# High-Fidelity Drone Simulation with Depth Camera Noise and Improved Air Drag Force Models
## 3 Rotor Drag Force Model Implementation
### Experiments
#### 3.2.2. Rotor Drag Force Experiments  
To validate our model, simulated experiments in an indoor environment at five different speeds of 1.5 m/s, 7.5 m/s, and 26 m/s have been carried out. These experiments included circular and lemniscata trajectories, which are commonly used to assess tracking error performance due to their balanced motion on all axes and trajectory aggressiveness.  
Figure 14. Experiment to find out the relationships between PWM (to ESC) and the rotor thrust.  
Figure 15. Box plot of the experiment results of PWM-thrust relationships using a battery of 14.8 V.  
The Root Mean Square Error (RMSE) results for both our model and the Airsim model are presented in Figure 16. Figure 16 clearly demonstrates that our proposed rotor drag force model exhibits significant improvements over the Airsim drag force model, especially at high speeds. On average, the RMSE error when using our model was 4.58 times smaller than that of the Airsim model across all speeds. The accuracy of our model relative to the Airsim model increased with higher speeds, reaching a reduction in RMSE errors by 7.3 times at a wind speed of 26 m/s. These results validate the effectiveness of our model in accurately representing the drag force. A real-world experiment was conducted to compare the rotor and air drag force at various speeds.  
The drone flew back and forth at a straight distance of 80 m at an altitude of 15 m, and at speeds of 5 m/s and 10 m/s. Data were acquired using IMU and GPS sensors as shown in Figure 17. The force acting on the drone during flight can be written as follows:  
\[\mathbf{a_{input}}=\mathbf{F_{input}}/m=\mathbf{R}\sum_{i=1}^{4}T_{i}/m \tag{14}\]  
where \(\mathbf{a_{input}}\) is acceleration calculated by thrust input from the quadrotor's controller, \(m\) is the mass of the quadrotor, and \(\mathbf{F_{input}}\) is the input force acting on the drone by the quadrotor's rotors.  
\[\mathbf{a_{cal}}=\mathbf{a_{input}}-\mathbf{F_{a}}/m\cong\mathbf{a_{measured}} \qquad\left\{\begin{aligned} \mathbf{F_{a}}&=\mathbf{F_{ airsim}}\\ \mathbf{F_{a}}&=\mathbf{F_{rotor}}\end{aligned}\right. \tag{15}\]  
where \(\mathbf{a_{measured}}\) is the measured acceleration of the quadrotor by IMU and GPS sensors and \(\mathbf{F_{Airsim}}\) is Airsim's air drag force as shown in Equation (5), \(\mathbf{F_{rotor}}\) is rotor drag force as shown in Equation (13). To figure out the accuracy of the two different air drag force models, RMSE error is used as below:  
\[\text{RMSE}=(\mathbf{a_{measured_{x}}}-\mathbf{a_{cal_{x}}})^{2}+\left( \mathbf{a_{measured_{y}}}-\mathbf{a_{cal_{y}}}\right)^{2}+\left(\mathbf{a_{ measured_{z}}}-\mathbf{a_{cal_{x}}}\right)^{2} \tag{16}\]  
The comparison of the rotor drag force model and the Airsim drag force model reveals significant differences in accuracy when the drone undergoes acceleration. We also conducted outdoor experiments at speeds of 5 m/s and 10 m/s, the results of which are presented in Table 2 and Figures 18-20. At a speed of 5 m per second, the two models exhibited similar accuracy, with only a minor discrepancy between predicted values and  
Figure 16: RMSE of drag force models at different speeds of 1.5 m/s (circular trajectory), 7.5 m/s (circular trajectory), 15 m/s (leminiscata trajectory), 20 m/s (leminiscata trajectory), 26 m/s (circular trajectory).