Ci a ion: Es e ez, J.; Lopez-Guede,
J.M.; Ga a e, G.; G aña, M. Hyb id
Modeling o De o mable Linea
Objec s o Thei Coope a i e
T anspo a ion by Teams o
Quad o o s. Appl. Sci. 2022,12, 5253.
h ps://doi.o g/10.3390/
app12105253
Academic Edi o : Alessand o
Gaspa e o
Recei ed: 9 Ma ch 2022
Accep ed: 17 May 2022
Published: 23 May 2022
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A ibu ion (CC BY) license (h ps://
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applied
sciences
A icle
Hyb id Modeling o De o mable Linea Objec s o Thei
Coope a i e T anspo a ion by Teams o Quad o o s
Julian Es e ez 1,* , Jose Manuel Lopez-Guede 2, Go ka Ga a e 1and Manuel G aña 3
1Facul y o Enginee ing o Gipuzkoa, Uni e si y o he Basque Coun y, 20018 San Sebas ian, Spain;
[email p o ec ed]
2Facul y o Enginee ing o Vi o ia, Uni e si y o he Basque Coun y, 01006 Vi o ia, Spain; [email p o ec ed]
3Facul y o Compu e Science, Uni e si y o he Basque Coun y, 20018 San Sebas ian, Spain;
[email p o ec ed]
*Co espondence: [email p o ec ed]
Abs ac :
This pape deals wi h he con ol o a eam o unmanned ai ehicles (UAVs), speci ically
quad o o s, o which hei mission is he anspo a ion o a de o mable linea objec (DLO), i.e.,
a cable, hose o simila objec in quasi-s a iona y s a e, while c uising owa ds des ina ion. Such
missions ha e s ong indus ial applica ions in he anspo a ion o hoses o powe cables o spe-
ci ic loca ions, such as he eme gency powe o wa e supply in haza d si ua ions such as i es
o ea hquake damaged s uc u es. This con ol mus be obus o wi hs and s ong and sudden
wind dis u bances and emain s able a e agg essi e maneu e s, i.e., sha p changes o di ec ion
o accele a ion. To cope wi h hese, we ha e p e iously de eloped he online adap a ion o he
p opo ional de i a i e (PD) con olle s o he quad o o s h us e s, implemen ed by a uzzy logic
ule sys em ha expe ienced adap a ion by a s ochas ic g adien ule. Howe e , sagging condi ions
appea ing when he anspo ing d ones a e oo close o oo a away induce singula i ies in he DLO
ca ena y models, b eaking apa he con ol sys em. The pape ’s main con ibu ion is he o mula ion
o he hyb id selec i e model o he DLO sec ions as ei he ca ena ies o pa abolas, which allows
us o o e come hese sagging condi ions. We p o ide he speci ic decision ule o shi be ween
DLO models. Simula ion esul s demons a e he pe o mance o he p oposed app oach unde
s ingen condi ions.
Keywo ds: quado o o ; de o mable linea objec s; payload anspo a ion
1. In oduc ion
Since 1970s, owed cable sys ems ha e been analyzed o a ious applica ions o
payload ae ial anspo a ion including payload deli e y, ki es, ae ial e ueling sys ems,
b ick anspo a ion and escue missions [
1
]. Recen ly, unmanned ae ial ehicles (UAVs)
capabili ies o he anspo a ion and manipula ion o objec s ha e caugh he a en ion
o esea che s o he anspo a ion o di e se ypes o objec s, o inspec ion and main e-
nance o indus ial elemen s and su aces and o o he indus ial- and eme gency- ela ed
applica ions [
2
–
4
]. Resea che s ha e in es ed a big e o in las yea s in de eloping di e -
en con ol models, ision sys ems and g asping o con ac mechanisms in o de o cope
wi h all he di icul ies ha hese sys ems ind [
5
]. In pa icula , mul i- o o s ha e become
inc easingly a o dable o e by indus ies o deli e y and inspec ion, wi h some s a -ups
becoming success ul companies. Coope a i e eams o quad o o s ha e g ea po en ial o
some applica ions such as suspended objec anspo a ion [6,7].
