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USC Computer Science Technical Reports, no. 543 (1993)
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USC Computer Science Technical Reports, no. 543 (1993)
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Content
Conict Resolution for Air T rac
Con trol
USCCS
Y uiBin Chen
y
A lfr e d Inselb er g
z
June COMPUTER SCIENCE DEP AR TMENT
UNIVERSITY OF SOUTHERN CALIF ORNIA
LOS ANGELES CALIF ORNIA This researchw as supp orted b y NASA Ames Researc hCen ter under gran t NCC
y
The authors curren t address is Computer Science Departmen t Univ ersit y of Southern California Los Angeles
CA Email address is yuibinchp oll ux us ce du z
The authors curren t addresses are Computer Science Departmen t Univ ersit y of Southern California Lo s
Angeles CA and IBM Scien tic Cen ter Colorado Av en ue San ta Monica CA Emai l
address is inselberp ol lux u sc edu
Conict Resolution for Air T rac Con trol
Y uiBin Chen
y
A lfr e d Inselb er g
z
June Abstract
The t w odimensional Conict Resolution problem for Air T rac Con trol corresp onds to the searc h
for collisionfree tra jectories with constrained maneuv ers for n mo ving disks A global rather than
partial ie subset b y subset algorithm for a sp ecialized v ersion of this problem ha ving O n
log n time complexit y without bac ktrac king is prop osed It is based on a normalized relativ e co ordinate
system An extension of this algorithm requiring O n
time can nd all p ossible resolutions with a
sp ecied input order
Dieren t p erm utations of the input sequence lead to dieren t resolutions Heuristics for c ho osing
promising p erm utations are giv en
In tro duction
Motion planning is a pro cedure deriving a sequence of con tin uous motions for leading ob jects from initial
placemen ts to goal placemen ts without violating sp ecied constrain ts and satisfying some optimalit y cri
terion P oin ts or disks are the simplest mo dels for mo ving ob jects in the plane and are also used in this
pap er F or the motion planning of a p oin t rob ot among stationary obstacles the problem is similar to the
visibilit y graph problem in Computational Geometry Prep Planning the motion with mo ving obstacles
is computationally harder than with stationary obstacles Cull Erdm F uji OD un Sc h w and
is kno wn as an NPhard problem with v elo cit y b ound on the rob ot Cann Ho w ev er if the rob ot can
b e mo deled as a p oin t and can mo v e faster than obstacles whic h are all con v ex p olygonal shap es F ujim ura
and Samet F uji sho w ed that there is an O n
log n algorithm to determine and nd the shortest path
where n is the total n um ber of v ertices in all obstacles
Here a m ultiple rob ots problem arising in Air T rac Con trol A TC is studied The goal of A TC is to
direct aircraft safely to their destinations while minim al ly in terfering with their in tended tra jectories This
in v olv es main taining a minim um separation d on the horizon tal plane
bet w een aircraft detecting conicts
and resolving the conicts according to certain maneuv ering priorities and constrain ts see Eic k Lee
Sp en T obi for examples
Tw o aircraft are considered in conict when the distance b et w een them is less than the minim um re
quired separation While conict detection can b e accomplished with O n
time complexit ybyc hec king
This researchw as supp orted b y NASA Ames Researc h Cen ter under gran t NCC
y
The authors curren t address is Computer Science Departmen t Univ ersit y of Southern California Los Angeles CA Email address is yuibinch pol lux us c edu z
The authors curren t addresses are Computer Science Departmen t Univ ersit y of Southern California Los Ange
les CA and IBM Scien tic Cen ter Colorado Av en ue San ta Monica CA Email address is
inselberp oll ux usc e du The minim um separation is usually nautical miles nmi on the horizon tal plane and to feet in altitude
dep ending on the altitude Sp en
T e chnic al R ep ort USCCS
r
s
i
s
i
s
i
i
i
turn angle P
i
P
i
P
i
A C
i
C C CW C C CO oset
d
Figure A parallel oset maneuv er and asso ciated notations
the tra jectories of eac h pair of aircraft conict resolution b eing an instance of the Asteroid Av oidance
