Noname manuscript No.
(will be inserted by the editor)

Evolutionary Design Method of Probabilistic Finite State
Machine for Swarm Robots Aggregation
Yoshiaki Katada

Received: date / Accepted: date

Abstract This paper proposes to use evolutionary computations to determine the parameters of probabilisticﬁnite state machine controllers for swarm robots. The
robots are evolved to perform an aggregation task. This
problem was formulated as an optimization problem
and solved by the PSO. Several computer simulations
were conducted to investigate the validity of the proposed method. The results obtained in this paper show
to us that the proposed method is useful for the aggregation problem and the best evolved controllers arefeasible as well as interpretable. This would be transferable to real swarm robots problems.
Keywords Evolutionary Swarm Robotics · Aggregation · Probabilistic Finite State Machine
1 INTRODUCTION
Swarm Robotics (SR) [1,2] have attracted much research interest in recent years. Generally, the tasks in
SR are diﬃcult or ineﬃcient for a single robot to copewith. Thus, SR and multi-robot systems overlap eachother. Şahin [3] enumerated several criteria1 for distinguishing swarm robotics as follows:– autonomy: Each robot should be physically embodied and situated.
– redundancy: Minimum group size accepted as swarms
is 10 to 20.
Y. Katada
Setsunan University, 17-8 Ikeda-Nakamachi, Neyagawa, Osaka 572-8508, JAPAN
Tel.: +81-72-839-9148
Fax: +81-72-839-9148
E-mail: katada @ ele.setsunan.ac.jp
1Şahin [3] claimed that these criteria should be used as ameasure of the degree of SR in a particular study.

– scalability: SR system should be able to operate under a wide range of group sizes.
– simplicity: Each robot should employ cheap design,that is, the structure of a robot would be simple andthe cost would be cheap.
– homogeneity: SR system should be composed of homogeneous individuals. This enhances the above 2nd
and 3rd criterion.
Following the last criterion, homogeneous controllers
for individuals are desirable for SR systems. Additionally, this approach does not assume the existence of an
explicit leader in swarm robots due to the above criteria. This results in that a collective behavior emerges
from the local interactions among robots and between
the robots and the environment. Therefore, individuals
in SR systems are required to show various behaviors
through those interactions although the individuals arehomogeneous.
Following the taxonomies proposed by Brambilla[2],we have the most common design methods to develop
swarm robotics systems: behavior-based design and automatic design. In behavior-based design methods, there
are further three categories: probabilistic ﬁnite statemachine design method, virtual physics-based design
method and others. On behavior-based design methods,
such swarm robotics systems are developed by handof the designer until the desired collective behavior is
obtained. Such design for collective behavior requires
more eﬀort than that for single robot behavior does because the interactions are so complex among robots andbetween the robots and the environment.
In automatic design methods, there are two categories: reinforcement learning and evolutionary robotics.
In reinforcement learning (RL), an agent aims at building value functions in the process of learning while getting rewards from the environment. The optimality is

2

Yoshiaki Katada

guaranteed based on Markovian decision process, e.g.,
for Q-learning[4]. The existence of other robots may violate such an assumption. From the viewpoint of a search
problem, RL is considered as a single point search. The
ability to search the optimal solution is not so high.
In evolutionary robotics (ER)[5], evolutionary computation (EC) is generally applied to design artiﬁcial
neural networks (ANNs) for robot control. In the caseof SR [1], controllers are commonly designed for all therobots or a controller is individually designed for eachrobot in the context of cooperative coevolution. The
best evolved neural controller may show good performance on robot control tasks in simulated environment.
However, such controllers suﬀer from the gap between
simulated and real environment when the controllers
are transferred to real environment[6–10]. This is due touncertainty derived from real environment: noise, complexity and nonlinear dynamics of ANNs. When we adjust the parameters of ANNs due to such uncertainty,
we ﬁnd it diﬃcult to interpret the collective behavior
emerged from the best evolved neural controller.
In order to overcome these problems, we propose touse probabilistic ﬁnite state machines instead of a neural controller on swarm robotics tasks in the context
of ER. That is, we apply evolutionary computations
to design a probabilistic ﬁnite state machine for swarm
robot control. Such automatic design of probabilistic ﬁnite state machines would drastically decrease the eﬀort
for the design of collective behavior of swarm robots.
The paper is organized as follows. The next section describes an EC, particle swarm optimization applied to determine the parameters of probabilistic ﬁnite
state machines. Section 3 describes probabilistic ﬁnite
state machines adopted in a swarm robots control problem. Section 4 deﬁnes the swarm robot control problem,
aggregation[11–13], where the probabilistic ﬁnite statemachines designed by the PSO are evaluated. Section
5 gives the results of our computer simulations. Section 6 studies scalability of the obtained probabilisticﬁnite state machines. Conclusions are given in the lastsection.

