0 10203040 0 10203040
(a) h˙0 = 0ft/min,h˙1 = 0ft/min,sRA = COC (b) h˙0 = 1500ft/min,h˙1 = 0ft/min,sRA = COC
Figure 20. Optimal action plots for .ve terminal cycles. Horizontal axis indicates τ (s) and the vertical axis indicates h (ft).
deterministic pilot response to advisories, but the next section will show how the MDP can be easily modi.ed to accommodate probabilistic responses.
The remainder of the report will use the decomposition approach introduced in this section to approximate the solution to the full three-dimensional MDP. Unless otherwise stated, dynamic programming will be used to estimate the entry time distribution.

6. PROBABILISTIC PILOT RESPONSE MODELS

Previous sections have discussed formulating the collision avoidance problem as a Markov decision process and solving for the optimal collision avoidance strategy using dynamic programming. The logic was optimized to a pilot response model in which the pilot responds deterministically to all advisories. The current TCAS logic also uses a deterministic pilot response model to predict the future paths of the aircraft. However, it is unlikely that in actual .ight the pilot will respond to advisories exactly according to the assumed deterministic model. As recorded radar data have shown, there is signi.cant variability in the delay and strength of the response of pilots to advisories [43].
This
section
extends
the
methodology
of
the
previous
sections
to
include
probabilistic
pilot
response models that capture the variability in pilot behavior in order to enhance robustness.
6.1 PILOT RESPONSE MODELS
Two models were constructed to capture the response of the pilot to resolution advisories. These models describe how the variable sRA changes in response to di.erent advisories. Each state in the Markov chain indicates the active advisory as well as the current response. The state “CL1500/COC,” for example, signi.es that the CL1500 advisory is currently on display but that the pilot is unresponsive to any advisory. In the .rst model, the number of states scales linearly with the number of advisories (13 states in total). The second model scales quadratically with the number
of
advisories
(49
states
in
total).
Figure
21,
while
not
explicitly
enumerating
all
state-action
pairs, captures the essential features of the models.
Before an advisory is issued, the Markov chain is in the “COC/COC” state. After a climb advisory is issued, for example, the pilot responds immediately with probability 1/6, transitioning to “CL1500/CL1500,” and remains unresponsive otherwise, transitioning to “CL1500/COC.” Should the climb advisory continue at the next time step, the pilot responds with probability 1/6 if he has not responded already. For a given advisory, therefore, the response delay is a geometric random variable; a success probability of 1/6 was chosen so that, on average, the pilot will respond in .ve seconds.
According to the linear model, if a descend advisory is subsequently issued, the pilot responds with probability 1/4 and neglects all advisories otherwise, regardless of whether he was responding to the climb advisory. The quadratic model di.ers in that, if the pilot is responding to the climb advisory, he will continue to respond to the climb advisory with probability 3/4. With a success probability of 1/4 the pilot will respond to the new advisory in three seconds on average.
If the advisory is discontinued, the Markov chain transitions to “COC/COC” with probability one in the linear model. However, in the quadratic model, the pilot retains some memory of the advisory he was previously executing and continues executing it with probability 3/4 even after the advisory is terminated.
　
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