Page 68 - Human Inspired Dexterity in Robotic Manipulation

64 Human Inspired Dexterity in Robotic Manipulation

T
ð 3 2
1 d x h
J¼ dt, (5.1)
2 0 dt 3
where x h stands for the hand coordinate. Minimization of this criterion
under the boundary conditions:

x h ð0Þ¼ 0, _x h ð0Þ¼ 0, €x h ð0Þ¼ 0, (5.2)
x h ðTÞ¼ L, _x h ðTÞ¼ 0, €x h ðTÞ¼ 0, (5.3)

leads to the following hand trajectory
3
2
x h ðtÞ¼ Lτ ð10 15τ +6τ Þ, (5.4)
where τ ¼ t/T is the nondimensional time.
We now consider reaching movements in a dynamic environment.
Assume that the environment is represented by two beads connected to
the hand by two parallel springs as shown in Fig. 5.1. The dynamics of
the composite object connected to the hand are given by

m 1 €x 1 + k 1 ðx 1 x h Þ¼ 0, (5.5)
m 2 €x 2 + k 2 ðx 2 x h Þ¼ 0, (5.6)
where x 1 and x 2 are coordinates of the beads, m 1 and m 2 are their masses, and
k 1 and k 2 are stiffness coefficients of the connecting springs.
Assume that a human subject is requested to transfer the hand according
to the boundary conditions (5.2), (5.3) and, at the same time, to transport the
two beads from the initial state
x 1 ð0Þ¼ 0, _x 1 ð0Þ¼ 0, x 2 ð0Þ¼ 0, _x 2 ð0Þ¼ 0, (5.7)

to the final state
x 1 ðTÞ¼ L, _x 1 ðTÞ¼ 0, x 2 ðTÞ¼ L, _x 2 ðTÞ¼ 0: (5.8)

Thus, the moving task is to generate a rest-to-rest motion command that
eliminates residual vibrations of the beads.
Note that if the dynamic environment contains only one spring (or a chain
of springs connected sequentially), one can always express the hand position
through the highest derivatives of the last bead, substitute it into Eq. (5.1),
write down the corresponding Euler-Lagrange equation, and solve the opti-
mization problem by using the standard technique of the calculus of variations
[16]. If, however, the springs are connected in parallel, this approach does not
work as it is not clear how to express the hand position through the highest
derivatives of the beads. Based on this consideration, the minimum hand-jerk