Guaranteed rates of convergence of a class of PD controllers for trajectory tracking problems of robotic manipulators with dynamic uncertainties

The paper provides a better understanding of the behaviour of a class of simple proportional plus derivative (PD) controllers applied to robotic manipulators and to highlight some useful design criteria. The stability and robustness of PD controllers for trajectory tracking problems of robotic manipulators with dynamic uncertainties is investigated. Based on Lvapunov´s second method it is shown that the composite velocity and position tracking error vector is guaranteed to exponentially converge from any initial condition to a closed ball, defined by its L₂ norm being less than a certain threshold provided that the PD controller gains are chosen greater than a specific bound depending on the dynamic parameters, desired trajectories and levels of external disturbances. Moreover, the size of the ball can be made arbitrarily small by increasing the controller gains wherever appropriate and possible. As a result, both transient and steady-state performance of the simple PD controllers for trajectory tracking is assured