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DYNAMIC MODELLING OF A DOUBLEPENDULUM GANTRY CRANE SYSTEM

INCORPORATING PAYLOAD

R. M. T. Raja Ismail, M. A. Ahmad, M. S. Ramli, R. Ishak, and M. A. Zawawi 

Citation: AIP Conference Proceedings 1337, 118 (2011); doi: 10.1063/1.3592452 

View online: http://dx.doi.org/10.1063/1.3592452 

View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1337?ver=pdfcov 

Published by the AIP Publishing 

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DYNAMIC MODELLING OF A DOUBLE-PENDULUM GANTRY CRANE SYSTEM

INCORPORATING PAYLOAD

 R. M. T. Raja Ismail, M. A. Ahmad, M. S. Ramli, R. Ishak and M. A. Zawawi  Faculty of Electrical and Electronics Engineering, Universiti Malaysia Pahang 

Lebuhraya Tun Razak , 26300, Kuantan, Pahang, Malaysia

ra jamohd@um p.edu.my, mashraf @um p.edu.my, syak irin@um p.edu.my, ruhaizad@um p.edu.my,

mohdanwar @um p.edu.my 

Abstract

The natural sway  of crane payloads is detrimental to

safe and efficient operation. Under certain conditions,

the problem is com plicated when the payloads create a

double pendulum  effect. This paper presents dynamic

modelling  of a double-pendulum  gantr y  crane system 

 based on closed-for m  equations of motion. The

Lagrangian method is used to derive the dynamic model

of the system. A dynamic model of the system 

incorporating  payload is developed and the effects of

 payload on the response of the system  are discussed.

Extensive results that validate the theoretical derivation

are presented in the time and frequency domains.

Keywords: Double-pendulum gantr y crane, Lagrangian

method, dynamic modelling.

1. IntroductionGantr y crane wor k s as a robot in many places such as

wor k shops and harbours to transport all k inds of

massive goods. It is desired for the crane to transport

the payloads to the required position as fast and as

accurately  as possible without collision with other

equipment. Research on the dynamic modelling  of a

system  is crucial before any  controller can be

im plemented to the system. Dynamic model andcontrols of double-pendulum  gantr y  crane has been

reported in [1,2]. The underactuated mechanical system 

model that exhibits double-pendulum  overhead crane

and its control design is also investigated in [3,4]. In

[5,6], the modelling and reduction technique of double-

 pendulum  oscillation has been studied. Moreover, the

study  of the dynamic modelling  of the other vibrator y 

system incorporating payload is also has been reported.

In [7], the investigation on the effects of payload on the

dynamic behaviour of the two-link  flexible manipulator

system has been presented.

This paper presents a generalised modelling framewor k  that provides a closed-for m  dynamic equation of

motion of a double-pendulum  gantr y  crane system.

Moreover, the wor k  presents the effect of payloads onthe dynamic behaviour of the system. The Euler-

Lagrange principle is used to derive the dynamic model

of the system. The simulation algorithm thus developed

is im plemented in Matlab. Cart position, hook   swing 

angle and load swing angle responses of the system and

the power spectral density  (PSD) are obtained in both

time and frequency  domains. To study  the effect of

 payloads, the results are evaluated with var ying 

 payloads of the gantr y  crane. Simulation results are

analysed in both time and frequency domains to assess

the accuracy  of the model in representing  the actual

system.

2. The Double-Pendulum Gantry Crane SystemThe double-pendulum  gantr y  crane system  with its

hook   and load considered in this wor k   is shown in

Figure 1, where  x  is the cart position, mc  is the cart

mass, mh is the hook  mass and m p is the payload mass.

 1 is the hook  swing angle,  2 is the load swing angle, l 1and l 2  are the cable length of the hook   and load,

respectively, and F  is the cart drive force. In this study,

the hook  and load can be considered as point masses.

Fig. 1: Description of the double-pendulum gantr y 

crane system.

For sim plicity, the cart friction force is ignored. The

cart, the hook   and the payload can be considered as

 point masses. The tension force that may  cause the

hook  and load cables elongate is also ignored. The cart

and the payload are assumed to move in two

dimensional only.