Mo e speci ically, coope a i e asks o quad o o anspo a ion o de o mable linea
objec s (DLO) (i.e., cables o hoses) is being p o ed o be e y use ul in eme gencies and
in ha dly accessible a eas [
8
], i e ex inc ion [
9
], windmill u bine cleaning [
10
], liquid
sp aying [
11
,
12
] o anspo a ion o payloads suspended om cables a ached o he
UAVs [13,14]
. In he la e case, mos e app oaches assume ha cables a e igid links
Appl. Sci. 2022,12, 5253. h ps://doi.o g/10.3390/app12105253 h ps://www.mdpi.com/jou nal/applsci
Appl. Sci. 2022,12, 5253 2 o 17
connec ing he UAV and he payload wi hou in insic dynamics. Howe e , in he o he
cases, accu a e DLO geome ical and dynamical modeling is essen ial in o de o achie e
p ecise and obus con ol o he en i e sys em encompassing he DLO and he anspo -
ing quad o o s. Recen wo ks deal wi h DLO modeling by ca ena ies [
13
], while o he s
decompose he DLO in o a sequence o connec ed igid links [
15
] in o de o build up he
con ol sys em coping wi h DLO anspo a ion and manipula ion. Ca ena y modeling has
been applied o design p opo ional de i a i e (PD) con olle s [
16
] ha achie e he ask
o DLO anspo a ion by a eam o quad o o s [
17
,
18
]; howe e , ca ena y models canno
cope wi h agg essi e maneu e s, in ol ing sha p changes in di ec ion and an accele a ion
o he d ones desi ed ajec o ies. Sudden changes in he ela i e posi ions o he d ones
can ab up ly change he shape o he DLO segmen s pushing hei geome ical models
o limi s. Such sagging condi ions appea due o dis ances ha a e oo sho o oo long
be ween he d ones, making he DLO ca ena y model all in o singula i ies so ha he
en i e con ol sys em ails. Al e na i ely, he pa abola may be used as an app oxima ion
o he ca ena y [19–21] ha does no su e om sagging condi ions.
In p e ious s udies [
18
], we ha e de eloped he online adap a ion o a uzzy logic
ule sys em ha se s he pa ame e s o he d one PD con olle s. This adap a ion is ca ied
ou independen ly o each d one in he eam anspo ing he DLO. Figu e 1p esen s
an o e all block diag am o he sys em. We ha e shown he e ec i eness o he uzzy
logic adap i e con ol sys em o achie e some agg essi e maneu e s wi h sha p di ec ion
changes, comp omising UAV eam o ma ions du ing DLO anspo a ion. The enhanced
con ol sys em pe mi s UAVs o ollow he mission pa h and o e ain a dis ance be ween
hem in a s able and smoo h manne .
Figu e 1. O e all block diag am.
The main con ibu ion o his pape is as ollows: We p esen a hyb id modeling sys em
swi ching be ween ca ena y and pa abola models o DLO segmen s hanging be ween
pai s o d ones in o de o achie e obus con ol in he p esence o wind dis u bances
and agg essi e maneu e s and o e coming sagging condi ions. We p o ide he decision
h eshold based on he ela ion be ween he dis ance among d ones and he ac ual leng h
o he DLO.
The a icle s uc u e is as ollows: Sec ion 2, e iews ela ed wo ks in he li e a u e.
Sec ion 3p esen s he hyb id DLO modeling swi ching be ween he ca ena y and
pa abola. Sec ion 4p esen s he ollow- he-leade o ma ion o he quad o o eam ha
will be ollowed in he expe imen . Sec ion 5desc ibes he quad o o con ol sys em,
including he no el online adap i e PD uning based on a uzzy adap i e g adien
descen ule. Sec ion 6desc ibes he expe imen al se ings. Sec ion 7 epo s he esul s
ha demons a e he e ec i eness o he p oposed con ol sys em. Finally, Sec ion 8
p o ides ou conclusions and lines o u u e wo k.
Appl. Sci. 2022,12, 5253 3 o 17
2. Rela ed Wo ks
Table 1o e s a summa y ca ego iza ion o he ele an li e a u e. Disc e e DLO models
ackle he p oblem by b eaking down he s uc u e in o a numbe o igid od elemen s
o ini e leng h, which can be physically simula ed as pendulums, cu e segmen s o by a
ne wo k o masses and sp ings, so called lumped-mass models. These models equi e he
ep esen a ion o o ces and momen s a each elemen so ha i is possible o model he
mo ion o each and all o hem. A s udy compa ing se e al modeling me hods concluded
ha he lumped-mass ep esen a ion is he mos e sa ile me hod, despi e he la ge amoun
o compu a ional esou ces equi ed o i s implemen a ion [
22
]. Al e na i ely, some
app oaches p opose he modeling o he DLO as a chain o igid links allowing he di e en-
ially la con ol [
23
] o he eam ca ying he DLO anspo a ion unde quasi-s a iona y
condi ions [
15
]. Ne e heless, compa a i e s udies on cable s uc u e modeling show
he supe io nume ical e iciency o ca ena ies o he s udy o di e en o ce si ua ions,
ib a ions and o sions [24–26].
