# Multi-Sensor Fusion Positioning Method Based on Batch Inverse Covariance Intersection and IMM
## 3 Fusion Algorithms Considering Accuracy and Robustness
### SICI Fusion Algorithm  
The SICI multi-sensor fusion algorithm is realized by the (\(H\)\(-\)1) two-sensor ICI method. The structure of the SICI fusion algorithm is displayed in Figure 1 as follows.  
The SICI multi-sensor fusion estimator is given by [22]  
\[\xi_{k}^{SICI(h)}= K_{k}^{SICI(h)}\xi_{k}^{SICI(h-1)}+\left(I-K_{k}^{SICI(h)}\right) \xi_{k}^{(h+1)} \tag{9}\] \[\left(P_{k|k}^{SICI(h)}\right)^{-1}= \left(P_{k|k}^{SICI(h-1)}\right)^{-1}+\left(P_{k|k}^{(h+1)}\right)^ {-1}\] \[-\left(\omega_{h}P_{k|k}^{SICI(h-1)}+(1-\omega_{h})P_{k|k}^{(h+1) }\right)^{-1}\] (10) \[\xi_{k}^{SICI}=\xi_{k}^{SICI(H-1)} \tag{11}\]  
where \(\hat{x}_{k}^{(h+1)}\) are local filters. The initial values are  
\[\xi_{k}^{SICI(0)}=\hat{x}_{k}^{(1)},\qquad\left(P_{k|k}^{SICI(0)}\right)^{-1}= \left(P_{k|k}^{(1)}\right)^{-1} \tag{12}\]  
where the filtering gains \(K_{k}^{SICI(h)}\) are obtained by  
\[K_{k}^{SICI(h)}=P_{k|k}^{SICI(h)}\bigg{(}\left(P_{k|k}^{SICI(h-1)}\right)^{-1 }-\omega_{h}\Big{(}\omega_{h}P_{k|k}^{SICI(h-1)}+(1-\omega_{h})P_{k|k}^{(h+1) }\Big{)}^{-1}\bigg{)} \tag{13}\]  
the optimal \(\omega_{h}\) minimizes trace of covariance matrix \(P_{k|k}^{SICI(h)}\) as  
\[objective:\min_{0\leq\omega_{h}\leq 1}tr(P_{k|k}^{SICI(h)}) \tag{14}\]  
where \(tr(\cdot)\) represents the matrix traces. It is necessary to solve the \(H-1\) one-dimensional optimization problems in one filtering cycle for SICI. The optimization problem can be solved by the MATLAB \(fminbnd\) function [16].  
Theorem 1: _SICI fusion algorithm \(\hat{x}_{k}^{SICI}\) can be represented as the batch form of a weighted mean_  
\[\hat{x}_{k}^{SICI}=\sum_{h}\theta_{h}^{(H)}\hat{x}_{k}^{h} \tag{15}\]  
\[\sum_{h}\theta_{h}^{(H)}=I \tag{16}\]  
Figure 1: SICI multi-sensor fusion algorithm structure.  
_where \(\hat{x}_{k}^{h}\) are local filters. Weighting coefficients \(\theta_{h}^{(H)}\) can be obtained recursively as_  
\[\theta_{h}^{(I)}=K_{I-1}\theta_{h}^{(I-1)},\quad h=1,2,\ldots,l-1 \tag{17}\]  
\[\theta_{h}^{(I)}=I-K_{I-1},\quad l=2,3,\ldots,H \tag{18}\]  
_with initial values are_  
\[\theta_{1}^{(2)}=K_{1},\quad\theta_{2}^{(2)}=I-K_{1} \tag{19}\]  
_where \(K_{I-1}\) and \(\omega_{I-1}\) can be calculated from Equations (13) and (14), respectively._  
Proof of Theorem 1.: We chose mathematical induction to prove the Theorem 1.  
Step 1: For \(H=2\), according to Equations (9)-(12), we can obtain  
\[\hat{x}_{k}^{SICI(2)}=K_{1}\hat{x}_{k}^{(1)}+(I-K_{1})\hat{x}_{k}^{(2)}\,. \tag{20}\]  
Therefore, when the number of sensors is 2, Theorem 1 holds.  
Step 2: Assuming that Theorem 1 holds when the number of sensors is \(H-1\), then  
\[\hat{x}_{k}^{SICI(H-1)}=\sum_{h=1}^{H-1}\theta_{h}^{(H-1)}\hat{x}_{k}^{h}\,. \tag{21}\]  
We have to prove that Theorem 1 holds when the number of sensors is \(H\). The fusion of the \(\hat{x}_{k}^{SICI(H-1)}\) and \(\hat{x}_{k}^{(H)}\) can be determined by  
\[\hat{x}_{k}^{SICI}= K_{H-1}\hat{x}_{k}^{SICI(H-1)}+(I-K_{H-1})\hat{x}_{k}^{H}\] \[= K_{H-1}\sum_{h=1}^{H-1}\theta_{h}^{(H-1)}\hat{x}_{k}^{h}+(I-K_{H -1})\hat{x}_{k}^{H}\] \[= \sum_{h=1}^{H-1}\theta_{h}^{(H)}\hat{x}_{k}^{h}+\theta_{h}^{(H)} \hat{x}_{k}^{H}\] \[= \sum_{h=1}^{H}\theta_{h}^{(H)}\hat{x}_{k}^{h}\,. \tag{22}\]  
It can be seen that when the number of sensors is \(H\), Theorem 1 still holds. The proof is completed.