Generation of Large Gymnastic Motions for a Family of Under-Actuated Robots by Zero-Moment-Manifold Control

This paper is concerned with a novel control method to generate any desired motion for a family of under-actuated robots such as gymnastics robots and acrobots having a single lower leg pivoted at the floor. This pivotal ankle joint can not be actuator-driven but may have only a passive viscous device such as a rotary damper. Therefore these robots are intrinsically unstable under the effect of gravity. In order to maneuver such an under-actuated robot with n joints under non-holonomic constraint without loosing stability of motion, we have proposed a zero moment manifold (ZMM) control method by introducing an (n - 1)-dimensional configuration manifold, on which a set of joint angle vectors satisfies zero of the rotational moment around the ankle joint. It could be shown theoretically that any motion starting from any initial posture on a subset of the ZMM and moving to a desired posture can be stabilized by using a coordinated control composed of gravity compensation and PD feedback. Though the theoretical proof can guarantee only stability of movements of joint angles in a narrow range, it has been observed that the robot can move in a wider range through preliminary numerical simulation results. Notwithstanding such a conservative result of theory, this paper shows that the ZMM control with constant PD gains can generate global gymnastics movements for a class of under-actuated robots by verifying through a variety of numerical simulations that most of given attractors (target postures) have corresponding large attraction regions. This is also verified experimentally by using a real gymnastics robot with four joints and a series of large movements can be produced consecutively as programmed by selecting several target postures enough separated each other and combining them