PSO-based Search Mechanism in DynamicEnvironments: Swarms in Vector Fields

Palina BartashevichFaculty of Computer Science

University of MagdeburgGermany

Email: palina.bartashevich@ovgu.de

Luigi GrimaldiFaculty of Computer Science

University of MagdeburgGermany

Email: luigi.grimaldi@st.ovgu.de

Sanaz MostaghimFaculty of Computer Science

University of MagdeburgGermany

Email: sanaz.mostaghim@ovgu.de

Abstract—This paper presents the Vector Field Map PSO(VFM-PSO) as a collective search algorithm for aerial micro-robots in environments with unknown external dynamics (such aswind). The proposed method is based on a multi-swarm approachand allows to cope with unknown disturbances arising by thevector fields in which the positions and the movements of theparticles are highly affected. VFM-PSO requires gathering theinformation regarding the vector fields and one of our goalsis to investigate the amount of the required information for asuccessful search mechanism. The experiments show that VFM-PSO can reduce the drift and improves the performance of thePSO algorithm despite incomplete information (awareness) aboutthe structure of considered vector fields.

I. INTRODUCTION

Due to its computationally simple characteristic, ParticleSwarm Optimization (PSO) can be used as an efficient searchmechanism in the context of Swarm Robotics, in which a largenumber of robotic systems are used to collectively search forcertain points in an unknown environment [26]. In this paperwe address the application of PSO on robotic search scenariosand focus on the dynamic of the environment. We intend tomodel the unknown dynamics of the environment (such aswind, flow, etc.) and propose a new PSO-based approach foraerial micro-robots which can cope with such dynamics.

PSO in robotic search scenarios has been studied in theliterature: Hereford et al. in [12] embedded PSO into a swarmof robots, where each robot behaves as one particle and movesbased on the update equations of PSO, makes a measurementand then broadcasts to the other robots in the swarm. Otherauthors have investigated PSO (e.g. [6], [15], [24], [23]) withmultiple (simple) robots for search applications. Pugh et al. in[24] explored using PSO on problems in noisy environments,focusing on unsupervised robotic learning. They used the PSOto evolve a robot controller that avoids obstacles and finds apre-defined source. Malakar and Sinha in [20] proposed PSOin Potential Fields for obstacle avoidance. Jatmiko et al. in[15], [14], [16] used mobile robots for plume detection, i.e.odor source localization problems, by using a modified PSOmethod with the goal to control the robots. In addition, theinfluence of turbulence and wind variations were studied. Akatand Gazi in [1] proposed a version of the PSO for robotswarms that uses dynamic neighborhoods and asynchronousupdates.

The influence of the environment (wind, flow, etc.) on therobotic search mechanism has a large impact on scenariosusing aerial micro-robots. Dealing with a noisy and windyenvironment has been studied in other context presented by[27], in which the authors proposed stable outdoor flight withaerial robots following the leader in tight formations. Duarteet al. in [7] demonstrated a swarm of aquatic surface robotswith evolved network-based controllers operating in a realand uncontrolled environment for such collective tasks ashoming, clustering, dispersion, and area monitoring. In [3], [2]a new optimization technique based on atmospheric motionof small air particles with additional terms in the velocityupdate equation following Newton’s second law of motionwas introduced, but not applied to robot scenarios. However,none of these approaches study collective search algorithmsin robotics.

PSO has been extensively studied in the context of dynamicoptimization (e.g. [4], [18], [13], [10], [25], [29], [22]). Theproposed approach in our paper is different from dynamic op-timization as we consider the external dynamics independentfrom the fitness landscape. Considering both dynamic opti-mization and external disturbances, Wang et al. in [28] studieda new approach in the context of multi-agent systems andcontrol theory, where optimization with disturbance rejectionis archived by internal-model and Lyapunov-based algorithm.PSO with disturbance term (PSO-DT) has been studied by [9]which modifies the velocity by adding a small random valueto overcome the stagnation in local optima.

The main goal of this paper is to model the dynamics ofthe environment by using Vector Fields which are inspired bythe nature model of wind. While this model is designed to beapplied in robotics search applications using aerial roboticssystems, this paper only addresses the PSO algorithm and theinfluence of vector fields. Application in robotics scenariosrequires further investigations regarding the robotic move-ments among the others. To our knowledge, this is the firsteffort to study PSO-based search mechanism under influenceof unknown external perturbations using Vector Fields.

