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Page 340 Sellers et al. Intell Robot 2022;2(4):333­54 I http://dx.doi.org/10.20517/ir.2022.21

4. IMPROVED PSO-BASED MULTI-WAYPOINT NAVIGATION
The particle swarm optimization (PSO) algorithm is a swarm-based bio-inspired algorithm based on the be-
havioral observation of birds. It uses an iterative methodology to optimize randomly initialized particles to
define a path from the initial position to the goal [27] . In this section an improved PSO (IPSO) algorithm by
introduction of a weighted particles is addressed to resolve the multi-waypoint sequence issue.

4.1. Multi­waypoint visiting sequence
Inreal-worldscenarios, oneimportantfactoristhattheGPScoordinatesprovideportionsforthemultipleway-
points. Traveling from one waypoint to another, the distance between them determines their associated cost.
Theprimarypurposeoftravelingfromonewaypointtoanotheristosimultaneouslyfindtheminimalcostofall
generated trajectories. Using the coordinates of each waypoint and the PSO algorithm, the minimal-distance
path can be found within the environment. The PSO algorithm finds the best waypoint visiting sequence by

initializing randomized particles. The algorithm denotes the local best position as and the global best posi-

tion as . Then by taking advantage of a fitness function, the algorithm guides each particle towards the local
and global best positions. The particle velocities are updated as follows:

( + 1) = ( ) + 1 1 [ ( ) − ( )] + 2 2 [ ( ) − ( )] (6)

( +1) ( ) ( +1) (7)
= +

where, ( ) representsthevelocityofparticle atinstant , ( ) isthepositionofparticle atinstant , 1 and

2 arethepositiveaccelerationconstantsused toscale thecontributionofcognitiveandsocialcomponents. 1
and 2 are the uniform random number between 0 and 1. ( ) is the best position the particle achieved up

to instant at current iteration, ( ) is the global position that any of ’s neighbors has reached up to instant

. However, if a particle lies close to the ( ) and ( ), only one term guides the to search the potential

solution.The optimization process in our navigation issue is more likely trapped in local minima. Thus, an
improved PSO algorithm is utilized to provide a more promising search direction for all particles during the
optimization process.

∑

= ¯ ( ) (8)

=1
ˆ
¯ (9)
= ∑
=1 ˆ

( ( )) ( )

max 1≤ ≤ F ( ) − F ( ) +

ˆ = ( ( )) ( ( )) , = 1, 2, . . . , , (10)

max 1≤ ≤ F ( ) − min 1≤ ≤ F ( ) +

where is a positive constant, ˆ is the weighted constant of each particle. F(·) is the fitness function. The
)
(

worst and the best fitness values of all personal best particles are represented by max 1≤ ≤ (F ) and

(
)

min 1≤ ≤ (F ), respectively. The order is optimized through this method, in which each waypoint is

visited. A sequence of particles are initialized to compose a population in the original PSO algorithm. A possi-
ble optimal solution to an optimization issue in our multi-waypoint sequence is discovered by one particle in