

Ci a ion: Wang, X.; Pan, J.-S.; Yang,
Q.; Kong, L.; Snášel, V.; Chu, S.-C.
Modi ied May ly Algo i hm o UAV
Pa h Planning. D ones 2022,6, 134.
h ps://doi.o g/10.3390/d ones
6050134
Academic Edi o : Oleg Yakimenko
Recei ed: 19 Ap il 2022
Accep ed: 18 May 2022
Published: 23 May 2022
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d ones
A icle
Modi ied May ly Algo i hm o UAV Pa h Planning
Xing Wang 1, Jeng-Shyang Pan 1, Qingyong Yang 1, Lingping Kong 2, Václa Snášel 2
and Shu-Chuan Chu 1,*
1College o Compu e Science and Enginee ing, Shandong Uni e si y o Science and Technology,
Qingdao 266590, China; [email p o ec ed] (X.W.); [email p o ec ed] (J.-S.P.);
[email p o ec ed] (Q.Y.)
2Facul y o Elec ical Enginee ing and Compu e Science, VŠB-Technical Uni e si y o Os a a,
70032 Os a a, Czech Republic; [email p o ec ed] (L.K.); acla [email p o ec ed] (V.S.)
*Co espondence: [email p o ec ed]
Abs ac :
The unmanned ae ial ehicle (UAV) pa h planning p oblem is p ima ily conce ned wi h
a oiding collision wi h obs acles while de e mining he bes ligh pa h o he a ge posi ion. This
pape i s es ablishes a cos unc ion o ans o m he UAV ou e planning issue in o an op imiza ion
issue ha mee s he UAV’s easible pa h equi emen s and pa h sa e y cons ain s. Then, his pape
in oduces a modi ied May ly Algo i hm (modMA), which employs an exponen dec easing ine ia
weigh (EDIW) s a egy, adap i e Cauchy mu a ion, and an enhanced c osso e ope a o o e ec i ely
sea ch he UAV con igu a ion space and disco e he pa h wi h he lowes o e all cos . Finally, he
p oposed modMA is e alua ed on 26 benchma k unc ions as well as he UAV ou e planning
p oblem, and he esul s demons a e ha i ou pe o ms he o he compa ed algo i hms.
Keywo ds:
pa h planning; modi ied may ly algo i hm; exponen dec easing ine ia weigh ; adap i e
Cauchy mu a ion; enhanced c osso e ope a o
1. In oduc ion
Unmanned ae ial ehicles (UAV), as opposed o manned ai c a , e e o unmanned
ai c a ha can be con olled by emo e adio con ol equipmen o ai bo ne compu e s.
UAVs a e less expensi e and mo e lexible han manned ai c a , and hey can each di icul
o dange ous places o humans o pe o m asks. The e o e, UAVs ha e been widely used
in ci il and mili a y ields. In he ci il ield, d ones ha e played an impo an ole in
ag icul u al plan p o ec ion en i onmen , ae ial pho og aphy, exp ess anspo a ion,
emo e sensing o he oceans, and so on. In he mili a y ield, mili a y d ones can eplace
pilo s in complex and dange ous missions such as in elligence ga he ing and econnaissance
su eillance. I he UAV is o comple e he abo e asks, one o he basic capabili ies ha he
UAV mus ha e is pa h planning capabili y [1].
The UAV pa h planning p oblem is o plan a collision- ee pa h o he UAV om he
s a ing poin o he a ge poin unde he gi en ligh condi ions and ligh en i onmen .
The planned pa h has he minimum cos and sa is ies he ele an cons ain s. This p oblem
can be simply desc ibed as an op imiza ion p oblem wi h mul iple cons ain s [
2
], so i is a
challenge o adi ional op imiza ion s a egies. The e a e al eady many ma u e solu ions
o his p oblem. In he di ec ion o pa h planning based on g aph sea ch,
Re . [3]
uses
he A* algo i hm o he UAV pa h planning p oblem wi h he goal o aking he leas
isk and consuming he leas amoun o uel. Re . [
4
] uses he D* Li e algo i hm o plan
an e icien pa h o he i e poin o i e igh ing UAVs. In he di ec ion o pa h planning
based on sampling, Re . [
5
] uses a hyb id algo i hm combined wi h he a i icial po en ial
ield me hod and he RRT-Connec algo i hm o he pa h planning p oblem o UAV. E en
hough he echniques desc ibed abo e a e ela i ely ma u e in ma hema ical heo y, hey
a e ine icien when wo king wi h discon inuous and non-de i a i e unc ions [
6
]. When
D ones 2022,6, 134. h ps://doi.o g/10.3390/d ones6050134 h ps://www.mdpi.com/jou nal/d ones
D ones 2022,6, 134 2 o 21
ackling he UAV ou e planning p oblem wi h many es ic ions [
7
], hey a e eadily
apped in o local op imum solu ions.
