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On the Common-Mode and Configuration-Dependent Stiffness Control of Multiple Degrees of Freedom Hands

Conference Paper

On the Common-Mode and Configuration-Dependent Stiffness Control of Multiple Degrees of Freedom Hands

Abstract

Object manipulation using hands is a compelling topic in which the interaction forces play a key role. These influence the stability of the grasp and the dexterity of the hand manipulation. A well-known technique to modulate these forces is through the grasp stiffness. Inspired by the observations on human motor behaviour, this paper proposes a novel control method that exploits the dominant contribution of the finger poses to the major axes directions of the grasp stiffness ellipsoid. This is achieved by the optimisation of the hand/fingers posture minimizing the error between the desired orientation of the grasp stiffness ellipsoid and the obtained one. The adjustment of the volume of the ellipsoid is achieved through the adaptation of the fingers joint stiffness in a coordinated way. The performance of the proposed technique is evaluated in transferring desired grasp stiffness features from an anthropo-metric hand model to two different robotic hand models. Results show that the method is able to obtain a new grasp configuration approximating the desired grasp stiffness. Moreover, it is capable of adapting the orientation of the achieved grasp stiffness to the required variations of the task.
On the Common-Mode and Configuration-Dependent Stiffness Control
of Multiple Degrees of Freedom Hands
Virginia Ruiz Garate1, Nikos Tsagarakis2, Antonio Bicchi3, and Arash Ajoudani1
Abstract Object manipulation using hands is a compelling
topic in which the interaction forces play a key role. These
influence the stability of the grasp and the dexterity of the hand
manipulation. A well-known technique to modulate these forces
is through the grasp stiffness. Inspired by the observations on
human motor behaviour, this paper proposes a novel control
method that exploits the dominant contribution of the finger
poses to the major axes directions of the grasp stiffness ellipsoid.
This is achieved by the optimisation of the hand/fingers posture
minimizing the error between the desired orientation of the
grasp stiffness ellipsoid and the obtained one. The adjustment
of the volume of the ellipsoid is achieved through the adaptation
of the fingers joint stiffness in a coordinated way.
The performance of the proposed technique is evaluated in
transferring desired grasp stiffness features from an anthropo-
metric hand model to two different robotic hand models. Results
show that the method is able to obtain a new grasp configuration
approximating the desired grasp stiffness. Moreover, it is
capable of adapting the orientation of the achieved grasp
stiffness to the required variations of the task.
I. INTRODUCTION
Object manipulation, in autonomous or teleoperated poly-
articulated robotic hands, is a complex problem with several
difficult aspects. One of the most complicated points is the
modulation of the interaction forces in contacts to guarantee
both the stability of the grasp and the versatility of the
operation [1], [2]. These can be achieved either by force
control techniques or modulation of the grasp compliance
[3], [4]. However, the latter presents an enhanced stability
to unexpected changes of the environment. This is due to
its dominance in the dynamic response to small disturbance
profiles when manipulating a tool or an object. Such con-
sideration has promoted the design of several multi degrees-
of-freedom (DoF) robotic hands with dedicated active [5],
passive [6], or hybrid [7] compliant joints. Still, the added
redundancy in the dynamic coordinates has contributed to
a raised complexity in the control of grasp compliance and
hand pose. Several studies have tried to address this problem
from a robotics point of view facing the trade-off between
the stability of the grasp and the dexterity of the operation
[8]–[10].
The authors are with the 1Human-Robot Interfaces and physical In-
teraction, 2Humanoids and Human Centred Mechatronics, and 3Soft
Robotics for Human Cooperation and Rehabilitation Laboratories of the
Advanced Robotics Department, Istituto Italiano di Tecnologia. Email:
virginia.ruiz@iit.it
This work is supported in part by the EU H2020 projects ‘SOMA: SOft
MAnipulation” (no. 645599), and “SoftPro: Synergy-based Open-source
Foundations and Technologies for Prosthetics and RehabilitatiOn”(no.
688857).
On the other hand, the complex biomechanical architecture
of the human hand raises challenging questions for under-
standing the control strategies that underlie the coordination
of movements and forces required for a wide variety of
tasks. These can range from the individual movements of
single joints to multi-DoF in-hand manipulation [11], [12].
The problem of redundancy in kinematic control of the hand
DoF has been tackled by reducing the state space of many di-
mensions to a control space of few, commonly known as the
postural hand synergies [13]. Embedding such coordinated
movements in the joint space has shown promising results,
either using software [14] or hardware [15] solutions.
