## 4 Simulation Results  
We use the OSPA error [26] and OSPA\({}^{(2)}\) error with \(p=q=2\), \(c=100\) m and \(L_{\varpi}=5\)[27] and cardinality error as the metrics to test the performance of the AMTB filter. \(\mathbf{\Phi}_{k-1}\) and \(\mathbf{Q}_{k-1}\) in (6) and (8) are given by:\[\mathbf{\Phi}_{k-1}= \begin{bmatrix}1&T&0&0\\ 0&1&0&0\\ 0&0&1&T\\ 0&0&0&1\end{bmatrix} \text{, }\mathbf{Q}_{k-1}= \begin{bmatrix}\frac{T^{2}}{2}&0\\ T&0\\ 0&\frac{T^{2}}{2}\\ 0&T\end{bmatrix}\begin{bmatrix}\frac{T^{2}}{2}&T&0&0\\ 0&0&\frac{T^{2}}{2}&T\end{bmatrix} \sigma_{v}^{2} \tag{56}\]  
where \(T=1\) s and \(\sigma_{v}=2\) ms\({}^{-2}\). The \(\mathbf{R}_{k}\) used in (54) is given by:  
\[\mathbf{R}_{k}= \begin{bmatrix}\sigma_{\theta}^{2}&0\\ 0&\sigma_{r}^{2}\end{bmatrix}  \tag{57}\]  
where \(\sigma_{r}=2.5\) m and \(\sigma_{\theta}=\frac{0.3\pi}{180}\) rad. The used detection probability is \(p_{D}=0.9\) and the radar is located at \( \begin{bmatrix}0&0\end{bmatrix} ^{\mathsf{T}}\). The average clutter density in the simulation measurements is \(\lambda_{c}=1.6883\times 10^{-3}\) rad\({}^{-1}\) m\({}^{-1}\) (i.e., the average number of clutter is \(N_{c}=15\)). Two examples were considered in the simulation. In first example, we compared the AMTB filter with the AGLMB filter [22], ACBMeMber filter [20], AMB filter [19] and APHD filter [17] in terms of OSPA and OSPA\({}^{(2)}\) errors and cardinality error to exhibit the tracking performance of the AMTB filter. In the second example, we compared the AMTB filter with the above four filters to demonstrate the performance of the AMTB filter for maneuvering object tracking. **Example 1**. : _We consider eleven objects in example 1. The initial states of objects 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 are \([-860,17,620,0]^{\mathsf{T}}\), \([-860,17,560,0]^{\mathsf{T}}\), \([-620,0,860,-17]^{\mathsf{T}}\), \([-560,0,860,-17]^{\mathsf{T}}\), \([-860,24,-620,0]^{\mathsf{T}}\), \([-860,19,200,-8]^{\mathsf{T}}\), \([-860,19,-50,0]^{\mathsf{T}}\), \([-860,19,-300,8]^{\mathsf{T}}\), \([620,0,860,-26]^{\mathsf{T}}\) and \([560,0,860,-26]^{\mathsf{T}}\), respectively. They appear at \(t=1\) s, 1 s, 4 s, 4 s, 9 s, 9 s, 9 s, 27 s, 27 s, 32 s and 32 s, respectively, and disappear at \(t=100\) s, \(100\) s, \(100\) s, \(68\) s, \(68\) s, \(68\) s, \(100\)s, \(100\) s, \(100\)s and \(100\) s, respectively. The eleven objects consist of close objects (such as objects 10 and 11, objects 8 and 9, objects 3 and 4, and objects 1 and 2) and crossing objects (such as objects 5, 6 and 7). Objects 5, 6 and 7 cross their paths at \(t=39.25\) s. The true trajectories of the eleven objects is given in Figure 1._  
In the experiment, the relevant parameters of the AMTB filter were set to \(\tau_{r}=0.005\), \(q_{a}=7.824\), \(v_{\text{max}}=50\) ms\({}^{-1}\) and \(v_{\text{min}}=5\) ms\({}^{-1}\). Using the five adaptive filters to handle the simulated measurements for 200 Monte Carlo runs, we obtained their tracking results. According to the OSPA error in Figure 2 and Table 1, the ACMeMber filter, AMB filter and APHD are inferior to the AMTB filter and AGLMB filter. This is because the former three filters require a high detecting probability, whereas the latter two filters avoid this requirement. As seen in Table 1 and Figures 2 and 3, the AGLMB filter had larger OSPA and OSPA\({}^{(2)}\) errors than the AMTB filter, which indicates that the AMTB filter performs better than the AGLMB filter although it obviates the requirement for clutter density. The cardinality error in Table 1 and cardinality estimation in Figure 4 reveal that the cardinality error of the AMTB filter is the lowest and its cardinality estimation is the most accurate. As seen in Table 1, the AMTB filter requires significantly less computation time than the AGLMB filter. Several peaks appear in Figures 2 and 3 because of the delayed response of the filter to the object appearing and disappearing.