya Tun Razak
26300, Kuantan, Pahang, Malaysia
E-mail:
Tel: +609-5492366; Fax: +609-5492377
R.M.T. Raja Ismail
Control and Instrumentation Research Group (COINS)
Faculty of Electrical and Electronics Engineering
Universiti Malaysia Pahang, Lebuhraya Tun Razak
26300, Kuantan, Pahang, Malaysia
E-mail:
Tel: +609-5492366; Fax: +609-5492377
M.S. Ramli
Control and Instrumentation Research Group (COINS)
Faculty of Electrical and Electronics Engineering
Universiti Malaysia Pahang, Lebuhraya Tun Razak
26300, Kuantan, Pahang, Malaysia
E-mail:
Tel: +609-5492366; Fax: +609-5492377
N.M. Abdul Ghani
Control and Instrumentation Research Group (COINS)
Faculty of Electrical and Electronics Engineering
Universiti Malaysia Pahang, Lebuhraya Tun Razak

26300, Kuantan, Pahang, Malaysia
E-mail:
Tel: +609-5492366; Fax: +609-5492377
M.A. Zawawi
Control and Instrumentation Research Group (COINS)
Faculty of Electrical and Electronics Engineering
Universiti Malaysia Pahang, Lebuhraya Tun Razak
26300, Kuantan, Pahang, Malaysia
E-mail:
Tel: +609-5492366; Fax: +609-5492377


Optimal Tracking with Sway Suppression Control for a Gantry Crane System

631

Abstract
This paper presents investigations into the development of hybrid control schemes
for input tracking and anti-swaying control of a gantry crane system. A nonlinear overhead
gantry crane system is considered and the dynamic model of the system is derived using the
Euler-Lagrange formulation. To study the effectiveness of the controllers, initially a Linear
Quadratic Regulator (LQR) control is developed for cart position control of gantry crane.
This is then extended to incorporate input shaper control schemes for anti-swaying control
of the system. The positive input shapers with the derivative effects are designed based on
the properties of the system. Simulation results of the response of the manipulator with the
controllers are presented in time and frequency domains. The performances of the hybrid
control schemes are examined in terms of level of input tracking capability, swing angle
reduction, time response specifications and robustness to parameters uncertainty in
comparison to the LQR control. The effects of derivative order of the input shaper on the
performance of the system are investigated. Finally, a comparative assessment of the

control techniques is presented and discussed.

Keywords: Gantry crane, sway control, input shaping, LQR controller.

1. Introduction
The main purpose of controlling a gantry crane is transporting the load as fast as possible without
causing any excessive swing at the final position. However, most of the common gantry crane results
in a swing motion when payload is suddenly stopped after a fast motion [1]. The swing motion can be
reduced but will be time consuming. Moreover, the gantry crane needs a skilful operator to control
manually based on his or her experiences to stop the swing immediately at the right position. The
failure of controlling crane also might cause accident and may harm people and the surrounding.
Various attempts in controlling gantry cranes system based on open loop system were
proposed. For example, open loop time optimal strategies were applied to the crane by many
researchers such as discussed in [2,3]. They came out with poor results because open loop strategy is
sensitive to the system parameters (e.g. rope length) and could not compensate for wind disturbances.
Another open loop control strategies is input shaping [4,5,6]. Input shaping is implemented in real time
by convolving the command signal with an impulse sequence. The process has the effect of placing
zeros at the locations of the flexible poles of the original system. An IIR filtering technique related to
input shaping has been proposed for controlling suspended payloads [7]. Input shaping has been shown
to be effective for controlling oscillation of gantry cranes when the load does not undergo hoisting [8,
9]. Experimental results also indicate that shaped commands can be of benefit when the load is hoisted
during the motion [10].
On the other hand, feedback control which is well known to be less sensitive to disturbances
and parameter variations [11] is also adopted for controlling the gantry crane system. Recent work on
gantry crane control system was presented by Omar [1]. The author had proposed proportionalderivative PD controllers for both position and anti-swing controls. Furthermore, a fuzzy-based
intelligent gantry crane system has been proposed [12]. The proposed fuzzy logic controllers consist of
position as well as anti-sway controllers. However, most of the feedback control system proposed
needs sensors for measuring the cart position as well as the load swing angle. In addition, designing the
swing angle measurement of the real gantry crane system, in particular, is not an easy task since there
is a hoisting mechanism.

