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An Evolutionary Approach for the Optimal Design of the iCub mk.3 Parallel Wrist

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The payload-to-weight ratio is one of the most important metrics when designing robotic wrists. A solution to maximize it and to reduce the share of the motive power required to drive the robot's links is to employ parallel kinematic mechanisms (PKMs). Indeed PKMs allow relocating distal masses closer to the robot's base actually increasing the overall payload. On the other hand, PKMs are often characterized by limited ranges of motion (RoM) and non-uniform motion in their workspace. In this article, we considered a class of 2-DOF spherical six-bar mechanisms. We first developed the kinematic model of the system. We then tackled both the workspace limitation and uniformity issues with a numerical optimization approach. Differential evolution (a multi-objective, multivariate, gradient-free optimization method) was applied to the model of the system to explore a large space of parameter combinations. The optimization algorithm allowed obtaining an almost uniform and large RoM (exceeding 50°on both axes). We then proceeded with the detailed design of the joint as we envision integrating it on the future releases of the iCub robot forearm-hand assemblies.
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An Evolutionary Approach for the Optimal Design of the iCub mk.3
Parallel Wrist
Raed Bsili+, Giorgio Metta, Alberto Parmiggiani
Abstract The payload-to-weight ratio is one of the most
important metrics when designing robotic wrists. A solution
to maximize it and to reduce the share of the motive power
required to drive the robot’s links is to employ parallel
kinematic mechanisms (PKMs). Indeed PKMs allow relocating
distal masses closer to the robot’s base actually increasing
the overall payload. On the other hand, PKMs are often
characterized by limited ranges of motion (RoM) and non-
uniform motion in their workspace.
In this article, we considered a class of 2-DOF spherical
six-bar mechanisms. We first developed the kinematic model
of the system. We then tackled both the workspace limitation
and uniformity issues with a numerical optimization approach.
Differential evolution (a multi-objective, multivariate, gradient-
free optimization method) was applied to the model of the
system to explore a large space of parameter combinations.
The optimization algorithm allowed obtaining an almost
uniform and large RoM (exceeding 50°on both axes). We then
proceeded with the detailed design of the joint as we envision
integrating it on the future releases of the iCub robot forearm-
hand assemblies.
I. INTRODUCTION
The payload-to-weight ratio is one of the most important
metrics for describing the physical capabilities of robots.
The payload-to-weight ratio of robots with serial kine-
matics can be improved by combining adjacent joints into
multi-DOF parallel kinematics mechanisms (PKMs). Some
interesting applications of this design approach can be found
in [1], [2], [3] and [4].
PKMs are particularly interesting for medium-size hu-
manoids such as the iCub [5] because of their compact
size and good stiffness characteristics. For this purpose, in
previous works, we focused on the numerical simulation of
N-UU class mechanisms [6] and on their comparison with
various closed spherical linkage mechanisms [7].
Indeed, PKMs are often characterized by limited ranges
of motion and complex kinematics if compared to serial
kinematics manipulators. Moreover, the motor-input/joint-
output Jacobian, i.e. the relation between the motion of
the actuators and the resulting joint motion, is often cou-
pled and configuration-dependent. This, in turn, significantly
complicates the control of the joint as the same motor
motion causes different joint motions in different regions
of the mechanism’s workspace. Further details about the
Giorgio Metta and Alberto Parmiggiani, are within
iCub, Fondazione Istituto Italiano di Tecnologia, Via Morego
30, 16163, Genoa, Italy. giorgio.metta@iit.it,
alberto.parmiggiani@iit.it
+Raed Bsili is within iCub, Fondazione Istituto Italiano di Tecnologia,
Via Morego 30, 16163, Genoa, Italy and Danieli telerbotlabs srl, Via Buccari
9, 16153, Genoa, Italy. raed.bsili@iit.it
characteristics comparison between the PKMs and serial
kinematic mechanisms (SKMs) can be found in [8].
When designing a manipulator, especially a PKM, an
interesting metric to evaluate the “uniformity” of motion in
the mechanism’s workspace is isotropy, as defined in [9].
In this article, we investigate the 2DOF 2DSPM parallel
mechanism introduced by Ogata et al. [10] and later refined
by Ueda et al. in [11].
