Exploiting output redefinition for the inverse dynamics of a non-minimum phase, underactuated gantry crane moving a spatial double-pendulum

This paper proposes a method for solving inverse dynamics of an underactuated gantry crane driving a spatial double pendulum, consisting of a cable moving a spatial rigid-body attached to the pendulum terminal mass. The outcomes of the method are both the control forces to be commanded to the torque (or current) control loops and the reference position of the actuated coordinates. The system has 6 degrees of freedom, and just two coordinates are actuated. The reference is imposed to the two Cartesian coordinates of the tip of the rigid-body, while the actuated coordinates are the two displacements of the moving platform. The goal is to ensure precise tracking during transients and at steady states, both in terms of time and space, by compensating for the oscillating attitude of the suspended load. This (non-collocated) definition of input and output leads to a non-minimum phase system. Additionally, inertial and viscous properties of the system make it non-flat; therefore, inverse dynamics should be solved through a differential-algebraic approach. Given the presence of an unstable internal dynamics, stabilization process is carried out in this paper through the concept of output redefinition, which is exploited to stabilize the computation. Numerical simulations are provided to show the effectiveness of the method, also by means of the comparison with a benchmark technique.