## 4 Simulation and Discussion
### Simulation Results Comparison and Discussion  
We using several different algorithms to track HGV maneuvering trajectory and run 100 Monte-Carlo simulations for every algorithm on each trajectory. We use root mean square error (RMSE) of position and velocity to compare the performances of different tracking algorithms. The RMSE, average RMSE (ARMSE) and peak RMSE (PRMSE) can be defined as  
\[RMSE\left(k\right)=\sqrt{\frac{1}{n_{MC}}\sum_{i=1}^{n_{MC}}\left(\widehat{X} _{k/k}^{\left(i\right)}-X_{k}\right)^{2}} \tag{45}\]  
\[ARMSE=\frac{1}{m}\sum_{k=n}^{n+m}RMSE(k) \tag{46}\]  
\[PRMSE=\max_{n\leq k\leq m+n}(RMSE(k)) \tag{47}\]where \(n_{MC}=100\) is the Monte-Carlo simulation number; \(\widehat{X}_{k/k}^{(i)}\) is the predicted state at discrete time \(k\) at \(i\)-th Monte-Carlo simulation; \(X_{k}\) is the real state at discrete time \(k\). As defenders, velocity tracking error and position tracking error are the main indicators for HGV tracking performance [2,6,8] and more general target tracking performance [37, 38, 39]. Thus, in following tracking performance comparison, we mainly use velocity RMSE and position RMSE. \(V\) RMSE and \(r\) RMSE can represent the RMSE of velocity and position, respectively. Moreover, the attack angle error and bank angle error at discrete time k have a certain effect on the velocity and position at discrete time \(k+1\). Thus, \(V\) RMSE, \(r\) RMSE, \(\alpha\) RMSE and \(\
u\) RMSE using different algorithms will be compared and analyzed as follows. To compare the tracking performance of MAM algorithm with that of FSMM (three-model IMM and nine-model IMM) algorithm and compare the tracking performance of MAM algorithm with that of VSMM (AGIMM and HGMM) algorithm, we choose a point in every 2 s to plot the former and the later as Figure 2 for tracking Scenario 1 and as Figure 3 for tracking Scenario 2. Furthermore, the average RMSE and peak RMSE using different algorithms are listed as Table 1 for tracking Scenario 1 and as Table 2 for tracking Scenario 2. From Tables 1 and 2, we can know that the MAM algorithm outperforms multiple model algorithms in almost every ARMSE and PRMSE of velocity and position. Furthermore, VSMM algorithm performs better than FSMM algorithm for tracking Scenario 1 and Scenario 2. In VSMM algorithms, HGMM algorithm performs better than AGIMM algorithm. Although nine-model IMM algorithms are more accurate than the MAM algorithm, the MAM algorithm performs better than the MAM algorithm. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Algorithm** & **V ARMSE (m/s)** & **V PRMSE (m/s)** & **r ARMSE (m)** & **r PRMSE (m)** & **a ARMSE (\(\cdot\))** & **v ARMSE (\(\cdot\))** \\ \hline MAM & 10.94 & 38.79 & 165.97 & 797.03 & 1.52 & 9.25 \\ Three-model IMM & 12.07 & 43.15 & 169.79 & 847.91 & 2.66 & 9.15 \\ Nine-model IMM & 12.27 & 48.61 & 167.27 & 832.13 & 2.91 & 11.49 \\ AGIMM & 12.09 & 40.09 & 172.12 & 862.06 & 2.17 & 9.25 \\ HGMM & 11.95 & 38.73 & 172.88 & 850.41 & 1.95 & 9.30 \\ \hline \end{tabular}
\end{table} 
Table 1: The performances for stably tracking Scenario 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Algorithm** & **V ARMSE (m/s)** & **V PRMSE (m/s)** & **r ARMSE (m)** & **r PRMSE (m)** & **a ARMSE (\(\cdot\))** & **v ARMSE (\(\cdot\))** \\ \hline MAM & 10.95 & 38.04 & 161.99 & 790.48 & 1.63 & 14.57 \\ Three-model IMM & 12.18 & 40.26 & 164.69 & 842.41 & 2.74 & 14.55 \\ Nine-model IMM & 12.27 & 41.50 & 165.02 & 833.08 & 2.98 & 13.13 \\ AGIMM & 12.20 & 38.50 & 167.31 & 824.74 & 2.25 & 14.61 \\ HGMM & 12.10 & 39.20 & 168.57 & 863.36 & 2.03 & 14.63 \\ \hline \end{tabular}
\end{table} 
Table 2: The performances for stably tracking Scenario 2. Figure 3: RMSE for tracking Scenario 2.  
algorithm uses more models, it does not perform better than three-model IMM algorithm. The tracking results using different algorithms will be analyzed and discussed in detail as follows.