Page 175 - Human Inspired Dexterity in Robotic Manipulation
P. 175
A Grasping and Manipulation Scheme 173
where K o > 0 is a positive constant; J ðtÞ2 3 N D denotes the Jacobian
Ω i
matrix for the orientational angular velocity of each fingertip with respect
to the angular velocity of each joint _q; and the desired orientation of the
virtual-object frame is expressed in the form of a rotational matrix
ðtÞ 2 SOð3Þ. The summation of each cross
R d vir ðtÞ¼ r xd vir ðtÞ,r yd vir ðtÞ,r zd vir
product in Eq. (9.12) implies the desired instantaneous rotational axis of
the virtual-object frame. In other words, the orientation error between
the present virtual frame and the desired-virtual frame can be reduced by
rotating the virtual frame around this axis. The desired orientation of the
ðtÞ is designed in the following manner:
virtual-object frame R d vir
T
ðtÞ¼ R d Rðt t delay Þ R vir ðt t delay Þ, (9.13)
R d vir
where the desired orientation of the measured-object frame is expressed in
the form of a rotational matrix R d ¼ r xd ,r yd ,r zd 2 SOð3Þ. The orientations
of both the measured- and virtual-object frames at t delay (s) or one time step
before the present time are denoted by R(t t delay ) and R vir (t t delay ),
respectively, and they are obtained for the same time as x(t t delay ).
9.4.3 Conditions for Convergence of a Desired State
For the control signal u(t) to ensure that the position and orientation of the
measured-object frame converge to their desired values in the final state, it is
necessary to fulfill the following inequalities.
For the object’s position:
j xðt B Þ x vir ðt B Þg xðt A Þ x vir ðt A Þgj jxðt B Þ xðt A Þj, (9.14)
f
f
For the object’s orientation:
1
1 1 1
cos cos tr½R β 1 , (9.15)
ð tr½R α 1Þ
2 2
where
T T T
R α ¼ Rðt B ÞR vir ðt B Þ Rðt A ÞR vir ðt A Þ , (9.16)
T
R β ¼ Rðt B ÞRðt A Þ , (9.17)
and both t A and t B denote an instantaneous arbitrary time between the one
time step and next time step of a visual image acquisition interval. These
times satisfy the following inequality:
1
0 t B t A : (9.18)
h
