## 6 Experimental Results
### Simulation Results  
The simulated step signal representations of the quadcopter's inputs, roll (\(\phi\)), pitch (\(\theta\)), yaw (\(\psi\)) and thrust (\(z\)), using a PID controller along with simulations of the amplitudes of square signals of the inputs' responses obtained by the GA and ZN methods, are illustrated in Figures 7, 8, 9 and 10, respectively. Figure 8: Simulated output response of the pitch input of a quadcopter using GA and ZN methods. Figure 9: Simulated output responses of the yaw input of the quadcopter using GA and ZN methods. It is clearly evident from Figures 7-10 that there are significant differences between GA and ZN step response curves, indicating that GA achieves a slight overshoot, a small rise time and a small settling time, compared to those arrived at by the ZN method. By comparing the simulated step responses from the GA- and ZN-based methods, the GA-based approach is faster and has a smaller overshoot for input yaw of the quadcopter. However, the GA-based approach has a longer settling time. Remarkably, the degree of separation between the GA and ZN curves is relatively large for all quadcopter inputs. The simulated step signals of the inputs--roll, pitch, yaw and thrust--using the PID controller of the quadcopter as well as the simulations of amplitudes of the input responses obtained by the PSO and ZN methods are presented in Figures 11, 12, 13 and 14, respectively. Figure 11: Simulated output responses of the roll input of a quadcopter using PSO and ZN methods. Figure 10: Simulated output responses of the thrust input of a quadcopter using GA and ZN methods. Figure 12: Simulated output responses of the pitch input of a quadcopter using PSO and ZN methods. Figure 13: Simulated output responses of the yaw input of a quadcopter using PSO and ZN methods. Obviously, Figures 11-14 confirm that PSO is substantially more effective than the ZN method in controlling the movement of the quadcopter. This is due to the fact that the signal representations obtained by PSO report small overshoot, modest rise and settling times, compared to those reported by the ZN method. The simulated step signal representations using the PID controller of the quadcopter inputs--roll, pitch, yaw and thrust--along with the simulations of amplitudes of the input responses obtained by the CSA and ZN methods are displayed in Figures 15, 16, 17 and 18, respectively. It is clearly observed from Figures 15-18 that there are appreciable differences between CSA and ZN step signal representations, indicating that CSA arrives at slight overshoot, small rise and small settling times compared to those reached by the ZN method. Significantly, there is a relatively large degree of separation between the CSA and ZN curves for all of the quadcopter inputs. The simulated step signals for the roll, pitch, yaw and thrust inputs of the quadcopter shown in Figure 1 based on the use of GA, PSO, CSA and ZN methods by optimizing the PID controller of the quadcopter to stabilize its movements are presented in Figure 19.  
The results shown in Figure 19a-d demonstrate the effectiveness of GA, PSO and CSA in reliably controlling the roll, pitch, yaw and thrust inputs of a quadcopter, respectively. As it is clear, the simulated signals for the roll, pitch, yaw and thrust inputs in Figure 19a-d, respectively, obtained based on GA, PSO and CSA are better than those corresponding responses obtained by the ZN method. Interestingly, the responses obtained based on GA, PSO and CSA for roll, pitch, yaw and thrust inputs of the quadcopter are almost similar to each other, with not much difference between them. However, PSO provides the best results, compared to the others. Figure 16: Simulated output responses of the pitch input of a quadcopter using CSA and ZN methods. Figure 17: Simulated output responses of the yaw input of a quadcopter using CSA and ZN methods. Figure 19: Simulated output responses of the roll, pitch, yaw and thrust inputs of a quadcopter using (**a**) GA, (**b**) PSO, (**c**) CSA and (**d**) ZN methods.