arXiv:1410.3864v2  [cs.NE]  16 Oct 2014
Multi-Agent Shape Formation and Tracking
Inspired from a Social Foraging Dynamics
Debdipta Goswami1, Chiranjib Saha2, Kunal Pal3, Nanda Dulal Jana4 and
Swagatam Das5
1,2,3 Dept. of Electronics and Telecommunication Engineering,
Jadavpur University, Kolkata
4 Department of Information Technology,
National Institute Of Technology, Durgapur
5 Electronics and Communication Sciences Unit,
Indian Statistical Institute, Kolkata
E-mail: 1debdipta goswami@yahoo.in, 2mail.chiranjib92@gmail.com,
3mail.kunalpal@gmail.com, 4nandadulal.jana@it.nitdgp.ac.in and
5swagatam.das@isical.ac.in
Abstract. Principle of Swarm Intelligence has recently found widespread
application in formation control and automated tracking by the auto-
mated multi-agent system. This article proposes an elegant and eﬀective
method inspired by foraging dynamics to produce geometric-patterns
by the search agents. Starting from a random initial orientation, it is
investigated how the foraging dynamics can be modiﬁed to achieve con-
vergence of the agents on the desired pattern with almost uniform den-
sity. Guided through the proposed dynamics, the agents can also track a
moving point by continuously circulating around the point. An analyti-
cal treatment supported with computer simulation results is provided to
better understand the convergence behaviour of the system.
1
Introduction
In recent years, pattern formation and control of multi-agent autonomous sys-
tems has the related research communities. With the tremendous advancement
of technology, the interest is also growing in this ﬁeld where a collection of agents
can communicate with each other and also with the environment and can do jobs
which are far beyond the capability of a single agent. Primary objective of multi-
robot pattern formation is to form a cohesive unit around the region of interest,
surveillance and move in a speciﬁed disciple. Real-life applications of formation
control can be found in multi-robot systems [1] [2], micro-satellite clusters [3],
unmanned land, sea, or air vehicles [4], [5], rendezvous [4], and mobile robotics
[6].
Common components of a generic shape-formation algorithm can be identi-
ﬁed as [7], 1. a homogeneous architecture i.e. components with same hardware
and software or heterogeneous with at least one single component having a dif-
ferent conﬁguration, 2. formation control strategy, 3. a behavior based model
2
that describes the nature of the system and environment. Formation control
strategy, the most decisive factor of an algorithm can be generally categorized in
two primitive classes, centralized and decentralized. In centralized control, the
agents follow a single controller that processes all the information while in later,
each agent is equipped with its own controller and autonomous decision making
capability. Both system has its advantages and disadvantages. Centralized con-
trol [8], [9], [10]., though is limited by the bandwidth of information exchange
or communication between the agents, is more structured for increasing number
of robots and complex situations while decentralized system [11], [12], [13], [14],
[7] having local data processing and less information sharing: no reliability cam
be attributed to a single robot or master, falls short when more sophisticated
planning is necessary for high level control. Hence, hybrid algorithms have been
developed employing both centralized and distributed control compromising the
communication cost between the agents to add more robustness to the system
[15], [16], [17], [18], [19].
