## 3 Experiment
### Experimental Result
Regarding the validity of Equation (16), i.e., whether or not the simplified yaw moment derived from Equations (7) and (13) is feasible, depends on the similarity of the angular velocities of each propeller. Since it is difficult to measure the angular velocity of each propeller directly, its PW signal was used to estimate its angular velocity. Although this indirect method does not provide the exact value, it gives a general understanding of their rotational speeds. CCW rotating propellers generate a CW moment, while CW rotating propellers generate a CCW moment. As shown in the 'Rotors Pulse Width' graph of Figure 7, the PW values of the CCW rotating propeller 1 and propeller 2 are similar to those of the CW rotating propeller 3 and propeller 4 in both experiments. This symmetry indicates that the angular velocity of each propeller is similar, and the total moment of the propellers is close to zero. Specifically, the \(c_{M}\xi b_{1}^{b}\omega_{1}^{b^{2}}\) and \(c_{M}\xi b_{2}^{b}\omega_{2}^{b^{2}}\) generated by the CCW rotating propellers 1 and 2 cancel out the \(-c_{M}\xi c_{3}^{b}\omega_{3}^{b^{2}}\) and \(-c_{M}\xi b_{4}^{b}\omega_{4}^{b^{2}}\) generated by the CW rotating propellers 3 and 4. Therefore, the simplified Equation (16) is valid. In the translation experiment shown in the 'Rotors Tiling Angle' graph of Figure 7a, the tilt angles of all four propellers were manually set to be 27 degrees and 51.4 degrees, respectively. However, there were slight deviations in the actual tilt angles of propellers 2 and 4. The actual tilt angles for propeller 2 were measured to be 27 degrees and 50.9 degrees, and those for propeller 4 were 27 degrees and 50.4 degrees. In this work, these small deviations are negligible. In the rotation experiment shown in the 'Rotors Tiling Angle' graph of Figure 7b, the tilt angles of all four propellers were set to be 29.8 degrees and 51.4 degrees, respectively. However, there were also slight deviations in the actual tilt angles of propellers 2, 3, and 4 of less than 1.0 degrees. The experimental data obtained from the F/T sensor and the filtered data are shown in Figure 8. Figure 8a presents the force data measured during the translation experiment, which includes the vertical component of thrust force \(F_{z}^{e}\), as shown in the 'Vertical Component of Thrust Force', the horizontal pulling force \(F_{x}^{e}\), as shown in the 'Horizontal Pulling Force', their comparison after smoothing with a Gaussian filter, as shown in the 'Filtered Force', and the comparison between the measured horizontal pulling force from the translation experiment and the theoretically calculated horizontal pulling force, as shown in the 'Horizontal Pulling Force Comparison'. Figure 8b shows the force data measured during the rotation experiment, which includes the vertical component of thrust force \(F_{z}^{e}\), as shown in the 'Vertical Component of Thrust Force', the yaw moment \(M_{z}^{e}\), as shown in the 'Yaw Moment', their comparison after smoothing with the Gaussian filter, as shown in the 'Filtered Force/Moment', and the comparison between the measured yaw moment from the rotation experiment and the theoretically calculated yaw moment, as shown in the 'Yaw Moment Comparison'. In both experiments, the tilting angle of the propellers varied while maintaining the throttle stick at 50%. As shown in the 'Vertical Component of Thrust Force' graph of Figure 8, the vertical component of the thrust of all four propellers in both experiments did not differ significantly. In the translation experiment, when the tilt angle of all four propellers was 27 degrees, the maximum horizontal pulling force was measured to be 5.74 N. When the tilt angle was 51.4 degrees, the maximum horizontal pulling force was measured to be 8.50 N. In the rotation experiment, when the tilt angle was 29.8 degrees, the maximum yaw moment was measured to be approximately 0.65 Nm. When the tilt angle was 51.4 degrees, the maximum yaw moment was measured to be approximately 0.81 Nm.