.
Veri.cation. It is important that a manufacturer’s implementation of the logic be veri.ed for correctness. TCAS implementations are currently veri.ed against a set of test encounters. The system proposed in this report can also be veri.ed against test encounters. Because the complexity of the logic can be encoded using numerical lookup tables that can be standardized and delivered to manufacturers, there is very little code for manufacturers to implement and hence less opportunity for errors to be introduced.

1.2 BACKGROUND
Developing a robust collision avoidance algorithm that reliably prevents collision without alerting excessively is challenging due to sensor error and uncertainty in the future behavior of the aircraft. A wide variety of di.erent collision avoidance algorithms have been proposed, and they di.er in how they predict the motion of aircraft. These methods may be categorized as follows:
.
Nominal trajectory propagation. Many collision avoidance systems, including TCAS, propa-gate the most likely positions of the aircraft into the future. If an intruder is predicted to come within the NMAC volume or some enlarged con.ict zone in the near future, the system decides that an alert is necessary. The system then predicts the e.ects of the available ad-visories and chooses the one that best resolves the NMAC. One problem with this approach is that it does not explicitly account for low-probability events that can lead to collision. To make the logic robust to deviations from the nominal predicted trajectories, algorithms that use
this
approach
must
rely
upon
complex
heuristics.
[1–4]

.
Worst-case trajectory propagation. Other collision avoidance systems examine a range of possible maneuvers and determine whether any of them results in NMAC. One disadvantage of this approach is that it can result in an excessively high alert rate. Excessive alerting can result in
NMACs
with
third-party
aircraft,
impaired
e.ciency,
and
decreased
pilot
compliance.
[5–7]

.
Probabilistic trajectory propagation. This method examines all possible future trajectories and accounts for their relative likelihood. Nominal and worst-case propagation are special cases of probabilistic propagation; nominal propagation assigns probability one to the most likely trajectory and worst-case propagation assigns equal probability to all trajectories. Accounting for the likelihood of di.erent trajectories can result in robustness against low-probability hazardous
events
while
maintaining
an
acceptable
alert
rate.
[8–10]

Figure 1. Trajectory propagation methods. Nominal trajectory propagation (green) only examines the most likely trajectory. Worst-case trajectory propagation (red) searches the space of possible trajectories for one that results in NMAC. Probabilistic trajectory propagation (blue) accounts for the relative likelihood of all trajectories.
These
trajectory
propagation
methods
are
illustrated
in
Fig.
1.
Surveys
of
these
methods
are
found
in
[11,12].

Although using a probabilistic model can result in signi.cantly more robust collision avoid-ance, the majority of the algorithms in the literature use nominal trajectory propagation. With nominal trajectory propagation, only a single trajectory needs to be modeled, but with probabilistic trajectory propagation, all trajectories (or a large sample) need to be modeled. Hence, methods adopting nominal trajectory propagation typically require much less computation.
Several di.erent probabilistic propagation methods have been proposed in the literature, including analytic, numerical approximation, Monte Carlo, and dynamic programming methods. The analytic and numerical approximation methods are fast, but they typically require strong assumptions to be made regarding the model of the aircraft dynamics. Monte Carlo methods are much more .exible, but many trajectory samples are typically required because NMACs are rare, although
　
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