+c2 · rand() · (gbestj − xkij ),
k+1
,
xk+1
= xkij + vij
ij

(2)
(3)

where rand() is a random number between [0, 1], w,
c1 and c2 are weight factors for the terms. xkij is thej-th variable in the position vector of the i-th particlekat iteration k, vij
is the j-th variable in the velocity
vector of the i-th particle. pbestij is the j-th variable inthe best position visited so far by the i-th particle andgbestj is the j-th variable in the best position visited
so far by all the particles. Each particle is evaluated by
the objective function f (·) to be optimized.

3 probabilistic finite state machine (PFSM)
3.1 Deﬁnition of the PFSM
There are many literatures in swarm robotics employing
a controller to be considered as a class of ﬁnite state machines with probabilistic transitions. Brambilla[2] classiﬁed those controllers as probabilistic ﬁnite state machines (PFSMs). There are several variants of the PFSMs employed in SR.
In the PFSMs employed in this study, we set several
states for an individual robot. Let S = {S1 , S2 , · · · , Sm }be a ﬁnite set of m states. Actions in each state may
correspond to a module[15], a primitive of the domain
ontology[16] or a basic behavior[12]. A state transits
to the next state with a probability (Fig. 1). We candescribe those probabilities as a state transition probability matrix. This is deﬁned as follows:m∑

pij = 1,(4)

j=1

Kennedy and Eberhart [14] introduced particle swarm
optimization (PSO) that was based on social behaviorof ﬁsh schooling or bird ﬂocking. The PSOs have been
applied to various ﬁelds so far due to the search performance and the simplicity of their algorithms.
The optimization problems solved by the PSOs inthis paper are formulated as follows:
f (x),

k+1
kvij= w · vij
+ c1 · rand() · (pbestij − xkij )P = {pij ∈ ℜ | 0 ≤ pij ≤ 1,

2 Particle Swarm Optimization (PSO)minxwhere x = (x1 , x2 , · · · , xL )T ∈ ℜL .
In the PSOs, a solution in the search space is called,“particle”. All the particles in a population have theirown positions and velocities. At each iteration, particles are updated. The update method of positions andvelocities in the PSO is as follows:(1)

i, j ∈ {1, 2, · · · , m}},where P ∈ ℜm×m is a matrix in which each element pij
is the transition probability from state i to state j.

3.2 Formulation as an optimization problem
In the proposed method, the elements of a state transition probability matrix P are optimized as the design

Evolutionary Design Method of Probabilistic Finite State Machine for Swarm Robots Aggregationp33

4 Swarm robot control problem and theobjective formula

p22
p32

S3

Swarm robot control problem in this paper is set to beaggregation[11] where robots aggregate to an arbitrary
position as closely as possible from their initial positions. Aggregation is considered to be a building block
for applications so that an aggregation task is often
used as a case study [12][13].
In this paper, two performance measures were employed according to the reference[13]: dispersion metricand cluster metric. The dispersion metric to be minimized in this paper is as follows:N
1 ∑ (t)||p − p(t) ||2 ,
(12)
f= 2
4r i=1 i

S2

p23
p13

p21
S1

p31

p12

p11
Fig. 1 State transition diagram of the PFSM

variables in an optimization problem by the PSO. The
optimization problem is formulated as follows:minf (pij ),(5)subject to 0 ≤ pij ≤ 1, i, j = 1, 2, · · · , m,m∑
pij = 1, i = 1, 2, · · · , m.

