IEEE Circuits and Systems Magazine - Q3 2019 - 9

Models for pattern formations can be separated
into two fashions (Fig. 1) based on different social
interactions: models for Line formations and models
for Cluster formations. In the following, some typical
models will be respectively described and discussed.
A. Line Formations
V-like formation is one of the most typical Line formations attracting eyes of investigators, we set it as an example to know the Line formations of birds.
In 2002, Seiler et  al. proposed a mathematical model [36], rather than a previous model based on observation
in which investigators used a radar technique measured
the V-like formations of Canada goose [37]. One year later,
they established a system interpretation for V-like formation of bird flocks [38]. To explain this particular formation of birds, there exist two predominant hypotheses in
their model:
1) In a V-like formation, birds will gain some aerodynamics [39].
2) In a V-like formation, visual communication (social interactions) among birds will be improved,
therefore-navigation capabilities of bird flocks
will be enhanced [6].
To model the V-like formation, only a pair of birds
were concerned in Ref. [38], just as Fig. 2(a) shows, then
one can deduce flight laws of other birds by the same
way. Fig. 2(b) displays an abstraction of Fig. 2(a). Suppose that each lateral of the V-like formation includes
N +1 birds, let x 0 (t ) be the position of the lead bird at
time t, while x i (t ) (i ! I ) denotes the ith follower bird's
position, note d as the lateral offset between the centers
of bird mass (see Fig. 2(b)). At time t, tracking error of
bird i is defined by

.

w i (0) = 0, x i (0) = 0, x i (0) = id.
By using the Laplace transform in control theory, the
individual bird model (Eq. (1) and Eq. (2)) is given by
X i (s ) = 12 H (s ) E i (s ) + id , i ! I,
s
s
where H (s) = C (sI - A)-1 B + D, and the space errors are
E 1 (s) = X 0 (s) - X 1 (s) + d
s
= X 0 (s) - L (s) E 1 (s) = S (s) X 0 (s),
E i (s) = X i -1 (s) - X i (s) + d = T (s) E i -1 (s),
s
L (s) = (1/s 2 ) H (s)
where * S (s) =1/ (1 + L (s))
T (s) = L (s)/(1 + L (s)) .
We are aware that the formation flights of birds or
tracking the positions of the preceding birds are inherently difficult. This model provides a relatively convenient way for researchers to study the birds' formations.
It used the systems theory to explain observations of
V-like formations. The simulation results of this model
indicate that even if there do not obtain more details of

Direction
of Travel

WTS
Depth
Wingspan, b

e i (t ) = (x i -1 (t ) + d ) - x i (t ), i ! I.
The goal of follower birds is to force their tracking
error to zero under the two hypotheses stated previously [38]. For simplicity, assuming that the generation
process of birds' force can be linearized, thus the dynamical system model of the lateral force can be described as

(a)
x

Direction
of Travel
y

z

b

.

w i = Aw i + Be i
i ! I,
Fi = Cw i + De i

(1)

where w i is the state of bird i (i ! I ), Fi is the lateral
force of bird i given by the Newton's second law:
..

mx i = Fi, i ! I.

(2)

And assume that the birds start at the correct position [7], that is, the initial conditions for all birds are
THIRD QUARTER 2019

0

Delta

x0
x1

1
1

(b)

Figure 2. Illustration of the V-like formation model [38]. (a) V
formation notation, (b) V formation coordinates

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IEEE Circuits and Systems Magazine - Q3 2019

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