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1American Institute of Aeronautics and Astronautics

Aircraft Conflict Resolution with an Arrival Time Constraint

Karl D. Bilimoria� and Hilda Q. Lee✝

NASA Ames Research Center, Moffett Field, CA 94035

Abstract

A conflict resolution maneuver consists of twophases: avoidance and recovery. The goal of theavoidance maneuver is to increase the missdistance until it meets the separation standard,while the recovery maneuver seeks to merge withthe nominal trajectory that was interrupted by theavoidance maneuver. Determination of therecovery maneuver can be non-trivial if there is anarrival time constraint at a downstream waypoint,because the conflict avoidance maneuvergenerally introduces a time delay. An analyticaltreatment of the integrated conflict resolution(avoidance plus recovery) problem has beendeveloped, building on a Geometric Optimizationapproach for solving the conflict avoidanceproblem. A family of solutions for conflictresolution with an arrival time constraint has beengenerated and presented. Of particular interest isa special conflict avoidance solution, called theDelay Compensated solution. This solution avoidsthe conflict by blending a heading change with aspeed change in such a way that the delayintroduced by the path stretching is exactlycancelled out by the increased speed; hence thecorresponding recovery maneuver is executed atthe nominal speed. Unlike the standard speedand heading change avoidance solutions, whichare often limited by aircraft performance, the DelayCompensated solutions remain feasible even forvery demanding constraints on arrival time.

� Research Scientist, Automation Concepts Research Branch;Mail Stop 210-10; E-mail: [email protected] Fellow, AIAA.

✝ Programmer/Analyst, Raytheon ITSS.

Introduction

There are generally two phases in conflictresolution: avoidance and recovery (see Fig. 1).The goal of the avoidance maneuver is to increasethe miss distance until it meets the separationstandard. Many methods for conflict avoidanceare available (Refs. 1 – 6); Ref. 1 describes aGeometric Optimization approach to this process.

The recovery maneuver is initiated after theavoidance maneuver has been completed. Thegoal of the recovery maneuver is to merge with thenominal trajectory that was interrupted by theavoidance maneuver. This typically requiresrejoining the nominal route at the next waypoint.

In cases where there is no Required Time ofArrival (RTA) constraint at the next waypoint,recovery simply requires adjusting the headingangle (to set a course for the waypoint) and/orresuming the nominal speed, after the avoidancemaneuver has been completed. Determination ofthe recovery maneuver can be non-trivial if there isan RTA constraint at the next waypoint. A conflictavoidance maneuver often utilizes path stretching(via heading changes), which effectivelyintroduces a time delay; if a speed change isutilized (alone or in combination with a headingchange), it can introduce a positive or negativedelay.

One possible approach to conflict resolution is todecouple the avoidance and recovery solutions.After the avoidance maneuver has beencompleted, the recovery maneuver can bedetermined by adjusting the heading angle toreturn to the appropriate waypoint, while alsoadjusting the speed to compensate for the net timedelay introduced by the avoidance maneuver.However, the recovery speed required to meet theRTA constraint may lie outside the performancelimits of the aircraft; for example, transport aircrafthave a very limited speed range at cruising

AIAA Guidance, Navigation, and Control Conference and Exhibit5-8 August 2002, Monterey, California

AIAA 2002-4444

Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code.The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes.All other rights are reserved by the copyright owner.


altitudes. In such cases, since the avoidancemaneuver has already been executed, the aircraftwould have no other options to satisfy the RTAconstraint, and may lose its time slot.

Hence it is generally desirable to determine therecovery maneuver before executing theavoidance maneuver. This permits an evaluationof the operational feasibility of the entire conflictresolution (avoidance plus recovery) maneuverbefore actually executing it. If a recovery problemis identified, the avoidance maneuver can be re-designed so that the corresponding recoverymaneuver is feasible.

An integrated approach to conflict resolution maybe desirable even when there is no RTAconstraint, and is generally necessary when anRTA constraint is active. This is particularlyrelevant in a Free Flight operational concept, suchas the Distributed Air/Ground Traffic Management(DAG-TM) concept of operations,7 when an aircraftis responsible for separating itself from otheraircraft while also conforming to its own RTAconstraint. A numerical solution to this problem isdescribed in Ref. 6. An analytical treatment of theintegrated conflict resolution (avoidance plusrecovery) problem with an RTA constraint ispresented in this paper, building on the analyticalsolution of the conflict avoidance problem reportedin Ref. 1.

