 
Abstract—The work reported in this paper is motivated 
towards the development of a mathematical model for 
swarm systems based on macroscopic primitives. A pattern 
formation and transformation model is proposed. The 
pattern transformation model comprises two general 
methods for pattern transformation, namely a macroscopic 
transformation and mathematical transformation method. 
The problem of transformation is formally expressed and 
four special cases of transformation are considered. 
Simulations to confirm the feasibility of the proposed models 
and transformation methods are presented. Comparison 
between the two transformation methods is also reported.  
I. INTRODUCTION 
WARM robotics research specifically deals with 
some classical problems which have been of 
concern to researchers over the years. An overall review 
of work done in swarm robotics presented in [1] 
identifies pattern formation as one challenge. The 
problem of pattern formation relates to the microscopic 
(influencing individual robot behavior) or macroscopic 
(influencing group behavior) properties of a swarm 
system.  
 Pattern formation has been addressed by researchers 
in swarm robotics using two approaches, namely 
potential fields and behavior based models. In [2] a 
control law is proposed for pattern formation that 
consists of the sum of a repulsive potential and an 
attractive potential field. The approach is useful for the 
classical problem of obstacle avoidance in patterns. The 
paper reports pattern formation control based on 
parameters describing microscopic behaviors and 
presents a mathematical model for the control strategy. 
In [3, 4] potential field and sliding mode control have 
been used for pattern formations. In [5] the authors 
study a general class of Attractive and Repulsive 
functions used to achieve swarm aggregation. These 
models lack a description of a swarm model describing 
the macroscopic properties.  
The behavior based approach has focused on pattern 
formation behaviors for multi-robot teams and patterns 
including the line, column, diamond and wedge 
geometric formations have been reported [6]. Obstacle 
avoidance and other navigational behaviors are 
 
  
integrated within the model. Though the microscopic 
properties of the system are defined, the group behavior 
of a system cannot be explicitly determined by the 
behavior based approach. Moreover this approach does 
not 
lend 
itself 
to 
mathematical analysis and 
formulations. Hence, the approach fails in articulating a 
swarm model with macroscopic parameters. 
Other approaches used by researchers in pattern 
formation include the dynamic window approach [7] – 
[9] and flow field method [10]. Both these approaches 
consider microscopic properties of the swarm system.   
 In short, pattern formation approaches do not 
consider macroscopic parameters of a swarm system. 
Hence there arises a need to develop a pattern formation 
model based on macroscopic parameters. There are five 
main benefits of using macroscopic parameters. Firstly, 
implicit coordination, which refers to the coordination 
of a pattern comprising of mobile robots, need not be 
specified externally. Coordination is achieved as a 
result of varying the macroscopic properties. Secondly, 
Group behavior definition, which refers to the 
collective behavior of the group, is possible by 
controlling the macroscopic parameters. The individual 
behavior of the units is affected by the variation in the 
macroscopic property. Thirdly, Adaptability, which 
refers to the ability of the group to adjust to change of 
internal or external circumstances, can be affected by 
macroscopic parameters. Fourthly, Stability, which 
refers to the factor by which the robot group maintains a 
pattern, can be controlled by using macroscopic 
parameter to dampen the propagation of errors. Fifthly, 
higher order parameters can control parameters of 
lower order. 
 Researchers tend to consider pattern transformation 
along with pattern formation without much distinction. 
Transformation, which refers to the reconfiguration of 
swarm patterns, is a little considered area within swarm 
robotics. Transformation of patterns is an appropriate 
response to obstacles for unhindered motion. Patterns are 
reconfigured by repositioning all or a subset of agents in a 
swarm. A transformation may result in a change of 
geometric orientation of a pattern and relationships 
between interacting units in the pattern. 
A Mathematical Model, Implementation and Study  
of a Swarm System 
Blesson Varghese and Gerard T. McKee 
S 
 
 
 
