IEEE Robotics & Automation Magazine - December 2015 - 75

volume (i.e., on the area in the case of planar grasp) of the
polyhedron can be considered.
Curvature Index
The curvature quality index I C is evaluated by summing an
integer score assigned to the curvature of each contact
region composing the current grasp configuration. As the
stability of the contact points is conditioned by the curvature
of the contact region, the following score is employed: 1) ten
for a concave-angular region, 2) five for a convex-angular
region, 3) two for a convex region, 4) one for a planar region,
and 5) zero for a concave region. As a result, grasp configurations with an index close to zero are characterized by a
more stable contact.

(a)

(b)

Grasp Isotropy Index
The grasp isotropy index I G proposed in [10] and [11] considers the closeness of the finger configurations to singular
values. Notice that only by avoiding the singular configurations can force control as well as a precise position be
achieved during the object manipulation with a multifingered hand. Moreover, the uniformity of the contact force
contributions to the total wrench applied to the object
depends on this quality measure. In fact, this index is close to
one for isotropic grasp configurations and close to zero for
singular grasp configurations.
Force-Closure Test
The property of a grasp to resist any external force or moment
applied to the object is known as force closure [35]. Since the
computation of this property is computationally expensive
[36], the force-closure test is computed only for the best grasp
configurations selected after the ranking procedure.
Hand and Task Kinematic Indices
The overall quality of a grasp configuration also depends on
the hand kinematic configuration and its effectiveness with
respect to an assigned task. Other aspects, such as joint limits
and environmental constraints, should also be considered [10].
Many quality indices have been proposed in the literature,
most of which are based on the analysis of the Jacobian matrix
of the system composed by the hand and the object G @T J h,
where J h is the hand Jacobian and G @ is the generalized
inverse of the grasp matrix. In particular, a weighted sum of a
measure of the distance with respect to singular configurations
and of the hand manipulability has been considered [26].
Notice that in the case of no redundant hand kinematics, the
imposition of a grasp configuration also fixes the joint positions. Instead, in case of redundancy, it is known that infinite
joint configurations correspond to the assigned grasp configuration, but it is always possible to choose the hand configuration maximizing the previous manipulability measure.
Grasp Ranking
The grasp ranking is performed by computing the angular
distribution, minimal inertia index, extension index, and cur-

(c)

(d)
Figure 8. (a) The CAD models of the objects employed in the
3-D case studies together with the corresponding (b) MIRs
(yellow), (c) UCRs (planar regions in gray, concave regions in
violet, convex regions in orange, concave corner regions in blue,
and convex corner regions in red), and (d) GRs (discs with the
fingertips size and the same color convention adopted for the
UCRs). The black sphere indicates the object's CoG.

vature index for all feasible grasp configurations in the sets
C n, with n = 2, f, 5. The dynamically consistent approach
adopted for the GR computation typically ensures a small
value for the size of C n .
First, the sets of possible n-fingered grasps are inserted
into the initial grasp ranking list L = C 2 , g , C 5, and the
following three-level sort algorithm is executed:
1) First Sort Level: All grasp configurations present in L
with a quality index I D close to the current one (e.g., differing less than 5%) are selected, resulting in an ordered
sublist LD 3 L for the next sort level; if none is found,
the current configuration is sorted into L using I D as an
ordering criterion and the elaboration of the current grasp
candidate ends.
2) Second Sort Level: All grasp configurations of L D with an
index I E close to the current one (e.g., differing less that
10%) are selected for the next sort level, resulting in an
ordered sublist L E 3 L D; if none is found, the candidate
grasp configuration is positioned into L D using I E as an
ordering criterion and the elaboration of the current grasp
candidate ends.
3) Third Sort Level: The current configuration is inserted into
L E using I C as an ordering criterion.
DECEMBER 2015

IEEE ROBOTICS & AUTOMATION MAGAZINE

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