SWARM 2019: The 3rd International Symposium on Swarm Behavior and Bio-Inspired Robotics

Swarm Robots Using Lévy Flight in Targets Exploration-Computer Simulation for Performance of Lévy Flight with Constant andUpdated Minimum Movement TimeYoshiaki Katada1†
1Department of Electrical and Electronic Engineering, Setsunan University, Osaka, Japan(Tel: +81-72-839-9148; E-mail: katada @ ele.setsunan.ac.jp)

Abstract: This study tackles the task for swarm robotics where robots explore the environment to detect many targets.
When a robot detects a target, the robot must be connected with a base station via intermediate relay robots for wireless communication. In our previous results, we confirmed that Lévy flight outperformed the usual random walk for
exploration strategy in real robot experiments. This paper investigated the performance of Lévy flight varying minimum
movement time in navigation through a series of computer simulations and proposed the update method of the minimum
movement time. The results suggest that the search efficiency of Lévy flight has an optimal value for minimum movement time and the performance of Lévy flight with the update method fell between the best and worst performance withconstant parameter setting for minimum movement time.
Keywords: Swarm Robotics, Lévy Flight, Navigation, Mobile Robots.

1. INTRODUCTION

each other via wireless communication networks (WCN)[3] due to the multi-hop transmission to achieve collective exploration. As soon as a robot detects a target, theinformation is sent from the robot to the base station via
intermediate relay robots. Therefore, all robots should be“connected” to the base station via WCN.

Swarm Robotics (SR) [1] have attracted much researchinterest in recent years. Generally, the tasks in SR are difficult or inefficient for a single robot to cope with. Thus,
you find overlap between SR and multi-robot systems.
Şahin [2] enumerated several criteria1 for distinguishingswarm robotics as follows:• autonomy: Each robot should be physically embodied
and situated.
• redundancy: Group sizes accepted as swarms is 10 to20.
• scalability: SR system should be able to operate undera wide range of group sizes.
• simplicity: Each robot should employ cheap design,that is, the structure of a robot would be simple and also
the cost of it would be cheap.
• homogeneity: SR system should be composed of homogeneous individuals. This enhances the above 2nd and3rd criterion.
According to 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 thelocal interactions among robots and between the robots
and the environment. Therefore, it is required for SR systems that individuals in swarm show various behaviors
through interactions although the individuals are homogeneous.
The typical control tasks in SR are navigation, aggregation, formation and transport requiring distributed collective strategies [1]. In this paper, we copes with a target
detection problem, which is one of navigation problems.
In this control task, several robots can communicate with

Our research group investigated communication range
and the number of robots required for a SR network toachieve connectivity based on percolation theory in computer simulation [4]. According to the results obtained in[4], we conducted a series of real experiments [5]. Also,
we must consider the performance of exploration as wellas connectivity of the SR network in a target detection
problem. Therefore, we improve exploration strategies toenhance the performance.
In a target detection problem which we cope with, we
assume that robots have no prior knowledge of the environment. In these scenarios, a random walk seems
to be appropriate for exploration strategy. It has beenreported in the literature [6, 7] that some species show
a specific random walk, Lévy flight [8], when prey aresparsely and randomly distributed. This condition might
correspond to a target detection problem. In our previous
works[9], we conducted a series of real experiments on
a target detection problem by swarm robotic network inorder to investigate the effect of the step size of the random walk and the number of robots. We confirmed thatLévy flight outperformed the usual random walk for exploration strategy in real robot experiments. In computer
simulation [10], we investigated which probability distribution for Lévy flight shows best performance in manytarget detection problem varying its parameters and confirmed that the probability distribution which adopted in[9] was best. For integrating Lévy flight to real robots, wemust set some parameters with regard to the experienceof the real experiments. Those parameters would affectthe performance of Lévy flight as well as the probability† Yoshiaki Katada is the presenter of this paper.

1 Şahin [2] claimed that these criteria should be used as a measure of thedegree of SR in a particular study.

