IEEE Robotics & Automation Magazine - September 2022 - 147

that is able to keep the system safe to synthesize a CBF that
guarantees safety in the presence of input bounds. However,
this formulation is very notationally dense and requires full
knowledge of the underlying system dynamics, so it will not
be showcased in this article. Instead, we will outline the more
recent approach introduced in [5].
First, rather than simply considering the kinematics, we
consider the Euler-Lagrange dynamic equation of the robot,
given by:
() (, )( ),
Dq qC qq qG qu++ =
po o
where u Rm
(13)
! is the vector of joint torques, D(q) and (, )Cq qo
Gq Rm
are respectively the inertia and Coriolis-centrifugal mm#
matrices, and ()
! is the gravity torque vector. By
incorporating the kinetic energy of the system into the barrier
formulation, we can generate a so-called energy-based safety
constraint, defined as:
hqqq Dq qh q $ 0
D oo o + ce
(, )()( ),
|=2
1
T
(14)
in
which ce 02 is an additional scalar parameter to be properly
tuned. Thanks to the kinetic energy term, we ensure that
the CBF has a relative degree of one with respect to the input,
meaning that a purely positional constraint can be enforced.
Due to the inclusion of the system dynamics through the
kinetic energy term, we can now guarantee the safety of the
true system described by (13). Since the kinetic energy term is
positive semidefinite, we guarantee the fact that
hh /,
$ c
De
meaning that safety is maintained ()h 0$ whenever hD 0$ .
Moreover, by taking ec large, we get a set that approaches the
nominal safety set described by h(q).
In many situations, it can be assumed that the embedded
controller of the industrial robot implements a feedback law
4
3
2
1
-1
02 46 81012
t (s)
Figure 8. The purely kinematic barrier function on the 6-DoF
manipulator. Safety depends on the choice of
c . The black line
in the figure is the safe set boundary. The simulated robotic tasks
and their execution are shown in the video (see http://youtube.
com/watch?v=ilLRA7fpVBI).
3.5
3
γ = 1
γ = 2
γ = 3
2.5
2
1.5
1
0.5
-0.5
0 0.51 1.5 2 2.5 3 3.5 4
t (s)
Figure 9. The energy-based kinematic CBF on the 6-DoF
manipulator: the values of the CBFs during a simulation with
ce 150 .0=
Obstacle avoidance is guaranteed since ()
γ = 2, γe = 1,500
γe
1
to compute the joint torques required to track the velocity
commands, which we may denote as
of a feedback law is (),uK qqd
q .d
o
=vel
ooThe
simplest choice
where Kvel
is an
mm# feedback gain matrix. Assuming further that the
underlying torque controller is known (or can be estimated),
the CBF constraint can be implemented at the kinematic level:
oo o<<2,
)
qq q
Jq qK qq Kq Gq h
d=- des
q !o d Rn
argmin
d
subjecttocc$
oo
eh oo oo oo+- +- D
12 3
TT
44444444 444444444
oDd
velvel
hq qq(, ,)
where the constraint embeds the knowledge of the Kvel
feedback
gain matrix coming from the feedback law.
By implementing the energy-based CBF constraint, safety
can be guaranteed for all values of c . To demonstrate this,
we apply the energy-based CBF condition to a value of c
that previously resulted in a safety violation. As can be seen
from Figure 9, plotting the values of the CBFs, obstacle
avoidance is guaranteed thanks to the CBF-based safety filter
because h(q) is always strictly positive. Moreover, the
added computation time is minimal, with each QP solution
taking roughly 19 μs on average (measured on a PC with an
Intel 9700 K 3.7-GHz processor and 32 GB of randomaccess
memory). Indeed, QPs are computationally efficient
and allow for implementation on the hardware in real time.
This is one of the main reasons why CBFs or QPs are highly
sought after in industry.
The guaranteed safe condition is valid for any value of
but increasing this value improves the performance of the
ce ,
d
T
,
hD (Energy Based)
h (Energy Based)
h (Kinematic)
hq 02 all
the time. See http://youtube.com/watch?v=ilLRA7fpVBI for the
video, and see https://github.com/DrewSingletary/manipulator_
asif_ros for the code.
SEPTEMBER 2022 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
147
CBF Value (h)
CBF Value (h)
http://www.youtube.com/watch?v=ilLRA7fpVBI http://youtube.com/watch?v=ilLRA7fpVBI https://www.github.com/DrewSingletary/manipulator_asif_ros https://www.github.com/DrewSingletary/manipulator_asif_ros http://youtube.com/watch?v=ilLRA7fpVBI
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