Collision A oidance o Mul iple UAVs using
Rolling-ho izon Policy
S. Ve a, J. A. Cobano, G. He edia and A. Olle o
Abs ac —This pape add esses he p oblem o collision
a oidance in scena ios wi h mul iple ae ial ehicles and
p oposes a me hod based on a Legend e pseudospec al
colloca ion in o de o compu e he solu ion ajec o ies
and gua an ee ha he sa e y dis ance be ween hem is
always main ained. The me hod uses a olling ho izon
policy in which ajec o ies a e planned up o a gi en
ime ho izon, hus conside ing a much smalle p oblem
space. Then, he sys em is applied i e a i ely. S udies ha e
been pe o med o se he alues o he look-ahead ime
and he numbe o colloca ions poin s. The compu a ional
load and scalabili y o he me hod a e also s udied in
andomly gene a ed scena ios o es i s applica ion in
eal ime. Expe imen s ha e been also ca ied ou in he
mul i ehicle ae ial es bed o he Cen e o Ad anced
Ae ospace Technologies (Se ille, Spain).
I. INTRODUCTION
Mul iple UAVs a e being coope a i ely used in
he las yea s o ca y ou coo dina ed missions [1]
[2]. Coo dina ion and collision a oidance a ises as
c i ically impo an aspec s in his kind o applica-
ions in dynamic en i onmen s. The e o e, a me hod
o plan collision- ee ajec o ies wi h low compu a-
ional load should be implemen ed in o de o ensu e
he sa e y in hese en i onmen s. Conc e ely, an
e icien e-planning is needed o sol e he collisions
de ec ed.
Many wo ks on planning algo i hms and colli-
sion a oidance me hods ha e been published. A
de ailed su ey on he o me is p esen ed in [3]
and [4] e iews pape s on he la e . Planning me h-
ods include Rapidly-explo ing Random T ees (RRT)
*This wo k was suppo ed by he Eu opean Commission FP7 ICT
P og amme unde he EC-SAFEMOBIL p ojec (288082) and he
CLEAR p ojec (DPI2011-28937-C02-01) unded by he Minis e io
de Ciencia e Inno acion o he Spanish Go e nmen . The au ho s
would like o hank M . Miguel Angel T ujillo o hei unsel ish
help du ing he de elopmen o he expe imen s in he es bed o
CATEC (Se ille).
1S. Ve a, J. A. Cobano, G. He edia and A. Olle o a e
wi h he Robo ics, Vision and Con ol G oup, Enginee ing
School, Uni e si y o Se ille, 41092 Se ille, Spain
{s e a,jcobano,guille ,aolle o}@us.es
[5], pa icle swa m op imiza ion [6], e olu iona y
compu a ion me hods [7], an colony op imiza ion
me hods [8]. The main d awback o hese me hods
is ha he compu a ion ime is no p edic able and
he con e gence o a solu ion is no ensu ed in a
ini e ime in e al. The e o e, hey a e no good
candida es o plan collision- ee ajec o ies in a
dynamic en i onmen and a small ime ho izon.
An in e es ing op ion could be o use colloca-
ion me hods o ajec o y gene a ion, which ha e
been inc easingly employed in he las yea s. In
colloca ion me hods ajec o y gene a ion is posed
as an op imal con ol p oblem and he solu ion is
app oxima ed by polynomials. Then, he di e en ial
equa ions and cons ain s a e en o ced in colloca ion
poin s, and he op imal con ol p oblem is ans-
o med in o a nonlinea p og amming p oblem.
