Shaping Stable Oscillation of a Pendulum on a Cart around the Horizontal

Abstract

Planning and stabilizing induced oscillations for underactuated mechanical systems are challenging tasks. Available analytical solutions are primary linked to formats of representation of feasible trajectories and can give a rather limited perception of a variety of possibilities for particular systems. The paper provides new insights to the tasks exploring the classical and popular robotic benchmark set-up. In particular, the case study illustrates the procedure for generating a periodic behaviour of a pendulum on a cart, when the pendulum oscillates around the horizontal. Planning such a behaviour requires novel arguments for establishing a presence of a forced cycle. Furthermore, if found, the orbital stabilization of the cycle requires an alternative set of transverse coordinates. Both assignments are successfully solved. The analytical contributions are discussed and supported by numerical simulations.