Integrated Conflict Resolution

The objective of this analysis is to determine avelocity-vector based solution for horizontal-planeconflict resolution that maintains separation andalso satisfies an RTA constraint at the nextwaypoint downstream of the conflict location.Such a constraint may arise in DAG-TMoperations, for example, from an en route flow-rateconstraint to be satisfied during cruise. A constantwind field is assumed in this initial analysis. Thiswork builds on the baseline GeometricOptimization approach described in Ref. 1.

In this paper, the vector V denotes ground-relative velocity, while the scalar V denotes groundspeed; χ denotes heading (course) anglemeasured clockwise from North. For these twovariables, the absence of a superscript denotesthe Ownship, while the superscript intr denotes theIntruder; the superscript rel is associated with therelative velocity vector V V Vrel = −( )int . Thesubscripts NOM , CA, and R E C denote Nominal(initial), Conflict Avoidance, and Recovery values,respectively. Fig. 2 below defines variousparameters used in the analysis. rLOS and χ LOS

are the length and azimuth of the Line Of Sightvector, respectively. dmin is the minimumhorizontal separation resulting from the nominalvelocities, while Dsep is the horizontal separation

standard (currently 5 nm in en route airspace).

Commands for heading and speed are initiallycomputed assuming instantaneous responses;the effects of turn and acceleration dynamics arethen compensated by appropriate corrections tothe heading and speed commands (details arepresented in Appendix A).

The following subsections present the analysis for

WPT

ConflictAvoidance

Recovery

Aircraft A(Ownship)

Aircraft B(Intruder)

Fig. 1: Conflict Resolution Trajectory

Intruder

Ownship

χ LOS Dsep

dmin

rLOS

χNOMrel

χCArel

−V int

V int

Desired Direction ofVCArel

V V rel

Fig. 2: Conflict Geometry and Parameters


determining a set of solutions for conflict resolutionwith an RTA constraint. A family of candidateavoidance solutions is first computed, and thecorresponding family of recovery solutions(satisfying the RTA constraint) is then determined.If a candidate set of avoidance and recoverysolutions is not feasible or desirable, alternativesets of solutions can be evaluated until allrequirements are satisfied.

Conflict AvoidanceA detailed analysis of conflict avoidance ispresented in Ref. 1. Some key results are givenbelow for completeness and convenience.

The azimuth of the nominal relative velocity vectoris given by Eq. (10) of Ref. 1, presented below in aslightly modified format.

χχ χ

χ χNOMrel NOM NOM NOM

iNOMi

NOM NOM NOMi

NOMi

V V

V V=

−

−

−tan

sin sin

cos cos1

ntr ntr

ntr ntr(1)

The azimuth of the relative velocity vector desiredfor conflict avoidance is given by Eq. (13) of Ref.1, presented below in a slightly modified format.

χ χCArel

LOSsep

LOS

D

r= ±

−sin 1 (2)

In this work, the conflict avoidance maneuvers areassumed to be cooperative; this is consistent withFree Maneuvering for DAG-TM operations. Henceboth the Ownship and Intruder contribute equallyto the conflict avoidance process, each rotatingthe relative velocity vector by an angle∆χ χ χCA

relCArel

NOMrel= −0 5. ( ) .

The fundamental requirement for horizontalseparation assurance is given by Eq. (17) in Ref.1, presented below in a slightly modified format.

V

V

CA NOMrel

CArel

CA

NOM NOMrel

CArel

NOM

sin ( )

sin ( )

χ χ χ

χ χ χ

+ −[ ]= + −[ ]

∆

∆intr intr(3)

In Eq. (3) above, VCA and χCA are the Ownship’sconflict avoidance solutions for speed andheading, respectively.

Horizontal plane conflict avoidance can beconducted by appropriate combinations of headingand speed change maneuvers. Eq. (3) defines a

family of Ownship solutions for conflict avoidancein the horizontal plane; each member of the familycorresponds to a particular combination of speedchange and heading change. This family has twobranches, corresponding to the frontside (passahead) and backside (pass behind) conflictavoidance solutions that arise from the twopossible solutions of χCA

rel in Eq. (2). For example,Fig. 2 shows the frontside avoidance solution.

