# Acceleration based PSO for Multi-UAV Source-Seeking

Adithya Shankar1, Harikumar Kandath1 and J. Senthilnath2 1 Robotics Research
Center, International Institute of Information Technology, Hyderabad-500032,
India. E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>for Infocomm Research, Agency for Science,
Technology and Research (A*STAR), Singapore, 138632. E-mail:
<EMAIL_ADDRESS>
###### Abstract

This paper presents a novel algorithm for a swarm of unmanned aerial vehicles
(UAVs) to search for an unknown source. The proposed method is inspired by the
well-known PSO algorithm and is called acceleration-based particle swarm
optimization (APSO) to address the source-seeking problem with no a priori
information. Unlike the conventional PSO algorithm, where the particle
velocity is updated based on the self-cognition and social-cognition
information, here the update is performed on the particle acceleration. A
theoretical analysis is provided, showing the stability and convergence of the
proposed APSO algorithm. Conditions on the parameters of the resulting third
order update equations are obtained using Jury’s stability test. High fidelity
simulations performed in CoppeliaSim 111https://www.coppeliarobotics.com/,
shows the improved performance of the proposed APSO algorithm for searching an
unknown source when compared with the state-of-the-art particle swarm-based
source seeking algorithms. From the obtained results, it is observed that the
proposed method performs better than the existing methods under scenarios like
different inter-UAV communication network topologies, varying number of UAVs
in the swarm, different sizes of search region, restricted source movement and
in the presence of measurements noise.

##  I INTRODUCTION

The objective of source-seeking problem in robotics is to locate a stationary
or moving source by measuring the signal strength that is emanated from the
source. In large and unknown outdoor environments, unmanned aerial vehicles
(UAVs) becomes the preferred robotic platform for such applications. Deploying
multiple UAVs for such a mission proves to be efficient than using a single
UAV [6]. However, source seeking based on multiple UAVs has its own
challenges, like to develop an efficient cooperative framework among UAVs to
complete the task. Stochastic search methods became a natural choice when
there is no a priori information available on the probability distribution of
the source location [5]. Swarm intelligence based algorithms, such as Particle
Swarm Optimization (PSO) [13], and Fruit Fly Optimization Algorithm (FOA) [9],
are widely used for solving the source-seeking problem. This study focuses on
the PSO algorithm and its variants for solving the source-seeking problem.

Particle swarm optimization (PSO) is a swarm-based algorithm developed by
Kennedy et al. [8], which is based on the collective social behavior of the
particles. The method performs well and is capable of optimizing the source-
seeking problem. Over the last decade, a lot of the research has focused on
using PSO-based algorithms for the source-seeking problem. This research has
led to many variants of PSO such as the Darwinian PSO (DPSO) [11] algorithm to
overcome the local optima problem of PSO. The algorithm does this by
initializing multiple swarms that compete with each other in order to simulate
a phenomenon that is similar to natural selection. PSO uses optimal computing
budget allocation to improve the performance in the presence of noise for
multiple UAVs performing required task [4]. This study focuses on modified PSO
to locate the source in the presence of noise.

Two other variations of PSO that have gained popularity are Robotic PSO (RPSO)
and Robotic Darwinian PSO (RDPSO) [1]. RPSO consists of a population of robots
that collectively searches source in the search space. RDPSO is an extension
of RPSO that introduces a “punish-reward” mechanism to reduce the shared
information between the UAVs since multiple swarms are formed. The main goal
behind this technique is to prevent robots from erroneously converging towards
local optima. In both of these algorithms, the team of robots is capable of
performing the source-seeking task in a distributed manner. The other popular
PSO based source seeking algorithm is the standard-PSO (SPSO) [14]. The SPSO
implements the optimized hyperparametrs for the multi-robot source seeking
problem and proves to be better than the conventional PSO based algorithm
[14]. It also evaluates the performance of the algorithm under three different
inter-robot communication network topologies. In the above-mentioned PSO
variants, the hyperparameters are kept constant during the entire robotic
search mission. Whereas, the hyperparameters are adaptively updated in
Adaptive-Robotic PSO algorithm [2]. In [2], the authors showed that ARPSO
performed better than RPSO and RDPSO. Hence, in this paper we use ARPSO and
SPSO for the comparative study with the proposed APSO algorithm.

