1. business and industrial
2. energy
3. nuclear power

1
Self-organized fission control for flocking system
Mingyong Liu, Panpan Yang, Xiaokang Lei, Yang Li
Abstract—This paper studies the self-organized fission
control problem for flocking system. Motivated by the
fission behavior of biological flocks, information coupling
degree (ICD) is firstly designed to represent the interaction
intensity between individuals. Then, from the information
transfer perspective, an “maximum-ICD” based pairwise
interaction rule is proposed to realize the directional
information propagation within the flock. Together with
the “separation/alignment/cohesion” rules, a self-organized
fission control algorithm is constructed that achieves the
spontaneous splitting of flocking system under conflict external stimuli. Finally, numerical simulations are provided
to demonstrate the effectiveness of the proposed algorithm.
Index Terms—Flocking system; fission control; information coupling degree; self-organization; pairwise interaction
I. I NTRODUCTION
The fission behavior of flocking system is a
widely observed phenomenon in biology, society
and engineering applications [1]. A bird flock may
sometimes split into multiple clusters for food foraging or predator escaping [2]. An unmanned ground
vehicle (UGV) swarm often needs to segregate
into small sub-groups for the multi-site surveillance
mission [3]. In these cases, members in the flock
are often identical ones with limited sensory ability
and only rely on the local interaction with their
neighbors. Therefore, the fission phenomenon of
flocking system is virtually an emergent behavior
that arises in a self-organized fashion [4] [5]. How
a cohesive flock splits and forms clusters remain
a fascinating issue of both theoretical and practical
interests.
Currently, the research on flocking system mainly
focuses on the consensus based problems such as
aggregation and formation [6][7], of which the central rules are separation, alignment and cohesion [8].
Mingyong Liu, Panpan Yang, Xiaokang Lei and Yang Li are
with the School of Marine Science and Technology, Northwestern
Polytechnical University, Xi’an, Shannxi, 710072, China.
Corresponding Author: Panpan Yang (E-mail address: [email protected]).
These rules have been extensively used in the distributed sensing of mobile sensor networks, formation keeping of satellite clusters, cooperative control
of unmanned ground/aerial/underwater vehicles and
etc [9]. On the other hand, the “average consensus”
property of these rules gives the flock a “collective
mind” [10] and leads to a group-level ability of
“consensus decision-making” [11][12], which may
dispel the conflict information and encumber the
process of group splitting [13].
At present, literatures that address the fission
control problem seem diverse. By predefining the
leaders/targets to different individuals, fission behavior emerges in the multi-objective tracking issue
[14][15][16]; in [17], Kumar et al. assign different
coupling strength to heterogeneous robot swarm that
leads weak coupling robots to separate and the
strong coupling robots form clusters; In addition, a
long range attractive, short range repulsive interaction as well as an intermediate range Gauss-shaped
interaction is employed for flock aggregation and
splitting in [18].
In this paper, we tend to study the self-organized
fission control problem for flocking system without
predefining the leaders or identifying the differences
between individuals. Motivated by the fact that
interaction intensity plays a key role in the fission
behavior of animal flocks [2][11][19], information
coupling degree (ICD) is used as an index to denote
the interaction intensity between individuals. Then,
a “maximum-ICD” based pairwise interaction rule
is proposed to achieve the effective information
transfer within the flock. Together with the traditional “separation/alignment/cohesion” rules, a selforganized fission control algorithm is formulated,
which realizes the spontaneous splitting of flocking
system under conflict external stimuli. Finally, numerical simulations are performed to illustrate the
effectiveness of the proposed method.
The remainder of this paper is organized as
follows: in Section 2, the fission control problem
for flocking system is formulated; in Section 3,
the biological principle for fission behavior is de-
2
scribed and an information coupling degree based
fission strategy is provided; in Section 4, a selforganized fission control algorithm that combines
the pairwise interaction rule is proposed and the
theoretical analysis is given; in section 5, various
numerical simulations are carried out to demonstrate
the effectiveness of the fission control algorithm;
Section 6 offers the concluding remarks.
