February 14, 2025 - PhD defense of mgr inż. Sandra Zarychta

Promotor of the Thesis is professor Jerzy Wojewoda, and the auxillary promoter dr Marek Balcerzak.

Abstract: Discontinuous capsule drives, commonly referred to as capsubots, are devices designed to navigate environments that are physically inaccessible to humans. A particularly interesting subset of capsubots includes mechanisms driven by the periodic motion a mechanical oscillator. The interaction of the oscillating mass and the main body generates inertia forces, which, combined with the frictional environment, enable the capsule to move in the desired direction without external moving parts. This feature shows the great potential of these devices in many applications, such as engineering, pipeline inspection, and medicine. Notably, the latter has garnered substantial interest among scientists, presenting the potential of capsubots as a promising alternative to classic endoscopy.

Despite advancements in technology enabling the development of pill-sized capsule devices, effectively reaching critical areas of the intestines remains a challenge. Therefore, the need for fully controlled active robots is vital. The primary issue with such systems is finding an efficient control strategy considering the  operational costs, such as minimizing energy consumption or maximizing the capsule’s average progression. Furthermore, the control function must consider real-world constraints, such as the maximum available torque of the driving motor, requiring a bounded set of admissible controls. The complexity of the task is further compounded by the system’s operation in an intricate environment. Unfortunately, classical approaches to optimal control, as well as other methods found in the literature, seem to be insufficient to be used in such kinds of systems due to the discontinuity of the vector field or may be time-consuming and complex to solve, necessitating sophisticated numerical analysis, which adds further challenges.

To overcome these obstacles, in my dissertation, I proposed a novel method of control optimization based on Fourier series to address the optimal control problem in discontinuous systems effectively. This approach is particularly relevant for capsule robots, for which considering the real-world constraints of the control function seems to be significant. The methodology involves transforming the control function optimization problem into a nonlinear programming task, where a finite number of Fourier series terms is optimized. Unlike conventional solutions found in the literature, where the control function’s profile is predefined, this technique optimizes the shape of the control function as a part of the process.

The Fourier series-based method of control optimization, when applied to any of the selected mechanisms (pendulum capsule drive, vibro-impact, inverted pendulum), demonstrates significant effectiveness, regardless of whether the forcing is periodic or non-periodic. The method proved to be simple and flexible, requiring minimal information about the control object, i.e., a well-defined set of admissible controls, and a unique performance measure for any admissible control are needed.

These findings confirm the thesis of the dissertation, demonstrating that the developed algorithm is both straightforward and easy to implement to efficiently solve the optimal control problem in non-smooth and discontinuous mechanical systems. The proposed control optimization algorithm has the potential to significantly enhance the performance of discontinuous systems, particularly capsubots, such as vibro-impact or pendulum-like driven systems, when operating in a complex environment. These findings could yield significant advantages and contribute to advancements in classical mechanics and control theory.