## 3 Passivity Based Robust Switching Control Design for Posture Stabilization
We know that \(\varphi_{v}(t_{3})<\varphi_{v}(t_{2})\), where \(t_{3}\) is a next switching time from the first control law to the second control law when \(\big{|}v(t_{3})\big{|}=\Phi\) again. This means \(\varphi_{v}(t_{3})<\varphi_{v}(t_{2})<\varphi_{v}(t_{1})<\varphi_{v}(t_{0})\). Similarly, this inequality is valid for \([t_{i}\,\ t_{i+1}]\cup[t_{i+1}\,\ t_{i+2}]\cup[t_{i+2}\,\ t_{i+3}]\). By induction, \(\varphi_{v}(t_{i+1})<\varphi_{v}(t_{i})\) is true for \(\forall i\in N\cup\{0\}\) as long as \(\big{|}v(t_{i+1})\big{|}=\big{|}v(t_{i})\big{|}\) except \(v(t_{0})\). Consequently, by repeating the switching of control law back and forth, the WMR will be able to reach the final destination (i.e., \(\underset{t\to\infty}{\lim}(x(t),y(t))\to E_{d}\)) because of \(\varphi_{v}(t_{i})\geq 0\) and \(\varphi_{v}(t_{i+1})<\varphi_{v}(t_{i})\). After \((x(t),y(t))\to E_{d}\), if the final correction of the robot orientation is still required, Equation (28) will be turned on to complete its last task (i.e., \(E_{d}\) and \(\theta\to 0\)). Proof completed. \(\Box\)  
**Proof of Claim 1. ** This is because applying Equation (27) into Equations (9) and (10) yields,  
\[\dot{L}_{v}(t)=\Big{[}[md-\gamma]\ w^{2}-\rho\ w^{2}sign(\frac{v}{\Phi})- \alpha\ sign(\frac{v}{\Phi})\Big{]}v \tag{34}\]  
\[\dot{L}_{w}(t)=\Big{[}[\gamma-md]\ v\ w-\rho\ v\ w\ sign(\frac{v}{\Phi})-\beta \ w+K_{\theta}\theta_{o}\Big{]}w \tag{35}\]  
Based on Equations (34) and (35), the derivative of total energy \(L_{t}(t)\) with respect to a time is given by,  
\[\dot{L}_{t}(t)=-2\rho\ v\ w^{2}\ sign(\frac{v}{\Phi})-\beta\ w^{2}-\alpha\ v \ sign(\frac{v}{\Phi})+K_{\theta}\theta_{o}w \tag{36}\]Integrating Equation (36) from \(t_{1}\) to \(t>t_{1}\),  
\[L_{t}(t)-L_{t}(t_{1})=-\int_{t_{1}}^{t}\left(\rho\left|v\right|w^{2}\right.+\beta \left.w^{2}+\alpha\left|v\right|\right.\right)d\tau+K_{\theta}\theta_{\sigma} \int_{t_{1}}^{t}wd\tau \tag{37}\]  
Specifically, Equation (37) becomes,  
\[ \begin{array}{l}\kappa_{v}(t)-\kappa_{v}(t_{1})=\cdots\\ \cdots=\varphi_{v}(t_{1})-\varphi_{v}(t)+\varphi_{w}(t_{1})-\varphi_{w}(t)+ \kappa_{w}(t_{1})-\kappa_{w}(t)-\int_{t_{1}}^{t}\left(\rho\left|v\right|w^{2} \right.+\beta\left.w^{2}+\alpha\left|v\right|\right.\right)d\tau+K_{\theta} \theta_{\sigma}\int_{t_{1}}^{t}wd\tau\end{array}  \tag{38}\]  
Since both \(v(t)\) and \(w(t)\) are bounded such that \(\left|v(t)\right|\leq v_{\max}\) and \(\left|w(t)\right|\leq w_{\max}\) along with the fact that \(\varphi_{v}(t)\leq\varphi_{v\max}\) is also bounded (Since the control, Equations (26) and (27), guarantees that the responses of system is bounded), the right side of Equation (38) has following inequalities,  
\[ \begin{array}{l}\varphi_{v}(t_{1})-\varphi_{v}(t)+\varphi_{w}(t_{1})- \varphi_{w}(t)+\kappa_{w}(t_{1})-\kappa_{w}(t)-\int_{t_{1}}^{t}\left(\rho \left|v\right|w^{2}\right.+\beta\left.w^{2}+\alpha\left|v\right|\right.)d \tau+K_{\theta}\theta_{\sigma}\int_{t_{1}}^{t}wd\tau\\ >H-\varphi_{v\max}-\varphi_{w}(t)-\kappa_{w}(t)-\int_{t_{1}}^{t}\left(\rho \left|\rho_{\max}\right|w_{\max}\right.^{2}+\left.w_{\max}^{2}+\left.w_{\max }\right|\right)d\tau+K_{\theta}\theta_{\sigma}(\theta(t)-\theta(t_{1}))\\ >H-\varphi_{v\max}-\varphi_{w}(t)-\kappa_{w}(t)-\Psi(t_{2}-t_{1})+K_{\theta} \theta_{\sigma}(\theta(t)-\theta(t_{1}))\\ >H-\varphi_{v\max}-1/2\ K_{\theta}[\theta(t)]^{2}-1/2\ I_{r}\ {w_{\max}}^{2}-\Psi(t_{2}-t_{1})+K_{\theta} \theta_{\sigma}(\theta(t)-\theta(t_{1}))\\ =-1/2\ K_{\theta}(\theta(t)-\theta_{\sigma})^{2}+H-C+1/2\ K_{\theta}\theta_{ \sigma}{}^{2}\end{array}  \tag{39}\]  
where, \(\Psi=\rho\left|v_{\max}\right|\ w_{\max}{}^{2}+\beta\ w_{\max}{}^{2}+\alpha \left|v_{\max}\right|\) and \(C=[\ 1/2\ I_{r}\ w_{\max}{}^{2}+\Psi(t_{2}-t_{1})+\varphi_{v\max}+K_{\theta} \theta_{\sigma}\theta(t_{1})\ ]\) as well as \(H=\kappa_{w}(t_{1})+\varphi_{w}(t_{1})\).