© SWARM 2019 All Rights Reserved.
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3. SIMULATED SWARM ROBOT AND
ENVIRONMENT

distribution.
This paper investigated the performance of Lévy flightin targets detection problem varying one of the parameters described above, minimum movement time, and proposed the update method of it through a series of computer simulations. The paper is organized as follows. The
next section explains Lévy flight. Sect. 3 shows the structure of the simulated mobile robots. Sect. 4 describes thecontroller of the robots and how to implement Lévy flightin the controller, respectively. Sect. 5 investigates the performance of Lévy flight with constant minimum movement time. Sect. 6 investigates the performance of Lévy
flight with the update method of the minimum movement
time. Conclusions are given in the last section.

According to our previous work using the swarm mobile robots [9], the setting of computer simulations was asfollows. Differential wheeled robots (Fig. 2) were used inthis experiment. The robot’s diameter and height are 0.17[m] and 0.075 [m], respectively. The robot is equippedwith four distance sensors located at the front of the bodyfor measuring the distance to other robots and walls, andtwo sensors located at both ends of the body for detecting targets (Fig. 3). The maximum detection ranges ofthe former distance sensor and the latter sensor are 0.3[m] and 0.2 [m], respectively. The robots are assumed tobe equipped with wireless devices, which can compose
wireless ad hoc networks and communicate with eachother via multi-hop path.
The simulated environment is a square arena withwalls (Fig. 4). The length of the wall was set to 20 [m]. At
the lower left corner, a wireless base station was placed.
The communication range of the wireless device was set
to 20 [m]. The connectivity of the wireless communication network is checked based on geometric model [?,4]. At the beginning of each trial, swarm robots were always placed 1 [m] apart at the same initial position, thelower right corner, next to the base station at random orientations (Fig. 4(e)). The cylindrical objects were placed
as targets. The radius of the cylinder is 0.11 [m] andthe height is 0.22 [m]. Targets were uniformly or nonuniformly distributed over the arena. The number of targets was set at T ∈ {84, 168, 336}.
Open Dynamics Engine (ODE) [13] was employed inorder to consider dynamics of robots and the interaction
between robots and environment.

2. LÉVY FLIGHT
A random walk with a constant step size is wellknown. On the other hand, Lévy flight is a random walkwhose step size varies according to a Lévy probability
distribution [8]. Lévy probability distribution for a stepsize, w, is formulated as follows:∫
1 ∞−γqα
Lα,γ (w) =ecos(wq)dq, γ > 0, w ∈ R (1)π 0
where γ is the scaling factor and α (0 < α < 2) is a parameter varying the shape of the probability distribution.
Lévy probability distribution for a step size can be approximated in the following [11]:L(w) ∝ w−α(2)According to the recommendation in [9, 12], we define it
as follows:
L(w) ≡ w−1.2(3)

Probability distribution

Fig. 1 shows Eq. (3) over the range 1 ≤ w ≤ 30. In computer simulation on many target detection problem[10],we compared the performance of Eq. (3) with those of
the other formulations of Lévy probability distribution
and confirmed that Eq. (3) shows the best performancein the control task. Thus, we employ Eq. (3) in this work.

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Fig. 2 Differential wheeled robots in an ODE simulation

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Fig. 3 Ray of the distance sensors (dash line) and thetarget detection sensors (dot-dash line)

Fig. 1 Lévy probability distribution

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distance sensor

obstacle avoidance layer(a)T = 84, uniform(b)T = 84, non-uniform

detectobstacleturn right㻵turn lefttarget exploration layer

explorebase station

stop

㻵

forwardmotorstargetdetectingsensortransmission layertransmit
message㼟

detecttargetturn rightturn left

Fig. 5 Layer structure of SSA

move forward

turnFig. 6 Transition between move phase and rotate phasein navigation(c)T = 168, uniform(d)T = 168, non-uniform

moving forward and turn right (left) rotating clockwise(counter clockwise) at the position. In the obstacle avoidance layer, the detect obstacle module sends messages toeither turn right or turn left according to the sensory inputs from distance sensors described in Sect. 3, in order
to avoid the obstacles which the robot faces. In the transmission layer, the detect target module sends messages tothe transmit messages module and the stop module when
the sensory inputs from the sensors for detecting targets
are beyond a threshold. The transmit messages module
transmits messages to the base station via intermediate
relay robots. The stop module sends messages to its own
motors to stop them.

