# Multi-Sensor Fusion Positioning Method Based on Batch Inverse Covariance Intersection and IMM
## 3 Fusion Algorithms Considering Accuracy and Robustness
### Simulation  
The goal of this section is to evaluate the performance of the BICI with uncertain time-varying noise variance and unknown covariance. The numerical example provides two representative cases, which are fixed and time-varying noise. The SCI, BCI, SICI and BICI multi-sensor positioning algorithms are taken for comparison. In the presentation of the simulation result, the root mean square (RMS) of positioning error is utilized for positioning accuracy and robustness evaluation.  
The simulation trajectory is shown in Figure 3, and the simulation time is 100 s. A 3-sensor fusion positioning system with uncertain time-varying noise variance can be represented as  
\[x_{k+1}= F_{k}x_{k}+B_{k}w_{k} \tag{30}\] \[z_{k}^{h}= C_{k}^{h}x_{k}+v_{k}^{h},\quad h=1,2,3 \tag{31}\]  
\[B_{k}=\left[\begin{array}{cccc}T^{2}/2&0&0\\ T&0&0\\ 0&T^{2}/2&0\\ 0&T&0\\ 0&0&T^{2}/2\\ 0&0&T\end{array}\right] \tag{32}\]  
\[F_{k}=\left[\begin{array}{cccccc}1&T&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&T&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&T\\ 0&0&0&0&0&1\end{array}\right] \tag{33}\]  
\[C_{k}^{h}=\left[\begin{array}{cccccc}1&0&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&0&1&0\end{array}\right] \tag{34}\]  
where \(T=100\) ms is the sampling period of sensor, \(x_{k}=\left[p_{x},v_{x},p_{y},v_{y},p_{z},v_{z}\right]^{T}\) is the positioning system state. \(p_{x}\), \(p_{y}\) and \(p_{z}\) are the position of X, Y and Z directions, respectively. \(v_{x}\), \(v_{y}\) and \(v_{z}\) are the velocity of X, Y and Z directions, respectively. \(w_{k}\) and \(v_{k}^{h}\) are white Gaussian noise with zero mean and variance matrix \(\bar{Q}_{k}\), \(\bar{R}_{k}^{h}\), respectively. In the simulation, we take \(Q=3I\), \(\bar{Q}_{k}=0.8\) and the conservative accuracy of three sensors as 1 m, 2 m and 3 m, respectively. The observation and process noise of sensors 1 and 2 have random correlation coefficients of \(\rho_{k}^{\omega_{k}}\) and \(\rho_{k}^{v_{k}^{(12)}}\), respectively. \(\rho_{k}^{\omega_{k}}\in[0.1,0.3]\), \(\rho_{k}^{v_{k}^{(12)}}\in[0.1,0.2]\).  
The true trace of the fixed noise variance matrix of about three sensors is 0.8 m, 1.6 m and 2.5 m, respectively. The trace of the time-varying noise variance matrix is \(0.8+0.2\sin(2\pi kT/50)\) m, \(1.6+0.4 sin(2\pi kT/50)\) m and \(2.5/3+0.5/3 sin(2\pi kT/50)\) m. The time-varying trace of three sensors is shown in Figure 4.  
Through 50 Monte Carlo simulations, the cumulative distribution functions (CDF) of SCI, BCI, SICI, BICI, SCI (fix), BCI (fix), SICI (fix) and BICI (fix) with uncertain time-varying noise are shown in Figure 5. It is indicated that SCI, BCI, SICI, and BICI multi-sensor fusion positioning algorithms have the same positioning accuracy under uncertain fixed and time-varying noise variance, respectively. Therefore, the four algorithms have good robustness under the time-varying noise variance. The positioning accuracy relation is \(SICI<BCI\approx SCI<BICI\) in Table 1. BICI has less conservativeness and improved the positioning accuracy by at least 26.7% compared with the other three algorithms.  
Figure 4: The true trace of time-varying noise variance matrix.  
Figure 3: The simulation trajectory of three-dimensional motion.