# Test Method for Single Satellite's Inter-Satellite Link Pointing and Tracking via Ground Station
## 3 Inter-Satellite Links Pointing and Tracking in-Orbit Test Method
### Virtual Target Satellite Based on Ground Stations  
In the present paper, the inter-satellite link pointing and tracking performance is tested using a virtual target satellite. A practical way to generate a virtual target satellite is by simulating the signal-receiving process and forming a virtual orbit around two ground stations. The geometry of the inter-satellite link via the virtual satellite is shown in Figure 4.  
Figure 3: Test method for inter-satellite link pointing and tracking. In Figure 4, the ground stations are defined as A and B to simulate the virtual target satellite orbit. Taking ground station A as an example, the position vector of ground station A \(\mathbf{r}_{dA}\) in the Earth-fixed coordinate system is given as follows:  
\[\mathbf{r}_{dA}= \begin{bmatrix}(H+h)\cos\epsilon\cos\lambda\\ (H+h)\cos\epsilon\sin\lambda\\ \big{[}\big{(}1-e_{E}^{2}\big{)}H+h\big{]}\sin\epsilon\end{bmatrix}  \tag{9}\]  
where \(\lambda\)\(\epsilon\), and \(h\) denote the longitude, latitude, and altitude of the ground station, respectively, and \(e_{E}=0.08182\) is the oblateness of the Earth. \(H\) is denoted as follows:  
\[H=\frac{R_{e}}{\sqrt{1-e_{E}^{2}\sin^{2}\epsilon}} \tag{10}\]  
where \(R_{e}\) is the radius of the Earth, and \(R_{e}=6378.145\) km. Then, the position vector of the ground station \(\mathbf{r}_{d}(t_{1})\) in the J2000 geocentric inertial frame at time \(t_{1}\) is obtained as follows:  
\[\mathbf{r}_{d}(t_{1})=A_{ie}(t_{1})\mathbf{r}_{dAs} \tag{11}\]  
where \(\mathbf{A}_{ie}(t_{1})\) defines the transformation from the Earth-fixed coordinate frame to the J2000 geocentric frame. Meanwhile, the position vector of the local satellite in the J2000 geocentric frame at time \(t_{1}\) is shown as follows:  
\[\mathbf{r}_{s}(t_{1})= \begin{bmatrix}r\big{(}\cos u\cos\Omega-\sin u\cos i\sin \Omega\big{)}\\ r\big{(}\cos u\sin\Omega+\sin u\cos i\cos\Omega\big{)}\\ r\sin u\sin i\end{bmatrix}  \tag{12}\]  
where \(a\), \(e\), \(i\), \(\Omega\), \(\omega\), and \(f\) represent the semi-major axis, eccentricity, inclination, right ascension of ascending node, argument of perigee, and true anomaly of the local satellite, respectively; \(u\) represents the argument of latitude, \(u=\omega+f\); and \(\mathbf{r}\) represents the geocentric distance of the local satellite, \(\mathbf{r}=\frac{p}{1+e\cos f}\), \(p=a\big{(}1-e^{2}\big{)}\). Figure 4: Schematic diagram of the inter-satellite link between the two satellites in-orbit. The relative position \(\mathbf{r}_{dAs}\) from the ground station A to the local satellite in the J2000 geocentric frame at time \(t_{1}\) can be calculated as follows:  
\[\mathbf{r}_{dAs}(t_{1})=\mathbf{r}_{s}(t_{1})-\mathbf{r}_{dAs}(t_{1}) \tag{13}\]  
Then, as shown in Figure 4, in order to simulate the position of the virtual target satellite using a ground station, the local satellite should be simultaneously linked to the ground station and the virtual satellite. In this situation, the local satellite, the virtual target satellite, and the ground station should be triple-collinear as follows:  
\[k^{2}\cdot{\mathbf{r}_{dAs}}^{2}(t_{1})+{\mathbf{r}_{dAs}}^{2}(t_{1})+2k\cdot\mathbf{r}_{ dAs}(t_{1})\mathbf{r}_{dAs}(t_{1})={\mathbf{r}_{A}}^{2}(t_{1}) \tag{14}\]  
where \(h_{o}\) denotes the orbital altitude of the virtual target satellite; \(\mathbf{r}_{A}(t_{1})\) denotes the position vector of the virtual satellite in the J2000 geocentric frame at time \(t_{1}\); and \(k>0\). In addition, it should be emphasized that in practical missions, the beam of the local satellite's antenna is a cone with a small angle. In this situation, the ground station can still receive the signal from the local satellite for an extended period. To solve Equation (14), the parameters are defined as follows:  
\[\left\{ \begin{array}{l}a_{1}={\mathbf{r}_{dAs}}^{2}(t_{1})\\ a_{2}=2k\cdot{\mathbf{r}_{dAs}}(t_{1}){\mathbf{r}_{dAs}}(t_{1})\\ a_{3}={\mathbf{r}_{dAs}}^{2}(t_{1})-{\mathbf{r}_{A}}^{2}(t_{1})\end{array} \right.