## 3 Passivity Based Robust Switching Control Design for Posture Stabilization
)_  
(2) _If \(\frac{\partial\varphi_{\triangledown}(x,y)}{\partial x}=0\), \(\sin\theta=0\) (i.e., \(\theta=\pm\pi n\) for \(\forall n\in\)), and \(y\
eq 0\)._  
(3) _If \(\frac{\partial\varphi_{\triangledown}(x,y)}{\partial y}=0\), \(\cos\theta=0\) (i.e., \(\theta=\pm\frac{n\pi}{2}\) for \(\forall n\in N\)), and \(x\
eq 0\)._  
(4) _If \(\tan\theta=-\frac{\varphi_{\triangledown}(x,y)}{\varphi_{\triangledown}(x,y)}\), where, \(\varphi_{\triangledown}(x,y)=\frac{\partial\varphi_{\triangledown}(x,y)}{ \partial x}\
eq 0\) and \(\varphi_{\triangledown,y}(x,y)=\frac{\partial\varphi_{\triangledown}(x,y)}{ \partial y}\
eq 0\)._  
The first condition of **Remark 3** is the desired position of robot. Thus, the given task is completed. Condition 3 implies that there exists a \(w\)-motion (\(\theta\
eq 0\)) so that the \(w\)-dynamics (rotational motion) induces the \(v\)-dynamics (\(v\
eq 0\)). Condition 4 induces both \(v\) and \(w\) dynamics (\(x\
eq 0\), \(y\
eq 0\), \(\theta\
eq 0\)). Thus, both Equations (3) and (4) are not invariant which leads that the WMR is under the motion. For Condition 2, if both \(v\) and \(w\) are \(0\) with \(\theta=0\) for \(n=0\), then closed-loop dynamics Equation (16) become \(m_{r}\dot{v}=0\) and \(I_{r}\dot{w}=0\). This implies that the robot will be stuck in the middle of \(y\)-axis (i.e., unwanted local minima). According to **Remark 3**, we can define a set (unwanted) local minima where the robot should avoid or pass through s.t  
\[E_{avoid}:=\left\{\mathbf{X}\in\Re\right|v=w=\theta=0,x=0,y\
eq 0\right\} \tag{25}\]  
By inspecting Equations (16) and (25), the WMR will begin to lose its own kinetic energies as it approaches \(E_{avoid}\) and finally be trapped at \(E_{avoid}\). Therefore, with an intuitive sense, we should not wait until robot is stuck at \(E_{avoid}\) (i.e., unwanted local minima) and have to utilize another control at the non-zero forward velocity such as the condition \(|v|=\Phi\
eq 0\) for better navigation performance. This way possibly preserves current \(v\)-kinetic energies (at least \(0.5\ m\ \Phi^{2}\)) for next switched control law. In this regard, the following switching law is presented to allow the robot to reach \(E_{d}\). **Theorem 1. ** (Robust switching control law) The following control law enables us to accomplish that the robot is able to pass through the forbidden region such as a set \(E_{avoid}\) and then reach the final desired position \(s.t\ E_{d}\). Based on Equation (12), the modified control laws are designed to be,  
\[\left[ \begin{array}{c}u_{v.1}\\ u_{w.1}\end{array} \right]:=\left[ \begin{array}{c}F_{p}-\frac{\partial v_{p} (x,y)}{\partial x}\cos\theta-\frac{\partial v_{p}(x,y)}{\partial y}\sin \theta-\alpha\ sign(\frac{v}{\Phi})\\ F_{w}-K_{1}w-K_{\theta}\theta\end{array} \right]\ hold\ until\ |v|\rightarrow\Phi \tag{26}\]  
\[\left[ \begin{array}{c}u_{v.2}\\ u_{w.2}\end{array} \right]:=\left[ \begin{array}{c}F_{p}-\frac{\partial v_{p} (x,y)}{\partial x}\cos\theta-\frac{\partial v_{p}(x,y)}{\partial y}\sin \theta-\alpha\ sign(\frac{v}{\Phi})\\ F_{w}-K_{1}w-K_{\theta}(\theta-\theta_{\alpha})\end{array} \right]\ hold\ until\ |v|\rightarrow\Phi \tag{27}\]  
where \(K_{\circ},K_{1}\in\Re\) are the positive constant gains. Let \(t_{1},\ t_{2},\cdots,t_{n}\in[t_{0}\ t_{f}]\) be each switching time between the first control [\(u_{v.1}\ u_{w.1}\)] and the second one [\(u_{v.2}\ u_{w.2}\)] or vice versa at the moment \(|v|=\Phi\
eq 0\). If we assume that the initial states of the WMR do not belong to \(E_{avoid}\), the first control law is initially activated to transport the robot until \(|v|\rightarrow\Phi\), where the (\(x\),\(y\))-navigation potential energy is monotonically decreased according to **Proposition 1**. Simultaneously, \(\theta\approx 0\) is possibly achieved via \(-K_{1}\ w-K_{\theta}\theta\) in Equation (26). If the WMR enters into \(E_{d}\) via a pure action of [\(u_{v.1}\ u_{w.1}\)] and then following control will be performed to ensure \(\theta\to 0\). \[\left[ \begin{array}{c}u_{v}\\ u_{w}\end{array} \right]=\left[