This paper presents investigations into the development of hybrid control schemes for input
tracking and anti-swaying control of a gantry crane system. Hybrid control schemes based on
feedforward with LQR controllers are investigated. To demonstrate the effectiveness of the proposed


632

M.A. Ahmad, R.M.T. Raja Ismail, M.S. Ramli, N.M. Abdul Ghani and M.A. Zawawi

control schemes, initially a LQR controller is developed for cart position control of gantry crane
system. This is then extended to incorporate the positive input shapers for swing control of the gantry
crane. This paper provides a comparative assessment of the performance of hybrid control schemes
with different derivative order of input shapers.

2. Gantry Crane System
The two-dimensional gantry crane system with its payload considered in this work is shown in Figure
1, where x is the horizontal position of the cart, l is the length of the rope, θ is the sway angle of the
rope, M and m is the mass of the cart and payload respectively. In this simulation, the cart and payload
can be considered as point masses and are assumed to move in two-dimensional, x-y plane. The tension
force that may cause the hoisting rope elongate is also ignored. In this study the length of the cart, l =
1.00 m, M = 2.49 kg, m = 1.00 kg and g = 9.81 m/s2 is considered.
Figure 1: Description of the gantry crane system

3. Modelling of the Gantry Crane
This section provides a brief description on the modelling of the gantry crane system, as a basis of a
simulation environment for development and assessment of the active sway control techniques. The
Euler-Lagrange formulation is considered in characterizing the dynamic behaviour of the crane system
incorporating payload.
Considering the motion of the gantry crane system on a two-dimensional plane, the kinetic
energy of the system can thus be formulated as

T=

1
1
Mx& 2 + m( x& 2 + l&2 + l 2 θ& 2 + 2 x&l& sin θ + 2 x&lθ& cos θ )
2
2

(1)

The potential energy of the beam can be formulated as

U = − mgl cos θ

(2)
To obtain a closed-form dynamic model of the gantry crane, the energy expressions in (1) and
(2) are used to formulate the Lagrangian L = T − U . Let the generalized forces corresponding to the
generalized displacements q = { x , θ } be F = { F x ,0} Using Lagrangian’s equation


Optimal Tracking with Sway Suppression Control for a Gantry Crane System

633

⎛ ∂L ⎞ ∂ L
⎜
⎟−
j = 1,2
= Fj

⎜ ∂q& j ⎟ ∂q j
⎝
⎠
the equation of motion is obtained as below,
d
dt

(3)

Fx = ( M + m ) &x& + ml (θ&&cos θ − θ& 2 sin θ ) + 2 ml&θ& cos θ + m&l&sin θ
lθ&& + 2l&θ& + &x& cos θ + g sin θ = 0

(4)

(5)
In order to eliminate the nonlinearity equation in the system, a linear model of gantry crane
system is obtained. The linear model of the uncontrolled system can be represented in a state-space
form as shown in equation (6) by assuming the change of rope and sway angle are very small.

x& = Ax + Bu
y = Cx

[

(6)

]

T
with the vector x = x θ x& θ&

0
⎡0
⎢0
0
⎢
mg
A = ⎢0
M
⎢
(
M
+ m) g
⎢
0
−
⎢⎣
Ml

and the matrices A and B are given by
1 0⎤
⎡ 0 ⎤
⎢ 0 ⎥
0 1⎥⎥
⎢ 1 ⎥
⎥, C = [1 0 0 0] ,
0 0⎥ , B = ⎢
⎢ M ⎥
⎥
⎢ 1 ⎥
⎥

0 0⎥
⎢⎣− Ml ⎥⎦
⎦

D = [0]

(7)

4. Linear Quadratic Regulator (LQR) Control Scheme
A more common approach in the control of manipulator systems involves the utilization linear
quadratic regulator (LQR) design [13]. Such an approach is adopted at this stage of the investigation
here. In order to design the LQR controller a linear state-space model of the gantry crane system was
obtained by linearising the equations of motion of the system. For a LTI system
x& = Ax + Bu
(8)
the technique involves choosing a control law u = ψ (x) which stabilizes the origin (i.e., regulates x to
zero) while minimizing the quadratic cost function
∞

J =

∫ x (t )

T

Qx (t ) + u (t ) T Ru (t ) dt

(9)

0

where Q = Q T ≥ 0 and R = R T > 0 The term “linear-quadratic” refers to the linear system dynamics
and the quadratic cost function.
The matrices Q and R are called the state and control penalty matrices, respectively. If the
components of Q are chosen large relative to those of R , then deviations of R from zero will be
penalized heavily relative to deviations of u from zero. On the other hand, if the components of R are
large relative to those of Q then control effort will be more costly and the state will not converge to
zero as quickly.
A famous and somewhat surprising result due to Kalman is that the control law which
minimizes J always takes the form u = ψ ( x) = − Kx The optimal regulator for a LTI system with
respect to the quadratic cost function above is always a linear control law. With this observation in
mind, the closed-loop system takes the form
x& = ( A − BK ) x
(10)
and the cost function J takes the form
∞

