# Angle-Only Cooperative Orbit Determination Considering Attitude Uncertainty
## 3 Attitude and Orbit Determination Method with Angle-Only Measurements
### Results and Discussion
#### 4.2.2 Case II: The Orbit of \(S_{1}\) Is Known, and the Attitude of \(S_{1}\) Is Unchanged
This is because aspheric perturbation of the celestial body (i.e., Earth in this paper) contains higher order terms. Moreover, the solar pressure, atmospheric drag and gravitational perturbation of the third body could also influence the orbiting spacecraft. However, the significance of the observability analysis is to state (as might be expected) that when the dynamics are very close to the two-body dynamics (e.g., when an orbit is high, for example, high-orbit GPS satellites), it is difficult to estimate the orbit because the influence of aspheric perturbation is weak. Figure 8: Observability simulation of case III (nonperturbation and with \(J_{2}\) perturbation). Figure 9: Observability simulation of case III (with \(J_{2}\), \(J_{3}\), and \(J_{4}\) perturbations). The simulation results of the subcase with nonperturbation and \(J_{2}\) perturbation are illustrated in Figure 10. It was shown that estimation quality is poor and appears to be diverging (the \(v_{z}\) for \(S_{1}\) and \(v_{x}\) for \(S_{2}\) are clearly divergent with time), meaning that the system is unobservable under the given dynamics and initial conditions, which validates the observability analysis listed in Table 4. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Epoch \(t_{k}\)** & **Rank(N)** & **Observability** & **1/cond(N)** & **ln[1/cond(N) ]** \\ \hline
0 & 3 & False & - & - \\
1 & 5 & False & - & - \\
2 & 7 & False & - & - \\
3 & 9 & False & - & - \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\
488 & 15 & False & - & - \\
489 & 16 & True & \(2.3821\times 10^{-13}\) & \(-29.0656\) \\
490 & 16 & True & \(2.4174\times 10^{-13}\) & \(-29.0509\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Observability test results of case III (with \(J_{2}\), \(J_{3}\), and \(J_{4}\) perturbations). Figure 10: Orbit determination errors of case III (nonperturbation and with \(J_{2}\) perturbation). (**a**) Orbit estimation errors of spacecraft \(S_{1}\). (**b**) Orbit estimation errors of spacecraft \(S_{2}\). Figures 11 and 12 depict the autonomous attitude and orbit determination results of the two spacecraft considering \(J_{2}\), \(J_{3}\), and \(J_{4}\) perturbations. Although exhibiting an obvious oscillation, the system still succeeds in converging. However, compared with the results in case II, the convergence is much weaker, meaning that this form is not stable enough. It is noted that although we have proven that case II under dynamics (5) is observable, this does not contradict the estimation results obtained here but implies that the system is higher-order locally weakly observable [24]. 2.4 Case IV: Both the Orbits of \(S_{1}\) and \(S_{2}\) Are Unknown, and the Angular Velocity of \(S_{1}\) Is Known  
In case IV, the results are demonstrated in Figures 13 and 14 and Table 5. As shown in Figure 13 and Table 5, when applying a known attitude maneuver to observer spacecraft \(S_{1}\), the system is completely observable even under the simplest two-body dynamics. As indicated in Table 5, only 344 measurements are needed to observe the 16 variables, which is less than that of case III (situation considering \(J_{2}\), \(J_{3}\), and \(J_{4}\) perturbations in Table 4). Furthermore, at time epoch \(t_{489}\), which is difficult to observe in case III, the CN of the system is larger than that of case IV (for case III, as shown in Table 4, \(\frac{1}{\text{cond}(\mathbf{N})}=2.3821\times 10^{-13}\), while for case IV, listed in Table 5, \(\frac{1}{\text{cond}(\mathbf{N})}=3.4291\times 10^{-12}\)), indicating that compared to the perturbation acceleration, a suitable attitude maneuver is more likely to attach obvious improvement to the observability of the two-spacecraft system. 2.5 Case V: Both the Orbits of \(S_{1}\) and \(S_{2}\) Are Unknown, and the Angular Velocity of \(S_{1}\) Is Unchanged  
The test results of case V are illustrated in Figure 17 and Table 6. When not considering any perturbation, the system is observable at epoch \(t_{1028}\).