In this paper, we propose to achieve exponentially stable periodic bipedal walking based on recharacterization of bipedal gait and generalization of walking encoding. To conveniently define an asymmetric walking pattern, a gait is characterized here in terms of the left and the right legs instead of the support and the swing legs. Another benefit of this characterization is that the joint positions become well-defined and continuous throughout a walking process even under impulse effects caused by impacts. A more general walking encoding method is then introduced, which not only includes walking pattern encoding but also enables upper-level task planning and control. Walking dynamics is then rewritten with the roles of the left and the right legs differentiated and with the biped's global position included. The desired walking pattern, as well as the desired global motion, is tracked exponentially fast through a controller designed using the output feedback linearization method. Stability of the hybrid dynamical control system is analyzed based on the construction of multiple Lyapunov functions. Finally, a fully actuated compass-gait biped is simulated to show that the proposed framework can realize exponentially stable walking, both symmetric and asymmetric, while satisfactorily tracking the desired walking pattern and the planned global motion.

Locomotion stability of a humanoid robot is closely related to the capacity to regulate its Center of Mass (CoM) motion. In this paper, the Feasible Center of Mass Dynamic Manipulability (FCDM) is introduced and analyzed as a measure of this capacity. The effects of posture, joint velocities and gravity on the torque-bounded dynamic manipulability ellipsoid are first analyzed on an n-DOF planar humanoid robot with single-foot support. The ellipse orientation has a linear relationship with the ankle angle, and its shape is independent on the ankle angle. Furthermore, three common and important ground-contact constraints - the unilateral contact-force constraint, the friction constraint, and the Center of Pressure constraint - are incorporated in the derivation of FCDM. It shows geometrically how each of the three constraints shrinks the original torque-bounded manipulability polytope and affects the maximum achievable CoM acceleration in different directions. Finally, a push recovery task was simulated to show that a robot's posture affects the feasible range of the CoM acceleration in a specific direction.