(6)
(7)

j=1

According to Eq. (4), the number of design variables is
L = m × m. In the process of the PSO, the variables of
the initial particles x0 s are generated in the range [0, 1].
According to Eqs. (2) and (3) for k > 0, xk s would
violate the constraints, Eqs. (6) or (7). Thus, we must
consider these two constraints in order not to leave xk
unfeasible.
These constraints can be replaced with the following
constraints.
0 ≤ pij ,pij ≤ 1 ∩(8)m∑

pij = 1,

i, j = 1, 2, · · · , m.

(9)

j=1

In order for pij to satisfy Eq. (8), we apply mirroringto pij as follows:{−pij : if pij < 0p′ij =(10)
pij : otherwise.
p′ij is kept in the design variables for next update. After
that, to satisfy Eq. (9), we normalize p′ij as follows:
p′ij
p′′ij = ∑m

′ .
j=1 pij

3(11)

p′′ij is used only for evaluations of the objective function
and is not kept in the design variables for next update.

where r is the radius of a robot, N is the number of(t)robots, pi is the position of the i-th robot in environment at time t, p(t) is the center of the positions of all
the robots. In two dimensional case, the minimum valueof f for each number of robots is given geometrically in[17].
The cluster metric is as follows:
number of robots in the largest cluster at time t
c=
,(13)
N
where a cluster is deﬁned as a maximal connected subgraph, where two robots are adjacent if the distance
between the centers of them is less than 4r.
The above two metrics are evaluated when t is theﬁnal time step in a trial.
5 Computer Simulation
5.1 Setting of computer simulations
5.1.1 Simulated swarm robot and environment
According to the work using the swarm mobile robots[18], the setting of computer simulations was as follows.
Diﬀerential wheeled robots (Fig. 2) were used in this experiment. The robot’s diameter (D) and height are 0.17[m] and 0.075 [m], respectively. The robot is equippedwith four infrared distance sensors located at the frontof the body for measuring the distance to other robots
and walls (Fig. 3 (right)), four light sensors located at
the center for detecting other robots’ light source (Fig.
3 (left)) and an omni-directional light source locatedat the center. The light source is always on. The maximum detection range of the infrared sensor is 0.3 [m].
The maximum detection range of the light sensor is R[m], which has directional characteristics (Fig. 4). The
light sensor outputs the following value:{cos θh · (R − dh )/(R − D/2) (dh ≤ R)
Ih =
(14)
0
(dh > R)

4

Yoshiaki Katada

Fig. 2 Setup for swarm mobile robots

Fig. 5 Set up for computer simulation: Initially, swarm
robots are always placed at the center.

Fig. 3 Sensor conﬁgurations: a triangle shows a sensor for
light sources, a rectangle shows a distance sensor for robots
and walls and a red line shows the sensor direction.

30°

20°

10° 0°

10°

20°
20

40°

30°
30
50°

q

60°

60°

d

70°

70°

80°

80°

90°
100

5.1.2 Controller

40°
40

50

90°
80

60

40

20

0

20

40

60

80

100 [%]

Fig. 4 Relative sensitivity of the light sensor: a red line shows
the light sensor direction, a green curve shows directional
characteristics and a yellow circle shows a light source.

Is =

∑

Ih

(15)

h∈H

where dh is the distance from the light sensor to thecenter of the h-th light source, θh is a relative angle
from the light sensor direction to the h-th light source(Fig. 4) and H is the number of light sources.
The simulated environment is a square arena withwalls (Fig. 5). The length of the wall was set to 10 [m].
A series of computer simulations have been conductedvarying the number of robots2 No ∈ {10, 20} and the2light sensor range R ∈ {1.0, 1.5, 2.0, . . . , 5.0}. At the beginning of each trial, swarm robots were always placed 1[m] apart at random orientations. One trial ends when
300 steps (300 sec) are performed. We conducted 10
independent runs varying initial orientations.
Open Dynamics Engine (ODE) [19] was employed inorder to consider dynamics of robots and the interaction
between robots and environment. Control cycle was set
to 1 sec and the physics is updated every 0.01 sec.

