## 3 Main Results
### FTESO-Based FXZNN Model Design  
An auxiliary velocity control input that achieves tracking for kinematic model (2) is given by (21), which is a uniformly asymptotically stable velocity command obtained and used in the study of tracking problems for mobile robots [37]. \[z_{c}=\left[ \begin{array}{c}v_{c}\\ w_{c}\end{array} \right]=\left[ \begin{array}{c}v_{d}\cos\theta_{e}+k_{1}x_{e} \\ \omega_{d}+k_{2}v_{d}y_{e}+k_{3}v_{d}\sin\theta_{e}\end{array} \right], \tag{21}\]  
where \(v_{d}\geq 0\), \(\omega_{d}\) are the desired linear and angular velocities, respectively, and \(k_{1}\), \(k_{2}\) are positive constants. Since the approach assuming \"perfect velocity tracking\" is unrealistic [38], we should find the torque input (\(\tau\)) to implement trajectory tracking, such that \(z\) converges to \(z_{c}\) in fixed time. Then, the FTESO-based FXZNN model is introduced. The schematic diagram of the FTESO-based FXZNN control system is shown in Figure 2.  
Considering the design process of ZNNs, the following design formula is introduced:  
\[\dot{e}(t)=-\gamma\Phi(e(t)), \tag{22}\]  
where \(e(t)\) represents the velocity error vector and \(\gamma>0\in R\) denotes the design parameter used to adjust the rate of convergence. \(\Phi(\cdot):R^{n}\to R^{n}\) is the vector of activation functions, any elements \(\phi(\cdot):R\to R\) of which can be any odd function with the monotonically increasing property [39]. Then, a vector error equation, which enables the estimated velocities \(\vartheta\), \(\hat{\omega}\) to follow the ones generated by (21) as soon as possible, is constructed as follows:  
\[e_{c}=z_{c}-\hat{z}, \tag{23}\]  
where \(\hat{z}=\left[ \begin{array}{cc}\hat{v}&\hat{\omega}\end{array} \right]^{T}\). By combining Equations (22) and (23), a neurodynamic model of the TMR dynamics equation can be obtained as follows:  
\[\dot{z}_{c}-\hat{z}+\gamma\phi(e_{c})=0, \tag{24}\]  
Figure 2: Schematic of the fixed-time control system for a TMR. where \(\hat{z}= \begin{bmatrix}\dfrac{1}{mr}\tau_{v}+\dfrac{1}{m}\hat{\rho}_{3}+\mu_{2} sig^{a_{2}}(\psi-\hat{\psi})+\varepsilon_{2}sig^{\hat{\rho}_{2}}(\psi-\hat{\psi})\\ \dfrac{R}{T\tau_{\omega}}+\dfrac{1}{1}\hat{\rho}_{4}+\mu_{2}sig^{a_{2}}(\theta- \hat{\theta})+\varepsilon_{2}sig^{\hat{\rho}_{2}}(\theta-\hat{\theta})\\ \end{bmatrix} ,\)  
\(\hat{z}_{c}= \begin{bmatrix}\dot{v}_{d}\cos(\theta_{e})-\dot{\theta}_{e}v_{d} \sin(\theta_{e})+k_{1}\hat{x}_{e}\\ \dot{\omega}_{d}+k_{2}\hat{v}_{d}y_{e}+k_{2}v_{d}\hat{y}_{e}+k_{3}\hat{v}_{d} \sin(\theta_{e})+k_{3}v_{d}\theta_{e}\cos(\theta_{e})\\ \end{bmatrix} \). The control input can be obtained from (24) as follows:  
\[ \begin{bmatrix}\tau_{v}\\ \tau_{\omega}\end{bmatrix} = \begin{bmatrix}mr\bigg{\{}(\hat{v}_{c}-\gamma\phi(e _{c1}))-\dfrac{1}{m}\hat{\rho}_{3}-\mu_{2}sig^{a_{2}}(\psi-\hat{\psi})- \varepsilon_{2}sig^{\hat{\rho}_{2}}(\psi-\hat{\psi})\bigg{\}}\\ \dfrac{Ir}{R}\bigg{\{}(\dot{\omega}_{c}-\gamma\phi(e_{c2}))-\dfrac{1}{T}\hat{ \rho}_{4}-\mu_{2}sig^{a_{2}}(\theta-\hat{\theta})-\varepsilon_{2}sig^{\hat{ \rho}_{2}}(\theta-\hat{\theta})\bigg{\}}\\ \end{bmatrix} , \tag{25}\]  
where \(e_{c1}=v_{c}-\hat{v},e_{c2}=\omega_{c}-\hat{\omega}\). Up until now, we have constructed the ZNN control scheme based on the FTESO. Different activation functions are used to obtain controllers with different performance, and the following activation function with the fixed-time-convergence property is designed:  
\[\phi(e_{ci})=(g_{1}sig^{l_{1}}x+g_{2}sig^{1-1/l_{1}}(x))^{l_{2}}+g_{3}x, \tag{26}\]  
where \(\{g_{1},g_{2},g_{3},l_{1},l_{2}\}\in R^{+},l_{1}l_{2}>1,l_{2}(1-1/l_{1})<1,i= 1,2\). **Remark 2**. : _The residual error of the conventional ZNN model exponentially converges to zero, indicating that the convergence rate is slower with a smaller residual error. In view of this, the activation function \(\phi(e_{ci})\) is designed to amplify the value of \(\dot{e}_{ci}/e_{ci}\) to achieve fixed-time convergence. Additionally, the linear part of the activation acts as the robust term to achieve noise suppression._  
**Theorem 2**.