_ Vasii_Claudiu_Conflict Detection and Resolution for 1 (1)

7/25/2019 _ Vasii_Claudiu_Conflict Detection and Resolution for 1 (1)

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American Institute of Aeronautics and Astronautics Paper ATIO-2007-7737

ATIO-2007-7737

7th AIAA Aviation Technology, Integration and Operations onference, !"#20 $eptem%er 2007, &elfast, 'orthern Ireland

onflict (etection and )esolution for !,2,* * *

,' Aircraft

+illes (oe .esar

/uno1

This paper presents a mathematical frameor for the formal specification of aircraft

conflict detection and resolution algorithms and the verification of their properties* The

frameor is illustrated ith original algorithms that detect conflicts ith multipleaircraft

and effectively solve all them via vertical speed-only maneuvers*

I* Introduction

ormal methods in computer science refers to the use of logic and mathematics to verify that a system

design and its implementation satisfy functional reuirements and safetyproperties*!

(espite the fact thatseveral onflict (etection and )esolution 4(5)6 systemshave %een proposed in the past fe years,2 very

fe of these systems have %een descri%ed and analy1ed using formal methods* Therefore, it is not alays

clear hat properties these algorithms satisfy* or instance, the notion of cooperative resolution isoften used in the (5) literature epressing different and, sometimes, contradictory concepts, e*g*, it

epresses echange of intent information,3#8 degree of coordination,2, 7 sharing of effort to achieve

separation," and temporary delegation of responsi%ility for separation*9 This confusion could %e avoided %yproviding rigorous definitions of the concepts and properties that are fundamental to all (5) systems*

This paper presents a mathematical frameor for the formal specification and analysis of conflict de-

tection and resolution algorithms and their properties* This frameor applies to state-%ased, pair,ise,

geometric, and distri%uted conflict detection and resolution algorithms* $tate-%ased 4or tactical6 refers

to the use of aircraft state information, e*g*, position and velocity vectors, as opposed to strategic

approaches that use intent information, e*g*, flight plan* Pairise refers the use of a conflict detection

and resolution logic for t,odistinguished aircraft: the onship and the traffic aircraft* +eometric refers tothe use of linear pro;ections to predict aircraft tra;ectories as opposed to pro%a%ilistic or performance-

%ased tra;ectories* inally, distri%uted refers to systems that are deployed on several aircraftdetecting andsolving conflicts for the onship as opposed to centrali1ed systems that detect and simultaneously solveconflicts for a large set of aircraft* The distri%uted approach that e consider in this frameor reuires

minimal communication %et,een the aircraft* In particular, the only information that is periodically

echanged %eteen the aircraft is the state information, e*g*, position and velocity vectors of the aircraft*


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American Institute of Aeronautics and Astronautics Paper ATIO-2007-7737

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aircraft tra;ectories can %e descri%ed %y a position, a velocity vector, and a time interval* e also assume

that in order to solve a conflict an aircraftcan instantaneously change course and speed* These assumptions

are typicalof tactical (5) systems ith short looahead times 4usually, @ minutes6*

The state information of an aircraft is represented %y an identifier, a position, and a velocity vector*

Identifiers are assumed to%e uniue and invariant, i*e*, 4!6 there are no t,odifferentaircraftin the airspace

ith the same identifier, and 426 the identifier of an aircraft does not change over time* &y a%use of notation,

e ill usually refer to an aircraft%y its state* Benceforth, e use letters A, &, * * * to denote aircraft states,%oldface letters to denote vector varia%les, e*g*, s denotes position and v denotes velocity, and typewriter

type for algorithms* hen a vector varia%le refers to a particular aircraft, the varia%le is su%-indicated%ythe aircraft state* or instance the current position and velocity vectors of an aircraft A are denoted sAand vA , respectively* As usual, su%-indices , y, and 1 denote ais values, e*g*, vA , vAy are the hori1ontal

components of the velocity vector vA *


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p2D y2

|1|

k4, y, 16k Gma74

, 6* 4>6( B

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Proposition ! ormulas 426, 436, and 4>6 are ell-defined vector norms, i*e*, they satisfy positivity, nullity,scala%ility, and triangular ineuality*

