A Novel Approach for Solving Time-Varying Linear Equations and Its Application in Motion Tracking of Redundant Robots

Solving time-varying linear equations is a common problem in many fields. In this article, a novel approach is proposed based on the cost function constructed by the mixed error, which is made up of the residual error and its integration. The merits of the novel approach include high accuracy, noise immunity, and asymptotic convergence. Since the derivative information of time-varying signals is used in the proposed method, a proportional-integral differentiator is designed based on the mixed estimation error for the first derivative estimation. The proposed differentiator has a high degree of precision, asymptotic convergence, and no chattering. In addition, theoretical analyses prove the convergence of the proposed optimal strategy and differentiator. Simulation results validate the effectiveness and superiority of the proposed strategy and the designed differentiator. Finally, the proposed approach is employed in the motion tracking of redundant robots.