# Practical Adaptive Sliding-Mode Control Approach for Precise Tracking of Robot Manipulators
## Appendix B Proof of Theorem  
To prove the overall system stability, the Lyapunov function, denoted by \(V_{t}\in\Re\), is defined as  
\[V_{t}=\frac{1}{2}\mathbf{s}_{t}^{T}\mathbf{s}_{t}+\frac{1}{2}\sum_{i=1}^{n}\frac {1}{\delta_{i}}\big{(}\tilde{K}_{i,t}\big{)}^{2}+\frac{1}{2}\sum_{i=1}^{n}\frac {1}{\tilde{\tau}_{i}}\big{(}\hat{\eta}_{i,t}\big{)}^{2} \tag{11}\]  
whose the time derivative can be computed as  
\[\dot{V}_{t} =\mathbf{s}_{t}^{T}\dot{\mathbf{s}}_{t}-\sum_{i=1}^{n}\frac{1}{ \delta_{i}}\tilde{K}_{i,t}\dot{\hat{K}}_{i,t}^{s}+\sum_{i=1}^{n}\frac{1}{ \tilde{\tau}_{i}}\hat{\eta}_{i,t}\hat{\eta}_{i,t}\] \[=\mathbf{s}_{t}^{T}\dot{\mathbf{s}}_{t}+\Lambda_{1,t}+\Lambda_{2,t} \tag{12}\]  
where  
\[\Lambda_{1,t} =-\sum_{i=1}^{n}\frac{1}{\delta_{i}}\hat{K}_{i,t}\dot{\hat{K}}_{ i,t}^{s}\] \[\Lambda_{2,t} =\sum_{i=1}^{n}\frac{1}{\tilde{\tau}_{i}}\hat{\eta}_{i,t}\hat{ \eta}_{i,t}.\]  
Substituting the time derivative of Equation (6) into Equation (12), we have  
\[\dot{V}_{t}=\mathbf{s}_{t}^{T}\big{[}\ddot{\mathbf{q}}_{d,t}-\ddot{\mathbf{q }}_{t}+\bar{\mathbf{K}}_{s}\big{]}+\Lambda_{1,t}+\Lambda_{2,t}. \tag{13}\]  
Substituting Equations (3) and (12) into Equation (13), Equation (13) can be represented as  
\[\dot{V}_{t} =\mathbf{s}_{t}^{T}\big{[}\ddot{\mathbf{q}}_{d,t}-\mathbf{\mathcal{L }}_{t}-\bar{\mathbf{M}}^{-1}\mathbf{\tau}_{t}+\bar{\mathbf{K}}_{s}\big{]}+\Lambda_ {1,t}+\Lambda_{2,t}\] \[=-\mathbf{s}_{t}^{T}\big{[}\mathbf{E}_{t}+\bar{\mathbf{K}}_{s} \mathbf{s}_{t}-\bar{\mathbf{K}}_{s}^{s}\text{sgn}(\mathbf{s}_{t})\big{]}+ \Lambda_{1,t}+\Lambda_{2,t} \tag{14}\]  
where \(\mathbf{E}_{t}=\mathbf{\mathcal{L}}_{t}-\mathbf{\mathcal{L}}_{t-L}\in\Re^{n}\) is the TDE error that is bounded according to Equation (11), e.g., \(|E_{i,t}|\leq E_{i}^{*}\) for all \(i=1,2,\cdots,n\). It follows then that  
\[\dot{V}_{t} \leq-\big{[}\bar{\mathbf{K}}_{s}\mathbf{s}_{t}\big{]}^{T}\mathbf{ s}_{t}+\sum_{i=1}^{n}|s_{i,t}|\big{(}|E_{i,t}|-\hat{K}_{i,t}^{s})+\Lambda_{1,t}+ \Lambda_{2,t}\] \[=-\big{[}\bar{\mathbf{K}}_{s}\mathbf{s}_{t}\big{]}^{T}\mathbf{s}_{ t}+\sum_{i=1}^{n}|s_{i,t}|\bar{K}_{i,t}-\sum_{i=1}^{n}\hat{\eta}_{i,t}|s_{i,t}| \hat{K}_{i,t}\chi_{t}+\sum_{i=1}^{n}\frac{1}{\tilde{\tau}_{i}}\hat{\eta}_{i,t} \hat{\eta}_{i,t}. \tag{15}\]  
If \(\|\mathbf{s}_{t}\|_{\infty}\) is larger than the boundedness \(\bar{\epsilon}\), i.e., \(\chi_{t}=1\), Equation (15) can be represented as follows,  
\[\dot{V}_{t} =\mathbf{s}_{t}^{T}\dot{\mathbf{s}}_{t}-\sum_{i=1}^{n}\frac{1}{ \delta_{i}}\tilde{K}_{i,t}\dot{\hat{K}}_{i,t}^{s}+\sum_{i=1}^{n}\frac{1}{ \tilde{\tau}_{i}}\hat{\eta}_{i,t}\hat{\eta}_{i,t}\]0  
If \(|\hat{K}_{i,t}^{s}|\) is less than the positive value \(\bar{\eta}\), Equation (16) yields  
\[\dot{V}_{t} =\mathbf{s}_{t}^{T}\dot{\mathbf{s}}_{t}-\sum_{i=1}^{n}\frac{1}{ \delta_{i}}\tilde{K}_{i,t}\dot{\hat{K}}_{i,t}^{s}+\sum_{i=1}^{n}\frac{1}{ \tilde{\tau}_{i}}\hat{\eta}_{i,t}\hat{\eta}_{i,t}\]1  
when \(\|\mathbf{s}_{t}\|_{\infty}\leq\bar{\epsilon}\) is satisfied. Equation (18) means that the Lyapunov function \(V_{t}\) will be decreased. Then, the time derivative of the Lyapunov function has a low-bounded value, not zero, as follows,  
\[\dot{V}_{t} =\mathbf{s}_{t}^{T}\dot{\mathbf{s}}_{t}-\sum_{i=1}^{n}\frac{1}{ \delta_{i}}\tilde{K}_{i,t}\dot{\hat{K}}_{i,t}^{s}+\sum_{i=1}^{n}\frac{1}{ \tilde{\tau}_{i}}\hat{\eta}_{i,t}\hat{\eta}_{i,t}\]2which means that the sliding variable \(\mathbf{s}_{t}\) reaches the boundedness \(\tilde{\varepsilon}\) within the finite time \(t_{\tilde{t}}=t_{1}>0\). In other words, given that the sliding variable \(\mathbf{s}_{t}\) enters the boundedness \(\tilde{\varepsilon}\), we are able to have more room for the improper tracking performance if the boundedness \(\tilde{\varepsilon}\) can be chosen as being smaller. However, the time derivative of the Lyapunov function does not always guarantee that the sliding variable \(\mathbf{s}_{t}\) the boundedness \(\tilde{\varepsilon}\), i.e., \(\|\mathbf{s}_{t}\|_{\infty}<\tilde{\varepsilon}\).