IEEE Robotics & Automation Magazine - December 2015 - 57

Tu represents a variation on the object reference frame position. The contact stiffness matrix can be defined in SynGrasp
with the function SGcontactStiffness().
Furthermore, within SynGrasp, the structural stiffness of
the links and the controllable servo compliance of the joints
can also be modeled [22]. The joint torque variation
Tx ! R nq is proportional to the difference between a reference Tq r and the actual Tq variations of the joint
displacements
Tx = K q ^Tq r - Tqh,

(2)

where K q ! R nq # nq is the joint stiffness matrix, symmetric
and positive definite, and can be defined in SynGrasp with the
function SGjointStiffness().
Finally, SynGrasp allows a compliant model to be defined
for the joint aggregation inputs defined by synergies, referred
to as a softly underactuated model in [17]
Tv = K z ^T zr - T z h,

(3)

nz # nz

where K z ! R
is a symmetric and positive definite
matrix that defines the synergy stiffness. The synergy stiffness
matrix is set by SGsynergyStiffness().
Grasp Analysis
The grasp analysis is the most important feature of SynGrasp.
The included functions are the results of different studies on
both fully and underactuated hand models, considering compliance at different levels. In this section, we report the main
results obtained considering a synergy-actuated human hand
model. The quasistatic model adopted to define the main
properties of a grasp is obtained by performing a linear
approximation of the kinematic and compliance equations in
the neighborhood of an initial static equilibrium configuration. More details of this model can be found in [8], [17], and
[23]. Starting from an equilibrium configuration and applying
a small variation to the input synergy reference values Tz r in
the hypotheses so that the system reaches a new equilibrium
configuration, the following linear equations hold:
R
0
S-G 0
S JT
0
0
S
0
0
0
S
S I K c G T -K c J
S 0
Kq
0
S
S 0
0
0
T

0
-I
ST
0
I
0

R 0 V
V
W
0
0 W R Tm V S
S W S 0 W
0
0 W S Tu W S
0 W
W
W
-I
0 W S Tq W S
S W = 0 W . (4)
0
0 W STx W SS
0 W
0 -K q SWW STvW S
W
S
W
K z Tz r W
I
K z W T Tz X SS
W
X
T 0 X

The solution of this linear system leads to the following
mapping between the input controlled variable, i.e., the synergy reference variation Tz r and the output variables
T u = VT z r ,
Tq = XSYTz r = Qdz r,
Tz = YTz r ,
Tm = PTz r,

(5)
(6)
(7)
(8)

where the transfer matrices V, Q, Y, and P depend on the
grasp characteristics. The solution of this linear system is
detailed in [9].
The SynGrasp function SGquasistatic() solves the
linear system in (4) for a given grasp and variation of the reference synergy values and evaluates the corresponding variation of grasp configuration, according to (5)-(8). It uses the
function SGquasistaticMaps(), which evaluates the
matrices mapping the input variation Tz r to the outputs.
From (8), a basis matrix E s for the subspace of controllable
internal forces, i.e., the internal forces Tm that can be produced by activating the synergy references Tz r [8], can be
defined as
E s = R (P) .

(9)

All of the internal forces that are controllable by synergy
actions can then be parameterized through a free vector
y ! R nh as Tm = E s y, where n h represents the dimension of
the controllable internal force subspace [20].
The SynGrasp function SGgraspAnalysis() analyzes
the given grasping configuration in terms of internal forces
and object motions.
Equation (5) shows how the object displacements Tu are
controlled from one equilibrium configuration to another by
small variations in synergy values Tz r . Among all the possible object motions, rigid body motions are those that do not
involve viscoelastic deformations in the contact points. Rigid
body motions that are controllable by synergies have to be
compatible with contact constraint (1), evaluated assuming
Tm = 0, and with (6), which relates to controlled postural
synergies and joint displacements. The synergy reference values that modify the hand and object configuration without
modifying the contact force values belong to the P matrix
nullspace, i.e., Tz rh ! N (P). The corresponding object displacement and hand configuration variation, according to
(5) and (6), are given by Tu h = VTz rh and Tq h = QTz rh,
respectively. The SynGrasp function SGrbMotions()evaluates, for a given grasp configuration, the subspace of the
hand and object rigid body motions.
Remark 2
When the hand structure is generated, if not specifically
defined, the S matrix is set to an identity matrix I ! R nq # nq .
This corresponds to a fully actuated hand, in which each
component of q can be independently controlled. Thus, all of
the grasp analysis functions can be used for fully actuated
hands. The stiffness of a grasp is defined as the linear relationship between a variation of the wrench applied on an object
and the resulting motion
T w = KT u .

(10)

As previously outlined, the prefix T indicates that we are considering a small variation of the grasp configuration with
respect to a reference equilibrium condition. This condition
allows for the linearization of the kinematic and equilibrium
DECEMBER 2015

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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