## Appendix A  
In this Appendix A, a robust gain estimation is firstly introduced, and then, the robustness stability analysis for \(w=0\) is further addressed. The parameters of the motion tracking control system subjected to friction were listed in Table 1. According to the analysis in Section 2, the desired error dynamics can be guaranteed if the reaching condition (20) is satisfied. Therefore, it is important to estimate a suitable \(w\) for suppressing the disturbance effect on the control process. In the tracking task, most of the tracking errors are induced in the pre-sliding region, and it is reasonable to conclude that the variation rate of the disturbance in the pre-sliding region is larger than that in the sliding region. In other words, the reaching condition in the motion process is guaranteed if it can be satisfied in the pre-sliding region. For this reason, the variation rate of \(F_{B}\) in the pre-sliding region should be estimated for determining a suitable \(w\). To fulfill the so-called approaching condition and attain the sliding condition, it has been shown in (20) that \(w>\eta\) should be satisfied. In other words, the derivation of the lumped perturbation needs to be bounded. In the following, the variation of the lumped perturbation is going to be addressed. From (3), the deflection of the bristles is bounded, and its maximum value is \(\kappa_{ss}=G(\dot{x}_{p})\text{sgn}(\dot{x}_{p})\). Moreover, \(G(\dot{x}_{p})\) is positive and bounded by \(f_{C}/\sigma_{0}\leq G(\dot{x}_{p})\leq f_{S}/\sigma_{0}\). For \(\dot{x}_{p}\geq 0\), the dynamic equation of the bristle can be represented as \(\dot{\kappa}=\dot{x}_{p}-\frac{\dot{x}_{p}}{G(\dot{x}_{p})}\kappa\), and the following inequality can be obtained. \[ \begin{array}{ll}\dot{\kappa}=\dot{x}_{p}-\frac{\dot{x}_{p}}{G(\dot{x}_{p})} \kappa&\leq\dot{x}_{p}+\frac{\dot{x}_{p}}{G(\dot{x}_{p})}\kappa\leq\dot{x}_{p }+\frac{\dot{x}_{p}}{f_{C}}\sigma_{0}\kappa\leq\dot{x}_{p}+\frac{\dot{x}_{p}}{ f_{C}}\sigma_{0}\kappa_{ss}\\ &\leq\dot{x}_{p}+\frac{\dot{x}_{p}}{f_{C}}\sigma_{0}G(\dot{x}_{p})\leq\dot{x}_{ p}+\frac{\dot{x}_{p}}{f_{C}}f_{S}\leq\dot{x}_{p}\Big{(}1+\frac{f_{S}}{f_{C}}\Big{)} \end{array}  \tag{10}\]  
For \(\dot{x}_{p}\geq 0\), \(\ddot{x}_{p}\geq 0\):  
\[ \begin{array}{ll}\ddot{\kappa}&=\ddot{x}_{p}-\frac{\ddot{x}_{p}G(\ddot{x}_{p}) -\dot{x}_{p}\dot{G}(\dot{x}_{p})}{G(\dot{x}_{p})^{2}}\kappa-\frac{\dot{x}_{p}} {G(\dot{x}_{p})}\dot{\kappa}\leq\ddot{x}_{p}+\frac{\ddot{x}_{p}G(\dot{x}_{p})- \dot{x}_{p}\dot{G}(\dot{x}_{p})}{\dot{G}(\dot{x}_{p})^{2}}\kappa+\frac{\dot{x} _{p}}{G(\dot{x}_{p})}\dot{\kappa}\\ &\leq\ddot{x}_{p}+\frac{\ddot{x}_{p}\frac{f_{S}}{f_{C}}-\dot{x}_{p}\Big{\{} \frac{1}{q_{0}}(f_{S}-f_{C})\exp[-(\frac{\dot{x}_{p}}{f_{S}})^{N}]\cdot\Big{[} -N\Big{(}\frac{\dot{x}_{p}}{f_{S}}\Big{)}^{N-1}\cdot\frac{\dot{x}_{p}}{f_{S}} \Big{]}}{\dot{\kappa}_{ss}+\frac{\dot{x}_{p}}{(f_{C}/\sigma_{0})}\dot{\kappa}} \\ &\leq\ddot{x}_{p}+\frac{\ddot{x}_{p}\frac{f_{S}}{f_{C}}+\dot{x}_{p}\frac{f_{S} -f_{C}}{\sigma_{0}}N\Big{(}\frac{\dot{x}_{p}}{f_{S}}\Big{)}^{N-1}\cdot\frac{ \dot{x}_{p}}{f_{S}}}{\dot{\kappa}_{ss}+\frac{\dot{x}_{p}}{(f_{C}/\sigma_{0})} \cdot\dot{x}_{p}\Big{(}1+\frac{f_{S}}{f_{C}}\Big{)}}\\ &\leq\ddot{x}_{p}+\frac{\ddot{x}_{p}\frac{f_{S}}{f_{C}}+\dot{x}_{p}\frac{f_{S} -f_{C}}{\sigma_{0}}N\Big{(}\frac{\dot{x}_{p}}{f_{S}}\Big{)}^{N-1}\cdot\frac{ \dot{x}_{p}}{f_{S}}}{\dot{\sigma}_{0}}\cdot\frac{f_{S}}{\sigma_{0}}+\frac{\dot{ x}_{p}^{2}}{(f_{C}/\sigma_{0})}\cdot\Big{(}1+\frac{f_{S}}{f_{C}}\Big{)}\\ &\leq\ddot{x}_{p}+\Bigg{[}\ddot{x}_{p}+\dot{x}_{p}\cdot\Big{(}1-\frac{f_{C}}{f _{S}}\Big{)}\cdot N\Big{(}\frac{\dot{x}_{p}}{f_{S}}\Big{)}^{N-1}\cdot\frac{ \ddot{x}_{p}}{f_{S}}\Bigg{]}\cdot\Big{(}\frac{f_{S}}{f_{C}}\Big{)}^{2}+\frac{ \dot{x}_{p}^{2}}{(f_{C}/\sigma_{0})}\cdot\Big{(}1+\frac{f_{S}}{f_{C}}\Big{)} \end{array}  \tag{11}\]  
where \(\hat{G}(\dot{x}_{p})=-N(f_{S}-f_{C})\exp[-\big{(}\dot{x}_{p}/v_{S}\big{)}^{N}] \cdot\Big{[}\dot{x}_{p}^{N-1}\cdot\ddot{x}_{p}/v_{S}^{N}\Big{]}/\sigma_{0}\).