Apart from architectural classiﬁcations, the algorithms, by nature can be
categorized in three broad classes, namely, leader/neighbor following algorithms,
potential ﬁeld based algorithms, and bio-inspired algorithms. Leader/neighbor-
following algorithms [20], [21] require the individual robots to follow a trailing
neighborhood of the leader or the leader itself, which contains all the global infor-
mation, eg. the target shape position along with maintaining a speciﬁc geometric
shape and avoiding collision with each other. This leader-follower approach was
modiﬁed in [22] by adding a control graph that deﬁnes the relative position
of each robot in the formation. Fredslund and Mataric [23] used local sensing
and minimal communication between agents to preserve a predetermined for-
mation. In a recent approach [24], feedback control and weight adaptation of
the interconnect structure of the communication graph topology enhancing the
Lyaponov stability has been proposed. The second category of the shape for-
mation algorithms are based on potential ﬁeld theory [25], [26], [3], [27], [28],
[29], [30], [31]. Typically, a force ﬁeld is created in the region of interest exert-
ing attractive forces on the agents and the obstacles repelling them, the agents
gradually through their directed motion converges to the target shape. Each
agent moves along the gradient of the potential ﬁeld which is the sum of these
virtual attractive and repulsive forces. In [26], the authors represented the de-
sired formation pattern in terms of queues and formation vertices. Some artiﬁcial
potential trenches were employed to represent the desired pattern and trajec-
tory for the group of robots. Although the method greatly improved on node
to robot formation structures, queues and vertices were still calculated based
on the formation and number of robots. Later a limited communication frame-
work was utilized to improve the work of [26] in [30]. In [31], a set of artiﬁcial
points is used as beacons that guide the robots to their goal. This approach is
based on the geometric relationship between beacon points to move the robots
in formation. In [3], a simple potential ﬁeld based on scalar sigmoid scaling
functions has been proposed where the diverging and converging forces with re-
spect to the origin of a radial coordinate system balances along the target shape
3
and a third curl ﬁeld exists along the pattern for further stability. The third
group of algorithms draw their inspiration from biological systems. Biological
systems, ranging from macroscopic swarms of social insects to microscopic cel-
lular systems, can give rise to robust and complex emerging behaviors through
much simpler local interactions in the presence of various kinds of uncertainties
[32]. Several bio-inspired methods for multi-agent shape formation and control
have been proposed over the past few years. Motivated by the cell structure,
Fukuda et al. [33] presented an optimal structure decision method to determine
cell type, arrangement, degree of freedom, and link length. They demonstrated
that through simulating the cells behavior, multiple agents can form and main-
tain an optimal structure. Shen et al. [34] proposed a Digital Hormone Model
(DHM) to control the tasking and executing of robot swarms based on local
communication, signal propagation, and stochastic reactions. The authors em-
ployed Turings reaction diﬀusion model [35] to describe the interactions between
the hormones. The DHM scheme integrated mechanisms of a dynamic network,
stochastic action selection, and hormone reaction-diﬀusion. Taylor [36] proposed
a Gene Regulatory Network (GRN) inspired real-time controller for a group of
underwater robots to perform a simple clustering task. In that system, the robots
can only adapt to local environmental changes without considering its inﬂuence
on global behaviours. Guo et al. [37] further utilized the GRN inspired dynamics
to devise a distributed self-organizing algorithm for swarm robot pattern forma-
tion. In [38], abstractions of morphogenesis in a network of molecular particles
has been employed to manage the spatial self-organization in an ensemble of
mobile robots without exploiting the common structural components of shape
formation algorithms like global perception, distance and direction sensing intel-
ligence. Apart from all these, few more approaches to formation control by using
graph algorithms [22], [39], models of visual perception [40], model predictive
control [41], reinforcement learning [42], and neural networks [43] have also been
proposed in literature.
Swarm Intelligence (SI) [44], [45] refers to a family of bio-inspired algorithms
imitating the collectively intelligent behavior of the groups of natural creatures
like bird ﬂocks, ﬁsh schools, and insect colonies. One of the natural traits of
such communal and cooperative systems is pattern formation. Gravagne and
Marks proposed a swarm model [46] with very simple rule to update the agents
and showed the emergent behaviors of aggressor, protector, and refugee swarms.
Andrews et al. [47] used the social foraging behavior of swarm to design robust
multi-agent systems. Particle Swarm Optimization (PSO) [48], [49] is one of
the leading SI algorithms of current interest. The basic PSO was devised and
is still most well-known as a function optimizer in the real-parameter space.