Problem is an NPhard problem Cann Resolution metho ds ha v e b een prop osed whic h resolv e the con icts subset b y subset leading to bac ktrac king Haus and v ery high complexit ySuc hsc hemes can lead
to w orse conicts than the ones they resolv ed and p ose some fundamen tal diculties in program pro ving
Sc h w artz and Sharir sho w ed an algorithm for planning conictfree tra jectories for k disks in Sc h w This algorithm runs in O n
time when k where n is the n um b er of edges in all p olygonal obstacles
But the time complexit y increases exp onen tially with k Other approac hes using Articial In telligence
tec hniques for A TC automation are not usually suitable for realtime application and require extensiv e
searc hing time when the n um b er of rules and input size b ecome large In this pap er a t w ostep priori tized planning algorithm for conict detection and resolution with O n
log n time complexit y based on a
Normalized Relativ e Co ordinate System NR CS is prop osed The algorithm itself do es not in v olvem uc h
geometric computation On the other hand there is a dra wbac k while simplifying the mo deling the rs t
step ma y eliminate p oten tial solutions A suggested solution requiring O n
time to a v oid this dra wbac k
is in tro duced in x Although the general problem is threedimensional in space it is preferable whenev er p ossible t o
resolv e conicts b et w een aircraft without an altitude c hange F or this reason the t w odimensional case i s
studied rst The allo w able maneuv ers considered illustrated in Figure are constrained b y
main taining constantspeed b efore during and after the maneuv er
starting the maneuv er no earlier than some sp ecied lead time prior to conicts
ha ving a maxim um allo w able turn angle ha ving a maxim um allo w able deviation oset from the original trac k and
up on completion returning aircraft to their original headings The lead time requiremen t is due to the uncertain t y in the radar trac king and for a v oiding unnecessary
maneuv ers in case the aircraft w ere going to turn an yw a y Returning to the same heading facilitates
main taining course using directional radio na vigation The constan t sp eed constraintma y b e relaxed Bu t
with the v aried sp eed the problem has three degrees of freedom for eac h aircraft W e will discuss the
v aried sp eed and v aried altitude cases in x
Main taining horizon tal separation b et w een aircraft is equiv alenttoha ving op enededge disks of diamete r
d mo ving with constan t sp eed and not b eing allo w ed to collide Collision ma ybe a voidedonlybythe
T e chnic al R ep ort USCCS H H H H H H H H H Hj ij
V
j
V
i
V
ij
Figure The v elo cityofA C
i
relativeto A C
j
maneuv ers outlined ab o v e Conictfree tra jectories of aircraft are hence equiv alen t to the collisionfree
tra jectories of disks cen tered at eac h aircraft
As sho wn in Figure the tra jectory of aircraft i or the related disk denoted b yA C
i
undergoing a
maneuv er is partitioned in to three parts
the original trac k denoted byP
i
the maneuv er at equal sp eed denoted byP
i
and
the completion denoted b yP
i
b y returning to the original heading
F or the purp oses of analysis it is con v enien t to consider the situation in terms of three aircraft existing
during distinct time p erio ds needed to complete P
i
P
i
and P
i
That is A C
i
can b e though t of spa wning
t w o disks A C
i
and A C
i
with tra jectories P
i
and P
i
resp ectiv ely corresp onding to t w o pseudoaircraft The existing time or activ e time of A C
i
is a subset of time p erio d and is denoted b y T
i
Normalized Relativ e Co ordinate System
Let
V hV
x
V
y
i be a v ector and V k
V k be the L
norm of
V where V
x
and V
y
are its x and y
comp onen ts resp ectiv ely The function Ang V
x
V
y
abbreviated b y
Vis the coun terclo c kwise angle
bet w een the p ositiv e xaxis and
V The v elo cityof A C
i
relativeto A C
j
is dened as
V
ij
V
i
V
j
with
V
ij
denoted b y ij
see Figure T o construct the normalized relativ e co ordinate system with resp ect to A C
i
the disk A C
i
is shrunk to
a p oin t and placed at the origin and the radii of all other disks are doubled note that this pro cedure is