No denotes the number of robots when the PSO was applied to design the PFSM.

Fig. 6 shows the controller employed in this experiment.
This controller is composed of the PFSM, exploration
layer, avoidance layer, approach layer and stop layer.
Some layers have the following modules: forward,
turn right or turn left, where forward means moving forward and turn right (left) rotating clockwise (counterclockwise) at the position. In the approach layer, theapproach module sends messages to one of the following modules: forward, turn right or turn left according to the sensory inputs from light sensors describedin Section 5.1.1 in order to approach to light sources.
In the obstacle avoidance layer, the avoidance module
sends messages to either turn right or turn left according to the sensory inputs from infrared distance sensors
in order to avoid the walls or other robots which therobot faces. The exploration layer can be explained as
the following; In the random walk module, the wholesteps are divided into the rotation phase and the moveforward phase [18]. In the rotation phase, the controller
determines the direction of rotation and selects an angleof rotation randomly from {45, 90, 135} degree. Then,a robot rotates until reaches the desired angle. In the
move-forward phase, a robot moves forward driving two
wheels for a few seconds.

Wait

Evolutionary Design Method of Probabilistic Finite State Machine for Swarm Robots Aggregation

p33

stop

p22
p32

Repel

forward

㻵

Wait

p23

turn rightturn left

p13

motorsapproach layer

lightsensorApproachstop layer

approach5

p21

obstacle
distance
sensorRepel

turn right
avoidance㻵avoidance layer

turn leftp31

㻵turn right

exploration layer

turn left

Fig. 6 Subsumption architecture[15] with the PFSM

p12

p11

forwardrandomwalk

ApproachFig. 7 State transition diagram of the PFSM for aggregation

In the PFSM (Fig.7), we have three states: {Approach, 5.2 Experimental Results
Wait, Repel}. This takes its inspiration from the referTable 1 shows the best objective function value at theence [12]. Thus, L = 3 × 3 = 9. The details are as
ﬁnal iteration for each light sensor range, where thefollows;ory denotes the geometrical minimum values of f givenin [17]. Thus, the nearly optimal values were obtained– Approach:when R ≥ 2.5 for No = 10 and No = 20.
– If the controller detects lights emitted by otherFig. 8 shows the results for each iteration with No =robots, the robot moves to the brightest direc10. In Fig. 8(b), the inverse of c is plotted for the cluster
tion based on Eq.(15).
metric according to the decrease of f in Fig. 8(a). In– If the controller detects not other robots but
Fig. 8, the lines for R ≥ 2.5 are overlapped so that only
walls, the avoidance module becomes active.
the upper line, which is for R = 5.0, can be seen. f s– If the controller detects neither robot or wall,decreased early iterations when R ≥ 2.0. f s were not
the random walk module becomes active.
improved when R = 1.0 and 1.5. The cluster metric,– W ait: the robot stops immediately.
1/c converged toward 1.0 when R ≥ 2.5. That is, robots– Repel:could aggregate into one cluster when R ≥ 2.5. These– If the controller detects other robots or walls,
tendencies are also conﬁrmed for No = 20 in Fig. 9.
the avoidance module becomes active.
For R, the lower bound for good performance was– If the controller detects neither robot or wall,
found to be 2.5. Robots aggregate into a few clusters
the random walk module becomes active.
through observing the behavior when R < 2.5. After
those aggregations, robots rarely change into one cluster due to the detection range limitation of the sensor.
5.1.3 Setting of the PSO
Computer simulations were conducted using particlesof size 20. The number of design variables was L = 9
as mentioned above. The PSO was employed to determine the PFSM parameters. The parameters of the
PSO adopted in this experiment are as follows: c1 =2.0, c2 = 2.0, w = 0.5 in Eq.(2). Each run lasted 200 iterations. f (Eq.(12)) was employed as an objective function. 1/c (Eq.(13)) was measured as the cluster metric.
We conducted 10 independent runs for each condition.
All results were averaged over 10 runs.

