r
2 rel cos(θ)
slant . h2
.
hrel The vector w is the zero-mean white Gaussian measurement noise vector with covariance
..
θ
r
σ2 00
slant
σ2
0
0
.. ..
(B-26)
R =
.
h
σ2
, assuming the relative bearing and own aircraft heading are
00
rel
h
θ
The variance σ2 independent. Similarly, σ2 +q2/12, where σ2
jitter
due to quantization error, q being the amount of quantization (e.g., 25ft or 100ft).
χ
h
is equal to σ2
rel
h
ψh
+ σ2 = σ2 + σ2
0
h
. The variance in the intruder altitude measurement, σ2 is the contribution due to random noise and q2/12 is the contribution
,
11
2is σh
jitter
The state-transition function is the following linear function:
x(k +1) = Fx(k)+ Bv(k)
..
..
1
2(Δt)2
100Δt 00
00
......... .........
x(k)+
.........
1
2(Δt)2
.........
010 0 Δt 0
0010 0Δt
0001 0 0
0000 1 0
0
0
(B-27)
1
2(Δt)2
00
v(k),
=
Δt 00
0Δt 0
000001 0 0Δt
TABLE B-1
Parameters for the horizontal unscented Kalman .lter
Parameter Description Typical values
σrslant std dev in the slant range 50 ft
σθ std dev in the relative bearing and own aircraft heading 10 deg
σhjitter std dev in intruder altitude due to random noise 0
q quantization in intruder altitude 25 ft
σh0 σ¨x std dev in own altitude std dev in the x position process 0 8 ft/s2 (white noise) / 30 ft/s2 (correlated)
σ¨y σ¨ h std dev in the y position process std dev in the h position process 8 ft/s2 (white noise) / 30 ft/s2 (correlated) 3 ft/s2
σv0 initial velocity std dev scale factor 250ft/s
where v(k) is the zero-mean white Gaussian process noise vector with covariance
..
2¨
xrel
00
σ
..
2¨
yrel
..
0
0
,
(B-28)
σ
Q =
2¨
00 σ
hrel
where x¨rel, y¨rel, and h¨ rel represent the accelerations in the relative x, y, and h dimensions, respec-tively. The variances of x¨rel, y¨rel, and h¨ rel are twice those of the individual aircraft, x¨, y¨, and h¨, respectively.
2
The .lter is initialized using the .rst measurement. The mean of the velocity components of the state is initialized to zero with an arbitrary covariance of the form σ
I3×3. The various
v0
parameters used in the horizontal tracker are listed in Table B-1.
It also lists some typical values used when tracking white-noise and correlated model encounters.
The posterior state distribution at time k, N (x(k); .x(k | k),P (k | k)), is a continuous dis-tribution. Because the logic accepts discrete distributions, the posterior state distribution must be sampled to produce a collection of samples with associated probabilities. Thirteen sigma-point samples were drawn from the posterior, and each sample was converted into the uncontrolled state representation of Section 5.3
as follows:
.
2
2
rel + yx˙
2
rel,
r = x
.
.
2
(B-29)
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