## 4 Experimental Results and Discussion
### Numerical Simulations and Analysis
#### 4.1.2 CPIMM-NAUKF Simulation  
To validate the effectiveness of CPIMM-NAUKF, the second set of comparisons, including the IMM-UKF, IMM-NAUKF, CPIMM-UKF, and CPIMM-NAUKF were simulated, assuming there are sensor faults, similar to Simulation Case 2 in the scenario design section.  
Figure 8 gives the true and estimated trajectories of the maneuvering target. Both CPIMM and IMM can track the trajectory of the target, but the estimated trajectory of CPIMM-NAUKF was closest to the true trajectory.  
The comparisons of the RMSE position and velocity curves of four filtering algorithms are shown in Figures 9 and 10, and the corresponding statistical data of RMSE are listed in Table 3. The sensor faults negatively influenced the filtering precision of the UKF and the filtering error of the standard IMM-UKF was the largest. The IMM-NAUKF has better filtering accuracy than the IMM-UKF, because the internal link NAUKF effectively handled the sensor fault problem, but the filtering convergence performance was poor. The CPIMM-UKF also has better filtering accuracy than the IMM-UKF because the internal link CPIMM has a better effect for maneuvering target tracking than IMM, but the filtering accuracy relationship between IMM-NAUKF and CPIMM-UKF does not exist. Ultimately, CPIMM-NAUKF has the best filtering accuracy, since it significantly improves the IMM-UKF through the external and internal links, which makes it extremely effective for maneuvering target tracking when sensor faults occur.  
In summary, as a combination of CPIMM and NAUKF, the CPIMM-NAUKF, compared with the existing IMM-UKF, makes a greater contribution to filtering performance in terms of accuracy and convergence speed. This is because it includes two important links, the external CPIMM and the internal NAUKF, which work simultaneously and effectively.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{2}{c}{**Position Error (m)**} & \multicolumn{2}{c}{**Velocity Error (m/s)**} \\ \cline{2-5}  & **Mean** & **Standard Deviation** & **Mean** & **Standard Deviation** \\ \hline IMM & 3.8245 & 2.0377 & 0.7102 & 0.6935 \\ CPIMM & 2.0630 & 1.1604 & 0.4596 & 0.4187 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Performance comparison of the algorithms for CPIMM for Case 1.  
Figure 8: True and estimated trajectories of the maneuvering target for CPIMM-NAUKF for Case 2.  
Figure 10: Velocity RMSE comparison for CPIMM-NAUKF for Case 2.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{2}{c}{**Position Error (m)**} & \multicolumn{2}{c}{**Velocity Error (m/s)**} \\ \cline{2-5}  & **Mean** & **Standard Deviation** & **Mean** & **Standard Deviation** \\ \hline IMM-UKF & 11.8533 & 4.3387 & 3.1026 & 1.9552 \\ IMM-NAUKF & 7.0473 & 2.9116 & 1.2143 & 1.0348 \\ CPIMM-UKF & 9.3558 & 3.0098 & 2.4961 & 1.2246 \\ CPIMM-NAUKF & 4.8671 & 0.9169 & 0.4294 & 0.3857 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Performance comparison for CPIMM-NAUKF for Case 2.  
Figure 9: Position RMSE comparison for CPIMM-NAUKF for Case 2.