Targets

Robots

4.2. Implementation of Lévy flight in the SSA
Lévy Flight (Sect. 2) is implemented in the target exploration layer (Fig. 5). The details are described in thefollowing subsections.
For the differential wheeled robots assumed to be usedin this study, it is difficult to simultaneously move forward with a regular step size and rotate in the predetermined direction. Therefore, the whole steps are divided into the rotation phase and the move-forward phase(Fig. 6), between which the transition occurs at 100%.
In the rotation phase, a robot determines the direction ofrotation and selects an angle of rotation randomly from{45, 90, 135} degree. Then, a robot rotates until reaches
the desired angle (the turn right or turn left module inFig. 5). In the move-forward phase, a robot moves forward driving two wheels (corresponding to the forward
module in Fig. 5). A step size in the move-forward phaseis determined according to a Lévy probability distribution
described in Sect. 2. In our previous experiment using theswarm mobile robots [9], the execution time of one step
size in the move-forward phase is set at 6 sec based on the
results in the preliminary experiment. This corresponds
to setting the minimum movement in the move-forward
phase for a constant angular velocity of the wheel. Practically, the execution time of one step size (minimum
movement time) must be determined suitable for the area

Base station(e)T = 336, uniform

Fig. 4 Set up for computer simulation: distribution oftargets

4. CONTROLLER
4.1. Subsumption architecture
Subsumption architecture (SSA) [14] is employed asa format to describe behavior of individuals which compose swarm robotic network according to the setting ofour previous work [9]. Fig. 5 shows a layer structureof SSA implemented in this swarm robots. The SSA toachieve the control task in this study is composed of thefollowing three layers: transmission, obstacle avoidance
and target exploration. A capital I in a circle in Fig. 5
indicates inhibition by which a lower layer is inhibitedwhen an upper layer is activated. Each layer is composedof some modules connected to each other.
Behavior of each layer can be explained as the following; In the target exploration layer, the explore module
sends messages to one of the following three modules:
forward, turn right and turn left, where forward means226

of the environment. In this work, we define w0 as theminimum movement time and investigate the effect of iton the performance of Lévy flight. Therefore, the execution time in the move-forward phase is w0 multiplied bya random value w according to a Lévy probability distribution(Eq. (3)). Precisely, the minimum movement canbe calculated as w0 w multiplied by both the radius of thewheel and the angular velocity.

of the designer in SR until the desired performance is obtained. From the viewpoint of automatic design methods,
automatic parameter setting would be possible as an alternative method. In this section, I propose an automatic
update method of w0 in the search process based on thefrequency of target detection. Additional computer simulations were conducted in order to evaluate the performance of the proposed update method.

5. EXPERIMENT FOR CONSTANTMINIMUM MOVEMENT TIME

6.1. Update rule of w0
w0 for each robot is updated in the following way:
 w0 − ∆w0 , if s = 1 and w0 > w0min
w0 + ∆w0 , if s = 0 and w0 < w0max (4)
w0 ←

w0 ,
otherwise,

5.1. Setting of computer simulations
A series of computer simulations have been conductedvarying the number of robots N ∈ {5, 10, 15, 20} andthe minimum movement time w0 ∈ {1, 5, 10, · · · , 30}[sec/step] described in Sect. 4. One trial ends when
360, 000 steps (3600 sec) are performed. This experiment investigated the performance of Lévy flight formulating target detection rate as T (t)/T , where T (t) is thenumber of targets detected by swarm robots at t time step.
We conducted 50 independent runs varying initial orientations. All results were averaged over 50 runs.