The pa h planning issue has been demons a ed o be an NP-ha d p oblem, and
he p oblem complexi y g ows wi h p oblem size [
8
]. When sol ing NP-ha d p ob-
lems, heu is ic algo i hms can p oduce high-quali y solu ions and a e simple o imple-
men . As a esul , some well-de eloped me a-heu is ics wi h good pe o mance ha e
been p oposed in ecen yea s, such as he Cuckoo Sea ch algo i hm (CS) [
9
,
10
], Pa icle
Swa m
Op imiza ion (PSO) [11–13]
, Gene ic Algo i hm (GA) [
14
,
15
], A i icial Bee Colony
(ABC)
algo i hm [16,17],
Di e en ial E olu ion (DE) algo i hm [
18
,
19
], G ey Wol Op i-
mize (GWO) [20,21], An Colony Op imiza ion (ACO) algo i hm [22,23], e c.
The May ly algo i hm (MA) [24] is a ecen ly sugges ed op imiza ion me hod, which
akes he ligh and ma ing beha io o he may ly as he model. Female and male may lies
make up he en i e may ly popula ion. The mo emen s o he male and emale may lies
o e he MA he capaci y o conduc local sea ches, and he p ocedu e o de eloping
o sp ing h ough may ly ma ing gi es he MA global sea ch capabili y.
A p esen , esea ch on MA is s ill ela i ely small. Inspi ed by he ba e-bones PSO
(BBPSO) [
25
], and hen, Juan Zhao [
26
] p oposed he ba e-bones may ly algo i hm. The
chao ic may ly algo i hm was sugges ed by Mohamed A. M. Shaheen [
27
], who used a
logical chao ic map o ini ialize he may ly popula ion. This pape p esen ed a modi-
ied May ly Algo i hm (modMA), which includes he exponen dec easing ine ia weigh
s a egy, adap i e Cauchy mu a ion, and an enhanced c osso e ope a o o analyze he
pa h planning issue o UAV om he g ound s a ion o he des ina ion, and i achie es
good esul s.
Many wo ks on imp o ing wo-dimensional pa hs using me aheu is ics ha e been
p oposed o his poin . Re . [
28
] used he ba and cuckoo algo i hm in pa h planning.
Re . [
29
] used he g ey wol op imize o ackle he UAV ou e planning issue. Re . [
30
] used
a chao ic cuckoo sea ch algo i hm o op imize d one pa hs in wo-dimensional (2D) space.
Besides, many esea che s used o he me a-heu is ic algo i hms o s udy UAV pa h plan-
ning p oblems [
31
–
33
]. Howe e , no one has p oposed he use o MA o ackle UAV pa h
planning in a 2D en i onmen . Hence, his pape will use he modi ied May ly Algo i hm
(modMA) o s udy his p oblem. The ollowing a e he con ibu ions o his pape .
(1)
I p oposes an enhanced c osso e ope a o based on he MA, which can imp o e
MA’s explo a ion capabili y and con e gence speed. Then i p oposes a modi ied
May ly Algo i hm (modMA), which combines he EDIW s a egy, adap i e Cauchy
mu a ion, and he enhanced c osso e ope a o o balance he MA’s p ocesses o
explo a ion and exploi a ion;
(2)
I in oduces h ee e sions o he modMA and compa es hei pe o mance on six
100 and 300 dimension benchma k unc ions;
(3)
Compa e he h ee e sions o he modi ied MA wi h o he algo i hms (MA, PSO,
GWO, BOA) on wen y-six 50 dimension benchma k unc ions, and analyze he esul s
o he expe imen s;
(4)
Use he h ee e sions o he modi ied MA o ackling he 2D pa h planning p oblem
o ag icul u al UAV.
This pape is o ganized as ollows: The s anda d MA, as well as he p oblem o UAV
pa h planning, a e b ie ly desc ibed in Sec ion 2. Sec ion 3p o ides h ee op imiza ion ideas
o imp o ing he MA. In Sec ion 4, wo expe imen s a e planned o compa e he MA’s
enhanced pe o mance using di e en op imiza ion ideas and apply he h ee e sions o
he modi ied MA o he pa h planning p oblem o he ag icul u al UAV. Sec ion 5concludes.
2. Rela ed Wo k
This sec ion will p o ide a quick o e iew o he s anda d MA and he wo-dimensional
pa h planning p oblem o UAV.
D ones 2022,6, 134 3 o 21
2.1. S anda d May ly Algo i hm
The p oposed MA’s undamen al concep is de i ed om he ligh and ma ing beha -
io o may lies. I combines he ad an ages o PSO [
11
], GA [
14
] and FA [
34
]. MA ini ially
p oduces a may ly popula ion comp ising males and emales a andom. The cu en
eloci y and posi ion o he i h may ly a e wo
n
-dimensional ec o s, hey a e deno ed
as
i=( i1, i2. . . , in)
and
xi=(xi1,xi2. . . , xin)
, espec i ely. Each may ly modi ies i s
posi ion depending on i s indi idual bes (
pbes
) posi ion and he bes (
gbes
) posi ion
ound by he whole may ly swa m hus a .
