H_∞ generation of periodic motion of mechanical systems of one degree of underactuation

In contrast to fully actuated systems, in underactuated systems, the presence of unactuated links, whose dynamics depend on those of actuated links, does not allow one to independently control each link of such a system. A powerful analytical tool of generating a periodic motion in underactuated mechanical systems relies on the virtual constraint approach to be presented in this chapter in terms of the joint generalized coordinates. Although the configuration is generally given in Cartesian coordinates and orientation, however, using inverse kinematics, they can be transformed into joint coordinates. Coupled with the virtual constraint approach, the H_∞ generation of periodic motion is further developed and applied to a three-degree-of-freedom (3-DOF) helicopter prototype of one degree of underactuation.