## 5 Simulations  
Two sets of simulation results are presented to validate the proposed image-based switched approach for mobile robots. The Egt-toolbox [33] is used to accomplish the simulation. The virtual camera parameters are set as \(f_{u}=829.77,f_{v}=826.59\), \(u_{0}=393.74\), \(v_{0}=285.87\). Two static points are selected as \(\mathcal{P}_{1}=[3,3,11]^{T}m,\mathcal{P}_{2}=[-0.5312]^{T}m\). The estimation error of the Principal point is less than 2.5 pixels, which is verified by multi-group experiments in [20]. Thus A random Gaussian noise with a deviation of 1.5 pixels, is added to the image points to verify the robustness.  
The control parameters are shown in Table 1. These control parameters are utilized in the following two sets of simulations.  
An initial pose of the mobile robot is selected as \(X_{0}=[-2,-6,10^{\circ}]^{T}\). Simulation results are shown in Figures 4 and 5. In Figure 3(a), the trajectories of two image points pass through the circular points at the starting and ending positions, respectively. The star points represent the desired image points. The circular points at the end positions coincide with the star points, which indicates that the mobile robot has reached the desired configuration. This also can be seen from the motion trajectory of the mobile robot in Figure 3(b). The state curves of the mobile robot are shown in Figure 5.  
At almost 41 seconds,  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \(\mathbf{k_{c1}}\) & \(\mathbf{k_{c2}}\) & \(\mathbf{k_{c3}}\) & \(\mathbf{k_{a}}\) & \(\mathbf{k_{w}}\) & \(\mathbf{k_{t}}\) & \(\mathbf{k_{v}}\) \\ \hline
0.3 & 0.3 & 1 & 5 & 1 & 0.001 & 0.0005 \\ \hline \end{tabular}
\end{table}
Table 1: Control parameters.  
the mobile robot was very close to the desired position, then a pure rotation control was applied to the robot. The threshold of the state \(\rho\) is set at 0.01. In this case, no switching control occurs for the mobile robot. The feature points are always in the field-of-view during the stabilization task of the mobile robot, which is desirable. Next, a case that is not so ideal is considered.  
In this case, an initial pose being \(X_{0}=[2,-6,10^{\circ}]^{T}\) is selected. Simulations are shown in Figures 6 and 7. As shown in Figure 6a, despite several round trips, the feature trajectories remain within the image boundaries. In Figure 7, the movement process of the mobile robot is divided into four stages. In stage 1, the control laws (23) and (24) are used. When an image feature falls into the left border area or the right border area of the image, it switches to stage 2 or stage 3 correspondingly. The controllers (34) and (35) are applied in stage 2 and stage 3, respectively. If the state \(\rho\) is smaller than the threshold, the last stage 4 is activated.  
Figure 4: Simulation of the initial poseture \(X_{0}=[-2,-6,10^{\circ}]^{T}\). (a) Feature trajectories. (b) Motion trajectory.  
Figure 5: Simulation of the initial pose \(X_{0}=[-2,-6,10^{\circ}]^{T}\) : Evolution of the mobile robot pose and the stage curve.  
In this part, simulation results are provided to verify the effectiveness of the proposed strategy. According to above two sets of simulations, it is known that whether or not to start the switching process is related to the initial pose of the mobile robot. If the initial pose is good, the mobile robot can be driven to the desired pose quickly as shown in the first simulation. However, there is no basis for judging whether an initial pose is good or not good at present. This paper focus on the control strategy to keep two marker points in the field-of-view, which is a preliminary work. The efficiency of the control strategy is sacrificed to remain image features in view. Based on this work, the efficiency should be improved in the future work, which will be verified on a real mobile robot in an actual scenario.