This paper presents a novel update rule for the particle’s position. The
original PSO can be portrayed as a second-order system with the particle’s
position being dependent on the particle’s velocity. In our proposed method,
the particle position is updated based on it’s velocity and acceleration
resulting in a third order dynamical system. We term this new algorithm as
”APSO” (Acceleration Based PSO).

The major contributions of this paper are the following.

  * •

A new PSO based source seeking algorithm is presented that utilizes a third
order update rule for particle’s position. The acceleration of the particle is
updated using the self-cognition and social-cognition component unlike the
existing PSO based methods where the particle’s velocity is updated through
the self-cognition and social-cognition component.

  * •

Detailed convergence and stability analysis of the proposed algorithm is
provided.

  * •

The condition on the values to be taken by the hyperparameters of the
algorithm to ensure stability is established through Jury’s stability test.

  * •

A detailed performance comparison is provided with the state-of-the-art PSO
based source seeking algorithms, showing the improved performance of the
proposed method quantified in terms of average source seeking time, average
number of iterations taken and the average swarm distance. A high fidelity
simulation is executed in CoppeliaSim for the performance comparison. The
improvement in performance of the proposed algorithm is shown to be consistent
across scenarios like different inter-UAV communication network topologies,
varying number of UAVs in the swarm, different sizes of search region,
restricted source movement and in the presence of measurements noise.

The paper is structured in the following manner. Section 2 briefly describes
PSO, SPSO, and ARPSO. Section 3 details the APSO algorithm along with its
convergence and stability analysis. The supporting results and discussions are
presented in section 4. In section 5, the paper is concluded with a mention of
the future work.

##  II Background

This section provides the details on the UAV motion, details of the sensor
carried by the UAV, and also a description of the existing PSO based multi-
robot source seeking algorithms.

###  II-A UAV and sensor description

The swarm of $n$ UAVs are randomly initialized at different locations in the
search space to locate the source position solely based on the sensor
information. The position of the UAVs are denoted with respect to an inertial
frame of reference as $X_{I}-Y_{I}-Z_{I}$. All the UAVs are assumed to be
flying at a fixed altitude different from each other to avoid inter-UAV
collision, and hence the $Z_{I}$ coordinate is not discussed here. The
position coordinate of the $i^{th}$ UAV at any given time $t$ is denoted by
$P_{i}(t)=[x_{i}(t),\,y_{i}(t)]^{T}$. The UAVs fly at a constant speed between
the successive waypoints.

The source is assumed to be located with in the search space and emits a
signal having a power that exponentially decays with the distance from it.
Each UAV carries a sensor that measures the signal strength emitted by the
source using the below model.

$S_{mi}=S_{s}e^{{-\alpha d_{i}^{2}}}+\delta S$ (1)

where $S_{mi}$ is the power measured by the $i^{th}$ UAV located at a distance
$d_{i}$ from the source, $S_{s}$ is the source power and $\alpha>0$ is a
constant. The measurement noise is denoted by $\delta S$ and is random with a
magnitude proportional to the measured power $S_{mi}$. The discrete time
version of the position coordinates is used in the subsequent sections and is
denoted by $P_{i}(k)=[x_{i}(k),\,y_{i}(k)]^{T}$, where $t=k\tau$, $k$ is the
discrete time instant and $\tau$ is the sampling time in seconds.