Towards the extension of this concept to the dynamic
coordinates while focusing on the mechanical stability of the
human grasp, the work in [16] investigated the existence of
a coordinated stiffening pattern in human fingertips during
a tripod grasp. In this study, the human fingertip stiffness
profiles were estimated using external perturbations and
illustrated by ellipsoids. The results suggested that the co-
activations of the forearm muscles contribute to a coordinated
stiffening of the fingers. This led to an increase in the
amplitudes of the major axes of the ellipsoids with minor
effects on their orientations. However, effective modulations
of the grasp forces in certain directions require the overall
geometry of the fingertip stiffness to be controllable. For
example, if an object slippage occurs, an effective way to
avoid it is to increase the contact normal forces, e.g. by
rotating the major axes of the fingertip stiffness ellipsoids in
the direction normal to the contact surface [17]. Similarly, if
high interaction forces are expected in certain directions of
a tool-tip hold by fingers (e.g. carving using a chisel), the
fingertip stiffness profiles must be modulated to provide a
resulting tool-tip stiffness ellipsoid with its major axis in the
direction of the highest interaction force.
To achieve this, humans explore the role of limb config-
uration [18] in modulating the geometry of the limb end-
point stiffness ellipsoid. The reason for that is the major and
quadratic effect of the limb geometry (through finger Jaco-
bians), in modifications of the orientation of the end-point
stiffness ellipsoid. In addition, self-selected postures are
more ergonomic solutions in comparison to co-contractions.
This may be the reason for some postures appearing as more
natural to the humans. For instance, the way a pencil or a
specific tool is held, or even the human body pose while
pushing a heavy object, contain very important information
on the expected physical interaction between the limb end-
point and the environment. The first attempt to explore
the relevance of this concept to robotic applications was
presented in [19] for the teleoperation of a redundant robotic
arm. The volume-adjusting component of the end-point stiff-
ness was achieved by using the Common-Mode Stiffness
(CMS) variable. CMS implemented a coordinated activation
across the arm joints. The control of redundant kinematic
DoF was achieved simultaneously using a Configuration-
Dependent Stiffness (CDS) variable. CDS controlled the
nullspace velocity of the manipulator to change the overall
geometry of the Cartesian stiffness ellipsoid.
The work presented here aims to transfer the CMS and
CDS principles to robotic hand manipulation of objects. This
presents the additional challenge of dealing with the multiple
DoF of the several fingers of the hand. In the proposed frame-
work, the CMS control implements a coordinated stiffening
across the finger joints (similar to [16]), a common behaviour
in robotic fingers with coupled joints [20]. At the same time,
the grasp pose and the finger configurations in redundant
space are controlled to achieve a desired grasp stiffness
geometry (CDS). This framework contributes to a reduction
in the dynamic control complexity of multi-DoF hands while
providing a solution for mapping dexterous manipulation
capabilities between hands with different kinematic and
dynamic characteristics (e.g. in teleoperation scenarios).
The performance of the proposed control method is eval-
uated by transferring grasp stiffness behaviours from an
anthropomorphic model of the human hand (20 DoF) to the
Schunk (7 DoF) and Allegro (16 DoF) robotic hands.
II. METHODOLOGY
When holding an object, the grasp stiffness matrix K
R6×6relates the wrench wR6applied to the object to
its displacement uR6:
w=Ku= (GKcGT)∆u, (1)
where Kcis the equivalent contact stiffness matrix by taking
into account all the system compliance sources [21]. The
dimension of Kcdepends on the numbers of contact points
on the object. Gis the grasp matrix relating the contact forces
and moments transmitted through the contact points, to the
set of wrenches that the hand can apply on the object [22]
[23]. The stiffness matrix Kcincorporates the fingers and
object structural elasticity. It is defined as in [23]:
Kc= (Cs+JK1
qJT)1,(2)
where Csrepresents the structural compliance matrix, and
Kqis a nq×nqdiagonal matrix representing the joint
stiffness. nqis the total number of finger joints in the hand1.
The grasp matrix Gis defined as G=e
GHT.His the
selection matrix representing the contact constraints [23],
i.e, the transmitted forces and moments between the hand
and the object [21]. If the contacts are modelled as hard
fingers, only the force components (and no moments) are
transmitted through the contacts [22]. According to the
notation in [23], the selection matrix is therefore defined
1In this work only finger joint movements are considered, keeping a fixed
wrist configuration.
as: H=I3n03n×3n, where nis the number of hard-
finger contacts. Similarly, e
Gcan be composed as:
e
G=I3I3. . . I303×3n
S(bc1)S(bc2). . . S(bcn)I3. . . I3,
(3)
where S(bci)stands for the skew-symmetric matrix of the
position of the ith contact point ciwith respect to the
grasped object centre b, both expressed in the world frame.