equiv alen t to obtaining a Minco wski sum with magnitude d to all other disks Then all other disks A C
j
are indep enden tly rotated around the origin and corresp ondingly normalized b y their relativev elo cities
V
ij
After the rotation and normalization
V
ij
are transformed to unit v ectors on the xaxis and the radii
of A C
j
are scaled appropriately b y their relativev elo cities This is done as follo ws
x
ij
y
ij
V
ij
cos ij
sin ij
sin ij
cos ij
x
j
x
i
y
j
y
i
r
ij
d
V
ij
T e chnic al R ep ort USCCS
s
A C
i
A C
j
A C
k
Time t
y
x
r
ij
r
ik
x
ij
y
ij
h i
s
Figure Normalized relativ e co ordinate system NR CS
i
t where the cen ters of A C
j
with radii r
ij
denoting aircraft j with resp ect to aircraft i are at x
ij
y
ij
I n
the co ordinates of A C
j
s cen ters are giv en in terms of a translation a rotation and a scaling applie d
sequen tially Clearly the NR CS is sp ecic to A C
i
and is time dep enden t so it should b e prop erly denote d
byNR CS
i
t An example of NR CS
i
t with aircraft is sho wn in Figure The dierence of r
ij
and r
ik
is due to the dierence of the relativ e sp eeds V
ij
and V
ik
W e formalize the n time construction of NR CS
i
t as follo ws
Construction Giv en tra jectories of A C
j
for j n a sp ecied index i and time t follo wing
steps construct NR CS
i
t Compute co ordinates of cen ter of A C
i
at time t F or eachA C
j
A C
i
compute the follo wing information
a the co ordinates of cen ter of A C
j
at time t b the relativ ev elo cit y
V
ij
c the new relativ e co ordinates of cen ter of A C
j
b y using F orm ula and
d the new radius of A C
j
b y using F orm ula Here w e assume that all arithmetic computations in Lines ad can b e done in constan t time Hence
NR CS
i
t can b e constructed in n time where n is the total n um b er of disks
Resolution Algorithm
The discussion of conict detection and resolution is based on the NR CS In x and x the algorith m
is applied pairwise and a global resolution for m ultiaircraft case is obtained subsequen tly in x
T e chnic al R ep ort USCCS s
A C
i
A C
j
Time t
y
x
r
ij
s s
t
j t
j x
ij
y
ij
V ij
V ij
Figure Conict of A C
i
and A C
j
in NR CS
i
P airwise conict detection
T o main tain the conictfree tra jectories for t w o aircraft the corresp onding t womo ving disks m ust not
o v erlap for all times
In NR CS
i
when the disk A C
j
in tersects the xaxis at t w o distinct p oin ts ie
y
ij
r
ij
there
is a collision b et w een A C
i
and A C
j
if A C
j
exists in the conict p erio d t
j t
j see Figure In other
w ords T
j
has some o v erlaps with t
j t
j Due to the normalization of
V
ij
the xaxis also represen ts the
timeaxis and the conict p erio d t
j t
j is found b y
t
j x
ij
q
r
ij
y
ij
t
j x
ij
q
r
ij
y
ij
So in NR CS
i
t for all t
j t t
j A C
j
m ust con tain the origin
Another example is giv en in Figure A C
i
will collide with A C
j
but not with A C
k
Again there is no
direct correlation b et w een A C
j
and A C
k
in NR CS
i
t ev en when these t w o disks o v erlap
Algorithm P airwise Conict Detection
Input NR CS
i
and index j Output The c onict p eriodof A C
i
and A C
j
if
y
ij
r
ij
then Use to compute in tersection p oin ts of xaxis and A C
j
return t
j t
j T
j
else return end
In NR CS when time go es on all disks mo v e in the negativ e x direction horizon tally It is clear that
the disk A C
j
whic h indicates aircraft jcon tains the origin if and only if the distance b et w een aircraft i
and j in the w orld co ordinate system is less than d ie aircraft i and j are in conict Hence w eha v e
follo wing observ ations
T e chnic al R ep ort USCCS
Observ ation F r om A lgorithm Two air cr aft i and jar einc onict if and only if
y
ij
is less
than r
ij
and the c onict p erio d is a nonempty set
Observ ation F r om A lgorithm Two air cr aft i and jar ein c onict at time t if and only if t is
in the c onict p erio d
Since the computation in Algorithm can b e done in constan t time to nd all conicts can therefore
b e done in n
time b y applying the algorithm for eac h pair of aircraft
P airwise conict resolution
As men tioned earlier the conicts are resolv ed using parallel oset maneuv er see Figure with constrain ts
the maxim um turn angle range of A C
i
!