 Work based on pattern transformation is reported in 
[11], where a stable virtual leader pattern transforms to 
a different pattern by the addition of a morphing force. 
Illustrations of transformation and mathematical 
notations for computation of forces in the pattern are 
also presented. The transformation technique facilitates 
pattern change by allowing participating agents to find 
their own equilibrium. However, the morphing 
procedure for transforming pattern is not defined. 
An algorithm reported in [12] is capable of 
transforming patterns in response to a command issued 
by a human operator. The command is issued to a single 
robot and causes a chain reaction in the neighboring 
robots resulting in a global transformation. Pattern 
transformation from a parabola to a sine curve is 
illustrated. Though the notion of transforming patterns 
is presented, the transformation method remains 
unaddressed.  
 A relative distance versus orientation model for 
transformation is reported in [13]. The strategy involves 
varying the orientation value to globally transform a 
triangle to a line formation. Though a positional 
transformation is not executed on all participating 
agents (one position remains unaltered), a global 
geometrical transformation is achieved. This strategy is 
specific to the scenario when a triangle to line 
transformation is performed. 
 The use of affine transformations, a mathematical 
tool, for transforming swarms patterns is presented in 
[14]. The target position of each member of the swarm 
is pre-determined by the mathematical transformation 
tool. The shortest path between the original and target 
positions is traced by considering the ant colony 
optimization 
[15] 
algorithm. 
Simulation 
results 
illustrate the transformation of a horizontal line pattern 
to a diagonal line pattern. Though a mathematical 
method is explored, a geometrical transformation 
between different shapes may not be possible using 
affine transformations. 
 In short, researchers have not concentrated on 
investigating pattern transformation in swarm robotics. 
General methods for reconfiguring patterns are not 
emphasized either. 
 The work reported in this paper is motivated towards 
the development of a swarm model based on primary 
and secondary macroscopic primitives. A pattern 
transformation model comprising two transformation 
methods that enable geometric transformation are 
proposed. Firstly, a swarm macroscopic parameter 
oriented method based on macroscopic (group behavior 
of a swarm pattern) parameters is proposed. Secondly, a 
mathematical 
tool 
based 
on 
macroscopic 
and 
microscopic (individual robot behavior) parameters is 
proposed. The problem of transformation is formally 
stated and four special cases of transformation are 
considered. Experimental studies confirming the 
feasibility of the proposed methods are presented.  
The remainder of this paper is organized as follows. 
Section II proposes the pattern formation model. 
Section III states the problem of transformation and 
presents the pattern transformation model. Sections IV 
and 
V 
proposes 
two 
transformation 
methods 
comprising the pattern transformation model. The 
feasibility of the proposed models is confirmed by 
simulation which is reported in Section VI. Section VII 
concludes this paper and reports future work.  
II. SWARM PATTERN FORMATION MODEL 
This section proposes a mathematical model for 
swarm pattern formation based on primary and 
secondary primitives. The mathematical model is 
formulated based on the foundations of the Complex 
Plane. The De Moivre’s formula to obtain roots of an 
equation is used to represent the model. If z = x + iy [16] 
and 
is 
represented 
in 
the 
polar 
form 
as 
)
sin
(cos


i
r
z


and r is called the absolute value or 
the modulus of z, then 
)
sin
(cos


n
i
n
r
z
n
n


 for n = 
0, 1, 2,… The nth root of z is obtained by 


















n
k
i
n
k
r
z
n
n




2
sin
2
cos
where k = 0, 1,… 
(n-1). The roots of the equation lie on a circle of radius 
 
Fig. 1.  The primitives of the multi-agent pattern formation model. 
 
 
 
 
n r with centre at the origin and constitute the vertices 
of a regular polygon of n sides. The result of joining the 
n roots is an n-sided polygon. The polygon is 
circumscribed by a circle otherwise referred as the 
circumcircle of the polygon. When mapping these 
results onto the area of multi-agent pattern formations, 
it is assumed that the robotic agents are positioned on 
the vertices of the polygon. Hence the robots form a 
closed polygonal pattern and the system is mobile with 
appropriate 
communication 
and 
coordination 
mechanism. 
The mathematical model is realized by considering 
macroscopic primitives (Figure 1). The term primitive 
in this paper refers to an element used as a building 
block to define aspects of the model. The macroscopic 
primitives are separated into primary and secondary 
primitives. Primary macroscopic primitives are basic or 
fundamental elements. They are considered as input 
variables to the model and are irreducible to simpler 
parameters or expressions and therefore termed as 
independent 
primitives. 
Secondary 
macroscopic 
primitives are derived from other primitives of the 
mathematical model. Hence, these primitives are 
termed as dependent primitives.  
The primary macroscopic primitives of the model 
proposed in this paper are the total number of robots, 
angular separation, formation radius and elongation. 
The total number of robots in a polygonal pattern, given 
by n, equates to the number of vertices of a polygon or 
the roots of the complex equation. Angular separation is 
an important factor that determines the coordinates for 
positioning robots in a polygonal pattern. Angular 
separation, denoted by, is a measure of the angular 
spacing between adjacent robots of a pattern. Formation 
radius, denoted by r, is the radius of the circumscribing 
circle of the polygonal pattern. This primitive 
determines the area occupied by the pattern. Elongation 
ratio of a pattern, denoted by e, is a ratio of magnitudes 
of the major and minor axis of the pattern and quantifies 
the shape transforming behavior of a pattern. The 
symmetry of a pattern can also be described by the 
elongation ratio. 
The secondary macroscopic primitives are linear 
distance and shaping radii. The distance between 
adjacent robots in the polygon is a constant if the 
polygon is regular. To compute the distance between 
robots, the coordinate positions of the robots need to be 
known. The centroid of the pattern, (h,k), is used to 
compute the coordinates of robots. Further, the 
Euclidean distance between adjacent robots A and B is 
given by
2
2
)
(
)
(
A
B
A
B
AB
y
y
x
x
d




