IEEE Robotics & Automation Magazine - December 2015 - 74

Three-Fingered Grasps
The three-fingered grasp configuration is achieved by locating the thumb on the current GR at point p c, similar to the
previous case. The contact points of the other two opposing
fingers are computed starting from the point p o as follows.
First, we assume that the angle j F characterizes the opposing finger aperture with respect to the thumb vertex (Figure 6). Without loss of generality, we assume that the angle
j F between two consecutive opposed fingers is constant
with the hand aperture and the number of fingers. In a more
general case, j F decreases with the hand aperture and the
number of fingers. Then, a circular cone with the apex at p c,
aperture j F /2 and p o lying on the symmetry axis is
employed to generate a set of contact point couples. The
cone directrix is sampled with a fixed angular step. For each
sampled point, the opposed point of the directrix with
respect to the cone base center is considered, thus achieving
a couple of half lines starting from p c and intersecting S in a
couple of candidate-opposed contact points, p o1 and p o2, as
shown in Figure 6. Small local adjustments of the contact
points can be applied in the case of proximity of GRs. Notice
that the number of grasp configuration candidates generated
for each GR depend linearly on the number of samples kept
on the directrix.
The set of candidates is finally verified from the accessibility and feasibility point of view as in the previous case.
First, the distances d ci = < p c - p oi <, with i = 1, 2, are compared with the maximum aperture for the available hand. In
addition, the distance d 12 =< p o1 - p o2 < between the two
opposed fingers is verified with respect to the kinematic limits of the hand. Then, the existence of a hand pose avoiding
contact of the palm with the object is tested in a similar way
as it was in the two-fingered case. In this case, two semielliptical test curves are considered, both starting from p c and
arriving at p o1 and p o2, respectively. Only the grasp configurations that overcome these tests are considered for the final
ranking step. The force-closure test and other computational
complex tests are performed only on a limited number of
best grasp configurations, as described in the "Four-Fingered
Grasps" section.

po4

po1
RGi

po2

c(pc)

po3
vf

po4

Figure 7. The four- and five-fingered grasp configurations.

IEEE ROBOTICS & AUTOMATION MAGAZINE

DECEMBER 2015

Four-Fingered Grasps
The four-fingered grasp configuration case is similar to the
previous one. The main difference is that the cone aperture
angle becomes j F and an additional contact point (middle
finger) is fixed at p o / p o2 (Figure 7). The accessibility and
feasibility tests are performed in a similar way to the threefingered case.
Five-Fingered Grasps
The selection of the fifth-finger contact point candidate is
made by adding a second cone with an aperture angle 2j F a,
as shown in Figure 7. Notice in this case that two different
grasp configurations are achieved for each directrix sample.
In fact, the second cone allows two possible positions for the
little finger. In addition, the accessibility and feasibility tests
are performed similarly to previous cases.
Grasp Configurations Ranking
Let C n, with n = 2, f, 5, be the set of n-fingered grasp configuration candidates computed with the previous procedure. Notice that if the shape and size of the object do not
allow a suitable grasp with multiple fingers, the corresponding C n is empty.
The optimal n-fingered grasp configuration is computed by ranking each grasp of C n through a number of
grasp measures in a hierarchical (i.e., serial) composition.
The chosen quality measures have been split into two
groups to keep the computational complexity limited. The
first group of indices is used in a hierarchical manner to
rank C n . Then, at the end of the first ranking phase, the
second group of indices (those with higher computational
complexity) is applied starting with the best-ranked configuration. All candidates that do not guarantee a suitable level
for properties such as force closure, hand kinematic constraints, and manipulability are discarded until an optimal
grasp is found. The priority composition method and a
brief description of the adopted grasp quality indices are
provided in the following sections.
Angular Distribution and Minimal Inertia Index
A uniform angular distribution of the grasp contact points
increases the capability of the candidate grasp to stand up
to external forces and disturbances [13], [33]. Moreover, if
the contact force vectors point toward the object's CoG, the
gravitational and inertial effects are also reduced, thus
achieving an isotropic dynamically consistent grasp. To
obtain this result, the composed quality index I D proposed
in [26] is employed, which ranks the solid-angular distribution of the n-fingered grasp as well as the capability to
reduce the gravitational and inertial effects.
Extension Index
The resistance capability of a grasp configuration with respect
to external moments increases with the volume of the polyhedron having the grasp contact points as vertexes [34]. Therefore, an extension quality measure I E that depends on the

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - December 2015