## 4 Simulation and Discussion
### Simulation Configuration  
We set the initial augmented state of the HGV as follows: \(V=6000\) m/s; \(\theta=-0.1\) rad; \(\sigma=0\) rad; \(r=6.378\times 10^{6}+50\times 10^{3}\) m; \(\lambda=0^{\circ}\); \(\phi=0^{\circ}\). To design a trajectory in a global strike mission, some constraints of real flight must be considered such as waypoints for reconnaissance, no-fly zones for threat avoidance, maximum dynamic pressure, maximum aerodynamic load and maximum heat flux. However, in this paper, we mainly study the HGV tracking algorithms and focus on the tracking performance of different algorithms. Thus, it is proper to design some simple but typical trajectories to test the tracking algorithms regardless of above constraints. To validate the effectiveness of the HGV maneuvering trajectory tracking algorithms proposed in this paper, two maneuvering methods are given from  
\[\left\{ \begin{array}{l}\left\{\begin{array}{c}\alpha=\alpha_{\max},V>V_{1 }\\ \alpha=\frac{\alpha_{\max}+\alpha_{\min}}{2}+\frac{(\alpha_{\max}-\alpha_{ \min})\sin\left[\frac{\pi(V-0S(V_{1}+V_{2}))}{V_{1}-V_{2}}\right]}{2},V_{2}<V <V_{1}\\ \alpha=\alpha_{\max},V<V_{2}\end{array}\right.\\ \left\{\begin{array}{l}\
u=20^{\circ},V>V_{1}\\ \
u=0^{\circ},V_{2}<V<V_{1}\\ \
u=-20^{\circ},V<V_{2}\end{array}\right.\\ \left\{\begin{array}{l}\left\{\begin{array}{c}\alpha=\alpha_{\max},V>V_{1 }\\ \alpha=\frac{\alpha_{\max}+\alpha_{\min}}{2}+\frac{(\alpha_{\max}-\alpha_{\min} )\sin\left[\frac{\pi(V-0S(V_{1}+V_{2}))}{V_{1}-V_{2}}\right]}{2},V_{2}<V<V_{1} \\ \alpha=\alpha_{\max},V<V_{2}\end{array}\right.\\ \left\{\begin{array}{l}\
u=20^{\circ}\sin\left(t\pi/40\right)\end{array} \right.\end{array}\right.\end{array} \right. \tag{37}\]  
where \(V_{1}\)= 5000 m/s, \(V_{2}\)= 4500 m/s, \(\alpha_{\max}=~{}12^{\circ}\) and \(\alpha_{\min}\)= \(~{}6^{\circ}\); \(t\) denotes the flight time. Equation (37) shows the HGV maneuvers laterally at the biggest degree. Furthermore, in the real world, controlling the bank angle cannot jump \(20^{\circ}\) immediately. The change law of Equation (37) is to challenge the tracking algorithm in simulation. Equation (38) shows the HGV maneuvers laterally by the bank angle changing in the sine law with period 80 s and amplitude \(20^{\circ}\). Then, two HGV trajectory scenarios can be generated as Figure 1.  
Figure 1: Hypersonic glide vehicle (HGV) maneuvering trajectory. In Figure 1, Scenario 1 is generated based on Equation (37) and Scenario 2 is generated based on Equation (38). From Figure 1 we can know that HGV maneuvers laterally more strongly when bank angle changes according to Equation (37). Assume the radar is located at the position: \(r=6.378\times 10^{6}+200\) m; \(\lambda=-\pi/32\) rad; \(\phi=\pi/16\) rad. The standard deviation of the radar measurement errors is set as  
\[\left[ \begin{array}{cc}\sigma_{R}&\sigma_{A}&\sigma_{E}\end{array} \right]= \left[ \begin{array}{cc}10\;\mbox{m}&0.5\;\mbox{mrad}&0.5\;\mbox{mrad}\end{array} \right] \tag{39}\]  
Every process noise covariance matrix in all tracking models is set as  
\[Q=diag\left(\left[5^{2},0.001^{2},0.001^{2},50^{2},0.00001^{2},0.00001^{2},0. 01^{2},0.01^{2}\right]\right) \tag{40}\]  
In the algorithm, simulation the time interval \(T\) is 0.1 s. The model switching probability \(\rho_{ij}\) for the multiple model algorithm is a constant matrix with a large value on the diagonal elements. The three-model switching probability can be set the same for three-model IMM, AGIMM and HGMM algorithm as  
\[\rho_{ij}=\left\{ \begin{array}{c}0.8,i=j\\ 0.1,i\
eq j\end{array} \right. \tag{41}\]  
The nine-model switching probability for nine-model IMM algorithm can be set as  
\[\rho_{ij}=\left\{ \begin{array}{c}0.6,i=j\\ 0.05,i\
eq j\end{array} \right. \tag{42}\]  
The initial model probability for the three-model tracking algorithm is taken assuming the second model is near to the real model of HGV; that is,  
\[\omega_{0}^{j}=\left\{ \begin{array}{c}0.6,j=2\\ 0.2,\mbox{otherwise}\end{array} \right.