RecoveryFig. 3 shows the conflict avoidance and recoverysegments of a conflict resolution trajectory. Therecovery maneuver begins when a conflict-freetrajectory back to the waypoint becomes available.An ( , )x y coordinate system is defined with thex-axis pointing North, and the y-axis pointing East.The heading angle for the recovery maneuver,χ REC , is simply the value that takes the aircraftback to the waypoint. Hence

χ RECWPT REC

WPT REC

y y

x x=

−

−

−tan 1 (4)

where ( , )x yWPT WPT are the position coordinates ofthe waypoint. ( , )x yREC REC are the positioncoordinates of the point at which the recoverymaneuver is initiated (end of conflict avoidance),given by

x x V tREC CA CA CA= +0 cos χ (5a)

y y V tREC CA CA CA= +0 sin χ (5b)

where ( , )x y0 0 are the position coordinates of theaircraft at the initiation of the conflict avoidancemaneuver, and tCA is the conflict avoidance time.

Noting that both Ownship and Intruder cooperateto avoid the conflict, the relative speed duringconflict avoidance, VCA

relCA CA= −V V intr , is given by

VV V

V VCArel CA CA

CA CA CA CA

=+

− −

( ) ( )

cos( )

2 2

2

intr

intr intrχ χ(6)

From the geometry of Fig. 2, it can be determinedthat the conflict avoidance time is given by

tr D d

VCA

LOS sep CAbuffer

CArel=

− +2 2

(7)


where d D rCAbuffer

sep LOS~ << is the additional (beyond

the minimum separation point) distance that mustbe traveled along the conflict avoidance path(before initiating recovery) in order to avoidpenetrating the protected zone (as a firstapproximation, d DCA

buffersep= is used in this work).

From the geometry of Fig. 3, using the cosine rule,it is clear that the recovery distance is:

l l l l lREC CA WPT CA WPT CA= + −2 2 2 cos∆χ (8)

where lWPT is the initial (start of conflict avoidance)distance of the Ownship from the waypoint atwhich the RTA is to be enforced, lCA is thedistance traveled during conflict avoidance, and∆χ χ χCA CA NOM= −( ) .

The RTA constraint can be expressed as

( )t t tCA REC RTA+ = (9)

where tRTA is the required time of arrival, relative tothe initial time. In some cases, tRTA is the timeassigned by the air traffic service provider (e.g.,sector controller). Otherwise, this is the time atwhich the aircraft would have arrived at thewaypoint if it had continued flying along its nominaltrajectory; in this case

t

VRTAWPT

NOM

=l

(10)

The recovery speed can then be calculated frombasic kinematics and Eq. (9) as

V

t t tRECREC

REC

REC

RTA CA

= =−

l l

( )(11)

Substituting l CA CA CAV t= into Eq. (8), and thensubstituting the result into Eq. (11) above yields

V

V t V t

t tRECCA CA WPT CA CA WPT CA

RTA CA

=+ −

−

( ) cos

( )

2 2 2l l ∆χ

(12)

If tRTA is not directly specified, it is determined fromEq. (10).

It is noted that the conflict avoidance solution( , )χCA CAV appears explicitly in Eq. (12) and alsoimplicitly through the term tCA which depends onVCA

rel . Furthermore, VCArel depends on the Intruder’s

conflict avoidance solution ( , )χCA CAVintr intr . Hence theOwnship’s recovery solution depends on theconflict avoidance solutions of both the Ownshipand the Intruder.

In order to facilitate exposition of results in thisstudy, it is assumed that the Intruder conductsconflict avoidance by executing a maneuvercomplementary to that of the Ownship. Henceboth aircraft cooperatively execute the same typeof avoidance maneuver (e.g., heading change orspeed change), resulting in V VCA

relNOMrel= .

