Discrete gradient-zeroing neural network algorithm for solving future Sylvester equation aided with left–right four-step rule as well as robot arm inverse kinematics

The temporal-variant Sylvester equation (TVSE) occupies a significant position in applied mathematics, particularly in the realms of optimal control theory and matrix optimization engineering applications. Within the framework of prediction modeling systems, the future Sylvester equation (FSE) emerges as the discrete manifestation of TVSE, characterized by unknown future information. Leveraging a novel left and right four-step (LRFS) rule, we propose a novel discrete gradient-zeroing neural network (DGZNN) algorithm with order-5 precision, which is developed from the continuous gradient-zeroing neural network (GZNN) model, for solving the FSE problem. The proposed algorithm is named as LRFS-DGZNN algorithm, which stands out as an inverse-less neurodynamic algorithm. Additionally, the convergence properties of the GZNN model in solving the TVSE problem are elucidated through Lyapunov stability theory and matrix spectral theory. Furthermore, the LRFS-DGZNN algorithm’s error pattern property in solving the FSE problem is established and proven using stability theory of linear multi-step methods and ordinary differential equation numerical approximation theory. Three numerical experiments are conducted to evaluate the performance of the proposed GZNN model for solving the TVSE problem and the LRFS-DGZNN algorithm for solving the FSE problem. Moreover, the study showcases the inverse-kinematics solutions and simulations involving planar robot arm with 2 degrees of freedom (DOFs), the Kinova Jaco2 robot arm with 6 DOFs, and the Franka Emika Panda robot arm with 7 DOFs, illustrating the high efficiency of the LRFS-DGZNN algorithm.