## 2 Theoretical Basis
### The Terminal Trajectory of Anti-Ship Missiles
\right., \tag{3}\]  
where \(H=\|\mathbf{p}_{m}\|-R\) represents the altitude variable, while \(\dot{\alpha}_{\max}=\frac{n_{\max}g}{V}\sqrt{1-\Big{(}\frac{\dot{\beta}V}{n_ {\max}g}\Big{)}^{2}}\) and \(\dot{\beta}_{\max}=\frac{n_{\max}g}{V}\sqrt{1-\Big{(}\frac{\dot{\alpha}V}{n_ {\max}g}\Big{)}^{2}}\) denote the boundary values for the azimuth angles' speed and elevation angles' speed, respectively. Additionally, \(\text{sat}(u,u_{\max})=\left\{ \begin{array}{l}u_{\max},\text{ if }u>u_{\max}\\ -u_{\max},\text{ if }u<-u_{\max}\\ u,\text{ else }\end{array} \right.\) signifies the saturation function, which is implemented to ensure compliance with overload constraints during missile operation. (2) In Phase 2, when the missile-target distance diminishes to a specific threshold, the anti-ship missile executes a rapid ascent to a higher altitude, thereby enhancing its ability to acquire a superior vantage point for ship target detection. During the concluding phase, the missile adopts a stabilized flight attitude, with its speed vector aligned parallel to the sea surface. As Phase 2 follows seamlessly from Phase 1, adjustments are required for only a limited number of state variables, as illustrated in the following equations:  
\[f_{2}(\mathbf{X})|_{f_{1}}=\left\{ \begin{array}{l}\dot{\alpha}=0\\ \dddot{\beta}=\left\{\begin{array}{l}sat\Big{(}K_{4}(H_{2}-H),\frac{n_{\max}g }{V}-\dot{\beta}\Big{)},\text{if }H<H_{1}+c_{1}(H_{2}-H_{1})\\ sat\Big{(}-K_{5}\dot{\beta}+K_{6}(H_{2}-H),\frac{n_{\max}g}{V}-\dot{\beta}\Big{)},\text{else }\end{array}\right.\end{array} \right. \tag{4}\]  
(3) In Phase 3, once the anti-ship missile's attitude is stabilized, its radar seeker is activated. It is established that the missile employs the two-dimensional (2D) sum  
Figure 1: An anti-ship missile’s terminal phase penetration trajectory. difference amplitude ratio of the radar seeker head to acquire the target angle information \(\theta^{\prime}\) and \(\phi^{\prime}\). Under the assumption of an ideal operational environment with a high signal-to-noise ratio (SNR) (\(\geq\)10 dB), the angle estimation model can be formulated as follows:  
\[\left\{ \begin{array}{l}\theta^{\prime}=\theta+\epsilon^{\prime}_{\theta}, \epsilon^{\prime}_{\theta}\sim\mathcal{N}\Big{(}0,\delta^{2}_{\epsilon^{ \prime}_{\theta}}\Big{)}\\ \phi^{\prime}=\phi+\epsilon^{\prime}_{\phi},\epsilon^{\prime}_{\phi}\sim \mathcal{N}\Big{(}0,\delta^{2}_{\epsilon^{\prime}_{\theta}}\Big{)}\end{array}  \right.. \tag{5}\]  
Under the control of \(\theta^{\prime}\) and \(\phi^{\prime}\), the anti-ship missile will adjust its flight path in accordance with the guidance law while maintaining altitude stabilization. The state equation for this phase follows from Phase 2 and is redefined as detailed below:  
\[f_{3}(\mathbf{X})|_{f_{2}}=\left\{ \begin{array}{l}\tilde{\alpha}=sat\big{(}-K_{ 1}\theta^{\prime}-K_{2}\hat{\alpha},\hat{\alpha}_{\text{max}}-\hat{\alpha} \big{)}\\ \tilde{\beta}=sat\Big{(}K_{3}(H_{2}-H),\hat{\beta}_{\text{max}}-\hat{\beta} \Big{)}\end{array} \right.. \tag{6}\]  
(4) In Phase 4, the anti-ship missile has successfully obtained the necessary target tracking information for the terminal engagement. Owing to its elevated altitude, the missile must execute a rapid descent to an ultra-low altitude to evade detection by shipboard radar systems. By the conclusion of this phase, the missile maintains a stabilized flight attitude, with its speed vector aligned parallel to the sea surface, akin to Phase 2. Given the symmetrical nature of the trajectory, the state equations governing Phase 4 are as follows:  
\[f_{4}(\mathbf{X})|_{f_{3}}=\left\{\begin{array}{l}\dot{\alpha}=0\\ \tilde{\beta}=\left\{ \begin{array}{l}sat\Big{(}K_{4}(H_{1}-H),\frac{n_{\text {max}}\mathcal{S}}{V}-\hat{\beta}\Big{)},\text{if }H>H_{1}+(1-c_{1})(H_{2}-H_{1})\\ sat\Big{(}-K_{5}\hat{\beta}+K_{6}(H_{1}-H),\frac{n_{\text{max}}\mathcal{S}}{V}- \hat{\beta}\Big{)},\text{else}\end{array} \right.. \tag{7}\]  
(5) In Phase 5, the anti-ship missile enters the final preparation stage for penetration.