Page 72 - Human Inspired Dexterity in Robotic Manipulation

68 Human Inspired Dexterity in Robotic Manipulation

t ¼ T/2. Also, the position x h (T/2) ¼ L/2, and for all even derivatives
ð2kÞ
x ðT=2Þ¼ 0 for k ¼ 1, 2, 3, …. It can also be shown that exactly the same
h
properties hold for the bead trajectories x 1 (t) and x 2 (t).

5.3 MINIMUM HAND-FORCE-CHANGE MODEL

Consider now the minimum hand-force-change model [19] with the per-
formance index

1 ð T
2
J¼ _ F dt, (5.25)
2 0
with the driving force F defined by the hand dynamics

m h €x h ¼ F + k 1 ðx 1 x h Þ + k 2 ðx 2 x h Þ, (5.26)
where m h is the mass of the hand.
T
Let x ¼ x 1 , _x 1 ,x 2 , _x 2 ,x h , _x h ,F½ Š be the state vector with the boundary
T T
conditions xð0Þ¼ 0,0,0,0,0,0,0½ Š , and xðTÞ¼ L,0,L,0,L,0,0Š . By set-
½
ting the hand-force change as the control input u ¼ _ F, the system dynamics
(5.5), (5.26) can be represented as the linear time-invariant system
_ x ¼ A + bu, where

0 1 0 0 0 0 0 0
2 3 2 3
ω 2 ω 2
0
1 1
6 0 0 0 0 0 7 6 7
6 7 6 7
0 0 0 1 0 0 0 0
6 7 6 7
6 2 2 7 6 7
A ¼ 0 0 ω 0 ω 0 0 7 , b ¼ 0 :
2 2
6
6 7
6 7 6 7
0 0 0 0 0 1 0 0
6 7 6 7
6 7 6 7
0
k 1 =m h 0 k 2 =m h 0 ðk 1 + k 2 Þ=m h 01=m h
4 5 4 5
0 0 0 0 0 0 0 1
(5.27)
p ffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffi
Here, as before, ω 1 ¼ k 1 =m 1 and ω 2 ¼ k 2 =m 2 , but they are not the nat-
ural frequencies of the combined (object and hand) system. To find the nat-
ural frequencies, consider the equations of free vibrations
2 3
2 3 2 3 2 3 2 3
m h 0 0 € x h k 1 + k 2 k 1 k 2 x h 0
0 m 1 0 + 0 x 1 ¼ 0 : (5.28)
4 5 4 € x 1 5 4 k 1 k 1 5 4 5 4 5
0 0 m 2 € x 2 k 2 0 k 2 x 2 0
The characteristic equation for the previous system has one zero root (cor-
responding to the rigid body mode) and two nonzero natural frequencies
defined as