. Hence, linear 
distance is dependent on the position coordinates of 
robots.  
The shaping radii along the x and y axis, sx and sy 
respectively determine the measure of deflation or 
inflation of a pattern laterally and longitudinally. The 
magnitudes of elongation and formation radius are 
useful to determine the shaping radii of a pattern and are 
given by 
re
sx =
 and
e
r
sy =
. The equations that define 
the 
shaping 
radii 
are 
also 
given 
by
cos
+
=
θ
s
h
x
x
B
and
θ
s
k
y
y
B
sin
+
=
. 
Hence, 
orientation radii are dependent on formation radius and 
elongation.  
Variation in the magnitude of the shaping radii results 
in a regular or irregular pattern. Regular patterns refer 
to swarm formations as regular polygons. The regularity 
of polygonal patterns is preserved by scaling (deflate or 
inflate) the pattern laterally and longitudinally in equal 
magnitudes. On the other hand, irregular patterns refer 
to swarm formations as irregular polygons. Irregular 
patterns can be obtained by scaling patterns laterally or 
longitudinally with unequal magnitudes.  
The swarm pattern formation model presented here is 
chosen for the study of transformation. Two pattern 
transformation methods are applied to the swarm model; 
these are discussed in Section IV and V.  
III. SWARM PATTERN TRANSFORMATION MODEL 
Considering the fact that pattern transformation is 
little addressed in research and general methods for 
transformations are not investigated, the problem of 
swarm pattern transformation is presented here. 
The term transformation is also associated with 
modular robotic systems. Algorithms to transform the 
shape of modular robots are reported in [17]-[19]. 
However, it is necessary to draw distinction between 
reconfiguration in modular systems and transformation 
of swarm patterns considered in this paper. Firstly, in 
modular robot systems physical connectivity between 
modules exists ensuring modules in close vicinity of 
adjacent modules. Secondly,  reconfiguring in modules 
is constrained by being able to reposition on the 
periphery 
of 
an 
attached 
module. 
Thirdly, 
reconfiguration in modules is not strictly geometry 
oriented. 
Definition: Consider a pattern P with geometric 
relationships 
represented 
as
P
G
. 
The 
pattern 
P comprises of N robots such that their positions are 
given by 
)
,
(
i
i
i
y
x
p
where 
2


ip
 and
N
i
,...,
2,1

. 
Pattern P transforms into the pattern Q with geometric 
 
 
 
constraints or relationships represented as
Q
G . The 
pattern Q also comprises of N robots such that the 
position of the robotic agents is given by 
)
,
(
i
i
i
y
x
q
 
where 
2


iq
and
N
i
,...,
2,1

.  
The function which enables the transformation of the 
pattern P to Q is given by
Q
P
f

)
(
. In other words, 


).
,
(
),...,
,
(
),
,
(
)
,
(
),...,
,
(
),
,
(
2
2
2
1
1
1
2
2
2
1
1
1
N
N
N
N
N
N
y
x
q
y
x
q
y
x
q
y
x
p
y
x
p
y
x
p
f

The application of an inverse transformation function 
on the transformed pattern Q  yields the pattern
,
P  
given by
.
=
)
(
1
P
Q
f
 The transformation on the pattern 
also results in a transformation of the geometrical 
relationships from 
P
G to 
Q
G  between the participating 
agents in the pattern. Four cases of transformation 
based on the above definition are derived by imposing 
restrictions on the geometrical constraints. 
Case 1: 
P
Q
G
G

 after a transformation that involves 
repositioning all agents. This case is relevant when 
robotic agents in the pattern have repositioned, yet the 
geometrical pattern has not changed. Such a 
transformation is termed as Elementary transformation 
in this paper. This term also refers to those 
transformations very basic in nature. For instance, a 
swarm could be rotated with respect to its centroid or 
translated such that all robotic agents have repositioned 
themselves. Though the orientation of the pattern has 
changed, the configuration of the pattern remains 
unaltered.  Mathematically, the case of elementary 
transformation would be such that 
P
Q
G
G

 and 


)
,
(
:
i
i
i
y
x
p
i
).
,
(
i
i
i
y
x
q
 
Case 2: 
P
Q
G
G

 after a transformation without 
repositioning all agents. This case considers the rotation 
or translation of the swarm with respect to some robotic 
agent whose position remains fixed. This case is also 
classified under Elementary transformation, yet 
repositioning of all agents has not occurred. 
Mathematically, this case of elementary transformation 
would 
be 
such 
that 
P
Q
G
G

