Controlled symmetries and passive walking

In this note, we investigate the relationship between nonlinear control and passive walking in bipedal locomotion for the general case of an n degree-of-freedom biped in three dimensional space. We introduce the notion of controlled symmetry to capture the effect of the control input on the invariance of the system Lagrangian under group action. We then show the existence of a controlled symmetry for general bipeds under the action of SO(3) taking into account not only the kinetic energy but also the potential energy and impact dynamics. We use this result to show the existence of a nonlinear control law that reproduces so-called passive gaits independent of the particular ground slope. Our contribution in this note is two-fold. First, our result contains the first rigorous proof of the existence of so-called passivity mimicking control laws that explicitly accounts for the impact dynamics. Second, whereas previous papers have studied only planar bipeds with and without knees, our result is completely general. Our results can be viewed as direct extensions of several previous results, such as passivity-based control, virtual gravity, and virtual passive dynamic walking from the planar case to general n-degrees-of-freedom (DOF) robots in three-dimensional space.