The use of a norm allos for definitions of aircraftdistance and loss of separation that are independent

of the shape of the protected 1one*

(efinition 2 4(istance6 The distance %eteen aircraft A and & is defined as

H4A, &6 G ksA s&k*4@6

(efinition 3 4oss of $eparation6 A loss of separation or violation occurs hen the protected 1ones ofto aircraftoverlap, e*g*,

PA P& G *

486 The notions of loss of separation and aircraftdistance are related %y the folloingproposition*

Proposition 2 Aircraft A and & are in loss of separation if and onlyif

H4A, &6 E !*

The linear tra;ectory of an aircraft ispredicted from the current state of the aircraft, here the velocity

of the aircraft is assumed to %e constant*

(efinition > 4Predicted $tate6 The predicted state of an aircraft A at future time t relative to acurrent time 0, denoted A4t6, satisfies:

sA4t6 G sA D t vA ,

vA4t6 G vA *

A conflict is a predicted loss of separation %eteen t,oaircraft ithin a looahead time T*

(efinition @ 4onflict6 Aircraft A and & are said to %e in conflict if there is a time t in the future

%ut%efore the looahead time, i*e*, 0 t T , such that the distance %eteen the aircraft at time t is

strictly less than !:

H4A4t6, &4t66 E !* 476

onflicts are usually checed %y comparing the predicted closest separation of t,o aircraft against

theminimum safe distance defined %y the protected 1one*

(efinition 8 4Time of losest $eparation6 The time of closest separation %eteen aircraft A and &ithin a looahead time T, denoted JT 4A, &6, is the smallest time J, 0 J T, such that for any

time t,0 t T:

H4A4J6, &4J 66 H4A4t6, &4t66*4"6

'ote that the time of closestseparation alays eists %ecause H is a continuous function and the timerange 0 t T defines a close interval*

(efinition 7 4(istance of losest $eparation6 The distance of closest separation %eteen to aircraft

A and &, ithin a looahead time T, is the distance at the time of closest separation, i*e*,

KT 4A, &6 G H4A4JT 4A, &66, &4JT 4A, &666* 496

The folloingpropositionprovides an alternative definition of conflict%ased on distance of closestsepa-

ration*


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Proposition 3 Aircraft A and & are in conflict if and only if KT 4A, &6 E !*

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In our frameor, a resolution maneuver is modeled as an instantaneous change of the velocity vector of

the currentaircraftstate*

(efinition " 4)esolution /aneuver6 A resolution maneuver RA is a ne state for an aircraft A that4only6 modifies the value of the current velocity vector* The ne velocity vector for aircraft A undermaneuver RA is denoted vRA *

)esolution maneuvers can %e classified according to the parameters of the velocity vector that they

modify* Assuming a 3-dimensional geometry:

RA is a trac-only maneuver if vRA7

D vRAy G vA2 D vAy

2 and vRA1 G vA1 *

RA is a ground speed-only maneuver if vRA1 G vA1 and there is a positive such that 4vRA7, vRAy 6

G4vA7, vAy 6*

RA is a vertical speed-only maneuver if vRA7 G vA and vRAy G vAy *

)esolution maneuvers can also %e classified according to the separation criteria that they satisfy* etRAand R& %e resolution maneuvers for aircraftA and &,

RA is an independent resolution for aircraft A 4ith respect to aircraft &6 if KT 4RA, &6 !*If

KT 4RA, &6 G !, RA is also tangential*

RA and R& are coordinated if KT 4RA,R&6 !* If KT 4RA,R&6 G !, they are also minimal*