 and 
)
,
(
)
,
(
:
i
i
i
i
i
i
y
x
q
y
x
p
i


 
Case 3: 
P
Q
G
G

 after a transformation that involves 
repositioning all agents. This relates to the case when 
the geometrical constraints of the pattern have changed 
and a new pattern has emerged.  It is termed a 
Geometric transformation. This concept is relevant 
when robotic agents in the pattern reposition to result in 
a geometry change. For instance, the shape of a swarm 
could be geometrically transformed from a polygon to a 
line. It is interesting to note that the scaling of a pattern 
would result in a geometric transformation, since the 
geometrical constraints are dissimilar in both cases. 
Mathematically, the case of geometrical transformation 
would 
be 
such 
that 
P
Q
G
G

 and 


)
,
(
:
i
i
i
y
x
p
i
).
,
(
i
i
i
y
x
q
 
Case 4: 
Q
G
≠
P
G
after transformation without 
repositioning all agents. This case considers the 
geometric transformation such that the position of one 
or more than one robotic agent remains fixed. It is 
classified 
under 
geometric 
transformation, 
yet 
repositioning of all agents has not occurred. 
Mathematically, the case of geometrical transformation 
would 
be 
such 
that 
P
Q
G
G

 and 
)
,
(
)
,
(
:
i
i
i
i
i
i
y
x
q
y
x
p
i


. 
Cases 1 and 2 relate to elementary transformation of 
the pattern. In these cases, the geometric constraint or 
relationship 
persists 
even 
after 
elementary 
transformation. Cases 3 and 4 consider geometric 
transformation. In these cases the geometrical 
relationships change after transformation.  
The swarm pattern formation model presented in the 
previous section is chosen for the study of 
transformation. Two feasible transformation methods, 
namely a macroscopic transformation method and a 
mathematical transformation method are proposed in 
this paper. Cases 1, 3 and 4 of transformation are 
considered in the transformation methods. Case 2 will 
be reported in a future paper.  
IV. METHOD 1: MACROSCOPIC TRANSFORMATION 
The transformation method proposed in this section is 
inclusive of elementary and geometric transformations 
applied on the swarm model. Transformations of cases 
1, 3 and 4 are achieved by varying the secondary 
macroscopic primitives, namely the shaping radii 
(along x and y axis) of the swarm model. It is interesting 
to note that a sequences of operations performed on the 
swarm model results in a transformation. The set of 
operations are:  
1) Rotation: The initial step of rotation of the model is 
performed to achieve collision avoidance during the 
next step. A predefined angle offset is used to rotate the 
swarm. Though the robots are repositioned, the 
operation results in the same polygonal pattern with a 
different orientation from the former. Here, the concept 
of elementary transformation is introduced. Though all 
robots were repositioned in this operation, a geometric 
transformation is not evident since the shape of the 
pattern is retained. Though a geometric transformation 
 
 
 
is not evident, yet an elementary transformation of case 
1 is achieved in this step.  
2) Macroscopic Parameter Operation: Following a 
rotation operation, the macroscopic parameter is set to 
be modified. Deflating the model along the y-axis 
would result in a deformed polygonal pattern. The 
deflation of the model is performed by decrementing 
the magnitude of the shaping radius along the y-axis. 
When deflation has reached its maximum value, the 
robotic agents have aligned themselves entirely along 
the x-axis. Maximum deflation is achieved when the 
shaping radius value along any axis vanishes. An 
inflation operation along the other perpendicular axis 
simultaneously while deflating would result in a pattern 
with larger inter-linear distance between the agents (a 
measure for avoiding collisions). This variation is 
possible due to the notion of flexibility in rigid patterns.  
3) Further Rotation: This step is performed to 
achieve equidistance between the participating agents. 
Though the pattern has transformed its shape by this 
step, the participating agents are still governed by the 
rules of the swarm model. A corrective rotation measure 
would ensure that the agents are loosely equidistant. 
V. METHOD 2: MATHEMATICAL TRANSFORMATION 
The method proposed in this section considers case 
3, which is achieved by using a mathematical 
transformation tool. Many mathematical tools are 
available for transformations which include stretching, 
rotating, reflecting and translating transformations. The 
linear fractional transformation is one such readily 
available mapping function that maps a set of points 
from one plane to another. The transformation is given 
by
d
cz
b
az
z
f