i
!
i
the maxim um oset distance D
i
and the xed sp ee d
V
i
If a lead time t
lead
is sp ecied the maneuv er is started no earlier than t
lead
b efore the conict time
Without loss of generalit y assume that the curren t time is In order to reduce the in terference of air
trac con trol the maneuv er is p erformed at exactly t
lead
time b efore the rst conict time F or this
reason NR CS
i
T
i is constructed simply b y translating A C
j
along the negativ e xaxis b y t
j t
lead
units
in NR CS
i
T
i t
j t
lead
x
ij
x
ij
T
i y
ij
y
ij
After time T
i A C
i
is considered to b e no longer existing and A C
i
follo wing P
i
tak es the place of A C
i
in the con tin uation of the maneuv er
In NR CS
i
T
i t wora ys from the origin tangen t to the new disk A C
j
cen tered at x
ij
y
ij
with radiu s
r
ij
describ e a forbidden area suc hthat to a v oid the conict during the maneuv er
V
i
j
m ust b e outside this
region The b oundary angles j and j without loss of generalit y considering j
j see Figure are
j Ang x
ij
y
ij
sin
r
ij
q
x
ij
y
ij
A
j Ang x
ij
y
ij
sin
r
ij
q
x
ij
y
ij
A
Toa v oid a conict
V
i
j
V
ij
j
j So the range of angles denoted b y R
ij
whic h causes
V
i
j
to
fall in to the forbidden region ha v e to b e remo v ed from the range of p ossible turn angles R
i !
i
!
i
dep ending on the ph ysical capabilit y of aircraft i The conictfree turn angle range R
i
is therefore
dened b y
R
i
R
i R
ij
T e chnic al R ep ort USCCS s
A C
i
A C
j
x
ij
y
ij
Time T
i t
j t
lead
y
x
Lead Time
j j V ij
V ij
Figure Conict angle range of
V
ij
in NR CS
i
T
i
where
R
ij
f j V
i
V
i
suc h that
V
i
j
V
ij
j
j where
V
i
j
V
i
V
j
and V
i
V
i
g Because of the constan t sp eed constrain t after c hanging the direction of
V
i
the head of the v ector
V
i
still
falls on a circle
C
i
of radius V
i
see Figure Hence R
i
R
i and R
ij
can b e equiv alen tly represen ted as
p ortions of the circle ie an arc or a collection of arcs Supp ose that w ec ho ose i
whic h will cause
V
ij
to rotate b y angle ij
from R
i
to b e the turn angle of A C
i
The mapping b et w een i
and ij
is
i
V
i
V
i
ij
V
i
j
V
ij
The next step is to decide the earliest time referred to as turnbac k time for A C
i
to turn backto
the original heading ie to followP
i
This turnbac k time also pro vides the minim um oset tra jectory
giv en a c hosen turn angle It can b e accomplished b y recomputing the relativ e co ordinates of A C
j
with
resp ect to A C
i
at time T
i ie constructing NR CS
i
T
i the transformation in v olv ed is the same as that
describ ed in x Once A C
j
in NR CS
i
T
i is computed and placed on its new co ordinates with justied
radius t w o parallel tangen t lines to the disk cen tered at x
i
j
y
i
j
with an angle of ij
from the p ositiv e
xaxis are dra wn see Figure Where these t w o lines in tersect the xaxis marks the in terv al t
j t
j that A C
i
m ust a v oid in considering the turnbac k p oin t
t
j x
i
j
"
j "
j t
j x
i
j
"
j "
j In x w e will discuss extensions of this algorithm to higher degrees of freedom and C
i
is corresp ondi ngl y mo died to an
ann ulus or a sphere
T e chnic al R ep ort USCCS
R
i R
ij
!
