Time-scaling control and passive walking of bipeds with underactuation degree one

The paper develops a feedback control law that results in periodic orbits on different slopes for an n degree-of-freedom biped robot with one degree of underactuation. This is achieved by introducing the time-scaling control approach to translate a periodic orbit for a passive robot, which exists on a specific shallow slope, to other slopes. The resulting periodic orbits on other slopes, if exist, are the same as those orbits generated by controlled symmetries for full actuated bipeds in geometry. The temporal solutions are influenced by a one-degree subsystem related to the evolution of virtual time that scales the passive gait. Necessary and sufficient conditions are investigated for the existence and stability properties of periodic orbits on different slopes with the proposed control law. Finally, the proposed method is illustrated by two underactuated compass biped robots.