Among he mos used colloca ion echniques a e
he di ec colloca ion and he pseudospec al me h-
ods. The o me di ides ime in se e al segmen s,
compu es he solu ion conside ing a ixed deg ee
polynomial s a e app oxima ion in each segmen
and he con e gence is achie ed by inc easing he
numbe o segmen s [9]. The la e uses a single
segmen and con e gence is achie ed by inc eas-
ing he deg ee o he polynomial. Bo h me hods
choose he colloca ion poin s based on accu a e
quad a u e ules and he basic unc ions a e yp-
ically Chebyshe o Lag ange polynomials. The
mo e commonly used pseudospec al me hods a e
he Gauss pseudospec al me hod (GPM) [10], he
Radau pseudospec al me hod [11] (RPM), and he
Loba o pseudospec al me hod [12] (LPM). Pseu-
dospec al and Di ec Colloca ion me hods ha e
been applied o compu e ai c a ajec o ies [13]
[14] [15] bu hei compu a ion imes a e oo la ge o
plan ajec o ies in dynamic and unce ain en i on-
men s. Mo eo e , mos o published wo ks conside
ajec o y gene a ion o a s andalone ehicle [13]
[14].
P ep in e sion o :
Ve a S, Cobano JA, He edia G and Olle o A (2016), Collision A oidance o Mul iple UAVs Using Rolling-Ho izon Policy,
Jou nal o In elligen & Robo ic Sys ems, Decembe 2016, Vol. 84
A modi ied pseudospec al me hod is called hp-
adap i e [16]. In his me hod, he numbe o seg-
men s and he deg ee o he polynomial can be
inc eased wi hin a segmen o achie e an e o less
han he ole ance e o allowed. Howe e , each
segmen adds colloca ion poin s in each i e a ion,
so he numbe o colloca ion poin s could quickly
g ow and ha p o okes la ge compu a ion imes.
This pape add esses he p oblem o collision
a oidance wi h mul iple UAVs in dynamic en i on-
men s using a olling ho izon policy. The goal is
o ensu e he sa e y and eliabili y o he mission.
The p oposed me hod is based on pseudospec al
colloca ion echniques and is i e a i e. The in o -
ma ion o all he UAVs is known up o a ligh ime
de ined by he look-ahead ime and he dynamics o
he UAVs is conside ed o compu e mo e ealis ic
ajec o ies. Look-ahead imes a e de e mined by a
olling ho izon app oxima ion in o de o quickly
compu e solu ion ajec o ies o he sub-p oblem
conside ed. The maneu e s allowed o sol e he de-
ec ed collisions a e changes o speed and heading.
The main cha ac e is ic o he p oposed me hod
is he low compu a ional load and he scalabili y.
The look-ahead ime and he numbe o colloca ion
poin s in luence he ime o compu a ion and he
sa e y o he solu ion. The easible alues o hese
pa ame e s a e s udied in his pape .
O he impo an imp o emen in he implemen-
a ion is he e alua ion o he whole solu ion a-
jec o ies because pseudospec al echniques only
compu e he solu ion alid in he colloca ion poin s.
Tha is, he minimum sepa a ion among ehicles is
main ained in hese poin s. The e o e, an e alua ion
conside ing a model o ehicle should be ca ied
ou in o de o ensu e ha he minimum sepa a ion
is no iola ed du ing he whole ajec o y o he
sub-p oblem conside ed in each ins an .
The pape is o ganized in o se en sec ions. Sec-
ion II desc ibes he p oblem o mula ion. The p o-
posed me hod is explained in Sec ion III. Simula-
ions and expe imen s pe o med a e showed in Sec-
ion IV and V, espec i ely. Finally, he conclusions
a e de ailed in Sec ion VI.
II. PATH PLANNING FOR MULTIPLE UAVS
The p oblem o collision a oidance o mul iple
UAVs o pe o m he coo dina ed missions is con-
side ed in his pape . The goal is o assu e ha he
UAVs do no collide wi h each o he . The p oposed
me hod o sol e he collisions allows changes o he
speed p o ile and he heading o he UAVs in ol ed
in he con lic .
The ajec o y o each UAV is gi en by an ini ial
waypoin and a inal waypoin . Each waypoin is
de ined by: 2D coo dina es (x, y), speed module
om ha waypoin ( ), and he Es ima ed Time o
A i al (ETA) o he waypoin , . I is assumed ha
all UAV ajec o ies a e known in he ime in e al
gi en by he look-ahead ime. We conside ha he
UAVs main ain he sa e y sepa a ion i hey a e
sepa a ed by a minimum dis ance, D.