###  II-B Algorithms for PSO based source-seeking

A brief overview of the popular PSO based source-seeking algorithms is
presented below. Equations are provided for a single-axis $X_{I}$ and
analogous equations follows for the $Y_{I}$ axis also.
a) Regular PSO: The conventional PSO used for optimization is implemented for
source-seeking problem in [3]. The velocity ($v_{i}(k)$) and the position
($x_{i}(k)$) updates for the $i^{th}$ robot for $k^{th}$ sampling time instant
is given by,

$\begin{split}v_{i}(k+1)=v_{i}(k)+R(0,c_{1})(x_{ib}(k)-x_{i}(k))\\\
+R(0,c_{2})(x_{gb}(k)-x_{i}(k))\end{split}$ (2)

$x_{i}(k+1)=x_{i}(k)+v_{i}(k+1)T$ (3)

Here $T$ is a scaling factor (typically $T$=1 is used), $c_{1}>0$, $c_{2}>0$,
$R(a,\,b)$ denotes a random number from uniform distribution within the
interval $[a,\,b]$, $x_{ib}(k)$ is the best individual position and
$x_{gb}(k)$ is the best global position.

b) Standard Particle Swarm Optimization (SPSO): SPSO is an extension to the
regular PSO algorithm [14] with an inertia factor added in the velocity update
of the $i^{th}$ particle is given by,

$\begin{split}v_{i}(k+1)=\omega v_{i}(k)+R(0,c_{1})(x_{ib}(k)-x_{i}(k))\\\
+R(0,c_{2})(x_{lb}(k)-x_{i}(k))\end{split}$ (4)

The parameter values used are $\omega$=0.721 and $c_{1}=c_{2}=1.193$ to
achieve the best performance. Another important difference from the regular
PSO is the incorporation of communication constraints among the particles. The
global best position $x_{gb}(k)$ is replaced by the local best $x_{lb}(k)$
considering the particles that can communicate with each other.
c) Adaptive-Robotic PSO (ARPSO): The ARPSO algorithm prevents the swarm from
converging at a local minima. The algorithm is also capable of handling
obstacles that are present in the environment [2].

$\begin{split}v_{i}(k+1)=\omega_{i}v_{i}(k)+R(0,c_{1})(x_{ib}(k)-x_{i}(k))\\\
+R(0,c_{2})(x_{gb}(k)-x_{i}(k))+R(0,c_{3})(x_{a}-x_{i}(k))\end{split}$ (5)

where $x_{a}$ is an attractive position that is located away from the
obstacle. The parameter $c_{3}$ is set to 0 for obstacle free environment.
Unlike in SPSO, the parameter $\omega_{i}$ in ARPSO is not constant. The value
of $\omega_{i}$ is different for different UAVs and also changes for each
iteration.

##  III Acceleration-based PSO (APSO) for source-seeking

In the proposed APSO, the particle acceleration is also updated apart from
velocity as in the above-mentioned PSO and its variants to generate a new
waypoint. This additional equation transforms the input to output relation to
a third-order system. The following equations from (6) to (8) shows the
waypoint update equations for the $i^{th}$ UAV.

$\begin{split}a_{i}(k+1)=w_{1}a_{i}(k)+R(0,c_{1})(x_{ib}(k)-x_{i}(k))\\\
+R(0,c_{2})(x_{gb}(k)-x_{i}(k))\end{split}$ (6)

$v_{i}(k+1)=w_{2}v_{i}(k)+a_{i}(k+1)T$ (7)

$x_{i}(k+1)=x_{i}(k)+v_{i}(k+1)T$ (8)

where $a_{i}(k)$ denotes the particle acceleration for the $i^{th}$ UAV.

###  III-A Convergence Analysis

By simplifying (6) to (8), the following relation is obtained with
$r_{1}=R(0,c_{1})$ and $r_{2}=R(0,c_{2})$.

$\begin{split}x_{i}(k+1)=(1+w_{1}+w_{2}-r_{1}T-r_{2}T)x_{i}(k)\\\
+(-w_{1}-w_{2}-w_{1}w_{2})x_{i}(k-1)+w_{1}w_{2}x_{i}(k-2)\\\
+T(r_{1}x_{ib}(k)+r_{2}x_{gb}(k))\end{split}$ (9)

A steady state condition for $x_{i}(k)$ (denoted by $x_{i}(ss)$) in (9) is
achieved when i) $r_{1}$ and $r_{2}$ are made constant or when ii)
$x_{ib}(ss)=x_{gb}(ss)$. When $r_{1}$ and $r_{2}$ are made constant (denoted
by $r_{1c}$ and $r_{2c}$ respectively), the following result is obtained.

