# Test Method for Single Satellite's Inter-Satellite Link Pointing and Tracking via Ground Station
## 2 Models of Inter-Satellite Link Pointing and Tracking  
### Pointing Vector Calculation  
The inter-satellite link transmits data between satellites, which can be simplified as multiple pointing vectors from point to point in the constellation. These pointing vectors are the relative position vectors from a local satellite with a projector to another target satellite with a receiver. By propagating the orbits of both satellites, the pointing needed to maintain the transmissions of the inter-satellite link can be calculated.  
In the present study, the inter-satellite link vector pointing from the local satellite to the target satellite in the J2000 geometric inertial frame of reference is \(\mathbf{r}_{link}^{i}\), in the orbital frame is \(\mathbf{r}_{link}^{o}\), in the satellite body frame is \(\mathbf{r}_{link}^{b}\), and in the antenna frame is \(\mathbf{r}_{link}^{a}\). The transforms between coordinate frames are shown as follows:  
\[\mathbf{r}_{link}^{a}=\mathbf{A}_{ab}\mathbf{r}_{link}^{b}=\mathbf{A}_{ab}\mathbf{A}_{bo}\mathbf{r}_{ link}^{o}=\mathbf{A}_{ab}\mathbf{A}_{bo}\mathbf{A}_{oi}\mathbf{r}_{link}^{i} \tag{1}\]  
where \(\mathbf{A}_{ab}\) is the transformation matrix from the satellite body frame to the antenna frame; \(\mathbf{A}_{bo}\) is the transformation matrix from the satellite orbital frame to the satellite body frame; and \(\mathbf{A}_{oi}\) is the transformation matrix from the J2000 inertial frame to the satellite orbital frame.  
Then, the position vector in the J2000 inertial frame of reference from the Earth's center to the local satellite is \(\mathbf{r}_{local}^{i}\), and the velocity vector of the local satellite is \(\mathbf{v}_{local}^{i}\). The vectors of the local satellite orbital frame reference are  
\[\left\{\begin{array}{l}\mathbf{Z}_{o}=-\frac{\mathbf{r}_{local}^{i}}{|\mathbf{r}_{local }^{i}|}\\ \mathbf{Y}_{o}=-\frac{\mathbf{r}_{local}^{i}\times\mathbf{r}_{local}^{i}}{|\mathbf{r}_{local} ^{i}\times\mathbf{r}_{local}^{i}|}\\ \mathbf{X}_{o}=\mathbf{Y}_{o}\times\mathbf{Z}_{o}\end{array}\right. \tag{2}\]  
Following these vectors, the transformation matrix from the inertial frame to the orbital frame \(\mathbf{A}_{oi}\) is given as  
\[\mathbf{A}_{oi}=\begin{bmatrix}\mathbf{X}_{o}^{T}&\mathbf{Y}_{o}^{T}&\mathbf{Z}_{o}^{T}\end{bmatrix} \tag{3}\]  
The transformation matrix from the satellite orbital frame to the satellite body frame is \(\mathbf{A}_{bo}\) given as in Equations (4) and (5), where \(\varphi\), \(\theta\), and \(\psi\) are the roll, pitch, and yaw angles of the satellite body frame relative to the satellite orbital frame, respectively.  
\[\mathbf{A}_{bo}=\mathbf{A}_{y}(\theta)\mathbf{A}_{x}(\varphi)\mathbf{A}_{z}(\psi) \tag{4}\]  
\[\mathbf{A}_{y}(\theta)=\begin{bmatrix}\cos\theta&0&-\sin\theta\\ 0&1&0\\ \sin\theta&0&\cos\theta\end{bmatrix},\mathbf{A}_{x}(\varphi)=\begin{bmatrix}1&0&0 \\ 0&\cos\varphi&\sin\varphi\\ 0&-\sin\varphi&\cos\varphi\end{bmatrix},\mathbf{A}_{z}(\psi)=\begin{bmatrix}\cos \psi&\sin\psi&0\\ -\sin\psi&\cos\psi&0\\ 0&0&1\end{bmatrix} \tag{5}\]The position vector of the target satellite in the J2000 inertial frame \(\mathbf{r}_{tag}^{i}\) and \(\mathbf{r}_{link}^{i}\) can be expressed as follows:  
\[\mathbf{r}_{link}^{i}=\frac{\mathbf{r}_{tag}^{i}-\mathbf{r}_{local}^{i}}{\left|\mathbf{r}_{tag} ^{i}-\mathbf{r}_{local}^{i}\right|} \tag{6}\]