# Biologically-Inspired Learning and Adaptation of Self-Evolving Control for Networked Mobile Robots
## 4 Simulations, Comparative Analysis, and Discussion  
This section aims to conduct several simulations to examine the effectiveness of the proposed methods. The proposed networked mobile robot system consists of three four-wheeled Swedish omnidirectional mobile robots, including one leader and two follower mobile robots in a broadcast communication environment. The desired formation shape is triangular as shown in Figure 8. The system parameters in the proposed networked mobile robot system are \(L=0.25\) m and \(r=5.08\) cm. The first simulation is conducted to demonstrate the performance of the circular formation control using the proposed GA-KF-RBFNN control approach. The number of iterations for the modified GA is 150, and the probabilities of crossover and mutation are determined by the Levy flight. The circular trajectory for the leader robot is expressed as \(\left[ \begin{array}{cc}x_{r}(t)&y_{r}(t)&\theta_{r}(t)\end{array} \right]^{T}= \left[ \begin{array}{cc}1.75\cos(\omega_{i}t)\text{ m}&1.75\sin(\omega_{i}t) \text{ m}&\pi/4\text{ rad }\end{array} \right]^{T}\), \(\omega_{i}=0.35\) rad/ sec. Figure 9 depicts the simulation result of the circular formation control. The three omnidirectional mobile robots are initially placed at different poses in the workspace. The desired trajectory for leader robot is a circular trajectory and the two follower robots aim to maintain a triangular formation. The tracking error of the leader mobile robot is presented in Figure 10. As shown in Figure 10, the leader mobile robot successfully tracks the desired circular trajectory in 4 s. Figure 11 depicts the formation error of follower robot #1 and Figure 12 depicts the formation error of follower robot #2. These simulation results demonstrate that the proposed GA-KF-RBFNN optimization is capable of accomplishing the self-learning formation control of networked mobile robotic systems. Figure 8: Leader-follower formation control with three mobile robots. Figure 11: Formation error of follower robot #1 for circular formation control. Figure 9: Simulation result of circular formation control. Figure 10: Tracking error of the leader mobile robot for circular formation control. The second simulation is provided to show the effectiveness of the proposed GA-KF-RBFNN formation control for daisy curve tracking. The desired trajectory of the leader robot is a special daisy curve with three petals, expressed by \(\left[ \begin{array}{cc}x_{r}(t)&y_{r}(t)&\theta_{r}(t)\end{array} \right]^{T}= \left[ \begin{array}{cc}\frac{1}{2}+b(a+r^{\prime}\cos(p\omega_{i}t))\cos( \omega_{i}t)\text{ m }&\frac{1}{2}+b(a+r^{\prime}\cos(p\omega_{i}t))\sin(\omega_{i}t)\text{ m }&\pi/4\text{ rad }\end{array} \right]^{T}\), \(b=0.2\) m, \(a=0.6\) m, \(r^{\prime}=0.42\) m, \(p=3\), and \(\omega_{i}=0.35\) rad/ sec. To illustrate the noise reduction capability of the proposed GA-KF-RBFNN self-learning controller, a Gaussian noise is added into the process and measurement phases. Figure 13 presents the simulation result of daisy curve formation control using the proposed GA-KF-RBFNN and Figure 14 depicts the tracking error of the leader robot. Moreover, Figures 15 and 16 present the formation error of follower robot #1 and follower robot #2, respectively, for daisy curve formation control. Both the tracking performance and formation behavior are guaranteed. These simulation results clearly indicate that the proposed metaheuristic GA-KF-RBFNN self-evolving control scheme with noise reduction achieves the formation control task for networked mobile robots. This approach outperforms the traditional consensus control methods where the uncertainty and self-adaptation are not considered. Figure 12: Formation error of follower mobile robot #2 for circular formation control. Figure 13: Simulation result for the daisy curve formation control. In order to provide a comparative analysis to illustrate the merits of the proposed methods over other existing approaches, Table 1 lists the comparison of the proposed GA-KF-RBFNN formation control and traditional controllers.