  This study presents a theoretical framework for planning and controlling
agile bipedal locomotion based on robustly tracking a set of non-periodic apex
states. Based on the prismatic inverted pendulum model, we formulate a hybrid
phase-space planning and control framework which includes the following key
components: (1) a step transition solver that enables dynamically tracking
non-periodic apex or keyframe states over various types of terrains, (2) a
robust hybrid automaton to effectively formulate planning and control
algorithms, (3) a phase-space metric to measure distance to the planned
locomotion manifolds, and (4) a hybrid control method based on the previous
distance metric to produce robust dynamic locomotion under external
disturbances. Compared to other locomotion frameworks, we have a larger focus
on non-periodic gait generation and robustness metrics to deal with
disturbances. Such focus enables the proposed control framework to robustly
track non-periodic apex states over various challenging terrains and under
external disturbances as illustrated through several simulations. Additionally,
it allows a bipedal robot to perform non-periodic bouncing maneuvers over
disjointed terrains.
