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Page 6 of 19 Mai et al. Intell Robot 2023;3(4):466-84 I http://dx.doi.org/10.20517/ir.2023.37

Figure 3. UAV maximum turning angle.

follow. The core principle of this algorithm involves representing potential solutions to a problem through
paths selected by the simulated ants. These paths collectively constitute the space of feasible solutions, and
within this space, the optimal solution corresponds to the shortest path.

3.1 State transition function
Theantstartsitsjourneyfromtheinitialpoint,calculatingthetransitionprobabilitiestovariousstates. Itselects
thenextnodebasedonthepheromonelevel andtheheuristicfunction untilitreachesthetargetpoint. In
theearlystagesoftheACOalgorithm, pheromoneaccumulationisminimal, andpathselectionprimarilyrelies
on the heuristic function. As pheromone concentrations reach a certain threshold, the heuristic function and
the pheromone level influence path selection. In the traditional ACO algorithm, the probability of a transition
from node to node for ant k can be expressed using Equation (7):

[ ] [ ]
 ( ) ( )

 ∈


( ) = ∑ ∈ [ ( )] [ ( )] (7)



 0 ℎ

where is the set of next nodes that the current node can choose, and are the weight parameters of
pheromone and heuristic function, respectively, reflecting the weight of the ant colony’s prior knowledge and
unexplored paths. represents the pheromone concentration, represents the heuristic function, repre-
sents the current ant, and represents the candidate node.
3.2 Pheromone update rules
Pheromone concentrations naturally diminish over time. The pheromone update process occurs once all ants
have finished their path search from the starting point to the target point. This pheromone update involves
two components: the residual pheromone and the freshly released pheromone. The update process is shown
in Equations (8)-(10):
(8)
( + 1) = (1 − ) ( ) + Δ ( )

∑
Δ = Δ (9)
=1
{ ( , )

Δ = / (10)

0 ℎ

In Equations (8)-(10), denotes the total amount of pheromones, is the evaporation rate, Δ represents
the increment of pheromone, Δ is the quantity of pheromone laid on edge ( , ) by ant k, Q is a constant,

and is the length of the tour constructed by ant k.