# 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  
The observability results of case II are illustrated in Figure 5 and Table 3. As indicated in Figure 4, the system of Equation (39) is observable after approximately 94 measurements (from epoch \(t_{0}\) to epoch \(t_{93}\)). Compared with case I, the system corresponding to case II is much more difficult to observe. Note that as containing the state vector of target spacecraft \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Epoch \(t_{k}\)** & **Rank(N)** & **Observability** & **1/cond(N)** & **In[1/cond(N) ]** \\ \hline
0 & 3 & False & - & - \\
1 & 4 & True & 0.0005 & \(-7.4740\) \\
2 & 4 & True & 0.0009 & \(-6.9835\) \\
3 & 4 & True & 0.0012 & \(-6.6692\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Observability test results of case I (nonperturbation). Figure 4: Attitude determination errors of case I (nonperturbation) and the corresponding larger plot. Figure 3: Observability simulation of case I (nonperturbation). \(S_{2}\), the dimension of the state to be estimated in case II is higher than that of case I, which implies more effort in measurements. The estimation results are illustrated in Figures 6 and 7. As shown in Figure 6, the attitude converges much faster than the orbit state, as the previous item converges within only a few minutes, while it takes approximately five hours for the orbit state of spacecraft \(S_{2}\) to converge. Note that measurement model (17) is much more sensitive to the attitude of observer \(S_{1}\) than the orbits of the observer and the target. Therefore, the attitude converges before the orbit state converges. Note that the observation interval has no effect on the observability, so the influence of shadow or light conditions are not considered in this paper. In most cases, the long-term observation in Figure 6 is impossible to realize due to eclipses, but this simplification is reasonable considering the research content of this paper. \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 & - & - \\... &... &... &... &... \\
92 & 9 & False & - & - \\
93 & 10 & True & \(5.1763\times 10^{-14}\) & \(-30.5921\) \\
94 & 10 & True & \(5.3998\times 10^{-14}\) & \(-30.5498\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Observability test results of case II (nonperturbation). Figure 5: Observability simulation of case II (nonperturbation). 2.3 Case III: Both the Orbits of \(S_{1}\) and \(S_{2}\) Are Unknown, and the Attitude of \(S_{1}\) Is Unchanged  
In this case, three kinds of dynamics (state model as Equation (3) and Equation (4), respectively) are considered, as illustrated in Figures 8 and 9. It can be concluded that for dynamics with nonperturbation and \(J_{2}\) perturbation, the systems are unobservable (as shown in Figure 8). For case III, the state vector with 16 variables is estimated. However, when considering two-body dynamics, the observable states are 13-dimensional, which means that only 13 variables (or variable combinations) of the 16-dimensional state vector can be observed. In addition, when taking the \(J_{2}\) perturbation into consideration, one more state variable (or variable combination) could be observed, suggesting that the system is still unobserved. Figure 6: Attitude determination errors of case II (nonperturbation) and the corresponding larger plot. Figure 7: Orbit determination errors of spacecraft \(S_{2}\) in case II (nonperturbation). Fortunately, as depicted in Figure 6 and Table 4, the system is observable when \(J_{2}\), \(J_{3}\) and \(J_{4}\) perturbations are considered. In this situation, a total of 16 variables are observed after 490 angle-only measurements. It should be noted that strictly two-body dynamics, or dynamics with a particular perturbation, are, of course, unlikely.