The mechanism’s RoM and the degree of variations in
the motor-input/joint-output Jacobian are a function of the
design parameters. We addressed both aspects by selecting
the design parameters with a methodology based on the
differential evolution numerical optimization method.
The article describes in detail the approach followed and
presents the definition and solutions of the optimization
process. In addition, we present qualitative and quantitative
analysis on the effect of the design parameters on the wrist
mechanism behavior.
II. APPROACH
The current iCub mk.3 wrist RoM requirements are ±56°
and ±38° on the pitch and yaw axes. The goal we pursued in
this work was to design a 2DSPM with equivalent and pos-
sibly larger range of motion and with a uniform mechanism
behaviour. The Fig. 1 presents the design model (at zero-
position) of the 2DSPM where the green part defines the end-
effector. The pitch rotation φpis defined as the end-effector’s
Fig. 1: CAD model of the investigated 2DSPM
rotation about the y-axis while yaw as the end-effector’s
rotation φyabout the z-axis. The distance between the end-
effector point Eand the center of rotation Odenoted by L, is
the scale of the wrist. The design configuration is described
by the design parameters, denoted by l1,l2,l3and α, that
define the wrist’s links geometry. There are two actuated
joints: a left and a right one, denoted respectively by θLand
θR. To achieve an optimal design, a proper mathematical
model describing completely the kinematics in function of
the design parameters was modeled. Then, the optimization
model describing the optimal design configuration from
mechanism uniformity and workspace width point of view
was set up and solved. To generate the workspace, a square
grid input to the actuated joints with a given amplitude Iis
generated, resulting in the generation of all different input
combinations of θL,θR[I,I]range. In this paper, the
square grid semi-length Iis called the amplitude.
A. Kinematics
A fixed coordinate frame O(x,y,z)in the center of rota-
tion Ois placed with the same origin of the end-effector
coordinate frame O(x0,y0,z0)as shown in Fig. 1. The vectors
uLand vLdescribing the rotation axis of the passive joints
of the left leg are expressed using the rotational matrix
Raxis(angle)SO(3):
uL=Ry(θL)·Rx(l1)·[0,1,0]T(1)
vL=Ry(φp)·Rzφyπ
2·Rxπ
2l3·Rz(α)·[0,1,0]T
(2)
where SO(3)denotes the 3Drotation group. The design
parameter l2imposes the following mathematical constraint:
uL·vLcos(l2) = 0(3)
The same mathematical operations have been applied on
the right leg. Thus, the kinematics problem became written as
a 2-dimensional system of highly non-linear equations, with
two unknowns φpand φyto solve the forward kinematics:
fL=uT
L·vLcos(l2) (4)
fR=uT
R·vRcos(l2) (5)
This system is solved using an optimization algorithm
meant to minimize the sum square of fLand fR. Due to the
theoretical non-unicity of solutions, mathematical constraints
were added to ensure the uniqueness of the realistic solution.
Therefore, the forward kinematics are reduced to the follow-
ing constrained and bounded optimization problem, with a
dimension N=2:
argmin
x=[φp,φy]
f(x) = f2
L+f2
R
(6)
where:
φp,φy[180°,180°]
and:
(gL(x) = [0,1,0]·(uL×vL)>0
gR(x) = [0,1,0]·(uR×vR)>0
The optimization problem has been solved numerically
by providing the exact analytic expressions of the objective
function’s Hessian Hfand the Jacobian of both the objective
and constraints functions Jfand Jg:
Jf=d f
dφp
,d f
dφyR1×2(7)
Hf=J(f)TR2×2(8)
The inverse kinematics are solved in the same way de-
scribed above by solving x= [θL,θR], for a desired pitch and
yaw [φp,φy]configuration.
