Comments on "Time-Varying Lyapunov Functions for Tracking Control of Mechanical Systems With and Without Frictions"

In the article$^a$, the authors introduced a time-varying Lyapunov function for the stability analysis of nonlinear systems whose motion is governed by standard Newton-Euler equations. The authors established asymptotic stability with the choice of two symmetric positive definite matrices restricted by certain eigenvalue bounds in the control law. Exponential stability in the sense of Lyapunov using integrator backstepping and Lyapunov redesign is established in this note using just one matrix in the derived controller. We do not impose minimum eigenvalue bound requirements on the symmetric positive definite matrix introduced in our analysis to guarantee stability. Reducing the parameters needed in the control law, our analysis improves the stability and convergence rates of tracking errors reported in the article$^a$. $^a$Ren, W., Zhang, B, Li, H, and Yan L. IEEE Access. vol. 8. pp. 51510-51517. 2020.