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ω1 [meters]
x 105
ω1 [meters]
x 105
(c) 1000 accepted states, J = 50
(d) 1000 accepted states, J = 100
Figure 4: Accepted states during MCMC simulation
exit point: either A1 arrives before A2 (top left and bottom right clouds) or A1 arrives after A2 (middle cloud). In this case the proposal distribution g was uniform over the parameter space and the ratio of accepted/proposed states was 0.27. This means that approximately 1100 · 10/0.27 = 40740 simulations were needed to obtain 1000 accepted states. At the average simulation speed of 5 simulations/second, the required computational time to obtain 1000 accepted states was then approximately 2 hours. In this simulation we actually extracted 5100 states. Figure 4(b) displays the last 2000 extracted states.
Figure 4(c) illustrates the case J = 50. In this case the proposal distribution g was a sum of 2000 Gaussian distributions N (μ, �2I) with variance �2 = 107 m2 . The means of Gaussian distributions were 2000 parameters randomly chosen from those accepted in the MCMC simulation for J = 10. The choice of this proposal distribution gives clear computational advantages since less computational time is spent searching over regions of
ω 2 [meters] ω2 [meters]
13
non optimal parameters. In this case the ratio accepted/proposed states was 0.34. This means that approximately 1100 · 50/0.34 = 161764 simulations were needed to obtain 1000 accepted states. At an average of 5 simulations/second, the required computational time to obtain 1000 states was approximately 9 hours.
Figure 4(d) illustrates the case J = 100 and a proposal distribution constructed as before from states accepted for J = 50. Here the ratio accepted/proposed states was 0.3. This means that approximately 1100·100/0.3 = 366666 simulations were needed to obtain 1000 accepted states. At an average of 5 simulations/second, the required computational time to obtain 1000 states was approximately 20 hours. Figure 4(d) indicates that a nearly optimal maneuver is ω1 = −40000 and ω2 = 40000. The probability of conflict for this maneuver, estimated by 1000 Monte Carlo runs, was zero. The estimated expected time separation between arrivals was 283 seconds.
4.4 Coordination of approach maneuvers
In this section, we optimize an approach maneuver with coordination between two aircraft. In Figure 5 several trajectory realizations, of an aircraft performing the approach maneuver described in Section 4.4, are displayed. Here, the aircraft is initially at FL100 and descends to 1500ft during the approach. Uncertainty in the trajectory is due to the action of the wind and randomness in the aircraft mass. In the final leg a function that emulates the localizer and so eliminates cross-track error is implemented.
The problem formulation is illustrated in Figure 6(a). We consider Aircraft One (A1) and Aircraft Two (A2) approaching the runway. The glide path towards the runway starts at the origin of the reference frame and coordinates are expressed in meters. The aircraft are initially in level flight. The parameters of the approach maneuver are the distance, from initial position, of the start of the final descent (ω1) and the length of the downwind leg (ω2).
　
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