## 3 Passivity Based Robust Switching Control Design for Posture Stabilization
\begin{array}{c}-K_{v}v\\ -K_{1}w-K_{\theta}\theta\end{array} \right]\ hold\ until\ \theta\to 0 \tag{28}\]  
However, if the WMR is still outside of \(E_{d}\), it will switch from the first control law to the second one until, again, \(|v|\rightarrow\Phi\). The second law induces the rotational motion (i.e., \(\theta\rightarrow\theta_{\alpha}\)) so that the forward velocity of WMR suddenly increases and, due to damping term \(-\alpha\ sign(v/\Phi)\), decreases eventually. This is because the \(v\)-dynamics is coupled with the \(w\)-dynamics. Due to this perturbation invoking the forward velocity \(v(t)\), the (\(x\),\(y\))-navigation potential energy of WMR is also diminished. Therefore, either control in (24) strictly reduces the (\(x\),\(y\))-navigation potential energy so that the WMR finally reaches \(E_{d}\) by conducting the switching back and forth between Equations (26) and (27). **Proof :** Based on **Proposition 1**, if \(\left|v(t_{0})\right|\leq\Phi\), the first control in Equation (26) guarantees the strict decrease of (\(x\),\(y\))-navigation potential energy until the time \(t_{1}\) when \(\left|v(t_{1})\right|=\Phi\). In other words, \(\varphi_{\triangleright}(t)\ <\ \varphi_{\triangleright}(t_{0})\) for \(\forall t\in[t_{0}\ t_{1}]\). If the WMR enters into \(E_{d}\) via Equation (26), switch from Equation (26) to Equation (28) for correcting the orientation of the WMR (i.e., \(\theta\to 0\)). Based on the following Lyapunov function \(L_{t}(t)=1/2 mv^{2}+1/2I_{t}w^{2}+1/2K_{\theta}\theta^{2}\), applying Equation (28) into Equation (2) yields \(L_{t}(t)=-(K_{v}v^{2}+K_{1}\ w^{2})\). This implies that \(\theta\to 0\). Therefore, the final correction of WMR orientation can be achieved via Equation (28). However, if the WMR is still outside of \(E_{d}\), the second control is initiated to gain a rotational energy for further navigation (i.e., \(\theta\rightarrow\theta_{a}\
eq 0\)). Here, the second control still diminishes the navigation potential energy. This is because Equation (27) yields,  
\[\dot{L}_{v}(t)=\Big{[}[md-\gamma]w^{2}-\rho w^{2}sign(\frac{v}{\Phi})-\alpha\ sign (\frac{v}{\Phi})\Big{]}v<-\alpha\Big{[}v(t)\Big{]}\leq-\alpha\ \Phi \tag{29}\]  
In other words,  
\[\dot{L}_{v}(t)<-\alpha\ \Phi\ \text{for}\ \forall t\in[t_{1}\ \ \ \ t_{2}] \tag{30}\]  
where \(t_{2}\) is a next switching time when the perturbed forward velocity due to the rotational energy of the WMR again becomes \(\Phi\) (i.e., \(\big{|}v(t_{2})\big{|}=\Phi\)). Integrating Equation (30) from \(t_{2}\) to \(t_{1}\) results in,  
\[L_{v}(t_{2})-L_{v}(t_{1})<-\alpha\ \Phi(t_{2}-t_{1}) \tag{31}\]  
Furthermore, applying \(\big{|}v(t_{1})\big{|}=\Phi\) and \(\big{|}v(t_{2})\big{|}=\Phi\) (i.e., \(k_{v}(t_{1})=k_{v}(t_{2})\)) to Equation (31) produces,  
\[\varphi_{v}(t_{2})-\varphi_{v}(t_{1})<-\alpha\ \Phi(t_{2}-t_{1})+k_{v}(t_{1})-k_{v}(t_{2 })=-\alpha\ \Phi(t_{2}-t_{1})<0\Rightarrow\varphi_{v}(t_{2})<\varphi_{v}(t_{1}) \tag{32}\]  
Thus, as long as the switching is intentionally enforced at times \(t_{1}\) and \(t_{2}\) such as \(\big{|}v(t_{1})\big{|}=\Phi\) and \(\big{|}v(t_{2})\big{|}=\Phi\), the decrease of (\(x\),\(y\))-navigation can be guaranteed (i.e., \(\varphi_{v}(t_{2})<\varphi_{v}(t_{1})<\varphi_{v}(t_{0})\)). Moreover, it is found that \(\theta\approx\theta_{a}\
eq 0\) is obtained via Equation (27) and the period \(t_{2}-t_{1}\) is bounded based on **Proposition 1**. **Claim 1. ** The perturbed \(v(t)\) due to action \(\theta\to\theta_{o}\
eq 0\) has the following scenario \(\big{|}v(t_{1})\big{|}\to\big{|}v(t)\big{|}\to\big{|}v(t_{2})\big{|}=\Phi\) with \(\big{|}v(t_{1})\big{|}=\big{|}v(t_{2})\big{|}<\big{|}v(t)\big{|}\) for \(\forall t\in(t_{1}\ t_{2})\) (see Figure 2a). If this is true and then, the following condition is also valid,  
\[\varphi_{v}(t)<\varphi_{v}(t_{1})\ \text{for}\ \forall t\in(t_{1}\ \ \ t_{2}) \tag{33}\]  
For proof of this, see Proof of Claim 1. Next, if the WMR is still outside of \(E_{d}\), switch back to Equation (26) from Equation (27).