# Multi-Sensor Fusion Positioning Method Based on Batch Inverse Covariance Intersection and IMM
## 3 Fusion Algorithms Considering Accuracy and Robustness
### BICI Fusion Algorithm  
The batch forms of ICI method and the BICI multi-sensor fusion positioning algorithm are proposed based on two-sensor ICI. Meanwhile, the noise variance upper bound of error variance matrix is derived to prove the robustness of BICI. The structure of the BICI fusion algorithm is displayed in Figure 2 as follows.  
Inspired by Theorem 1, the batch form of ICI is a convex combination of local filters, given as  
Figure 2: BICI multi-sensor fusion algorithm structure.  
\[\hat{x}_{k}^{BICI}=\sum_{h=1}^{H}K_{h}\hat{x}_{k}^{(h)} \tag{23}\]  
where \(\hat{x}_{k}^{(h)}\) is local filter, \(\sum_{h=1}^{H}K_{h}=I\). The error variance matrix is  
\[\left(P_{k}^{BICI}\right)^{-1}=\sum_{h}(P_{k}^{h})^{-1}-\left(\sum_{h}\omega_{h} P_{k}^{h}\right)^{-1} \tag{24}\]  
where filtering gains \(K_{h}\) are  
\[K_{h}=P_{k}^{BICI}\left((P_{k}^{h})^{-1}-\omega_{h}\left(\sum_{h}\omega_{h}P_{k }^{h}\right)^{-1}\right) \tag{25}\]  
where \(\omega_{h}\) fulfills the condition \(0\leq\omega_{h}\leq 1,\sum_{h}\omega_{h}=1\). The selection of the coefficients \(\omega_{h}\) is the minimum trace of the variance matrix \(P_{k}^{BICI}\) as [15]  
\[objective:\min_{\omega}tr(P_{k|k}^{BICI(h)}),\quad s.t:0\leq\omega_{h}\leq 1, \sum_{h}\omega_{h}=1\,. \tag{26}\]  
The above optimization problem can also be solved by the \(fmincon\) function in the MATLAB optimization toolbox.  
**Theorem 2**.: _The error variance matrix of BICI fusion algorithm is \(\hat{P}_{k}^{BICI}=E\left(\hat{x}_{k}^{BICI}\left(\hat{x}_{k}^{BICI}\right)^{T }\right)=\sum_{m}\sum_{h}K_{m}P_{k}^{mm}K_{n}^{T},\hat{x}_{k}=x_{k}-\hat{x}_{k }^{BICI}\), and noise variance upper bound is \(P^{BICI}\)._  
**Proof of Theorem 2.**  
\[E\left(\hat{x}_{k}^{BICI}\left(\hat{x}_{k}^{BICI}\right)^{T}\right)= E\left(\left(\sum_{h}K_{h}\hat{x}_{k}^{(h)}-x_{k}\right)\left(\sum_{h}K_{h} \hat{x}_{k}^{(h)}-x_{k}\right)^{T}\right)\] \[= E\left(\left(\sum_{h}K_{h}\hat{x}_{k}^{(h)}\right)\left(\sum_{h} K_{h}\hat{x}_{k}^{(h)}\right)^{T}\right)\] \[= \sum_{m}\sum_{n}K_{m}P_{k}^{mm}K_{n}^{T}\,. \tag{27}\]  
Using the inequality is  
\[K_{m}P^{mm}K_{n}^{T}+K_{n}P^{mm}K_{m}^{T}\leq rK_{m}P^{m}K_{m}^{T}+\frac{1}{r }K_{n}P^{n}K_{n}^{T} \tag{28}\]  
where \(r>0\). The noise variance matrix upper bound \(P_{k}^{BICI}\) can be subtracted by  
\[E\left(\hat{x}_{k}^{BICI}\left(\hat{x}_{k}^{BICI}\right)^{T}\right)\leq\sum_{ h}K_{h}P_{k}^{h}K_{n}^{T}+\sum_{m}\sum_{n}^{m}\left(r_{m}K_{m}P_{k}^{m}K_{m}^{T}+ \frac{1}{r_{m}}K_{n}P_{k}^{n}K_{n}^{T}\right)=P^{BICI}\,. \tag{29}\]  
The proof is completed.  
Using Theorem 2, we demonstrated that the BICI multi-sensor fusion positioning method with unknown covariance is robust. Combined with Theorem 1, the robustness of the SICI fusion positioning method is indirectly verified as  
**Corollary 1**.: _The error variance matrix of the SICI fusion algorithm is \(\hat{P}_{k}^{SICI}=E\left(\hat{x}_{k}^{SICI}\left(\hat{x}_{k}^{SICI}\right)^{T }\right)=\sum_{m}\sum_{n}\theta_{m}^{(H)}P_{k}^{mm}\left(\theta_{m}^{(H)} \right)^{T}\),\(\hat{x}_{k}=x_{k}-\hat{x}_{k}^{SICI}\), and noise variance upper bound is_\[P^{SICI}=\sum_{h}\theta_{h}^{(H)}P_{k}^{h}\Big{(}\theta_{h}^{(H)}\Big{)}^{T}+ \sum_{m}\sum_{n}^{m}\left(r_{m}\theta_{m}^{(H)}P_{k}^{m}\Big{(}\theta_{m}^{(H)} \Big{)}^{T}+\frac{1}{r_{m}}\theta_{n}^{(H)}P_{k}^{n}\Big{(}\theta_{n}^{(H)} \Big{)}^{T}\right).\]