2.1.1. Male May ly Fligh
Male may lies clus e oge he , which sugges s ha hei posi ions a e upda ed based
on social and pe sonal expe ience.
x
i
is de ined as he p esen posi ion o he i h male
may ly a i e a ion
. Assuming ha
x
i
is he cu en posi ion o he i h may ly a i e a ion
,
he posi ion can be upda ed by adding o a eloci y
i
. Thus he male may ly’s posi ion
upda e o mula is:
x +1
i=x
i+
i. (1)
Conside hose male may lies a e always pe o ming nup ial dances no a om he
wa e . The male may ly’s eloci y upda e o mula is as ollows:
+1
ij =g
ij +a1e−β 2
ppbes ij −x
ij+a2e−β 2
ggbes j−x
ij, (2)
whe e
β
is he isibili y coe icien ,
a1
and
a2
a e de ined as he posi i e cons an s ep esen -
ing he a ac ion.
pbes ij
is he bes posi ion ob ained by he i h male may ly in dimension
j
.
p
and
g
a e he Euclidean dis ances be ween
xi
and
pbes i
and be ween
xi
and
gbes
,
espec i ely. The g a i y coe icien is gi en by
g
, which can be a ixed numbe be ween 0
and 1, o i can be s eadily lowe ed o e i e a ions as in Equa ion (3).
g=gmax −gmax −gmin
i e max ×i e , (3)
whe e
gmax
and
gmin
espec i ely ep esen he smalles and la ges weigh s. The p esen
and he o al numbe o i e a ions a e gi en by i e and i e max, espec i ely.
The pe sonal bes posi ion pbes ia i e a ion +1 is de e mined by Equa ion (4).
pbes i=(x +1
i,i x +1
i< (pbes i)
same as be o e,o he wise.(4)
The bes male may lies keep pe o ming up and down mo ions h ough di e en
eloci ies. The eloci ies o hese male may lies a e de e mined by Equa ion (5).
+1
ij =
ij +d× , (5)
whe e he nup ial dance coe icien is gi en by
d
, and
means a andom alue in he ange
−1 o 1.
2.1.2. Female May ly Fligh
Female may lies do no clus e oge he , howe e , hey mo e owa ds male may lies.
y
i
is de ined as he p esen posi ion o he i h emale may ly a i e a ion
. The change o
he i h emale may ly in posi ion is calcula ed as:
y +1
i=y
i+
i. (6)
In he MA, male and emale may lies wi h he same indi idual i ness anking will
a ac each o he , and he emale may ly’s posi ion changes in esponse o he loca ion o a
male may ly wi h he same anking. The emale may ly’s eloci y upda e o mula is:
D ones 2022,6, 134 4 o 21
+1
ij =


g
ij +a2e−β 2
m x
ij −y
ij,i (yi)> (xi)
g
ij + l × ,i (yi)≤ (xi),(7)
whe e
ij
and
y
ij
a e espec i ely de ined as he i h emale may ly’s eloci y and posi ion in
dimension
j
a i e a ion
.
a2
and
β
a e de ined as he a ac ion cons an and he isibili y
coe icien , espec i ely.
m
is he Euclidean dis ance be ween he i h emale may ly and
he i h male may ly,
l
is a andom walk coe icien which sugges s ha a emale is no
a ac ed o a male, and means a andom alue in he ange −1 o 1.
2.1.3. Ma ing P ocedu e
The ma ing is ep esen ed by he c osso e ope a o which p o ides he global sea ch
capabili y o he MA. In he MA, male and emale may lies wi h he same indi idual i ness
anking will ma e wi h each o he o p oduce o sp ing may lies. The ma ing ope a ion
o each pai o male and emale may lies p oduces wo o sp ing, and he o mula o he
c osso e ope a o is:
γ1=L×male +(1−L)× emale
γ2=L× emale +(1−L)×male,(8)
whe e
L
means a andom alue in he ange 0 o 1. Mo eo e ,
male
and
emale
deno e he
pa en s. Ini ially, he o sp ing’s eloci y is se o ze o.
2.1.4. Mu a e he Genes o O sp ing
The mu a ion ope a o p o ides pa ial local sea ch capabili y o MA. The chosen
o sp ing can be upda ed by adding a andom alue o he a iable h oughou his p ocess,
so he o sp ing a e changed o
γ0
n=γn+σNn(0,1), (9)
whe e he s anda d de ia ion and he s anda d no mal dis ibu ion a e ep esen ed by
σ
and Nn, espec i ely.
2.1.5. Reduc ion o Nup ial Dance and Random Walk
Al hough hese ope a o s p o ide he MA wi h powe ul local sea ch capabili y,
d
and
l
mus be g adually educed. Bo h alues can be upda ed o e he i e a ions
using
Equa ions (10) and (11), espec i ely.
d =d0δ (10)
l = l0δ , (11)
whe e δis a ixed numbe be ween 0 and 1.
As p e iously s a ed, Algo i hm 1depic s he MA’s pseudo-code.
D ones 2022,6, 134 5 o 21
Algo i hm 1 Pseudocode o MA.
