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12
shows
the
results
when
the
environment
noise
parameter
of
the
white-noise
encounter
model was varied from 0ft/s2 to 15ft/s2. The resulting RDP logic signi.cantly reduced the number of NMACs compared to TCAS but did so at the expense of more alerts. Logic derived with an alert cost of 0.1 cut the alert probability in half while maintaining the NMAC rate well below that of
TCAS.
Compared
to
the
DP
logic
optimized
for
a
single
model
(Fig.
9),
the
RDP
logic
provides
a .atter response, reducing the sensitivity of the optimized logic to environment noise.
The
robust
logic
was
also
evaluated
on
the
correlated
encounter
model.
Table
5
summarizes
the results. The DP logic optimized using an alert cost of 0.01 achieves a minimum NMAC rate of 0.000875,
as
shown
in
Fig.
11,
lower
than
that
of
TCAS
(0.0113), while alerting at approximately the same rate. The RDP logic using the same alert cost of 0.01 is able to reduce the NMAC rate even further (0.000295) while only marginally increasing the alert rate. By increasing the alert cost, the RDP logic has the potential to reduce the alert rate by 10% and the NMAC rate by an order of magnitude compared to TCAS.
TABLE 5
Robust logic performance on correlated encounter model

Logic Alert Cost Pr(NMAC) Pr(Alert)
TCAS N/A
DP 0.01
8.75 · 10.4 9.32 · 10.1 RDP 0.01 RDP 0.10

4.6 STATE-BASED ROBUSTNESS ANALYSIS
The results in this section have been using Monte Carlo to estimate the NMAC and alert rates assuming a distribution over initial states. Each run of one million encounters took approximately 10min on a 3GHz processor, which is manageable when evaluating overall performance. However, one might be interested in estimating performance for a large collection of individual initial states. This is useful, for example, when identifying regions of the state space where the logic has di.culty resolving NMAC. Monte Carlo simulation may not, in such situations, be feasible. This section presents an alternative approach based on dynamic programming for e.ciently evaluating metrics on the full state space. This approach can be used to identify regions of the state space where safety is degraded due to modeling error.
The objective is to compute M(s), the expected value of some metric, for every starting state s in the discrete state space. The metric evaluated on a trajectory given by the state-action sequence s1,a1,...,sN ,aN is calculated by
N
m(sn,an), (19)
n=1
where m is the immediate metric function. For example, to estimate the probability of NMAC, m(s, a) would be one if s is an NMAC state and zero otherwise.
Let π be the policy to be evaluated and T be the transition model used for evaluation. When evaluating the robustness of π, the transition model T need not be the model used for deriving π. An iterative process, similar to value iteration, is used to compute M from every discrete state. First, M0 is set to zero for all states. The values for Mk are determined from Mk.1 as follows:
Mk(s)= m(s, π(s)) + T (s, π(s),s )Mk.1(s ). (20)
s
This iteration, as with value iteration, is repeated to the desired horizon.
Computing the probability of NMAC for the full discrete state space requires eight seconds on a single processor. Running Monte Carlo on the 8.7 million states would require 17 years, even if each state only required one minute instead of ten.
Figure
13
shows
the
probability
of
NMAC
across
the
　
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