Delay-induced bifurcations in collocated position control of an elastic arm

The interference of the elasticity of a single robotic arm and the unavoidable time delay of its position control is analysed from nonlinear vibrations viewpoint. The simplified mechanical model of two blocks and a connecting spring considers the first vibration mode of the arm, while the collocated proportional-derivative (PD) control uses the state of the first block only and actuates also there. It is assumed that the relevant nonlinearity is the saturation of the delayed control force. The linear stability analysis proves that stabilizable and non-stabilizable parameter regions follow each other periodically even for large spring stiffnesses and for tiny time delays. Hopf bifurcation calculation is carried out after an infinite-dimensional centre manifold reduction, and closed-form algebraic expressions are given for the amplitudes of the emerging oscillations. These results support the experimental tuning of the control gains since the parameters of the arising and often unexpected self-excited vibrations can serve as a guide for this practical procedure.