## 3 Passivity Based Robust Switching Control Design for Posture Stabilization
On the other hand, although we cannot individually ensure \(\dot{L}_{v}(t)\leq 0\) and \(\dot{L}_{w}(t)\leq 0\) within the boundary layer, we still have \(\dot{L}_{t}(t)\leq 0\). This is because if \(|v|>\Phi\), applying (12) into Equations (9) and (10), it is found that,  
\[ \begin{array}{c}\dot{L}_{v}(t)=\left[[md-\gamma]w^{2}-\rho\;w^{2}sign(\frac{ v}{\Phi})-\alpha\;sign(\frac{v}{\Phi})\right]v\leq-v\left[\alpha\;sign(\frac{v}{ \Phi})\right]\leq 0\\ \dot{L}_{w}(t)=\left[[\gamma-md]v\;w-\rho\;v\;w\;sign(\frac{v}{\Phi})-\beta\;w \right]w\leq-\beta\;w^{2}\leq 0\end{array}  \tag{13}\]  
Also, the closed-loop dynamics of WMR for \(|v|>\Phi\) is given by,  
\[ \begin{array}{c}m\dot{v}+[\gamma-md]w^{2}+\rho\;w^{2}sign(\frac{v}{\Phi})+ \alpha\;sign(\frac{v}{\Phi})+\frac{\partial\varphi_{v}(x,y)}{\partial x}\cos \theta+\frac{\partial\varphi_{v}(x,y)}{\partial y}\sin\theta=0\\ \;l\dot{w}+[md-\gamma]v\;w+\rho\;v\;w\;sign(\frac{v}{\Phi})+\beta\;w+\frac{ \partial\varphi_{w}(\theta)}{\partial\theta}=0\end{array}  \tag{14}\]  
On the other hand, if \(|v|\leq\Phi\),  
\[ \begin{array}{c}\dot{L}_{t}(t)=\dot{L}_{v}(t)+\dot{L}_{w}(t)\leq 0\;\mbox{ with }\dot{L}_{v}(t)=\left[-mdw^{2}-\alpha\frac{v}{\Phi}\right]v\;\mbox{ and }\\ \dot{L}_{w}(t)=[mdv\;w-\beta\;v]w\end{array}  \tag{15}\]  
The corresponding closed-loop dynamics of the WMR is given by,  
\[ \begin{array}{c}m\dot{v}-mdw^{2}+\alpha\frac{v}{\Phi}+\frac{\partial\varphi_ {v}(x,y)}{\partial x}\cos\theta+\frac{\partial\varphi_{v}(x,y)}{\partial y} \sin\theta=0\\ \;l\dot{w}+mdv\;w+\beta w+\frac{\partial\varphi_{w}(\theta)}{\partial\theta}=0 \end{array}  \tag{16}\]  
**Proposition 1. ** (Decrease of (\(x\),\(y\))-navigation potential energy and finite control time) Under Equation (12), if the initial velocity of WMR is \(\left[v(t_{0})\right]\leq\Phi\), the (\(x\),\(y\))-navigation potential energy of WMR is strictly decreased from \(t_{0}\) to \(t^{*}\) such that,  
\[\varphi_{v}(\;t^{*}\;)\;<\;\varphi_{v}(t_{0}) \tag{17}\]  
where \(t^{*}\) is defined as another particular time when the perturbed \(v(t)\) meets this condition \(\left[v(t^{*})\right]=\Phi\) for the first time. According to Equation (14), the WMR initiates its navigation by accelerating itself until losing its own kinetic energies thus the transient forward velocity \(|v(t)|\) is suddenly increased from \(v(t_{0})\) and eventually decreased to \(\left[v(t^{*})\right]=\Phi\). In other words, we have such a scenario: \(\left[v(t_{0})\right]\rightarrow\left[v(t)\right]\rightarrow\left[v(t^{*}) \right]=\Phi\) for \(\forall t\in(\;t_{0}\;t^{*})\) with \(\left[v(t_{0})\right]\leq\left[v(t^{*})\right]<\left[v(t)\right]\) (see Figure 2a). By investigating Equation (14), it is obvious based on the property of second order Lagrangian dynamics that the potential force and moment \(\frac{\partial\varphi_{v}(x,y)}{\partial x}\cos\theta+\frac{\partial\varphi_ {v}(x,y)}{\partial y}\sin\theta\) and \(\frac{\partial\varphi_{w}(\theta)}{\partial\theta}\) will perturb both \(v\) and \(w\) of WMR initially and eventually the perturbed \(v\) and \(w\) will be decreased by damping terms \(\alpha\;sign(v/\Phi)\) and \(\beta\;w\). In addition, the control duration to enforce WMR to move from the initial position to the particular one with \(|v|=\Phi\) is bounded by,  
\[\Delta t<\frac{\varphi_{v}(t_{0})+0.5m\Phi^{2}}{\alpha\;\Phi} \tag{18}\]  
where \(\Delta t=t^{*}-t_{0}>0\) is the control duration.