PITCHFORK-TYPE BIFURCATIONS IN A PARAMETRICALLY EXCITED, PD-CONTROLLED PENDULUM OR MANIPULATOR

In this paper, the effect of a vibrating support base upon the behavior of a simple robotic manipulator with PD control is examined. The physical system to be controlled, i.e., the plant, is modelled initially as an inverted pendulum. The vibrating support generates a time-periodic parametric excitation of the controller–plant system that is shown, under certain operating conditions, to produce a bifurcation in the steady state motion, whereby a globally stable limit cycle encircling the target position is replaced by two offset, and competing, periodic attractors. The basins of attraction of each are sketched, and predictive criteria for the bifurcation to occur are presented in terms of key system parameters. As the resulting steady state error can be quite large, the post-bifurcation regime of motion often implies a total control failure and is therefore of significant interest. The extrapolation of the results obtained for a higher ordered model is discussed, and the bifurcation boundary is mapped in the disturbance parameter plane for a two-degrees-of-freedom manipulator model.