CHAPTER A. Appendix
which is a measure of the grasp ability to resist unit moments normal to the grip
The optimal grasp proposed in  is defined as the grasp that maximizes Q
among all grasps which maximize Q
It can be proven (see ) that the optimum grasp with three fingers in a 2-
D case under the above optimal criterion is reached when the normal forces are
symmetric, with directions spaced 120
apart. Moreover, this grasp maximizes
also the size of the outer triangle, defined as the triangle formed by the three lines
perpendicular to the normal finger forces passing through the respective contact
points. Under the same criterion, the optimum grasp with three fingers in a 3-D
case is achieved when the maximum circumscribing prism-shaped grasp, that has
the largest outer triangle, is selected among the grasps where the normal finger
forces lie within the same grip plane and are in an equilateral configuration.
Therefore, to reach the optimum in the 3-D case with three fingers, the planner
has to seek three points in equilateral configuration on the object's surface, so that
the normal forces lie in the same grip plane, and for which the circumscribing prism
grasp is maximum.
Re-arrangement of the Mirtich and Canny
Since the reconstructed surface of the object is sampled by points/masses, the
above method cannot be directly applied. Differently from the continuous case,
due to the presence of a finite set of sampled points, the existence of a "grip plane"
containing all the normal forces is not guaranteed. This is mainly due to the fact
that, because of the discretization, the normals to the surface are an approximation
of the real ones. Considering that the optimal criterion requires that the desired
normals have to be spaced 120
apart, a discretized implementation of the method
of  is hence here proposed.
For each candidate configuration of three grasp points, the normal directions
are estimated on the basis of the available point-wise approximation of the surface.
Then, the unit vector normal to the grip plane containing the three points is
evaluated. Denoting with
the angle between the direction of the normal force
applied to point j and the direction normal to the grip plane, a Coplanarity Error
Index (CEI) can be defined as follow:
Obviously, the closer CEI to zero, the more the normal forces lie in the same plane.
The definition of a threshold
allows discarding all those configurations having