Contents I. Introduction
Stable bipedal walking has been studied by many researchers; furthermore, studies for stable bipedal running have also been conducted. The hopping motion is a prerequisite for running motion, and thus, there have been various approaches for the bipedal running and/or hopping motion. Through the expansion of the zero-moment point, hopping and running motions have been studied [1], [2] , and online balance controllers have been developed for stable hopping and running [3], [4]. However, these methods do not consider the efficiency of the motion. For efficient hopping and running, the motion-generation methods have been developed for bipedal robots with passively compliant structures [5], [6], and with new leg architectures inspired by an ostrich [7] or a velociraptor [8]. A planar bipedal robot with a drivetrain system that uses motors, cable differentials, and springs also has been developed for energy-efficient running [9]. These methods can be applied to the bipedal robots with the specific structures. In addition, for human-like running and jumping, the data from human motion have been used [10], [11], of which the effectiveness was verified through simulations and does not guarantee an optimally efficient motion. None of the aforementioned methods provide the optimal hopping motion for general humanoid robots, although the hopping motions have been implemented successfully.
