**Length:** 3 hours

**Intended Audience and Prerequisites:** The tutorial is intendend for listeners interested in non-linear state estimation problems and fundamental aspects of solving
them (by other means than sequential Monte-Carlo methods). Prerequisite is a knowledge of probability theory on graduate
level plus (linear and non-linear) algebra.

**Description:** The Kalman filter is well-known as the workhorse of recursive state estimation. For linear state estimation problems
under noisy measurements, the Kalman filter provides consistent estimates being optimal in various senses. In many tracking
applications however, the equations describing the state propagation and/or that for the measurement process are non-linear. As
long as those non-linearities remain mild in a certain sense, relatives of the Kalman filter suffice to provide suitable estimates
with at least approximate consistency.

The proposed tutorial focusses on a variety of those relatives. It starts by recalling the basic relationships and properties of
the Kalman filter. As a motivation for the later topics, its different optimality properties are shortly discussed as are various
ways to write in particular its update equation (e. g., the Joseph form and the information form). The latter then leads over to
the aspect of data fusion with convex combination (CC) and covariance intersection (CI) as two prominent approaches. The
interacting multiple model (IMM) is presented as a method to deal with jump-Markovian linear systems.

Starting from those results for linear systems, the generalization to non-linear systems is discussed. The best linear unbiased
estimator (BLUE) filter is introduced and general-purpose approximations like the extended Kalman filter (EKF), the unscented
Kalman filter (UKF), and the Gauss filter (GF) are put in its context, detailed for polar measurements. For that same example,
converted measurement (CM) filters are examined. While being straight-forward to implement, they come with the inherent
problem of generating estimation bias. This problem and proposed solutions are investigated.

An alternative to converting the measurements to state-(sub-)space is given by a conversion of the states to measurement
space. For the example of angular-only measurements, log-spherical coordinates (LSCs) are presented as such a suitable statespace
representation that is, in particular, capable to deal with the fact that part of the state-space is not observable by the
measurements. Filter initialization, propagation, and update are handled. The discussion on how to use those coordinates for
fusing state estimates from different types of sensors in sense-and-avoid-applications wraps up the proposed tutorial.

**Presenter:** Dietrich Fränken

**Dietrich Fränken** is systems engineer and expert for ground target tracking at Airbus DS Electronics and Border Security
GmbH, where he is responsible for tracking and data fusion in various national and international projects. Additionally, he
is a lecturer (Privatdozent) at Ulm University. He presents on a regular basis at conferences, universities, and in front of
customers. He is member of the technical program comittee for symposia and serves as a reviewer for scientific journals.
Research interests include modeling, simulation, and estimation of physical systems, nonlinear filtering, graph and system
theory, and digital signal processing. Dr. Fr¨anken holds a Dipl.-Ing. degree from Bochum University and a Dr.-Ing. degree as
well as a Habilitation degree from Paderborn University, all of them in electrical engineering.