II. P ROBLEM FORMULATION
Consider a flocking system consisting of N identical individuals moving in the n-dimensional Euclidean space with the following dynamics
½
p˙i = qi
(1)
q˙i = ui , i = 1, 2, · · · , N
where pi ∈ Rn is the position vector of individual
i, qi ∈ Rn is its velocity vector and ui ∈ Rn is the
acceleration vector (control input) acting on it. For
notational convenience, we let
 
 
p1
q1
 p2 
 q2 
Nn

 
p=
(2)
 ...  , q =  ...  ∈ R
pN
qN
The neighboring set of individual i is defined as
Ni (t) = {j : kpi − pj k ≤ R, j = 1, · · · , N, j 6= i},
where k · k is the Euclidean norm and R is the
sensory range of each individual. Also, we define
the neighboring graph G = (V, E, A) [20] to be
an undirected graph consisting of a set of nodes
V = {v1 , v2 , · · · , vN }, a set of edges E = {(vi , vj ) ∈
V × V : vi ∼ vj } and an adjacency matrix A = [aij ]
with aij > 0 if (vi , vj ) ∈ E and aij = 0, otherwise.
The adjacency matrix A of G is symmetric and the
corresponding Laplacian matrix is L = D − A,
N ×N
where D = diag{d1 , d2 , · · · , dN } ∈ RP
is the
in-degree matrix of graph G and di = N
j=1 aij is
the in-degree of node vi .
Essentially speaking, flocking behavior is a selforganized emergent phenomenon of large numbers
of individuals by interacting with its neighbors and
surrounding environment [21]. Correspondingly, the
control input acting on an individual member can be
written as
out
ui = uin
(3)
i + gi ui
where uin
i is the internal interaction force between
individual i and its neighbors, uout
i is the force acting
on individuals from external environment. Usually,
not all the members are directly influenced by the
environment, here we utilize gi to denote whether
individual i can sense environment information,
where gi = 1 means the environment information
can be directly obtained by individual i and gi = 0,
otherwise.
Fission behavior often occurs when a cohesive
flock encounters obstacles/danger on their moving
path or observes multiple targets for tracking [2][15]
[22][23]. In such occasions, only a small portion
of individuals (lie on the edge of the flock) are
directly influenced by the environment information and often response with fast maneuvering like
abrupt accelerating or turning [2][22], the motion
of other members are only governed by their local interaction force uin
i . Therefore, environment
information can only be seen as the trigger event
of fission behavior, whether the whole flock has
the ability to split is essentially determined by the
internal interaction force uin
i [5].
The objective of this paper is to design the
distributed local interaction rules and synthesize the
control input ui for a flocking system such that
when conflict environment information acts on part
of individuals in the flock, it can segregate into
clustered sub-groups spontaneously.
To better describe the fission behavior in a quantitative way, we first give the mathematical definition
of fission behavior as follows:
Definition 1: A flocking system is said to be
segregated (or a fission behavior occurs) if and only
if it satisfies the following conditions:
(1) The distance between individuals in the same
sub-group remain bounded, i.e., kpi (t) − pj (t)k ≤
δ, i, j ∈ Gk , where δ is a constant value and Gk
denotes the sub-group k.
(2) For individuals in the same sub-group, their
velocities will asymptotically converge to the same
value, i.e., kqi (t) − qj (t)k → 0, i, j ∈ Gk .
(3) For any distance D > R, there exists a
time after which the distance between individuals in
different sub-groups is at least D, i.e., min kpi (t) −
pj (t)k ≥ D, i ∈ Gk , j ∈ Gl , which means that
the sub-groups will ultimately lose connection with
each other and the fission behavior emerges.
It’s worth mentioning that the fission behavior
studied in this paper is a spontaneous response to
external stimuli, during which only a small portion
of members can directly sense the external stimuli
A new paragraph is added to highlight the main
difference of our work from existing literatures.
3
and the whole flock governed by the fission control
algorithm will segregate autonomously in a selforganized fashion. Therefore, there is an essential
difference from the aforementioned fission control
approaches like assignment, identification or centralized control [14][15][16] [17].
III. I NFORMATION COUPLING
FISSION RULE
Some
facts are
added to
support
our
method.
DEGREE BASED
individuals. As individuals are usually sensitive to
some specific behavior of their neighbors like abrupt
accelerating or turning [19], we design ωij as
k(qi − qj ) × q¯i k
¯i k
j∈Ni (t) k(qi − qj ) × q
ωij = µij P
(6)
P
where q¯i = N1i j∈Ni (t) qj is the average velocity of
the neighbors of individual i, Ni is the number of
its neighbors, µij > 0 is the coefficient of velocity
coupling term. Here, k(qi − qj ) × q¯i k reflects the
degree of the difference between (qi − qj ) and q¯i ,
with (qi − qj ) being the relative velocity between
two individuals. The bigger k(qi − qj ) × q¯i k is, the
motion of individual j is more different from the
neighbors of individual i, the more attention will be
paid by individual i to individual j and they will
tend to have a tighter correlation, correspondingly.
