An inverse-free Getz-Marsden dynamic system and its eleven-instant discrete model for time-variant linear equations solving

Abstract

Time-variant linear equations (TVLEs) are fundamental problems appeared in many practical applications. There is an increasing demand for algorithms that can solve TVLEs with high accuracy. This paper proposes an inverse-free continuous-time Getz–Marsden dynamic system (CTGMDS) for solving TVLEs. Additionally, a general eleven-instant finite-difference method (FDM) is proposed to discretize the CTGMDS, resulting in a general eleven-instant discrete-time GMDS (11-DTGMDS) model. Theoretically, the proposed 11-DTGMDS has truncation error of seven-order precision that delivers higher precision than the existing FDMs. This paper also compares five DTGMDS models using other FDMs to discretize the CTGMDS. The convergence of the CTGMDS and 11-DTGMDS is proven by theoretical analysis. Numerical validations are performed by comparing the 11-DTGMDS and other DTGMDSs, showing that the 11-DTGMDS provides higher solution accuracy. Finally, a path-tracking task is completed by applying the 11-DTGMDS model to a Franka Emika Panda robot, validating the practicality of the 11-DTGMDS.