In PSO, each trial solution is modelled as a particle and several such particles
collectively form a swarm. Particles ﬂy through the multi-dimensional search
space following a typical dynamics and searching for the global optima. The
PSO dynamics has been improved and rigorously analyzed by several researchers
from an optimization point of view. However, very little research work has so
far been undertaken to use the dynamics for some purpose like shape formation
4
and tracking, which is completely diﬀerent from optimization on a functional
landscape. At this point we would like to mention a more or less related work in
[50], where a swarm of virtual particles, denoted as marching pixels, are crawling
according to simple mathematical rules (though these are diﬀerent from the
basic PSO-dynamics) within a 2D pixel image to ﬁnd the centroids of arbitrary
geometrical ﬁgures.
In this paper, we propose a simple method for formation of geometric pattern
that utilizes a foraging dynamics equipped with a repulsion between nearest
neighbours. The intended shape is given as input which we call the target vector
to the swarm. The target vector is a parametric vector that deﬁnes the shape of
the intended pattern and helps the swarm to converge on the desired shape. The
stable region of the dynamics is used so that the agents can converge properly.
The theoretical results are substantiated using computer simulations.
The organization of the paper is as follows: the section II the proposed system
is outlined; in section III the analytical treatment is carried out and the nature
of the target vector for each shape is discussed; in section IV the experimental
results with simple geometric shapes are shown, the importance of the repulsion
between nearest neighbours is explained and the ability of the proposed model for
automated tracking is brieﬂy demonstrated. Section V deals with the conclusion
and the future possibilities of research.
2
Proposed System
The proposed system for automated shape formation has taken a cue from the
foraging dynamics suggested by Das et al. in [51]. The basic position update
mechanism of the foraging dynamics was given by,
˙xi =
M
X
j=1,j̸=i
k(xj −xi)
[σ2 + ∥xj −xi∥2]β +
k(p −xi)
[σ2 + ∥p −xi∥2]β
,
(1)
where p ∈ℜn is the optimum of the artiﬁcial potential ﬁeld considered to sim-
ulate the real world application. Here the swarm is being pushed or attracted
towards p to achieve the desired goal. For shape formation purpose we want
to modify this, dynamics so that the agents converge on a pre-deﬁned shape
with uniform spacing. k, σ and β are positive constant parameters as deﬁned
in [51]. One basic requirement for this is to attract each particle towards the
boundary of the shape that we are intended to form. It is also desired that the
particles will be equally spaced along the shape and remain uniformly separated
from each other as much as possible. Hence, each particle should repel its nearest
neighbour. Thus the directing equation of the swarm that helps to form diﬀerent
shapes are given by,
˙xi =
M
X
j=1,j̸=i
k1(xj −xi)
[σ2 + ∥xj −xi∥2]β +
k2(xT −xi)
[σ2 + ∥xT −xi∥2]β
+r xi(t) −xn(t)
∥xi(t) −xn(t)∥.
(2)
5
where xi(t) is the position vector of an individual in the swarm. xT (t) denotes
the target vector which controls the motion of the individual and depends on
the geometric structure to be formed and the position of the individual at time
t. xn(t) is the position vector of the nearest individual of xi(t) ; || • || denotes
the Euclidean distance function; k1, k2 and r are constants.
This ﬁrst term is inspired from the mutual interaction of the agents in forag-
ing dynamics, the second term is responsible to form a geometric structure based
on an attraction. The third term is a mutual repulsion term introduced to ensure
that the agents of the swarm do not agglomerate or collide with each other and
thus maintaining a uniform density over the entire shape. If the repulsion term
is absent, the swarm will still form the desired structure, however, the symmetry
in the structure will be very poor. This has been illustrated in Section 4 through
computer simulations.
3
Analytical Treatment
The foraging dynamics proposed by [51] has distinct damped, limit-cyclic and
chaotic behaviour in its stable region. For shape formation, it’s important that
the agents will converge in the desired formation quickly. So the damped oscil-
latory behaviour is more appropriate to use. The aparameters of the dynamics
have been so chosen that it remains within the limit of stable damped oscillatory
behaviour.
To form diﬀerent shapes, diﬀerent types of target vectors are necessary ac-
cordingly. In this section we discuss about the target-vectors needed to produce
the patterns common in literature.
