This article presents the design, development, and implementation of walking control for the bipedal exoskeleton prototype Co-Ex. The main objective in developing this prototype is to take a successive step towards ambulatory support via an exoskeleton with self-stabilization capability. To attain this goal, Co-Ex is equipped with 8 torque-controllable active joints to provide ambulatory support while ensuring improved environmental interaction. The development of Co-Ex led to three contributions: i) self-stabilization capability in 3D against external disturbances, ii) a locomotion control framework that provides dynamically balanced walking behavior in 3D despite the underactuated leg configuration, iii) a power-aware leg design in which most actuators are deployed around the waist for reduced leg inertia. To verify the self-stabilization and locomotion capabilities of Co-Ex, we conducted a series of experiments using a dummy manikin. As a result, Co-Ex showed self-stabilization behavior against disturbances and exhibited favorable locomotion characteristics that validated the proposed approach.