With the development of more sophisticated robots that are increasingly aimed to venture outside of the lab, the higher dimensionality, complexity and nonlinearities of the underlying robotic systems as well as environmental uncertainty pose challenges to reliance on nominal models of robots obtained from first principles. Koopman operator theory, a tool first introduced for data-driven model extraction and global linearization, and its various extensions have been widely implemented in robot modeling and control. This dissertation contributes fundamental theory and practical algorithms for the implementation of the Koopman operator theory across distinctive types of robots.

Specific contributions of this dissertation span over four key sub-phases. These include data collection, model extraction, controller design, and physical implementation. First, the practical scenario of working with noisy data for modeling is considered. Prediction errors because of noisy measurements when estimating the model via the data-driven Koopman operator-based approaches for control are derived. Then, the explicit quantification of the error is embedded into an existing Koopman-based data-driven robot modeling and control architecture to enhance its robustness without making significant changes to current parts of the underlying structure. Second, considering the importance of the space-lifting process in the Koopman theory, we propose a general and analytical algorithm to formalize the construction of the lifting functions based on characteristic properties of robots - namely their configuration space or, in the case of soft robots, their workspace. The resulting design of the lifting functions is proven to be complete and leads to an approximated Koopman operator with provable guarantees of convergence to the true one. Finally, we develop and present Koopman-based controllers and implement them to drive and/or improve the performance of robots. The first design is an online modeling and control approach via the Koopman operator theory for grasping tasks using soft multi-fingered robotic grippers. Then, a hierarchical structure that refines the reference signal sent to an existing high-rate, pre-tuned low-level controller to deal with uncertainty is presented, which aims to decrease the effect of environmental disturbance in real time.

The advancements in data analysis, model extraction, controller design, and their applications to robots, as presented in this dissertation, lay the groundwork for the practical implementation of the Koopman operator theory in physical robotic platforms. The theoretical and experimental explorations, along with the enhancements made, open up new possibilities for modeling and controlling nonlinear robots operating in uncertain environments. These advancements enable the achievement of more intricate objectives and can contribute to the overall progress in the field of robotic control.

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