## 4 Implementation
### Simulation Results  
#### 4.2.1 String Formation When Moving along a Circular Trajectory  
In Figure 3, the multi-robot platform carries out the task of string formation when moving along a circular trajectory, where the red means the leader robot and the green and blue delegate the two followers. The initial postures of the three robots are allocated at  
\[\mathbf{q}_{1}=\left[\begin{array}{cc}0.5\mathrm{m}&0\mathrm{m}&0.5\pi \mathrm{rad}\end{array}\right]^{T}\!\!,\ \mathbf{q}_{2}=\left[\begin{array}{cc}0.8\mathrm{m}&-0.4\mathrm{m}&0 \mathrm{rad}\end{array}\right]^{T}\!\!,\ \mathbf{q}_{3}=\left[\begin{array}{cc}1\mathrm{m}&-0.5\mathrm{m}&\pi \mathrm{rad}\end{array}\right]^{T} \tag{58}\]  
According to the initial postures and the formation task, the initial states of the formation dynamics (8) can be calculated as  
\[\mathbf{x}_{12}(0)=\left[\begin{array}{cc}0.525\mathrm{m}&0\mathrm{m}/ \mathrm{s}&0.8\pi\mathrm{rad}&0\mathrm{rad}/\mathrm{s}\end{array}\right]^{T}\! \!,\ \mathbf{x}_{13}(0)=\left[\begin{array}{cc}0.70\mathrm{m}&0\mathrm{m}/ \mathrm{s}&0.75\pi\mathrm{rad}&0\mathrm{rad}/\mathrm{s}\end{array}\right]^{T} \tag{59}\]Similarly, the desired states of the two followers are assigned by  
\[\mathbf{x}_{12}^{d}=\left[\begin{array}{cc}0.13\text{m}&0\text{m/s}&0.5\pi\text{rad }&0\text{rad/s}\end{array}\right]^{T}\text{, }\mathbf{x}_{13}^{d}=\left[\begin{array}{cc}0.26\text{m}&0\text{m/s}&0.5\pi\text{rad }&0\text{rad/s}\end{array}\right]^{T} \tag{60}\]  
The leader's linear speed and its angular velocity are set as \(v_{1}=0.5\text{m/s}\) and \(\omega_{1}=1\text{rad/s}\).  
Figure 4 demonstrates the state variables in (8) when the multi-robot system fulfills the formation task in Figure 3. For the purpose of comparisons, the other three classic control methods were also implemented on the same platform to accomplish the same formation task besides the presented control method (short for AST-SMC in Figure 4). These control methods are listed as the derivative-integral terminal sliding mode control [14] (short for DI-TSMC in Figure 4) and the sole super-twisting sliding mode control without any adaptive gains (short for ST-SMC in Figure 4).  
Figure 4: Comparisons of the state variables by different control methods. (**a**) \(l_{12}\), (**b**) \(\Psi_{12}\), (**c**) \(l_{13}\), (**d**) \(\Psi_{13}\).  
Figure 3: String formation of the platform when moving along a circular trajectory.  
Shown in Figures 5 and 6, the adaptive-gain super-twisting sliding mode control method can decrease the chattering phenomenon effectively. In theory, the adaptive-gain super-twisting sliding mode control can completely compensate the disturbances and uncertainties entering the formation control system by the control channel. However, the formation dynamics (8) contain some disturbances and uncertainties that enter the control system by other channels. Therefore, the control inputs have to frequently switch to resist their adverse effects.  
Figure 7 illustrate the sliding surfaces of the adaptive-gain super-twisting sliding mode control method. The gains \(\alpha_{2}\), \(\beta_{2}\), \(\alpha_{3}\), and \(\beta_{3}\) governed by the designed adaptive law (24) are demonstrated in Figure 8. As proven in Theorem 1, the sliding surfaces are convergent in infinite time although the time instants cannot be known in advance. Further, the curves of the gains in Figure 8 are not convergent