i
C
i
V
i
V
j
V
ij
j j Figure Conictfree turn angle range Shaded area describ es the forbidden area
x
y
Time T
i A C
i
V
i
j
V
i
j
r
A C
j
x
i
j
y
i
j
T T T T T r
i
j
"
j "
j t
j t
j T
iM
ij
F
i
Figure Conictfree turnbac k p erio d where T
iM
D i
jV i sin i j
T e chnic al R ep ort USCCS ID Mo ving Disk P ath Sp eed Direction Activ e Time P erio d
i A C
i
P
i
j
V
i
j i
T
i i
A C
i
P
i
j
V
i
j j
V
i
j i
i
T
i T
i i
A C
i
P
i
j
V
i
j j
V
i
j i
T
i T able The data for aircraft i and its spa wned pseudoaircraft i i
where
"
j r
i
j
sin ij
"
j y
i
j
tan
ij
In other w ords the leftmost p oin t of this p erio d on the p ositiv e xaxis is the earliest time A C
i
can turn
bac k to the original heading and still a v oid the collision with A C
j
Hence the conictfree turnbac k range
F
i
is dened as
F
i
T
iM
t
j t
j where T
iM
D i
jV i sin i j
the latest turnbac k time suc h that the oset is no more than D
i
The earliest time
to turn bac k to the original heading T
i is dened b y
T
i min F
i
T
i Notation F x ft x j t F g W e summarize the conict resolution algorithm as follo ws and sho w the data for aircraft i and its t w o
spa wned pseudoaircraft after the motion of aircraft i to a v oid the conict with aircraft j has b een decided
in T able Algorithm P airwise Conict Resolution
Input The tr aje ctories of A C
i
A C
j
and the c onict time t
j Output The p ar al lel oset maneuver for c onict r esolution satisfying c onstr aints
Use to compute the maneuv er time T
i Construct NR CS
i
T
i Find i
the turn angle of A C
i
Find T
i the turnbac k timeofA C
i
with resp ect to i
The conictfree tra jectory of A C
i
is constructed b y three p ortions P
i
P
i
and P
i
Spa wn t w o pseudodisks A C
i
and A C
i
from A C
i
Prop erly up date their data see T able end
In Algorithm the motion is planned at lead time b efore the conict When aircraft i turns its
new tra jectory is c hosen from one of the conictfree tra jectories and when it turns bac k to the original
heading it still remains on a conictfree tra jectory Therefore w eha v e follo wing observ ations
Observ ation F r om A lgorithm A t time T
i A C
i
makes a turn with angle R
i such that A C
i
c an avoid the c onict with A C
j
if is in R
i
T e chnic al R ep ort USCCS
Observ ation A C
i
wil l not bein c onict with A C
j
and the oset D
i
if A C
i
turns b ack to its
original he ading at time t F
i
T
i No w w e giv e a formal pro of that Algorithm deriv es a conictfree tra jectory for aircraft i Theorem The p ar al lel oset tr aje ctory of air cr aft i as determine dby A lgorithm and the origina
tr aje ctory of air cr aft j arec onictfr e e
Pro of
T o pro v e that the parallel oset tra jectory of aircraft i and the tra jectory of aircraft j are conictfre e
is same as to pro v e that aircraft i i
and i
during their existing p erio ds are not in conict wit h
aircraft j First w ekno w that aircraft i only exists till lead time b efore the conict Therefore aircraf t
i during its existing p erio d is not in conict with aircraft jF rom Lines and in Algorithm th e
tra jectories of aircraft i
and i
are c hosen to a v oid the conict with aircraft j Hence the paralle
oset tra jectory of aircraft i and the tra jectory of aircraft j are conictfree It is clear that the pairwise conict resolution algorithm tak es constan t time for eac h pair of aircraft
Ho w ev er when w e generalize it to the global conict resolution algorithm due to the computation of se t
op erations the global algorithm ma ytak e O n
log n time
Global conict resolution
No w the general case in v olving n aircraft mo ving disks in a monitoring sector is considered Global
rather than pairwise conict detection can b e accomplished b y successiv ely applying Algorithm t o
eac h pair Similarly global resolution is found b y generalizing the pairwise algorithm
Algorithm Global Conict Resolutio n
Input T r aje ctories of All A Cs
Output The glob al c onstant sp e e d singlemaneuver c onictfr eep aths
