## 4 Simulation Results and the Practical Implementation
### Set Up of Simulation Environments  
We will apply the parameters of practical hardware of a swarm of WMRs as \(m=5(\,\mathrm{kg})\), \(d=7(\,\mathrm{cm})\), \(R=8.9(\,\mathrm{cm})\), and \(r=3.25(\,\mathrm{cm})\) to build up a simulation environment. This simulation scenario is close to the real situation, and one desired trajectory tracking circular and S-type for the simulation and verification is generated by the following equations. Circular Trajectory:  
\[\left\{ \begin{array}{l}x=x_{0}+r_{d}\cos(\theta_{d})\\ y=y_{0}+r_{d}\sin(\theta_{d})\end{array} \right.\]  
where \(r_{d}\) and \(\theta_{d}=\int_{0}^{t}\omega_{d}dt\) are the radius of the desired circular trajectory and desired rotation angle with a constant predefined angular velocity \(\omega_{d}\), respectively. The initial conditions of the desired trajectory are \(x_{0}=0(\mathrm{m})\), \(y_{0}=0(\mathrm{m})\), and \(\omega_{d}=3^{\circ}/s\). The control of a swarm of WMRs, which is driven by the proposed method, starts from initial conditions, as below. Circular Scenario:  
\(x_{c}(0)=1.9(\mathrm{m})\), \(y_{c}(0)=0(\mathrm{m})\), \(\theta(0)=0^{\circ}\); \(x_{c}(0)=-1.9(\mathrm{m})\), \(y_{c}(0)=0(\mathrm{m})\), and \(\theta(0)=0^{\circ}\)  
\(x_{c}(0)=0(\mathrm{m})\), \(y_{c}(0)=1.9(\mathrm{m})\), \(\theta(0)=0^{\circ}\); \(x_{c}(0)=0(\mathrm{m})\), \(y_{c}(0)=-1.9(\mathrm{m})\), and \(\theta(0)=0^{\circ}\)S type Trajectory:  
\[\left\{ \begin{array}{l}x=x_{0}+r_{d}\cos(2\theta_{d})\\ y=y_{0}+2r_{d}\sin(\theta_{d})\end{array} \right.\]  
where \(r_{d}\) and \(\theta_{d}=\int_{0}^{t}\omega_{d}dt\) are the radius of the desired S-type trajectory and desired rotation angle with a constant predefined angular velocity \(\omega_{d}\), respectively. As a circular scenario, the initial conditions of the desired trajectory are \(x_{0}=0(\text{m})\), \(y_{0}=0(\text{m})\), and \(\omega_{d}=3^{\circ}/s\). This case will also verify the trajectory tracking ability of this proposed \(H_{2}\) control design. The control of a swarm of WMRs, which is driven by the proposed method, starts with the following initial locations. S type Scenario:  
\(x_{c}(0)=1.9(\text{m})\), \(y_{c}(0)=0(\text{m})\), \(\theta(0)=0^{\circ};x_{c}(0)=-1.9(\text{m})\), \(y_{c}(0)=0(\text{m})\), and \(\theta(0)=0^{\circ}\)  
Figure 4: Results of circular trajectory tracking errors concerning initial point (\(x_{c}=-1.9\) m, \(y_{c}=0\) m). Figure 5: Results of circular trajectory tracking errors concerning initial point (\(x_{c}=0\) m, \(y_{c}=1.9\) m). Figure 3: Results of circular trajectory tracking errors concerning initial point (\(x_{c}=1.9\) m, \(y_{c}=0\) m). Figure 8: Results of circular trajectory tracking torques concerning initial point (\(x_{c}=-1.9\) m, \(y_{c}=0\) m). Figure 6: Results of circular trajectory tracking errors concerning initial point (\(x_{c}=0\) m, \(y_{c}=-1.9\) m). Figure 7: Results of circular trajectory tracking torques concerning initial point (\(x_{c}=1.9\) m, \(y_{c}=0\) m). Finally, the \(H_{2}\) control design of a desired S-type trajectory with a 1.6 m radius gives the simulation results of Figures 11-19. Figure 11 reveals the trajectory tracking performance of the \(H_{2}\) control design in the desired S-type trajectory. From the significant effect, the trajectory tracking performance of this proposed \(H_{2}\) design to the reference S-type trajectory also demonstrates an excellent outcome. From Figures 12-19, it points out noticeable results, and this intended \(H_{2}\) design can track the desired S-type trajectory fast, and errors and torques convergence rate approach zero quickly. Figure 10: Results of circular trajectory tracking torques concerning initial point (\(x_{c}=0\) m, \(y_{c}=-1.9\) m). Figure 9: Results of circular trajectory tracking torques concerning initial point (\(x_{c}=0\) m, \(y_{c}=1.9\) m).