# Joint Probabilistic Hypergraph Matching Labeled Multi-Bernoulli Filter for Rigid Target Tracking
## 4 Simulation
### Scenario 1
The effect of the turning rate \(\omega\) and detection probability \(p_{D}\) on the tracking performance is analyzed. Note that while analyzing the effect of one parameter, the other parameters remain unchanged. The OSPA of these filters with different detection probability \(p_{D}\) and \(\omega=3^{\circ}\)/s are shown in Figure 6.  
When targets maneuvering during the interval 20 s-40 s, there is a mismatch between actual motion model and dynamic motion model. Consequently, the estimation accuracy of target states degrades even if the data association is correct. Furthermore, it is susceptible to obtain wrong association results because of the model mismatching. From Figure 6, we can see that the OSPA distances of three filters increase when targets maneuvering. It can be observed that JPHGM-LMB filter outperforms HGM-LMB and LMB filter when the detection probability varies because not only the structure information of targets is used, but also all of the undetected cases are considered. However, the performance of JPHGM -LMB also degrades as detection probability decreases. The reason is that missed detections broken the structure of targets to some extent. An extreme case is that only one target is detected and thus no structure information can be used. In order to further validate the superiority of JPHGM -LMB, clear mot metrics of these filters with different detection probability \(p_{D}\) are shown in Table 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & **MOTP** & **Miss Rate** & **False Positive Rate** & **Mismatches** & **MOTA** \\ \hline \multirow{3}{*}{\(p_{D}=0.98\)} & LMB & **0.166** & 8.84\% & 8.17\% & 6.63 & 80.35\% \\  & HGM-LMB & 0.1892 & 1.36\% & 0.69\% & 0.64 & 97.68\% \\  & JPHGM-LMB & 0.1901 & **1.34**\% & **0.65**\% & **0.04** & **98**\% \\ \hline \multirow{3}{*}{\(p_{D}=0.9\)} & LMB & **0.1562** & 11.93\% & 10.87\% & 9.43 & 73.27\% \\  & HGM-LMB & 0.1753 & 5.94\% & 4.75\% & 2.92 & 88.09\% \\  & JPHGM-LMB & 0.1797 & **4.2**\% & **3.54**\% & **0.95** & **91.86**\% \\ \hline \multirow{3}{*}{\(p_{D}=0.8\)} & LMB & **0.1467** & 16.07\% & 14.86\% & 12.75 & 63.75\% \\  & HGM-LMB & 0.1525 & 12.4\% & 11.05\% & 8.54 & 72.98\% \\  & JPHGM-LMB & 0.1654 & **8.56**\% & **7.17**\% & **4.35** & **82.46**\% \\ \hline \multirow{3}{*}{\(p_{D}=0.7\)} & LMB & **0.1281** & 20.87\% & 18.92\% & 16.64 & 53.28\% \\  & HGM-LMB & 0.1312 & 18\% & 15.74\% & 13.35 & 60.7\% \\ \cline{1-1}  & JPHGM-LMB & 0.1407 & **14.33**\% & **12.2**\% & **11.55** & **68.83**\% \\ \hline \hline \end{tabular}  
* MOTP is the multiple object tracking precision and MOTA is the multiple object tracking accuracy.  
\end{table} 
Table 1: Clear mot metrics of filters with different detection probability. Figure 6: The OSPA of three filters with different detection probability: (**a**) \(p_{D}=0.98\), (**b**) \(p_{D}=0.9\), (**c**) \(p_{D}=0.8\) and (**d**) \(p_{D}=0.7\). From Table 1, we can see that JPHGM-LMB achieves lower miss rate and false positive rate than HGM-LMB and LMB. The number of mismatches of JPHGM-LMB is less than other filters as detection probability varies. These results further demonstrate the effectiveness of the proposed method. However, it can be observed that LMB achieves smaller MOTP than other filters and MOTP of these filters decrease as detection probability decreases. The reason is that MOTP only takes the difference between true target states and estimated states into account when there is no mismatch. However, mismatches are more likely to happened when targets maneuvering and the estimation accuracy of target states degrades when target maneuvering as mentioned above. The OSPA of these filters with different turning rate \(\omega\) and \(p_{D}=0.9\) are shown in Figure 7. From Figure 7, we can see that turning rate \(\omega\) has significant effect on the performance of targets tracking, the tracking performance degrades a lot as the turning rate increases. Highly unreliable estimates are obtained with larger turning rate and appear in the OSPA curve.