From the information transfer perspective, fission
behavior can be generalized as the conflict stimulus information propagation process among members within the flock [2][23][29]. Individuals which
change their direction of travel in response to the direction taken by their nearest neighbors can quickly
transfer information about the predator or food
source [29]. Therefore, we propose a “maximumICD” based strategy to maximize the stimulus information transfer, where individuals tend to have
closer relation with the neighbor that has maximum
ICD with it [19][30].
Based on the above description, we formulate the
“maximum-ICD” strategy as
Fission behavior is the result of “collective decision making” in the presence of motion differences
(conflict information) of individuals in the flock
[2][24]. Couzin et al. revealed that for significant
differences in the preferences of individuals, the decision dynamics may bifurcate away from consensus
and lead to the emergence of fission behavior [11].
In addition, research from biologists also suggests
that fission behavior, to a large extent, depends on
the mutual interaction intensity between individuals
[1]. Individuals with larger interaction intensity tend
to have tighter correlation and form clusters in the
presence of significant differences in the preferences
of individuals [2][11].
Accordingly, we construct a new index named
information coupling degree (ICD) to denote the
mutual interaction intensity between individuals.
ICD is a motion dependent variable that is relevant
to many factors, e.g., individuals are usually more
influenced by the close neighbors (distance) [25],
they tend to be more sensitive to fast moving neighbors (velocity) [19][26] and individual with more
neighbors are usually more dominated (number of
neighbors) [27]. In particular, we choose the two
fi = {j| max cij , cij > c∗ , j ∈ Ni (t)}
(7)
most dominating factors, the relative position and
∗
value of the fission berelative velocity between individuals, to design ICD where c is the threshold
∗
havior.
When
c
>
c
,
fission
behavior occurs,
ij
in the following form
otherwise, it is not disturbed by conflict environment
cij = ξij · ωij
(4) stimuli and moves in stable formation. In this paper,
∗
where ξij is the position coupling term determined we choose c to be a appropriate value to prevent
by the relative position between individuals. As the the unexpected fission behavior due to random flucinfluence of each individual is decreasing with the tuation or other unknown factors.
Utilizing the “maximum-ICD” strategy, we proincrease of their relative distance due to the sensory
ability [28], we write the position coupling term as pose a pairwise interaction rule to realize the directional information flow among individuals. The infollows
rij
(5) ternal interaction mechanisms of individuals during
ξij =
1 + kpi − pj k2
the fission process is illustrated in Fig. 1. Assume
t−∆t
where rij > 0 is the coefficient of position coupling that at time t − ∆t, individual fi (locates at pfi
with velocity qft−∆t
) is propelled by external stimuterm.
i
In addition, ωij is the velocity coupling term lus and changes its motion rapidly to a new position
that is relevant to the relative velocity between ptfi with velocity qft i in a small time interval ∆t.
4
t
i
u
p
t
u tf'
i
p
R
t
fi
, q tfi
t 't
fi
t
, q tf'
i
¦u
t 't
i
i
Reply 2:
Give a
specific
form of
external
stimuli
movement change of individuals and evokes the
fission behavior, gi denotes whether individual i can
sense external stimuli. Usually, fie has diverse forms
according to different environment stimuli, such as
the direction to a known source or a segment of a
migration route [11][31]. Here, for the convenience
of analysis, we design the external stimulus term as
the following simple position feedback form
fie = −(pi − pei )
(10)
where pei is the desired position driven by external
Fig. 1. Internal interaction mechanisms of individuals based on stimuli and it is supposed to be a constant value at
every sampling time interval.
“maximum-ICD” strategy Reply 3: The desired position is
supposed to be a constant value
Obviously, the internal interaction force uin
i conat every sampling time interval.
4: 1) Explain the reason why
sists of three components: Reply
employ the attraction/repulsion force
According to (4), individual i is more influenced by
(1) fip is used to regulate the position between
fi and tends to have a relatively tighter correlation
individual i and its neighbors. This term is responwith it. Therefore, at time t the force of neighbors
sible for collision avoidance and cohesion in the
acting on individual i can be written as
flock, that is, when individuals are too far away
N
i −1
X
from each other, the attraction force will make them
ui (t) = ut−∆t
+
uit−∆t
(8) move together; when individuals are too near to
fi
i=1
each other, the repulsion force will propel them
where ut−∆t
is
the
pairwise
interaction
force
of
the
away to avoid collision. It is derived from the field
fi
most correlated
neighbor
f
acting
on
individual
produced by a collective function which depends
i
PNi −1 t−∆t
denotes the sum of the forces on the relative distance between individual i and its
i and i=1 ui
of other neighbors acting on it, with Ni being the neighbors, and is defined as
number of neighbors.
