IEEE Circuits and Systems Magazine - Q3 2019 - 11

respectively [47]. The possible "blind volume" is behind
an agent with the angle of ( 360 - a)°, as shown in Fig. 4,
where a denotes the field of perception. The collective
behavior of the flock is analyzed by changing system parameters, including the number of agents, turning rate,
the width of the zones, etc [47]. For all cases, the following two properties are calculated [47]:
p group (t ) = 1
N
m group (t ) = 1
N

N

/ v i (t ) ,

i ! I,

(3)

i =1
N

/ ric (t ) # v i (t ) ,

i ! I,

(4)

i =1

where
ric (t ) = x i (t ) - x group (t ),

have emerged substantially. For example, in 1998, Mogilner and Edelstein-Keshet proposed a continuum model
for swarming behavior based on non-local interactions
among agents [50]. In 2010, Eriksson et al. introduced a
flock swarms model determined by local interactions in
animals flock [51]. In 2004, Topaz and Bertozzi studied
the flock swarms in two dimensional space [52]. In 2009,
Yates et al. added noise to the swarm model, and found
that the inherent noise in system can promote swarm
motion of flocks [53]. In 2010, Li and Xiao exhibited a new
swarm model in the homogeneous environments  [54].
Researchers also have observed flock swarms in various perspectives, such as mathematical [55] and kinetic
theory [56]. Very recently, flock swarm has been applied
to humans intelligence [57], microtubules field [58] and
robotics [59].

and
x group (t ) = 1
N

N

/ x i (t ).

i =1

p group is equivalent with the order parameter which
measures the agent alignment degree and it is determined by the absolute value of the average normalized
velocity [48]. While m group measures the rotation degree
of the group relative to its center, i.e., angular momentum. When setting different parameters, various simulation results will emerge, as summarized by Fig. 5. The
four sharp phase transitions of agents' collective behaviors are labeled as follows [47]:
1) Swarm. This case occurs when both the level of
parallel alignment and angular momentum are
low, which describes little or no parallel orientation (Fig. 5(a)).
2) Torus. This case occurs when the level of parallel alignment is low, but angular momentum is
high, which denotes that agents' rotations are
around an empty core with random directions
(Fig. 5(b)).
3) Dynamic parallel group. This case arises when
the level of parallel alignment is high, but angular
momentum is low, which is more mobile than the
two cases mentioned above (Fig. 5(c)).
4) Highly parallel group. This case arises when the
level of parallel alignment is high while angular
momentum is low, at the same time, the relative
length of zoo and zor increases. In this case, the
group self-organized into a highly aligned arrangement (Fig. 5(d)).
Swarm of bird flocks was first introduced by Okubo
in 1980, in his classical book [49], he put forward some
mathematical concepts which can be applied to studying flock swarms. From then on, literatures about swarm
THIRD QUARTER 2019

zoa
zoo

360-α
zor

Figure 4. Illustration of representation of an agent in the
Zonal interaction model.

(a)

(b)

(c)

(d)

Figure 5. The collective behaviors exhibited by the Zonal
interaction model. (a) Swarm, (b) torus, (c) dynamic parallel
group, (d) highly parallel group.

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IEEE Circuits and Systems Magazine - Q3 2019

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