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Figure
16
shows
two
slices
of
the
entry
distribution
computed
using
DP,
Monte
Carlo,
and
the
simple method. The plots show the mean of the entry distribution for di.erent x and y horizontal displacements between the aircraft conditioned on certain values for the relative horizontal speed, rv. The relative horizontal velocity is pointing directly left. The plots can be understood by imagining the own aircraft as lying stationary at the origin and the axes as indicating the location of the intruder. The DP entry distribution is more di.use and smoother than that produced by Monte Carlo. The simple entry distribution has a mean under the horizon for much of the plot. It tends to overestimate the collision risk. Use of the DP entry distribution over the simple entry distribution, which only uses range and range rate similar to TCAS, allows for automatic horizontal miss distance .ltering.

5.4 ONLINE SOLUTION
After J0,...,JK ,J ˉ and D0,...,DK ,D ˉ have been computed o.ine, they are used together online
KK
to determine the approximately optimal action to execute from the current state. For any discrete state s in the original state space, the expected cost J. (s, a) may be computed as follows
K
K J . (s, a)= D ˉ (su)J ˉ (sc,a)+ Dk(su)Jk(sc,a), (27)
K KK
k=0
where su is the discrete uncontrolled state and sc is the discrete controlled state associated with
s. Combining the controlled and uncontrolled solutions online in this way requires time linear in the size of the horizon. Multilinear interpolation can be used to estimate J. (x,a) for an arbitrary
K
state x, and from this the optimal action may be obtained.
The memory requirements for directly storing the true J. (s, a) is O(|A||Sc||Su|). However,
K
the hybrid o.ine-online method presented in this section allows the solution to be represented using O(K|A||Sc| + K|Su|) storage, which can be a tremendous savings when |Sc| and |Su| are large. For the collision avoidance problem, this method allows the cost table to be stored in 500MB instead of over 1 TB. The o.ine computational savings are even more signi.cant.
Simple  Monte Carlo Dynamic Programming
×104  ×104
1

40 1

40 0.5
30 0.5
30
0
20 0
20
.0.5
10 .0.5
10
.1
0 .1
0
00.511.52  00.511.52
×104 ×104 ×104 ×104
1

40 1

40 0.5
30 0.5
30
0
20 0
20
.0.5
10 .0.5
10
.1
0 .1
0
00.511.52  00.511.52
×104 ×104 ×104 ×104
1

40 1

40 0.5
30 0.5
30
0
20 0
20
.0.5
10 .0.5
10
.1
0 .1
0
0 0.511.52 00.511.52 ×104 ×104
rv = 250ft/s  rv = 500ft/s

 

 

Figure 16. Mean of the entry distribution for two slices of the state space when the relative horizontal velocity is pointing directly left. Horizontal axis represents the relative x displacement (ft) and the vertical axis the relative y displacement (ft).

5.5 EXAMPLE ENCOUNTER

Figure
17
shows
an
example
encounter
comparing
the
behavior
of
the
system
using
the
DP
entry
time distribution against the TCAS logic. The encounter was produced using the white-noise encounter
model.
Figure
18
shows
the
entry
time
distribution
computed
using
the
three
methods
of
Section
　
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