RA and R& are repulsive if KT 4RA,R&6 L KT 4A, &6*

III* onflict (etection

This section presents our mathematical frameor for the specification of conflict detection algorithms

and their safety properties* It illustrates the use of the frameor ith a three dimensional conflictdetection

algorithm for multiple aircraft*

III*A* Pairise onflict (etection

A 4pairise6 conflict detection algorithm taes as parameters the currentstate of t,oaircraft, and returns

a &oolean value that indicates a predicted loss of separation %eteen them*

(efinition 9 4onflict (etection orrectness and ompleteness6 +iven a looahead time T , acon- flict detection algorithm, namely cd, is correct if only conflicts are detected, i*e*, for all aircraftstates A and &, if cd4A, &6 returns true then A and & are in conflict* The algorithm is complete if allconflicts are detected, i*e*, if A and & are in conflict then cd4A, &6 returns true*

'ote that according to the previous definition, an algorithm that alays returns true is complete,%utnot correct, and an algorithm that alays returns false if correct, %ut not complete*

Proposition > If cd is correct and complete then cd is symmetric, i*e*, for all aircraft states A and

&,cd4A, &6 if and only if cd4&,A6*

III*&* onflict (etection for /ultiple Aircraft

A detection algorithm for multiple aircraft cMd taes as parameters the currentstate of the onship A anda set of aircraft states N here A 6 N, and returns a &oolean value that indicates a predicted loss ofseparation %et,een A and any aircraft in N*

(efinition !0 4onflict (etection orrectness and ompleteness for /ultiple Aircraft6 +iven a

looahead time T, a conflict detection algorithm for multiple aircraft, namely cMd, is correct if for allaircraft state A and set of aircraft states N, here A 6 N, if cMd4A, N6 returns true then A is in

conflict ith an aircraft & N* The algorithm is complete if every time that A 6 N and & N are inconflict then cMd4A, N6 returns true*

22


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(efinition !! A pairise conflict detection algorithm cd can %e transformed into a conflict detectionalgo- rithm for multiple aircraft cMd as follos:

cMd(AN!" #$$l %if N G t&e'

falseelse let & N i'

cd(A&! $r

e'dif

cMd(AN {&}!

The algorithm cMdis recursive* Boever, it is ell-defined since the cardinalityof the setof aircraftstrictly decreases at each iteration*

Proposition @ If cdis a pairise correct and complete conflict detection algorithm then cMdis a correctand complete conflict detection algorithm for multiple aircraft*

III** A 3-(imensional (etection Algorithm for /ultiple Aircraft

=&3( is a pairise state-%ased (5) system co-developed %y the authors!! and formally verified in the

Program ?erification $ystem 4P?$6*!@ In this section, e illustrate our conflict detection frameor ith a

simplified version of the conflictdetection algorithm used %y =&3(, hich e call cd)d*The protected 1one in =&3( is cylinder of diameter ( and height B * Therefore, the norm is defined as

in ormula 4>6* The conflict detection logic considers the folloing cases:

The relative hori1ontal velocity of the aircraft is 1ero, i*e*, v72 D vy

2 G 0, and the aircraft

have already losthori1ontal separation, i*e*, s72 D sy

2 E (2 * In this case, the aircraft are in conflict

if they have already lost vertical separation, i*e*, |s1 | E B , or if they ill lose vertical separationithin the looahead time T , i*e*, v1 s1 E 0 and B E si*'4v1 64T v1 D s1 6*

The relative vertical speed of the aircraft is 1ero, i*e*, v1 G 0, %ut the aircraft are hori1ontallyseparated* irst, e compute the time interval of hori1ontal conflict Ot(in , t(out P from the uadratic

euation on t:

4s7 D tv762

D 4sy D tvy 62

G (2

*

In this case, there is a conflictif the aircrafthave already lostvertical separation, i*e*, |s1 | E B , andif the time interval of hori1ontal conflict is ithin the looahead time T, i*e*, t(out L 0 and t(in E T

*

In the general case, e compute the time interval of hori1ontal conflict Ot(in , t(out P as in theprevious case, and the time interval of vertical conflictOtBin, tBout P from the euation on t:

|s1 D tv1 | GB*

rom these intervals, e compute the time interval of conflict Otin, tout, here

tin G ma4t(in, tBin 6,

tout G min4t(out

, tBout

6*

In this case, there is a conflict if this interval is ithin the looahead time T, i*e*, tin E tout and tout L

