Stability of zero-moment-manifold control for a family of under-actuated robots

This paper proposes a novel control scheme called the ZMM control for generating any desired motion for a family of under-actuated robots with instability such as gymnastics robots and robotic puppets having a single leg pivoted at the floor. This pivotal ankle joint can not be actuator-driven but is passive with a rotary damper. Therefore these robots are intrinsically unstable under the effect of gravity and considered to be under-actuated and nonholonomically constrained. This paper first introduces an (n $1)-dimensional configuration manifold for such robots with n joints, which is called the ZMM (zero moment manifold) and defined as a set of joint angle vectors that satisfy zero of the rotational moment around the first ankle joint. It is shown theoretically that any motion starting from any given posture on a subset of the ZMM and targeting to a desired form of motion can be stabilized by using a coordinated control composed of gravity compensation for other joints except the ankle and PD feedback despite of the existence of a nonholonomic constraint. The effectiveness of the proposed control scheme is verified by numerical simulation and experimental results using a gymnastics robot with four joints.