Table 1 f at the ﬁnal iteration for each No and R in 10 runs
R = 1.0
R = 1.5
R = 2.0
R = 2.5
R = 3.0
R = 3.5
R = 4.0
R = 4.5
R = 5.0
theoryNo = 10432.47
192.95
25.78
13.61
13.63
13.67
13.76
13.66
13.67
13.50

No = 201640.34725.30
69.40
55.35
55.36
55.34
55.19
55.23
55.28
54.65

6

Yoshiaki Katada

600
500

f

400
300
200

f

R=1.0
R=1.5
R=2.0
R=2.5
R=3.0
R=3.5
R=4.0
R=4.5
R=5.0

100
0
50

100
150
Iterations(a) Dispersion metric

50

100
150
Iterations(a) Dispersion metric

2.5
2.0

R=1.0
R=1.5
R=2.0
R=2.5
R=3.0
R=3.5
R=4.0
R=4.5
R=5.0

3.5
3.0

1/c

1/c

3.0

2.5
2.0

1.5

1.5

1.0

1.0
50

100
Iterations(b) Cluster metric

150

200

Fig. 8 Objective function value and cluster ratio for each
iteration with No = 1050100
Iterations(b) Cluster metric

0.8

0.8
Probability

1.0

0.6
0.4
0.2

200

0.6
0.4
0.2

p12

p13

p21

p22

p23

p31

p32

0.0
p11

p33

p12

p13

(a) R = 2.5

p21

p22

p23

p31

p32

p33

(a) R = 2.5

1.0

1.0

0.8

0.8
Probability

Probability

150

Fig. 9 Objective function value and cluster ratio for each
iteration with No = 20

1.0

0.6
0.4
0.2
0.0
p11

200

4.0
R=1.0
R=1.5
R=2.0
R=2.5
R=3.0
R=3.5
R=4.0
R=4.5
R=5.0

3.5

0.0
p11

R=1.0
R=1.5
R=2.0
R=2.5
R=3.0
R=3.5
R=4.0
R=4.5
R=5.0

200

4.0

Probability

2000
1800
1600
1400
1200
1000
800
600
400
200
0

0.6
0.4
0.2

p12

p13

p21

p22

p23

p31

p32

p33

(b) R = 5
Fig. 10 State transition probabilities obtained for No = 10
in 10 runs

0.0
p11

p12

p13

p21 p22 p23 p31 p32 p33(b) R = 5

Fig. 11 State transition probabilities obtained for No = 20
in 10 runs

Evolutionary Design Method of Probabilistic Finite State Machine for Swarm Robots Aggregation

In the rest of the paper, R = 2.5 and 5 were employed for further analyses. Figs. 10 and 11 show thestate transition probabilities of the PFSM obtained for
No = {10, 20} and R = {2.5, 5} in 10 runs. The indices of p correspond to those in Fig. 7. The darker theline is, the smaller its f is. The red line shows thoseof the PFSM in the best run. We can ﬁnd a tendency
among those graphs: pi1 is high. We explain this fromthe viewpoint of the aggregation task because pi1 is a
probability at which a state transits to the state, Approach. For the aggregation task, Approach is executed
frequently. Additionally, p22 and p32 in some runs are
relatively high compared with the other probabilities,
p23 , p33 . The PFSMs obtained in all the runs are not
the same for each (No , R) although there were no large
diﬀerences among the runs in the objective function
values (Table 2). This means that the nearly optimal
PFSM is not unique.
Figs. 12 and 13 show the state transition diagramof the best evolved PFSMs for No = {10, 20} and R =

