## 3 Results and Discussion  
Figure 1 represents the flow diagram of the MCCEKF with a nonlinear regression process and the algorithm is summerized above. The simulation analyses aim to verify the suggested strategy's efficacy in dealing with the effects of a time-varying satellite signal quality by comparing its performance to that of the traditional GPS navigation procedure. The commercially available software programs, Satellite Navigation (SatNav) Toolbox [36] and Inertial Navigation System (INS) Toolbox [37] with MATLAB(r) codes are used for this study. The INS Toolbox is utilized to generate the vehicle's trajectory, while the SatNav Toolbox is employed to generate the satellite orbits and pseudoranges. To investigate the estimation performance of different types of filters for navigation, filters are designed, including EKF, MCCEKF, MCCEKF-NR, and Mixture-LRMCCEKF, among others. Figure 2 exhibits the test trajectory from the simulation. Seven GPS satellites (marked in red color dots) with the GPS ID number may be reached, as shown in Figure 3. The related Geometry Dilution of Precision (GDOP) is approximately 2.4, which ensures that the positional measurement precision is accurate due to the proper navigation satellite geometry. We concentrated on examining the location errors brought about by pseudorange entities, and these represent the caliber of the measurement model within the filters. The simulated vehicle is situated nearly 100 m above the earth's surface. To evaluate the functionality concerning the outlier level of multipath interruptions, the randomly generated inaccuracies are purposely included in the GPS pseudorange observation data while the vehicle is moving. The ionospheric delay, tropospheric delay, multipath, and receiver noise were the error factors that distorted the GPS observables. Furthermore, most errors were avoided by assuming that the differential GPS (DGPS) mode was used. A Gaussian sequence with a zero median and unity variance was assumed to represent the error caused by the thermal noise in the receiver. In addition, some additional deliberately created and randomly generated situations were used to assess the effectiveness of the outlier kind of multipath interference. These events introduce inaccurate data that interfere with the GPS sensor's ability to precisely position as a whole. In this study, different types of error calculations were carried out to obtain an error-free signal for GPS navigation; for example, the general error, mean error (ME), mean square error (MSE), root mean square error (RMSE), root mean square error (RMSE), and the  
Figure 3: The skyplot for satellite location. Figure 2: The test trajectory for the simulated vehicle. average root mean square error (ARMSE) at each time step are all measured statistically during the 100 Monte Carlo simulations. The following are the calculation formulas:  
\[\text{ME(k)}= \frac{1}{M}\sum_{m=1}^{M}\;(x_{k}-\hat{x}_{k}),\;\text{m}\;=\;1,\; \ldots,\;M \tag{38}\] \[\text{MSE(k)}= \frac{1}{K}\sum_{k=1}^{K}(x_{k}-\hat{x}_{k})\;^{2},\;\text{k}\;= \;1,\;\ldots,\;K\] (39) \[\text{RMSE(k)}= \sqrt{\frac{1}{K}\sum_{k=1}^{K}(x_{k}-\hat{x}_{k})^{2}}\] (40) \[\text{ARMSE(k)}= \frac{1}{K}\sum_{k=1}^{K}\text{RMSE}(k) \tag{41}\]  
where \(M\) and \(K\) denote the total number of Monte Carlo runs and total time steps in a single Monte Carlo run, respectively. In the current scenario, the position-velocity (PV) model suitable for low/medium dynamic motion is adopted from [37]. With a sampling period of \(\Delta\)T, the model is obtained using the rectangle integral, and data are recorded from the radar every 1 s.  
The six different filtering approaches' precision in terms of errors for different positioning comparisons, such as EKF, MCCEKF-BC, MCCEKF-DS, MCCEKF-GL, MCCEKF-NR, and Mixture-LRMCCEKF, are presented in Figure 4a-c.