3. Dynamic Modelling of the Double-Pendulum

Gantry Crane SystemTo derive the dynamic equations of motion of the

double-pendulum gantr y crane system, the total ener gy 

associated with the crane system needs to be com puted

using  the Lagrangian approach [8,9]. Then, the Euler-

Lagrange for mulation is considered in characterizing 

the dynamic behaviour of the crane system.

Based on Figure 1, the cart, hook  and payload position

vectors, respectively are given by 

 Proceedings of the Fourth Global Conference on Power Control and Optimization

AIP Conf. Proc. 1337, 118-122 (2011); doi: 10.1063/1.3592452© 2011 American Institute of Physics 978-0-7354-0893-7/$30.00

118

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]0,[ xc   =r

]cos,sin[ 1111   θ θ    l l  xh   −+=r

]coscos,sinsin[22112211

  θ θ θ θ    l l l l  x p

  −−++=r

The total k inetic ener gy, K and potential ener gy, P  of

the whole system are given by 

 payloadhook cart   KKKK   ++=

 payloadhook    PPP   +=

where

2

cart2

1ccr m   =K , 2

hook 2

1hhr m   =K , 2

 payload2

1 p pr m   =K ,

11hook  cosθ  gl mh−=P ,

)coscos( 2211 payload   θ θ    l l  g m p   +−=P .

where  g   is the gravity  acceleration. Then, the total

k inetic ener gy  and the total potential ener gy  are

obtained as

212121

2

2

2

2

2

1

2

1222

111

2

)cos(

2

1)(

2

1cos

cos)()(2

1

θ θ θ θ 

θ θ θ θ 

θ θ 

−+

++++

++++=

l l m

l ml mm xl m

 xl mm xmmm

 p

 p ph p

 ph phcK

(1)

2211 coscos)(   θ θ    gl m gl mm  p ph   −+−=P   (2)

Using  the Lagrangian approach, the Lagrangian of the

system is obtain as

PKL   −=   (3)

Then, the Euler-Lagrange for mulation is used that is

i

ii

 F qqdt 

d =

∂

∂−

 

  

 

∂

∂   LL

, .3,2,1=i   (4)

where q  is the state vector and F  is the control vector

which are defined as

[ ]T  x 21   θ θ =q

[ ]T  F  00=F

where  F   denotes the control force inputs acting on the

 x-direction of motion. From (4), a set of three equations

is obtained that is

0,0

,

2211

=∂∂− 

  

 

∂∂=∂∂−

 

  

 

∂∂

=∂

∂−

 

  

 

∂

∂

θ θ θ θ LLLL

LL

dt d 

dt d 

 F  x xdt 

d 

  (5)

From  (5), the dynamic model of the double-pendulum 

gantr y  crane system  can be ex pressed in the for m  of

matrices such that

Fqqqqqq   =++ )(G),(C)(M     (6)

where the matrices33

)(M   ×ℜ∈q ,33

),(C   ×ℜ∈qq   , and

3)(G   ℜ∈q  represent the inertia, Centrifugal-Coriolis

ter ms, and gravity, respectively, and are defined as

−

+++++

=

)cos(cos

)(cos)(cos)(

)(M

212122

2

111

11

θ θ θ 

θ 

θ 

l l ml m

l mml mml mmmmm

 p p

 ph ph

 ph phc

q

−2

2

2121

22

)cos(

cos

l m

l l m

l m

 p

 p

 p

θ θ 

θ 

(7)

−−

+−

=

12121

111

)sin(0

00

sin)(0

),(C

θ θ θ 

θ θ 

l l m

l mm

 p

 ph

qq

 

−

−

0)sin(

sin

12121

222

θ θ θ 

θ θ 

l l m

l m

 p

 p

  (8)

+=

22

11

sin

sin)(

0

)(G

θ 

θ 

 gl m

 gl mm

 p

 phq   (9)

After rearranging  (6) and multiplying  both sides by 1M− , one obtains

)GC(M 1Fqq   +−−=   −

  (10)

where1

M−

 is guaranteed to exist due to 0)Mdet(   > .

In this study the values of the parameters are defined as

tabulated in Table 1 [1]. In order to investigate the

dynamic behaviour of the system  incorporating 

 payload, the value of m p is varied from 1 kg to 10 kg.

