## 4 Simulation Results  
The simulation was conducted to illustrate the performance of the robust switching control strategy proposed here. The numerical simulation program was constructed in Matlab/Simulink based on the fourth order Runge-Kutta integration algorithm with a fixed time step interval of 0.001. We assumed \(m=1.5\) (kg) \(d=0.1\)\((m)\), \(I_{c}=0.55\)\((\)kgm\({}^{2})\), \(r=0.035\)\((\)m\()\), \(c=0.05\)\((\)m\()\), and \(\Phi=0.01\) for this simulation, unless specified otherwise. For \(E_{d}:=\{x_{d},y_{d}\in\Re\big{|}\ dist(x(t)\ y(t),\ x_{d}\ y_{d})\leq\beta\}\), we set \(B=0.005\)\((\)m\()\). The perturbation angle \(\theta_{\sigma}\) was selected as 1.6 (rad) based on Equations (35) and (36). Figure 3 shows the (\(x\),\(y\))-trajectory of the WMR, the rotational angle \(\theta(t)\) and the (\(x\),\(y\))-navigation potential energy \(\varphi_{v}(t)\), as well as the forward velocity \(\big{|}v(t)\big{|}\). The initial condition was assumed as \((x_{i},y_{i},\theta_{i})=(2,\ 1,\ 45^{\circ})\). According to results in Figure 3, we observed that the WMR quickly reached the desired set (\(x\),\(y\)) = (0,0) by performing four times of switching (specified by black arrows). In particular, it can be seen from Figure 3c that the WMR took approximately 2 s to consume the majority of \(\varphi_{v}(t)\) in a monotonically decreasing manner. Figure 4 also investigates the performance of the WMR based on the initial condition \((x_{i},y_{i},\theta_{i})=(-1,\ 0,\ -45^{\circ})\). The results show that the WMR experienced eight times of switching to reach the desired set with strict decrease of \(\varphi_{v}(t)\) and it took about 12 s to complete its given task. In addition, it is commonly found from Figures 3 and 4 that the WMR reached near forbidden set (i.e., \(E_{avoid}\)) and the first switching had occurred, which clearly shows the intent of the switching control law in **Theorem 1**. From Figures 3b and 4b, it can be seen that the navigation energy of the WMR was sufficiently stored although the perturbation angle of the second law had been maintained almost identically. This was because the proposed technique chooses that the WMR do not wait until the forward velocity had become completely insignificant. Therefore, **Remark 4** is addressed along with those simulation outcomes. However, we cannot deny that the perturbed transient forward velocity between switching gradually decreased as the WMR came closer to \(E_{d}\) (see both Figures 3d and 4d). Moreover, we can see from Figures 3b and 4b that the range of \(\theta(t)\) was approximately 0.03 to 1.7 after the control was switched from the first control to the second one until the moment from the second control to the first one, which implies \(\theta_{\sigma\min}\approx 0.03\) according to Equation (36). From Figures 3d and 4d, we can see the following transient scenario of forward velocity \(\big{|}v(t_{i})\big{|}=\Phi\rightarrow\big{|}v(t)\big{|}\rightarrow\big{|}v( t_{i+1})\big{|}=\Phi\) for \(i=0,1,2,\cdots\) with \(\big{|}v(t)\big{|}>\Phi\) for \(t\in(t_{i}\ t_{i+1})\) between each switching. This response is exactly synchronized with the scenario described in Figure 2a. Figure 5 presents the estimation results of model uncertainty for Figures 3 and 4. Here, \(\gamma\) and \(\rho\) are selected as 0.13 and 0.01 for \(d=0.1\). Due to the action of model uncertainty compensation term, \(F_{v}\) it was found that the estimates \(|F_{v}|\) were well synchronized with the true values, resulting in cancelling the Coriolis effect of the WMR. In Figure 6, the (\(x\),\(y\))-navigation potential energies for three different tuning parameters have been presented. As the main parameters F and a of robust control increase, the convergence rate of potential energy became superior.