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Guan et al. Intell Robot 2024;4(1):61-73 I http://dx.doi.org/10.20517/ir.2024.04 Page 65

Figure 2. Map and compass operator in PIO. PIO: Pigeon-inspired optimization.

2.2. PIO principles
Drawing inspiration from the distinctive navigational behavior of pigeon flocks during their homing process,
we put forth a bionic population intelligence algorithm, termed as the PIO algorithm [14] . This algorithm
employs distinct operators at different stages, specifically the map and compass operator and the landmark
operator [15] . The map and compass operator encapsulates the influence of the magnetic field on the pigeon’s
flight direction, while the landmark operator represents the impact of landmarks on its navigational trajectory.

Suppose a flock of pigeons numbered N searches for a target in a d-dimensional space. When is considered,
the map and compass operator identifies each unique pigeon as “i”, which stands for the maximum number of
iterations that the map and compass operator can perform. Its spatial coordinates and speed are symbolized
as follows:

= [ 1 , 2 , ..., ],
(9)
= [ 1 , 2 , ..., ].

when the number of iterations is ; each pigeon updates its position and velocity according to the
following formulas:

= −1 − × + ( − −1 ), (10)
= −1 + .

Inthiscontext, Rrepresentsthemapandcompassoperator, whichhasavaluerangefrom01. Similarly, ()
is a function that generates a random number within the same range, 0 1. is the global optimal position
obtained by comparing the positions of all pigeons after − 1 iterative loop. In accordance with Equation 9,
each pigeon has the ability to modify its flight direction. This is represented by the blue arrow, while the yellow
one signifies the original direction. The subsequent direction of the pigeon is determined by the vector sum
of the blue and yellow directions. Figure 2 gives an example of the action of pigeons in the map and compass
operator phase.

As shown in Figure 3, when the above loop reaches the maximum number of iterations of the phase, i.e.,
1 < < 2 the map and compass operator stops working and enters the land-mark operator phase,