The inpu s o he me hod a e he ollowing:
•Model o each UAV
•Look-ahead ime
•Numbe o nodes pe segmen
Numbe o nodes pe segmen , also known as
colloca ion poin s, de ines he deg ee o he polyno-
mial o in e pola ion used in each segmen . Look-
ahead ime is he ime in which each UAV knows
he in o ma ion o he es o UAVs. This ime
in e al de ines he sub-p oblem o he ajec o y
planning o each UAV. Bo h look-ahead ime and he
numbe o colloca ion poin s should be analyzed.
The bes alues o bo h pa ame e s should ensu e
a sa e solu ion in each compu a ion and wi h low
compu a ional load o i e a i ely pe o m ajec o y
e-planning in dynamic en i onmen .
Di e en op imiza ion c i e ia can be conside ed.
In his pape , wo c i e ia ha e been implemen ed:
minimize he changes o he heading angle o each
UAV and minimize he changes o he con ol inpu s
(pi ch and oll angles).
III. LEGENDRE PSEUDOSPECTRAL METHOD
The pseudospec al me hod nume ically sol es
op imal con ol p oblems. The basic app oach is o
ans o m he op imal con ol p oblem in o a se-
quence o nonlinea cons ained op imiza ion p ob-
lems by disc e izing he s a e and con ol a iables.
I compu es a se o colloca ion poin s ha p o ides
an accu a e app oxima ion o he solu ion o he
op imal con ol p oblem. The p oposed me hod is
based on a Legend e pseudospec al me hod known
as DIDO [17]. I s no el y is how an i e a i e im-
plemen a ion is ca ied ou in o de o ob ain an
e icien ajec o y e-planning.
The op imal p oblem is modeled as a Bolza p ob-
lem in τ[−1,1] domain and he objec i e is o ind
he con ol inpu ec o u(τ)and he co esponding
s a e χ(τ)which minimize he cos unc ion:
J=φ(χ(−1), χ(+1)) + Z1
−1
L(χ(τ), u(τ), τ)dτ
(1)
subjec o he dynamic cons ain s
˙χ= (χ, u)(2)
inequali y pa h cons ain s
C(χ, u)≤0(3)
and he bounda y condi ions
E(χ(−1), χ(+1)) = 0 (4)
whe e φ,Cand Ea e unc ions. The no malized
ime τ[−1,1] and he ime [ 0, ]a e ela ed by:
= − 0
2τ+ − 0
2(5)
Eq. (1) should be app oxima ed by applying
quad a u e ules. In his pape Legend e-Gauss-
Loba o (LGL) quad a u e ule is used, so:
J=φ(χ1, χN) +
N
X
j=1
L(χj, uj)wj(6)
whe e wja e he LGL quad a u e weigh s, and N
is he numbe o nodes o colloca ion poin s. In he
used no a ion, he o e line means disc e e a iables
and he supe sc ip means he colloca ion poin used
χj=χ(τj).
LGL nodes a e de ined in he no malized ime
domain τ[−1,1] as τ0=−1< τ1< τ2< ... <
τN= 1 whe e 1,2, ..., N −1a e he oo s o he
de i a i e o he N- h o de Legend e polynomial.
The oo s o he de i a i e o he Legend e poly-
nomials a e ze os in hese nodes because hey a e
o hogonal polynomials.
The e o e, χ(τ)and u(τ)could be app oxima ed
by χ(τ)and u(τ):
χ(τ)≈χ(τ) =
N
X
j=0
χjLj(τ)(7)
u(τ)≈u(τ) =
N
X
j=0
ujLj(τ)(8)
whe e Lj(τ)a e he basis unc ions o he La-
g ange in e pola ing polynomials o o de N.
The p oposed me hod is i e a i e in such a way
ha i compu es an op imal solu ion o a sub-
p oblem in e e y i e a ion. The look-ahead ime, Tla,
is de e mined by he knowledge o he en i onmen
in e e y i e a ion and he maximum speed o he e-
hicle ha is he wo s case o ajec o y e-planning.