B. The Global Isotropy and Workspace Numbers
An important criterion to evaluate a parallel manipulator
is to evaluate its mechanism uniformity using a metric called
isotropy, denoted by . It is a measure that indicates how
well over the workspace a wrist is manipulable and how close
or far it is from singularity at the same time. The isotropy
is defined as:
(J) =
m
pdet(JJT)
tr(JJT)/m[0,1] (9)
where JRm×nis the manipulator Jacobian. The isotropy
could be written equivalently in terms of the JJTeigenvalues,
denoted by {λi}iN:
(J) = (m
i=1λi)1/m
(m
i=1λi)/m[0,1] (10)
where mdenotes the number of DoF of a manipulator’s
end-effector. In our case, m=2. The isotropy’s numerator
and denominator are defined respectively as the geometric
mean, and the arithmetic mean of the JJTeigenvalues. The
geometrical mean describes the wrist manipulability. When
=0, the manipulator is in a singular configuration (i.e.
iN,λi=0). Using the AM-GM Inequality, one can
conclude that the isotropy is indeed bounded [0,1]as JJTis
positive definite. Further, is maximum, only, and only if all
the manipulator’s Jacobian eigenvalues are non-zeros and are
equal to each other, implying the full Jacobian m- spheroid
which we intend to realize i.e. a perfectly posed Jacobian
matrix. The isotropy is a local property as it depends on the
end-effector pose. Therefore, we evaluate the isotropy over
the whole workspace with its weighted mean:
=RW(J)dW
RWdW [0,1] (11)
where Wdenotes implicitly the workspace area spanned
by the end-effector’s orientation angles φpand φy. In this
work, we loosely refer to as the Global Isotropy Number.
It is a metric used to compare the uniformity of different
manipulators mechanisms, independently from their scale L.
Another measure of motion uniformity commonly used in
the literature is the Global Conditioning Index, introduced
by Gosselin and Angeles in [12], which is a measure of
the Jacobian condition number over the whole workspace.
The condition number of the Jacobian matrix has been
first introduced in [13] and it is used as well as a metric
to describe how ill-posed a matrix is. In this work, we
nevertheless preferred to use the measure of isotropy, denoted
by , for the following reasons:
The index is faster to compute numerically than
the condition number. This could be explained by the
fact that the condition number requires computing the
matrix inverse, which is relatively more computationally
expensive.
The index, unlike the condition number, can be
expressed analytically in function of the manipulator’s
joints angles. This speeds up further its computation.
In order to evaluate the workspace width of the manipula-
tor and compare it to other manipulators, independently from
their scale L, we propose the following Workspace Number:
=RWdW
SJ
=Rφpmax
φpmin Rφymax
φymin dφydφp
RθLmax
θLmin RθRmax
θRmin
dθLdθR
R>0(12)
where here SJdenotes the area spanned by the input
actuated joints θRand θL. When the input is a square grid
with an amplitude I, as in our case, the joints angular
workspace area denoted by SJbecomes simplified as SJ=
4I2. More details are explained in Fig. 3.
For example, in the case of an “ideal” gimbal, ==1.
Finally, the mapped cartesian workspace area denoted by SC
is defined as the area of the end-effector workspace envelope
(see Appendix).
C. Optimization
The optimal design here requires finding the best de-
sign parameters that ensure the maximum isotropy and the
widest workspace. The maximization of the isotropy and
the workspace was expressed as an N=4 dimensional
optimization problem:
argmin
X=[l1,l2,l3,α]
f(X) = 1(min..max ).
(13)
where:
l1,l2,l3,α[,85°]
and:
g(X) = Sl1Sα(Cl2SαCl1)>0(14)
In which the above abbreviations are Sx=sin(x)and Cx=
cos(x). In our case, the Workspace Number has been re-
stricted to be bounded i.e. (X)[0,1]. The above bounds
have been added to ensure a realizable potential design. The
above constraint g(X)has been added to ensure that the
design parameters describe a design where the zero pitch
and zero yaw positions of the manipulator are defined from
a symmetric zero position of the actuated joints. Due to the
complexity of the objective function f(X)R, and the error-
prone numerical approximation of its Jacobian JfR1×4, a
gradient-free optimization technique is required. Therefore,
meta-heuristic evolutionary optimization algorithms were
needed and the Differential Evolution DE [14] optimizer is
selected.
initialize the first generation of individuals;
G=1
while Stop criterion not met do
for each target vector Xi,G;
for i=1 to NP do
Mutation: generate a donor vector Vi,G;
Vi,G =Xri1,G +F(Xri2,G Xri3,G)
for j=1to N do
Crossover: generate a trial vector ui,G;
uj,i,G =(Vj,i,G, if Uj(0,1)6CR or j=jrnd
Xj,i,G, otherwise
end
Selection: accept the trial vector if not worse
than the target vector;
Xi,G+1 =(Ui,G, if f(Ui,G)6f(Xi,G)
Xi,G, otherwise
end
G+ = 1
end
Algorithm 1: Differential Evolution pseudo-code
The DE optimization algorithm requires three parameters
to tune: the mutation factor F[0,2], the crossover prob-
ability CR [0,1]and the population size NP. Algorithm.
