## 4 Simulation Experiments
### Tracking Performance in Noise-Free Environment  
To verify the proposed FTESO's performance, it was compared with the finite-time extended state observer (FESO) proposed in [26] and the linear extended state observer (LESO) proposed in [41]. The observer gains and initial conditions of the LESO and FESO are the same as those in this article. The comparison results are shown in Figure 3.  
\begin{table}
\begin{tabular}{c c c} \hline
**No.** & **Noise Item** & **Expression** \\ \hline
1 & Constant noise & \(n_{i}(t)=0.5\) \\
2 & Periodic noise & \(n_{i}(t)=0.5\cos(\pi t)\) \\
3 & Disappearing noise & \(n_{i}(t)=0.5 exp(-t)\) \\ \hline \end{tabular}
\end{table}
Table 2: Various noises.  
\begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Value** & **Unit** \\ \hline \(m\) & 150 & Kg \\ \(I\) & 35 & Kg\(\cdot\)m\({}^{2}\) \\ \(h\) & 0.25 & m \\ \(r\) & 0.1 & m \\ \(\xi\) & 0.01 & - \\ \hline \end{tabular}
\end{table}
Table 3: TMR parameters.  
Two indices, integrated time absolute error (\(ITAE=\int_{0}^{t_{final}}t|e_{ci}|dt,i=1,2\)) and integrated absolute error (\(IAE=\int_{0}^{t_{final}}|e_{ci}|dt,i=1,2\)), were utilized to evaluate the transient- and steady-state performance of the observer, where \(t_{final}=120\) s is the running time of the simulation. Small performance index values represent good performance. The comparisons of the performance indices of the scheme are shown in Table 4. Obviously, the performance of the proposed FTESO is better than that of the FESO and LESO.  
In the following, simulation results of the TMR are presented to demonstrate the effectiveness of the proposed FTESO-based FXZNN model.  
The simulation results are shown in Figures 4-7. Figure 4 presents the overall tracking performance of the model and the control inputs of the TMR. It can be observed that the TMR can reach the desired trajectory under the proposed control scheme. In Figure 5, we can observe the TMR's performance in detail, which confirms the TMR's correct behavior. It is observed in Figure 5c that the angle suddenly changes at \(t=15\pi\) and \(t=25\pi\). When conducting real experiments, it is recommended to change the angle defined at \(t\in[15\pi,25\pi]\) to avoid potential risk, though it does not affect the simulation result.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Index**} & **FTESO (Proposed** & \multirow{2}{*}{**LESO**} & \multirow{2}{*}{**FESO**} \\  & **Observer)** & & \\ \hline \(\int_{0}^{t_{final}}|e_{c1}|dt\) & \(5.2872\times 10^{-4}\) & \(0.02994\) & \(0.13891\) \\ \(\int_{0}^{t_{final}}|e_{c2}|dt\) & \(5.39\times 10^{-3}\) & \(0.05634\) & \(0.60107\) \\ \(\int_{0}^{t_{final}}t|e_{c1}|dt\) & \(2.61\times 10^{-3}\) & \(1.4943\) & \(5.17001\) \\ \(\int_{0}^{t_{final}}t|e_{c2}|dt\) & \(1.4131\times 10^{-4}\) & \(0.03808\) & \(39.38684\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparisons of performance indices of observers.  
Figure 3: Comparison results of the FTESO, the FTESO proposed in [26], and LESO proposed in [41].  
Figure 4: Tracking performance of the proposed model and control inputs in noise-free environment.  
Figure 5: Evolution of TMR’s position (\(x\),\(y\),\(\theta\)).  
Figure 6: Evolution of TMR linear velocity \(v\) and angular velocity \(\omega\) in noise-free environment.  
Figure 6 displays the curves of the TMR's velocities. Obviously, the linear and angular velocity observation values can converge to the ones generated by the kinematic controller in fixed time. Figure 7 illustrates that the proposed observer can accurately estimate the system state and compensate for unknown lumped disturbances. It is observed that the proposed control scheme drove the TMR to follow the desired trajectory under the conditions of unknown lumped disturbances. Thus, the proposed control scheme is effective and efficient.