## 3 Distributed Cubature Kalman Filter
### Distributed Bayesian Filter
The estimate \(\tilde{x}_{i,k}\) by sensor \(i\) can be obtained by letting \(\triangledown_{x_{k}}f_{i,k}\) be equal to zero, and we get:  
\[\tilde{x}_{i,k}=\tilde{x}_{k}+N(\tilde{P}_{k}^{-1}+NH_{i,k}^{\top}R_{i,k}^{-1} H_{i,k})^{-1}H_{i,k}^{\top}R_{i,k}^{-1}(z_{i,k}-h_{i}(\tilde{x}_{k})). \tag{23}\]  
By the matrix inverse lemma, we have:  
\[N(\tilde{P}_{k}^{-1}+NH_{i,k}^{\top}R_{i,k}H_{i,k})^{-1}H_{i,k}^ {\top}R_{i,k}^{-1}= (I+N\tilde{P}_{k}H_{i,k}^{\top}R_{i,k}H_{i,k})^{-1}N\tilde{P}_{k} H_{i,k}^{\top}R_{i,k}^{-1} \tag{24}\] \[= N\tilde{P}_{k}H_{i,k}^{\top}(NH_{i,k}\tilde{P}H_{i,k}^{\top}+R_{ i,k})^{-1}. \tag{25}\]  
Substituting (25) into (23), we obtain:  
\[\tilde{x}_{i,k}=\tilde{x}_{k}+N\tilde{P}_{k}H_{i,k}^{\top}(NH_{i,k}\tilde{P}H _{i,k}^{\top}+R_{i,k})^{-1}(z_{i,k}-h_{i}(\tilde{x}_{k})). \tag{26}\]  
The estimate error covariance can be computed as follows,  
\[\tilde{P}_{i,k}=\tilde{P}_{k}+P_{i,xz,k}P_{i,xz,k}^{-1}P_{i,xz,k}^{\top}, \tag{27}\]  
where:  
\[P_{i,zz,k} =NH_{i,k}\tilde{P}H_{i,k}^{\top}+R_{i,k}, \tag{28}\] \[P_{i,xz,k} =N\tilde{P}_{k}H_{i,k}^{\top}. \tag{29}\]  
**Remark 3**. : _It should be noted that \(P_{i,zz,k}\) and \(P_{i,xz,k}\) are a little different with the standard extended Kalman filter, even though they can be obtained by each sensor individually. With \(N=1\) in (26) and (27), it will reduce to the standard Kalman filter._  
Up to now, we obtain the optimal solution of \(\min_{x_{k}}\ f_{i,k}\) as \(\tilde{x}_{i,k}\) and \(\tilde{P}_{i,k}\), which follows the Gaussian distribution \(\mathbb{N}(\tilde{x}_{i,k},\tilde{P}_{i,k})\). As discussed in (20), the global optimal solution can be approximated by averaging local estimate \(\mathbb{N}(\tilde{x}_{i,k},\tilde{P}_{i,k})\). However, the traditional average consensus algorithm in Euclidean space may not be suitable to compute the average of PDFs. Therefore, we use the consensus of PDFs described in Section 2.2 to compute the global solution to Problem (19). The natural parameter of Gaussian distribution \(p_{i}(x|\tilde{x}_{i},\tilde{P}_{i})\) is \(\lambda_{i}= \begin{bmatrix}\tilde{P}_{i}^{-1}\tilde{x}_{i}\\ -\frac{1}{2}\tilde{P}_{i}^{-1}\end{bmatrix} \)[38]. Then, the global posterior distribution can be approximated by the consensus of probability densities (12) as follows,  
\[(\tilde{P}_{i}^{s+1})^{-1}\tilde{x}_{i}^{s+1}=\sum_{j\in\mathcal{N }_{i}}a_{ij,k}(\tilde{P}_{j}^{s})^{-1}\tilde{x}_{j}^{s}, \tag{30}\] \[(\tilde{P}_{i}^{s+1})^{-1}=\sum_{j\in\mathcal{N}_{i}}a_{ij,k}( \tilde{P}_{i}^{s})^{-1}, \tag{31}\]  
where \(s=1,\dots,S\) is the step of the consensus. Then, the estimates of each node can be achieved by \(P_{i,k}=(\tilde{P}_{i,k}^{S})^{-1}\) and \(\hat{x}_{i,k}=(P_{i,k})^{-1}(\tilde{P}_{i}^{S})^{-1}\tilde{x}_{i}^{S}\). **Remark 4**. : _In [19], each sensor performed a standard Kalman filter, then the fusion estimation was obtained based on consensus of PDFs. It should be noted that, if \(N=1\), (26)-(27) will reduce to the measurement update of the standard Kalman filter. We derive an optimal solution of each sensor for Problem (19), which may achieve better performance compared to [19]. However, we should highlight that [19] provided meaningful information the theoretical expression for the distributed filter._  
Equations (26)-(31) provide a general framework of the distributed Bayesian filter for posterior estimation. Based on the measurement update Equations (26)-(31), we can construct distributed nonlinear filtering by combining the existing method for state propagation. For example, in [29], the ensemble Kalman filter (EnKF) was used for state propagation, which uses the Monte Carlo technique for integral operation in the Bayesian filtering. In this paper, we use cubature rules for state propagation and measurement update, which we will discuss in the following.