J = ∫ x(t )T Qx(t ) + (− Kx(t ))T R(− Kx(t ))dt
0

(11)


634

M.A. Ahmad, R.M.T. Raja Ismail, M.S. Ramli, N.M. Abdul Ghani and M.A. Zawawi
∞

J = ∫ x(t )T (Q + K T RK ) x(t )dt

(12)

0

Assuming that the closed-loop system is internally stable, which is a fundamental requirement
for any feedback controller, the following theorem allows the computation value of the cost function
for a given control gain matrix K.

5. Input Shaping Control Schemes
The design objectives of input shaping are to determine the amplitude and time locations of the
impulses in order to reduce the detrimental effects of system flexibility. These parameters are obtained
from the natural frequencies and damping ratios of the system. The input shaping process is illustrated
in Figure 2. The corresponding design relations for achieving a zero residual single-mode swaying of a
system and to ensure that the shaped command input produces the same rigid body motion as the
unshaped command yields a two-impulse sequence with parameter as
(13)
t1 = 0, t2 = π , A1 = 1 , A2 = K
ωd

1+ K

1+ K

where
− ζπ
1−ζ
K =e
, ωd = ωn 1 − ζ 2
( ω n and ζ representing the natural frequency and damping ratio respectively) and tj and Aj are the
time location and amplitude of impulse j respectively. The robustness of the input shaper to errors in

natural frequencies of the system can be increased by solving the derivatives of the system swaying
equation. This yields a four-impulse sequence with parameter as
3π
π
2π
, t4 =
, t3 =
t1 = 0, t2 =
2

ωd

A1 =

A3 =

ωd

ωd

1
1 + 3K + 3K + K
3K 2
2

3

1 + 3K + 3K 2 + K 3
where K as in (13).

, A2 =

3K
1 + 3 K + 3K 2 + K 3

, A4 =

K3

(14)

1 + 3K + 3K 2 + K 3

A1

*

A2

Time
Unshaped Input

Input Shaper

Amplitude

Amplitude

Figure 2: Illustration of input shaping technique.

Time
Shaped input

6. Implementation and Result
In this investigation, hybrid control schemes for tracking capability and sway angle suppression of the
gantry crane system are examined. Initially, a Linear Quadratic Regulator (LQR) is designed. This is
then extended to incorporate input shaping scheme for control of sway angle of the hoisting rope.
The tracking performance of the Linear Quadratic Regulator applied to the gantry crane system
was investigated by firstly setting the value of vector K and N which determines the feedback control


Optimal Tracking with Sway Suppression Control for a Gantry Crane System

635

law and for elimination of steady state error capability respectively. Using the lqr function in the
Matlab, both vector K and N were set as
K = [700 428.3885 3.8267 1.4066 ] and N = [700]
The natural frequencies were obtained by exciting the gantry crane system with an unshaped
reference input under LQR controller. The input shapers were designed for pre-processing the
trajectory reference input and applied to the system in a closed-loop configuration, as shown in Figure
3.
Figure 3: Block diagram of the hybrid control schemes configuration.
LQR Controller
Desired
input

Input
shaper

Shaped
input

N

+

Gantry Crane
System

-

Output
responses

K

6.1. LQR Controller

In this work, the input is applied at the cart of the gantry crane. The cart position of the gantry crane is
required to follow a trajectory within the range of ± 4 m. The first three modes of swing angle
frequencies of the system are considered, as these dominate the dynamic of the system.
The responses of the gantry crane system to the unshaped trajectory reference input were
analyzed in time-domain and frequency domain (spectral density). These results were considered as the
system response to the unshaped input under tracking capability and will be used to evaluate the
performance of the input shaping techniques. The steady-state cart position trajectory of +4 m for the
gantry crane was achieved within the rise and settling times and overshoot of 1.372 s, 2.403 s and 0.20
% respectively. It is noted that the cart reaches the required position from +4 m to -4 m within 3 s, with
little overshoot. However, a noticeable amount of swing angle occurs during movement of the cart. It is
noted from the swing angle response with a maximum residual of ±1.4 rad. Moreover, from the PSD of

the swing angle response the swaying frequencies are dominated by the first three modes, which are
obtained as 0.3925 Hz, 1.177 Hz and 2.06 Hz with magnitude of 33.02 dB, -9.929 dB and -22.86 dB
respectively.
6.2. Hybrid Controller