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Table 1: Parameter of the double-pendulum gantr y 

crane

Sym bol Parameter Value

mc  Cart mass 5 kg 

mh  Hook  mass 2 kg 

m p  Payload mass 1-10 kg 

l 1  Hook  pendulum length 2 m 

l 2  Load pendulum length 1 m  g Gravity acceleration 9.8 m-s –2

4. Simulation Results

In this section, simulation results of the dynamic

 behaviour of the double-pendulum gantr y crane system 

are presented in the time and frequency  domains. A

 bang-bang  signal of am plitude 10 N and 1 s width is

used as an input force, applied at the cart of the gantr y 

crane. A bang-bang  force has a positive (acceleration)

and negative (deceleration) period allowing  the cart to,

initially, accelerate and then decelerate and eventually 

stop at a tar get location.

System responses are verified by undertak ing com puter

simulation using  the fourth-order Runge-Kutta

integration method for duration of 20 s with a sam pling 

time of 1 ms. The hook   swing  angle and load swing 

angle responses of the system with the power spectral

density (PSD) are obtained and evaluated.

To demonstrate the effects of payload on the dynamic

 behaviour of the system, various payloads of up to 10

kg weight were simulated. Figure 2 shows the response

of the cart position for various payloads. It is noted that

the average final position of the cart decreases and the

chattering of the final position increases with increasing 

 payloads. Table 2 summarizes the relation between

 payload and the cart position response. Figure 3 and

Figure 4 show the hook   swing  angle and load swing 

angle responses respectively with various payloads. It is

shown that the oscillations of the hook  swing angle and

the load swing angle decrease with increasing payloads.

Table 3 shows the relation between payload and both

hook   and load swing  angles. Figure 5 and Figure 6

show the corresponding  PSDs for both hook   swing 

angle and load swing  angle. It demonstrates that the

resonance modes of vibration of the system  shift to

higher frequencies with increasing  payloads. Table 4

summarizes the relation between payload and the

resonance frequencies of the system.

By  com paring  the results presented in Table 4, it is

noted that the resonance frequency of the system which

deter mines the frequencies of both hook  and load swing 

angles was affected by variations in the payload. It also

shows that both hook   and load swing  angles have the

same resonance frequencies of mode 1. It is due to thesway of the payload is always follow the oscillation of

the hook . Besides, the system  has the same resonance

frequencies of mode 1 that is 0.3662 Hz, even though

the payload is varied from  1 kg  to 4 kg  and has the

same frequency  of 0.4883 Hz when the payload is

varied from 5 kg to 10 kg. This shows that, in order to

decrease the oscillation of the system, a same control

design can be used for several systems although they 

have different payloads. This result is significant since

one does not have to redesign a controller if the payload

is varied within the specific range. Besides, the hook  and the load swing  angles have different resonance

frequencies of mode 2. However, these resonance

frequencies do not affect much on the system since the

mode 1 frequency is the dominant mode to the system.

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (s)

   C  a  r   t  p  o  s   i   t   i  o  n   (  m   )

mp = 1 kg

mp = 5 kg

mp = 10 kg

Fig. 2: Response of the cart position.

0 2 4 6 8 10 12 14 16 18 20-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time (s)

   H  o  o   k  s  w   i  n  g  a  n  g   l  e   (  r  a   d   )

mp = 1 kg

mp = 5 kg

mp = 10 kg

Fig 3: Response of the hook  swing angle.

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0 2 4 6 8 10 12 14 16 18 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

   L  o  a   d  s  w   i  n  g  a  n  g   l  e   (  r  a   d   )

mp = 1 kg

mp = 5 kg

mp = 10 kg

Fig 4: Response of the load swing angle.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

Frequency (Hz)

   P   S   D

  o   f   t   h  e   h  o  o   k  s  w   i  n  g  a  n  g   l  e   (   d

   B   )

mp = 1 kg

mp = 5 kg

mp = 10 kg

Fig 5: Power spectral density of the hook  swing angle.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

Frequency (Hz)

   P   S   D

  o   f   t   h  e   l  o  a   d  s  w   i  n  g  a  n  g   l  e   (   d   B   )

mp = 1 kg

mp = 5 kg

mp = 10 kg

Fig 6: Power spectral density of the load swing angle.

