# Joint Probabilistic Hypergraph Matching Labeled Multi-Bernoulli Filter for Rigid Target Tracking
## 4 Simulation
### Scenario 1  
In the first scenario 1, the surveillance region is \([-100\) m, \(1000\) m\(]\times[-100\) m, \(1000\) m\(]\), we set totally 60 time steps and the sampling time is 1 s. These targets move along a straight line with a nearly constant velocity (CV) from 0 s to 20 s and follow a nearly constant speed turn with turning rate \(\omega\) during the interval 20 s-40 s and move along a straight line with a nearly constant velocity (CV) during the interval 40 s-60 s.  
The initial states of four targets are given  
\[ \begin{split}&\mathbf{x}_{0}^{(1)}=[0\;\text{m},10\;\text{m},20\; \text{m}/\text{s},0\;\text{m}/\text{s}],t_{s}^{(1)}=0\;\text{s},t_{e}^{(1)}=60 \;\text{s};\\ &\mathbf{x}_{0}^{(2)}=[0\;\text{m},0\;\text{m},20\;\text{m}/\text{s}, 0\;\text{m}/\text{s}],t_{s}^{(2)}=0\;\text{s},t_{e}^{(2)}=60\;\text{s};\\ &\mathbf{x}_{0}^{(3)}=[5\;\text{m},11\;\text{m},20\;\text{m}/\text{s}, 0\;\text{m}/\text{s}],t_{s}^{(3)}=0\;\text{s},t_{e}^{(3)}=60\;\text{s};\\ &\mathbf{x}_{0}^{(4)}=[5\;\text{m},7\;\text{m},20\;\text{m}/\text{s}, 0\;\text{m}/\text{s}],t_{s}^{(4)}=0\;\text{s},t_{e}^{(4)}=60\;\text{s};\end{split}  \tag{46}\]  
where \(\mathbf{x}_{0}^{(i)}\) is the initial state of \(i\)-th target, \(t_{s}^{(i)}\) is the born time of \(i\)-th target, \(t_{e}^{(i)}\) is the end time of \(i\)-th target. The birth intensity in the simulation is  
\[D_{b}(\mathbf{\xi})=\sum_{j=1}^{4}w_{b}\mathcal{N}(\mathbf{x};\mathbf{x}_{0}^{(j)},\mathbf{P}_ {b}), \tag{47}\]  
where \(w_{b}=0.03\), \(\mathbf{P}_{b}=\text{diag}([100,100,100,100])\). The true trajectories of these targets are shown in Figure 2.  
In order to test the superiority of the proposed approach, LMB, HGM-LMB and JPHGM-LMB are applied to track these four targets with detection probability \(p_{D}=0.98\), the obtained trajectories by these three filters are shown in Figure 3. From Figure 3, it can be seen that the trajectories of these filters are almost identical to the true trajectory during the intervals 0 s-20 s and 40 s-60 s. However, the trajectories of LMB cross each other a lot and that of other filters are maintained correctly during the interval 20 s-40 s. The reason is that the actual motion model does not coincide with the transition motion model during 20 s-40 s. Therefore, measurements generated by one target may not appear near this target, the data association may produce a wrong association result in LMB. However, structure information facilitates HGM-LMB and JPHGM-LMB to produce more accurate trajectories. HGM-LMB and JPHGM-LMB achieve similar performance with high detection probability, HGM-LMB and JPHGM-LMB maintain correct trajectories as shown in Figure 3b,c. Figure 2: True trajectories of four targets. To further illustrate this, data association results of LMB and JPHGM-LMB at \(k=33\) are shown in Figure 4.  
Figure 4: Data association results of LMB and JPHGM-LMB: (**a**) LMB and (**b**) JPHGM-LMB. Figure 3: Trajectories obtained by three filters: (**a**) LMB, (**b**) HGM-LMB and (**c**) JPHGM-LMB. As we can see from Figure 4, the measurement generated by target 4 is associated with target 3 in LMB since the distance between measurement 4 and target 3 is smaller than others. When the structure information is used, the correct association result can be obtained, see Figure 4b. In order to illustrate that JPHGM-LMB achieves better data association performance than HGM-LMB with missed detections and clutter, the detection probability \(p_{D}\) is set to 0.9. Then these two filters are used to track these four targets, and the data association results of HGM-LMB and JPHGM-LMB at are shown in Figure 5.  
From Figure 5, we can see that target 2 is not detected and the clutter is associated with target 2 in HGM-LMB since HGM-LMB is built based on the assumption that there is no missed detection. Therefore, the clutter is considered as a valid measurement and associated with target 2. However, target 2 can be identified as the undetected target in JPHGM-LMB, then better association performance can be achieved.