Prior TCAS safety analyses used Monte Carlo simulation to evaluate performance. Running an adequate number of Monte Carlo simulations from each discrete state would be prohibitively expensive. For example, even if Monte Carlo required only one second to compute the probability of NMAC from each state, it would require 100 days of continuous computation to estimate the probability of NMAC from all 8.7 million states. DP can do this computation in under a minute.
3.10 DISCUSSION
This section showed that, compared to the current TCAS logic in head-on encounters with .xed clo-sure rates, the optimized logic provides greater safety while alerting much less frequently. Although the model only represents motion in two spatial dimensions, the next section will demonstrate how well it does in three-dimensional encounters, even when the model used for optimization does not match the one used for evaluation. Performance in three dimensions, however, can be improved even further by explicitly modeling motion in the horizontal plane. This enhancement to the logic is
presented
in
Section
5.
0 5 10152025303540
(a) Probability of NMAC.
0 5 10152025303540
(b) Probability of alert.
Figure 7. Metric evaluation on the state slice where h˙0 = 0ft/min,h˙1 = 0ft/min,sRA = COC. Horizontal axis indicates τ (s) and the vertical axis indicates h (ft).
4. ROBUSTNESS ANALYSIS
The previous section presented a way for computing the optimal collision avoidance strategy as-suming a simple, head-on model. It presented performance results from simulations that used the same dynamic model that was used for optimization. Because it is unlikely that the model used for optimizing the logic will exactly match the real world, it is important to quantify the robustness of the logic to modeling errors. This section analyzes how the performance of the logic, in terms of safety and operational metrics, degrades as the model used for evaluating the logic diverges from the model used for optimizing the logic.
This section uses the same simple model and cost function from the previous section to optimize the logic. When evaluating the logic in three spatial dimensions,
.
. if ˙
.r/ r˙r< 0,
τ = (14).∞ otherwise,
where r is the horizontal range to the intruder and r˙is the horizontal closure rate. The next section will present a more principled method for handling three-dimensional dynamics, but this simple method will be used to provide a rough estimate of the robustness of the approach.
This section describes two encounter models for evaluating the robustness of the optimal logic to modeling errors. These models are described in the next two sections.
4.1 WHITE-NOISE ENCOUNTER MODEL
The
white-noise
encounter
model
uses
the
dynamic
model
of
Section
3.3
to
model
the
vertical
dynamics. The process noise parameter σ¨ controls the amount of vertical accelerating the aircraft
h
do. The horizontal dynamics are also modeled by a white-noise acceleration model. Each aircraft moves in the horizontal plane in response to independent random accelerations generated from a zero-mean Gaussian with a standard deviation of 3ft/s2.
Encounters are initialized by randomly generating the initial ground speeds of both aircraft, s0 and s1, from a uniform distribution from 100ft/s to 500ft/s. The horizontal range between the aircraft is set to
r = ttarget(s0 + s1)+ ur, (15)
where ur is selected from a zero-mean Gaussian with 500ft standard deviation. The parameter ttarget, set to 40s in the experiments, controls the expected time to NMAC.
The bearing of the intruder aircraft with respect to the own aircraft is sampled from a zero-mean Gaussian distribution with a 2° standard deviation. The heading of the intruder relative to the heading of the own aircraft is sampled from a Gaussian distribution with a mean of 180° and a standard deviation of 2°.
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