  Achieving stable hopping has been a hallmark challenge in the field of
dynamic legged locomotion. Controlled hopping is notably difficult due to
extended periods of underactuation combined with very short ground phases
wherein ground interactions must be modulated to regulate global state. In this
work, we explore the use of hybrid nonlinear model predictive control paired
with a low-level feedback controller in a multi-rate hierarchy to achieve
dynamically stable motions on a 3D hopping robot. In order to demonstrate
richer behaviors on the manifold of rotations, both the planning and feedback
layers must be designed in a geometrically consistent fashion; therefore, we
develop the necessary tools to employ Lie group integrators and appropriate
feedback controllers. We experimentally demonstrate stable 3D hopping, as well
as trajectory tracking and flipping in simulation.
