General description of our course on Optimal Filtering
Scope
The MP208 course provides a solid introductory research background for students with an interest in Kalman filtering theory as well as its aerospace applications, such as navigation, attitude determination, and object/target tracking. Most of the course is dedicated to the theoretical aspects of optimal state estimation problems from which one can say that it is useful for students interested in developing novel related estimation techniques or for applications in other kinds of dynamic systems (e.g., biological, financial, populational, and in infectology).
More on our main applications
Navigation can be considered the main application with which this course is concerned. In general, we can say that it consists of estimating the position and attitude of a platform (e.g., airplane, eVTOL, drone, robot, ship, and car) from the measurements provided by the onboard sensors. Usual navigation sensors include inertial ones (accelerometers, rate gyroscope), magnetometers, GNSS receivers (GPS, Galileo, Glonass, and Compass), different kinds of cameras, and Lidar. These sensors can be differently combined, but the schemes involving GNSS receivers and inertial sensors are the most common ones in flying vehicles.
A Graduate Course
MP208 is usually offered to the ITA’s Graduate Program on Aeronautical and Mechanical Engineering.
Methodology
The course is organized in 8 + 8 weeks, with weekly classes of 3 hours, adding up to 48 face-to-face hours. The methodology is based on lectures, discussions, and simulation exercises in MATLAB.
For more details about the course structure and methodology, see the syllabus.
Sequence of topics
Introduction
This chapter starts by introducing the concepts of filtering, prediction, and smoothing as well as the four optimality criteria adopted here (LS, ML, MMSE, and MAP). Then it presents a brief history of the optimal estimation problem, talks about some aerospace applications, and finishes with the formulation, design, and simulation of a simple linear estimator (the Luenberger observer) applied to the estimation of a drone’s vertical motion using measurements of a downward pointing sonar – Slides.
Fundamentals review
For completeness, this chapter is presented to support a comprehensive review of the required mathematical background. It is divided into seven sections, respectively concerned with linear algebra and matrices, linear dynamic systems, set theory, probability theory, random variables, random vectors, and stochastic processes – Sec 2.1, Sec 2.2, Sec 2.3, Sec 2.4, Sec 2.5, Sec 2.6, Sec 2.7.
Parameter estimation
This chapter formulates some parameter estimators based on the four adopted optimality criteria: least squares (LS), maximum likelihood (ML), minimum mean square error (MMSE), and maximum a posteriori probability (MAP). Moreover, the main properties of the studied estimators are also derived. They include bias, covariance, mean square error, consistency, and efficiency (related to the Cramér-Rao lower bound) – Slides.
Kalman filter
This chapter finally presents the Kalman filter (KF), which is the MMSE optimal state estimator, in a filtering sense, of a linear Gaussian system. Slides.
Computational aspects
This chapter is concerned with different formulations of the Kalman filter that make it better in terms of computations. In particular, we present the information filter, the square root filter, and sequential updates – Slides.
Extended Kalman filter
This chapter presents the extended Kalman filter (EKF), which is the most popular approximation of the Kalman filter for nonlinear systems. It is based on Taylor series linearization around the most recent state estimates. The chapter is divided into two parts, the first for the discrete formulation and the second for the continuous-discrete one – Part I, Part II.
Unscented Kalman filter
This chapter presents the unscented Kalman filter (UKF), another approximation of the KF, which instead of using Taylor series linearization, adopts a “statistical” linearization based on a sample of sigma points. Like the previous chapter, the present one is also divided into two parts, the first for the discrete formulation and the second for the continuous-discrete one – Part I, Part II.
Ensemble Kalman filter
This chapter presents the ensemble Kalman filter (EnKF), which is a particle filter based on the KF structure. The present chapter is also divided into two parts, the first for the discrete formulation and the second for the continuous-discrete one – Part I, Part II.
An attitude determination problem for drones
This chapter formulates an attitude determination problem based on measurements of inertial sensors (accelerometers and rate gyros) and magnetometers. This formulation is useful for attitude determination of drones – Slides.
An image-based navigation problem for drones
This chapter formulates a navigation problem based on measurements of inertial sensors (accelerometers and rate gyros) and fiducial markers acquired from a downward-facing camera. This formulation is useful for estimating the translational states of drones in a GNSS-denied scenario – Slides.
Besides this course, I am currently offering at ITA two more subjects, one on linear control systems and the other one on the dynamic modeling, control, and guidance of multirotor aerial vehicles (drones and eVTOL). You can find more information about these courses at the following links:
MP282: Dynamic Modeling and Control of MAVs
Do you like this course? If you want to know more about it, send me an e-mail. It will be a pleasure for me to share and exchange knowledge on this subject!