Table 1. Summa y compa ison o DLO ela ed esea ch. D C = Disc e e e sus con inuous model.
Re . Yea D C Pa adigm Applica ions
[27] 1971 con inuous ca ena y cable owed by ai c a
[22] 1973 bo h Analy ical me hods (su ey) Ocean Science
[28] 1981 con inuous ca ena y cable s uc u es
[29] 1995 con inuous ca ena y cable owed by ai c a
[30] 1999 bo h ca ena y s. od elemen s –
[31] 2000 con inous ca ena y unde wa e owing
[32] 2001 con inuous ca ena y ocean sciences, moo ing
[33] 2001 con inuous ca ena y unde wa e owing
[24] 2006 con inuous ca ena y designs o ne s o cables
[34] 2007 con inuoys igid link e he ed UAV
[35] 2008 con inuous pa abola cable s uc u es
[36] 2008 con inuous ca ena y ocean sciences, moo ing
[37] 2009 con inuous igid link coope a ing UAV payload anspo
[38] 2010 con inuous igid link coope a ing UAV payload anspo
[39] 2012 con inuous ca ena y cable-d i en pa allel obo
[40] 2012 con inuous igid link coope a ing UAV payload anspo
[41] 2013 con inuous igid link e he ed UAV
[42] 2013 con inuous ca ena y cable-d i en pa allel obo
[43] 2015 con inous igid link e he ed su illance UAV
[44] 2015 disc e e se ies o igid links coope a ing UAV payload anspo
[18,45] 2015, 2017 con inuous ca ena y DLO anspo a ion by n≥2 UAVs
[46] 2016 con inuous ca ena y e he ed UAV
[47] 2017 con inuous ca ena y e he ed UAVs
[48] 2017 con inuous ca ena y cable anspo a ion by 2 UAVs
[25] 2018 con inuous ca ena y suspension b idges
[26] 2018 con inuous ca ena y suspension b idges
[49] 2018 con inuous ca ena y and pa abola cable ib a ions
[15] 2020 disc e e chain o igid links hose anspo a ion by 2 UAVs
[13] 2021 con inuous ca ena y cable anspo a ion by 2 UAVs
Ca ena ies a e widely accep ed as accu a e cable models, o ins ance, in moo ing cable
simula ions [
32
,
36
,
39
], and in he design o ci il cable s uc u es [
28
,
50
]. Con inuous cable
Appl. Sci. 2022,12, 5253 4 o 17
modeling by ca ena ies is mo e accu a e and less compu a ionally demanding [
30
] han dis-
c e iza ion app oaches. The cable modeling by a ca ena y elies on he ollowing assump ions
and simpli ica ions [
51
]: The mass pe uni leng h o cable is cons an , he e is no o sion,
he cable canno inc ease i s leng h and he c oss-sec ion o he cable is much smalle han
he longi udinal dimension, co esponding o a 2D solid a any ime, hence esul ing in he
gene al e m o de o mable linea objec (DLO) ha we use in his pape . In some s udies, he
geome y o cable segmen s has been app oxima ed by pa abolas [
35
] and o he second deg ee
polynomials [
24
], which allow as e compu a ions and a e obus enough o accu a ely model
he sagged cables [
49
]. Following his backg ound, we p opose ou hyb id DLO modeling
me hods, as discussed below.
Te he ed UAVs, also known as au e he s, a e a special case whe e he UAV has i s cen e
o he mass a ached o he global coo dina e ame o igin by a DLO, which is a ense cable
wi h negligible mass. Mos esea ch s udies conside ha , in his con igu a ion, he cables a e
igid links. Di e en a ia ions and e olu ions o his model ha e been p oposed in he las
yea s o di e en asks in g ound obo ics o ship ope a ions [
34
,
41
,
43
,
52
]. Ca ena ies usage
o e he ed UAVs ha e no been deeply s udied despi e ea ly p omising esul s [29,46].