In vector fields, the movement and the positions of theindividuals are highly affected by additional costs comingfrom the environment. In certain vector field environmentswith strong flow in the opposite direction of the gradient, it is

978-1-5090-4601-0/17/$31.00 c�2017 IEEE

almost impossible to reach an optimal solution by PSO. Wepropose a new PSO-based search mechanism called VectorField Map PSO (VFM-PSO) which additionally provides theindividuals the ability to cope with unknown disturbancesarising by the vector fields. The proposed method uses theconcepts of a multi-swarm approach ([5], [21]) with two sub-swarms: explorer and optimizer swarms. The explorer swarmis intended to obtain information regarding the unknown vectorfields by following the flow and saving the flow behaviorin a matrix, called Information Map. The optimizer swarmuses the gathered information in the Information Map andcorrects the positions of the individuals accordingly. In theexperiments we analyze the influence of different vector fieldsand the performance of the explorer swarm in gatheringthe information about the vector fields. The main researchquestions to investigate are in which vector fields can theVFM-PSO deliver reasonable results and what percentage ofthe information regarding the vector fields is required in orderto get good results from VFM-PSO. The results confirm someof our hypotheses and reveal that with VFM-PSO we canachieve a faster convergence. The proposed approach doesnot require 100 percentage of the information regarding thevector fields in order to find a good approximation of theglobal optimal solution.

This paper is structured as follows. We describe the back-ground in Section II. Afterwards, we introduce our modelin Section III. Section IV contains the experiments and theresults. The paper is concluded in Section V.

II. BACKGROUND

In this section, we briefly describe the background aboutVector Fields, Interpolation and Particle Swarm Optimization.

A. Vector FieldsThe environment which is studied in this paper is modeled

using a two-dimensional search space. The dynamics aremodeled by vector fields which are considered to be definedon a Cartesian Grid of M ⇥M cells as shown in Figure 1.

Definition 1. A Cartesian Grid is a special case where theelements are unit squares and the vertices are integer points.

More in details, we model the dynamics of the environmentas a two-dimensional constant time-independent vector field.

Definition 2. A planar Vector Field (V F ) is a function thattakes any point (x1, x2) in the space and assigns a vector~

V F (x1, x2) to it:

V F : R2 � D ! R2 (1)(x1, x2) 7! ~

V F (x1, x2).

D is called domain of the vector field and is usually givenas a grid consisting of cells with the vector data either givenat the vertices, edges or cell centers. An interpolation methodis used to create a continuous vector field description for theentire domain. For this reason, we assume that the vector fieldconsists of a finite number of analytic pieces (one for each cell)that are glued together along the grid edges, thus defining acontinuous vector field [8].

B. Interpolation

In order to calculate a continuous vector field, we need touse interpolation on the grid environment [17]. Let us considera point p = (x1, x2) inside one cell of the Cartesian Grid.This cell has four corner points p0,0, p1,0, p0,1, p1,1, each ofwhich has an associated vector ~

V F 0,0 = ~

V F (p0,0), ~

V F 1,0 =~

V F (p1,0), ~

V F 0,1 = ~

V F (p0,1), ~

V F 1,1 = ~

V F (p1,1), respec-tively (as shown in Figure 1).

𝑝0,0

𝑝0,1 𝑝1,1

𝑝1,0

𝑝 𝑉𝐹

𝑉𝐹0,1 𝑉𝐹1,1

𝑉𝐹1,0

𝑉𝐹0,0

𝑢1 1

𝑢2

Fig. 1: Interpolating vectors in a two-dimensional space in-cluding vector field on a Cartesian Grid of 3⇥ 3 cells.