Inpu :
male and emale popula ion sizes,
N1
and
N2
; maximum i e a ions,
i e max
; isibili y
coe icien ,
β
; lea ning ac o s,
a1
and
a2
; nup ial dance coe icien ,
d
; andom ligh
coe icien , l; objec i e unc ion, (x)
Ou pu : Op imal solu ion gbes
1: Ini ialize he male and emale eloci ies mand
2: E alua e popula ion based on (x)
3: Find he global bes (gbes )
4: o i e = 1 o i e max do
5: Adjus he speed and posi ion o each emale may ly using Equa ions (6) and (7)
6: Adjus he speed and posi ion o each male may ly using Equa ions (1), (2) and (5)
7: So he may lies and ank hem based on (x)
8: Pe o m c osso e and gene a e male and emale o sp ing
9: Mu a e he o sp ing
10: Di ide o sp ing in o male and emale a andom
11: Upda e he wo s old indi iduals wi h he ines new ones
12: Upda e pbes and gbes
13: end o
2.2. UAV Pa h Planning P oblem
UAV pa h planning aims o disco e he bes ligh ou e o he UAV gi en a a ie y
o complica ed lying es ic ions such as h ea cons ain s and uel limi s [
35
]. The lying
heigh o he d one migh be cons an along i s ligh pa h om he s a poin o he a ge
poin , so his pape only ocuses on he UAV’s ho izon al lying dis ance. The ma hema ical
model o UAV pa h planning and he cos unc ion used o e alua e he planned pa h a e
desc ibed as ollows.
2.2.1. Ma hema ical Model o Ua Pa h Planning
In his model, we de ine
S
and
T
o be he s a poin and a ge poin o he UAV,
espec i ely. I is c ucial o p e en collisions wi h nume ous obs acles du ing he ligh o
he d one. As a esul , his pape employs ci cles o a ying adii o subs i u e ba ie s o
a ying h ea le els. So he ask o a UAV ligh is o ind a ligh pa h om
S
o
T
wi h he
smalles o e all cos while a oiding collisions wi h obs acles. Figu e 1depic s a schema ic
ep esen a ion o he en i onmen modeling o he UAV’s wo-dimensional pa h planning.
The pa h connec ed by he ed line segmen s is he easible pa h.
Figu e 1. Two-dimensional ep esen a ion o planning space.

D ones 2022,6, 134 6 o 21
The segmen
ST
be ween he
S
and
T
in he o iginal coo dina e sys em
OXY
is spli
in o
D+
1 equal po ions by
D
e ical lines
{Lk,k=1, 2, . . . , D}
. We can ge he pa h om
S
o
T
by picking a poin on each e ical line. Thus, he planned pa h can be exp essed as:
L={S,(x(1),y(1)), . . . , (x(D),y(D)),T}. (12)
To accele a e he p ocessing speed, le he segmen
ST
be he ho izon al axis and use
Equa ion (13) o ans e each poin
(x(k),y(k))
in he
OXY
o
x0(k),y0(k)
in he new
coo dina e sys em OX0Y0.
"x0(k)
y0(k)#=cos θsin θ
−sin θcos θ x(k)−xs
y(k)−ys,(13)
whe e
θ
deno es o he o a ion angle o he
OXY
, and he poin
(xs,ys)
is he coo dina e in
he OXY.
As shown in Figu e 1, he abscissa o each node in he
OX0Y0
can be calcula ed using
o mula x0(k)=|ST|
D+1×k, so he planned pa h can be simpli ied and exp essed as:
L0=n(0,0),x0(1),y0(1), . . . . . ., x0(D),y0(D),(|ST|,0)o. (14)
2.2.2. Fi ness Func ion Modeling
A i ness unc ion can be used o calcula e he cos o a planned pa h. The i ness
unc ion alue can be used o judge he quali y o he planned pa h. The i ness unc ion
es ablished in his pape is de ined as he sum o wo cos unc ions, gi en as ollows:
FL0=w1F1L0+w2F2L0, (15)
whe e
FL0
deno es he o al cos unc ion o he planned pa h;
F1L0
deno es he cos
o he planned pa h’s leng h; F2L0deno es he smoo hness cos o he planned pa h. ω1
and ω2a e he weigh coe icien s o he abo e cos unc ion, and hey a e de ined as:
ωi≥0
∑2
i=1wi=1. (16)
Gi en he UAV’s es ic ed ene gy supply, he sho e he planned pa h leng h, he
less ime and ene gy he UAV spends, and he be e i is o he UAV. Assume ha he
opog aphy o he ligh en i onmen and in o ma ion on he h ea ened a ea a e known.
The s a s a e and he a ge s a e a e
L0
0
and
L0D+1
, espec i ely. Hence, he cos
F1
ela ed
o he pa h leng h can be compu ed as:
F1L0=
D
∑
k=0
dL0
k,L0
k+1, (17)
whe e
dL0
k,L0
k+1
is he dis ance be ween
x0(k),y0(k)
and
x0(k+1),y0(k+1)
. I he
waypoin alls in o he obs acle, we choose he in e sec ion o he e ical line wi h he
obs acle, he poin closes o he waypoin , as he new sea ch agen . Figu e 2a,b depic s he
pos -collision p ocessing low. The pa h connec ed by he ed line segmen s is he modi ied
easible pa h.
D ones 2022,6, 134 7 o 21
(a) (b)
Figu e 2. The pos -collision p ocessing low. (a)Scena io 1; (b)Scena io 2.
The smoo hness cos e alua es he u ning a e c i ical o gene a ing easible pa hs.
The smoo h cos F2can be compu ed as:











F2L0=∑D
k=1∆k
∆k=cos ϕ−cos θk,ϕ≥θi
ϕ,ϕ<θi
cos θ=aT
iai+1
|ai||ai+1|
, (18)
whe e he
ϕ
is he maximum u ning angle,
θ
is he cu en u ning angle, and
ai
is a ec o
ep esen ing he i h segmen o he en i e ou e.
