IEEE Robotics & Automation Magazine - September 2022 - 129

abilities and their robustness to perturbations. The bimanual
coordination problem is formulated using the Extended
Cooperative Task Space representation [14]. We subsequently
combine this flexible planning trajectory with a QP to handle
the balancing of force constraints. Assuming prior but
approximate knowledge of the object's mass and the friction
coefficient, the QP generates online interaction wrenches that
achieve stable grasp subject to contacts' constraints.
The scope of this article being motion generation, we leave
aside the impact dynamics and assume that the associated
states' jumps remain within the robots' limits. The interested
reader is referred to [15] for the control of impact with states
jump mitigation and to [13] or [16] for explicit enforcement
of hardware limits during impact generation.
The Proposed Approach
In the considered dual-arm task, the robotic system is
required to reach and swiftly grasp an object and then either
toss it, by releasing it at a desired position and with a specific
velocity, or place it at a desired location. A schematic illustrating
the considered task is given in Figure 2, where the tasks
phases are depicted.
To induce the desired motion on the object when fast
picking and when tossing requires that the two robots' arms
adopt the required velocity just prior to impact (for fast picking)
and prior to release (when tossing). To obtain this behavior,
the robot arms must transit, at contact and release time,
through desired states expressed in both position and velocity
simultaneously. Unlike attractors, these transitory states are
not equilibrium points, and therefore, the robotic system can
only transit through such states. Thus, to realize the desired
task in a robust way, we proposed an approach based on modulated
DSs (MDSs), where state-dependent functions shape
locally the generated motion of the robot-prior to contact or
release of the object-such that the motion aligns first with
the desired velocity while moving toward the desired contact
or release position. Therefore, for a dual-arm system that
requires coordination, to realize fast grabbing and afterward a
tossing task, we formulate at the position level (the control of
orientation is described in " Orientation Control " ) the following
MDS.
xx xx () () (),
where x = !8 B
xR
x
x
R
L
o=+ (1)
R6 is the state vector of the DS with xL
Mf f
ng
and
representing respectively the position of the left and right
robot of the dual-arm system. ()f x R6
n
! is the nominal DS
that generates the coordinated motion toward transitory
attractors located in the vicinity of the desired positions.
Orientation Control
To control the orientation task, which consists of driving the
current orientation of the hth end effector represented by the
rotation matrix R Rc
h
!
Hence, pi_ ni
With i
ph
c
h
33
#
we define a state vector i
toward its desired value
h R3
p !
where n ! R3
dR .c
h
representation of the relative orientation,
h (),Rd
dRR R () .
c
h
d
h
c
h
R Rd
h
!
33
#
,
using the axis/angle
_ <
and i ! R represent
respectively the axis and the angle associated with the
rotation matrix
value is located at the origin, that is p =i
h
d
defined as mentioned previously, its desired
0 . Thus, similarly
to the position task, if we assume a linear or LPV DS for the
orientation, we can write
pp pp=- =ii
o
hh h
where A R33
#
i !
negative definite ()A 01i
ph
i AAd() h,iii
is the dynamic matrix chosen to be
to ensure asymptotic converges of
ph
indicates the matching Rc
h
i toward its attractor 0 (). Such convergence
with
limt " 3 i = 0
R .d
h
The angular velocity
associated with the orientation DS is obtained as follows
i
Σl
Σabs
Σol
Σvl
Σo
Σor
Σvr
ΣW
X
where sinsin= i and [] Rh
symmetric matrix associated with nh .
iic
Figure 2. A schematics illustration of the considered dual-arm
task, which can be seen as succession of two main phases:
a free motion phase when reaching (red dashed line) and a
constrained motion phase when in contact and executing the
placing or tossing motion (blue and green dashed lines). w
R are the world and the object frame. l
R
and o
R and or
configuration on the object side. i
R and r
R denote
respectively the end-effector frames of the left and right robot,
while ol
R denote respectively their desired grasping
R and oi
/ left, right).
R are respectively the
ith end effector's and object's grasp configuration frames, while
vi
R denotes the ith frame of a virtual or auxiliary attractor that
shapes the trajectory for impact, with index (i
i
Rd
h
n !#
33 denotes a skew#
To
coordinate the position and the orientation task,
the latter was coupled to the position task using a statedepend
coefficient ()xh
position: ()x 1 -- +
is a scalar that tunes how fast ()xh
h =- vf (( /<< )), where v2 0
varies within [, ].01
exp xxd
abs
pi_ )
n
h (),Rc
h
with RR .R
the rotation matrix computed from the spherical interpolation
between a resting orientation Rr
c _ ((h))
h
)
when () , RR .
h x 1 "
) h "
()
d
h
as function of ().xh When () , RR .
h
h x 0 "
) h " and
()
h
r
and the desired orientation
h
c
h
Here ()Rh h)
function of the error on the absolute
abs
The orientation state vector [, ]01 is now computed as
) hh <
denotes
Z
LI
Y
h hh
33
in=- +-##
1
2 fp
[]
inn2
ci
sin
sin
c
2 i
2
[]#,
Σr
where LL ()Rh
p
h _ in
c
h
~p p==
< with L Rh
pp i
-hho
i
in
!
33
#
h
LL A ,hh
11
a matrix mapping the
angular velocity to the time derivative of orientation state
vector hp and given by [20]
SEPTEMBER 2022 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
129
IEEE Robotics & Automation Magazine - September 2022

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