The vector ~

V F at p can be calculated by using the valuesof u1 and u2 and bilinearly interpolating the vectors at thecorners:

~

V F (x1, x2) = (1� u2)(1� u1) ~

V F 0,0 +

(1� u2)u1~

V F 1,0 + u2(1� u1) ~

V F 0,1 +

u1u2~

V F 1,1 (2)

C. Particle Swarm Optimization

PSO is a search mechanism based on the movement ofa population of N particles, each of which is consideredas a potential solution to the optimization problem in a d-dimensional search space defined by S [19]. Each particle i

has a position ~x

i(t) 2 S and a velocity ~v

i(t) at time stept. Furthermore, it has a local memory indicated by ~

P

best(t)which is used to keep the best so-far obtained position whichit has experienced. A globally shared memory ~x

g(t) keepsthe best position found by the particles in the populationand is denoted as global best position. Both local and globalmemories contribute to the velocity of each particle i usingthe following standard PSO update equations:

~v

i(t+ 1) = w~v

i(t) + C1~

�1(~Pbest(t)� ~x

i(t)) + (3)+ C2

~

�2(~xg(t)� ~x

i(t))

~x

i(t+ 1) = ~x

i(t) + ~v

i(t+ 1) (4)

where ~

�1 and ~

�2 are two random vectors uniformly distributed2 [0, 1]n. C1 and C2 are two positive constants called accel-eration coefficients and determine the relative influence of thepersonal and social experiences.

III. VECTOR FIELD PSO

In this section, we introduce our model which is designedas a collective search mechanism based on PSO in the en-vironments with unknown external dynamics. The dynamicenvironment is modeled using vector fields as explained inSection II-A. As the dynamics of environment influences thepositions of the particles, we can calculate the real positionof each particle by adding the value of vector fields to itsmovement as follows:

~x

i(t+ 1) = ~x

i(t) + ~v

i(t+ 1) +KX

k=0

~

V F (pk) (5)

Where pk denotes the intermediate particle positions in thecells along the movement, p0 is dedicated to the position of theparticle at time t: ~xi(t). The value for ~

V F (pk) is calculatedusing interpolation described in Section II-B. As a result notonly the position ~x

i(t) is influenced by the vector fields, butalso the movement of the particle along a certain vector asshown in Figure 2.

𝑣 𝑖(𝑡 + 1)

𝑥 𝑖 𝑡 = 𝑉𝐹(𝑝0)

𝑉𝐹(𝑝1)

𝑝0

𝑝1 𝑝2 = 𝑥 𝑖 𝑡 + 1

Fig. 2: Influence of the vector field on the position and themovement of a particle.

We denote the movement described by Equation 5, VectorField PSO (VF-PSO).

A. VFM-PSO

In order to overcome the influence of the unknown externaldynamics, we propose to use two populations which work inparallel. A modified PSO population (called optimizer swarm)which is dedicated to search the search space for possibleoptimal solutions and a second so-called explorer populationwhich is responsible to gather information about the unknownvector fields in a Map accessible by the PSO population.

B. Explorer Population

The task of the explorer swarm is to gather informationabout the disturbance factor in the external environment,performed by vector fields defined on the Cartesian GridG, which consists of n ⇥ n cells over the search space S.

Obtained information will be used as the correction values forthe modified PSO population.

The explorer population is initialized randomly in the en-vironment and their movement is kept to be simple, i.e. theyjust have to follow the flow. The velocity of each particle isdefined by the corresponding value of the vector field wherethe particle is located, i.e. ~

V F (~xe

i

(t)), and calculated usinginterpolation as described in Section II-B. In order to overcomethe multiple gaps in trajectory, the individuals suppose tomove on a normalized vector || ~V F (~xe

i

(t))||, what providescontinuous moving of the particle from cell to cell. Thus themovement of the explorers is calculated by following equation:

~x

e

i

(t+ 1) = ~x

e

i

(t) +~

V F (~xe

i

(t))

| ~V F (~xe

i

(t))|, (6)

where | ~V F (~xe

i

(t))| is the length of the corresponding vector.In case the particles reach or exceed the boundaries of thesearch space, they are replaced to its boundary [11].

The non-normalized values of the velocity vectors~

V F (~xe

i

(t)) are saved in the matrix M , called InformationMap, and used as the correction value for the optimizer popu-lation. The Map M consists of two matrices M = (Mx

,M

y),which store the correction values in x- and y-direction respec-tively. Each index (i, j) of the matrices Mx

,M

y refers to a cellof the Grid G ⇢ S. Thus a correction value m

i,j

, saved in theInformation Map M under the index (i, j), also determines thecorrection value in the corresponding cell’s area of the searchspace (i, j) 2 G ⇢ S, i.e. the value m

i,j

refers to the cell ini

th row and j

th column of the grid G. Algorithm 1 shows themain concept for the explorer swarm.

