## 4 Results and Discussion  
Simulation experiments were conducted to evaluate the performance of the proposed approach compared to the conventional methods of GPS navigation processing to validate the effectiveness of the proposed approach in the presence of a time-varying satellite signal quality. The various types of navigation filters, including the EKF, IMM-EKF, MCEKF, and IMM-MCEKF, were developed by the authors using the MATLAB(r) codes with the Inertial Navigation System (INS) Toolbox [24] and the Satellite Navigation (SatNav) Toolbox [25] commercial software programs, both produced by GPSoft LLC. The vehicle's trajectory was produced using the INS Toolbox, while the satellite orbits and pseudoranges were created using the SatNav Toolbox. The test trajectory from the simulation is shown in Figure 4. As shown in Figure 5, eight GPS satellites (red colored dots followed by the GPS ID number) are reachable and the corresponding Geometry Dilution of Precision (GDOP) is around 2.4, which is good to assure appropriate navigation satellite geometry on positional measurement precision. Thus, we focused on the investigation of the position errors induced by the pseudorange observables, which reflect the quality of the measurement model in the filters. The simulated vehicle is located at an altitude of 100 m, which is comparable to \(\left[-3042329.2\quad 4911080.2\quad 2694074.3\right]^{\mathrm{T}}\) m in the WGS-84 ECEF coordinate system. It starts at East 121.7775 degrees and North 25.1492 degrees. The origin is defined as the (0, 0, 0) m point in the local tangent East-North-Up (ENU) frame. Multiple extras and randomly generated errors were purposefully added to the GPS pseudorange observation data while the vehicle was in motion to assess performance involving outlier type of multipath interferences. The error factors that distort the GPS observables were receiver noise, multipath, ionospheric delay, and tropospheric delay. Further, by assuming that the differential GPS (DGPS) mode was used, most errors were corrected; however, the multipath effect and receiver thermal noise could not be eliminated in this proposed study. The error induced by the receiver thermal noise was considered to be a Gaussian sequence with zero mean and unity variance. In addition, the efficacy of the outlier kind of multipath interference was further evaluated using several other intentionally produced and randomly generated problems. For example, urban areas or environments with NLOS and multipath receptions make it difficult for vehicles to navigate using GNSS data. Furthermore, the introduction of inaccurate data by these occurrences interferes with the precise positioning of the GPS sensor. Figure 6 depicts the sequences of outlier interferences added to the pseudorange observables during the experiment. These  
Figure 3: Flow chart of the maximum correntropy-extended Kalman filter (MCEKF). The IMM approach described in this work handles measurement uncertainty by modifying the \(\mathbf{R}_{k}\). Since this study avoided outlier types of multipath interference or NLOS reception, the influence of measurement noise \(\mathbf{R}_{k}\) was essential. Therefore, in environments with high levels of multipath, a more significant value of measurement noise covariance \(\mathbf{R}_{k}\) should be used; in environments with low levels of multipath, a less significant value of measurement noise covariance \(\mathbf{R}_{k}\) should be used. This modeling approach allows various model estimates to address the challenge of nonlinear filtering, given the noise level and distribution uncertainty. A collection of switching models describes the uncertainty of the noise strength. In this work, the idea of several models interacting was intended to reduce measurement uncertainty by modifying \(\mathbf{R}_{k}\). The following Markov chain transition probability matrix was used:  
\[\Pi= \begin{bmatrix}\pi_{11}&\pi_{12}\\ \pi_{21}&\pi_{22}\end{bmatrix} = \begin{bmatrix}0.7&0.3\\ 0.3&0.7\end{bmatrix}