On trajectory tracking control of prismatic and revolute joined robotic manipulators

The purpose of this paper is to construct a trajectory tracking feedback controller for prismatic and revolute joined multi-link robotic manipulators using a new form of sliding modes. Methods. In this paper, the Lyapunov functions method has been applied to establish the stability property of the closed-loop system. Results. Due to the presence of rotational joints, the motion equations of the manipulator are periodic in the angular coordinates of the corresponding links. A control law is constructed which is also periodic in the angular coordinates of the links. Thus, a closed-loop system has not one, but a whole set of equilibrium positions that differ from each other by a multiple of the system period. The dynamics mathematical model of a complex five-link manipulator with cylindrical and prismatic joints has been constructed on the basis of the Lagrange equations. Simulation results on a 5-degree-of-freedom robotic arm demonstrate the applicability of the proposed control scheme. Conclusion. We obtain a relay controller such that the set of all equilibrium positions of the closed-loop system is uniformly asymptotically stable. The novelty of the obtained control law is based on a new approach that takes into account the periodicity of the model in angular variables with the solution of the tracking problem in the cylindrical phase space. The simulation results for a 5-degree-of-freedom robotic manipulator clearly show the good performance of our controller. The applied significance of the results obtained in the paper is as follows. At present, in connection with the widespread introduction and mass production of manipulators, it seems important to develop the mathematical foundations for designing a control structure that has a universal character, namely, allowing to perform the required process without additional adjustment of control parameters with simple and convenient algorithms and programs of their implementation.