Aerospace and Electronic Systems - October 2018 - 43

Góez-Sánchez, Jaramillo-Garzón, and Velásquez
d 2 = ρdst − x j

(6)

d 3 = ρdst − ρ i

(7)

The turn ratio cost Cθ intends to favour particles that produce
soft direction changes and guarantees that the direction change
does not exceed the airship maximum turn ratio. First, the angle
change between the current direction vector and the direction vector associated to the particle is computed as shown in (8):

(

)

 μ i −1 ⋅ x j − ρ i −1 



x j − ρ i −1



θ μ = arccos 

(8)

(9)

where θg is the maximum turn ratio of the plane and depends on
the maximum roll angle supported by the airship ϕmax and the cruise
speed vCR km/h, as shown is (10), taken from [20].
6355 ⋅ tan φmax
θg =
π vCR

(10)

The last component of the objective function is the obstacle cost
Cω, which considers the unexpected obstacles that an airship may
find in its path. In this work, only geographical obstacles are considered. A landform is considered an obstacle if the flight altitude
is less or equal to the landform height plus a safety gap. Hence, Cω
assigns an infinite cost to those particles closer than that predefined
safety gap Rmin to a geographic obstacle as shown in (11):
0 when Rg > Rmin
Cω = 
∞ when Rg ≤ Rmin

(12)

To generate new solutions xg+1 for the ith cuckoo nest a Lévy
flight is generated using (13).

CUCKOO SEARCH OPTIMIZATION (CK)
The CK algorithm is a relatively new population-based stochastic
algorithm proposed by [17]. Since then, several papers have shown
the superiority of CK over other metaheuristic algorithms such as
PSO, GA, and Artificial Bee Colony algorithms in the solution of
common benchmark optimization problems [18], [21].
The cuckoos are generally medium-sized slender birds.
Many cuckoo species are brood parasites, laying their eggs
in the nests of other species to raise their young. The cuckoo
search optimization method is inspired by the parasitic behavior
of cuckoos. Each possible solution to a problem is associated
with an occupied nest [18]. Hence, the algorithm starts with
a set of nests where eggs have been placed. From all of them,

(13)

where β is the step size which should be related to the search space
[23]. Lévy flights are characterized by long straight flights followed by sudden changes on direction. The direction turn is uniformly distributed and the step size λ is established according to
the Lévy distribution on (14), [17].

λ=

υ
ϑ

(14)

1

β

where υ and ϑ are random variables from a normal distribution
with mean zero and standard deviations συ and σϑ, respectively (see
(15)). Equation 16 provides the value for those standard deviations, where Γ stands for the Gamma distribution [24].

υ ~  ( 0,σ υ2 )

(15)

ϑ ~  ( 0,σ ϑ2 ) )

(11)

where Rg represents the distance from the particle to any geographical obstacle.

OCTOBER 2018

Levy ∼ u = t − λ , (1 < λ  3)

x gj +1 = x gj + β ⊕ Levy ( λ )

Then, Cθ is computed according to (8):
θ μ when θ μ ≤ θ g
Cθ = 
∞ when θ μ > θ g

the best eggs go unnoticed while the less fit eggs have a high
probability of being discovered by the nest owner and being discarded. To select new nests, and to complete the search space,
CK algorithm imitates the behavior of some fruit flies and birds
exhibiting random flight patterns similar to the so called Lévy
Flights [22]. Lévy flights essentially provide a random walk
while their random steps are drawn from a Lévy Distribution
for large steps, (12), [17].

1

β

 β 
 Γ (1 + β ) ⋅ sin  Π.  
 2 
σ υ ~ = 
 1+ β 
β −1  
 Γ
⋅β ⋅2

  2 
2  
 
σϑ ~= 1

(16)

PARTICLE SWARM OPTIMIZATION (PSO)
PSO is an optimization algorithm inspired by the swarm behavior
of large groups of insects, bird flocks, or shoal of fish [25]. PSO
emulates the social behavior of these groups of animals assuming
some sort of communication among the individuals, particles in
PSO notation. The movement of each particle is driven by three
components: self-experience on their own best position, the group
experience on the best position achieved by any members in the
swarm, and an inertial component that softens the particles speed
[16], [26].

IEEE A&E SYSTEMS MAGAZINE

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Aerospace and Electronic Systems - October 2018

Table of Contents for the Digital Edition of Aerospace and Electronic Systems - October 2018