1 presents the DE pseudo-code where a rand/1/bin strat-
egy was used to generate the donor vector [15], in which
U(a,b)Rdenotes a random number between aand b. The
residual, defined as well as the candidate energy, is denoted
by f(X)R,XRN.
The algorithm works by iterating over every generation G
and executing mainly three phases: mutation, crossover and
finally the selection phase.
III. COM PUTATIONAL PERSPECTIVES
To compute the Global Isotropy Number of a de-
sign parameters set, first, the forward kinematics over the
whole workspace are solved using the optimization prob-
lem described by Eq. (6). Secondly, the Jacobian matrix
Jand the local isotropy are calculated by feeding the
forward kinematics information to the analytic expressions
of J([θR,θL],[φp,φy]) and (J). Lastly, the Global Isotropy
Number and Workspace Number are computed numerically
using Eq. (11) and Eq. (12), respectively. All analytic ex-
pressions were derived using symbolic computational meth-
ods. The DE optimization algorithm allowed to explore
a significant amount of parameter combinations. To make
a ballpark comparison, a brute force exploration of the
parameter set with a 1° resolution would have required more
than 5000 times the computations that were performed for
this study. The whole algorithm is developed in the Python
programming language [16]. All computations have been
executed on a Dual AMD Opteron computer server with a
6328 @3.2 GHz processor and 64 GB RAM.
IV. RESULTS
In the beginning, a square grid input with an amplitude
I=65° was used. After tuning the DE optimizer parameters,
the crossover factor was set to CR =0.1 to have a more
stable candidates population. The mutation factor was taken
arbitrary from F=U(1.5,1.9)to randomize it for every
generation. This increased the search radius at the cost of
a slower convergence rate. Population size is set to NP =60
candidate individuals. To initialize the first population of
individuals, a Latin Hypercube Sampling (LHS) method
[17] was used to maximize the coverage of the available
parameters space i.e. R4. Finally, a best/1/bin strategy has
been selected to formulate the donor vector gene, taking
the best candidate to mutate instead of a random individual
vector Xri1,G.
As a result, the optimum design parameters vector was found
to be OPT1= [33.7°,83°,32.7°,10.7°], whose isotropy plot
in the Y Z -plane is shown in Fig. 2. All plots were performed
using a normalized scale L=1, so that all the results are
bounded between 1 and 1. Furthermore, all colorbars are
scaled equally from the corresponding min to max, in order
to help the reader appreciate the differences between the
isotropy distributions for the different design parameters
candidates. The design parameters vector OPT1is shown
Fig. 2: OPT12D isotropy contour on Y Z-plane (I=65°)
performing a full isotropic spheroid where (min,,max ) =
(0.978,0.998,1)and with rotational ranges consisting in
a pitch: φp[65°,65°]and a yaw: φy[63.,63.].
This parameter combination yields the required ranges,
as shown in Fig. 3, and improves the design parameters
P= [42°,74°,50°,20.4°]proposed by Ueda et al. in [11].
The Workspace Number was computed as the contour
angular area mapped by the end-effector’s angles, divided
by the square grid input area defined by the black rectangle,
as shown in Fig. 3. Extending the input square grid to
an amplitude I=80° causes the OPT1isotropy to drop
significantly to min =0.11, which makes it not the best
Fig. 3: OPT12D isotropy contour on the Euler plane (I=65°)
solution, from isotropy point of view. The optimization
process has been adjusted to optimize only the isotropy
because an input of I>65° is enough to fulfill the iCub
mk.3 ranges of motion requirement.
