Page 174 - Human Inspired Dexterity in Robotic Manipulation

172 Human Inspired Dexterity in Robotic Manipulation

uðtÞ¼ u s ðtÞ + u p ðtÞ + u o ðtÞ: (9.7)

The control input for stable grasping u s (t) is designed such that the center
of each fingertip approaches the same point, and it achieves stable grasping of
an arbitrary polyhedral object [6]. It is given as follows:
N
X T
ð
u s ðtÞ¼ K s J ðtÞ x vir ðtÞ x 0i ðtÞÞ C_qðtÞ + gðtÞ, (9.8)
0i
i¼1
f d
,
K s ¼ X N (9.9)
r j
j¼1
where J ðtÞ2  3 N D denotes the Jacobian matrix for the velocity of the posi-
0i
tion of the center of each fingertip x 0i with respect to the angular velocity of
each joint _q; C 2  N D N D > 0 is a positive-definite diagonal matrix that plays
the role of a damping gain for each joint; gðtÞ2  N D is a gravity compensation
term for the robot; and f d represents the nominal desired grasping force.
The control input for regulating the position of the virtual-object frame,
u p (t), is designed as follows:

N
T
X
ð
u p ðtÞ¼ K p J ðtÞ x d vir ðtÞ x vir ðtÞÞ, (9.10)
0i
i¼1
ðtÞ denotes the desired position of the
where K p is a positive constant and x d vir
virtual-object frame. The latter is designed by considering the time delays in
the following way:

ðtÞ¼ x vir ðt t delay Þ + x d xðt t delay Þ , (9.11)
x d vir
where x d denotes the desired position of the measured-object frame;
x(t t delay ) represents the measured-object position that was obtained
by the visual sensor t delay (s) previously; and x vir (t t delay )represents the
position of the virtual-object frame that was obtained at the same time that
x(t t delay ) was obtained. The relationship between these values is shown
in Fig. 9.3.
The control input for regulating the orientation of the virtual-object
frame, u o (t), is designed as follows:
N
T
X
u o ðtÞ¼K o J ðtÞ r x vir ðtÞ r xd vir ðtÞ + r y vir ðtÞ r yd vir ðtÞ + r z vir ðtÞ r zd vir ðtÞ ,
Ωi
i¼1
(9.12)