## 4 Study Cases
### Robot Mobile Controller  
The second control problem is a unicycle mobile robot, and a Sugeno type controller was used, which has a higher level of complexity. The main goal of this controller is to make the robot follow a reference trajectory. The robot is composed of two drive wheels located on the same axis and a front free wheel that is used only for stability. The graphical representation of the robot mobile is illustrated in Figure 6, in which we can observe the two wheels that make up the robot, the two torques, and the angle with respect to the \(Y\) axis. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Input 1** & **Input 2** & **Output 1** & **Output 2** \\ \hline Low MF: & Low MF: & Low MF: & Low MF: \\ \(a_{0}=0\) & \(a_{0}=0\) & \(0\leq a_{0}\leq 0.17\) & \(0\leq a_{0}\leq 0.17\) \\ \(0\leq b_{0}\leq 0.5\) & \(0\leq b_{0}\leq 0.5\) & \(b_{0}=0.17\) & \(b_{0}=0.17\) \\ \(c_{0}=0.5\) & \(c_{0}=0.5\) & \(0.17\leq c_{0}\leq 0.35\) & \(0.17\leq c_{0}\leq 0.35\) \\ \hline Medium MF: & Medium MF: & Medium Low MF: & Medium Low MF: \\ \(0\leq a_{1}\leq 0.5\) & \(0\leq a_{1}\leq 0.5\) & \(0.17\leq a_{1}\leq 0.35\) & \(0.17\leq a_{1}\leq 0.35\) \\ \(b_{1}=0.5\) & \(b_{1}=0.5\) & \(b_{1}=0.35\) & \(b_{1}=0.35\) \\ \(0.5\leq c_{1}\leq 1\) & \(0.5\leq c_{1}\leq 1\) & \(0.35\leq c_{1}\leq 0.5\) & \(0.35\leq c_{1}\leq 0.5\) \\ \hline \multirow{3}{*}{MFs Parameters} & High MF: & Medium MF: & Medium MF: \\ \(a_{2}=0.5\) & \(a_{2}=0.5\) & \(0.35\leq a_{2}\leq 0.5\) & \(0.35\leq a_{2}\leq 0.5\) \\ \(0.5\leq b_{2}\leq 1\) & \(0.5\leq b_{2}\leq 1\) & \(b_{2}=0.5\) & \(b_{2}=0.5\) \\ \(c_{2}=1\) & \(c_{2}=1\) & \(0.5\leq c_{2}\leq 0.7\) & \(0.5\leq c_{2}\leq 0.7\) \\ \hline \multirow{3}{*}{-} & \multirow{3}{*}{-} & Medium High MF: & Medium High MF: \\ \cline{3-4}  & & \(0.5\leq a_{3}\leq 0.7\) & \(0.5\leq a_{3}\leq 0.7\) \\ \cline{3-4}  & & \(b_{3}=0.7\) & \(b_{3}=0.7\) \\ \cline{3-4}  & & \(0.7\leq c_{3}\leq 0.85\) & \(0.7\leq c_{3}\leq 0.85\) \\ \hline \multirow{3}{*}{-} & \multirow{3}{*}{-} & High MF: & High MF: \\ \cline{3-4}  & & \(0.7\leq a_{4}\leq 0.85\) & \(0.7\leq a_{4}\leq 0.85\) \\ \cline{3-4}  & & \(b_{4}=0.85\) & \(b_{4}=0.85\) \\ \cline{3-4}  & & \(0.85\leq c_{4}\leq 1\) & \(0.85\leq c_{4}\leq 1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Limits of the parameters of the membership functions. Equations (15) and (16) represent the operation of the mathematical model of the mobile robot, and for each equation a brief explanation is given of each of the variables that constitute the equations is presented. \[M(q)\dot{v}+C\big{(}q,\dot{q}\big{)}v+Dv=\ \tau+\mathrm{P}(\mathrm{t}) \tag{15}\]  
where  
\(\mathrm{q}=(\mathrm{x},\mathrm{y},\ \theta)^{\mathrm{T}}\) is the vector of the configuration coordinates;  
\(\mathrm{v}=(\mathrm{v},\mathrm{v})^{\mathrm{T}}\) is the vector of velocities;  
\(\tau=(\tau_{1},\ \tau_{2})\) is the vector of torques applied to the wheels of the robot where \(\tau_{1}\) and \(\tau_{2}\) denote the torques of the right and left wheel, respectively;  
\(P\in\ R^{2}\) is the uniformly bounded disturbance vector;  
\(M(q)\in\ R^{2\times 2}\) is the positive-definite inertia matrix;  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Search Domain** & **F Min** & **Equation** \\ \hline Rosenbrock & \([-5,10]\) n & 0 & \(f(x)=\sum\limits_{i=1}^{n-1}\!\!\left[100\!\left(x_{i-1}-x_{i}^{2}\right)^{2}+ \left(1-x_{1}\right)^{2}\right]\) \\ Sphere & \([-5.12,5.12]\) n & 0 & \(f(x)=\sum\limits_{i=1}^{n}x_{i}^{2}\) \\ Rastrigin & \([-5.12,5.12]\) n & 0 & \(f(x)=10n+\sum\limits_{i=1}^{n}\!\!\left[x_{i}^{2}-10\cos(2x_{i})\right]\) \\ Schwefel & \([-500,500]\) n & 0 & \(f(x)=\sum\limits_{i=1}^{n}\!\left[-x_{i}sin\left(\sqrt{|x_{i}|}\right)\right]\) \\ Sum squares & \([-10,10]\) n & 0 & \(f(x)=\sum\limits_{i=1}^{n}ix_{i}^{2}\) \\ [PERSON] & \([-5,10]\) n & 0 & \(f(x)=\sum\limits_{i=1}^{n}x_{i}^{2}+\frac{\pi}{i+1}\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!