plane。 The following dynamic equation of a crane system

e5 

d 

 

can be obtained by using Lagrange method as below。

where  xd, ld  and θd  are the reference input trajectory    of

trolley position, rope length and swing angle, respectively, in generally, θd=0。 In addition, the reference input trajectory xd, ld can satisfy the assumption that the first and second time derivatives were assumed to be uniformly bounded internally, and that ld is not close to zero to avoid l=0 throughout the entire control。 Then, the system error model is as follows:

e1   e2

e2   xd   ( f1  g1u1  h1u2 )

e3   e4

 

(4)

Fig。 1 Simplified model of crane

x( fx Dx xfl sinDl lsin) / M



e4   ld   ( f2   g2u1  h2u2 )

e5   e6

e6    f3   g3u1  h3u2

where

lg cosl2  (D xf



sinf  

x x l

 sinDl l)(sin/ M ) ( fl Dl l) / m

(1)

u1=fx;

u =f ;

2     l

(g sin2l) / l cos( f sin D D

 Dx xDl lsinfx ) / lM

f1   l  x4 sin x5   x  x2 ;

M M

where x, l and θ are the trolley position, the length of the suspension rope and the swing angle of load, respectively; M is the trolley mass; m is the load mass; Dx and Dl are

g1 

h

1 ;

M

sin x5 ;

the viscous damping coefficients associated with the X

1  M

2

traveling  and  l  hoisting  down  motions  respectively; fx

f   g cos x



x  x 2  Dx  x

sin x



(sin

x5  1 )D x  ;

and fl are the driving forces in the X and l directions, respectively; g denotes the gravitational acceleration。

2 5 3  6 M   2 5 M m l   4

1 sin2 x 1

  5  

Suppose  that  x1=x,  x2= x,   x3=l,  x4= l,  x5=θ,  x6= 。

Then, the Eq。 (1) can be equivalent to the following state

g2

f3 

sin x5 ; h2 ;

M M m

equation:

g sin x



2x x