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• Range Tracker. An unscented Kalman filter is used to estimate the relative ground range and
ground range rate of the target aircraft. This filter was designed as a third-order UKF using a
range squared motion model to handle the nonlinear property that range is always a positive
value while its first and second derivatives, range rate and range acceleration, can be negative
or positive values. The states are ordered in the state estimate as [r dr ddr]. While this third-
order filter maintains an estimate of range acceleration, only the range and range rate estimates
are provided to the TRM.
When an STMREPORT is generated for consumption by the TRM, the azimuth and cross ground
range rate components of the Cartesian Track are combined with the Range Track to create a single
distribution that represents the relative horizontal state of the target. This process is described in
COMBINEANDSAMPLE (Algorithm 102). The approach of separating the problem of tracking the
relative horizontal state of a target aircraft into range and azimuth components was found by 
to yield superior tracker level and system level performance when compared against a stand alone
18.104.22.168 Cartesian Tracker
The Cartesian Tracker is initialized with INITIALIZECARTESIANTRACKER (Algorithm 16). This
algorithm takes as input the associated MODESTRACKFILE or MODECTRACKFILE (trk), mea-
sured slant-range (r_slant), absolute bearing (Chi_abs), and relative altitude (z_rel). If the absolute
bearing to the intruder is invalid (i.e., evaluates to a NaN), the algorithm produces a no-bearing Track
File. This special condition is represented by setting all elements of the tracked state (trk.mu_cart)
and covariance (trk.Sigma_cart) to NaN, resetting the number of track updates (trk.updates_cart) to
zero and setting the Cartesian Track’s validity flag (trk.valid_cart) to false.
If the absolute bearing to the intruder is valid, a check is performed to make sure that the range mea-
surement is always at least as big as the relative altitude measurement in magnitude to ensure that
certain matrices remain positive semi-definite. The initial position distribution is obtained through
the non-linear transformation of the observation from its native coordinate frame to Cartesian coor-
dinates. This process is done in the following steps:
1. The observation distribution represented by the observed values (r_slant, Chi_abs and z_rel)
and the observation noise matrix, R_k, is sampled using SIGMAPOINTSAMPLE (Algorithm
2. For each sample, the slant range is verified to be larger than the absolute relative altitude, oth-
erwise the zero_ground_range flag is set. Each sample is converted into Cartesian coordinates
(0_xy) using CONVERTTOCARTESIAN (Algorithm 18).
3. WEIGHTEDMEANANDCOVARIANCE (Algorithm 19) is used to calculate the weighted mean
and covariance in the Cartesian coordinate frame (mu_xy, Sigma_xy) from the samples.
In the event that any sample corresponds to states for which the ground range is zero, a constant
gamma is added to all diagonal elements of the position covariance to ensure numerical stability.
The initial velocity distribution is set to a zero-mean Gaussian with covariance (U) read from the
parameters file since it cannot be determined from the first measurement alone. Also for numerical
stability, kappa is added to the diagonal of the full covariance matrix.
© 2018 RTCA, Inc.
ACAS Xa/Xo MOPS, Vol. II
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