Iis the identity matrix. If the rolling between the object and
the fingers at the contact points is neglected, Gis constant
as long as the same grasp is held [21].
The joint stiffness matrix in (2) is defined as Kq=
α·1,Γ2, ..., Γnf], being α[Nm/rad]the synergistic joint
stiffness variable and nfthe number of fingers of the
hand. Γiis a constant vector implementing the coordinated
stiffening of the hand fingers (CMS) [16].
As mentioned earlier, the grasp stiffness matrix (K) is
an important quantity to characterise the stability and the
versatility of the grasp. In particular, its translational com-
ponents (KtR3×3, given that the rolling between the
object and the fingers is negligible) can provide meaningful
insights into the underlying physical interactions between the
fingers, the object, and the environment. This behaviour can
be graphically represented by an ellipsoid [16].
The purpose of the present study is to achieve the desired
features of a stiffness ellipsoid, i.e., volume and orientation,
in various hands with different kinematic and dynamic prop-
erties. The proposed method is divided in two consecutive
steps: (i) finding the optimal finger pose that generates the
best grasping stiffness orientation by means of the CDS
control (Section II-A), (ii) finding the optimal joint stiffness
(through α) to match the ellipsoid volume to the desired
one by means of the CMS control (Section II-B). Finally,
the grasp stability is evaluated for the resulting hand poses
(Section II-C).
A. Finding the optimal finger pose: CDS
To achieve the optimal finger pose, the hand workspace
must be explored to find a grasp that best approximates the
desired stiffness ellipsoid orientation. This is achieved by
first searching within an off-line computed mesh of possible
locations of the object centre in the hand workspace. The
hand workspace is defined based on the finger joint limits.
The object centre is usually considered at its centre of mass.
This broad check is needed to avoid falling in a possible
local minima. That would be the case if the optimization is
constrained to start from a particular grasp configuration in
such a high dimensional space. During the search, the contact
points are constrained to remain in the same position with
respect to the object centre.
Once a first stiffness ellipsoid approximation is obtained
from an object position and its corresponding hand pose,
a local on-line optimisation is performed. This second step
serves to adjust the finger poses in the redundant space
around the solution provided by the mesh, while minimizing
the difference between the desired and obtained stiffness
ellipsoid orientations.
Starting with the exploration of the mesh positions, from
a initial randomly pre-defined hand configuration q0, and
provided the initial contact points c0and an object centre
position b0of a stable grasp, the initial grasp matrix e
G0can
be obtained using (3). Then, by moving the object to the
new centre locations bnew defined in the mesh, subsequent
e
Gnew matrices are constructed. At the same time, all the
contact points at the fingers must be kept constant, that
is, for every contact point ci:e
G0=e
Gnew. This implies
that the new desired positions of each contact point in the
world reference frame can be obtained from the new object
positions as ci,new =ci,0b0+bnew.
Next, inverse kinematics is performed for every finger
to find the new joint configurations that are capable of
grasping the object in the different mesh positions with the
desired contact points locations. In case of joint redundancy,
the Levenberg-Marquardt algorithm [24] [25] is used to
minimize ||ci,des ci,new||2. At every new mesh position, the
iterative algorithm starts from the Jacobian pseudo-inverse
using the previous position: q1=J#(cnew c0) + q0.
Obtained results are discarded if: (i) there are more/less
contact points than desired, (ii) contacts are placed in differ-
ent links of the finger, (iii) the obtained joint angles are above
their limits, (iv) the relative distance error from the contact
points to the centre of the object is bigger than 10% of the
desired one: ||bcnew bcdes||2<0.1||bcdes ||2. However, the
lack of a feasible solution can be due to a poor starting
condition. Therefore, in cases (i), (ii), and (iii), the iterative
process is repeated, starting from a known stable initial grasp
configuration of the hand as q1. If still no solution is found,
then the position is finally dismissed.
As this procedure is performed for all the pre-defined mesh
points, the total computational time depends on the size and
resolution of such mesh. To avoid a long processing, the
computation is executed off-line once for the grasped object,
and the results are stored. Then, for every movement, the
previously obtained values just need to be loaded.
