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:
e1 e2
e2 xd ( f1 g1u1 h1u2 )
e3 e4
(4)
Fig。 1 Simplified model of crane
x( fx Dx xfl sinDl lsin) / M
e4 ld ( f2 g2u1 h2u2 )
e5 e6
e6 f3 g3u1 h3u2
where
lg cosl2 (D xf
sinf
x x l
sinDl l)(sin/ M ) ( fl Dl l) / m
(1)
u1=fx;
u =f ;
2 l
(g sin2l) / l cos( f sin D D
Dx xDl lsinfx ) / 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
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