Revisit Controlled Lagrangians for Spherical Inverted Pendulum

Recently, passivity based control theory (energy approaches) has undergone a breakthrough in dealing with underactated mechanical systems with two successful and similar tools, controlled Lagrangians and controlled Hamiltanians (IDAPBC). Because of the complexity, successful case studies are lacking, for example, MIMO, underactuated and unstable systems. The seminal paper of controlled Lagrangians proposed by Bloch and his colleagues presented a benchmark example–a simplified spherical inverted pendulum on a cart but the detailed design and its verification were neglected. To compensate this ignorance, the note revisits their design idea by addressing explicit control functions for a similar device motivated by real applications. Some observations are given through computer simulation. At the courtesy of the original idea, the case study is known to be the first MIMO, underactuated and unstable system stabilized in full state space via energy approaches.