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Cost Function. An effective collision avoidance system must satisfy competing objec-
tives, including maximizing safety and minimizing the rate of unnecessary alerts. These
objectives are encoded in the cost function. In addition to incurring a cost for conflict,
it is desirable to incur a cost for other events such as alerting or changing an advisory,
as shown in Table 2. A small negative cost is awarded at every step the system is not
alerting to provide an incentive to discontinue alerting after resolution of the encounter.
Ta b l e 2 . Event Costs
Conflict
Alert
Strengthening
Reversal
Clear of Conflict
1 · 10 4
1
0.001
0.009
0.01
Optimal Policy. The optimal cost-to-go tables J 0 ,...,J K ,J K were computed offline
in less than two minutes on a single 3 GHz Intel Xeon core using a horizon of K =39
steps. Storing only the values for the valid state-action pairs requires 263 MB using a
64-bit floating point representation. Figure 1 shows a plot of the optimal policy through
a slice of the state space where the own aircraft is initially climbing at 1500 ft/min, the
intruder is level, and no alert has been issued. The blue region indicates where the logic
will issue a descend advisory, and the green region indicates where the logic will issue a
climb advisory. The optimal policy will sometimes issue a climb even when the intruder
is above. This occurs when the aircraft are closely separated in altitude and little time
remains until potential conflict. Because the own aircraft is already climbing, there is
insufficient time to accelerate downward to avoid conflict. Climbing above the intruder
is more effective. Another notable feature of the plot is that no advisory is issued when
τ
5 s. Because an advisory has no effect until five seconds after it is issued, alerting
less than five seconds prior to conflict is ineffective.
1000
500
Descend
0
Climb
500
No advisory
1000
0
10
20
30
40
τ (s)
Fig. 1. Optimal action plot for h 0 = 1500 ft/min, h 1 =0ft/min, s RA = “no advisory”
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