X
¡
¢
fip = −
∇pi ψij kpij − pfij kσ
(11)
Remarks 1: It can be seen from (4)∼(6) that information coupling degree combines both the position
j∈Ni (t)
and velocity information between individuals and
the σ-norm k · kσ
their neighbors, meanwhile they also tend to have where ∇ is the gradient operator,
n
of
a
vector
is
a
map
R
→
R+ defined by
p
closer relation with the fast maneuvering neighbors
2
a parameter
that are near to them, which is consistent with the kzkσ = (1/²)[ 1 + ²kzk − 1] with
T
² > 0 and z = [z1 , z2 , · · · , zn ] ∈ Rn . Note
property of real flocking system [19].
that thep
map kzkσ is differentiable everywhere, but
kzk = z12 + z22 + · · · + zn2 is not differentiable at
IV. S ELF - ORGANIZED FISSION C ONTROL
z = 0. This property of σ-norm is used for the
A LGORITHM
Based on the above fission control principle, we construction of smooth collective potential functions
take the most correlated neighbor to form a pairwise for individuals. To construct a smooth pairwise
interaction with individual i. By integrating the potential with finite cut-off, we follow the work of
motion information of the most correlated neighbor [7] and integrate an action function φα (z) that varies
into the coordinated law and together with the “sep- for all z ≥ rα . Define this action function as
aration/alignment/repulsion” rules, the distributed
(12)
φα (z) = ρh (z/rα )φ(z − dα )
fission control algorithm is formulated as
1
φ(z) =
[(a + b)σ1 (z + c) + (a − b)] (13)
ui = fip + fiv + fif + gi fie
(9)
2
|
{z
} |{z}
√
uout
uin
i
i
where σ1 (z) = z/ 1 + z 2 and φ(z) is an uneven
is represented by sigmoidal function with
where the environment force uout
i
√ parameters that satisfy 0 <
the external stimulus term fie , it causes the rapid a ≤ b, c = |a − b|/ 4ab to guarantee φ(0) = 0.
Reply 1: We integrate the external
stimuli force into the fission control
algorithm, which is a universal control
law for all the individuals
Reply 4: 2) Explain the reason why use such $\sigma$ norm
and $\psi_{ij}$.
Give the detailed process of constructing a smooth potential
function.
5
Reply 4:
2)
Construct
the
smooth
adjacency
matrix
The pairwise attractive/repulsive potential ψα (z) is
Remarks 3: Note that in our fission control algodefined as
rithm (9), the most correlated neighbor of individual
Z z
ψα (z) =
φα (s)ds
(14) i is chosen according to (7), and it is dynamically
updating during the fission process of the flock,
dα
which realizes the implicit stimulus information
Additionally, pij = pi − pj is the relative position transfer among members in the flock. Thus, there
vector between individual i and j, pfij = pfi − pfj is an essential difference between our work and the
is the relative position vector between the most research of [14] [15] and [16], as the latter assume
the leader (virtual leader) or target of each member
correlated neighbors of individual i and j.
v
(2) fi is the velocity related term that regulates is predefined and fixed during the fission process.
Remarks 4: From (7) we can also see that if cij <
the velocity of individuals
X
c∗ , the most correlated neighbor of individual i does
aij (qij − qfij )
(15) not exist. In this case, the fission control algorithm
fiv = −
j∈Ni (t)
(9) degrades into
X
where qij = qi − qj is the relative velocity vector
ui = −
∇pi ψij (kpi − pj kσ )
between individual i and j, qfij = qfi − qfj is the
j∈Ni (t)
X
relative velocity vector between the most correlated
−
aij (qi − qj ) + gi fie
(19)
neighbors of individual i and j. Moreover, A(t) =
j∈N
(t)
i
[aij (t)] is the adjacency matrix which is defined as
½
which is equivalent to the first flocking control law
0,
if j = i
aij (t) =
(16) proposed in [7]. This situation occurs when the difρh (kpj − pi kσ /krkσ ), if j 6= i
ferences of motion preference between individuals
are relatively small and hence the fission behavior
with the bump function ρh (z), h ∈ (0, 1), being
dose not occur. Therefore, algorithm (19) in [7]

z ∈ [0, h)
 1,
can be seen as a special case of ours. However,
1
z−h
[1
+
cos(π
)],
z
∈
[h,
1]
ρh (z) =
(17)
algorithm (19) is not able to achieve the spontaneous
1−h
 2
0,
otherwise
fission behavior due to its average consensus property. A detailed comparative simulation and analysis
(3) fif is the pairwise interaction term that pro- is demonstrated in part 5.