The numbe o colloca ion poin s conside ed in each
i e a ion, Nc, also in luences he p oposed me hod.
This numbe a ec s he quali y o he solu ion and
he compu a ion ime. The quali y means ha he
solu ion could be lown and he minimum sep-
a a ion dis ance should be me by all UAVs. I
cons ain s a e me in e e y piece o he pa h, i
would ensu e he minimum sepa a ion dis ance in
he whole ajec o y. Mo eo e , a low compu a ional
ime is equi ed. Also, he equency o compu a ion
should be de e mined in o de o ensu e ha each
UAV does no ly he whole ajec o y compu ed
in he p e ious i e a ion du ing he co esponding
ime be ween wo i e a ions, T . This ime should
be la ge han he compu a ion ime o he UAV
ajec o ies, Tc:
T > Tc(9)
Figu e 1 illus a es he pe o mance o he p o-
posed me hod. Fi e colloca ion poin s ha e been
conside ed (blue poin s) and a look-ahead ime is
conside ed om he knowledge o he en i onmen
in e e y i e a ion and he maximum speed o he
ehicle. Fi s , he ajec o y o each UAV is com-
pu ed wi hin he look-ahead ime. An i e a ion is
compu ed e e y T , so he compu a ion ime, Tcin
he second i e a ion should ul ill:
Tla −T > Tc(10)
Figu e 1. Desc ip ion o he p oposed me hod.
Finally, a model o UAV by conside ing a sim-
pli ied dynamics is used o compu e mo e ealis ic
ajec o ies. The al i ude is assumed o be cons an .
The s a e ec o is de ined by (xi, yi,Φi,Θi, i)
whe e xi, yi he 2D posi ion o he ae ial ehicle,
Φi,Θia e he pi ch and oll angles and i he ime o
a i al in each colloca ion poin . The con ol inpu s
a e he pi ch and oll o ques uΦi, uΘi.
The model conside ed is:
¨xi=T
m·sin(Φi)(11)
¨yi=T
m·sin(Θi)(12)
¨
Θi=uΘi
Iy
(13)
¨
Φi=uΦi
Ix
(14)
whe e Tis he h us needed o main ain he
cons an al i ude, mis he mass, Θiis he oll angle,
Φiis he pi ch angle, Ixand Iya e he momen s o
ine ia wi h espec o he axes xand y, espec i ely.
The mul i-UAV sys em is de ined by conca ena -
ing he s a e o all he UAVs. The e o e, he s a e
ec o and con ol ec o a e de ined as ollows:
X= [x1, y1,Φ1,˙
Φ1,Θ1,˙
Θ1, 1, ....,
xn, yn,Φn,˙
Φn,Θn,˙
Θn, n](15)
U= [uΦ1, uΘ1, uΦ2, uΘ2...., uΦn, uΘn](16)
whe e nis he numbe o UAVs.
The solu ion should sa is y cons ain s aking in o
accoun he physical limi a ions o he inpu s o each
UAV:
uθmin ⩽uθi⩽uθmax (17)
uΦmin ⩽uΦi⩽uΦmax (18)
And he sepa a ion be ween UAViand UAVj
should mee :
dis ance(UAVi, UAVj)≥D(19)
whe e Dis he sa e y dis ance.
IV. SIMULATIONS
The me hod has been es ed by ca ying ou many
simula ions. Di e en scena ios andomly gene a ed
wi h se e al UAVs ha e been conside ed.
The p oblems ha e been sol ed wi h he pseu-
dospec al LGL colloca ion me hod, using he op-
imal con ol so wa e named DIDO [17]. These
algo i hms ha e been un in a PC wi h a CPU In el
Co e i7-3770 @ 3.4 Ghz and 16 GB o RAM.
The ope a ing sys em used in he simula ions was
Windows 7 OS and he code has been implemen ed
in Ma lab.