$x_{i}(ss)=\lim_{k\to\infty}x_{i}(k)=\frac{r_{1c}x_{ib}(ss)+r_{2c}x_{gb}(ss)}{r_{1c}+r_{2c}}$
(10)

When $x_{ib}(ss)=x_{gb}(ss)$,

$x_{i}(ss)=x_{ib}(ss)=x_{gb}(ss)$ (11)

###  III-B Stability Analysis

The equation describing the evolution of $x_{i}(k)$ given in (9) represents a
third order dynamical system. Applying $z$-transform on (9) yields the
following transfer function.

$G(z)=\frac{X(z)}{U(z)}=\frac{z^{-1}}{1+a_{1}z^{-1}+a_{2}z^{-2}+a_{3}z^{-3}}$
(12)

where the coefficients $a_{1}=-1-w_{1}-w_{2}+r_{1}T+r_{2}T$,
$a_{2}=w_{1}+w_{2}+w_{1}w_{2}$, $a_{3}=-w_{1}w_{2}$ and the input
$U(z)=T(r_{1}X_{ib}(z)+r_{2}X_{gb}(z))$. The stability of $G(z)$ depends upon
the roots of the polynomial given below.

$H(z)=1+a_{1}z^{-1}+a_{2}z^{-2}+a_{3}z^{-3}$ (13)

Since $r_{1}$ and $r_{2}$ are random numbers varying between $[0,\,c_{1}]$ and
$[0,\,c_{2}]$ respectively, the coefficient
$a_{1}\in[-1-w_{1}-w_{2},\,-1-w_{1}-w_{2}+T(c_{1}+c_{2})]$. Since $H(z)$ is a
third order uncertain polynomial, it is enough to confirm that the extreme
polynomials $H(z)$ at $a_{1}=-1-w_{1}-w_{2}$ (the coefficient and
corresponding polynomial denoted by $a_{1l}$ and $H(z,a_{1l})$ respectively)
and $H(z)$ at $a_{1}=-1-w_{1}-w_{2}+T(c_{1}+c_{2})$ (the coefficient and
corresponding polynomial denoted by $a_{1u}$ and $H(z,a_{1u})$ respectively)
does not have roots outside the unit circle [7]. Applying the Jury’s stability
criterion for $H(z,a_{1l})$ and $H(z,a_{1u})$, the following conditions are
obtained on the parameters $w_{1}$, $w_{2}$, $T$, $c_{1}$ and $c_{2}$.

$\frac{2}{T}(1+w_{1}+w_{2}+w_{1}w_{2})>c_{1}+c_{2}$ (14) $|w_{1}w_{2}|<1$ (15)
$|(1-w_{1}w_{2})(w_{1}+w_{2})+(w_{1}w_{2})(c_{1}T+c_{2}T)|<|1-(w_{1}w_{2})^{2}|$
(16) $|w_{1}+w_{2}|<|1+w_{1}w_{2}|$ (17)

##  IV Results and Discussion

In this section, the performance of APSO is compared against that of ARPSO and
the standard PSO (SPSO). The obtained results are verified by simulation
performed in the CoppeliaSim simulator (www.youtube.com/watch?v=ad_GD3NQR-w).
Coppeliasim (formerly known as ”V-REP”) is high fidelity and scalable robotic
simulator used for testing and verification of different algorithms before
deployment in the actual hardware [10]. All the three search algorithms (APSO,
ARPSO, and SPSO) were implemented using python 3.7 in a conda based
environment. A typical search scenario with 5 UAVs is shown in Fig. 1. The
search space considered here is in the shape of a square. For each simulation,
the UAVs starts from randomly initialized locations along the boundary of the
search space. All the UAVs move from a waypoint to the next with a constant
speed of 10 $m/s$. For all the simulations, the source is considered to be at
the center of the search space. The algorithm is terminated when any of the
UAV reaches a distance of 0.1 $m$ from the source, in the case of measurement
noise this value is increased to 0.5 $m$.

