Section: New Results
Robust state estimation (Sensor fusion)
This research is the follow up of Agostino Martinelli's investigations carried out during the last five years, which are in the framework of the visual and inertial sensor fusion problem and the unknown input observability problem.
Visualinertial structure from motion
Participant : Agostino Martinelli.
We have continued our study on the visual inertial sensor fusion problem in the cooperative case, with a special focus on the case of two agents. During this year, we have carried out an exhaustive analysis of all the singularities and minimal cases of this cooperative sensor fusion problem. As in the case of a single agent and in the case of other computer vision problems, the key of the analysis is the establishment of an equivalence between the cooperative visualinertial sensor fusion problem and a Polynomial Equation System (PES). In the case of a single agent, the PES consists of linear equations and a single polynomial of second degree. In the case of two agents, the number of second degree equations becomes three and, also in this case, a complete analytic solution can be obtained [19], [20]. The power of the analytic solution is twofold. From one side, it allows us to determine the state without the need of an initialization. From another side, it provides fundamental insights into all the structural properties of the problem. The research of this year has focused on this latter issue. Specifically, we have obtained all the minimal cases and singularities depending on the number of camera images and the relative trajectory between the agents. The problem, when non singular, can have up to eight distinct solutions. The usefulness of this analysis has also been illustrated with simulations. In particular, we have quantitatively obtained how the performance of the state estimation worsens near a singularity. The results of this research will be published by the Robotics and Automation Letter (RAL) journal [18].
Unknown Input Observability
Participant : Agostino Martinelli.
The Unknown Input Observability problem (UIO) in the nonlinear case was an open problem since the sixties years, when it was solved only in the linear case. In the last five years, I have obtained its general analytic solution. So far, I only published the solution for systems characterized by driftless dynamics. In particular, this solution was published as a full paper on the IEEE Transaction on Automatic Control [17]. In December 2018, I was invited by the Society for Industrial and Applied Mathematics (SIAM) to write a book with the general solution. This has been the main work of this year. Since this general solution is based on tensorial calculus (Ricci algebra) and many mathematics procedures and tricks borrowed from theoretical physics, the scope of book has gone much more beyond the presentation of the solution. Basically, by writing this book, I've obtained a new theory of observability.
The current theory of nonlinear observability, does not capture/exploit the key features that are intimately related to the concept of observability. This results in two important limitations:

The theory, although simple and based on elementary mathematics, can be sometimes burdensome with the risk of easily loosing the meaning of the results and losing the meaning of their assumptions.

More complex observability problems (e.g., the unknown input observability problem to which this book provides the complete analytic solution) remained unsolved for half a century.
The key to overcome the two above limitations, consists in building a new theory of observability that accounts for the group of invariance that is inherent to the concept of observability. This is the typical manner the research in physics has always proceeded. To this regard, I wish to emphasize that the derivation of the basic equations of any physics theory (e.g., the General Relativity, the Yang Mills theory, the Quantum Chromodynamics) starts precisely from the characterization of the group of invariance of the theory.
One of the major novelties introduced by this book is the characterization of the group of invariance of observability and, regarding the case of unknown inputs, the characterization of a subgroup that was called the Simultaneous Unknown Input Output transformations' group.
In summary, the book provides several novelties with respect to the existing literature in control theory. Specifically, the reader will learn the following:

The solution of two open problems in control theory (the book provides separately the solution and the derivation), which are:

A new and more palatable derivation of the existing results in nonlinear observability.

A new manner of approaching scientific and technological problems, borrowed from theoretical physics (a chapter summarizes in a very intuitive and quick manner the basic mathematics, which includes tensorial calculus).

A new manner of dealing with the variable time in system theory, which is obtained by introducing a new framework, which was called the chronospace.
I believe this book could be an opportunity for control and information theory communities to borrow basic mathematics, tricks, types of reasoning from theoretical physics to revisit many aspects of control and information theory.