## 4 Experiment Verification
### Simulation Results  
In the experimental verification, we tested the step response and frequency response of the control system. The numerical simulation included two parts. The first part tested the MRAC and \(\mathcal{L}_{1}\) control performance of \(r=[0.2\,0.3\,0.78\,0.25\,0.5\,1.0]\), as illustrated in Figure 7. The second part tests tested the MRAC and \(\mathcal{L}_{1}\) control performance of \(r=[0.2 sin(\pi t),\;0.3 sin(\pi t),\;0.48+0.14 sin(\pi t),\;\frac{\pi}{\Omega}sin (\pi t),\;\frac{\pi}{6}sin(\pi t),\;\frac{\pi}{3}sin(\pi t)]\), as illustrated in Figure 8. Because of space limitation, we only give the simulation data of position DOFs, and the attitude DOFs are similar to it.  
The simulation results of the MRAC are shown in Figure 7a,c,e and the \(\mathcal{L}_{1}\) adaptive controller shown in Figure 7b,d,f for step reference inputs \(r=[0.2\,0.3\,0.78\,\pi/12\,\pi/6\,\pi/3]\). We found that both of MRAC and the \(\mathcal{L}_{1}\) adaptive controller had the same instantaneous response characteristics. Therefore, we could improve the transient response time by changing the state matrix \(A_{m}\) to meet the requirements of abnormal attitude, large angular velocity, and acceleration changes of the aircraft in the abnormal flight state.  
From Figure 7g,h, it can be seen that the instantaneous output peak of \(\mathcal{L}_{1}\) adaptive controller was significantly lower than that of MRAC. Taking the output of No.2 leg driver (u2) of the Stewart simulator as an example, the instantaneous output peak value of MRAC controller was 438 mNm, while the instantaneous output peak value of leg 2 corresponding to \(\mathcal{L}_{1}\) adaptive controller was 263 mNm, which reduced 39.95% of the peak output. The results show that the \(\mathcal{L}_{1}\) adaptive controller improved the instantaneous performance of the actuator and reduced the requirement for the performance of the actuator without changing the response speed.  
It can be seen from Figure 8 that the \(\mathcal{L}_{1}\) adaptive controller can filter out the high-frequency signals in the control variables and add the low-frequency signals to enhance the smoothness of flight simulator motion and improve the motion sense effect of abnormal state recovery training on the flight simulator.  
Figure 6: Curve trajectory of parameter \(\sigma\) in complex state.  
Figure 7: Performance of MRAC and \(\mathcal{L}_{1}\) adaptive controller for \(r=[0.2\ 0.3\ 0.78\ 0.25\ 0.5\ 1.0]\). (**a**) X-position curve of the platform when using MRAC controller; (**b**) X-position curve of the platform when using \(\mathcal{L}_{1}\) adaptive controller; (**c**) Y-position curve of the platform when using MRAC controller; (**d**) Y-position curve of the platform when using \(\mathcal{L}_{1}\) adaptive controller; (**e**) Z-position curve of the platform when using MRAC controller; (**f**) Z-position curve of the platform when using \(\mathcal{L}_{1}\) adaptive controller; (**g**) Torque **u** curve of the platform when using MRAC controller; (**h**) Torque **u** curve of the platform when using \(\mathcal{L}_{1}\) adaptive controller.  
Figure 8: Performance of MRAC and \(\mathcal{L}_{1}\) adaptive controller for \(r=[0.2 sin(\pi t)\), \(0.3 sin(\pi t)\), \(0.48+0.14 sin(\pi t)\), \(\frac{\pi}{12}sin(\pi t)\), \(\frac{\pi}{8}sin(\pi t)\), \(\frac{\pi}{8}sin(\pi t)\). **(a)** X-position curve of the platform when using MRAC controller; **(b)** X-position curve of the platform when using \(\mathcal{L}_{1}\) adaptive controller; **(c)** Y-position curve of the platform when using MRAC controller; **(d)** Y-position curve of the platform when using \(\mathcal{L}_{1}\) adaptive controller; **(e)** Z-position curve of the platform when using MRAC controller; **(f)** Z-position curve of the platform when using \(\mathcal{L}_{1}\) adaptive controller; **(g)** Torque **u** curve of the platform when using MRAC controller; **(h)** The torque **u** curve of the platform with \(\mathcal{L}_{1}\) adaptive controller.