IEEE Aerospace and Electronic Systems Magazine - December 2020 - 21

Duník et al.
happen that the state domain is subject to certain constraints,
e.g., estimated pressure cannot be negative, estimated vehicle position needs to be aligned with a road. Such constraints
(belonging to a priori knowledge) cannot be incorporated in
the state-space model to be straightforwardly used by the
introduced state estimators. Therefore, various purposely
designed approaches for constrained state estimation have
been proposed [9], [49].

MULTIPLE MODEL STATE ESTIMATION AND
PARAMETER ESTIMATION
The nonlinear functions fk ðÁÞ and hk ðÁÞ of the model (1), (2)
need not necessarily be smooth, but they can be viewed as a
set of possibly simpler functions. As an example, consider
(radar-based) tracking of aircraft, a highly manoeuvrable
object. Aircraft dynamics can be described by a set of models, where each is suitable for the description of a different
phase of flight (e.g., an almost constant velocity or acceleration model, a constant turn-rate model). During tracking,
however, a suitable model can hardly be determined a priori as it is difficult to predict aircraft future manoeuvrers.
Therefore, the concept of the (interacting) multiple model
(MM) has been developed [23]. The MM approach is based
on the determination of a set of possible models of a system
under different " working conditions. " For each model,
then, a filter (typically an LF) is constructed and its likelihood w.r.t. available measurements is computed. The output of the MM filter can thus be either a weighted sum of
all particular local estimates or the estimate with the highest likelihood. Note that the MM filter is algorithmically
similar to the GSF. The MM approach has been significantly developed over recent years and MM-based state
estimation approaches capable of tackling data association,
clutter measurements, and estimating set variables have
been proposed. Examples of these are the multiple hypothesis tracking filter and probability hypothesis density filter [101]-[104].
The MM approach is also suitable for tasks, where the
state-space model contains parameter(s) that are unknown
but can acquire a value from an a priori known set. Then, a
set of filters is designed for each potential parameter value
and is used in the MM filter [50]. Note that alternative
approaches for concurrent estimation of the state and parameters are the joint and dual estimation approaches [51]. The
former approach is based on extension of the state vector
with the unknown parameters (resulting in the extended
state-space model) and their simultaneous estimation by,
typically, a nonlinear filter. The latter approach is based
on a definition of two filters, which are regularly
switched. The first one estimates the unknown states
under the assumption of given parameters and the second
one estimates the unknown parameters under the
assumption of a given state.
DECEMBER 2020

SOFTWARE TOOLS
An extensive number of state estimation algorithms have
been proposed so far. It is, therefore, a challenging task to
choose a suitable estimator for a given task or application.
Fortunately, many of the estimators have been implemented
and can be used for assessment (or prediction) of a filters'
performance. Most of the methods and toolboxes are
designed for the MATLAB environment. The estimators are
available in MATLAB proprietary toolboxes or in publicly
available toolboxes. The latter includes, e.g., the Nonlinear
Estimation Framework toolbox available at http://nft.kky.
zcu.cz/nef, the EKF/UKF Toolbox available at https://
github.com/EEA-sensors/ekfukf, or the DynaEst Toolbox
available at http://www.codeforge.com/article/41828. A
wide portfolio of routines for tracking is available in the
Tracking Component Library available at https://github.
com/USNavalResearchLaboratory/TrackerComponentLibrary [93]. Besides the toolboxes for the MATLAB, there
are also early toolboxes in Python, a modern widely used
programming language. An example is the International
Society of Information Fusion (ISIF) StoneSoup initiative,
which can be found at https://stonesoup.readthedocs.io/en/
latest/index.html. Note that some books also come accompanied with sample implementations, e.g., [2], [4], [23].

LITERATURE ON STATE ESTIMATION AND NAVIGATION
Because of the scope of this article, it was not possible to
mention, discuss, and address all topics, details, and recent
advancements in state estimation methods. Nevertheless, in
the literature, there is an extensive number of comprehensive books and survey papers on state estimation and navigation system design, which offer an in-depth description of
these areas. To name a few, the following references focused
on estimation theory [1], [2], [5], [9], [12], [16], [29], [34],
[52] and navigation system design [3], [4], [23], [60], [105].

ESTIMATION IN NAVIGATION SYSTEMS
The origin of state estimation methods is closely associated with the development of navigation and tracking systems. Indeed, any modern navigation system uses a state
estimation algorithm for optimal processing of data from a
variety of sensors. In this section, recent developments
and examples of application of state estimation methods
are provided with the stress on the area of expertise of the
AESS NSP members. Besides recent navigation applications, this section also discusses their expected level of
nonlinearity/non-Gaussianity and error sources of the sensors and maps typically used in the navigation system.
Knowledge of the considered task nonlinearity/non-Gaussianity and correct treatment of the sensor (or map) error
sources is essential for the selection of the appropriate

IEEE A&E SYSTEMS MAGAZINE

http://www.nft.kky.zcu.cz/nef http://www.nft.kky.zcu.cz/nef https://github.com/EEA-sensors/ekfukf https://github.com/EEA-sensors/ekfukf http://www.codeforge.com/article/41828 https://github.com/USNavalResearchLaboratory/TrackerComponentLibrary https://github.com/USNavalResearchLaboratory/TrackerComponentLibrary https://github.com/USNavalResearchLaboratory/TrackerComponentLibrary https://stonesoup.readthedocs.io/en/latest/index.html https://stonesoup.readthedocs.io/en/latest/index.html

IEEE Aerospace and Electronic Systems Magazine - December 2020

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