Algorithm 1: Area Exploration by the Explorer SwarmInput : number of N

e

explorer individualst = 0Initialize the Explorers in the Search Space S:for i = 1 to N

e

do~x

e

i

(t) = StartPosition(S)endInitialize the Map matrices M

x = M

y = n⇥ n:for i = 1 to n do

for j = 1 to n dom

x

ij

= m

y

ij

= 0end

endwhile Stopping Criterion not fulfilled do

for i = 1 to N

e

dom

ij

= UpdateMap(~xe

i

(t), Map)~x

e

i

(t+ 1) = ~x

e

i

(t) +~

V F (~xei (t))

| ~

V F (~xei (t))|

endSend(Map)

end

The exploration starts with defining random positions ofthe individuals in integer points of the search space S, i.e.

on the vertices of the grid G ⇢ S, where the vectors arealready known. The Information Map M = (Mx

,M

y) isinitialized with zero values (mx,y

i,j

)ni,j

= 0 in both of itsmatrices. By every movement of the explorers, the values ofthe Map matrices are updated. The position of the particle~x

e

i

(t+1) determines the index in the matrices of the Map andthe corresponding cell of the Grid, to which the correction isdefined. In fact, it additionally determines the calculated valueof the correction, i.e. ~

V F (~xe

i

(t)), which we assume to set tothe whole cell of the Grid in the search space.

In order the updates in the Information Map will be acces-sible by the PSO population during the whole optimizationprocess, both populations are run in parallel. The functionSend(Map) transfers the new information gathered by theexplorers to the optimizer swarm at each iteration time step.

C. Optimizer PopulationThis population which works based on a modified PSO,

is dedicated to search the search space for possible optimalsolutions. The particles of this population refer to the gathereddata by the explorer population about the vector fields in theInformation Map. Every vector from the Map is used as acorrection value for the calculated velocity in the PSO. Thecorrections are considered in the following way:

~v

i(t+ 1) = w~v

i(t) + C1�1(~Pbest(t)� ~x

i(t)) +

+ C2�2(~xg(t)� ~x

i(t))�

�KX

k=0

Map(pk), (7)

where pk denotes the intermediate particle positions in thecells along the movement. p0 is dedicated to the position ofthe particle at time t: ~xi(t). Each position pk is dedicated tothe corresponding cell of the Grid G ⇢ S, i.e. pk 2 (i, j) ⇢ G,where (i, j) also defines the corresponding matrices index inthe Map. Therefore, Map(pk) is a correction value, saved inthe matrix M under the index (i, j), which corresponds to thecell where pk is placed.

Thus due to the Explorer Population, PSO particles areaware about some disturbances in the environment, whereMap(pk) 6= 0, and can show resistance to them. In case ifthe corresponding area (denoted to cell) was not investigatedby the explorer, i.e. Map(pk) = 0, the particle behaves as inthe VF-PSO.

Algorithm 2 shows the main concept of our proposedapproach called the Vector Field Map PSO (VFM-PSO):

The Algorithm 2 starts with the random initialization of thepopulation in the search space S. Before moving to the nextpositions, particles receive the information about environmentfrom the Explorer Population through the Map. Accordingto the values in the Map, the particles can correct theirvelocities using Equation 7, i.e. denoted at ComputeVelocity.The positions of the particles are updated as in the Equation5, i.e. UpdatePosition. Additionally, we assign a limit of themaximum velocity value v

max

as a threshold. In case thevelocity vector is larger than v

max

, it will be set to vmax

.

Algorithm 2: Vector Field Map PSOInput : Number of N

p

PSO Individualst = 0Initialize the PSO individuals:for i = 1 to N

p

do~x

i

(t) = StartPosition(S)~v

i

(t) = 0endwhile Stopping Criterion not fulfilled do

Wait for the Map from the Explorer Population:Map = Receive(Map)for i = 1 to N do

~x

g

(t) = FindGlobalBest()~v

i

(t+ 1) = ComputeVelocity (~xi

(t), ~xg

(t), Map)~x

i

(t+ 1) = UpdatePosition(~vi

(t), ~xi

(t))endt = t+ 1

end

In the case that the particles reach or exceed the boundariesof the search space, they get reflected from the boundary intothe search space as described by [11]. We set the maximumnumber of iterations as the Stopping Criterion. As the explorerand optimizer populations are run in parallel, the maximumnumber of iterations for both Algorithm 1 and Algorithm 2 isthe same.