After all positions in the mesh have been evaluated, the one
giving the closest stiffness ellipsoid orientation to the desired
one is chosen. For this, the algorithm checks the normalized
difference (eθ) between the major axis of the desired and
obtained translational stiffness matrices.
The second step is to locally optimise the redundant DoF
of the fingers to better approximate the desired orientation.
This exploration leads to a new object location and finger
pose within the vicinity of the previously found one. To do
so, the method exploits the direct kinematics of the fingers:
˙cf=Jf˙qf, being cthe contact point position, and fthe
corresponding finger. In this work we consider the object
to be grasped by the fingertips, being clocated at the most
distal link of each finger. Similar to previous step, the new
configuration needs to keep the contact point locations with
respect to the centre of the object constant (bci=cte).
Therefore:
b˙cf= ˙cf˙
b=Jf˙qf˙
b= 0,(4)
where, Jfdenotes the Jacobian matrix of the finger. Ex-
panding (4) by putting all finger equations together and
transforming into a matrix form:
J10 0 · · · I3×3
0J20· · · I3×3
.
.
....· · · .
.
.
0 0 · · · JnfI3×3
×
˙q1
.
.
.
˙qnq
˙
bT
=
b˙c1
b˙c2
.
.
.
b˙cn
,
(5a)
A3n×nqf×˙pnqf×1=b˙c3nf×1,(5b)
where ˙q1... ˙qnqare the joints of all fingers. Solving (5):
˙p=A# b ˙c+N(A)·σ, (6)
where N(A)is the nullspace of matrix A. As bc=cte,
b˙c= 0, hence ˙p=N(A)·σ. This equation can be put
down numerically into an iterative process minimizing σ.
Therefore, this parameter is derived from the difference
between the desired stiffness ellipsoid orientation and the
actual one: σ=0.5(θd,k)
∂p , where θd,k =θdes θk
represents the difference between the desired orientation and
the one at the kth iteration, and pis the vector defined in
(5). The three last elements of σthat correspond to the partial
derivative with respect to small object displacements, are set
to 0. As we focus on the translational stiffness components
and not the rotational, small changes in the object centre
location do not generate a change in the translational part of
the stiffness. For the angular displacements, differences of -
0.01 rad are considered and a maximum of 100 iterations
are allowed. Clearly, hands with more DoF are able to
iterate more within the nullspace of (5), further reducing the
stiffness orientation error.
B. Finding the optimal joint stiffness: CMS
Once a hand pose and the object centre position are
estimated, the algorithm minimises the difference between
the volumes of the desired stiffness ellipsoid and the achieved
one. Due to the coupling between the different finger joint
stiffness profiles of the hand (Kq=α·Γi), the only way to
vary the stiffness is by modifying α.
To optimise α, we use fminsearchbnd, a bound constrained
optimization function provided by Matlab (The MathWorks
Inc.), with α[0.001,50] Nm/rad. Since Ktis a sym-
metric positive-definite matrix, only the diagonal and upper
triangular values of the desired (Ktd,des) and actual (Ktd)
translational matrices are evaluated. Then, the error function
eKtd (%) = 100||Ktd,des Ktd||2
||Ktd,des||2is minimised.
C. Grasp stability check
For every final configuration the obtained grasp stability
is checked. To do so, the algorithm examines if the ratio
between the tangential and normal forces at all the contact
points is lower than the surface friction coefficient by using
the procedure suggested in [26]. A tolerance ktol =0.01,
and an object friction coefficient of µ= 0.8are stablished.
For the examples presented here, an object weight similar to
that of a tennis ball (59.4 g) is considered. We further use
the concept of Potential Contact Robustness (P CR) defined
in [27]. This parameter is used to compare the robustness of
the initial and the final achieved grasps.
III. MODELS OF THE HANDS
To validate the proposed method, a series of virtual
hands are generated to perform the required simulations.
These hands have different kinematic and dynamic proper-
ties. Moreover, selected grasps are performed with spherical
objects of random sizes.
As exploring an actual human hand stiffness is not
straightforward, it is modelled departing from the anthro-
pometric example of 20 DoF (4 DoF per finger) provided in
the Syngrasp toolbox [26]. Some modifications are applied
based on joint definitions and limits described in [28] [29].
Finger lengths are based on those described in [30]. The
final simulation model is displayed in Fig. 1a. An initial
grasp on a spherical object of 30 mm radius is then defined
(Fig. 1b). The structural diagonal stiffness of (2) is set to
Ks=C1
s= 1000 N/m. The joint stiffness is set initially
to Kq=α·thumb,Γindex ,Γmiddle,Γring ,Γlittle ], being
α= 2 Nm/rad, and Γi= [1,1,0.8,0.6] for every finger.
