  In this paper, we holistically present a Hybrid-Linear Inverted Pendulum
(H-LIP) based approach for synthesizing and stabilizing 3D foot-underactuated
bipedal walking, with an emphasis on thorough hardware realization. The H-LIP
is proposed to capture the essential components of the underactuated and
actuated part of the robotic walking. The robot walking gait is then directly
synthesized based on the H-LIP. We comprehensively characterize the periodic
orbits of the H-LIP and provably derive the stepping stabilization via its
step-to-step (S2S) dynamics, which is then utilized to approximate the S2S
dynamics of the horizontal state of the center of mass (COM) of the robotic
walking. The approximation facilities a H-LIP based stepping controller to
provide desired step sizes to stabilize the robotic walking. By realizing the
desired step sizes, the robot achieves dynamic and stable walking. The approach
is fully evaluated in both simulation and experiment on the 3D underactuated
bipedal robot Cassie, which demonstrates dynamic walking behaviors with both
high versatility and robustness.