where w0min , w0max and s are the lower, upper value of
w0 and the flag to detect a new target, respectively. ∆w0
was set at 1. According to the results obtained in theprevious section, w0min and w0max were set at 1 and 30,
respectively. At the beginning of each trial, w0 for eachrobot is initialized as w0min . s is defined as follows;s =5.2. Experimental results
Figs. 7 to 9 show the average detection rate at eachtime step for w0 and T with N = 5, 15, 20. The final
detection rate converged to 100 % for all the w0 except
for N = 5. No significant differences in the final detection rate were observed for w0 with N = 15, 20. The
detection rate in the search process (we call this a search
speed in the reminder of this paper) for w0 = 5 was thelargest among all the w0 . The speed was smallest whenw0 = 1 or 30. It becomes smaller with the increase of w0
over the range 10 ≤ w0 ≤ 30. Thus, the optimal value
is identified, which results in the fastest speed over therange of w0 . Increasing w0 had the same effect for thetarget distribution and N .
Fig. 10 shows the average detection rate at each time
step for N and T with w0 = 5, which was the optimalvalue in this experiment. The final detection rate converged to 100 % for all the T except for N = 5. The
speed increased with the increase of T for each N . The
speed for uniform were larger than those for non-uniform.
This result would be valid because targets are relatively
widely distributed for larger T with uniform due to nooverlap between targets in this simulation. This resultsin early detection of targets; more targets are distributednear the initial positions of the swarm robots. The speed
increased with the increase of N for each T .


if a target detected and ∆t > 5
 1,
0,
if ∆t > 10
−1, otherwise,

where ∆t [sec] is the time passed since when the robot
detected the previous target or when ∆t is reset. ∆t isreset at 0 when w0 is updated according to Eq. (4).
6.2. Experimental results
The setting of computer simulations were the sameas those of the previous section, except for updating
w0 . Figs. 11 to 13 show the average detection rate
at each time step for the constant and updating w0 andT with N = 5, 15, 20. The results with the constantw0 ∈ {1, 5, 10, 30} are those in the previous section. The
final detection rate converged to 100 % for the updating
w0 except for N = 5. The speed for the updating w0
was between those of wo = 5 and w0 = 30. They outperformed the worst results for the constant wo although
they did not outperform the best results for the constant
wo at any time step.

7. CONCLUSIONSThis paper investigated the performance of Lévy flightin targets detection problem varying the minimum movement time of Lévy flight and proposed the update methodof it through a series of computer simulations. The results suggest that the search efficiency of Lévy flight has
an optimal value for minimum movement time and theperformance of the update method fell between those of
the best and worst parameter setting for minimum movement time.
We could improve the update method of minimum
movement time of Lévy flight with regard to setting initial values, e.g., setting not constant values but random
values. Future work will investigate the performance ofthe propose method in a dynamic environment.

6. EXPERIMENT FOR AUTOMATIC
UPDATE OF MINIMUM MOVEMENTTIME
In the previous section, we found that the target search
speed has the optimal value for w0 . Probably, the optimal value depends on the area of environment. The optimal value for w0 would be determined based on the intuition and experience of the designer or tuned by hand

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Detection rate

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Fig. 9 Average detection rate for each time step with
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w0=10
w0=30
updating w00

100000200000
Step

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

3000000100000200000Step

Detection rate

Detection rate

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

3000000100000200000Step

Detection rate

Detection rate

300000100000200000Step

300000100000200000Step

300000(d)T = 168, non-uniform

Detection rate

Detection rate

200000
Step

w0=1
w0=5
w0=10
w0=30
updating w0
0

w0=1
w0=5
w0=10
w0=30
updating w00

100000

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

(d)T = 168, non-uniform
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

300000(c)T = 168, uniform

w0=1
w0=5
w0=10
w0=30
updating w00200000
Step

w0=1
w0=5
w0=10
w0=30
updating w0(c)T = 168, uniform
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

100000(b)T = 84, non-uniform

w0=1
w0=5
w0=10
w0=30
updating w00300000

w0=1
w0=5
w0=10
w0=30
updating w0(b)T = 84, non-uniform
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

200000Step(a)T = 84, uniform

Detection rate

Detection rate(a)T = 84, uniform

100000300000

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

w0=1
w0=5
w0=10
w0=30
updating w00

100000200000Step

300000(e)T = 336, uniform(e)T = 336, uniform

Fig. 12 Average detection rate for constant and updating
w0 with N = 15

Fig. 13 Average detection rate for constant and updating
w0 with N = 20231