## 3 Experiment
### Experimental Result  
To compare the differences between theoretical and experimental values, in both experiments, all four propellers were tilted at specific angles at 20 s and 40 s. For the theoretical values, the horizontal pulling force was calculated using Equation (15) and the yaw moment was calculated using Equation (16). When the attitude angles of Tilt-X were very small, the horizontal pulling force was calculated using Equation (17) and the yaw moment was calculated using Equation (18). For the experimental values, both the horizontal pulling force and the yaw moment were directly measured by the F/T sensor. Figure 7 shows experimental data obtained from the Pixhawk 4. Figure 7a presents the results of the translation experiment, with the propellers tilted as illustrated in Figure 5a. Figure 7b displays the results of the rotation experiment, with the propellers tilted as illustrated in Figure 5b. Both experiments were conducted in AltHold mode. Since the Tilt-X is connected to the telemetry, and the telemetry communicates with the ground station, i.e., Mission Planner, throughout both experiments, the flight log file '.tlog' was converted to a '.mat' file via Mission Planner. In MATLAB, the real-time roll, pitch, and yaw radians measured by the IMU in Pixhawk 4 are obtained from roll_mavlink_attitude_t, pitch_mavlink_attitude_t, and yaw_mavlink_attitude_t, respectively. After converting radians to degrees in units, the results are shown in the 'Roll Angle and Pitch Angle' and 'Yaw Angle' graphs of Figure 7. The PW of rotors 1 to 4 are obtained from servo1_raw_mavlink_servo_output_raw_t, servo2_raw_mavlink_servo_amvlink_raw_t, servo3_raw_mavlink_servo_output_raw_t, and servo4_raw_mavlink_servo_output_raw_t, respectively, and can be directly exported as shown in the 'Rotors Pulse Width' graph of Figure 7. The PW of rotor 1 to 4 are obtained from servo9_raw_mavlink_servo_output_raw_t, servo10_raw_mavlink_servo_output_raw_t, servo11_raw_mavlink_servo_output_raw_t, and servo12_raw_mavlink_servo_output_raw_t, respectively. Using these \begin{table}
\begin{tabular}{c c c} \hline \hline
**Component** & **Part Number** & **Quantity** \\ \hline Power Module & [Holybro] PM07 Power Module (14S) & 1 \\ Flight Controller & [Holybro] Pixhawk 4 & 1 \\ Telemetry & [Holybro] SiK Telemetry Radio V3 & 1 \\ Battery & [Poly-Tronics] B1500N-UKP60 & 1 \\ Receiver & [FrSky] R-XSR & 1 \\ ESC & [Cyclone] 45A BLHeli-S 2-6S Lipo & 4 \\ Rotor & [T-Motor] P2207 V3.0 1950 Kv & 4 \\ Propeller & [Gemfan] Hurricane 51466-3 MCK V2 & 4 \\ BEC & [Hobbywing] UBEC 10A HV Air 3-14S & 1 \\ Servo Motor & [Savox] SB-2290 SG & 4 \\ Servo Arm & [Futaba] Full Arm M3 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The components of Tilt-X. Figure 6: Experimental environment. (**a**) Initial calibration of the F/T sensor without the load of Tilt-X. (**b**) Tilt-X placed on the F/T sensor. PW values, the tilt angles of rotors 1 to 4 can be calculated as shown in the 'Rotors Tilting Angle' graph of Figure 7.  
Figure 7: Experimental data from Pixhawk 4, including roll angle, \(\phi\), pitch angle, \(\theta\), yaw angle, \(\psi\), each rotor’s PW, which indirectly reflects each propeller’s rotational speed, i.e., each propeller’s angular velocity, \(\omega_{i}^{p}\), and each rotor’s tilting angle, \(\beta_{i}^{b}\). (**a**) Experimental results of translation. (**b**) Experimental results of rotation. As shown in the 'Roll Angle and Pitch Angle' graph of Figure 7, the roll, pitch, and yaw angles in both experiments were all small. Consequently, the impact of the attitude angle of the Tilt-X on the horizontal pulling force and yaw moment is limited. To calculate the theoretical values of the horizontal pulling force and yaw moment, Equations (17) and (18), which were simplified from Equations (15) and (16), can be used when the attitude angles can be ignored.