IEEE Robotics & Automation Magazine - September 2022 - 145

oo o
o oo
qq q
hq qJ qq
subjectto
q
)=- ,
o !R6
argmin<<2
des
(, )( )( ()),
=h
where
(( ))hq hq withac=
()
$ a hq
(12)
c ! R , as we did in the previis
perfecto
q
.des
o
This safety requireous
examples. In this section and in the related simulations on
Code Ocean, we assume that the velocity vector q)
ly tracked by the robot joints.
The solution of the QP forces the center of the PUMA
+
wrist to stay at a distance from pobs higher than RR ,w
whatever is the desired joint velocity
ment is rather simple and may be acceptable for a large number
of applications (e.g., pick and place operations in an
industrial scenario). However, the CBF-based safety controller
can be easily extended to avoid collisions on any part of
the manipulator by considering multiple constraints (e.g.,
constraints on the distance between the middle point of each
link and the obstacle) and, possibly, multiple obstacles.
The result of a simulation performed with the companion
code, in which the desired input of the manipulator was set as
a constant vector applying a velocity of 1 rad/s at the base
joint and zero velocities for all the other joints, is shown in
Figure 6. The function ()$a is chosen as in the previous examples,
here considering only a value of
c 5 .= The obstacle is
represented as a red sphere, while the safety volume around
the wrist is depicted as a green one. Both R and Rw
were set
equal to 0.1 m for the simulation. The 3D model of the
PUMA 560 is drawn in the initial configuration, while the
trajectory followed by the wrist when the safe control input is
applied is shown as a blue line.
The plot in Figure 7 shows the velocity commands, limited
to the first three joints, computed by the safety controller.
Since the constraint is related to the wrist center, the safe
velocities of the last three joints are always equal to their
desired values, i.e., zero in this simulation; thus, they are not
shown. Running the simulation code on Code Ocean, it is
possible to visualize the values of the CBF of (9), which is
always strictly positive, meaning that the green and the red
spheres shown in Figure 6 never intersect each other. Even
though the formulation of the CBF is not, strictly speaking,
equal to the geometric distance between the nearest two
points on the surfaces of the spheres, it is also easy to relate the
two quantities from
dh () () ().
=+ +- +
mi 13 10-3
n = #
is equivalent to 3.2 cm.
qR RR Rww
2
Indeed, given the setting of R and Rw we can compute that
the minimal distance reached during the simulation, corresponding
to .,h
The Peculiarities of Industrial Manipulators
Extending the CBF framework from the basic examples
shown so far to a realistic industrial environment requires
one to address a number of additional issues, both theoretical
and practical. First, it is important to remark that the
safety objectives achieved through CBFs in the proposed
examples rely on the robot control hardware and software
operating properly. From the industry point of view, instead,
safety standards related to human-robot collaborations, like
the ISO 10218-1 (http://www.iso.org), require industrial
robotic systems to be safe even in the presence of hardware/
software failures. However, fault-tolerant, CBF-based control
is a highly advanced research topic, outside of the scope
of this article.
On the other hand, the safety functions embedded in
industrial robots, either certified as collaborative robots or
not, are basically related to triggering an immediate stop, limiting
the end-effector velocity, or reducing the usable workspace
within bounds fixed a priori and are executed on the
0.6
0.4
0.2
-0.2
-0.4
-0.6
-0.2 0 0.2 0.40.6 0.5
y (m)
x (m)
Figure 6. A simulation of the 6-DoF manipulator avoiding an
obstacle; the green sphere is the safety volume around the wrist
with radius
R ,w and the red sphere, centered in pobs
R , is the obstacle.
-0.5
1.2
1
0.8
0.6
0.4
0.2
-0.2
0 0.5 1 1.52 2.5 3
t (s)
with radius
Figure 7. The desired (unsafe) velocity of the first joint and
actuated (safe) velocities of the first three joints of the
manipulator considered in the example (the values of u ,2des
and u ,3des
are set to zero; thus, they are not shown).
SEPTEMBER 2022 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
145
u1,des
u1,act
u2,act
u3,act
z (m)
(rad/s)
http://www.iso.org
IEEE Robotics & Automation Magazine - September 2022

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