Table 2: Relation between payload and cart position

response

Payload

(kg)

Average cart

 position (m)

Oscillation

(m)

0 - -

1 1.9920 ±0.3630

2 1.9151 ±0.3831

3 1.9145 ±0.4228

4 1.8521 ±0.4929

5 1.8419 ±0.50496 1.7219 ±0.5057

7 1.6940 ±0.5075

8 1.6063 ±0.5091

9 1.5472 ±0.5100

10 1.5154 ±0.5102

Table 3: Relation between payload and hook  and load

swing angles

Payload

(kg)

Hook  swing 

angle (°)

Load swing 

angle (°)

0 - -

1 ±0.4132 ±0.8826

2 ±0.4063 ±0.7418

3 ±0.3770 ±0.6140

4 ±0.3493 ±0.5319

5 ±0.3080 ±0.3982

6 ±0.3049 ±0.3791

7 ±0.2919 ±0.3431

8 ±0.2813 ±0.3244

9 ±0.2333 ±0.3007

10 ±0.2305 ±0.2902

Table 4: Relation between payload and resonance

frequencies of the double-pendulum gantr y crane

system 

Payload

(kg)

Resonance frequency (Hz)

Hook  swing angle Load swing angle

Mode 1 Mode 2 Mode 1 Mode 2

0 - - - -1 0.3662 1.343 0.3662 1.099

2 0.3662 1.587 0.3662 1.221

3 0.3662 1.709 0.3662 1.221

4 0.3662 1.709 0.3662 1.221

5 0.4883 1.099 0.4883 1.221

6 0.4883 1.221 0.4883 1.221

7 0.4883 1.343 0.4883 1.343

8 0.4883 1.343 0.4883 1.343

9 0.4883 1.465 0.4883 1.465

10 0.4883 1.465 0.4883 1.465

5. ConclusionInvestigation into the development of a dynamic model

of a double-pendulum  gantr y  crane system incorporating  payload has been presented. A closed-

for m  finite dimensional dynamic model of the system 

has been developed using the Euler-Lagrange approach.

The derived dynamic model has been simulated with

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 bang-bang  force input. The cart position, hook   swing 

angle and load swing  angle responses of the gantr y 

system  have been obtained and analysed in time and

frequency domains. Moreover, the effects of payload on

the dynamic characteristic of the system  have been

studied and discussed. These results are ver y  helpful

and im portant in the development of effective control

algorithms for a double-pendulum gantr y crane system with var ying payload.

6. References

[1] D. T. Liu, W. P. Guo and J. Q. Yi and D. B. Zhao,

Double-pendulum-ty pe Overhead Crane Dynamics

and its Adaptive Sliding Mode Fuzzy Control, Proc.

Of the Third International Conference on Machine

Learning and Cy bernetics, 2004, pp. 423-428.

[2] D. T. Liu, W. P. Guo and J. Q. Yi, GA-based

Com posite Sliding  Mode Fuzzy  Control for

Double-pendulum-ty pe Overhead Crane, Lecture

 Notes in Com puter Science, Springer Berlin, 2005,

 pp. 792-801.

[3] D. T. Liu, W. P. Guo and J. Q. Yi, Dynamics and

GA-based Stable Control for a Class ofUnderactuated Mechanical Systems, International

Journal of Control, Automation, and System, Vol.

6, No. 1, 2008, pp. 35-43.

[4] D. T. Liu, W. P. Guo and J. Q. Yi, Dynamics and

Stable Control for a Class of Underactuated

Mechanical Systems, Acta Automatica Sinica, Vol.

32, No. 3, 2006, pp. 422-427.

[5] D. Kim  and W. Singhose, Reduction of Double-

Pendulum  Bridge Crane Oscillations, The 8

th

International Conference on Motion and Vibration

Control, 2006, pp. 300-305.

[6] M. Kenison and W. Singhose, Input Shaper Design

for Double-pendulum Planar Gantr y  Cranes, 1999

IEEE Conference on Control Applications, 1999.

[7] M. A. Ahmad, Z. Mohamed and N. Ham bali,

Dynamic Modelling  of a Two-link   Flexible

Manipulator System  Incorporating  Payload, 3rd

IEEE Conference on Industrial Electronics and

Applications, 2008, pp. 96-101.

[8] M. W. Spong, S. Hutchinson and M. Vidyasagar,

Robot Modeling  and Control. New Jersey: John

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[9] M. W. Spong, Underactuated Mechanical Systems,

Control Problems in Robotics and Automation.London: Springer-Verlag, 1997.

122

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