## 3 Proposed Methodology
### Case 2 Fuzzy Controller  
In this case, we propose the optimization for a fuzzy controller of an autonomous mobile robot, using the multi-metaheuristic competitiveness model to find out which is the satisfactory optimization algorithm for this specific problem; below, the fuzzy controller is explained in more detail.  
This autonomous robot mobile controller is presented in [42; 43]; in [44], some applications of the method are described. The controller aims at accurately tracking a given desired trajectory, and it has nine rules that allow it to create a relationship between the linguistic variables of the fuzzy system. The linguistic variables are as follows: in the first input, it is the linear velocity (ev); in the second and last input, it is the angular velocity (ew). The two inputs have the same linguistic values: positive (P), zero (Z), and negative (N), with the same membership functions in each one. The outputs are two: torque 1 (t1) and torque 2 (t2) represent the movement of the wheels when the robot rotates, and each of these outputs has three triangular functions.  
Figure 3 shows the desired trajectory that the robot has to follow; this figure has the Y axis of the desired displacement line, and the X axis shows the displacement time.  
\begin{table}
\begin{tabular}{c} \hline
**Benchmark functions \(f_{\textbf{9}}\)-\(f_{\textbf{11}}\)** \\ \hline \(f_{\textbf{9}}(x)=\sum_{i=1}^{n}\left[x_{i}^{2}-10\cos(2\pi x_{i})+10\right]\) \\ _Search space \(x_{j}\in[-5.12,\ 5.12]\) and \(f(x^{\prime})=0\)_ \\ \(f_{\textbf{10}}(x)=-20\exp\left(-0.2\sqrt{\frac{1}{2}}\sum_{i=1}^{n}x_{i}^{2} \right)-exp\left(\frac{1}{2}\sum_{i=1}^{n}\cos(2\pi x_{i})\right)+20+e\) \\ _Search space \(x_{j}\in[-32,\ 32]\) and \(f(x^{\prime})=0\)_ \\ \(f_{\textbf{11}}(x)=\frac{1}{200}\sum_{i=1}^{n}x_{i}^{2}-\frac{1}{2}\cos\left( \frac{x_{i}}{\sqrt{3}}\right)+1\) \\ _Search space \(x_{j}\in[-600,\ 600]\) and \(f(x^{\prime})=0\)_ \\ \hline \end{tabular}
\end{table}
Table 3: Multimodal benchmark functions.  
Figure 3: Desired trajectory in normalized axis.  
Figure 4 shows the red line as the desired trajectory, and the actual trajectory of the robot in blue, and as can be seen, the robot is very much lost, since it generates a large error of \(\mathbf{3.8541\times 10^{+03}}\).  
Figure 5 illustrates the membership functions of input 1, which is called the linear speed, where its linguistic variables represent negative, zero, and positive values.  
Figure 6, we can find the membership functions; the blue color corresponds to a trapezoidal, the orange represents the zero linguistic variable and is a triangular membership function, and the yellow line is a trapezoidal membership function of input 2 of the fuzzy controller.  
Figure 7 shows the three triangular membership functions of output 1, which represents the movement of wheel 1 of the fuzzy control, as illustrated by the functions that do not look well defined, because the controller is not optimized in this figure.  
Figure 4: Obtained mean square error (MSE).  
Figure 5: Input linear velocity (ev), not optimized.  
Figure 8 represents that output 2 of the fuzzy controller (T2) represents the movement of wheel 2, and has negative, zero, and positive as linguistic variables. In this case, the three membership functions are triangular.  
Figure 6: Input angular velocity (ew), not optimized.  
Figure 7: Output T1, not optimized.  
Table 5 shows the representation of the if-then rules of the fuzzy controller that describe the relationship between the linguistic variables, where \(\mathrm{e}_{\mathrm{v}}\) represents the linear velocity and \(\mathrm{e}_{\mathrm{w}}\) is the angular velocity.