Fi s , he alues o he look-ahead ime, Tla, and
he numbe o colloca ion poin s, Nc, should be se .
A s udy conside ing one hund ed andom scena ios
wi h i e UAVs has been pe o med o analyze he
bes alues. The chosen alues should ensu e a sa e
solu ion wi h a low compu a ion ime. The possi-
ble alues a e: Tla = [1.0,1.5,2.0,2.5,3.0]sand
Nc= [3,4,5,6,7,8] poin s. All he combina ions
o hese alues a e explo ed in each scena io. The
ob ained mean compu a ion ime and i s s anda d
de ia ion o each combina ion which ensu es a sa e
solu ion in all he scena ios ( ha is, sa is ies all
he cons ain s) is shown in Table I. Only eigh
combina ions me i . In o de o mee (9), T should
be g ea e han 1.107s( he la ge alue o Tc+σT c in
Table I), so T = 1.25sis conside ed. Fi s , second,
ou h and six h cases only mee (10). Among hese
op ions, Tla = 2.5sand Nc= 4 a e chosen because
hey ensu e a minimum compu a ion ime.
Table I
COMBINATIONS THAT ENSURES A SAFE SOLUTION.
Tla(s) NcTc(s) σT c (s)
2.5 4 0.808 0.207
2.5 3 0.819 0.212
1.0 4 0.815 0.258
3.0 3 0.824 0.304
1.5 3 0.837 0.197
2.5 6 0.847 0.219
1.0 7 0.851 0.207
2.0 6 0.873 0.234
Once bo h pa ame e s ha e been se , he scalabil-
i y is analyzed, ha is, how he compu a ion ime
depends on he numbe o UAVs. Figu e 2 shows
he scena io conside ed wi h up o eigh UAVs. No e
ha in case o 7 o 8 UAVs, he densi y o UAVs
in he scena io is e y high.
Figu e 2. Scena io S1 conside ed in he simula ions wi h up o eigh
UAVs: UAV1 in blue, UAV2 in ed, UAV3 in black, UAV4 in g een,
UAV5 in pink, UAV6 in clea blue, UAV7 in yellow and UAV8 in
dashed blue.
Table II shows he compu a ion mean ime and
s anda d de ia ion o he scalabili y es . No e ha
scena io S1 is conside ed. The p oposed me hod
adap s well conside ing up o se en UAVs because
(9) is me .
Table II
MEAN COMPUTING TIME WHEN THE NUMBER OF UAVS
INCREASES BY CONSIDERING THE SCENARIO S1.
Numbe o UAVs UAVs Time (s) σ (s)
3 1-3 0.308 0.150
4 1-4 0.374 0.204
5 1-5 0.480 0.269
6 1-6 0.689 0.279
7 1-7 0.961 0.294
8 1-8 1.297 0.472
Figu e 3 p esen s he scena io S2 conside ed
wi h ou UAVs o show how he p oposed me hod
compu es a solu ion e e y T . Figu es 4, 5, 6
and 7 shows ou ins an s which co espond o
= 2.5,7.5,10.0,16.5s. The sub-p oblems sol ed
a e p esen ed in e e y i e a ion. No e ha each UAV
ajec o y con e ges o i s goal waypoin .
Figu e 3. Scena io S2 wi h ou UAVs: UAV1 in blue, UAV2 in ed
and UAV3 in black and UAV4 in g een.
Figu e 4. T ajec o y compu ed by he me hod a he ins an = 2.5s.
Figu e 5. T ajec o y compu ed by he me hod a he ins an = 7.5s.
The whole ajec o ies a e shown in Figu e 8. The
speed p o ile ob ained is p esen ed in Figu e 9.

Figu e 6. T ajec o y compu ed by he me hod a he ins an =
10.0s.
Figu e 7. T ajec o y compu ed by he me hod a he ins an =
16.5s.
Figu e 8. Final ajec o y o scena io S2: UAV1 in blue, UAV2 in
ed and UAV3 in black and UAV4 in g een.
