IEEE Aerospace and Electronic Systems Magazine - December 2020 - 18

State Estimation Methods in Navigation: Overview and Application
with respect to validity of the estimates, into local and
global methods [6]-[8].

EXACT METHODS
The exact methods have been typically designed for a set of
the linear models. For this set, the solution to the BRRs
results in reproducible conditional PDFs, i.e., the conditional PDFs at subsequent time instants share the same distribution and, thus, recursive conditional PDF computation
reduces to recursive computation of conditional PDF parameters only. The exact methods are represented, e.g., by the
Kalman filter (KF) or the Gaussian sum filter (GSF) [5],
[94]. The KF, developed in the sixties, is an optimal4 estimator for the linear Gaussian models, i.e., for the linear model
(1), (2) with the state noise, measurement noise, and the initial condition described by Gaussian PDFs. The recursive
solution to the functional BRRs then collapses to the recursive computation of the conditional mean and covariance
matrices only, which fully describe the Gaussian conditional
PDFs. The GSF is an optimal estimator for the linear Gaussian sum models and can be imagined as a bank of concurrently running KFs [94]. Consequently, the conditional
PDFs are in the form of Gaussian sums and the solution of
the BRRs, then, lies in computation of the weights, means,
and covariance matrices of the particular terms of the Gaussian sum conditional PDF.

LOCAL METHODS
Local methods are based on two approximations; first, the
joint conditional predictive state and measurement PDF is
assumed to be Gaussian; second, the nonlinear functions in
(1) and (2) are linearized. The former approximation results
in a linear structure of a local filter (LF) with respect to the
measurement, and, together with the latter approximation,
it allows use of the (linear) KF design technique also for
nonlinear models. All LFs, therefore, share the same algorithm structure, but they differ in which linearization of the
nonlinear functions in (1) and (2) is used. In particular, two
different types of linearization can be found in the literature: derivative-based and derivative-free.
The derivative-based LFs, developed in the seventies,
approximate the nonlinear function by the Taylor expansion
(TE). Whereas utilization of the first-order TE leads to the
extended Kalman filter (EKF) or the linearized KF (depending on the selection of the linearization point) [5], [9],
approximation based on the second-order TE results in the
second-order filter (SOF). In the literature, several versions
of the SOF have been proposed [9], [10], [12] and, also, utilization of a higher order TE in the LF design has been
discussed [13].
4

The term " optimal " is, in this article, related to the exact solution to
the BRR.

18

The derivative-free LFs appeared at the beginning of the
century. They are based either on a polynomial expansion of
the nonlinear functions or on the approximation of the state
estimate by a weighted set of deterministically or stochastically selected points. The former approximation in the LF
design is represented by Stirling's interpolation (SI) of the
first or second order, which results in the divided difference
filters of the first or second order (DD1, DD2), respectively [14], [15]. The SI can be understood as the TE, where
derivatives are substituted with differences [8]. The latter
approximation takes advantage of a different idea, where
the nonlinear function is preserved, but the conditional
(Gaussian-assumed) PDF is approximated by a set of
points. This approximation is represented by the unscented
transformation5 (UT) [2], [9], [16], deterministic quadrature and cubature integration rules [7], [11], [17]-[19], and
stochastic integration rules [20], which results in the set of
the LFs including the unscented Kalman filter (UKF), cubature Kalman filter, the stochastic integration filter, or the
ensemble Kalman filter. Note that last mentioned filter,
propagating the set of randomly drawn samples instead of
the moments, is a suitable algorithm for a high-dimensional
state-space model [20], [98]. It is worth noting that
although the point-based approximations use a different
basic idea, they can be interpreted as examples of the statistical linear regression of the nonlinear functions [8], [21].
Examples of approximation of a scalar nonlinear
function fk ðxk Þ by the derivative-based first-order TE and
the derivative-free first-order Stirling's interpolation are
shown in Figure 2. It can be seen that the TE-based linearization f^k ðxk ; x^k Þ is more accurate in a close vicinity of
the linearization point x^k , whereas the SI-based is better
in the wider vicinity. The reason can be found in the fact
that the derivative-free approximation is computed over
an interval defined by the set of (transformed) points.
Independent of which nonlinear function approximation is used, all the LFs provide estimates in the form of
the first two moments of an approximate Gaussian conditional PDF, i.e., in the form of the conditional mean and
covariance matrix. The moments do not represent a full
description of the immeasurable state and are valid if and
only if the filter is working in the close vicinity of the true
state (thus, the name local), which is, however, not known
in practice. Therefore, significant attention has been
devoted to the theoretical analysis and monitoring of the
conditions under which a LF provides accurate6 and

5

The UT should be understood as a class of approximations rather
than one single approach. In the literature, various versions of the
UT have been proposed with different strategies to point
selection [16].
6
The LFs can be divided into first-order filters (e.g., the EKF, DD1)
and SOFs (e.g., the SOF, DD2, UKF). The latter are expected to
provide more accurate estimates, but it is not a rule (due to
unknown impact of neglected terms in the nonlinear function
approximation).

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DECEMBER 2020
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