Figure 1: Search scenario showing 5 UAVs looking for the source (shown with in
a red circle). Note that each UAV maintain an altitude different from the
others to avoid inter-UAV collision.

In the implementation of APSO, the values of $w_{1}$ and $w_{2}$ are taken as
0.675 and -0.285 respectively. Both of the parameters $c_{1}$ and $c_{2}$ are
selected as 1.193, and the value of T is taken as 1. It can be verified that
the above-mentioned parameter setting satisfies the stability conditions
mentioned in (14) to (17). The values of the parameters of SPSO and ARPSO
algorithm are selected from [14] and [2] respectively.

###  IV-A Metrices for the performance comparison of source-seeking algorithms

The metrics used for performance comparison are given below. The metrices are
averaged over 1000 Monte Carlo runs.

  * •

Average source seeking time ($\mu(T_{s})$): This metric quantifies the time
efficiency of the underlying source seeking algorithm and is defined in (18).
Source seeking time is taken as the time duration from the start of the search
till the source has been located by one or more members of the UAV swarm.

$\mu(T_{s})=\frac{\sum_{i=1}^{N}T_{si}}{N}$ (18)

where $N$ is the total number of Monte Carlo runs and $T_{si}$ is the source
seeking time for the $i^{th}$ run.

  * •

Average number of iterations ($\mu(I)$): This metric quantifies the sampling
efficiency of the algorithm and is defined in (19) as the average number of
waypoints generated to locate the source.

$\mu(I)=\frac{\sum_{i=1}^{N}W_{si}}{N}$ (19)

where $W_{si}$ is the number of waypoints generated for the $i^{th}$ run to
locate the source.

  * •

Average Swarm Distance ($\mu(SD)$): This parameter is related to the fuel
efficiency when UAVs are deployed for the search and is defined in (20). Swarm
distance is defined as the total distance travelled by the UAVs in the swarm,
till the source has been located.

$\mu(SD)=\frac{\sum_{i=1}^{N}\sum_{j=1}^{n}D_{ji}}{N}$ (20)

where $n$ is the size of UAV swarm and $D_{ji}$ is the distance travelled by
the $j^{th}$ UAV in the $i^{th}$ simulation run.

The performance comparison is done under different operating conditions as
given below.
a) Inter-UAV communication network topology: The exchange of information
between the UAVs depends on the inter-UAV communication network topology. We
consider three scenarios viz. i) fully connected, i) ring and iii) adaptive
toplogies. For fully connected topology, any UAV can communicate with any
other UAV in the swarm and is the best case scenario. Ring topology is a
restricted case where each UAV can communicate only to it’s two adjacent
neighbours. For adaptive topology, each UAV communicates with randomly chosen
UAVs from the swarm for each waypoint computation. More details of these three
topologies are available in [zou2015particle]. The other operating conditions
considered are b) different size of the search region, c) varying number of
UAVs in the swarm, d) restricted source movement, and e) measurement noise.

###  IV-B Performance comparison of source-seeking algorithms

The average is computed for $N=1000$, for a swarm size $n=5$ for a search
space of dimensions 100 $m$ $\times$100 $m$ unless otherwise stated
explicitly.

####  IV-B1 Effect of inter-UAV communication network topology

: The results shown in Table I indicate that APSO is performing better than
SPSO and ARPSO in all the three metrices for all the three topologies. The
fully connected topolgy gives the best performance as expected. The swarm
under adaptive topology performs better than the ring topology. Due to its
superior performance, for all the remaining experiments the Fully Connected
(FC) topology is used for all the three algorithms.