ExampleAs an illustrative example, consider the two-aircraft conflict scenario shown in Fig. 1,characterized by an encounter angle of 90 degand zero miss distance (collision). Both aircraftare cruising (at the same altitude) at a speed of450 kts, and will lose separation (violate the 5 nmseparation standard) in 5 minutes. It is recalledthat both aircraft contribute equally to conflictavoidance in this study. An RTA constraint isimposed at a waypoint 100 nm downstream of theinitial position of Aircraft A. Hence, Aircraft A mustdo its part to avoid the conflict with Aircraft B, andis also required to arrive at the waypoint at thesame time that it would have arrived had itcontinued along its nominal trajectory. In thisstudy, Aircraft A is the Ownship, and Aircraft B isthe Intruder.

The complete solution set for conflict resolutionwith an RTA constraint is given by ( χCA , VCA , χREC ,VREC ). It is noted that the recovery componentshave a strong dependence on the avoidancecomponents. The family of integrated conflict

Intruder

Ownship WPT lWPT

lCA lREC

χNOM

χCA

χ REC

Fig. 3: Conflict Resolution Parameters


resolution solutions can be conceptualized as apair of 1-dimensional manifolds (corresponding tofrontside and backside avoidance) in4-dimensional state space. These solutions canbe computed using Eqs. (3), (4), and (12).

The recovery speed VREC can be significantlylimited by aircraft performance, while this isgenerally not the case for the recovery headingχREC . Therefore the rest of this work focuses onthe subset of the integrated conflict resolutionsolution given by ( χCA , VCA , VREC ). Thecorresponding family of integrated conflictresolution sub-solutions can be visualized as twocurves (corresponding to frontside and backsideavoidance) in 3-dimensional state space. Figures4a and 4b are derived from projections of thesecurves on the ( χCA , VCA ) plane and the ( VCA , VREC )plane respectively. All data shown in Fig. 4include corrections for speed and headingdynamics, implementing the approach presentedin Appendix A of this paper. It is noted thatsolutions requiring large changes in speed maynot be flyable due to aircraft performance(stall/buffet or thrust) limits. Typical speedbounds, arising from stall/buffet and thrust limits,8

permit an increase of about 25 knots and adecrease of about 50 knots, relative to the nominalspeed of 450 knots.

Fig. 4a depicts the two sets of conflict avoidancesolutions (expressed as changes relative to thenominal speed and heading) for the Ownship;operating at any point on either of the two curveswill satisfy its requirement to provide half of thetotal conflict avoidance effort. Assuming that theIntruder provides the other half, the resultingminimum horizontal separation will be 5 nm.

Some points of special interest are indicated onFig. 4a. For example, the two Optimal Avoidancesolutions are indicated; these solutions attempt tominimize deviations from the nominal trajectoryduring conflict avoidance, by minimizing themagnitude of the vector ∆V V VCA CA NOM= −( ). Itshould be noted that the Optimal Avoidancesolutions provide efficient conflict resolution forcases without any RTA constraint; no optimalityclaims are made about these solutions when anRTA constraint is active. Also indicated on Fig. 4aare two Heading Change solutions thatcorrespond to conflict avoidance with zero speedchange, and two Speed Change solutions that

correspond to conflict avoidance with zero headingchange. It is noted that closed-form analyticalsolutions for three special conflict avoidance casesare available in Ref. 1.

Fig. 4b depicts conflict resolution solutions in the(VCA , VREC ) plane. Ownship recovery speeds andavoidance speeds, expressed as changes relativeto the nominal speed, are shown for the two setsof conflict avoidance solutions. Special solutionpoints of interest are indicated on Fig. 4b. For

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–15 –10 –5 0 5 10 15Incremental Avoidance Heading (degrees)

FRONTSIDESOLUTIONS

BACKSIDESOLUTIONS

SpeedChangeSolution

HeadingChangeSolution

SpeedChangeSolution

HeadingChangeSolution

OptimalAvoidance

Solution

OptimalAvoidance

Solution

Fig. 4a: Conflict Avoidance Solutions

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Incremental Avoidance Speed (knots)

SpeedChangeSolution

Heading Change Solutions

SpeedChangeSolution

FRONTSIDESOLUTIONS

BACKSIDESOLUTIONS

OptimalAvoidance

Solution

OptimalAvoidanceSolution

Fig. 4b: Avoidance and Recovery Speeds


example, it can be seen that the backside optimalavoidance solution requires a speed change ofabout –22 kts (relative to the nominal speed of 450kts); it is noted that Fig. 4a indicates a headingchange of about +2.5 deg is also required forconflict avoidance. Fig. 4b indicates that in orderto meet the waypoint RTA, the correspondingrecovery maneuver must be executed with aspeed change of about +22 kts (relative to thenominal speed of 450 kts). The increase inrecovery speed compensates for the pathstretching and lower speed during the conflictavoidance maneuver. Another interesting specialcase (not indicated on Fig. 4) is the DelayCompensated avoidance solution described in thefollowing section.

