## 6 Discussion
### Two Improved Algorithms
#### 6.1.2 The Addition of Inter-Satellite Angle Data  
The Osiris-Rex mission uses the angle-measurement data from the spacecraft to the particles near Bennu's surface to recover the asteroid's gravity to high orders and degrees [10]. Inspired by this, we added the inter-satellite angle measurement data into our study as a supplementary to the inter-satellite range data to improve the accuracy. For inter-satellite angle data, \(\widetilde{H_{x}}=[\frac{\partial\delta}{\partial X},\frac{\partial\alpha}{ \partial X}]\) will be  
\[\frac{\partial\delta}{\partial X}=[\delta_{x},\delta_{y},-\delta_{Z},0,0,0,- \delta_{x},-\delta_{y},\delta_{Z},0,0,0] \tag{20}\]  
\[\frac{\partial\alpha}{\partial X}=[\frac{y}{x^{2}+y^{2}},-\frac{x}{x^{2}+y^{2 }},0,0,0,0,-\frac{y}{x^{2}+y^{2}},\frac{x}{x^{2}+y^{2}},0,0,0,0] \tag{21}\]  
where \(\delta_{x}\), \(\delta_{y}\) and \(\delta_{z}\) can be calculated using Equation (22). \[\delta_{x}=\frac{xz}{\rho^{3}\sqrt{1-(z/\rho)^{2}}}\qquad\delta_{y}=\frac{yz}{ \rho^{3}\sqrt{1-(z/\rho)^{2}}}\qquad\delta_{z}=\frac{1}{\sqrt{1-(z/\rho)^{2}} }(\frac{1}{r}-\frac{z^{2}}{\rho^{3}}) \tag{22}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Initial Estimate** & **Test 4** \\ \hline \(C_{20}\) & \(-5\times 10^{-2}\) & \(-5.0871\times 10^{-2}\) \\ \hline \(C_{22}\) & \(9\times 10^{-2}\) & \(8.2667\times 10^{-2}\) \\ \hline
**Initial position error of** & & \\
**Chief Satellite** & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Recovery result of test 5. Figure 8: Residual in the observation data (**left**), norm of the position error of the determined orbit from the true one (**middle**) and orbits of spacecraft (**right**) (test 5 with 8 links). Firstly, our study indicates that the addition of the inter-satellite angle measurement data can significantly improve the accuracy of the orbit determination. As test 6 (Figure 9) shows, all of the initial conditions are the same as those of test 2 in Section 3.2. Compared with test 2, the accuracy of test 6 is much better. The range residuals are less than 0.1 m and the angle residuals are less than 10 arc-seconds. Both of them reach the accuracy level of observation data, just like test 1. The position errors of test 6 are less than 0.04 m and the position errors of test 2 are about 14 m. Thus, it is clear that the addition of angle data can provide more constraints to improve the accuracy. When both initial orbital elements and gravity-field coefficients are erroneous, the addition of angle measurement data also works. In test 7, all of the initial conditions are the same as test 4 in Section 5 and the result is shown in Figure 10 and Table 7.  
Similarly to test 5, the accuracy of result has improved a lot. The range residuals are about one meter and the angle residuals are about 40 arc-seconds. Position errors are only 3.5 m and the initial errors are less than 3 m. The value of gravity-field coefficients is close \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Initial Estimate** & **Test 4** \\ \hline \(C_{20}\) & \(-5\times 10^{-2}\) & \(-5.4093\times 10^{-2}\) \\ \hline \(C_{22}\) & \(9\times 10^{-2}\) & \(8.2626\times 10^{-2}\) \\ \hline
**Initial position error of** & \(50\) m & \(2.89\) m \\
**Satellite 1** & \(50\) m & \(0.08\) m \\ \hline
**Initial position error of** & \(50\) m & \(0.08\) m \\
**Satellite 2** & \(50\) m & \(0.08\) m \\ \hline
**RMS** & \(39.9917\) & \(5.5752\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Recovery result of test 7. Figure 10: Residual in the range data (**left**), residual in the angle data (**middle**) and norm of the position error of the determined orbit from the true one (**right**) (test 7). Figure 9: Residual in the range data (**left**), residual in the angle data (**middle**) and norm of the position error of the determined orbit from the true one (**right**) (test 6). to the real value. Their errors are 3% and 0.1%, respectively. Compared with test 4, this result is excellent. The improvement in accuracy is significant.