Image Processing Reference
In-Depth Information
m
θ
mg
F
x
u
(
t
)
F
y
M
x
FIGURE 4.5
Inverted pendulum.
d
2
dt
2
(l cos
d
dt
(l u
¼ mg þ ml(u
u þ u
2
cos
F
y
¼ mg m
u
)
¼ mg þ m
sin
u
)
sin
u
)
(4
:
35)
Appling Newton
s law of motion to the cart results in
'
M
d
2
x
x þ lu
u l u
2
sin
dt
2
¼ u(t)
F
x
¼ u(t)
m(
cos
u
)
(4
:
36)
Equation 4.36 can be written as
x þ mlu
u ml u
2
sin
(M þ m)
cos
u ¼ u(t)
(4
:
37)
Newton
s law applied to the motion of the pendulum yields
'
u ¼ mgl sin
ml
2
u þ F
y
l sin
u F
x
l cos
u
(4
:
38)
Substituting F
x
and F
y
into Equation 4.38 results in
u ¼ mgl sin
u þ m(g þ lu
u þ l u
ml
2
2
cos
sin
u
)l sin
u
x þ lu
u l u
2
sin
l cos
u m(
cos
u
)
(4
:
39)
This equation can be simpli
ed as
)u l( sin 2
) u
2
l(1
þ
cos 2
u
u
þ
cos
ux ¼
2g sin
u
(4
:
40)
x and u
Equations 4.38 and 4.40 are used to solve for
. The results are
u
(2M þ m) cos
mg sin 2
u
(2M þ m) cos
1
þ
cos 2
x ¼
u
þ
u(t)
(4
:
41)
u
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