Modeling the Dynamics of Flexible Aerospace Vehicles Using the Theory of Functional Connections

Modeling and control of flexible vehicles is a topic of high interest in the aerospace field and a key challenge lies in finding accurate mathematical representations of the flexible dynamics of continuous elastic structures that allow simple integration into estimation and control algorithms. The answer was found in approximating the dynamics of these systems with sets of coupled Ordinary Differential Equations (ODE) for which a well-established estimation and control theory is available. Each of the techniques employed to achieve this goal is characterized by its own strengths and limitations. Hence, the main objective of this research is to develop the essential tools to enable a modeling methodology capable to represent the coupled structural and attitude dynamics of flexible spacecrafts in a simple and compact form while addressing the limitations in existing techniques. The aim of the research is pursued by employing the Theory of Functional Connections (TFC), that is a recently developed mathematical framework to perform functional interpolation in combination with the Lagrangian mechanics. Specifically, after providing an overview of this two fundamental tools, the key principles of the developed theory are outlined, and the resulting modeling strategy is presented in its general logic. Selected applications of the theory to cases of practical interest will be shown. In each case, a full explanation of the theoretical derivation of the solution is provided to show how the theory can be applied in the practice. Finally, numerical simulations were run to validate the resulting models and show the potential of the proposed theory in developing accurate and efficient representations of the dynamics of flexible structures and vehicles. The results show effectiveness of the methodology and open the possibility of extending it to more complex objects and investigating the interaction between the resulting models and some control and estimation algorithms.