# Distributed Localization of Non-Cooperative Targets in Non-Coplanar Rendezvous Processes
## 2 Distributed Localization Method
### System Observability  
Since the accuracy of state estimation is closely related to the system's observability [16], it is necessary to analyze the observability to assess the performance of a multi-line-of-sight collaborative positioning system.  
The observability matrix [17] can be constructed as follows:  
\[\mathbf{O}_{k}=\left[\begin{array}{c}H_{k}\\ H_{k+1}\mathbf{\Phi}_{k}\\ H_{k+2}\mathbf{\Phi}_{k+1}\mathbf{\Phi}_{k}\\ :\\ H_{k+n-1}\mathbf{\Phi}_{k+n-2}\ldots\mathbf{\Phi}_{k}\end{array}\right],n\geq 2 \tag{10}\]At any given time, the rank of the observability matrix \(\mathbf{O}_{k}\) is 6\(n\); to make it greater than the dimension of the state vector \(\mathbf{X}_{k}\), set \(n\geq 2\).  
According to the literature [18], this paper uses the reciprocal of the condition number of the observability matrix as the standard parameter for evaluating the system's observability. The observability degree is defined as follows [19]:  
\[D(\mathbf{O})=\frac{1}{\textit{cond}(\mathbf{O})},\textit{cond}(\mathbf{O})=\frac{\sigma_ {\text{max}}(\mathbf{O})}{\sigma_{\text{min}}(\mathbf{O})} \tag{11}\]  
where \(\sigma_{\text{max}}(\mathbf{O})\) and \(\sigma_{\text{min}}(\mathbf{O})\) represent the maximum and minimum singular values of the observability matrix \(\mathbf{O}\), respectively, and \(0\leq D(\mathbf{O})<1\). If \(D(\mathbf{O})=0\), this indicates that the system is unobservable; if \(D(\mathbf{O})>0\), this indicates that the system is observable, and the greater the observability degree, the higher the accuracy of the system's state estimation.  
A group of orbit parameters is used to verify the system observability, which is shown in Table 1, below. And the simulation results are shown in Figure 2.  
The variation in observability during the rendezvous process for double-LOS and triple-LOS collaborative positioning systems was compared, as shown in Figure 2a. The observation level is represented by the reciprocal of the condition number of the observability matrix mentioned in Equation (11). The value reveals that triple-LOS positioning significantly outperforms dual LOS throughout the entire rendezvous process. Figure 2b shows the change in the distance between the tracking satellites and the target in this rendezvous process; at about 1820 s, tracking satellites 1 and 2 intercepted the target. Figure 2c demonstrates that, as the rendezvous progressed, the angle formed between the two chasers and the target came closer to 180 degrees, which means that double LOS deteriorated into single LOS and therefore caused the observability degree to gradually decrease. When the LOS of the two chasers almost recombined into single LOS, the observation degree dropped to the lowest level. At the interception point, as satellite 3 became too distant from the target, the observability level was mainly determined by the nearby satellites 1 and 2; therefore, the observability of triple-LOS and double-LOS systems tends to be consistent.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  &
\begin{tabular}{c} **Orbital** \\ **Altitude** \\ \end{tabular} & **Eccentricity** & **Inclination** & **Asceding** & **Argument** & **True** \\  & **Altitude** & **Eccentricity** & **Inclination** & **Node** & **of Perigee** & **Anomaly** \\ \hline Satellite 1 & 1103.05 km & 0 & 58\({}^{\circ}\) & 0\({}^{\circ}\) & 0\({}^{\circ}\) & 102.70\({}^{\circ}\) \\ Satellite 2 & 1088.36 km & 0 & 58\({}^{\circ}\) & 0\({}^{\circ}\) & 0\({}^{\circ}\) & 105.24\({}^{\circ}\) \\ Satellite 3 & 1100.00 km & 0 & 58\({}^{\circ}\) & 0\({}^{\circ}\) & 0\({}^{\circ}\) & 100.00\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Orbit parameters of satellites.  
Figure 2: System observability. (**a**) System observability. (**b**) Distance between tracking satellites and target. (**c**) Observation angle between satellites 1 and 2 and target.  
Nevertheless, observability remained at a high level, indicating that the system maintained high observability throughout the whole process.