## 3 Preliminaries
### Relative Motion of the Flight Vehicle with a Moving Object  
Firstly, as shown in Figure 3, we denote the inertial coordinate system by \(MX_{g}Y_{g}Z_{g}\), the body coordinate system by \(MX_{b}Y_{b}Z_{b}\), the LOS coordinate system by \(MX_{L}Y_{L}Z_{L}\), and the body-LOS coordinate system by \(MX_{L}Y_{BL}Z_{BL}\). Moreover, the ballistic and velocity coordinate systems are denoted by \(MX_{k}Y_{k}Z_{k}\) and \(MX_{p}Y_{p}Z_{p}\) respectively. The orientations between these coordinate systems are illustrated in Figure 4, showing the definition of the Euler angles, angle of attack, angle of sideslip, angle of bank, LOS angles, and body-LOS angles [29].  
Figure 4: The transformation of the coordinate systems.  
Figure 3: Three-dimensional geometry of the flight vehicle and the object.  
The three-dimensional vehicle-object pursuit model can be described by the following nonlinear differential equation [8; 30]:  
\[\left[\begin{matrix}\ddot{R}-R\dot{q}_{y}^{2}\cos^{2}q_{z}-R\dot{q}_{z}^{2}\\ R\ddot{q}_{z}+2\dot{R}\dot{q}_{z}+R\dot{q}_{y}^{2}\sin q_{z}\cos q_{z}\\ -2\dot{R}\dot{q}_{y}\cos q_{z}-R\ddot{q}_{y}\cos q_{z}+2R\dot{q}_{y}\dot{q}_{z} \sin q_{z}\end{matrix}\right]=\left[\begin{matrix}a_{Txs}-a_{xs}\\ a_{Tys}-a_{ys}\\ a_{Tzs}-a_{zs}\end{matrix}\right] \tag{5}\]  
where \(R\) is relative distance. \(q_{z}\) and \(q_{y}\) are the elevation and azimuth angles of the LOS and their time derivatives, and \(\dot{q}_{z}\) and \(\dot{q}_{y}\) are the LOS angular rates. \(\mathbf{a}_{T\mathrm{s}}=\left[a_{Txs}\quad a_{Tys}\quad a_{Tzs}\right]^{T}\) and \(\mathbf{a}_{s}=\left[a_{xs}\quad a_{ys}\quad a_{zs}\right]^{T}\) represent the acceleration vectors of the object and the flight vehicle, expressed in the LOS frame.  
According to the parallel approaching method [31], keeping the LOS direction ultimately constant, i.e., \(\lim_{t\to\infty}(\dot{q}_{z},\dot{q}_{y})=(0,0)\), is one of the sufficient conditions to acquire an interception of the object [13]. In addition, the relative distance R is usually uncontrollable in the terminal guidance phase. Thus, the last two equations of (5) can be rearranged as  
\[\left\{\begin{array}{l}\ddot{q}_{z}=-\frac{2\dot{R}}{R}\dot{q}_{z}-\dot{q}_{ y}^{2}\cos q_{z}\sin q_{z}+\frac{a_{Tys}-a_{ys}}{R}\\ \ddot{q}_{y}=-\frac{2\dot{R}}{R}\dot{q}_{y}+2\dot{q}_{y}\dot{q}_{z}\tan q_{z}- \frac{a_{Txs}-a_{xs}}{R\cos q_{z}}\end{array}\right. \tag{6}\]  
As for the flight vehicle, its acceleration in the LOS frame can be given by  
\[\mathbf{a}_{s}=\frac{1}{m}\mathbf{L}_{sv}\begin{bmatrix}-X\\ Y\\ Z\end{bmatrix}+\mathbf{L}_{sg}\begin{bmatrix}0\\ -g\\ 0\end{bmatrix} \tag{7}\]  
where \(\mathbf{L}_{sv}\) is the rotation matrix from the velocity frame to the LOS frame, while \(\mathbf{L}_{se}\) is that from the inertial frame to the LOS frame. Supposing that the Skid-to-Turn (STT) maneuver is adopted to improve the performance, the angle of bank, \(\gamma_{v}\), is expected to approximate zero. Thus, by combining (3) and (7), through simple calculations, it can be seen that  
\[\begin{bmatrix}a_{ys}\\ a_{zs}\end{bmatrix}=\frac{qS}{m}\begin{bmatrix}L_{sk}(2,2)&L_{sk}(2,3)\\ L_{sk}(3,2)&L_{sk}(3,3)\end{bmatrix}\begin{bmatrix}C_{y}^{s}&0\\ 0&C_{z}^{\beta}\end{bmatrix}\begin{bmatrix}\alpha\\ \beta\end{bmatrix}+\begin{bmatrix}-g\cos q_{z}\\ 0\end{bmatrix}+\Delta_{1} \tag{8}\]  
where \(\mathbf{L}_{sk}\) is the rotation matrix from the ballistic frame to the LOS frame.