Efficient and Guaranteed Geometric Methods for Motion Generation and Perception Taosha Fan Even though a number of techniques have been developed for motion generation and perception, few of them focus on the computational efficiency and theoretical guarantees at the same time. Typically, improved guarantees come with increased complexity, making theoretically guaranteed methods challenging use in real-time applications. Thus, existing methods usually have to ignore either efficiency or

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... es in practical implementation. Nevertheless, numerous problems in motion generation and perception require computational efficiency as well as theoretical guarantees, making the implementation of existing techniques strictly limited. To address this issue, we present efficient and guaranteed methods for motion generation and perception by utilizing geometry and optimization. In this thesis, we develop fast algorithms for higher-order variational integrators with linearand quadratic-time complexity for integration and linearization, respectively; we make use of the complex number representation to solve the planar graph-based SLAM that is not only certifiably correct but also more efficient and robust; we propose majorization