IEEE Circuits and Systems Magazine - Q3 2020 - 54

uniform circular motion with the same direction and radius R = 1/|~| for ~ 0 ! 0 and K 1 0; (d) a uniform circular motion with the same radius R = 1/|~| and separate
directions for ~ 0 ! 0 and K 2 0. Several control laws
have been designed for achieving parallel and circular
formations [14], which is generalized by investigating
the collective motion with limited communication [15].
The time-invariant, undirected, and uniformly directed
communication topologies have been considered, respectively, which also allows for directed and timevarying communication [15]. Klein et al. [16] have used
the Kuramoto model in discrete time for multivehicle
coordination over broadcast networks. By using measurements of the relative position only, an observerbased distributed control algorithm, theoretically justified for the second-order vehicle model, is described
for the stabilization of parallel and circular motion [17].
Seyboth et al. [18] have extended the previous efforts
of [14], [15] by taking nonidentical cruising speeds
into consideration in collective circular motion, which
includes the case of common (or different) angular
frequency and different (or common) radius for each
agent. On the basis of [18], Sun et al. [19] have designed
novel controllers recently to achieve a desired formation shape instead of the circular motion stabilization
for a group of unicycle type vehicles with fixed cruising
and nonidentical speeds. Recently, the steering control
law has been designed as a directed Kuramoto-oscillator model with a pacemaker [20]:
	

N

u k (t) = ~ +

/ a kj

sin (i j - i k) + fk sin (i 0 - i k), (16)

j=1

5
4
3

where a kj 2 0 (k ! j ) if and only if the k-th agent receives the heading angle from the j-th agent; otherwise, a kj = 0. fk 2 0 is the strength of the k-th agent
that influenced by the pacemaker. For ~ = 0, all
agents almost globally converge to a uniform linear
motion. For ~ ! 0, all agents almost globally converge
to a circular motion with the same radius R = 1/|~| as
illustrated in Fig. 8.
3.4 Opinion Synchronization in Social Networks
In order to find the conditions under which a group
of agents with different natural tendencies (rates) to
change opinion can achieve an agreement, Pluchino
et al. [21], [149] propose the Opinion Changing Rate
(OCR) model by modifying the classical Kuramoto
model (1) into
	

xo i (t) = ~ i + K
N

(y )

	

0
-1
-2
-3
-4
-4

-2

0

(x )

2

4

6

Figure 8. Cooperative formation for kinematic model (12)
with the steering control law (16). The green circle and the
red curves denote the movement of the pacemaker and
those of the five controlled agents, respectively. The black
and blue arrows denote their initial velocity vectors and those
after a period of movements, respectively [20].
54 	

a |x j - x i|

, (17)

j=1

where x i (t) ! R represents the opinion of the ith individual at time t. ~ i is the natural opinion changing rate,
which can be randomly chosen from a given symmetric, unimodal distribution g (~) , such as Gaussian distribution centered at ~ 0 . Then, ~ i 1 ~ 0 corresponds
to the conservative people who change their opinions
very slowly; ~ i . ~ 0 corresponds to more flexible individuals who change their ideas more easily to follow
fashions; and ~ i 2 ~ 0 corresponds to individuals who
anticipate the others with new ideas and insights. The
parameter a tunes the effect of opinion difference for
reciprocal distance.
Due to the nonperiodicity of the opinion values xi in
OCR model (17), the order parameter (2) of the classical Kuramoto model (1) can not be used here to measure the degree of synchronization. An alternative order
parameter R(t) related to the standard deviation of the
opinion changing rate xo j (t) is defined as [21]:

2
1

N

/ a sin (x j - x i) e -

R (t) = 1 -

1
N

N

/ (xo j (t) - Xo (t)) 2 , (18)

j=1

where Xo (t) is the average of all individuals xo j (t). In this
way, when K & K c, fully synchronization emerges, i.e.,
R = 1, probably corresponding to the appearance of
a dictatorial regime in social network, or a globalized
world, in which social cultural differences disappears
and all opinions are enrolled into a single way of thinking [21]. When K 1 K c, opinions and their changing rates
are different, namely, all individuals are in the incoherent state, which can be interpreted as anarchy in a society. Only when K . K c, individuals arrive at partially
synchronized state, where bipolarism and democracy
are possible in this intermediate regime.

IEEE CIRCUITS AND SYSTEMS MAGAZINE 		

THIRD QUARTER 2020
IEEE Circuits and Systems Magazine - Q3 2020

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