)
(
, where z, a, b, c and d are complex 
numbers satisfying ad -

bc
 0. The linear fractional 
transformation 
is 
also 
known 
as 
a 
Moebius 
transformation [16].  
The transformation functions are applied onto the 
swarm pattern which is polygonal in shape. Since the 
vertices of the polygonal pattern lie on the circle 
circumscribing the pattern, a circle to line and a line to 
circle transformations of the complex plane are used. 
However, the transformation function cannot be applied 
directly to the multi-robot pattern. This is due to the fact 
that the multi-robot pattern is defined on a global frame 
of reference while the mathematical function is 
applicable on the local frame of reference. Hence, the 
sequence of operations performed on the multi-robot 
pattern is: 
1) Transformation from global to local frame of 
reference: The frame of reference of the multi-robot is 
temporarily transformed from the global to a local 
frame. The local frame of reference considered is such 
that the circumscribing circle is divided into four equal 
quadrants. Hence the centroid of the pattern lies on the 
origin position of the local frame.  
2) Discrete Transformation: This step applies the 
mathematical 
transformation 
function 
on 
the 
multi-robot model. The transformation of a circle to a 
line is obtained from
)
1(
)
1(
z
z
i
w



. Applying the equation 
on the Euclidean plane, the mapping function is 
deduced 
as 














2
2
1
2
2
1
,
2
2
1
2
y
x
y
x
y
x
y
. 
The 
transformation from a line to a circle is applied by 
considering 
a 
special 
case 
of 
the 
Moebius 
transformation. The transformation 
z
w
1

  maps every 
straight line or circle onto a circle or straight line. It is 
also known as the inversion in the unit circle or 
reciprocal transformation. Applying the equation on the 
Euclidean plane, the mapping function is otherwise 
written as










2
2
,
2
2
y
x
y
y
x
x
. The destination 
coordinates obtained by the mathematical functions are 
the coordinates to which individual robot agents need to 
reposition while the pattern transforms. However, it is 
evident that these transformation functions are discrete 
in nature yielding only one set of destination coordinate 
rather than sub-goals or intermediate destination 
coordinates. 
3) Transformation from local to global frame of 
reference 
with 
magnification: 
The 
destination 
coordinates are obtained on the local frame of 
reference. Hence, the local frame needs to be shifted to 
the global frame of reference. Since the mathematical 
functions considered in step (ii) are reducing functions 
(destination coordinates reduce the span of the pattern), 
a magnification ratio is used in the local frame to 
achieve gain in the destination coordinates. 
4) Path planning by discretization: Since the 
achieved destination coordinate set is discrete, the 
major challenge in repositioning agents is to plan their 
path to the destination coordinates. In this paper, the 
technique adopted to reposition robots is along straight 
line trajectories without collisions. The straight line 
path between the agent and its estimated destination is 
discretized. A straight line discretization process is 
done by slicing the domain values to extrapolate the 
 
 
 