The decreasing values from proximal to distal joints are
based on experiments with actual human hands, like the one
in [18]. This pattern is followed for the rest of the hand
models. The projection of the resulting translational grasp
stiffness Ktis displayed in Fig. 1c for the 3 main planes.
10010
20
0
20
40
0
50
100
150
x (mm)
Defined Configuration
y (mm)
z (mm)
(a) (b)
2000 0 2000
2000
0
2000
XY Stiffness (N/m)
2000 0 2000
2000
0
2000
XZ Stiffness (N/m)
2000 0 2000
2000
0
2000
YZ Stiffness (N/m)
(c)
Fig. 1: (a) Anthropometric hand model. Ball shapes define the fingertips.
Blue arrows represent the Z axis, rotation axis for the finger joints. (b) Initial
configuration. (c) Initial grasp stiffness in the 3 translational planes.
The goal in the proposed simulations is two-fold: first to
mimic the initial modelled anthropometric hand stiffness into
a robotic hand, and then to follow possible variations in the
obtained robotic grasp stiffness. For robotic hands we choose
the Schunk and Allegro hands and model them in Matlab
(The MathWorks Inc., Natick, MA, 2000). The Schunk hand
has 8 joints with 7 DoF, while the Allegro hand has 16
independent DoF. These hands together with their virtual
models are displayed in Fig. 2. It must be pointed out that,
in the model of the Allegro hand an extra joint has been
added to each finger (Fig. 2e) to account for the division
of the last phalanges from the fingertips (white part in the
photo, Fig. 2d). In simulations, these joints remain inactive
and positioned at 0 rad. Additionally, they are assigned a
joint stiffness of 5 Nm/rad. Fig. 2c and Fig. 2f show the
initial grasp of the Schunk and Allegro hands while holding
spherical objects of 50 mm and 34.3 mm radius, respectively.
Based on the finger joint limits, we define the workspace
of possible object center locations for the Schunk hand
in (x, y, z)ranging from (252.90,0,193.08) mm to
(22.90,0,186.92) mm, with a resolution of 10 mm. In
the case of the Allegro hand, the workspace is defined from
(134.62,99.25,140.58) mm to (85,38140.75,139,42)
mm, with a resolution of 20 mm.
In the case of the Schunk hand, the thumb inverse kine-
matics can be directly solved. For the other fingers, as well
as for those of the Allegro hand, the redundant problem is
solved as explained in Section II-A. For both hands, the
structural diagonal stiffness is set to 1000 N/m and the joint
stiffness initially to α= 5 Nm/rad. For the Schunk hand
thumb Γthumb = [0.8,0.6], whereas Γi= [1,0.8,0.6]
for the other two fingers. In the case of the Allegro hand,
Γi= [1,1,0.8,0.6] for all the fingers. Both initial grasps of
the Schunk and Allegro hands satisfy the contact constraints,
being P CR = 10.94 and P CR = 4.971, respectively.
(a)
200
100
0
40
20
0
20
40
50
0
50
x (mm)
y (mm)
z (mm)
(b) (c)
(d)
020
40
0
100
200
100
50
0
50
100
150
x (mm)
y (mm)
z (mm)
(e) (f)
Fig. 2: (a) Photo of the Schunk hand. (b) Schunk hand model. Ball shapes
define the fingertips. Blue lines represent the Z axis, rotation axis for the
finger joints. (c) Initial configuration of Schunk hand grasp. (d) Allegro hand
model. (e) Photo of the Allegro hand. (f) Initial configuration of Allegro
hand grasp.
IV. RESULTS AND DISCUSSION
When analysing the pre-defined mesh for the Schunk and
Allegro hands, the algorithm provided 98 and 50 feasible
hand configurations respectively for different object loca-
tions. At these positions, the contact points are held as in
the initial configuration (see Fig. 2c and Fig. 2f). Fig. 3
shows the set of grasp stiffness ellipsoids that are generated
in such positions projected in the 3 main planes. Although
the evaluation of all these points is time consuming, once
performed results can be stored and loaded for the rest of
trials.