0 and tin E T*

The pairise algorithm cd)d is defined as follos*

cd)d(A&!" #$$l %let s % sA s& i'let v % vA v& i'

if v72 D vy

2 G 0 a'd s72 D sy

2 E (2 t&e'|s1 | E B $r4v1 s1 E 0 a'd B E si*'4v1 6 4T v1 D s1 66

elselet d % 2 s7 v7 sy vy D (

2 4v72 D vy

26 4s72 vy

2 D sy2 v7

26 i'


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if d L 0 t&e'let a % v7

2 D vy2 i'

let% % s7 v7 D sy vy i'+let t(in % 4%let t(out % 4% Dif v1 G 0 t&e'

d6Qa i'+d6Qa i'

|s1 | E B a'd t(out L 0 a'd t(in E T

elselet tBin % 4si*'4v16 B s1 6Qv1i' let tBout % 4si*'4v1 6 B s1 6Qv1i' let tin % ma4t(in, tBin 6 i'let tout % min4t(out, tBout 6 i'

tin E tout a'd tout L 0 a'd tin E Te'dif

else falsee'dif

e'dif

Proposition 8 The conflict detection algorithm cd)d is correct andcomplete*

The folloing theorem is a corollary of Propositions @and 8*

Theorem 7 The conflict detection algorithm for multiple aircraft cdM)d, from (efinition !!, is correctand complete*

I?* onflict )esolution

This section presents our mathematical frameor for the specification of conflict resolution algorithms

and their safety properties* &ased on this frameor, e propose an original vertical-only resolution algo-

rithm for multiple aircraft*

I?*A* Pairise onflict )esolution

A 4pairise6 conflict resolution algorithm taes as parameters the currentstate of t,oaircraft, the onship

and the traffic aircraft, and returns a set of possi%le resolution maneuvers for the onship* Benceforth, e

assume that conflictresolution algorithms are only applied to conflicting aircraft, i*e*, aircraftA and & that

are in conflictithin a looahead time T: KT 4A, &6 E !,%ut not in violation: H4A, &6 !*ompleteness of a conflictresolution algorithm refers to the eistence of at least one resolution maneuver

for every pair of conflicting aircraft*

(efinition !2 4onflict )esolution ompleteness6 A conflict resolution algorithm, namely cr, is com-plete if for all conflicting aircraft A and &, cr4A, &6 is non-empty*

e define several notions of conflict resolution correctness depending on the num%er of aircraft thatmaneuver to resolve a predicted conflict* or one aircraft, the correctness property is called independence,

for t,oor more aircraftis called coordination*

I?*&* onflict )esolution for One Aircraft

A %asic assumption in a distri%uted conflict resolution approach is that only one aircraft, e*g*, the onship,maneuvers* In this case, the traffic aircraft is assumed to eep its currenttra;ectory*

(efinition !3 4Independent onflict )esolution6 A conflict resolution algorithm, namely cr, is inde-pendent if and only if all resolutions computed %y the algorithm are independent for the onship, i*e*,

for all conflicting aircraft A and &:

RA cr4A, &6 implies KT 4RA, &6 !*

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'ote that a conflict resolution algorithm that is independent guarantees a minimum safe separation

distance %eteen the onship and traffic aircraft hen only the onship maneuvers*

(efinition !> 4Tangential )esolution6 A conflict resolution algorithm cris tangential if all resolutionscomputed %y the algorithm are tangential, i*e*, for all conflicting aircraft A and &:

RA cr4A, &6 implies KT 4RA, &6 G!*

The folloingproperty is a corollary of (efinitions !3 and !>*

Proposition " All tangential algorithms are independent*

I?** onflict )esolution for To Aircraft

In contrast to independent conflict resolution algorithms, here the resolution effort relies on the onship,

a coordinated algorithm assumes that %oth the onship and traffic aircraftpotentially maneuver to avoid

an impending conflict*

(efinition !@ 4oordinated onflict )esolution6 A conflict resolution algorithm cr is coordinated ifall resolutions computed %y the algorithm are pairise coordinated, i*e*, for all conflicting aircraft A and