range values. This relates to the underlying principle of 
Discrete Event Simulations (DEVS). The potential of 
DEVS in path planning for robots is reported in [20]. 
VI. SIMULATION STUDIES 
Simulation studies were developed to validate and 
visualize the proposed geometric approaches for pattern 
formation and transformation. Most robotic simulators 
proved ineffective for incorporating the geometric 
approach. Hence, a non-robotic particle physics 
simulation engine was employed. The remainder of this 
section 
is 
organised 
into 
understanding 
the 
experimental environment, studies on pattern formation 
and transformation. 
A. Experimental Environment 
The feasibility of the proposed approach was 
validated on the Processing [21] and Traer Physics [22] 
environment. 
Processing 
is 
an 
open 
source 
programming language and environment enabling 
visualizations for learning and prototyping. Traer 
Physics is a particle physics simulation engine for 
Processing. 
The traer physics library has provisions for modeling 
a particle system, particles, springs and attractive or 
repulsive forces [22]. The particle system enables 
prototyping particles and forces. Particles represent 
objects having four properties, namely mass, position, 
velocity and age. Particles can be stationary or dynamic 
in an environment. Springs can connect two particles 
and prevent collisions. Springs are characterized by 
three properties, namely rest length, strength and 
damping. Attractions or repulsions pull particles 
together or apart and have two properties, namely 
strength and minimum distance. The simulations 
reported in this paper employ particle system, particles 
and attractive or repulsive forces. 
The swarm pattern is designed as particles in an open 
environment with forces, namely macro and micro level 
forces of attraction and repulsion acting on the pattern. 
The macro level forces include repulsive forces, which 
act on the centroid of the swarm. The forces of 
repulsion are generated from obstacles (modeled as 
forces) in the environment. All robotic agents align 
themselves around the centroid with respect to the 
forces forming a virtual structure polygonal pattern. 
Obstacles in the path of the pattern are detected by the 
computation of the net force acting on the group of 
robots. Beyond a maximum threshold value of force, the 
pattern reacts appropriately by transforming its shape to 
avoid obstacles. The pattern regains its polygonal shape 
when the net force acting on the centroid decreases 
below a minimum threshold value, such as when the 
pattern has escaped from obstacles. The inter-agent 
bonding force and the forces of interaction with the 
centroid contribute to the micro level forces. The 
pattern generates a propulsive force to trace paths 
against repulsive forces.  
The experimental setup comprised a tunnel through 
which the swarm had to displace. The walls of the 
tunnel generated repulsive forces and acted as the 
obstacle. The swarm initiated its motion from the left of 
the tunnel and aimed to reach a goal beyond the tunnel 
on the right side. 
B. Studies on Pattern Formation 
Regular pattern formations were studied when the 
robot pattern displaced through obstacles similar to 
bridges and tunnels. The pattern was expected to deflate 
when subjected to a potential above a minimum 
threshold force, continue motion and inflate beyond the 
obstacles. Simulation results for regular pattern 
formation with 5 and 6 robots are presented in figure 2 
(left columns). 
Irregular pattern formations were studied when the 
geometric robot patterns displaced through obstacles 
that converged and hence offered a narrowed path of 
movement. This replicated motion through a funnelled 
path. The pattern was expected to deflate laterally or 
longitudinally when subjected to a potential above a 
minimum threshold. Simulation results for irregular 
pattern formation with 5 and 6 robots are presented in 
figure 2 (right columns). 
It is observed from both the regular and irregular 
pattern formation simulations that the patterns deflated 
to traverse through obstacle paths. Beyond obstacle 
paths, the patterns inflated to achieve their original 
configuration. It is notable that obstacle avoidance is an 
inherent property of the system and hence implicitly 
guaranteed since obstacles are modelled as forces. 
These observations are consistent with the theoretical 
studies of Section 3 and according to the authors 
expectations. Hence, the simulation studies confirm the 
formulations of the geometric approach for polygonal 
robot configurations. 
C. Studies on Pattern Transformation 
Both transformation methods discussed in Section 
IV and V were implemented. The macroscopic method 
of Section IV, consisting of a sequence of three 
operations, was implemented. Firstly, the swarm model 
was rotated to avoid collisions while deflating. Table I 
illustrates the different rotation angles that were applied 
on the swarm. Higher value angles resulted in collisions 
for most patterns. Angles less than 15 degrees proved 
 
 
 
effective for collision avoidance. Secondly, the 
macroscopic parameters were varied. This variation 
resulted in deflation or inflation of the pattern (along 
the x or y axis). Thirdly, a corrective rotation was 
applied to avoid agents from colliding against each 
other. Hence by transformation of the pattern, the 
swarm successfully displaced itself through the obstacle 
path. Figure 3 is a snapshot of the simulation studies for 
n = 3 to 6, 10 and 20 robots transforming in accordance 
with the first method. 
 
TABLE I 
PRE DEFINED ROTATION VALUES & ESTIMATED 
COLLISIONS 
 
ANGLE 
OFFSET 
15º 
30 º 
45 º 
60 º 
NO. OF 
ROBOTS 
3 
- 
- 
- 
1 
4 
- 
- 
2 
- 
5 
- 
1 
- 
- 
6 
- 
3 
- 
3 
 
The mathematical method of Section V which 
consists of a sequence of four steps was implemented. 
Firstly, the swarm pattern was transformed from the 
global to a local frame such that the centroid of the 
pattern lies on the origin of local frame of reference.  
Hence, the pattern is equally spanned over the four 
quadrants in the local frame of reference, which was 
necessary 
for 
proper 
implementation 
of 
the 
transformation functions.  
Secondly, the discrete transformation function was 
applied on the microscopic property, namely the 
position coordinates of the individual robots in the 
pattern. The transformation from a circle to line was 
employed in order for the pattern to pass through the 
tunnel in the environment. Beyond the obstacles, the 
transformation from a line to circle was employed. Both 
transformation operations yield a set of discrete 
destination coordinates for each robot. 
Thirdly, transformation from the local to global frame 
of 
reference 
was 
performed. 
The 
destination 
coordinates obtained in the local frame of reference 
were such that the pattern radius is reduced. Hence a 
magnification of the coordinates in the local frame was 
performed and further mapped on to the global frame of 
reference. 
Fourthly, path planning by discretization was 
executed. This step is essential to determine the sub 
goals 
or 
intermediate 
position 
coordinates. 
Repositioning the robots to sub-goals or intermediate 
coordinates is a computationally expensive process. 
Straight line trajectories from agents to calculated 
destination 
coordinates 
without 
collisions 
were 
considered in the work reported in this paper. Figure 4 
is a snapshot of the simulation studies for 17 robots that 
transform shape in accordance with the second method.  
It is observed that the circle to line transformation 
yielded a pattern in which robotic agents were loosely 
equidistant. The line to circle transformation employing 
 
Fig. 2. Left columns (n = 5 and 6) Regular pattern formation and obstacle avoidance through a tunnel. Right columns (n = 5 and 6) Irregular pattern 
 formation and obstacle avoidance through a funneled path. 
 
