Planar Bipedal Walking with Anthropomorphic Foot Action

This paper investigates the key problem of walking with both fully actuated and underactuated phases. The studied robot is planar, bipedal, and fully actuated in the sense that it has feet with revolute, actuated ankles. The desired walking motion is assumed to consist of three successive phases: a fullyactuated phase where the stance foot is flat on the ground, an underactuated phase where the stance heel lifts from the ground and the stance foot rotates about the toe, and an instantaneous double support phase where leg exchange takes place. The main contribution of the paper is to provide a provably asymptotically stabilizing controller that integrates the fully-actuated and underactuated phases of walking. By comparison, existing humanoid robots, such as Asimo and Qrio, use only the fully-actuated phase (i.e., they only execute flatfooted walking), or RABBIT, which uses only the underactuated phase (i.e., it has no feet, and hence walks as if on stilts). The controller proposed here is organized around the hybrid zero dynamics of Westervelt et al. (2003) in order that the stability analysis of the closed-loop system may be reduced to a one-dimensional Poincaré map that can be computed in closed form. The ubiquitous Zero Moment Point (ZMP) is used to establish conditions under which the foot will not rotate about its extremities; the ZMP is not used for stability analysis. An example is given to show that a periodic walking motion can be unstable while the ZMP remains strictly within the footprint of the robot.