Delay Compensated Avoidance

The Delay Compensated (DC) avoidance solutionis a special combination of heading and speed,such that the delay introduced by path stretchingduring conflict avoidance is exactly compensatedby an appropriate increase in avoidance speed.Consequently, the corresponding recovery speedis equal to the nominal speed. Mathematically,

( , ) arg( , )

χχ

DC DCV

REC NOMV V VCA CA

= ={ } (13)

where VREC is given by Eq. (12), and ( , )χCA CAV

satisfy the conflict avoidance constraint of Eq. (3).It is recalled that the conflict avoidance solution( , )χCA CAV appears explicitly in Eq. (12), and alsoappears implicitly through the term tCA whichdepends on VCA

rel .

ExampleContinuing the example scenario presented in theprevious section, Figure 5 shows a close-up viewof the conflict avoidance and recovery solutionspresented in Fig. 4b. Indicated on this graph aretwo (frontside and backside) solution points ofspecial interest. Each corresponds to a recoverysolution with zero speed change relative to thenominal speed, i.e., the recovery speed is thesame as the nominal speed (which is presumablya user-preferred speed). For the backsidesolution, it can be seen from Fig. 5 that thisrequires a special conflict avoidance solution witha speed change of about +4 kts, relative to thenominal speed (and a corresponding heading

change of about +5.5 deg, as observed from Fig.4a). This is a delay compensated conflictavoidance solution for which the waypoint arrivaltime delay due to path stretching for conflictavoidance is exactly compensated by theincreased conflict avoidance speed. A recoverymaneuver at the nominal speed of 450 kts willmeet the waypoint RTA (same arrival time asnominal trajectory). A frontside delaycompensated avoidance solution is also available(see Fig. 5).

Due to the trigonometric complexity of the conflictavoidance and recovery equations, a closed-formanalytical expression for the delay compensatedconflict avoidance solution is not available.However, it can be obtained by simultaneousnumerical solution of Eqs. (3) and (12). A simpleiterative algorithm is given below:

1. Pick a candidate value of χCA by incrementingthe current iteration value by a small value∆χCA

* (e.g., ±0.01 deg). Use ( )*χ χNOM CA+ ∆ asthe initial candidate.

2. Compute associated value of VCA from Eq. (3).

3. Compute resulting value of VREC from Eq. (12),using tCA calculated from Eq. (7).

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2

4

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d (k

nots

)

–6 –4 –2 0 2 4 6Incremental Avoidance Speed (knots)

Delay CompensatedSolutions

FRONTSIDESOLUTIONSBACKSIDE

SOLUTIONS

Fig. 5: Delay Compensated Avoidance


4. If V VREC NOM− < ε , where ε is some small

number (e.g., 0.1 kt), then the candidateavoidance solution is the delay compensatedsolution; else, go back to Step 1 to update thecandidate value and continue until successful.

More sophisticated algorithms, such as gradient-based or grid-refinement searches, could beemployed to determine the recovery speed in amore efficient way.

Parametric Study of RTA Conformance

A parametric study of integrated conflict resolution(avoidance and recovery) solutions was conductedby varying the value of RTA for various conflictscenarios. The location of the waypoint wasvaried (resulting in various values of RTA) togenerate families of solutions. The fundamentalparameter used in this study is defined as

τ =t

tFLS

RTA

(14)

where tFLS is the time to first loss of separation,relative to the initial time. For the integratedconflict resolution problem to be meaningfullydefined, 0 1< <τ . For example, τ = 0.4 indicatesthat the conflict time is 40% of the required arrivaltime.