TABLE I: Effect of Network Topology Algorithm | Topology | $\mu(I)$ |  $\mu(T_{s})$ ($s$) |  $\mu(SD)$ ($m$)
---|---|---|---|---
SPSO | Ring | 43.22 | 34.97 | 387.98
ARPSO | Ring | 41.91 | 34.45 | 343.24
APSO | Ring | 27.36 | 17.31 | 175.52
SPSO | FC | 19.44 | 22.94 | 286.34
ARPSO | FC | 15.28 | 19.59 | 217.2
APSO | FC | 10.81 | 11.03 | 116.58
SPSO | Adaptive | 38.08 | 32.09 | 371.41
ARPSO | Adaptive | 38.24 | 31.78 | 320.69
APSO | Adaptive | 22.3 | 16.12 | 167.41

####  IV-B2 Different size of search region

The dimensions of the search space is varied from 10 $m$ $\times$ 10 $m$ to
100 $m$ $\times$ 100 $m$. Table II shows the performance of all three
algorithms with UAVs communicating under fully connected topology. The APSO
performs the best among the three algorithms. The improvement in the
performance of APSO is higher with an increase in the search area when
compared to SPSO and ARPSO algorithm.

TABLE II: Effect of Search Area Dimensions Algorithm | Area ($m^{2}$) | $\mu(I)$ |  $\mu(T_{s})$ ($s$) |  $\mu(SD)$ ($m$)
---|---|---|---|---
SPSO |  | 6.92 | 1.29 | 14.77
ARPSO | 10 $m$ $\times$ 10 $m$ | 6.17 | 1.25 | 14.19
APSO |  | 5.22 | 0.80 | 8.34
SPSO |  | 11.41 | 4.63 | 54.65
ARPSO | 25 $m$ $\times$ 25 $m$ | 8.85 | 3.74 | 43.22
APSO |  | 7.26 | 2.37 | 25.42
SPSO |  | 15.63 | 10.67 | 131.71
ARPSO | 50 $m$ $\times$ 50 $m$ | 11.55 | 8.67 | 97.60
APSO |  | 9.01 | 5.33 | 56.28
SPSO |  | 17.82 | 16.93 | 209.34
ARPSO | 75 $m$ $\times$ 75 $m$ | 13.58 | 14.20 | 155.29
APSO |  | 10.18 | 8.21 | 86.74
SPSO |  | 19.44 | 22.94 | 286.34
ARPSO | 100 $m$ $\times$ 100 $m$ | 15.28 | 19.59 | 217.2
APSO |  | 10.81 | 11.03 | 116.58

####  IV-B3 Number of UAVs in a swarm

Here the UAV swarm size is increased from 5 till 30 under fully connected
topology. The average source seeking time for the three algorithms is plotted
in Fig. 2. The $\mu(T_{s})$ for APSO is almost twice better than SPSO and
ARPSO for $n=5$ and later the improvement decreases. The $\mu(T_{s})$
saturates after a certain swarm size for all the three algorithms. The
comparison for average number of iterations is shown in Fig. 3. The APSO
outperforms SPSO irrespective of the number of UAVs in a swarm. The
performance of ARPSO converges to that of APSO for $n\geq 20$. The APSO
outperforms the other two algorithms for the case of $\mu$(SD) for all the UAV
swarm size as shown in Fig. 4.

Figure 2: Effect of number of UAVs on $\mu(T_{s})$ Figure 3: Effect of number
of UAVs on $\mu$(I) Figure 4: Effect of number of UAVs on $\mu$(SD)

####  IV-B4 Restricted source movement

: In this experiment, the source is moving randomly with in a circle of radius
4$m$ and the search space considered here is of dimensions 200 $m$ $\times$
200 $m$. The speed of the source is varied from 0 to 0.3 $m/s$ and the
performance of all the three algorithms is compared. Figure 5 shows the effect
of source speed on the average source seeking time indicating the better
performance of APSO when compared to ARPSO and SPSO. The value of $\mu(T_{s})$
increases slowly with source speed for APSO, whereas it increases rapidly for
ARPSO and SPSO. Similar result can be observed for the case of $\mu(SD)$ as
shown in Fig. 7. The increase in $\mu(I)$ with the source speed is noted for
all the algorithms from Fig. 6 with APSO performing the best among the three
algorithms. For higher source speed, the difference between the performance of
APSO and SPSO reduces.