&:

RA cr4A, &6 and R& cr4&, A6 implies KT 4RA,R&6!*

(efinition !8 4/inimal )esolution6 A conflict resolution algorithm cris minimal if all resolutions com-puted %y the algorithm are pairise minimal, i*e*, for all conflicting aircraft A and &:

RA cr4A, &6 and R& cr4&, A6 implies KT 4RA,R&6 G!*

The folloingproperty is a corollary of (efinitions !@ and !8*

Proposition 9 All minimal algorithms arecoordinated*

The concepts of independent and coordinated conflict resolution are orthogonal* $ection ? presentseamples of resolution algorithms that are independent ithout %eing coordinated, coordinated ithout

%eing independent, and coordinated and independent simultaneously* urthermore, any independent

conflict resolution algorithm can %e transformed into a coordinated algorithm %y using a priority relation

%et,een aircraft* If the priority relation is non to all aircraft, coordination can %e achieved ithout a

central authority* The priority relation could %e as simple as an order over aircraft identifiers* It does not

need to %e constant under time* Boever, at a given time all aircraft have to use the same priority

relationship*ormally, apriority relation is a strict order over aircraftstates* e rite A , & to denote that aircraft

A has priority over aircraft & at currenttime*

(efinition !7 et cr%e a conflict resolution algorithm, the resolution algorithm cr, is defined as follos*

cr, (A&!" set %if A , & t&e'{A}

elsecr(A&!

e'dif

Proposition !0 If the pairise conflict resolution cr is independent, then cr, is coordinated*

(efinition !" 4)epulsive )esolution6 A conflict resolution algorithm cr is repulsive if all resolutionscomputed %y the algorithm are pairise repulsive, i*e*, for all conflicting aircraft A and &:

RA cr4A, &6 and R& cr4&, A6 implies KT 4RA,R&6 L KT 4A,&6*


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In contrast to coordinated resolutions, repulsive resolutions separate%ut not necessarily solve a conflict*

Boever, the iteration of a repulsive algorithm is epectedto eventually solve the conflict*

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(efinition !9 et cr %e a conflict resolution algorithm, e define the algorithm crn recursively on n 0:

crn(A&!" set %if n G 0 t&e'{A}

elseR "% -

f$reac& RA cr4A, &6 d$f$reac& R& cr4&, A6d$

R "% R . crn!(RAR&!d$'e

d$'eR

e'dif

(espite the epectation, the iteration of a repulsive conflict resolution algorithm does not alays yield

a coordinated algorithm* This is not even the case if the iteration is performed infinitely many times*

e provide sufficient conditions for finite and infinite convergence of repulsive algorithms into coordinated

algorithms*

Proposition !! 4(efinite onvergence6 et cr%e a repulsive conflict resolution algorithm and %e astrictly positive constant, i*e*, L 0* If for any pair of conflicting aircraft A and &, and resolution

maneuversRA cr4A, &6 and R& cr4&, A6, it holds that

KT 4RA,R&6 KT 4A, &6

,

then there eists a natural num%er ' such that cr' is a coordinated conflict resolution algorithm*

Proposition !2 4


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e note thatpairise resolution algorithms are a special case of resolution algorithms for multiple aircraft

here the set of aircraft is a singleton* Indeed, the pairise coordinated algorithms are the coordinated

algorithms for 2 aircraft*

As in the case of pairise conflict resolution, resolution algorithms for multiple aircraft can %e iterated*

(efinition 23 et cMr%e a resolution algorithm for multiple aircraft, e define the algorithm cMrn

%y

induc- tion on n 0 as follos*

cMr

n

(A{A0, * * * , Am}!" set% let N % {A0, * * * , Am}i' ifn G 0 t&e'{A}

elseR "% -f$reac& RA cMr4A, N6 d$

f$reac& RA0 cMr4A0, N A0 . {A}6 * * * RAm cMr4Am, N Am . {A}6 d$R "% R .

d$'ed$'eR

e'dif

cMrn!

