## 3 Main Results
### FTESO-Based FXZNN Model Design
: _Using FTESO (13) and (18), if the controller in (24) and the activation function in (25) are adopted, then the TMR can accurately follow the desired velocity generated by the kinematic controller in (21) in fixed time \(T_{e_{ci}}\), that is, \(\hat{z}\equiv z_{c}\). The upper bound of convergence time \(T_{e_{ci}}(i=1,2)\) satisfies_  
\[T_{e_{ci}}\leq\dfrac{1}{\gamma S_{1}^{l_{2}}(l_{1}l_{2}-1)}+\dfrac{1}{\gamma S _{2}^{l_{2}}(1-l_{2}(1-1/l_{1}))}. \tag{27}\]  
Proof. : In the first step, we will verify that these states do not escape to infinity in any time interval \([0,T_{2})\). Since the analyses of the two subsystems (11) and (12) are relatively similar, we take (12) as an example for analysis, and the other subsystem can be analyzed according to it. \[ \begin{cases}\dot{\omega}_{e}{=}\dot{\omega}-\omega_{c}\\ \omega_{e}{=}\omega-\omega_{c}\end{cases} \Rightarrow\dot{\omega}_{e}-\omega_ {e}=-\tilde{\omega}\Rightarrow\dot{\omega}_{e}+\tilde{\omega}=\dot{\theta}_{e}. \tag{28}\]  
We take the bounded function  
\[F(\theta_{e},\tilde{\omega},\tilde{\rho}_{4})=\dfrac{1}{2}(\theta_{e}^{2}+ \tilde{\omega}^{2}+\tilde{\rho}_{4}^{2}). \tag{29}\]  
The derivative of the above equation can be obtained as  
\[ \begin{split}\dot{F}(\theta_{e},\tilde{\omega},\tilde{\rho}_{4}) =\theta_{e}\dot{\theta}_{e}+\tilde{\omega}\dot{\omega}+\tilde{\rho}_{4}\dot{ \tilde{\rho}}_{4}\\ =\theta_{e}(\hat{\omega}_{e}+\tilde{\omega})+\tilde{\omega}\dot{ \tilde{\omega}}+\tilde{\rho}_{4}\dot{\tilde{\rho}}_{4}.\end{split}  \tag{30}\]\[\text{Considering } \begin{cases}\hat{\omega}{=}\frac{1}{\bar{1}}\bar{\rho}_{4}-\mu_{2}( sig^{\alpha_{2}}(\theta-\hat{\theta})-\varepsilon_{2}sig^{\beta_{2}}(\theta-\hat{ \theta})\\ \hat{\rho}_{4}{=}-\mu_{3}sig^{\alpha_{3}}(\theta-\hat{\theta})-\varepsilon_{3} sig^{\beta_{3}}(\theta-\hat{\theta})-\text{Y}sign(\theta-\hat{\theta})\end{cases} ,\text{ we have } \tag{31}\] \[\hat{F}(\theta_{e},\hat{\omega},\tilde{\rho}_{4}) =\theta_{e}\hat{\rho}_{e}+\tilde{\omega}\hat{\omega}+\tilde{\rho} _{4}\tilde{\rho}_{4}\] \[=\theta_{e}(\tilde{\omega}_{e}+\tilde{\omega})+\tilde{\omega} \Big{[}\frac{1}{\bar{1}}\bar{\rho}_{4}-\mu_{2}(sig^{\alpha_{2}}(\tilde{\theta })-\varepsilon_{2}sig^{\beta_{2}}(\tilde{\theta})\Big{]}\] \[+\tilde{\rho}_{4}\Big{[}-\mu_{3}sig^{\alpha_{3}}(\tilde{\theta}) -\varepsilon_{3}sig^{\beta_{3}}(\tilde{\theta})-\text{Y}sign(\tilde{\theta}) \Big{]}\] \[\leqslant\theta_{e}\hat{\omega}_{e}+\theta_{e}\tilde{\omega}+ \frac{1}{\bar{1}}\tilde{\omega}\tilde{\rho}_{4}+2\text{Y}|\tilde{\rho}_{4}|+G,\]  
where \(\tilde{\theta}=\theta-\hat{\theta}\), \(G=\tilde{\omega}\big{[}-\mu_{2}(sig^{\alpha_{2}}(\tilde{\theta})-\varepsilon_ {2}sig^{\beta_{2}}(\tilde{\theta})\big{]}+\tilde{\rho}_{4}\big{[}-\mu_{3}sig^ {\alpha_{3}}(\tilde{\theta})-\varepsilon_{3}sig^{\beta_{3}}(\tilde{\theta}) \big{]}\)  
Because \(\tilde{\theta},\tilde{\omega},\tilde{\rho}_{4}\) converge to zero in fixed time, it can be seen that they are bounded, and we can obtain \(|G|\leqslant L_{1}\). According to [PERSON]'s inequality, we have  
\[\hat{F}(\theta_{e},\tilde{\omega},\tilde{\rho}_{4}) \leqslant\frac{1}{2}\theta_{e}^{2}+\frac{1}{2}\tilde{\omega}_{e }^{2}+\frac{1}{2}\tilde{\omega}^{2}+\frac{1}{2}\tilde{\omega}^{2}+\frac{1}{2 }\tilde{\eta}_{4}^{2} \tag{32}\] \[+2\text{Y}^{2}+\frac{1}{2}\bar{\rho}_{4}^{2}+L_{2}\] \[=\theta_{e}^{2}+\tilde{\omega}^{2}+\frac{1}{\bar{1}}\tilde{\rho}_ {4}^{2}+\frac{1}{2}\tilde{\omega}_{e}^{2}+2\text{Y}^{2}+L_{1}.\]  
Since \(\tilde{\omega}_{e}\) is bounded, \(|\tilde{\omega}_{e}|\leqslant L_{3}\); therefore,  
\[\hat{F}(\theta_{e},\tilde{\omega},\tilde{\rho}_{4})\leqslant\theta_{e}^{2}+ \tilde{\omega}^{2}+\frac{1}{\bar{1}}\bar{\rho}_{4}^{2}+\bar{L}, \tag{33}\]  
where \(\bar{L}=\frac{1}{2}L_{2}^{2}+2\text{Y}^{2}+L_{1}\).