duces a closer relation between individual i and its
Here, we define the sum of the total artificial
most correlated neighbor fi , which is used to guide potential energy and the total relative kinetic energy
the motion preference of individual i. Specifically, between all individuals and the external stimuli as
we employ an attractive force and velocity feedback follows:
approach to generate fif as
¶
N µ
1X
T
Ui (p) + (qi − qfi ) (qi − qfi ) (20)
fif = −k1 ∇pfi Ufi − k2 (qi − qfi )
(18) Q(p, q) = 2
i=1
Reply 5: Energy function is defined before Theorem 1
Reply 6: 1) Redefined the energy function according to
the newly designed control input
where k1 , k2 > 0 are the constant feedback gains, where
Ufi = 21 (pi − pfi )2 is the potential energy of its
Ui (p) = Vi (p) + k1 (pi − pfi )T (pi − pfi )
most correlated neighbor. This term guarantees the
X
¡
¢
velocity convergence of individual i to its most
=
ψij kpij − pfij kσ
correlated neighbor fi and ultimately induces the
j∈Ni (t)
fission behavior. Reply 4: Address the reason why design
+k1 (pi − pfi )T (pi − pfi )
such potential function and adjacency
matrix
(21)
+gi (pi − pei )T (pi − pei )
Remarks 2: The specifically designed pairwise attractive/repulsive artificial potential ψα and adjacent
matrix aij here are to guarantee the smoothness of
the potential function and Laplacian matrix, which
will facilitate the theoretical analysis with the traditional Lyapunov-based stability analysis method.
Obviously, Q is a positive semi-definite function.
We have the following result:
Theorem 1: Consider a flocking system consisting
of N individuals with dynamics (1). Suppose the
6
Prove the
stability
according
to the
newly
designed
fission
control
algorithm
and the
energy
function
initial energy Q0 (Q0 = Q(p(0), q(0))) of the flock
Due to the symmetry of the artificial potential
is finite and the initial graph G is connected before function ψij and adjacency matrix A, we have
the fission process, when conflict external stimuli
∂ψij (k˜
pij kσ )
∂ψij (k˜
pij kσ )
∂ψij (k˜
pij kσ )
cause the rapid maneuvering of individuals that lie
=
=−
(26)
∂ p˜ij
∂ p˜i
∂ p˜j
on the edge of the flock, under the fission control
law (9), a cohesive flocking system will segregate
Take the derivative of Q, we can obtain
Ã
into clustered sub-groups due to the differences of
Ni
X
X
information coupling degree between individuals. Q˙ = 1
ψ˙ ij (k˜
pi − p˜j kσ )
2 i=1
Then, the following statements hold:
j∈Ni (t)
!
(i) The distance between individuals in the same
sub-group
is not larger than δ, where δ = (Ni −
p
+2k1 q˜iT p˜i + 2˜
qiT ui + 2gi (p˜˙ei )T p˜ei
1) 2Q0 /k1 and Ni is the number of individuals in
Ã
the sub-group.
Ni
X
X
1
(ii) The velocities of individuals in the same
=
p˜˙Ti ∇p˜i ψij (k˜
pi − p˜j kσ )
2
sub-group will asymptotically converge to the same
i=1
j∈Ni (t)
X
value.
−
p˜˙Tj ∇p˜j ψij (k˜
pi − p˜j kσ ) + 2k1 q˜iT p˜i
(iii) The distance between different sub-groups is
j∈Ni (t)
larger than R as time goes by.
µ X
Proof 1: We take a cluster of Ni individuals in the
T
∇p˜i ψij (k˜
pi − p˜j kσ )
+2˜
qi −
neighborhood of individual i into consideration and
j∈Ni
¶
assume that graph GNi is connected in the sampling
X
e
time interval from t to t1 .
−
aij (˜
qi − q˜j ) − k1 p˜i − k2 q˜i − gi p˜i
(i) Let p˜ei = pi − pei be the position difference
j∈Ni (t)
!
between individual i and the external stimuli, p˜i =
pi −pfi , q˜i = qi −qfi be the position difference vector
+2gi q˜iT p˜ei
and velocity difference vector between individual
Ã
!
Ni
i and its most correlated neighbor fi , respectively.
