IEEE Robotics & Automation Magazine - September 2022 - 131

with their origin at xt
!
h in the basis ().E xh
are measured respectively in 3D, in 2D normal to ,eh
1D along e .h
tangent} as () /,e11x
(( ))
ha x
ii i
CR -
h hh h hh
t
xx EE )] .
R !
i R33
#
i =-xx xx xt
are diagonal matrices that select the considered
<
hh h
<
/
i
azradial
xxh
= R =1
12
2
C
These distances
1 and in
m = () () ()f i
ij
h
1 Accordingly, we define associated activation
zi xh
i
where ()xi
h
h repreh
12
functions () [, ]01 with i = {radial, normal,
z =+ d C-- h
sents state-dependent distances of the robot given
by () [( )( )( ())(
directions of ().E xh
The elements of iR are mainly zero but
h
are one at index(es) of the desired direction(s). Hence, ()xa
in (3), is designed such that () /( ).
The behavior of the DS as it generates the desired motion
is show in Figure 4 along with the activation of the modulation
functions ().xiz
Modulation-Based Coordinated Control
The motion coordination of the dual-arm system, in this
work, exploits the cooperative task space representation [14],
which relates the states of each robot to the cooperative coordinates
formed by the absolute and relative states of the dualarm
system. The coordination is achieved by controlling the
two robots' cooperative coordinates and mapping the resulting
motion to each robot.
Thus, assuming that the nominal DS ()f xn
is linear, the
coordinated motion that it generates can be written as:
()
fT-1 (),
A
n xx ATb xxb==o
>l
where Tb = I3
6A
R66
x
x
!
rel
abs
x
x
R
L
(/ )( /)I12 3@ ! R #
12 I3
I3
robot positions (xL
and )xR
and relative position x Rrel3
88= T bBB and where I3
x .)
- )
Hence, for each hL , R= "
(4)
66 is a matrix that maps the two
to the absolute position x Rabs3
! of the dual-arm system, such that
is a 3 × 3 unit matrix. In (4),
!
# denotes the dynamics or gain matrix, which is negative
definite ()A 01 to ensure stability and convergence to a
given attractor
dynamics of xabs
The coordination is thus achieved by controlling the
and x ,rel
and xrel
is then mapped through Tb
-1
which amounts to controlling
respectively the two robots' joint motion and their relative displacement
and thereby their synchronization. The resulting
motion of xabs
to each
robot's motion.
The modulation shapes the behavior of the nominal DS in
the region where it is active. This shaping must preserve the
DS stability and convergence properties of the nominal DS
f ()xn
toward its equilibrium points x .)
Stability and Convergence to Attractors
Based on the previous work [17] on single robot, we can state
the following proposition for dual-arm system.
Proposition 1
For any given state {(), () }
xx 10f x
!;a =
R
K as
3
n
h
!
, setting
the state-dependent coefficients of the modulation matrix
h ()x
()
()
e
e
L <
i
R <
i
x
x
.
.
L
R =
Finally, substituting (4) for fm
h
() () (),
ee xxxlA
h < . h
i
=- )
<
h
i
h
which gives in terms of the left and right components
() [( )( )]
() [( )( )]
e
e
L
i
R
i
<
<
ll
ll
AA
AA
LL
RL
xx xx
xx xx
LL
LL
-+ -
-+ -
))
))
LR
RR
RR
RR
.
(S3)
Clearly, (S3) shows the two robots' interaction, which is
necessary to preserve the coordination. At the same time,
all robots converge toward their attractors since at the
equilibrium (x 0
. L
= and
0 = ; cm l().
A xx)
l
l
A
A
LL
RL
l
l
A
A
LR
RR
E
This implies that xx 0-=)
xx
xx
-
-
LL
RR
) =)
given
that Al is full rank.
Y
SEPTEMBER 2022 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
131
x 0
. R
= we have
),
i in (S1) yields
(S2)
()i
ee
e
h < . h
xx x
x
= m
=
j=1
3
() () /
j=1
h <
i
f
= ei() (),
h <
f
m
h
i x
where we use the following simplification
3
3
/
j=1
ff
f
n
h
n
h
<
() ()
()
xx
x
<
e
n
h
j
h
() ()
e ==I.
ff
h <
i
f
n
h
x
ff
() ()
n
h
() / () ()
<<
xx
j=1
n
h
xx
ee
<
j
h
n
h
j
h
n
h
m
h
i
ff
f
n
h
n
h
<
() ()
()
xx
x
<
e
n
h
j
h
()< ()f x
e
j
h
n
h
12 3
44444444444444
I
(S1)
/ ()() ()
<
ij
h
j
h
f
n
h
3
, component, we have
xxi
e
h <
m
h
ff
f
n
h
n
h
<
h
i
Moreover, if ()f xm
h
i
() ()
()
xx
x
<
e
n
h
j
h
,
(5)
the motion generated by (1) will be governed by the DS
f ().xm
lim0t xx " 3 -=)
is a stable linear or linear paramewhile
main.
ter-varying
(LPV) DS, for instance, of the form of (4), the
state x will asymptotically reach its attractor x)
taining the coordination between the robots of the dual-arm
system. That is,
Proof: See " Proof of Proposition 1. "
Generation of Impact Velocity
To generate the desired grabbing impact velocities with the
dual-arm system, we introduce the following proposition.
Proof of Proposition 1
Substituting (5) in (2) and then in (1) and multiplying by
diag () ,( )eei
"
L <
R <
i
()
()
e
e
L <
i
R <
i
x
x
.
.
, gives
L
R =
() () ()(( )) ()
() () ()(( )) ()
e
e
L LL L
i
R RR R
i
xx xx
xx xx
<<
<<
K
K
EE f
EE f
r
L
R
r
.
IEEE Robotics & Automation Magazine - September 2022

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