Fo UAV anspo a ion o payloads hanging om a cable [
13
], cable dynamics model-
ing is a key ac o o unde s anding he sys em’s dynamics and pe mi he compu a ion o
o ces exe ed on he quad o o , whe e he mo ion o he en i e cable is ep esen ed as a
con inuous s uc u e wi h app op ia e bounda y condi ions. Thei main ad an age is ha
he simul aneous conside a ion o each poin in he ma e ial pe mi s he calculus o mo e
accu a e cable dynamics [27,29,31,33].
In he coope a i e anspo a ion o a load by a eam o UAVs, he cables a e o en
modeled as a igid link [
37
,
38
]. Coope a i e ae ial owing p oblem is simila o he p oblem
o con olling cable-ac ua ed pa allel manipula o s in h ee dimensions. In hese sys ems,
he a ia ion o he leng hs o cable a achmen s de e mines he payload o ien a ion and
posi ion. Following his line o esea ch, [
44
] modeled he sys em as a se ially connec ed
links sys em o he coope a i e anspo a ion and o ien a ion o a igid wo-dimensional
payload. The anspo a ion o a DLO a ached by cables o he UAVs unde se e e dynamic
limi a ions and quasi-s a ic condi ions was also achie ed [40].
In p e ious s udies dealing wi h he coope a i e anspo a ion o DLOs by eams
o UAVs, DLO sec ions we e modeled as ca ena ies in an equiload e ical con igu a-
ion [
18
,
45
]. Addi ional sou ces con i m ha ca ena ies a e a good modeling al e na i e
o he coope a i e ae ial anspo a ion o DLOs [
13
], including isual se oing ap-
p oaches [
47
], and collision a oidance [
39
,
48
]. Howe e , ca ena y cu es a e hype bolic
unc ions ha su e om nume ical singula i ies, which may lead he con ol sys em
o collapse when he dis ances be ween quad o o s a e oo sho o oo la ge ela i e o
he DLO’s sec ion leng h due o cable sagging [
42
]. This si ua ion occu s when he eam
o quad o o s mus pe o m agg essi e maneu e s consis ing o sudden sha p changes
o di ec ion and/o accele a ions. In o de o deal wi h hese ex eme condi ions, we
p opose he hyb id modeling o DLOs ha shi s be ween ca ena y and pa abola models
acco ding o he sys em s a e.
Finally, he anspo a ion o a cable by pai s o UAVs has been p oposed o he g asping
and anspo a ion o objec s ea u ing some kind o hook, such as umb ellas [
13
]. A e he
hooking maneu e , he shape o he cable can be modeled by s aigh sec ions, and he en i e
sys em can be ea ed as coope a i e payload anspo a ion om suspended cables.
Du ing ecen yea s, he e has been a la ge e o de o ed o he de elopmen o a
lexible dynamic model o low compu a ional cos o DLO payloads, because, despi e hei
passi e na u e, payload con igu a ions migh a ec he pe o mance o he con ol o he
obo ca ying ou he anspo a ion ask. Tau cables modeled as a me al ba a e alid only
o a small spec um o applica ions. Disc e e cable modeling emains compu a ionally
oo cos ly. Ca ena y models eme ged as a possible ep esen a ion model wi h p omising
esul s and ha e al eady been es ed in simula ions o simple obo ic expe imen s. In
o de o cap u e he bes possible eali y, by aking in o accoun he bibliog aphy on cable
Appl. Sci. 2022,12, 5253 5 o 17
s uc u es [
22
,
24
,
30
,
53
], we p opose a hyb id ca ena y–pa abola cable model so ha he
con ol sys ems o eams o d ones o ae ial anspo a ion o long cables can cope wi h
demanding maneu e s. As a as he au ho s know, his ype o swi ching model has ne e
been applied in cable anspo a ion asks wi h quad o o s.
