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