3. Modi ided May ly Algo i hm
The s anda d MA has a be e con e gence speed han o he swa m algo i hms when
dealing wi h low-dimensional si ua ions. Howe e , due o he in luence o eloci y luc-
ua ion, he s abili y o he MA is poo , hus i leads o poo esul s. Fu he mo e, when
dealing wi h high-dimensional nonlinea complica ed si ua ions, he MA canno me ely
ely on i s mechanism o ge ou o he local op imal zone, so he MA pe o ms poo ly
on he mul imodal unc ions. This sec ion p oposes a modi ied may ly algo i hm named
modMA, which combines h ee s a egies o imp o e he MA. One imp o emen is o use
he exponen dec easing ine ia weigh s a egy o imp o e MA, ano he uses he adap i e
Cauchy mu a ion s a egy o imp o e MA, and a hi d uses an enhanced c osso e ope a o
o imp o e MA.
3.1. Imp o emen 1: Exponen Dec easing Ine ia Weigh S a egy
The e a e many dec easing ine ia weigh s a egies, such as linea [
36
], adap i e [
37
],
loga i hmic [
38
], and o he s [
39
]. The la ge ine ia weigh in he ea lie pe iod is a o able
o he explo a ion o pa icles, and he pa icles can sea ch a la ge space. The smalle
ine ia weigh in he la e pe iod is bene icial o he exploi a ion o pa icles. This pape
in oduces he exponen dec easing ine ia weigh [40] in o MA, o mula is as ollows:
g=gmin +exp1−i e max
i e max −i e +1∗(gmax −gmin), (19)
whe e gmax,gmin,i e , and i e max a e he same o Equa ion (3).
D ones 2022,6, 134 8 o 21
3.2. Imp o emen 2: Adap i e Cauchy Mu aion S a egy
When u ilizing Equa ions (1) and (2) in he s anda d MA o change he eloci y o each
male may ly, he di e si y o he en i e may ly popula ion may decline. To mi iga e his
p oblem, he posi ion o each male may ly should be eadjus ed by using Cauchy-based
mu a ion [
41
,
42
]. The s anda d Cauchy dis ibu ion’s p obabili y densi y unc ion can be
desc ibed as:
(x)=1
π×1
1+x2, (20)
whe e −∞<x<∞, he co esponding dis ibu ion unc ion is exp essed as:
F(x)=1
πa c an(x)+1
2. (21)
Random numbe s gene a ed ia he s anda d Cauchy dis ibu ion a e shown below:
CM = anπ×ε−1
2, (22)
whe e
CM
deno es a andom alue p oduced by he in e se unc ion o he s anda d
Cauchy dis ibu ion unc ion,
ε
means a andom alue in he ange 0 o 1 p oduced by a
uni o m dis ibu ion.
Acco ding o Foga y [
43
], imp o ed pe o mance is a ained when he mu a ion a e
alls exponen ially wi h he numbe o i e a ions. Re . [
44
] poin ed ou ha he concep is
ela ed o ha o he simula ed annealing (SA) algo i hm [
45
]. So a e u ilizing Equa ions (1)
and (2) o change he posi ion o each male may ly, he posi ion o each male may ly needs
o be eadjus ed using he Equa ion (22), which combines Cauchy mu a ion s a egy wi h
an adap i e mu a ion.
X +1
i=x +1
i+x +1
i×CM ×exp((1− )×α), (23)
whe e
x +1
i
is he i h indi idual calcula ed using Equa ions (1) and (2), and
X +1
i
ep esen s
he mu a ed indi idual.
α
is a con ac ion-expansion coe icien ha con ols he speed o
con e gence and i is de ined as 0.15.
In his sec ion, we combine he Cauchy mu a ion s a egy wi h an adap i e mu a ion
which can make he MA gene a e pe u ba ions wi h a la ge s ep size in he ea ly i e a ions
so ha he pa icles can jump om hei cu en posi ion o ano he and help MA escape
he local op ima. The pe u ba ions wi h a smalle s ep size in he la e i e a ions can speed
up he con e gence o he MA. I is wo h men ioning ha we do no use Cauchy-based
mu a ion o eadjus he posi ions o he pa icles calcula ed using Equa ions (1) and (5).
3.3. Imp o emen 3: Enhanced C osso e Ope a o
Al hough he c osso e and ep oduc ion ope a ions o may lies p o ide he MA wi h
explo a ion abili y as well as enhance he MA’s con e gence speed, he MA has a no
s ong explo a ion abili y. Hence, we use he ho izon al c osso e sea ch [
46
] o imp o e
i . The enhanced c osso e ope a o o he MA employing ho izon al c osso e sea ch is
as ollows:
γ1=L×male +(1−L)× emale +c1×(male − emale)
γ2=L× emale +(1−L)×male +c2×( emale −male),(24)
whe e
c1
and
c2
a e expansion coe icien s, and hey a e bo h andom alues be ween
−
1
and 1 p oduced by a uni o m dis ibu ion.
