1
Lagrange equations  Example 3
Figure 1.1(a) is a schematic representation of an open kinematic chain (robot
arm) consisting of three links 1, 2, 3, and a rigid body
RB
. Link 1 can
be rotated at
A
in a “ﬁxed” cartesian reference frame (0) of unit vectors
[
ı
0
,
0
,
k
0
] about a vertical axis
ı
0
. The unit vector
ı
0
is ﬁxed in link 1. Link
1 is connected to link 2 through pin joints
B
and
B
0
. The link 2 rotates
relative to 1 about an axis ﬁxed in both 1 and 2, passing through
B
, and
B
0
. The link 3 is connected to 2 by means of a slider joint 2’. The slider
joint is rigidly attached to link 2. The last link 3 holds rigidly the rigid body
RB
. The mass centers of links 1, 2, 2’ and 3 are
C
1
,
C
2
=
C
2
0
, and
C
3
,
respectively. The mass center of
RB
is
C
R
. The mass if the link 1 is
m
1
, the
masses of the bars 2 and 3 are
m
2
and
m
3
, the mass of the slider 2’ is
m
2
0
and the mass of
RB
is
m
R
. The length of 2 is
l
and the length of 3 is
L
.
Find the equations of motion for the robotic system.
Solution
A reference frame (1) of unit vectors [
ı
1
,
1
,
k
1
] is attached to body 1,
with
ı
1
=
ı
0
.
A reference frame (2) of unit vectors [
ı
2
,
2
,
k
2
] is attached to link 2, as it
is shown in Fig. 1.1. The unit vector
2
is parallel to the axis of link 2,
BB
0
,
and
2
=
1
. The unit vector
k
2
is parallel to the axis of link 3,
C
2
C
R
.
To characterize the instantaneous conﬁguration of the arm, the
general
ized coordinates
q
1
(
t
)
, q
2
(
t
)
, q
3
(
t
) are employed. The generalized coordinates
are quantities associated with the position of the system.
The ﬁrst generalized coordinate
q
1
denotes the radian measure of the
angle between the axes of (1) and (0) , Fig. 1.1(b). The second generalized
coordinate
q
2
designates also a radian measure of rotation angle between (1)
and (2), Fig. 1.1(c). The last generalized coordinate
q
3
is the distance from
C
2
to
C
3
.
Angular velocities
Next the angular velocities of the links and the rigid body will be ex
pressed in the ﬁxed reference frame (0). One can express the angular velocity
of link 1 in (0) as
ω
10
= ˙
q
1
ı
1
= ˙
q
1
ı
0
.
(1.1)
The angular velocity of link 2 with respect to (1) is
ω
21
= ˙
q
2
2
= ˙
q
2
1
,
(1.2)
1
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View Full Documentand the angular velocity of link 2 with respect to the ﬁxed reference frame
(0) is
ω
20
=
ω
10
+
ω
21
= ˙
q
1
ı
1
+ ˙
q
2
2
.
(1.3)
The unit vector
ı
1
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 Summer '11
 Marghitu
 Kinetic Energy, Special Relativity, Moment Of Inertia, Lagrangian mechanics, reference frame

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