 
 
the reciprocal transformation yielded a polygon irregular in 
nature. This was due to the nature of the reciprocal 
transformation, which was anticipated. 
It was observed that in both methods, the swarm 
successfully displaced itself through the obstacle path 
by transforming shape. The transformed patterns were 
loosely equidistant. Collision avoidance between 
repositioning agents is not implicitly guaranteed. Hence, 
at least one operation in both methods ensured collision 
avoidance. The geometrical transformation of a circle to 
a line in both cases was achieved by transforming a 
regular polygonal pattern to an irregular pattern by 
repositioning agents. The observations are consistent 
with the theoretical studies in Section IV and V and 
according to the authors expectation.  
D. Results 
The time taken to transform a pattern in the two 
transformation methods for different number of robots in 
the configuration was measured. This experiment was 
carried out for different number of robots varying from 3 to 
25 and keeping the initial formation radius of the swarm a 
constant. Figure 5 (left) and Figure 5 (right) are graphs 
plotted using MATLAB and are based on the results 
obtained from simulation for the first and second 
transformation method respectively. The graphs show the 
time taken for transformation versus the number of robots 
in the pattern. 
The average time taken to transform a pattern in the first 
transformation method was computed as 20.35 seconds. It 
is noted that the values plotted on the graph can be 
divided into two bands. Firstly, the set of times that lie 
below 20 seconds and secondly, those that lie above 20 
seconds. The first band follows a linear trend with a 
steady rise. Though the second set of times is scattered, 
they form three different clusters for n = 4 to 8, 11 to 15 
and 22 to 25. 
Working from left to right on the robot axis, the 
smallest time taken for transforming the pattern was 
observed when three robots constituted the pattern. This 
was due to the fact that a corrective rotation step was 
not required for obtaining equidistance between the 
patterns. Hence, the robots traversed lesser distances to 
achieve the transformed pattern. There is a steep rise in 
the time taken to transform the patterns for n = 4 to 8. 
This can be accounted for by the fact that the initial 
rotation and further rotation steps cause the robots to 
trace further distances to keep themselves equidistant. 
 
Fig. 3. Simulation results on Traer Physics and Processing simulator for the first method. (i) Rotated swarm model for various number of robots, (ii) 
Deflation of the model along the y – axis (For n = 10 and 20, inflation along x – axis performed), (iii) Transformed pattern without corrective rotation 
measure, (iv) Transformed pattern after corrective rotation measure is applied (Except for n = 3 and 4, since equidistance is more or less achieved), (v) 
Inverse transformation by inflation back to original pattern. 
 
 
 
For n = 9 and 10, however the time taken has a steep 
fall. It is likely that the robots trace less distances for 
collision avoidance in these cases. The highest time for 
transformation was noted for n = 11 to 15. This was due 
to the fact that a higher degree of rotation was required 
initially to avoid collisions. Hence the robots took 
longer times to reposition. In the further rotation step, 
more agents compared to the previous cases had to 
reposition. Hence, this accounts for this cluster having 
the largest time. For n = 16 to 21 there is a gradual 
increase in the time taken to transform, but much lower 
than the time taken by the previous cluster since the 
corrective rotation step was not necessary. An increase 
in the time taken for n = 22 to 25 is noted. This was due 
to the fact that during transformation an inflation 
operation was performed along the x-axis to 
accommodate all the robots. Hence, this led to an 
increase in time. 
The average time taken to transform a pattern in the 
second method was computed as 21.40 seconds, slightly 
higher than the first method. It is noted that the values 
plotted on the graph can be divided into two bands. Firstly, 
the set of times that lie above 21 seconds and secondly, 
those that lie below 21 seconds. The first band follows a 
linear trend with a steady decrease. Though the second 
band lies scattered, all set of times in this band lie close to 
20 seconds. 
Working from left to right on the robot axis, the longest 
time taken to transform a pattern was observed when 3 
robots constituted the pattern. This was due to the nature of 
the mathematical transformation in which the 3 robots 
traverse longer distances within the pattern to reach their 
destination coordinates. The transformation time decreases 
steadily until n = 11 due to the fact that the distance 
traversed within the pattern decreases. For n > 11, the set of 
times is scattered closely around the 20 second time line. It 
is evident that the mathematical transformation method is 
consistent and effective for patterns with more than 11 
constituting robots. This is due to the fact that robots 
traversed lesser distances in the pattern. However, the time 
taken for a few configurations (n > 11) is greater than 20 
seconds. This can be accounted for the fact that collision 
avoidance in robots was achieved by temporarily 
 
 
Fig. 4. Simulation studies on Processing and Traer Physics for the second transformation method presented in two stages. . Firstly, a circle to a line 
transformation (First three rows excluding the fourth sequence in the third row). Secondly, a line to circle transformation (Fourth sequence of the third row 
and last two rows).  
 