X
X
Then the following equations hold
=
q˜iT −
aij (˜
qi − q˜j ) − k2 q˜i
pij − pfij = p˜i − p˜j = p˜ij
(22)
qij − qfij = q˜i − q˜j = q˜ij
(23)
i=1
T
j∈Ni (t)
= −˜
q [(L(˜
p) + k2 INi ) ⊗ In ] q˜
(27)
where L(˜
p) is the Laplacian matrix of graph
G(˜
p), p˜ = col(˜
p1 , p˜2 , · · · , p˜Ni ) ∈ RNi ×n , q˜ =
Accordingly, the control input ui can be rewritten
col(˜
q1 , q˜2 , · · · , q˜Ni ) ∈ RNi ×n , INi is the Ni dimenas
sional identity matrix.
X
As L(˜
p) is positive semi-definite, Q˙ < 0, hence
ui = −
∇p˜i ψij (k˜
pi − p˜j kσ )
Q is a non-increasing function. As the initial energy
j∈Ni (t)
X
Q0 of the sub-group is finite, for any time interval
−
aij (˜
qi − q˜j )
t ∼ t1 , Q < Q0 . Form (25), we have k1 p˜Ti p˜i ≤
j∈Ni (t)
2Q0 . Accordingly, the distance between individual
−k1 p˜i − k2 q˜i − gi p˜ei
(24) i and its most correlated neighbor is not greater than
p
2Q0 /k1 . Due to the connectivity of the sub-group,
there exists a joint connected path from individual i
Substituting (22) and (23) into (20) yields
to Ni in the small time internal t ∼ t1 , the distance
Ni µ X
X
between
any two individuals is less than δ = (Ni −
1
p
ψij (k˜
pij kσ ) + q˜iT q˜i
Q =
2Q
/k
1)
0
1.
2 i=1
j∈Ni (t)
(ii) From above, Q < Q0 , the set {˜
p, q˜} is closed
¶
and bounded. Therefore, the set
T
e T e
+k1 p˜i p˜i + gi (˜
pi ) p˜i
(25)
o
n£
¤
(28)
Ω = p˜T , q˜T ∈ R2Ni ×n |Q ≤ Q0
7
is a compact invariant set. According to the
LaSalle’s invariant set principle, all the trajectories
of individuals that start from Ω will converge to
the largest invariant set inside the region Ω =
{[˜
pT , q˜T ] ∈ R2Ni ×n |Q˙ = 0}. Therefore, the velocity
of individual i will converge to that of its most
correlated neighbor fi , i.e.,
A. Simulation Setup
Suppose the 40 individuals lie randomly within
the region of 20 × 20m2 and their initial distribution
is connected. The initial velocities are set with
arbitrary directions and magnitudes within the range
of [0 10]m/s. The sensory range of each individual
is constrained to R = 5m. The other simulation
parameters are chosen as: rij = 0.5, µij = 5,
qi = qf i
(29) c∗ = 0.3, ∆t = 0.005s, ² = 0.2, k1 = k2 = 1.
In particular, we consider the case where a flock
Without loss of generality, we assume the Ni th segregates into two clustered sub-groups by tracking
individual in the sub-group is the only one that two moving targets towards different directions.
senses the external stimulus and it responses with Before the fission behavior takes place, we first let
fast maneuvering. According to (4), Ni is the most individuals aggregate and move in formation with
correlated neighbor with largest information cou- the same velocity of [5 0]T m/s. Suppose that at
pling degree. In the small time internal t ∼ t1 , there t = 7s, two individuals (e.g., lie on the edge of
exists a joint connected path from individual i to the flock. Without loss of generality, we let them
Ni , the velocities of all the individuals in the sub- be individual 1 and individual 2) sense two targets
group will converge to that of the Ni th individual moving towards different directions, other members
asymptotically. Hence, we have
are not able to sense the external stimuli and hence
we have g1 = g2 = 1, gi = 0, i 6= 1, 2. Meanwhile,
q1 = q2 = · · · = qNi −1 = qNi
(30) the desired positions of individual 1 and individual
2 are updating according to (1) with the updating
where qNi is the velocity of individual Ni that senses speed q e = [5 3]T m/s and q e = [5 − 5]T m/s,
2
1
the external stimuli.
respectively.
(iii) For individuals i and j in different subgroups, as has been illustrated in (i) and (ii), their
motion is substantially determined by their most B. Simulation Results
Steered by the fission control algorithm (9), a
correlated neighbors. Therefore, if the most correlated neighbors of individual i and j move in self-organized fission behavior will occur and the
opposite directions under different external stimuli, simulation results are shown in Fig. 2∼5.