(RA{RA , * * * , RA }!

The rest of this section gives a sufficient condition that guarantees that the iteration of a

resolution algorithm for multiple aircraft is coordinated*

(efinition 2> 4,-$egment6 et , %e a priority relation that is sta%le %y velocity changes* The ,-segments of a set of aircraft states N are the donard close su%sets of N* That is, S is a ,-segment of Nif and only if

!* S 0 N, and

2* for all A N and & S, if A , & then A S*

The cardinality of the greatest solved ,-segment of N is denotedONP*

(efinition 2@ 4Progressive )esolution6 A resolution algorithm for multiple aircraft cMr is progressiveif it increases the si1e of solved ,-segments, i*e*, for all N and al l sets of the form {RA0 , * * * , RAm},,hereRAi cMr4Ai, N {Ai}6, it holds that ONP E O{RA0, * * * ,RAm}P*

Proposition !3 If the conflict resolution algorithm for multiple aircraft cMr is progressive, then cMr'

is coordinated for ' aircraft*

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A tangential independentvertical-only resolution maneuver RO for the onship satisfies

KT 4RO, I6 G !,

vRO7 G vO7, and

vROy G vOy *

Therefore,

RO4t6 G 4sO7 D tvO7, sOy D tvOy, sRO 1D tvRO 16 for all t* 4!06

rom ormulas 496, H4RO4J 6, I4J 66 G !, here J is the timeof closestseparation for aircraft states

ROand I* ets G sO sI , v G vO vI , v

1 G vRO vI ,

H4RO4J 6, I4J 66 G

!

ksI4J6 s k G !, from ormula 4@63

s2 2

ma474J 6 D sy 4J6

(

,| s1 D J v1 |

6 G !, from ormulas 4>6 and 4!06B

ma4

p4s7 D J v 6

2 D 4sy D J

vy 62,(

+4s

DJ v

62D4s

yDJ

vy

62

| s1 D J v0 |

6 G !*B

| s1DJ v0

Thus, %oth epressions ( and B are less than or eual to ! and one of them is

eual to !* $tudying the variation of these epressions in the neigh%orhood of J e can prove that actually%oth epressions are eual to !* Therefore, the solutions touch the top or the %ottom circle of theprotected

1one, i*e*,

p4s7 D Jv76

2 D 4sy D Jvy 62

(|s1 D J v

1 |

G !, and 4!!6

G !* 4!26

Bormula 4!!6 is a uadratic euation ith t,o possi%le solutions for J:

J2

Gs7 v7 sy vy 2 discr4s7 , sy , v , vy 6

,

here

7 D vy

discr4s7, sy , v7, vy 6 G (2

4v2

D v2

6 4s7vy sy v762*7 y

$ince the aircraft are in conflict %ut not in violation, there is a time t here the aircraft lose hori1ontal

separation, i*e*,

p4s7 D tv76

2 D 4sy D tvy 6

2

(

E !*

Thus, discr4s7, sy , v7, vy 6 L 0* If the relative ground speed of the aircraft is non-1ero, i*e*, v2 D v2 G 0,7 y

the t,o solutions for J eists and J E JD* onsider the solution for J that touches the top circle ofthe

protected 1one* igures ! and 2 illustrate the cases s1 B and s1 E B , respectively* The time J thatyields a vertical solution is given %y

4 JD if s B, JG

Jotherise*

hen the relative ground speed of the aircraft is non-1ero, ormula 4!26 yields the folloing vertical

resolution maneuver that alays touches the top circle of the protected 1one:

s

RO4J6

1

1

1|

1

p

1


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B s1 22vRO1 G vI1 D if v7 D vy G 0* 4!36J

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igure !* s1 B

igure 2* s1 E B

In the case here the relative ground speed is 1ero, a possi%le vertical solution maneuver is:

