# Joint Probabilistic Hypergraph Matching Labeled Multi-Bernoulli Filter for Rigid Target Tracking
## 4 Simulation
### Scenario 1
The larger the turning rate is, the more serious the model mismatch is. As mentioned above, the estimation accuracy of target states degrades even if the data association is correct when model is mismatched. If one target is undetected at one moment, there is no measurement available to update its prediction state which is obtained through preset dynamic motion model. Because of the model mismatch, the state estimates of this target will be far away from the true state, the more serious model mismatch is, the worse target state estimation is, the target may be eliminated if target state estimation is worse enough, then the targets cardinality will be underestimated. The structure of targets is destroyed in some degree, the state estimates of this target can converge to true state again after several time steps if it is detected, then the structure of targets can be recovered. The more time is needed with larger turning rate. If missed detection occurs again before the structure is restored, the structure of targets may be destroyed permanent, the poor performance of JPHGM-LMB and HGM-LMB will be achieved. Clear mot metrics of these filters with different turning rate \(\omega\) are in Table 2.  
Figure 7: _Cont._  
From Table 2, it is shown that larger turning rate leads to poor performance. The performance of these filters degrades dramatically with increasing turning rate. However, JPHGM-LMB always outperforms LMB and HGM-LMB as turning rate varies. The effectiveness of the proposed method is validated again. The average computation time for tracking these four targets of three filter with \(p_{D}=0.9\) is shown in Figure 8. As mentioned in Section 3, JPHGM-LMB follows some idea of JPDA, it suffers the problem of JPDA more or less, the number of association events grows exponentially with the number of targets. The better tracking performance is achieved at the sacrifice of computation time. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & **MOTP** & **Miss Rate** & **False Positive Rate** & **Mismatches** & **MOTA** \\ \hline \multirow{3}{*}{\(\omega=1^{\circ}/\mathrm{s}\)} & LMB & **0.0844** & 1.97\% & 0.8\% & 1.13 & 96.75\% \\  & HGM-LMB & 0.0851 & 1.5\% & 0.31\% & 0.3 & 98.07\% \\  & JPHGM-LMB & 0.0852 & **0.95**\% & **0.31**\% & **0.12** & **98.69**\% \\ \hline \multirow{3}{*}{\(\omega=3^{\circ}/\mathrm{s}\)} & 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}{*}{\(\omega=5^{\circ}/\mathrm{s}\)} & LMB & **0.2369** & 14.11\% & 12.65\% & 13.13 & 67.77\% \\  & HGM-LMB & 0.2379 & 10.81\% & 9.32\% & 9.59 & 75.88\% \\  & JPHGM-LMB & 0.2664 & **5.89**\% & **4.92**\% & **6.51** & **86.48**\% \\ \hline \multirow{3}{*}{\(\omega=8^{\circ}/\mathrm{s}\)} & LMB & **0.076** & 39.35\% & 29.65\% & 18.52 & 23.28\% \\  & HGM-LMB & 0.0847 & 35.39\% & 22.55\% & 16.74 & 35.08\% \\ \cline{1-1}  & JPHGM-LMB & 0.0938 & **32.6**\% & **20.47**\% & **13.5** & **41.3**\% \\ \hline \hline \end{tabular}  
* MOTP is the multiple object tracking precision and MOTA is the multiple object tracking accuracy.  
\end{table} 
Table 2: Clear mot metrics of filters with different turning rate. where \(\mathbf{x}_{0}^{(i,j)}\) is the initial state of \(i\)-th scattering center of \(j\)-th rigid target, \(t_{s}^{(i,j)}\) is the born time of \(i\)-th scattering center of \(j\)-th rigid target, \(t_{e}^{(i)}\) is the end time of \(i\)-th scattering center of \(j\)-th rigid target. The birth intensity in the simulation is  
\[D_{b}(\mathbf{\xi})=\sum_{i=1}^{2}\sum_{j=1}^{4}w_{b}\mathcal{N}(\mathbf{x};\mathbf{x}_{0}^ {(i,j)},\mathbf{P}_{b}), \tag{49}\]  
where \(w_{b}=0.03\), \(\mathbf{P}_{b}=\text{diag}([100,100,100])\). The true trajectories of these targets are shown in Figure 9.  
In this section, the detection probability is \(p_{D}=0.9\) and the acceleration of CA model is \(\mathbf{a}=1.5\) m/s\({}^{2}\).