The ho izon al c osso e ope a ion di ides he p oblem-sol ing space o mul i-
dimensional in o hal -popula ion o hype cubes, allowing new spo s on he hype cube’s
pe iphe y o be sampled wi h a low p obabili y, and educing he blind egion ha a e un-
able o explo e by wo pai ed pa en indi iduals. O sp ing gene a ed using he ho izon al
D ones 2022,6, 134 9 o 21
c osso e ope a ion need o be compa ed wi h hei pa en s, and indi iduals wi h g ea e
i ness alues a e selec ed o he ollowing i e a ion. I causes he MA o con inuously
con e ge o he bes solu ion and ensu e he con e gence e iciency wi hou a ec ing he
op imiza ion accu acy h ough he ho izon al c osso e ope a ion.
We can also app op ia ely sh ink o expand he sea ch space gene a ed by he wo
pai ed pa en indi iduals. The o mula o he c osso e ope a o o he MA imp o ed by
sh inking he sea ch space is as ollows:
γ1=d1×(L×male +(1−L)× emale)
γ2=d3×(L× emale +(1−L)×male),(25)
whe e
d1
and
d3
a e he con ac ion coe icien s, and hey a e bo h andom alues be ween
0.7 and 1 p oduced by a uni o m dis ibu ion.
The o mula o he c osso e ope a o o he MA imp o ed by expanding he sea ch
space is as ollows:
γ1=d2×(L×male +(1−L)× emale)
γ2=d4×(L× emale +(1−L)×male),(26)
whe e
d2
and
d4
a e he expansion coe icien s, and hey a e bo h andom alues be ween 1
and 1.3 p oduced by a uni o m dis ibu ion.
The e o e, he enhanced c osso e ope a o o mula o MA can be desc ibed as:
γ1=






L×male +(1−L)× emale, and1<pone
L×male +(1−L)× emale +c1×(male − emale), and1>pone, and2<p wo
d1×(L×male +(1−L)× emale), and1>pone, and2<p wo, and3<p h ee
d3×(L×male +(1−L)× emale), and1>pone, and2<p wo, and3>p h ee.
(27)
γ2=






L× emale +(1−L)×male, and1<pone
L× emale +(1−L)×male +c2×( emale −male), and1>pone, and2<p wo
d2×(L× emale +(1−L)×male), and1>pone, and2<p wo, and3<p h ee
d4×(L× emale +(1−L)×male), and1>pone, and2<p wo, and3>p h ee,
(28)
whe e
and1
,
and2
, and
and3
a e he andom inpu numbe s be ween [0,1].
pone
,
p wo
, and
p h ee
a e he swi ching p obabili ies, which con ol he me hod o o sp ing gene a ion.
Fo he alue o swi ching p obabili y
pone
, we e e o he pa ame e uning echnique
o swi ching p obabili y in he Bu e ly Op imiza ion Algo i hm (BOA) [
47
]. F om ou
simula ions, we ound ha
pone
= 0.8,
p wo
= 0.5, and
p h ee
= 0.5 wo k be e o mos
applica ions.
4. Expe imen al Resul s and Applica ion
In his pa , we selec a a ie y o nume ical op imiza ion unc ions om CEC bench-
ma k unc ions o assess he e ec o he modi ied MA p esen ed in his a icle. Tables A1
and A2 in Appendix Ashow he exp ession, dimension, domain o a a iable, and mini-
mum alue o each benchma k unc ion. These benchma k unc ions used in his s udy
can be ound in [
48
], and hey a e classi ied in o wo g oups. The unimodal benchma k
unc ions (F1–F14) and mul imodal benchma k unc ions (F15–F26) a e applied o calcula e
he abili ies o exploi and explo e o he algo i hm benchma ked by benchma k unc ions,
espec i ely.
To assess he e icacy o he h ee imp o emen echniques desc ibed in his s udy o
MA, we i s es and analyze he h ee e sions o he imp o ed MA h ough compa ison
expe imen 1 by six benchma k unc ions om Tables A1 and A2 wi h
Dim
= 100 and
Dim = 300
. I is wo h men ioning ha modMA-1 is cha ac e ized in his s udy as an MA
ha is imp o ed by an adap i e Cauchy mu a ion s a egy and he o iginal c osso e
ope a o , modMA-2 is de ined as an MA modi ied using only he enhanced c osso e
ope a o , and modMA is de ined as an MA ha inco po a es h ee imp o ed s a egies.
D ones 2022,6, 134 16 o 21
Table 7. Space en i onmen se ings.
