Low-input accurate periodic motion of an underactuated mechanism: Mass distribution and nonlinear spring shaping

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.