Free-falling paper shapes exhibit rich, complex and varied behaviours which are extremely challenging to model analytically. Physical experimentation aids in system understanding but is time-consuming, sensitive to initial conditions and reliant on subjective visual behavioural classification. In this study robotics, computer vision and machine learning are used to autonomously fabricate, drop, analyse and classify the behaviours of hundreds of shapes. The system is validated by reproducing results for falling disks, which exhibit four falling styles: tumbling, chaotic, steady and periodic. A previously-determined mapping from a non-dimensional parameter space to behaviour groups is shown to be consistent with these new experiments for tumbling and chaotic behaviours. Steady or periodic behaviours, however, are observed in previously unseen areas of the parameter space. More complex hexagon, square and cross shapes are investigated, showing that the non-dimensional parameter space generalises to these shapes. The system highlights the potential of robotics for the investigation of complex physical systems, of which falling paper is one example, and provides a template for future investigation of such systems.