## 3 Distributed Cubature Kalman Filter
### Distributed Cubature Kalman Filter
(**a**) MSE of position; (**b**) MSE of velocity. Figure 7: Distributed cubature information filter (DCIF) for the switching topology. (**a**) MSE of position; (**b**) MSE of velocity. of the position of DCKF and DCIF are almost the same. However, there is a large velocity overshoot in DCIF, which indicates that the DCKF enjoys a stronger stability property than the DCIF. The computational time of a single satellite of different filters is given in Table 3. All tests are operated on a notebook with an Intel central processing unit (i7 4510U) and MATLAB. Note that the computational time of the DCKF is longer than that of the DEKF, and the computational time of the DCIF is twice as long as the DCKF. ## 5 Discussion

The distributed Bayesian filter design has been researched, and a distributed cubature Kalman filter was proposed to deal with the time-varying topology and weak observability of sensors. It can be seen from Figures 2 and 3 that the standard EKF cannot provide good results for weak observability of a single sensor. When it comes to the distributed setting, for the nodes with weak observability, both DCKF and DEKF can obtain stable estimation, and DCKF performs better than DEKF. From Figure 7, it also can be seen that DCKF is suitable for the case of switching topology. Namely, the proposed DCKF can handle the problem of blockage of the communication channel in time. Figure 9 illustrates that the proposed DCKF enjoys stronger stability properties than DCIF for the case of weak observability of some sensors, by noting that large velocity overshoot in the DCIF. A possible reason for such satisfactory performance of DCKF is that the global posterior PDF is considered for distributed estimation rather than just the innovations, as in DCIF. Moreover, the number of consensus in the DCKF could be one, which will conserve communication resources. However, we should highlight that the DCIF in [26] can approach the centralized solution as the number of consensus tends to infinity, and the DCKF in our paper cannot. The K-L divergence for average consensus can be treated as a convex combination of the information matrices and vectors. This convex combination is well known as covariance intersection (CI) in the literature [41; 42]. It is well known that the CI scheme provides an information fusion that is robust with respect to the unknown correlations among the information sources. The stability of such \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Filters** & **DCKF, \(\boldsymbol{S=1}\)** & **DCKF, \(\boldsymbol{S=10}\)** & **DEKF, \(\boldsymbol{S=1}\)** & **DEKF, \(\boldsymbol{S=10}\)** & **DCIF, \(\boldsymbol{S=1}\)** & **DCIF, \(\boldsymbol{S=10}\)** \\ \hline Time (s) & 0.2002 & 0.2421 & 0.0428 & 0.0916 & 0.5087 & 0.5827 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average computational time of the filters. Figure 9: Comparison between DCIF and DCKF under weak observability for some sensors. (**a**) MSE of position; (**b**) MSE of velocity. a fusion strategy for distributed estimation has been proven in [19] for linear time-invariant dynamics, and the results were extended to distributed EKF in [40]. A key point in our paper is that the global cost function (19) has a form of \"sum-of-cost\", which is amenable to distributed implementations [32; 43; 44]. Information geometric optimization approaches can be used to construct the formulation, in which the natural gradient descent method is used to seek the optimal estimation. For example, in [29; 30], the natural gradient descent method was used to construct the Bayesian nonlinear filter. In the distributed setting, a global optimal estimate can be obtained under the structure of the distributed convex optimization ([43]) by natural gradient descent. In summary, the proposed DCKF has the advantages of strong stability and being more suitable for the cooperative object tracking problem compared to DEKF and DCIF. Future research issues mainly include the problems of measurement and communication delay, which will broaden the application of DCKF.