Reciprocal-Kind Zhang Neurodynamics Method for Temporal-Dependent Sylvester Equation and Robot Manipulator Motion Planning

It is noteworthy that the Sylvester equation plays a pivotal role in the field of industrial intelligence control. To meet the demands of real-time applications, temporal-dependent Sylvester equations (TDSEs) are employed to formulate motion planning problems for robot manipulators. Traditionally, the classical Zhang neurodynamics (ZN) method is utilized to address the TDSE problems, which encounters challenges associated with temporal-dependent inverse matrix computations. In this article, we introduce an inverse-free approach based on energy zeroing, termed the reciprocal-kind ZN (RKZN) model, specifically designed to tackle the TDSE problem. Additionally, we propose a discrete RKZN (DRKZN) algorithm to address future Sylvester equation (FSE) problems and the motion planning challenges of robot manipulators. Furthermore, we conduct a thorough analysis of the convergence property and robustness of the RKZN method for addressing the TDSE problem. This analysis is grounded in the Lyapunov stability theory of nonlinear systems and a comparative method for nonlinear systems with temporal-dependent error-feedback-related uncertainty disturbances. Numerical experiments, simulations, and physical experiments substantiate the effectiveness and superiority of the developed RKZN method in addressing both the TDSE problem and the motion planning challenges of robot manipulators.