Vibration-induced bifurcations and chaos in PD controlled robotic manipulators

Manipulator linkages are commonplace in a variety of industrial, manufacturing, and medical settings. High precision is a typical performance requirement and is achieved by means of a control implementation. As a result, the robustness of the controller-plant system with respect to external disturbances is an area of high practical interest.; In this dissertation, driving, parametric, time periodic, and quasi-periodic disturbances are considered, and their potential effects upon the performance of robotic manipulators utilizing PD control with gravity compensation are assessed. As the governing ordinary differential equation is both nonlinear and non-autonomous, a combination of analytical and numerical methods is employed. One and two degree of freedom models are specifically considered, following a change in variable to a dimensionless equation of motion.; Local analysis and a modified form of the method of multiple scales are applied to obtain a frequency-amplitude relationship for the single degree of freedom case of a driving disturbance input. The results indicate that for a certain range of nominally allowable control parameters a significant amplification of disturbance inputs may occur. Numerical simulation confirms this result for both one and two degree of freedom systems.; A non-trivial Hopf bifurcation is documented under odd parametric excitation caused by a vertical periodic oscillation of the support base. Under small deflection at lowest order, this system reduces to the damped Mathieu equation, where a rough correlation is observed between the damped Mathieu stability boundary and the bifurcation curve. A pitchfork-type bifurcation occurs under horizontal oscillation of the support base, and the basins of attraction of the competing limit cycles that result are determined. In both cases, numerical simulation is employed to map the bifurcations in the multiple degree of freedom parameter space and to identify the impact of payload inertia.; Quasi-periodic disturbance combinations can result in chaotic oscillations, predicable by application of Melnikov's method. Diagnostic techniques including Poincare mappings and Fourier spectrum analysis are employed to describe the resulting behavior in both one and two degrees of freedom. The purely parametric case is identified as particularly susceptible to chaotic onset.