Case Numbe Se ial Numbe Obs acle Cen e Obs acle Radius
1
1 (50, 105) 70
2 (125, 250) 35
3 (304, 400) 45
4 (404, 320) 50
5 (440, 440) 20
6 (280, 310) 25
7 (230, 220) 25
8 (230, 100) 50
2
1 (160, 160) 15
2 (50, 105) 70
3 (275, 185) 80
4 (400, 425) 40
5 (125, 250) 35
6 (275, 325) 28
7 (450, 250) 45
8 (175, 410) 70
9 (35, 325) 50
10 (330, 300) 25
The expe imen al esul s a e lis ed in Table 8. The co esponding con e gence cu es
a e shown in Figu es 7a– and 8a– . F om Figu es 7a– and 8a– , all he se en algo i hms
can ind collision- ee ligh pa hs, bu hey pe o m di e en ly in e ms o pa h leng h
and smoo hness. I can be seen om Table 8 ha in he case o
D
= 30, ha is, in a lowe
dimension, each algo i hm can achie e good esul s, and he pa h leng h planned by
modMA is sho e and he pa h is smoo he . In he case o
D
= 50, ha is, in a highe
dimension, modMA, modMA-1, and modMA-2 g ea ly imp o e he pe o mance o MA.
modMA and modMA-1 can achie e be e esul s in mos cases, and bo h ha e good ini ial
solu ions and con e gence a es, bu he luc ua ion o modMA is smalle han ha o
modMA-1. Among all he compa ison algo i hms, he o e all cos o modMA planning is
he smalles in mos cases, which shows ha he pa h planned by he modMA can each
he bes when he UAV a oids collision, which p o es he e ec i eness o he modMA.
Table 8. The a e age and s anda d de ia ion alues o he o al cos unc ion o e hi y uns.
Case Numbe D Resul s MA modMA-1 modMA-2 modMA BOA PSO GWO
1
30 A g 705.432 690.146 692.234 689.532 711.309 698.138 693.970
S d 6.199 1.146 3.313 1.014 7.455 6.897 2.749
50 A g 753.834 699.756 723.285 698.312 720.847 739.353 710.557
S d 18.143 2.822 11.390 1.419 13.549 18.421 9.951
2
30 A g 717.855 692.493 693.226 691.735 711.219 694.774 694.458
S d 10.718 1.729 5.087 1.359 4.180 3.273 1.675
50 A g 781.436 703.315 747.785 702.119 721.602 732.321 710.133
S d 28.557 3.123 23.130 1.978 6.189 12.167 7.973
The pa h planning p oblem has many local solu ions, his makes classical op imiza ion
echniques di icul o sol e. modMA shows excellen pe o mance on his complex p ob-
lem, i.e., he quali y o he solu ion and he s abili y o he solu ion. In addi ion, modMA
also has good ini ial solu ion and as con e gence speed. Howe e , he model es ablished
in his pape s ill has some sho comings. A e collision handling, ha is, he ope a ion o
mo ing he waypoin o he edge o he obs acle, he planned pa h leng h may be longe
han he heo e ical op imum.

D ones 2022,6, 134 17 o 21
(a) (b) (c)
(d) (e) ( )
Figu e 7.
Pa h diag am o a single un in Case 1. (
a
) Compa a i e pa h planning esul s in Case
1,
Dim
= 30; (
b
) E olu ion cu es o ou algo i hms in Case 1,
Dim
= 30; (
c
) E olu ion cu es o
di e en algo i hms in Case 1,
Dim
= 30; (
d
) Compa a i e pa h planning esul s in Case 1,
Dim
=
50; (
e
) E olu ion cu es o ou algo i hms in Case 1,
Dim
= 50; (
) E olu ion cu es o di e en
algo i hms in Case 1, Dim = 50.
(a) (b) (c)
(d) (e) ( )
Figu e 8.
Pa h diag am o a single un in Case 2. (
a
) Compa a i e pa h planning esul s in Case
2,
Dim
= 30. (
b
) E olu ion cu es o ou algo i hms in Case 2,
Dim
= 30. (
c
) E olu ion cu es o
di e en algo i hms in Case 2,
Dim
= 30. (
d
) Compa a i e pa h planning esul s in Case 2,
Dim
=
50. (
e
) E olu ion cu es o ou algo i hms in Case 2,
Dim
= 50. (
) E olu ion cu es o di e en
algo i hms in Case 2, Dim = 50.
5. Conclusions
A new algo i hm modMA which inco po a es a mixed s a egy consis ing o h ee
imp o ed echniques is p oposed h oughou his esea ch o sol ing he UAV pa h
planning p oblem. By in oducing he EDIW s a egy o balance he pa icles’ p ocesses
o explo a ion and exploi a ion. By pe u bing he posi ion o each male pa icle wi h an
adap i e Cauchy mu a ion ope a o , he di e si y o he swa m is inc eased. By in oducing
an enhanced c osso e ope a o , he explo a ion abili y o he modMA is enhanced. To
D ones 2022,6, 134 18 o 21
assess he e icacy o hese h ee imp o emen echniques on MA, wo expe imen s a e
designed. The sugges ed modMA is es ed on 26 benchma k unc ions, and i has ob ious
ad an ages in inding bo h he bes alue and con e gence speed. Finally, he p oposed
modMA, oge he wi h PSO, GWO, and BOA, a e applied o he UAV pa h planning
p oblem; simula ion expe imen s show ha he e ec o modMA which is applied o he
UAV pa h planning p oblem is ela i ely s able, and he quali y o he planned pa h is high.
The ime cos o modMA in he pa h planning p oblem o UAV is highe han ha o PSO,
BOA and GWO, which is de e mined by i s algo i hm s uc u e, bu he he con e gence
speed is as in he p ocess o sol ing he p oblem.
In u u e wo k, hyb id [
49
,
50
] can be in oduced o u he enhance he pe o mance
o he modMA. The cos o he d one collision h ea can also be conside ed in he UAV
pa h planning p oblem.