0.034

0.484

Table 2 f at the ﬁnal iteration for each No and R = 2.5, 5.0
in 10 runs

statistics
R = 2.5
R=5

best
13.56
13.54

No = 10avg
worst
13.61 13.67
13.67 13.92

0.208

0.047

0.047

0.159
0.030

Wait

Repel

0.084

Wait

0.088

0.469
Approach
0.057

0.757
Approach
0.012

0.923

0.939

0.900

(a) R = 2.5(a) R = 2.5

0.713

0.087

0.727

0.269

No = 20avg
worst
55.35 55.69
55.28 55.91

0.004

0.004
Repel

best
55.12
55.00

{2.5, 5}, respectively. These correspond to the red lines
in Figs. 10 and 11. Commonly, the PFSMs have the highprobability p11 , where Approach transits to Approach.
Besides this, they have the low probabilities p12 and
p13 , where Approach rarely transits to Wait or Repel.
This would make it possible to aggregate more closely.
These PFSMs have their unique characteristics and nocommon structure was found for No with the same R.
By seeing the detail of the PFSM obtained by evolutionary design, we can explain the characteristics of thecontroller and be convinced by the collective behaviorof the swarm.

0.758
Repel

7

0.011

0.363
0.773

Wait

Repel

0.044

Wait

0.032

0.031

0.276
Approach
0.007

0.593
Approach
0.043

0.140

0.961

0.926

(b) R = 5(b) R = 5

Fig. 12 State transition diagram of the best evolve PFSM
with No = 10

Fig. 13 State transition diagram of the best evolved PFSM
with No = 20

8

Yoshiaki Katada

6 Scalability of the best evolved controller

important to achieve robot control tasks in the PFSM
approach. The concept called domain ontology and low
This section investigates the scalability of the best evolved level specification by Pfeifer in [16] would be helpful.
controllers obtained in the previous section. Additional
For basic collective behavior [2, 20–22], several primicomputer simulations were conducted using the same
tive actions have been proposed. These could be also
setting as the one in Section 5.1.1. The best evolved
useful for the proposed method.
controllers for No ∈ {10, 20} were evaluated varying
As conﬁrmed in the last two sections, the best evolved
the number of robots N ∈ {10, 20, 30, 40}. We conPFSM is interpretable as well as scalable. This means
ducted 20 independent runs for each parameter setthat the PFSM would be transferable to real environting. Tables 3 and 4 show the average values of f inment and readjustable after its transfer. Future work
20 trials. In each table, theory denotes the geometrical
will investigate these potential abilities of the propose
minimum value for each N [17], as referred in Section
method in real robot experiments. In literature men5.2. Robots aggregated into a single cluster and thentioned above, there are several swarm robotics control
f s show the nearly minimum values for almost all Ntasks where the controller with the PFSM was employed
except for (N, R) = (40, 2.5). Even when N is more
[20–22]. The proposed method will be applied to thosethan No , the controller shows good performances. Surcontrol tasks.
prisingly, there were no large diﬀerences between when
No = 10 and 20 in f s. When R = 2.5 and N = 40,
References
robots aggregated into a few clusters.
Table 3 Performance of the best evolved controllers with
No = 10 on fN
R = 2.5
R = 5.0
theory

10
13.86
13.82
13.50

20
57.44
56.39
54.65

30
131.8
127.7
123.5

40
4625.97241.14220.33

Table 4 Performance of the best evolved controllers with
No = 20 on fN
R = 2.5
R = 5.0
theory

10
14.29
13.95
13.50

20
56.44
56.71
54.65

30
129.4
128.1
123.5

40
4868.77239.71220.337 Conclusions
This paper proposed to use evolutionary computations
to determine the parameters of a probabilistic ﬁnite
state machine for a swarm robot controller. Several
computer simulations were conducted to investigate thevalidity of the proposed method. The results obtainedin this paper show that the proposed method is useful for the aggregation problem and the best evolved
controllers are feasible.
The proposed method depends much on primitive
actions initially prepared for states of the PFSMs. Generally, determination or selection of such primitives is

