## 4 Simulation Results  
In this section, the results of the conducted simulation studies are presented to demonstrate the effectiveness of the proposed formation control protocols. MATLAB R2014a/Simulink software was used for simulation purposes, and the ode45 (Dormand-Prince) solver was used for the differential calculations with a relative tolerance value of \(0.001\). The considered system consisted of four mobile robots and a virtual leader moving along a parameterized path. Only one robot was assumed to have the ability to access the information of the desired path. The information exchange among the mobile robots was directed, and the communication topology between the mobile robots is shown in Figure 3.  
The parameters of the mobile robots used in the simulations are the same as those in [35] and presented in Table 1. The initial conditions of the mobile robots were chosen as \([x_{1}(0),y_{1}(0),\theta_{1}(0)]^{\mathrm{T}}=[-1.3,10.5,0.2]^{\mathrm{T}}\), \([x_{2}(0),y_{2}(0),\theta_{2}(0)]^{\mathrm{T}}=[-1.4,10.5,0.2]^{\mathrm{T}}\), \([x_{3}(0),y_{3}(0),\theta_{3}(0)]^{\mathrm{T}}=[-1,9,0.2]^{\mathrm{T}}\), \([x_{4}(0),y_{4}(0),\theta_{4}(0)]^{\mathrm{T}}=[-0.1,5,0.2]^{\mathrm{T}}\). The desired path was generated by \(p_{r}(\varphi)=[10\sin(0.5\varphi),10\cos(0.5\varphi)]^{\mathrm{T}}\), with the commanded speed \(v_{s}=0.5\). The desired geometry position vectors were chosen as \(p_{1d}=[0,0]^{\mathrm{T}},p_{2d}=[-3,0]^{\mathrm{T}}\), \(p_{3d}=[0,-3]^{\mathrm{T}}\), and \(p_{4d}=[-3,-3]^{\mathrm{T}}\). The extended state observer gains were set as \(b_{11}=90\), \(b_{22}=2700\), \(b_{33}=27000\), \(\gamma_{i1}=90\), \(\gamma_{i2}=2700\), \(\gamma_{i3}=27000\), and \(\alpha=0.95\). The control gains of the path-updating law were set as \(\rho_{1}=\rho_{3}=0.3\) and \(\beta^{s}=0.5\). The control gains were set as \(k_{i}^{p}=15\), \(k_{i}^{p}=15\), \(k_{i}^{\theta}=5\), \(k_{i}^{\omega}=10\), \(\beta_{i}^{p}=0.95\), \(\beta_{i}^{\varphi}=0.95\), \(\beta_{i}^{\theta}=0.95\), and \(\beta_{i}^{\omega}=0.95\). To demonstrate the effectiveness of the proposed FFTESO-based formation control protocols, the bounded external disturbances were given as follows:  
\[d_{1}(t)= \begin{bmatrix}4\cos(t)\\ 2\sin(t)\end{bmatrix} \]  
Figure 3: The communication topology of the NMRs. \[d_{2}(t)= \begin{bmatrix}2+4\sin(2t)\\ 2\cos(2t)\end{bmatrix} \]  
\[d_{3}(t)= \begin{bmatrix}-2+3\sin(t)\\ -1+2\cos(3t)\end{bmatrix} \]  
\[d_{4}(t)= \begin{bmatrix}2+3\sin(4t)\\ 1+2\cos(4t)\end{bmatrix} \]  
The comparisons between the different observers are shown in Figure 4. For clarity, we only show the comparison results of one mobile robot. The other mobile robots resulted in the same conclusions, because the applied observers were distributed. By employing the proposed FFTSO (7) and (15), the estimation errors of the unmeasured velocities \(v_{i}\) and \(\omega_{i}\) and the disturbance signals \(\sigma_{i}^{v}\) and \(\sigma_{i}^{\omega}\) are shown in Figure 4. To demonstrate the superiority of the proposed FFTSO, the comparisons with the linear extended state observer (LESO) proposed in [29] and the finite-time extended state observer (FTESO) proposed in [26] were made. The observer gains and initial conditions of the linear extended state observer in [29] and the finite-time extended state observer in [26] are the same as those in this article. It can be observed from Figure 4 that the settling time and estimation error of the proposed FFTSO is less than those under the methods proposed in [29] and [26]. The following performance indices were used to evaluate the performance of the proposed FFTSO: the integral of the absolute value of the error (IAE), the integral of the square error (ISE), the integral of the time multiplied by the absolute value of the error (ITAE), and the integral of the time multiplied by the square error (ITSE). The comparisons of the performance indices of the scheme are shown in Tables 2-5. Small performance index values represent good performance.