Page 158 - Human Inspired Dexterity in Robotic Manipulation

Dynamic Manipulation Based on Thumb Opposability 155

Fingertips

x 01 Object
x 02
(A) (B)
Fig. 8.4 Effect of the opposed motion in precision grasp. (A) Stable equilibrium.
(B) Disturbed motion.

Fig. 8.4B. The opposable strategy generates contact forces parallel to the line
between connecting both the centers of the fingertips, and the force gener-
ates the counter rotational torque to pull the object back to the equilibrium
point, and the rotation torque becomes zero only when the system is in equi-
librium, as shown in Fig. 8.4A. Therefore, this strategy can stabilize the
object.
The strategy can be expressed as an impedance control method to min-
imize the distance between the position x 0i (q i )(i ¼ 1, 2) of the fingertip cen-
ter, where q i is the joint angle of the ith finger. The force f i applied to the ith
fingertip can be expressed as follows:
i
f ¼ð 1Þ γðx 01 x 02 Þ for i ¼ 1,2, (8.4)
i
where γ is a spring constant, and the sign in front of γ is changed because
direction of the force applied to the left and right fingertips are opposite.
Then, a passivity-based controller u sgbi to generate f i is given as
follows [8]:
T
u sgbi ¼ J f C i _q , for i ¼ 1,2 (8.5)
0i i i
where J 0i is the fingertip Jacobians and C i is the damping coefficient. The
first term in the right-hand side of Eq. (8.5) is the relative impedance

between the fingertips, and the second term is the joint damping. u sgbi
can be generated by the kinematic parameter on the link length and the sens-
ing of q i and _q , and it does not require any external sensor and object infor-
i
mation. Thus, the opposed control is called blind grasping, and this blind
property is very useful in implementation. By substituting Eq. (8.5) into
Eq. (8.1) and introducing the special Lyapunov-like function, the stability
of the closed-loop system can be proven [8]. Intuitively, the closed-loop sys-
tem satisfies Eq. (8.3) at the equilibrium point of the object.