This project focuses on the controllability and observability of systems governed by high-order partial differential equations. The project scope is twofold. On the one side, the analysis of controllability and observability for such class of infinite-dimensional systems, along with the synthesis of optimal control laws and distributed state observers, will be provided from a purely theoretical viewpoint. On the other side, the obtained results will be instrumental for pursuing advances and improvements in challenging applications within the biomedical domain. In particular we will address the task of controlling and simulating the motion of flexible catheters, the problem of controlling continuum-robots by means of magnetic actuation and the operation of automatic positioning of probes for thermal ablation procedures.
The novelty of the proposed research relies on the use of advanced theoretical tools from the mathematical theory of partial differential equations applied to the control of some classes of high-order infinite-dimensional systems, together with the development of efficient and accurate approximation methods and simulation tools.
The foreseen advances with respect to the state-of-the-art include a deep analysis of the optimal control problem for the Euler-Bernoulli beam equations with dynamic boundary conditions and singularly distributed actuators as well as the study of the somewhat dual problem of observer design with distributed sensors.
The obtained theoretical results may lead, in turn, to improvements in the application to the control of flexible catheters.
For instance, in the current version of the demonstrator described in Fig.1, needle-tissue interaction forces are computed according the to scene geometry, the robot state and a theoretical model of the interaction dynamics at the needle tip. The deformation of the needle and the tissues, however, are not taken into account. Using the studies in this project, we expect to be able to compute the error with respect to the target in real-time and use it in a closed loop control scheme with formally proven stability properties. The insertion progress with associated deformation of needle and tissues will be simulated and visualized in the virtual environment.
A strong requirement for this proposal is in fact to have simulation models that are able to produce outputs that can be super-imposed to real data and thus do not diverge from what is expected. This is useful not only for training but also to augment the visual feedback provided to the user in real interventions.
In the case of magnetic actuation of a continuum-robot, motion control can be obtained by means of an additional robot manipulator with a magnetic source attached to its end-effector. By moving the end-effector, the induced magnetic field will vary according to some reference value entailing the motion of the controlled robot. Our goal is to use Maxwell equations for tracking electromagnetic field references in order to increase the degrees of actuation by allowing the electromagnetic field to be shaped dynamically. The result will apply also to the field of micro-robotics that is gaining an ever growing attention in the medical domain.
Finally, regarding the application to thermal ablation, the use of a new theoretical tool for the automatic positioning of the probe taking into account all the forces acting on the probe (e.g. mechanical deformation of the tissues, elastic forces, etc) as well as the electromagnetic field distribution as achieved by solving Maxwell's equations is
expected to provide enhanced performances in comparison with traditional methods such as the classical finite difference approach.
Overall, at least 5 journal papers and 3 conference papers collecting the project results are expected to be published.