# Systematic Error Correction for Geo-Location of Airborne Optoelectronic Platforms
## 2 Materials and Methods
### Ground Target Geolocation Error Analysis and Systematic Error Correction  
#### 2.4.1 Analysis of Errors Affecting Geo-Location  
Geo-location accuracy is an important factor that measures the ability of airborne optoelectronic platform to obtain target geographic location information. Therefore, it is very important to analyze the individual sources of measurement uncertainty of each part on geo-location process. Sensor measurement errors and installation errors are two major factors that affect the geo-location accuracy. For this platform, the position and attitude errors of the aircraft are mainly derived from the measurement errors of the integrated navigation system. The azimuth and pitch angle errors of the platform are derived from the measurement errors of the encoders, and the laser ranging errors are derived from the measurement errors of the laser rangefinder. These are normally distributed random errors, shown as sensor measurement error in Figure 4, which cannot be eliminated but can only be estimated with filters. Here we focus on the systematic error that affects the positioning accuracy shown in Figure 6 as installation error.  
(1) POS installation error  
In order to reduce the impact of aircraft vibration on target positioning, an independent POS can be installed on the base of the airborne optoelectronic platform to directly obtain the position and attitude information of the platform. During the installation of the POS, calibration is required to ensure that the coordinate system of the aircraft coordinate frame coincides with the navigation coordinate frame in inertial state. After calibration, the error of pitch and roll direction is generally about1 mrad, and the error of heading direction is about 1.5 mrad. Use \(\delta_{\phi}\), \(\delta_{\theta}\), and \(\delta_{\phi}\) to represent the installation errors of the heading, pitch and roll directions respectively. Since these errors are all small values, conversion matrix from the geographic coordinate system to the aircraft coordinate system can be approximated by:  
\[\left(M_{p}^{b}\right)^{\prime}=\left[\begin{array}{ccc}1&-\delta_{\psi}& \delta_{\theta}\\ \delta_{\psi}&1&-\delta_{\phi}\\ -\delta_{\theta}&\delta_{\phi}&1\end{array}\right]\cdot M_{p}^{b} \tag{11}\]  
(2) Payload installation error  
The optical system of the airborne optoelectronic platform is installed in a two-axis frame. Ideally, when the pitch and azimuth angles of the platform are zero, the imaging coordinate frame should coincide with the aircraft system, and the pitch axis of the imaging coordinate frame should be aligned with the pitch axis of the aircraft coordinate frame, and the azimuth axis should be aligned with the yaw axis of the aircraft coordinate frame. The error generated during the assembly and installation will cause the yaw and pitch directions of the two frame axes to be inconsistent, resulting in an error in the angle measurement of the imaging coordinate frame. Under normal circumstances, the installation error of this frame is about 1.5 mrad. We use \(\delta_{\alpha}\) and \(\delta_{\beta}\) represent the installation error of the azimuth and pitch, respectively. Since these two error angles are both small, conversion matrix from imaging coordinate frame to the aircraft coordinate frame can be approximated by the following matrix:  
\[\left(M_{b}^{v}\right)^{\prime}=\left[\begin{array}{ccc}1&-\delta_{\beta}& \delta_{\alpha}\\ \delta_{\beta}&1&0\\ -\delta_{\alpha}&0&1\end{array}\right]\cdot M_{b}^{v} \tag{12}\]