3. Pa abola–Ca ena y Hyb id DLO Geome ical Model
The ca ena y equa ion
y=acosh x
a
is de i ed om well-known undamen al equa-
ions o applied mechanics as he shape ha akes a lexible bu non-elas ic DLO hanging
om wo ex emes unde i s own weigh . Pa abola equa ion
y=ax2
has been used as a
su oga e geome ic model app oxima ion o he shape o hanging cables o bo h s a ic
o kinema ics analysis [
30
,
54
,
55
], because i is mo e obus o ex eme condi ions ha
induce singula i ies in he ca ena y equa ion. In gene al applica ions, such as modeling
he dynamic beha io o cables o b idge s uc u es, he dynamic simula ions o objec s
modeled al e na i ely as a pa abola o he ca ena y a e e y simila , excep unde e y
hea y payloads whe e he di e ences among hese unc ions migh in oduce subs an ial
di e ences on analysis esul s [
24
]. Mo eo e , he lowe compu a ional cos o a pa abola
is ano he eason o i s use in he ma hema ical modeling o cables [
25
]. In he case o
obo ics, his simpli ica ion has been con es ed in some applica ions [
47
]. Figu e 2 isual-
izes he app oxima ion o a ca ena y by a pa abola, which may be good enough o some
applica ions [55–58], especially in he de elopmen o cable-d i en obo s [59].
Figu e 2. Visual compa ison o he ca ena y and pa abola cu es wi h pa ame e a.
This a icle p oposes a hyb id be ween ca ena y and pa abola geome ic models o a
DLO sec ion hanging be ween wo d ones. Au oma ed swi ching om one model o he o he
occu s when he Euclidean dis ance be ween he d ones is oo sho ela i e o he ac ual leng h
o he DLO sec ion. In his si ua ion, he ca ena y is no longe a good app oxima ion o he
shape o he DLO. Heu is ically, we ha e se he h eshold o he shi be ween models a
d<L/
3, whe e
L
is he leng h o he DLO sec ion and
d
he Euclidean ho izon al dis ance
be ween he d ones suppo ing i . In d one eam ope a ion modeling, he e a e some p e ious
s udies on hyb id modeling o con ol s a egies and hei o ma ion [
60
–
62
], bu he e a e no
p e ious s udies on hyb id payload modeling.
4. Quad o o Team Fo ma ion S a egy
The quad o o eam anspo ing he DLO is a ollow- he-leade column pla oon
o ma ion [
63
] ha o e s ad an ages o obs acle a oidance and needs only he speci ica ion
o he ajec o y o he leade o guide he en i e eam. In ae ial anspo , his con igu a ion
ep esen s a no el y, as mos o he published esea ch s udies s udy he collabo a i e
anspo a ion o a hea y load using di e en app oaches o calcula e he payload’s posi ion
ela i e o he he quad o o s a any momen , such as he Udwadia–Kalaba me hod o
modeling [
6
] ocused on he es ima ion o he posi ion o each UAV wi h espec o he
payload in a dynamic equa ion minimiza ion me hod. O he s udies’ use geome ic c i e ia
o calcula ing he UAVs’ desi ed posi ion o accomplish he ask. Fo ins ance, e . [
64
] se s
he o ma ion wi h Delaunay iangles, and [65] uses ec o ial condi ions.
Appl. Sci. 2022,12, 5253 6 o 17
Ou app oach is inspi ed in g ound obo ics [
63
], adding he ex a cons ain o
main aining a ho izon al Euclidean dis ance be ween obo s. O ien a ion and posi ion
o each obo a e calcula ed a each momen . The g aphical ep esen a ion o he eam
con igu a ion o e he (X,Y) plane can be seen in Figu e 3, whe e
ρ
co esponds o he
desi ed dis ance be ween obo s, and he
L
and
F
subindices deno e he leade ’s and
ollowe ’s a iables, espec i ely. Vec o s
VL
and
VF
a e he mo ion di ec ions o he leade
and ollowe d ones, espec i ely.
Figu e 3. Follow- he-leade pla oon model.
Finally, he equa ions o he posi ion and o ien a ion o he ollowe UAV ela i e o
he posi ion o he leade UAV [64,66], a e as ollows.
xF=xL−ρcos(α+ψF)
yF=yL+ρsin(α+ψF)
ψF=ϕ+ψL−π
. (1)
5. Sys em Con ol
The con ol sys em o each quad o o in he eam is composed o an inne and ou e
loop wi h p opo ional de i a i e (PD) con olle s o each deg ee o eedom. Figu e 4
depic s he s uc u e o he sys em. The inne con ol loop is in cha ge o con olling he
a i ude o he quad o o by p o iding o o commands o achie e desi ed a i ude angles.
