rv =
rel +˙yrel,
.
.
θv = atan2( ˙yrel,x˙rel) . atan2(yrel,xrel).
Figure B-3
shows the root-mean-square error in r, rv, and θv over the course of 50 seconds, averaged over 100,000 encounters from both the white-noise encounter model and the correlated en-counter model. The white-noise encounters were produced using an 8ft/s2 horizontal acceleration, 3ft/s2 vertical acceleration. The own aircraft was equipped with the TCAS-like sensor of Sec-tion 7.1.
The performance of the unscented Kalman .lter is compared to a simple .nite-di.erence
6000
1000 UKF (correlated)
r RMS error (ft)
FD (correlated) UKF (white noise)
FD (white noise)
500
r v RMS error (ft/s)
4000
2000
00  10 20 30 Time (s) (a) r RMS error.  40  50  00  10 20 30 Time (s) (b) rv RMS error.  40  50 
200 

θv RMS error (deg)
150 
100 
50 
00  10 20 30 Time (s) (c) θv RMS error.  40  50 
Figure B-3. Horizontal tracking performance.

approach. The .nite-di.erence approach produces better estimates of the horizontal range because it can be uniquely determined from the slant range, intruder altitude, and own altitude measure-ments, which are not very noisy. However, when making use of the intruder bearing measurement, which can be very inaccurate, the intruder can be mislocated, leading to large range errors. Given su.cient time, however, the horizontal range error becomes small.

B.4 VERTICAL TRACKER
The altitude and vertical rate of the intruder are estimated using a regular Kalman .lter. The tracker receives quantized measurements of the intruder altitude, z = h1, from which it estimates the altitude and vertical rate, x =[h1 h˙1]. The measurement function is
z = Hx + w = [1 0] x + w. (B-30)
TABLE B-2
Parameters for the vertical Kalman .lter

Parameter  Description  Typical values 
σhjitter q σ¨ h1  std dev in intruder altitude due to random noise quantization in intruder altitude std dev in intruder altitude process  0 25 ft 3 ft/s2 
σ˙h1  std dev in initial vertical rate  100 ft/s 

0 1020304050 0 1020304050
Time (s) Time (s)
(a) h1 RMS error. (b) h˙1 RMS error.
Figure B-4. Vertical tracking performance.
The variance of w is σ2 = σ2 +q2/12, as in the horizontal tracker. The state-transition function,
whjitter
similar
to
Eq.
(B-27),
is
given
by

x(k +1) = Fx(k)+ Gv(k)
1
1Δt 2(Δt)2 (B-31)
= x(k)+ v(k),
01 Δt
where σv2(k) = σ2.
¨
h1
The .lter is initialized using the .rst measurement. The mean of the vertical rate is initialized to zero with an arbitrary variance σ2. The various parameters used in the vertical tracker are listed
˙
h1
in
Table
B-2.
It
also
lists
some
typical
values.

Figure
B-4
shows
the
root-mean-square
error
in
h1 and h˙1 over the course of 50 seconds, averaged over 100,000 encounters from both the white-noise encounter model and the correlated encounter model. Both aircraft in the encounters were not equipped with collision avoidance systems; therefore their vertical motion did not change in response to resolution advisories. The performance of .nite-di.erence tracking is showed as a baseline.

B.5 DISCUSSION
This
appendix
presented
details
of
the
horizontal
and
vertical
trackers
used
in
Section
7.
Although
the horizontal and vertical estimation was decoupled, in reality there may be correlation between the horizontal and vertical variables that can only be captured by performing estimation of the entire state given all the measurements. This is unlikely to have a signi.cant impact on the results. The trackers can be enhanced in various ways. For example, the vertical tracker can explicitly model the
　
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