2 2vRO1

G vI1

if v7

D vy

G 0* 4!>6

The vertical speed-only resolution algorithm k3)d is defined %y ormulas 4!36 and 4!>6 as follos*

k3)d(OI!" set %let s % sO sI i'let v % vO vI i'let vRO1 %

if v72 D vy

2 G 0 t&e'vI1

elsevI1 D 4B s1 6QJ

e'dif i'{4v7, vy , vRO1 6}

Proposition !> The pairise algorithm k3)d is a tangential independent vertical speed-only resolutional- gorithm*

inally, e define the resolution algorithm for multiple aircraft k3M)d as follos*

k3M)d(ON!" set %let S % {A | A , O a'd KT 4O, A6 E !}i' if S G t&e'{O}

elselet I S i'

let O1 % k3)d(OI! i'k3M)d(O1 N {I}!

e'dif

The algorithm k3M)d is recursive* Boever, it is ell-defined since the cardinality of the set of aircraftstrictly decreases at each iteration* /oreover, the algorithm is non-deterministic as an ar%itrary aircraft I is

chosen at each iteration* Once this conflict is solved, solving a conflict ith another aircraft ill only

increase the vertical speed further* Therefore, hen the recursion terminates the onship has solved all

conflicts ith the aircraft%elo it* rom this, it can %e proved that k3M)d isprogressive*

Proposition !@

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!* k3M)d is independent for the onship and the set of aircraft that are %elo it*

2* k3M)d is progressive*

The folloing theorem is a corollary of Propositions !3 and !@*

Theorem !8 The conflict resolution algorithm for multiple aircraft

k3M)dnated for ' aircraft*

?* onclusion

, from (efinition 23, is coordi-

e have presented a frameor, i*e*, a setof mathematical definitions, that ena%les the characteri1ation

of state-%ased conflict detection and resolution algorithms for different assumptions on the num%er of

aircraft that maneuver to resolve a conflict* urthermore, e identify sufficient conditions that

guarantee that these algorithms satisfy properties such as correctness and completeness* Theframeor is illustrated ith eamples of original conflict detection and conflict resolution algorithms for

multiple aircraft that are completeand correct*

The characteri1ation of (5) algorithms that e propose in this or, complements less rigorous, %ut

more general, characteri1ations as the one proposed %y =uchar and ang*2, !7 or instance, Ta%le !classifies the folloing pairise state-%ased conflict resolution algorithms according to the properties they

satisfy*

?P: $elf-organi1ational voltage potential algorithm*!0

/?P: /odified voltage potential algorithm*7

+O: +eometricoptimi1ation algorithm, Unon-cooperativeV setting*"

+O-coop: +eometric optimi1ation algorithm, UcooperativeV setting*"

=&3(: Original version*!!

=&3(-coord: oordinated version*!3

Property ?P /?P +O +O-coop =&3( =&3(-coord

ompleteness+ +

Independence 4+

4+ +

Tangentiality 4+

4+ +

oordination 4 4+

4+

/inimality 4 4+

4 4

)epulsion+ +

4+

4+

Ta%le !* Properties of (5) algorithms

The chec sym%ol 4+

6 represents properties that are claimed to %e satisfied %y the developers of the

algorithms, hile the cross sym%ol 446 represents properties that are non not to hold* &lansrepresentproperties hose satisfia%ility is unnon to the authors of this paper*

inally, e stress that the mathematical developmentpresented in this paper has %een formally specified

and verified in a mechanical theorem prover* +iven the critical nature of (5) systems,e %elieve that the

use of formal methods is an essential step in the analysis, design, and implementation of the ne generation

of distri%uted air traffic managementconcepts*

)eferences

! /onin, W*-*, Cnderstanding ormal /ethods , AIT 4ormal Approaches to omputing and Information

Technology6, $pringer-?erlag, ondon, C=, 2003*

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_ Vasii_Claudiu_Conflict Detection and Resolution for 1 (1)

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