0 5 10152025303540
τ (s)
(a) Linear pilot response model.
0 5 10152025303540
τ (s)
(b) Quadratic pilot response model.
Figure 22. Optimal action plots for h˙0 = 0ft/min, h˙1 = 0ft/min, and sRA = DES1500/DES1500.

belief state is updated as follows:
bt(sRA) ∝ p(h˙0 | h˙0,sRA,a) T (sRA, a, sRA) bt.1(sRA), (29)
sRA
where p(·| h˙0,sRA,a) represents the probability density of the own aircraft vertical rate at time t conditioned on the previously observed vertical rate, h˙0, the action taken, a, and the advisory state, sRA. This density is evaluated at the observed vertical rate at the current time, h˙0. After each iteration of the .ltering process, the action to take accounts for the likelihood of di.erent responses
using
the
QMDP
approximation
[30]:

π .(b) = arg min b(s)J .(s, a). (30)
a
s
This .ltering approach relies on a model of how pilots respond to advisories. If the model is not re.ective of reality, the .ltering may become unstable as the model fails to explain unexpected observations. It is important, therefore, to design and vet a pilot response model that captures a wide spectrum of pilot behavior.
Advisory belief updating using a Bayesian framework is one way of monitoring the progression of the encounter. TCAS, similarly, makes accommodations to handle situations when the own aircraft is moving counter to the current advisory. In such a situation, if certain requirements are met, the advisory will be reversed to prevent potentially dangerous vertical chases. One strength of the current approach, however, is that the introduction of specialized heuristics is unnecessary.

6.4 EXAMPLE ENCOUNTER
Figure
23
shows
an
example
encounter
comparing
the
behavior
of
the
DP
logic
using
probabilistic
pilot response with that of TCAS. The logic was optimized using the quadratic pilot response model. The encounter was generated using the correlated encounter model. In this example encounter, although the logic is optimized using a probabilistic pilot response model, the pilot responds to all initial advisories in exactly .ve seconds and to all subsequent advisories in exactly three seconds.
Figure
24
shows
the
sRA belief state throughout the course of the example encounter. Fig-ure
25
shows
the
evolution
of
the
entry
time
distribution.
The
entry
time
distribution
was
computed
using the white-noise horizontal model with an acceleration standard deviation of 3ft/s2 and an expanded horizontal NMAC region of 1000ft.
Nineteen seconds into the encounter, the DP logic issues a descend to pass below the intruder. The expected cost for issuing a descend advisory is approximately 0.043, lower than the expected cost for issuing a climb advisory (0.054) or for not issuing an advisory (0.046). After the descend advisory
is
issued,
as
Fig.
24
illustrates,
the
probability
distribution
over
sRA indicates that with probability 5/6 the own aircraft is not executing the descend advisory and with probability 1/6 it is executing it. These are the probabilities of not executing and executing an initial advisory according to the model, respectively.
3000
2500
2000
1500
1000

Altitude (ft)
North (ft)
Time (s)
(a) Vertical pro.le.
×104
0.2
0 .0.2 .0.4 .0.6

.0.8
.1 East (ft) ×104
(b) Horizontal pro.le.
Figure 23. Example encounter comparing the performance of the DP logic optimized using the quadratic pilot response model with that of TCAS.
　
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