  Interconnection and damping assignment passivity-based control scheme has
been used to stabilize many physical systems such as underactuated mechanical
systems through total energy shaping. In this method, some partial differential
equations (PDEs) arisen by kinetic and potential energy shaping, shall be
solved analytically. Finding a suitable desired inertia matrix as the solution
of nonlinear PDEs related to kinetic energy shaping is a challenging problem.
  In this paper, a systematic approach to solve this matching equation for
systems with one degree of underactuation is proposed. A special structure for
desired inertia matrix is proposed to simplify the solution of the
corresponding PDE. It is shown that the proposed method is more general than
that of some reported methods in the literature. In order to derive a suitable
desired inertia matrix, a necessary condition is also derived. The proposed
method is applied to three examples, including VTOL aircraft, pendubot and 2D
SpiderCrane system.