From the example results presented earlier it isclear that, for a given waypoint and associatedRTA, the recovery speed is strongly dependent onthe avoidance maneuver ( χCA , VCA ). Fromintuition it is apparent that, for a given avoidancemaneuver, the recovery speed is also dependenton the time remaining for RTA conformance; it ishypothesized that this dependence is weak if τ issmall, and grows stronger as τ approaches unity.

The objective of the parametric study is todetermine the sensitivity of the recovery speeds tothe parameter τ, for various avoidance solutions(Heading Change, Speed Change, OptimalChange). Another objective is to determine thesensitivity of the Delay Compensated avoidancespeed to the parameter τ.

Conflict scenarios were designed with equalOwnship/Intruder speeds and a miss distance ofzero, for crossing angles of 30, 90, and 150 deg.

Integrated conflict resolution solutions werecomputed for various values of waypoint distance( lWPT ), holding all other parameters fixed. Theresults for Ownship maneuvers are presented inFigs. 6 – 8. It is recalled that the Intruder (whichhas no RTA constraint) cooperatively participatesin conflict avoidance.

For an avoidance or recovery maneuver, let ∆Vrepresent the required speed change assuminginstantaneous dynamics, and let ∆Vdyn represent

∆V compensated for the dynamic effects of finiteacceleration (see Appendix A). The maneuver isspeed limited if ∆V lies outside the incrementalspeed bounds [ , ]∆ ∆V Vmin max arising from aircraftperformance (stall/buffet and thrust) limits. Themaneuver is acceleration limited if ∆V is within[ , ]∆ ∆V Vmin max , but ∆Vdyn is outside [ , ]∆ ∆V Vmin max .

Feasibility of both avoidance and recoverymaneuvers was determined by a simple model ofaircraft performance that limits speed changes tovalues from –50 to +25 knots, and accelerations tovalues of ±0.4 knot/sec.

Fig. 6 presents Ownship results for conflicts with acrossing angle of 90 deg. Fig. 6a showsincremental recovery speeds (for feasibleavoidance solutions) for various values of τ ; it isnoted that the example discussed in the previoussections corresponds to τ = 0.375. The recoveryspeed change associated with the DelayCompensated avoidance solution is zero bydefinition, and is not shown in Fig. 6a. It can beseen from Fig. 6a that all recovery solutions areeventually limited (at various values of τ) by speedor acceleration capabilities of the aircraft.

Fig. 6b shows the Delay Compensated avoidancesolution as a function of τ ; other specialavoidance solutions, Speed Change and OptimalChange, are also shown for reference (HeadingChange is not shown because the associatedavoidance speed change is zero by definition).The avoidance speeds for Optimal Change andSpeed Change are invariant over τ because theydepend only on tFLS which is held constant (5 minfor the 90 deg crossing angle conflict). It is notedthat the variations in τ arise from variations in tRTA

for various values of l WPT . A key observationmade from Fig. 6b is that at least one DelayCompensated solution (frontside / backside) isfeasible for all values of τ examined.


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d (k

nots

)

0.1 0.2 0.3 0.4 0.5 0.6 0.7

τ = (tFLS / tRTA)

Optimal Change

Heading Change

Speed Change

Optimal Change

Heading Change

Speed LimitedAcceleration Limited

Acceleration Limited


Frontside SC Avoidance infeasibleRecovery solution undefined

Backside Avoidance Frontside Avoidance

Identical HC solutions forFrontside & Backside

Fig. 6a: Sensitivity of Recovery Speeds(Crossing Angle = 90 deg; Equal Speeds)

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Spe

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0.1 0.2 0.3 0.4 0.5 0.6 0.7

τ = (tFLS / tRTA)

Delay Compensated

Speed Change

Optimal Change

Delay Compensated

Speed Change

Optimal Change

Frontside SC Avoidance solution is Speed Limited


Speed Limited

Fig. 6b: Sensitivity of Avoidance Speeds(Crossing Angle = 90 deg; Equal Speeds)

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

τ = (tFLS / tRTA)

Optimal Change

Heading Change

Optimal Change

Heading Change

Speed Limited


Frontside Avoidance


Backside SC Avoidance infeasibleRecovery solution undefined



Backside Avoidance


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τ = (tFLS / tRTA)

