  Application of IDA-PBC to mechanical systems has received much attention in
recent decades, but its application is still limited by the solvability of the
so-called matching conditions. In this work, it is shown that total
energy-shaping control of under-actuated mechanical systems has a
control-by-interconnection interpretation. Using this interpretation, alternate
matching conditions are formulated that defines constraints on the added
energy, rather then the total closed-loop energy. It is additionally shown
that, for systems that are under-actuated degree one with the mass matrix
depending on a single coordinate, the kinetic energy matching conditions
resolve to ODEs which can be evaluated numerically. Using this approach
controllers are proposed for the benchmark cart-pole and acrobot systems.