IV. EXPERIMENTS

A. Parameter SettingsThe main idea of this paper is motivated by a real case

scenario of an aerial micro-robotic swarm. The proposedmodel and the corresponding features are meant to providean algorithmic design for the search behavior of the aerialswarms in environments with unknown external dynamics(such as wind). Therefore the goal of the experiments is toprovide a baseline for further realistic tests. We compare threedifferent algorithms, standard PSO [19] (denoted as PSO) asa baseline algorithm without considering vector fields, PSOin the vector fields but without any corrections using theinformation from explorer (denoted as VF-PSO) and VFM-PSO which is the proposed method in this paper. For all ofthe algorithms we select the parameters shown in Table I. Thenumber of explorers N

e

is set for the VFM-PSO approach. Asthe simplification of the model we assume that the particle canat most move two cells, i.e. the velocity limit v

max

is set to2. The influence of the vector fields with this restriction wasillustrated in Figure 2. According to our extensive preliminarytests, inertia weight w is selected to be 0.6.

The movements particles are modeled in a d = 2 di-mensional search space. Three test problems such as Sphere,Ackley and Rosenbrock from the literature are used for theexperiments. These test problems can very well simulate theterrain in which aerial swarms can move and search, whileSphere is only meant for simple tests, Ackely and Rosenbrock

TABLE I: Parameter values

Parameter ValuePopulation size N

p

30Number of explorers N

e

20Dimension d 2Iterations N 100

Inertia weight w 0.6Velocity limit v

max

2Acceleration coefficients C1, C2 1

respectively capture search terrains with lots of local optimaand a flat plateau. For better analysis of the vector fields, wehave shifted the optimal solutions of these problems to (-10,10). The value of the optimal solution is zero the same as thestandard test problems. In the experiments we have selected5 various vector fields (denoted by VF1 to VF5) as shown inFigure 3. The function descriptions of considered vector fieldsare given in Table II.

TABLE II: Function descriptions of the vector fields

”Cross” ~

V F1(x1, x2) = (x2, x1)

”Rotation” ~

V F2(x1, x2) = (�x2, x1)

”Sheared” ~

V F3(x1, x2) = (x1 + x2, x2)

”Wave” ~

V F4(x1, x2) = (� sin(x2), cos(x1 · x2 � x

21))

”Tornado” ~

V F5(x1, x2) = (�x1 � x2, x1)

-15 -10 -5 0 5 10 15

x-values

-15

-10

-5

0

5

10

15

y-va

lues

"Cross"

(a) ~

V F1(x1, x2)

-15 -10 -5 0 5 10 15

x-values

-15

-10

-5

0

5

10

15

y-va

lues

"Rotation"

(b) ~

V F2(x1, x2)

-15 -10 -5 0 5 10 15

x-values

-15

-10

-5

0

5

10

15

y-va

lues

"Sheared"

(c) ~

V F3(x1, x2)

-15 -10 -5 0 5 10 15

x-values

-15

-10

-5

0

5

10

15

y-va

lues

"Wave"

(d) ~

V F4(x1, x2)

-15 -10 -5 0 5 10 15

x-values

-15

-10

-5

0

5

10

15

y-va

lues

"Tornado"

(e) ~

V F5(x1, x2)

Fig. 3: Vector fields

All the experiments are run for 100 times for each testproblem in each of the vector fields. The Map is set to 31by 31 units with the center point being the coordinate (0,0).The search space is defined as x1, x2 2 [�15, 15]. The initialpositions for the particles, for both optimizers and explorers,are randomly generated in the search space.

B. Evaluation Metrics

The results are compared in terms of convergence rate, theso-called Awareness Rate and Success Rate. The AwarenessRate captures the information about the ratio of the particleslocated on units in the Map with known values regardingthe vector field. The Success Rate measures the percentageof successful runs among all simulations. The simulation isconsidered successful, i.e. a near-optimal solution is found, if|F � F⇤| < 0.01, where F⇤ = 0. In other words it indicatesthe percentage of the runs, in which the function value of thebest individual is smaller than a certain threshold (e.g., 0.01)by the end of iterations.

C. Results

Figure 4 illustrates the results of convergence rates andawareness rates for all the test problems in 5 vector fields.The results of the success rates for VFM-PSO and VF-PSOare shown in Table III.

