MIT-LIDS Analysis of intrersecting flows of agents Eric Feron & David Dugail Mini MURI 03/02/02.
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Transcript of MIT-LIDS Analysis of intrersecting flows of agents Eric Feron & David Dugail Mini MURI 03/02/02.
MIT-LIDS MIT-LIDS
Analysis of intrersecting flows of agents
Eric Feron & David Dugail
Mini MURI 03/02/02
MIT-LIDS MIT-LIDS
Research motivation & goals
• Analyses of conflict resolution usually involve pairs or a finite number of aircraft
• Need to address the fear of domino effect– one conflict resolution triggers a new conflict elsewhere possibly leading
to divergence in the system
• Analysis of aircraft flows– worst-case standpoint
• Stability & performance– simulations for insight on the system dynamics– analytical proofs
MIT-LIDS MIT-LIDS
Background• A very rich literature on management of conflicts involving 2,3 or
more but finite number of aircraft - Erzberger, Krozel, Kuchar, Niedringhaus, Sastry, Tomlin, Zeghal ….
• An equally rich literature on conflict management with aircraft flows - eg gas models - Bakker, Blom, Simpson… Open-loop probabilistic models.
• Very little available from current robotics literature (Recent research by Ruspini, Devasia, Meyer,…otherwise computational complexity results, eg Reif & Sharir)
• How does one prove stability, and bound required aircraft deviations, for conflict resolution over a class of closed-loop aircraft interactions in a deterministic setting?
MIT-LIDS MIT-LIDS
A "Control Volume" approach
•Motivation: Infinite # of aircraft flow in and out•Analysis of completely random aircraft flows is difficult•Need to structure the flows & flow behaviors
MIT-LIDS MIT-LIDS
-100-80-60-40-20020406080100
-100
-80
-60
-40
-20
0
20
40
60
80
100
x (nm )
y (n
m)
A control volume approach
• 2-D
• Structured converging flows– pre-determined points of entry– regular or random entry
• Aircraft make 1 maneuver when entering– maneuver is minimal – offset, heading change maneuvers
N
MIT-LIDS MIT-LIDS
Heading afterconflict
resolution
Originalheading
Position afterconflict resolution
Original position
W
w
Conflict area
Conflict area Conflict area
Heading Change vs. Offset Maneuver Models
MIT-LIDS MIT-LIDS
d
d tan
Offset maneuver (2/2) (2 flows, decentralized)
• 2 successive heading changes
• Proof by contradiction• Upper bound on lateral
displacement
dsep : min. separation dist.
N
sepdL 22
MIT-LIDS MIT-LIDS
Conflict geometry
Eastboundflow
Southboundflow
N
S
W E
z
z
T
T
(b, )
(a, )
b
a
O
R
+
Airw ays
Control volume
Entry points
Cone of decision
Decision bound
• 2-D
• Structured converging flows– pre-determined
points of entry– regular or random
entry
• Aircraft make 1 maneuver when entering– maneuver is
minimal – offset, heading
change maneuvers
MIT-LIDS MIT-LIDS
Conflicting flows
-1 0 1 2 3 4 5 60
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Deviation angle (deg)
Po
pu
latio
n
Population of deviation angles taken by aircraft
Distribution of deviations
MIT-LIDS MIT-LIDS
Conflict analysis
N
W
Protection-providingzone "U"
Protection-providingaircraft
New comer
An
Ai
PCZ of Ai
PCZ of An
Deviation angle is
bounded by T:
2
2
12
Ttan
where =Dsep/R
MIT-LIDS MIT-LIDS
Other results• Bounds on lateral displacements for arbitrary encounter angle,
speed, distance to conflict. Define: : Encounter angle, v2/v1 Then lateral displacement maneuver amplitude for stream 1is less than
with
• Bounds on displacement for longitudinal/lateral maneuvers(Independent utility functions for each aircraft)
)tan)tancos(sin,0max(cos 12
ddd
cos2
cos1cos 1
MIT-LIDS MIT-LIDS
-2 0 0 -1 5 0 -1 0 0 -5 0 0 5 0 1 0 0 1 5 0 2 0 0-2 0 0
-1 5 0
-1 0 0
-5 0
0
5 0
1 0 0
1 5 0
2 0 0
x (nm )
y (nm )
F low 3
F low 1 F low 2
Three flows (1/4)
• Decentralized resolution– diverges
N
• Centralized resolution (using mixed integer programming)
– stable– exhibits particular structure
MIT-LIDS MIT-LIDS
• Idea – create a control structure– independent of flow– optimize it to decrease
lateral distance from original flight path
• Concept– three-flow compatible– aircraft are assigned
conflict free spots
Three flows (2/4)
Flow 1
Flow 2Flow 3
MIT-LIDS MIT-LIDS
– a systematic way to deal with conflicts
– only 30% higher deviation compared with MIP
Three flows (3/4)
Structured solutionMixed Integer Prog. solution
Results
MIT-LIDS MIT-LIDS
• Control structure– flow independent– optimized
Flow 1
Flow 2Flow 3
Three flows (4/4)Flow 1
Flow 2Flow 3
MIT-LIDS MIT-LIDS
Analysis of robustness
• Maneuver imprecision– leads to divergence in
some scenarios
• Aircraft position uncertainties
• 3-D geometrical tool may help