The ou e con ol loop is in cha ge o ollowing he desi ed ajec o y by p o iding he
desi ed a i ude angles o he inne con ol loop. Tuning o he con olle pa ame e s by an
o line Pa icle Swa m Op imiza ion algo i hm achie ed he e ical equiload con igu a ion
in he inne loop [
17
,
45
] minimizing he inal heigh adjus men o e sho and p o ing o be
a scalable sys em.
Figu e 4. Con ol sys em o each UAV in he sys em.
Appl. Sci. 2022,12, 5253 7 o 17
Online Adap a ion o PD Con olle s
O line uning o he PD con olle s [
17
] in he ou e loop is unable o cope wi h agg essi e
maneu e s, such as sho adius cu es and sha p changes o di ec ion, and nei he p o ides
adap a ions o di e en leng hs and weigh s o DLOs. The e o e, we p oposed [
18
] and hei
online adap i e uning ollowing an Adap i e Fuzzy Modula ion (AFM) app oach, combining
a g adien descen adap a ion ule and a uzzy membe ship unc ion ac i a ion [
67
,
68
]. The
membe ship unc ions ac on he PD con olle pa ame e s i a uzzy logic exp ession is sa is ied,
ollowing he Takagi–Sugeno con olle design pa adigm [69].
The pe cei ed e o
Pe
(c . Equa ion (2)) measu es he ela i e e o be ween he eal
posi ion o he UAV Y eal and i s e e ence Y e a each momen .
Pe =Y e −Y eal
Y e
·100, (2)
The p oposed uzzy uning ules con empla e ou e o condi ions dependen on
Pe
alue modeled by co esponding ou iangle-shaped membe ship unc ions
{µi(Pe)}4
i=1
,
o which i s membe ship suppo s a e p o ided by he ollowing.
Dµ1=(−2, −9),Dµ2=(−1, −5),Dµ3=(1, 5),Dµ4=(4, 9). (3)
The adap a ion o pa ame e
Kp
is modula ed by unc ions
µ1(Pe( ))
and
µ4(Pe( ))
h ough Equa ion (4), while he adap a ion o pa ame e
Kd
is modula ed by unc ions
µ2(Pe( )) and µ3(Pe( )) wi h Equa ion (5):
Kp( +1)=Kp( )+αe( )(µ1(Pe( )) +µ4(Pe( ))) (4)
Kd( +1) = Kd( )+αe( )(µ2(Pe( )) +µ3(Pe( ))) (5)
whe e
α
is he adap a ion ac o , which akes a cons an alue be ween 0 and 1 du ing
he en i e expe imen . The
e( )
unc ions compu e he ins an aneous e o ela i e o he
desi ed alues
θd
and
φd
o he angles ha de e mine he mo ion in he
XY
plane. The
online adap a ion Equa ions (4) and (5) ollow a s ochas ic g adien descen algo i hm The
con e gence o he con inuously adap i e p ocess o he uzzy logic con ol app oach has
been p o en [18].
6. Expe imen s
We ha e ca ied ou h ee compu a ional simula ion expe imen s ha equi e a eam
o h ee quad o o s anspo ing a DLO a ached o hem in a ollow- he-leade s a egy,
whe e ollowe s y o mimic he mo ion o he leade , i.e., ollowing pa allel pa hs o he
one o he leade ying o p ese e he dis ance among quad o o s. In bo h expe imen al
simula ions, ime is disc e ized in s eps o 0.1 s, he DLO is modeled by he ca ena y-
pa abola app oxima ion and we apply a uzzy logic app oach o he adap i e uning o
he PD con olle . Bo h expe imen s ea u e sha p pa h changes ha we e unmanageable
wi h p e ious e sions o he con olle [17].
Expe imen 1:
In his expe imen , he nominal pa h se o he leade quad o o has
sudden changes o di ec ion, as shown in Figu e 5. The objec i e o he expe imen is
o check whe he he d one eam’s o ma ion emains s able and is able o cope wi h he
di e en pa h co ne s, pa icula ly wi h he sha p angle loca ed a
x=
400 cm. Fo he
expe imen , bo h wi h and wi hou wind dis u bance condi ions in he X di ec ion a e used
o model he ollowing dynamic equa ion d( ) = 5+5 sin(π
2 ).