TABLE III: Average values of success rate for VFM-PSO andVF-PSO

Sphere Rosenbrock AckleyVFM- VF-PSO VFM- VF-PSO VFM- VF-PSO

VF1 12% 9% 10% 7% 8% 12%VF2 54% 14% 15% 5% 13% 8%VF3 11% 6% 10% 10% 7% 5%VF4 89% 82% 21% 20% 11% 3%VF5 49% 11% 9% 7% 7% 12%

Considering VFM-PSO and the shifted Sphere function(Figure 4), we observe that the convergence rate in VF2 isthe best followed by VF4, VF5, VF3 and VF1 delivers theworst. We observe a similar convergence behavior for theshifted Ackley function (in the order VF2, VF4, VF5, VF3and VF1). For the shifted Rosenbrock problem, the resultslook different. We have much better results in VF3 and thensimilar behavior as in Sphere and Ackley (in the order VF2,VF5, VF4 and VF1). The results look different for VF-PSO. Theses results are used to analyze the performance andinfluence of the proposed approach. According to the figures,we observe that the order in terms of the convergence rateis VF4, VF3, VF5, VF2 and VF1. VF-PSO cannot obtain theoptimal solutions in all of these vector fields. The results on theAckley test problem look similar. There is a slight differencefor Rosenbrock: VF3 (found the optimal solution), VF2, VF5,VF4 and VF1. In terms of Success Rate for VFM-PSO, weobserve the following (from high to low values): Sphere (VF4,VF2, VF5, VF1 and VF3), Ackley (VF2, VF4, VF1, VF3and VF5) and Rosenbrock (VF4, VF2, VF1, VF3 and VF5).And corresponding results for VF-PSO: Sphere(the same as

in VFM-PSO: VF4, VF2, VF5, VF1 and VF3), Ackley (VF1,VF5, VF2, VF3 and VF4) and Rosenbrock (VF4, VF3, VF1,VF5 and VF2). These results are going to be analyzed anddiscussed in the following:

D. Influence of the Vector Fields

a) Vector Field 1 (”Cross”): This vector field has a verystrong opposite force in the area close to the minimal solution(i.e., shifted optimal solution at (-10, 10)). We expect that boththe VF-PSO and VFM-PSO encounter difficulties to find theoptimal solutions. The convergence rate and the success rateas well confirm this fact for all objective functions (fitnesslandscapes). It is the most complex vector field among theothers, as the standard deviation for the both algorithms ishigh and does not decrease by the iterations.

b) Vector Field 2 (”Rotation”): This environment repre-sents counterclockwise rotation dispersing by circles aroundthe center of the search space at (0,0) with increasing mag-nitude of the vectors. According to this, the particles locatedat the upper-right corner of the search domain have benefitfrom the flow and converge faster to the optimal solution, so alarge portion of particles can set close to the minimum. Thusit must deliver better results than in VF1 and it is. Sphereand Rosenbrock find the optimal solution, Ackley tends toconverge (the variance decreases by the iterations). We expecta better success rate in this case and VF-PSO confirms it.

c) Vector Field 3 (”Sheared”): In this vector field closeto the optimum we have a weak opposite/same direction windwhich helps to reach the optimal solution for the particlesconcentrated in the left-upper quarter of the search domain.The weak vectors spread along the left-right diagonal of thesearch domain. In other cases particles have to deal withstronger flow pushing them away from the optimum area tothe borders of the space. We expect to have a low success ratehere, because only those particles who are by chance in thevicinity of the the optimum are successful and the rest - not. Incase of Rosenbrock we obtain the best results in both VF-PSOand VFM-PSO even faster than usual PSO. That takes placebecause the particles in a flat plateau region deal with weakvectors, which in addition are blowing towards the optimumin the half space of the plateau.

d) Vector Field 4 (”Wave”): This environment performsupside-down zigzag movements. It helps to get out of localoptimum. That confirms one of the best results in obtainedconvergence rate for Ackley and Rosenbrock, as well as interms of Success rate. For Sphere it delivers the best resultsin Success rate, what suppose that it brings many particlesclose to the optimum. And VF-PSO also confirms it.

e) Vector Field 5 (”Tornado”): This vector field is verysimilar to VF2 in the sense of vectors directions and VF3in the sense of magnitude. The particles, wherever they are,are twirled counterclockwise to the center of the domain andpushed from the borders to the left-right diagonal of the searchspace, where the weak wind is. At the optimum we have gotweak opposite direction wind. So here we suppose to have

a good convergence as in VF2, and a low success rate as inVF3.

