Generation of 3-D motion of under-actuated gymnastics robots having two free joints

In the previous paper, a zero moment manifold (ZMM) control method effective for control of a class of under- actuated robots has been proposed and some experimental results have been presented. However, those robots can exercise only in the sagittal plane. This paper extends this control method to generate three-dimensional motion for gymnastics robots having an ankle joint pivoted at the floor. This ankle joint has two orthogonal axes and can not be actuator-driven. In order to maneuver intrinsically unstable robots, we propose an extended ZMM method by redefining the ZMM as a (n-2)- dimensional configuration manifold for such robots with n joints. It is shown theoretically that any motion toward the target pose starting from any given posture on a subset of the ZMM can be stabilized by using this extended ZMM control. Some simulation results of a 10-DOF gymnastics robot model show the effectiveness of the proposed method.