Page 265 - Mathematical Techniques of Fractional Order Systems

254 Mathematical Techniques of Fractional Order Systems

From (9.6),
_ _ _
ð
_ x 2 52 l 1 sin θ 1 θ 1 2 l 2 sin θ 1 1 θ 2 Þ θ 1 1 θ 2 ð9:10Þ
From (9.7),
_ _ _
ð
_ y 5 l 1 cos θ 1 θ 1 1 l 2 cos θ 1 1 θ 2 Þ θ 1 1 θ 2 ð9:11Þ
2
Now, from (9.9) (9.11),
_
2
2 2 2 _ 2 2 _ _ 2 2 _ _

v 5 l S θ 1 1 l S θ 1 1θ 2 1 2l 1 l 2 S 1 S 12 θ 1 1 θ 1 θ 2
2 1 1 2 12
ð9:12Þ
_

2

2 2 _ 2 2 _ _ 2 2 _ _

1 l C θ 1 1 l C θ 1 1θ 2 1 2l 1 l 2 C 1 C 12 θ 1 1 θ 1 θ 2
1 1 2 12
To shorten the calculation, some abbreviations are used in (9.12) where,
S 1 5 sin θ 1 , C 1 5 cos θ 1 , S 2 5 sin θ 2 , C 2 5 cos θ 2 , S 12 5 sin θ 1 1 θ 2 Þ, and
ð
C 12 5 cos θ 1 1 θ 2 Þ
ð
Further (9.12) can be expressed as:
2
_
2 2 _ 2 _ _ 2 2 _ _

v 5 l θ 1 1 l θ 1 1θ 2 1 2l 1 l 2 C 2 θ 1 1 θ 1 θ 2 ð9:13Þ
2 1 2
The kinetic energy of second link is expressed as:
1
K 2 5 m 2 v 2 2 ð9:14Þ
2
1 h 2 _ 2 _ _ 2 2 _ _ i
_
2

K 2 5 m 2 l θ 1 1 l θ 1 1θ 2 1 2l 1 l 2 C 2 θ 1 1 θ 1 θ 2 ð9:15Þ
2
1
2
The potential energy of link-2 can be written as:
P 2 5 m 1 l 1 gS 1 1 m 2 l 2 gS 12 ð9:16Þ
A scalar function, called the Lagrangian function is defined as the differ-
ence between the total kinetic energy and total potential energy of a mechan-
ical system.
Lagrangian; L 5 K 2 PÞ ð9:17Þ
ð
where, K 5 K 1 1 K 2 Þ is represented as the total kinetic energy, whereas
ð
P 5 P 1 1 P 2 Þ is represented as the total potential energy of the two-link
ð
manipulator system.
For the present considered system, the Lagrangian can be written as:
1 2 1 h 2 2 2 i
2 _
_ _
2 _
_
2 _
L 5 m 1 l θ 1 1 m 2 l θ 1 1 l θ 1 1θ 2 1 2l 1 l 2 C 2 _ θ 1 1 θ 1 θ 2
1
2
1
2 2
2 m 1 l 1 gS 1 2 m 1 l 1 gS 1 2 m 2 l 2 gS 12
ð9:18Þ