## 3 Distributed Cubature Kalman Filter
### Distributed Cubature Kalman Filter
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Six Orbital Parameter** & \(a_{h}\) **(km)** & \(e\) & \(u\) **(Deg)** & \(\gamma\) **(Deg)** & \(\Gamma\) **(Deg)** & \(m_{h}\) **(Deg)** \\ \hline Object & 8667.13 & 0 & 73.9116 & 14.108 & 0 & 52.632 \\ Satellite 1 & 9067.13 & 0 & 73.9116 & 128.495 & 0 & 52.942 \\ Satellite 2 & 8067.1 & 0 & 73.9116 & 91.0768 & 0 & 18.88 \\ Satellite 3 & 8667.13 & 0 & 73.9116 & 103.658 & 0 & 44.818 \\ Satellite 4 & 8467.13 & 0 & 73.9116 & 116.24 & 0 & 70.756 \\ Satellite 5 & 8267.13 & 0 & 73.9116 & 88.8216 & 0 & 96.694 \\ Satellite 6 & 9067.13 & 0 & 73.9116 & 88.495 & 0 & 112.942 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Six orbital parameters of the observation satellites and object. Figure 4: Topology of the network. illustrates that the estimations of different satellite almost reach a consensus, which can increase the robustness of tracking. More importantly, the estimate of each sensor is stable and converges even if the number of consensus is \(S=1\), which can reduce the communication rate compared with DCIF in [26]. Simulation Case 2: In this case, we test the performance of the proposed DCKF for the switching topology, i.e., the communication is time-varying. To be specific, the communication topology is switching among the given topologies as shown in Figure 6. The setting of initialization and noise variances is the same as Simulation Case 1. As shown in Figure 7, the filtering precision and stability of the proposed DCKF for different satellites are also demonstrated. It should be noted that, for the switching case, the estimate of each node only needs the information of its neighbors. However, the algorithm in [16] needs global information \(\Delta_{\max}\) (\(\Delta_{\max}=3,2,3\) in Figure 6), where \(\Delta_{\max}=\max_{i}d_{i}\), \(d_{i}\) is the degree of node \(i\), which is not easy to obtain in time. Simulation Case 3: In this case, we compare the performance of DCKF with the distributed extend Kalman filter (DEKF) in [40] and the distributed cubature information filter (DCIF) in [26]. The discrete model (52) is adopted for the time update in the DEKF algorithm. We assume that each satellite can only obtain either azimuth \(\alpha\) or elevation \(\beta\) for the tested filters. To be specific, Satellites 1, 3 and 5 can only obtain the azimuth \(\alpha\), and Satellites 2, 4 and 6 can only obtain the elevation \(\beta\) of the object. In this case, we assume the measurement noises of satellite \(i\) to be \(v_{i,k}\)\(\
eg\mathbb{N}(0,10^{-4})\). Figure 5: MSE curves of distributed cubature Kalman filter (DCKF) by different satellites under a fixed topology. (**a**) MSE of position; (**b**) MSE of velocity. Figure 6: Switching topology of the network. Figure 8 shows the comparison between DEKF and DCKF for different numbers of consensus. It can be seen that, for the cases \(S=1\), \(S=3\) and \(S=10\), the DCKF performs more accurately than the DEKF, since the DEKF suffers form linearization errors due to numerically-linearizing the nonlinear dynamics and measurement map, which will enlarge the estimate errors. As \(S\) increases, the DEKF is more accurate. This is due to the fact that the local information will spread to the whole network as \(S\rightarrow\infty\). Despite the weak observability, the proposed DCKF algorithm provides reasonable performance even for \(S=1\), which indicates that the proposed DCKF is more robust and suitable for real-time applications with weak observability of some sensors than the DEKF. Figure 9 gives the comparison between DCIF and DCKF with different consensus numbers \(S\) under the weak observability condition. We also assume that Satellites 1, 3 and 5 can only obtain the azimuth \(\alpha\) and Satellites 2, 4 and 6 can only obtain the elevation \(\beta\) of the object. It can be seen that, under the weak observability condition, all test filters can successfully track, and the estimation errors

Figure 8: Comparison between DEKF and DCKF under weak observability for some sensors.