In the case of hybrid control schemes, a ZV (two-impulse sequence) and ZVDD (four-impulse
sequence) shapers were designed for three modes utilising the properties of the system. With the exact
natural frequencies of 0.3925 Hz, 1.177 Hz and 2.06 Hz, the time locations and amplitudes of the
impulses were obtained by solving equations (13) and (14). For evaluation of robustness, input shapers
with error in natural frequencies were also evaluated. With the 30% error in natural frequency, the
system swaying frequencies were considered at 0.5103 Hz, 1.5301 Hz and 2.678 Hz for the three
modes of swaying frequencies. Similarly, the amplitudes and time locations of the input shapers with
30% erroneous natural frequencies for both the ZV and ZVDD shapers were calculated. For digital
implementation of the input shapers, locations of the impulses were selected at the nearest sampling
time.


636

M.A. Ahmad, R.M.T. Raja Ismail, M.S. Ramli, N.M. Abdul Ghani and M.A. Zawawi

The system responses of the gantry crane system to the shaped trajectory input with exact
natural frequencies using LQR control with ZV and ZVDD shapers are shown in Figure 4. Table 1
summarises the levels of swaying reduction of the system responses at the first three modes in
comparison to the LQR control. Higher levels of swaying reduction were obtained using LQR control
with ZVDD shaper as compared to the case with ZV shaper. However, with ZVDD shaper, the system
response is slower. Hence, it is evidenced that the speed of the system response reduces with the
increase in number of impulse sequence. The corresponding rise time, setting time and overshoot of the
cart trajectory response using LQR control with ZV and ZVDD shapers with exact natural frequencies
is depicted in Table 1. It is noted that a slower cart trajectory response with less overshoot, as

compared to the LQR control, was achieved.
To examine the robustness of the shapers, the shapers with 30% error in swaying frequencies
were designed and implemented to the gantry crane system. The analysis from Figure 5 shows that the
swaying angles of the system were considerable reduced as compared to the system with LQR control.
However, the level of swaying reduction is slightly less than the case with exact natural frequencies.
Table 1 summarises the levels of swaying reduction with erroneous natural frequencies in comparison
to the LQR control. The time response specifications of the cart trajectory with error in natural
frequencies are summarised in Table 1. It is noted that the response is slightly faster for the shaped
input with error in natural frequencies than the case with exact frequencies. However, the overshoot of
the response is slightly higher than the case with exact frequencies. Significant swaying reduction was
achieved for the overall response of the system to the shaped input with 30% error in natural
frequencies, and hence proved the robustness of the input shapers.


Optimal Tracking with Sway Suppression Control for a Gantry Crane System
Figure 4: Response of the gantry crane with exact natural frequencies.
(a): Cart trajectory.
5
LQR-ZV Shaper
LQR-ZVDD Shaper

Horizontal position of the cart (m)

4
3
2
1
0
-1
-2

-3
-4
-5
0

5

10

15

20

25

30

Time (s)

(b): Sway angle.
2
LQR-ZV Shaper
LQR-ZVDD Shaper

Swing angle of the rope (rad)

1.5
1
0.5
0

-0.5
-1
-1.5
-2
0

5

10

15

20

25

30

Time (s)

(c): Power spectral density.
40
LQR
LQR-ZV Shaper
LQR-ZVDD Shaper

Power Spectral Density (dB)

20
0

-20
-40
-60
-80
-100
0

0.5

1

1.5

Frequency (Hz)

2

2.5

637


M.A. Ahmad, R.M.T. Raja Ismail, M.S. Ramli, N.M. Abdul Ghani and M.A. Zawawi
Figure 5: Response of the gantry crane with erroneous natural frequencies.
(a): Cart trajectory.
5
LQR-ZV Shaper
LQR-ZVDD Shaper

Horizontal position of the cart (m)

4
3
2
1
0
-1
-2
-3
-4
-5
0

5

10

15

20

25

30

Time (s)

(b): Sway angle.
2
LQR-ZV Shaper

LQR-ZVDD Shaper

Swing angle of the rope (rad)

1.5
1
0.5
0
-0.5
-1
-1.5
-2
0

5

10

15

20

25

30

Time (s)