Run the global conict detection algorithm to detect all p ossible conicts
Mo dify Algorithms b y replacing equations and b y
T
i min
j i
t
j t
lead
R
i
R
i j i
R
ij
and
F
i
T
iM
j i
t
j t
j resp ectiv ely for eac h A C
i
Use mo died algorithm deriv ed from Line to obtain the conictfree path of A C
i
and
if necessary create t w o pseudodisks corresp onding t w o pseudoaircraft
if A C
i
exists
then Insert A C
i
and A C
i
to A C set
Up date the activ e time of A C
i
end
T e chnic al R ep ort USCCS In equations to T
i is the time t
lead
units b efore the earliest conict of A C
i
R
i
sp ecies
the turn angle range of A C
i
to a v oid all of the conicts with the rest and F
i
indicates the conictfree
turnbac k range of A C
i
Giv en n aircraft in solving this problem based on the parallel oset maneuv er with giv en constrain ts
the total n um b er of aircraft and spa wned pseudoaircraft is no more than n The last aircraft do es
not ha veto c hange its tra jectory Therefore the space complexit y is b ounded by n T ond T
i tak es
O n time but to nd R
i
and F
i
tak es O n log n time Because set union op eration tak es O n log n time
b y rst sorting all endp oin ts Hence resolving this problem without considering bac ktrac king requires
O n
log n time Indeed this t w ostep global conict resolution algorithm can blo c k some p oten tial
resolutions b ecause the rst step ma y screen out resolutions in order to a v oid the future conict whic h
can b e a v oided when an earlier turnbac k time is c hosen In x w e prop ose a dieren t idea to nd all
p ossible resolutions Ho w ev er the time to compute them ma yha vetobeextended to O n
Situations ma y arise where no resolution is p ossible for some aircraft due to the previous c hoice of
tra jectories already committed for others A resolution ma y then b e found with bac ktrac king Of course
this drastically increases the time complexit y A go o d strategy for ordering the input sequence is necessary
to a v oid or to reduce bac ktrac king
Ordering Heuristics with Examples
In connection with the design of the new automated A TC system the F ederal Aviation Administration
pro vided a n um b er of complex scenarios Leje to test conict resolution algorithms Resolutions of
one of these scenarios scenario obtained with the prop osed algorithm are sho wn They are dieren t
resolutions than those pro vided earlier for the same scenario Inse
The input is ordered byn um b er of conicts highest n um b er rst In the rst case aircraft are
resolv ed with lead time of seconds maxim um turn angles of b oth righ t and left turns and
maxim um oset of nmi# in the second case aircraft can b e resolv ed when the maxim um oset is
relaxed to nmi# and without maxim um oset constrain t all aircraft
can b e resolv ed see Figures and resp ectiv ely By con trast an exp ert air trac con troller w as able to resolv e only aircraft and
he needed o v er min utes to accomplish it
In Figure the aircraft example is resolv ed with t w o dieren t strategies for ordering the input
sequence Figure a is ordered b yn um b er of conicts highest n um b er rst and Figure b the
rev ersed order In order to obtain the resolution b y using the rev ersed order without bac ktrac king the
maxim um oset has to b e relaxed to nmi
In this w ork the n um b er of conicts highest n um b er rst is the k ey comp onen t in deciding the input
sequence order Exp erimen tation suggests that this w as a good c hoice In the real case w e should not
need to rearrange the input order unless sev eral aircraft en ter the monitoring sector at almost the same
time
Cases of Higher Degrees of F reedom
This algorithm ma y b e generalized to allowfor c hanging sp eed or c hanging altitude or b oth While the
sp eed v ariation is tak en in to consideration the head of v ector
V
i
will no w fall in an ann ulus b ounded b y
t w o concen tric circles see Figure where the outer circle indicates the case of sp eedup and the inner
