Page 73 - Human Inspired Dexterity in Robotic Manipulation

Modeling and Human Performance in Manipulating Parallel Flexible Objects 69

1 1
2
ω 2 ¼ ð1+ μ Þω + ð1+ μ Þω 2
2
1,2
1
2 1 2 2
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
1 2 1 2
2
2
ω ω ð1+ μ + μ Þ + ð1+ μ Þω + ð1+ μ Þω 2 ,
1
2
1
1
2
2
1
2 2
(5.29)
where
m 1 m 2
μ 1 ¼ , μ 2 ¼ : (5.30)
m h m h
Note that matrix A in Eq. (5.37) is not in block upper triangular form as
in Eq. (5.11). To facilitate the computations, define the center of mass
m h x h + m 1 x 1 + m 2 x 2
x c ¼ (5.31)
m h + m 1 + m 2
and rearrange the system dynamics (5.5), (5.26)as
2
2
2
€ x 1 + ð1+ μ Þω x 1 + μ ω x 2 ¼ð1+ μ + μ Þω x c , (5.32)
1 1 2 1 1 2 1
2
2
2
€ x 2 + ð1+ μ Þω x 2 + μ ω x 1 ¼ð1+ μ + μ Þω x c , (5.33)
1
1
2
2
2
2
2
€ x x ¼ F=M, (5.34)
where M ¼ m h + m 1 + m 2 is the total mass. With the state vector
T
x ¼ x 1 , _x 1 ,x 2 , _x 2 ,x c , _x c ,F½ Š (5.35)
and control u ¼ _ F, the state dynamics
_ x ¼ A + bu (5.36)
have the block upper triangular form
0 1 0 0 0 0 0 0
2 3 2 3
ð1+ μ Þω 2 0 μ ω 2 0 ð1+ μ + μ Þω 2 0
6 1 1 2 1 1 2 1 0 7 6 07
6 7 6 7
0 0 0 1 0 0
6 0 7 6 0 7
6 7
6
A ¼ μ ω 2 0 ð1+ μ Þω 2 0 ð1+ μ + μ Þω 2 0 0 7 , b ¼ 0 :
2
7
6 7
6
1 2
2
1
6 2 2 7 6 7
0 0 0 0 0 1 0 0
6 7 6 7
6 7 6 7
0 0 0 0 0 0 1=M 0
4 5 4 5
0 0 0 0 0 0 0 1
(5.37)
The minimum hand-force-change model can now be stated as the fol-
lowing optimization problem with fixed end-points: Minimize the
minimum-effort-type criterion