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In-Depth Information
5.3
Uncontrolled Subproblem
The uncontrolled subproblem involves estimating the distribution over
τ
(i.e., the time
until the aircraft are separated less than 500 ft horizontally) given the current state. This
section describes the horizontal dynamics and three methods for estimating the entry
time distribution.
Dynamic Model.
The aircraft move in the horizontal plane in response to independent
random accelerations generated from a zero-mean Gaussian with a standard deviation of
3 ft/s
2
. The motion can be described by a three-dimensional model, instead of the typical
four-dimensional (relative positions and velocities) model, due to rotational symmetry
in the dynamics. The three state variables are as follows:
-
r
: horizontal range to the intruder,
-
r
v
: relative horizontal speed, and
-
θ
v
: difference in the direction of the relative horizontal velocity and the bearing of
the intruder.
These variables are illustrated in Figure 2.
Relative
velocity vector
θ
v
r
v
Intruder
r
Own
Fig. 2.
Three-variable model of horizontal dynamics
Dynamic Programming Entry Time Distribution.
The entry time distribution can
be estimated offline using dynamic programming as discussed in Section 4.3. The state
space was discretized using the scheme in Table 3, resulting in 730 thousand discrete
states. The offline computation required 92 seconds on a single 3 GHz Intel Xeon core.
Storing
D
0
,...,D
39
in memory using a 64-bit floating point representation requires
222 MB.
Ta b l e 3 .
Uncontrolled Variable Discretization
Variable Grid Edges
r
0
,
50
,...,
1000
,
1500
,...,
40000 ft
r
v
0
,
10
,...,
1000 ft/s
θ
v
−
180
◦
, −
175
◦
,...,
180
◦
