(23)
The probability that τ = k for k> 0 is computed from Dk.1 as follows:
..
0 if su ∈ G,Dk(su)=
.
T (su,s)Dk.1(s) otherwise.
suu
u
(24)
Depending on su, there may be some probability that su does not enter G within K steps. This probability is denoted D ˉ (su) and may be computed from D0,...,DK :
K K
D ˉ (su)=1 . Dk(su). (25)
K
k=0
The sequence D0,...,DK ,D ˉ is stored in a table with O(K|Su|) entries. Multilinear interpolation
K
of the distributions may be used to determine Dτ (xu) at an arbitrary continuous state xu.
The dynamics of the aircraft in the horizontal plane are assumed to follow those of the white-noise
encounter
model
described
in
Section
4.1.
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. The three state variables are as follows:
Relative velocity vector
TABLE 6
Uncontrolled variable discretization
Variable Grid Edges
r 0, 50,..., 1000, 1500,..., 40000ft rv 0, 10,..., 1000ft/s θv .180°, .175°,..., 180°
.
r: horizontal range to the intruder,
.
rv: relative horizontal speed, and
.
θv: di.erence in the direction of the relative horizontal velocity and the bearing of the intruder.
These
variables
are
illustrated
in
Fig.
15.
The entry time distribution can be estimated o.ine using dynamic programming. The state space
was
discretized
using
the
scheme
in
Table
6,
resulting
in
730,000 discrete states. The o.ine computationrequired92sonasingle3GHzIntelXeoncore. Storing D0,...,D39 in memory using a 64bit .oating point representation requires 222MB. Storing D ˉ
39 is unnecessary because it can be inferred from the other tables. The standard deviation of the noise in the horizontal accelerations was set to 3ft/s2 when generating the tables for the experiments in this section.
An alternative to using DP for computing the entry time distribution o.ine is to use Monte Carlo to estimate the entry time distribution online. A Monte Carlo approach does not require the uncontrolled variables to be discretized and does not require D0,...,DK ,D ˉ to be stored in
K
memory. However, using Monte Carlo increases the amount of computation required online. Since the NMAC region is small in the collision avoidance problem, the number of samples required to produce an adequate distribution may be large. Importance sampling and other sampling methods may
be
used
to
help
improve
the
quality
of
the
estimates
of
the
entry
time
distribution
[13].
The
experiments in this section use 100 trajectory samples to estimate the Monte Carlo entry time distribution.
Another alternative to DP is to use the simple point estimate of τ , originally suggested in Section
4.
The
range
rate
r˙may be computed directly from the horizontal state variables:
r˙= rv cos(θv). (26)
If the aircraft are converging in range, then τ can be approximated by .r/ r˙. Otherwise, τ is set beyond the horizon. This approach to estimating τ is very fast, but it does not take into account the uncertainty of the estimate. In the experiments, this method is referred to as the simple entry distribution.
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