hh
σ¨ parameter is small, the aircraft tend to .y in straight lines. Large values for σ¨
signi.cant variability in the paths of the aircraft. This section assumes σ¨ 3 ft/s2, but this

h
can lead to

=

parameter may be estimated from radar data. Following the response delay to an advisory, it is assumed that the pilot maneuvers with exactly the prescribed acceleration to come within the target vertical rate range. Once within the target vertical rate range, the aircraft resumes white-noise vertical accelerations.
In order to apply DP, the dynamic model must be discretized into a discrete transition model T (s, a, s ). One way to do this is to de.ne Pr(s | x), a probability distribution over discrete states given a continuous state. Several di.erent schemes can be used; this report uses multilinear interpolation because it was shown to work well for the collision avoidance problem in ATC-360. In this scheme, probability is assigned to the discrete states associated with the vertices of the grid cell that encloses the continuous state x. The probability assigned to a state s is based on how close it is to x. If the probabilities are treated as weights, the weighted average of the vertices enclosing x is exactly x. Once Pr(s | x) is speci.ed using multilinear interpolation or some other method, then the discrete transition model can be computed as follows:3

T (s, a, s ) = Pr(s | s, a) (10)
= Pr(s | x )Pr(x | s, a). (11)
x
Given a state and action, Pr(x | s, a) can be nonzero for a non.nite set of continuous next states. To build the discrete transition function, this report relies upon sigma-point sampling. Other sampling methods may be used, but sigma-point sampling has been shown to work well for the collision avoidance problem in ATC-360, and it has the desirable property of being determin-istic. The only noise in the dynamic model is due to sampling h¨ 0 and h¨ 1, the accelerations of the aircraft when an advisory is not being followed. The following acceleration pairs are deterministi-
h
h
h
h
cally generated using sigma-point sampling: (0, 0), (σ¨ , 0), (.σ¨ , 0), (0,σ¨), (0, .σ¨
is assigned weight 1/6, except for the .rst one, which is assigned weight 1/3. When executing

¨
action a from state s with acceleration pair (h¨ 0,h1), the resulting state is uniquely given by the model
¨¨
x = f(s, a, h0,h1). (12)
3In reality, Pr(x | s, a) is a density, but the probability density function is approximated by a probability mass function through sampling.
). Each sample

TABLE 3
Event costs

NMAC Alert Strengthening Reversal Clear of Con.ict
1 0.01 0.009 0.01 .1 × 10.4
Given N samples of the accelerations (.ve in the sigma-point scheme just described), Pr(x |s, a) is approximated by
N
(n)(n)(n)(n)
¨¨ ¨
Pr(x | s, a)= δ(x ,f(s, a, h0 ,h1 ))Pr(h¨ 0 ,h1 ), (13)
n=1 (n)(n)
¨
where δ(X, Y )=1 if X = Y and 0 otherwise. The probability Pr(h¨ ,h) is simply the weight
01
assigned
to
the
sample.
Equation
(13)
may
be
combined
with
Eq.
(11)
to
de.ne
the
discrete
transition probabilities.
Computing the distribution over discrete next states scales O(N2D), where N is the number of samples and D is the number of dimensions in the state space. Both N and D are 5 for the model in this section, so the required computation is quite modest. If the dimensionality of the state space is required to grow signi.cantly larger to accommodate additional state variables not currently modeled, simplex interpolation may be a better option than multilinear interpolation because its computation
　
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