# Systematic Error Correction for Geo-Location of Airborne Optoelectronic Platforms
## 3 Results
### Simulation of System Error Correction
#### 3.1.2 Systematic Error Correction and Results  
Assuming that the installation error of the integrated navigation system is \(0.30^{\circ}\) for the heading direction, \(-0.05^{\circ}\) for the pitch direction, and \(0.20^{\circ}\) for the roll direction, the installation error of the image coordinate frame is \(-0.20^{\circ}\) for the azimuth angle and \(0.10^{\circ}\) for the pitch angle, and the control point is at (33.980849\({}^{\circ}\) N, 107.523239\({}^{\circ}\) E, 3132.10 m). In the simulation, the aircraft takes multiple imaging measurements on the control points at different positions and in different attitudes to get the installation error. The systematic error and measurement errors are shown in Table 1.  
After performing 256 measurements on the control point, we got the installation error shown in Figure 7. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Error Type** & **Name of Error Variable** & **Symbol** & **Error Value** \\ \hline \multirow{4}{*}{Systematic error} & \multicolumn{2}{c}{Roll} & 0.2\({}^{\circ}\) \\  & POS installation error & Pitch & \(-0.05^{\circ}\) \\  & \multicolumn{2}{c}{Yaw} & 0.3\({}^{\circ}\) \\  & Payload installation error & Pitch & \(0.1^{\circ}\) \\  & \multicolumn{2}{c}{Azimuth} & \(-0.2^{\circ}\) \\ \hline \multirow{4}{*}{Random error} & \multirow{4}{*}{Platform position} & Latitude & 0.0001\({}^{\circ}\) (10 m) \\  & & Altitude & 0.00012\({}^{\circ}\) (10 m) \\  & & Altitude & 10 m \\  & & Roll & 0.02\({}^{\circ}\) \\  & \multicolumn{2}{c}{Platform attitude} & Pitch & \(0.02^{\circ}\) \\  & \multicolumn{2}{c}{Yaw} & 0.05\({}^{\circ}\) \\  & \multicolumn{2}{c}{Pitch} & 0.027\({}^{\circ}\) \\  & \multicolumn{2}{c}{Azimuth} & 0.027\({}^{\circ}\) \\  & \multicolumn{2}{c}{Laser range} & Laser & 5 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Systematic error and random measurement error in the geo-location. From Figure 7, it is not difficult to see that after 100 imaging measurements on the control points, the five installation errors are calculated as follows: the integrated navigation system heading direction is \(0.349^{\circ}\), the pitch direction is \(-0.047^{\circ}\), the roll direction is \(0.199^{\circ}\). The azimuth angle of the imaging coordinate frame is \(-0.244^{\circ}\), and the pitch angle is \(0.095^{\circ}\). Compared with the actual installation error, the calculated errors are \(0.049^{\circ}\), and \(-0.003^{\circ}\), \(0.001^{\circ}\), \(0.044^{\circ}\) and \(-0.005^{\circ}\), which are only 1/6 of the original installation error. After 100 measurements, the calculated installation errors have not changed significantly. Therefore, the actual application can consider more than 100 measurements on the control points to estimate the installation error for the platform. Using the measurement error and systematic error data in Table 1, using the control point and aircraft positions, the geo-location results are shown in the Figure 8a,c,e. The root mean square error of the geo-location at this time is calculated to be 53.28 m, and the mean value of the positioning error is \((0.0005126^{\circ},0.0001104^{\circ},4.67\) m) or 22.13 m. Due to the installation error, the mean error of the target positioning result is not zero. After the installation error is corrected with the systematic error obtained above, the geo-location data is shown in Figure 8b,d,f. It is calculated that the root mean square error of the positioning at this time is \(13.61\) m, and the mean value of the geo-location error at this time is \((0.0000076^{\circ},0.0000011^{\circ},-0.03\) m), which corresponds to \(6.37\) m. The comparison between before and after correction is shown in Table 2. After the systematic error were corrected, the root mean square error is reduced to \(1/3\) of the original, and the mean value of the geo-location error is closer to zero. The use of methods such as a Kalman filter, etc. for geo-location would be more accurate after the systematic errors were corrected. Figure 7: The iteration curve of the installing errors for simulation data.