# Modified Uncertainty Error Aware Estimation Model for Tracking the Path of Unmanned Aerial Vehicles
## 3 Simulation Analysis and Result
### Nonlinear UAV Tracking Environment  
Here we present a nonlinear tracking environment where the position of UAV considering two-dimensional (2D) space is measured using two parameters such as radius and azimuth angle. A sample 2D process obtained of uncertainty maneuvering of UAV is shown in Figure 3. The corresponding operation, which represents the Cartesian position of the UAV at different intervals, is represented as a discrete-time [PERSON] equation as follows. \[\Delta\theta_{n+1}^{(1)}=-\frac{1}{2}\Big{(}\theta_{n}^{(1)}-1\Big{)}^{3}+ \Big{(}\theta_{n}^{(1)}-1\Big{)}+\sqrt{2}u_{n}^{(1)} \tag{42}\]  
\[\Delta\theta_{n+1}^{(2)}=-\frac{1}{2}\Big{(}\theta_{n}^{(2)}-6\Big{)}^{3}+ \Big{(}\theta_{n}^{(2)}-6\Big{)}+\sqrt{2}u_{n}^{(2)} \tag{43}\]  
Here, \(\theta_{n}^{(1)}\) and \(\theta_{n}^{(2)}\) represent Gaussian noise, and the drift terms define the double-well potentials. The UAV positions are calculated in polar coordinates using radius and azimuth as follows. \[\varnothing_{n}=arc\ tan\Bigg{(}\frac{\theta_{n}^{(1)}}{\theta_{n}^{(2)}} \Bigg{)},r_{n}=\sqrt{\Big{(}\theta_{n}^{(1)}\Big{)}^{2}+\Big{(}\theta_{n}^{( 1)}\Big{)}^{2}} \tag{44}\]  
Figure 3: A sample two-dimensional space of UAV path. and the UAV tracking model measurement is generated by introducing Gaussian noise as follows. \[Z_{n}=\Big{[}\varnothing_{n}+v_{n}^{(\varnothing)},r_{n}+v_{n}^{(r)}\Big{]}, \tag{45}\]  
where \(v_{n}^{(\varnothing)}\) Gaussian noise with variance \(\sigma_{\varnothing}^{2}\) and \(v_{n}^{(r)}\) Gaussian noise operations with variance \(\sigma_{r}^{2}\). Here, we generated 1000 samples of trajectories with the interval time of \(\Delta t=0.01\) and varied the signal-to-noise ratio (SNR), by changing \(\sigma_{\varnothing}^{2}\) and \(\sigma_{r}^{2}\). Then, KFUEA, PF, and DMK were applied simultaneously to evaluate the tracking performance of the different models for \(z_{n}\). Figure 4 shows the normalized root mean square error (\(nRMSE\)) results of the clean measurement estimation obtained by KFUEA and existing filtering methods such as PF and DMK. The \(nRMSE\) is measured on a logarithmic scale; the \(\varnothing_{n}\) and \(r_{n}\) obtained by the corresponding filtering algorithms and the measurement error (\(Meas\)) provide an upper limit. The \(nRMSE\) outcomes are obtained by varying the SNR level, and the mean and standard deviation of the \(nRMSE\) are computed for 50 samples considering the different SNR individually. Figure 4 shows that KFUEA performs better than conventional filtering methods such as PF and DMK at different noise levels. For both low and high SNR values, KFUEA outperforms PF and DMK. Figure 5 shows the KFUEA predicted clean measurement (in red), PF (in gray), DMK (in blue), clean (dashed in black), and noisy measurement (in gray \"x\") with SNR value set to 1. Figure 5a shows the measurement of \(r_{n}\) and Figure 5b shows the measurement of \(\varnothing_{n}\). However, we obtain all values from the radar, the azimuth angle, and the radius. The values will always be noisy, so we do not obtain accurate values. The gray poses are all the values going into the system as input, and the black values are the clean measure. Therefore, our job is to protect the black value because those gray poses are contaminated (attack values/noisy values). The closer to the black values we obtain, the better results we obtain in the system. From the results, we can see that the KFUEA estimation is close to the true value, while the estimates of DMK and PF vary so much due to errors presented in their models (higher noise). Figure 4: The clean measurement estimations’ \(nRMSE\) mean and standard deviation. The result of the \(nRMSE\) were averaged over 50 samples of process and noise trajectories. For evaluating the filtering algorithm under the presence of noise experiment is conducted. The corresponding \(nRMSE\) considering different SNR levels are shown in Figure 6, where the mean and standard deviation of respective \(nRMSE\) is computed for 50 samples considering different SNR level individually.