Page 159 - Human Inspired Dexterity in Robotic Manipulation

156 Human Inspired Dexterity in Robotic Manipulation

If the shape of a grasped object is not a rectangular object (more precisely,
the object does not have parallel surfaces), the tangential element of the con-
tact force generates the rotational torque to the fingertips. Nevertheless, the
net force to the object is zero, as shown in Fig. 8.5. Then, the following
integral term compensates the rotational torque without changing the net
force on the object to stop the rotation of the fingers:

u sgi ¼ u sgbi + κ i e i , for i ¼ 1,2, (8.6)
T
where e i ¼ (1, 1, …,1) and κ i is defined as follows:
Z t
κ i ¼ h i ðq ðτÞ, _q ðτÞÞdτ: (8.7)
i i
0
The kernel h i () usually depends only on the kinematic structure of the fingers
[9, 10]. The blind-grasping controller can be extended to multifinger cases
[11], and the relative spring force in Eq. (8.4) can be replaced to a feed-
forward opposed force [5]. To clarify the difference between the proposed
and conventional controllers, the stiffness expressions are drawn in Fig. 8.6.

Fingertips

Object

(A) (B)
Fig. 8.5 Effect of the opposed force applied to an object with nonparallel surfaces.
(A) Net force on the object. (B) Net force on the fingertips. The tangential elements
generate the rotational torque to disturb the fingers.

Object Joint Blind grasping
impedance impedance (opposed impedance)

Equivalent

(A) (B)
Fig. 8.6 Impedance controllers. (A) Absolute impedance. (B) Relative impedance.