CALVO-FERNÁNDEZ ET AL.
Downloaded by UNIV. OF ARIZONA on March 14, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.G000691
with its flight plan, the aircraft trajectory passes through fixed points
p
1
; p
2
; : : : ; p
n
, for which the geographical coordinates are known in
advance. The 4-D coordinates
φ; λ; h; t are calculated between each
waypoint pair (Fig. 3).
We used pairwise conflict detection to calculate the distance
between each pair of 4-D trajectories. Where loss of separation was
detected, we registered the conflict parameters: that is, the geometric
points (GPCs) and the conflict times (TCs).
The pairwise conflict detection algorithm was described in
[27,28]. In these, a sampling time of 10 s was used. We reduced this to
5 s to improve the accuracy.
Figure 4 shows two pairwise-processed trajectories, corre-
sponding to the trajectories of aircraft 1 and 2. A cross indicates a
conflict. The dotted lines indicate the 4-D coordinates at the same
time instant.
The results indicate that over half of the operational conflicts,
which require controller deconfliction, occur when both aircraft are
in the cruise phase. Therefore, this study will focus on the en route
environment. This is to avoid the effects of other parameters, such as
distance to destination or possible climb, which mainly affect the
climb and descent phases and may mask the results obtained due to
induced complexity.
3.
Conflict Resolution Based on Historical Data
Possible conflict-resolution maneuvers performed by controllers
include 1) speed changes, i.e., temporal deviation of radar tracks vs
flight plans; 2) vertical maneuvers, i.e., vertical deviation of radar
tracks vs flight plans; and 3) two-dimensional turns, i.e., horizontal
deviation of radar tracks with respect to the flight plans. The conflict-
resolution model in this study deals with the first two maneuvers.
We identified the historical conflicts and then categorized them
according to the tactical resolution applied in each case. Equations (4)
and (5) were used to assess the adherence of the actual trajectory
flown to the updated flight plan, using radar track data. So, for each
conflict, we calculated the following:
1) We first calculated the temporal deviation (TD) of the trajectory
flown (radar track) compared to the latest updated flight plan at the
geometric point where the conflict was detected (GPC); see Fig. 5:
TD t
GPC
FP
− t
GPC
RT
(4)
where t
GPC
FP
is the time estimated by the flight plan in the GPC; and
t
GPC
RT
is the actual time indicated by the radar track.
A temporal deviation of 0 min means that the flight passed over the
geometric point, where the conflict was detected, at the same time as
that estimated by the flight plan.
2) We then calculated the vertical deviation (VD) of the trajectory
flown (radar track) compared to the latest updated flight plan at the
time when the conflict was detected (TC); see Fig. 6;
VD h
TC
FP
− h
TC
RT
(5)
where h
TC
FP
is the altitude estimated by the flight plan at TC; and h
TC
RT
is
the actual altitude indicated by the radar track.
A vertical deviation of 0 ft means that the flight passed over the
geometric point, where the conflict was detected, at the same altitude
as that estimated by the flight plan. Any conflict with a vertical
deviation of within 1000 ft would not be resolved vertically.
These criteria were contrasted with the experience of experts
involved in day-to-day ATC operations and found to have a valid
theoretical and empirical basis.
4.
Development of Model
This section describes the methodology used to develop the
conflict-resolution model. The model uses the large database of
conflicts and controller actions to resolve them, and it performs two
fundamental tasks: classification and clustering [29].
We classified the conflict resolutions according to the relationship
between the aircraft trajectories involved in the conflict. Although we
are dealing with a radar-controlled environment, we considered it
valid to classify the conflicts using the International Civil Aviation
Organization
’s document 4444-ATM/501 [30] criteria of “applicable
for procedural control in the en-route phase.
” Specifically, for the
purpose of application of longitudinal separation, the terms
“same
track,
” “reciprocal tracks,” and “crossing tracks” are consid-
ered (Fig. 7).
