Page 264 - Mathematical Techniques of Fractional Order Systems

Design of Fractional Order Fuzzy Sliding Mode Controller Chapter | 9 253

FIGURE 9.1 Two-link planar rigid robotic manipulator diagram.
1
K 5 mv 2 ð9:1Þ
2
where, v is considered as the linear velocity of the of the link. Now, the
linear velocity of link-1 can be expressed as:
_
v 1 5 l 1 θ 1 ð9:2Þ
For link-1, the kinetic energy can be written as:

1
K 1 5 m 1 v 2 ð9:3Þ
2 1
From (9.2) and (9.3)
1 2 _
2
K 1 5 m 1 l θ 1 ð9:4Þ
1
2
For link-1, the potential energy can be expressed as:
P 1 5 m 1 l 1 g sin θ 1 ð9:5Þ
To find the linear velocity of link-2, the Cartesian coordinates of the end-
point of link-2 are needed. After taking the first derivative, linear velocity of
link-2 can be easily obtained. Now, the Cartesian coordinates of the end-
point of second link are:
x 2 5 l 1 cos θ 1 1 l 2 cos θ 1 1 θ 2 Þ ð9:6Þ
ð
y 2 5 l 1 sinθ 1 1 l 2 sin θ 1 1 θ 2 Þ ð9:7Þ
ð
Further, the linear velocity of link-2 can be written as:
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
v 2 5 _ x 2 1 _ y 2 ð9:8Þ
2
From (9.8),
2
2
v 5 _ x 2 1 _ y 2 ð9:9Þ
2 2