# Line-of-Sight Initial Pointing Model of Space Dynamic Optical Network and Its Verification
## 3 Multi-Link LOS Pointing Model
### The Principle of LOS Pointing
#### 3.1.2 Coordinate Conversion and Transfer Principle  
1. Geodetic Coordinate-Rectangular Coordinate Conversion  
Before performing coordinate conversion, it is necessary to convert the geodetic polar coordinates into rectangular coordinates, that is, to convert the longitude, latitude, and altitude into rectangular coordinate values. The ellipsoid parameters are selected according to the WGS-84 ellipsoid model, and the conversion formula is shown in Formula (1):  
\[ \begin{pmatrix}X_{o,i}\\ Y_{o,i}\\ Z_{o,j}\end{pmatrix} = \begin{pmatrix}(N+H_{o,i})\cos B_{o,i}\cos L_{o,i}\\ (N+H_{o,j})\cos B_{o,i}\sin L_{o,i}\\ (N(1-e^{2})+H_{o,j})\sin B_{o,j}\end{pmatrix}  \tag{1}\]  
where \(N=\frac{a}{\sqrt{1-e^{2}(\sin B_{o,i})^{2}}}\); ellipsoid long radius a \(=6378137\)m; and eccentricity squared \(e^{2}=0.006694379995\). \(B_{o,i},L_{o,i},H_{o,i}\) are the longitude, latitude, and altitude values, obtained in real time. The subscript o represents one's own side, and the subscript \(i\) represents the other party of the \(i\) link, where \(i=1,2,3,4\ldots\)  
2. Matrix transfer principle [9]  
The \(i\)-th LOS conversion and transmission principle is shown in the Figure 2.  
The conversion matrix used to convert the coordinates in the opponent's WGS-84 coordinate system to the coordinates in the east-north-up coordinate system with one's own side as the origin is as follows:  
\[C_{e}^{n}= \begin{pmatrix}-\sin L_{o}&\cos L_{o}&0\\ -\sin B_{o}\cos L_{o}&-\sin B_{o}\sin L_{o}&\cos B_{o}\\ \cos B_{o}\cos L_{o}&\cos B_{o}\sin L_{o}&\sin B_{o}\end{pmatrix}  \tag{2}\]  
\(C_{e}^{n}\) real-time compensation for the impact of changes in one's own position on the pointing model. Among them, \(B_{o},L_{o},H_{o}\) are the longitude, latitude, and altitude values of the origin in the east-north-up coordinate system, which are obtained in real time by one's own position sensor. The conversion matrix used to convert the coordinates in the opponent's east-north-up coordinate system into the coordinates in the carrier coordinate system, with one's own side as the origin, is as follows:  
\[C_{n}^{\mathrm{b}}= \begin{pmatrix}\cos r\cos y-\sin r\sin p\sin y&\cos r\sin y+ \sin r\sin p\cos y&-\sin r\cos p\\ -\cos p\sin y&\cos p\cos y&\sin p\\ \sin r\cos y+\cos r\sin p\sin y&\sin r\sin y-\cos r\sin p\cos y&\cos p\cos r \end{pmatrix}  \tag{3}\]  
\(C_{u}^{\mathrm{b}}\) is real-time compensation for the impact of changes in one's own heading angle, pitch angle, and roll angle on the pointing model, where \(y\), \(p\), and \(r\), respectively, represent one's own real-time heading angle, pitch angle, and roll-angle values, which are obtained in real time from one's own attitude sensor. The angle range of heading angle rotation is from 0 degrees to 360 degrees. The angle range of pitch angle rotation is from \(-90\) degrees to \(+90\) degrees. The angle range of roll-angle rotation is from \(-90\) degrees to \(+90\) degrees. Definition of the conversion matrix that converts the coordinates in the opponent's carrier coordinate system into the coordinates in the \(i\)-th visual-axis coordinate system, with one's own side as the origin, is as follows:  
\[C_{bi}^{a}= \begin{pmatrix}\cos\theta_{xi}\cos\theta_{yi}-\sin\theta_{xi}\sin \theta_{zi}\sin\theta_{yi}&\cos\theta_{xi}\sin\theta_{yi}+\sin\theta_{xi}\sin \theta_{zi}\cos\theta_{yi}&-\sin\theta_{xi}\cos\theta_{zi}\\ -\cos\theta_{zi}\sin\theta_{yi}&\cos\theta_{zi}\cos\theta_{yi}&\sin\theta_{zi }\\ \sin\theta_{xi}\cos\theta_{yi}+\cos\theta_{xi}\sin\theta_{zi}\sin\theta_{yi}& \sin\theta_{xi}\sin\theta_{yi}-\cos\theta_{xi}\sin\theta_{zi}\cos\theta_{yi} &\cos\theta_{zi}\cos\theta_{xi}\end{pmatrix}  \tag{4}\]  
\(C_{bi}^{a}\) compensates for the effect of differences between the different axes of view on the pointing model. \(\theta_{xi}\), \(\theta_{yi}\), and \(\theta_{zi}\) are the angles between the three coordinate axes of the \(i\)-th LOS coordinate system and the carrier coordinate system. These are known values which are measured and calibrated during installation.