Stability of time-varying control for an underactuated biped robot based on choice of controlled outputs

This paper studies the effect of controlled outputs selection on the walking stability for an under-actuated planar biped robot. The control is based on tracking reference motions expressed as function of time. First, the reference motions are adapted at each step in order to create an hybrid zero dynamic system. Second, the stability of the walking gait under closed-loop control is evaluated with the linearization of the restricted Poincaré map of the hybrid zero dynamics. It shows that for the same robot, and for the same reference trajectory, the stability of the walking can be modified by some pertinent choices of controlled outputs. Third, we find that, at the desired moment of impact for one step, the height of swing foot is nearly zero for all the stable walking, though the configurations of the robot are not the desired configurations. Based on this, we propose a new method to choose the controlled outputs to obtain the stable walking for robot. As a result, two stable domains for the controlled outputs selection are obtained. Furthermore, we point out that, this kind of control law, which based on reference motion as a function of time, can produce better convergent property than that based on reference motion as a function of a state of robot via pertinent choices of controlled outputs.