3.4 COST FUNCTION
The
costs
of
various
events
are
summarized
in
Table
3.
NMAC, in this section, occurs when |h| < 100ft and τ = 0s. The 100ft threshold was chosen because it corresponds to the vertical separation of an NMAC used historically as a metric for evaluating TCAS. The costs were chosen somewhat arbitrarily, but the cost of alerting was made low compared to the cost of NMAC. The cost of strengthening was made lower than that of reversing because the required maneuvering would be less severe. A small negative cost, called the clear-of-con.ict reward, is awarded at every time step the system is not alerting to provide some incentive to discontinue alerting after the encounter has been resolved.
This report focuses on a relatively small number of cost factors, but additional factors can be incorporated to address other safety or operational considerations. It may be desirable to add another state variable that represents the altitude above ground and then add a large penalty for issuing down-sense advisories at low altitude to prevent collision with terrain. Operationally, it may be desirable to discourage issuing advisories when aircraft are .ying level and separated by more than 500ft in altitude. If the state space does not require expanding, the computation required to construct the expected cost table grows linearly with the number of cost factors and the storage required remains constant. Online execution of the logic also remains constant.

3.5 OPTIMAL POLICY

DP,
as
introduced
in
Section
2,
can
be
applied
directly
to
the
discretized
model
to
obtain
a
discrete
expected cost table J(s, a).
The
discretization
in
Table
2
leads
to
an
expected
cost
table
of
263MB
using
the
sparse
action
representation
described
in
Appendix
A.
Computing
the
expected
cost
table
requires less than a minute on a single 3 GHz Intel Xeon core.
Since the collision avoidance logic is critical to safety, it is important for humans to understand and be able to anticipate the behavior of the system. Because the logic makes decisions based on values in an expected cost table, which is not directly informative to a human, it is necessary to develop ways to visualize the logic. Visualization is also important in building con.dence that the logic produced through computer optimization is sensible.
Figure
4
shows
plots
that
indicate
the
optimal
action
for
di.erent
slices
of
the
state
space.
In the .rst plot, both aircraft are initially level and no advisory has been issued. The blue region indicates where the logic will issue a descend advisory, and the green region indicates where the logic will issue a climb advisory. The colored region (either green or blue in this .rst plot) is called the alerting region.
There are two notable properties of the shape of the alerting region. The .rst is that no advisory is issued when τ ≤ 5s,whichisduetothe5spilotresponsedelay. Alertinglessthan5s prior to collision makes no di.erence in preventing collision, so the optimized logic chooses not to accrue the cost of alerting. In reality, there is likely to be some nonzero chance that the pilot will respond sooner. If the model is adjusted to allow some probability of the pilot responding in fewer than .ve seconds, then the alerting region will extend further to the left. This will be discussed in Section
　
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