Figure 5: Effect of source speed on $\mu(T_{s})$ Figure 6: Effect of source
speed on $\mu$(I) Figure 7: Effect of source speed on $\mu$(SD)

####  IV-B5 Sensor Noise

In real scenarios, the sensor measurement is corrupted by noise as modeled in
(1). Gaussian noise with zero mean and standard deviation proportional to the
power measured by the UAV is considered as the sensor noise. The percentage of
noise added is increased from 0 to 10 $\%$ to see its effect on the three
source seeking algorithms. Additionally, the minimum distance of any UAV from
the source for which the algorithm terminates is increased from 0.1 $m$ to 0.5
$m$ to account for the measurement noise. The results are tabulated in Table
III.

TABLE III: Effect of Sensor Noise Algorithm | Sensor Noise ($\%$) | $\mu(I)$ |  $\mu(T_{s})$ ($s$) |  $\mu(SD)$ ($m$)
---|---|---|---|---
SPSO |  | 10.51 | 18.23 | 213.38
ARPSO | 0 | 8.33 | 15.59 | 174.80
APSO |  | 6.94 | 9.78 | 102.63
SPSO |  | 22.93 | 25.88 | 278.38
ARPSO | 0.05 | 16.81 | 19.91 | 205.31
APSO |  | 10.66 | 11.49 | 114.49
SPSO |  | 24.33 | 28.34 | 295.44
ARPSO | 0.1 | 19.06 | 20.04 | 201.43
APSO |  | 12.44 | 13.17 | 118.61

The performance of the three algorithms degrades when the Gaussian noise is
introduced. However, APSO is capable of handling the noise much better than
the other two algorithms.

###  IV-C DISCUSSION

The selection of the parameters $w_{1}$, $w_{2}$, $c_{1}$ and $c_{2}$ plays a
crucial role in the performance of APSO. The parameters $c_{1}$ and $c_{2}$
are selected to be the same as the case with the SPSO algorithm. The stability
conditions limits the choice of $w_{1}$ and $w_{2}$ to be with in $(-1,\,1)$.
The values of $w_{1}$ and $w_{2}$ used here are the ones yielding the best
performance. A detailed analysis on the performance of APSO with varying
$w_{1}$, $w_{2}$, $c_{1}$ and $c_{2}$ is not presented here due to the page
limit.
The performance of ARPSO algorithm remained closer to the proposed APSO
algorithm when compared to the SPSO algorithm. This is mainly due to the
adaptive hyperparameters of the ARPSO algorithm when compared to the SPSO
algorithm. The improved measurement noise handling capability of APSO
algorithm can be attributed to the underlying third order dynamical system in
contrast to the second order dynamical system for the ARPSO and SPSO
algorithms.

##  V CONCLUSION AND FUTURE WORK

This paper presented a new algorithm based on PSO for the multi-UAV source
seeking problem. The proposed algorithm implemented a third-order update rule
when compared to the existing second-order update rule. The stability analysis
provided a guideline for selecting the hyperparamter values of the algorithm.
The convergence of the proposed algorithm is similar to the existing PSO-based
source-seeking algorithms [12]. The proposed APSO algorithm showed improved
performance when compared to SPSO and ARPSO algorithms under different
operating conditions. Obstacle avoidance, dynamic target tracking, and
hardware implementation will be interesting extensions of this research work.

##  VI ACKNOWLEDGMENT

The authors thank the financial support provided from the SEED grant of the
KCIS project titled ”RoboMate: Towards 24/7 Companionship and Assistance,”
IIIT Hyderabad to carry out this research work.

##  VII References

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