Aerospace and Electronic Systems Magazine May 2017 - 22

The Tracker Component Library
The filters are generally implemented as in the literature,
though generalizations are often made. One deviation from the
literature worth noting is the implementation of the pure propagation filter of [21]. This filter is specifically tailored to a particular
set of cubature points and requires that the points have a certain
symmetry, which might be broken after a few measurement update
steps. We have generalized the result to work with an arbitrary set
of cubature points by scaling the cubature points to account for
the effects of process noise rather than by adding various values
to the points in the filter. The change is simple. Given a set of
cubature points with covariance matrix P, we want to obtain a set
with covariance matrix P + Q. Thus, we solve the equation P + Q =
CPC′ for C, which is the factor by which a centered set of cubature
points must be scaled. The same type of scaling is also used in the
measurement update function purePropUpdate.
Most of these filters can be used in multiple model filters by
passing them as function handles to multipleModelUpdate and
multipleModelPred. The aforementioned multiple model functions implement the interacting multiple model estimator, the autonomous multiple model estimator, and the generalized pseudoBayesian estimators 1 and 2. Moreover, the implementation is
generic, allowing mixing of models having different numbers of
components that can mix as in [12].
As shown in [6], the cubature Kalman filter, the unscented Kalman filter, the sigma-point Kalman filter, and many others are all
essentially the same, differing in the choice of cubature points used
for numerical integration. All of the filters having "generic cubature/unscented/sigma-point" in their descriptions take basic cubature points and weights for the standard multivariate normal distribution as their inputs, allowing all of the aforementioned filters to
be used, as well as a great many forms that are not in the literature.
Cubature points for the filters are in the Mathematics/Numerical
Integration/Cubature Points folder of the library. Therein, routines
for many different types of weighting functions are given. The most
relevant ones for the filters are in the Gaussian Weight subfolder.
The notion of "weighting functions" stems from the general
concept of cubature integration, which is briefly overviewed in [6].
Basically, cubature points are chosen such that one can write



x∈

f (x) w(x)dx = ωi f (ξi )



f (x) w(x)dx = ωi f (ξi )
i =1

(1)

=1 and equality holds for all real
x∈ is some region in real ispace
where
multivariate polynomial functions f up to a specified total (the order)
degree. The ξi are cubature points, and the ωi are cubature weights,
which are chosen based on the real weighting function w. For nonpolynomial functions, the use of cubature points is an approximation,
as is generally the case in problems involving target tracking.
For track filtering, the most relevant functions are inside the
Gaussian Weight subfolder. These functions are

1. secondOrderCubPoints implement second-order simplex sigma points used in unscented Kalman filters.
2. thirdOrderCubPoints implement five types of cubature points,
including those most commonly used in unscented/cubature
filters and two from outside the tracking literature.
22

Figure 1.

A plot of a single Monte Carlo run of a 2D angle-only tracking problem
when used with 12 different filtering algorithms, as implemented in the
sample code given by the demoNonlinearMeasFilters function.

3. fifthOrderCubPoints implement 14 types of fifth-order cubature points, most from outside the tracking literature.
4. seventhOrderCubPoints implement nine types of seventh-order
cubature points, all from outside the tracking literature.
5. arbOrderGaussCubPoints implement arbitrary-order cubature
points and allow for weighting functions that are a Gaussian
times norm(x)β.
6. GaussianLCDSamples generate arbitrary-order approximate
cubature points using a stochastic method based on localized
cumulative distributions.
7. GenzKeisterPoints use one of two algorithms to generate cubature points over a number of orders.
8. spherSurfPoints2GaussPoints transform cubature points for a
spherical shell (in the Sphere subfolder) to work with a multivariate Gaussian distribution.
9. transformCubPoints transform the cubature points to be used
with Gaussian distributions other than a standard normal distribution.
In addition to the aforementioned functions, outside of the Gaussian Weight folder, there are also the functions quadraturePoints1D
and linCubPoints2MultiDim. The function quadraturePoints1D
generates one-dimensional cubature points of a desired order, and
the function linCubPoints2MultiDim allows those points to be
made to work over multiple dimensions by using one of two algorithms.
Thus, with the wide choice of available cubature points, one
can construct far more types of cubature/unscented/sigma-pointstyle filters than currently exist in the literature.
In the Sample Code folder provided with the library, 12 different measurement update filters are compared on a simple twodimensional (2D) tracking scenario demoNonlinearMeasFilters
function, where angle-only measurements from a moving platform
are used in a 2D scenario, as shown in Figure 1. The use of multi-

IEEE A&E SYSTEMS MAGAZINE

MAY 2017

Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine May 2017