# Systematic Error Correction for Geo-Location of Airborne Optoelectronic Platforms
## 2 Materials and Methods
### Ground Target Geo-Location  
When the airborne optoelectronic platform in the initial position, azimuth \(\beta\) and pitch \(\alpha\) angle both equal to zero, and the imaging coordinate frame coincides with the aircraft coordinate frame. The matrix transforms from the frame \(V(X_{p}Y_{p}Z_{p})\) to the frame \(B(X_{b}Y_{b}Z_{b})\) can be expressed as:  
\[M_{b}^{v}=\left[\begin{array}{ccc}\cos\beta&-\sin\beta&0\\ \sin\beta&\cos\beta&0\\ 0&0&1\end{array}\right]\cdot\left[\begin{array}{ccc}\cos\alpha&0&\sin \alpha\\ 0&1&0\\ -\sin\alpha&0&\cos\alpha\end{array}\right] \tag{3}\]  
The position and orientation system (POS), which is composed of the global positioning system (GPS) and inertial measurement unit, can measure the position and attitude information of the airborne platform accurately. The position information of the airborne optoelectronic platform includes the latitude, longitude, and height \((B,L,H)\) and the attitude information includes roll, pitch and yaw angles \((\phi,\theta,\psi)\). The matrix transforms from the frame \(B(X_{b}Y_{b}Z_{b})\) to the frame \(P(X_{p}Y_{p}Z_{p})\) can be expressed as:  
\[M_{p}^{b}=\left[\begin{array}{ccc}\cos\psi&-\sin\psi&0\\ \sin\psi&\cos\psi&0\\ 0&0&1\end{array}\right]\cdot\left[\begin{array}{ccc}\cos\theta&0&\sin \theta\\ 0&1&0\\ -\sin\theta&0&\cos\theta\end{array}\right]\cdot\left[\begin{array}{ccc}1&0&0 \\ 0&\cos\phi&-\sin\phi\\ 0&\sin\phi&\cos\phi\end{array}\right] \tag{4}\]  
The matrix transforms from the frame \(P\big{(}X_{p}Y_{p}Z_{p}\big{)}\) to the frame \(G\big{(}X_{g}Y_{g}Z_{g}\big{)}\) can be expressed as:  
\[M_{g}^{p}=\left[\begin{array}{ccc}\cos L&-\sin L&0\\ \sin L&\cos L&0\\ 0&0&1\end{array}\right]\cdot\left[\begin{array}{ccc}\cos(-B-0.5\pi)&0&\sin (-B-0.5\pi)\\ 0&1&0\\ -\sin(-B-0.5\pi)&0&\cos(-B-0.5\pi)\end{array}\right] \tag{5}\]  
and the matrix transform from frame imaging coordinate frame to the geodetic coordinate frame can be expressed as:  
\[M_{g}^{v}=M_{g}^{p}\cdot M_{p}^{b}\cdot M_{b}^{v} \tag{6}\]  
The distance (\(Rng\)) between the airborne platform and the target can be provided by the laser range finder. Since the laser beam is parallel with the LOS, the vector from the imaging system to the target can be expressed in geodetic coordinate frame as:  
\[\overrightarrow{R_{g}}=M_{g}^{v}\cdot\left[\begin{array}{ccc}0&0&Rng\end{array} \right]^{T} \tag{7}\]Then the target position \(\overrightarrow{T_{g}}\) in geodetic coordinate frame can be expressed as:  
\[\overrightarrow{T_{g}}=\overrightarrow{P_{g}}+\overrightarrow{R_{g}} \tag{8}\]  
where \(\overrightarrow{P_{g}}\) is a shift of origin from the center of gravity of the aircraft to the origin of the imaging system coordinate frame.  
According to the elliptical Earth model, the latitude, the longitude and the geodetic height of the target can be solved according to [6]. The latitude of the northern hemisphere is positive, the latitude of the southern hemisphere is negative, and the target latitude and the geodetic height can be solved by the following iteration equations:  
\[\left\{\begin{array}{l}R_{0}=R_{E}\\ H_{0}=\sqrt[2]{x_{g}^{2}+y_{g}^{2}+z_{g}^{2}}-\sqrt[2]{R_{E}R_{P}}\\ B_{0}=\text{atan}\left[\frac{z_{g}}{\sqrt[2]{x_{g}^{2}+y_{g}^{2}}}\cdot\left(1- \frac{e^{2}R_{0}}{R_{0}+H}\right)^{-1}\right]\\ R_{k+1}=\frac{R_{E}}{\sqrt{1-e^{2}\sin^{2}{B_{k}}}}\\ H_{k+1}=\frac{\sqrt{x_{g}^{2}+y_{g}^{2}}}{\cos{B_{k}}}-R_{k}\\ B_{k+1}=\text{atan}\left[\frac{z_{g}}{\sqrt[2]{x_{g}^{2}+y_{g}^{2}}}\cdot \left(1-\frac{e^{2}R_{k+1}}{R_{k+1}+H}\right)^{-1}\right]\end{array}\right. \tag{9}\]  
According to the elliptical Earth model, the longitude of the eastern hemisphere is positive, the longitude of the western hemisphere is negative, and the target longitude can be solved by the following equation:  
\[L=\left\{\begin{array}{l l}l&x_{g}>0\\ l+\pi,&x_{g}<0,l<0\\ l-\pi,&x_{g}<0,l>0\end{array}\right.,\;l=\text{atan}\frac{y_{g}}{x_{g}} \tag{10}\]