3.1
Straight Line
For forming a straight line with equation: x1 = mx2 + c, the target vector xT (t)
is given by:
xT (t) =
(x1 + mx2) −mc
1 + m2
, m(x1 + mx2) + mc
1 + m2
T
.
(3)
For simulation purpose we use the following values to ensure the stable behaviour
of the dynamics: k1 = 0.1, k2 = 2.0, σ = 3.5, β = 1.2, m = 1, c = 1. The initial
values for the positions are taken randomly between -5 and 5. The solution for
a single agent is shown in Figure 2a.
3.2
Ellipse
To form an ellipse with the equation x2
1
a2 + x2
2
b2 , the target vector needs to be
xT (t) = [a cosθ, b sin θ]T ,
(4)
6
where θ = tan−1
ax2
bx1

. The swarm parameter values are kept same as that
of the straight line case and the ellipse parameter values are set at a = 4 and
b = 2.
3.3
Circle
A circle is a special kind of ellipse where a = b = r0 and x2
1 + x2
2 = r2
0, , being
the equation of the circle, the target vector can be simply written as:
xT (t) = [r0 cos θ, r0 sin θ]T ,
(5)
where θ = tan−1
x2
x1

. In this case r0 is taken as 4. The plot for a single agent’s
position w.r.t. time is shown in Figure 2b.
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(a)
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(b)
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(c)
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(f)
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(g)
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(h)
Fig. 1: 1a-1e Formation of a straight line; 1b-1f formation of a circle; 1c-1g for-
mation of an ellipse; 1d-1h formation of a rectangle.
3.4
Square
We consider a simple square with length of side 2a, the sides are parallel to the
coordinate axes and the square is centered at origin. Mathematically it can be
represented by two pair of parallel straight lines |x1| = a and |x2| = a within
7
the ranges |x1| < a and |x2| < a respectively. In this case, the target vector can
be written as follows:
xT (t) = [a, ma]T ,
for −π
4 < φ < π
4
(6)
xT (t) = [ a
m, a]T ,
forπ
4 < φ < 3π
4
(7)
xT (t) = [−a
m, −a]T , for −3π
4 < φ < −π
4
(8)
xT (t) = [−a, −ma]T
otherwise;
(9)
where m = x2
x1
and φ is the angle obtained when [x1, x2] is transformed into
polar coordinates. The simulated position coordinates of a single agent is shown
in Figure 2c while forming a square.
0
20
40
60
80
100
120
140
160
180
200
−2
−1
0
1
2
3
4
5
Time
Position
 
 
1st dimension
2nd dimension
(a)
0
20
40
60
80
100
120
140
160
180
200
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−6
−5
−4
−3
−2
−1
0
1
2
Time
Position
 
 
1st dimension
2nd dimension
(b)
0
20
40
60
80
100
120
140
160
180
200
−2
−1.5
−1
−0.5
0
0.5
1
Position
Time
 
 
1st dimension
2nd dimension
(c)
Fig. 2: Variation of position of an agent with respect time for formation of straight
line (2a), circle (2b) and rectangle (2c). a circle
4
Experimental Results
4.1
Formation of Simple Patterns
The proposed method is used to form simple structures like straight-lines, el-
lipses, circles and squares. For that purpose we have ﬁxed our parameter values
in the stable region of the dynamics as k1 = 0.1, k2 = 2.0, σ = 3.5, β = 1.2 and
r = 0.1. Formation speciﬁc parameters are set as: m = 1, c = 5 for straight line;
a = 4, b = 2 for ellipse and r0 = 4 for circle. For the case of formation of square,
the used parameter a = 5. The ﬁnal formations are presented in Figures 1a- 1h.
The ﬁnal positions of the agents are shown as the tiny ﬁlled circles and the
targeted geometrical formations are drawn in dashed lines.
