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Figure 6(c) illustrates the case J = 50. In this case the proposal distribution g was a sum of 100 Gaussian distributions N (μ, �2I) with means selected randomly among the maneuvers accepted for J = 10 and variance �2 = 105 m2 . In this case the ratio between accepted and proposed parameters was 0.25. This means that approximately 1100 · 50/0.25 = 220000 simulations were needed to obtain 1000 accepted states, which required approximately 12 hours of computation.
Figure 6(d) illustrates the case J = 100 and proposal distribution constructed as before from states accepted for J = 50. In this case the ratio between accepted and proposed parameters was 0.3. This means that approximately 1100 · 100/0.3 = 366666 simulations were needed to obtain 1000 accepted states (approximately 20 hours). Figure 6(d) indicates that a nearly optimal maneuver is ω1 = 35000 and ω2 = 35000. The probability of conflict for this maneuver, estimated by 1000 Monte Carlo runs, was zero.
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Figure 6: Accepted states during MCMC simulation
5 Conclusions
In this paper we illustrated an approach to air traffic conflict resolution in a stochastic setting based on Monte Carlo methods. The main motivation for our approach is to enable the use of realistic stochastic hybrid models of aircraft flight; Monte Carlo methods appear to be the only ones that allow such models. We have formulated conflict resolution as the optimization of an expected value criterion with probabilistic constraints. Here, a penalty formulation of the problem was developed, which guarantees constraint satisfaction but delivers suboptimal solutions. A side effect of the optimization procedure is that structural differences between maneuvers are highlighted as “clouds” of maneuvers accepted by the algorithm, with the density of the clouds related to the cost of the particular maneuver.
We presented the application of this method to two realistic scenarios inspired by terminal area and final approach maneuvering respectively. The solutions proposed by
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