  This paper presents a hierarchical control framework that enables robust
quadrupedal locomotion on a dynamic rigid surface (DRS) with general and
unknown vertical motions. The key novelty of the framework lies in its higher
layer, which is a discrete-time, provably stabilizing footstep controller. The
basis of the footstep controller is a new hybrid, time-varying, linear inverted
pendulum (HT-LIP) model that is low-dimensional and accurately captures the
essential robot dynamics during DRS locomotion. A new set of sufficient
stability conditions are then derived to directly guide the controller design
for ensuring the asymptotic stability of the HT-LIP model under general,
unknown, vertical DRS motions. Further, the footstep controller is cast as a
computationally efficient quadratic program that incorporates the proposed
HT-LIP model and stability conditions. The middle layer takes the desired
footstep locations generated by the higher layer as input to produce
kinematically feasible full-body reference trajectories, which are then
accurately tracked by a lower-layer torque controller. Hardware experiments on
a Unitree Go1 quadrupedal robot confirm the robustness of the proposed
framework under various unknown, aperiodic, vertical DRS motions and
uncertainties (e.g., slippery and uneven surfaces, solid and liquid loads, and
sudden pushes).
