  This work presents a control-oriented structural design approach for a 2-DOF
underactuated mechanical system, with the purpose of generating an optimal
oscillatory behavior of the end-effector. To achieve the desired periodic
motion, we propose to adjust the dynamic response of the mechanism by selecting
its mass distribution and the characteristic of a nonlinear spring. In
particular, we introduce a two-step optimization strategy to shape the system's
zero dynamics, obtained via input-output linearization. The first part of the
procedure aims to minimize the root-mean-square value of the input torque by
optimizing the mechanism's mass distribution. In this context, we show that a
perfect matching with the desired trajectory can be reached by assuming the
ability to design an arbitrary shape of the system's elastic properties. Then,
in order to favor a simpler physical implementation of the structure, we
dedicate the second optimization step to the piecewise linear approximation of
the previously defined stiffness characteristic. The proposed procedure is
finally tested in detailed numerical simulations, confirming its effectiveness
in generating a complex and efficient periodic motion.
