Page 267 - Mathematical Techniques of Fractional Order Systems

256 Mathematical Techniques of Fractional Order Systems

1
d @L 5 m 2 l 2θ 1 2θ ̈ ̈ _ _
̈
2

1
_ 2 2 1 1 2l 1 l 2 C 2 θ 2 S 2 θ 1 θ 2 ð9:28Þ
dt @θ 2 2
From (9.25) (9.28), the dynamic equation of manipulator for link-2 can
be expressed as:
2 2 2 2
dθ 1 d θ 1 2 d θ 1 d θ 2
ðÞ
ðÞ
τ 2 5 m 2 l 1 l 2 sin θ 2 1 m 2 l 1 l 2 cos θ 2 2 1 m 2 l 2 2 1 2
dt dt dt dt
1 m 2 l 2 g cos θ 1 1 θ 2 Þ
ð
ð9:29Þ
The dynamic equation of two-link rigid robotic manipulator system can
also be expressed as:
€ _
τ 5 M θ ðÞθ 1 V θ;θ 1 G θ ðÞ ð9:30Þ
T

where θ 5 θ 1 θ 2 is the angular position of end point of two links.
2 2 m 2 l 1 l 2 cosθ 2 1 m 2 l 2

1
M 5 l ðm 1 1 m 2 Þ 1 l m 2 1 2m 2 l 1 l 2 cosθ 2 2 2 ð9:31Þ
2
2
m 2 l 1 l 2 cosθ 2 1 m 2 l 2 m 2 l 2
0 10 1
2 3
2

dθ 1 dθ 2
dθ 2
2m 2 l 1 l 2 sinθ 2 @ A@ A m 2 l 1 l 2 sinθ 2
6 dt dt dt 7
_

V θ;θ 5 6 7 ð9:32Þ
7
6
6 7
2
4 5
dθ 1
m 2 l 1 l 2 sinθ 2
dt

G θðÞ 5 ðm 1 1 m 2 Þl 1 g cosθ 1 1 m 2 gl 2 cos ðθ 1 1 θ 2 Þ ð9:33Þ
m 2 gl 2 cosðθ 1 1 θ 2 Þ
_
where MðθÞ, Vðθ;θÞ, and GðθÞ are inertia, centripetal, and gravitational matri-
ces respectively.
The parameters used for the manipulator model for simulation is described
in Table 9.1 as,
TABLE 9.1 Parameter of Two-Link Manipulator System
Parameter Symbol Value Unit
Length of link-1 l 1 0.8 m
Length of link-2 l 2 0.4 m
Mass of link-1 m 1 0.1 kg
Mass of link-2 m 2 0.1 kg
Acceleration due to gravity g 9.8 m/s 2