IEEE Robotics & Automation Magazine - September 2021 - 96

are constrained to two performance and safety requirements.
The minimum coordination requirement seeks the fastest
coordination corresponding to an astrobot: a target is
assigned to an astrobot whose initial ferrule coordination is
closest to the projection of the objective on the host focal
plane when compared to the distance of other astrobots.
Moreover, safety and completeness are addressed using the
maximum distribution requirement. A target has to be
assigned to the astrobot that is the farthest from it. This stipulation
produces the maximum dispersion of assigned astrobots
across a focal plane so that the robots' interactions, i.e.,
deadlocks and collisions, are minimized. One may note that
the optimality criterion formulation proposed in [11], based
on standard optimization challenges, makes the problemsolving
process very inefficient because both the cost function
and the class of constraints will be extremely nonlinear. Then,
it becomes unlikely that global optimal solutions may be efficiently
obtained.
Coordination Control
The configurations of optical fibers are unique with respect to
any observation because each observation consists of different
targets. Thus, a reconfiguration procedure is a must to move
fibers from one observation to another. The manual reconfiguration
of fibers was extensively taken into account during
the first generations of spectroscopic surveys, e.g., SDSS I-IV
[12]. In those projects, for each observation, a new plate had
to be assembled so that fibers pointed at the targets of an
observation. Then, the plate was located in the focal plane to
perform a planned observation. The cost of manufacturing
multiple plates and the substantial labor to manually transfer
astrobots from one plate to another were not negligible. The
SDSS-V project, however, uses fully automated focal planes
whose astrobots have to be coordinated. The coordination
must suffice for collision freeness and the complete convergence
requirements.
Collision Avoidance
Each coordination operation seeks a particular configuration
of astrobots. Thus, one expects a coordination process to stop
only when its desired configuration is reached. For this purpose,
distributed navigation functions (DNFs) [13] inherently
avoid collisions. In safety-critical applications, such as astrobotics,
one enjoys the intrinsic safety and fast convergence of
the control synthesized by this class of coordinators compared
to other strategies [14]. In particular, each astrobot is assigned
to a DNF as follows [15]:
} mm min >0,
2
() :=- +qq q
ii i
12 3
444444
Here, qi
12
attractive term
T
j Ni
!
12 3
repulsive term
44444444 44444444
/
qq
qq
ij
ij
--2
T
is the target position of astrobot i, and Ni
d
D
(2)
denotes
the set of the neighboring astrobots. The radius of the safety
envelope around each astrobot is D, and d < D represents the
96 * IEEE ROBOTICS & AUTOMATION MAGAZINE * SEPTEMBER 2021
2
2 H .
2
safety region's radius. A safe-control law can be immediately
derived from the function (2):
uqk
i :=- d} (),
i
(3)
where k is a position step parameter that determines the
intensity of each motion step.
The preceding control law, interpreted as a velocity profile,
successfully avoids collisions. However, it cannot resolve
many deadlock cases in which two or more astrobots block
one another's way, resulting in infinite oscillating motions. To
handle deadlocks, a priority coordination mechanism is taken
into account. This mechanism is implemented by a finitestate
machine (FSM) [16]. The resulting nonlinear hybrid
controller is constituted by two decision layers. A low-level
navigator, i.e., a DNF, first governs the coordination process
until it faces a problematic deadlock case. Then, a high-level
decision maker, i.e., an FSM, exploits the priority order of
astrobots and their targets to manage the deadlock. This strategy
improves the convergence rate of a typical astrobots
swarm by up to ~85%. Specifically,
this framework controls
each astrobot in a selfish manner; once an astrobot reaches its
target, it refuses any extra motion despite the potential of
occluding the paths of some of its neighbors and preventing
them from reaching their targets. Put differently, the aforementioned
formulation is not able to completely coordinate a
general astrobot swarm associated with a particular observation
in a guaranteed fashion. However, complete coordination
is particularly interesting, according to which the information
throughput of an observation is maximized.
Complete, Safe Coordination
A DNF is unable to completely coordinate a swarm of astrobots.
One may note that astrobot coordination may be better
done in a cooperative, rather than a competitive, manner.
With a competitive approach, an astrobot does not care about
the convergence of its counterparts. Thus, the convergence rate
may be dramatically reduced if many astrobots in a locality
need to cross an area corresponding to an already-converged
astrobot. Alternatively, a cooperative perspective requires
astrobots to seek not only their own convergences but the convergences
of their neighbors as well. So the overall completeness
of the swarm is taken into account. For this purpose, the
idea of cooperative artificial potential fields (CAPFs) [17] is
proposed. Each CAPF includes an extra cooperative-attractive
term, as follows:
z =- +
+ii
i
qq qT
12 3
444444attractive term
3
j Ni
!
12 3cooperative attractive term
44444444
The cooperative term injects extra dynamics into the
velocity profile of its corresponding astrobot. So, if an astrobot
is converged yet blocking some peers' paths, it may
() 12: mm min >0,
2
j Ni
!
m / qqT
i
j
2
.
12 3
repulsive term
4444444444444444
/
qq
qq
ij
ij
--2
d
D
(4)
2
2
H
2

IEEE Robotics & Automation Magazine - September 2021

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