## 6 Numerical Example: Trajectory Tracking of a Wheeled Mobile Robot
### Proposed Controller Design  
In this subsection, we construct the proposed controller (64) for the error systems (84) subject to the convex constraint (85).  
Figure 1: Model of the wheeled mobile robot.  
We employ the following function \(V(t,e)\) as a TCLF for (84) [32]:  
\[V(t,e) =\frac{3}{2}(e_{1}^{2}+e_{2}^{2})+\frac{1}{2}h^{2}(t,e), \tag{87}\] \[h(t,e) =(2\dot{x}_{1d}-e_{1})\sin(e_{3}+x_{3d}(t))-(2\dot{x}_{2d}-e_{2}) \cos(e_{3}+x_{3d}(t)). \tag{88}\]  
By using \(V(t,e)\) and the mappings \(\tilde{f}(t,e)\) and \(\tilde{g}(t,e)\) of (84), we can calculate \(\omega_{1}(t,e)\) and \(L_{\tilde{g}}V(t,e)\) as follows:  
\[\omega_{1}(t,e) =3u_{1r}(t)[e_{1}I_{1}(t,e)+e_{2}I_{2}(t,e)]+h(t,e)[u_{1r}(t)\sin e _{3}+J(t,e)], \tag{89}\] \[L_{\tilde{g}}V(t,e) =\big{[}L_{\tilde{g}_{1}}V\quad L_{\tilde{g}_{2}}V\big{]}\] (90) \[=\bigg{[}3e_{1}\cos(e_{3}+x_{3d}(t))+3e_{2}\sin(e_{3}+x_{3d}(t)) \quad h(t,e)\cdot\frac{\partial h}{\partial e_{3}}(t,e)\bigg{]}\]  
where  
\[I_{1}(t,e) =\cos(e_{3}+x_{3d}(t))-\cos x_{3d}(t),\ I_{2}(t,e)=\sin(e_{3}+x_{ 3d}(t))-\sin x_{3d}(t), \tag{91}\] \[J(t,e) =\dot{h}(t,e)=\frac{\partial h}{\partial x_{3d}}(t,e)\dot{x}_{3 d}(t)+[2\dot{x}_{1d}(t)+2\dot{x}_{2d}(t)\dot{x}_{3d}(t)]\sin(e_{3}+x_{3d}(t))\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad+[-2\dot{x}_{2d}(t)+2 \dot{x}_{1d}(t)\dot{x}_{3d}(t)]\cos(e_{3}+x_{3d}(t)),\] (92) \[\frac{\partial h}{\partial e_{3}}(t,e) =\frac{\partial h}{\partial x_{3d}}(t,e)=(2\dot{x}_{1d}-e_{1})\cos (e_{3}+x_{3d}(t))+(2\dot{x}_{2d}-e_{2})\sin(e_{3}+x_{3d}(t)). \tag{93}\]  
Then, we design the minimizing input \(\tilde{k}(t,e)\). Since \(L_{\tilde{g}}V(t,e)=0\) implies \(\tilde{k}(t,e)=0\) (see (54)), we fix \((t,e)\) such that \(L_{\tilde{g}}V(t,e)\
eq 0\). According to Lemma 3 and the function \(\tilde{G}(t,e)\) given by (86), the necessary and sufficient condition of the optimal solution \(\bar{u}_{t,e}\) is  
\[L_{\tilde{g}}V(t,e)+\lambda_{t,e}\frac{\partial\tilde{G}}{\partial\bar{u}}(t,\bar{u}_{t,e})=L_{\tilde{g}}V(t,e)+2\lambda_{t,e}[\bar{u}_{t,e}-u_{r}(t)]^{ \top}=0 \tag{94}\]  
By solving (94) with respect to \(\bar{u}_{t,e}\), we have  
\[\bar{u}_{t,e}=-\bigg{(}\frac{1}{2\lambda_{t,e}}L_{\tilde{g}}V^{\top}(t,e)+u_{ r}(t)\bigg{)}. \tag{95}\]  
Substituting (95) into the condition \(\tilde{G}(t,\bar{u}_{t,e})=0\), we can obtain  
\[\lambda_{t,e}=\frac{\|L_{\tilde{g}}V(t,e)\|}{2\beta}. \tag{96}\]  
Hence, the minimizing input (54) is calculated as follows:  
\[\tilde{k}(t,e)=\begin{cases}-\bigg{(}\frac{\beta}{\|L_{\tilde{g}}V(t,e)\|}L_ {\tilde{g}}V^{\top}(t,e)+u_{r}(t)\bigg{)}&(L_{\tilde{g}}V(t,e)\
eq 0)\\ 0&(L_{\tilde{g}}V(t,e)=0)\end{cases}. \tag{97}\]  
Then, we can construct the proposed tracking controller \(\bar{u}=\tilde{k}(t,e)\) by using (62)-(64), (89)-(93), and (97). Finally, the trajectory-tracking controller \(u=k(t,x)\) for the original systems (81) by applying the input transformation (65).  
To characterize the asymptotically stabilizable domain \(W\) of Definition 4, for example, the following function is available as \(\overline{V}(e)\):  
\[\overline{V}(e)=2(e_{1}^{2}+e_{2}^{2})+c_{1}\sqrt{e_{1}^{2}+e_{2}^{2}}+c_{2} \sin^{2}e_{3}, \tag{98}\]_where \(c_{1}=2\sup_{t\in[0,\infty)}u_{1r}(t)\) and \(c_{2}=c_{1}^{2}/2\). Note, however, that we do not calculate \(W\) itself in this article. A specific computation method for \(W\) based on \(\overline{V}(e)\) and \(\bar{k}(t,e)\) is a future issue._