E. Analysis of the Awareness rate

As it is mentioned in Section IV-B the calculation ofAwareness rate depends on the behavior of explorers, as theinformation from the explored area is used by the optimizers.In view of the fact that exploration is done by following theflow, the amount of gathered information from the same num-ber of explorers during the same iteration time differs withinvector fields as shown in Table IV. The highest percentage ofthe explored area is expected to be in VF2 and VF4, and thelowest in VF3. The reported values in Table IV confirm thisfact as most of the particles get stuck at the borders of thesearch space.

TABLE IV: Average explored area

Explored areaVF1 VF2 VF3 VF4 VF5

20,56% 55,50% 12,01% 40,61% 22,14%

The obtained average awareness rate for each VF after 100iterations is shown in Table V. We observe that the awarenessrates for VF2 is the highest followed by VF5 and VF1.Considering the awareness rates in Figure 4, the optimizingparticles tend to concentrate mostly in known area after about25 iterations, what promotes the tendency to convergence inVFM-PSO in contrast to unaware VF-PSO.

TABLE V: Average Awareness Rate after 100 iterations

VF1 VF2 VF3 VF4 VF5Sphere 64,57% 92,27% 47,07% 31,97% 69,13%

Rosenbrock 74,10% 91,53% 50,77% 22,17% 72,47%Ackley 67,43% 93,80% 46,47% 29,40% 68,53%

avg. 68,70% 92,53% 48,10% 27,84% 70,04%

TABLE VI: The required percentage of the AwarenessRate to get better results in VFM-PSO than in VF-PSO.LIGHT GRAY indicates that VFM-PSO found the optimal

solution, i.e. converged. DARK GRAY indicates the trend toconvergence.

VF1 VF2 VF3 VF4 VF5Sphere 63,47% 80,73% 16,03% 29,53% 61,87%

Rosenbrock - 78,43% 36,37% - 63,30%Ackley - 84,23% 36,87% - 63,07%

avg. - 81,13% 29,76% - 62,74%

According to Figure 4, in case of VF4, due to the highstandard error, VFM-PSO and VF-PSO overlap during theentire iterations time (as a consequence the results do notdepend on the awareness rates), but both perform reasonablegood performance. In case of VF1 the same holds but theperformance is opposite, e.g. both VFM-PSO and VF-PSOtend to converge. For Rosenbrock-VF3 both VFM-PSO andVF-PSO find global optimum within 10 iterations and outper-form standard PSO. In all other cases, we have VFM-PSO

better than VF-PSO. Referring to the Table V and Table VIthe visible divergence of the algorithms traces with setting thestable rate of awareness, e.g. approx. within 25 iterations, andshould be more than 50%. This result is obtained from theTable VI as the average Awareness Rate among all the vectorfields, where convergence takes place.

V. CONCLUSION AND FUTURE WORK

In this paper we presented VFM-PSO (Vector Field MapPSO) method as a search mechanism for swarm of aerialmicro-robots in dynamic environments with unknown externalperturbations modeled by vector fields. To deal with uncertain-ties two sub-swarms populations are used: explorers and op-timizers. The paper is the first attempt which contains severalrestrictions and assumptions. The experiments on five vectorfields for three objective functions show that our approachcan reduce the influence of disturbances on the search processand VFM-PSO always produces better performance than VF-PSO regardless of the structure of considered vector field.Based on our experiments we can conclude that the qualityof search for VFM-PSO depends on both the vector field andthe fitness landscape. In almost most of the cases VFM-PSOis better than VF-PSO, but cannot be as efficient as standardPSO (with no vector field). This is because the informationabout the vector field obtained by explorers is relative, due tothe discrete motion of the particles, but continuous influenceof the vector fields. According to this restriction, we cannotfully compensate the drift at each point of the investigatedsearch space. The next step is to work on the assumptionsand to implement the search mechanism with continuous-time motion of the particles what will allow to use theproposed approach in real-robot scenarios. As the part of thefuture work, also more tests in more difficult vector fields onmore benchmark functions will be held along with statisticalsignificance analysis of the differences in performance.

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