A Structure-Preserving Learning Scheme on SO(3)

Gaussian process regression has been extensively studied due to its ability for learning and predicting a dynamic process. Though this data-driven approach has received considerable attention, it shows the drawback of ignoring the physical prior of the system. In this paper, a structure-preserving Gaussian process regression is proposed for learning and predicting the $\mathrm{SO}(3)$ dynamics. The dynamics of the satellite is represented under the framework of geometric mechanics, which results in a reduced equations of motion. As such, by using variational mechanics the priori estimation is constructed to determine the posteriori mean during the Gaussian process regression. The obtained learning method inherits the advantages of the numerical integrator, such as energy-conserving and constraint-preserving. The benefits of the proposed approach are highlighted in a numerical simulation with a real cuboid satellite.