1. Trianni V (2008) Evolutionary swarm robotics. SpringerVerlag, Berlin
2. Brambilla M, Ferrante E, Birattari M, Dorigo M (2013)
Swarm robotics: a review from the swarm engineering perspective. Swarm Intelligence 7(1):1-41
3. Şahin E (2005) Swarm robotics: from sources of inspiration to domains of application. In: E. Şahin, WM. Spears(Eds.) Swarm Robotics, Springer, pp.10-20 (Lecture Notes
in Computer Science 3342)
4. Sutton RS, Barto AG (1998) Reinforcement learning: anintroduction. MIT Press, Cambridge
5. Nolﬁ S, Floreano D (2000) Evolutionary robotics: the biology, intelligence, and technology of self-organizing machines. MIT Press, Cambridge
6. Brooks RA (1992) Artiﬁcial life and real robots. In: F.J.
Varela and P. Bourgine (Eds.) Toward a Practice of Autonomous Systems, Proceedings of the First European Conference on Artiﬁcial Life, MIT Press, pp.3-10
7. Jakobi N (1997) Half-baked ad-hoc and noisy: minimal
simulation for evolutionary robotics. In Proceedings of the
Fourth European Conference on Artiﬁcial Life:348-357
8. Miglino O, Lund HH, Nolﬁ S (1995) Evolving mobile
robots in simulated and real environments. Artiﬁcial Life
2:417-434
9. Keymeulen D, Iwata M, Konaka K, Suzuki R, Kuniyoshi
Y, Higuchi T (1998) Oﬀ-line model-free and on-line modelbased evolution for tracking navigation using evolvablehardware. In: P. Husbands, JA. Meyer (Eds.) Evolutionary Robotics, First European Workshop EvoRobot 98,
Springer, pp.211-226 (Lecture Notes in Computer Science
1468)
10. Katada Y, Ohkura K (2006) An update method of computer simulation for evolutionary robotics. In: T. Arai, et
al (Eds.) Intelligent Autonomous Systems 9 IOS Press,
pp.357-364
11. Garnier S, Jost C, Jeanson R, Gautrais J, Asadpour M,
Caprari G, Theraulaz G. (2005) Aggregation behaviour asa source of collective decision in a group of cockroachlike-robots. In: M.S. Capcarrere, A.A Freitas, P.J. Bentley,
C.G. Johnson, J. Timmis (Eds.) Advances in Artiﬁcial Life,
Springer, pp.169-178 (Lecture notes in Computer Science3630)

Evolutionary Design Method of Probabilistic Finite State Machine for Swarm Robots Aggregation
12. Soysal O, Bahçeci E, Şahin E (2007) Aggregation inswarm robotic systems: evolution and probabilistic control.
Turkish Journal of Electrical Engineering and Computer
Sciences 15(2):199-225
13. Gauci M, Chen J, Li W, Dodd TJ, Groß R (2014) Selforganised aggregation without computation. The International Journal of Robotics Research 33(9):1145-1161
14. Kennedy J, Eberhart RC (1995) Particle swarm optimization. In Proceedings of IEEE International Conference
on Neural Networks, pp.1942-1948
15. Brooks R (1986) A robust layered control system for amobile robot. IEEE Journal of Robotics and Automation
2(1):14-23
16. Pfeifer R, Scheier C (1999) Understanding intelligence.
MIT Press, Cambridge
17. Graham RL, Sloane NJA (1990) Penny-packing and twodimensional codes. Discrete & Computational Geometry5(1):1-11
18. Katada Y, Nishiguchi A, Moriwaki K, Watakabe R (2016)
Swarm robotic network using Lévy ﬂight in target detection
problem. Artiﬁcial Life and Robotics 21(3):295-301
19. Open Dynamics Engine (ODE). http://ode.org/.
20. Nouyan S, Campo A, Dorigo M (2008) Path formation ina robot swarm – self-organized strategies to ﬁnd your way
home. Swarm Intelligence 2(1):1-23
21. Liu W, Winﬁeld A, Sa J (2009) A macroscopic probabilistic model of adaptive foraging in swarm robotics systems.
In Proceedings of 6th Vienna International Conference onMathematical Modelling
22. Labella TH, Dorigo M, Deneubourg JL (2006) Divisionof Labor in a group of robots inspired by ants’ foraging
behavior. ACM Transactions on Autonomous and Adaptive
Systems 1(1):4-25

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