As a result, the optimum OPT2= [41°,76.3°,41.1°,25.7°],
(see Fig. 4) was found to be the best solution scoring a
(min,,max ) = (0.88,0.97,1). It is performing a nearly full
isotropic spheroid, improving OPT1and design parameters
Pplotted in Fig. 6. In Fig. 5, we show the 2Disotropy
Fig. 4: OPT23DIsotropy on the cartesian workspace (I=
80°)
contour of OPT1design parameter on Y Z plane overlapped
by the workspace of OPT2. As shown in Fig. 5, the OPT2has
thinner workspace than OPT1, but better isotropy distribution
due to the considerable OPT1isotropy drop in the workspace
lateral corners. This means that in such design configura-
tion, approaching these regions would result in a significant
amplification of the end-effector velocity when compared to
other regions in the workspace. This is a behavior to avoid
Fig. 5: OPT2-OPT12D isotropy and workspace comparison
(I=80°)
to ensure more robust control of the wrist.
To validate the results, using PTC Creo Parameteric 4.0, a
Fig. 6: P- 2D isotropy contour on YZ -plane (I=80°)
CAD model with the design parameters OPT1,OPT2and P
were modeled. A simulation with the same square grid input
Ihas been performed, showing an approximate matching
between the mathematical model and the CAD simulation
with an average error equal to ε105on the isotropy
(see Fig. 7).
V. DISCUSSION
To increase the likelyhood of obtaining global optimality,
different optimization runs were performed over G=100
generations. The optimizer converged in all the trials to the
same optimum, as shown in Fig. 8. The optimization runs
Fig. 7: OPT1Isotropy comparison between anatylical and
CAD models: ε=|Creo |, (I=65°)
Fig. 8: Convergence of the fittest individual per generation
G
were all performed at the same time, in parallel, and on the
same computer server. The computations for the six trials
required approximately seven days.
One can notice that such a 2DSPM is characterized by always
having a symmetric ranges of motion, a constant isotropy
along the pitch direction, and a variable symmetric isotropy
along the yaw direction. In particular, OPT1and OPT2
designs isotropies tend to oscillate along the yaw direction
with variable amplitudes, however, Pdesign isotropy tends to
decrease along it, as shown in Fig. 9. This is mathematically
consistent with the fact that we have always φpmax =I,
as shown in Tab I. This is interpreted by the fact that an
angular variation on the actuated joints would induce the
same angular variation on the end-effector’s pitch angle,
implying a perfectly posed Jacobian matrix along the pitch
Fig. 9: Isotropy variation along yaw direction (I=80°)
direction, i.e.(J)'1.
It should be noted that tuning the input amplitude Iwould
result in finding different optima. This depends on how the
optimum was defined, from isotropy or workspace width
point of view, or both of them. For example, during the
optimization process, we count several candidates found
to have a wider workspace than OPT2, but lower Global
Isotropy Number. Alternative loss functions could be used
to define the objective function f(X)in order to speed up
the convergence rate and control the optimization process.
In the end, OPT1is shown to be likely the global optimum
when I=65°, validating the iCub mk.3 wrist range of motion
requirement. At last, for general applications, in which
wider ranges are required, for example, I=80°, OPT2is
recommended to use as design parameters. The scatter matrix
Fig. 10: Scatter matrix for the objective functions and the
design parameters
shown in Fig. 10 presents the inter-dependencies between
the decision variables during the optimization process, when
I=65°, obtained from the Pareto front, by sorting the best
50 individuals. It is shown that during optimization, through
successive generations, the parameters αand l2have evolved
inversely proportional to each other, with a correlation factor
equal to 0.973. Table II summarizes the different design
parameters candidates that were discussed above. Table I
TABLE I: Comparison of different design parameters
I=65°(min,,max ) (,SC[]) (φpmax ,φymax)
OPT1(0.978, 0.998, 1) (0.5, 4.92) (65°, 63.1°)
OPT2(0.90, 0.97, 1) (0.41, 4.17) (65°, 50.6°)
P(0.80, 0.95, 1) (0.41, 4.15) (65°, 49.3°)
I=80°
OPT1(0.11, 0.92, 1) (0.49, 6.58) (80°, 72.5°)
OPT2(0.88, 0.97, 1) (0.41, 6.02) (80°, 64°)
P(0.63, 0.93, 1) (0.39, 5.72) (80°, 56°)
summarizes the results found so far and their comparison.
Finally, one notes that the dynamics behavior of such 2DSPM
TABLE II: Design parameters candidates
[l1,l2,l3,α]
OPT1[33.,83°,32.,10.]