V. EXPERIMENTS
Expe imen s ha e been ca ied ou in he indoo
mul i-UAV es bed o he CATEC wi h ou Hum-
Figu e 9. Final speed p o ile o scena io S2: UAV1 in blue, UAV2
in ed and UAV3 in black and UAV4 in g een.
mingbi d quad o o s (see Figu e 10) wi h up o 20
minu es ligh au onomy. The es bed has an indoo
localiza ion sys em based on 20 VICON came as.
This sys em is able o p o ide, in eal ime, he
posi ion and a i ude o each UAV wi h cen ime e
accu acy.
Figu e 10. Indoo mul i-UAV es bed o he CATEC wi h Hum-
mingbi d quad o o om Ascending echnologies.
This sec ion shows an expe imen o demon-
s a e he pe o mance o he p oposed me hod.
The minimum sepa a ion is 1.0m. In his case, he
op imiza ion c i e ion is o minimize he heading
changes. Figu e 11 shows he scena io conside ed.
Two collisions a e de ec ed. Fou quad- o o s in
hei ini ial posi ion a e shown in Figu e 12.
The whole solu ion ajec o ies compu ed a e
shown in Figu e 13. The speed p o ile ob ained is
p esen ed in Figu e 14.
Figu e 11. Scena io conside ed in he expe imen wi h ou UAVs:
UAV1 in blue, UAV2 in ed and UAV3 in black and UAV4 in g een.
Figu e 12. Fou quad- o o s conside ed in he expe imen . Each
quad- o o is in i s ini ial posi ion.
Figu e 13. Solu ion ajec o ies de ined by he waypoin s compu ed:
UAV1 in blue, UAV2 in ed and UAV3 in black and UAV4 in g een.
Each UAV eal ajec o y is ep esen ed in Figu e
15 and Figu e 16 shows he sepa a ion among
Figu e 14. Speed p o ile o each UAV: UAV1 in blue, UAV2 in ed
and UAV3 in black and UAV4 in g een.
UAVs. Each UAV main ains he minimum sepa a-
ion dis ance.
Figu e 15. UAV eal ajec o ies in he expe imen : UAV1 in black,
UAV2 in blue, UAV3 in ed and UAV4 in g een.
VI. CONCLUSIONS
This pape add esses he p oblem o collision
a oidance wi h mul iple UAVs in coo dina ed mis-
sions by ensu ing he sa e y and eliabili y o he
mission in dynamic en i onmen s. A me hod is
p oposed o e icien ly e-plan each UAV ajec o y
and pe o m he mission. The me hod akes in o
accoun he dynamics o he ehicles o compu e
mo e ealis ic ajec o ies. I is based on a Legend e
pseudospec al colloca ion o gene a e ajec o ies
and a olling ho izon policy is used. I is applied
i e a i ely and collision- ee ajec o ies a e planned
in a sub-p oblem de ined by he ime ho izon.
Figu e 16. Sepa a ion be ween UAVs in he expe imen : UAV1-
UAV2 in black, UAV1-UAV3 in blue, UAV1-UAV4 in ed,UAV2-
UAV3 in g een, UAV2-UAV4 in clea blue, UAV3-UAV4 in pink and
he minimum sepa a ion in dashed black line.
Two maneu e s a e allowed: changes o speed and
heading.
The main ad an age o he p oposed me hod is
i s low compu a ional load. Ano he no el aspec
is o conside mul iple UAVs and wo possible
maneu e s. Mos o wo ks published on colloca ion
echniques o UAVs conside one ehicle [13] [16].
Look-ahead ime and numbe o colloca ion
poin s a e he mo e ele an pa ame e s o he
me hod because o hei in luence on he compu-
a ion ime. A lo o es s ha e been ca ied ou o
ensu e sa e solu ions and low compu a ion imes by
conside ing andom scena ios. The scalabili y o he
me hod has been analyzed.
Finally, eal expe imen s ha e been ca ied ou
in he mul i ehicle ae ial es bed o he Cen e o
Ad anced Ae ospace Technologies (Se ille, Spain).
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