## 2 Theoretical Basis  
### The Terminal Trajectory of Anti-Ship Missiles  
Terminal trajectories of anti-ship missiles generally refer to precise strikes achieved through motor-driven means within a limited approach distance to a target ship. If we define the launch coordinate system as \(Ox_{m}y_{m}z_{m}\) and the projectile coordinate system as \(Ox_{y}y_{b}z_{b}\), then the transformation matrix from \(Ox_{m}y_{m}z_{m}\) to \(Ox_{y}y_{b}z_{b}\) is as follows:  
\[\mathbf{C}_{m}^{b}= \begin{bmatrix}\cos\beta_{m}\cos\alpha_{m}&\cos\beta_{m}\sin \alpha_{m}&\sin\beta_{m}\\ -\sin\alpha_{m}&\cos\alpha_{m}&0\\ -\sin\beta_{m}\cos\alpha_{m}&-\sin\beta_{m}\sin\alpha_{m}&\cos\beta_{m}\\ \end{bmatrix} \in\mathbb{R}^{3\times 3}. \tag{1}\]  
The missile and ship position vectors are defined as \(\mathbf{p}_{m}=[x_{m},y_{m},z_{m}]^{\mathrm{T}}\) and \(\mathbf{p}_{s}=[x_{s},y_{s},z_{s}]^{\mathrm{T}}\), respectively, under the coordinate system \(Ox_{m}y_{m}z_{m}\). The line-of-sight vector \(I=\mathbf{p}_{s}-\mathbf{p}_{m}\) is defined as the vector from the missile to the target ship at time \(t\). The missile-target distance is denoted by \(d=\|I\|\). The radar seeker's observation value, \(d^{\prime}\), is assumed to follow a normal distribution characterized by \(d^{\prime}=d+\varepsilon_{d},\varepsilon_{d}\sim\mathcal{N}\Big{(}0,\delta_{ \varepsilon_{d}}^{2}\Big{)}\). Within the projectile coordinate system, the target's azimuth angles and elevation angles relative to the anti-ship missile are determined as follows:  
\[\left\{ \begin{array}{l}\theta=\mathrm{arctan}_{2}\Big{(}l_{by},l_{bx}\Big{)} \in(-\pi,\pi]\\ \phi=\mathrm{arcsin}\Big{(}\frac{l_{by}}{d}\Big{)}\in(-\frac{\pi}{2},\frac{ \pi}{2}]\end{array} \right., \tag{2}\]  
where \(\Big{[}l_{bx},l_{by},l_{bz}\Big{]}^{\mathrm{T}}=\mathbf{C}_{m}^{b}I\) and \(\mathrm{arctan}_{2}(\cdot)\) represent four-quadrant arctangent functions, ensuring consistency between the angle signs and their respective quadrants. As illustrated in Figure 1, an anti-ship missile enhances its terminal penetration capability by employing ultra-low altitude, sea-skimming flight and a multi-stage propulsion system, thereby evading detection by shipborne radar during its terminal approach. Furthermore, the radar seeker is strategically activated only during Phase 3 and Phase 7 of the terminal trajectory to acquire the target ship's kinematic data, mitigating the risk of jammer interference. In all other stages, the missile relies on pre-programmed intelligence and its onboard tracking system for target information. The definition of the state variable \(\mathbf{X}=[\mathbf{p}_{m},\dot{x}_{m},\dot{y}_{m},\dot{z}_{m},\alpha_{m},\beta_{m}, \dot{\alpha}_{m},\dot{\beta}_{m},\ddot{\alpha}_{m},\ddot{\beta}_{m}]^{\mathrm{T}}\) has been provided, with subsequent sections elaborating the trajectory expressions for each phase. (1) In Phase 1, when the missile-target distance is relatively large, the anti-ship missile maintains a fixed altitude and maximum cruising speed, and its desired course in the \(x_{m}\)O\(y_{m}\)-plane always aligns with the target. Based on the proportional-derivative (PD) feedback adjustment concept, the state equation for the motor-driven process of the anti-ship missile is designed as follows:  
\[f_{1}(\mathbf{X})=\left\{ \begin{array}{l}\dot{x}_{m}=V\cos\beta\cos\alpha\\ \dot{y}_{m}=V\cos\beta\sin\alpha\\ \dot{z}_{m}=V\sin\beta\\ \ddot{\alpha}=sat\big{(}-K_{1}\dot{\theta}-K_{2}\dot{\alpha},\dot{x}_{\max}- \dot{\alpha}\big{)}\\ \ddot{\beta}=sat\big{(}K_{3}(H_{1}-H),\dot{\beta}_{\max}-\dot{\beta}\big{)}\\ \dot{\theta}=\theta+\epsilon_{\theta},\epsilon_{\theta}\sim\mathcal{N}\Big{(} 0,\delta_{\epsilon_{\theta}}^{2}\Big{)}\end{array}