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Page 66 Guan et al. Intell Robot 2024;4(1):61-73 I http://dx.doi.org/10.20517/ir.2024.04

Figure 3. Landmark operator in PIO. PIO: Pigeon-inspired optimization.

where 2 is the maximum number of iterations of the landmark operator. As depicted in the figure, the

landmark operator reduces the number of pigeons (gray) by half after each iteration. Pigeons distant from the
destination lose their path-distinguishing ability due to unfamiliarity with the landmarks and are consequently
discarded. Conversely, those near the destination (white) quickly orient themselves towards their target.

In this phase, the center position of the flock, constituted by the remaining pigeons, serves as a landmark.
This landmark provides a reference direction for the flight path of the remaining pigeons. The position update
equation for pigeon i is as follows:

( )

= ,
2
( )
∑ −1 −1
=1 (11)
−1 = ∑ ( −1 ) ,
=1
( )
= −1 + −1 − −1 .

( )
where is the ceiling function.
2
( )
 −1
 , for maximization problems,


( ) = 1 (12)
 ( ) , for minimization problems.
 −1
 +

( )
where −1 is the cost function of pigeon i at the sub-iteration.

2.3. PIO principles
Inthissection, weintroduceanovelapproach, termedIMCPIO,formanagingthePIOalgorithm. Thismethod
is grounded in the work of Sun and Duan [16] . Although the PIO base algorithm has advantages such as higher
robustness, it still faces problems of being prone to falling into local optimal solutions and slower convergence
and it is not applicable to UAV formation scenarios. Inspired by the SA algorithm, the improved Metropolis
criterion(IMC)preventstheparticlesfromfallingintolocaloptimalsolutions. TheMetropoliscriterionmakes