Expe imen 2:
In his expe imen , he h ee quad o o s anspo he DLO in a s aigh
line pa h. When he leade d one has a e sed a dis ance o
x=
350 cm, i su e s a sudden
la e al dis u bance consis ing o a push displacing i 80 cm in he Y posi i e di ec ion, as
shown in Figu e 6. The expe imen aims o obse e how he leade and ollowe s eco e
he nominal pa h a e he dis u bance.
Expe imen 3:
Now, he h ee quad o o s mus ollow a spi al 3D ascending pa h
speci ied by
x=
100
sin( )
,
y=
100
cos( )
,
z=
5
( )
, wi h
= [
0
:
3
π]
, as seen in Figu e 7.
Appl. Sci. 2022,12, 5253 8 o 17
This es aims o check he capaci y o he d ones con ol sys em o cope wi h he h ee
di ec ion pa hs a he same ime, wi h no-wind condi ions, and conside ing only he hyb id
DLO model o ca ena y and pa abola. The leade d one’s s a ing posi ion is a (0, 0, 0).
Figu e 5. Nominal pa h o he d one leade in Expe imen 1, ea u ing sha p changes in di ec ion.
Figu e 6.
Expe imen 2. Nominal pa h ollowed by he leade d one su e ing a sudden la e al dis u -
bance.
Figu e 7. Spi al 3D ascending pa h in Expe imen 3.
In he h ee expe imen s, he ini ial dis ance in he ho izon al axis be ween d ones a
he ex emes and he cen al d one is he same and is se as
ρd=70
cm. Mo eo e , in o de
o ensu e balanced ene gy consump ion, he sys em is in equiload condi ions [
18
,
45
]. As a
consequence, no u he co ec ion o al i ude is applied, al hough he ho izon al Euclidean
dis ance be ween obo s migh change. In pla oon o ma ion, he maximum e ical h us
was limi ed o each quad o o o 20 N ollowing s anda d ha dwa e speci ica ions. The
leng h o he ca ena ies was se o
L0=
240 cm; he mass densi y o he DLO was se o
w= 0.005 [kg/cm]
. Fo online uzzy uning o he PD con olle , he adap a ion ac o was
se o
α=
0.5. Dynamic pa ame e s o each quad o o appea in Table 2. We se he ini ial
PD pa ame e alues as ollows: Kpx =Kpy =0.22; Kdx =Kdy =0.76.
Appl. Sci. 2022,12, 5253 9 o 17
Expe imen s ha e been coded in house in Scilab 5.4. No o he public o p i a e
so wa e solu ions ha e been used. The code o he implemen a ion has been published in
an online eposi o y (h ps://gi hub.com/Jules e ez/Quad o o -simula o / ee/mas e /
ca ena y%20and%20pa abola%20hyb id%20modelling, accessed on 10 May 2022).
Table 2. Quad o o s uc u al and dynamic pa ame e s.
Pa ame e Value
mass, m0.5 kg
a m leng h, l25 cm
ine ia momen s, Ixx =Iyy 5×10−3[Nms2]
ine ia momen , Izz 1×10−2[Nms2]
p opelle h us coe icien , b3×10−6[Ns2]
d ag, d1×10−7[Nms2]
7. Resul s
7.1. Expe imen 1
Figu e 8shows he pa hs ollowed by he eam o quad o os unde wind condi ions in
a simula ion las ing 60 s o simula ed ime. We ound ha he d ones ollowed he same
pa h in he epe i ions wi hou wind dis u bances. Figu e 9shows he posi ion o he h ee
quad o o s and he DLO a di e en momen s o he simula ion, whe e we can obse e
how he ollowe d ones a emp o keep hei linea o ma ion by p ese ing, as much as
possible, he DLO con igu a ion and he dis ance among quad o o s. In he ollowing, le
us deno e
D1
,
D2
and
D3
as he leade , mid and ea d ones, espec i ely. Figu es 10 and 11
show he plo in ime o he h us o
D1
wi hou and wi h wind dis u bances, espec i ely.
I can be app ecia ed ha he esponse o he wind pe u ba ions in oduces some changes
in he h us p o ile in o de o ollow he nominal pa h as close as possible, hanks o he
online uning o he PD con olle by he AFM algo i hm.
Figu e 8.
T ajec o y o he quad o o s in Expe imen 1 unde wind pe u ba ions. Colo code: ed,
g een and blue co espond o leade , mid and ea d ones, espec i ely.
Appl. Sci. 2022,12, 5253 16 o 17
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