Fig. 2 (a) is the snapshot of the flocking systhe motion of the two individuals will diverge as
tem
at t = 6s, from which it can be seen that
time goes by. Hence, we will ultimately have
individuals aggregate from random distribution and
lim D > R
(31) form a cohesive formation; Fig. 2 (b) shows the
t→∞
response of flock when individual 1 and individual
where D = kpi −pj k is the relative distance between 2 (denoted by red solid dots) sense the moving
targets, individuals in the flock tend to change
individual i and j.
their movements to different directions due to the
From the above proof, we can conclude that under
pairwise interaction rule; In Fig. 2 (c), the flock
the fission control algorithm (9), the sub-groups are
segregates into sub-groups and two clusters emerge;
gradually moving out of the sensory range of each
In Fig. 2 (d), the sub-groups run out of the sensory
other. Meanwhile, the sub-group itself will keep a
range of each other and the segregated sub-groups
cohesive whole and move in formation separately.
move in formation separately.
Fig. 3 (a)∼(b) give the plot of the velocities of
individuals
during the simulation in both X and Y
V. S IMULATION S TUDY
axis. From which we can see that before two moving
To verify the effectiveness of the proposed fis- targets appear (t < 7s), individuals move in forsion control algorithm, we choose 40 individuals to mation and their velocities asymptotically converge
perform the simulation in two dimensional plane.
to [5 0]T m/s; At t = 7s, under the fission control
8
30
30
25
20
20
y/m
y/m
15
10
10
5
0
0
−5
−10
−10
10
20
30
40
50
60
10
20
30
40
x/m
(a) t=6s
60
70
(b) t=9s
40
40
30
30
20
20
y/m
y/m
50
x/m
10
10
0
0
−10
−10
−20
−20
10
20
30
40
50
60
x/m
70
80
20
40
60
80
x/m
(d) t=15s
(c) t=12s
Fig. 2. Snapshots of the trajectories of individuals during the fission process, where ‘◦’ represents the initial position of individuals, ‘•’
denotes the finally position of each individual. Specifically, ‘•’ are the two individuals that sense external stimuli with ‘→’ being their
desired directions.
vx/m ⋅ s−1
30
20
10
0
0
5
10
time/s
(a) Velocity in X axis
15
20
vy/m ⋅ s−1
10
0
−10
−20
algorithm, fission behavior occurs and the velocities
of the two sub-groups tend to that of the individuals
who sense the moving targets. Consequently, the
velocities of individuals in sub-group 1 converge
to [5 3]T m/s while the velocities of individuals in
sub-group 2 converge to [5 − 5]T m/s.
In addition, we utilize the average distance of
individuals to illustrate the fission behavior in a
more intuitional way. The average distance davg is
represented by
davg =
0
5
10
time/s
(b) Velocity in Y axis
Fig. 3. Velocity of individuals during the fission process
15
Ni X
Ni
1 X
kpi − pj k
Ni i=1 j=1
(32)
where Ni is the number of individuals in the flock.
Fig. 4 shows the average distance between individuals during the simulation, where blue line
denotes the average distance of all the individuals
while red dash line and black dash dot line represent
the average distance of individuals in sub-group 1
9
0.8
0.7
0.6
1
0.5
ICD
and sub-group 2, respectively. We can easily see
that the three lines tend to converge to a fixed
value before segregation (t < 7s). When the fission
behavior occurs, the average distance of all the individuals increases with time, denoting the divergence
of the two sub-groups. On the contrary, the average
distances of the two sub-groups tend to maintain a
constant value (about 3m), which demonstrates that
the two sub-groups are moving in stable formation.
0.4
0.3
0.2
0.1
0
35
Average Distance of all individuals
Average Distance of individuals in Sub−group 1
Average Distance of individuals in Sub−group 2
30
0
5
10
15
t/s
12
25 10
Fig. 5. Information coupling degree between individual 1 and its
neighbors
davg/m
8
20
6
fission behavior of animal flocks, which is also
more feasible, robust and efficient in engineering
2
7
7.5
8
8.5
9
10 6.5
application than the centralized approach.
To better understand the size distribution of sub5
groups, we define the size ratio S as a specification
0
to evaluate the fission behavior
0
5
10
15
t/s
Nmin
(33)
S=
Nmax
Fig. 4. Average distance of the whole group and two separated subgroups
where Nmin and Nmax are the size of the smallest sub-group and biggest sub-group, respectively.
Fig. 5 gives the information coupling degree Obviously, S ∈ [0, 1], the bigger S is, the size
between individual 1 and its neighbors during the difference between the two sub-groups is smaller.
fission process, which clearly demonstrates that be- Particularly, when S = 1, the numbers of individufore the two moving targets appear (t < 7s), the ICD als in the two sub-groups are the same.
between individual 1 and its neighbors converge to a
constant value. This is because in the steady forma90
tion state, the relative position and velocity between
80
individual 1 and its neighbors remain stable. With
70
the introduction of the two moving targets, the ICD
between individual 1 and its neighbors begins to
60
vary due to their rapid movement variation, and the
50
ICD between individual 1 and its neighbors in the
40
same sub-group converge to a steady state, while
30
ICD of individual 1 and other individuals in the
20
other sub-group quickly decreases to 0 for the loss
10
of connection between each other.