Finally, we took the tactical component of the controller conflict-
resolution strategy into account by using an unsupervised learning
method (clustering) [31,32]. The goal of clustering is to determine the
intrinsic grouping in a set of data. Thus, clustering identifies
distinctive groups/clusters in the data, based on the hypothesis that a
cluster in a data space is a contiguous region of high point
density [33].
The k-means algorithm [34–36], which is one of the most
frequently used clustering algorithms, is classified as a partitional
clustering method, presenting two distinct advantages:
1) It is easy to implement and works with any of the
standard norms.
2) It allows straightforward parallelization.
Flight Plan
4-D Trajectory
p
1
(
1
,
1
,h
1
,t
1
)
p
2
(
2, 2,
h
2,
t
2
)
p
1
p
3
p
2
p
4
p
5
p
6
5s
Fig. 3
Four-dimensional trajectory based on the flight plan.
Aircraft 1
Aircraft 2
Conflict
Fig. 4
Sample pairwise algorithm.
The flight passes over the
GPC 1 min earlier than
the time estimated by the FP
The flight passes over the
GPC 1 min later than the
time estimated by the FP
-2 min
-1 min
0 min
1 min
2 min
Temporal
deviation
Fig. 5
Temporal deviation.
The
flight passes over at the
TC 1000 ft above the altitude
estimated by the FP
The flight passes over at the
TC 1000 ft below the altitude
estimated by the FP
-2,000 ft
-1,000 ft
0 ft
1,000 ft
2,000 ft
No vertical resolution
Vertical
deviation
Fig. 6
Vertical deviation.
618
CALVO-FERNÁNDEZ ET AL.
Downloaded by UNIV. OF ARIZONA on March 14, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.G000691
A drawback of the k-means algorithm is that the solution reached
often depends on the starting points. Therefore, it is possible to reach
a local minimum, where reassigning any one point to a new cluster
would increase the total sum of point-to-centroid distances when, in
fact, a better solution may exist. To avoid local minima, a parameter
called
“replicates” is used to overcome the problem.
¶
For each of the
replicates, the k-means algorithm starts from a different randomly
selected set of initial centroids. Sometimes, the algorithm finds more
than one local minimum. However, the final solution returned by k
means is the one with the lowest total sum of distances.
To determine the number of clusters, we used Sturges
’s rule
[37,38], which is a rule for determining the desirable number of
groups into which a distribution of observations should be classified.
As Sturges
’s rule (or formula) is frequently applied to histograms, it is
suitable for use with the k-means algorithm.
Sturges
’s formula [38] is derived from a binomial distribution and
implicitly assumes an approximately normal distribution:
k 1 log
2
o
(6)
where k is the number of classes (clusters); and o is the number of
observations used to build the histogram.
In the end, we produced a conflict-resolution model based on 1) the
relationship between the aircraft tracks involved in the conflict; 2) the
t
n
t
i
t
f
t
x
t’
n
=t
n
t’
x
=t
x
t’
f
=t’
f
-2 min
t’
i
=t’
i
-2 min
New FP
(Conflict-free)
Actual FP
(Conflicted)
Fig. 8
Illustration of temporal (speed) conflict resolution.
h’
CR
=h
CR
+2,000 ft
h
CR
h’
CR
=h
CR
-2,000 ft
h
CR
New FP
(Conflict-free)
Actual FP
(Conflicted)
New FP
(Conflict-free)
Actual FP
(Conflicted)
Fig. 9
Illustration of vertical conflict resolution.
Fig. 10
Possible resolutions to an individual conflict (based on the data-
driven model).
Likelihood of ATC resolution (%)
0
2
4
6
8
10
12
Additional fuel consumed (kg)
-100
-50
0
50
100
150
200
250
300
f
1
(x)
f
2
(x)
Solutions ‘out of
bounds’ due to
i
constraint
Optimal
solution
Fig. 11
ε-constraint method.
Fig. 12
Pareto frontier of an individual conflict. Intermediate solutions
as a function of
tan α.
45º
45º
90º
135º
90º
135º
Less
than 45º
More than
135º
45º to
90º
90º to
135º
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