# Multi-Spacecraft Tracking and Data Association Based on Uncertainty Propagation
## 4 Numerical Simulation
### Simulation Results  
First, assume that all the four targets are visible during the 3 h tracking. The initial position and velocity estimated errors are respectively set to be 10 km per axis and 1 m/s per axis. The initial covariance is given as:  
\[\mathbf{P}_{\rm TI}(t_{0})=\left[ \begin{array}{cc}100\mathbf{I}_{3}\ \mathrm{km}^{2}&\mathbf{0}_{3 \times 3}\\ \mathbf{0}_{3\times 3}&1\mathbf{I}_{3}\ \mathrm{m}^{2}/\mathrm{s}^{2}\end{array}  \right],\quad i\in\{1,2,3,4\} \tag{38}\]  
where \(\mathbf{I}_{3}\) is the three-dimensional identified matrix and \(\mathbf{0}_{3\times 3}\) is the zero matrix. The covariance of the measurement noise is set to be:  
\[\mathbf{R}=\left[ \begin{array}{cc}10^{-4}\ \mathrm{deg}^{2}&\mathbf{0}_{3\times 3}\\ \mathbf{0}_{3\times 3}&10^{-4}\ \mathrm{deg}^{2}\end{array} \right] \tag{39}\]  
The computation follows the flow chart in Figure 2. Table 2 and Table 3 illustrate the data association results when \(t_{k}=0\) and \(t_{k}=129.5\) min, respectively. \(t_{k}=0\) denote the initial epoch and \(t_{k}=129.5\) min is the epoch when the orbits of the four targets cross with each other. In Table 2, \(\approx 0\) means that the value is smaller than \(10^{-18}\). The real observations and the targets are well matched. Take the epoch \(t_{k}=129.5\) min as an example. When the association matrix \(\mathbf{\lambda}\) is given as \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \(\mathbf{a}\)/\(\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ }}}}}}}}}}}}}}}\) & \(\mathbf{e}\) & \(\mathbf{i}\)/\({}^{\circ}\) & \(\mathbf{\Omega}\)/\({}^{\circ}\) & \(\mathbf{\omega}\)/\({}^{\circ}\) & \(\mathbf{n}\)/\({}^{\circ}\) \\ \hline Observer & S\({}_{\rm O}\) & 8362 & 0.1007 & 53.60 & 100.60 & 33.00 & 21.51 \\ \hline \multirow{4}{*}{Target} & S\({}_{\rm T1}\) & 9381 & 0.1498 & 43.60 & 103.1 & 42.00 & 22.02 \\  & S\({}_{\rm T1}\) & 9406 & 0.1398 & 48.60 & 103.6 & 42.00 & 22.02 \\ \cline{1-1}  & S\({}_{\rm T2}\) & 9456 & 0.1198 & 58.60 & 104.6 & 42.00 & 22.02 \\ \cline{1-1}  & S\({}_{\rm T2}\) & 9481 & 0.1098 & 63.60 & 105.1 & 42.00 & 22.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Nominal orbit elements of the target and the observer. Figure 4: The nominal angular measurements. (**a**) time history of the two angular measurements; (**b**) projection of the two angular measurements. \[\lambda=\left[ \begin{array}{ccccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right], \tag{40}\]  
the objective function has the largest value, i.e., \(\sum_{i=1}^{N}\sum_{j=1}^{M}\lambda_{ij}\ln(p_{ij})=-0.5839\). All the measurements are associated correctly. The estimated errors are shown in Figure 5, where \(\mathbf{e}\) and \(\dot{\mathbf{e}}\) represent the position and velocity estimated error:  
\[\mathbf{e}=\hat{\mathbf{r}}_{Ti}-\mathbf{r}_{Ti} \tag{41}\]  
\[\dot{\mathbf{e}}=\hat{\mathbf{v}}_{Ti}-\mathbf{v}_{Ti} \tag{42}\]  
It can be seen from Figure 5 that the estimated errors converge at the epoch around 1.5 h. The converged position and velocity estimated errors are around 0.1 km and 0.1 m/s. To further investigate the performance of the proposed method, 300 Monte Carlo runs are performed. For each target, the initial estimation \(\hat{\mathbf{x}}_{Ti}(t_{0})\) is randomly generated from