0
From the above simulation results (Fig. 2 ∼ Fig.
0
0.2
0.4
0.6
0.8
1
S
5) we can conclude that the proposed fission control
algorithm is capable of realizing the self-organized
Fig. 6. Histogram statistics for the size ratio of the separated subfission behavior when the introduction of the mov- groups
ing targets cause the motion information conflicts
Fig. 6 is the histogram of the size ratio of 40
in the flock. Particularly, our approach is neither
negotiation nor appointment based but mimics the individuals for 100 times of simulation. From which
15
4
Further explain the shortcomings
of the traditional "average
consensus" based control rules
in the fission behavior
10
we can see that about 80% of the results show the
size ratio equal to 1 and nearly 10% are close to
1 , only a very small portion of the size ratio are
randomly located from 0 to 1. Therefore, it can
be concluded that, from a probabilistic perspective,
the algorithm proposed in this paper can realize the
equal-sized fission control.
Finally, we carry out a comparative simulation
using the first algorithm (19) proposed in [7] based
on “separation/alignment/repulsion” rules. At t =
7s, we introduce two moving targets the same as
the above simulation and the flocking system are
then following the algorithm
formation state. The simulation result demonstrate
that the flocking algorithm (34) decrease the flexibility and maneuverability of flocking system especially in the case of danger/obstacle avoidance
or multiple objects tracking. As a matter of fact,
under algorithm (34), fission behavior only occurs
when all the individuals can directly sense external
stimuli, which is essentially different from our work.
The main reason for the above result lies in
the “average consensus” approach adopted in (34),
which counteracts the external stimuli and leads the
flock move towards the average direction of the
stimulus signal. Therefore, the traditional “separation/alignment/cohesion” based coordination rules
X
X
for flocking system may dispel the conflict inforui =
φα (kpj − pi kσ )nij +
aij (p)(qj − qi )
mation and encumber the fission behavior, which, as
j∈Ni
j∈Ni
a result, decreases the maneuverability of flocking
(34) system.
+gi (pi − pei ), (g1 = g2 = 1)
With the same simulation parameters, the moving
trajectories of individuals are shown in Fig. 7.
VI. C ONCLUSION AND FUTURE WORK
This paper addresses the fission control problem
of flocking system. To overcome the shortcomings
10
25
of the “average consensus” based interaction rules
8
20
that encumber the fission behavior, a new pairwise
6
40
60
80
interaction rule is proposed to implement the fis15
sion behavior. Firstly, we propose an information
10
coupling degree based approach to describe the
5
internal interaction intensity between individuals.
0
Then, by choosing the neighbors with largest in−5
formation coupling degree as the most correlated
−10
0
10
20
30
40
50
60
70
80
90
100
neighbor, we integrate the motion information of
x/m
the most correlated neighbor into the distributed
fission control algorithm to form a strong correlated
Fig. 7. Trajectories of individuals under the external stimuli with the
pairwise interaction, which realizes the directional
algorithm (34)
information flow of external stimuli and induces
Fig. 7 demonstrates the simulation results with the fission behavior of the flock in a self-organized
the flocking algorithm (34), where two blue solid fashion. Moreover, theoretical analysis proves that
circles represent the moving targets with the arrow a flock will segregate into clustered sub-groups
being their moving direction, red lines are the when external stimuli cause the motion information
trajectories of individual 1 and individual 2 after conflict in it. Finally, simulation studies demonstrate
they sense the moving target. From which we can the effectiveness of the proposed fission control
clearly see that only two individuals that sense the algorithm.
moving target can split from the group while the
rest move in a cohesive formation along its previous
trajectory. It can also be seen from the amplified
C ONFLICT OF I NTERESTS
figure that when individual 1 and individual 2 split
The authors declare that there is no conflict of
from the flock, the motion of other individuals tend
to fluctuate, but they finally return to the cohesive interests regarding the publication of this paper.
35
y/m
30
14
12
11
ACKNOWLEDGEMENTS
This work was partially supported by the National
Natural Science Foundation of China under Grants
51179156 and 51379179. The authors are also grateful for the constructive suggestions from anonymous
reviewers that significantly enhance the presentation
of this paper.
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