OPT2[41°,76.,41.,25.]
P[42°,74°,50°,20.]
would be similarly isotropic as its kinematics behavior. This
is justified by the fact that, when neglecting gravitation and
frictions, the task force space is mapped to the joints force
space by JTand that (J) = (JT), implying a robust hybrid
control of such wrist.
VI. DESIGN
The OPT1design parameters were selected for the iCub
mk.3 wrist, and used for the preliminary assembly design
represented in Fig. 11. Because of the small size of the
Fig. 11: (OPT1) - iCub mk.3 hand and forearm preliminary
design
iCub forearm, a Four-bars linkage was selected to actuate the
active joints θLand θR. One of the main advantages of using
the planar Four-bars linkages is to ensure an efficient power
transmission between the two joints (crank and follower)
while keeping the assembly relatively simple and compact.
On the other hand, the Four-bars linkage has a singular
configuration (when all linkages are aligned) that shall be
avoided. This limitation needs to be taken into account since
passing through the singular configuration shall be avoided
in all wrist configurations. In order to avoid the singularities
Fig. 12: (OPT1) - iCub mk.3 hand reaching its required
ranges of motion
TABLE III: Design’s range of motion summary
Pitch [°] Yaw [°]
iCub mk.3 Wrist RoM Requirement ±56 ±38
OPT1Theoretical RoM ±65 ±63.1
OPT1Mechanical RoM ±58 ±50
and ensure that the follower moves inside a RoM of an
amplitude I=±65°, an offset angle has been given to it
and to the crank. The two active DoF are actuated by two
Faulhaber 1724T012 DC motors coupled to two CSF-5-30-
2HX-F-1 Harmonic Drive compact speed reducers. To avoid
mechanical interferences and collisions among some design
elements, we have constrained our mechanism to have a
range of motion RoM consisting of ±58 on the pitch and
±50 on the yaw axis. Table III summarizes the design
ranges of motion. Finally, Fig. 12 shows the iCub mk.3
wrist reaching 56° on the pitch axis and 38° on the yaw
axis, fulfilling the required iCub mk.3 wrist range of motion
requirement.
VII. CONCLUSIONS
A full mathematical model has been designed to ensure
the global optimal design of a 2DSPM wrist from isotropy
and workspace point of view. All models have been tested
on PTC Creo Parameteric 4.0 in which a simulation was
performed to validate the results. The simulation showed a
smooth and realistic motion within the input range’s ampli-
tude I, with results that match the developed mathematical
model with an ε105average error on the isotropy.
Finally, the OPT1design has been selected for rapid
prototyping to test it on the iCub mk.3 forearm, as shown
in Fig. 11. The selected wrist design has shown promising
results with a maximum attained isotropy '1=max, over
a symmetric workspace (φpmax ,φymax) = (65°,63.).
APPENDIX
The wrist Jacobian JR2×2was computed as J=J1
φJθ
where:
f= [ fL,fR];Jθ=f
∂ θ ;Jφ=f
∂ φ
In which θ= [θL,θR]and φ= [φp,φy]. Once the forward
kinematics solved, the end-effector cartesian position de-
scribed by the point Ecoordinates on the base coordinate
frame O(x,y,z)was found in function of the wrist scale L
and the solved Euler angles:
xE=L·cos(φp)·cos(φy)
yE=L·sin(φy)
zE=L·sin(φp)·cos(φy)
The mapped cartesian workspace area SCwas then com-
puted by integrating over the xE,yEand zEvalues.
ACKNOWLEDGMENT
This study was conducted as a part of the Deictic Com-
munication (DComm) project and has received funding from
the European Union’s Horizon 2020 research and innovation
programme under the Marie Skodowska-Curie Actions grant
agreement No 676063.
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... T HE two-degrees-of-freedom spherical parallel link mechanism (2-DOF SPM) is designed to limit the degree of freedom of the end-effector to only two-directional motion around a fixed center [1]. It is suitable for applications that need to change the end-effector posture while always aimed at the control target at the fixed center [2], [3], such as tracking [4], [5], wrist or ankle joints of humanoid robots [6]- [9], surgical robots [10], and stabilization platforms [11]- [13]. ...
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