I. Intoroduction
Inverted pendulums are typical examples of nonlinear and underactuated mechanical systems and well known in control engineering for verification and practice of various kinds of control theories. Inverted pendulum systems have several types, e.g., a single pendulum, a parallel double pendulum, a serial double pendulum and a two dimensional pendulum, etc. And many control methods have been proposed to control the inverted pendulum systems, such as feedback stabilization[1] [4], energy based control[1] [4], bang-bang control[3], sliding mode control[6], robust control[2], hybrid control[4] [5], partial linearization[7] [8] (see [1] for more details). In spite of these existing methods, to control the inverted pendulums is still an open research topic. In particular, the serial double inverted pendulum is strongly nonlinear and highly underactuated than a single inverted pendulum and the control of the serial double pendulum is a difficult problem. And a solution of this problem is applicable to other nonlinear and underactuated control problems. Hence, this paper considers the swing-up control problem for the serial double inverted pendulum.