Page 266 - Mathematical Techniques of Fractional Order Systems

Design of Fractional Order Fuzzy Sliding Mode Controller Chapter | 9 255

The dynamic model of the two-link manipulator system can be obtained
by Lagrangian-Euler formula which is expressed as:

d @L @L
τ i 5 _ 2 ð9:19Þ
dt @θ i @θ i
where, i represents the number of links present in the manipulator system.
To find the torque for link-1, (9.19) can be expressed as:
d @L @L
τ 1 5 _ 2 ð9:20Þ
dt @θ 1 @θ 1
Further, from (9.18),

@L
5 2m 1 gl 1 C 1 1 m 2 gl 2 C 12 ð9:21Þ
@θ 1
@L 2 _ 2 _ 2 _ 2 _ _ _
_ 5 m 1 l θ 1 1 m 2 l θ 1 1 l θ 1 1 l θ 2 1 l 1 l 2 C 2 2θ 1 1 θ 2 ð9:22Þ
1
2
2
1
@θ 1
d @L 5m 1 l θ 1m 2 l θ 1l θ 1l θ 2l 1 l 2 S 2 θ 2 2θ 1 1θ 2 1l 1 l 2 C 2 2θ 1θ ̈
_
2 ̈
2 ̈
2 ̈
̈
_
2 ̈
_

_ 1 1 1 1 2 1 2 2 1 2
dt @θ 1
ð9:23Þ
Further, from (9.20) (9.23), the dynamic equation of manipulator for
link-1 can be calculated which can be expressed as:
2 2 2

2 d θ 1 2 d θ 2 2 d θ 1
τ 1 5 l m 2 1 l m 2 1 l ðm 1 1 m 2 Þ
2 2 2 2 1 2
dt dt dt
2
dθ 2
ð
ðÞ
ðÞ
1 m 2 gl 2 cos θ 1 1 θ 2 Þ 1 ðm 1 1 m 2 Þl 1 g cos θ 1 2 m 2 l 1 l 2 sin θ 2
dt
2

dθ 1 dθ 2 d θ 1 d2θ 2
ðÞ 2
ðÞ
2 2m 2 l 1 l 2 sin θ 2 1 m 2 l 1 l 2 cos θ 2 2 1 2
dt dt dt dt
ð9:24Þ
To find the torque for link-2, (9.19) can be written as:

d @L @L
τ 2 5 _ 2 ð9:25Þ
dt @θ 2 @θ 2
Now, from (9.18),
@L 2 _ _
_
52m 2 gl 2 C 12 2 2l 1 l 2 S 2 θ 1 1 θ 1 θ 2 ð9:26Þ
@θ 2
@L 1 2 _ _ _
2
_ 5 2 m 2 l 2θ 2 1 2θ 1 1 2l 1 l 2 C 2 θ 1 ð9:27Þ
@θ 2