 
 
decelerating alternative robots in the configuration and 
hence led to a small increase in time. The lowest time taken 
to transform is for n = 20. In this case, the distance 
traversed by robots and collision avoidance deceleration of 
alternative robots in the pattern is minimal.  
It is understood from the graphs that the mathematical 
transformation method employing both macroscopic and 
microscopic parameters is not advantageous for small 
number of robots. For smaller number of robots, the robots 
in the mathematical method traverse more distance within 
the pattern. As the number of robots increase, the distance 
traversed by a robot within the pattern decreases. The 
mathematical 
method 
performs 
better 
than 
the 
macroscopic method for higher number of robots. As the 
number of robots increase, the mathematical method tends 
to be effective since the time taken to transform decreases. 
However, the first method performs consistently for any 
number of robots. 
E. Comparing the methods 
The transformation methods presented in this paper are 
feasible methods for reconfiguring patterns. However, it is 
noted that the mathematical method employing Moebius 
transformation is not strictly macroscopic in nature. The 
microscopic properties of the swarm units are taken into 
consideration. For example, path planning of individual 
robots is necessary to reposition the robots. The method is 
not advantageous for small number of robots in the pattern. 
Moreover the mathematical transformation method is a 
discrete transformation method. Hence, discretizing and 
quantizing the path to reposition are required. This is a 
computationally expensive process unsupported and 
unwarranted on minimal processing swarm units. 
Therefore global planning is required thereby increasing 
wireless communication overheads. A high bandwidth for 
communication 
and 
synchronized 
and 
consistent 
communication with a centralized unit are challenges in 
realizing the mathematical method in real time. 
On the other hand, the macroscopic method considers the 
group behaviour of the swarm system. Hence, individual 
robots need not be addressed, eliminating microscopic 
parameter operations. For example, transformation in the 
first method is obtained by a sequence of operations 
performed on the entire swarm pattern rather than 
considering 
individual 
robot 
path 
planning. 
The 
macroscopic method is observed to be consistent in the 
time taken for transformation, and is also a continuous 
method thereby reducing computations for individual 
robot planning. This method would hence offer better 
synchronization between the swarm units since local 
planning is sufficient. Hence wireless communication 
overheads are relatively less compared to the mathematical 
method. 
By implementing a macroscopic method in a real time 
robot system, planning overheads for individual robots 
could 
be 
minimized. 
However, 
a 
mathematical 
transformation function is advantageous since it belongs to 
an analytical class of tools and mathematical analysis is 
possible. 
In summary, the simulation studies confirm the 
feasibility of the proposed methods. The transformation 
cases discussed in Section III are considered in the 
transformation methods. A brief comparison between the 
method employing only macroscopic parameters and the 
method employing both microscopic and macroscopic 
parameters is presented based on the results obtained.  
VII. CONCLUSIONS 
The swarm pattern formation model presented in this 
paper 
considers 
macroscopic 
and 
microscopic 
primitives. The swarm pattern transformation model 
 
 
Fig. 5. Graphs obtained from experiments. (Left) Graph plotted based on the first transformation method. (Right) Graph plotted based on the second transformation  
method.  
 
 
 
comprising two transformation methods, namely a 
macroscopic 
transformation 
and 
mathematical 
transformation method are proposed. The first method is 
a macroscopic parameter method while the second 
considers both, macroscopic and microscopic parameters. 
A formal definition for transformation is presented with 
four special cases of transformation. Elementary and 
geometrical 
transformations 
are 
considered 
by 
repositioning agents. Transformation using both methods 
is achieved by a sequence of operations performed on the 
swarm pattern. The proposed methods are implemented on 
the Processing and Traer Physics environment. A 
comparison between the two methods considering 
transformation time from one pattern to another is 
presented. The simulation studies confirm the feasibility of 
the proposed methods. 
 Future work will include the real time implementation of 
the proposed transformation methods on a swarm robot 
system. The challenges in mapping simulation studies to 
real time robot systems will be studied. Efforts will be 
made to explore continuous mathematical transformation 
methods which are expected to minimize individual robot 
path planning. 
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