It can be observed that while a wide range of stiffness
orientations are attainable, not all of them are feasible. In the
case of the Schunk hand the larger stiffness is always in the
’Y’ or ’Z’ direction, whereas no large stiffness is achievable
in the diagonal direction of the ’YZ’ plane. Likewise, for
the Allegro hand, with the defined contact points no large
stiffness can be obtained in the negative diagonal ’XZ’
direction. Therefore, comparing these ellipsoids to the grasp
stiffness of the anthropometric hand (see in Fig. 1c), we
estimate that the Schunk hand will better perform when
mimicking the orientation of the stiffness ellipsoid in the
’XZ’ plane. Reversely, the Allegro hand may be able to better
represent the ’XY’ and ’YZ’ stiffness.
(a)
(b)
Fig. 3: Grasp stiffness ellipsoids and their main axes projected in the 3 main
planes for the (a) 98 possible object positions for the Schunk hand, (b) 50
possible object positions for the Allegro hand.
A. Approximating the anthropometric stiffness
Using the methodology described in Section II, our first
objective is to achieve the anthropometric grasp stiffness
(Fig. 1c) by the robotic hands. Results are presented in Fig.
4 for the Schunk hand and Fig. 5 for the Allegro hand.
Table I displays all numerical results of the different
steps of the proposed method for all the trials. Among the
variables, it shows the difference between the desired and
achieved main orientation of the grasp stiffness ellipsoid
along the different phases of the proposed optimization
method (eθ,mesh, eθ ,null). The initial errors for the Schunk
and Allegro hands, are reduced at the end of the process
to 0.1281 and 0.7531, respectively. As already mentioned,
the optimization of the joint stiffness parameter αre-sizes
the ellipsoid volume without re-orienting it. Therefore, the
orientation error before and after this optimization remains
the same. In any case, a reduction in this error is observed for
both hands, being the decrease larger for the Schunk hand.
As expected, from Fig. 4 and Fig. 5 it can be observed that
the Schunk hand is able to better match the orientation of the
grasp stiffness in the ’XZ’ plane than the Allegro hand. This
is also the stiffest grasp direction, thus resulting in a bigger
error on the orientation of the Allegro hand. Conversely, the
Allegro hand performs better in the ’XY’ plane whereas the
differences in the ’YZ’ plane result to be quite similar. On
the other hand, the nullspace optimization is able to iterate
more in the case of the Allegro hand than the Schunk hand
due to the more redundant DoF. This translates into a slight
reduction of the error (see Table I), as the Allegro hand is
able to better exploit its redundant DoF.
Initial Desired Oriented Final
Fig. 4: Upper row: Initial Schunk hand grasp, after mesh search of ellipsoid
orientation, and after nullspace optimization. Lower row: Grasp stiffness
projected in the 3 main planes. Oriented and final stiffness are obtained
before and after the optimization of the joint stiffness Kqrespectively.
Fig. 5: Upper row: Initial Allegro hand grasp, after mesh search of ellipsoid
orientation, and after nullspace optimization. Lower row: Grasp stiffness
projected in the 3 main planes. Oriented and final stiffness are obtained
before and after optimization of the joint stiffness Kqrespectively.
The final optimized αfor the Schunk and the Allegro
hands are 24.64 Nm/rad and 18.49 Nm/rad, respectively.
Therefore, a larger joint stiffness is needed with the robotic
hands than with the anthropometric hand. A possible rea-
son for this could be the higher amount of contact points
restricting the object possible movements in the case of
TABLE I: Initial values and results of the optimization process. eθ: normalized difference in the main axis orientation of the grasp stiffness ellipsoid
after mesh search eθ,mesh, and nullspace optimization eθ ,null.eKtd (%): relative difference between the translational stiffness components. t(s) is the
computational time taken for each of the method main steps.
Schunk α(Nm/rad) eθ,mesh eθ,null eKtd (%) PCR tmesh (s) tnull (s) tKq(s)
Initial Anthropometric 5 1.321 46.69 10.94
Optimized (Section IV-A) 24.64 0.1281 0.1281 25.02 10.92 0.008087 0.8874 0.1351
Rotation (Section IV-B) 28.47 0.1133 0.1133 9.816 10.92 0.02203 0.5074 0.02974
Allegro α(Nm/rad) eθ,mesh eθ,null eKtd (%) PCR tmesh (s) tnull (s) tKq(s)
Initial Anthropometric 5 1.348 56.05 4.971
Optimized (Section IV-A) 18.49 0.7534 0.7531 38.70 1.694 0.005221 15.84 0.1702
Rotation (Section IV-B) 7.529 0.3586 0.3586 17.27 4.744 0.01445 2.371 0.05481
Initial Anthropometric (Section IV-C) 5 1.309 48.87 4.971
Optimized (Section IV-C) 11.05 0.6362 0.6357 23.69 4.912 0.005256 4.584 0.1535
the anthropometric hand. This would be supported by the
fact that a lower joint stiffness is needed with the Allegro
hand (with 4 fingertip contacts) than with the Schunk hand
(3 fingertip contacts). However, this could also be due to
the different phalanges length of the hands fingers. Further
simulations would be needed to evaluate the influence of
both factors.
