Page 69 - Human Inspired Dexterity in Robotic Manipulation

Modeling and Human Performance in Manipulating Parallel Flexible Objects 65

model was claimed to be inapplicable to modeling of reaching movements in
dynamic environments [17]. This conclusion, however, appears to be
premature.
To show the validity of the minimum hand-jerk model, one can formu-
late the optimization problem in the optimal control settings instead of the
standard calculus of variations. As the acceleration of the hand enters the
boundary conditions (5.2), (5.3), it must be a part of the state vector. Taking
into account the object dynamics (5.5), (5.6), one can define the state vector

T
x ¼ x 1 , _x 1 ,x 2 , _x 2 ,x h , _x h , €x h Š , (5.9)
½
...
and set the hand jerk as the control signal u ¼x h . The state dynamics are
then represented by the following linear time-invariant system:
_ x ¼ Ax + bu, (5.10)

where
2 3 2 3
0 1 0 000 0 0
2
2
ω 0 0 0 ω 00 0
1 1
6 7 6 7
6 7 6 7
0 0 0 100 0 0
6 7 6 7
6 2 2 7 6 7
A ¼ 0 0 ω 0 ω 00 , b ¼ 0 : (5.11)
6 7
6
7
6 2 2 7 6 7
0 0 0 001 0 0
6 7 6 7
6 7 6 7
0 0 0 000 1 0
4 5 4 5
0 0 0 000 0 1
p ffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffi
Here, ω 1 ¼ k 1 =m 1 and ω 2 ¼ k 2 =m 2 are the natural frequencies of the
composite dynamic object. As
2 3 4 5 6 4 4 2 2
det b,Ab,A b,A b,A b,A b,A b½ Š ¼ ω ω ðω ω Þ, (5.12)
1 2 2 1
the system (5.10) is controllable as long as the natural frequencies are not zero
and ω 1 6¼ω 2 .
The minimum hand-jerk model can now be stated as the following opti-
mization problem with fixed end-points: Minimize the minimum-effort-
type criterion
1 ð T 2
J¼ u ðtÞ dt, (5.13)
2 0
under the dynamic constraint (5.10) and the boundary conditions
T T
xð0Þ¼ 0,0,0,0,0,0,0½ Š and xðTÞ¼ L,0,L,0,L,0,0Š . To find the optimal
½
solution, define the Hamiltonian