8
4.2
Usefulness of the repulsion term
The usefulness of the repulsion term in equation (2) is also investigated. For this
purpose, we again simulate the algorithm with the parameter settings r = 0 and
in the other case r = 0.1; keeping the other parameters ﬁxed at k1 = 0.1, k2 =
2.0, σ = 3.5 and β = 1.2, for a 12 agent swarm forming an ellipse and a square.
The ﬁnal formed structures are shown in Figure 3. In the ﬁrst case, the repulsion
term is absent, hence the agents do not communicate with each other and the
ﬁnal formation lacks from the problem of symmetry which can be overcome by
introducing the repulsion term. This can be observed from Figure 3 very well.
4.3
Tracking Ability
The proposed system can be used to track a moving point in real-time by sur-
rounding it. This important property of continuously surrounding a moving point
can be put to use in real life in case of the automatic protection of a convoy by
swarm robots. The formation of circle can be used in this case for surrounding
the moving point and the moving point can be treated as the center of the circle
to be formed. To accomplish this the target vector has to be modiﬁed slightly
as,
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(a) Ellipse
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(b) Rectangle
Fig. 3: Shape formation without repulsion
xT (t) = [C1(t) + r0 cos φ, C2(t) + r0 sin φ]T ,
(10)
where C(t) = [C1(t), C2(t)]T is the coordinate of the moving center. For experi-
mental purposes, we considered two types of movements of the center:
9
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  t = 40
t = 80
t = 120
t = 160
t = 200
(a)
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t = 40
t = 80
t = 120
t = 160
t = 200
(b)
Fig. 4: Circle formed around the center which is moving in a straight line (4a)
and circle (4b).
– Straight Line
The center moves along the straight line y = x with the initial position
C(0) = [10, 10]T and with the uniform velocity vector vC(0) = [0.1, 0.1]T .
The circle formation of a 10 agent swarm is considered with k1 = 0.1, k2 =
2.0, σ = 3.5, β = 1.2, r = 0.1 and r0 = 2 The snapshots of the swarms
at iterations, t = 40, 80, 120, 160, and 200 are super-imposed in Figure 4a.
The ﬁlled small circles denote the agents and the unﬁlled circles denote the
position of the moving center at that point of time. It is clearly seen from
the ﬁgure that the formed circle tracks the moving center or in other words
the swarm is surrounding the center. In Figure 5a the movement of a single
agent during tracking is shown along with the snapshots of the swarm. The
trajectory of the agent closely resembles the trajectory of the moving point
itself as expected.
– Circle
In this case, we consider that the center is moving along the circle x2+y2 = 62
with initial position C(0) = [6, 0]T and with a uniform angular velocity such
that it completes a full rotation in T iterations. The parameter values of a 10
agent swarm are taken as k1 = 0.1, k2 = 2.0, σ = 3.5, β = 1.2, r = 0.1, r0 = 2
and T = 200. We got similar results as found in the case of the center
following a line. The simulation results are shown in Figure 4b. The Figure 5b
shows the trajectory of a single agent during tracking for circular motion of
the moving point.
10
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(a)
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(b)
Fig. 5: Trajectory of an agent during tracking the center which is moving in a
straight line (5a) and circle (5b).
5
Conclusion
This article proposed a simple but elegant method to form elementary geometric
structure with multi-agent system. This method is partly inspired from a for-
aging dynamics where the agents communicate with each other to accomplish a
common goal. The attractant-repellent proﬁle and the convergence point of the
dynamics is modiﬁed using a parametric target vector and an extra repulsion
term among the nearest neighbours to meet the demand of target-shape and
nearly even density of agents over it.
Only simple geometric structures like straight line, ellipse, circle and square
have been considered to keep the understanding of the swarm-mechanism eas-
ier and vivid. The tracking ability has also been demonstrated point moving
along a straight line or around a circle and surrounding it with agents in desired
shape. The future work in this ﬁeld can extend to the formation of more com-
plex geometric structures with their Cartesian or polar equation supplied. The
emergence of deterministic chaos in the perspective of shape formation can also
be investigated.
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