## 3 Distributed Cubature Kalman Filter
### Distributed Cubature Kalman Filter
In [26], the authors proposed DCIF for cooperative space object tracking. For comparison, we briefly summarize the main steps of DCIF in Table 1, where the prediction step is omitted due to it being the same as the prediction step of DCKF. In DCIF, \(\tilde{z}_{i,k}\) and \(P_{i,xz,k}\) can be obtained by (39) and (43), and \(0<\epsilon<\frac{1}{\Delta_{max}}\), \(\Delta_{max}=\max_{i}\{d_{i}\}\). \begin{table}
\begin{tabular}{l} \hline \hline
**Local estimation:** \\ \(\tilde{y}_{i,k}=\tilde{P}_{i,k}^{-1}\tilde{x}_{i,k},\tilde{Y}_{i,k}=\tilde{P}_ {i,k}^{-1}\) \\ \(i_{i,k}\approx\tilde{P}_{i,k}^{-1}P_{i,xz,k}R_{i,k}^{-1}\left[(z_{i,k}-\tilde{ z}_{i,k})+P_{i,xz,k}^{\top}(\tilde{P}_{i,k}^{-1})^{\top}\tilde{x}_{i,k}\right]\), \\ \(I_{i,k}\approx\tilde{P}_{i,k}^{-1}P_{i,xz,k}R_{i,k}^{-1}P_{i,xz,k}^{-1}(P_{i,k} ^{-1})^{\top}\), \\ \(y_{i,k}=\frac{\tilde{y}_{i,k}}{N}+i_{j,k}\), \(Y_{i,k}=\frac{\tilde{y}_{i,k}}{N}+l_{j,k}\); \\
**Consensus:** \\
**for**\(s=1,\ldots,S\) **do** \\ \(y_{i,k}^{s+1}=y_{i,k}^{s}-\epsilon\sum_{j\in\mathcal{N}_{i}}(y_{i,k}^{s}-y_{j,k }^{s})\), \\ \(\gamma_{i,k}^{s+1}=Y_{i,k}^{s}-\epsilon\sum_{j\in\mathcal{N}_{i}}(\gamma_{i,k} ^{s}-\gamma_{j,k}^{s})\), \\
**end for** \\
**Estimate of sensor \(i\)**: \\ \(P_{i,k}=(N\gamma_{i,k}^{s})^{-1},\ \ \hat{x}_{i,k}=(Y_{i,k})^{-1}(y_{i,k}^{s})^{-1}\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Local estimation and consensus steps of DCIF. ```
0: At time \(k\), a prior information \(P_{i,k-1}=S_{i,k-1}S_{i,k-1}^{\top}\) and \(\hat{x}_{i,k-1}\); * Prediction \[X_{i,t,k-1}= S_{i,k-1}\xi_{t}+\hat{x}_{i,k-1},\ \ X_{i,t,k}^{*}=f(X_{i,t,k-1}),\ \ \hat{x}_{i,k}=\frac{1}{m}\sum_{t=1}^{m}X_{i,t,k}^{*},\] \[\hat{P}_{i,k}= \frac{1}{m}\sum_{t=1}^{m}X_{i,t,k}^{*}X_{i,t,k}^{*\top}-\hat{x}_{i,k}\xi_{i,k}^{\top}+Q_{k-1}.\]
* Local estimation
* Measurement prediction \[\hat{z}_{i,k}=\frac{1}{m}\sum_{t=1}^{m}\hat{Z}_{i,t,k},\ \ \hat{X}_{i,t,k}=\tilde{S}_{i,k}\xi_{t}+\hat{x}_{i,k},\ \ \hat{Z}_{i,t,k}=h_{i}(\tilde{X}_{i,t,k}).\]
* Local estimate and estimate error covariance \[P_{i,zz,k}= N\left[\frac{1}{m}\sum_{t=1}^{m}\hat{Z}_{i,t,k}\hat{Z}_{i,t,k}^{ \top}-\hat{z}_{i,k}\hat{z}_{i,k}^{\top}\right]+R_{i,k},\] \[P_{i,xz,k}= N\left[\frac{1}{m}\sum_{t=1}^{m}\hat{X}_{i,t,k}\hat{Z}_{i,t,k}^{ \top}-\hat{x}_{i,k}\hat{z}_{i,k}^{\top}\right],\] \[\hat{x}_{i,k}= \hat{x}_{i,k}+P_{i,xz,k}P_{i,xz,k}^{-1}(z_{i,k}-\hat{z}_{i,k}),\] \[\hat{P}_{i,k}= \hat{P}_{k}+P_{i,k,xz}P_{i,k,xz}^{-1}P_{i,k,xz}^{\top}.\]

* for\(s=1\) to \(S\)do \[(\tilde{P}_{i}^{s+1})^{-1}\hat{x}_{i}^{s+1}=\sum_{j\in\hat{X}_{i}} a_{ijk,k}(\tilde{P}_{j}^{s})^{-1}\hat{x}_{j}^{s},\ \ \ s=1,2,\ldots\] \[(\tilde{P}_{i}^{s+1})^{-1}=\sum_{j\in\hat{X}_{i}}a_{ijk,k}(\tilde{P }_{i}^{s})^{-1},\ \ \ s=1,2,\ldots\]
* Compute the estimate \(\hat{x}_{i,k}\) and covariance matrix \(P_{i,k}\), \[X_{i,t,k-1}= S_{i,k-1}\xi_{t}+\hat{x}_{i,k-1},\ \ X_{i,t,k}^{*}=f(X_{i,t,k-1}),\ \ \hat{x}_{i,k}=\frac{1}{m}\sum_{t=1}^{m}X_{i,t,k}^{*},\]0

**Algorithm 1** DCKF at node \(i\) at time \(k\)

**Remark 5**. : _The algorithm in [26] can approach the centralized solution, which is achieved by performing a consensus on information pairs, i.e., \(\tilde{H}_{i,k}^{\top}R_{i,k}^{-1}(\tilde{z}_{i,k}+\tilde{H}_{i,k}\tilde{x}_{ i,k})\) and \(\tilde{H}_{i,k}^{\top}R_{i,k}^{-1}\tilde{H}_{i,k}\), where \(\tilde{H}_{i,k}\approx\tilde{P}_{i,k}^{-1}P_{i,xz,k}\) is the pseudo measurement matrix and \(\tilde{z}_{i,k}=z_{i,k}-\tilde{z}_{i,k}\). The main limitation of [26] is that it needs a sufficiently large number of consensus steps at each time step, so that the local information pairs can spread to the whole network. It should be noted that we do not limit the range of \(S\). In [19], the convergence properties were proven for such a fusion principle, even \(S=1\) in the linear dynamics case, which used the fact that the whole posterior PDFs are combined rather than the state or the information pairs. A distributed extended Kalman filter was discussed in [20], which needs the linearization of nonlinear dynamics and measurement mapping.