The final error eKtd is notably reduced for both hands
at the end of the process, remaining higher for the Allegro
hand than the Schunk hand (see Table I). This is in agreement
with the previous results on orientation adjustments. In both
obtained configurations, the imposed contact constraints are
respected, with P CR = 10.92 and P C R = 1.694 for the
Schunk and Allegro hands, respectively. Therefore, while
for the Schunk hand there is no big difference in terms of
grasp robustness, in the case of the Allegro hand there is a
significant decrease.
Interestingly, from Fig. 4 and Fig. 5, it can be noticed that
the object position with respect to the Schunk hand is more
similar to that of the anthropometric hand grasp (see Fig.
1b). However, the Allegro hand provides a finger pose more
similar to that of the anthropometric configuration.
B. Modelling the achieved robot hand grasp stiffness
Once the anthropometric grasp stiffness is approximated,
or directly copied from an already existing grasp of the
robotic hand, some changes of the stiffness ellipsoid might
be required. For example, in the case of grasping a heavier
object or applied disturbances in the vertical axis of the world
frame, it would be necessary to re-orient the ellipsoid to
become stiffer in the vertical ’Z’ direction. This could be
achieved by rotating the stiffness ellipsoids of the Schunk
and Allegro hands from previously approximated ones by
35 and 90 deg, respectively (see Fig. 6b and Fig. 7b).
To do so, from the previous existing robotic grasp stiffness
K, a new desired stiffness matrix is created by applying
a rotation to its eigenvectors. Then, the method defined
in Section II is again used to find the best fitting grasp
configuration and the joint stiffness. Figs. 6 and 7 show
the results for the Schunk and Allegro hands, respectively.
An evident change in the finger joint configuration can be
observed. In the case of the Schunk hand, the joint stiffness
coupling value found to be optimal is 28.47 Nm/rad, while
(a)
Initial Rot 35° Final
2000 0 2000
2000
0
2000
XY Stiffness (N/m)
2000 0 2000
2000
0
2000
XZ Stiffness (N/m)
2000 0 2000
2000
0
2000
YZ Stiffness (N/m)
(b)
Fig. 6: (a) Schunk initial hand configuration and configuration achieved to
match the 35 deg rotation in Y. (b) Projection of grasp stiffness ellipsoids
in the 3 main planes.
(a)
2000 0 2000
2000
0
2000
XY Stiffness (N/m)
2000 0 2000
2000
0
2000
XZ Stiffness (N/m)
2000 0 2000
2000
0
2000
YZ Stiffness (N/m)
Initial Rot 90° Final
(b)
Fig. 7: (a) Allegro initial hand configuration and configuration achieved to
match the 90 deg rotation in Y. (b) Projection of grasp stiffness ellipsoids
in the 3 main planes.
for the Allegro hand is 7.529 Nm/rad. This result is in
agreement with the ones before, needing larger joint stiffness
for the Schunk hand. The difference in alignment between
desired and achieved main orientation of the grasp stiffness
ellipsoid is of 0.1133 for the Schunk hand and 0.3586 for the
Allegro hand, with the total difference being eKtd = 9.816%
and eKtd = 17.27% respectively. Again, error remains higher
for the Allegro hand. In any case, all differences have
been reduced with respect to those obtained when trying to
approximate the stiffness given by the anthropometric hand
(see Table I). This is coherent with the fact that the desired
stiffness is now obtained from a known achievable grasp
stiffness by the robotic hands. Moreover, contact constraints
for both hands at the new found scenarios are satisfied in both
cases for all contact points. P CR = 10.92 for the Schunk
hand, meaning that the grasp is as robust as before. In the
case of the Allegro hand, P CR = 4.744, which means that
robustness is regained to a similar value as with the initial
configuration.
Best approximation results obtained with the Schunk hand
can be again explained by the higher number of pre-defined
mesh positions for the object centre that are found to be
graspable while maintaining the contact point configuration.
Even if with the Allegro hand the redundant DoF allow for
a better exploration of the nullspace, again if the departing
position is further from the global optimal position, the
performed optimization will only reach a local minima.
