## 4 Navigation Integration Processing and Performance Evaluation
Furthermore, it conducted circular motion with radius 2,000 meters during 301-700 s (counterclockwise) and 801-900 s (counterclockwise). The vehicle conducted the third circular motion with radius 1,000 meters (clockwise) during 1,201-1,400 s, where medium and high dynamic maneuverings are involved. The following parameters were used: the values of noise standard deviation are 3e-3 m/s\({}^{2}\) for accelerometers and gyroscopes. Figure 8 shows the trajectories for the simulated vehicle and the INS derived position, where the error grows unbounded. Assume that the differential GPS (DGPS) mode is used and most of the errors can be corrected, but the multipath and receiver thermal noise cannot be eliminated. The measurement noise variances \(r_{pi}\) are  
Figure 5: Illustration of the two-dimensional inertial navigation. assumed _a priori_ known, which is set as 4 \(m^{2}\). The measurement noise covariance matrix is given by \(\text{R}_{k}=100\ \times I_{m\ \times m}\). The following model transition probability matrices of the Markov chain \(\pi_{ij}\) were assumed:  
\[\pi_{ij}= \begin{cases}0.99&\text{if }i=j\\ \frac{1-0.99}{N-1}&otherwise\end{cases}  \tag{35}\]  
In this paper, two models are employed, therefore \(N=2\) and:  
\[\pi_{ij}= \begin{bmatrix}0.99&0.01\\ 0.01&0.99\end{bmatrix}  \tag{36}\]  
The initial model probability for each sub-model is chosen as:  
\[\boldsymbol{\mu}_{0}^{\ j}= \begin{cases}0.5&\text{if }j=1\\ \frac{1-0.5}{N-1}&otherwise\end{cases}  \tag{37}\]  
Figure 6: Simulated sensor outputs for the accelerometers and gyroscope. Figure 7: Trajectories for the simulated vehicle (solid) and the INS derived position (dashed). The process noise covariance matrix is given by:  
\[\mathbf{Q}_{k}= \begin{bmatrix}0&0&0&0&0\\ 0&4e^{-4}&0&0&0\\ 0&0&0&0&0\\ 0&0&0&4e^{-4}&0\\ 0&0&0&0&4e^{-5}\end{bmatrix}  \tag{38}\]  
and the parameters utilized in the UKF are given as follows: \(\alpha=1e-4\), \(\beta=2\) and \(k=0\). In addition, the parameters utilized in the STA of the STUKF are: \(\eta=0.6\), \(\rho=0.1\), and the softening factor \(\varepsilon=4.5\). The sigma points capture the same mean and covariance irrespective of the choice of matrix square root which is used. The numerical efficient and stable method such as the Cholesky factorization has been used in obtaining the sigma points. The membership functions (MFs) of input fuzzy variable DOD parameters as shown in Figure 9 are triangle MFs. The presented FLAS is the _If-Then_ form and consists of nine rules: \begin{table}
\begin{tabular}{c c c} \hline
**Segment number** & **Time interval (sec)** & **Motion** \\ \hline
1 & [0–300] & Constant velocity straight line \\
2 & [301–700] & Circular motion (counter-clockwise) \\
3 & [701–800] & Constant velocity straight line \\
4 & [801–900] & Counter-clockwise turn \\
5 & [901–1,200] & Constant velocity straight line \\
6 & [1,201–1,400] & Clockwise turn \\
7 & [1,401–1,800] & Constant velocity straight line \\ \hline \end{tabular}
\end{table} 
Table 1: Description of the vehicle motion. Figure 8: East and north components of INS navigation errors. 1. IF \(\xi\) is zero and \(\zeta\) is zero THEN \(\mathbf{Q}_{k}\) is 1
2. IF \(\xi\) is zero and \(\zeta\) is small THEN \(\mathbf{Q}_{k}\) is 1
3. IF \(\xi\) is zero and \(\zeta\) is large THEN \(\mathbf{Q}_{k}\) is 1
4. IF \(\xi\) is small and \(\zeta\) is zero THEN \(\mathbf{Q}_{k}\) is 1
5. IF \(\xi\) is small and \(\zeta\) is small THEN \(\mathbf{Q}_{k}\) is \(\xi+\zeta+4\)
6. IF \(\xi\) is small and \(\zeta\) is large THEN \(\mathbf{Q}_{k}\) is \(\xi+\zeta+4\)
7. IF \(\xi\) is large and \(\zeta\) is zero THEN \(\mathbf{Q}_{k}\) is \(6\zeta+6\zeta+12\)
8. IF \(\xi\) is large and \(\zeta\) is small THEN \(\mathbf{Q}_{k}\) is \(6\zeta+6\zeta+12\)
9. IF \(\xi\) is large and \(\zeta\) is large THEN \(\mathbf{Q}_{k}\) is \(6\zeta+6\zeta+12\)  
Figures 10-13 provide the navigation results for the UKF, IMMUKF, STUKF and FUZZY-IMMUKF approaches. Before the FLAS was incorporated, preliminary evaluation on the adaptation algorithms were carried out.