Stability of PID control for industrial robot arms

PID control is the most widespread technique for the control of industrial robot arms. However, the adoption of PID control is not adequately supported by a theoretical basis, since the results presented in the literature are of dubious interpretation and difficult, when not impossible, to verify. Motivated by this lack of theoretical support, this paper presents a novel proof for the stability of rigid robot arms controlled by PID algorithms: the proof is based on a model of the robot where the nominal decoupled linear part is emphasized. The main result consists in a simple condition between the exponential stability degree of the nominal closed loop system and the parameters of a bound on the nonlinear terms in the dynamic model of the mechanical manipulator. Some considerations are also worked out on the relations between the eigenvalues of the nominal system and the extension of the stability region. The theoretical results are finally verified on a simple two DOF example.