Au ho Con ibu ions:
Concep ualiza ion, X.W. and J.-S.P.; Fo mal analysis, X.W., J.-S.P., Q.Y. and
S.-C.C.; Me hodology, X.W., J.-S.P., L.K., V.S. and S.-C.C.; Valida ion, J.-S.P., Q.Y., L.K. and V.S.;
W i ing—o iginal d a , X.W.; W i ing— e iew & edi ing, X.W., J.-S.P., Q.Y., L.K., V.S. and S.-C.C. All
au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding: This esea ch ecei ed no ex e nal unding.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : No applicable.
Con lic s o In e es : The au ho s decla e no con lic o in e es o his pape .
Appendix A
Table A1. Unimodal benchma k unc ions.
Name Fo mula o Func ions Dim Range min
Sphe e F1(x)=Dim
Σ
i=1x2
i50 [−100, 100] 0
Schwe el 2.22 F2(x)=Dim
Σ
i=1|xi|+∏Dim
i=1|xi|50 [−10, 10] 0
Schwe el 1.2 F3(x)=∑Dim
i=1∑i
j=ixj250 [−100, 100] 0
Schwe el 2.21 F4(x)=max{|xi|,1 ≤i≤Dim}50 [−10, 10] 0
S ep F5(x)=∑Dim
i=1(xi+0.5)250 [−10, 10] 0
Qua ic F6(x)=∑Dim
i=1Dim ·x2
i+ and(0,1)50 [−1.28, 1.28] 0
Exponen ial F7(x)=exp0.5 ∑Dim
i=1xi50 [−10, 10] 0
Sum powe F8(x)=∑Dim
i=1|xi|(i+1)50 [−1, 1] 0
Sum squa e F9(x)=∑Dim
i=1Dim ·x2
i50 [−10, 10] 0
Rosenb ock F10(x)=∑Dim
i=1100xi+1−x2
i+(xi−1)250 [−5, 10] 0
Zakha o F11(x)=∑Dim
i=1x2
i+∑Dim
i=10.5ixi2+∑Dim
i=10.5ixi450 [−5, 10] 0
T id F12(x)=(xi−1)2+∑Dim
i=1i·2x2
i−xi−1250 [−10, 10] 0
Ellip ic F13(x)=∑Dim
i=1106(i−1)/(Dim−1)·x2
i50 [−100, 100] 0
Ciga F14(x)=x2
1+106∑Dim
i=1x2
i50 [−100, 100] 0
D ones 2022,6, 134 19 o 21
Table A2. Mul imodal benchma k unc ions.
Name Fo mula o Func ions Dim Range min
Ras igin F15(x)=∑Dim
i=1x2
i−10cos(2πxi)+1050 [−5.12, 5.12] 0
NCRas igin
F16(x)=
Dim
∑
i=1hy2
i−10cos(2πyi)+10i,
yi=xi,|xi|<0.5
ound(2xi)/2, |xi|>0.5
50 [−5.12, 5.12] 0
Ackley
F17(x)=−20 exp
−0.2
u
u
1
Dim
Dim
∑
i=1
x2
i

+exp 1
Dim
Dim
∑
i=1
cos(2πxi)!+20 +exp(1)
50 [−50, 50] 0
G iewank F18(x)=1
4000 ∑Dim
i=1x2
i−∏Dim
i=1cosxi
√i+1 50 [−600, 600] 0
Alpine F19(x)=∑Dim
i=1|xi·sin(xi)+0.1xi|50 [−10, 10] 0
Penalized 1
F20(x)=π
Dim (Dim−1
∑
i=1
(yi−1)2h1+10sin2(πy1)i+(yDim −1)2+10 sin2(πy1))+
Dim
∑
i=1
u(xi,10,100,4)
yi=1+(xi+1)/4, uyi,a,k,m=


k(xi−a)m,xi>a
0, −a≤xi≤a
k(−xi−a)m,xi<a
50 [−100, 100] 0
Penalized 2 F21(x)=1
10(sin2(πx1)+
Dim−1
∑
i=1
(xi−1)2h1+sin2(3πxi+1)i+(xDim−1)21+sin2(2πxi+1))+
Dim
∑
i=1
u(xi,5,100,4)50 [−100, 100] 0
Schwe el F22(x)=∑Dim
i=1xi·sinp|xi|50 [−100, 100] 0
Le y F23(x)=sin2(3πxi)+∑Dim−1
i=1(xi−1)21+sin2(3πxi+1)+|xDim −1|·1+sin2(2πxDim)50 [−10, 10] 0
Weie s ass F24(x)=∑Dim
i=1∑kmax
k=0akcos2πbk(xi+0.5)−Dim ·∑kmax
k=0akcosπbk,a=0.5, b=3, kmax =20 50 [−1, 1] 0
Solomon F25(x)=1−cos2πq∑Dim
i=1x2
i+0.1q∑Dim
i=1x2
i50 [−100, 100] 0
Bohache sky F26(x)=∑Dim
i=1x2
i+2x2
i+1−0.3 ·cos(3πxi)50 [−10, 10] 0
D ones 2022,6, 134 20 o 21
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