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).

Accurate control of a humanoid robot's global position (i.e., its three-dimensional position in the world) is critical to the reliable execution of high-risk tasks such as avoiding collision with pedestrians in a crowded environment. This paper introduces a time-based nonlinear control method that achieves accurate global-position tracking (GPT) for multi-domain bipedal walking. Deriving a tracking controller for bipedal robots is challenging due to the highly complex robot dynamics that are time-varying and hybrid, especially for multi-domain walking that involves multiple phases/domains of full actuation, over actuation, and underactuation. To tackle this challenge, we introduce a continuous-phase GPT control law for multi-domain walking, which provably ensures the exponential convergence of the entire error state within the full and over actuation domains and that of the directly regulated error state within the underactuation domain. We then construct sufficient multiple-Lyapunov stability conditions for the hybrid multi-domain tracking error system under the proposed GPT control law. We illustrate the proposed controller design through both three-domain walking with all motors activated and two-domain gait with inactive ankle motors. Simulations of a ROBOTIS OP3 bipedal humanoid robot demonstrate the satisfactory accuracy and convergence rate of the proposed control approach under two different cases of multi-domain walking as well as various walking speeds and desired paths.