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7.1 SENSING
The sensor model used in this report is based on the current TCAS sensor. The sensor measures the slant range and bearing of all nearby intruder aircraft. The slant range error is modeled as a zero-mean Gaussian with 50ft standard deviation. The bearing error is modeled as a zero-mean Gaussian with 10° standard deviation. There is no jitter in the intruder altitude measurements, but they are quantized at 25 ft intervals. It is assumed that the own altitude, vertical rate, and heading are provided by the onboard avionics without error. The model does not account for biases.

7.2 STATE ESTIMATION PROCESS
The belief state is updated recursively using all measurements up to and including the current time.
Figure
27
outlines
the
steps
in
the
state
estimation
process.
Upon
receipt
of
the
.rst
set
of
measurements, the state estimator, using the information provided in the .rst measurements and any prior knowledge about the state, computes the posterior state distribution, which re.ects all present knowledge about the state given all information up to and including the present time. The state estimation component uses a sensor model together with a dynamic model to perform the estimation. As new information becomes available, a new posterior distribution is calculated, using the previous posterior distribution as the prior. State estimation using the Kalman .lter follows this general .ow. A Kalman .lter performs Bayesian recursive estimation using Gaussian posterior distributions
[44–46].

state distribution update
The belief state is constructed using a composition of several di.erent processes. Each process produces a collection of samples with associated probabilities representing di.erent portions of the total belief state. The belief state is the Cartesian product of the collection of samples from all processes, representing the joint distribution over the following variables: h, h˙0, h˙1, sRA, r, rv, and θv. These individual processes are enumerated below.
1.
Using observations of the slant range, bearing, and altitude of the intruder, as well as obser-vations of the own altitude and heading, an unscented Kalman .lter is used to estimate the Gaussian posterior distribution over the variables descriptive of the relative horizontal motion. Sigma-point samples from this distribution are drawn with associated weights, which are in-terpreted as probabilities. Each sample is converted into the three-dimensional representation consisting of r, rv, and θv.

2.
Another Kalman .lter is used to estimate the Gaussian posterior of the intruder altitude and vertical rate given observations of intruder altitude. Because it is assumed that the own altitude and vertical rate are noiseless, no estimation is required for those quantities. The relative altitude h is estimated by the maximum a posteriori estimate of the intruder altitude from the Kalman .lter minus the own altitude measurement. The intruder vertical rate h˙1 is estimated by the maximum a posteriori estimate of the intruder vertical rate from the Kalman .lter. Because uncertainty in the intruder altitude and vertical rate is not projected to be signi.cant, the maximum a posteriori estimates were used instead of the entire distribution in order to reduce computation time.

3.
The own vertical rate is used to estimate the distribution over the advisory state variable sRA according
to
Eq.
(29).
Because
the
posterior
distribution
of
sRA is discrete, the set of points on the support of the distribution serves as a collection of samples with their associated probabilities.

Further details regarding Kalman .ltering as well as the details of the .lters particular to this work can
be
found
in
Appendix
　
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