Use of a Simple Mechanical Analogy to Analytically Tune the PD Controller of a Flexible Manipulator System

A study is presented in this paper that uses a simple mechanical analogy to analytically tune the PD (proportional-derivative) controller of a linear flexible manipulator system. More specifically, the aim is to give simple closed-form solutions of the optimal P and D gains to yield the maximum bandwidth under a given damping requirement or conversely the maximum damping under a given bandwidth requirement. The idea of this study is based on the observation that the performance of the complete manipulator system is largely determined by the operational dynamics of the fundamental vibration mode. A lumped element method is thus applied to model this dynamics in terms of simple lumped mechanical elements. It subsequently turns out that the original servo control problem is analogous to a conventional Zener mount design problem, that is, mathematically, to optimize a third-order dynamic system consisting of the Zener model of a viscoelastic mount and an inertial object upon it. A design methodology is finally established to analytically determine the optimal elements of the mount, corresponding to the optimal control gains. Simulations and experiments were also conducted with a single-link flexible beam to support the model and the design methodology developed.