  This paper seeks insight into stabilization mechanisms for periodic walking
gaits in 3D bipedal robots. Based on this insight, a control strategy based on
virtual constraints, which imposes coordination between joints rather than a
temporal evolution, will be proposed for achieving asymptotic convergence
toward a periodic motion. For planar bipeds with one degree of underactuation,
it is known that a vertical displacement of the center of mass---with downward
velocity at the step transition---induces stability of a walking gait. This
paper concerns the qualitative extension of this type of property to 3D walking
with two degrees of underactuation. It is shown that a condition on the
position of the center of mass in the horizontal plane at the transition
between steps induces synchronization between the motions in the sagittal and
frontal planes. A combination of the conditions for self-synchronization and
vertical oscillations leads to stable gaits. The algorithm for
self-stabilization of 3D walking gaits is first developed for a simplified
model of a walking robot (an inverted pendulum with variable length legs), and
then it is extended to a complex model of the humanoid robot Romeo using the
notion of Hybrid Zero Dynamics. Simulations of the model of the robot
illustrate the efficacy of the method and its robustness.
