Gaussian Process Modeling on Directional Manifolds for Data-Driven Estimation of Rigid Body Motion

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The project aims to develop novel schemes for recursive estimation of rigid body motion by adapting Gaussian processes to domains of directional quantities. Rigid body motion incorporates both rotation and translation. Conventional modeling and filtering schemes tackle the nonlinear group structure of rigid body motions based on linearization with the assumption of local perturbations. The DFG project "Recursive Estimation of Rigid Body Motions" of the applicant has shown promising results to apply directional statistics for stochastic filtering over rigid body motions. By directly modeling uncertain directional states on the (hyper)spheres, local linearization is no longer required. Thus, a better filtering performance under large uncertainties and fast motions is expected. As a systematic extension, the proposed project considers rigid body motion estimation with complex systems that are hard to model. Gaussian processes (GPs) are to be modified adaptively to the nonlinear group structure for non-parametric modeling to further enhance motion estimation in a data-driven manner. We first focus on GP regression techniques with only the kernels interpreting the geometric structure of the underlying manifolds. This will further be combined with deterministic sampling schemes to incorporate uncertain inputs during online inference for GP-based filtering. Moreover, we extend the approaches to functions with both input and target sets on (hyper)spheres. Then, GP-based filters are to be extended further to manifolds parameterizing rigid body motion. In summary, the proposal focuses on the systematic development of GP non-parametric modeling on domains involving rigid body motion. The proposed framework is expected to benefit many robotic pose estimation and control tasks in which high-fidelity models of complex systems are required.