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Although manipulation capabilities of aerial robots greatly improved in the last decade, only few works addressed the problem of aerial physical interaction with dynamic environments, proposing strongly model-based approaches. However, in real scenarios, modeling the environment with high accuracy is often impossible. In this work we aim at developing a control framework for OMAVs for reliable physical interaction tasks with articulated and movable objects in the presence of possibly unforeseen disturbances, and without relying on an accurate model of the environment. Inspired by previous applications of energy-based controllers for physical interaction, we propose a passivity-based impedance and wrench tracking controller in combination with a momentum-based wrench estimator. This is combined with an energy-tank framework to guarantee the stability of the system, while energy and power flow-based adaptation policies are deployed to enable safe interaction with any type of passive environment. The control framework provides formal guarantees of stability, which is validated in practice considering the challenging task of pushing a cart of unknown mass, moving on a surface of unknown friction, as well as subjected to unknown disturbances. For this scenario, we present, evaluate and discuss three different policies.