(c): Power spectral density.
40

LQR
LQR-ZV Shaper
LQR-ZVDD Shaper

20

Power Spectral Density dB

638

0
-20
-40
-60
-80
-100
0

0.5

1

1.5

Frequency Hz

2

2.5

Optimal Tracking with Sway Suppression Control for a Gantry Crane System

639

6.3. Comparative Performance Assessment

By comparing the results presented in Table 1, it is noted that the higher performance in the reduction
of swaying of the system is achieved using LQR control with ZVDD shaper. This is observed and
compared to the LQR control with ZV shaper at the first three modes of swaying frequency. For
comparative assessment, the levels of swaying reduction of the hoisting rope using LQR control with
both ZV and ZVDD shapers are shown with the bar graphs in Figure 6. The result shows that, highest
level of swaying reduction is achieved in hybrid control schemes using the ZVDD shaper, followed by
the ZV shaper for all modes of swaying frequency. Therefore, it can be concluded that the LQR control
with ZVDD shapers provide better performance in swaying reduction effect as compared to the LQR
control with ZV shapers in overall.
Comparisons of the specifications of the cart trajectory responses of hybrid control schemes
using both ZV and ZVDD shapers are summarised in Figure 7 for the rise times and settling times. It is
noted that the differences in rise times of the cart trajectory response for the LQR control with ZV and
ZVDD shapers are negligibly small. However, the settling time of the cart trajectory response using the
LQR control with ZV shaper is faster than the case using the ZVDD shaper. It shows that, by
incorporating more number of impulses in hybrid control schemes resulted in a slower response.
Comparison of the results shown in Table 1 for the shaping techniques with error in natural
frequencies reveals that the higher robustness to parameter uncertainty is achieved with the LQR
control with ZVDD shaper. For both case of the ZV and ZVDD shapers, errors in natural frequencies
can successfully be handled. This is revealed by comparing the magnitude level of swaying frequency
of the system in Figure 6. Comparisons of the cart trajectory response using LQR control with ZV and
ZVDD shapers with erroneous natural frequencies are summarised in Figure 7. The results show a
similar pattern as the case with exact natural frequencies. The system response with ZVDD shaper
provides slightly slower responses than the ZV shaper.

Figure 6: Level of swaying reduction with exact and erroneous natural frequencies using ZV and ZVDD
shapers.
70

Level of reduction (dB)

60
50

LQR-ZV Shaper

40
LQR-ZVDD Shaper

30
LQR-ZV Shaper
(error)

20
LQR-ZVDD Shaper
(error)

10
0
Mode 1

Mode 2
Mode of Vibration

Mode 3

640

M.A. Ahmad, R.M.T. Raja Ismail, M.S. Ramli, N.M. Abdul Ghani and M.A. Zawawi

Figure 7: Rise and settling times of the cart trajectory with exact and erroneous natural frequencies using ZV
and ZVDD shapers.
7

Time (s)

6

5

LQR

4

LQR-ZV Shaper

3

LQR-ZVDD Shaper

2

LQR-ZV Shaper
(error)

1

LQR-ZVDD Shaper
(error)

0
Rise time

Table 1:

Frequenc
y
Exact
Error

Settling time

Level of swaying reduction of the hoisting rope and specifications of cart trajectory response for
hybrid control schemes.
Attenuation (dB) of swaying angle

Types of
shaper

Mode 1

Mode 2

Mode 3

ZV
ZVDD
ZV
ZVDD

12.99
38.45
3.94
18.97

20.29
51.20
0.09
25.91

26.26
64.18
12.26
40.17

Specifications of cart trajectory response
Rise time
Settling
Overshoot
(s)
time (s)
(%)
2.287
3.707

0.08
3.449
6.285
0.00
2.004
3.333
0.15
2.786
5.241
0.03

7. Conclusion
The development of hybrid control schemes for input tracking and swaying suppression of a gantry
crane system has been presented. The hybrid control schemes have been developed based on LQR with
feedforward control using the ZV and ZVDD input shapers technique. The proposed control schemes
have been implemented and investigated within the simulation environment of the overhead gantry
crane system. The performances of the control schemes have been evaluated in terms of level of input
tracking capability, swaying reduction, time response specifications and robustness. Acceptable input
tracking capability and swaying suppression have been achieved with both control strategies. A
comparison of the results has demonstrated that the LQR control with input shaping using ZVDD
shapers provide higher level of swaying reduction as compared to the cases using ZV shapers. In term
of speed of the responses, ZVDD shapers results in a slower tracking response with less overshoot. It
has also demonstrated that input shaping technique is very robust to error in natural frequencies
especially with higher number of impulses. It is noted that the proposed hybrid controllers are capable
of reducing the system swaying while maintaining the input tracking performance of the crane.


Optimal Tracking with Sway Suppression Control for a Gantry Crane System

641

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