# Systematic Error Correction for Geo-Location of Airborne Optoelectronic Platforms
## 2 Materials and Methods
### Establishment of the Basic Coordinates
#### 2.2.1 The Imaging Coordinate Frame \(V(X_{v}Y_{v}Z_{v})\)  
This frame has its origin at the rotation center of the payload. The \(X_{v}\) axis points upwards if the camera is along vehicle x-axis, and the \(Z_{v}\) axis is along the light of sight (LOS) of the imaging system and pointing to the target, and \(Y_{v}\) forms an orthogonal right-handed frame set. This imaging system is installed in a two-axis' gimbal, while the outer gimbal angle rotates around the \(Z_{b}\) axis, represents the azimuth angle \(\beta\), having the initial position as front, and positive to the right. The inner gimbal angle rotates around the \(y_{v}\) axis, represents the pitch angle \(\alpha\), having initial position as the LOS point down along the \(Z_{v}\) axis and positive while rotate to the front. The imaging coordinate frame and the transition process between the imaging coordinate frame and aircraft coordinate frame are shown in Figure 3.  
Figure 3: The imaging coordinate frame and aircraft coordinate frame.  
Figure 2: The geo-location process of system.  
2.2.2. The Aircraft Coordinate Frame \(\mathrm{B}(\mathrm{X}_{\mathrm{b}}\mathrm{Y}_{\mathrm{b}}\mathrm{Z}_{\mathrm{b}})\)  
This frame is the standard ARINC frame with origin shifted to the center of the optical platform center. The \(X_{b}\) axis points to the nose of the aircraft, \(Z_{b}\) axis points to the bottom of the aircraft, and \(Y_{b}\) forms an orthogonal right-handed set. POS were mounted on this frame, and the roll, pitch and yaw angles \((\phi,\theta,\psi)\) rotate around the \(X_{b}\) axis, \(Y_{b}\) axis and \(Z_{b}\) axis. The aircraft coordinate frame and the transition between the aircraft coordinate frame and navigation coordinate frame are shown in Figure 4.  
2.2.3. The Navigation Coordinate Frame \(\mathrm{P}\big{(}\mathrm{X}_{\mathrm{p}}\mathrm{Y}_{\mathrm{p}}\mathrm{Z}_{ \mathrm{p}}\big{)}\)  
This frame is the standard NED (north, east, down) reference frame and has its origin at the aircraft's center; the \(X_{p}\) axis points to true north, \(Y_{p}\) axis points to the east, and \(Z_{p}\) axis lies along the local geodetic vertical and is positive down. The transition between the geodetic coordinate frame and the navigation coordinate frame is shown in Figure 5.  
2.2.4. The Geodetic Coordinate Frame \(\mathrm{G}\big{(}\mathrm{X}_{\mathrm{g}}\mathrm{Y}_{\mathrm{g}}\mathrm{Z}_{ \mathrm{g}}\big{)}\)  
This frame is defined in WGS-84 and has its origin at the Earth's geometric center. The \(X_{g}\) is in the equatorial plane at the prime meridian, \(Z_{g}\) points north through the polar axis,  
Figure 4: The aircraft coordinate frame and navigation coordinate frame.  
Figure 5: The geodetic coordinate frame and the navigation coordinate frame.  
and \(Y_{g}\) forms an orthogonal right-handed set. The elliptical Earth model can be expressed as Figure 5:  
\[\frac{x_{g}^{2}}{R_{E}^{2}}+\frac{y_{g}^{2}}{R_{E}^{2}}+\frac{z_{g}^{2}}{R_{P}^{2 }}=1 \tag{1}\]  
where \(R_{E}=6,378,137\) m is the semi-major axis and \(R_{P}=6,356,752\) m is the semi-minor axis.  
The geographical position of a point can be expressed as the longitude, latitude, and geodetic height (\(B\), \(L\), and \(H\)). The point in the geodetic coordinate frame can be expressed as:  
\[P_{g}=\left[\begin{array}{c}x_{g}\\ y_{g}\\ z_{g}\end{array}\right]=\left[\begin{array}{c}\left(R_{N}+H\right)\;\cos B \;\cos L\\ \left(R_{N}+H\right)\;\cos B\;\sin L\\ \left[R_{N}\left(1-e^{2}\right)\;+H\right]\sin B\end{array}\right] \tag{2}\]  
where \(e=\sqrt{\frac{R_{E}^{2}-R_{P}^{2}}{R_{E}}}\) the first eccentricity of the Earth ellipsoid is, \(R_{N}=\frac{R_{E}}{\sqrt{1-e^{2}sin^{2}B}}\) is the prime vertical of curvature.