## 6 Problem Statement  
To test the proposed methods with dynamic parameter adaptation, a complex problem was selected. In this case, the optimization of a fuzzy system design used for controlling an autonomous mobile robot, the task of the fuzzy controller is to provide a way to control the two motors of the robot in order to minimize the trajectory error. The robot body illustrated in Figure 7 has two wheel motors and one passive wheel for stabilization. The motion of the passive wheel can be ignored in the dynamics represented by the following set of equations from [65]:  
\[M(q)\dot{v}\;+\;C\big{(}q,\dot{q}\big{)}v\;+\;Dv\;=\;\tau\;+\;P(t) \tag{12}\]  
\[\dot{q}\;=\;\left[ \begin{array}{ccc}\cos\;\theta&0\\ \sin\;\theta&0\\ 0&1\end{array} \right]\left[ \begin{array}{c}v\\ w\end{array} \right] \tag{13}\]  
where \(q\;=\;(x,y,\theta)^{T}\) is the vector of the configuration coordinates; \(v\;=\;(v,w)^{T}\) 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; \(M(q)\in R^{2x2}\) is the positive-definite inertia matrix; \(C\big{(}q,\dot{q}\big{)}v\) is the vector of centripetal and Coriolis forces; and \(D\in R^{2x2}\) is a diagonal positive-definite damping matrix. Equation (13) represents the kinematics of the system, where \((x,y)\) is the position in the \(X\)-\(Y\) (world) reference frame; \(\theta\) is the angle between the heading direction and the \(x\)-_axis_; and \(v\) and \(w\) are the linear and angular velocities, respectively. Figure 8 shows the complex plant for the autonomous mobile robot, where a fuzzy system is used to control the trajectory of a robot to follow a desired reference trajectory. The fuzzy system is shown in Figure 9 and is used for the complex plant to control the trajectory of the autonomous mobile robot. This is the fuzzy system that the bio-inspired optimization algorithms will optimize; in this case, only the parameters of the membership functions are optimized. The fuzzy controller of Figure 9 has two inputs, the Linear error and Angular error, and two outputs, Torque 1 and Torque 2, for each motorized wheel of the robot. The inputs are granulated into two trapezoidal membership functions in the ends and one triangular membership function in the middle, the outputs are granulated into three triangular membership functions. Also this fuzzy controller uses the fuzzy rule set from Table 4. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{2}{c}{**Inputs**} & \multicolumn{2}{c}{**Outputs**} \\ \cline{2-5}  & **Iteration** & **Diversity** & **Beta** & **Pulse Rate** \\ \hline
1 & Low & Low & Low & High \\
2 & Low & Medium & Low & High \\
3 & Low & High & Low & MediumHigh \\
4 & Medium & Low & MediumLow & MediumHigh \\
5 & Medium & Medium & MediumLow & Medium \\
6 & Medium & High & Medium & MediumLow \\
7 & High & Low & MediumHigh & MediumLow \\
8 & High & Medium & MediumHigh & MediumLow \\
9 & High & High & High & Low \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fuzzy rules for dynamic parameter adaptation in BA (Bat Algorithm). The desired reference trajectory is illustrated in Figure 10, where it starts from the (0,0) point and makes turns creating that \"U\" shape trajectory, in order to create a complex problem. The main difference between the type-1 and interval type-2 fuzzy systems is the ability of the latter to handle uncertainty, thus, to simulate this problem and to provide better tools to perform the comparison, a modification of the original plant is added, where a level of noise is added to the desired reference trajectory and the same noise added to the inputs of the fuzzy controller. The modified plant is illustrated in Figure 11, where two random number generators were added to simulate noise; the level of noise used is 5%.