PID Controllers in the Trajectory Tracking Control Problem of Robotic Manipulators with Time-Delayed Feedback*

Proportional-Integral-Derivative (PID) controllers are widely used in the control problems of many mechanical systems. Most of the studies are limited to the stabilization problem of the steady motions and positions based on the analysis of the model equations within the framework of a linear approximation. One of the urgent problems of the controlled motion mechanics continues to be the problem of using PID controllers in the semi-global and global trajectory tracking of multi-link robotic manipulators in a non-linear statement. Practically, the problem on applicability of PID controllers is little studied taking into account the possible delay in the feedback structure. This paper deals with the trajectory tracking control problem of multi-link robotic manipulators with a delayed feedback. The mathematical simulation of the controlled robotic manipulators is carried out in a nonlinear statement based on the Lagrange equations. The control model can be characterized as a non-linear PID controller with a variable delay in the feedback structure. We use the Lyapunov functional method as well as the limiting equations to prove the asymptotic stability property of the closed-loop system. Sufficient conditions for the trajectory tracking control are defined in the form of restrictions on control parameters and inequalities with respect to the mass-inertial parameters of the system, and admissible delay. As a solution to an applied problem, the motion controller of an industrial six-link robot manipulator is obtained.