## 3 Passivity Based Robust Switching Control Design for Posture Stabilization
Furthermore, taking derivative of Equation (3) with respect to a time yield,  
\[\dot{L}_{t}(t)=\dot{\kappa}_{v}(t)+\frac{\partial\varphi_{v}(x,y)}{\partial x} \frac{dx}{dt}+\frac{\partial\varphi_{v}(x,y)}{\partial y}\frac{dy}{dt}+\dot{ \kappa}_{w}(t)+\frac{\partial\varphi_{w}(\theta)}{\partial\theta}\frac{d\theta }{dt} \tag{5}\]  
By the admissible (i.e., not violating the non-holonomic constraints [5; 19]) velocity of WMRs, we have \(\dot{x}=v\cos\theta\) and \(\dot{y}=v\sin\theta.\) Thus,  
\[\dot{L}_{t}(t)=\dot{\kappa}_{v}(t)+\frac{\partial\varphi_{v}(x,y)}{\partial x} v\cos\theta+\frac{\partial\varphi_{v}(x,y)}{\partial y}v\sin\theta+\dot{ \kappa}_{w}(t)+\frac{\partial\varphi_{w}(\theta)}{\partial\theta}w \tag{6}\]  
Here, the total energy can be decomposed into \(\dot{L}_{v}(t)\) and \(\dot{L}_{w}(t)\),  
\[\dot{L}_{v}(t)=v\bigg{[}m\dot{v}+\frac{\partial\varphi_{v}(x,y)}{\partial x} \cos\theta+\frac{\partial\varphi_{v}(x,y)}{\partial y}\sin\theta\bigg{]} \tag{7}\]  
\[\dot{L}_{\dot{0}}(t)=w\bigg{[}l_{r}\,\dot{w}+\frac{\partial\varphi_{w}(\theta )}{\partial\theta}\bigg{]} \tag{8}\]  
Substituting Equation (2) into Equations (7) and (8) yields  
\[\dot{L}_{v}(t)=v\bigg{[}mdw^{2}+u_{v}+\frac{\partial\varphi_{v}(x,y)}{ \partial x}\cos\theta+\frac{\partial\varphi_{v}(x,y)}{\partial y}\sin\theta \bigg{]} \tag{9}\]  
\[\dot{L}_{w}(t)=w\bigg{[}-mdw\,v+u_{w}+\frac{\partial\varphi_{w}(\theta)}{ \partial\theta}\bigg{]} \tag{10}\]  
To ensure the negative semi-definitive of Equations (9) and (10) individually, we can suggest the following control law,  
\[\left[ \begin{array}{c}u_{v}\\ u_{w}\end{array} \right]=\left[ \begin{array}{c}-md\;w^{2}\\ md\;w\;v\end{array} \right]-\left[ \begin{array}{c}\frac{\partial\varphi_{v} (x,y)}{\partial x}\cos\theta+\frac{\partial\varphi_{v}(x,y)}{\partial y}\sin \theta+\alpha\;sat(\frac{v}{\Phi})\\ \frac{\partial\varphi_{v}(\theta)}{\partial\theta}+\beta\;w\end{array} \right] \tag{11}\]  
where \(sat\;(v/\Phi)=sign\;\;(v/\Phi)\) if \(|v|>\Phi\) and \(sat\;(v/\Phi)=v/\Phi\) if \(|v|\leq\Phi\) is the positive boundary layer thickness to eliminate the chattering on the sliding phase [26] and \(\alpha\) and \(\beta\in\Re\) are the positive constant gains. The control law, Equation (11), ensures both \(\dot{L}_{v}(t)=-\alpha\;v\;sat(v/\Phi)\leq 0\) and \(\dot{L}_{w}(t)=-\beta\;w^{2}\leq 0\) and the derivative of the overall Lyapunov function is also \(\dot{L}_{t}(t)=-\alpha\;v\;sat(v/\Phi)-\beta\;w^{2}\leq 0\). **Remark 2**: (_model uncertainty) The Coriolis terms of the control law Equation (11) contains md which is usually difficult to measure its exact value. In this regard, the model uncertainty should be considered to achieve better navigation performance of WMR in reality. In this study, we denote the estimation of \(\,md\) as \(\gamma\), and assume to know a bounded constant \(\exists\rho\in\Re\;s.t\;|md-\gamma|\leq\rho.\)_  
Based on Equation (11), the control law including the compensation of the model uncertainty is then designed to be  
\[\left[ \begin{array}{c}u_{v}\\ u_{w}\end{array} \right]=\left[ \begin{array}{c}F_{v}\\ F_{w}\end{array} \right]+\left[ \begin{array}{cc}-\frac{\partial\varphi_{v}(x, y)}{\partial x}\cos\theta-\frac{\partial\varphi_{v}(x,y)}{\partial y}\sin \theta-\alpha\;sat(\frac{v}{\Phi})\\ -\frac{\partial\varphi_{w}(\theta)}{\partial\theta}-\beta\;w\end{array} \right] \tag{12}\] \[F_{v}=\left\{ \begin{array}{cc}-\gamma\;w^{2}-\rho w^{2}\;sign( \frac{v}{\Phi})&if\;|v|>\Phi\\ 0&if\;|v|\leq\Phi\end{array} \right.\text{ and }\] \[F_{w}=\left\{ \begin{array}{cc}\gamma\;w\;v-\rho w\;v\;sign( \frac{v}{\Phi})&if\;|v|>\Phi\\ 0&if\;|v|\leq\Phi\end{array} \right.\]Here, the robot system is governed by two separate control laws according to \(|v|>\Phi\) and \(|v|\leq\Phi\). This control law Equation (12) ensures that \(\dot{L}_{v}(t)\leq 0\) and \(\dot{L}_{w}(t)\leq 0\) individually while compensating the model uncertainty of system out of boundary layer, \(\Phi\).