Input shaping for non-zero initial conditions and arbitrary input signals with an application to overhead crane control

Input shaping is a well-established approach for suppressing oscillations and vibrations in systems with flexible modes. When applying input shaping, it is common to assume the system to be at rest initially, i.e. the initial conditions of the oscillatory part of the system have to be at zero. In this paper, we propose a method that allows relaxing the aforementioned assumption for a large class of input signals. The approach relies on the standard input shaper structure but re-parameterizes the shaper such that non-zero initial conditions are cancelled out by the inputs, resulting in zero residual oscillation. Natural physical limitations of the concept are discussed and the application to overhead cranes is presented. The method is validated both in simulation as well as in experiments using a small scale crane.