# Multi-Spacecraft Tracking and Data Association Based on Uncertainty Propagation
## 4 Numerical Simulation
### Simulation Results
\begin{table}
\begin{tabular}{c c c c c} \hline  & & \(\mathbf{z_{1}}\) & \(\mathbf{z_{2}}\) & \(\mathbf{z_{3}}\) & \(\mathbf{z_{4}}\) \\ \cline{2-5}  & & \(\mathbf{[-0.5569,43.1841]}\) & \(\mathbf{[-16.6373,49.4247]}\) & \(\mathbf{[-51.2908,76.0587]}\) & \(\mathbf{[-52.8157,-70.0077]}\) \\ \hline \(\hat{\mathbf{z}}_{T1}\) & \(\mathbf{[-0.8393,43.1550]}\) & \(0.6922\) & \(\approx 0\) & \(\approx 0\) & \(\approx 0\) \\ \(\hat{\mathbf{z}}_{T2}\) & \(\mathbf{[-17.0700,49.4502]}\) & \(\approx 0\) & \(0.4117\) & \(\approx 0\) & \(\approx 0\) \\ \(\hat{\mathbf{z}}_{T3}\) & \(\mathbf{[-51.7809,76.4379]}\) & \(\approx 0\) & \(\approx 0\) & \(0.1785\) & \(\approx 0\) \\ \(\hat{\mathbf{z}}_{T4}\) & \(\mathbf{[-53.1513,-69.3724]}\) & \(\approx 0\) & \(\approx 0\) & \(\approx 0\) & \(0.1933\) \\ \hline \end{tabular}
\end{table} 
Table 2: Data association results when \(t_{k}=0\). Figure 5: Time history of the estimated errors. \begin{table}
\begin{tabular}{c c c c c c} \hline  & & \(\mathbf{z_{1}}\) & \(\mathbf{z_{2}}\) & \(\mathbf{z_{3}}\) & \(\mathbf{z_{4}}\) \\ \cline{2-5}  & & \(\mathbf{[42.3691,61.2692]}\) & \(\mathbf{[42.2580,63.1710]}\) & \(\mathbf{[41.5619,66.5737]}\) & \(\mathbf{[44.3305,64.0263]}\) \\ \hline \(\hat{\mathbf{z}}_{T1}\) & \(\mathbf{[42.3522,61.2866]}\) & \(0.8767\) & \(0.1544\) & \(7.7160\times 10^{-7}\) & \(0.0032\) \\ \(\hat{\mathbf{z}}_{T2}\) & \(\mathbf{[42.2656,63.1609]}\) & \(0.1499\) & \(0.8682\) & \(0.0022\) & \(0.0721\) \\ \(\hat{\mathbf{z}}_{T3}\) & \(\mathbf{[41.5623,66.5585]}\) & \(6.3340\times 10^{-7}\) & \(0.0023\) & \(0.8544\) & \(0.0008\) \\ \(\hat{\mathbf{z}}_{T4}\) & \(\mathbf{[44.3387,64.0293]}\) & \(0.0029\) & \(0.0709\) & \(0.0008\) & \(0.8576\) \\ \hline \end{tabular}
\end{table} 
Table 3: Data association results when \(t_{k}=129.5\) min. the normal distribution \(N(\hat{x}_{Tl}(t_{0});\mathbf{x}_{Ti}(t_{0}),\mathbf{P}_{Tl}(t_{0}))\), in which \(\mathbf{x}_{Ti}(t_{0})\) is the true initial value listed in Table 1 and \(\mathbf{P}_{Ti}(t_{0})\) is the initial covariance. All the 324,000 data associations (each Monte Carlo run contains 1080 data associations), only two measurements are incorrectly associated. The data association accuracy is 99.99%. The root mean square error (RMSE) is taken as the metrics to evaluate the performance of the proposed method. The position and velocity RMSEs are expressed as follows:  
\[\text{RMSE}_{pos}(t_{k})=\sqrt{\frac{1}{100}\sum_{p=1}^{100}\left\|\hat{\mathbf{r} }_{Tl,p}(t_{k})-\mathbf{r}_{Tl,p}(t_{k})\right\|} \tag{43}\]  
\[\text{RMSE}_{vel}(t_{k})=\sqrt{\frac{1}{100}\sum_{p=1}^{100}\left\|\hat{\mathbf{v} }_{Tl,p}(t_{k})-\mathbf{v}_{Tl,p}(t_{k})\right\|} \tag{44}\]  
where \(\left\{\hat{\mathbf{r}}_{Tl,p}(t_{k}),\hat{\mathbf{v}}_{Tl,p}(t_{k})\right\}\) and \(\left\{\mathbf{r}_{Tl,p}(t_{k}),\mathbf{v}_{Tl,p}(t_{k})\right\}\) are the estimations and true values at the \(p\)-th Monte Carlo run. The RMSEs of the 300 Monte Carlo runs are shown in Figure 6, and the mean RMSE is listed in Table 4. As shown in Table 4, the position and velocity RMSEs are smaller than 0.71 km and 0.46 m/s. All the four targets are well tracked. The second target (S\({}_{T2}\)) has the best estimation accuracy, with the position and velocity RMSEs being 0.6108 km and 0.4197 m/s. The third target (S\({}_{T3}\)) has the poorest estimation precision, with the position and velocity RMSEs being 0.7004 km and 0.4594 m/s. The differences in estimation accuracy are related to the observation configuration, which is not discussed in this paper. The readers with the interest on this issue can refer to [4, 44, 45, 46]. The simulations are performed using MATLAB R2018b on a computer with a 3.6 GHz AMD Ryzen processer and a 16 GB RAM. The mean one-step run time of the proposed method is 25.2532 ms, which means that the proposed method is efficient and can be used for real-time tracking. Then, the case considering invisible targets are investigated. Assume that the first target (S\({}_{T1}\)) is invisible from the epoch \(t=1\) h to the epoch \(t=1.5\) h and the second target