## Appendix A
From (25) and (26), the maximum variation rate of the friction is:  
\[ \begin{array}{ll}\dot{F}_{B}=\sigma_{0}\dot{\kappa}+\sigma_{1}\ddot{\kappa}\\ \leq\sigma_{0}\Big{[}\dot{x}_{p}\Big{(}1+\frac{f_{S}}{f_{C}}\Big{)}\Big{]}+ \sigma_{1}\left\{\begin{array}{ll}\ddot{x}_{p}+\Bigg{[}\ddot{x}_{p}+\dot{x}_{ p}\cdot\Big{(}1-\frac{f_{C}}{f_{S}}\Big{)}\cdot N\Big{(}\frac{\dot{x}_{p}}{f_{S}} \Big{)}^{N-1}\cdot\frac{\ddot{x}_{p}}{f_{S}}\Bigg{]}\cdot\Big{(}\frac{f_{S}}{f_ {C}}\Big{)}^{2}\\ +\frac{\dot{x}_{p}^{2}}{(f_{C}/\sigma_{0})}\cdot\Big{(}1+\frac{f_{S}}{f_{C}} \Big{)}\end{array}\right\}\end{array}  \tag{12}\]  
Based on the limitation of the possible maximum velocity and acceleration of the servo mechanism, (12) is bounded, and therefore, exists. According to the desired motion trajectory, we can substitute its maximum velocity and maximum acceleration into (12) to estimate the required gain \(w\). However, for practical implementation, if the reference command is a sine wave as shown in Figure 1,the corresponding desired velocity in the pre-sliding region is not zero at the initial instant, such that if we adopt its value in (A3), it requires an extremely large gain \(w\) for achieving the approaching condition, but it may not be realizable in practice. Alternatively, a reference signal with zero velocity is preferred. Moreover, according to the friction model, Stribeck velocity determines which friction region occurs, so that \(v_{S}\) and \(\left|\ddot{x}_{d}\right|_{\text{max}}\) are used for \(\dot{x}_{p}\) and \(\ddot{x}_{p}\), respectively, to obtain the maximum required \(w\). In Remark 2, the system stability is briefly addressed by means of BIBO for the critical condition \(w=0\). In the following, the closed-loop robust stability issue is going to be interpreted as a feasibility problem of a linear matrix inequality (LMI). According to (24), when applying zero robust gain, the sliding controller (22) turns into a PD control plus a feed-forward compensation, which leads to the following error dynamics:  
\[ \begin{array}{ll}\dot{e}_{2}&=-(1+k_{1}k_{2})e_{1}-(k_{1}+k_{2})e_{2}+\frac{ \widetilde{M}}{M}(1+k_{1}k_{2})e_{1}+\frac{\widetilde{M}}{M}(k_{1}+k_{2})e_{2 }-\frac{\widetilde{\varphi}}{M}e_{2}\\ &+\frac{\widetilde{\varphi}}{M}\dot{x}_{d}+\left(1-\frac{\widetilde{M}}{M} \right)\ddot{x}_{d}+\frac{1}{M}F_{B}\big{(}\kappa,\dot{\kappa}\big{)}+\frac{ 1}{M}d\end{array} \] (A4)  
where \(\widetilde{M}=M-\hat{M}\) and \(\widetilde{\varphi}_{2}=\sigma_{2}-\hat{\sigma}_{2}\). System (A4) can then be further represented by the following state space form:  
\[\dot{\mathbf{e}}=(\mathbf{A}+\Delta\mathbf{A})\mathbf{e}+\mathbf{D}\] (A5)  
where:  
\[ \begin{array}{ll}\mathbf{A}=\left[\begin{array}{cc}0&1\\ -(1+k_{1}k_{2})&-(k_{1}+k_{2})\end{array}\right]\!,&\Delta\mathbf{A}=\left[ \begin{array}{cc}0&0\\ \frac{\widetilde{M}}{M}(1+k_{1}k_{2})&\frac{\widetilde{M}}{M}(k_{1}+k_{2})- \frac{\widetilde{\varphi}_{2}}{M}\end{array}\right]\\ &0\\ &\mathbf{D}=\left[\begin{array}{c}\widetilde{\varphi}_{M}\dot{x}_{d}+\left( 1-\frac{\widetilde{M}}{M}\right)\ddot{x}_{d}+\frac{1}{M}F_{B}\big{(}\kappa, \dot{\kappa}\big{)}+\frac{1}{M}d\end{array}\right]\end{array} \] (A6)  
For (A5), consider a [PERSON] candidate as:  
\[V_{e}=\mathbf{e}^{T}\mathbf{P}\mathbf{e}\] (A7)  
where \(\mathbf{P}^{T}=\mathbf{P}>0\). Taking the time derivative of (A7) yields:  
\[\dot{V}_{e}=\mathbf{e}^{T}\Big{(}\mathbf{A}^{T}\mathbf{P}+\mathbf{P}\mathbf{A} +\Delta\mathbf{A}^{T}\mathbf{P}+\mathbf{P}\Delta\mathbf{A}\Big{)}\mathbf{e}+2 \mathbf{e}^{T}\mathbf{P}\mathbf{D}\] (A8)  
Since the last term in (A8) is bounded, the closed-loop system is said to be robust stable if the following inequality exists:  
\[\mathbf{A}^{T}\mathbf{P}+\mathbf{P}\mathbf{A}+\Delta\mathbf{A}^{T}\mathbf{P}+ \mathbf{P}\Delta\mathbf{A}+\mathbf{Q}<0\] (A9)  
where \(\mathbf{Q}^{T}=\mathbf{Q}>0\). According to the structure of \(\Delta\mathbf{A}\), it can be factorized by:  
\[\Delta\mathbf{A}=\left[ \begin{array}{cc}0&0\\ (1+k_{1}k_{2})&(k_{1}+k_{2})\end{array} \right]\left[