one slo wdo wn instead of on a circle C
i
with the remaining parts of the algorithm b eing the same In
In the original scenario the last aircraft are on the dieren tlev el as others
T e chnic al R ep ort USCCS
Air T rac Con trol at Time Figure A TC resolution example with aircraft a initial situation
T e chnic al R ep ort USCCS Air T rac Con trol at Time Figure con tin ue A TC resolution example with aircraft b unresolv ed result at time
T e chnic al R ep ort USCCS
Air T rac Con trol at Time Figure con tin ue A TC resolution example with aircraft c global resolution at time
T e chnic al R ep ort USCCS Air T rac Con trol at Time Figure con tin ue A TC resolution example with aircraft d resolv ed result at time
T e chnic al R ep ort USCCS
Air T rac Con trol at Time Figure A TC resolution example with aircraft a initial situation
T e chnic al R ep ort USCCS Air T rac Con trol at Time Figure con tin ue A TC resolution example with aircraft b resolv ed result
T e chnic al R ep ort USCCS
Air T rac Con trol at Time Figure A TC resolution example with aircraft a initial situation
T e chnic al R ep ort USCCS Air T rac Con trol at Time Figure con tin ue A TC resolution example with aircraft b resolv ed result
T e chnic al R ep ort USCCS
Air T rac Con trol at Time Figure A TC resolution example with dieren t ordering strategies a highest conict n um b er rst
T e chnic al R ep ort USCCS Air T rac Con trol at Time Figure con tin ue A TC resolution example with dieren t ordering strategies b rev ersed order
T e chnic al R ep ort USCCS
B B B B B B B B B B B BB c c c c c i
C
i
V i
V j
V ij
j j sp eedup
slo wdo wn
Figure Conictfree angle range shaded area with v aried sp eed
Figure w esho w the example of c hanging sp eed Altitude c hanges require the head of v ector
V
i
to b e
on the surface of a sphere
When w e allo w the sp eed or altitude to b e c hanged during the maneuv er to compute R
i
and F
i
ma y
require O n
time for aircraft i Then it will bring up the total time complexit yto O n
for the globa
resolution In the next section w e prop ose a onestep resolution algorithm whic htak es O n
time an d
can nd all p oten tial resolutions with a giv en ordering
Extensions
In the t w ostep global resolution algorithm when a turn angle is xed for the parallel oset maneuv er
the minim um oset of this aircraft is determined Unfortunately w e are unable to observean y relation
bet w een the turn angle and the maxim um oset for global resolution Ho w ev er it is p ossible to deriv e a
onestep algorithm to eliminate this problem
In a parallel oset maneuv er since the maneuv er starts at lead time b efore the rst conict the onl y
parameters to decide the tra jectory are the turn angle and turnbac k time If w ew ork on an S R
conguration space where S indicates a unit circle the follo wing onestep algorithm can nd all p oten tial
resolutions with a giv en ordering
Algorithm Onestep Conict Resolution Input T r aje ctories of All A Cs
Output The glob al c onstant sp e e d singlemaneuver c onictfr eetr aje ctories
for eac h A C
i
T e chnic al R ep ort USCCS Use to compute the maneuv er time T
i Construct NR CS
i
T
i S
i f t j R
i and t T
iM
g
for eac h A C
j
A C
i
S
ij
f t j R
ij
and t t
j t
j g
S
i
S
i S
j i
S
ij
if S
i
is not empt y
then Deriv e the conictfree path of A C
i
Create t w o pseudoaircraft A C
i
and A C
i
Insert A C
i
and A C
i
to ACset Up date the activ e time of A C
i
end
In this algorithm to nd all conictfree paths for eac h aircraft is essen tially to b ound the conictfree
region in the S R conguration space In the w orst case it tak es O n
time to compute S
i
Therefore
o v erall the algorithm requires O n
time
Conclusions
F rom algorithms describ ed in previous sections w ekno w that the resolution is dep ended on the order of
input It is also ob vious that the higher the densit y
the few er the n um b er of resolutions Therefore it is
imp ortantto ha v e a good w a y to decide the order of input and to dene the resolv able densit y whic h
