Choice of output for time-variant walking control for a five-link underactuated planar biped robot

This paper presents a feedback controller that achieve an asymptotically stable, periodic walking gait for an under-actuated planar biped robot. A biped studied consists of five links, connected to form two legs with knees and a torso, it have point feet without actuation between the feet and ground. In single support, the configuration of the robot is defined by n independent configuration variables and m=n-1 actuators only, thus the degree of under-actuation is one. The control is based on the tracking of reference motions expressed as function of time. The reference motions are adapted at each step in order to create an hybrid zero dynamic system. The stability of the walking gait under closed-loop control is evaluated with the linearization of the restricted Poincare¿ map of the hybrid zero dynamics. For a system with one degree of under-actuation, the resulted Jacobian involves in the stability test is a square matrix of dimension two, thus this analysis can be easily done numerically. Most periodic walking gaits for this robot are unstable when the controlled outputs are selected to be the actuated coordinates. Here the effect of output selection on the zero dynamics is discussed and a pertinent choice of outputs is proposed, leading to stabilization without the use of a supplemental controller.