IEEE Robotics & Automation Magazine - December 2015 - 71

introduced for demonstration purposes (e.g., the hand kinematic description and the accessibility test), the main optimization algorithm is analogous to the C++ version. We
experienced that this code could be eight-to-ten times
slower than the corresponding C++ version. A MATLAB
demo toolbox endowed with wizards to generate both suitable 3-D MATLAB models from STL files and optimal grasp
configurations has also been developed (more details are
provided in the rest of this article).
Object Surface Discretization
The discretization of the object surface is crucial for the
grasp synthesis because it severely affects the effectiveness, in
terms of the final grasp quality, and the computational complexity. A rough discretization characterized by a low number of graspable regions decreases the computational complexity, but there is the risk that the best grasps could be cut
off. In fact, a fine discretization results in a large number of
regions that should endure the capability of the algorithm to
find the best grasp, but it exponentially increases the computational time (see the "Computational Complexity" section)
and, therefore, reduces the possibility to compute the best
grasp online.
The proposed solution is based on the adoption of some
grasp quality measures and hand properties to lead the
object surface discretization and selection process [27]. The
main idea consists of the selection and aggregation of object
surface parts that are capable of generating grasp configurations that are optimal for dynamic object grasping and
manipulation, i.e., optimal from an inertial point of view, and
that are suitable to be reached by the fingertips. This goal is
achieved in three steps.
First, the minimal inertia regions (MIRs) of the object
surface, i.e., those parts of the object surface that can generate grasp configurations suitable from an inertial point of
view, are found. Since this step is computationally efficient,
a high resolution of the object surface discretization can be
used without significantly affecting the algorithm performances. Then, the MIR set is clustered into uniform curvature regions (UCRs). The result of this step is a very small
number of connected regions that are characterized by
similar geometrical and inertial properties. Finally, the
UCR set is decomposed into a number of grasp regions
(GRs) that are suitable to be reached by the fingertips of
the employed hand and are dimensionally adequate. These
steps are further explained in the "MIRs," "UCRs," and
"GRs" sections.
Note that the 3-D model of the object surface, e.g., a mesh
representation as well as the corresponding mass distribution
if the object density is not uniform, is assumed to be known.
MIRs
The minimization of the grasp forces required for the
compensation of the gravitational and inertial force is
chosen as a main criterion for the MIR elements selection. This goal can be achieved by minimizing the dis-

tance between the object's center of gravity (CoG) and
the center of grip [13].
However, by adopting this criterion, only the positions
of the grasp points are considered, while the direction of
the action lines of the contact forces are ignored. In fact, the
ideal condition to compensate the gravity and inertial
forces is achieved when the lines of action of the grasp
forces have an isotropic angular distribution and are
directed toward the object's CoG. These criteria, known as
focus centering and force arrangement, are quality indices
for the grasp assessment, and real focus centering is a quality
index for the configuration assessment [13], [28]. Without
loss of generality, the case of hard fingers and point contact
with friction is considered. Hence, when the friction cone
(FC) of each grasp point contains the CoG of the object,
then the corresponding center of grip remains close to the
CoG. Under this favorable condition, stable grasp configurations with respect to gravitational and inertial wrenches
can be computed. Taking these considerations into account,
the MIRs are defined as those portions of the object surface
where the corresponding FC contains the CoG c m for a
given friction coefficient n.
Let S be a 3-D representation of the object surface that
can be extracted from an available computer-aided design
(CAD) model or built with a visual system [7], [8] or a tactile
inspection system [29], [30]. The finite MIR set is defined as
the set of all of the connected regions
RI = {RI1, f, RI-} 3 S,

(1)

n T (p)c (p) # cos (arctan (n)) 6p ! RI,

(2)

such that

where c ^ ph is the unit central vector, here defined as the unit
vector pointing from the contact point p to c m, and n ^ ph is
the inward unit vector normal to the object surface at p, as
shown in Figure 1.

Object Surface

MIR
RI

S
c

Figure 1. The MIRs R I (in yellow) for a portion of an object
surface S (in gray).

DECEMBER 2015

IEEE ROBOTICS & AUTOMATION MAGAZINE

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