Delay Compensated

Optimal Change

Delay Compensated

Optimal Change


B ackside SC Avoidance solution is Speed Limited




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Optimal Change

Heading Change

Speed Change

Optimal Change

Heading Change

Backside Avoidance

Speed LimitedAcceleration Limited


Frontside Avoidance


Frontside OPT Avoidance infeasibleRecovery solution undefined



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0.1 0.2 0.3 0.4 0.5 0.6τ = (tFLS / tRTA)

Delay Compensated

Optimal Change

Speed Change

Optimal Change



Backside DC Avoidance solution not found

Frontside OPT Avoidance solution is Speed Limited


Figures 7a,b show the sensitivity of recovery andavoidance speeds, respectively, for a 150-degcrossing angle conflict. Head-on type conflictsgenerally require small maneuvers for avoidanceand the recovery solutions corresponding toHeading Change and Optimal Change remainfeasible for larger values of τ (compared to the 90-deg encounter). However, Speed Changemaneuvers are not feasible because theiravoidance solutions are speed limited; this istypical for head-on type conflict geometries. TheDelay Compensated solution is feasible for allvalues of τ examined.

Figures 8a,b show the sensitivity of recovery andavoidance speeds, respectively, for a 30-degcrossing angle conflict. Shallow angle conflictsgenerally require large maneuvers for avoidance,and the recovery solutions corresponding toSpeed Change, Heading Change and OptimalChange become infeasible at smaller values of τ(compared to the 90-deg encounter). It is notedthat many frontside maneuvers are infeasible; thisis typical for shallow angle conflict geometries. ADelay Compensated solution is feasible for allvalues of τ examined.

Conclusions

An integrated approach to conflict resolution hasbeen presented, addressing the dependence ofthe recovery phase on the avoidance phase. Afamily of such solutions has been presented for anexample scenario, and some special avoidancesolutions have been highlighted. Of particularinterest is the Delay Compensated solution thatavoids the conflict by blending a heading changewith a speed change in such a way that the delayintroduced by the path stretching is exactlycancelled out by the increased speed.

A parametric study was conducted to examine theinfluence of required arrival time on conflictavoidance and recovery speeds. For the conflictgeometries examined, it was observed thatavoidance maneuvers (other than HeadingChange) are generally constrained by speedlimits, while recovery maneuvers can beconstrained by speed or acceleration limits.These maneuver constraints can be correlatedwith significant nonlinearities that appear for caseswhere the time to conflict is less than half therequired time to the waypoint (τ > 0.5). Recovery


solutions corresponding to Speed Change,Optimal Change, and Heading Change maneuversbecome infeasible at various values of τ,depending on encounter angle. It is noted that thedomain (in terms of τ) of feasibility for thesesolutions is severely limited for shallow angleconflicts. A preliminary study indicates that theDelay Compensated avoidance solution is feasibleover the domain of practical interest, and mayprovide the solution of choice for horizontal planeconflict resolution with an arrival time constraint.

References

1. Bilimoria, K.D., “A Geometric OptimizationApproach to Aircraft Conflict Resolution,”Paper No. 2000-4265, AIAA Guidance,Navigation, and Control Conference, August2000.

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3. Krozel, J. and Peters, M., “Conflict Detectionand Resolution for Free Flight”, Air TrafficControl Quarterly, Vol. 5, No. 3, 1997, pp.181– 212.

4. Kirk, D.B., Bowen, K.C., Heagy, W.S., Viets,K.J., “Problem Analysis, Resolution andRanking (PARR) Development andAssessment,” 4th USA/Europe Air TrafficManagement R&D Seminar, Santa Fe, NM,December 2001.

5. McNally, B.D., Bach, R., and Chan, W., “FieldTest of the Center-TRACON AutomationSystem Conflict Prediction and Trial PlanningTool,” Paper No. 98-4480, AIAA Guidance,Navigation, and Control Conference, August1998.

6. Mondoloni, S. and Conway, S., “An AirborneConflict Resolution Approach using a GeneticAlgorithm,” Paper No. 2001-4054, AIAAGuidance, Navigation, and ControlConference, August 2001.

