For the development of a future TCAS system, it appears that the best option is to use dynamic programming. As with Monte Carlo, dynamic programming can accommodate a wide variety of di.erent models, but it requires relatively little computation during .ight. Dynamic programming does all the intense computation o.ine during development and stores the results in a lookup table to be used later. Since dynamic programming requires the state space to be discretized, one potential limitation is that the number of state variables must be kept manageable to prevent the number of discrete states from exploding. However, this report shows that by carefully choosing the representation of the model, the number of discrete states can be kept manageable.
Collision avoidance algorithms that use probabilistic dynamic models di.er in how they use the relative likelihood of di.erent trajectories to make decisions about when to alert and which advisory to issue. Many of the existing algorithms convert the trajectory likelihoods into a prob-ability
of
NMAC
that
is
then
compared
against
a
threshold
[14,
15].
Some
algorithms
alert
the
moment the probability of NMAC without evasive maneuvering exceeds some threshold, but there are other strategies for delaying alerts. For example, one strategy is to delay alerting until a single advisory remains whose probability of NMAC is below some threshold.
An alternative to approaches based on NMAC probability thresholds is to de.ne a cost metric for trajectories and optimize the collision avoidance logic to minimize the expected cost. Under certain assumptions to be discussed later, dynamic programming can be applied to compute the expected cost for each action (e.g., no alert, climb, or descend) at each state o.ine and store the results in a table. The table can then be used online to determine the best action from the current state. This report adopts a dynamic programming approach.

1.3 PROPOSED APPROACH
The proposed approach uses a probabilistic model of aircraft behavior and a cost metric to be optimized. In order to quickly decide what action is best given the history of sensor measurements, a signi.cant amount of processing is done o.ine during the development phase. The results are summarized in large numerical tables that can be distributed to manufacturers and incorporated into their avionics. During .ight, the collision avoidance system consults these tables in real time to decide which advisory to issue or whether to delay alerting.
1.3.1 O.ine Development
The o.ine development of the system involves specifying a probabilistic dynamic model that cap-tures the behavior of aircraft during close encounters and the response of pilots to resolution advisories. The dynamic model may be constructed based on a combination of expert judgment and recorded data. Because of the in.uence the dynamic model has on the optimized logic, it is critical that the model be thoroughly vetted with the development community.
A cost metric must also be speci.ed that balances di.erent safety and operational considera-tions, such as the alert rate and the NMAC rate. The optimal collision avoidance logic is de.ned to be the one that provides the lowest expected cost. The choice of cost function strongly in.uences the behavior of the logic. Optimizing the system according to di.erent cost metrics and examining the resulting behavior in simulation may guide the choice of cost parameters, as will be discussed later in this report.
If the model and the cost metric meet certain criteria to be discussed later, dynamic pro-gramming may be applied to recursively compute the expected cost for each action from each state. These costs are what are stored in the numerical lookup tables to be used in real time to select actions. Computing these tables is relatively e.cient because dynamic programming lever-ages the structure of the model and cost metric so that every possible future trajectory does not require explicit enumeration.
　
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