guaran tees that with densitylo w er than the resolv able densit y there is alw a ys a resolution
In the real case when considering acceleration constrain ts the tra jectory of an aircraft is no longer a
p olyline Instead of ha ving an extreme turn p oin t w e shall ha v e an arc or some smo oth curv e As one
ma y exp ect this c hange can increase the algebraic complexit y when computing the conictfree zones
Ho w ev er it is p ossible to relax the acceleration constrain ts b y using v aried sp eed to mak e up the dierence
while follo wing a curv ed path
The densit y can b e dened as total n um b er of aircraft in a certain area
T e chnic al R ep ort USCCS
References
Cann J Cann y and J Reif New lo w er b ound tec hniques for rob ot motion planning problems I n
Pr o c e e dings of the th A nnual IEEE Symp osium on F oundations of Computer Scienc e page s
W ashington DC Cull RK Culley and KG Kempf A collision detection algorithm based on v elo cit y and dis tance b ounds In Pr o c e e dings of the IEEE International Confer enceon R ob otics and
A utomation pages San F rancisco CA April Eic k JS Eic k emey er A Inselb erg and B Dimsdale Visualizing pats in nspace using paralle
co ordinates T ec hnical Rep ort G IBM P alo Alto Scien tic Cen ter July Erdm M Erdmann and T LozanoP erez On m ultiple mo ving ob jects A lgorithmic a F uji K F ujim ura and H Samet Timem inim al paths among mo ving obstacles In Pr o c e e dings
of the IEEE International Confer enceon R ob otics and A utomation pages Scottsdale AZ Ma y Haus S Hauser AE Gross and RA T ornese En route conict resolution advisories MTR W R ev MITRE Co Inse A Inselb erg M Boz and B Dimsdale Planar conict resolution algorithm for air tra c
con trol and oneshot problem T ec hnical Rep ort G IBM P alo Alto Scien tic Cen ter
No v em b er Lee HQ Lee An advisory system for predicting and resolving airspace violations based on four dimensional guidance tec hniques In Pr o c e e dings of AIAA Guidanc e Navigation and Contr o
Confer enc e Leje RO Lejeune Go v ernmen tpro vided complex scenarios for the adv anced automated system
design comp etition phase MTRW MITRE Co OD un C
OD unlaing Motion planning with inertial constrain ts A lgorithmic a Prep FP Preparata and MI Shamos Computational Ge ometryc hapter SpringerV erlag Sc h w JT Sc h w artz and M Sharir On the piano mo v ers problem Iii co ordinating the motio n
of sev eral indep enden t b o dies The sp ecial case of circular b o dies mo ving amidst p olynomia
barriers International Journal of R ob otics R ese ar ch Sc h w JT Sc h w artz and M Sharir A surv ey of motion planning and related geometric algorithms
A rticial Intel ligenc e Sp en DA Sp encer Applying articial in telligence tec hniques to air trac con trol automation The
Linc oln L ab or atory Journal T obi L T obias and Jr JL Scoggins Timebased air trac managemen t using exp ert systems
IEEE Contr ol System Magazine pages
Abstract (if available)
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Description
Yui-Bin Chen and Alfred Inselberg. "Conflict resolution for air traffic control." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 543 (1993).
Asset Metadata
Creator
Chen, Yui-Bin
(author),
Inselberg, Alfred
(author)
Core Title
USC Computer Science Technical Reports, no. 543 (1993)
Alternative Title
Conflict resolution for air traffic control (
title
)
Publisher
Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA
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1991/2017
Repository Email
csdept@usc.edu
Repository Name
USC Viterbi School of Engineering Department of Computer Science
Repository Location
Department of Computer Science. USC Viterbi School of Engineering. Los Angeles\, CA\, 90089
Publisher
Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA
(publisher)
Copyright
In copyright - Non-commercial use permitted (https://rightsstatements.org/vocab/InC-NC/1.0/