C. Mimicking the evolution of the anthropometric grasp
Finally, another point of analysis is the capability of the
method to not only approximate an initial grasp stiffness and
modify it, but also to follow the configuration and stiffness
of the human-like hand to more extreme positions. For this
last part, we evaluate the method only using the Allegro hand
due to its similarity to the anthropomorphic hand in terms
of the number of DoF.
A closed grasp position of the anthropometric hand is
chosen with the same contact points as in the initial example.
The difference lies on the object location, which is displaced
in the positive ’X’ direction and negative ’Z’. This new grasp
configuration also reflects in the obtained grasp stiffness (see
Fig. 8). The ’XZ’ plane is more balanced in both directions
than in the initial configuration (compare with Fig. 1c). There
is actually a decrease in the ’X’ direction making the grasp
to be the stiffest in the ’Y’ direction.
Starting from the same initial configuration of the Allegro
hand as before (Fig. 2f), the algorithm is able to find the solu-
tion displayed in Fig. 9 with a joint stiffness synergy value of
α= 11.05 Nm/rad. This value remains quite below the ones
of the Schunk hand in previous configurations. The final error
regarding the orientation of the grasp stiffness ellipsoid is
reduced from an initial 1.309 to 0.6357, while eKtd decreases
from 48.87% to 23.69%. Again, in this configuration all
contact constraints are fulfilled, being P CR = 4.912.
Interestingly, from Fig. 8 and Fig. 9, it can be noticed
that even if some error remains, the Allegro hand moves
towards a finger pose that mimics the one produced by the
anthropometric hand.
XY XZ YZ
Fig. 8: Modified configuration of the anthropometric hand and object
position, and grasp stiffness projection in the 3 main translational planes.
Fig. 9: Upper row: Initial Allegro hand grasp, after mesh search of ellipsoid
orientation, and after nullspace optimization. Lower row: Grasp stiffness
projected in the 3 main planes. Oriented and final stiffness are obtained
before and after optimization of the joint stiffness Kqrespectively.
D. Computational time
Computational time is an important factor if this method
is planned to be used with actual teleoperated robotic hands.
There are three main computation timings: (i) search of the
best approximation of the stiffness orientation among the
mesh positions, (ii) optimization around the nullspace, (iii)
optimization of the synergistic joint stiffness. Computations
are made with an Intel(R)Core(TM) i5-6200U 2.30 GHz,
with 4.00 GB of RAM.
The average computational time for the search among the
mesh positions is of 0.0110 s. In the case of the nullspace
optimization, it highly depends on the number of iterations,
being 1.049 s the average computational time per iteration.
This means an average of 3.146 s for 3 iterations and
8.388 s for 8 iterations (the maximum the method executed).
Finally, for the optimization of the joint stiffness synergy, the
process takes 0.1087 s in average. While the mesh search and
optimization of the joint stiffness is done in a relatively rapid
way, the nullspace optimization might take a relevant amount
of time depending on the number of iterations performed.
Therefore, in the case of a real-time application this nullspace
optimization could be done using an analytical solution rather
than numerical. This would potentially result in an efficient
computational time.
V. CONCLUSION
In this work, inspired by the observations in human motor
behaviour on the coordination of the fingers stiffness and
pose of the hand, a new method for the control of robot
grasping compliance was proposed. The method exploited
the dominant contribution of the finger poses to the grasp
stiffness ellipsoid geometry. This was achieved by the op-
timisation of the DoF of the hand (CDS) to minimise the
error between the desired orientation of the grasp stiffness
ellipsoid and the obtained one. The adjustment of the volume
of the ellipsoid was achieved through the adaptation of the
fingers joint stiffness (CMS) in a coordinated way, a common
design concept in robot hands with coupled finger joints [20].
The use of a synergistic stiffness activation across the
finger joints of a hand is actually in concordance with
the stiffening patterns found in humans [16]. Moreover, it
drastically reduces the actuation complexity of the robotic
hand, as having coupled joints allows to control all their
stiffness with just one motor. Stiffness boundaries with such
joint coupling have begun to be studied in works like [20].
The proposed method was evaluated based on the ability
to transfer desired stiffness profiles from an anthropomorphic
model of the human hand to two robotic hands with different
kinematic and dynamic properties. In addition, the efficacy of
the technique in tracking the required on-line modulations of
the grasp stiffness matrix of the robotic hands was assessed.
It ought to be mentioned that the simulation software used
in this study [26] is known to include detailed kinematic and
dynamic considerations to make the simulation environment
as realistic as possible. Nevertheless, future work will aim
at the experimental validation of the method on a multi-DoF
robotic hand.
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