## 3 Main Results
### The Designed Control Methodology  
Let us take the time derivative of Equation (8):  
\[ \begin{split}\dot{s}&=\dot{x}_{de}+\frac{2\gamma_{1 }}{1+e^{-\mu_{1}(|x_{de}|-\phi)}}x_{de}+\frac{2\gamma_{1}\mu_{1}x_{de}\text{ sgn}(x_{de})e^{-\mu_{1}(|x_{de}|-\phi)}}{\big{(}1+e^{-\mu_{1}(|x_{de}|-\phi)} \big{)}^{2}}x_{e}\\ &\quad+\frac{2\gamma_{2}a}{1+e^{\mu_{2}(|x_{de}|-\phi)}}|x_{e}|^{ \alpha-1}x_{de}-\frac{2\gamma_{2}\mu_{2}x_{de}e^{\mu_{2}(|x_{de}|-\phi)}}{\big{(}1 +e^{\mu_{2}(|x_{de}|-\phi)}\big{)}^{2}}|x_{e}|^{\alpha}\end{split}  \tag{17}\]  
With \(\dot{x}_{de}=\ddot{x}_{2}-\ddot{x}_{d}\), the time derivation of Equation (17) gets along with the system in Equation (5) as follows:  
\[ \begin{split}\dot{s}&=\Pi(x,t)+\Phi(x,t)u+D(\theta,\Delta u)-\ddot{x}_{d}+\frac{2\gamma_{1}}{1+e^{-\mu_{1}(|x_{de}|-\phi)}}x_{de }+\frac{2\gamma_{1}\mu_{1}x_{de}\text{sgn}(x_{de})e^{-\mu_{1}(|x_{de}|-\phi)}} {\big{(}1+e^{-\mu_{1}(|x_{de}|-\phi)}\big{)}^{2}}x_{e}\\ &\quad+\frac{2\gamma_{2}\alpha}{1+e^{\mu_{2}(|x_{de}|-\phi)}}|x_{ e}|^{\alpha-1}x_{de}-\frac{2\gamma_{2}\mu_{2}x_{de}e^{\mu_{2}(|x_{de}|-\phi)}}{ \big{(}1+e^{\mu_{2}(|x_{de}|-\phi)}\big{)}^{2}}|x_{e}|^{\alpha}\end{split}  \tag{18}\]  
In order to facilitate controller design, there is the following assumption:  
**Assumption 1**. : _The lumped uncertain terms, \(D(\theta,\Delta u)=[D_{1}(\theta,\Delta u),\dots,D_{n}(\theta,\Delta u)]\), need to satisfy the following standard condition:_  
\[\|D_{i}(\theta,\Delta u)\|\leq K_{i}|s_{i}|^{\frac{1}{2}};i=1,\dots,n \tag{19}\]  
_where \(K_{i}>0\)._  
In order to achieve the stabilization target of the robot system, the following control action is proposed:  
\[u=-\Phi^{-1}(x,t)\Big  
**Proof. ** Applying control torque in Equation (20)-(22) to Equation (19) gains:  
\[\left\{ \begin{array}{l}\dot{s}=D(\theta,\Delta u)-\Sigma_{1}|s|^{1/2}\text{sgn }(s)-\eta\\ \dot{\eta}=-\Sigma_{2}\text{sgn}(s)\end{array} \right. \tag{24}\]  
Based on the assumption in Equation (19), and the selection condition of the sliding gains in Equation (23), it can be verified that the sliding manifold and its time derivative will converge to zero in finite time. Now, considering one of the elements in Equation (24) as follows:  
\[\left\{ \begin{array}{l}\dot{s}_{i}=D_{i}(\theta,\Delta u)-\Sigma_{1i}|s_{i}|^ {1/2}\text{sgn}(s_{i})-\eta_{i}\\ \dot{\eta}_{i}=-\Sigma_{2i}\text{sgn}(s_{i})\end{array} \right. \tag{25}\]  
The following Lyapunov function is defined for the system in Equation (25):  
\[L_{2}=\sigma^{T}\Gamma\sigma \tag{26}\]  
Here, \(\sigma=\left[s_{i}^{1/2},\lambda_{i}\right]^{T}\), \(\Gamma=\frac{1}{2}\left[ \begin{array}{cc}4\Sigma_{2i}+\Sigma_{1i}^{2}&- \Sigma_{1i}\\ -\Sigma_{1i}&2\end{array} \right]\). If \(\Sigma_{2i}>0\), so, according to [PERSON]'s inequality:  
\[\lambda_{\text{min}}(\Gamma)\|\sigma\|^{2}\leq L_{2}\leq\lambda_{\text{max}} (\Gamma)\|\sigma\|^{2} \tag{27}\]  
with \(\|\sigma\|^{2}=|s_{i}|+\eta_{i}^{2}\). The time derivation of Equation (26) is:  
\[\dot{L}_{2}=-\frac{1}{|s_{i}|^{1/2}}\sigma^{T}\Phi\sigma+\frac{1}{|s_{i}|^{1/2 }}[D_{i}(\theta,\Delta u),0]\Gamma\sigma \tag{28}\]  
with \(\Phi=\frac{\Sigma_{1i}}{2}\left[ \begin{array}{cc}2\Sigma_{2i}+\Sigma_{1i}^{2 }&-\Sigma_{1i}\\ -\Sigma_{1i}&1\end{array} \right]\). Based on the assumption in Equation (19), we can gain:  
\[ \begin{array}{l}\dot{L}_{2}\leq-\frac{1}{|s_{i}|^{1/2}}\sigma^{T}\overline{ \Phi}\sigma\\ \leq-\frac{1}{|s_{i}|^{1/2}}\lambda_{\text{min}}\Big{(}\overline{\Phi}\Big{)} \|\sigma\|^{2}\end{array}  \tag{29}\]  
where \(\overline{\Phi}=\frac{\Sigma_{1i}}{2}\left[ \begin{array}{cc}2\Sigma_{2i}+ \Sigma_{1i}^{2}-(4\Sigma_{2i}+\Sigma_{1i})K_{i}&-(\Sigma_{1i}+2K_{i})\\ -(\Sigma_{1i}+2K_{i})&1\end{array} \right]\). \(\overline{\Phi}\) is selected to be greater than zero. Consequently, \(\dot{L}_{2}<0\). Applying the Equation (27) gives:  
\[|s_{i}|^{1/2}\leq\|\sigma\| \tag{30}\]  
It follows that:  
\[\dot{L}_{2}\leq vL_{2}^{1/2} \tag{31}\]  
with \(v=\frac{\lambda_{\text{min}}(\overline{\Phi})}{\alpha_{\text{max}}^{1/2}( \Gamma)}\).