## 2 Proposition of EBSMC  
The LuGre model, which is capable of describing most friction phenomena, was proposed by [PERSON] et al. [26]. This model states that once relative motion occurs between two surfaces, friction force is caused by the deflection of bristles on those surfaces. Consider the following motion system with friction:  
\[M\ddot{x}_{p}=u-F-d \tag{1}\]  
where \(M\) is the system mass, \(x_{p}\) is the system position and \(u\) is the applied force. Moreover, \(F\) and \(d\) are the friction force and an external load, respectively. The friction force described by the LuGre model is comprised of two components. One originates from the bristle behavior defined as \(F_{B}\big{(}\kappa,\dot{\kappa}\big{)}\), and the other is due to the viscous effect \(F_{V}\), that is:  
\[F=F_{B}\big{(}\kappa,\dot{\kappa}\big{)}+F_{V} \tag{2}\]where \(F_{B}(\kappa,\dot{x})=\sigma_{0}\kappa+\sigma_{1}\dot{\kappa}\) and \(F_{V}=\sigma_{2}\dot{x}_{p}\). \(\sigma_{0}\) is the stiffness coefficient; \(\sigma_{1}\) is the damping coefficient; and \(\sigma_{2}\) is the viscous coefficient. \(\kappa\) is the deflection of the bristle, and its variation is defined as follows:  
\[\dot{\kappa}=\dot{x}_{p}-\frac{\left|\dot{x}_{p}\right|}{G(\dot{x}_{p})}\kappa \tag{3}\]  
where \(\sigma_{0}G(\dot{x}_{p})=f_{C}+(f_{S}-f_{C})\exp[-(\dot{x}_{p}/v_{S})^{N}]\). \(f_{C}\) is the Coulomb friction, and \(f_{S}\) is the maximum static friction; \(v_{S}\) is the Stribeck velocity; and the positive factor \(N\) is employed in modifying the transition curve of friction at low velocity. Define tracking error as \(e_{1}=x_{d}-x_{p}\) and \(e_{2}=\dot{e}_{1}=\dot{x}_{d}-\dot{x}_{p}\), where \(x_{d}\) is the desired position. Then, the system in the form of error dynamics becomes:  
\[ \begin{array}{l}\dot{e}_{1}=e_{2}\\ \dot{e}_{2}=-\frac{\sigma_{0}}{M}e_{2}-\frac{1}{M}u+\frac{\sigma_{0}}{M}\dot {x}_{d}+\ddot{x}_{d}+\frac{1}{M}F_{B}\big{(}\kappa,\dot{x}\big{)}+d\end{array}  \tag{4}\]  
Define a virtual state \(e_{3}=\dot{e}_{2}\); the system (4) can then be represented as the following extended third order extended dynamic system [29]:  
\[ \begin{array}{l}\dot{e}_{1}=e_{2}\\ \dot{e}_{2}=e_{3}\\ \dot{e}_{3}=-\frac{\sigma_{2}}{M}e_{3}-\frac{1}{M}\dot{u}+\frac{\sigma_{2}}{M} \ddot{x}_{d}+\dddot{x}_{d}+\frac{1}{M}\dot{F}_{B}\big{(}\kappa,\dot{\kappa} \big{)}+\dot{d}\end{array}  \tag{5}\]  
The proposed EBSMC design procedure for System (5) is described as follows. Step 1:  
By choosing the nominal control effort \(\dot{u}\) of the extended system as:  
\[\dot{u}=-\sigma_{2}e_{3}+\sigma_{2}\ddot{x}_{d}+\hat{M}\ddot{x}_{d}-\hat{M} \ddot{u}_{C} \tag{6}\]  
where \(\hat{M}\) and \(\sigma_{2}\) are the nominal values of system mass and viscous coefficient, respectively. Then, the extended system in (5) can be formulated as:  
\[ \begin{array}{l}\dot{e}_{1}=e_{2}\\ \dot{e}_{2}=e_{3}\\ \dot{e}_{3}=\dot{u}_{C}+\dot{\hat{D}}\Big{(}e_{3},\ddot{x}_{d},\dddot{u}_{C}, \dot{F}_{B}\big{(}\kappa,\dot{x}\big{)},\dot{d}\Big{)}\end{array}  \tag{7}\]  
where \(\dot{\hat{D}}(e_{3},\ddot{x}_{d},\dddot{x}_{d},\dot{u}_{C},\dot{F}_{B}(\kappa, \dot{x}))\)\(=\)\(\frac{1}{M}\Big{[}-\widetilde{\sigma}_{2}e_{3}+\widetilde{\sigma}_{2}\ddot{x}_{d}+ \widetilde{M}\dddot{x}_{d}-\widetilde{M}\dot{u}_{C}+\dot{F}_{B}\big{(}\kappa, \dot{x}\big{)}+\dot{d}\Big{]}\) denotes the lumped system perturbations. Note that \(M=\hat{M}+\widetilde{M}\) and \(\sigma_{2}=\partial_{2}+\widetilde{\sigma}_{2}\). For the servo motor, since the maximum applied force is limited, the resulting speed, acceleration and jerk are also limited. Therefore, it is reasonable to assume that there exists a positive constant \(\eta\) that satisfies \(\eta=\sup\Bigl{|}\dot{\hat{D}}(e_{3},\ddot{x}_{d},\dddot{x}_{d},\dot{u}_{C}, \dot{F}_{B}\big{(}\kappa,\dot{x}\big{)},\dot{d})\Bigr{|}>0\). The detailed proof can be found in the Appendix A.