## 3 Passivity Based Robust Switching Control Design for Posture Stabilization
If we choose \(\theta_{\sigma}>0\) such that,  
\[\theta_{\sigma}>\sqrt{2(C-H)/K_{\theta}}\ \text{and}\ \theta_{\sigma}<-\sqrt{2(C-H)/K_{ \theta}} \tag{40}\]  
Based on Equation (40), the last outcome in Equation (39) becomes,  
\[-1/2\ K_{\theta}(\theta(t)-\theta_{\sigma})^{2}-C+1/2\ K_{\theta}\theta_{ \sigma}{}^{2}+H>0\ \text{for}\ \theta_{\sigma\min}<\theta(t)<\theta_{\sigma\max} \tag{41}\]  
where, \(\theta_{\sigma\min}=\theta_{\sigma}-\sqrt{{\theta_{\sigma}}^{2}-2(C-H)/K_{ \theta}}\) and \(\theta_{\sigma,\max}=\theta_{\sigma}+\sqrt{{\theta_{\sigma}}^{2}-2(C-H)/K_{ \theta}}\). To have a clear understanding of Equations (35) and (36), the corresponding graphical representation is provided in Figure 2a. Consequently, we know from Equations (39) to (41) that,  
\[ \begin{array}{l}\kappa_{v}(t)-\kappa_{v}(t_{1})>-1/2\ K_{\theta}(\theta(t)- \theta_{\sigma})^{2}-C+1/2\ K_{\theta}\theta_{\sigma}{}^{2}>0\ \text{for}\\ \theta_{\sigma\min}<\theta(t)<\theta_{\sigma\max}\end{array}  \tag{42}\]  
Equation (42) implies \(\kappa_{v}(t)>\kappa_{v}(t_{1})\) for \(\theta_{\sigma\min}<\theta(t)<\theta_{\sigma\max}\). Furthermore, Equation (42) can be extended to \(\kappa_{v}(t)>\kappa_{v}(t_{1})\) for \(\forall t\in(t_{1}\ \ \ \ \ t_{2})\) (i.e., \(\left|v(t_{1})\right|\rightarrow\left|v(t)\right|\rightarrow\left|v(t_{2}) \right|=\Phi\) with \(\left|v(t_{1})\right|=\left|v(t_{2})\right|<\left|v(t)\right|\) for \(\forall t\in(t_{1}\ \ t_{2})\)), if we design proper perturbation angle \(\theta_{\sigma}\) and gain \(K_{\theta}\). Since the size of window \(\left|\theta_{\sigma\min}-\theta_{\sigma\max}\right|\) relies on the parameters \(\theta_{\sigma}\) and gain \(K_{\theta}\), we can intentionally construct that \(\Psi(t)\) during \(\theta\rightarrow\theta_{\sigma}\
eq 0\) can be fallen into the window \(\left|\theta_{\sigma\min}-\theta_{\sigma\max}\right|\) by manipulating \(\theta_{\sigma}\) and gain \(K_{\theta}\). For instance, if \(\theta_{\sigma}=\pi/2\), \(\theta_{\sigma\min}\approx 0\) and \(\theta_{\sigma\max}\approx\pi\) are possible according to Equation (41). Therefore, \(\Psi(t)\) during \(\theta\rightarrow\pi/2\) can be possibly within the domain \((0,\ \pi)\). In actual simulation or implementation of \(\theta_{\sigma}\), the condition \(\theta_{\sigma\min}<\theta(t)<\theta_{\sigma\max}\) is anticipated to be more flexible because Equation (39) used the extreme measure for the inequalities. On the other hands, from Equation (30),  
\[\dot{L}_{v}(t)<-\alpha\ \Phi\ \text{for}\ \forall t\in[t_{1}\ t] \tag{43}\]Integrating Equation (43) from \(t_{1}\) to \(t>t_{1}\) results in,  
\[L_{v}(t)-L_{v}(t_{1})<-\alpha\;\Phi(t-t_{1}) \tag{44}\]  
Equation (44) becomes based on Equation (42),  
\[\varphi_{v}(t)-\varphi_{v}(t_{1})<-\alpha\;\Phi(t-t_{1})+k_{v}(t_{1})-k_{v}(t)<- \alpha\;\Phi(t-t_{1})<0\Rightarrow\varphi_{v}(t)<\varphi_{v}(t_{1}) \tag{45}\]  
Therefore, we can guarantee that \(\varphi_{v}(t)<\varphi_{v}(t_{1})\) for \(\forall t\in(t_{1}\;t_{2})\). Proof is completed. In summary of Theorem1, the first control guides the WMR into the neighborhood of \(E_{\textit{avoid}}\) because we decide the switching time when \(|v|=\Phi\). This means that the WMR is still under motion. However, the WMR is losing its navigation potential energy due to the fact that the WMR is very close to \(E_{\textit{avoid}}\). Based on this issue, the second control is required because the WMR needs to regain further navigation energy via a rotational motion which induces that the forward velocity is re-perturbed. In addition, the period of each control between switching is a finite. Based on **Proposition 1**, the total time (\(\Delta T\)) of control for \(n\)-times switching in **Theorem 1** is bounded by,  
\[\Delta T<\sum_{i=0}^{n-1}\frac{\varphi_{v}(t_{i})+0.5m\Phi^{2}}{\alpha\;\Phi} \tag{46}\]  
**Remarks 4**. : (_Convergence rate in the vicinity of \(E_{d}\)) the convergence rate became slower as the robot approached the desired set, which is a common issue for the posture stabilization.