7. Green, S.M., Bilimoria, K.D., and Ballin, M.G.,“Distributed Air/Ground Traffic Management

for En Route Flight Operations,” Air TrafficControl Quarterly, Vol. 9, No. 4, 2001, pp.259 – 285.

8. Mueller, K.T., Schleicher, D., and Bilimoria,K.D, “Conflict Detection and Resolution withTraffic Flow Constraints,” Paper No. 2002-4445, AIAA Guidance, Navigation, andControl Conference, August 2002.

Appendix A: Maneuver Dynamics

Let ∆V denote a maneuver solution over the timeinterval t tF∈ ( )0, , where tF is the total maneuver

time. Define

∆ ∆ ∆V z z VT T= [ ] = [ ]1 2 χ (A1)

Let CCCC represent a piece-wise constant velocityprofile corresponding to a maneuver solutiondetermined under the assumption ofinstantaneous changes in the velocity vector. LetLLLL represent a piece-wise linear velocity profilederived from CCCC, under the assumption of constantacceleration, such that the average velocitychange over the interval t tF∈ ( )0, is the same asthat for CCCC. These velocity profiles are illustrated inFig. A1.

It is noted that ∆V appears linearly in thekinematic equations for position changes, while∆χ appears nonlinearly (sin and cos terms);however, for small values of ∆χ (up to about 15deg) the relationship is almost linear. It is evidentfrom linearity that the aircraft will arrive at the

L

C

timetFtTR

ziavg

zicom

zi

0

Fig. A1: Velocity Component Profiles


same location at time tF for both velocity profiles ifthe areas under the profiles CCCC and LLLL are identical.Let zi

com indicate the commanded change (in avelocity component) that results in an averagechange zi

avg over the time interval t tF∈ ( )0, ,

subject to a maximum acceleration z i( )max. Then

z t z t z t tiavg

F icom

TR icom

F TR= + −12 ( ) (A2)

where tTR is the transient response time given by

tz

ztTR

icom

iF=

( )≤

˙max

(A3)

The maneuver is not feasible if t tTR F> .Substituting Eq. (A3) into Eq. (A2) yields

1

20

2

zz t z z t

iicom

F icom

iavg

F( ) ( ) − ( )+ [ ] =max

(A4)

Solving the quadratic equation,

z z t tz t

zicom

i F Fiavg

F

i

= ( ) ± −( )

˙

˙maxmax

2 2(A5)

Of the two solutions, only the “minus” solution willsatisfy the requirement that z zi

comiavg→ in the limit

as z i( ) → ∞max

. Hence

z z t tz t

zicom

i F Fiavg

F

i

= ( ) − −( )

˙

˙maxmax

2 2(A6)

It is clear from Eq. (A6) that in order to obtain avalid solution it is necessary that

z z tiavg

i F≤ ( )12 ˙

max(A7)

This condition ensures that t tTR F≤ .

Figs. A2 and A3 show solutions obtained from Eq.(A6) for an example case with tF = 5 min,

˙ ˙z V1( ) = =max max 0.4 knot/sec, and ˙ ˙z2( ) = =

max maxχ1 deg/sec. It can be seen that the effect of turndynamics is not very significant for the range ofheading changes typically required for conflictavoidance. However, the effects of speeddynamics are quite significant. In particular, it isnoted that Eq. (A7) requires that the desired

average speed change not exceed 60 knots forthe example case presented here.

Conflict avoidance speed (or heading) changesare computed relative to the nominal value of theconflict avoidance speed (or heading). However,recovery speed (or heading) changes should becomputed relative to the commanded value of theconflict avoidance speed (or heading).

0

20

40

60

80

100

120

Com

man

ded

Spee

d C

hang

e (k

nots

)

0 10 20 30 40 50 60

Average Speed Change (knots)

Maneuver Time = 5 min

Acceleration = 0.4 knot/sec

InstantaneousDynamics

Fig. A2: Commanded vs. Average Speeds

0

5

10

15

20

25

30

35

Com

man

ded

Hea

ding

Cha

nge

(deg

)

0 5 10 15 20 25 30

Average Heading Change (deg)

Turn Rate = 1 deg/sec

InstantaneousDynamics

Maneuver Time = 5 min

Fig. A3: Commanded vs. Average Headings

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