## 4 Numerical Simulations  
Considering the dynamic model of the leader-follower formation system (6), the derivative and integral terminal sliding mode controller is proposed. There are three robots in the leader-follower framework, where the two follower robots track along with the leader robot. The radius of each robot is 0.05 m. The parameter fluctuations in formation system are determined by  
\[\Delta_{i}=\Delta_{k}=\left[ \begin{array}{cc}0.3 rand()-0.15&0\\ 0&0.3 rand()-0.15\end{array} \right], \tag{35}\]  
where \(i=1\) denotes the leader robot, and \(k=2\), 3 represent the two follower robots. The uncertainties and external disturbances in lumped model (6) are depicted by  
\[ \begin{array}{c}\pi_{ix}=\pi_{iy}=\pi_{i\theta}=0.5\sin(2\pi t)\\ \pi_{kx}=\pi_{ky}=\pi_{k\theta}=0.2\sin(\pi t)\end{array}  \tag{36}\]  
The parameters in control design are set as \(\gamma_{12,1}=\gamma_{12,2}=\gamma_{13,1}=\gamma_{13,2}=4\), \(\lambda_{12,1}=\lambda_{12,2}=\lambda_{13,1}=\lambda_{13,2}=1\), \(p_{11}=p_{12}=9\), \(q_{11}=q_{12}=7\), \(q_{21}=3\), \(p_{21}=5\), \(q_{22}=7\), \(p_{22}=9\), \(\lambda_{e12}=k_{e13}=20\), \(\varepsilon_{12}=\varepsilon_{13}=4\), \(k_{12}=k_{13}=2\), \(\delta_{12}=\delta_{13}=4\), \(\eta_{12}=\eta_{13}=0.2\). Considering the circle trajectory for the formation system in Figure 3, the initial state vector is respectively set as \(x_{12}^{0}\) = [0.5 m 0 m/s\({}^{-1}\) 3.2\(\pi\)/4 rad 0 rad/s], \(x_{13}^{0}\) = [0.707 m 0 m/s\({}^{-1}\) 3\(\pi\)/4 rad 0 rad/s]. The desired state vectors are respectively designated as \(x_{12}^{d}\) = [0.13 m 0 m/s\({}^{-1}\) \(\pi\)/2 rad 0 rad/s], \(x_{13}^{d}\) = [0.26 m 0 m/s\({}^{-1}\) \(\pi\)/2 rad 0 rad/s]. The desired linear and angular velocities of leader robot are designated as \(v_{1}^{d}=0.5\) m, \(\omega_{1}^{d}=1\) rad/s. The simulation results are shown in Figures 4-6. Figure 3 displays the moving curve of the formation system, which shows the robots are in a line while moving along the circular trajectory. In Figure 3, the solid point denotes the initial position of the formation robots. The arrows are the moving directions of the three robots. It is seen form the Figure 3 that the follower can track the leader robot with the desired distance and angle, while the leader robot tracks the circular trajectory. Figure 3: Moving trajectory of leader–follower formation system. Figure 5 denotes the control input of follower robot 2 using different control method. In Figure 5a, the acceleration of follower robot 2 are shown, while the angular accelerations of follower robot 2 are displayed in Figure 5b. From Figure 5a,b, the control input of DIT-SMC based NDOB is smoother than other control methods, which denotes the acceleration and angular acceleration are more stable. Figure 5: The acceleration and angular acceleration of follower robot 2; (a) \(\alpha_{2}\); (b) \(\beta_{2}\). Figure 6: _Cont._  
The control inputs of follower robot 3 are shown in the Figure 6, which denotes the acceleration and angular acceleration of follower robot 3. The accelerations of follower robot of follower robot using the three control methods are displayed in Figure 6a, while the angular acceleration using three methods are shown in Figure 6b. From Figures 5 and 6, the combination of the DIT-SMC and NDOB can benefit the decrease of the chattering phenomenon that is an inherent drawback of the SMC methodology. Figure 7 denotes the sliding mode vectors of two follower robots. As proven in the Theorem 1, the reaching time of sliding surface will be eliminated, and the error of formation system will reach to the equilibrium point in the finite time. From Figure 7, the formation system can enter the sliding mode in the beginning, which can guarantee the system stability. Figure 6: The acceleration and angular acceleration of follower robot 3; (a) \(\alpha_{3}\); (b) \(\beta_{3}\). Figure 7: The sliding surface.